diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/-OyjdPkiDt8_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/-OyjdPkiDt8_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1e23c102521bfe1ab182c3f3de4a5a7a7d8a46ab
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+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/-OyjdPkiDt8_postprocess.srt
@@ -0,0 +1,2176 @@
+1
+00:00:00,590 --> 00:00:04,670
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,670 --> 00:00:10,770
+وبركاته هذه هي المحاضرة رقم خمسة عشرين مساقة فاضل
+
+3
+00:00:10,770 --> 00:00:15,530
+وتكامل جيم وهي المحاضرة السابعة في الشبطر الأخير
+
+4
+00:00:15,530 --> 00:00:19,770
+من المادة اللي هي خمساش سبعة وهو بعنوان triple
+
+5
+00:00:19,770 --> 00:00:24,010
+integrals in cylindrical and spherical coordinates
+
+6
+00:00:24,010 --> 00:00:27,570
+حكينا المرة الماضية في ال section السابق عن triple
+
+7
+00:00:27,570 --> 00:00:28,410
+integrals
+
+8
+00:00:33,750 --> 00:00:40,370
+الحديث ليس موسيقى موسيقى موسيقى موسيقى موسيقى
+
+9
+00:00:40,370 --> 00:00:43,810
+موسيقى موسيقى موسيقى موسيقى موسيقى موسيقى موسيقى
+
+10
+00:00:43,810 --> 00:00:47,650
+موسيقى
+
+11
+00:00:47,650 --> 00:00:52,650
+موسيقى
+
+12
+00:00:53,730 --> 00:00:57,630
+أحنا في الأول بدنا نشوف كيف .. اللي هو .. ال .. ال
+
+13
+00:00:57,630 --> 00:01:01,070
+.. ال .. كيف نصل للنقطة بواسطة .. اللي هو النقطة
+
+14
+00:01:01,070 --> 00:01:04,130
+في الفرار طبعا بواسطة الـ cylindrical coordinates
+
+15
+00:01:04,130 --> 00:01:07,810
+احنا في ال .. في ال plane أخدنا اللي هو ال
+
+16
+00:01:07,810 --> 00:01:10,890
+rectangular .. ال .. ال .. في ال XY plane ال
+
+17
+00:01:10,890 --> 00:01:14,850
+rectangular و أخدنا ال .. ال polar الآن هذا اللي
+
+18
+00:01:14,850 --> 00:01:19,270
+هو بيستعمل ال polar بالاضافة لل Z axis نشوف إيش
+
+19
+00:01:19,270 --> 00:01:22,340
+اللي بنحكيهلأن المرة الماضية كانت اللي هو الوصول
+
+20
+00:01:22,340 --> 00:01:26,120
+للنقطة عبر الـ X Axis بعدين الـ Y Axis بعدين الـ Z
+
+21
+00:01:26,120 --> 00:01:30,880
+Axis اللي هي كلها إيش؟ Horizontal اللي هي أو
+
+22
+00:01:30,880 --> 00:01:34,920
+Vertical عن قطوط اللي هي موازية للـ Z Axis أو الـ
+
+23
+00:01:34,920 --> 00:01:38,200
+X Axis أو الـ Y Axis أو اللي هي على الـ X Axis
+
+24
+00:01:38,200 --> 00:01:41,900
+بعدين نمثل على الـ X Axis بعدين الـ Y Axis بعدين
+
+25
+00:01:41,900 --> 00:01:45,600
+الـ Z Axis إلى خير بقى اللي هي ال distance هذه
+
+26
+00:01:45,600 --> 00:01:50,820
+اللي حكينا عنها سابقاالان الوضع مختلف شوية الان
+
+27
+00:01:50,820 --> 00:01:55,900
+الـ cylindrical نصل لنقطة هذه نصل في ال X Y plane
+
+28
+00:01:55,900 --> 00:02:01,580
+نمشي بالبولر يعني الان بدي اللي هو بمشي بزاوية θ
+
+29
+00:02:01,580 --> 00:02:06,780
+المطلوبةوبعمشي بـ Directed Distance R اللي هي اللي
+
+30
+00:02:06,780 --> 00:02:10,200
+في الـ Polar العاديات و بطلع اللي هو ب .. اللي هي
+
+31
+00:02:10,200 --> 00:02:13,820
+موازنة الـ Z Axis اللي هي بمعنى أفرع الـ Z Axis زي
+
+32
+00:02:13,820 --> 00:02:17,080
+ال .. ال rectangle العادى بمقدار اللي هي الـ Z
+
+33
+00:02:17,080 --> 00:02:21,320
+اللي هيها فبتكون إحداثية النقطة هي عبارة عن R و θ
+
+34
+00:02:21,320 --> 00:02:25,900
+و Z يعني هدول ال Polar و هد ال Z العادى هذا بنسميه
+
+35
+00:02:25,900 --> 00:02:28,990
+اللي هو الـ Cylindrical Coordinatesالسيلندريكال
+
+36
+00:02:28,990 --> 00:02:35,130
+كواردنايت ريبريزينت ا بوينت بي in space by order
+
+37
+00:02:35,130 --> 00:02:39,870
+of the triple R و θ و z in which يعني R و θ ر بورر
+
+38
+00:02:39,870 --> 00:02:43,510
+كواردنايت for the vertical projection of B on the
+
+39
+00:02:43,510 --> 00:02:48,150
+xy plane و
+
+40
+00:02:48,150 --> 00:02:53,850
+ال z is the rectangular vertical coordinates
+
+41
+00:02:53,850 --> 00:02:57,970
+العادىالان ليش سموها cylindrical ؟ cylindrical
+
+42
+00:02:57,970 --> 00:03:03,970
+أقرب لإنه لو أخدنا اللي هو R ثابت لما R ثابت بتصير
+
+43
+00:03:03,970 --> 00:03:07,310
+تعملني إيه؟ Circle في ال .. اللي أخدناها Circle في
+
+44
+00:03:07,310 --> 00:03:11,070
+ال بولر لكن ال Z مختلفة تتارك في ال space فبتصير
+
+45
+00:03:11,070 --> 00:03:14,870
+ليه Circles فوق بعض بتعملني إيه؟ سيلندر عشان هيك
+
+46
+00:03:14,870 --> 00:03:18,950
+اللي هو R بتساوي A اللي هي ثابت بتعملني عبارة عن
+
+47
+00:03:18,950 --> 00:03:24,670
+سيلندر اللي هي بكل سهولة نيجي الآننشوف العلاقة بين
+
+48
+00:03:24,670 --> 00:03:30,710
+equations relating rectangular, x,y,z and
+
+49
+00:03:30,710 --> 00:03:34,790
+cylindrical, R, θ,و,z هدولة اللي هي إحداثيات اللي
+
+50
+00:03:34,790 --> 00:03:38,190
+هي ال rectangular, x,y,z اللي بنعرفها و هاد
+
+51
+00:03:38,190 --> 00:03:41,890
+الإحداثيات ال cylindrical اللي عرفناها سابقا
+
+52
+00:03:41,890 --> 00:03:46,550
+بالبولر والزد اللي هيهاطبيعي العلاقة بين R و θ و X
+
+53
+00:03:46,550 --> 00:03:51,170
+و Y علاقة سابقة أخدناها قبل ذلك X بيسوي R Cos θ و
+
+54
+00:03:51,170 --> 00:03:55,790
+Y بيسوي R Sin θ و Z هي Z طبعا و بناء عليه اللي هو
+
+55
+00:03:55,790 --> 00:04:00,110
+لو أجينا اعملنا X تربيع زياد Y تربيع بيسوي R تربيع
+
+56
+00:04:00,110 --> 00:04:03,750
+Cos تربيع زياد R تربيع Sin تربيع داخد ال R تربيع
+
+57
+00:04:03,750 --> 00:04:06,670
+من المشترك بيصير R تربيع في Cos تربيع زي ال Sin
+
+58
+00:04:06,670 --> 00:04:10,150
+تربيع اللي هي واحد بتظلها R تربيعيعني X تربيع زواد
+
+59
+00:04:10,150 --> 00:04:13,170
+Y تربيع بسوار التربيع اللي احنا عارفينها و X مو
+
+60
+00:04:13,170 --> 00:04:18,130
+اللي هي ال Y على ال X فبصير Sine على Cos لإن ال R
+
+61
+00:04:18,130 --> 00:04:21,810
+بتبقى احنا ال R فتطلع tan θ و tan θ بيساوي Y على X
+
+62
+00:04:21,810 --> 00:04:26,330
+او اللي هو من خلال اللي هي هذه الرسمة بتقدر تاخد
+
+63
+00:04:26,330 --> 00:04:31,270
+اللي هو هاي θ و هاي R و هاي X و هاي Y من هنا توجد
+
+64
+00:04:31,270 --> 00:04:34,550
+العلاقات اللي قلنا عنها قبل و شوية Sine of θ
+
+65
+00:04:34,550 --> 00:04:38,710
+بيساوي Y على R Cosine of θ بيساوي X على Rوتالت
+
+66
+00:04:38,710 --> 00:04:44,110
+FETA وهي Y على X بكل سهولة هذه العلاقات القديمة
+
+67
+00:04:44,510 --> 00:04:49,370
+الان الـ Dv زي ما حكينا الماضية دي ايه كانت بدل دي
+
+68
+00:04:49,370 --> 00:04:54,330
+اكس دي واي دي ار دي ار دي ثتا وDz زي ما هو اذا الـ
+
+69
+00:04:54,330 --> 00:04:59,370
+Dz دي اكس دي واي اش بيصير Dz ار دي ار دي ثتا اللى
+
+70
+00:04:59,370 --> 00:05:04,850
+هو عبارة عن مين عن الـ Dv اللى بيكمل ال volume
+
+71
+00:05:04,850 --> 00:05:09,350
+للتجارب يعني ال triple integral Dv اللى عبارة عن
+
+72
+00:05:09,350 --> 00:05:10,010
+Dz
+
+73
+00:05:13,460 --> 00:05:18,840
+هذه الترتيبات بين كامل الهواتف التي نريد ان نقوم
+
+74
+00:05:18,840 --> 00:05:22,520
+باستخدامها الان نأتي لبعض الشغلات السهلة والمشهورة
+
+75
+00:05:22,520 --> 00:05:25,400
+في المعادلات
+
+76
+00:05:27,240 --> 00:05:32,120
+X بتساوي 1 بلين و Y بتساوي 1 بلين برضه و 1 مواظلة
+
+77
+00:05:32,120 --> 00:05:38,360
+X Z بلين و 1 مواظلة Y Z بلين و Z بتساوي 1 مواظلة X
+
+78
+00:05:38,360 --> 00:05:41,720
+Y بلين هذا زمان في ال .. ال .. ال .. ال .. ال ..
+
+79
+00:05:41,720 --> 00:05:41,940
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+80
+00:05:41,940 --> 00:05:41,960
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+81
+00:05:41,960 --> 00:05:42,000
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+82
+00:05:42,000 --> 00:05:42,080
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+83
+00:05:42,080 --> 00:05:42,540
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+84
+00:05:42,540 --> 00:05:43,520
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+85
+00:05:43,520 --> 00:05:43,540
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+86
+00:05:43,540 --> 00:05:43,580
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+87
+00:05:43,580 --> 00:05:47,420
+ال .. ال ..
+
+88
+00:05:47,420 --> 00:05:53,380
+ال
+
+89
+00:05:53,380 --> 00:06:06,570
+.. الالتي تأخذ تتا نوت هي عبارة عن كل الذد
+
+90
+00:06:06,570 --> 00:06:10,890
+المختلفة الخط هذا وهي الخط هي وهي وفرضنا الخط هي
+
+91
+00:06:10,890 --> 00:06:14,890
+هيكمن هنا إلى هنا ومن هنا إلى هنا ومن هنا إلى هنا
+
+92
+00:06:14,890 --> 00:06:15,230
+إلى هنا ومن هنا إلى هنا إلى هنا ومن هنا إلى هنا
+
+93
+00:06:15,230 --> 00:06:18,510
+إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا
+
+94
+00:06:18,510 --> 00:06:19,310
+إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا
+
+95
+00:06:19,310 --> 00:06:24,370
+إلى هنا إلى هنا إلى هنا إلى هنا إلى
+
+96
+00:06:24,370 --> 00:06:24,850
+هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى
+
+97
+00:06:24,850 --> 00:06:27,990
+هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى
+
+98
+00:06:27,990 --> 00:06:29,330
+هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى
+
+99
+00:06:29,330 --> 00:06:29,850
+هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى
+
+100
+00:06:29,850 --> 00:06:30,730
+هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى
+
+101
+00:06:30,730 --> 00:06:31,370
+هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى
+
+102
+00:06:31,370 --> 00:06:32,770
+هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى هنا إلى
+
+103
+00:06:32,770 --> 00:06:32,830
+هنا إلى هنا إلى هنا إلى هنا
+
+104
+00:06:40,260 --> 00:06:46,240
+التي هي مادام z بيساوي z note يعني اللي عند z note
+
+105
+00:06:46,240 --> 00:06:50,700
+اللي هي هنا اللي بيصير اللي هي نقطة بس الآن اللي
+
+106
+00:06:50,700 --> 00:06:55,060
+هي θ في ال R اللي هو مختلف ر بيجيهك و ر بيجيهك و ر
+
+107
+00:06:55,060 --> 00:06:56,840
+بيجيهك في كل الاتجاهات
+
+108
+00:07:02,220 --> 00:07:06,180
+كامل موازنة الـ X Y Plane من كل أطرافه إلى مالة
+
+109
+00:07:06,180 --> 00:07:09,520
+نهاية هذا Z بيساوي Z باتنين نعود للأسواق اللي قبل
+
+110
+00:07:09,520 --> 00:07:14,920
+قليل قلناها الـ R بتساوي R من هان لهان في الـ X Y
+
+111
+00:07:14,920 --> 00:07:20,160
+Plane في عبداللي هو بيعمل دائرة لإياش دائرة ليش
+
+112
+00:07:20,160 --> 00:07:25,940
+لأن فتحها متغيرة فبتمسح كل اللي هو إياش الاتنين by
+
+113
+00:07:25,940 --> 00:07:29,980
+فبتصير R بيتساوي إيه في ال plane اللي هو دائرة و
+
+114
+00:07:29,980 --> 00:07:33,280
+لما نقول في ال space برضه ذات تاخد قيمها فبتعمل ال
+
+115
+00:07:33,280 --> 00:07:39,050
+cylinder الزرقى اللي أمامي هذه R بتساوي إيهطيب
+
+116
+00:07:39,050 --> 00:07:42,350
+ديجي ناخد example find the limits of integration
+
+117
+00:07:42,350 --> 00:07:46,370
+in cylindrical coordinates for integrating a
+
+118
+00:07:46,370 --> 00:07:50,030
+function f of r و θ و z over the region D bounded
+
+119
+00:07:50,030 --> 00:07:53,610
+below by بوصف جاعد عساس انه لو انت حتى ماعرفتش اش
+
+120
+00:07:53,610 --> 00:07:59,410
+ترسم من الوصف تبعه ممكن تقدر تحصل على اللي هي حدود
+
+121
+00:07:59,410 --> 00:08:02,430
+ال integration over the region D bounded below by
+
+122
+00:08:02,430 --> 00:08:07,730
+ايش بقولك من تحت bounded by z بساوة 0طيب ..
+
+123
+00:08:07,730 --> 00:08:11,970
+laterally by .. by the .. the .. the circular
+
+124
+00:08:11,970 --> 00:08:15,510
+cylinder X تربيع زائد Y minus واحد كل تربيع بساوة
+
+125
+00:08:15,510 --> 00:08:18,490
+واحد يعني الآن اللي هو هذا اللي هو فيه عند ال
+
+126
+00:08:18,490 --> 00:08:23,830
+cylinder اللي هو X تربيع زائد Y تربيع زائد Y minus
+
+127
+00:08:23,830 --> 00:08:28,210
+واحد كل تربيع بساوة واحد طبعا هذه الآن اللي هي في
+
+128
+00:08:28,210 --> 00:08:31,650
+ال play في ال XY play circle .. شو اسمه circle؟
+
+129
+00:08:31,680 --> 00:08:37,680
+سيركل مركزها 01 وهي 01 مركزها على هذه و بتاخد الـ
+
+130
+00:08:37,680 --> 00:08:42,300
+circle بالشكل اللي أمامي هذه إذا في المنطقة هذه و
+
+131
+00:08:42,300 --> 00:08:46,500
+واحد طول قيمة نصف القطر إذا بيصير اللي هو بي miss
+
+132
+00:08:46,500 --> 00:08:53,220
+ال y-axis أو بيمر بال origin عنده نقطة zero عنده
+
+133
+00:08:53,220 --> 00:09:00,170
+نقطة 00 اللي هو بتكون عنده .. بتحقق المعادلةإذا
+
+134
+00:09:00,170 --> 00:09:04,450
+هاي اللي هي الـ circle هذه لكن هو cylinder يعني
+
+135
+00:09:04,450 --> 00:09:08,010
+الـ Z مختلفة إذا بتضلها ماشية إلى ما لنهاية هذه
+
+136
+00:09:08,010 --> 00:09:11,550
+اللي هي الـ cylinder لكن بيكمل الوصف and above by
+
+137
+00:09:11,550 --> 00:09:15,770
+هذا من الأسفل يعني أو من حواليها and above by the
+
+138
+00:09:15,770 --> 00:09:19,070
+paraboloid Z بساوي كتربيع زد واي تربيع هنا اللي هو
+
+139
+00:09:19,070 --> 00:09:23,290
+paraboloid بيجي هيك الآن بقطع هذه المنطقة في
+
+140
+00:09:23,290 --> 00:09:27,330
+المنطقة هذهاللي هي هذه اللي هي مبتع اللي هو Z
+
+141
+00:09:27,330 --> 00:09:30,550
+بساوي 6 تربيع زائد Y تربيع طبعا الرسمات الصعبة
+
+142
+00:09:30,550 --> 00:09:35,850
+يأدى بتلزم بنرسملكم إياها في الامتحان طيب هو اللي
+
+143
+00:09:35,850 --> 00:09:39,830
+بتلزمش هنفعل اسمه تعتمد على الوصف اللي موجود نيدي
+
+144
+00:09:39,830 --> 00:09:43,850
+الآن بدنا نحدد حدود ال integration بدنا نكامل F of
+
+145
+00:09:43,850 --> 00:09:50,160
+R وθ وZ DZ R DR Dθ بهذا الترتيبلان انا انا انا انا
+
+146
+00:09:50,160 --> 00:09:51,260
+انا انا انا انا انا انا انا انا انا انا انا انا
+
+147
+00:09:51,260 --> 00:09:54,040
+انا انا انا انا انا انا انا انا انا انا انا انا
+
+148
+00:09:54,040 --> 00:09:56,160
+انا انا انا انا انا انا انا انا انا انا انا انا
+
+149
+00:09:56,160 --> 00:09:57,220
+انا انا انا انا انا انا انا انا انا انا انا انا
+
+150
+00:09:57,220 --> 00:10:02,240
+انا انا انا انا انا انا انا انا انا انا انا انا
+
+151
+00:10:02,240 --> 00:10:05,300
+انا انا
+
+152
+00:10:06,700 --> 00:10:09,500
+اللي هو X تربيع زي Y تربيع لكن أنا بدي إياها باشي
+
+153
+00:10:09,500 --> 00:10:13,440
+يا جماعة بالبولر و X تربيع زي Y تربيع إيش بتساوي
+
+154
+00:10:13,440 --> 00:10:17,080
+قبل شوية قولنا R تربيع إذا بتيجي من عند السفر و
+
+155
+00:10:17,080 --> 00:10:22,940
+بتطلع وين لعند اللي هي R تربيع هذا قيمة ال Z إذا
+
+156
+00:10:22,940 --> 00:10:27,840
+من سفر لمين لل R تربيعنأتي الآن لما عدلتنا هذه الـ
+
+157
+00:10:27,840 --> 00:10:31,080
+X تربيع زي Y ماينوس واحد تربيع بسوء واحد خلّينا
+
+158
+00:10:31,080 --> 00:10:37,760
+نحولها للسلندر نحولها لبولر عشان هيتلزمنا طيب حولي
+
+159
+00:10:37,760 --> 00:10:44,620
+الآن X تربيع زي Y تربيع اللي هو فكات نقص اتنين Y
+
+160
+00:10:44,620 --> 00:10:48,370
+زي واحد بسوء واحدلأن ما كان X تربيع زاد Y تربيع
+
+161
+00:10:48,370 --> 00:10:52,930
+إيش حطيت؟ اللي هي R تربيع الـ Y إيش ما لها؟ R sin
+
+162
+00:10:52,930 --> 00:10:56,790
+θ نقص 2R sin θ و الواحد بروح مع الواحد فبصير هذا
+
+163
+00:10:56,790 --> 00:10:59,970
+المقدر بساوة 0 وبنجي لهذا على الجهة هذه بصير R
+
+164
+00:10:59,970 --> 00:11:05,630
+تربيع بساوة 2R sin θ بنختصر R مع الـ R بصير R
+
+165
+00:11:05,630 --> 00:11:11,070
+بساوة 2 sin θ هذه هي معادلة 8وطبعا الـ R اعتبرناها
+
+166
+00:11:11,070 --> 00:11:14,410
+لأ بيساوي سفر لإن السفر هيكون متضمن هنا اللي هو R
+
+167
+00:11:14,410 --> 00:11:18,930
+بسفر عن θ ابسي في الفترة الششكالية R بيساوي 2 sin
+
+168
+00:11:18,930 --> 00:11:24,210
+θ ماشي هذه معادلة 2 هي الدائرة الأمامى لو جينا
+
+169
+00:11:24,210 --> 00:11:27,970
+رسمنا هذه الدائرة كلها هي لاحظ الدائرة هذه اللي هي
+
+170
+00:11:27,970 --> 00:11:33,850
+الأمامى هذهعندى لما θ بصفر اللى هى اشمالها بيصير
+
+171
+00:11:33,850 --> 00:11:36,830
+طبعا الـ 0 مع الـ x axis بيصير اللى هى r بالساوية
+
+172
+00:11:36,830 --> 00:11:43,370
+0 لأنه صينة 00 الآن θ بيساوية πاى على 2 لما نتصل
+
+173
+00:11:43,370 --> 00:11:47,350
+لها باى على 2اللي هي بيصير R بيساو 2 بي على 2 يعني
+
+174
+00:11:47,350 --> 00:11:50,970
+2 سين ال باي على 2 يعني واحد يصير اتنين اذا هنا
+
+175
+00:11:50,970 --> 00:11:57,070
+الان لما انتصل ل باي بتيجي ايش R بيساو 2 سين ال
+
+176
+00:11:57,070 --> 00:12:02,100
+باي هي zero اذا ال هادي كلها الدائرة بترتسممن عند
+
+177
+00:12:02,100 --> 00:12:08,800
+θ اللى هو بصفر لعند θ اشمالها بتساوي باي ندى الآن
+
+178
+00:12:08,800 --> 00:12:12,180
+نحدد حدود ال integration زى ما قلنا حددنا Z دلوقت
+
+179
+00:12:12,180 --> 00:12:14,800
+بنعمل projection لهذا واضح ان ال projection لما
+
+180
+00:12:14,800 --> 00:12:20,520
+ننزل بكون اشماله في اللى هى العمود وهى العمود
+
+181
+00:12:20,520 --> 00:12:24,030
+نزلها دى بتقعد داخل الدائرة اللى أنا لهالأن
+
+182
+00:12:24,030 --> 00:12:28,030
+الدائرة اللي أمامي هذه هي اللي بتتحدد للـR والـθ
+
+183
+00:12:28,030 --> 00:12:32,490
+كيف بنحدد R؟ بنطلع اللي هو Ray من اللي هو
+
+184
+00:12:32,490 --> 00:12:36,810
+الـorigin إلى اللي هو الجسم لأن واضح كل الريزي
+
+185
+00:12:36,810 --> 00:12:42,350
+بتيجي من عند نقطة الأسر Zero وبتطلع من من عند
+
+186
+00:12:42,350 --> 00:12:48,290
+الدائرة اللي هي 2Sinθ اذا قيمة R من 0 ل2Sinθ هي
+
+187
+00:12:48,290 --> 00:12:52,970
+2Sinθ من 2Sinθ الاخرىأذا R من صفر لعين 2 ثانية
+
+188
+00:12:52,970 --> 00:12:53,270
+ثانية ثانية ثانية ثانية ثانية ثانية ثانية ثانية
+
+189
+00:12:53,270 --> 00:12:56,810
+ثانية ثانية ثانية ثانية
+
+190
+00:12:56,810 --> 00:12:58,350
+ثانية ثانية ثانية ثانية ثانية ثانية ثانية ثانية
+
+191
+00:12:58,350 --> 00:13:02,750
+ثانية ثانية ثانية ثانية ثانية ثانية ثانية ثانية
+
+192
+00:13:02,750 --> 00:13:03,130
+ثانية ثانية ثانية ثانية ثانية ثانية ثانية ثانية
+
+193
+00:13:03,130 --> 00:13:03,870
+ثانية ثانية ثانية ثانية ثانية ثانية ثانية ثانية
+
+194
+00:13:03,870 --> 00:13:07,050
+ثانية ثانية ثانية ثانية ثانية ثانية ثانية ثانية
+
+195
+00:13:07,050 --> 00:13:08,250
+ثانية ثانية ثانية ثانية ثانية ثانية ثانية ثانية
+
+196
+00:13:08,250 --> 00:13:08,290
+ثانية ثانية ثانية ثانية ثانية ثانية ثانية ثانية
+
+197
+00:13:08,290 --> 00:13:14,150
+ثانية ثانيةإذا من سفر إلى باى و بكون هذه اللى هو
+
+198
+00:13:14,150 --> 00:13:18,070
+حدود ال integration لدى اللى اللى عندنا اللى
+
+199
+00:13:18,070 --> 00:13:22,910
+المنطقة اللى عملناها إذا مختصر الحديث بحول كل شيء
+
+200
+00:13:22,910 --> 00:13:26,630
+إلى polar الآن بجيب أولا الشيء ال z بعدين برعمله
+
+201
+00:13:26,630 --> 00:13:30,070
+projection projection بالزي التحت اللى تحت بنحددها
+
+202
+00:13:30,070 --> 00:13:35,060
+زى ما كنا نحدد زمان اللى هو في ال polarالان بأكدلك
+
+203
+00:13:35,060 --> 00:13:37,100
+اللي هو how to integrate in cylindrical
+
+204
+00:13:37,100 --> 00:13:39,540
+coordinates مفروض هذا قبل اللي حكيناه لكن هي اللي
+
+205
+00:13:39,540 --> 00:13:43,700
+حطوا هيك خلينا نشوف الآن هي تدي المنطقة هي المنطقة
+
+206
+00:13:43,700 --> 00:13:46,760
+اللي بحكي عنها بدنا نحدد كيف حدود ال integration
+
+207
+00:13:46,760 --> 00:13:50,620
+ايش بنسوي زي ما قلنا هي المنطقة مرسومة هيها
+
+208
+00:13:50,620 --> 00:13:56,440
+بنعملها ل production تنزل هيه الآنبأخد الـ Ré اللي
+
+209
+00:13:56,440 --> 00:13:59,920
+هو Asif Vertical Line موازية الـ Z Axis من وين دخل
+
+210
+00:13:59,920 --> 00:14:02,420
+المنطقة من هنا وخرج منها من هنا إذا حدود الـ
+
+211
+00:14:02,420 --> 00:14:06,160
+Integration من الـ Surface هذا اللي هو Z بساوة G1
+
+212
+00:14:06,160 --> 00:14:11,860
+وRθ هو المعطنية و بطلع من Z بساوة G2 وRθ إذا هذه
+
+213
+00:14:11,860 --> 00:14:16,840
+حدود Zالان نذهب الى حدود R هنا وهي حدود R هنا وهي
+
+214
+00:14:16,840 --> 00:14:28,860
+حدود R هنا وهي حدود R هنا وهي حدود R هنا
+
+215
+00:14:29,030 --> 00:14:33,450
+اللي هو مسميها هو H1 of θ وخرج من المنطقة هذه اللي
+
+216
+00:14:33,450 --> 00:14:38,890
+هو مسميها H2 of θ فبصير R من H1 of θ لعند H2 of θ
+
+217
+00:14:38,890 --> 00:14:42,690
+الآن هذه المنطقة كيف ارتسمت ؟ ارتسمت من عند θ هذه
+
+218
+00:14:42,690 --> 00:14:48,360
+لعند θ هذهماشي لأن θ هدم سميها Alpha هيها و θ
+
+219
+00:14:48,360 --> 00:14:52,680
+الأخيرة هدم سميها كلها Beta إذا قيمة θ هتكون من
+
+220
+00:14:52,680 --> 00:14:57,040
+Alpha لعند مين يا جماعة لعند Beta و هيك بنحدد زي
+
+221
+00:14:57,040 --> 00:15:00,700
+ما قلنا في المثال و زي ما قلنا هي نظريا حدود ال
+
+222
+00:15:00,700 --> 00:15:04,760
+integrationناخد مثل آخر بيقولي find the centroid
+
+223
+00:15:04,760 --> 00:15:07,760
+احنا مش هنحكي عن ال centroid لإنه ال centroid
+
+224
+00:15:07,760 --> 00:15:09,600
+ماخدناش اللي هو في ال section اللي جابله هذا اللي
+
+225
+00:15:09,600 --> 00:15:13,940
+هو الملغة ال centroid و ال moment of inertia و ال
+
+226
+00:15:13,940 --> 00:15:17,900
+mass و ال mass و مش عارف ايش هذا ماخدناش التطبيقات
+
+227
+00:15:17,900 --> 00:15:20,740
+اللي هو في ال chapter اللي جابله عشان هيك بدأ
+
+228
+00:15:20,740 --> 00:15:22,000
+أقولكم find the volume
+
+229
+00:15:26,140 --> 00:15:30,440
+هذه الدائرة مغلقة
+
+230
+00:15:30,440 --> 00:15:34,300
+بواسطة السليندر X تربيع Y تربيع X تربيع Y تربيع X
+
+231
+00:15:34,300 --> 00:15:35,020
+تربيع Y تربيع X تربيع Y تربيع X تربيع Y تربيع X
+
+232
+00:15:35,020 --> 00:15:35,980
+تربيع Y تربيع X تربيع Y تربيع X تربيع Y تربيع X
+
+233
+00:15:35,980 --> 00:15:39,000
+تربيع Y تربيع X تربيع Y تربيع X تربيع Y تربيع X
+
+234
+00:15:39,000 --> 00:15:47,080
+تربيع Y تربيع X تربيع Y تربيع X تربيع Y تربيع X
+
+235
+00:15:47,080 --> 00:15:53,940
+تربيعالان بيقول
+
+236
+00:15:53,940 --> 00:15:57,200
+ان البرابولويد
+
+237
+00:15:57,200 --> 00:16:02,840
+مجموعة من البرابولويد ومن فوق من أعلى
+
+238
+00:16:06,820 --> 00:16:10,300
+and bounded below by the x-wavelength إذا على طول
+
+239
+00:16:10,300 --> 00:16:14,300
+حتى لو ما رسمناش جاي من عند اللي هي اللي هي من عند
+
+240
+00:16:14,300 --> 00:16:20,440
+السفر ورايحالمين لاللي هي ال Z اطلع هيك Vertical
+
+241
+00:16:20,440 --> 00:16:23,600
+line Vertical line كلها جاي من عند السفر و جاي لك
+
+242
+00:16:23,600 --> 00:16:29,260
+من أعلى ال paraboloid لعند اللي هوز تربيع ز وي
+
+243
+00:16:29,260 --> 00:16:31,880
+تربيع ز وي تربيع ز وي تربيع ز وي تربيع ز وي تربيع
+
+244
+00:16:31,880 --> 00:16:36,780
+ز وي تربيع ز وي تربيع ز وي تربيع ز وي تربيع ز وي
+
+245
+00:16:36,780 --> 00:16:39,860
+تربيع ز وي تربيع ز وي تربيع ز وي تربيع ز وي تربيع
+
+246
+00:16:39,860 --> 00:16:41,340
+ز وي تربيع ز وي تربيع ز وي تربيع ز وي تربيع ز وي
+
+247
+00:16:41,340 --> 00:16:50,500
+ز وي تربيع ز وي تربيع ز وي تربيع ز وي تربيع ز وي
+
+248
+00:16:50,500 --> 00:16:57,420
+تربيع ز وي تربيع ز وي تربيع ز وي تربيع ز وعن ال
+
+249
+00:16:57,420 --> 00:17:03,940
+volume دي
+
+250
+00:17:03,940 --> 00:17:09,700
+زد دي ار دي ار دي ثتا بدنا نوجد ايه احنا الآن بدنا
+
+251
+00:17:09,700 --> 00:17:12,820
+نوجد يا جماعة اللي هو حدود ال integration زي ما
+
+252
+00:17:12,820 --> 00:17:18,340
+قلنا ال z بنطلع من هنا في داخل المنطقة من هنا كله
+
+253
+00:17:18,340 --> 00:17:24,090
+بيجي من ال zeroلعند مين ؟ لعند الـR تربيع نجي الأن
+
+254
+00:17:24,090 --> 00:17:27,830
+ايش نكمل نكمل اللي هو بدنا نعمل projection لهذه
+
+255
+00:17:27,830 --> 00:17:32,310
+واضح ال projection هنا هذا كله بنزل و بقعد وين في
+
+256
+00:17:32,310 --> 00:17:36,410
+داخل اللي هو الدائرة اللي عندي هذا الجسم كله نزله
+
+257
+00:17:36,410 --> 00:17:40,270
+هيك هتلاقيه في داخل الدائرة الآن الدائرة اللي عندي
+
+258
+00:17:40,270 --> 00:17:44,910
+هذه اللي هي بتحدد لR وθسهل الأمور هي X تربيع زي Y
+
+259
+00:17:44,910 --> 00:17:48,150
+تربيع بسوء أربعة يعني R تربيع بسوء أربعة يعني R
+
+260
+00:17:48,150 --> 00:17:52,210
+بتسوء اتنين اذا هذه دائرة R بتسوء اتنين مكتملة
+
+261
+00:17:52,210 --> 00:17:55,650
+الاركان وهذا ما اتعلمناها مزام بيسار بتسوء اتنين
+
+262
+00:17:55,650 --> 00:18:00,210
+اذا اللي هي ال ال ال ال بندخل المنطقة من Zero لعند
+
+263
+00:18:00,210 --> 00:18:03,410
+اللي هي اتنين ال R هذه من Zero لعند اتنين من Zero
+
+264
+00:18:03,410 --> 00:18:06,450
+لعند اتنين في كل الاتجاهات اذا كلها على نفس
+
+265
+00:18:06,450 --> 00:18:10,510
+المنوال تبدأ من Zero لعند اتنين اذا حدود R من سفر
+
+266
+00:18:10,510 --> 00:18:14,940
+لعند مين لعند اتنينالآن واضح أنه اللي هو θ عنده
+
+267
+00:18:14,940 --> 00:18:18,680
+اللي هي .. عشان ترسم الدائرة كاملة عشان تاخد θ
+
+268
+00:18:18,680 --> 00:18:22,720
+واضحة طبيقه كامل بدها تمشي اللي هو من 0 لعندمين
+
+269
+00:18:22,720 --> 00:18:26,840
+اتنين باي اذا هذه اللي هي حدود التكامل الان ال
+
+270
+00:18:26,840 --> 00:18:30,470
+volume بالكامل لاقلش بالنسبة لل z بيصير zتكمل دي
+
+271
+00:18:30,470 --> 00:18:33,810
+زد زد عبارة عن زد من صفر لأر تربيع بنعوض بيصير R
+
+272
+00:18:33,810 --> 00:18:38,230
+تربيع و في R بيصير R تكعيد تكملها من صفر لأتنين دي
+
+273
+00:18:38,230 --> 00:18:41,410
+R اللي هي R أقصى أربعة على أربعة من صفر لأتنين
+
+274
+00:18:41,410 --> 00:18:44,370
+بيصير اتنين أقصى أربعة اللي هو اتنين في اتنين
+
+275
+00:18:44,370 --> 00:18:46,950
+أربعة في أربعة في ستة عشرة على أربعة بتطلع أربعة
+
+276
+00:18:46,950 --> 00:18:51,210
+من صفر لتو باي دي ثتا بنوجد اللي هو بيصير أربعة
+
+277
+00:18:51,210 --> 00:18:55,470
+ثتا من صفر لأتنين باي بتطلع اش تمانية باي هذه اللي
+
+278
+00:18:55,470 --> 00:19:00,260
+هي ال volumeالان نجي نلاقش الـ Spherical
+
+279
+00:19:00,260 --> 00:19:02,920
+Coordinates and Integration
+
+280
+00:19:09,350 --> 00:19:11,970
+النوع الثالث طبعا هذا اللي هو spherical
+
+281
+00:19:11,970 --> 00:19:15,630
+coordinates and integration اللي انا بدنا نشوف
+
+282
+00:19:15,630 --> 00:19:22,290
+اللي هو ايش اللي بحدد مكان النقطة كنا نحدد سابقا
+
+283
+00:19:22,290 --> 00:19:26,990
+بواسطة اللي هو الـZ و بواسطة الـR و الـθتا لما نصل
+
+284
+00:19:26,990 --> 00:19:32,370
+لنقطة Phi بواسطة اللي هو directed distance ρهو
+
+285
+00:19:32,370 --> 00:19:37,170
+زاوية فاي التي يصنعها الـ Raw مع مين التي تأخذ من
+
+286
+00:19:37,170 --> 00:19:45,650
+0 إلى أرقام كمان تريدها التي هي زاوية الـ positive
+
+287
+00:19:45,650 --> 00:19:50,410
+z-axis التي تتحرك بالشكل هذا الشكل لما تصل الـ z
+
+288
+00:19:50,410 --> 00:19:55,930
+-axis الثاني يعني فاي التي تأخذ قيمة بس من صفرلعند
+
+289
+00:19:55,930 --> 00:20:00,250
+Y طبعا الان اللي هي ابتصل لأي نقطة اللي هو بال
+
+290
+00:20:00,250 --> 00:20:03,270
+Direct distance روه طب لو كانت النقطة على الجهة
+
+291
+00:20:03,270 --> 00:20:06,750
+الثانية مين اللي بيساعدها فتا اللي هي فتا بتدور
+
+292
+00:20:06,750 --> 00:20:11,010
+بمقدار تلكتون ستين درجة فبتدور في كل الفراغ اللي
+
+293
+00:20:11,010 --> 00:20:16,130
+هو من أعلى و بكل اللي هوبقدش طول هذا فبتصل لكل
+
+294
+00:20:16,130 --> 00:20:20,790
+النقاط و نفس الاشي لما تنزل هنا اللي هي Raw لتحت
+
+295
+00:20:20,790 --> 00:20:28,130
+اللي هي بقى برضه من 0 ل 2π بلزمش أكتر من هي الان
+
+296
+00:20:28,130 --> 00:20:31,830
+اللي بدورها كامل اللي هو بتدورها ثقة كاملة وينما
+
+297
+00:20:31,830 --> 00:20:36,220
+بدها هان وطولها بصل لأي رقم في الدنياعشان هي
+
+298
+00:20:36,220 --> 00:20:41,680
+أحداكيات الـ spherical coordinates هي عبارة عن
+
+299
+00:20:41,680 --> 00:20:45,440
+روه، directed distance، هذا كبير أو سفر اللي عنده
+
+300
+00:20:45,440 --> 00:20:49,480
+اللي هي فاي اللي هي زاوية من سفر لإتنين باى بس نحن
+
+301
+00:20:49,480 --> 00:20:53,940
+نعمل محور z أكسس مع ال positive z أكسس بس بتاخد من
+
+302
+00:20:53,940 --> 00:20:57,540
+سفر لباى وثم احنا عارفين اللي في البولر بتاخد من
+
+303
+00:20:57,540 --> 00:21:03,750
+سفر لعند اتنين باىالان هناك علاقات بين رو و فاي و
+
+304
+00:21:03,750 --> 00:21:10,430
+تاتا و Z و R
+
+305
+00:21:10,430 --> 00:21:16,030
+و X و Y و Z دعونا نشوف هذه العلاقات بعد قليل الان
+
+306
+00:21:16,030 --> 00:21:19,810
+الوضع الاسفريكالي يمثل نقطة B في المخصص بمعرفة
+
+307
+00:21:19,810 --> 00:21:23,670
+order of the triple كما قلناRow is the distance
+
+308
+00:21:23,670 --> 00:21:26,490
+from B to the origin زي ما قلنا فاي is the angle
+
+309
+00:21:26,490 --> 00:21:30,850
+OB mix with the positive z axis زي ما حكينا اللي
+
+310
+00:21:30,850 --> 00:21:33,870
+هي theta is the angle from cylindrical coordinates
+
+311
+00:21:33,870 --> 00:21:37,770
+اللي احنا عارفينها هذه اللي هي زاوية اللي بتنصنع
+
+312
+00:21:37,770 --> 00:21:42,570
+من هذهطيب الان نجي نشوف العلاقات بين هدولة بعد
+
+313
+00:21:42,570 --> 00:21:45,550
+شوية خلينا نشوف العلاقات بس على السريع لو اتطلعنا
+
+314
+00:21:45,550 --> 00:21:50,990
+عندي اللي هو ال raw هي ال raw لو اجينا نزلنا عمود
+
+315
+00:21:50,990 --> 00:21:54,750
+هنا هيك فبصير هذه الزاوية قائمة و الآن الزاوية
+
+316
+00:21:54,750 --> 00:21:59,090
+القائمة هذه بيصير عند الان هذا ال هنا هي العمود
+
+317
+00:21:59,090 --> 00:22:03,130
+هيك بالشكل هذا فبيصير الان لو اجينا أخدنا اللي هي
+
+318
+00:22:03,130 --> 00:22:09,600
+cosine phi cosine phi هتصير ال rawالتي هي الوطر
+
+319
+00:22:09,600 --> 00:22:15,600
+على مجاور قيمة z روح على z فهكذا نربط z تصبح z
+
+320
+00:22:15,600 --> 00:22:20,660
+روكو صين فاى لأن هذا هيساوي هذا لأنه لما نزل
+
+321
+00:22:20,660 --> 00:22:24,640
+عمودها تصبح هذا الشكل كله مستطيل فهكذا
+
+322
+00:22:28,370 --> 00:22:34,430
+الان لو أخدنا هذا هى هك اللى هو هذا عمودي هى موازي
+
+323
+00:22:34,430 --> 00:22:39,030
+هذا اللى هو فى الواقع هو R فبكون قيمة R هنا هى هك
+
+324
+00:22:39,030 --> 00:22:44,070
+اللى هو عبارة عن نفس الأسلوب اللى هو رو ساين فايد
+
+325
+00:22:44,070 --> 00:22:48,090
+لإن ساين فايد هسير رو على اللى هو R هذا اللى بيسير
+
+326
+00:22:48,090 --> 00:22:52,410
+رو
+
+327
+00:22:52,410 --> 00:22:58,450
+ساين فايدالان العلاقات الأخرى هي العلاقات الرئيسية
+
+328
+00:22:58,450 --> 00:23:02,350
+العلاقات الأخرى لو جينا أخدنا اللي هو هذا رو اللي
+
+329
+00:23:02,350 --> 00:23:07,910
+هو هنجده بعد شوية رو تربيع بساوية اللي هو عبارة عن
+
+330
+00:23:07,910 --> 00:23:12,050
+اللي هي هذه زاوية قائمة تصير هذه قائمة هيها تصير
+
+331
+00:23:12,050 --> 00:23:15,540
+رو تربيع بساوية زي التربيع زي ال R تربيعيعني لو
+
+332
+00:23:15,540 --> 00:23:20,800
+بدنا نحوّلها لكارتيزيان بسيرة R تربيع بسيرة Z
+
+333
+00:23:20,800 --> 00:23:25,980
+تربيع وR تربيع بسيرة Y تربيع بسيرة X تربيع بسيرة Z
+
+334
+00:23:25,980 --> 00:23:29,760
+تربيع بسيرة Y تربيع بسيرة Z تربيع بسيرة Z تربيع
+
+335
+00:23:29,760 --> 00:23:34,420
+بسيرة Y تربيع بسيرة Z تربيع بسيرة Z تربيع بسيرة Z
+
+336
+00:23:34,420 --> 00:23:34,460
+تربيع بسيرة Z تربيع بسيرة Z تربيع بسيرة Z تربيع
+
+337
+00:23:34,460 --> 00:23:44,880
+بسيرة Z تربيع بسيرة Z تربيع بسيرة Z تربيع
+
+338
+00:23:45,620 --> 00:23:53,300
+اللي هي رو فاى بتساوي فاى نوت يعني فاى ثابتة الان
+
+339
+00:23:53,300 --> 00:23:57,200
+فاى نوت هي قيمة فاى نوت ماشي الحال الان مين
+
+340
+00:23:57,200 --> 00:24:00,700
+المطلوقة المطلوقة اللي هي رو بتضل هذا ماشي إلى ما
+
+341
+00:24:00,700 --> 00:24:05,240
+لنهاية هيكوثيتا مطلوقة إذا ثيتا بتلف لفتة كاملة
+
+342
+00:24:05,240 --> 00:24:09,860
+فالآن هذا الخط اللي هو عندى لما يلف لفة كاملة
+
+343
+00:24:09,860 --> 00:24:13,660
+حوالينى اللى هو الـ z axis بعملى كون الكون هذا
+
+344
+00:24:13,660 --> 00:24:17,360
+ببدأ من نقطة الأصل و بظله إلى ما لها إذا فاى
+
+345
+00:24:17,360 --> 00:24:20,860
+بتساوي فاى نوت هو عبارة عن معادلة كون شايفين مثله
+
+346
+00:24:20,860 --> 00:24:24,900
+لو بدنا نحوّلها لـ Cartesian فيها غلبة جصد فيها ده
+
+347
+00:24:24,900 --> 00:24:29,800
+فاشا لدك لكن أخدناها قبل ذلك المهمالان لو جينا
+
+348
+00:24:29,800 --> 00:24:35,710
+اخدنا θ بالثاوية اللى هو عندى θ بالثاوية θ نوتالان
+
+349
+00:24:35,710 --> 00:24:38,950
+theta بالساوية theta note هي ال theta note هي التي
+
+350
+00:24:38,950 --> 00:24:44,830
+تبع ال R التي هي theta note هيها الآن ال R تبعها
+
+351
+00:24:44,830 --> 00:24:48,550
+هي و فوق ال R طبعا بيسير اللي هو بيمتد هيك لما
+
+352
+00:24:48,550 --> 00:24:52,590
+لانهية لكن شغله من هنا لعند ال هنا بس عشان هيك
+
+353
+00:24:52,590 --> 00:24:55,470
+theta بالساوية theta note في هذه الحالة بس هو هذا
+
+354
+00:24:55,470 --> 00:24:58,590
+اللي بالأزرق اللي بيمتد هيك لما لانهية هيك لما
+
+355
+00:24:58,590 --> 00:25:01,530
+لانهية و لا جاي لما لانهية لكن بيروش على الجهة
+
+356
+00:25:01,530 --> 00:25:06,920
+التالية طيب الآنلأن θ مثلا في المنطقة دي لو θ
+
+357
+00:25:06,920 --> 00:25:09,680
+بيساوي مثلا في الجهة التانية بيصير الشكل في الجهة
+
+358
+00:25:09,680 --> 00:25:13,360
+التانية حسب قيمة θ اللي هو الـ Plane بيكون إذا θ
+
+359
+00:25:13,360 --> 00:25:17,520
+بيساوي θ نوت الـ Plane من الـ Z و بيظل طالع بالشكل
+
+360
+00:25:17,520 --> 00:25:22,340
+العادى الآن ضال عندي رو بتساوي .. رو بتساوي إيه؟
+
+361
+00:25:22,340 --> 00:25:25,660
+الرو ثابتة هي رو بتساوي إيه؟ هذا رو بتساوي إيه
+
+362
+00:25:25,660 --> 00:25:32,230
+لهنا مثلا؟الان فاي هتكون متغيرة فبتيجي الفاي بيمشي
+
+363
+00:25:32,230 --> 00:25:37,910
+هيك هيك هيك بيرسم ليه نص دائرة هذا نص الدائرة اللي
+
+364
+00:25:37,910 --> 00:25:40,770
+بيرسمها روه اللي هو نص قطرها روه اللي هي نص
+
+365
+00:25:40,770 --> 00:25:45,900
+الدائرة اللي بيرسمها روه الان هي ماشيها بس فايالان
+
+366
+00:25:45,900 --> 00:25:51,300
+فاى المتغيرة ترسم لراو بساوي اللى هو عبارة عن مثلا
+
+367
+00:25:51,300 --> 00:25:57,520
+مقدار رونالد اللى هو ايه قولنا الان هذا الان فاى
+
+368
+00:25:57,520 --> 00:26:01,140
+متغيرة و θ متغيرة مدام تت متغيرة ده نص الدائرة ده
+
+369
+00:26:01,140 --> 00:26:05,640
+هتدور دورة كاملة حوالين ال z axis ده هتعمل لكرة
+
+370
+00:26:05,640 --> 00:26:10,200
+عشان هيك راو بتساوي ايه عبارة عن الاسفير اللى لها
+
+371
+00:26:14,160 --> 00:26:17,320
+العلاقة بين equations related spherical
+
+372
+00:26:17,320 --> 00:26:19,740
+coordinates to cartesian and cylindrical
+
+373
+00:26:19,740 --> 00:26:23,420
+coordinates نجي الآن اللي وعدناكم قبل شوية فيها
+
+374
+00:26:23,420 --> 00:26:27,910
+علاقة بينالـ Z بيساوي رو كوساين فاى و رو بيساوي
+
+375
+00:26:27,910 --> 00:26:30,030
+الجدر التربيع ال X تربيع زي ال Y تربيع زي ال Z
+
+376
+00:26:30,030 --> 00:26:36,430
+تربيع و ال X بيساوي R كوساين تيتا سابقا و ال Y
+
+377
+00:26:36,430 --> 00:26:40,290
+بيساوي R ساين تيتا سابقا ال R بنشيله و بنعوض عنه
+
+378
+00:26:40,290 --> 00:26:44,290
+بيصير رو ساين فاى كوساين تيتا و ال R بنشيله و
+
+379
+00:26:44,290 --> 00:26:48,230
+بنعوض عنها من قيمتها بيصير رو أو ساين فاى كوساين
+
+380
+00:26:48,230 --> 00:26:52,930
+تيتاوهي علاقات
+
+381
+00:26:52,930 --> 00:27:01,050
+بين X و Y و Z و ρو و فاي و فتا و R صحيح؟ ناخد مثال
+
+382
+00:27:01,050 --> 00:27:02,130
+مثال تلاتة
+
+383
+00:27:09,380 --> 00:27:14,300
+هذه الاسفير مركزها 0 و0 و1 على الـ z إذاً فوق
+
+384
+00:27:14,300 --> 00:27:17,960
+قليلا و نصف قطرها واحد إذاً على ال z فوق اللي يعني
+
+385
+00:27:17,960 --> 00:27:21,920
+بتمس اللي هي ال origin لما نقل من نصف قطر واحد و
+
+386
+00:27:21,920 --> 00:27:26,460
+البعد واحد فالسفير بتمس ال originهذه هي الاسفير
+
+387
+00:27:26,460 --> 00:27:31,560
+اللي بيقول عنها هي X تربيع زائد Y تربيع زائد زائد
+
+388
+00:27:31,560 --> 00:27:33,840
+minus واحد كل تربيع يتساوى واحد اللي هي سفير
+
+389
+00:27:33,840 --> 00:27:39,920
+مركزها 001 عند الان نشوف ايش اللي هي معدلتها الان
+
+390
+00:27:39,920 --> 00:27:43,100
+نحولها ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل
+
+391
+00:27:43,100 --> 00:27:43,180
+.. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل
+
+392
+00:27:43,180 --> 00:27:43,500
+.. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل
+
+393
+00:27:43,500 --> 00:27:43,700
+.. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل
+
+394
+00:27:43,700 --> 00:27:43,780
+.. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل
+
+395
+00:27:43,780 --> 00:27:45,140
+.. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل
+
+396
+00:27:45,140 --> 00:27:46,160
+.. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل .. ل
+
+397
+00:27:46,160 --> 00:27:46,200
+.. ل
+
+398
+00:27:51,030 --> 00:27:56,170
+الـ Y تربيع هي رو تربيع صين تربيع فاي صين تربيع
+
+399
+00:27:56,170 --> 00:28:00,570
+تيتا زائد الـ Z نفسي بإيشي رو كوسين فاي اللي هي
+
+400
+00:28:00,570 --> 00:28:04,330
+نقص واحد لكل تربيع بساوي واحد لان خد من الجهتين
+
+401
+00:28:04,330 --> 00:28:08,540
+هدول اللي هو رو تربيع صين تربيعفاى بيبقى cos تربيع
+
+402
+00:28:08,540 --> 00:28:12,660
+تتا زائد sin تربيع تتا هذا طبعا واحد وهذا فكه
+
+403
+00:28:12,660 --> 00:28:17,060
+بيصير رو تربيع cos تربيع نقص اتنين رو cos فاى زائد
+
+404
+00:28:17,060 --> 00:28:20,920
+الواحد فكنا هذا المقدار كله بيساوي واحد الا ان
+
+405
+00:28:20,920 --> 00:28:25,020
+الواحد مع الواحد بروح وهذا بيصير واحد فبيصير رو
+
+406
+00:28:25,020 --> 00:28:29,120
+تربيعاللي هي sin تربيع زي رو تربيع كو سين تربيع
+
+407
+00:28:29,120 --> 00:28:32,500
+نقص اتنين رو كو سين فاى بنجيها على الجهة دى بيصير
+
+408
+00:28:32,500 --> 00:28:35,580
+اللى هو رو تربيع في sin تربيع زي كو سين تربيع اللى
+
+409
+00:28:35,580 --> 00:28:39,020
+بيساوي اتنين رو كو سين فاى الى ان هذه بيصير واحد
+
+410
+00:28:39,020 --> 00:28:41,700
+بيصير رو تربيع بيساوي اتنين رو كو سين فاى والرو
+
+411
+00:28:41,700 --> 00:28:44,860
+اكبر من سفر اذا بقدر اختصر بيصير رو بيساوي اتنين
+
+412
+00:28:44,860 --> 00:28:51,380
+كو سين فاى وهذه عبارة عن معادلة السفير اللى أمامى
+
+413
+00:28:53,960 --> 00:28:56,820
+لنجد مثال أخر مثال أربعة find the spherical
+
+414
+00:28:56,820 --> 00:28:59,780
+coordinate equation for the cone z بيستوي الجدر
+
+415
+00:28:59,780 --> 00:29:03,640
+التربيع ل X تربيع زي Y تربيع نجي نشوف الآن ال Z
+
+416
+00:29:03,640 --> 00:29:07,580
+إيش هي رو كسين فاي بيستوي الجدر التربيع ل X تربيع
+
+417
+00:29:07,580 --> 00:29:12,160
+زي Y تربيع قبل شوية عملناها اللي هو رو تربيع سين
+
+418
+00:29:12,160 --> 00:29:12,880
+تربيع
+
+419
+00:29:16,000 --> 00:29:19,760
+هذه التربية هي X تربيع زائد Y تربيع وهي X تربيع
+
+420
+00:29:19,760 --> 00:29:23,340
+زائد Y تربيع حيها وهي تتطلع تقن دي رو تربيع ساين
+
+421
+00:29:23,340 --> 00:29:26,660
+تربيع فاي وهذا طبعا كانت اي شمالة واحد دم اخدنا
+
+422
+00:29:26,660 --> 00:29:27,360
+عنه اشترك
+
+423
+00:29:32,720 --> 00:29:37,080
+عن طريق الوصول للجدر
+
+424
+00:29:37,080 --> 00:29:41,880
+التربيعي الوصول
+
+425
+00:29:41,880 --> 00:29:45,860
+للجدر التربيعي الوصول للجدر التربيعي الوصول للجدر
+
+426
+00:29:45,860 --> 00:29:49,250
+التربيعياللي هنصر رو كوساين فاي بيساوي رو ساين فاي
+
+427
+00:29:49,250 --> 00:29:51,630
+الراو بتروح مع الراو ان انا أكبر من سفر بيبقى رو
+
+428
+00:29:51,630 --> 00:29:54,850
+كوساين فاي بيساوي رو ساين فاي الان ال كوساين
+
+429
+00:29:54,850 --> 00:29:57,190
+بيساوي ال sign عند اللي هو رو كوساين فاي بيساوي رو
+
+430
+00:29:57,190 --> 00:30:00,450
+ساين فاي على أربعة لإنه بس اللي هي فاينة من سفر
+
+431
+00:30:00,450 --> 00:30:02,810
+لعين باي و بيكون رو كوساين فاي بتساوي رو ساين فاي
+
+432
+00:30:02,810 --> 00:30:09,090
+على أربعة هي معادلة الكون اللي عندنا هذا اللي هو
+
+433
+00:30:09,090 --> 00:30:13,010
+ده طبعا عبارة عن الكون رو كوساين فاي بتساوي أربعة
+
+434
+00:30:13,930 --> 00:30:17,370
+الان نتكلم عن كيفية انتجار الاسفريكال الاسفريكال
+
+435
+00:30:17,370 --> 00:30:22,750
+كالسابق الروح الاسفريكال الاسفريكال الاسفريكال
+
+436
+00:30:22,750 --> 00:30:32,730
+الاسفريكال الاسفريكال الاسفريكال
+
+437
+00:30:36,700 --> 00:30:46,340
+DV بيساوي DX DY DZ اللي هي نفسها DV بيساوي DR Dθ
+
+438
+00:30:46,340 --> 00:30:48,380
+DZ
+
+439
+00:30:54,320 --> 00:30:58,360
+مش مش معنية بالاشتقاق لكن مش صعب الاشتقاق خلجت
+
+440
+00:30:58,360 --> 00:31:02,100
+بتقولكوا كيف الواروتر بي assign فاي دي فاي دي رو
+
+441
+00:31:02,100 --> 00:31:07,390
+دي ثتا دي فاي دي إيش دي ثتاالان الـ DV هيكون عبارة
+
+442
+00:31:07,390 --> 00:31:14,910
+عن D ρو في رو D فاي في رو سين فاي في D ثتا الان
+
+443
+00:31:14,910 --> 00:31:20,670
+هذا اللي هو عبارة عن روتر بيه سين فاي D رو D فاي D
+
+444
+00:31:20,670 --> 00:31:24,570
+ثتا طبعا فيه التفاصيلة ده فيه اللي هو ان احنا نحكي
+
+445
+00:31:24,570 --> 00:31:28,770
+كيف نجده لكن البرهان اللي هو مش مطلوب فعشان هي ده
+
+446
+00:31:28,770 --> 00:31:33,130
+اللي هو بس DV بيستوي روتر بيه سين فاي D رو D فاي D
+
+447
+00:31:33,130 --> 00:31:39,730
+ثتا و بعينكم اللهاللي هو بتحفظوها طبعا اللي بنسا
+
+448
+00:31:39,730 --> 00:31:43,330
+بتروح للاشتقاقات العلاقات اللي قبل بشوية بتجيب هذه
+
+449
+00:31:43,330 --> 00:31:45,750
+العلاقات اللي حكيناها قبل بشوية لكن مابديش
+
+450
+00:31:45,750 --> 00:31:51,220
+التفاصيلالان كيف بدنا نعمل integration لـ
+
+451
+00:31:51,220 --> 00:31:54,280
+spherical coordinates عند المنطقة دى اول اشي برسم
+
+452
+00:31:54,280 --> 00:31:59,760
+المنطقة دى اللى عندي الان عندي اللى هو روه كامل
+
+453
+00:31:59,760 --> 00:32:03,420
+بالنسبة لروه بعدين five بعدين بالنسبة لفيتا بنشوف
+
+454
+00:32:03,420 --> 00:32:06,860
+كيف بدنا نسمع لان فرضنا ان هذا اللى هو عبارة عن
+
+455
+00:32:06,860 --> 00:32:10,000
+روه بالساويةهذه هي المجسم اللي عندي بده أشوف كيف
+
+456
+00:32:10,000 --> 00:32:13,880
+بدنا أعمله طبعا أكيد هاخد رو رو من نقطة الأصل بطلع
+
+457
+00:32:13,880 --> 00:32:16,600
+Directed Distance من وين بدخل المنطقة و من وين
+
+458
+00:32:16,600 --> 00:32:21,620
+بخرج المنطقة بدخلها من أندي 1 فاتا دي 1 فيو و فاتا
+
+459
+00:32:21,620 --> 00:32:24,800
+دي لعندي 2 فيو و فاتا هي اللي هو Directed Distance
+
+460
+00:32:24,800 --> 00:32:28,020
+و الرو بطلع من نقطة الأصل و بمشي معها هذا بدخل من
+
+461
+00:32:28,020 --> 00:32:31,420
+هنا و بخرج من هنا إذا من المنطقةكلهم على نفس
+
+462
+00:32:31,420 --> 00:32:35,660
+المنوال وهي هنا أوضح للصورة هذه عندي ال row بيدخل
+
+463
+00:32:35,660 --> 00:32:39,900
+هيك ويطلع اذا بيبدأ من عندي one of phi of theta او
+
+464
+00:32:39,900 --> 00:32:42,980
+بتل عندي اتنين phi of theta هذه حدود ال
+
+465
+00:32:42,980 --> 00:32:46,920
+integration الا طبعا لل F of row فاي وثيتا رو
+
+466
+00:32:46,920 --> 00:32:50,320
+تربيع صين تربيع فاي هذه كلها عبارة عن اللي هو ال
+
+467
+00:32:50,320 --> 00:32:55,580
+DV طيب دي جيه الآن لمين لل phi الان الفي الفي لو
+
+468
+00:32:55,580 --> 00:32:59,440
+جينا فرد الفي من هنا الفي تاخد في الفي تبدأ من
+
+469
+00:32:59,440 --> 00:33:05,280
+عندها دهوبتنتهي وين ؟ عند هنا اذا الفايل اللي عند
+
+470
+00:33:05,280 --> 00:33:10,420
+هذه فيه اللي بتبدأ اللي هي من عند اللي هي المنطقة
+
+471
+00:33:10,420 --> 00:33:15,400
+اللي هي الجدة اللي هي المنطقة اللي هي الجدة اللي
+
+472
+00:33:15,400 --> 00:33:19,580
+هي المنطقة اللي هي الجدة اللي هي المنطقة اللي هي
+
+473
+00:33:19,580 --> 00:33:20,180
+الجدة اللي هي المنطقة اللي هي الجدة اللي هي
+
+474
+00:33:20,180 --> 00:33:20,790
+اللي هي الجدة اللي هي الجدة اللي هي الجدة اللإذا
+
+475
+00:33:20,790 --> 00:33:25,750
+هذه هي الـ minimum لها و الآن هذا بيجهت لعند أخرها
+
+476
+00:33:25,750 --> 00:33:29,210
+هذه اللي هو عبارة عن اللي عندها ناسفة اللي عندها
+
+477
+00:33:29,210 --> 00:33:33,610
+ناسفة هي اللي هي ايش فاي الأخيرة إذا هذه فاي
+
+478
+00:33:33,610 --> 00:33:39,130
+الأولى لهنا ايها و فاي الأخيرة لهنا فبيصير اللي هي
+
+479
+00:33:39,130 --> 00:33:43,050
+فاي من وين لوينطبعا الفهمين كيف تعمل من ال minimum
+
+480
+00:33:43,050 --> 00:33:50,810
+لعند ال maximum هذه قيمة في هي تبحث للجسم اللي
+
+481
+00:33:50,810 --> 00:33:54,130
+بدها و هو بآخر الجسم هيه فتكون من ال minimum
+
+482
+00:33:54,130 --> 00:33:57,460
+للماكسماالان نجي لمين؟ لكيه؟ طب نعمل ال projection
+
+483
+00:33:57,460 --> 00:34:00,960
+هذا تحت لازم نعمله طلع عند ال projection زي ما
+
+484
+00:34:00,960 --> 00:34:05,100
+قلنا هنا الان هنا بدأت theta هي عند alpha node وهي
+
+485
+00:34:05,100 --> 00:34:09,500
+نهاية theta لعند beta من alpha لعند beta هذه اللي
+
+486
+00:34:09,500 --> 00:34:14,960
+هي بتمسح كل اللي هو ال projection تبع الجسم فبيكون
+
+487
+00:34:14,960 --> 00:34:18,240
+دي theta الأخيرة هي عبارة عن من alpha لعند beta
+
+488
+00:34:19,260 --> 00:34:22,680
+نأخد مثلًا find the volume of the ice cream cone
+
+489
+00:34:35,010 --> 00:34:39,610
+الان اللى هى المنطقة محصورة بين جزء من الاسفير
+
+490
+00:34:39,610 --> 00:34:45,810
+الراوه من اللى هى ال solid sphere يعنى الجشرة مع
+
+491
+00:34:45,810 --> 00:34:49,550
+الداخل روه أصغر بتساوي واحد هى روه بتساوي واحد
+
+492
+00:34:49,550 --> 00:34:53,590
+هيها اللى هى بترسملى ايش الآن روه بتساوي واحد هيها
+
+493
+00:34:53,590 --> 00:34:59,030
+بترسملى اللى هى عبارة عن كرةمركزها نقطة الأصل هيها
+
+494
+00:34:59,030 --> 00:35:03,990
+الآن هي الآن بدنا الجزء هذا اللي من الكرة مع مين
+
+495
+00:35:03,990 --> 00:35:07,750
+مع الكون اللي تحت اللي هو five تساوية على تلاتة
+
+496
+00:35:07,750 --> 00:35:12,830
+فالان هي الكون وهي اللي هو الاسفير هي المنطقة اللي
+
+497
+00:35:12,830 --> 00:35:16,510
+بين الكون والاسفير هي المنطقة المطلوبة اللي هي ال
+
+498
+00:35:16,510 --> 00:35:19,430
+ice cream cone بدنا نجد ال volume منها ال volume
+
+499
+00:35:19,430 --> 00:35:23,240
+اللي هو عبارة عن ال integration لل DVDاللي هو لو
+
+500
+00:35:23,240 --> 00:35:25,900
+تربيع sin فعال جي فعال جي رو دي دي فعال ايش بنيجي
+
+501
+00:35:25,900 --> 00:35:30,500
+لقه بناخد ال distance من نقطة الاصل و بنمشي لدخول
+
+502
+00:35:30,500 --> 00:35:33,760
+الجسم طبعا الجسم مدخول منه من عند ال zero و بيبقى
+
+503
+00:35:33,760 --> 00:35:38,400
+ماشي لما تطلع من ال sphere اذا رو من سفر لعند ايش
+
+504
+00:35:38,400 --> 00:35:44,940
+واحد لان رو تابتة واحد على ال sphereالان بدنا نيجي
+
+505
+00:35:44,940 --> 00:35:49,060
+لمين ؟ لأ اللى فاى فاى الان هذه اللى بتبقى ماشية
+
+506
+00:35:49,060 --> 00:35:54,590
+منها لها لما تصل عند اخر الكونالان عشان تنسح
+
+507
+00:35:54,590 --> 00:35:57,550
+المنطقة هذه كلها يامن سفر بتبقى ال five بتنتهي
+
+508
+00:35:57,550 --> 00:36:00,810
+لعين داخل الكون اللي هو ايش بيساوي بايع تلاتة هذه
+
+509
+00:36:00,810 --> 00:36:05,710
+اللي هي الكون اللي عنده اللي بتخلص اللي هي اعلى كل
+
+510
+00:36:05,710 --> 00:36:09,870
+المنطقة هذه فبيسير عنده من سفر لمين لبايع تلاتة
+
+511
+00:36:09,870 --> 00:36:13,170
+اللي هو الكون الان اعمل projection لهذا واضح ال
+
+512
+00:36:13,170 --> 00:36:16,930
+projection ايش هيطلع دائرة دائرة مكتملةلأن باسفير
+
+513
+00:36:16,930 --> 00:36:19,850
+تنزل تطلع عايش الدائرة اللي هي واحد تتغطى وحده
+
+514
+00:36:19,850 --> 00:36:25,730
+بنزل من كله من كل مكان في المنطقةفتح فيها من 0 ل 2
+
+515
+00:36:25,730 --> 00:36:31,670
+باى و بنجي من كامل ل 1 روتر بيئي بيصير روتر قعيبة
+
+516
+00:36:31,670 --> 00:36:36,710
+تلاتة من 0 ل 1 بيصير عبارة عن تلت sin فاى دى فاى
+
+517
+00:36:36,710 --> 00:36:41,750
+تلت sin فاى اللى هو ناقص تلت cos فاى من 0 لعند باى
+
+518
+00:36:41,750 --> 00:36:49,010
+ع تلاتة و بعوض بيصير عبارة عن cos فاى ع تلاتة اللى
+
+519
+00:36:49,010 --> 00:37:00,790
+عبارة عناللي هي نص في نقصه لPC Rhesusناقص تلت ناقص
+
+520
+00:37:00,790 --> 00:37:04,310
+ناقص
+
+521
+00:37:04,310 --> 00:37:08,290
+تلت ناقص تلت ناقص تلت ناقص تلت ناقص تلت ناقص تلت
+
+522
+00:37:08,290 --> 00:37:08,370
+ناقص تلت ناقص تلت ناقص تلت ناقص تلت ناقص تلت ناقص
+
+523
+00:37:08,370 --> 00:37:08,630
+تلت ناقص تلت ناقص تلت ناقص تلت ناقص تلت ناقص تلت
+
+524
+00:37:08,630 --> 00:37:09,170
+ناقص تلت ناقص تلت ناقص تلت ناقص تلت ناقص تلت ناقص
+
+525
+00:37:09,170 --> 00:37:18,850
+تلت ناقص تلت ناقص تلت ناقص تلت ناقص تلت ناقص تلت
+
+526
+00:37:18,850 --> 00:37:24,170
+ناقص تلت ناقص تلت ناقفتنين باي بيصير عبارة عن بايع
+
+527
+00:37:24,170 --> 00:37:27,530
+تلاتة و يكون هذا هو ال volume لل ice cream اللي
+
+528
+00:37:27,530 --> 00:37:32,590
+أمامنا و الآن تلخيص بس لكل اللي عملناها من
+
+529
+00:37:32,590 --> 00:37:36,490
+العلاقات coordinate convergence formula اللي هان
+
+530
+00:37:36,490 --> 00:37:39,350
+في ال cylindrical to rectangular من cylindrical ل
+
+531
+00:37:39,350 --> 00:37:43,350
+rectangular Ax بسوار cos θ و Y بسوار sin θ و Z
+
+532
+00:37:43,350 --> 00:37:46,570
+بالساوة Z و تقريبا احفظناهامن spherical to
+
+533
+00:37:46,570 --> 00:37:50,930
+rectangular X بتساوي رو صين فاي كوسين فاي وY
+
+534
+00:37:50,930 --> 00:37:52,070
+بتساوي رو صين فاي كوسين فاي وY بتساوي رو صين فاي
+
+535
+00:37:52,070 --> 00:37:53,590
+رو صين فاي كوسين فاي وY بتساوي رو صين فاي كوسين
+
+536
+00:37:53,590 --> 00:37:55,550
+فاي وY بتساوي رو صين فاي كوسين فاي وY بتساوي رو
+
+537
+00:37:55,550 --> 00:37:56,930
+صين فاي كوسين فاي وY بتساوي رو صين فاي كوسين فاي
+
+538
+00:37:56,930 --> 00:38:01,670
+وY بتساوي رو صين فاي كوسين فاي وY بتساوي رو صين
+
+539
+00:38:01,670 --> 00:38:06,970
+فاي كوسين فاي كوسين فاي وY بتساوي رو صين فاي كوسين
+
+540
+00:38:06,970 --> 00:38:10,580
+فايوهذا في الـ Cartesian في الـ Polar دي زد .. في
+
+541
+00:38:10,580 --> 00:38:14,720
+ال .. ال cylindrical دي زد ار دي ار دي تيتا في ال
+
+542
+00:38:14,720 --> 00:38:17,920
+spherical ال rotor بي assign فاي دي فاي دي رو دي
+
+543
+00:38:17,920 --> 00:38:21,580
+فاي دي تيتا او كده كده احنا انهينا المحاضرة اللي
+
+544
+00:38:21,580 --> 00:38:26,620
+هي خلال خلاص عشت سابعة و الى لقاء اخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE.srt
new file mode 100644
index 0000000000000000000000000000000000000000..17dd4e0f83931e99867eec4abf959138902c759d
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE.srt
@@ -0,0 +1,1964 @@
+1
+00:00:00,710 --> 00:00:04,770
+بسم الله الرحمن الرحيم هذه هي المحاضرة الثانية
+
+2
+00:00:04,770 --> 00:00:10,230
+مساق تفاضل وتكامل جيم لطلاب وطالبات الجامعة
+
+3
+00:00:10,230 --> 00:00:16,000
+الإسلامية، كليتي العلوم والهندسة. اليوم هنبدأ إن شاء
+
+4
+00:00:16,000 --> 00:00:21,640
+الله، اللي هو في 12 2، اللي هو vectors، اللي هي
+
+5
+00:00:21,640 --> 00:00:28,960
+المتجهات. الموضوع طبعاً موضوع تم قبل هيك، أخد فكرة
+
+6
+00:00:28,960 --> 00:00:34,480
+عنه، أو حتى يعني بعض الشغلات تم دراستها في اللي هي
+
+7
+00:00:34,480 --> 00:00:39,260
+مساق الفيزياء، في اللي هي اللي برضه لكليتي العلوم
+
+8
+00:00:39,260 --> 00:00:44,700
+والهندسة. نعرف شو معناه متجه أو شو معناه vector الآن.
+
+9
+00:00:44,700 --> 00:00:48,980
+بنجي عنا initial point، نقطة بداية a، و terminal
+
+10
+00:00:48,980 --> 00:00:53,260
+point، نقطة نهاية b. الآن الـ line segment اللي بيمتد
+
+11
+00:00:53,260 --> 00:00:59,120
+من a وبتجه لعند b، هذا بنسميه الـ vector a,b. إذا
+
+12
+00:00:59,120 --> 00:01:03,360
+اللي له نقطة بداية وله نقطة نهاية، بتجه من البداية
+
+13
+00:01:03,360 --> 00:01:06,880
+إلى النهاية، هذا اللي هو the directed line segment.
+
+14
+00:01:06,880 --> 00:01:11,220
+a,b is called a vector. بنسميه إيش يا جماعة؟ بنسميه
+
+15
+00:01:11,220 --> 00:01:15,570
+vector. الآن عنا بدنا نشوف شو معناتها إن two vectors
+
+16
+00:01:15,570 --> 00:01:20,050
+are equal. يعني كيف اللي هو عندي two vectors أو
+
+17
+00:01:20,050 --> 00:01:27,370
+متجهين متساوين. قبل نعرف شو معناتها طول الـ vector AB،
+
+18
+00:01:27,370 --> 00:01:32,390
+طول الـ vector AB هو طول القياس، الطولة ده بنسميه الـ
+
+19
+00:01:32,390 --> 00:01:36,790
+magnitude للـ vector، وبنررمزه بالرمز magnitude للـ
+
+20
+00:01:36,790 --> 00:01:43,130
+AB. الآن بنقول الـ vector AB اللي هو بيساوي الـ vector
+
+21
+00:01:43,130 --> 00:01:47,990
+CD، إذا حدث أمرين، الأمر الأول أن هذا له نفس
+
+22
+00:01:47,990 --> 00:01:52,910
+الاتجاه، يعني هذا بوازيه ومشيّات في اتجاه السهم في
+
+23
+00:01:52,910 --> 00:01:56,690
+اتجاه واحد. اتنين، يكون طول الـ vector هذا وطول الـ
+
+24
+00:01:56,690 --> 00:01:59,950
+vector هذا إيه؟ مالهم متساوية. في هذه الحالة بتقول
+
+25
+00:01:59,950 --> 00:02:03,330
+عن هذا الـ vector وهذا الـ vector إنهم are equal.
+
+26
+00:02:03,330 --> 00:02:06,130
+إذا الـ vector represented by the directed line
+
+27
+00:02:06,130 --> 00:02:10,590
+segment AB has initial point A and terminal point B
+
+28
+00:02:10,590 --> 00:02:15,430
+and its length is denoted by |AB|. بنقول إنه هذا
+
+29
+00:02:15,430 --> 00:02:22,310
+مساوي للـ vector CD، إذا كان اللي هو اتجاه الـ CD نفس
+
+30
+00:02:22,310 --> 00:02:27,180
+اتجاه الـ AB، والاثنين اللي هي نفس الـ magnitude. Two
+
+31
+00:02:27,180 --> 00:02:30,660
+vectors are equal if they have the same length and
+
+32
+00:02:30,660 --> 00:02:35,820
+the same direction. لاحظوا أن هذا وهذا وهذا، الأربعة
+
+33
+00:02:35,820 --> 00:02:39,680
+vectors هذول متوازيات، اللي هي نفس الاتجاه، وكلهم
+
+34
+00:02:39,680 --> 00:02:45,360
+اتجاه اللي أمامنا، والأربعة اللي هي نفس الطول، هذول
+
+35
+00:02:45,360 --> 00:02:48,900
+بنسميه are equal vectors. لاحظوا الـ vector هذا OB،
+
+36
+00:02:49,390 --> 00:02:53,410
+هذا الـ vector جاي من نقطة الأصل، ومنطلق للنقطة من
+
+37
+00:02:53,410 --> 00:02:57,410
+B. يعني الـ initial point له نقطة الأصل، هذا بنسميه
+
+38
+00:02:57,410 --> 00:03:01,290
+position vector. يعني الآن كل الـ vectors الموازية
+
+39
+00:03:01,290 --> 00:03:05,450
+له، كلهم بيساووه، إذا كانوا في نفس الاتجاه، واللي هو
+
+40
+00:03:05,450 --> 00:03:08,710
+نفس الطول. إذا هنتعامل مع الـ position vector بعد
+
+41
+00:03:08,710 --> 00:03:14,880
+هيك كمتجه بيمثل كل الـ vectors الأخرى. نأتي الآن
+
+42
+00:03:14,880 --> 00:03:22,740
+اللي هو نشوف إيش معناه، إيش كيف بدنا نكتب الـ vector
+
+43
+00:03:22,740 --> 00:03:28,440
+على صورة، على حاجة اسمها الـ component form. طيب، صلوا
+
+44
+00:03:28,440 --> 00:03:33,580
+على النبي عليه الصلاة والسلام. الآن عندي اللي هو if
+
+45
+00:03:33,580 --> 00:03:38,620
+V is a two-dimensional vector in the plane, equal to
+
+46
+00:03:38,620 --> 00:03:41,240
+the vector with initial point at the origin and
+
+47
+00:03:41,240 --> 00:03:45,880
+terminal point at (v1, v2). يعني الآن لو فرضنا عندي
+
+48
+00:03:45,880 --> 00:03:49,100
+الآن في الـ plane، في عندي vector v في الـ two
+
+49
+00:03:49,100 --> 00:03:53,660
+dimensions. هذا الـ vector v جاي من النقطة، النقطة
+
+50
+00:03:53,660 --> 00:03:57,500
+الأصل، الـ initial point له نقطة الأصل اللي هي الـ
+
+51
+00:03:57,500 --> 00:04:01,680
+origin، والـ terminal point اللي هي نقطة (v1, v2).
+
+52
+00:04:02,150 --> 00:04:08,190
+Then the component form of V هو عبارة عن V بيساوي
+
+53
+00:04:08,190 --> 00:04:13,930
+(v1, v2). إذا لما نكتب هذه الصورة بنقصد
+
+54
+00:04:13,930 --> 00:04:18,410
+فيها الـ vector اللي جاي من النقطة (0, 0) ورايح
+
+55
+00:04:18,410 --> 00:04:21,930
+لمين؟ لـ (v1, v2). هذا لو كان في الـ two
+
+56
+00:04:21,930 --> 00:04:25,070
+dimensions. طب لو كان اللي هو الـ vector في الـ three
+
+57
+00:04:25,070 --> 00:04:30,090
+dimensions. الآن الـ vector اللي قصدناه، نوعرفنا إن
+
+58
+00:04:30,090 --> 00:04:34,700
+هذا هو الـ vector اللي جاي من نقطة الأصل، ورايح
+
+59
+00:04:34,700 --> 00:04:39,280
+للنقطة (v1, v2, v3). لأن نحكي عن الـ three
+
+60
+00:04:39,280 --> 00:04:43,820
+dimensions. فبيصير عندي الآن هذا الـ vector اللي هو
+
+61
+00:04:43,820 --> 00:04:49,960
+من النقطة هذه للنقطة هذه، الـ component form له هو
+
+62
+00:04:49,960 --> 00:04:56,360
+عبارة عن V بتساوي (v1, v2, v3). إذا من الآن وطالع لما
+
+63
+00:04:56,360 --> 00:05:01,440
+نكتب V بتساوي (v1, v2, v3) نقصد بها الـ vector اللي الـ
+
+64
+00:05:01,440 --> 00:05:06,660
+initial point له (0, 0, 0) والـ terminal point هي (v1,
+
+65
+00:05:06,660 --> 00:05:16,100
+v2, v3). طيب، الآن لو قدّينا للـ vector اللي هو PQ
+
+66
+00:05:16,100 --> 00:05:21,900
+من (x1, y1, z1) و(x2, y2, z2). هذا الـ vector أخذناه
+
+67
+00:05:21,900 --> 00:05:25,560
+موازي لهذا الـ vector، هو موازي لهذا الـ vector، و
+
+68
+00:05:25,560 --> 00:05:30,500
+لنفس الطول. فبيصير عندي وكأن الـ V بتساوي (v1, v2,
+
+69
+00:05:30,500 --> 00:05:36,860
+v3). هي الـ vector PQ. الآن ده نتعرف كيف نربط العلاقة
+
+70
+00:05:36,860 --> 00:05:41,480
+بين هذا وبين هذا، وكيف نكتب هذا الـ vector PQ
+
+71
+00:05:41,480 --> 00:05:46,060
+برضه على صورة اللي هو مين؟ الـ component form.
+
+72
+00:05:46,060 --> 00:05:50,980
+اتطلعوا كيف. الآن عندي لو كان في عندي given the
+
+73
+00:05:50,980 --> 00:05:55,100
+points P(x1, y1, z1) وQ(x2, y2, z2). الآن الـ standard
+
+74
+00:05:55,100 --> 00:05:58,220
+position vector to V equal to PQ. زي ما شفنا قبل
+
+75
+00:05:58,220 --> 00:06:03,970
+بشوية، بنقول إن الـ vector هذا بيساوي الـ vector هذا
+
+76
+00:06:03,970 --> 00:06:08,610
+للأسباب اللي ذكرناها، اللي هو إن هذا الـ vector
+
+77
+00:06:08,610 --> 00:06:13,510
+موازي لهذا الـ vector ويساوي طوله. الآن PQ هذا الـ
+
+78
+00:06:19,530 --> 00:06:22,990
+vector، بنكتبه على صورة الـ component form. إيش الـ
+
+79
+00:06:22,990 --> 00:06:27,650
+component form في هذه الحالة؟ اللي هو x2 - x1،
+
+80
+00:06:27,650 --> 00:06:34,650
+اللي هتمثلنا v1، y2 - y1 هتمثلنا v2، z2 - z1
+
+81
+00:06:34,650 --> 00:06:40,530
+هتمثلنا مين؟ v3. إذا الآن صار عندي الـ component
+
+82
+00:06:40,530 --> 00:06:49,140
+form للـ vector PQ الذي واصل من نقطة P لنقطة Q هو
+
+83
+00:06:49,140 --> 00:06:56,680
+عبارة عن الـ vector V بتساوي (x2 - x1, y2 - y1, z2 - z1). هذا
+
+84
+00:06:56,680 --> 00:07:01,140
+اللي هو عبارة عن الـ
+
+85
+00:07:01,140 --> 00:07:05,480
+component form لهذا الـ vector. الآن لو بدنا نجي
+
+86
+00:07:05,480 --> 00:07:09,580
+نعرف كده طول هذا الـ vector، الـ magnitude للـ vector
+
+87
+00:07:09,580 --> 00:07:13,560
+هذا. انتبهوا عليها شوية، خلينا نرجع للرسمة ونشوف
+
+88
+00:07:13,560 --> 00:07:19,620
+كيف نوجد اللي هو الطول. نيجي الآن لو أخذنا من عند O
+
+89
+00:07:19,620 --> 00:07:23,040
+لعند النقطة هذه، وصلنا، طبعاً هذا بيصير مثلث قائم،
+
+90
+00:07:23,040 --> 00:07:26,360
+الزاوية في هذا. فبيصير هذا تربيع، الطول هذا، طوله هذا
+
+91
+00:07:26,360 --> 00:07:31,180
+من هذا لهذا، هذا طوله تربيع بيساوي هذا تربيع، v1
+
+92
+00:07:31,180 --> 00:07:37,680
+تربيع + v2 تربيع. لإيش؟ لإنه الآن هي الـ
+
+93
+00:07:37,680 --> 00:07:41,780
+vector، الأضلاع عندنا v اللي بنوجه طوله هذه اللي هو
+
+94
+00:07:41,780 --> 00:07:46,500
+من هنا لهنا، عبارة عن v3، اللي هو قيمة z، هيها من
+
+95
+00:07:46,500 --> 00:07:51,190
+هنا لهنا، اللي هي قيمة x، هيها. الإحداثيات، لأن إحداثيات الـ
+
+96
+00:07:51,190 --> 00:07:57,550
+v1 من هنا لهنا، الإحداثيات، اللي هي v2. الآن
+
+97
+00:07:57,550 --> 00:08:02,230
+طول هذا الوتر من هنا لعندي هنا إيش بيساوي؟ بيساوي
+
+98
+00:08:02,230 --> 00:08:07,050
+تربيعه بيساوي هذا تربيع، v1 تربيع + هذا التربيع
+
+99
+00:08:07,050 --> 00:08:11,980
+اللي هو v2 تربيع. الآن هذا في المثلث هذا، هي هيظهر
+
+100
+00:08:11,980 --> 00:08:16,640
+هذا، واللي أضلاعه، طوله، هي بيصير هذا الوتر تربيع
+
+101
+00:08:16,640 --> 00:08:20,680
+اللي هو الطول المطلوب، بيساوي v3 تربيع +
+
+102
+00:08:20,680 --> 00:08:24,780
+هذا تربيع، اللي هو عبارة عن v1 تربيع + v
+
+103
+00:08:24,780 --> 00:08:28,000
+2 تربيع. الأضلاع هنا، يعني الآن طول الـ vector
+
+104
+00:08:28,000 --> 00:08:32,200
+هذا تربيع بيساوي v1 تربيع + v2 تربيع
+
+105
+00:08:32,200 --> 00:08:38,100
++ v3 تربيع. بيصير الطول v هو عبارة عن الجذر
+
+106
+00:08:38,100 --> 00:08:42,380
+التربيعي لـ v1 تربيع + v2 تربيع + v3 تربيع.
+
+107
+00:08:42,380 --> 00:08:46,460
+هذا عبارة عن الـ magnitude or length of the vector
+
+108
+00:08:46,460 --> 00:08:50,320
+V. بتساوي، equal is the magnitude
+
+109
+00:08:50,320 --> 00:08:54,200
+لـ V بتساوي الجذر التربيعي v1 تربيع + v2 تربيع
+
+110
+00:08:54,200 --> 00:09:00,180
++ v3 تربيع. الآن أو هو بيساوي بالـ ... بالـ ... بالـ ...
+
+111
+00:09:00,180 --> 00:09:03,700
+بالإحداثيات اللي عندنا، أخذنا PQ زي ما قلنا، هاي
+
+112
+00:09:03,700 --> 00:09:09,380
+x2 - x1، y2 - y1، بيصير x2 - x1 الكل تربيع، زي y2
+
+113
+00:09:09,380 --> 00:09:15,740
+- y1 تربيع، زي z2 - z1 تربيع. هذا اللي هو مين؟
+
+114
+00:09:15,740 --> 00:09:20,750
+اللي هو طول الـ vector V، اللي هو عبارة عن الـ vector
+
+115
+00:09:20,750 --> 00:09:27,350
+من B لعند Q. نأخذ أمثلة عددية على اللي حكيناه.
+
+116
+00:09:27,350 --> 00:09:31,650
+Find the component form and the length of the
+
+117
+00:09:31,650 --> 00:09:35,570
+vector with the initial point P(-3, 4, 1) and terminal
+
+118
+00:09:35,570 --> 00:09:41,110
+point Q(-5, 2, 2). يعني الآن بدنا الـ component form للـ
+
+119
+00:09:41,110 --> 00:09:44,740
+vector هذا PQ. بعد ما نوجد الـ component form، نوجد
+
+120
+00:09:44,740 --> 00:09:47,920
+طوله. الـ component form، اتفقنا كيف نوجدها، هاي نقطة
+
+121
+00:09:47,920 --> 00:09:50,800
+البداية، وهي نقطة النهاية. الـ component form، الـ x
+
+122
+00:09:50,800 --> 00:09:55,820
+component، ناقص خمسة ناقص ناقص تلاتة، بتطلع مين؟ اللي
+
+123
+00:09:55,820 --> 00:10:00,480
+هي v1. نفرض إن إحنا اللي هي إحداثيات v اللي هو PQ
+
+124
+00:10:00,480 --> 00:10:06,870
+عبارة عن (v1, v2, v3). لأن كيف نوجدها؟ ناقص خمسة ناقص
+
+125
+00:10:06,870 --> 00:10:10,650
+ناقص تلاتة، اللي هي x2 - x1، ناقص خمسة ناقص ناقص
+
+126
+00:10:10,650 --> 00:10:15,350
+تلاتة، بيطلع ناقص اثنين. لأن v2 اللي هو اثنين ناقص
+
+127
+00:10:15,350 --> 00:10:18,990
+أربعة، اثنين ناقص أربعة اللي هي سالب اثنين. هذه اللي
+
+128
+00:10:18,990 --> 00:10:23,970
+سميناها الإحداثي y. الآن الأخيرة، الـ z، اثنين ناقص
+
+129
+00:10:23,970 --> 00:10:27,970
+واحد، بتطلع v3، اثنين ناقص واحد، بيساوي واحد. إذا صار
+
+130
+00:10:27,970 --> 00:10:39,130
+عندي الآن الـ component form of V، PQ، (-2,
+
+131
+00:10:39,130 --> 00:10:43,830
+-2, 1). هذا الـ
+
+132
+00:10:43,830 --> 00:10:48,770
+component form. الآن بدنا نجد الـ length للـ V أو الـ
+
+133
+00:10:48,770 --> 00:10:49,750
+length للـ PQ.
+
+134
+00:10:52,610 --> 00:10:56,450
+الجذر التربيعي لـ v1 تربيع + v2 تربيع + v3
+
+135
+00:10:56,450 --> 00:11:00,930
+تربيع، بيساوي الجذر التربيعي لـ 4 + 4 + 1. من هنا جبت هذه، ربعت هذا وربعت هذا وربعت اللي
+
+136
+00:11:00,930 --> 00:11:06,670
+هو هذا. طالع عندي إيش؟ بيساوي 3. كويس، هيك
+
+137
+00:11:12,370 --> 00:11:18,370
+ماشي الحال. إذا هو الـ length of V هيطلع عندي للـ
+
+138
+00:11:18,370 --> 00:11:20,850
+vector بيساوي إيش؟ بيساوي 3.
+
+139
+00:11:23,610 --> 00:11:28,270
+الآن نتعرف على vector algebra operations، أو بمعنى
+
+140
+00:11:28,270 --> 00:11:32,580
+آخر، العمليات الجبرية على المتجهات. الآن بدنا نشوف
+
+141
+00:11:32,580 --> 00:11:37,200
+جبرياً كيف بنجمع اللي هو two vectors. هي عندي الـ vector
+
+142
+00:11:37,200 --> 00:11:43,040
+U، هي الـ component form (u1, u2, u3)، وV بتساوي (v1,
+
+143
+00:11:43,040 --> 00:11:48,520
+v2, v3). اللي أنا، عشان نجمع جبرياً U وV، بنيجي بنقول
+
+144
+00:11:48,520 --> 00:11:52,660
+الـ vector الجديد اللي اسمه U + V، إيش بيساوي؟ u1
+
+145
+00:11:52,660 --> 00:12:02,920
++ v1، u2 + v2، u3 + v3. هذا اللي هو مجموع two vectors.
+
+146
+00:12:02,920 --> 00:12:07,300
+إذا بتجمع الـ x للـ x، والـ y للـ y، والـ z للـ z، بتطلع
+
+147
+00:12:07,300 --> 00:12:12,560
+عندك اللي هو الـ vector U + V. الآن scalar
+
+148
+00:12:12,560 --> 00:12:16,340
+multiplication، اللي هي عملية ضرب الـ vector بـ
+
+149
+00:12:16,340 --> 00:12:25,070
+scalar، هي K في U بتساوي K في u1، يعني بتدخل الآن K
+
+150
+00:12:25,070 --> 00:12:30,970
+على u1، K u2، K u3. الـ vector الجديد هذا اسمه اللي هو
+
+151
+00:12:30,970 --> 00:12:37,290
+عبارة عن مين؟ عن الـ vector K فيه. الآن هذا جبرياً هيك
+
+152
+00:12:37,290 --> 00:12:41,330
+اللي هو الجمع الجبري. لما نشوف إيش معناه هندسياً أو
+
+153
+00:12:41,330 --> 00:12:46,470
+كيف هندسياً ونقارنه بحاجات إحنا بنعرفها قبل هيك،
+
+154
+00:12:46,470 --> 00:12:52,270
+اللي هو في اللي هو جمع المتجهات هندسياً. الآن لو كان
+
+155
+00:12:52,270 --> 00:12:56,410
+عندي الـ vector U هي هذا الـ position vector اللي هو
+
+156
+00:12:56,410 --> 00:13:01,330
+عبارة عن إحداثياته u1 وu2. باجي برسم هي U،
+
+157
+00:13:01,330 --> 00:13:04,490
+قاعدة المثلث يا جماعة اللي أنتو عارفينها، أو قاعدة
+
+158
+00:13:04,490 --> 00:13:10,290
+المتوازي الأضلاع. هي U، اللي إن من عند نهاية U باجي
+
+159
+00:13:10,290 --> 00:13:15,310
+برسم الـ vector V، اللي هو اللي إحداثياته V
+
+160
+00:13:15,310 --> 00:13:19,610
+واحد وV اثنين. يعني بمشي هيك V واحد وبطلع فوق
+
+161
+00:13:19,610 --> 00:13:23,420
+اللي هو مين؟ V اثنين. فبطلع عند الـ vector هذا اللي
+
+162
+00:13:23,420 --> 00:13:27,960
+عندي، هذا هو الـ vector V. الآن مجموع U + V زي ما
+
+163
+00:13:27,960 --> 00:13:31,460
+أنتوا عارفين اللي هو الـ vector اللي بيغلق المثلث
+
+164
+00:13:31,460 --> 00:13:36,860
+هذا، اللي بالأحمر. هذا هو عبارة عن U + V. طيب،
+
+165
+00:13:36,860 --> 00:13:40,300
+لو اجينا اتطلعنا، هل هذا بتناسق مع اللي حكيناها قبل
+
+166
+00:13:40,300 --> 00:13:46,240
+وشوية؟ اه، طبعاً، هذه الآن هي عبارة عن U (u1, u2)، وV،
+
+167
+00:13:46,240 --> 00:1
+
+223
+00:18:01,300 --> 00:18:06,400
+الـ u إذا بقولك the length of ku is the absolute
+
+224
+00:18:06,400 --> 00:18:09,420
+value of the scalar k times the length of u. ده
+
+225
+00:18:09,420 --> 00:18:14,080
+vector ناقص واحد في u إيش هيساوي؟ ناقص u زي ما
+
+226
+00:18:14,080 --> 00:18:18,750
+شوفنا هنا. لأن ناقص u هيكون عكس اتجاه، يعني نفس
+
+227
+00:18:18,750 --> 00:18:23,030
+الطول، و لكن عكس الاتجاه. يعني ناقص واحد في u بيساوي
+
+228
+00:18:23,030 --> 00:18:27,790
+ناقص u، اللي هي نفس الطول لكن عكس الاتجاه عن الـ u
+
+229
+00:18:27,790 --> 00:18:31,950
+has the same length as u but in the opposite
+
+230
+00:18:31,950 --> 00:18:34,330
+direction.
+
+231
+00:18:36,780 --> 00:18:44,000
+معرفة U ناقص V of two vectors is defined by U ناقص
+
+232
+00:18:44,000 --> 00:18:50,280
+V بتساوي U ناقص V بتساوي U زائد ناقص V. طبعا احنا
+
+233
+00:18:50,280 --> 00:18:55,380
+ناقص V عرفناها، هي عبارة عن ناقص واحد في V. الآن إذا
+
+234
+00:18:55,380 --> 00:18:59,980
+هذه معرفة والجمع معرفة، إذا هذه الجديدة عرفناها
+
+235
+00:18:59,980 --> 00:19:04,500
+بإيش بنعرفها الآن؟ اللي هو جمع two vectors وهذا ناقص
+
+236
+00:19:04,500 --> 00:19:09,220
+الـ V هو ناقص واحد في V. نشوف هذا الكلام هل هي
+
+237
+00:19:09,220 --> 00:19:14,300
+تلائم جبريا أو هندسيا أو لأ، ونشوف كيف اللي هو مع
+
+238
+00:19:14,300 --> 00:19:18,280
+معلوماتنا السابقة. هتطلع الآن هي عندي هذا اللي هو
+
+239
+00:19:18,280 --> 00:19:23,510
+الـ vector U وهذا الـ vector V. ماشي الحال. الآن توا
+
+240
+00:19:23,510 --> 00:19:30,170
+عرفنا اللي هو الآن u ناقص v، إيش بيقول؟ الـ u
+
+241
+00:19:30,170 --> 00:19:35,570
+ناقص v هو عبارة عن اللي هو الـ vector اللي بيصل من
+
+242
+00:19:35,570 --> 00:19:40,690
+عند الـ v، اللي هو اللي إلها ناقصها، وبنمتد لعند مين؟
+
+243
+00:19:40,690 --> 00:19:44,030
+الـ u اللي هي الموجبة. طب ليش هي كده؟ أنتَ قُلتَ عنها
+
+244
+00:19:44,030 --> 00:19:52,280
+عشان لما نقول u زائد v، عشان نقول v زائد u ناقص v هي
+
+245
+00:19:52,280 --> 00:19:56,040
+عملية الجمع اللي قبل شوية. هي هذا وهي هذا، وجمعين
+
+246
+00:19:56,040 --> 00:20:00,100
+اللي بيغلقه. يعني لما يكون هذا زائد هذا، الناقص V مع
+
+247
+00:20:00,100 --> 00:20:04,760
+الـ V بيطلع منين؟ الـ U. فعلا متناسق. هذا الخط الأحمر
+
+248
+00:20:04,760 --> 00:20:10,040
+عبارة عن U ناقص V. الآن نيجي نطلع عليها نظرة أخرى.
+
+249
+00:20:10,040 --> 00:20:18,480
+هي عند الـ U وهي عند الـ V. ماشي. الآن الـ V هذا ناقصه
+
+250
+00:20:18,480 --> 00:20:25,170
+هو نفسه الطول له، ولكن اتجاهه عكسه تماما. هي ناقص V.
+
+251
+00:20:25,170 --> 00:20:32,720
+فالآن لما نجمع U زائد ناقص V، اللي بيغلقه. إذاً هذا
+
+252
+00:20:32,720 --> 00:20:36,900
+عبارة عن اللي أغلق هنا هو عبارة عن U زائد ناقص V.
+
+253
+00:20:36,900 --> 00:20:43,060
+هو اللي هذا زائد الجديد اللي هو ناقص V، فبيصير U
+
+254
+00:20:43,060 --> 00:20:46,700
+زائد ناقص V هو اللي بالأحمر. لاحظ اللي بالأحمر هذا
+
+255
+00:20:46,700 --> 00:20:50,700
+له نفس الطول اللي فوق ونفس الـ direction اللي
+
+256
+00:20:50,700 --> 00:20:55,200
+بالأحمر. عشان هيك الـ U ناقص V بيساوي الـ U زائد ناقص
+
+257
+00:20:55,200 --> 00:20:59,120
+V، اللي هو فعلا كلامنا اللي قبل شوية منطقي.
+
+258
+00:20:59,120 --> 00:21:03,880
+والتعريف منطقي، وهي تفسيره الهندسي ويتلاءم معاه.
+
+259
+00:21:03,880 --> 00:21:08,620
+يعني الـ U ناقص V هو عبارة عن U زائد ناقص V زي ما
+
+260
+00:21:08,620 --> 00:21:14,390
+شفنا اللي هو في الرسم اللي حكيناه. طيب الآن ناخد بعض
+
+261
+00:21:14,390 --> 00:21:18,730
+الأمثلة العددية. let u بيساوي ناقص واحد و ثلاثة و
+
+262
+00:21:18,730 --> 00:21:23,090
+واحد، و v بيساوي أربعة و سبعة و صفر. هدول الـ two
+
+263
+00:21:23,090 --> 00:21:27,270
+vectors. الآن بدي بيقول لي بيسألني كمان، يا اللي طبعا
+
+264
+00:21:27,270 --> 00:21:31,170
+مكتوبات الاتنين في صورة مين يا جماعة؟ الـ component
+
+265
+00:21:31,170 --> 00:21:36,630
+فبيقول لي find the components of. اوجد الـ components
+
+266
+00:21:36,630 --> 00:21:42,370
+لمين؟  2u زائد 3v. عرفنا شو معناه 2U
+
+267
+00:21:42,370 --> 00:21:46,270
+وعرفنا شو معناه 3V، وعرفنا شو معناه جمع الـ two
+
+268
+00:21:46,270 --> 00:21:51,370
+vectors U ناقص V، وبرضه عرفنا الـ magnitude لنصف الـ U.
+
+269
+00:21:51,370 --> 00:21:55,610
+إذا الـ N اللي هو find the components of جزئها عن
+
+270
+00:21:55,610 --> 00:21:59,830
+2A وB وC، بدك إيش؟ توجد لها الـ magnitude
+
+271
+00:21:59,830 --> 00:22:06,310
+لها، أو نصف القيمة طول نصف الـ U. طيب خلونا نشوف و
+
+272
+00:22:06,310 --> 00:22:11,600
+نصل لعندها الآن. 2U زائد 3V هي 2U زائد
+
+273
+00:22:11,600 --> 00:22:15,460
+3V. شيل الـ U وحط مكانها قيمتها، هي قيمتها، والـ V
+
+274
+00:22:15,460 --> 00:22:20,020
+شيلها وحط قيمتها هيها. الآن ابدا اللي هو
+
+275
+00:22:20,020 --> 00:22:25,360
+اشتغل ضرب لداخل. ضرب كلنا كيف نضرب احنا scalar في
+
+276
+00:22:25,360 --> 00:22:32,860
+الـ vector. بيصير 2 في ناقص واحد. نفس الشيء بالنسبة
+
+277
+00:22:32,860 --> 00:22:35,540
+لـ 3 في 4، و3 في 7، و3 في صفر، و
+
+278
+00:22:35,540 --> 00:22:39,500
+نجمعها. نعرف كيف نجمع هذا مع هذا، وهذا مع هذا، وهذا
+
+279
+00:22:39,500 --> 00:22:40,640
+مع هذا، وهذا مع هذا، وهذا مع هذا، وهذا مع هذا، و
+
+280
+00:22:40,640 --> 00:22:42,400
+هذا مع هذا، وهذا مع هذا، وهذا مع هذا، وهذا مع
+
+281
+00:22:42,400 --> 00:22:42,700
+هذا، وهذا مع هذا، وهذا مع هذا، وهذا مع
+
+282
+00:22:42,700 --> 00:22:42,940
+هذا، وهذا مع هذا، وهذا مع هذا، وهذا مع
+
+283
+00:22:42,940 --> 00:22:42,960
+هذا، وهذا مع هذا، وهذا مع هذا، وهذا مع
+
+284
+00:22:42,960 --> 00:22:47,520
+هذا. ناقص 2، و3 في 4 بـ 12، ناقص 2
+
+285
+00:22:47,520 --> 00:22:53,180
+و12 هو 10. 2 في 3 بـ 6، 6 زائد 3
+
+286
+00:22:53,180 --> 00:22:56,480
+في 7 بـ 21، 6 و21 بيطلع 27.
+
+287
+00:22:56,480 --> 00:23:02,360
+و2 زائد صفر بيطلع 2. إذا هذا قيمة
+
+288
+00:23:02,360 --> 00:23:06,740
+اللي هو اللي هي 2U زائد 3V. نيجي للـ U
+
+289
+00:23:06,740 --> 00:23:11,460
+ناقص V. هي الـ U وهي الـ V، وهي الناقص طبعا ناقص
+
+290
+00:23:11,460 --> 00:23:15,580
+معناها اللي هو ناقص 4 وناقص 7 وصفر. يعني
+
+291
+00:23:15,580 --> 00:23:21,240
+ناقص 1 ناقص 4 ناقص 5، 3 ناقص 7
+
+292
+00:23:21,240 --> 00:23:26,280
+يعني ناقص 4، 1 ناقص صفر يعني 1. الآن بدنا
+
+293
+00:23:26,280 --> 00:23:29,380
+نيجي، نيجي ل سؤال، هو بدنا نوجد magnitude لنصف U.
+
+294
+00:23:29,900 --> 00:23:34,900
+الآن هي U نصه، نصف الـ U اللي هو عبارة عن نصف في ناقص
+
+295
+00:23:34,900 --> 00:23:39,160
+واحد، ونصف في 3، ونصف في 1. يعني الآن الـ
+
+296
+00:23:39,160 --> 00:23:43,520
+length اللي هو نصف في ناقص واحد الكل تربيع زائد نصف في
+
+297
+00:23:43,520 --> 00:23:47,380
+3 الكل تربيع زائد نصف في 1 الكل تربيع. هذا اللي
+
+298
+00:23:47,380 --> 00:23:51,400
+هو عبارة عن الجذر التربيعي اللي بتطلع الـ length لمن؟
+
+299
+00:23:51,400 --> 00:23:58,240
+للـ vector نصف U، ويساوي احسب إن الدولة هذه بتطلع نص
+
+300
+00:23:58,240 --> 00:24:03,080
+هذه 9 على 2، وهذه 1 على 2، يعني بيصير
+
+301
+00:24:03,080 --> 00:24:06,820
+1، 1، 2 و9، 11، يعني 11 على 2
+
+302
+00:24:06,820 --> 00:24:09,580
+على 4، آسف، لأن أنا بتربع، 2 تحت بيصير 4.
+
+303
+00:24:09,580 --> 00:24:13,940
+بيصير جذر التربيعي لـ 11 على 4. طلع الربع برا
+
+304
+00:24:13,940 --> 00:24:17,520
+الجذر، بيطلع من نص بيصير نصف الجذر الـ 11. إذا
+
+305
+00:24:17,520 --> 00:24:21,420
+magnitude أو طول الـ vector نصف U بيساوي نصف الجذر الـ
+
+306
+00:24:21,420 --> 00:24:24,900
+11. واحد ثاني قال لي ليش تغلبني زي هي؟ كانت باستخدام
+
+307
+00:24:24,900 --> 00:24:27,980
+الخاصية اللي قبل شوية، قُلتُ له مش مشكلة، استخدامها كده
+
+308
+00:24:27,980 --> 00:24:30,700
+قال لي بأخذ النص برا بيصير نصف في الـ magnitude ده
+
+309
+00:24:30,700 --> 00:24:35,160
+U. الـ magnitude value هي الجذر التربيعي لواحد زائد
+
+310
+00:24:35,160 --> 00:24:36,180
+تسعة زائد واحد
+
+311
+00:24:48,420 --> 00:24:56,900
+نيجي الآن لبعض خواص الـ vector operations.
+
+312
+00:24:56,900 --> 00:25:03,640
+أو بمعنى آخر خواص العمليات الجبرية على المتجهات.
+
+313
+00:25:03,640 --> 00:25:08,120
+خواص العمليات الجبرية على المتجهات. انتبهوا. let u و
+
+314
+00:25:08,120 --> 00:25:12,140
+v و w be vectors and a و b scalars. يعني نفترض أن
+
+315
+00:25:12,140 --> 00:25:17,760
+الـ u والـ v والـ w جمع، إيش هي؟ عبارة عن متجهات، والـ
+
+316
+00:25:17,760 --> 00:25:21,920
+a والـ b هي عبارة عن أعداد، الـ scalars. نيجي
+
+317
+00:25:21,920 --> 00:25:25,760
+للخاصية الأولى. إيش الخاصية الأولى بتقول؟ بتقول u
+
+318
+00:25:25,760 --> 00:25:29,940
+زائد V بيساوي V زائد U. يعني اللي هي عملية الجمع
+
+319
+00:25:29,940 --> 00:25:35,110
+على الـ vectors أو المتجهات إبدالية. الخاصية الثانية
+
+320
+00:25:35,110 --> 00:25:40,150
+اللي بتقول لي عملية الجمع عبارة عن عملية دامجة أو
+
+321
+00:25:40,150 --> 00:25:43,970
+تجميعية، أو اللي بيسميها associative. يعني عملية
+
+322
+00:25:43,970 --> 00:25:48,930
+الجمع على الـ vectors هي عملية دامجة أو تجميعية أو
+
+323
+00:25:48,930 --> 00:25:52,690
+associative. يعني لو قلت U زائد V وبعدين جمعت لها W
+
+324
+00:25:52,690 --> 00:25:57,630
+برضه  بنفس النتيجة تساوي V زائد W ولما نقول U زائد
+
+325
+00:25:57,630 --> 00:26:01,950
+اللي بيطلع هنا بيطلع نفس الشيء. الآن 3 بيقول لي أن
+
+326
+00:26:01,950 --> 00:26:08,830
+المحايد الجمعي للـ vectors u اللي هو من عملية
+
+327
+00:26:08,830 --> 00:26:11,370
+الجمع طبعا، اللي هو المحايد الجمعي اللي هو
+
+328
+00:26:11,370 --> 00:26:14,690
+الـ  صفر بيساوي صفر يعني 0 0 0 اللي هو الـ
+
+329
+00:26:14,690 --> 00:26:18,210
+vector صفر. إذا دائما u زائد zero vector طبعا دي
+
+330
+00:26:18,210 --> 00:26:21,540
+vector ودي vector بيساوي u. الآن أنا بقول الـ inverse
+
+331
+00:26:21,540 --> 00:26:26,700
+أو المعكوس الجمعي للـ U هو عبارة عن ناقص U كـ vector
+
+332
+00:26:26,700 --> 00:26:31,340
+طبعا. يعني U زائد معكوس الجمعي اللي هو ناقص الـ U
+
+333
+00:26:31,340 --> 00:26:36,260
+بيساوي Zero vector صفر. الآن صفر في U بيساوي صفر.
+
+334
+00:26:36,260 --> 00:26:39,840
+صفر هنا scalar يا جماعة، والـ U vector. صفر scalar
+
+335
+00:26:39,840 --> 00:26:46,040
+في U vector بيساوي Zero vector. طيب الواحد scalar في
+
+336
+00:26:46,040 --> 00:26:49,800
+U بيطلع إيش؟ بيساوي U. يعني إنك ضمن تضرب واحد في أي
+
+337
+00:26:49,800 --> 00:26:54,480
+vector بيظل نفس الـ vector، لا يتغير لا اتجاهه ولا
+
+338
+00:26:54,480 --> 00:27:00,160
+طوله. طيب الآن خاصية السابعة. عملية اللي هي ضرب الـ
+
+339
+00:27:00,160 --> 00:27:04,540
+scalars في الـ vector عملية دامجة أو تجميعية. يعني
+
+340
+00:27:04,540 --> 00:27:08,530
+بمعنى آخر a في b في u، يعني اضرب a في b في u وبعدين
+
+341
+00:27:08,530 --> 00:27:12,130
+اضرب a فيهم هو نفس الشيء، a في b وبعدين في مين؟ في
+
+342
+00:27:12,130 --> 00:27:15,890
+u. a في b ضرب عادي، بتبقى
+
+343
+00:27:15,890 --> 00:27:20,630
+أعداد، اللي هي من ضرب في u بيكون ضرب بين scalar و
+
+344
+00:27:20,630 --> 00:27:25,680
+vector وعرفناه. الخاصية الثانية a في u زائد v بيساوي
+
+345
+00:27:25,680 --> 00:27:29,400
+a u زائد a v. يعني يتوزع عملية الـ scalar
+
+346
+00:27:29,400 --> 00:27:33,500
+multiplication على الـ vector addition. بنوزع عملية
+
+347
+00:27:33,500 --> 00:27:36,980
+اللي هي ضرب اللي هو الـ scalar في الـ vector على
+
+348
+00:27:36,980 --> 00:27:40,060
+اللي هو الجمع بين المتجهات. بيطلع a u زائد a v.
+
+349
+00:27:40,060 --> 00:27:45,900
+الآن الجمع بين الـ scalars، الجمع العادي تبعنا، الـ
+
+350
+00:27:45,900 --> 00:27:50,220
+u لما تنضرب فيها يتوزع، يعني كـ vector بيصير u في a
+
+351
+00:27:50,930 --> 00:27:56,830
+زائد U في b، يعني aU زائد b في U. هذه برضه الخاصية
+
+352
+00:27:56,830 --> 00:28:01,290
+اللي لها متحققة. طيب احنا  بنخلي نبرهن  واحدة من
+
+353
+00:28:01,290 --> 00:28:04,250
+هدول، اختارت عشوائية. أنا اختارت الخاصية الثامنة. قلت
+
+354
+00:28:04,250 --> 00:28:07,690
+خاصية الثامنة، أصعب واحدة فيهم، و.. و.. و تاخدوا
+
+355
+00:28:07,690 --> 00:28:11,900
+فكرة عن كيف اللي هو بينبرهن. أولا  لت U
+
+356
+00:28:11,900 --> 00:28:12,660
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+357
+00:28:12,660 --> 00:28:13,320
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+358
+00:28:13,320 --> 00:28:14,580
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+359
+00:28:14,580 --> 00:28:15,260
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+360
+00:28:15,260 --> 00:28:19,220
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+361
+00:28:19,220 --> 00:28:20,920
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+362
+00:28:20,920 --> 00:28:25,060
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+363
+00:28:25,060 --> 00:28:30,450
+أولا لـ V. هي component formula V1, V2, V3. والآن إذا
+
+364
+00:28:30,450 --> 00:28:34,250
+أنا أحسب a في U زائد V، الـ a is scalar، وU زائد V
+
+365
+00:28:34,250 --> 00:28:39,390
+جمع Vectors، بيصير a في أو ضد الـ U زائد V. الـ U زائد
+
+366
+00:28:39,390 --> 00:28:50,640
+V إيش هيكون؟ U1 زائد V1، U2 زائد V2، U3 زائد V3.
+
+367
+00:28:50,640 --> 00:28:54,400
+(Repeating sequence of U2 V3, likely a transcription error)
+
+368
+00:28:54,400 --> 00:28:57,340
+(Repeating sequence of U2 V3, likely a transcription error)
+
+369
+00:28:57,340 --> 00:28:58,280
+(Repeating sequence of U2 V3, likely a transcription error)
+
+370
+00:28:58,280 --> 00:29:05,220
+(Repeating sequence of U2 V3, likely a transcription error)
+
+371
+00:29:05,220 --> 00:29:15,560
+(Repeating sequence of U2 V3, likely a transcription error)
+
+372
+00:29:15,560 --> 00:29:24,600
+(Repeating sequence of U2 V3, likely a transcription error)
+
+373
+00:29:24,600 --> 00:29:25,560
+(Repeating sequence of U2 V3, likely a transcription error)
+
+374
+00:29:25,560 --> 00:29:26,120
+(Repeating sequence of U2 V3, likely a transcription error)
+
+375
+00:29:26,120 --> 00:29:27,720
+(Repeating sequence of U2 V3, likely a transcription error)
+
+376
+00:29:27,720 --> 00:29:28,420
+(Repeating sequence of U2 V3, likely a transcription error)
+
+377
+00:29:28,420 --> 00:29:29,400
+(Repeating sequence of U2 V3, likely a transcription error)
+
+378
+00:29:29,400 --> 00:29:31,760
+طالعها برضه عامل مشترك اللي هو scalar
+
+379
+00:29:31,760 --> 00:29:34,760
+multiplication. بقصده عامل مشترك بده.. بده يرجّعها
+
+380
+00:29:34,760 --> 00:29:37,840
+لـ scalar multiplication اللي هي عملية ضرب اللي هو
+
+381
+00:29:37,840 --> 00:29:43,800
+scalar في vector. a في V1، V2، V3. هذا اللي هو U
+
+382
+00:29:43,800 --> 00:29:48,200
+وهذا V، فبيصير aU زائد aV. يعني بدأت a في U زائد V
+
+383
+00:29:48,200 --> 00:29:52,560
+واتسلسلت لما وصلت لـ a في U زائد aV حيث استخدمت
+
+384
+00:29:52,560 --> 00:29:56,680
+عملية،  مع عملية الجمع بين two vectors وعملية الـ
+
+385
+00:29:56,680 --> 00:30:00,360
+scalar multiplication اللي عرفناها سابقا. هذا هو
+
+386
+00:30:00,360 --> 00:30:05,700
+برهان هذه، والباقي اللي هي بشكل مشابه. الآن نيجي إن
+
+387
+00:30:05,700 --> 00:3
+
+445
+00:34:44,000 --> 00:34:49,560
+Similarly, OB2 هو عبارة
+
+446
+00:34:49,560 --> 00:34:54,340
+عن X2, Y2, و Z2. يعني بمعنى آخر، بنقدر نكتب OB2 عبارة
+
+447
+00:34:54,340 --> 00:34:59,180
+عن X2 في اتجاه I، و Y2 في اتجاه J، و Z2 في اتجاه K.
+
+448
+00:34:59,180 --> 00:35:03,800
+الآن، هذا الـ vector B1 B2 اللي من هنا لهنا، اللي هو
+
+449
+00:35:03,800 --> 00:35:08,360
+واضح أنه هو جاي من عند هذا، ورايح هذا، زي ما عملنا
+
+450
+00:35:08,360 --> 00:35:13,600
+قبل شوية. عبارة عن b2 ناقص b1، الـ vector اللي هو ob2
+
+451
+00:35:13,600 --> 00:35:16,920
+هذا ناقص هذا الـ vector، زي ما عرفنا عملية الطرح قبل
+
+452
+00:35:16,920 --> 00:35:21,940
+شوية. فبيصير هذا، لإنه هذا زائد هذا بيصير هذا. لأجل
+
+453
+00:35:21,940 --> 00:35:29,020
+الوضوح، أو بمعنى آخر، هذا عبارة عن x2 ناقص x1 في اتجاه
+
+454
+00:35:29,020 --> 00:35:33,720
+الـ I، Y2-Y1 في اتجاه الـ J، Y2-Y1 في اتجاه الـ J، Z2
+
+455
+00:35:33,720 --> 00:35:37,660
+-Z1 في اتجاه الـ K، Z2-Z1 في اتجاه الـ K، Z2-Z1 في
+
+456
+00:35:37,660 --> 00:35:38,840
+اتجاه الـ K، Z2-Z1 في اتجاه الـ K، Z2-Z1 في اتجاه
+
+457
+00:35:38,840 --> 00:35:42,460
+الـ K، Z2-Z1 في اتجاه الـ K، Z2-Z1 في اتجاه الـ K، Z2
+
+458
+00:35:42,460 --> 00:35:46,800
+-Z1 في اتجاه الـ K، Z2-Z1 في اتجاه الـ K، Z2-Z1 في
+
+459
+00:35:46,800 --> 00:35:47,640
+اتجاه الـ K، Z2-Z1 في اتجاه الـ K، Z2-Z1 في اتجاه
+
+460
+00:35:47,640 --> 00:35:49,640
+الـ K، Z2-Z1 في اتجاه الـ K، Z2-Z1 في اتجاه الـ K، Z2
+
+461
+00:35:49,640 --> 00:35:53,880
+-Z1 في اتجاه
+
+462
+00:35:53,880 --> 00:35:57,270
+الـ K. كيف كتبت اللي ob1 بيساوي هذا الكلام؟
+
+463
+00:35:57,270 --> 00:36:02,550
+خلّينا نشوف الآن كيف نكتبه، و اللي هي أي vector v
+
+464
+00:36:02,550 --> 00:36:06,270
+بيساوي v1, v2, v3, can be written as a linear
+
+465
+00:36:06,270 --> 00:36:09,710
+combination of the standard unit vectors as
+
+466
+00:36:09,710 --> 00:36:17,100
+follows. يعني الآن، عند أي vector v اللي هو مكتوب في
+
+467
+00:36:17,100 --> 00:36:20,800
+الـ component form، اللي هو هيه، بدنا نكتب هذا على
+
+468
+00:36:20,800 --> 00:36:25,340
+صورة I، وJ، وK. سهل يا جماعة، هتلاقوه سهل، وممكن بعضكم
+
+469
+00:36:25,340 --> 00:36:28,810
+يقول أنا على طول بعرفه. خلّيني أخد الـ V بيساوي V واحد،
+
+470
+00:36:28,810 --> 00:36:33,370
+V اثنين، و V ثلاثة. هذي الآن، افردليها كالتالي، و بدك
+
+471
+00:36:33,370 --> 00:36:37,250
+تبقى حاصل جمع ثلاثة اللي هي vectors، الـ vector الأول
+
+472
+00:36:37,250 --> 00:36:42,110
+V واحد، وهدول صفر، الـ vector الثاني صفر، و V اثنين،
+
+473
+00:36:42,110 --> 00:36:46,230
+و هذا صفر، صفر، صفر، V ثلاثة. اجمعهم مع بعض، هتلاقي V
+
+474
+00:36:46,230 --> 00:36:50,320
+واحد زائد صفر زائد صفر = V واحد. زيرو زيرو زيرو زيرو
+
+475
+00:36:50,320 --> 00:36:53,100
+زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو
+
+476
+00:36:53,100 --> 00:36:54,000
+زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو
+
+477
+00:36:54,000 --> 00:36:54,260
+زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو
+
+478
+00:36:54,260 --> 00:36:55,380
+زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو
+
+479
+00:36:55,380 --> 00:36:59,340
+زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو
+
+480
+00:36:59,340 --> 00:37:03,440
+زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو زيرو
+
+481
+00:37:03,440 --> 00:37:07,860
+زيرو. اخذ V1 برا، V2 برا، V3 برا، V4 برا، V5 برا، V6
+
+482
+00:37:07,860 --> 00:37:10,420
+برا، V7 برا، V8 برا، V9 برا، V10 برا، V11 برا، V12 برا،
+
+483
+00:37:10,420 --> 00:37:12,720
+V13 برا، V14 برا، V15 برا، V16 برا، V17 برا، V18 برا،
+
+484
+00:37:12,720 --> 00:37:20,320
+V19 برا، V20 برا، V21 برا، V21 برا، V22 برا، V23 برا،
+
+485
+00:37:20,320 --> 00:37:23,960
+V21 برا، V23 برا، V21 برا، V21 برا، V21 برا، V21 برا،
+
+486
+00:37:23,960 --> 00:37:23,980
+V21 برا، V21 برا، V21 برا، V21 برا، V21 برا، V21 برا،
+
+487
+00:37:23,980 --> 00:37:24,040
+V21 برا، V21 برا، V21 برا، V21 برا، V21 برا، V21 برا،
+
+488
+00:37:24,040 --> 00:37:24,060
+V21 برا، V21 برا، V21 برا، V21 برا، V21 برا، V21 برا،
+
+489
+00:37:24,060 --> 00:37:31,260
+V21 برا، V21 برا، V21 برا، V21 برا، V21 برا، V21 برا،
+
+490
+00:37:31,260 --> 00:37:42,620
+V21 برا، V21. قيمته بدلالة I، و J، و K هي V1، V2، V3، V4،
+
+491
+00:37:42,620 --> 00:37:45,640
+V5، V6، V7، V8، V9، V10، V11، V12، V13، V12، V13، V13، V13، V13،
+
+492
+00:37:45,640 --> 00:37:48,120
+V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13،
+
+493
+00:37:48,120 --> 00:37:50,820
+V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13،
+
+494
+00:37:50,820 --> 00:37:50,900
+V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13،
+
+495
+00:37:50,900 --> 00:37:52,940
+V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13،
+
+496
+00:37:52,940 --> 00:37:55,620
+V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13، V13،
+
+497
+00:37:55,620 --> 00:37:59,340
+V13، V13، V13، V...
+
+498
+00:38:04,140 --> 00:38:10,000
+إذا كان الـ V عبارة عن non-zero vector، إذا كان الـ V
+
+499
+00:38:10,000 --> 00:38:10,120
+عبارة عن non-zero vector، إذا كان الـ V عبارة عن non
+
+500
+00:38:10,120 --> 00:38:10,540
+إذا كان الـ V عبارة عن non-zero vector، إذا كان الـ V
+
+501
+00:38:10,540 --> 00:38:10,600
+-zero vector، إذا كان الـ V عبارة عن non-zero vector
+
+502
+00:38:10,600 --> 00:38:10,620
+إذا كان الـ V عبارة عن non-zero vector، إذا كان الـ V
+
+503
+00:38:10,620 --> 00:38:11,140
+-zero vector، إذا كان الـ V عبارة عن non-zero vector
+
+504
+00:38:11,140 --> 00:38:13,160
+إذا كان الـ V عبارة عن non-zero vector، إذا كان الـ V
+
+505
+00:38:13,160 --> 00:38:16,500
+عبارة عن non-zero vector، إذا كان الـ V عبارة عن non
+
+506
+00:38:16,500 --> 00:38:19,920
+-zero vector، إذا
+
+507
+00:38:19,920 --> 00:38:26,030
+كان الـ V عبارة عن non-zero vector، هو عبارة عن is a
+
+508
+00:38:26,030 --> 00:38:30,170
+unit vector in the direction of v، أو اختصارًا،
+
+509
+00:38:30,170 --> 00:38:35,070
+بنسميه the direction of the vector v. إذا the
+
+510
+00:38:35,070 --> 00:38:37,970
+direction of the vector v عبارة عن unit vector
+
+511
+00:38:37,970 --> 00:38:44,840
+حاصل من قسمة الـ V على magnitude، و على طوله. طيب الآن،
+
+512
+00:38:44,840 --> 00:38:48,240
+الـ equation v بتساوي magnitude v في v على
+
+513
+00:38:48,240 --> 00:38:53,260
+magnitude of v. إيه ده؟ بتروح، بيصير v. آه، أقصد
+
+514
+00:38:53,260 --> 00:38:57,480
+أكتبها هيك. ليش؟ أقصد أكتبها على صورة اللي هو طول الـ
+
+515
+00:38:57,480 --> 00:39:01,860
+vector في اتجاهه. إذا أي vector في الدنيا، أقدر أكتب
+
+516
+00:39:01,860 --> 00:39:06,300
+على صورة طوله ضرب اتجاهه. اتجاهه = V على magnitude V.
+
+517
+00:39:06,300 --> 00:39:14,080
+Expresses V as its length and its direction.  ناخد
+
+518
+00:39:14,080 --> 00:39:18,880
+أمثلة عليها اللي أنا بقوله. نيجي
+
+519
+00:39:18,880 --> 00:39:21,960
+نشوف المثال اللي عندنا. Find a unit vector u in the
+
+520
+00:39:21,960 --> 00:39:26,560
+direction of the vector b1 to b2. بقول: أوجد الـ unit
+
+521
+00:39:26,560 --> 00:39:30,920
+vector u اللي هي main اللي في اتجاه الـ vector b1
+
+522
+00:39:30,920 --> 00:39:34,740
+b2. يعني بدي أوجد الـ vector b1 b2، وبعدين أوجد
+
+523
+00:39:34,740 --> 00:39:39,000
+اتجاهه، بتكون هي الـ u. طيب، شوف كيف b1 b2. تعلمنا كيف
+
+524
+00:39:39,000 --> 00:39:44,340
+نحسبه، اللي هو ثلاثة ناقص واحد. هي بتطلع في اتجاه
+
+525
+00:39:44,340 --> 00:39:47,400
+الـ I، صفر، صفر، صفر، صفر، صفر، صفر، صفر، صفر، صفر،
+
+526
+00:39:47,400 --> 00:39:48,000
+صفر، صفر، صفر، صفر، صفر، صفر، صفر، صفر، صفر، صفر،
+
+527
+00:39:48,000 --> 00:39:51,150
+صفر، صفر، صفر، في اتجاه الـ J، صفر ناقص
+
+528
+00:39:51,150 --> 00:39:56,030
+واحد في اتجاه الـ K. هذا اللي هو B1 B2، المتجهة بيصير
+
+529
+00:39:56,030 --> 00:40:02,230
+عبارة عن 2I زائد 2J ناقص K. عشان إذا
+
+530
+00:40:02,230 --> 00:40:07,430
+هذا B1 B2، هو المتجه اللي عندي. كتبته على صورة 2I
+
+531
+00:40:07,430 --> 00:40:14,140
+زائد 2J ناقص K. على صورة I زائد J زائد K. and then the unit
+
+532
+00:40:14,140 --> 00:40:17,740
+vector in the direction of b1 b2 is. يعني الآن بدّي
+
+533
+00:40:17,740 --> 00:40:21,580
+الآن الـ unit vector هو عبارة عن الـ unit vector in
+
+534
+00:40:21,580 --> 00:40:25,520
+the direction of b1 b2. إيش بيكون يا جماعة؟ هو عبارة
+
+535
+00:40:25,520 --> 00:40:29,800
+عن الـ vector على magnitude، و على طوله. هاي الـ
+
+536
+00:40:29,800 --> 00:40:34,380
+vector 2i زائد 2j ناقص k، هاي يا جماعة، على طوله.
+
+537
+00:40:34,380 --> 00:40:38,460
+طوله = الجذر التربيعي لمين؟ لإثنين تربيع = أربعة،
+
+538
+00:40:38,460 --> 00:40:41,920
+و اثنين تربيع = أربعة، هي ثمانية، و واحد تربيع = واحد، هي
+
+539
+00:40:41,920 --> 00:40:45,760
+تسعة. إذا الجذر التربيعي لكل هذا، طبعًا الجذر لكله،
+
+540
+00:40:45,760 --> 00:40:48,730
+هذا، بيظل الجذر. ماشي. الجذر التربيعي للتسعة، إيش
+
+541
+00:40:48,730 --> 00:40:52,030
+بيساوي؟ ثلاثة. إذا صار عندي الـ direction اللي بدّي إياه،
+
+542
+00:40:52,030 --> 00:40:56,870
+اللي هو الـ U in the direction of B1 B2 هو عبارة عن
+
+543
+00:40:56,870 --> 00:41:02,450
+2I زائد 2J ناقص K على ثلاثة. طيب، صلّوا على النبي عليه
+
+544
+00:41:02,450 --> 00:41:05,850
+الصلاة والسلام.  ناخد مثال آخر، تطبيق يقول: express the
+
+545
+00:41:05,850 --> 00:41:10,800
+velocity vector V. الآن عبّر لي عن اللي هو الـ
+
+546
+00:41:10,800 --> 00:41:15,820
+velocity vector V، اللي هو 3I-4J، على صورة as a
+
+547
+00:41:15,820 --> 00:41:20,060
+product of its speed times a unit vector in the
+
+548
+00:41:20,060 --> 00:41:22,660
+direction of motion. يعني بيقول لي عبّر لي عن الـ
+
+549
+00:41:22,660 --> 00:41:26,100
+vector V، الـ velocity vector، على صورة اللي هو الـ
+
+550
+00:41:26,100 --> 00:41:30,980
+speed تبعته مضروبة في مين؟ في الـ unit vector في الـ
+
+551
+00:41:30,980 --> 00:41:33,460
+direction لل motion. direction لل motion هو الـ
+
+552
+00:41:33,460 --> 00:41:37,090
+direction لل velocity. هذا الـ vector، هو: أوجد أول
+
+553
+00:41:37,090 --> 00:41:41,270
+شيء الـ direction للحركة، وأوجد السبيد. السبيد هو
+
+554
+00:41:41,270 --> 00:41:42,610
+المجموع. المجموع المجموع المجموع المجموع المجموع
+
+555
+00:41:42,610 --> 00:41:46,290
+المجموع المجموع المجموع المجموع المجموع المجموع
+
+556
+00:41:46,290 --> 00:41:46,950
+المجموع المجموع المجموع المجموع المجموع المجموع
+
+557
+00:41:46,950 --> 00:41:48,130
+المجموع المجموع المجموع المجموع المجموع المجموع
+
+558
+00:41:48,130 --> 00:41:49,310
+المجموع المجموع المجموع المجموع المجموع المجموع
+
+559
+00:41:49,310 --> 00:41:49,370
+المجموع المجموع المجموع المجموع المجموع المجموع
+
+560
+00:41:49,370 --> 00:41:51,530
+المجموع المجموع المجموع المجموع المجموع المجموع
+
+561
+00:41:51,530 --> 00:42:00,810
+المجموع المجموع المج...
+
+562
+00:42:02,150 --> 00:42:05,210
+لأن الـ velocity اللي هو طبعًا هي اللي بتمثل اتجاه
+
+563
+00:42:05,210 --> 00:42:10,310
+الحركة. The direction of V is V على magnitude V. V
+
+564
+00:42:10,310 --> 00:42:14,550
+ثلاثة I ناقص أربعة J على خمسة. إذًا، Hence V بتاعتنا هذه
+
+565
+00:42:14,550 --> 00:42:18,130
+كتبناها على صورة The speed خمسة مضروبة في الـ
+
+566
+00:42:18,130 --> 00:42:24,340
+direction ثلاثة I ناقص أربعة J على خمسة. هنا نأخذ مثال
+
+567
+00:42:24,340 --> 00:42:29,520
+آخر، يقول: إن قوة
+
+568
+00:42:29,520 --> 00:42:35,380
+6 نيوتن، قيمة القوة جديش 6 نيوتن، is applied in the
+
+569
+00:42:35,380 --> 00:42:36,660
+direction of the vector
+
+570
+00:42:41,300 --> 00:42:48,500
+التي تعمل في اتجاه الـ vector هذا. أمامنا بيقول
+
+571
+00:42:48,500 --> 00:42:53,140
+express the force F as a product of its magnitude
+
+572
+00:42:53,140 --> 00:42:55,440
+and direction.  مجموعة معروفة
+
+573
+00:42:58,830 --> 00:43:04,570
+الـ direction للفورس هو الـ direction لهذا الـ V. فالـ direction
+
+574
+00:43:04,570 --> 00:43:08,650
+للفورس هو الـ direction لهذا الـ V. فالـ direction لهذا الـ V
+
+575
+00:43:08,650 --> 00:43:13,570
+هو V على المجموع للـ V، وساوي 2I زائد 2J ناقص K على جذر
+
+576
+00:43:13,570 --> 00:43:14,050
+التربيع
+
+577
+00:43:17,600 --> 00:43:21,500
+بتطلع  ثلاثة. إذًا، صار عندي الآن هي الـ
+
+578
+00:43:21,500 --> 00:43:25,560
+direction لل force. إذا الـ force بتساوي الستة،
+
+579
+00:43:25,560 --> 00:43:30,420
+اللي هو الـ magnitude لل force اللي معطينا إياها في
+
+580
+00:43:30,420 --> 00:43:35,200
+الـ direction اللي هو باللي الـ direction الـ V اللي
+
+581
+00:43:35,200 --> 00:43:38,420
+عندي أمامي. فصارت مكتوبة الـ force على صورة
+
+582
+00:43:38,420 --> 00:43:39,980
+magnitude في direction.
+
+583
+00:43:43,500 --> 00:43:48,240
+نيجي لآخر نقطة، اللي هي شغلة بسيطة وسريعة. الآن بدنا
+
+584
+00:43:48,240 --> 00:43:51,180
+نعرف شو معناه the midpoint M of the line segment
+
+585
+00:43:51,180 --> 00:43:56,060
+joining B1 وB2. يعني موجود عند B1 وB2 نقطتين في
+
+586
+00:43:56,060 --> 00:44:00,180
+الفراغ، وفي خط واصل B2. ما حكيتش عن متجهة، الآن بحكي عن
+
+587
+00:44:00,180 --> 00:44:03,980
+خط واصل، line segment بين B1 وB2. بدّه النقطة اللي في
+
+588
+00:44:03,980 --> 00:44:07,580
+منتصفه بالضبط. النقطة اللي في منتصفه اللي هو B1 وB2،
+
+589
+00:44:07,580 --> 00:44:15,340
+هي منها له نصه. إحداثياته هي:  X1 زائد X2، هي X1
+
+590
+00:44:15,340 --> 00:44:21,860
+زائد X2 على 2، Y1 زائد Y2 على 2، Z1 زائد Z2 على 2.
+
+591
+00:44:21,860 --> 00:44:26,380
+هذا قانون. ما نريد أن نشوفه هو القانون اللي موجود
+
+592
+00:44:26,380 --> 00:44:31,380
+عندنا، اللي هي الـ midpoint بين B1 وB2. نأخذ مثال
+
+593
+00:44:31,380 --> 00:44:36,060
+عددي. example 7: the midpoint of the line segment
+
+594
+00:44:36,060 --> 00:44:42,180
+joining B1 and B2 is. إيش هي الـ midpoint؟ إحداثياتها
+
+595
+00:44:42,500 --> 00:44:48,240
+ثلاثة وسبعة، على اثنين، يعني بتطلع خمسة. ناقص اثنين،
+
+596
+00:44:48,240 --> 00:44:53,980
+واربعه، اثنين على اثنين، بتطلع واحد. صفر، و أربعة على
+
+597
+00:44:53,980 --> 00:44:58,380
+اثنين، بتطلع اثنين. هذه اللي هي الـ midpoint اللي
+
+598
+00:44:58,380 --> 00:45:03,560
+مطلوبة. الآن هيك بيكون إحنا يعني خلصنا المحاضرة
+
+599
+00:45:03,560 --> 00:45:07,880
+الثانية، وهاي الـ exercises بتحلوها، واللي أنتو
+
+600
+00:45:07,880 --> 00:45:11,740
+يعني حاسين مش عارفين تحلوها أو كده، ممكن عبر
+
+601
+00:45:11,740 --> 00:45:13,980
+التواصل معنا على الـ email أو على الـ whatsapp اللي
+
+602
+00:45:13,980 --> 00:45:18,160
+هنقوم عليها إن شاء الله نحاول.. هنكونها معاكم إن
+
+603
+00:45:18,160 --> 00:45:20,580
+شاء الله. تبعتوا الأسئلة اللي بتكونوا معاها. هنا
+
+604
+00:45:
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..dbe1bef1ab49a54b0a76a663a87d6733be5f44c3
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE_postprocess.srt
@@ -0,0 +1,2424 @@
+1
+00:00:00,710 --> 00:00:04,770
+بسم الله الرحمن الرحيم هذه هي المحاضرة التانية
+
+2
+00:00:04,770 --> 00:00:10,230
+مساق تفاضل وتكامل جيم لطلاب و طالبات الجامعة
+
+3
+00:00:10,230 --> 00:00:16,000
+الإسلامية كليتي العلوم والهندسةاليوم هنبدأ ان شاء
+
+4
+00:00:16,000 --> 00:00:21,640
+الله اللي هو في 12 اتنين اللي هو vectors اللي هي
+
+5
+00:00:21,640 --> 00:00:28,960
+المتجهات الموضوع طبعا موضوع تم قبل هيك اخد فكرة
+
+6
+00:00:28,960 --> 00:00:34,480
+عنه او حتى يعني بعض الشغلات تم دراستها في اللي هي
+
+7
+00:00:34,480 --> 00:00:39,260
+مساق فيزيا في اللي هي اللي برضه لكليتي العلوم
+
+8
+00:00:39,260 --> 00:00:44,700
+والهندسة نعرف شو معناه متجه او شو معناه vectorالان
+
+9
+00:00:44,700 --> 00:00:48,980
+بنجي عنا initial point نقطة بداية a و terminal
+
+10
+00:00:48,980 --> 00:00:53,260
+point نقطة نهاية b الان ال line segment اللى بيمتد
+
+11
+00:00:53,260 --> 00:00:59,120
+من a و بتجه لعند b هذا بنسميه ال vector a,b اذا
+
+12
+00:00:59,120 --> 00:01:03,360
+اللى له نقطة بداية و له نقطة نهاية بتجه من البداية
+
+13
+00:01:03,360 --> 00:01:06,880
+الى النهاية هذا اللى هو the directed line segment
+
+14
+00:01:06,880 --> 00:01:11,220
+a,b is called a vector بنسميه ايش يا جماعة بسميه
+
+15
+00:01:11,220 --> 00:01:15,570
+vectorالان عنا بدنا نشوف شو معنات ان two vectors
+
+16
+00:01:15,570 --> 00:01:20,050
+are equal يعني كيف اللي هو عندي two vectors او
+
+17
+00:01:20,050 --> 00:01:27,370
+متجهين متساوين قبل نعرف شو معنات طول ال vector AB
+
+18
+00:01:27,370 --> 00:01:32,390
+طول ال vector AB هو طول القياس الطولة ده بنسميه ال
+
+19
+00:01:32,390 --> 00:01:36,790
+magnitude لل vector و بنرمزه بالرمز magnitude لل
+
+20
+00:01:36,790 --> 00:01:43,130
+AB الان بنقول ال vector ABاللي هو بيساوى ال vector
+
+21
+00:01:43,130 --> 00:01:47,990
+CD إذا حدث أمرين الأمر الأول أن هذا اله نفس
+
+22
+00:01:47,990 --> 00:01:52,910
+الاتجاه يعني هذا بوازيه ومشيات في اتجاه السهم في
+
+23
+00:01:52,910 --> 00:01:56,690
+اتجاه واحد اتنين يكون طول ال vector هذا وطول ال
+
+24
+00:01:56,690 --> 00:01:59,950
+vector هذا إيه شمالهم متساوية في هذه الحالة بتقول
+
+25
+00:01:59,950 --> 00:02:03,330
+عن هذا ال vector و هذا ال vector انهم are equal
+
+26
+00:02:03,330 --> 00:02:06,130
+إذا ال vector represented by the directed line
+
+27
+00:02:06,130 --> 00:02:10,590
+segment ABHas initial point A and terminal point B
+
+28
+00:02:10,590 --> 00:02:15,430
+and its length is denoted by AB بنقول أنه هذا
+
+29
+00:02:15,430 --> 00:02:22,310
+مساوي لل vector CD إذا كان اللي هو اتجاه ال CD نفس
+
+30
+00:02:22,310 --> 00:02:27,180
+اتجاه ال AB والتنين اللي هي نفس ال magnitudetwo
+
+31
+00:02:27,180 --> 00:02:30,660
+vectors are equal if they have the same length and
+
+32
+00:02:30,660 --> 00:02:35,820
+the same direction لاحظوا أن هذا وهذا وهذا الأربع
+
+33
+00:02:35,820 --> 00:02:39,680
+vectors هذوله متوازيات اللي هي نفس الاتجاه وكلهم
+
+34
+00:02:39,680 --> 00:02:45,360
+اتجاه اللي أمامنا والاربعة اللي هي نفس الطول هذوله
+
+35
+00:02:45,360 --> 00:02:48,900
+بنسميه are equal vectors لاحظوا ال vector هذا OB
+
+36
+00:02:49,390 --> 00:02:53,410
+هذا الـ vector جاي من نقطة الأصل و منطلق للنقطة من
+
+37
+00:02:53,410 --> 00:02:57,410
+B يعني ال initial point له نقطة الأصل هذا بنسميه
+
+38
+00:02:57,410 --> 00:03:01,290
+position vector يعني الآن كل ال vectors الموازية
+
+39
+00:03:01,290 --> 00:03:05,450
+له كلهم بساوينه إذا كانوا في نفس الاتجاه و اللي هو
+
+40
+00:03:05,450 --> 00:03:08,710
+نفس الطول إذا هنتعامل مع ال position vector بعد
+
+41
+00:03:08,710 --> 00:03:14,880
+هيك كمتجه بيمثل كل ال vectors الأخرياتنأتي الآن
+
+42
+00:03:14,880 --> 00:03:22,740
+اللي هو نشوف اشو معناه اشو كيف بدنا نكتب ال vector
+
+43
+00:03:22,740 --> 00:03:28,440
+على صورة على حاجة اسمها ال component form طيب صلوا
+
+44
+00:03:28,440 --> 00:03:33,580
+على النبي عليه الصلاة والسلام الان عندي اللي هو if
+
+45
+00:03:33,580 --> 00:03:38,620
+V is a two dimensional vector in the planeequal to
+
+46
+00:03:38,620 --> 00:03:41,240
+the vector with initial point at the origin and
+
+47
+00:03:41,240 --> 00:03:45,880
+terminal point at v1 وv2 يعني الان لو فرضنا عندي
+
+48
+00:03:45,880 --> 00:03:49,100
+الان في البلين في عندي vector v في ال two
+
+49
+00:03:49,100 --> 00:03:53,660
+dimension هذا ال vector v جاي من النقطة النقطة
+
+50
+00:03:53,660 --> 00:03:57,500
+الأصل ال initial point له نقطة الأصل اللي هي ال
+
+51
+00:03:57,500 --> 00:04:01,680
+origin و ال terminal point اللي هي نقطة من v1 وv2
+
+52
+00:04:02,150 --> 00:04:08,190
+then the component form of V هو عبارة عن V بيساوي
+
+53
+00:04:08,190 --> 00:04:13,930
+V واحد و V اتنين اذا لما نكتب هذه الصورة بنقصد
+
+54
+00:04:13,930 --> 00:04:18,410
+فيها ال vector اللي جاي من النقطة صفر صفر و رايح
+
+55
+00:04:18,410 --> 00:04:21,930
+لمين V واحد و V اتنين هذا لو كان في ال two
+
+56
+00:04:21,930 --> 00:04:25,070
+dimensions طب لو كان اللي هو ال vector في ال three
+
+57
+00:04:25,070 --> 00:04:30,090
+dimensions الان ال vector اللي قصدناها نوعرفنا ان
+
+58
+00:04:30,090 --> 00:04:34,700
+هذا هو ال vector اللي جايمن نقطة الأصل و رايح
+
+59
+00:04:34,700 --> 00:04:39,280
+للنقطة V1 و V2 و V3 لان نحكي عن الـ three
+
+60
+00:04:39,280 --> 00:04:43,820
+dimensions فبصير عندي الآن هذا ال vector اللي هو
+
+61
+00:04:43,820 --> 00:04:49,960
+من النقطة هذه للنقطة هذه ال component form له هو
+
+62
+00:04:49,960 --> 00:04:56,360
+عبارة عن Vبتساوى V1 V2 V3 إذا من الآن و طالع لما
+
+63
+00:04:56,360 --> 00:05:01,440
+نكتب V بيساوى V1 V2 V3 نقصد بها ال vector اللي ال
+
+64
+00:05:01,440 --> 00:05:06,660
+initial point له 0 0 0 و ال terminal point هي V1 و
+
+65
+00:05:06,660 --> 00:05:16,100
+V2 و V3 طيب الآنلو قدّينا للـ vector اللي هو P EQ
+
+66
+00:05:16,100 --> 00:05:21,900
+من X1 Y1 و Z1 و X2 Y2 و Z2 هذا ال vector أخدناه
+
+67
+00:05:21,900 --> 00:05:25,560
+موازي لهذا ال vector هيو موازي لهذا ال vector و
+
+68
+00:05:25,560 --> 00:05:30,500
+لنفس الطول فبصير عندي و كأن ال V بتساوي V1 و V2 و
+
+69
+00:05:30,500 --> 00:05:36,860
+V3 هي ال vector P EQالان ده نتعرف كيف نربط العلاقة
+
+70
+00:05:36,860 --> 00:05:41,480
+بين هذا وبين هذا وكيف نكتب هذا ال vector بي كيو
+
+71
+00:05:41,480 --> 00:05:46,060
+برضه على صورة اللي هو مين ال component form
+
+72
+00:05:46,060 --> 00:05:50,980
+اتطلعوا كيف الان عندى لو كان في عندى given the
+
+73
+00:05:50,980 --> 00:05:55,100
+points بي x واحد و y واحد و z واحد و q هو بقى ال x
+
+74
+00:05:55,100 --> 00:05:58,220
+اتنين و y اتنين و z اتنين الان ال standard
+
+75
+00:05:58,220 --> 00:06:03,970
+position vector to vequal to bq زي ما شفنا قبل
+
+76
+00:06:03,970 --> 00:06:08,610
+بشوية بنقول ان ال vector هذا بساوي ال vector هذا
+
+77
+00:06:08,610 --> 00:06:13,510
+للأسباب اللي ذكرناها اللي هو ان هذا ال vector
+
+78
+00:06:13,510 --> 00:06:19,530
+موازي لهذا ال vector ويساوي طولهالان bq هذا ال
+
+79
+00:06:19,530 --> 00:06:22,990
+vector بنكتبه على صورة ال component form ايش ال
+
+80
+00:06:22,990 --> 00:06:27,650
+component form في هذه الحالة اللي هو x2 ناقص x1
+
+81
+00:06:27,650 --> 00:06:34,650
+اللي هتمثلنا v1 y2 ناقص y1 هتمثلنا v2 z2 ناقص z1
+
+82
+00:06:34,650 --> 00:06:40,530
+هتمثلنا مين؟ v3 اذا الان صار عندي ال component
+
+83
+00:06:40,530 --> 00:06:49,140
+form لل vector bqالذي واصل من نقطة P لنقطة Q هو
+
+84
+00:06:49,140 --> 00:06:56,680
+عبارة عن الـ vector V بساوية X2-X1Y2Y1Z2-Z1 هذا
+
+85
+00:06:56,680 --> 00:07:01,140
+اللي هو عبارة عن الـ
+
+86
+00:07:01,140 --> 00:07:05,480
+component form لهذا الـ vectorالان لو بدنا نجي
+
+87
+00:07:05,480 --> 00:07:09,580
+نعرف كده طول ال vector هذا ال magnitude لل vector
+
+88
+00:07:09,580 --> 00:07:13,560
+هذا انتبهوا عليها شوية خلينا نرجع للرسمة و نشوف
+
+89
+00:07:13,560 --> 00:07:19,620
+كيف نوجد اللي هو الطول نيجي الان لو أخدنا من عند O
+
+90
+00:07:19,620 --> 00:07:23,040
+لعند النقطة هذا وصلنا طبعا هذا بيصير متلف قائم
+
+91
+00:07:23,040 --> 00:07:26,360
+الزاوية في هذا فبصير هذا تربيع الطول هذا طوله هذا
+
+92
+00:07:26,360 --> 00:07:31,180
+من هذا لهذا هذا طوله تربيع بساوي هذا تربيع V1
+
+93
+00:07:31,180 --> 00:07:37,680
+تربيع حيهازائد V2 تربيه لإيش لإنه الآن هي ال
+
+94
+00:07:37,680 --> 00:07:41,780
+vector الأصدر عندنا V اللي بنوجه طوله هذه اللي هو
+
+95
+00:07:41,780 --> 00:07:46,500
+من Han ل Han عبارة عن V3 اللي هو قيمة Z هيها من
+
+96
+00:07:46,500 --> 00:07:51,190
+Han ل Han اللي هي قيمة X هيهاالحداتي لان احداث ال
+
+97
+00:07:51,190 --> 00:07:57,550
+V1 من هان لهان الاحداث الصادر اللي يمين V2 فالان
+
+98
+00:07:57,550 --> 00:08:02,230
+طول هذا الوطر من هان لعندي الهان ايش بيساوي بيساوي
+
+99
+00:08:02,230 --> 00:08:07,050
+تربيعه بيساوي هذا تربيع V1 تربيع زائد هذا التربيع
+
+100
+00:08:07,050 --> 00:08:11,980
+اللي هو V2 تربيعالان هذا في المثلث هذا هيه هيظه
+
+101
+00:08:11,980 --> 00:08:16,640
+هذا واللي اودادنا طوله هيه بيصير هذا الوطن تربيع
+
+102
+00:08:16,640 --> 00:08:20,680
+اللي هو الطول المطلوب بيساوي V تلاتة تربيع زائد
+
+103
+00:08:20,680 --> 00:08:24,780
+هذا تربيع اللي هو عبارة عن V واحد تربيع زائد V
+
+104
+00:08:24,780 --> 00:08:28,000
+اتنين تربيع الأوداد هنا يعني الان طول ال vector
+
+105
+00:08:28,000 --> 00:08:32,200
+هذا تربيع بيساوي V واحد تربيع زائد V اتنين تربيع
+
+106
+00:08:32,200 --> 00:08:38,100
+زائد V تلاتة تربيعبسير الطول V هو عبارة عن الجدر
+
+107
+00:08:38,100 --> 00:08:42,380
+التربيعي لـ V1 تربيع زائد V2 تربيع زائد V3 تربيع
+
+108
+00:08:42,380 --> 00:08:46,460
+هذا عبارة عن الـ magnitude or length of the vector
+
+109
+00:08:46,460 --> 00:08:50,320
+V بساوة بـ Equal is the negative number magnitude
+
+110
+00:08:50,320 --> 00:08:54,200
+لV بساوة الجدر التربيعي V1 تربيع زائد V2 تربيع
+
+111
+00:08:54,200 --> 00:09:00,180
+زائد V3 تربيعالان او هو بساوي بال .. بال .. بال ..
+
+112
+00:09:00,180 --> 00:09:03,700
+بالإحداثيات اللي عنا أخدنا بي كيو زي ما قولنا هاي
+
+113
+00:09:03,700 --> 00:09:09,380
+X2 نقص X1 Y2 نقص Y1 بصير X2 نقص X1 لكل تربيع زي Y2
+
+114
+00:09:09,380 --> 00:09:15,740
+نقص Y1 تربيع زي Z2 نقص Z1 تربيع هذا اللي هو مين
+
+115
+00:09:15,740 --> 00:09:20,750
+اللي هو طول ال vector Vاللي هو عبارة عن الـ vector
+
+116
+00:09:20,750 --> 00:09:27,350
+من B لعند AQ نأخد الأمثل العددية على اللي حكيناه
+
+117
+00:09:27,350 --> 00:09:31,650
+find the component form and the length of the
+
+118
+00:09:31,650 --> 00:09:35,570
+vector with the initial point B-3,4,1 and terminal
+
+119
+00:09:35,570 --> 00:09:41,110
+point Q-5,2,2 يعني الآن بدنا ال component form لل
+
+120
+00:09:41,110 --> 00:09:44,740
+vector هذا BQبعد ما نوجد ال component form نوجد
+
+121
+00:09:44,740 --> 00:09:47,920
+طوله ال component form اتفجنا كيف نوجدها هاي نقطة
+
+122
+00:09:47,920 --> 00:09:50,800
+البداية وهي نقطة النهاية ال component form ال X
+
+123
+00:09:50,800 --> 00:09:55,820
+component ناقص خمسة ناقص ناقص تلاتة بتطلع مين اللي
+
+124
+00:09:55,820 --> 00:10:00,480
+هي V1 نفترض ان احنا اللي هي احداثيات V اللي هو PQ
+
+125
+00:10:00,480 --> 00:10:06,870
+عبارة عن V1 V2 V3لأن كيف نوجدها ناقص خمسة ناقص
+
+126
+00:10:06,870 --> 00:10:10,650
+ناقص تلاتة اللي هي X2 ناقص X1 ناقص خمسة ناقص ناقص
+
+127
+00:10:10,650 --> 00:10:15,350
+تلاتة بيصول ناقص اتنين لأن V2 اللي هو اتنين ناقص
+
+128
+00:10:15,350 --> 00:10:18,990
+اربعة اتنين ناقص اربعة اللي هي سالب اتنين هذه اللي
+
+129
+00:10:18,990 --> 00:10:23,970
+سمناها الاحداث ال Y الآن الأخيرة ال Z اتنين ناقص
+
+130
+00:10:23,970 --> 00:10:27,970
+واحد بتطلع V3 اتنين ناقص واحد Y ساو واحد اذا صار
+
+131
+00:10:27,970 --> 00:10:39,130
+عندي الآنالـ component form of V بيكيو ناقص اتنين
+
+132
+00:10:39,130 --> 00:10:43,830
+ال X و ناقص اتنين ال Y و واحد ال Z هذا ال
+
+133
+00:10:43,830 --> 00:10:48,770
+component form الان بدنا نجد ال link لل V أو ال
+
+134
+00:10:48,770 --> 00:10:49,750
+link لل بيكيو
+
+135
+00:10:52,610 --> 00:10:56,450
+الجدر التربيعي ل V1 تربيع زائد V2 تربيع زائد V3
+
+136
+00:10:56,450 --> 00:11:00,930
+تربيع يساوي الجدر التربيعي ل 4 زائد 4 زائد 4 زائد
+
+137
+00:11:00,930 --> 00:11:06,670
+1 من هنا جبت هذه ربعت هذا و ربعت هذا و ربعت اللي
+
+138
+00:11:06,670 --> 00:11:12,370
+هو هذا طالع عندي ايش بساوي بساوي تلاتة كويس هيك
+
+139
+00:11:12,370 --> 00:11:18,370
+ماشي الحالة اذا هو ال length of V هيطلع عندي لل
+
+140
+00:11:18,370 --> 00:11:20,850
+vector بساوي ايش بساوي تلاتة
+
+141
+00:11:23,610 --> 00:11:28,270
+الان نتعرف على vector algebra operations أو بمعنى
+
+142
+00:11:28,270 --> 00:11:32,580
+آخر العمليات الجبرية على المتجهاتالان بدنا نشوف
+
+143
+00:11:32,580 --> 00:11:37,200
+جبريا كيف بنجمع الهو two vectors هي عند ال vector
+
+144
+00:11:37,200 --> 00:11:43,040
+U هي ال component formula U1 U2 U3 و V بيساوي V1
+
+145
+00:11:43,040 --> 00:11:48,520
+V2 V3 اللي انا عشان نجمع جبريا U و V بنيجي بنقول
+
+146
+00:11:48,520 --> 00:11:52,660
+ال vector الجديد اللي اسمه U زائد V ايش بيساوي U1
+
+147
+00:11:52,660 --> 00:12:02,920
+زائد V1U2 V2 U3 V3 هذا اللي هو مجموع two vectors
+
+148
+00:12:02,920 --> 00:12:07,300
+إذا بتجمع ال X لل X و ال Y لل Y و ال Z لل Z بتطلع
+
+149
+00:12:07,300 --> 00:12:12,560
+عندك اللي هو ال vector U زائد V الان scalar
+
+150
+00:12:12,560 --> 00:12:16,340
+multiplication اللي هي عملية ضرب ال vector ب
+
+151
+00:12:16,340 --> 00:12:25,070
+scalar هي K في U بتساويك في U1 يعني بتدخل الان ك
+
+152
+00:12:25,070 --> 00:12:30,970
+على U1 ك U2 ك U3 ال vector الجديد هذا اسمه اللي هو
+
+153
+00:12:30,970 --> 00:12:37,290
+عبارة عن مين عن ال vector K فيه الان هذا جبريا هيك
+
+154
+00:12:37,290 --> 00:12:41,330
+اللي هو الجمع الجبري لما نشوف ايش معناته هندسيا او
+
+155
+00:12:41,330 --> 00:12:46,470
+كيف هندسيا و نقارنه بحاجات احنا بنعرفها قبل هيك
+
+156
+00:12:46,470 --> 00:12:52,270
+اللي هو في اللي هو جمع المتجهات هندسياالان لو كان
+
+157
+00:12:52,270 --> 00:12:56,410
+عندى ال vector U هى هذا ال position vector اللى هو
+
+158
+00:12:56,410 --> 00:13:01,330
+عبارة عن احداثياته U واحد و U اتنين باجي برسم هى U
+
+159
+00:13:01,330 --> 00:13:04,490
+قاعدة المثلثة جماعة اللى انتوا عارفينها او قاعدة
+
+160
+00:13:04,490 --> 00:13:10,290
+الدواز الأضلاع هى U اللى ان من عند نهاية U باجي
+
+161
+00:13:10,290 --> 00:13:15,310
+برسم ال vector V واحد اللى هو اللى احداثياته V
+
+162
+00:13:15,310 --> 00:13:19,610
+واحد و V اتنين يعني بمشي هيك V واحد و بطلع فوق
+
+163
+00:13:19,610 --> 00:13:23,420
+اللى هو مين V اتنينفبطلع عند ال vector هذا اللي
+
+164
+00:13:23,420 --> 00:13:27,960
+عندي هذا هو ال vector V الان مجموح U زائد V زي ما
+
+165
+00:13:27,960 --> 00:13:31,460
+انتوا عارفين اللي هو ال vector اللي بيغلق المثلف
+
+166
+00:13:31,460 --> 00:13:36,860
+هذا هي اللي بالأحمر هذا هو عبارة عن U زائد V طيب
+
+167
+00:13:36,860 --> 00:13:40,300
+لو اجينا اتطلعنا هل هذا بتناسق مع اللي حكيناها قبل
+
+168
+00:13:40,300 --> 00:13:46,240
+و شوية اه طبعا هذه الان هي عبارة عن U U1 و U2و V
+
+169
+00:13:46,240 --> 00:13:48,380
+هذا عبارة عن V واحد و V اتنين
+
+170
+00:14:01,160 --> 00:14:09,780
+إحداثيات الصادر هي U2 و V2 إذا فعلا ال vector
+
+171
+00:14:09,780 --> 00:14:16,480
+بالأحمر U زائد V هو فعلا إحداثيات U1 زائد V1 و U2
+
+172
+00:14:16,480 --> 00:14:20,440
+زائد V2 و كأنه بيقول لنا انه يا عم جمعكم الأولاني
+
+173
+00:14:20,440 --> 00:14:26,070
+بتناسق مع اللي بنحكي كيف نجمعته vectors هندسياالان
+
+174
+00:14:26,070 --> 00:14:29,230
+ممكن نظرة ايه لو نظرة اخرى واحد يقول لك انا مابديش
+
+175
+00:14:29,230 --> 00:14:31,710
+المتلف خلينا نرجع لقاعدة متوازلة اطلع و احنا
+
+176
+00:14:31,710 --> 00:14:35,790
+عارفينها اقول له ماشي الحال ارسم من عند ال U ارسم
+
+177
+00:14:35,790 --> 00:14:40,350
+ال position vector له U هيا و ارسم ال V برضه من
+
+178
+00:14:40,350 --> 00:14:44,180
+نقطة الاصل position vector Vالان كميلي متوازي
+
+179
+00:14:44,180 --> 00:14:50,180
+الأضلاح هذا خد القطر هذا هذا القطر هو اللي هيمثل
+
+180
+00:14:50,180 --> 00:14:54,220
+لك U زائد V وهذا طبعا بتنسخ مع اللي هان ليه؟ لأن
+
+181
+00:14:54,220 --> 00:14:57,870
+أصلا هذا V هيلو نفس الطول نفس الاتجاه نبغى زائله
+
+182
+00:14:57,870 --> 00:15:01,690
+في نفس الاتجاه و لا نفس الطول فبصير U زائد V هو U
+
+183
+00:15:01,690 --> 00:15:06,410
+زائد V إذن الآن عملية الجمع بين two vectors هندسيا
+
+184
+00:15:06,410 --> 00:15:11,810
+و جبريا عملية متناسقة و ملائمة و تولائم تولائم
+
+185
+00:15:11,810 --> 00:15:15,970
+اللي اتعلمنا سابقا نيجي الآن يا جماعة عملية scalar
+
+186
+00:15:15,970 --> 00:15:19,270
+multiplication نشوف المعنى الهندسيلها أكيد كلكم
+
+187
+00:15:19,270 --> 00:15:23,730
+قال إنه هى لو كان عندنا ال vector U لو ضربنا U في
+
+188
+00:15:23,730 --> 00:15:29,500
+حدداللي هو موجب هيظل محافظ على الاتجاه لكن لو
+
+189
+00:15:29,500 --> 00:15:35,120
+ضربناه في عدد سالم هيعكس الاتجاه طيب .. الان لو
+
+190
+00:15:35,120 --> 00:15:38,960
+كان العدد اللي ضربناه فيه اللي هو اكبر من واحد
+
+191
+00:15:38,960 --> 00:15:42,040
+هيزيده طول ال vector مثلا ضربناه في واحد و نص
+
+192
+00:15:42,040 --> 00:15:45,600
+هيصير طول ال vector جد اللي جابه المرة و نص هيحافظ
+
+193
+00:15:45,600 --> 00:15:48,880
+على نفس الاتجاه لو ضربناه في اتنين هيحافظ على نفس
+
+194
+00:15:48,880 --> 00:15:52,520
+الاتجاه و هيصير عندنا ضعف طول ال vector لأن لو
+
+195
+00:15:52,520 --> 00:15:57,000
+ضربناه في نقص اتنينهيناكس ال vector و هيكون طول
+
+196
+00:15:57,000 --> 00:16:01,020
+هذا ال vector هو نفسه طوله هذا ولكن ايه شماله اللي
+
+197
+00:16:01,020 --> 00:16:06,500
+هو في اتجاهه معاكس اذا الان الان عملية الضرب K في
+
+198
+00:16:06,500 --> 00:16:12,190
+U لل vector لما كان U صار K في U الان ال vectorيا
+
+199
+00:16:12,190 --> 00:16:16,670
+إما في نفس الاتجاه أو معاكس حسب الـ K أصغر من 0 أو
+
+200
+00:16:16,670 --> 00:16:21,210
+K أكبر من 0 أكبر من 0 نفس الاتجاه أصغر من 0 اللي
+
+201
+00:16:21,210 --> 00:16:25,590
+هو عكس الاتجاه K كانت اللي هي لو كان ضربنا في نص
+
+202
+00:16:25,590 --> 00:16:29,510
+حيصغر ال vector هذا يعني لو ضربناها في كسر حيصغرلو
+
+203
+00:16:29,510 --> 00:16:33,830
+ضربناه يعني كسر ازرع من واحد و اكبر من سفر لو
+
+204
+00:16:33,830 --> 00:16:37,310
+ضربناه في اشي اكبر من واحد هيكبر طول ال vector
+
+205
+00:16:37,310 --> 00:16:43,890
+الان بدنا نقارن بيقولي comparing the length of U
+
+206
+00:16:43,890 --> 00:16:50,390
+and K U جبريا خلينا نقارن
+
+207
+00:16:50,390 --> 00:16:53,090
+بين الجهتين شوفوا صلوا حنبي عليه الصلاة والسلام
+
+208
+00:16:53,850 --> 00:16:59,230
+عندك احسب ليه ال magnitude لكيف U ؟ احنا اتفجنا شو
+
+209
+00:16:59,230 --> 00:17:03,490
+معناه كيف U ال U لو كانت U واحد و U اتنين و U
+
+210
+00:17:03,490 --> 00:17:08,050
+تلاتة ال component لها ال كيف U قلنا كيف بيصير ك U
+
+211
+00:17:08,050 --> 00:17:13,240
+واحد ك U اتنين ك U تلاتةالـ Link لـ KU اللي هو
+
+212
+00:17:13,240 --> 00:17:17,560
+عبارة عن جدر التربيعي الإحداث السيني هو KU 1 تربيع
+
+213
+00:17:17,560 --> 00:17:21,620
+زاد الإحداث الصادر تربيع اللي هو KU 2 تربيع زاد
+
+214
+00:17:21,620 --> 00:17:27,110
+اللي هي إحداث الصادر KU 3 تربيعهذا طول مين اللي هو
+
+215
+00:17:27,110 --> 00:17:31,790
+ال vector k في u الان طلعلي ال k تربيع برا مالحساب
+
+216
+00:17:31,790 --> 00:17:36,350
+بضل ان هدول حاليا طلعلي جدر ال k تربيع برا ضل في
+
+217
+00:17:36,350 --> 00:17:39,370
+هذا لان جدر ال k تربيع زي ما انتوا عارفين هو عبارة
+
+218
+00:17:39,370 --> 00:17:43,050
+عن absolute value لكيه وهذا الجدر التربيعي هو
+
+219
+00:17:43,050 --> 00:17:46,050
+عبارة عن طول ال vector u لان ال components ليهم لل
+
+220
+00:17:46,050 --> 00:17:50,910
+u u واحد و u اتنين و u تلاتة اذا بطلع عندي دائما
+
+221
+00:17:50,910 --> 00:17:55,960
+ال absolute value او ال magnitudeأو طول اللي هو k
+
+222
+00:17:55,960 --> 00:18:01,300
+في u بساوي ال absolute value لل k في main في طول
+
+223
+00:18:01,300 --> 00:18:06,400
+ال u إذا بقولك the length of k u is the absolute
+
+224
+00:18:06,400 --> 00:18:09,420
+value of the scalar k times the length of u ده
+
+225
+00:18:09,420 --> 00:18:14,080
+vector ناقص واحد في u إيش هيساوي ناقص u زي ما
+
+226
+00:18:14,080 --> 00:18:18,750
+شوفنا هنا uلأن ناقص u هيكون عكس اتجاه يعني نفس
+
+227
+00:18:18,750 --> 00:18:23,030
+الطول و لكن عكس الاتجاه يعني ناقص واحد في u بساوي
+
+228
+00:18:23,030 --> 00:18:27,790
+ناقص u اللي هي نفس الطول لكن عكس الاتجاه عن ال u
+
+229
+00:18:27,790 --> 00:18:31,950
+has the same length as u but in the opposite
+
+230
+00:18:31,950 --> 00:18:34,330
+direction
+
+231
+00:18:36,780 --> 00:18:44,000
+معرفة U ناقص V of two vectors is defined by U ناقص
+
+232
+00:18:44,000 --> 00:18:50,280
+V بتساوي U ناقص V بتساوي U زائد ناقص V طبعا احنا
+
+233
+00:18:50,280 --> 00:18:55,380
+ناقص V عرفناها هي عبارة عن ناقص واحد في V الآن إذا
+
+234
+00:18:55,380 --> 00:18:59,980
+هذه معرفة والجامع معرفة إذا هذه الجديدة عرفناها
+
+235
+00:18:59,980 --> 00:19:04,500
+بإيش بنعرفه الآن اللي هو جامع two vectorsوهذا ناقص
+
+236
+00:19:04,500 --> 00:19:09,220
+ال V هو ناقص واحد في V تنشوف هذا الكلام هل هي
+
+237
+00:19:09,220 --> 00:19:14,300
+تلائم جدريا او هندسيا او لأ ونشوف كيف اللي هو مع
+
+238
+00:19:14,300 --> 00:19:18,280
+معلوماتنا السابقة هتطلع الآن هي عندي هذا اللي هو
+
+239
+00:19:18,280 --> 00:19:23,510
+ال vector U وهذا ال vector V ماشي الحالةالان توا
+
+240
+00:19:23,510 --> 00:19:30,170
+عرفنا اللى هو الان الان u ناقص v ايش بيقول ال u
+
+241
+00:19:30,170 --> 00:19:35,570
+ناقص v هو عبارة عن اللى هو ال vector اللى بيصل من
+
+242
+00:19:35,570 --> 00:19:40,690
+عند ال v اللى هو اللى إلها ناقصها وبنتى لعند مين
+
+243
+00:19:40,690 --> 00:19:44,030
+ال u اللى هى الموجبة طب ليش هي كده انت قولت عنه
+
+244
+00:19:44,030 --> 00:19:52,280
+عشان لما نقول u زائد v عشان نقول v زائد u ناقص vهي
+
+245
+00:19:52,280 --> 00:19:56,040
+عملية الجمع اللى قبل شوية هى هذا و هى هذا و جمحين
+
+246
+00:19:56,040 --> 00:20:00,100
+اللى بيغلقل يعني لما يكون هذا زائد هذا الناقص V مع
+
+247
+00:20:00,100 --> 00:20:04,760
+ال V بيطلع منين ال U فعلا متناسق هذا الخط الأحمر
+
+248
+00:20:04,760 --> 00:20:10,040
+عبارة عن U ناقص V الان نيجي نطلع عليها نظرة أخرى
+
+249
+00:20:10,040 --> 00:20:18,480
+هى عند ال U و هى عند ال V ماشي الان ال V هذا ناقصه
+
+250
+00:20:18,480 --> 00:20:25,170
+هو نفسهالطول له و لكن اتجاهه عكسه تماما هي ناقص V
+
+251
+00:20:25,170 --> 00:20:32,720
+فالان لما نجمع U زائد ناقص V اللي بيغلقهإذاً هذا
+
+252
+00:20:32,720 --> 00:20:36,900
+عبارة عن اللي أغلق هنا هو عبارة عن U زائد ناقص V
+
+253
+00:20:36,900 --> 00:20:43,060
+هو اللي هذا زائد الجديد اللي هو ناقص V فبصير U
+
+254
+00:20:43,060 --> 00:20:46,700
+زائد ناقص V هو اللي بالأحمر لاحظ اللي بالأحمر هذا
+
+255
+00:20:46,700 --> 00:20:50,700
+له نفس الطولة اللي فوق و نفس ال direction اللي
+
+256
+00:20:50,700 --> 00:20:55,200
+بالأحمرفعشان هيك ال U ناقص V بيساوي ال U زائد ناقص
+
+257
+00:20:55,200 --> 00:20:59,120
+V اللي هو فعلا كلامنا اللي قبل بشوية منطقي
+
+258
+00:20:59,120 --> 00:21:03,880
+والتعريف منطقي وهي تفسيره الهندسي ويتلاءم معاه
+
+259
+00:21:03,880 --> 00:21:08,620
+يعني ال U ناقص V هو عبارة عن U زائد ناقص V زي ما
+
+260
+00:21:08,620 --> 00:21:14,390
+شفنا اللي هو في الرسم اللي حكيناطيب الان ناخد بعض
+
+261
+00:21:14,390 --> 00:21:18,730
+الأمثلة العددية let u بالساوية ناقص واحد و تلاتة و
+
+262
+00:21:18,730 --> 00:21:23,090
+واحد و v بالساوية اربعة و سبعة و سفر هدولة ال two
+
+263
+00:21:23,090 --> 00:21:27,270
+vectors الان بدي بيقوللي بيسألي كمان يالي طبعا
+
+264
+00:21:27,270 --> 00:21:31,170
+مكتوبات التنتين في صورة مين يا جماعة ال component
+
+265
+00:21:31,170 --> 00:21:36,630
+فبقولي find the components of اوجد ال components
+
+266
+00:21:36,630 --> 00:21:42,370
+لمين اتنين u زي التلاتة vعرفنا شو معناه اتنين U
+
+267
+00:21:42,370 --> 00:21:46,270
+وعرفنا شو معناه تلاتة V وعرفنا شو معناه جمعتي و
+
+268
+00:21:46,270 --> 00:21:51,370
+vectors U نقص V وبرضه عرفنا ال magnitude لنص ال U
+
+269
+00:21:51,370 --> 00:21:55,610
+اذا ال N اللي هو find the components of جزده عن
+
+270
+00:21:55,610 --> 00:21:59,830
+اتنين عن A وB و ال C بدك ايش توجد لها ال magnitude
+
+271
+00:21:59,830 --> 00:22:06,310
+لها او نص النص قيمة طول نص ال U طيب خلونا نشوف و
+
+272
+00:22:06,310 --> 00:22:11,600
+نصل لعنديالان اتنين U زي التلاتة V هي اتنين U زي
+
+273
+00:22:11,600 --> 00:22:15,460
+التلاتة V شيل ال U و حط مكانها قيمتها هي قيمتها و
+
+274
+00:22:15,460 --> 00:22:20,020
+ال V شيلها و حط قيمتها هيها الان ابدا اللي هو
+
+275
+00:22:20,020 --> 00:22:25,360
+اشتغل ضرب لهد جوا ضرب كلنا كيف نضرب احنا scalar في
+
+276
+00:22:25,360 --> 00:22:32,860
+ال vector بصير اتنين في ناقص واحدنفس الشيء بالنسبة
+
+277
+00:22:32,860 --> 00:22:35,540
+لتلاتة في أربعة و تلاتة في سبعة و تلاتة في سفرة و
+
+278
+00:22:35,540 --> 00:22:39,500
+نجمحها نعرف كيف نجمح هدا مع هدا و هدا مع هدا و هدا
+
+279
+00:22:39,500 --> 00:22:40,640
+مع هدا و هدا مع هدا و هدا مع هدا و هدا مع هدا و
+
+280
+00:22:40,640 --> 00:22:42,400
+هدا مع هدا و هدا مع هدا و هدا مع هدا و هدا مع هدا
+
+281
+00:22:42,400 --> 00:22:42,700
+و هدا مع هدا و هدا مع هدا و هدا مع هدا و هدا مع
+
+282
+00:22:42,700 --> 00:22:42,940
+مع هدا و هدا مع هدا و هدا مع هدا و هدا مع هدا و
+
+283
+00:22:42,940 --> 00:22:42,960
+و هدا مع هدا و هدا مع هدا و هدا مع هدا و هدا مع
+
+284
+00:22:42,960 --> 00:22:47,520
+هدانقص اتنين و تلاتة فى أربعة باطن عاش نقص اتنين
+
+285
+00:22:47,520 --> 00:22:53,180
+واطن عاش هو عشرة اتنين فى تلاتة ستة ستة زائد تلاتة
+
+286
+00:22:53,180 --> 00:22:56,480
+فى سبعة بواحد وعشرين ستة وواحد وعشرين بيطلع سبعة
+
+287
+00:22:56,480 --> 00:23:02,360
+وعشرين اتنين زائد سفر بيطلع اش اتنين اذا هذا قيمة
+
+288
+00:23:02,360 --> 00:23:06,740
+اللى هو اللى هى اتنين U زائد تلاتة بيه نيجي لل U
+
+289
+00:23:06,740 --> 00:23:11,460
+نقص فيه هي ال Uوهي ال V وهي الناقص طبعا ناقص
+
+290
+00:23:11,460 --> 00:23:15,580
+معناته اللي هو ناقص أربعة و ناقص سبعة و سفر يعني
+
+291
+00:23:15,580 --> 00:23:21,240
+ناقص واحد ناقص أربعة ناقص خمسة تلاتة ناقص سبعة
+
+292
+00:23:21,240 --> 00:23:26,280
+يعني ناقص أربعة واحد ناقص سفر يعني واحد الآن بدنا
+
+293
+00:23:26,280 --> 00:23:29,380
+نيجي نيجي ل سيل هو بدنا نوجد magnitude ل نص يوم
+
+294
+00:23:29,900 --> 00:23:34,900
+الان هى U نصه نص ال U اللى هو عبارة عن نص فى نقص
+
+295
+00:23:34,900 --> 00:23:39,160
+واحد و نص فى تلاتة و نص فى واحد يعنى الان ال
+
+296
+00:23:39,160 --> 00:23:43,520
+length اللى نص فى نقص واحد لكل تربيع زائد نص فى
+
+297
+00:23:43,520 --> 00:23:47,380
+تلاتة لكل تربيع زائد نص فى واحد لكل تربيع هذا اللى
+
+298
+00:23:47,380 --> 00:23:51,400
+هو عبارة عن الجدر التربيع اللى بتطلع ال length لمن
+
+299
+00:23:51,400 --> 00:23:58,240
+لل vector نص U ويساوي احسب ان دولة هدى بتطلع نصو
+
+300
+00:23:58,240 --> 00:24:03,080
+هذه تسعة على اتنين وهذه واحد على اتنين يعني بيصير
+
+301
+00:24:03,080 --> 00:24:06,820
+واحد واحد اتنين وتسعة احد عاش يعني احداش عالمين
+
+302
+00:24:06,820 --> 00:24:09,580
+على اربعة اسف لان انا بتربع اتنين تحت بيصير اربعة
+
+303
+00:24:09,580 --> 00:24:13,940
+بيصير جدر التربيع ال 11 على اربعةطلعني الربع برا
+
+304
+00:24:13,940 --> 00:24:17,520
+الجذر بيطلع من نص بيصير نص الجذر ال 11 اذا
+
+305
+00:24:17,520 --> 00:24:21,420
+magnitude او طول ال vector نص U بيساوي نص الجذر ال
+
+306
+00:24:21,420 --> 00:24:24,900
+11 واحد تاني قال لي ليش تغلبني زي هي كانت استخدام
+
+307
+00:24:24,900 --> 00:24:27,980
+الخاص اللي قبله شوية قوله مش مشكلة استخدامها كده
+
+308
+00:24:27,980 --> 00:24:30,700
+قال لي باخده النص برا بيصير نص في ال magnitude ده
+
+309
+00:24:30,700 --> 00:24:35,160
+Uالـ magnitude value هي الجذر التربيعي الواحد زائد
+
+310
+00:24:35,160 --> 00:24:36,180
+تسعة زائد واحد
+
+311
+00:24:48,420 --> 00:24:56,900
+نجي الآن لبعض خواص ال vector operations
+
+312
+00:24:56,900 --> 00:25:03,640
+أو بمعنى آخر خواص العمليات عالميا على المتجهات
+
+313
+00:25:03,640 --> 00:25:08,120
+خواص العمليات الجبرية على المتجهات انتبهوا let u و
+
+314
+00:25:08,120 --> 00:25:12,140
+v و w be vectors and a و b scalars يعني نفترض ان
+
+315
+00:25:12,140 --> 00:25:17,760
+ال u و ال v و ال w جمع ايش هي عبارة عن متجهاتو ال
+
+316
+00:25:17,760 --> 00:25:21,920
+a و ال b هى عبارة عن اعداد ال scholars نيجى
+
+317
+00:25:21,920 --> 00:25:25,760
+للخاصية الاولى ايش الخاصية الاولى بتقول بتقول يو
+
+318
+00:25:25,760 --> 00:25:29,940
+زائد V بيساوي V زائد U يعنى اللى هى عملية الجمع
+
+319
+00:25:29,940 --> 00:25:35,110
+على ال vectors او المتجهات إبداليةالخاصية التانية
+
+320
+00:25:35,110 --> 00:25:40,150
+اللى بتقولى عملية الجمع عبارة عن عملية دامجة أو
+
+321
+00:25:40,150 --> 00:25:43,970
+تجميعية أو اللى بيسميها associative يعنى عملية
+
+322
+00:25:43,970 --> 00:25:48,930
+الجمع ع ال vectors هي عملية دامجة أو تجميعية أو
+
+323
+00:25:48,930 --> 00:25:52,690
+associative يعنى لو قلت U زائد V و بعدين جمعتلها W
+
+324
+00:25:52,690 --> 00:25:57,630
+برضه منفع حد ساوى ال V زائد W و لما نقول U زائد
+
+325
+00:25:57,630 --> 00:26:01,950
+اللى بطلع هنا بطلع نفس الاشياءالان تلاتة بيقولي ان
+
+326
+00:26:01,950 --> 00:26:08,830
+المحايد الجامعي لل vectors u اللي هو من عملية
+
+327
+00:26:08,830 --> 00:26:11,370
+الجامعة طبعا اللي حامل المحايد الجامعي اللي هو
+
+328
+00:26:11,370 --> 00:26:14,690
+جداش بساوي سفر يعني zero zero zero اللي هو ال
+
+329
+00:26:14,690 --> 00:26:18,210
+vector zero إذا دائما u زائد zero vector طبعا دي
+
+330
+00:26:18,210 --> 00:26:21,540
+vector و دي vector بساوي uالان انا بقول ال inverse
+
+331
+00:26:21,540 --> 00:26:26,700
+او المعكوس الجمعى لل U هو عبارة عن ناقص U ك vector
+
+332
+00:26:26,700 --> 00:26:31,340
+طبعا يعني U زائد معكوس الجمعى اللى هو ناقص ال U
+
+333
+00:26:31,340 --> 00:26:36,260
+بساوية Zero vector Zero الان Zero في U بساوية Zero
+
+334
+00:26:36,260 --> 00:26:39,840
+Zero هنا scalar يا جماعة و ال U vector Zero scalar
+
+335
+00:26:39,840 --> 00:26:46,040
+في U vector بساوية Zero vectorطيب الواحد scalar في
+
+336
+00:26:46,040 --> 00:26:49,800
+U بيطلع ايش بيساوي U يعني انك ضمن تضرب واحد في اي
+
+337
+00:26:49,800 --> 00:26:54,480
+vector بيظله نفس ال vector لا يتغير لا اتجاهه ولا
+
+338
+00:26:54,480 --> 00:27:00,160
+طوله طيب الان خاصية السابعة عملية اللي هي ضرب ال
+
+339
+00:27:00,160 --> 00:27:04,540
+scalars في ال vector عملية دامجة او تجميعية يعني
+
+340
+00:27:04,540 --> 00:27:08,530
+بمعنى اخرأ في ب في يو يعني اضرب في ب في يو و بعدين
+
+341
+00:27:08,530 --> 00:27:12,130
+اضرب ا فيهن هو نفس الاشي ا في ب و بعدين في مين في
+
+342
+00:27:12,130 --> 00:27:15,890
+يو ا في ب ضرب هن ضرب مين اللي هو ضرب عادي تبقى
+
+343
+00:27:15,890 --> 00:27:20,630
+اعداد اللي انا من ضرب في يو بيكون ضرب بين scalar و
+
+344
+00:27:20,630 --> 00:27:25,680
+vector وعرفناهالخاصية الثانية a في u زائد v بساوي
+
+345
+00:27:25,680 --> 00:27:29,400
+a u زائد a v يعني يتوزع عملية ال scalar
+
+346
+00:27:29,400 --> 00:27:33,500
+multiplication على ال vector addition بنوزع عملية
+
+347
+00:27:33,500 --> 00:27:36,980
+اللي هي ضرب اللي هو ال scalar في ال vector على
+
+348
+00:27:36,980 --> 00:27:40,060
+اللي هو الجامع بين المتجهات بيطلع a u زائد a v
+
+349
+00:27:40,060 --> 00:27:45,900
+الآن الجامع بين ال scalars الجامع العادي تبعنا ال
+
+350
+00:27:45,900 --> 00:27:50,220
+u لما تنضرب فيها توزع يعني ك vector بيصير u في a
+
+351
+00:27:50,930 --> 00:27:56,830
+زائد U في B يعني AU زائد B في U هذه برضه الخاصية
+
+352
+00:27:56,830 --> 00:28:01,290
+اللي لها متحققة طيب احنا بخلنا نبرهلنا واحدة من
+
+353
+00:28:01,290 --> 00:28:04,250
+هدول اختارت عشوائية انا اختارت رتارية تمانية قلت
+
+354
+00:28:04,250 --> 00:28:07,690
+تمكن تمانية أصعب واحدة فيهم و .. و .. و تاخدوا
+
+355
+00:28:07,690 --> 00:28:11,900
+فكرة عن كيف اللي هو بينبرهلأولا لت U أولا لت U
+
+356
+00:28:11,900 --> 00:28:12,660
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+357
+00:28:12,660 --> 00:28:13,320
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+358
+00:28:13,320 --> 00:28:14,580
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+359
+00:28:14,580 --> 00:28:15,260
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+360
+00:28:15,260 --> 00:28:19,220
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+361
+00:28:19,220 --> 00:28:20,920
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+362
+00:28:20,920 --> 00:28:25,060
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+363
+00:28:25,060 --> 00:28:30,450
+أولو V هي كمبيوتر فورميله V1, V2, V3 و الان اذا
+
+364
+00:28:30,450 --> 00:28:34,250
+انا احسب A في U زائد V الـ A is scalar و U زائد V
+
+365
+00:28:34,250 --> 00:28:39,390
+جمع Vectors بيصير A في او ضد ال U زائد V ال U زائد
+
+366
+00:28:39,390 --> 00:28:50,640
+V ايش هيكون U1 زائد V1U2 V2 U3 V3 U2
+
+367
+00:28:50,640 --> 00:28:54,400
+V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3
+
+368
+00:28:54,400 --> 00:28:57,340
+U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2
+
+369
+00:28:57,340 --> 00:28:58,280
+V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3
+
+370
+00:28:58,280 --> 00:29:05,220
+U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2
+
+371
+00:29:05,220 --> 00:29:15,560
+V3 U2 V3 U2 V3أي و اتنين اي و تلاتة اللي بيظل اي و
+
+372
+00:29:15,560 --> 00:29:24,600
+احد اي و اتنين اي و تلاتة اللي بيظل
+
+373
+00:29:24,600 --> 00:29:25,560
+اي و احد اي و اتنين اي و تلاتة اللي بيظل اي و احد
+
+374
+00:29:25,560 --> 00:29:26,120
+اي و تلاتة اللي بيظل اي و احد اي و تلاتة اللي بيظل
+
+375
+00:29:26,120 --> 00:29:27,720
+اي و اتنين اي و تلاتة اللي بيظل اي و احد اي و
+
+376
+00:29:27,720 --> 00:29:28,420
+تلاتة اللي بيظل اي و اتنين اي و تلاتة اللي بيظل اي
+
+377
+00:29:28,420 --> 00:29:29,400
+اللي بيظل اي و اتنين اي و تلاتة اللي بيظلأيه
+
+378
+00:29:29,400 --> 00:29:31,760
+طالعها برضه عام المشترك اللي هو scalar
+
+379
+00:29:31,760 --> 00:29:34,760
+multiplication بقصود عام المشترك بده .. بده رجحها
+
+380
+00:29:34,760 --> 00:29:37,840
+ل scalar multiplication اللي هي عملية ضرب اللي هو
+
+381
+00:29:37,840 --> 00:29:43,800
+scalar في vector A في V و 1 V 2 V 3 هذا اللي هو U
+
+382
+00:29:43,800 --> 00:29:48,200
+و هذا V فبصير A U زائد A V يعني بدأت A في U زائد V
+
+383
+00:29:48,200 --> 00:29:52,560
+و اتسلسلت لما وصلت لك A في U زائد A V حيث استخدمت
+
+384
+00:29:52,560 --> 00:29:56,680
+عمليةما عن عملية الجمع بين two vectors وعملية ال
+
+385
+00:29:56,680 --> 00:30:00,360
+scale of multiplication اللي عرفناها سابقاً هذا هو
+
+386
+00:30:00,360 --> 00:30:05,700
+برهان هذه والباقية اللي هي بشكل مشارع الان نيجي ان
+
+387
+00:30:05,700 --> 00:30:09,820
+اعرفلكم هالتعريف البسيط بنقول ان احنا عندي three
+
+388
+00:30:09,820 --> 00:30:15,240
+vectors or more space vectors are called coplanar
+
+389
+00:30:15,240 --> 00:30:18,940
+يعني لو كان في عندي تلاتة من ال vector تلاتة من
+
+390
+00:30:18,940 --> 00:30:21,020
+التجاهات او اربعة من التجاهات او خمس من التجاهات
+
+391
+00:30:21,020 --> 00:30:27,140
+او زي ما بدكبنقول عنهم ان كو بلانر بلانر إذا كان
+
+392
+00:30:27,140 --> 00:30:31,660
+كو بلانر vectors بنسميهم in the case that they are
+
+393
+00:30:31,660 --> 00:30:36,340
+in the same plane يعني لما نكون كلهم مالهم في نفس
+
+394
+00:30:36,340 --> 00:30:43,240
+ال plane مثال for example U و V و U زائد Vهذول ايش
+
+395
+00:30:43,240 --> 00:30:48,660
+مالهين co-planner لإن احنا كنا نقول عشان ال U و ال
+
+396
+00:30:48,660 --> 00:30:51,860
+V ارسم ال U و ارسم ال V زي ما رسمنا قبل شوية
+
+397
+00:30:51,860 --> 00:30:57,090
+المثلث اللي بيجفلهمبكون هو u زائد v إذا التلاتة
+
+398
+00:30:57,090 --> 00:31:01,310
+بكون نفس المثلث ال u ال v و ال u زائد v يعني هنا
+
+399
+00:31:01,310 --> 00:31:06,310
+in the same plane يعني قبلها نفسها طيب الان بدنا
+
+400
+00:31:06,310 --> 00:31:09,950
+نعرف حاجة اسمها ال unit vectors و منها نصل لحاجة
+
+401
+00:31:09,950 --> 00:31:13,070
+اسمة standard unit vectors اللي هتسهل علينا عملية
+
+402
+00:31:13,070 --> 00:31:18,810
+التعامل مع المتجهات نشوف ايش مقصوده الان a vector
+
+403
+00:31:18,810 --> 00:31:23,230
+v of length one is called a unit vectorيعني اي
+
+404
+00:31:23,230 --> 00:31:26,350
+vector يا جماعة في الدنيا طول واحد ال magnitude
+
+405
+00:31:26,350 --> 00:31:30,570
+الواحد بنسميه unit vector اذا في عندي عدد لانهائم
+
+406
+00:31:30,570 --> 00:31:33,570
+ال vectors اي vector طول واحد اي زي و vector تاني
+
+407
+00:31:33,570 --> 00:31:36,610
+و vector اتنين و هذا طول واحد طول اسف واحد و هذا
+
+408
+00:31:36,610 --> 00:31:40,850
+واحد واحد ارسم عدد لانهائم ال vectors طول واحد في
+
+409
+00:31:40,850 --> 00:31:45,370
+اتجاهات مختلفة كلهم بنسميهم ايه شمالهم unit
+
+410
+00:31:45,370 --> 00:31:52,050
+vectors اذا ال unit vector هو vectorطول واحد في
+
+411
+00:31:52,050 --> 00:31:55,310
+عندي vectors انا مهتم فيها جدا لان هذه ال vectors
+
+412
+00:31:55,310 --> 00:32:00,930
+هتكون اداء لتسهيل عملية التعامل مع ال vectors مين
+
+413
+00:32:00,930 --> 00:32:04,930
+هن دولة بنسميها the standard unit vectors او اللي
+
+414
+00:32:04,930 --> 00:32:11,570
+هي ال unit vectors معتمدات او اللي هن احنا يعني
+
+415
+00:32:11,570 --> 00:32:16,770
+هنتعامل معها على طول اللي بنسميها ال I و ال J و ال
+
+416
+00:32:16,770 --> 00:32:21,060
+Kماذا يعني ال vector I؟ ال I للمجزم له component
+
+417
+00:32:21,060 --> 00:32:25,640
+form هو واحد زر و زر يعني هو على محور السينات يعني
+
+418
+00:32:25,640 --> 00:32:29,120
+جزء من محور السينات مقطوع واحد وهذا سفر وهذا سفر
+
+419
+00:32:29,120 --> 00:32:32,540
+ال J اللي هو على محور الصدات لأن هذا سفر وهذا سفر
+
+420
+00:32:32,540 --> 00:32:36,120
+و المقطوع واحد من محور الصدات ال K على محور الزد
+
+421
+00:32:36,120 --> 00:32:39,760
+ذات اللي هو سفر و سفر الجزم المقطوع بساوية واحد
+
+422
+00:32:39,760 --> 00:32:44,820
+ماشي الحال طيب محور الزد المقصود يعني طيب شوفوا
+
+423
+00:32:44,820 --> 00:32:51,630
+عليهاالان هى الامشي على ال X axis بمقدار واحد
+
+424
+00:32:51,630 --> 00:32:56,890
+وجفان اللى هو المتجه اللى جاى من نقطة الاصل واقف
+
+425
+00:32:56,890 --> 00:33:01,430
+عند هذا النقطة بنسميه اللى هو ال I الان امشي على
+
+426
+00:33:01,430 --> 00:33:05,790
+محور الصدات بمقدار واحد وجفان اللى هو هذا ال
+
+427
+00:33:05,790 --> 00:33:10,890
+vector بنسميه Jأطلع من هنا في الاتجاه هذا و اقف
+
+428
+00:33:10,890 --> 00:33:15,070
+هنا بمقدار واحد هذا بنسمي ash k ايش ال component
+
+429
+00:33:15,070 --> 00:33:18,930
+form له طبيعي واحد و سفر و سفر هذا ال component
+
+430
+00:33:18,930 --> 00:33:22,990
+form له سفر و سفر و واحد هذا ال component form له
+
+431
+00:33:22,990 --> 00:33:27,010
+سفر و واحد و سفر هدول الان ال three vectors بدي
+
+432
+00:33:27,010 --> 00:33:30,410
+احاول اكتب كل ال vectors اللي في الدنيا في دلالتهم
+
+433
+00:33:30,670 --> 00:33:38,770
+طيب الان عندي لو اجيت ل ال vector اللي هو OB1
+
+434
+00:33:38,770 --> 00:33:45,450
+شايفينه جماعي OB1 هذا ال vector اللي هو احداثيات
+
+435
+00:33:45,450 --> 00:33:53,200
+النقطة هذه نفترض انها X1 و Y1 و Z1 معاياأذا ال O B
+
+436
+00:33:53,200 --> 00:33:59,240
+1 X 1 مقطوع من مين الإحداث ال سيني تبعتها X 1 في
+
+437
+00:33:59,240 --> 00:34:06,580
+اتجاه مين ال I 1 اللي هو الإحداث الصادر Y 1 في
+
+438
+00:34:06,580 --> 00:34:11,700
+اتجاه ال J و Z 1 في اتجاه مين ال K فبصير هذا O B 1
+
+439
+00:34:11,700 --> 00:34:18,380
+اكتب و اكتب و X 1 I A Y A 1 J Z 1 Kماشي بعد شوية
+
+440
+00:34:18,380 --> 00:34:22,280
+هقولكم كيف بالظبط احنا كيف استدلنا ان نكتب ال
+
+441
+00:34:22,280 --> 00:34:26,680
+vector لما يكون ال position vector كيف نكتبه اللي
+
+442
+00:34:26,680 --> 00:34:29,660
+هو x واحد في اتجاه ال I و Y واحد في اتجاه ال J زي
+
+443
+00:34:29,660 --> 00:34:32,460
+ال Z واحد في اتجاه ال K بعد شوية ان شاء الله بس
+
+444
+00:34:32,460 --> 00:34:33,880
+خلينا نفترض انه بنعرفها الآن
+
+445
+00:34:44,000 --> 00:34:49,560
+Similarly OB2 هو عبارة
+
+446
+00:34:49,560 --> 00:34:54,340
+عن X2 Y2 و Z2 يعني بمعنى آخر بنقدر نكتب OB2 عبارة
+
+447
+00:34:54,340 --> 00:34:59,180
+عن X2 في اتجاه I و Y2 في اتجاه J و Z2 في اتجاه K
+
+448
+00:34:59,180 --> 00:35:03,800
+الان هذا ال vector B1 B2 اللي من هنا لهنا اللي هو
+
+449
+00:35:03,800 --> 00:35:08,360
+واضح انههيو جاي من عند هذا ورايح هذا زي ما عملنا
+
+450
+00:35:08,360 --> 00:35:13,600
+قبل بشوية عبارة عن b2 نقص b1 ال vector اللي هو ob2
+
+451
+00:35:13,600 --> 00:35:16,920
+هذا ناقص ال vector هذا زي ما عرفنا عملية الطرح قبل
+
+452
+00:35:16,920 --> 00:35:21,940
+بشوية فبصير هذا لإنه هذا زائد هذا بصير هذا لأمغ
+
+453
+00:35:21,940 --> 00:35:29,020
+القرن أو بمعنى آخر هذا عبارة عن x2 ناقص x1 في درجة
+
+454
+00:35:29,020 --> 00:35:33,720
+ال IY2-Y1 في اتجاه الـ J Y2-Y1 في اتجاه الـ J Z2
+
+455
+00:35:33,720 --> 00:35:37,660
+-Z1 في اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في
+
+456
+00:35:37,660 --> 00:35:38,840
+اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في اتجاه
+
+457
+00:35:38,840 --> 00:35:42,460
+الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2
+
+458
+00:35:42,460 --> 00:35:46,800
+-Z1 في اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في
+
+459
+00:35:46,800 --> 00:35:47,640
+اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في اتجاه
+
+460
+00:35:47,640 --> 00:35:49,640
+الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2
+
+461
+00:35:49,640 --> 00:35:53,880
+-Z1 في اتجاه
+
+462
+00:35:53,880 --> 00:35:57,270
+الـ K Z2-كيف كتبت اللي ob1 بساوية هذا الكلام؟
+
+463
+00:35:57,270 --> 00:36:02,550
+خلّينا نشوف الآن كيف نكتبه و اللي هي أي vector v
+
+464
+00:36:02,550 --> 00:36:06,270
+بساوية v1, v2, v3 can be written as a linear
+
+465
+00:36:06,270 --> 00:36:09,710
+combination of the standard unit vectors as
+
+466
+00:36:09,710 --> 00:36:17,100
+follows يعني الآن عند أي vector vاللي هو مكتوب في
+
+467
+00:36:17,100 --> 00:36:20,800
+ال component form اللي هو هيه بدنا نكتب هذا على
+
+468
+00:36:20,800 --> 00:36:25,340
+صورة I وJ وK سهل يا جماعة هتلاقوه سهل وممكن بعضكم
+
+469
+00:36:25,340 --> 00:36:28,810
+جال انا على طول بعرفخلّيني أخد ال V بيساوي V واحد
+
+470
+00:36:28,810 --> 00:36:33,370
+V اتنين و V تلاتة هذي الآن افردليها كالتالي و بدك
+
+471
+00:36:33,370 --> 00:36:37,250
+تبقى حاصل جمع تلت اللي هي vectors ال vector الأول
+
+472
+00:36:37,250 --> 00:36:42,110
+V واحد و هدول أصفر ال vector التاني Zero و V اتنين
+
+473
+00:36:42,110 --> 00:36:46,230
+و هذا Zero Zero Zero V تلاتة اجمحل مع بعض هتلاقي V
+
+474
+00:36:46,230 --> 00:36:50,320
+واحد زائد سفر زائد Zero V واحدزي رو زي ات زي ات زي
+
+475
+00:36:50,320 --> 00:36:53,100
+ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات
+
+476
+00:36:53,100 --> 00:36:54,000
+زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي
+
+477
+00:36:54,000 --> 00:36:54,260
+ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات
+
+478
+00:36:54,260 --> 00:36:55,380
+زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي
+
+479
+00:36:55,380 --> 00:36:59,340
+ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات
+
+480
+00:36:59,340 --> 00:37:03,440
+زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي
+
+481
+00:37:03,440 --> 00:37:07,860
+ات زي اتخد V1 برا V2 برا V3 برا V4 برا V5 برا V6
+
+482
+00:37:07,860 --> 00:37:10,420
+برا V7 برا V8 برا V9 برا V10 برا V11 برا V12 برا
+
+483
+00:37:10,420 --> 00:37:12,720
+V13 برا V14 برا V15 برا V16 برا V17 برا V18 برا
+
+484
+00:37:12,720 --> 00:37:20,320
+V19 برا V20 برا V21 برا V21 برا V22 برا V23 برا
+
+485
+00:37:20,320 --> 00:37:23,960
+V21 برا V23 برا V21 برا V21 برا V21 برا V21 برا
+
+486
+00:37:23,960 --> 00:37:23,980
+V21 برا V21 برا V21 برا V21 برا V21 برا V21 برا
+
+487
+00:37:23,980 --> 00:37:24,040
+V21 برا V21 برا V21 برا V21 برا V21 برا V21 برا
+
+488
+00:37:24,040 --> 00:37:24,060
+V21 برا V21 برا V21 برا V21 برا V21 برا V21 برا
+
+489
+00:37:24,060 --> 00:37:31,260
+V21 برا V21 برا V21 برا V21 برا V21 برا V21 برا
+
+490
+00:37:31,260 --> 00:37:42,620
+V21 برا V21قيمته بدلالة I و J و K هي V1 V2 V3 V4
+
+491
+00:37:42,620 --> 00:37:45,640
+V5 V6 V7 V8 V9 V10 V11 V12 V13 V12 V13 V13 V13 V13
+
+492
+00:37:45,640 --> 00:37:48,120
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+493
+00:37:48,120 --> 00:37:50,820
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+494
+00:37:50,820 --> 00:37:50,900
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+495
+00:37:50,900 --> 00:37:52,940
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+496
+00:37:52,940 --> 00:37:55,620
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+497
+00:37:55,620 --> 00:37:59,340
+V13 V13 V13 V
+
+498
+00:38:04,140 --> 00:38:10,000
+إذا كان ال V عبارة عن non-zero vector اذا كان ال V
+
+499
+00:38:10,000 --> 00:38:10,120
+عبارة عن non-zero vector اذا كان ال V عبارة عن non
+
+500
+00:38:10,120 --> 00:38:10,540
+اذا كان ال V عبارة عن non-zero vector اذا كان ال V
+
+501
+00:38:10,540 --> 00:38:10,600
+-zero vector اذا كان ال V عبارة عن non-zero vector
+
+502
+00:38:10,600 --> 00:38:10,620
+اذا كان ال V عبارة عن non-zero vector اذا كان ال V
+
+503
+00:38:10,620 --> 00:38:11,140
+-zero vector اذا كان ال V عبارة عن non-zero vector
+
+504
+00:38:11,140 --> 00:38:13,160
+اذا كان ال V عبارة عن non-zero vector اذا كان ال V
+
+505
+00:38:13,160 --> 00:38:16,500
+عبارة عن non-zero vector اذا كان ال V عبارة عن non
+
+506
+00:38:16,500 --> 00:38:19,920
+-zero vector اذا
+
+507
+00:38:19,920 --> 00:38:26,030
+كان ال V عبارة عن non-zero vectorهو عبارة عن is a
+
+508
+00:38:26,030 --> 00:38:30,170
+unit vector in the direction of v او اختصارا
+
+509
+00:38:30,170 --> 00:38:35,070
+بنسميه the direction of the vector v اذا the
+
+510
+00:38:35,070 --> 00:38:37,970
+direction of the vector v عبارة عن unit vector
+
+511
+00:38:37,970 --> 00:38:44,840
+حصلة من قسمة ال V على magnitude و على طولهطيب الان
+
+512
+00:38:44,840 --> 00:38:48,240
+ال equation v بتساوي magnitude v فى v على
+
+513
+00:38:48,240 --> 00:38:53,260
+magnitude of v ايه ده بتروح محدد بيصير v اه اقصد
+
+514
+00:38:53,260 --> 00:38:57,480
+اكتبها هيك ليش اقصد اكتبها على صورة اللى هو طول ال
+
+515
+00:38:57,480 --> 00:39:01,860
+vector فى اتجاههإذا أي vector في الدنيا مقدر أكتب
+
+516
+00:39:01,860 --> 00:39:06,300
+على صورة طوله ضرب اتجاهه اتجاهه V على magnitude V
+
+517
+00:39:06,300 --> 00:39:14,080
+Expresses V as its length and its direction ناخد
+
+518
+00:39:14,080 --> 00:39:18,880
+أمثال عليها اللي أنا بقوله نيجي
+
+519
+00:39:18,880 --> 00:39:21,960
+نشوف المثال اللي عندنا Find a unit vector u in the
+
+520
+00:39:21,960 --> 00:39:26,560
+direction of the vector b1 to b2بقول لأوجد ال unit
+
+521
+00:39:26,560 --> 00:39:30,920
+vector u اللي هي main اللي في اتجاه ال vector b1
+
+522
+00:39:30,920 --> 00:39:34,740
+b2 يعني بدي اوجد ال vector b1 b2 و بعدين اوجد
+
+523
+00:39:34,740 --> 00:39:39,000
+اتجاهه بتكون هي ال u طيب شوف كيف b1 b2 تعلمنا كيف
+
+524
+00:39:39,000 --> 00:39:44,340
+نحوها اللي هو تلاتة ناقص واحد هي بتطلع في اتجاه
+
+525
+00:39:44,340 --> 00:39:47,400
+الاي سفر اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر
+
+526
+00:39:47,400 --> 00:39:48,000
+اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر
+
+527
+00:39:48,000 --> 00:39:51,150
+اثر اثر اثر اثر اثر اثرفى اتجاه الـ J سفر ناقص
+
+528
+00:39:51,150 --> 00:39:56,030
+واحد فى اتجاه ال K هذا اللى هو B1 B2 المتجهة بيصير
+
+529
+00:39:56,030 --> 00:40:02,230
+عبارة عن 2I زائد 2J هذا مش زائد K ناقص K عشان اذا
+
+530
+00:40:02,230 --> 00:40:07,430
+هذا B1 B2 هو المتجه اللى عندى كتبته على صورة 2I
+
+531
+00:40:07,430 --> 00:40:14,140
+زائد 2J على صورة I زائد J زائد K and thenthe unit
+
+532
+00:40:14,140 --> 00:40:17,740
+vector in the direction of b1 b2 is يعني الان بد
+
+533
+00:40:17,740 --> 00:40:21,580
+الآن ال unit vector هو عبارة عن ال unit vector in
+
+534
+00:40:21,580 --> 00:40:25,520
+the direction of b1 b2 ايش هيكون يا جماعة هو عبارة
+
+535
+00:40:25,520 --> 00:40:29,800
+عن ال vector على magnitude و على طوله هاي ال
+
+536
+00:40:29,800 --> 00:40:34,380
+vector b2i زاد 2j نقص كيها دي يا جماعة على طوله
+
+537
+00:40:34,380 --> 00:40:38,460
+طوله الجدر التربية على مينلأ اتنين تربيع ليه اربعة
+
+538
+00:40:38,460 --> 00:40:41,920
+و اتنين تربيع اربعة هي تمانية و واحد تربيع واحد هي
+
+539
+00:40:41,920 --> 00:40:45,760
+تسعة اذا الجدر التربيعي لكل هذا طبعا الجدر لكله
+
+540
+00:40:45,760 --> 00:40:48,730
+هذا بيظل الجدر ماشيالجدر التربيه على التسعة أيش
+
+541
+00:40:48,730 --> 00:40:52,030
+بيساوي تلاتة إذا صار عندي ال direction اللي بدويا
+
+542
+00:40:52,030 --> 00:40:56,870
+اللي هو ال U in the direction of B1 B2 هو عبارة عن
+
+543
+00:40:56,870 --> 00:41:02,450
+2I زاد 2J ناقص K على تلاتة طيب صلوا على النبي عليه
+
+544
+00:41:02,450 --> 00:41:05,850
+الصلاة والسلام ناخد مثل آخر تطبيق يقول express the
+
+545
+00:41:05,850 --> 00:41:10,800
+velocity vector Vالان اعبّرلي عن اللي هو الـ
+
+546
+00:41:10,800 --> 00:41:15,820
+Velocity vector V اللي هو 3I-4J على صورة as a
+
+547
+00:41:15,820 --> 00:41:20,060
+product of its speed times a unit vector in the
+
+548
+00:41:20,060 --> 00:41:22,660
+direction of motion يعني بيقوللي اعبّرلي عن ال
+
+549
+00:41:22,660 --> 00:41:26,100
+vector V ال velocity vector على صورة اللي هو ال
+
+550
+00:41:26,100 --> 00:41:30,980
+speed تبعته مضروبة في مين؟ في ال unit vector في ال
+
+551
+00:41:30,980 --> 00:41:33,460
+direction لل motion direction لل motion هو ال
+
+552
+00:41:33,460 --> 00:41:37,090
+direction لل velocityهذا الـ Vector هو اوجد اول
+
+553
+00:41:37,090 --> 00:41:41,270
+شيء الـ Direction للحركة و اوجد السبيد السبيد هو
+
+554
+00:41:41,270 --> 00:41:42,610
+ما هو المجموعه المجموعه المجموعه المجموعه المجموعه
+
+555
+00:41:42,610 --> 00:41:46,290
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+556
+00:41:46,290 --> 00:41:46,950
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+557
+00:41:46,950 --> 00:41:48,130
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+558
+00:41:48,130 --> 00:41:49,310
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+559
+00:41:49,310 --> 00:41:49,370
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+560
+00:41:49,370 --> 00:41:51,530
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+561
+00:41:51,530 --> 00:42:00,810
+المجموعه المجموعه المج
+
+562
+00:42:02,150 --> 00:42:05,210
+لأن الـ Velocity اللي هو طبعا هي اللي بتمثل اتجاه
+
+563
+00:42:05,210 --> 00:42:10,310
+الحركة The direction of V is V على magnitude V V
+
+564
+00:42:10,310 --> 00:42:14,550
+تلتة I نقص أربعة D على خمسة إذن Hence V تبعتنا هذه
+
+565
+00:42:14,550 --> 00:42:18,130
+كتبناها على سكورة The Speed خمسة مضروبة في ال
+
+566
+00:42:18,130 --> 00:42:24,340
+direction تلتة I نقص أربعة D على خمسةهنا ناخد مثال
+
+567
+00:42:24,340 --> 00:42:29,520
+اخر يقول ان قوة
+
+568
+00:42:29,520 --> 00:42:35,380
+6 نيوتن قيمة القوة جديش 6 نيوتن is applied in the
+
+569
+00:42:35,380 --> 00:42:36,660
+direction of the vector
+
+570
+00:42:41,300 --> 00:42:48,500
+التي تعمل في اتجاه الـ vector هذا الأمامها بيقول
+
+571
+00:42:48,500 --> 00:42:53,140
+express the force F as a product of its magnitude
+
+572
+00:42:53,140 --> 00:42:55,440
+and direction مجموعة معروفة
+
+573
+00:42:58,830 --> 00:43:04,570
+الدركشن للفورس هو الدركشن لهذا الـ V فالدركشن
+
+574
+00:43:04,570 --> 00:43:08,650
+للفورس هو الدركشن لهذا الـ V فالدركشن لهذا الـ V
+
+575
+00:43:08,650 --> 00:43:13,570
+هو V ع المجموعه للـ V و سوى 2I زي 2D نقص K على جدر
+
+576
+00:43:13,570 --> 00:43:14,050
+التربيع
+
+577
+00:43:17,600 --> 00:43:21,500
+بتطلع السبسيرت تلاتة اذا اندي صار الان هي ال
+
+578
+00:43:21,500 --> 00:43:25,560
+direction لل force اذا ال force of بتساوي الستة
+
+579
+00:43:25,560 --> 00:43:30,420
+اللي هو ال magnitude لل force اللي معطينا إياها في
+
+580
+00:43:30,420 --> 00:43:35,200
+ال direction اللي هو بالي ال direction ال V اللي
+
+581
+00:43:35,200 --> 00:43:38,420
+عندي أمامي فصارت مكتوبة ال force على صورة
+
+582
+00:43:38,420 --> 00:43:39,980
+magnitude في direction
+
+583
+00:43:43,500 --> 00:43:48,240
+نجي لآخر نقطة اللي هي شغلة بسيطة وسريعة الان بدنا
+
+584
+00:43:48,240 --> 00:43:51,180
+نعرف شو معناه the midpoint M of the line segment
+
+585
+00:43:51,180 --> 00:43:56,060
+joining B1 وB2 يعني موجود عند B1 وB2 نقطين في
+
+586
+00:43:56,060 --> 00:44:00,180
+الفراغ وفي خط واصل B2 ماحكيش عن متجهة الان بحكي عن
+
+587
+00:44:00,180 --> 00:44:03,980
+خط واصل line segment بين B1 وB2 بده النقطة اللي في
+
+588
+00:44:03,980 --> 00:44:07,580
+منتصفه بالظبط النقطة اللي في منتصفه اللي هو B1 وB2
+
+589
+00:44:07,580 --> 00:44:15,340
+هي منها لها نصه إحداثياته هي كتابةX1 زائد X2 هي X1
+
+590
+00:44:15,340 --> 00:44:21,860
+زائد X2 على 2 Y1 زائد Y2 على 2 Z1 زائد Z2 على 2
+
+591
+00:44:21,860 --> 00:44:26,380
+هذا قانون ما نريده ان نشوفه هو القانون اللي موجود
+
+592
+00:44:26,380 --> 00:44:31,380
+عندنا اللي هي ال midpoint بين B1 وB2 ناخد مثال
+
+593
+00:44:31,380 --> 00:44:36,060
+عددي example 7 the midpoint of the line segment
+
+594
+00:44:36,060 --> 00:44:42,180
+joining B1 and B2 is ايش هي ال midpoint احداثياتها
+
+595
+00:44:42,500 --> 00:44:48,240
+تلاتة وسبعة عشرة ع تنين يعني تطلع خمسة نقص اتنين
+
+596
+00:44:48,240 --> 00:44:53,980
+واربعة اتنين على اتنين تطلع واحد سفر واربعة على
+
+597
+00:44:53,980 --> 00:44:58,380
+اتنين تطلع اتنين هذه اللي هي ال midpoint اللي
+
+598
+00:44:58,380 --> 00:45:03,560
+مطلوبةالان هيك بيكون احنا يعني خلصنا المحاضرة
+
+599
+00:45:03,560 --> 00:45:07,880
+الثانية و هاي ال exercises بتحلوها و اللي انتوا
+
+600
+00:45:07,880 --> 00:45:11,740
+يعني حاسين مش عارفين تحلوها او كده ممكن عبر
+
+601
+00:45:11,740 --> 00:45:13,980
+التواصل معنا على ال model او على ال whatsapp اللي
+
+602
+00:45:13,980 --> 00:45:18,160
+هنقوم عها ان شاء الله نحاول .. هنكوّنها معاكم ان
+
+603
+00:45:18,160 --> 00:45:20,580
+شاء الله تبعتوا الأسئلة اللي بتكونوا معاهاهنا
+
+604
+00:45:20,580 --> 00:45:23,700
+تواصل ان شاء الله يا اما بنصورها يا اما بنحلها
+
+605
+00:45:23,700 --> 00:45:28,280
+بطريقة اللى بنلتقيها بنلاقيها مناسبة مع بعض
+
+606
+00:45:28,280 --> 00:45:30,820
+والسلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..25dc452b95dbbad09f2c04a2146947d08dc69537
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 1449, "start": 0.71, "end": 14.49, "text": "بسم الله الرحمن الرحيم هذه هي المحاضرة التانية مساق تفاضل وتكامل جيم لطلاب و طالبات الجامعة الإسلامية كليتي العلوم والهندسة", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 29538, 39896, 9673, 5016, 46958, 25720, 16712, 7649, 10632, 47524, 995, 4587, 6055, 5172, 46958, 1211, 34683, 4117, 10943, 1211, 10874, 32640, 5296, 9566, 15040, 3555, 4032, 23032, 6027, 3555, 9307, 25724, 10943, 27884, 33688, 3794, 37440, 10632, 9122, 20292, 31371, 18863, 1211, 20498, 16070, 3224, 41260, 3794, 3660], "avg_logprob": -0.12129237111342156, "compression_ratio": 1.6691176470588236, "no_speech_prob": 4.172325134277344e-07, "words": [{"start": 0.71, "end": 0.99, "word": "بسم", "probability": 0.898193359375}, {"start": 0.99, "end": 1.25, "word": " الله", "probability": 0.9462890625}, {"start": 1.25, "end": 1.77, "word": " الرحمن", "probability": 0.9803059895833334}, {"start": 1.77, "end": 2.35, "word": " الرحيم", "probability": 0.9957682291666666}, {"start": 2.35, "end": 3.33, "word": " هذه", "probability": 0.8330078125}, {"start": 3.33, "end": 3.55, "word": " هي", "probability": 0.52099609375}, {"start": 3.55, "end": 4.11, "word": " المحاضرة", "probability": 0.98828125}, {"start": 4.11, "end": 4.77, "word": " التانية", "probability": 0.8933919270833334}, {"start": 4.77, "end": 6.11, "word": " مساق", "probability": 0.8141276041666666}, {"start": 6.11, "end": 6.73, "word": " تفاضل", "probability": 0.9471435546875}, {"start": 6.73, "end": 7.29, "word": " وتكامل", "probability": 0.8968505859375}, {"start": 7.29, "end": 7.69, "word": " جيم", "probability": 0.60302734375}, {"start": 7.69, "end": 9.15, "word": " لطلاب", "probability": 0.9053955078125}, {"start": 9.15, "end": 9.27, "word": " و", "probability": 0.99365234375}, {"start": 9.27, "end": 9.81, "word": " طالبات", "probability": 0.9105224609375}, {"start": 9.81, "end": 10.23, "word": " الجامعة", "probability": 0.9871419270833334}, {"start": 10.23, "end": 10.99, "word": " الإسلامية", "probability": 0.9688720703125}, {"start": 10.99, "end": 12.33, "word": " كليتي", "probability": 0.6922200520833334}, {"start": 12.33, "end": 13.73, "word": " العلوم", "probability": 0.9085286458333334}, {"start": 13.73, "end": 14.49, "word": " والهندسة", "probability": 0.93994140625}], "temperature": 1.0}, {"id": 2, "seek": 4348, "start": 14.88, "end": 43.48, "text": "اليوم هنبدأ ان شاء الله اللي هو في 12 اتنين اللي هو vectors اللي هي المتجهات الموضوع طبعا موضوع تم قبل هيك اخد فكرة عنه او حتى يعني بعض الشغلات تم دراستها في اللي هي مساق فيزيا في اللي هي اللي برضه لكليتي العلوم والهندسة نعرف شو معناه متجه او شو معناه vector", "tokens": [6027, 1829, 20498, 8032, 1863, 44510, 10721, 16472, 13412, 16606, 21984, 13672, 1829, 31439, 8978, 2272, 1975, 2655, 1863, 9957, 13672, 1829, 31439, 18875, 13672, 1829, 39896, 9673, 2655, 7435, 3224, 9307, 9673, 2407, 11242, 45367, 23032, 3555, 3615, 995, 3714, 2407, 11242, 45367, 46811, 12174, 36150, 39896, 4117, 1975, 9778, 3215, 6156, 4117, 25720, 18871, 3224, 1975, 2407, 11331, 49975, 37495, 22653, 45030, 11242, 25124, 17082, 1211, 9307, 46811, 11778, 23557, 14851, 11296, 8978, 13672, 1829, 39896, 47524, 995, 4587, 8978, 11622, 25528, 8978, 13672, 1829, 39896, 13672, 1829, 4724, 43042, 3224, 5296, 4117, 20292, 31371, 18863, 1211, 20498, 16070, 3224, 41260, 3794, 3660, 8717, 3615, 28480, 13412, 2407, 20449, 8315, 3224, 44650, 7435, 3224, 1975, 2407, 13412, 2407, 20449, 8315, 3224, 8062], "avg_logprob": -0.18625000715255738, "compression_ratio": 1.9521739130434783, "no_speech_prob": 0.0, "words": [{"start": 14.88, "end": 15.24, "word": "اليوم", "probability": 0.8487955729166666}, {"start": 15.24, "end": 15.66, "word": " هنبدأ", "probability": 0.8922119140625}, {"start": 15.66, "end": 15.76, "word": " ان", "probability": 0.487060546875}, {"start": 15.76, "end": 16.0, "word": " شاء", "probability": 0.97998046875}, {"start": 16.0, "end": 16.36, "word": " الله", "probability": 0.9541015625}, {"start": 16.36, "end": 17.28, "word": " اللي", "probability": 0.598876953125}, {"start": 17.28, "end": 17.48, "word": " هو", "probability": 0.96044921875}, {"start": 17.48, "end": 17.66, "word": " في", "probability": 0.89501953125}, {"start": 17.66, "end": 18.04, "word": " 12", "probability": 0.32861328125}, {"start": 18.04, "end": 18.7, "word": " اتنين", "probability": 0.742919921875}, {"start": 18.7, "end": 19.56, "word": " اللي", "probability": 0.832763671875}, {"start": 19.56, "end": 20.02, "word": " هو", "probability": 0.9189453125}, {"start": 20.02, "end": 20.78, "word": " vectors", "probability": 0.93359375}, {"start": 20.78, "end": 21.32, "word": " اللي", "probability": 0.933349609375}, {"start": 21.32, "end": 21.64, "word": " هي", "probability": 0.810546875}, {"start": 21.64, "end": 22.46, "word": " المتجهات", "probability": 0.944140625}, {"start": 22.46, "end": 23.38, "word": " الموضوع", "probability": 0.9210205078125}, {"start": 23.38, "end": 23.72, "word": " طبعا", "probability": 0.9783935546875}, {"start": 23.72, "end": 24.3, "word": " موضوع", "probability": 0.9503173828125}, {"start": 24.3, "end": 25.36, "word": " تم", "probability": 0.97509765625}, {"start": 25.36, "end": 26.98, "word": " قبل", "probability": 0.9453125}, {"start": 26.98, "end": 27.64, "word": " هيك", "probability": 0.86376953125}, {"start": 27.64, "end": 28.58, "word": " اخد", "probability": 0.6925455729166666}, {"start": 28.58, "end": 28.96, "word": " فكرة", "probability": 0.9762369791666666}, {"start": 28.96, "end": 29.44, "word": " عنه", "probability": 0.994384765625}, {"start": 29.44, "end": 29.78, "word": " او", "probability": 0.89404296875}, {"start": 29.78, "end": 30.28, "word": " حتى", "probability": 0.855712890625}, {"start": 30.28, "end": 30.58, "word": " يعني", "probability": 0.86376953125}, {"start": 30.58, "end": 30.88, "word": " بعض", "probability": 0.987548828125}, {"start": 30.88, "end": 31.54, "word": " الشغلات", "probability": 0.9913330078125}, 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121.21, "end": 121.59, "word": " vector", "probability": 0.9912109375}, {"start": 121.59, "end": 122.39, "word": " انهم", "probability": 0.4984130859375}, {"start": 122.39, "end": 122.85, "word": " are", "probability": 0.27783203125}, {"start": 122.85, "end": 123.33, "word": " equal", "probability": 0.89111328125}, {"start": 123.33, "end": 123.65, "word": " إذا", "probability": 0.553466796875}, {"start": 123.65, "end": 123.83, "word": " ال", "probability": 0.59423828125}, {"start": 123.83, "end": 124.15, "word": " vector", "probability": 0.9775390625}, {"start": 124.15, "end": 124.77, "word": " represented", "probability": 0.48486328125}, {"start": 124.77, "end": 125.07, "word": " by", "probability": 0.96728515625}, {"start": 125.07, "end": 125.25, "word": " the", "probability": 0.564453125}, {"start": 125.25, "end": 125.77, "word": " directed", "probability": 0.89208984375}, {"start": 125.77, "end": 126.13, "word": " line", "probability": 0.9384765625}, {"start": 126.13, "end": 126.59, "word": " segment", "probability": 0.95703125}, {"start": 126.59, "end": 126.97, "word": " AB", "probability": 0.68310546875}], "temperature": 1.0}, {"id": 6, "seek": 14616, "start": 127.61, "end": 146.17, "text": "Has initial point A and terminal point B and its length is denoted by AB بنقول أنه هذا مساوي لل vector CD إذا كان اللي هو اتجاه ال CD نفس اتجاه ال AB والتنين اللي هي نفس ال magnitude", "tokens": [39, 296, 5883, 935, 316, 293, 14709, 935, 363, 293, 1080, 4641, 307, 1441, 23325, 538, 13838, 44945, 39648, 14739, 3224, 23758, 47524, 995, 45865, 24976, 8062, 6743, 11933, 15730, 25961, 13672, 1829, 31439, 1975, 2655, 7435, 40294, 2423, 6743, 8717, 36178, 1975, 2655, 7435, 40294, 2423, 13838, 16070, 2655, 1863, 9957, 13672, 1829, 39896, 8717, 36178, 2423, 15668], "avg_logprob": -0.27630209028720853, "compression_ratio": 1.4529411764705882, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 127.61, "end": 128.07, "word": "Has", "probability": 0.50640869140625}, {"start": 128.07, "end": 128.63, "word": " initial", "probability": 0.564453125}, {"start": 128.63, "end": 128.99, "word": " point", "probability": 0.974609375}, {"start": 128.99, "end": 129.19, "word": " A", "probability": 0.5390625}, {"start": 129.19, "end": 129.41, "word": " and", "probability": 0.85205078125}, {"start": 129.41, "end": 129.79, "word": " terminal", "probability": 0.833984375}, {"start": 129.79, "end": 130.17, "word": " point", "probability": 0.9716796875}, {"start": 130.17, "end": 130.59, "word": " B", "probability": 0.9677734375}, {"start": 130.59, "end": 131.37, "word": " and", "probability": 0.72412109375}, {"start": 131.37, "end": 131.67, "word": " its", "probability": 0.73583984375}, {"start": 131.67, "end": 132.05, "word": " length", "probability": 0.77001953125}, {"start": 132.05, "end": 132.31, "word": " is", "probability": 0.912109375}, {"start": 132.31, "end": 132.63, "word": " denoted", "probability": 0.97607421875}, {"start": 132.63, "end": 133.07, "word": " by", "probability": 0.8818359375}, {"start": 133.07, "end": 133.63, "word": " AB", "probability": 0.70361328125}, {"start": 133.63, "end": 134.85, "word": " بنقول", "probability": 0.5823974609375}, {"start": 134.85, "end": 135.17, "word": " أنه", "probability": 0.4256591796875}, {"start": 135.17, "end": 135.43, "word": " هذا", "probability": 0.79931640625}, {"start": 135.43, "end": 136.31, "word": " مساوي", "probability": 0.745361328125}, {"start": 136.31, "end": 136.93, "word": " لل", "probability": 0.80810546875}, {"start": 136.93, "end": 137.29, "word": " vector", "probability": 0.385498046875}, {"start": 137.29, "end": 137.97, "word": " CD", "probability": 0.7705078125}, {"start": 137.97, "end": 138.87, "word": " إذا", "probability": 0.94580078125}, {"start": 138.87, "end": 139.37, "word": " كان", "probability": 0.99609375}, {"start": 139.37, "end": 139.69, "word": " اللي", "probability": 0.856201171875}, {"start": 139.69, "end": 140.07, "word": " هو", "probability": 0.98828125}, {"start": 140.07, "end": 140.87, "word": " اتجاه", "probability": 0.958251953125}, {"start": 140.87, "end": 141.05, "word": " ال", "probability": 0.908203125}, {"start": 141.05, "end": 141.47, "word": " CD", "probability": 0.65869140625}, {"start": 141.47, "end": 142.31, "word": " نفس", "probability": 0.99169921875}, {"start": 142.31, "end": 142.85, "word": " اتجاه", "probability": 0.9822998046875}, {"start": 142.85, "end": 143.05, "word": " ال", "probability": 0.97021484375}, {"start": 143.05, "end": 143.47, "word": " AB", "probability": 0.79931640625}, {"start": 143.47, "end": 144.65, "word": " والتنين", "probability": 0.722900390625}, {"start": 144.65, "end": 144.89, "word": " اللي", "probability": 0.91943359375}, {"start": 144.89, "end": 145.05, "word": " هي", "probability": 0.62548828125}, {"start": 145.05, "end": 145.39, "word": " نفس", "probability": 0.99609375}, {"start": 145.39, "end": 145.65, "word": " ال", "probability": 0.62255859375}, {"start": 145.65, "end": 146.17, "word": " magnitude", "probability": 0.97119140625}], "temperature": 1.0}, {"id": 7, "seek": 16890, "start": 146.64, "end": 168.9, "text": "two vectors are equal if they have the same length and the same direction لاحظوا أن هذا وهذا وهذا الأربع vectors هذوله متوازيات اللي هي نفس الاتجاه وكلهم اتجاه اللي أمامنا والاربعة اللي هي نفس الطول هذوله بنسميه are equal vectors لاحظوا ال vector هذا OB", "tokens": [20534, 18875, 366, 2681, 498, 436, 362, 264, 912, 4641, 293, 264, 912, 3513, 20193, 5016, 19913, 14407, 14739, 23758, 37037, 15730, 37037, 15730, 16247, 25513, 3615, 18875, 8032, 8848, 12610, 3224, 44650, 2407, 31377, 1829, 9307, 13672, 1829, 39896, 8717, 36178, 2423, 9307, 7435, 40294, 4032, 28820, 16095, 1975, 2655, 7435, 40294, 13672, 1829, 5551, 2304, 10943, 8315, 16070, 9640, 3555, 27884, 13672, 1829, 39896, 8717, 36178, 41950, 12610, 8032, 8848, 12610, 3224, 44945, 38251, 1829, 3224, 366, 2681, 18875, 20193, 5016, 19913, 14407, 2423, 8062, 23758, 35538], "avg_logprob": -0.2678819358348846, "compression_ratio": 1.8097560975609757, "no_speech_prob": 1.1920928955078125e-06, "words": [{"start": 146.64, "end": 147.18, "word": "two", "probability": 0.0694580078125}, {"start": 147.18, "end": 147.84, "word": " vectors", "probability": 0.94384765625}, {"start": 147.84, "end": 148.06, "word": " are", "probability": 0.935546875}, {"start": 148.06, "end": 148.38, "word": " equal", "probability": 0.86962890625}, {"start": 148.38, "end": 148.68, "word": " if", "probability": 0.9404296875}, {"start": 148.68, "end": 148.88, "word": " they", "probability": 0.93017578125}, {"start": 148.88, "end": 149.12, "word": " have", "probability": 0.9423828125}, {"start": 149.12, "end": 149.28, "word": " the", "probability": 0.85546875}, {"start": 149.28, "end": 149.56, "word": " same", "probability": 0.900390625}, {"start": 149.56, "end": 149.94, "word": " length", "probability": 0.399658203125}, {"start": 149.94, "end": 150.66, "word": " and", "probability": 0.8798828125}, {"start": 150.66, "end": 150.8, "word": " the", "probability": 0.53466796875}, {"start": 150.8, "end": 151.12, "word": " same", "probability": 0.875}, {"start": 151.12, "end": 151.98, "word": " direction", "probability": 0.5751953125}, {"start": 151.98, "end": 153.32, "word": " لاحظوا", "probability": 0.780120849609375}, {"start": 153.32, "end": 153.46, "word": " أن", "probability": 0.4775390625}, {"start": 153.46, "end": 153.98, "word": " هذا", "probability": 0.80615234375}, {"start": 153.98, "end": 154.92, "word": " وهذا", "probability": 0.72265625}, {"start": 154.92, "end": 155.34, "word": " وهذا", "probability": 0.850830078125}, {"start": 155.34, "end": 155.82, "word": " الأربع", "probability": 0.77001953125}, {"start": 155.82, "end": 156.28, "word": " vectors", "probability": 0.82373046875}, {"start": 156.28, "end": 156.82, "word": " هذوله", "probability": 0.67108154296875}, {"start": 156.82, "end": 158.34, "word": " متوازيات", "probability": 0.97529296875}, {"start": 158.34, "end": 158.5, "word": " اللي", "probability": 0.770751953125}, {"start": 158.5, "end": 158.58, "word": " هي", "probability": 0.28857421875}, {"start": 158.58, "end": 158.84, "word": " نفس", "probability": 0.871826171875}, {"start": 158.84, "end": 159.34, "word": " الاتجاه", "probability": 0.83856201171875}, {"start": 159.34, "end": 159.68, "word": " وكلهم", "probability": 0.6416015625}, {"start": 159.68, "end": 160.04, "word": " اتجاه", "probability": 0.932861328125}, {"start": 160.04, "end": 161.08, "word": " اللي", "probability": 0.768310546875}, {"start": 161.08, "end": 161.72, "word": " أمامنا", "probability": 0.89208984375}, {"start": 161.72, "end": 163.18, "word": " والاربعة", "probability": 0.68927001953125}, {"start": 163.18, "end": 163.44, "word": " اللي", "probability": 0.970947265625}, {"start": 163.44, "end": 163.52, "word": " هي", "probability": 0.892578125}, {"start": 163.52, "end": 163.88, "word": " نفس", "probability": 0.984619140625}, {"start": 163.88, "end": 164.42, "word": " الطول", "probability": 0.919921875}, {"start": 164.42, "end": 165.36, "word": " هذوله", "probability": 0.8756103515625}, {"start": 165.36, "end": 166.2, "word": " بنسميه", "probability": 0.83984375}, {"start": 166.2, "end": 166.6, "word": " are", "probability": 0.62353515625}, {"start": 166.6, "end": 167.02, "word": " equal", "probability": 0.8759765625}, {"start": 167.02, "end": 167.44, "word": " vectors", "probability": 0.94091796875}, {"start": 167.44, "end": 167.9, "word": " لاحظوا", "probability": 0.9212646484375}, {"start": 167.9, "end": 167.98, "word": " ال", "probability": 0.79931640625}, {"start": 167.98, "end": 168.24, "word": " vector", "probability": 0.94140625}, {"start": 168.24, "end": 168.54, "word": " هذا", "probability": 0.88720703125}, {"start": 168.54, "end": 168.9, "word": " OB", "probability": 0.54296875}], "temperature": 1.0}, {"id": 8, "seek": 19295, "start": 169.39, "end": 192.95, "text": "هذا الـ vector جاي من نقطة الأصل و منطلق للنقطة من B يعني ال initial point له نقطة الأصل هذا بنسميه position vector يعني الآن كل ال vectors الموازية له كلهم بساوينه إذا كانوا في نفس الاتجاه و اللي هو نفس الطول إذا هنتعامل مع ال position vector بعد هيك كمتجه بيمثل كل ال vectors الأخريات", "tokens": [3224, 15730, 2423, 39184, 8062, 10874, 47302, 9154, 8717, 47432, 3660, 16247, 36520, 4032, 9154, 9566, 1211, 4587, 24976, 1863, 47432, 3660, 9154, 363, 37495, 22653, 2423, 5883, 935, 46740, 8717, 47432, 3660, 16247, 36520, 23758, 44945, 38251, 1829, 3224, 2535, 8062, 37495, 22653, 6024, 48506, 28242, 2423, 18875, 9673, 2407, 31377, 10632, 46740, 28242, 16095, 4724, 3794, 995, 2407, 9957, 3224, 11933, 15730, 25961, 14407, 8978, 8717, 36178, 2423, 9307, 7435, 40294, 4032, 13672, 1829, 31439, 8717, 36178, 41950, 12610, 11933, 15730, 8032, 29399, 3615, 10943, 1211, 20449, 2423, 2535, 8062, 39182, 39896, 4117, 9122, 2304, 2655, 7435, 3224, 4724, 32640, 12984, 1211, 28242, 2423, 18875, 16247, 34740, 1829, 9307], "avg_logprob": -0.24204798894269125, "compression_ratio": 1.884297520661157, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 169.39, "end": 169.89, "word": "هذا", "probability": 0.899169921875}, {"start": 169.89, "end": 169.99, "word": " الـ", "probability": 0.5863037109375}, {"start": 169.99, "end": 170.25, "word": " vector", "probability": 0.71533203125}, {"start": 170.25, "end": 170.57, "word": " جاي", "probability": 0.6143798828125}, {"start": 170.57, "end": 170.71, "word": " من", "probability": 0.9765625}, {"start": 170.71, "end": 171.19, "word": " نقطة", "probability": 0.98388671875}, {"start": 171.19, "end": 171.73, "word": " الأصل", "probability": 0.8818359375}, {"start": 171.73, "end": 172.15, "word": " و", "probability": 0.896484375}, {"start": 172.15, "end": 172.67, "word": " منطلق", "probability": 0.81427001953125}, {"start": 172.67, "end": 173.27, "word": " للنقطة", "probability": 0.9071044921875}, {"start": 173.27, "end": 173.41, "word": " من", "probability": 0.185546875}, {"start": 173.41, "end": 173.71, "word": " B", "probability": 0.2388916015625}, {"start": 173.71, "end": 173.95, "word": " يعني", "probability": 0.7939453125}, {"start": 173.95, "end": 174.03, "word": " ال", "probability": 0.8525390625}, {"start": 174.03, "end": 174.39, "word": " initial", "probability": 0.489990234375}, {"start": 174.39, "end": 174.87, "word": " point", "probability": 0.98583984375}, {"start": 174.87, "end": 175.09, "word": " له", "probability": 0.5380859375}, {"start": 175.09, "end": 175.49, "word": " نقطة", "probability": 0.9892578125}, {"start": 175.49, "end": 175.83, "word": " الأصل", "probability": 0.959716796875}, {"start": 175.83, "end": 176.43, "word": " هذا", "probability": 0.87744140625}, {"start": 176.43, "end": 177.41, "word": " بنسميه", "probability": 0.740478515625}, {"start": 177.41, "end": 177.89, "word": " position", "probability": 0.908203125}, {"start": 177.89, "end": 178.83, "word": " vector", "probability": 0.97802734375}, {"start": 178.83, "end": 179.53, "word": " يعني", "probability": 0.942626953125}, {"start": 179.53, "end": 179.85, "word": " الآن", "probability": 0.630615234375}, {"start": 179.85, "end": 180.15, "word": " كل", "probability": 0.9541015625}, {"start": 180.15, "end": 180.33, "word": " ال", "probability": 0.96875}, {"start": 180.33, "end": 180.67, "word": " vectors", "probability": 0.9287109375}, {"start": 180.67, "end": 181.29, "word": " الموازية", "probability": 0.9122314453125}, {"start": 181.29, "end": 181.59, "word": " له", "probability": 0.7900390625}, {"start": 181.59, "end": 182.41, "word": " كلهم", "probability": 0.65625}, {"start": 182.41, "end": 183.75, "word": " بساوينه", "probability": 0.7230631510416666}, {"start": 183.75, "end": 184.01, "word": " إذا", "probability": 0.74267578125}, {"start": 184.01, "end": 184.31, "word": " كانوا", "probability": 0.821533203125}, {"start": 184.31, "end": 184.41, "word": " في", "probability": 0.96435546875}, {"start": 184.41, "end": 184.61, "word": " نفس", "probability": 0.9951171875}, {"start": 184.61, "end": 185.09, "word": " الاتجاه", "probability": 0.9195556640625}, {"start": 185.09, "end": 185.21, "word": " و", "probability": 0.74072265625}, {"start": 185.21, "end": 185.33, "word": " اللي", "probability": 0.58172607421875}, {"start": 185.33, "end": 185.45, "word": " هو", "probability": 0.38671875}, {"start": 185.45, "end": 185.73, "word": " نفس", "probability": 0.855224609375}, {"start": 185.73, "end": 186.21, "word": " الطول", "probability": 0.89111328125}, {"start": 186.21, "end": 186.79, "word": " إذا", "probability": 0.812744140625}, {"start": 186.79, "end": 187.51, "word": " هنتعامل", "probability": 0.831689453125}, {"start": 187.51, "end": 187.65, "word": " مع", "probability": 0.97998046875}, {"start": 187.65, "end": 187.77, "word": " ال", "probability": 0.74560546875}, {"start": 187.77, "end": 188.05, "word": " position", "probability": 0.94091796875}, {"start": 188.05, "end": 188.47, "word": " vector", "probability": 0.98681640625}, {"start": 188.47, "end": 188.71, "word": " بعد", "probability": 0.97021484375}, {"start": 188.71, "end": 189.35, "word": " هيك", "probability": 0.718017578125}, {"start": 189.35, "end": 190.07, "word": " كمتجه", "probability": 0.9681640625}, {"start": 190.07, "end": 190.73, "word": " بيمثل", "probability": 0.837890625}, {"start": 190.73, "end": 191.03, "word": " كل", "probability": 0.90478515625}, {"start": 191.03, "end": 191.15, "word": " ال", "probability": 0.94091796875}, {"start": 191.15, "end": 191.61, "word": " vectors", "probability": 0.96142578125}, {"start": 191.61, "end": 192.95, "word": " الأخريات", "probability": 0.864013671875}], "temperature": 1.0}, {"id": 9, "seek": 21738, "start": 193.76, "end": 217.38, "text": "نأتي الآن اللي هو نشوف اشو معناه اشو كيف بدنا نكتب ال vector على صورة على حاجة اسمها ال component form طيب صلوا على النبي عليه الصلاة والسلام الان عندي اللي هو if V is a two dimensional vector in the plane", "tokens": [1863, 10721, 31371, 6024, 48506, 13672, 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24168, "start": 217.94, "end": 241.68, "text": "equal to the vector with initial point at the origin and terminal point at v1 وv2 يعني الان لو فرضنا عندي الان في البلين في عندي vector v في ال two dimension هذا ال vector v جاي من النقطة النقطة الأصل ال initial point له نقطة الأصل اللي هي ال origin و ال terminal point اللي هي نقطة من v1 وv2", "tokens": [12816, 304, 281, 264, 8062, 365, 5883, 935, 412, 264, 4957, 293, 14709, 935, 412, 371, 16, 4032, 85, 17, 37495, 22653, 2423, 7649, 45164, 6156, 43042, 8315, 18871, 16254, 2423, 7649, 8978, 29739, 1211, 9957, 8978, 18871, 16254, 8062, 371, 8978, 2423, 732, 10139, 23758, 2423, 8062, 371, 10874, 47302, 9154, 28239, 47432, 3660, 28239, 47432, 3660, 16247, 36520, 2423, 5883, 935, 46740, 8717, 47432, 3660, 16247, 36520, 13672, 1829, 39896, 2423, 4957, 4032, 2423, 14709, 935, 13672, 1829, 39896, 8717, 47432, 3660, 9154, 371, 16, 4032, 85, 17], "avg_logprob": -0.2585851687651414, "compression_ratio": 1.9702970297029703, "no_speech_prob": 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نقطة الأصل و رايح للنقطة V1 و V2 و V3 لان نحكي عن الـ three dimensions فبصير عندي الآن هذا ال vector اللي هو من النقطة هذه للنقطة هذه ال component form له هو عبارة عن V", "tokens": [27842, 8717, 47432, 3660, 16247, 36520, 4032, 12602, 47302, 5016, 24976, 1863, 47432, 3660, 691, 16, 4032, 691, 17, 4032, 691, 18, 5296, 7649, 8717, 5016, 4117, 1829, 18871, 2423, 39184, 1045, 12819, 6156, 3555, 9381, 13546, 18871, 16254, 6024, 48506, 23758, 2423, 8062, 13672, 1829, 31439, 9154, 28239, 47432, 3660, 29538, 24976, 1863, 47432, 3660, 29538, 2423, 6542, 1254, 46740, 31439, 6225, 3555, 9640, 3660, 18871, 691], "avg_logprob": -0.23686594030131464, "compression_ratio": 1.4831460674157304, "no_speech_prob": 1.728534698486328e-05, "words": [{"start": 272.28, "end": 272.52, "word": "من", "probability": 0.6884765625}, {"start": 272.52, "end": 273.0, "word": " نقطة", "probability": 0.97314453125}, {"start": 273.0, "end": 273.48, "word": " الأصل", "probability": 0.8017578125}, {"start": 273.48, "end": 274.1, "word": " و", "probability": 0.86376953125}, {"start": 274.1, "end": 274.7, "word": " رايح", "probability": 0.73828125}, {"start": 274.7, "end": 275.48, "word": " للنقطة", "probability": 0.8831787109375}, {"start": 275.48, "end": 276.08, "word": " V1", "probability": 0.6259765625}, {"start": 276.08, "end": 276.26, "word": " و", "probability": 0.92529296875}, {"start": 276.26, "end": 276.84, "word": " V2", "probability": 0.924072265625}, {"start": 276.84, "end": 277.42, "word": " و", "probability": 0.443603515625}, {"start": 277.42, "end": 278.28, "word": " V3", "probability": 0.9541015625}, {"start": 278.28, "end": 278.7, "word": " لان", "probability": 0.3922119140625}, {"start": 278.7, "end": 278.9, "word": " نحكي", "probability": 0.704864501953125}, {"start": 278.9, "end": 279.04, "word": " عن", "probability": 0.97314453125}, {"start": 279.04, "end": 279.12, "word": " الـ", "probability": 0.630615234375}, {"start": 279.12, "end": 279.28, "word": " three", "probability": 0.529296875}, {"start": 279.28, "end": 279.96, "word": " dimensions", "probability": 0.8896484375}, {"start": 279.96, "end": 281.08, "word": " فبصير", "probability": 0.86279296875}, {"start": 281.08, "end": 281.4, "word": " عندي", "probability": 0.74169921875}, {"start": 281.4, "end": 281.7, "word": " الآن", "probability": 0.72705078125}, {"start": 281.7, "end": 282.3, "word": " هذا", "probability": 0.892578125}, {"start": 282.3, "end": 282.52, "word": " ال", "probability": 0.9716796875}, {"start": 282.52, "end": 283.0, "word": " vector", "probability": 0.73876953125}, {"start": 283.0, "end": 283.62, "word": " اللي", "probability": 0.893798828125}, {"start": 283.62, "end": 283.82, "word": " هو", "probability": 0.95703125}, {"start": 283.82, "end": 283.98, "word": " من", "probability": 0.98681640625}, {"start": 283.98, "end": 284.5, "word": " النقطة", "probability": 0.97119140625}, {"start": 284.5, "end": 284.76, "word": " هذه", "probability": 0.92724609375}, {"start": 284.76, "end": 285.4, "word": " للنقطة", "probability": 0.8714599609375}, {"start": 285.4, "end": 285.8, "word": " هذه", "probability": 0.95556640625}, {"start": 285.8, "end": 286.56, "word": " ال", "probability": 0.88623046875}, {"start": 286.56, "end": 287.32, "word": " component", "probability": 0.9013671875}, {"start": 287.32, "end": 288.04, "word": " form", "probability": 0.94677734375}, {"start": 288.04, "end": 288.42, "word": " له", "probability": 0.4208984375}, {"start": 288.42, "end": 289.96, "word": " هو", "probability": 0.60498046875}, {"start": 289.96, "end": 290.5, "word": " عبارة", "probability": 0.995849609375}, {"start": 290.5, "end": 290.74, "word": " عن", "probability": 0.99609375}, {"start": 290.74, "end": 291.08, "word": " V", "probability": 0.94775390625}], "temperature": 1.0}, {"id": 13, "seek": 30992, "start": 292.32, "end": 309.92, "text": "بتساوى V1 V2 V3 إذا من الآن و طالع لما نكتب V بيساوى V1 V2 V3 نقصد بها ال vector اللي ال initial point له 0 0 0 و ال terminal point هي V1 و V2 و V3 طيب الآن", "tokens": [3555, 2655, 3794, 995, 2407, 7578, 691, 16, 691, 17, 691, 18, 11933, 15730, 9154, 6024, 48506, 4032, 23032, 6027, 3615, 5296, 15042, 8717, 4117, 2655, 3555, 691, 4724, 1829, 3794, 995, 2407, 7578, 691, 16, 691, 17, 691, 18, 8717, 4587, 9381, 3215, 4724, 11296, 2423, 8062, 13672, 1829, 2423, 5883, 935, 46740, 1958, 1958, 1958, 4032, 2423, 14709, 935, 39896, 691, 16, 4032, 691, 17, 4032, 691, 18, 23032, 1829, 3555, 6024, 48506], "avg_logprob": -0.19849917214167745, "compression_ratio": 1.4379084967320261, "no_speech_prob": 7.158517837524414e-05, "words": [{"start": 292.32, "end": 293.02, "word": "بتساوى", "probability": 0.7740885416666666}, {"start": 293.02, "end": 293.4, "word": " V1", "probability": 0.704345703125}, {"start": 293.4, "end": 293.92, "word": " V2", "probability": 0.758056640625}, {"start": 293.92, "end": 294.5, "word": " V3", "probability": 0.904052734375}, {"start": 294.5, "end": 295.24, "word": " إذا", "probability": 0.647705078125}, {"start": 295.24, "end": 295.44, "word": " من", "probability": 0.8837890625}, {"start": 295.44, "end": 295.76, "word": " الآن", "probability": 0.7232666015625}, {"start": 295.76, "end": 295.9, "word": " و", "probability": 0.943359375}, {"start": 295.9, "end": 296.16, "word": " طالع", "probability": 0.8922526041666666}, {"start": 296.16, "end": 296.36, "word": " لما", "probability": 0.886962890625}, {"start": 296.36, "end": 296.72, "word": " نكتب", "probability": 0.9881591796875}, {"start": 296.72, "end": 296.86, "word": " V", "probability": 0.94189453125}, {"start": 296.86, "end": 297.24, "word": " بيساوى", "probability": 0.7700602213541666}, {"start": 297.24, "end": 297.62, "word": " V1", "probability": 0.959716796875}, {"start": 297.62, "end": 298.04, "word": " V2", "probability": 0.97412109375}, {"start": 298.04, "end": 298.62, "word": " V3", "probability": 0.98779296875}, {"start": 298.62, "end": 299.6, "word": " نقصد", "probability": 0.96142578125}, {"start": 299.6, "end": 299.86, "word": " بها", "probability": 0.909423828125}, {"start": 299.86, "end": 300.02, "word": " ال", "probability": 0.9404296875}, {"start": 300.02, "end": 300.48, "word": " vector", "probability": 0.734375}, {"start": 300.48, "end": 301.28, "word": " اللي", "probability": 0.776611328125}, {"start": 301.28, "end": 301.44, "word": " ال", "probability": 0.92041015625}, {"start": 301.44, "end": 301.9, "word": " initial", "probability": 0.71728515625}, {"start": 301.9, "end": 302.8, "word": " point", "probability": 0.97314453125}, {"start": 302.8, "end": 303.1, "word": " له", "probability": 0.343994140625}, {"start": 303.1, "end": 303.5, "word": " 0", "probability": 0.2388916015625}, {"start": 303.5, "end": 303.86, "word": " 0", "probability": 0.57763671875}, {"start": 303.86, "end": 304.36, "word": " 0", "probability": 0.98486328125}, {"start": 304.36, "end": 304.88, "word": " و", "probability": 0.693359375}, {"start": 304.88, "end": 305.06, "word": " ال", "probability": 0.81298828125}, {"start": 305.06, "end": 305.44, "word": " terminal", "probability": 0.7373046875}, {"start": 305.44, "end": 305.86, "word": " point", "probability": 0.98046875}, {"start": 305.86, "end": 306.02, "word": " هي", "probability": 0.63720703125}, {"start": 306.02, "end": 306.52, "word": " V1", "probability": 0.97021484375}, {"start": 306.52, "end": 306.66, "word": " و", "probability": 0.658203125}, {"start": 306.66, "end": 307.32, "word": " V2", "probability": 0.913330078125}, {"start": 307.32, "end": 307.72, "word": " و", "probability": 0.97705078125}, {"start": 307.72, "end": 308.64, "word": " V3", "probability": 0.933349609375}, {"start": 308.64, "end": 309.58, "word": " طيب", "probability": 0.97705078125}, {"start": 309.58, "end": 309.92, "word": " الآن", "probability": 0.778564453125}], "temperature": 1.0}, {"id": 14, "seek": 33404, "start": 312.46, "end": 334.04, "text": "لو قدّينا للـ vector اللي هو P EQ من X1 Y1 و Z1 و X2 Y2 و Z2 هذا ال vector أخدناه موازي لهذا ال vector هيو موازي لهذا ال vector و لنفس الطول فبصير عندي و كأن ال V بتساوي V1 و V2 و V3 هي ال vector P EQ", "tokens": [1211, 2407, 12174, 3215, 11703, 1829, 8315, 24976, 39184, 8062, 13672, 1829, 31439, 430, 33580, 9154, 1783, 16, 398, 16, 4032, 1176, 16, 4032, 1783, 17, 398, 17, 4032, 1176, 17, 23758, 2423, 8062, 5551, 9778, 3215, 8315, 3224, 3714, 2407, 31377, 1829, 46740, 15730, 2423, 8062, 39896, 2407, 3714, 2407, 31377, 1829, 46740, 15730, 2423, 8062, 4032, 5296, 1863, 36178, 41950, 12610, 6156, 3555, 9381, 13546, 18871, 16254, 4032, 9122, 33456, 2423, 691, 39894, 3794, 995, 45865, 691, 16, 4032, 691, 17, 4032, 691, 18, 39896, 2423, 8062, 430, 33580], "avg_logprob": -0.251188848329627, "compression_ratio": 1.6761363636363635, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 312.46, "end": 312.8, "word": "لو", "probability": 0.807861328125}, {"start": 312.8, "end": 313.16, "word": " قدّينا", "probability": 0.635302734375}, {"start": 313.16, "end": 313.5, "word": " للـ", "probability": 0.5958251953125}, {"start": 313.5, "end": 313.8, "word": " vector", "probability": 0.5126953125}, {"start": 313.8, "end": 313.98, "word": " اللي", "probability": 0.6839599609375}, {"start": 313.98, "end": 314.38, "word": " هو", "probability": 0.9697265625}, {"start": 314.38, "end": 315.58, "word": " P", "probability": 0.2061767578125}, {"start": 315.58, "end": 316.1, "word": " EQ", "probability": 0.12139892578125}, {"start": 316.1, "end": 316.46, "word": " من", "probability": 0.880859375}, {"start": 316.46, "end": 317.02, "word": " X1", "probability": 0.700927734375}, {"start": 317.02, "end": 317.6, "word": " Y1", "probability": 0.67236328125}, {"start": 317.6, "end": 317.74, "word": " و", "probability": 0.77294921875}, {"start": 317.74, "end": 318.12, "word": " Z1", "probability": 0.799560546875}, {"start": 318.12, "end": 318.26, "word": " و", "probability": 0.7841796875}, {"start": 318.26, "end": 318.7, "word": " X2", "probability": 0.98388671875}, {"start": 318.7, "end": 319.28, "word": " Y2", "probability": 0.83544921875}, {"start": 319.28, "end": 319.44, "word": " و", "probability": 0.9736328125}, {"start": 319.44, "end": 319.98, "word": " Z2", "probability": 0.958984375}, {"start": 319.98, "end": 320.76, "word": " هذا", "probability": 0.84130859375}, {"start": 320.76, "end": 320.9, "word": " ال", "probability": 0.97265625}, {"start": 320.9, "end": 321.12, "word": " vector", "probability": 0.728515625}, {"start": 321.12, "end": 321.9, "word": " أخدناه", "probability": 0.9232421875}, {"start": 321.9, "end": 322.44, "word": " موازي", "probability": 0.9080810546875}, {"start": 322.44, "end": 322.82, "word": " لهذا", "probability": 0.9033203125}, {"start": 322.82, "end": 322.96, "word": " ال", "probability": 0.94970703125}, {"start": 322.96, "end": 323.28, "word": " vector", "probability": 0.974609375}, {"start": 323.28, "end": 323.72, "word": " هيو", "probability": 0.39739990234375}, {"start": 323.72, "end": 324.3, "word": " موازي", "probability": 0.8870849609375}, {"start": 324.3, "end": 324.46, "word": " لهذا", "probability": 0.92333984375}, {"start": 324.46, "end": 324.46, "word": " ال", "probability": 0.96240234375}, {"start": 324.46, "end": 324.7, "word": " vector", "probability": 0.9921875}, {"start": 324.7, "end": 325.56, "word": " و", "probability": 0.1988525390625}, {"start": 325.56, "end": 326.08, "word": " لنفس", "probability": 0.91064453125}, {"start": 326.08, "end": 326.72, "word": " الطول", "probability": 0.94921875}, {"start": 326.72, "end": 327.8, "word": " فبصير", "probability": 0.9295654296875}, {"start": 327.8, "end": 328.18, "word": " عندي", "probability": 0.73779296875}, {"start": 328.18, "end": 328.34, "word": " و", "probability": 0.93701171875}, {"start": 328.34, "end": 328.62, "word": " كأن", "probability": 0.90625}, {"start": 328.62, "end": 328.8, "word": " ال", "probability": 0.57958984375}, {"start": 328.8, "end": 328.94, "word": " V", "probability": 0.904296875}, {"start": 328.94, "end": 329.36, "word": " بتساوي", "probability": 0.71258544921875}, {"start": 329.36, "end": 329.86, "word": " V1", "probability": 0.919189453125}, {"start": 329.86, "end": 330.0, "word": " و", "probability": 0.90771484375}, {"start": 330.0, "end": 330.36, "word": " V2", "probability": 0.93408203125}, {"start": 330.36, "end": 330.5, "word": " و", "probability": 0.9970703125}, {"start": 330.5, "end": 331.1, "word": " V3", "probability": 0.95556640625}, {"start": 331.1, "end": 332.14, "word": " هي", "probability": 0.90380859375}, {"start": 332.14, "end": 332.44, "word": " ال", "probability": 0.98291015625}, {"start": 332.44, "end": 332.84, "word": " vector", "probability": 0.9873046875}, {"start": 332.84, "end": 333.34, "word": " P", "probability": 0.91845703125}, {"start": 333.34, "end": 334.04, "word": " EQ", "probability": 0.7939453125}], "temperature": 1.0}, {"id": 15, "seek": 36022, "start": 334.34, "end": 360.22, "text": "الان ده نتعرف كيف نربط العلاقة بين هذا وبين هذا وكيف نكتب هذا ال vector بي كيو برضه على صورة اللي هو مين ال component form اتطلعوا كيف الان عندى لو كان في عندى given the points بي x واحد و y واحد و z واحد و q هو بقى ال x اتنين و y اتنين و z اتنين الان ال standard position vector to v", "tokens": [6027, 7649, 11778, 3224, 8717, 2655, 3615, 28480, 9122, 33911, 8717, 25513, 9566, 18863, 15040, 28671, 49374, 23758, 46599, 9957, 23758, 4032, 4117, 33911, 8717, 4117, 2655, 3555, 23758, 2423, 8062, 4724, 1829, 9122, 1829, 2407, 4724, 43042, 3224, 15844, 20328, 13063, 3660, 13672, 1829, 31439, 3714, 9957, 2423, 6542, 1254, 1975, 2655, 9566, 1211, 3615, 14407, 9122, 33911, 2423, 7649, 43242, 7578, 45164, 25961, 8978, 43242, 7578, 2212, 264, 2793, 4724, 1829, 2031, 36764, 24401, 4032, 288, 36764, 24401, 4032, 710, 36764, 24401, 4032, 9505, 31439, 4724, 4587, 7578, 2423, 2031, 1975, 2655, 1863, 9957, 4032, 288, 1975, 2655, 1863, 9957, 4032, 710, 1975, 2655, 1863, 9957, 2423, 7649, 2423, 3832, 2535, 8062, 281, 371], "avg_logprob": -0.21674680149453318, 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"probability": 0.97607421875}, {"start": 354.5, "end": 354.8, "word": " بقى", "probability": 0.779052734375}, {"start": 354.8, "end": 354.92, "word": " ال", "probability": 0.677734375}, {"start": 354.92, "end": 355.1, "word": " x", "probability": 0.791015625}, {"start": 355.1, "end": 355.38, "word": " اتنين", "probability": 0.83013916015625}, {"start": 355.38, "end": 355.5, "word": " و", "probability": 0.9462890625}, {"start": 355.5, "end": 355.7, "word": " y", "probability": 0.77734375}, {"start": 355.7, "end": 355.92, "word": " اتنين", "probability": 0.9642333984375}, {"start": 355.92, "end": 356.0, "word": " و", "probability": 0.9892578125}, {"start": 356.0, "end": 356.16, "word": " z", "probability": 0.92578125}, {"start": 356.16, "end": 356.72, "word": " اتنين", "probability": 0.9736328125}, {"start": 356.72, "end": 357.56, "word": " الان", "probability": 0.8388671875}, {"start": 357.56, "end": 357.7, "word": " ال", "probability": 0.87890625}, {"start": 357.7, "end": 358.22, 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0.94580078125}, {"start": 510.28, "end": 510.62, "word": " واحد", "probability": 0.985107421875}, {"start": 510.62, "end": 511.12, "word": " تربيع", "probability": 0.9766845703125}, {"start": 511.12, "end": 511.4, "word": " زائد", "probability": 0.9095052083333334}, {"start": 511.4, "end": 511.56, "word": " V", "probability": 0.99072265625}, {"start": 511.56, "end": 511.84, "word": " اتنين", "probability": 0.9482421875}, {"start": 511.84, "end": 512.2, "word": " تربيع", "probability": 0.9720458984375}, {"start": 512.2, "end": 512.48, "word": " زائد", "probability": 0.88916015625}, {"start": 512.48, "end": 512.64, "word": " V", "probability": 0.99169921875}, {"start": 512.64, "end": 513.16, "word": " تلاتة", "probability": 0.951904296875}, {"start": 513.16, "end": 513.68, "word": " تربيع", "probability": 0.9847412109375}], "temperature": 1.0}, {"id": 23, "seek": 53586, "start": 514.68, "end": 535.86, "text": "بسير الطول V هو عبارة عن الجدر التربيعي لـ V1 تربيع زائد V2 تربيع زائد V3 تربيع هذا عبارة عن الـ magnitude or length of the vector V بساوة بـ Equal is the negative number magnitude لV بساوة الجدر التربيعي V1 تربيع زائد V2 تربيع زائد V3 تربيع", "tokens": [3555, 3794, 13546, 41950, 12610, 691, 31439, 6225, 3555, 9640, 3660, 18871, 25724, 3215, 2288, 16712, 2288, 21292, 3615, 1829, 5296, 39184, 691, 16, 6055, 2288, 21292, 3615, 30767, 16373, 3215, 691, 17, 6055, 2288, 21292, 3615, 30767, 16373, 3215, 691, 18, 6055, 2288, 21292, 3615, 23758, 6225, 3555, 9640, 3660, 18871, 2423, 39184, 15668, 420, 4641, 295, 264, 8062, 691, 4724, 3794, 995, 2407, 3660, 4724, 39184, 15624, 304, 307, 264, 3671, 1230, 15668, 5296, 53, 4724, 3794, 995, 2407, 3660, 25724, 3215, 2288, 16712, 2288, 21292, 3615, 1829, 691, 16, 6055, 2288, 21292, 3615, 30767, 16373, 3215, 691, 17, 6055, 2288, 21292, 3615, 30767, 16373, 3215, 691, 18, 6055, 2288, 21292, 3615], "avg_logprob": -0.2815217287644096, "compression_ratio": 2.0282485875706215, "no_speech_prob": 1.7881393432617188e-07, "words": 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"text": "الان او هو بساوي بال .. بال .. بال .. بالإحداثيات اللي عنا أخدنا بي كيو زي ما قولنا هاي X2 نقص X1 Y2 نقص Y1 بصير X2 نقص X1 لكل تربيع زي Y2 نقص Y1 تربيع زي Z2 نقص Z1 تربيع هذا اللي هو مين اللي هو طول ال vector V", "tokens": [6027, 7649, 1975, 2407, 31439, 4724, 3794, 995, 45865, 20666, 4386, 20666, 4386, 20666, 4386, 20666, 28814, 24401, 5718, 104, 1829, 9307, 13672, 1829, 6225, 8315, 5551, 9778, 3215, 8315, 4724, 1829, 9122, 1829, 2407, 30767, 1829, 19446, 12174, 12610, 8315, 8032, 47302, 1783, 17, 8717, 4587, 9381, 1783, 16, 398, 17, 8717, 4587, 9381, 398, 16, 4724, 9381, 13546, 1783, 17, 8717, 4587, 9381, 1783, 16, 5296, 28820, 6055, 2288, 21292, 3615, 30767, 1829, 398, 17, 8717, 4587, 9381, 398, 16, 6055, 2288, 21292, 3615, 30767, 1829, 1176, 17, 8717, 4587, 9381, 1176, 16, 6055, 2288, 21292, 3615, 23758, 13672, 1829, 31439, 3714, 9957, 13672, 1829, 31439, 23032, 12610, 2423, 8062, 691], "avg_logprob": -0.20422148965952688, "compression_ratio": 1.8876404494382022, 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جمعكم الأولاني بتناسق مع اللي بنحكي كيف نجمعته vectors هندسيا", "tokens": [28814, 24401, 5718, 104, 1829, 9307, 31767, 18513, 2288, 39896, 624, 17, 4032, 691, 17, 11933, 15730, 6156, 3615, 15040, 2423, 8062, 20666, 10721, 5016, 29973, 624, 30767, 16373, 3215, 691, 31439, 6156, 3615, 15040, 11933, 24401, 5718, 104, 1829, 9307, 624, 16, 30767, 16373, 3215, 691, 16, 4032, 624, 17, 30767, 16373, 3215, 691, 17, 4032, 9122, 33456, 3224, 4724, 1829, 39648, 5296, 8315, 16472, 3224, 35186, 6225, 2304, 10874, 2304, 3615, 24793, 16247, 12610, 7649, 1829, 39894, 8315, 3794, 4587, 20449, 13672, 1829, 44945, 5016, 4117, 1829, 9122, 33911, 8717, 7435, 2304, 34268, 3224, 18875, 8032, 41260, 3794, 25528], "avg_logprob": -0.33639706554366094, "compression_ratio": 1.668421052631579, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 841.16, "end": 842.04, "word": "إحداثيات", "probability": 0.7464396158854166}, {"start": 842.04, "end": 842.54, "word": " الصادر", "probability": 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{"start": 1071.17, "end": 1071.63, "word": " absolute", "probability": 0.93896484375}, {"start": 1071.63, "end": 1072.49, "word": " value", "probability": 0.990234375}, {"start": 1072.49, "end": 1072.87, "word": " او", "probability": 0.921875}, {"start": 1072.87, "end": 1073.05, "word": " ال", "probability": 0.75341796875}, {"start": 1073.05, "end": 1073.57, "word": " magnitude", "probability": 0.97412109375}], "temperature": 1.0}, {"id": 48, "seek": 109488, "start": 1074.24, "end": 1094.88, "text": "أو طول اللي هو k في u بساوي ال absolute value لل k في main في طول ال u إذا بقولك the length of k u is the absolute value of the scalar k times the length of u ده vector ناقص واحد في u إيش هيساوي ناقص u زي ما شوفنا هنا u", "tokens": [10721, 2407, 23032, 12610, 13672, 1829, 31439, 350, 8978, 344, 4724, 3794, 995, 45865, 2423, 8236, 2158, 24976, 350, 8978, 2135, 8978, 23032, 12610, 2423, 344, 11933, 15730, 4724, 39648, 4117, 264, 4641, 295, 350, 344, 307, 264, 8236, 2158, 295, 264, 39684, 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{"start": 1077.62, "end": 1077.94, "word": " ال", "probability": 0.52294921875}, {"start": 1077.94, "end": 1078.32, "word": " absolute", "probability": 0.417724609375}, {"start": 1078.32, "end": 1078.84, "word": " value", "probability": 0.990234375}, {"start": 1078.84, "end": 1079.08, "word": " لل", "probability": 0.46826171875}, {"start": 1079.08, "end": 1079.48, "word": " k", "probability": 0.306884765625}, {"start": 1079.48, "end": 1080.12, "word": " في", "probability": 0.88330078125}, {"start": 1080.12, "end": 1080.62, "word": " main", "probability": 0.41162109375}, {"start": 1080.62, "end": 1080.96, "word": " في", "probability": 0.399169921875}, {"start": 1080.96, "end": 1081.3, "word": " طول", "probability": 0.993896484375}, {"start": 1081.3, "end": 1081.46, "word": " ال", "probability": 0.85205078125}, {"start": 1081.46, "end": 1081.72, "word": " u", "probability": 0.60107421875}, {"start": 1081.72, "end": 1083.58, "word": " إذا", "probability": 0.638427734375}, {"start": 1083.58, "end": 1084.76, "word": " بقولك", "probability": 0.7713216145833334}, {"start": 1084.76, "end": 1084.9, "word": " the", "probability": 0.59375}, {"start": 1084.9, "end": 1085.2, "word": " length", "probability": 0.83203125}, {"start": 1085.2, "end": 1085.42, "word": " of", "probability": 0.974609375}, {"start": 1085.42, "end": 1085.62, "word": " k", "probability": 0.51171875}, {"start": 1085.62, "end": 1085.8, "word": " u", "probability": 0.2442626953125}, {"start": 1085.8, "end": 1085.94, "word": " is", "probability": 0.931640625}, {"start": 1085.94, "end": 1086.12, "word": " the", "probability": 0.8046875}, {"start": 1086.12, "end": 1086.4, "word": " absolute", "probability": 0.90478515625}, {"start": 1086.4, "end": 1086.92, "word": " value", "probability": 0.97509765625}, {"start": 1086.92, "end": 1087.08, "word": " of", "probability": 0.97216796875}, {"start": 1087.08, "end": 1087.26, "word": " the", "probability": 0.83837890625}, {"start": 1087.26, "end": 1087.56, "word": " scalar", "probability": 0.92919921875}, {"start": 1087.56, "end": 1087.84, "word": " k", "probability": 0.91357421875}, {"start": 1087.84, "end": 1088.26, "word": " times", "probability": 0.8642578125}, {"start": 1088.26, "end": 1088.44, "word": " the", "probability": 0.759765625}, {"start": 1088.44, "end": 1088.68, "word": " length", "probability": 0.84814453125}, {"start": 1088.68, "end": 1088.9, "word": " of", "probability": 0.9765625}, {"start": 1088.9, "end": 1089.12, "word": " u", "probability": 0.95166015625}, {"start": 1089.12, "end": 1089.42, "word": " ده", "probability": 0.605712890625}, {"start": 1089.42, "end": 1089.86, "word": " vector", "probability": 0.87353515625}, {"start": 1089.86, "end": 1090.66, "word": " ناقص", "probability": 0.9078369140625}, {"start": 1090.66, "end": 1091.08, "word": " واحد", "probability": 0.972412109375}, {"start": 1091.08, "end": 1091.24, "word": " في", "probability": 0.99560546875}, {"start": 1091.24, "end": 1091.56, "word": " u", "probability": 0.900390625}, {"start": 1091.56, "end": 1092.26, "word": " إيش", "probability": 0.7549641927083334}, {"start": 1092.26, "end": 1092.92, "word": " هيساوي", "probability": 0.7989501953125}, {"start": 1092.92, "end": 1093.68, "word": " ناقص", "probability": 0.8492431640625}, {"start": 1093.68, "end": 1093.8, "word": " u", "probability": 0.6669921875}, {"start": 1093.8, "end": 1094.02, "word": " زي", "probability": 0.788818359375}, {"start": 1094.02, "end": 1094.08, "word": " ما", "probability": 0.97802734375}, {"start": 1094.08, "end": 1094.4, "word": " شوفنا", "probability": 0.8390299479166666}, {"start": 1094.4, "end": 1094.58, "word": " هنا", "probability": 0.982421875}, {"start": 1094.58, "end": 1094.88, "word": " u", "probability": 0.7724609375}], "temperature": 1.0}, {"id": 49, "seek": 111433, "start": 1095.35, "end": 1114.33, "text": "لأن ناقص u هيكون عكس اتجاه يعني نفس الطول و لكن عكس الاتجاه يعني ناقص واحد في u بساوي ناقص u اللي هي نفس الطول لكن عكس الاتجاه عن ال u has the same length as u but in the opposite direction", "tokens": [1211, 33456, 8717, 995, 4587, 9381, 344, 39896, 30544, 6225, 4117, 3794, 1975, 2655, 7435, 40294, 37495, 22653, 8717, 36178, 41950, 12610, 4032, 44381, 6225, 4117, 3794, 2423, 9307, 7435, 40294, 37495, 22653, 8717, 995, 4587, 9381, 36764, 24401, 8978, 344, 4724, 3794, 995, 45865, 8717, 995, 4587, 9381, 344, 13672, 1829, 39896, 8717, 36178, 41950, 12610, 44381, 6225, 4117, 3794, 2423, 9307, 7435, 40294, 18871, 2423, 344, 575, 264, 912, 4641, 382, 344, 457, 294, 264, 6182, 3513], "avg_logprob": -0.2230468735098839, "compression_ratio": 1.7730061349693251, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1095.35, "end": 1095.79, "word": "لأن", "probability": 0.59375}, {"start": 1095.79, "end": 1096.11, "word": " ناقص", "probability": 0.87548828125}, {"start": 1096.11, "end": 1096.25, "word": " u", "probability": 0.304931640625}, {"start": 1096.25, "end": 1096.67, "word": " هيكون", "probability": 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"probability": 0.791748046875}, {"start": 1102.27, "end": 1102.41, "word": " في", "probability": 0.951171875}, {"start": 1102.41, "end": 1102.61, "word": " u", "probability": 0.79345703125}, {"start": 1102.61, "end": 1103.03, "word": " بساوي", "probability": 0.681640625}, {"start": 1103.03, "end": 1103.89, "word": " ناقص", "probability": 0.976318359375}, {"start": 1103.89, "end": 1104.05, "word": " u", "probability": 0.77734375}, {"start": 1104.05, "end": 1104.19, "word": " اللي", "probability": 0.6837158203125}, {"start": 1104.19, "end": 1104.33, "word": " هي", "probability": 0.6904296875}, {"start": 1104.33, "end": 1104.59, "word": " نفس", "probability": 0.992919921875}, {"start": 1104.59, "end": 1104.97, "word": " الطول", "probability": 0.985595703125}, {"start": 1104.97, "end": 1105.35, "word": " لكن", "probability": 0.80517578125}, {"start": 1105.35, "end": 1105.85, "word": " عكس", "probability": 0.9889322916666666}, {"start": 1105.85, "end": 1106.67, "word": " الاتجاه", "probability": 0.950927734375}, {"start": 1106.67, "end": 1107.45, "word": " عن", "probability": 0.9501953125}, {"start": 1107.45, "end": 1107.59, "word": " ال", "probability": 0.86669921875}, {"start": 1107.59, "end": 1107.79, "word": " u", "probability": 0.7529296875}, {"start": 1107.79, "end": 1108.69, "word": " has", "probability": 0.66845703125}, {"start": 1108.69, "end": 1108.89, "word": " the", "probability": 0.92919921875}, {"start": 1108.89, "end": 1109.21, "word": " same", "probability": 0.89697265625}, {"start": 1109.21, "end": 1109.53, "word": " length", "probability": 0.8388671875}, {"start": 1109.53, "end": 1109.85, "word": " as", "probability": 0.94482421875}, {"start": 1109.85, "end": 1110.15, "word": " u", "probability": 0.9384765625}, {"start": 1110.15, "end": 1110.61, "word": " but", "probability": 0.8984375}, {"start": 1110.61, "end": 1111.45, "word": " in", "probability": 0.45947265625}, {"start": 1111.45, "end": 1111.63, "word": " the", "probability": 0.77734375}, {"start": 1111.63, "end": 1111.95, "word": " opposite", "probability": 0.955078125}, {"start": 1111.95, "end": 1114.33, "word": " direction", "probability": 0.96826171875}], "temperature": 1.0}, {"id": 50, "seek": 114292, "start": 1116.78, "end": 1142.92, "text": "معرفة U ناقص V of two vectors is defined by U ناقص V بتساوي U ناقص V بتساوي U زائد ناقص V طبعا احنا ناقص V عرفناها هي عبارة عن ناقص واحد في V الآن إذا هذه معرفة والجامع معرفة إذا هذه الجديدة عرفناها بإيش بنعرفه الآن اللي هو جامع two vectors", "tokens": [2304, 3615, 28480, 3660, 624, 8717, 995, 4587, 9381, 691, 295, 732, 18875, 307, 7642, 538, 624, 8717, 995, 4587, 9381, 691, 4724, 2655, 3794, 995, 45865, 624, 8717, 995, 4587, 9381, 691, 4724, 2655, 3794, 995, 45865, 624, 30767, 16373, 3215, 8717, 995, 4587, 9381, 691, 23032, 3555, 3615, 995, 1975, 5016, 8315, 8717, 995, 4587, 9381, 691, 6225, 28480, 8315, 11296, 39896, 6225, 3555, 9640, 3660, 18871, 8717, 995, 4587, 9381, 36764, 24401, 8978, 691, 6024, 48506, 11933, 15730, 29538, 20449, 28480, 3660, 16070, 7435, 10943, 3615, 20449, 28480, 3660, 11933, 15730, 29538, 25724, 16254, 41891, 6225, 28480, 8315, 11296, 4724, 28814, 1829, 8592, 44945, 3615, 28480, 3224, 6024, 48506, 13672, 1829, 31439, 10874, 10943, 3615, 732, 18875], "avg_logprob": -0.23217975600691867, "compression_ratio": 1.9896907216494846, "no_speech_prob": 2.980232238769531e-07, "words": [{"start": 1116.78, "end": 1117.7, "word": "معرفة", "probability": 0.5651092529296875}, {"start": 1117.7, "end": 1118.24, "word": " U", "probability": 0.197998046875}, {"start": 1118.24, "end": 1119.02, "word": " ناقص", "probability": 0.7801513671875}, {"start": 1119.02, "end": 1119.22, "word": " V", "probability": 0.84814453125}, {"start": 1119.22, "end": 1120.12, "word": " of", "probability": 0.1331787109375}, {"start": 1120.12, "end": 1120.38, "word": " two", "probability": 0.88037109375}, {"start": 1120.38, "end": 1120.96, "word": " vectors", "probability": 0.9345703125}, {"start": 1120.96, "end": 1121.72, "word": " is", "probability": 0.533203125}, {"start": 1121.72, "end": 1122.18, "word": " defined", "probability": 0.8515625}, {"start": 1122.18, "end": 1122.44, "word": " by", "probability": 0.205078125}, {"start": 1122.44, "end": 1123.52, "word": " U", "probability": 0.427001953125}, {"start": 1123.52, "end": 1124.0, "word": " ناقص", "probability": 0.9329833984375}, {"start": 1124.0, "end": 1124.34, "word": " V", "probability": 0.86279296875}, {"start": 1124.34, "end": 1125.54, "word": " بتساوي", "probability": 0.736474609375}, {"start": 1125.54, "end": 1125.94, "word": " U", "probability": 0.81884765625}, {"start": 1125.94, "end": 1126.52, "word": " ناقص", "probability": 0.93310546875}, {"start": 1126.52, "end": 1126.64, "word": " V", "probability": 0.921875}, {"start": 1126.64, "end": 1127.1, "word": " بتساوي", "probability": 0.9310546875}, {"start": 1127.1, "end": 1127.66, "word": " U", "probability": 0.94970703125}, {"start": 1127.66, "end": 1128.58, "word": " زائد", "probability": 0.763427734375}, {"start": 1128.58, "end": 1129.08, "word": " ناقص", "probability": 0.9901123046875}, {"start": 1129.08, "end": 1129.34, "word": " V", "probability": 0.9775390625}, {"start": 1129.34, "end": 1130.02, "word": " طبعا", "probability": 0.82537841796875}, {"start": 1130.02, "end": 1130.28, "word": " احنا", "probability": 0.7674967447916666}, {"start": 1130.28, "end": 1130.98, "word": " ناقص", "probability": 0.972900390625}, {"start": 1130.98, "end": 1131.28, "word": " V", "probability": 0.92431640625}, {"start": 1131.28, "end": 1132.06, "word": " عرفناها", "probability": 0.9490966796875}, {"start": 1132.06, "end": 1132.28, "word": " هي", "probability": 0.58154296875}, {"start": 1132.28, "end": 1132.58, "word": " عبارة", "probability": 0.97802734375}, {"start": 1132.58, "end": 1132.72, "word": " عن", "probability": 0.97412109375}, {"start": 1132.72, "end": 1133.04, "word": " ناقص", "probability": 0.9930419921875}, {"start": 1133.04, "end": 1133.5, "word": " واحد", "probability": 0.959716796875}, {"start": 1133.5, "end": 1133.7, "word": " في", "probability": 0.93310546875}, {"start": 1133.7, "end": 1134.02, "word": " V", "probability": 0.93505859375}, {"start": 1134.02, "end": 1135.08, "word": " الآن", "probability": 0.56488037109375}, {"start": 1135.08, "end": 1135.38, "word": " إذا", "probability": 0.5069580078125}, {"start": 1135.38, "end": 1135.72, "word": " هذه", "probability": 0.63427734375}, {"start": 1135.72, "end": 1136.28, "word": " معرفة", "probability": 0.9391276041666666}, {"start": 1136.28, "end": 1136.8, "word": " والجامع", "probability": 0.7529296875}, {"start": 1136.8, "end": 1137.48, "word": " معرفة", "probability": 0.9518229166666666}, {"start": 1137.48, "end": 1137.78, "word": " إذا", "probability": 0.82177734375}, {"start": 1137.78, "end": 1138.08, "word": " هذه", "probability": 0.92724609375}, {"start": 1138.08, "end": 1138.76, "word": " الجديدة", "probability": 0.9503580729166666}, {"start": 1138.76, "end": 1139.98, "word": " عرفناها", "probability": 0.96923828125}, {"start": 1139.98, "end": 1140.3, "word": " بإيش", "probability": 0.805419921875}, {"start": 1140.3, "end": 1140.84, "word": " بنعرفه", "probability": 0.931884765625}, {"start": 1140.84, "end": 1141.2, "word": " الآن", "probability": 0.82861328125}, {"start": 1141.2, "end": 1141.54, "word": " اللي", "probability": 0.877685546875}, {"start": 1141.54, "end": 1141.78, "word": " هو", "probability": 0.98046875}, {"start": 1141.78, "end": 1142.2, "word": " جامع", "probability": 0.90234375}, {"start": 1142.2, "end": 1142.4, "word": " two", "probability": 0.84716796875}, {"start": 1142.4, "end": 1142.92, "word": " vectors", "probability": 0.9501953125}], "temperature": 1.0}, {"id": 51, "seek": 116228, "start": 1143.5, "end": 1162.28, "text": "وهذا ناقص ال V هو ناقص واحد في V تنشوف هذا الكلام هل هي تلائم جدريا او هندسيا او لأ ونشوف كيف اللي هو مع معلوماتنا السابقة هتطلع الآن هي عندي هذا اللي هو ال vector U وهذا ال vector V ماشي الحالة", 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zero إذا دائما u زائد zero vector طبعا دي vector و دي vector بساوي u", "tokens": [6027, 7649, 6055, 1211, 9307, 3660, 4724, 1829, 39648, 1829, 16472, 9673, 5016, 995, 25708, 25724, 10943, 3615, 1829, 24976, 18875, 344, 13672, 1829, 31439, 9154, 6225, 42213, 10632, 25724, 10943, 27884, 23032, 3555, 3615, 995, 13672, 1829, 11331, 10943, 1211, 9673, 5016, 995, 25708, 25724, 10943, 3615, 1829, 13672, 1829, 31439, 10874, 3215, 33599, 4724, 3794, 995, 45865, 8608, 5172, 2288, 37495, 22653, 4018, 4018, 4018, 13672, 1829, 31439, 2423, 8062, 4018, 11933, 15730, 11778, 16373, 15042, 344, 30767, 16373, 3215, 4018, 8062, 23032, 3555, 3615, 995, 11778, 1829, 8062, 4032, 11778, 1829, 8062, 4724, 3794, 995, 45865, 344], "avg_logprob": -0.2948638531241086, "compression_ratio": 2.0786516853932584, "no_speech_prob": 0.0, "words": [{"start": 1559.99, "end": 1560.37, "word": "الان", "probability": 0.3402099609375}, {"start": 1560.37, "end": 1560.99, "word": " تلاتة", "probability": 0.8759765625}, 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{"start": 1836.92, "end": 1837.9, "word": " مثال", "probability": 0.810546875}, {"start": 1837.9, "end": 1838.42, "word": " for", "probability": 0.8115234375}, {"start": 1838.42, "end": 1839.04, "word": " example", "probability": 0.9677734375}, {"start": 1839.04, "end": 1840.1, "word": " U", "probability": 0.50244140625}, {"start": 1840.1, "end": 1840.46, "word": " و", "probability": 0.83740234375}, {"start": 1840.46, "end": 1840.74, "word": " V", "probability": 0.591796875}, {"start": 1840.74, "end": 1841.04, "word": " و", "probability": 0.97412109375}, {"start": 1841.04, "end": 1841.24, "word": " U", "probability": 0.83935546875}, {"start": 1841.24, "end": 1841.54, "word": " زائد", "probability": 0.6268717447916666}, {"start": 1841.54, "end": 1841.8, "word": " V", "probability": 0.98779296875}], "temperature": 1.0}, {"id": 83, "seek": 185430, "start": 1842.4, "end": 1854.3, "text": "هذول ايش مالهين co-planner لإن احنا كنا نقول عشان ال U و ال V ارسم ال U و ارسم ال V زي ما رسمنا قبل شوية المثلث اللي بيجفلهم", "tokens": [3224, 8848, 12610, 1975, 1829, 8592, 3714, 6027, 3224, 9957, 598, 12, 564, 9805, 5296, 28814, 1863, 1975, 5016, 8315, 9122, 8315, 8717, 39648, 6225, 8592, 7649, 2423, 624, 4032, 2423, 691, 1975, 2288, 38251, 2423, 624, 4032, 1975, 2288, 38251, 2423, 691, 30767, 1829, 19446, 12602, 38251, 8315, 12174, 36150, 13412, 2407, 10632, 9673, 12984, 1211, 12984, 13672, 1829, 4724, 1829, 7435, 5172, 1211, 16095], "avg_logprob": -0.31553171641791045, "compression_ratio": 1.4927536231884058, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1842.4, "end": 1842.92, "word": "هذول", "probability": 0.6244710286458334}, {"start": 1842.92, "end": 1843.24, "word": " ايش", "probability": 0.6166178385416666}, {"start": 1843.24, "end": 1843.92, "word": " مالهين", "probability": 0.8468017578125}, {"start": 1843.92, "end": 1844.14, "word": " co", "probability": 0.08233642578125}, {"start": 1844.14, "end": 1844.6, "word": "-planner", "probability": 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0.974609375}], "temperature": 1.0}, {"id": 85, "seek": 190969, "start": 1882.17, "end": 1909.69, "text": "يعني اي vector يا جماعة في الدنيا طول واحد ال magnitude الواحد بنسميه unit vector اذا في عندي عدد لانهائم ال vectors اي vector طول واحد اي زي و vector تاني و vector اتنين و هذا طول واحد طول اسف واحد و هذا واحد واحد ارسم عدد لانهائم ال vectors طول واحد في اتجاهات مختلفة كلهم بنسميهم ايه شمالهم unit vectors اذا ال unit vector هو vector", "tokens": [40228, 22653, 1975, 1829, 8062, 35186, 10874, 15042, 27884, 8978, 32748, 22653, 995, 23032, 12610, 36764, 24401, 2423, 15668, 2423, 14407, 24401, 44945, 38251, 1829, 3224, 4985, 8062, 1975, 15730, 8978, 18871, 16254, 6225, 3215, 3215, 5296, 7649, 3224, 16373, 2304, 2423, 18875, 1975, 1829, 8062, 23032, 12610, 36764, 24401, 1975, 1829, 30767, 1829, 4032, 8062, 6055, 7649, 1829, 4032, 8062, 1975, 2655, 1863, 9957, 4032, 23758, 23032, 12610, 36764, 24401, 23032, 12610, 24525, 5172, 36764, 24401, 4032, 23758, 36764, 24401, 36764, 24401, 1975, 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ناخد أمثال عليها اللي أنا بقوله نيجي نشوف المثال اللي عندنا Find a unit vector u in the direction of the vector b1 to b2", "tokens": [28814, 15730, 36632, 8062, 8978, 32748, 22653, 995, 3714, 28543, 2288, 5551, 4117, 2655, 3555, 15844, 20328, 13063, 3660, 23032, 12610, 3224, 48812, 25513, 1975, 2655, 7435, 40294, 3224, 1975, 2655, 7435, 40294, 3224, 691, 15844, 15668, 691, 20212, 279, 691, 382, 1080, 4641, 293, 1080, 3513, 8717, 47283, 3215, 5551, 2304, 12984, 6027, 25894, 11296, 13672, 1829, 41850, 4724, 39648, 3224, 8717, 1829, 7435, 1829, 8717, 8592, 38688, 9673, 12984, 6027, 13672, 1829, 43242, 8315, 11809, 257, 4985, 8062, 344, 294, 264, 3513, 295, 264, 8062, 272, 16, 281, 272, 17], "avg_logprob": -0.2563844009112286, "compression_ratio": 1.52863436123348, "no_speech_prob": 0.0, "words": [{"start": 2339.6, "end": 2340.08, "word": "إذا", "probability": 0.584228515625}, {"start": 2340.08, "end": 2340.36, "word": " أي", "probability": 0.64111328125}, {"start": 2340.36, "end": 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vector", "probability": 0.97998046875}, {"start": 2421.16, "end": 2421.58, "word": " in", "probability": 0.94482421875}, {"start": 2421.58, "end": 2421.74, "word": " the", "probability": 0.92919921875}, {"start": 2421.74, "end": 2422.24, "word": " direction", "probability": 0.96630859375}, {"start": 2422.24, "end": 2422.48, "word": " of", "probability": 0.96923828125}, {"start": 2422.48, "end": 2422.88, "word": " b1", "probability": 0.9326171875}, {"start": 2422.88, "end": 2423.4, "word": " b2", "probability": 0.925537109375}, {"start": 2423.4, "end": 2423.64, "word": " ايش", "probability": 0.735595703125}, {"start": 2423.64, "end": 2423.94, "word": " هيكون", "probability": 0.9111328125}, {"start": 2423.94, "end": 2424.04, "word": " يا", "probability": 0.96337890625}, {"start": 2424.04, "end": 2424.48, "word": " جماعة", "probability": 0.9915364583333334}, {"start": 2424.48, "end": 2425.08, "word": " هو", "probability": 0.68115234375}, {"start": 2425.08, "end": 2425.52, "word": " 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التربية", "probability": 0.7506510416666666}, {"start": 2435.52, "end": 2435.7, "word": " على", "probability": 0.5283203125}, {"start": 2435.7, "end": 2436.04, "word": " مين", "probability": 0.986328125}], "temperature": 1.0}, {"id": 109, "seek": 244694, "start": 2436.48, "end": 2446.94, "text": "لأ اتنين تربيع ليه اربعة و اتنين تربيع اربعة هي تمانية و واحد تربيع واحد هي تسعة اذا الجدر التربيعي لكل هذا طبعا الجدر لكله هذا بيظل الجدر ماشي", "tokens": [1211, 10721, 1975, 2655, 1863, 9957, 6055, 2288, 21292, 3615, 32239, 3224, 1975, 25513, 27884, 4032, 1975, 2655, 1863, 9957, 6055, 2288, 21292, 3615, 1975, 25513, 27884, 39896, 46811, 7649, 10632, 4032, 36764, 24401, 6055, 2288, 21292, 3615, 36764, 24401, 39896, 6055, 3794, 27884, 1975, 15730, 25724, 3215, 2288, 16712, 2288, 21292, 3615, 1829, 5296, 28820, 23758, 23032, 3555, 3615, 995, 25724, 3215, 2288, 5296, 28820, 3224, 23758, 4724, 1829, 19913, 1211, 25724, 3215, 2288, 3714, 33599, 1829], "avg_logprob": -0.21875000075448917, "compression_ratio": 2.0476190476190474, "no_speech_prob": 1.3113021850585938e-06, "words": [{"start": 2436.48, "end": 2436.96, "word": "لأ", "probability": 0.6060791015625}, {"start": 2436.96, "end": 2437.5, "word": " اتنين", "probability": 0.754150390625}, {"start": 2437.5, "end": 2437.9, "word": " تربيع", "probability": 0.9066162109375}, {"start": 2437.9, "end": 2438.12, "word": " ليه", "probability": 0.519012451171875}, {"start": 2438.12, "end": 2438.46, "word": " اربعة", "probability": 0.8507486979166666}, {"start": 2438.46, "end": 2438.54, "word": " و", "probability": 0.32568359375}, {"start": 2438.54, "end": 2438.82, "word": " اتنين", "probability": 0.9300537109375}, {"start": 2438.82, "end": 2439.22, "word": " تربيع", "probability": 0.9609375}, {"start": 2439.22, "end": 2439.7, "word": " اربعة", "probability": 0.86572265625}, {"start": 2439.7, "end": 2439.92, "word": " هي", "probability": 0.8017578125}, {"start": 2439.92, "end": 2440.4, "word": " تمانية", "probability": 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"probability": 0.9404296875}, {"start": 2675.4, "end": 2676.06, "word": " segment", "probability": 0.94921875}, {"start": 2676.06, "end": 2676.86, "word": " joining", "probability": 0.84130859375}, {"start": 2676.86, "end": 2677.48, "word": " B1", "probability": 0.9609375}, {"start": 2677.48, "end": 2677.72, "word": " and", "probability": 0.5732421875}, {"start": 2677.72, "end": 2678.26, "word": " B2", "probability": 0.99169921875}, {"start": 2678.26, "end": 2678.62, "word": " is", "probability": 0.9111328125}, {"start": 2678.62, "end": 2679.5, "word": " ايش", "probability": 0.7049153645833334}, {"start": 2679.5, "end": 2679.82, "word": " هي", "probability": 0.9482421875}, {"start": 2679.82, "end": 2680.64, "word": " ال", "probability": 0.61328125}, {"start": 2680.64, "end": 2681.14, "word": " midpoint", "probability": 0.918701171875}, {"start": 2681.14, "end": 2682.18, "word": " احداثياتها", "probability": 0.7899344308035714}], "temperature": 1.0}, {"id": 120, "seek": 269914, "start": 2682.5, "end": 2699.14, "text": "تلاتة وسبعة عشرة ع تنين يعني تطلع خمسة نقص اتنين واربعة اتنين على اتنين تطلع واحد سفر واربعة على اتنين تطلع اتنين هذه اللي هي ال midpoint اللي مطلوبة", "tokens": [2655, 1211, 9307, 3660, 4032, 35457, 27884, 6225, 8592, 25720, 6225, 6055, 1863, 9957, 37495, 22653, 6055, 9566, 1211, 3615, 16490, 2304, 3794, 3660, 8717, 4587, 9381, 1975, 2655, 1863, 9957, 4032, 9640, 3555, 27884, 1975, 2655, 1863, 9957, 15844, 1975, 2655, 1863, 9957, 6055, 9566, 1211, 3615, 36764, 24401, 8608, 5172, 2288, 4032, 9640, 3555, 27884, 15844, 1975, 2655, 1863, 9957, 6055, 9566, 1211, 3615, 1975, 2655, 1863, 9957, 29538, 13672, 1829, 39896, 2423, 2062, 6053, 13672, 1829, 3714, 9566, 1211, 37746, 3660], "avg_logprob": -0.1305147086872774, "compression_ratio": 1.8450704225352113, "no_speech_prob": 1.3113021850585938e-06, "words": [{"start": 2682.5, "end": 2683.36, "word": "تلاتة", "probability": 0.7525634765625}, {"start": 2683.36, "end": 2684.36, "word": " وسبعة", "probability": 0.7840169270833334}, {"start": 2684.36, "end": 2685.02, "word": " عشرة", "probability": 0.900390625}, {"start": 2685.02, "end": 2685.26, "word": " ع", "probability": 0.61083984375}, {"start": 2685.26, "end": 2685.78, "word": " تنين", "probability": 0.7137858072916666}, {"start": 2685.78, "end": 2686.04, "word": " يعني", "probability": 0.789794921875}, {"start": 2686.04, "end": 2686.38, "word": " تطلع", "probability": 0.909912109375}, {"start": 2686.38, "end": 2686.98, "word": " خمسة", "probability": 0.971435546875}, {"start": 2686.98, "end": 2687.86, "word": " نقص", "probability": 0.8873697916666666}, {"start": 2687.86, "end": 2688.24, "word": " اتنين", "probability": 0.9251708984375}, {"start": 2688.24, "end": 2688.82, "word": " واربعة", "probability": 0.9051513671875}, {"start": 2688.82, "end": 2689.34, "word": " اتنين", "probability": 0.9676513671875}, {"start": 2689.34, "end": 2689.66, "word": " على", "probability": 0.5126953125}, {"start": 2689.66, "end": 2690.04, "word": " اتنين", "probability": 0.963134765625}, {"start": 2690.04, "end": 2690.4, "word": " تطلع", "probability": 0.975341796875}, {"start": 2690.4, "end": 2690.86, "word": " واحد", "probability": 0.98876953125}, {"start": 2690.86, "end": 2692.14, "word": " سفر", "probability": 0.8728841145833334}, {"start": 2692.14, "end": 2693.32, "word": " واربعة", "probability": 0.9478759765625}, {"start": 2693.32, "end": 2693.98, "word": " على", "probability": 0.80029296875}, {"start": 2693.98, "end": 2694.52, "word": " اتنين", "probability": 0.987548828125}, {"start": 2694.52, "end": 2695.0, "word": " تطلع", "probability": 0.9859619140625}, {"start": 2695.0, "end": 2695.42, "word": " اتنين", "probability": 0.9830322265625}, {"start": 2695.42, "end": 2696.34, "word": " هذه", "probability": 0.70458984375}, {"start": 2696.34, "end": 2696.48, "word": " اللي", "probability": 0.842041015625}, {"start": 2696.48, "end": 2696.78, "word": " هي", "probability": 0.85009765625}, {"start": 2696.78, "end": 2697.04, "word": " ال", "probability": 0.89013671875}, {"start": 2697.04, "end": 2697.72, "word": " midpoint", "probability": 0.799072265625}, {"start": 2697.72, "end": 2698.38, "word": " اللي", "probability": 0.985595703125}, {"start": 2698.38, "end": 2699.14, "word": " مطلوبة", "probability": 0.9439453125}], "temperature": 1.0}, {"id": 121, "seek": 271994, "start": 2699.48, "end": 2719.94, "text": "الان هيك بيكون احنا يعني خلصنا المحاضرة الثانية و هاي ال exercises بتحلوها و اللي انتوا يعني حاسين مش عارفين تحلوها او كده ممكن عبر التواصل معنا على ال model او على ال whatsapp اللي هنقوم عها ان شاء الله نحاول .. هنكوّنها معاكم ان شاء الله تبعتوا الأسئلة اللي بتكونوا معاها", "tokens": [6027, 7649, 39896, 4117, 4724, 1829, 30544, 1975, 5016, 8315, 37495, 22653, 16490, 1211, 9381, 8315, 9673, 5016, 46958, 25720, 6024, 104, 7649, 10632, 4032, 8032, 47302, 2423, 11900, 39894, 5016, 1211, 2407, 11296, 4032, 13672, 1829, 16472, 2655, 14407, 37495, 22653, 11331, 32277, 9957, 37893, 6225, 9640, 5172, 9957, 6055, 5016, 1211, 2407, 11296, 1975, 2407, 9122, 3215, 3224, 3714, 43020, 6225, 26890, 16712, 2407, 33546, 1211, 20449, 8315, 15844, 2423, 2316, 1975, 2407, 15844, 2423, 29625, 1746, 13672, 1829, 8032, 1863, 4587, 20498, 6225, 11296, 16472, 13412, 16606, 21984, 8717, 5016, 995, 12610, 4386, 8032, 1863, 4117, 2407, 11703, 1863, 11296, 20449, 995, 24793, 16472, 13412, 16606, 21984, 6055, 3555, 34268, 14407, 16247, 3794, 19986, 37977, 13672, 1829, 39894, 30544, 14407, 20449, 995, 11296], "avg_logprob": -0.24397145739690526, "compression_ratio": 1.83984375, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 2699.48, "end": 2700.08, "word": "الان", "probability": 0.747802734375}, {"start": 2700.08, "end": 2700.6, "word": " هيك", "probability": 0.740966796875}, {"start": 2700.6, "end": 2700.96, "word": " بيكون", "probability": 0.6131998697916666}, {"start": 2700.96, "end": 2701.38, "word": " احنا", "probability": 0.9300130208333334}, {"start": 2701.38, "end": 2701.76, "word": " يعني", "probability": 0.79443359375}, {"start": 2701.76, "end": 2703.0, "word": " خلصنا", "probability": 0.9888916015625}, {"start": 2703.0, "end": 2703.56, "word": " المحاضرة", "probability": 0.983154296875}, {"start": 2703.56, "end": 2704.06, "word": " الثانية", "probability": 0.848388671875}, {"start": 2704.06, "end": 2704.54, "word": " و", "probability": 0.449951171875}, {"start": 2704.54, "end": 2704.74, "word": " هاي", "probability": 0.676025390625}, {"start": 2704.74, "end": 2704.82, "word": " ال", "probability": 0.96435546875}, {"start": 2704.82, "end": 2705.5, "word": " exercises", "probability": 0.95263671875}, {"start": 2705.5, "end": 2706.8, "word": " بتحلوها", "probability": 0.9876953125}, {"start": 2706.8, "end": 2707.14, "word": " و", "probability": 0.8359375}, {"start": 2707.14, "end": 2707.38, "word": " اللي", "probability": 0.849853515625}, {"start": 2707.38, "end": 2707.88, "word": " انتوا", "probability": 0.8805338541666666}, {"start": 2707.88, "end": 2708.14, "word": " يعني", "probability": 0.8798828125}, {"start": 2708.14, "end": 2708.54, "word": " حاسين", "probability": 0.8976236979166666}, {"start": 2708.54, "end": 2708.94, "word": " مش", "probability": 0.81884765625}, {"start": 2708.94, "end": 2709.46, "word": " عارفين", "probability": 0.98046875}, {"start": 2709.46, "end": 2709.92, "word": " تحلوها", "probability": 0.96484375}, {"start": 2709.92, "end": 2710.06, "word": " او", "probability": 0.884765625}, {"start": 2710.06, "end": 2710.38, "word": " كده", "probability": 0.9816080729166666}, {"start": 2710.38, "end": 2711.1, "word": " ممكن", "probability": 0.961181640625}, {"start": 2711.1, "end": 2711.74, "word": " عبر", "probability": 0.9912109375}, {"start": 2711.74, "end": 2712.18, "word": " التواصل", "probability": 0.79754638671875}, {"start": 2712.18, "end": 2712.5, "word": " معنا", "probability": 0.804443359375}, {"start": 2712.5, "end": 2712.72, "word": " على", "probability": 0.86279296875}, {"start": 2712.72, "end": 2712.82, "word": " ال", "probability": 0.8681640625}, {"start": 2712.82, "end": 2713.06, "word": " model", "probability": 0.708984375}, {"start": 2713.06, "end": 2713.28, "word": " او", "probability": 0.82861328125}, {"start": 2713.28, "end": 2713.38, "word": " على", "probability": 0.60546875}, {"start": 2713.38, "end": 2713.46, "word": " ال", "probability": 0.96728515625}, {"start": 2713.46, "end": 2713.82, "word": " whatsapp", "probability": 0.602783203125}, {"start": 2713.82, "end": 2713.98, "word": " اللي", "probability": 0.93212890625}, {"start": 2713.98, "end": 2714.66, "word": " هنقوم", "probability": 0.398101806640625}, {"start": 2714.66, "end": 2714.86, "word": " عها", "probability": 0.3387451171875}, {"start": 2714.86, "end": 2714.94, "word": " ان", "probability": 0.7783203125}, {"start": 2714.94, "end": 2715.14, "word": " شاء", "probability": 0.91455078125}, {"start": 2715.14, "end": 2715.24, "word": " الله", "probability": 0.94384765625}, {"start": 2715.24, "end": 2715.68, "word": " نحاول", "probability": 0.667022705078125}, {"start": 2715.68, "end": 2716.02, "word": " ..", "probability": 0.428955078125}, {"start": 2716.02, "end": 2717.38, "word": " هنكوّنها", "probability": 0.7879464285714286}, {"start": 2717.38, "end": 2717.86, "word": " معاكم", "probability": 0.8818359375}, {"start": 2717.86, "end": 2718.16, "word": " ان", "probability": 0.91845703125}, {"start": 2718.16, "end": 2718.36, "word": " شاء", "probability": 0.99072265625}, {"start": 2718.36, "end": 2718.54, "word": " الله", "probability": 0.95166015625}, {"start": 2718.54, "end": 2719.06, "word": " تبعتوا", "probability": 0.8564453125}, {"start": 2719.06, "end": 2719.32, "word": " الأسئلة", "probability": 0.9534912109375}, {"start": 2719.32, "end": 2719.46, "word": " اللي", "probability": 0.894287109375}, {"start": 2719.46, "end": 2719.62, "word": " بتكونوا", "probability": 0.619384765625}, {"start": 2719.62, "end": 2719.94, "word": " معاها", "probability": 0.7129720052083334}], "temperature": 1.0}, {"id": 122, "seek": 273082, "start": 2720.28, "end": 2730.82, "text": "هنا تواصل ان شاء الله يا اما بنصورها يا اما بنحلها بطريقة اللى بنلتقيها بنلاقيها مناسبة مع بعض والسلام عليكم ورحمة الله وبركاته", "tokens": [3224, 8315, 33427, 33546, 1211, 16472, 13412, 16606, 21984, 35186, 1975, 15042, 44945, 9381, 13063, 11296, 35186, 1975, 15042, 44945, 5016, 1211, 11296, 4724, 9566, 16572, 28671, 13672, 7578, 44945, 1211, 2655, 38436, 11296, 44945, 15040, 38436, 11296, 3714, 8315, 35457, 3660, 20449, 45030, 11242, 16070, 3794, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224], "avg_logprob": -0.15881147052420944, "compression_ratio": 1.7185185185185186, "no_speech_prob": 2.980232238769531e-07, "words": [{"start": 2720.28, "end": 2720.58, "word": "هنا", "probability": 0.6124267578125}, {"start": 2720.58, "end": 2721.06, "word": " تواصل", "probability": 0.9532877604166666}, {"start": 2721.06, "end": 2721.16, "word": " ان", "probability": 0.60205078125}, {"start": 2721.16, "end": 2721.4, "word": " شاء", "probability": 0.9892578125}, {"start": 2721.4, "end": 2721.56, "word": " الله", "probability": 0.951171875}, {"start": 2721.56, "end": 2721.72, "word": " يا", "probability": 0.3671875}, {"start": 2721.72, "end": 2721.8, "word": " اما", "probability": 0.722412109375}, {"start": 2721.8, "end": 2722.46, "word": " بنصورها", "probability": 0.951171875}, {"start": 2722.46, "end": 2722.66, "word": " يا", "probability": 0.82470703125}, {"start": 2722.66, "end": 2722.86, "word": " اما", "probability": 0.971435546875}, {"start": 2722.86, "end": 2723.7, "word": " بنحلها", "probability": 0.9810791015625}, {"start": 2723.7, "end": 2724.42, "word": " بطريقة", "probability": 0.9295654296875}, {"start": 2724.42, "end": 2725.04, "word": " اللى", "probability": 0.7763671875}, {"start": 2725.04, "end": 2726.5, "word": " بنلتقيها", "probability": 0.8458984375}, {"start": 2726.5, "end": 2727.28, "word": " بنلاقيها", "probability": 0.68463134765625}, {"start": 2727.28, "end": 2727.82, "word": " مناسبة", "probability": 0.9744873046875}, {"start": 2727.82, "end": 2727.94, "word": " مع", "probability": 0.99658203125}, {"start": 2727.94, "end": 2728.28, "word": " بعض", "probability": 0.99560546875}, {"start": 2728.28, "end": 2729.0, "word": " والسلام", "probability": 0.8047688802083334}, {"start": 2729.0, "end": 2729.34, "word": " عليكم", "probability": 0.921630859375}, {"start": 2729.34, "end": 2729.76, "word": " ورحمة", "probability": 0.9852294921875}, {"start": 2729.76, "end": 2730.02, "word": " الله", "probability": 0.96826171875}, {"start": 2730.02, "end": 2730.82, "word": " وبركاته", "probability": 0.97861328125}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2732.3849375, "duration_after_vad": 2700.6374999999903}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..13075a773889ba83f4de4d105b22738cfffabe02
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/AKkGetUIktE_raw.srt
@@ -0,0 +1,2468 @@
+1
+00:00:00,710 --> 00:00:04,770
+بسم الله الرحمن الرحيم هذه هي المحاضرة التانية
+
+2
+00:00:04,770 --> 00:00:10,230
+مساق تفاضل وتكامل جيم لطلاب و طالبات الجامعة
+
+3
+00:00:10,230 --> 00:00:16,000
+الإسلامية كليتي العلوم والهندسةاليوم هنبدأ ان شاء
+
+4
+00:00:16,000 --> 00:00:21,640
+الله اللي هو في 12 اتنين اللي هو vectors اللي هي
+
+5
+00:00:21,640 --> 00:00:28,960
+المتجهات الموضوع طبعا موضوع تم قبل هيك اخد فكرة
+
+6
+00:00:28,960 --> 00:00:34,480
+عنه او حتى يعني بعض الشغلات تم دراستها في اللي هي
+
+7
+00:00:34,480 --> 00:00:39,260
+مساق فيزيا في اللي هي اللي برضه لكليتي العلوم
+
+8
+00:00:39,260 --> 00:00:44,700
+والهندسة نعرف شو معناه متجه او شو معناه vectorالان
+
+9
+00:00:44,700 --> 00:00:48,980
+بنجي عنا initial point نقطة بداية a و terminal
+
+10
+00:00:48,980 --> 00:00:53,260
+point نقطة نهاية b الان ال line segment اللى بيمتد
+
+11
+00:00:53,260 --> 00:00:59,120
+من a و بتجه لعند b هذا بنسميه ال vector a,b اذا
+
+12
+00:00:59,120 --> 00:01:03,360
+اللى له نقطة بداية و له نقطة نهاية بتجه من البداية
+
+13
+00:01:03,360 --> 00:01:06,880
+الى النهاية هذا اللى هو the directed line segment
+
+14
+00:01:06,880 --> 00:01:11,220
+a,b is called a vector بنسميه ايش يا جماعة بسميه
+
+15
+00:01:11,220 --> 00:01:15,570
+vectorالان عنا بدنا نشوف شو معنات ان two vectors
+
+16
+00:01:15,570 --> 00:01:20,050
+are equal يعني كيف اللي هو عندي two vectors او
+
+17
+00:01:20,050 --> 00:01:27,370
+متجهين متساوين قبل نعرف شو معنات طول ال vector AB
+
+18
+00:01:27,370 --> 00:01:32,390
+طول ال vector AB هو طول القياس الطولة ده بنسميه ال
+
+19
+00:01:32,390 --> 00:01:36,790
+magnitude لل vector و بنرمزه بالرمز magnitude لل
+
+20
+00:01:36,790 --> 00:01:43,130
+AB الان بنقول ال vector ABاللي هو بيساوى ال vector
+
+21
+00:01:43,130 --> 00:01:47,990
+CD إذا حدث أمرين الأمر الأول أن هذا اله نفس
+
+22
+00:01:47,990 --> 00:01:52,910
+الاتجاه يعني هذا بوازيه ومشيات في اتجاه السهم في
+
+23
+00:01:52,910 --> 00:01:56,690
+اتجاه واحد اتنين يكون طول ال vector هذا وطول ال
+
+24
+00:01:56,690 --> 00:01:59,950
+vector هذا إيه شمالهم متساوية في هذه الحالة بتقول
+
+25
+00:01:59,950 --> 00:02:03,330
+عن هذا ال vector و هذا ال vector انهم are equal
+
+26
+00:02:03,330 --> 00:02:06,130
+إذا ال vector represented by the directed line
+
+27
+00:02:06,130 --> 00:02:10,590
+segment ABHas initial point A and terminal point B
+
+28
+00:02:10,590 --> 00:02:15,430
+and its length is denoted by AB بنقول أنه هذا
+
+29
+00:02:15,430 --> 00:02:22,310
+مساوي لل vector CD إذا كان اللي هو اتجاه ال CD نفس
+
+30
+00:02:22,310 --> 00:02:27,180
+اتجاه ال AB والتنين اللي هي نفس ال magnitudetwo
+
+31
+00:02:27,180 --> 00:02:30,660
+vectors are equal if they have the same length and
+
+32
+00:02:30,660 --> 00:02:35,820
+the same direction لاحظوا أن هذا وهذا وهذا الأربع
+
+33
+00:02:35,820 --> 00:02:39,680
+vectors هذوله متوازيات اللي هي نفس الاتجاه وكلهم
+
+34
+00:02:39,680 --> 00:02:45,360
+اتجاه اللي أمامنا والاربعة اللي هي نفس الطول هذوله
+
+35
+00:02:45,360 --> 00:02:48,900
+بنسميه are equal vectors لاحظوا ال vector هذا OB
+
+36
+00:02:49,390 --> 00:02:53,410
+هذا الـ vector جاي من نقطة الأصل و منطلق للنقطة من
+
+37
+00:02:53,410 --> 00:02:57,410
+B يعني ال initial point له نقطة الأصل هذا بنسميه
+
+38
+00:02:57,410 --> 00:03:01,290
+position vector يعني الآن كل ال vectors الموازية
+
+39
+00:03:01,290 --> 00:03:05,450
+له كلهم بساوينه إذا كانوا في نفس الاتجاه و اللي هو
+
+40
+00:03:05,450 --> 00:03:08,710
+نفس الطول إذا هنتعامل مع ال position vector بعد
+
+41
+00:03:08,710 --> 00:03:14,880
+هيك كمتجه بيمثل كل ال vectors الأخرياتنأتي الآن
+
+42
+00:03:14,880 --> 00:03:22,740
+اللي هو نشوف اشو معناه اشو كيف بدنا نكتب ال vector
+
+43
+00:03:22,740 --> 00:03:28,440
+على صورة على حاجة اسمها ال component form طيب صلوا
+
+44
+00:03:28,440 --> 00:03:33,580
+على النبي عليه الصلاة والسلام الان عندي اللي هو if
+
+45
+00:03:33,580 --> 00:03:38,620
+V is a two dimensional vector in the planeequal to
+
+46
+00:03:38,620 --> 00:03:41,240
+the vector with initial point at the origin and
+
+47
+00:03:41,240 --> 00:03:45,880
+terminal point at v1 وv2 يعني الان لو فرضنا عندي
+
+48
+00:03:45,880 --> 00:03:49,100
+الان في البلين في عندي vector v في ال two
+
+49
+00:03:49,100 --> 00:03:53,660
+dimension هذا ال vector v جاي من النقطة النقطة
+
+50
+00:03:53,660 --> 00:03:57,500
+الأصل ال initial point له نقطة الأصل اللي هي ال
+
+51
+00:03:57,500 --> 00:04:01,680
+origin و ال terminal point اللي هي نقطة من v1 وv2
+
+52
+00:04:02,150 --> 00:04:08,190
+then the component form of V هو عبارة عن V بيساوي
+
+53
+00:04:08,190 --> 00:04:13,930
+V واحد و V اتنين اذا لما نكتب هذه الصورة بنقصد
+
+54
+00:04:13,930 --> 00:04:18,410
+فيها ال vector اللي جاي من النقطة صفر صفر و رايح
+
+55
+00:04:18,410 --> 00:04:21,930
+لمين V واحد و V اتنين هذا لو كان في ال two
+
+56
+00:04:21,930 --> 00:04:25,070
+dimensions طب لو كان اللي هو ال vector في ال three
+
+57
+00:04:25,070 --> 00:04:30,090
+dimensions الان ال vector اللي قصدناها نوعرفنا ان
+
+58
+00:04:30,090 --> 00:04:34,700
+هذا هو ال vector اللي جايمن نقطة الأصل و رايح
+
+59
+00:04:34,700 --> 00:04:39,280
+للنقطة V1 و V2 و V3 لان نحكي عن الـ three
+
+60
+00:04:39,280 --> 00:04:43,820
+dimensions فبصير عندي الآن هذا ال vector اللي هو
+
+61
+00:04:43,820 --> 00:04:49,960
+من النقطة هذه للنقطة هذه ال component form له هو
+
+62
+00:04:49,960 --> 00:04:56,360
+عبارة عن Vبتساوى V1 V2 V3 إذا من الآن و طالع لما
+
+63
+00:04:56,360 --> 00:05:01,440
+نكتب V بيساوى V1 V2 V3 نقصد بها ال vector اللي ال
+
+64
+00:05:01,440 --> 00:05:06,660
+initial point له 0 0 0 و ال terminal point هي V1 و
+
+65
+00:05:06,660 --> 00:05:16,100
+V2 و V3 طيب الآنلو قدّينا للـ vector اللي هو P EQ
+
+66
+00:05:16,100 --> 00:05:21,900
+من X1 Y1 و Z1 و X2 Y2 و Z2 هذا ال vector أخدناه
+
+67
+00:05:21,900 --> 00:05:25,560
+موازي لهذا ال vector هيو موازي لهذا ال vector و
+
+68
+00:05:25,560 --> 00:05:30,500
+لنفس الطول فبصير عندي و كأن ال V بتساوي V1 و V2 و
+
+69
+00:05:30,500 --> 00:05:36,860
+V3 هي ال vector P EQالان ده نتعرف كيف نربط العلاقة
+
+70
+00:05:36,860 --> 00:05:41,480
+بين هذا وبين هذا وكيف نكتب هذا ال vector بي كيو
+
+71
+00:05:41,480 --> 00:05:46,060
+برضه على صورة اللي هو مين ال component form
+
+72
+00:05:46,060 --> 00:05:50,980
+اتطلعوا كيف الان عندى لو كان في عندى given the
+
+73
+00:05:50,980 --> 00:05:55,100
+points بي x واحد و y واحد و z واحد و q هو بقى ال x
+
+74
+00:05:55,100 --> 00:05:58,220
+اتنين و y اتنين و z اتنين الان ال standard
+
+75
+00:05:58,220 --> 00:06:03,970
+position vector to vequal to bq زي ما شفنا قبل
+
+76
+00:06:03,970 --> 00:06:08,610
+بشوية بنقول ان ال vector هذا بساوي ال vector هذا
+
+77
+00:06:08,610 --> 00:06:13,510
+للأسباب اللي ذكرناها اللي هو ان هذا ال vector
+
+78
+00:06:13,510 --> 00:06:19,530
+موازي لهذا ال vector ويساوي طولهالان bq هذا ال
+
+79
+00:06:19,530 --> 00:06:22,990
+vector بنكتبه على صورة ال component form ايش ال
+
+80
+00:06:22,990 --> 00:06:27,650
+component form في هذه الحالة اللي هو x2 ناقص x1
+
+81
+00:06:27,650 --> 00:06:34,650
+اللي هتمثلنا v1 y2 ناقص y1 هتمثلنا v2 z2 ناقص z1
+
+82
+00:06:34,650 --> 00:06:40,530
+هتمثلنا مين؟ v3 اذا الان صار عندي ال component
+
+83
+00:06:40,530 --> 00:06:49,140
+form لل vector bqالذي واصل من نقطة P لنقطة Q هو
+
+84
+00:06:49,140 --> 00:06:56,680
+عبارة عن الـ vector V بساوية X2-X1Y2Y1Z2-Z1 هذا
+
+85
+00:06:56,680 --> 00:07:01,140
+اللي هو عبارة عن الـ
+
+86
+00:07:01,140 --> 00:07:05,480
+component form لهذا الـ vectorالان لو بدنا نجي
+
+87
+00:07:05,480 --> 00:07:09,580
+نعرف كده طول ال vector هذا ال magnitude لل vector
+
+88
+00:07:09,580 --> 00:07:13,560
+هذا انتبهوا عليها شوية خلينا نرجع للرسمة و نشوف
+
+89
+00:07:13,560 --> 00:07:19,620
+كيف نوجد اللي هو الطول نيجي الان لو أخدنا من عند O
+
+90
+00:07:19,620 --> 00:07:23,040
+لعند النقطة هذا وصلنا طبعا هذا بيصير متلف قائم
+
+91
+00:07:23,040 --> 00:07:26,360
+الزاوية في هذا فبصير هذا تربيع الطول هذا طوله هذا
+
+92
+00:07:26,360 --> 00:07:31,180
+من هذا لهذا هذا طوله تربيع بساوي هذا تربيع V1
+
+93
+00:07:31,180 --> 00:07:37,680
+تربيع حيهازائد V2 تربيه لإيش لإنه الآن هي ال
+
+94
+00:07:37,680 --> 00:07:41,780
+vector الأصدر عندنا V اللي بنوجه طوله هذه اللي هو
+
+95
+00:07:41,780 --> 00:07:46,500
+من Han ل Han عبارة عن V3 اللي هو قيمة Z هيها من
+
+96
+00:07:46,500 --> 00:07:51,190
+Han ل Han اللي هي قيمة X هيهاالحداتي لان احداث ال
+
+97
+00:07:51,190 --> 00:07:57,550
+V1 من هان لهان الاحداث الصادر اللي يمين V2 فالان
+
+98
+00:07:57,550 --> 00:08:02,230
+طول هذا الوطر من هان لعندي الهان ايش بيساوي بيساوي
+
+99
+00:08:02,230 --> 00:08:07,050
+تربيعه بيساوي هذا تربيع V1 تربيع زائد هذا التربيع
+
+100
+00:08:07,050 --> 00:08:11,980
+اللي هو V2 تربيعالان هذا في المثلث هذا هيه هيظه
+
+101
+00:08:11,980 --> 00:08:16,640
+هذا واللي اودادنا طوله هيه بيصير هذا الوطن تربيع
+
+102
+00:08:16,640 --> 00:08:20,680
+اللي هو الطول المطلوب بيساوي V تلاتة تربيع زائد
+
+103
+00:08:20,680 --> 00:08:24,780
+هذا تربيع اللي هو عبارة عن V واحد تربيع زائد V
+
+104
+00:08:24,780 --> 00:08:28,000
+اتنين تربيع الأوداد هنا يعني الان طول ال vector
+
+105
+00:08:28,000 --> 00:08:32,200
+هذا تربيع بيساوي V واحد تربيع زائد V اتنين تربيع
+
+106
+00:08:32,200 --> 00:08:38,100
+زائد V تلاتة تربيعبسير الطول V هو عبارة عن الجدر
+
+107
+00:08:38,100 --> 00:08:42,380
+التربيعي لـ V1 تربيع زائد V2 تربيع زائد V3 تربيع
+
+108
+00:08:42,380 --> 00:08:46,460
+هذا عبارة عن الـ magnitude or length of the vector
+
+109
+00:08:46,460 --> 00:08:50,320
+V بساوة بـ Equal is the negative number magnitude
+
+110
+00:08:50,320 --> 00:08:54,200
+لV بساوة الجدر التربيعي V1 تربيع زائد V2 تربيع
+
+111
+00:08:54,200 --> 00:09:00,180
+زائد V3 تربيعالان او هو بساوي بال .. بال .. بال ..
+
+112
+00:09:00,180 --> 00:09:03,700
+بالإحداثيات اللي عنا أخدنا بي كيو زي ما قولنا هاي
+
+113
+00:09:03,700 --> 00:09:09,380
+X2 نقص X1 Y2 نقص Y1 بصير X2 نقص X1 لكل تربيع زي Y2
+
+114
+00:09:09,380 --> 00:09:15,740
+نقص Y1 تربيع زي Z2 نقص Z1 تربيع هذا اللي هو مين
+
+115
+00:09:15,740 --> 00:09:20,750
+اللي هو طول ال vector Vاللي هو عبارة عن الـ vector
+
+116
+00:09:20,750 --> 00:09:27,350
+من B لعند AQ نأخد الأمثل العددية على اللي حكيناه
+
+117
+00:09:27,350 --> 00:09:31,650
+find the component form and the length of the
+
+118
+00:09:31,650 --> 00:09:35,570
+vector with the initial point B-3,4,1 and terminal
+
+119
+00:09:35,570 --> 00:09:41,110
+point Q-5,2,2 يعني الآن بدنا ال component form لل
+
+120
+00:09:41,110 --> 00:09:44,740
+vector هذا BQبعد ما نوجد ال component form نوجد
+
+121
+00:09:44,740 --> 00:09:47,920
+طوله ال component form اتفجنا كيف نوجدها هاي نقطة
+
+122
+00:09:47,920 --> 00:09:50,800
+البداية وهي نقطة النهاية ال component form ال X
+
+123
+00:09:50,800 --> 00:09:55,820
+component ناقص خمسة ناقص ناقص تلاتة بتطلع مين اللي
+
+124
+00:09:55,820 --> 00:10:00,480
+هي V1 نفترض ان احنا اللي هي احداثيات V اللي هو PQ
+
+125
+00:10:00,480 --> 00:10:06,870
+عبارة عن V1 V2 V3لأن كيف نوجدها ناقص خمسة ناقص
+
+126
+00:10:06,870 --> 00:10:10,650
+ناقص تلاتة اللي هي X2 ناقص X1 ناقص خمسة ناقص ناقص
+
+127
+00:10:10,650 --> 00:10:15,350
+تلاتة بيصول ناقص اتنين لأن V2 اللي هو اتنين ناقص
+
+128
+00:10:15,350 --> 00:10:18,990
+اربعة اتنين ناقص اربعة اللي هي سالب اتنين هذه اللي
+
+129
+00:10:18,990 --> 00:10:23,970
+سمناها الاحداث ال Y الآن الأخيرة ال Z اتنين ناقص
+
+130
+00:10:23,970 --> 00:10:27,970
+واحد بتطلع V3 اتنين ناقص واحد Y ساو واحد اذا صار
+
+131
+00:10:27,970 --> 00:10:39,130
+عندي الآنالـ component form of V بيكيو ناقص اتنين
+
+132
+00:10:39,130 --> 00:10:43,830
+ال X و ناقص اتنين ال Y و واحد ال Z هذا ال
+
+133
+00:10:43,830 --> 00:10:48,770
+component form الان بدنا نجد ال link لل V أو ال
+
+134
+00:10:48,770 --> 00:10:49,750
+link لل بيكيو
+
+135
+00:10:52,610 --> 00:10:56,450
+الجدر التربيعي ل V1 تربيع زائد V2 تربيع زائد V3
+
+136
+00:10:56,450 --> 00:11:00,930
+تربيع يساوي الجدر التربيعي ل 4 زائد 4 زائد 4 زائد
+
+137
+00:11:00,930 --> 00:11:06,670
+1 من هنا جبت هذه ربعت هذا و ربعت هذا و ربعت اللي
+
+138
+00:11:06,670 --> 00:11:12,370
+هو هذا طالع عندي ايش بساوي بساوي تلاتة كويس هيك
+
+139
+00:11:12,370 --> 00:11:18,370
+ماشي الحالة اذا هو ال length of V هيطلع عندي لل
+
+140
+00:11:18,370 --> 00:11:20,850
+vector بساوي ايش بساوي تلاتة
+
+141
+00:11:23,610 --> 00:11:28,270
+الان نتعرف على vector algebra operations أو بمعنى
+
+142
+00:11:28,270 --> 00:11:32,580
+آخر العمليات الجبرية على المتجهاتالان بدنا نشوف
+
+143
+00:11:32,580 --> 00:11:37,200
+جبريا كيف بنجمع الهو two vectors هي عند ال vector
+
+144
+00:11:37,200 --> 00:11:43,040
+U هي ال component formula U1 U2 U3 و V بيساوي V1
+
+145
+00:11:43,040 --> 00:11:48,520
+V2 V3 اللي انا عشان نجمع جبريا U و V بنيجي بنقول
+
+146
+00:11:48,520 --> 00:11:52,660
+ال vector الجديد اللي اسمه U زائد V ايش بيساوي U1
+
+147
+00:11:52,660 --> 00:12:02,920
+زائد V1U2 V2 U3 V3 هذا اللي هو مجموع two vectors
+
+148
+00:12:02,920 --> 00:12:07,300
+إذا بتجمع ال X لل X و ال Y لل Y و ال Z لل Z بتطلع
+
+149
+00:12:07,300 --> 00:12:12,560
+عندك اللي هو ال vector U زائد V الان scalar
+
+150
+00:12:12,560 --> 00:12:16,340
+multiplication اللي هي عملية ضرب ال vector ب
+
+151
+00:12:16,340 --> 00:12:25,070
+scalar هي K في U بتساويك في U1 يعني بتدخل الان ك
+
+152
+00:12:25,070 --> 00:12:30,970
+على U1 ك U2 ك U3 ال vector الجديد هذا اسمه اللي هو
+
+153
+00:12:30,970 --> 00:12:37,290
+عبارة عن مين عن ال vector K فيه الان هذا جبريا هيك
+
+154
+00:12:37,290 --> 00:12:41,330
+اللي هو الجمع الجبري لما نشوف ايش معناته هندسيا او
+
+155
+00:12:41,330 --> 00:12:46,470
+كيف هندسيا و نقارنه بحاجات احنا بنعرفها قبل هيك
+
+156
+00:12:46,470 --> 00:12:52,270
+اللي هو في اللي هو جمع المتجهات هندسياالان لو كان
+
+157
+00:12:52,270 --> 00:12:56,410
+عندى ال vector U هى هذا ال position vector اللى هو
+
+158
+00:12:56,410 --> 00:13:01,330
+عبارة عن احداثياته U واحد و U اتنين باجي برسم هى U
+
+159
+00:13:01,330 --> 00:13:04,490
+قاعدة المثلثة جماعة اللى انتوا عارفينها او قاعدة
+
+160
+00:13:04,490 --> 00:13:10,290
+الدواز الأضلاع هى U اللى ان من عند نهاية U باجي
+
+161
+00:13:10,290 --> 00:13:15,310
+برسم ال vector V واحد اللى هو اللى احداثياته V
+
+162
+00:13:15,310 --> 00:13:19,610
+واحد و V اتنين يعني بمشي هيك V واحد و بطلع فوق
+
+163
+00:13:19,610 --> 00:13:23,420
+اللى هو مين V اتنينفبطلع عند ال vector هذا اللي
+
+164
+00:13:23,420 --> 00:13:27,960
+عندي هذا هو ال vector V الان مجموح U زائد V زي ما
+
+165
+00:13:27,960 --> 00:13:31,460
+انتوا عارفين اللي هو ال vector اللي بيغلق المثلف
+
+166
+00:13:31,460 --> 00:13:36,860
+هذا هي اللي بالأحمر هذا هو عبارة عن U زائد V طيب
+
+167
+00:13:36,860 --> 00:13:40,300
+لو اجينا اتطلعنا هل هذا بتناسق مع اللي حكيناها قبل
+
+168
+00:13:40,300 --> 00:13:46,240
+و شوية اه طبعا هذه الان هي عبارة عن U U1 و U2و V
+
+169
+00:13:46,240 --> 00:13:48,380
+هذا عبارة عن V واحد و V اتنين
+
+170
+00:14:01,160 --> 00:14:09,780
+إحداثيات الصادر هي U2 و V2 إذا فعلا ال vector
+
+171
+00:14:09,780 --> 00:14:16,480
+بالأحمر U زائد V هو فعلا إحداثيات U1 زائد V1 و U2
+
+172
+00:14:16,480 --> 00:14:20,440
+زائد V2 و كأنه بيقول لنا انه يا عم جمعكم الأولاني
+
+173
+00:14:20,440 --> 00:14:26,070
+بتناسق مع اللي بنحكي كيف نجمعته vectors هندسياالان
+
+174
+00:14:26,070 --> 00:14:29,230
+ممكن نظرة ايه لو نظرة اخرى واحد يقول لك انا مابديش
+
+175
+00:14:29,230 --> 00:14:31,710
+المتلف خلينا نرجع لقاعدة متوازلة اطلع و احنا
+
+176
+00:14:31,710 --> 00:14:35,790
+عارفينها اقول له ماشي الحال ارسم من عند ال U ارسم
+
+177
+00:14:35,790 --> 00:14:40,350
+ال position vector له U هيا و ارسم ال V برضه من
+
+178
+00:14:40,350 --> 00:14:44,180
+نقطة الاصل position vector Vالان كميلي متوازي
+
+179
+00:14:44,180 --> 00:14:50,180
+الأضلاح هذا خد القطر هذا هذا القطر هو اللي هيمثل
+
+180
+00:14:50,180 --> 00:14:54,220
+لك U زائد V وهذا طبعا بتنسخ مع اللي هان ليه؟ لأن
+
+181
+00:14:54,220 --> 00:14:57,870
+أصلا هذا V هيلو نفس الطول نفس الاتجاه نبغى زائله
+
+182
+00:14:57,870 --> 00:15:01,690
+في نفس الاتجاه و لا نفس الطول فبصير U زائد V هو U
+
+183
+00:15:01,690 --> 00:15:06,410
+زائد V إذن الآن عملية الجمع بين two vectors هندسيا
+
+184
+00:15:06,410 --> 00:15:11,810
+و جبريا عملية متناسقة و ملائمة و تولائم تولائم
+
+185
+00:15:11,810 --> 00:15:15,970
+اللي اتعلمنا سابقا نيجي الآن يا جماعة عملية scalar
+
+186
+00:15:15,970 --> 00:15:19,270
+multiplication نشوف المعنى الهندسيلها أكيد كلكم
+
+187
+00:15:19,270 --> 00:15:23,730
+قال إنه هى لو كان عندنا ال vector U لو ضربنا U في
+
+188
+00:15:23,730 --> 00:15:29,500
+حدداللي هو موجب هيظل محافظ على الاتجاه لكن لو
+
+189
+00:15:29,500 --> 00:15:35,120
+ضربناه في عدد سالم هيعكس الاتجاه طيب .. الان لو
+
+190
+00:15:35,120 --> 00:15:38,960
+كان العدد اللي ضربناه فيه اللي هو اكبر من واحد
+
+191
+00:15:38,960 --> 00:15:42,040
+هيزيده طول ال vector مثلا ضربناه في واحد و نص
+
+192
+00:15:42,040 --> 00:15:45,600
+هيصير طول ال vector جد اللي جابه المرة و نص هيحافظ
+
+193
+00:15:45,600 --> 00:15:48,880
+على نفس الاتجاه لو ضربناه في اتنين هيحافظ على نفس
+
+194
+00:15:48,880 --> 00:15:52,520
+الاتجاه و هيصير عندنا ضعف طول ال vector لأن لو
+
+195
+00:15:52,520 --> 00:15:57,000
+ضربناه في نقص اتنينهيناكس ال vector و هيكون طول
+
+196
+00:15:57,000 --> 00:16:01,020
+هذا ال vector هو نفسه طوله هذا ولكن ايه شماله اللي
+
+197
+00:16:01,020 --> 00:16:06,500
+هو في اتجاهه معاكس اذا الان الان عملية الضرب K في
+
+198
+00:16:06,500 --> 00:16:12,190
+U لل vector لما كان U صار K في U الان ال vectorيا
+
+199
+00:16:12,190 --> 00:16:16,670
+إما في نفس الاتجاه أو معاكس حسب الـ K أصغر من 0 أو
+
+200
+00:16:16,670 --> 00:16:21,210
+K أكبر من 0 أكبر من 0 نفس الاتجاه أصغر من 0 اللي
+
+201
+00:16:21,210 --> 00:16:25,590
+هو عكس الاتجاه K كانت اللي هي لو كان ضربنا في نص
+
+202
+00:16:25,590 --> 00:16:29,510
+حيصغر ال vector هذا يعني لو ضربناها في كسر حيصغرلو
+
+203
+00:16:29,510 --> 00:16:33,830
+ضربناه يعني كسر ازرع من واحد و اكبر من سفر لو
+
+204
+00:16:33,830 --> 00:16:37,310
+ضربناه في اشي اكبر من واحد هيكبر طول ال vector
+
+205
+00:16:37,310 --> 00:16:43,890
+الان بدنا نقارن بيقولي comparing the length of U
+
+206
+00:16:43,890 --> 00:16:50,390
+and K U جبريا خلينا نقارن
+
+207
+00:16:50,390 --> 00:16:53,090
+بين الجهتين شوفوا صلوا حنبي عليه الصلاة والسلام
+
+208
+00:16:53,850 --> 00:16:59,230
+عندك احسب ليه ال magnitude لكيف U ؟ احنا اتفجنا شو
+
+209
+00:16:59,230 --> 00:17:03,490
+معناه كيف U ال U لو كانت U واحد و U اتنين و U
+
+210
+00:17:03,490 --> 00:17:08,050
+تلاتة ال component لها ال كيف U قلنا كيف بيصير ك U
+
+211
+00:17:08,050 --> 00:17:13,240
+واحد ك U اتنين ك U تلاتةالـ Link لـ KU اللي هو
+
+212
+00:17:13,240 --> 00:17:17,560
+عبارة عن جدر التربيعي الإحداث السيني هو KU 1 تربيع
+
+213
+00:17:17,560 --> 00:17:21,620
+زاد الإحداث الصادر تربيع اللي هو KU 2 تربيع زاد
+
+214
+00:17:21,620 --> 00:17:27,110
+اللي هي إحداث الصادر KU 3 تربيعهذا طول مين اللي هو
+
+215
+00:17:27,110 --> 00:17:31,790
+ال vector k في u الان طلعلي ال k تربيع برا مالحساب
+
+216
+00:17:31,790 --> 00:17:36,350
+بضل ان هدول حاليا طلعلي جدر ال k تربيع برا ضل في
+
+217
+00:17:36,350 --> 00:17:39,370
+هذا لان جدر ال k تربيع زي ما انتوا عارفين هو عبارة
+
+218
+00:17:39,370 --> 00:17:43,050
+عن absolute value لكيه وهذا الجدر التربيعي هو
+
+219
+00:17:43,050 --> 00:17:46,050
+عبارة عن طول ال vector u لان ال components ليهم لل
+
+220
+00:17:46,050 --> 00:17:50,910
+u u واحد و u اتنين و u تلاتة اذا بطلع عندي دائما
+
+221
+00:17:50,910 --> 00:17:55,960
+ال absolute value او ال magnitudeأو طول اللي هو k
+
+222
+00:17:55,960 --> 00:18:01,300
+في u بساوي ال absolute value لل k في main في طول
+
+223
+00:18:01,300 --> 00:18:06,400
+ال u إذا بقولك the length of k u is the absolute
+
+224
+00:18:06,400 --> 00:18:09,420
+value of the scalar k times the length of u ده
+
+225
+00:18:09,420 --> 00:18:14,080
+vector ناقص واحد في u إيش هيساوي ناقص u زي ما
+
+226
+00:18:14,080 --> 00:18:18,750
+شوفنا هنا uلأن ناقص u هيكون عكس اتجاه يعني نفس
+
+227
+00:18:18,750 --> 00:18:23,030
+الطول و لكن عكس الاتجاه يعني ناقص واحد في u بساوي
+
+228
+00:18:23,030 --> 00:18:27,790
+ناقص u اللي هي نفس الطول لكن عكس الاتجاه عن ال u
+
+229
+00:18:27,790 --> 00:18:31,950
+has the same length as u but in the opposite
+
+230
+00:18:31,950 --> 00:18:34,330
+direction
+
+231
+00:18:36,780 --> 00:18:44,000
+معرفة U ناقص V of two vectors is defined by U ناقص
+
+232
+00:18:44,000 --> 00:18:50,280
+V بتساوي U ناقص V بتساوي U زائد ناقص V طبعا احنا
+
+233
+00:18:50,280 --> 00:18:55,380
+ناقص V عرفناها هي عبارة عن ناقص واحد في V الآن إذا
+
+234
+00:18:55,380 --> 00:18:59,980
+هذه معرفة والجامع معرفة إذا هذه الجديدة عرفناها
+
+235
+00:18:59,980 --> 00:19:04,500
+بإيش بنعرفه الآن اللي هو جامع two vectorsوهذا ناقص
+
+236
+00:19:04,500 --> 00:19:09,220
+ال V هو ناقص واحد في V تنشوف هذا الكلام هل هي
+
+237
+00:19:09,220 --> 00:19:14,300
+تلائم جدريا او هندسيا او لأ ونشوف كيف اللي هو مع
+
+238
+00:19:14,300 --> 00:19:18,280
+معلوماتنا السابقة هتطلع الآن هي عندي هذا اللي هو
+
+239
+00:19:18,280 --> 00:19:23,510
+ال vector U وهذا ال vector V ماشي الحالةالان توا
+
+240
+00:19:23,510 --> 00:19:30,170
+عرفنا اللى هو الان الان u ناقص v ايش بيقول ال u
+
+241
+00:19:30,170 --> 00:19:35,570
+ناقص v هو عبارة عن اللى هو ال vector اللى بيصل من
+
+242
+00:19:35,570 --> 00:19:40,690
+عند ال v اللى هو اللى إلها ناقصها وبنتى لعند مين
+
+243
+00:19:40,690 --> 00:19:44,030
+ال u اللى هى الموجبة طب ليش هي كده انت قولت عنه
+
+244
+00:19:44,030 --> 00:19:52,280
+عشان لما نقول u زائد v عشان نقول v زائد u ناقص vهي
+
+245
+00:19:52,280 --> 00:19:56,040
+عملية الجمع اللى قبل شوية هى هذا و هى هذا و جمحين
+
+246
+00:19:56,040 --> 00:20:00,100
+اللى بيغلقل يعني لما يكون هذا زائد هذا الناقص V مع
+
+247
+00:20:00,100 --> 00:20:04,760
+ال V بيطلع منين ال U فعلا متناسق هذا الخط الأحمر
+
+248
+00:20:04,760 --> 00:20:10,040
+عبارة عن U ناقص V الان نيجي نطلع عليها نظرة أخرى
+
+249
+00:20:10,040 --> 00:20:18,480
+هى عند ال U و هى عند ال V ماشي الان ال V هذا ناقصه
+
+250
+00:20:18,480 --> 00:20:25,170
+هو نفسهالطول له و لكن اتجاهه عكسه تماما هي ناقص V
+
+251
+00:20:25,170 --> 00:20:32,720
+فالان لما نجمع U زائد ناقص V اللي بيغلقهإذاً هذا
+
+252
+00:20:32,720 --> 00:20:36,900
+عبارة عن اللي أغلق هنا هو عبارة عن U زائد ناقص V
+
+253
+00:20:36,900 --> 00:20:43,060
+هو اللي هذا زائد الجديد اللي هو ناقص V فبصير U
+
+254
+00:20:43,060 --> 00:20:46,700
+زائد ناقص V هو اللي بالأحمر لاحظ اللي بالأحمر هذا
+
+255
+00:20:46,700 --> 00:20:50,700
+له نفس الطولة اللي فوق و نفس ال direction اللي
+
+256
+00:20:50,700 --> 00:20:55,200
+بالأحمرفعشان هيك ال U ناقص V بيساوي ال U زائد ناقص
+
+257
+00:20:55,200 --> 00:20:59,120
+V اللي هو فعلا كلامنا اللي قبل بشوية منطقي
+
+258
+00:20:59,120 --> 00:21:03,880
+والتعريف منطقي وهي تفسيره الهندسي ويتلاءم معاه
+
+259
+00:21:03,880 --> 00:21:08,620
+يعني ال U ناقص V هو عبارة عن U زائد ناقص V زي ما
+
+260
+00:21:08,620 --> 00:21:14,390
+شفنا اللي هو في الرسم اللي حكيناطيب الان ناخد بعض
+
+261
+00:21:14,390 --> 00:21:18,730
+الأمثلة العددية let u بالساوية ناقص واحد و تلاتة و
+
+262
+00:21:18,730 --> 00:21:23,090
+واحد و v بالساوية اربعة و سبعة و سفر هدولة ال two
+
+263
+00:21:23,090 --> 00:21:27,270
+vectors الان بدي بيقوللي بيسألي كمان يالي طبعا
+
+264
+00:21:27,270 --> 00:21:31,170
+مكتوبات التنتين في صورة مين يا جماعة ال component
+
+265
+00:21:31,170 --> 00:21:36,630
+فبقولي find the components of اوجد ال components
+
+266
+00:21:36,630 --> 00:21:42,370
+لمين اتنين u زي التلاتة vعرفنا شو معناه اتنين U
+
+267
+00:21:42,370 --> 00:21:46,270
+وعرفنا شو معناه تلاتة V وعرفنا شو معناه جمعتي و
+
+268
+00:21:46,270 --> 00:21:51,370
+vectors U نقص V وبرضه عرفنا ال magnitude لنص ال U
+
+269
+00:21:51,370 --> 00:21:55,610
+اذا ال N اللي هو find the components of جزده عن
+
+270
+00:21:55,610 --> 00:21:59,830
+اتنين عن A وB و ال C بدك ايش توجد لها ال magnitude
+
+271
+00:21:59,830 --> 00:22:06,310
+لها او نص النص قيمة طول نص ال U طيب خلونا نشوف و
+
+272
+00:22:06,310 --> 00:22:11,600
+نصل لعنديالان اتنين U زي التلاتة V هي اتنين U زي
+
+273
+00:22:11,600 --> 00:22:15,460
+التلاتة V شيل ال U و حط مكانها قيمتها هي قيمتها و
+
+274
+00:22:15,460 --> 00:22:20,020
+ال V شيلها و حط قيمتها هيها الان ابدا اللي هو
+
+275
+00:22:20,020 --> 00:22:25,360
+اشتغل ضرب لهد جوا ضرب كلنا كيف نضرب احنا scalar في
+
+276
+00:22:25,360 --> 00:22:32,860
+ال vector بصير اتنين في ناقص واحدنفس الشيء بالنسبة
+
+277
+00:22:32,860 --> 00:22:35,540
+لتلاتة في أربعة و تلاتة في سبعة و تلاتة في سفرة و
+
+278
+00:22:35,540 --> 00:22:39,500
+نجمحها نعرف كيف نجمح هدا مع هدا و هدا مع هدا و هدا
+
+279
+00:22:39,500 --> 00:22:40,640
+مع هدا و هدا مع هدا و هدا مع هدا و هدا مع هدا و
+
+280
+00:22:40,640 --> 00:22:42,400
+هدا مع هدا و هدا مع هدا و هدا مع هدا و هدا مع هدا
+
+281
+00:22:42,400 --> 00:22:42,700
+و هدا مع هدا و هدا مع هدا و هدا مع هدا و هدا مع
+
+282
+00:22:42,700 --> 00:22:42,700
+هدا و هدا مع هدا و هدا مع هدا و هدا مع هدا و هدا
+
+283
+00:22:42,700 --> 00:22:42,940
+مع هدا و هدا مع هدا و هدا مع هدا و هدا مع هدا و
+
+284
+00:22:42,940 --> 00:22:42,940
+هدا مع هدا و هدا مع هدا و هدا مع هدا و هدا مع هدا
+
+285
+00:22:42,940 --> 00:22:42,960
+و هدا مع هدا و هدا مع هدا و هدا مع هدا و هدا مع
+
+286
+00:22:42,960 --> 00:22:47,520
+هدانقص اتنين و تلاتة فى أربعة باطن عاش نقص اتنين
+
+287
+00:22:47,520 --> 00:22:53,180
+واطن عاش هو عشرة اتنين فى تلاتة ستة ستة زائد تلاتة
+
+288
+00:22:53,180 --> 00:22:56,480
+فى سبعة بواحد وعشرين ستة وواحد وعشرين بيطلع سبعة
+
+289
+00:22:56,480 --> 00:23:02,360
+وعشرين اتنين زائد سفر بيطلع اش اتنين اذا هذا قيمة
+
+290
+00:23:02,360 --> 00:23:06,740
+اللى هو اللى هى اتنين U زائد تلاتة بيه نيجي لل U
+
+291
+00:23:06,740 --> 00:23:11,460
+نقص فيه هي ال Uوهي ال V وهي الناقص طبعا ناقص
+
+292
+00:23:11,460 --> 00:23:15,580
+معناته اللي هو ناقص أربعة و ناقص سبعة و سفر يعني
+
+293
+00:23:15,580 --> 00:23:21,240
+ناقص واحد ناقص أربعة ناقص خمسة تلاتة ناقص سبعة
+
+294
+00:23:21,240 --> 00:23:26,280
+يعني ناقص أربعة واحد ناقص سفر يعني واحد الآن بدنا
+
+295
+00:23:26,280 --> 00:23:29,380
+نيجي نيجي ل سيل هو بدنا نوجد magnitude ل نص يوم
+
+296
+00:23:29,900 --> 00:23:34,900
+الان هى U نصه نص ال U اللى هو عبارة عن نص فى نقص
+
+297
+00:23:34,900 --> 00:23:39,160
+واحد و نص فى تلاتة و نص فى واحد يعنى الان ال
+
+298
+00:23:39,160 --> 00:23:43,520
+length اللى نص فى نقص واحد لكل تربيع زائد نص فى
+
+299
+00:23:43,520 --> 00:23:47,380
+تلاتة لكل تربيع زائد نص فى واحد لكل تربيع هذا اللى
+
+300
+00:23:47,380 --> 00:23:51,400
+هو عبارة عن الجدر التربيع اللى بتطلع ال length لمن
+
+301
+00:23:51,400 --> 00:23:58,240
+لل vector نص U ويساوي احسب ان دولة هدى بتطلع نصو
+
+302
+00:23:58,240 --> 00:24:03,080
+هذه تسعة على اتنين وهذه واحد على اتنين يعني بيصير
+
+303
+00:24:03,080 --> 00:24:06,820
+واحد واحد اتنين وتسعة احد عاش يعني احداش عالمين
+
+304
+00:24:06,820 --> 00:24:09,580
+على اربعة اسف لان انا بتربع اتنين تحت بيصير اربعة
+
+305
+00:24:09,580 --> 00:24:13,940
+بيصير جدر التربيع ال 11 على اربعةطلعني الربع برا
+
+306
+00:24:13,940 --> 00:24:17,520
+الجذر بيطلع من نص بيصير نص الجذر ال 11 اذا
+
+307
+00:24:17,520 --> 00:24:21,420
+magnitude او طول ال vector نص U بيساوي نص الجذر ال
+
+308
+00:24:21,420 --> 00:24:24,900
+11 واحد تاني قال لي ليش تغلبني زي هي كانت استخدام
+
+309
+00:24:24,900 --> 00:24:27,980
+الخاص اللي قبله شوية قوله مش مشكلة استخدامها كده
+
+310
+00:24:27,980 --> 00:24:30,700
+قال لي باخده النص برا بيصير نص في ال magnitude ده
+
+311
+00:24:30,700 --> 00:24:35,160
+Uالـ magnitude value هي الجذر التربيعي الواحد زائد
+
+312
+00:24:35,160 --> 00:24:36,180
+تسعة زائد واحد
+
+313
+00:24:48,420 --> 00:24:56,900
+نجي الآن لبعض خواص ال vector operations
+
+314
+00:24:56,900 --> 00:25:03,640
+أو بمعنى آخر خواص العمليات عالميا على المتجهات
+
+315
+00:25:03,640 --> 00:25:08,120
+خواص العمليات الجبرية على المتجهات انتبهوا let u و
+
+316
+00:25:08,120 --> 00:25:12,140
+v و w be vectors and a و b scalars يعني نفترض ان
+
+317
+00:25:12,140 --> 00:25:17,760
+ال u و ال v و ال w جمع ايش هي عبارة عن متجهاتو ال
+
+318
+00:25:17,760 --> 00:25:21,920
+a و ال b هى عبارة عن اعداد ال scholars نيجى
+
+319
+00:25:21,920 --> 00:25:25,760
+للخاصية الاولى ايش الخاصية الاولى بتقول بتقول يو
+
+320
+00:25:25,760 --> 00:25:29,940
+زائد V بيساوي V زائد U يعنى اللى هى عملية الجمع
+
+321
+00:25:29,940 --> 00:25:35,110
+على ال vectors او المتجهات إبداليةالخاصية التانية
+
+322
+00:25:35,110 --> 00:25:40,150
+اللى بتقولى عملية الجمع عبارة عن عملية دامجة أو
+
+323
+00:25:40,150 --> 00:25:43,970
+تجميعية أو اللى بيسميها associative يعنى عملية
+
+324
+00:25:43,970 --> 00:25:48,930
+الجمع ع ال vectors هي عملية دامجة أو تجميعية أو
+
+325
+00:25:48,930 --> 00:25:52,690
+associative يعنى لو قلت U زائد V و بعدين جمعتلها W
+
+326
+00:25:52,690 --> 00:25:57,630
+برضه منفع حد ساوى ال V زائد W و لما نقول U زائد
+
+327
+00:25:57,630 --> 00:26:01,950
+اللى بطلع هنا بطلع نفس الاشياءالان تلاتة بيقولي ان
+
+328
+00:26:01,950 --> 00:26:08,830
+المحايد الجامعي لل vectors u اللي هو من عملية
+
+329
+00:26:08,830 --> 00:26:11,370
+الجامعة طبعا اللي حامل المحايد الجامعي اللي هو
+
+330
+00:26:11,370 --> 00:26:14,690
+جداش بساوي سفر يعني zero zero zero اللي هو ال
+
+331
+00:26:14,690 --> 00:26:18,210
+vector zero إذا دائما u زائد zero vector طبعا دي
+
+332
+00:26:18,210 --> 00:26:21,540
+vector و دي vector بساوي uالان انا بقول ال inverse
+
+333
+00:26:21,540 --> 00:26:26,700
+او المعكوس الجمعى لل U هو عبارة عن ناقص U ك vector
+
+334
+00:26:26,700 --> 00:26:31,340
+طبعا يعني U زائد معكوس الجمعى اللى هو ناقص ال U
+
+335
+00:26:31,340 --> 00:26:36,260
+بساوية Zero vector Zero الان Zero في U بساوية Zero
+
+336
+00:26:36,260 --> 00:26:39,840
+Zero هنا scalar يا جماعة و ال U vector Zero scalar
+
+337
+00:26:39,840 --> 00:26:46,040
+في U vector بساوية Zero vectorطيب الواحد scalar في
+
+338
+00:26:46,040 --> 00:26:49,800
+U بيطلع ايش بيساوي U يعني انك ضمن تضرب واحد في اي
+
+339
+00:26:49,800 --> 00:26:54,480
+vector بيظله نفس ال vector لا يتغير لا اتجاهه ولا
+
+340
+00:26:54,480 --> 00:27:00,160
+طوله طيب الان خاصية السابعة عملية اللي هي ضرب ال
+
+341
+00:27:00,160 --> 00:27:04,540
+scalars في ال vector عملية دامجة او تجميعية يعني
+
+342
+00:27:04,540 --> 00:27:08,530
+بمعنى اخرأ في ب في يو يعني اضرب في ب في يو و بعدين
+
+343
+00:27:08,530 --> 00:27:12,130
+اضرب ا فيهن هو نفس الاشي ا في ب و بعدين في مين في
+
+344
+00:27:12,130 --> 00:27:15,890
+يو ا في ب ضرب هن ضرب مين اللي هو ضرب عادي تبقى
+
+345
+00:27:15,890 --> 00:27:20,630
+اعداد اللي انا من ضرب في يو بيكون ضرب بين scalar و
+
+346
+00:27:20,630 --> 00:27:25,680
+vector وعرفناهالخاصية الثانية a في u زائد v بساوي
+
+347
+00:27:25,680 --> 00:27:29,400
+a u زائد a v يعني يتوزع عملية ال scalar
+
+348
+00:27:29,400 --> 00:27:33,500
+multiplication على ال vector addition بنوزع عملية
+
+349
+00:27:33,500 --> 00:27:36,980
+اللي هي ضرب اللي هو ال scalar في ال vector على
+
+350
+00:27:36,980 --> 00:27:40,060
+اللي هو الجامع بين المتجهات بيطلع a u زائد a v
+
+351
+00:27:40,060 --> 00:27:45,900
+الآن الجامع بين ال scalars الجامع العادي تبعنا ال
+
+352
+00:27:45,900 --> 00:27:50,220
+u لما تنضرب فيها توزع يعني ك vector بيصير u في a
+
+353
+00:27:50,930 --> 00:27:56,830
+زائد U في B يعني AU زائد B في U هذه برضه الخاصية
+
+354
+00:27:56,830 --> 00:28:01,290
+اللي لها متحققة طيب احنا بخلنا نبرهلنا واحدة من
+
+355
+00:28:01,290 --> 00:28:04,250
+هدول اختارت عشوائية انا اختارت رتارية تمانية قلت
+
+356
+00:28:04,250 --> 00:28:07,690
+تمكن تمانية أصعب واحدة فيهم و .. و .. و تاخدوا
+
+357
+00:28:07,690 --> 00:28:11,900
+فكرة عن كيف اللي هو بينبرهلأولا لت U أولا لت U
+
+358
+00:28:11,900 --> 00:28:12,660
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+359
+00:28:12,660 --> 00:28:13,320
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+360
+00:28:13,320 --> 00:28:14,580
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+361
+00:28:14,580 --> 00:28:15,260
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+362
+00:28:15,260 --> 00:28:19,220
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+363
+00:28:19,220 --> 00:28:20,920
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+364
+00:28:20,920 --> 00:28:25,060
+أولا لت U أولا لت U أولا لت U أولا لت U أولا لت U
+
+365
+00:28:25,060 --> 00:28:30,450
+أولو V هي كمبيوتر فورميله V1, V2, V3 و الان اذا
+
+366
+00:28:30,450 --> 00:28:34,250
+انا احسب A في U زائد V الـ A is scalar و U زائد V
+
+367
+00:28:34,250 --> 00:28:39,390
+جمع Vectors بيصير A في او ضد ال U زائد V ال U زائد
+
+368
+00:28:39,390 --> 00:28:50,640
+V ايش هيكون U1 زائد V1U2 V2 U3 V3 U2
+
+369
+00:28:50,640 --> 00:28:50,640
+V2 U3 V3 U2 V3 U3 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3
+
+370
+00:28:50,640 --> 00:28:50,640
+U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2
+
+371
+00:28:50,640 --> 00:28:54,400
+V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3
+
+372
+00:28:54,400 --> 00:28:57,340
+U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2
+
+373
+00:28:57,340 --> 00:28:58,280
+V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3
+
+374
+00:28:58,280 --> 00:29:05,220
+U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2 V3 U2
+
+375
+00:29:05,220 --> 00:29:15,560
+V3 U2 V3 U2 V3أي و اتنين اي و تلاتة اللي بيظل اي و
+
+376
+00:29:15,560 --> 00:29:24,600
+احد اي و اتنين اي و تلاتة اللي بيظل
+
+377
+00:29:24,600 --> 00:29:25,560
+اي و احد اي و اتنين اي و تلاتة اللي بيظل اي و احد
+
+378
+00:29:25,560 --> 00:29:26,120
+اي و تلاتة اللي بيظل اي و احد اي و تلاتة اللي بيظل
+
+379
+00:29:26,120 --> 00:29:27,720
+اي و اتنين اي و تلاتة اللي بيظل اي و احد اي و
+
+380
+00:29:27,720 --> 00:29:28,420
+تلاتة اللي بيظل اي و اتنين اي و تلاتة اللي بيظل اي
+
+381
+00:29:28,420 --> 00:29:28,420
+و اتنين اي و تلاتة اللي بيظل اي و اتنين اي و تلاتة
+
+382
+00:29:28,420 --> 00:29:29,400
+اللي بيظل اي و اتنين اي و تلاتة اللي بيظلأيه
+
+383
+00:29:29,400 --> 00:29:31,760
+طالعها برضه عام المشترك اللي هو scalar
+
+384
+00:29:31,760 --> 00:29:34,760
+multiplication بقصود عام المشترك بده .. بده رجحها
+
+385
+00:29:34,760 --> 00:29:37,840
+ل scalar multiplication اللي هي عملية ضرب اللي هو
+
+386
+00:29:37,840 --> 00:29:43,800
+scalar في vector A في V و 1 V 2 V 3 هذا اللي هو U
+
+387
+00:29:43,800 --> 00:29:48,200
+و هذا V فبصير A U زائد A V يعني بدأت A في U زائد V
+
+388
+00:29:48,200 --> 00:29:52,560
+و اتسلسلت لما وصلت لك A في U زائد A V حيث استخدمت
+
+389
+00:29:52,560 --> 00:29:56,680
+عمليةما عن عملية الجمع بين two vectors وعملية ال
+
+390
+00:29:56,680 --> 00:30:00,360
+scale of multiplication اللي عرفناها سابقاً هذا هو
+
+391
+00:30:00,360 --> 00:30:05,700
+برهان هذه والباقية اللي هي بشكل مشارع الان نيجي ان
+
+392
+00:30:05,700 --> 00:30:09,820
+اعرفلكم هالتعريف البسيط بنقول ان احنا عندي three
+
+393
+00:30:09,820 --> 00:30:15,240
+vectors or more space vectors are called coplanar
+
+394
+00:30:15,240 --> 00:30:18,940
+يعني لو كان في عندي تلاتة من ال vector تلاتة من
+
+395
+00:30:18,940 --> 00:30:21,020
+التجاهات او اربعة من التجاهات او خمس من التجاهات
+
+396
+00:30:21,020 --> 00:30:27,140
+او زي ما بدكبنقول عنهم ان كو بلانر بلانر إذا كان
+
+397
+00:30:27,140 --> 00:30:31,660
+كو بلانر vectors بنسميهم in the case that they are
+
+398
+00:30:31,660 --> 00:30:36,340
+in the same plane يعني لما نكون كلهم مالهم في نفس
+
+399
+00:30:36,340 --> 00:30:43,240
+ال plane مثال for example U و V و U زائد Vهذول ايش
+
+400
+00:30:43,240 --> 00:30:48,660
+مالهين co-planner لإن احنا كنا نقول عشان ال U و ال
+
+401
+00:30:48,660 --> 00:30:51,860
+V ارسم ال U و ارسم ال V زي ما رسمنا قبل شوية
+
+402
+00:30:51,860 --> 00:30:57,090
+المثلث اللي بيجفلهمبكون هو u زائد v إذا التلاتة
+
+403
+00:30:57,090 --> 00:31:01,310
+بكون نفس المثلث ال u ال v و ال u زائد v يعني هنا
+
+404
+00:31:01,310 --> 00:31:06,310
+in the same plane يعني قبلها نفسها طيب الان بدنا
+
+405
+00:31:06,310 --> 00:31:09,950
+نعرف حاجة اسمها ال unit vectors و منها نصل لحاجة
+
+406
+00:31:09,950 --> 00:31:13,070
+اسمة standard unit vectors اللي هتسهل علينا عملية
+
+407
+00:31:13,070 --> 00:31:18,810
+التعامل مع المتجهات نشوف ايش مقصوده الان a vector
+
+408
+00:31:18,810 --> 00:31:23,230
+v of length one is called a unit vectorيعني اي
+
+409
+00:31:23,230 --> 00:31:26,350
+vector يا جماعة في الدنيا طول واحد ال magnitude
+
+410
+00:31:26,350 --> 00:31:30,570
+الواحد بنسميه unit vector اذا في عندي عدد لانهائم
+
+411
+00:31:30,570 --> 00:31:33,570
+ال vectors اي vector طول واحد اي زي و vector تاني
+
+412
+00:31:33,570 --> 00:31:36,610
+و vector اتنين و هذا طول واحد طول اسف واحد و هذا
+
+413
+00:31:36,610 --> 00:31:40,850
+واحد واحد ارسم عدد لانهائم ال vectors طول واحد في
+
+414
+00:31:40,850 --> 00:31:45,370
+اتجاهات مختلفة كلهم بنسميهم ايه شمالهم unit
+
+415
+00:31:45,370 --> 00:31:52,050
+vectors اذا ال unit vector هو vectorطول واحد في
+
+416
+00:31:52,050 --> 00:31:55,310
+عندي vectors انا مهتم فيها جدا لان هذه ال vectors
+
+417
+00:31:55,310 --> 00:32:00,930
+هتكون اداء لتسهيل عملية التعامل مع ال vectors مين
+
+418
+00:32:00,930 --> 00:32:04,930
+هن دولة بنسميها the standard unit vectors او اللي
+
+419
+00:32:04,930 --> 00:32:11,570
+هي ال unit vectors معتمدات او اللي هن احنا يعني
+
+420
+00:32:11,570 --> 00:32:16,770
+هنتعامل معها على طول اللي بنسميها ال I و ال J و ال
+
+421
+00:32:16,770 --> 00:32:21,060
+Kماذا يعني ال vector I؟ ال I للمجزم له component
+
+422
+00:32:21,060 --> 00:32:25,640
+form هو واحد زر و زر يعني هو على محور السينات يعني
+
+423
+00:32:25,640 --> 00:32:29,120
+جزء من محور السينات مقطوع واحد وهذا سفر وهذا سفر
+
+424
+00:32:29,120 --> 00:32:32,540
+ال J اللي هو على محور الصدات لأن هذا سفر وهذا سفر
+
+425
+00:32:32,540 --> 00:32:36,120
+و المقطوع واحد من محور الصدات ال K على محور الزد
+
+426
+00:32:36,120 --> 00:32:39,760
+ذات اللي هو سفر و سفر الجزم المقطوع بساوية واحد
+
+427
+00:32:39,760 --> 00:32:44,820
+ماشي الحال طيب محور الزد المقصود يعني طيب شوفوا
+
+428
+00:32:44,820 --> 00:32:51,630
+عليهاالان هى الامشي على ال X axis بمقدار واحد
+
+429
+00:32:51,630 --> 00:32:56,890
+وجفان اللى هو المتجه اللى جاى من نقطة الاصل واقف
+
+430
+00:32:56,890 --> 00:33:01,430
+عند هذا النقطة بنسميه اللى هو ال I الان امشي على
+
+431
+00:33:01,430 --> 00:33:05,790
+محور الصدات بمقدار واحد وجفان اللى هو هذا ال
+
+432
+00:33:05,790 --> 00:33:10,890
+vector بنسميه Jأطلع من هنا في الاتجاه هذا و اقف
+
+433
+00:33:10,890 --> 00:33:15,070
+هنا بمقدار واحد هذا بنسمي ash k ايش ال component
+
+434
+00:33:15,070 --> 00:33:18,930
+form له طبيعي واحد و سفر و سفر هذا ال component
+
+435
+00:33:18,930 --> 00:33:22,990
+form له سفر و سفر و واحد هذا ال component form له
+
+436
+00:33:22,990 --> 00:33:27,010
+سفر و واحد و سفر هدول الان ال three vectors بدي
+
+437
+00:33:27,010 --> 00:33:30,410
+احاول اكتب كل ال vectors اللي في الدنيا في دلالتهم
+
+438
+00:33:30,670 --> 00:33:38,770
+طيب الان عندي لو اجيت ل ال vector اللي هو OB1
+
+439
+00:33:38,770 --> 00:33:45,450
+شايفينه جماعي OB1 هذا ال vector اللي هو احداثيات
+
+440
+00:33:45,450 --> 00:33:53,200
+النقطة هذه نفترض انها X1 و Y1 و Z1 معاياأذا ال O B
+
+441
+00:33:53,200 --> 00:33:59,240
+1 X 1 مقطوع من مين الإحداث ال سيني تبعتها X 1 في
+
+442
+00:33:59,240 --> 00:34:06,580
+اتجاه مين ال I 1 اللي هو الإحداث الصادر Y 1 في
+
+443
+00:34:06,580 --> 00:34:11,700
+اتجاه ال J و Z 1 في اتجاه مين ال K فبصير هذا O B 1
+
+444
+00:34:11,700 --> 00:34:18,380
+اكتب و اكتب و X 1 I A Y A 1 J Z 1 Kماشي بعد شوية
+
+445
+00:34:18,380 --> 00:34:22,280
+هقولكم كيف بالظبط احنا كيف استدلنا ان نكتب ال
+
+446
+00:34:22,280 --> 00:34:26,680
+vector لما يكون ال position vector كيف نكتبه اللي
+
+447
+00:34:26,680 --> 00:34:29,660
+هو x واحد في اتجاه ال I و Y واحد في اتجاه ال J زي
+
+448
+00:34:29,660 --> 00:34:32,460
+ال Z واحد في اتجاه ال K بعد شوية ان شاء الله بس
+
+449
+00:34:32,460 --> 00:34:33,880
+خلينا نفترض انه بنعرفها الآن
+
+450
+00:34:44,000 --> 00:34:49,560
+Similarly OB2 هو عبارة
+
+451
+00:34:49,560 --> 00:34:54,340
+عن X2 Y2 و Z2 يعني بمعنى آخر بنقدر نكتب OB2 عبارة
+
+452
+00:34:54,340 --> 00:34:59,180
+عن X2 في اتجاه I و Y2 في اتجاه J و Z2 في اتجاه K
+
+453
+00:34:59,180 --> 00:35:03,800
+الان هذا ال vector B1 B2 اللي من هنا لهنا اللي هو
+
+454
+00:35:03,800 --> 00:35:08,360
+واضح انههيو جاي من عند هذا ورايح هذا زي ما عملنا
+
+455
+00:35:08,360 --> 00:35:13,600
+قبل بشوية عبارة عن b2 نقص b1 ال vector اللي هو ob2
+
+456
+00:35:13,600 --> 00:35:16,920
+هذا ناقص ال vector هذا زي ما عرفنا عملية الطرح قبل
+
+457
+00:35:16,920 --> 00:35:21,940
+بشوية فبصير هذا لإنه هذا زائد هذا بصير هذا لأمغ
+
+458
+00:35:21,940 --> 00:35:29,020
+القرن أو بمعنى آخر هذا عبارة عن x2 ناقص x1 في درجة
+
+459
+00:35:29,020 --> 00:35:33,720
+ال IY2-Y1 في اتجاه الـ J Y2-Y1 في اتجاه الـ J Z2
+
+460
+00:35:33,720 --> 00:35:37,660
+-Z1 في اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في
+
+461
+00:35:37,660 --> 00:35:38,840
+اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في اتجاه
+
+462
+00:35:38,840 --> 00:35:42,460
+الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2
+
+463
+00:35:42,460 --> 00:35:46,800
+-Z1 في اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في
+
+464
+00:35:46,800 --> 00:35:47,640
+اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في اتجاه
+
+465
+00:35:47,640 --> 00:35:49,640
+الـ K Z2-Z1 في اتجاه الـ K Z2-Z1 في اتجاه الـ K Z2
+
+466
+00:35:49,640 --> 00:35:53,880
+-Z1 في اتجاه
+
+467
+00:35:53,880 --> 00:35:57,270
+الـ K Z2-كيف كتبت اللي ob1 بساوية هذا الكلام؟
+
+468
+00:35:57,270 --> 00:36:02,550
+خلّينا نشوف الآن كيف نكتبه و اللي هي أي vector v
+
+469
+00:36:02,550 --> 00:36:06,270
+بساوية v1, v2, v3 can be written as a linear
+
+470
+00:36:06,270 --> 00:36:09,710
+combination of the standard unit vectors as
+
+471
+00:36:09,710 --> 00:36:17,100
+follows يعني الآن عند أي vector vاللي هو مكتوب في
+
+472
+00:36:17,100 --> 00:36:20,800
+ال component form اللي هو هيه بدنا نكتب هذا على
+
+473
+00:36:20,800 --> 00:36:25,340
+صورة I وJ وK سهل يا جماعة هتلاقوه سهل وممكن بعضكم
+
+474
+00:36:25,340 --> 00:36:28,810
+جال انا على طول بعرفخلّيني أخد ال V بيساوي V واحد
+
+475
+00:36:28,810 --> 00:36:33,370
+V اتنين و V تلاتة هذي الآن افردليها كالتالي و بدك
+
+476
+00:36:33,370 --> 00:36:37,250
+تبقى حاصل جمع تلت اللي هي vectors ال vector الأول
+
+477
+00:36:37,250 --> 00:36:42,110
+V واحد و هدول أصفر ال vector التاني Zero و V اتنين
+
+478
+00:36:42,110 --> 00:36:46,230
+و هذا Zero Zero Zero V تلاتة اجمحل مع بعض هتلاقي V
+
+479
+00:36:46,230 --> 00:36:50,320
+واحد زائد سفر زائد Zero V واحدزي رو زي ات زي ات زي
+
+480
+00:36:50,320 --> 00:36:53,100
+ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات
+
+481
+00:36:53,100 --> 00:36:54,000
+زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي
+
+482
+00:36:54,000 --> 00:36:54,260
+ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات
+
+483
+00:36:54,260 --> 00:36:55,380
+زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي
+
+484
+00:36:55,380 --> 00:36:59,340
+ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات
+
+485
+00:36:59,340 --> 00:37:03,440
+زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي ات زي
+
+486
+00:37:03,440 --> 00:37:07,860
+ات زي اتخد V1 برا V2 برا V3 برا V4 برا V5 برا V6
+
+487
+00:37:07,860 --> 00:37:10,420
+برا V7 برا V8 برا V9 برا V10 برا V11 برا V12 برا
+
+488
+00:37:10,420 --> 00:37:12,720
+V13 برا V14 برا V15 برا V16 برا V17 برا V18 برا
+
+489
+00:37:12,720 --> 00:37:20,320
+V19 برا V20 برا V21 برا V21 برا V22 برا V23 برا
+
+490
+00:37:20,320 --> 00:37:23,960
+V21 برا V23 برا V21 برا V21 برا V21 برا V21 برا
+
+491
+00:37:23,960 --> 00:37:23,980
+V21 برا V21 برا V21 برا V21 برا V21 برا V21 برا
+
+492
+00:37:23,980 --> 00:37:24,040
+V21 برا V21 برا V21 برا V21 برا V21 برا V21 برا
+
+493
+00:37:24,040 --> 00:37:24,060
+V21 برا V21 برا V21 برا V21 برا V21 برا V21 برا
+
+494
+00:37:24,060 --> 00:37:31,260
+V21 برا V21 برا V21 برا V21 برا V21 برا V21 برا
+
+495
+00:37:31,260 --> 00:37:42,620
+V21 برا V21قيمته بدلالة I و J و K هي V1 V2 V3 V4
+
+496
+00:37:42,620 --> 00:37:45,640
+V5 V6 V7 V8 V9 V10 V11 V12 V13 V12 V13 V13 V13 V13
+
+497
+00:37:45,640 --> 00:37:48,120
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+498
+00:37:48,120 --> 00:37:48,120
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+499
+00:37:48,120 --> 00:37:48,120
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+500
+00:37:48,120 --> 00:37:50,820
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+501
+00:37:50,820 --> 00:37:50,900
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+502
+00:37:50,900 --> 00:37:52,940
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+503
+00:37:52,940 --> 00:37:55,620
+V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13 V13
+
+504
+00:37:55,620 --> 00:37:59,340
+V13 V13 V13 V
+
+505
+00:38:04,140 --> 00:38:10,000
+إذا كان ال V عبارة عن non-zero vector اذا كان ال V
+
+506
+00:38:10,000 --> 00:38:10,120
+عبارة عن non-zero vector اذا كان ال V عبارة عن non
+
+507
+00:38:10,120 --> 00:38:10,120
+-zero vector اذا كان ال V عبارة عن non-zero vector
+
+508
+00:38:10,120 --> 00:38:10,540
+اذا كان ال V عبارة عن non-zero vector اذا كان ال V
+
+509
+00:38:10,540 --> 00:38:10,540
+عبارة عن non-zero vector اذا كان ال V عبارة عن non
+
+510
+00:38:10,540 --> 00:38:10,600
+-zero vector اذا كان ال V عبارة عن non-zero vector
+
+511
+00:38:10,600 --> 00:38:10,620
+اذا كان ال V عبارة عن non-zero vector اذا كان ال V
+
+512
+00:38:10,620 --> 00:38:10,620
+عبارة عن non-zero vector اذا كان ال V عبارة عن non
+
+513
+00:38:10,620 --> 00:38:11,140
+-zero vector اذا كان ال V عبارة عن non-zero vector
+
+514
+00:38:11,140 --> 00:38:13,160
+اذا كان ال V عبارة عن non-zero vector اذا كان ال V
+
+515
+00:38:13,160 --> 00:38:16,500
+عبارة عن non-zero vector اذا كان ال V عبارة عن non
+
+516
+00:38:16,500 --> 00:38:19,920
+-zero vector اذا
+
+517
+00:38:19,920 --> 00:38:26,030
+كان ال V عبارة عن non-zero vectorهو عبارة عن is a
+
+518
+00:38:26,030 --> 00:38:30,170
+unit vector in the direction of v او اختصارا
+
+519
+00:38:30,170 --> 00:38:35,070
+بنسميه the direction of the vector v اذا the
+
+520
+00:38:35,070 --> 00:38:37,970
+direction of the vector v عبارة عن unit vector
+
+521
+00:38:37,970 --> 00:38:44,840
+حصلة من قسمة ال V على magnitude و على طولهطيب الان
+
+522
+00:38:44,840 --> 00:38:48,240
+ال equation v بتساوي magnitude v فى v على
+
+523
+00:38:48,240 --> 00:38:53,260
+magnitude of v ايه ده بتروح محدد بيصير v اه اقصد
+
+524
+00:38:53,260 --> 00:38:57,480
+اكتبها هيك ليش اقصد اكتبها على صورة اللى هو طول ال
+
+525
+00:38:57,480 --> 00:39:01,860
+vector فى اتجاههإذا أي vector في الدنيا مقدر أكتب
+
+526
+00:39:01,860 --> 00:39:06,300
+على صورة طوله ضرب اتجاهه اتجاهه V على magnitude V
+
+527
+00:39:06,300 --> 00:39:14,080
+Expresses V as its length and its direction ناخد
+
+528
+00:39:14,080 --> 00:39:18,880
+أمثال عليها اللي أنا بقوله نيجي
+
+529
+00:39:18,880 --> 00:39:21,960
+نشوف المثال اللي عندنا Find a unit vector u in the
+
+530
+00:39:21,960 --> 00:39:26,560
+direction of the vector b1 to b2بقول لأوجد ال unit
+
+531
+00:39:26,560 --> 00:39:30,920
+vector u اللي هي main اللي في اتجاه ال vector b1
+
+532
+00:39:30,920 --> 00:39:34,740
+b2 يعني بدي اوجد ال vector b1 b2 و بعدين اوجد
+
+533
+00:39:34,740 --> 00:39:39,000
+اتجاهه بتكون هي ال u طيب شوف كيف b1 b2 تعلمنا كيف
+
+534
+00:39:39,000 --> 00:39:44,340
+نحوها اللي هو تلاتة ناقص واحد هي بتطلع في اتجاه
+
+535
+00:39:44,340 --> 00:39:47,400
+الاي سفر اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر
+
+536
+00:39:47,400 --> 00:39:48,000
+اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر
+
+537
+00:39:48,000 --> 00:39:48,000
+اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر اثر
+
+538
+00:39:48,000 --> 00:39:51,150
+اثر اثر اثر اثر اثر اثرفى اتجاه الـ J سفر ناقص
+
+539
+00:39:51,150 --> 00:39:56,030
+واحد فى اتجاه ال K هذا اللى هو B1 B2 المتجهة بيصير
+
+540
+00:39:56,030 --> 00:40:02,230
+عبارة عن 2I زائد 2J هذا مش زائد K ناقص K عشان اذا
+
+541
+00:40:02,230 --> 00:40:07,430
+هذا B1 B2 هو المتجه اللى عندى كتبته على صورة 2I
+
+542
+00:40:07,430 --> 00:40:14,140
+زائد 2J على صورة I زائد J زائد K and thenthe unit
+
+543
+00:40:14,140 --> 00:40:17,740
+vector in the direction of b1 b2 is يعني الان بد
+
+544
+00:40:17,740 --> 00:40:21,580
+الآن ال unit vector هو عبارة عن ال unit vector in
+
+545
+00:40:21,580 --> 00:40:25,520
+the direction of b1 b2 ايش هيكون يا جماعة هو عبارة
+
+546
+00:40:25,520 --> 00:40:29,800
+عن ال vector على magnitude و على طوله هاي ال
+
+547
+00:40:29,800 --> 00:40:34,380
+vector b2i زاد 2j نقص كيها دي يا جماعة على طوله
+
+548
+00:40:34,380 --> 00:40:38,460
+طوله الجدر التربية على مينلأ اتنين تربيع ليه اربعة
+
+549
+00:40:38,460 --> 00:40:41,920
+و اتنين تربيع اربعة هي تمانية و واحد تربيع واحد هي
+
+550
+00:40:41,920 --> 00:40:45,760
+تسعة اذا الجدر التربيعي لكل هذا طبعا الجدر لكله
+
+551
+00:40:45,760 --> 00:40:48,730
+هذا بيظل الجدر ماشيالجدر التربيه على التسعة أيش
+
+552
+00:40:48,730 --> 00:40:52,030
+بيساوي تلاتة إذا صار عندي ال direction اللي بدويا
+
+553
+00:40:52,030 --> 00:40:56,870
+اللي هو ال U in the direction of B1 B2 هو عبارة عن
+
+554
+00:40:56,870 --> 00:41:02,450
+2I زاد 2J ناقص K على تلاتة طيب صلوا على النبي عليه
+
+555
+00:41:02,450 --> 00:41:05,850
+الصلاة والسلام ناخد مثل آخر تطبيق يقول express the
+
+556
+00:41:05,850 --> 00:41:10,800
+velocity vector Vالان اعبّرلي عن اللي هو الـ
+
+557
+00:41:10,800 --> 00:41:15,820
+Velocity vector V اللي هو 3I-4J على صورة as a
+
+558
+00:41:15,820 --> 00:41:20,060
+product of its speed times a unit vector in the
+
+559
+00:41:20,060 --> 00:41:22,660
+direction of motion يعني بيقوللي اعبّرلي عن ال
+
+560
+00:41:22,660 --> 00:41:26,100
+vector V ال velocity vector على صورة اللي هو ال
+
+561
+00:41:26,100 --> 00:41:30,980
+speed تبعته مضروبة في مين؟ في ال unit vector في ال
+
+562
+00:41:30,980 --> 00:41:33,460
+direction لل motion direction لل motion هو ال
+
+563
+00:41:33,460 --> 00:41:37,090
+direction لل velocityهذا الـ Vector هو اوجد اول
+
+564
+00:41:37,090 --> 00:41:41,270
+شيء الـ Direction للحركة و اوجد السبيد السبيد هو
+
+565
+00:41:41,270 --> 00:41:42,610
+ما هو المجموعه المجموعه المجموعه المجموعه المجموعه
+
+566
+00:41:42,610 --> 00:41:46,290
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+567
+00:41:46,290 --> 00:41:46,950
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+568
+00:41:46,950 --> 00:41:48,130
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+569
+00:41:48,130 --> 00:41:49,310
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+570
+00:41:49,310 --> 00:41:49,370
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+571
+00:41:49,370 --> 00:41:51,530
+المجموعه المجموعه المجموعه المجموعه المجموعه
+
+572
+00:41:51,530 --> 00:42:00,810
+المجموعه المجموعه المج
+
+573
+00:42:02,150 --> 00:42:05,210
+لأن الـ Velocity اللي هو طبعا هي اللي بتمثل اتجاه
+
+574
+00:42:05,210 --> 00:42:10,310
+الحركة The direction of V is V على magnitude V V
+
+575
+00:42:10,310 --> 00:42:14,550
+تلتة I نقص أربعة D على خمسة إذن Hence V تبعتنا هذه
+
+576
+00:42:14,550 --> 00:42:18,130
+كتبناها على سكورة The Speed خمسة مضروبة في ال
+
+577
+00:42:18,130 --> 00:42:24,340
+direction تلتة I نقص أربعة D على خمسةهنا ناخد مثال
+
+578
+00:42:24,340 --> 00:42:29,520
+اخر يقول ان قوة
+
+579
+00:42:29,520 --> 00:42:35,380
+6 نيوتن قيمة القوة جديش 6 نيوتن is applied in the
+
+580
+00:42:35,380 --> 00:42:36,660
+direction of the vector
+
+581
+00:42:41,300 --> 00:42:48,500
+التي تعمل في اتجاه الـ vector هذا الأمامها بيقول
+
+582
+00:42:48,500 --> 00:42:53,140
+express the force F as a product of its magnitude
+
+583
+00:42:53,140 --> 00:42:55,440
+and direction مجموعة معروفة
+
+584
+00:42:58,830 --> 00:43:04,570
+الدركشن للفورس هو الدركشن لهذا الـ V فالدركشن
+
+585
+00:43:04,570 --> 00:43:08,650
+للفورس هو الدركشن لهذا الـ V فالدركشن لهذا الـ V
+
+586
+00:43:08,650 --> 00:43:13,570
+هو V ع المجموعه للـ V و سوى 2I زي 2D نقص K على جدر
+
+587
+00:43:13,570 --> 00:43:14,050
+التربيع
+
+588
+00:43:17,600 --> 00:43:21,500
+بتطلع السبسيرت تلاتة اذا اندي صار الان هي ال
+
+589
+00:43:21,500 --> 00:43:25,560
+direction لل force اذا ال force of بتساوي الستة
+
+590
+00:43:25,560 --> 00:43:30,420
+اللي هو ال magnitude لل force اللي معطينا إياها في
+
+591
+00:43:30,420 --> 00:43:35,200
+ال direction اللي هو بالي ال direction ال V اللي
+
+592
+00:43:35,200 --> 00:43:38,420
+عندي أمامي فصارت مكتوبة ال force على صورة
+
+593
+00:43:38,420 --> 00:43:39,980
+magnitude في direction
+
+594
+00:43:43,500 --> 00:43:48,240
+نجي لآخر نقطة اللي هي شغلة بسيطة وسريعة الان بدنا
+
+595
+00:43:48,240 --> 00:43:51,180
+نعرف شو معناه the midpoint M of the line segment
+
+596
+00:43:51,180 --> 00:43:56,060
+joining B1 وB2 يعني موجود عند B1 وB2 نقطين في
+
+597
+00:43:56,060 --> 00:44:00,180
+الفراغ وفي خط واصل B2 ماحكيش عن متجهة الان بحكي عن
+
+598
+00:44:00,180 --> 00:44:03,980
+خط واصل line segment بين B1 وB2 بده النقطة اللي في
+
+599
+00:44:03,980 --> 00:44:07,580
+منتصفه بالظبط النقطة اللي في منتصفه اللي هو B1 وB2
+
+600
+00:44:07,580 --> 00:44:15,340
+هي منها لها نصه إحداثياته هي كتابةX1 زائد X2 هي X1
+
+601
+00:44:15,340 --> 00:44:21,860
+زائد X2 على 2 Y1 زائد Y2 على 2 Z1 زائد Z2 على 2
+
+602
+00:44:21,860 --> 00:44:26,380
+هذا قانون ما نريده ان نشوفه هو القانون اللي موجود
+
+603
+00:44:26,380 --> 00:44:31,380
+عندنا اللي هي ال midpoint بين B1 وB2 ناخد مثال
+
+604
+00:44:31,380 --> 00:44:36,060
+عددي example 7 the midpoint of the line segment
+
+605
+00:44:36,060 --> 00:44:42,180
+joining B1 and B2 is ايش هي ال midpoint احداثياتها
+
+606
+00:44:42,500 --> 00:44:48,240
+تلاتة وسبعة عشرة ع تنين يعني تطلع خمسة نقص اتنين
+
+607
+00:44:48,240 --> 00:44:53,980
+واربعة اتنين على اتنين تطلع واحد سفر واربعة على
+
+608
+00:44:53,980 --> 00:44:58,380
+اتنين تطلع اتنين هذه اللي هي ال midpoint اللي
+
+609
+00:44:58,380 --> 00:45:03,560
+مطلوبةالان هيك بيكون احنا يعني خلصنا المحاضرة
+
+610
+00:45:03,560 --> 00:45:07,880
+الثانية و هاي ال exercises بتحلوها و اللي انتوا
+
+611
+00:45:07,880 --> 00:45:11,740
+يعني حاسين مش عارفين تحلوها او كده ممكن عبر
+
+612
+00:45:11,740 --> 00:45:13,980
+التواصل معنا على ال model او على ال whatsapp اللي
+
+613
+00:45:13,980 --> 00:45:18,160
+هنقوم عها ان شاء الله نحاول .. هنكوّنها معاكم ان
+
+614
+00:45:18,160 --> 00:45:20,580
+شاء الله تبعتوا الأسئلة اللي بتكونوا معاهاهنا
+
+615
+00:45:20,580 --> 00:45:23,700
+تواصل ان شاء الله يا اما بنصورها يا اما بنحلها
+
+616
+00:45:23,700 --> 00:45:28,280
+بطريقة اللى بنلتقيها بنلاقيها مناسبة مع بعض
+
+617
+00:45:28,280 --> 00:45:30,820
+والسلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/B5_wN_rcAT0.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/B5_wN_rcAT0.srt
new file mode 100644
index 0000000000000000000000000000000000000000..eb8a4a71ca44a77b2f8a2a8e006f496b95c334df
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/B5_wN_rcAT0.srt
@@ -0,0 +1,1999 @@
+1
+00:00:01,080 --> 00:00:05,600
+بسم الله الرحمن الرحيم هذه هي المحاضرة الأولى
+
+2
+00:00:05,600 --> 00:00:10,080
+لمساق تفاضل وتكامل جيم لطلاب وطالبات الجامعة
+
+3
+00:00:10,080 --> 00:00:15,020
+الإسلامية الفصل
+
+4
+00:00:15,020 --> 00:00:25,500
+الصيفي 2020 الذي اعتمد فيه التعليم الإلكتروني وهي
+
+5
+00:00:25,500 --> 00:00:32,560
+المحاضرة الأولى التي سنبدأ بها بإذن الله الحديث
+
+6
+00:00:32,560 --> 00:00:39,300
+بدايةً على... على الـ course outline لهذا الفصل
+
+7
+00:00:39,300 --> 00:00:47,570
+فأهلًا وسهلًا بجميع الطلاب والطالبات في هذا الفصل
+
+8
+00:00:47,570 --> 00:00:52,870
+الـ course outline هو كالعادة كما هو في الفصول الماضية
+
+9
+00:00:52,870 --> 00:00:56,210
+من نفس الكتاب اللي هو textbook اللي هو calculus
+
+10
+00:00:56,210 --> 00:01:02,870
+Thomas اللي هي الطبعة 12 اللي هي اللي احنا بنكمل
+
+11
+00:01:02,870 --> 00:01:08,250
+فيها اللي أخذنا في تفاضل ألف وتفاضل باء الفصول
+
+12
+00:01:08,250 --> 00:01:15,570
+الماضية، الشباتر المطلوبة اللي هو هنبدأ بشابتر 12 ثم
+
+13
+00:01:15,570 --> 00:01:23,170
+شابتر 13 ثم 14 ثم 15. شابتر 12 اللي هو عبارة عن اللي
+
+14
+00:01:23,170 --> 00:01:27,290
+هو هنتحدث فيه على three-dimensional coordinate
+
+15
+00:01:27,290 --> 00:01:32,940
+systems. أحنا طبعًا تحدثنا في تفاضل ألف وتفاضل باء
+
+16
+00:01:32,940 --> 00:01:37,780
+في الـ two dimensions اللي هو في لبنان كنا أخذنا كل
+
+17
+00:01:37,780 --> 00:01:44,580
+متعلقات التفاضل والتكامل في اللي هي لبنان أو الـ
+
+18
+00:01:44,580 --> 00:01:50,160
+two dimensions. الآن هنبدأ الحديث على اللي هو في
+
+19
+00:01:50,160 --> 00:01:54,360
+تفاضل جيم على الـ three-dimensional coordinate
+
+20
+00:01:54,360 --> 00:01:58,820
+systems، ونبني على هذا الكلام هنحكي برضه عن الـ
+
+21
+00:01:58,820 --> 00:02:05,580
+vectors and dot والـ cross product، وهنيجي إن شاء
+
+22
+00:02:05,580 --> 00:02:10,060
+الله بالتفصيل لكل هذه الأمور. lines and planes in
+
+23
+00:02:10,060 --> 00:02:14,580
+space اللي هي معادلة الخط المستقيم طبعًا أخذناها في
+
+24
+00:02:14,580 --> 00:02:14,820
+الـ
+
+25
+00:02:17,640 --> 00:02:21,020
+معادلة الخط المستقيم في الـ space، في الفراغ
+
+26
+00:02:21,020 --> 00:02:24,880
+ومعادلة المستوى في الفراغ وإن شاء الله هنشوف
+
+27
+00:02:24,880 --> 00:02:30,820
+الكلام هذا، وإن شاء الله بكون كلام سهل. الآن أيضًا الـ
+
+28
+00:02:30,820 --> 00:02:35,580
+section اللي هو الأخير في الـ chapter 12 هو
+
+29
+00:02:35,580 --> 00:02:40,910
+cylinders and quadratic surfaces، اللي هي حاجة اسمها
+
+30
+00:02:40,910 --> 00:02:45,530
+تعميم للسيلندر العادية واللي هي الـ quadratic
+
+31
+00:02:45,530 --> 00:02:51,430
+surfaces أو السطوح الناجمة عن اللي هي المعادلات
+
+32
+00:02:51,430 --> 00:02:55,750
+التربيعية، وحنشوف هذا الكلام أيضًا بالتفصيل. بعد هيك
+
+33
+00:02:55,750 --> 00:02:59,350
+هنبدأ في شابتر 13 اللي هو curves in space and their
+
+34
+00:02:59,350 --> 00:03:03,630
+tangent. الـ curves in space and their tangent اللي
+
+35
+00:03:03,630 --> 00:03:08,010
+هو هنتحدث عنها إن شاء الله، وهنتحدث عن integrals
+
+36
+00:03:08,010 --> 00:03:13,410
+of vector functions اللي هي الدوال اللي... اللي هي
+
+37
+00:03:13,410 --> 00:03:20,850
+قيمتها عبارة عن vectors. هنتحدث عنها أيضًا، هنتحدث عن
+
+38
+00:03:20,850 --> 00:03:25,870
+الـ arc length in space. بعد هيك هنكون احنا وصلنا
+
+39
+00:03:25,870 --> 00:03:29,770
+لمرحلة اللي هو حاجة اسمها الـ first exam. طبعًا زي ما
+
+40
+00:03:29,770 --> 00:03:33,550
+انتوا عارفين اللي هو الفصل الصيفي هذا هيكون عندي
+
+41
+00:03:33,550 --> 00:03:38,770
+اللي هو الـ first exam والـ second exam والـ final
+
+42
+00:03:38,770 --> 00:03:44,350
+exam، كلهم هيكون عبارة عن امتحانات اللي هو عبر الـ
+
+43
+00:03:44,350 --> 00:03:49,190
+model، وهتكون إلكترونية أو في البيت إن شاء الله.
+
+44
+00:03:49,190 --> 00:03:53,550
+الآن الـ chapter اللي... اللي... اللي هنكمل فيه
+
+45
+00:03:53,550 --> 00:03:56,970
+chapter 13، هنحكي عن الـ curve vectors and normal
+
+46
+00:03:56,970 --> 00:04:02,330
+vectors of curve. هنحكي عن الـ tangential and normal
+
+47
+00:04:02,330 --> 00:04:07,310
+component of acceleration، وبنكون هيك خلصنا شابتر 13.
+
+48
+00:04:07,310 --> 00:04:11,990
+بنبدأ في شابتر 14 اللي هو functions of several
+
+49
+00:04:11,990 --> 00:04:16,710
+variables، اللي هو الدوال فيه عدد من المجاهيل. طبعًا
+
+50
+00:04:16,710 --> 00:04:19,630
+احنا في الـ... الـ two dimensions وفي calculus ألف
+
+51
+00:04:19,630 --> 00:04:22,370
+و calculus باء حكينا عن function of one variable
+
+52
+00:04:22,370 --> 00:04:26,030
+اللي هنحكي عن functions of two variables، three
+
+53
+00:04:26,030 --> 00:04:31,660
+variables and so on. هنحكي ما يتعلق بها من الـ limits
+
+54
+00:04:31,660 --> 00:04:34,920
+والـ continuity in higher dimensions. نفس اللي هو
+
+55
+00:04:34,920 --> 00:04:40,540
+الأسلوب اللي كان في تفاضل ألف وباء، ولكن بما يتعلق
+
+56
+00:04:40,540 --> 00:04:44,820
+بالدوال الجديدة. الآن اللي هنحكينا برضه في تفاضل ألف
+
+57
+00:04:44,820 --> 00:04:48,060
+وتفاضل باء عن اللي هو الـ derivatives. الآن هنحكي عن
+
+58
+00:04:48,060 --> 00:04:51,340
+الـ partial derivatives، لأنه صار في عندي أكثر من
+
+59
+00:04:51,340 --> 00:04:55,260
+مجهول، فهيصير عندي اللي هو حاجة اسمها التفاضل
+
+60
+00:04:55,260 --> 00:05:00,770
+الجزئي. هناخد برضه الـ chain rule في هذه الحالات
+
+61
+00:05:00,770 --> 00:05:04,670
+زي ما أخذنا chain rule سابقًا في الحالة اللي هي الـ
+
+62
+00:05:04,670 --> 00:05:08,050
+function of one variable. بعدين هنحكي عن حاجة اسمها
+
+63
+00:05:08,050 --> 00:05:11,490
+directional derivatives and gradient vectors، و
+
+64
+00:05:11,490 --> 00:05:15,170
+هنجيل كل هذا الكلام بالتفصيل إن شاء الله. tangent
+
+65
+00:05:15,170 --> 00:05:18,210
+planes and differentials أيضًا اللي هو زي ما كان
+
+66
+00:05:18,210 --> 00:05:24,100
+فيه مماسات في الحالة الـ curves العادية. في عندي اللي
+
+67
+00:05:24,100 --> 00:05:30,300
+هي مستويات تمثل اللي هي الأجسام اللي عندنا أو
+
+68
+00:05:30,300 --> 00:05:34,540
+الرسمات اللي عندنا، الرسمات اللي عندنا أيًّا كانت، و
+
+69
+00:05:34,540 --> 00:05:37,920
+هنيجي نحكي عن شو اللي هو معادلة الـ tangent planes
+
+70
+00:05:37,920 --> 00:05:40,400
+وكيف نوجدها، وهنحكي عن حاجة اسمها برضه
+
+71
+00:05:40,400 --> 00:05:44,300
+differentials، وبرضه ممكن نسمع عنها بس في حالة
+
+72
+00:05:44,300 --> 00:05:46,800
+اللي هو الـ two dimensions أو function of one
+
+73
+00:05:46,800 --> 00:05:50,760
+variable. الآن هنحكي أيضًا عن extreme values and
+
+74
+00:05:50,760 --> 00:05:54,400
+saddle points، وبرضه حكينا عن مقابلها في اللي هو
+
+75
+00:05:54,400 --> 00:05:59,100
+تفاضل باء أو تفاضل ألف، اللي هو اللي هي النقاط العظمى
+
+76
+00:05:59,100 --> 00:06:03,220
+والنقاط الصغرى، وقولها نحدد اسمها الـ saddle point، و
+
+77
+00:06:03,220 --> 00:06:08,010
+هنيجي إليها إن شاء الله لاحقًا. عند هنا في نهاية
+
+78
+00:06:08,010 --> 00:06:12,070
+شابتر هذا، شابتر طويل. 14 بيجي عندنا اللي هو second
+
+79
+00:06:12,070 --> 00:06:14,910
+exam أو الناس اللي بيقول second exam، بيقول second
+
+80
+00:06:14,910 --> 00:06:19,290
+quiz اللي هو برضه هيكون زي ما اتفقنا على اللي هو
+
+81
+00:06:19,290 --> 00:06:23,700
+الـ model. هذا اللي بحكي فيه، بحكي في الـ course
+
+82
+00:06:23,700 --> 00:06:30,140
+outline لسه لهذا المساق. نجي إلى اللي هو الجزء
+
+83
+00:06:30,140 --> 00:06:32,800
+الثالث من المادة اللي هو عبارة عن الـ chapter
+
+84
+00:06:32,800 --> 00:06:36,900
+الأخير اللي هو chapter 15 اللي هو ما يتعلق بالـ
+
+85
+00:06:36,900 --> 00:06:40,360
+integration. هنجي على نحكي عن الـ double and
+
+86
+00:06:40,360 --> 00:06:44,080
+iterated integrals over rectangles اللي هي الـ
+
+87
+00:06:44,080 --> 00:06:47,080
+double integral اللي هو الـ integration اللي هي two
+
+88
+00:06:47,080 --> 00:06:53,000
+integrations. اللي هنا نشوف إيش معناه integral
+
+89
+00:06:53,000 --> 00:06:57,220
+integral، وبعدين iterated integrals، وبعدين اللي
+
+90
+00:06:57,220 --> 00:07:02,080
+هو هنبدأ نحكي عن الـ double integrals over الآن
+
+91
+00:07:02,080 --> 00:07:06,780
+general region. هتلاقوا نفس المنهجية اللي تحدثنا
+
+92
+00:07:06,780 --> 00:07:11,630
+فيها في تفاضل ألف عن الـ integration. هنتحدث فيها في
+
+93
+00:07:11,630 --> 00:07:16,210
+اللي هو الـ integration أو اللي هو integration of
+
+94
+00:07:16,210 --> 00:07:18,910
+function of two variables أو three variables حسب
+
+95
+00:07:18,910 --> 00:07:21,910
+اللي بدنا إياه، وهيكون عندي double integral أو triple
+
+96
+00:07:21,910 --> 00:07:25,370
+integral حسب عدد الـ variables. كله هنجيب وهتفهم
+
+97
+00:07:25,370 --> 00:07:29,570
+عليّ إيش اللي بقوله. area by double integration
+
+98
+00:07:29,570 --> 00:07:33,780
+هنجيب الـ area اللي هي تحت منحنى by double
+
+99
+00:07:33,780 --> 00:07:36,880
+integration. كنا نجيبها عادي بالقبل الـ integration
+
+100
+00:07:36,880 --> 00:07:39,520
+العادي. الآن هنجيبها by double integration، وهنشوف
+
+101
+00:07:39,520 --> 00:07:42,860
+كيف، وهنجي الكلام سهل لا تقلقوا. double
+
+102
+00:07:42,860 --> 00:07:46,380
+integrations in polar forms. هنستخدم الـ double
+
+103
+00:07:46,380 --> 00:07:50,280
+integrations وسط البولر بتحل لنا بعض المشاكل، بعض
+
+104
+00:07:50,280 --> 00:07:54,280
+الدوال بتكون صعبة التكامل بالعادي، بنحولها للبولر
+
+105
+00:07:54,280 --> 00:07:57,880
+ولا تقلقوا برضه في التحويل للبولر إن شاء الله كله
+
+106
+00:07:57,880 --> 00:08:01,460
+مُجَدَّر عليه. لأن triple integrals in rectangular
+
+107
+00:08:01,460 --> 00:08:05,080
+coordinates. هيصير عندنا بدل ما هو double integral
+
+108
+00:08:05,080 --> 00:08:09,660
+integration اتنين، واحد واحد. هنشوف إيش بقصد فيه. بيصير
+
+109
+00:08:09,660 --> 00:08:13,460
+عندي ثلاثة integration، إيش شو معناه، وإيش كيف بدنا
+
+110
+00:08:13,460 --> 00:08:16,780
+نشتغل عليه. ده بيبقى triple integral in rectangular
+
+111
+00:08:16,780 --> 00:08:23,000
+coordinates. الجزء اللي قبل الأخير، triple integral
+
+112
+00:08:23,000 --> 00:08:27,140
+in cylindrical and spherical يعني حق نيجي اللي هو
+
+113
+00:08:27,140 --> 00:08:32,160
+باستخدام cylindrical و spherical coordinates، بتسهل
+
+114
+00:08:32,160 --> 00:08:36,180
+علينا أحيانًا بعض الـ integrations بواسطة الـ
+
+115
+00:08:36,180 --> 00:08:39,480
+rectangular coordinates، وحنا عارفين إيش معناه اللي
+
+116
+00:08:39,480 --> 00:08:43,060
+هو بتكون أسهل لنا في استخدام cylindrical و
+
+117
+00:08:43,060 --> 00:08:46,610
+spherical. وبعدين هنحكي عن الـ substitution in
+
+118
+00:08:46,610 --> 00:08:51,330
+multiple integrals، ومنكون هيك احنا أنهينا اللي هو
+
+119
+00:08:51,330 --> 00:08:56,430
+الـ course مع اللي هو... اللي هو chapter 15، و
+
+120
+00:08:56,430 --> 00:09:00,230
+هيجينا الـ final exam. الـ final exam هيكون في كل
+
+121
+00:09:00,230 --> 00:09:08,270
+المادة، وهيكون اللي هو الـ final exam برضه إلكتروني
+
+122
+00:09:08,270 --> 00:09:12,160
+وكله عبر الـ model. يعني التقييمات اللي عندنا في
+
+123
+00:09:12,160 --> 00:09:15,140
+الـ course، عشان نكون واضحين، هيكون اللي هو عندي تلاتة
+
+124
+00:09:15,140 --> 00:09:19,580
+شغلات. الـ course، الـ quiz الأول أو الـ first exam، الـ quiz
+
+125
+00:09:19,580 --> 00:09:23,320
+الثاني اللي هو الـ second exam، والـ quiz الـ final exam.
+
+126
+00:09:23,320 --> 00:09:31,900
+الآن ال... ال... هيكون عند التقييم كما يلي: التقييم
+
+127
+00:09:31,900 --> 00:09:37,560
+هيكون عندي اللي هو ال... الـ term exams هيكون عندي
+
+128
+00:09:37,560 --> 00:09:42,590
+أربعين في المائة. يعني الأول اللي هو عشرين والثاني
+
+129
+00:09:42,590 --> 00:09:46,550
+عشرين، أو حسب ما نتفق يعني إن شاء الله مبدئيًا
+
+130
+00:09:46,550 --> 00:09:50,170
+أربعين في المائة اللي هو أعطيهم الـ term exams اللي
+
+131
+00:09:50,170 --> 00:09:54,330
+هو كل واحد عشرين في المائة، وممكن نحكي في غير هيك إن
+
+132
+00:09:54,330 --> 00:10:00,720
+شاء الله. النهائي هيكون ستين في المائة. الآن يا جماعة
+
+133
+00:10:00,720 --> 00:10:06,640
+الطالب طبعًا حسب قرار الجامعة مخيّر، بده pass يعني بده
+
+134
+00:10:06,640 --> 00:10:10,980
+pass، pass بدهوش علامة pass أو fail، بده pass أو fail
+
+135
+00:10:10,980 --> 00:10:17,380
+بدخل اللي هو الـ first والثاني والـ final كلهم
+
+136
+00:10:17,380 --> 00:10:21,240
+إلكتروني وفي البيت، مش مشكلة في البيت أو وين ما بده
+
+137
+00:10:21,670 --> 00:10:25,710
+لأن لو الطالب بده اللي هو علامة في الصيف برضه
+
+138
+00:10:25,710 --> 00:10:29,870
+هنعطيه علامة، كيف هنعطيه علامة؟ بدخل طبعًا عادي اللي
+
+139
+00:10:29,870 --> 00:10:33,750
+هو الـ first و second عادي، بدخل في البيت اللي هو
+
+140
+00:10:33,750 --> 00:10:38,090
+بياخد الامتحان أو الـ quiz في البيت على الـ model
+
+141
+00:10:38,090 --> 00:10:45,890
+لكن الـ final اللي هو بيكون في الجامعة أو عبر الـ
+
+142
+00:10:45,890 --> 00:10:49,580
+model أو عبر اللي هو الوجهيّة، هي حسب ما نتفق مش مشكلة
+
+143
+00:10:49,580 --> 00:10:53,260
+بس إنهم بيكونوا في الـ final في الجامعة طبعًا، وكيف
+
+144
+00:10:53,260 --> 00:10:58,500
+بيحصل على هذه العلامة؟ بياخد غير مكتمل قبل ما نعمل
+
+145
+00:10:58,500 --> 00:11:02,490
+اللي هو لباقي الطلاب اللي بدهم الـ pass. يعني الآن
+
+146
+00:11:02,490 --> 00:11:07,050
+خيار العلامة، تاخد غير مكتمل بتكون مقدّم الـ first
+
+147
+00:11:07,050 --> 00:11:12,090
+ومقدّم الـ second، ومحسب لك اللي هو من الـ 40% اللي
+
+148
+00:11:12,090 --> 00:11:18,050
+بيظل 60% الـ final، بتقدمه في داخل الجامعة بالـ model
+
+149
+00:11:18,050 --> 00:11:23,330
+أو بال... اللي هي الوجهيّة، هي حسب ما نتفق، وغالباً
+
+150
+00:11:23,330 --> 00:11:27,650
+احنا هنعمل إن شاء الله بالـ model، لو حتى كان في
+
+151
+00:11:27,650 --> 00:11:32,570
+عندي اللي هو علامة، لكن في الجامعة. هذا اللي هو الـ
+
+152
+00:11:32,570 --> 00:11:39,890
+course outline بالنسبة للمساق اللي عندنا. نيجي الآن
+
+153
+00:11:39,890 --> 00:11:44,810
+نبدأ في اللي هو chapter 12. chapter 12 اللي هو
+
+154
+00:11:44,810 --> 00:11:50,050
+بالضبط اسمه vectors and the geometry of space، التي
+
+155
+00:11:50,050 --> 00:11:54,190
+هي المتجهات والهندسة اللي هي طبعًا المقصودة عن
+
+156
+00:11:54,190 --> 00:11:57,370
+الهندسة التحليلية أو ما يقابل الهندسة التحليلية في
+
+157
+00:11:57,370 --> 00:12:00,930
+الفراغ. يعني الآن بدل ما نحكي عن اللي هي اللي
+
+158
+00:12:00,930 --> 0:12:05,530
+حكيناه في calculus ألف وcalculus باء، كنا نحكي عن
+
+159
+00:12:05,530 --> 00:12:10,420
+الـ geometry in a plane، الآن بنحكي عن Geometry in
+
+160
+00:12:10,420 --> 00:12:13,940
+Space. ما حدش يروح في كل هندسة فراغية، لأ لأ احنا مش
+
+161
+00:12:13,940 --> 00:12:16,880
+هنحكي عن الهندسة الإقليدية الفراغية، لأ احنا هنحكي
+
+162
+00:12:16,880 --> 00:12:21,120
+عن ما يتعلق بأمور الدوال والتفاضل والتكامل والآخر
+
+163
+00:12:21,120 --> 00:12:25,100
+يعني حاجة اللي كنا بنسميها أو تسمى بالهندسة
+
+164
+00:12:25,100 --> 00:12:31,050
+التحليلية. نجي الآن لـ 12-1، الـ section الأول 12-
+
+165
+00:12:31,050 --> 00:12:35,010
+واحد اللي هو عبارة عن اسمه three-dimensional
+
+166
+00:12:35,010 --> 00:12:39,250
+coordinate system. لأن three-dimensional coordinate
+
+167
+00:12:39,250 --> 00:12:43,970
+system اللي هو خلينا نتطلع نشوف بالضبط إيش اللي
+
+168
+00:12:43,970 --> 00:12:49,330
+كنا احنا كنا هنا إيش اللي مقصود فيه، لأن زمان كنا
+
+169
+00:12:49,330 --> 00:12:53,290
+نحكي عن اللي هو لبنان، إذا في كان عندي X وY لأن كنا
+
+170
+00:12:53,290 --> 00:12:58,490
+على العرض نشتغل. الآن هنشتغل في الفراغ، فطبيعي هيكون
+
+171
+00:12:58,490 --> 00:13:04,250
+عندي X وY وفيه coordinate ثالث، إيه إسمه؟ اسمه Z
+
+172
+00:13:04,250 --> 00:13:11,600
+Axis. بالضبط هي عندي هي X وهي Y وهي Z. الآن كيف بدنا
+
+173
+00:13:11,600 --> 00:13:16,160
+نحدد النقاط اللي هي في الفراغ؟ قبلها طبعًا الـ
+
+174
+00:13:16,160 --> 00:13:19,240
+coordinates. الـ coordinates معناته في positive
+
+175
+00:13:19,240 --> 00:13:23,480
+coordinate، وعلى الجهة الثانية في الـ negative اللي هي
+
+176
+00:13:23,480 --> 00:13:28,220
+الـ x axis. الآن هي الـ
+
+223
+00:17:32,790 --> 00:17:37,090
+جعدنا في الـ X Z بلين سيرنا وين جاعدين في الـ X Z
+
+224
+00:17:37,090 --> 00:17:42,410
+بلين يعني بيصير الـ X متغيرة والـ Z متغيرة لكن فش Y
+
+225
+00:17:42,410 --> 00:17:47,490
+فكادنا جاعدين وين هنا طيب الآن لو اجينا وقلنا
+
+226
+00:17:47,490 --> 00:17:55,300
+والله خد لأ انا بدي اخد الثابت Z، الآن اخدت ثابت z
+
+227
+00:17:55,300 --> 00:18:00,380
+هي y و شايفين يا جماعة هي ثابت z، صارت الـ y هي
+
+228
+00:18:00,380 --> 00:18:05,160
+بتاخد حريتها و الـ x بتاخد حريتها فبتعملها plane
+
+229
+00:18:05,160 --> 00:18:10,200
+هذا بنسميه z بالثاوض ثابت هيكون موازلة الـ x y
+
+230
+00:18:10,200 --> 00:18:16,300
+plane الأرض هذه، اذا z ثابت عبارة عن مستوى موازية الـ
+
+231
+00:18:16,300 --> 00:18:22,180
+x y plane، Y ثابت معناته مستوى موازي للـ X وZ
+
+232
+00:18:22,180 --> 00:18:28,280
+بلين اللي هي X ثابت معناته مستوى موازي للـ Y Z
+
+233
+00:18:28,280 --> 00:18:32,660
+بلين و هكذا، طب لو أخدنا Z بصفر يعني لو قلنا Z
+
+234
+00:18:32,660 --> 00:18:37,730
+بساوا صفر، هذه تعني انه صار عند كلنا على الأرضية تبع
+
+235
+00:18:37,730 --> 00:18:42,130
+الزمان اللي هو الـ X Y plane تبع الزمان، إذا Z
+
+236
+00:18:42,130 --> 00:18:45,790
+بساوا صفر هو عبارة عن X Y plane اللي هو Y
+
+237
+00:18:45,790 --> 00:18:49,770
+بساوا صفر عبارة عن X Z plane، X بالساوية صفر
+
+238
+00:18:49,770 --> 00:18:56,240
+عبارة عن Z Y plane و هكذا، طب لو الآن اللي هو عندي
+
+239
+00:18:56,240 --> 00:19:01,920
+النقطة x و y و 0، بنصير هذا قيمة x وهذا قيمة y
+
+240
+00:19:01,920 --> 00:19:04,520
+ومافي ز، z على الأرض احنا قاعدين على الأرض فبصير
+
+241
+00:19:04,520 --> 00:19:09,000
+صفر و هكذا، طب لو كانت الـ .. الـ .. الـ y صفر بنصير
+
+242
+00:19:09,000 --> 00:19:13,060
+قاعدين في وين؟ في الـ xz اللي عندنا نقطة x و 0 و z
+
+243
+00:19:13,060 --> 00:19:18,780
+هيها، و هكذا تستطيع تكتب كل النقاط اللي بدك هيها
+
+244
+00:19:18,780 --> 00:19:22,500
+وتحصلها على اللي هو في الـ three dimensional space
+
+245
+00:19:22,500 --> 00:19:31,240
+تصل أي نقطة في الـ space، نأتي الآن لبعض المعادلات
+
+246
+00:19:31,240 --> 00:19:37,600
+اللي ناخدها عنفلة بمعنى آخر هاي الـ space هاي الـ
+
+247
+00:19:37,600 --> 00:19:40,240
+coordinates، هدا الـ x coordinate هاي المنطقة
+
+248
+00:19:40,240 --> 00:19:44,460
+الموجبة وهي المنطقة السالبة وهذا الـ y coordinate
+
+249
+00:19:44,460 --> 00:19:49,540
+وهي الـ y المنطقة الموجبة وهي السالبة وهي الـ z
+
+250
+00:19:49,540 --> 00:19:53,360
+coordinate الموجبة وهي الـ z coordinate اللي هي
+
+251
+00:19:53,360 --> 00:19:58,860
+axis اللي هي السالبة، ماشي يا جماعة طيب الآن زي ما
+
+252
+00:19:58,860 --> 00:20:03,000
+قلنا قبل شوية هذا النقطة الآن الـ origin اللي
+
+253
+00:20:03,000 --> 00:20:07,940
+عبارة عن صفر صفر صفر، قبل شوية قلنا X بتساوي صفر
+
+254
+00:20:07,940 --> 00:20:13,540
+إيش هتكون؟ عبارة عن الـ X الـ Y Z بلين، قبل شوية
+
+255
+00:20:13,540 --> 00:20:17,930
+ذكرتها أنا في الرسم اللي فات، لأن لو أخدت y بتساوي
+
+256
+00:20:17,930 --> 00:20:23,110
+صفر بيصير x متغيرة و z متغيرة بكون قعدت في الـ xz
+
+257
+00:20:23,110 --> 00:20:30,130
+plane، هاي الـ xz plane هذا البردجاني، الآن لو أخدت و
+
+258
+00:20:30,130 --> 00:20:34,990
+z بتساوي صفر، هاي أخدت هذا المستوى، هاي هذا هذا هذا
+
+259
+00:20:35,330 --> 00:20:40,770
+أيو هيك هيك هذا، z بتساوي 0، الآن هدول الـ planes لو
+
+260
+00:20:40,770 --> 00:20:45,290
+جينا قلنا الـ planes التلاتة z بتساوي 0 و y بتساوي
+
+261
+00:20:45,290 --> 00:20:49,490
+0 و x بتساوي 0، في إيش هيتقاطعن؟ هيتقاطعن في نقطة
+
+262
+00:20:49,490 --> 00:20:54,270
+واحدة اللي هي مين؟ هيتقاطعن في نقطة اللي هي الـ 000
+
+263
+00:20:54,270 --> 00:20:58,290
+لما أقول z بتساوي 0 و y بتساوي 0 و x بتساوي 0 في
+
+264
+00:20:58,290 --> 00:21:03,500
+نفس الوقت، بس عند الـ origin هذه، الآن لاحظوا زمان
+
+265
+00:21:03,500 --> 00:21:09,020
+كانت الـ X Y plane تتجسم إلى أربع إيش؟ مناطق اللي
+
+266
+00:21:09,020 --> 00:21:11,800
+هو الربع الأول وربع الثاني وربع الثالث وربع الرابع
+
+267
+00:21:11,800 --> 00:21:16,520
+الآخر، هنا لأ احنا عندنا الآن لما نيجي نجسم هي
+
+268
+00:21:16,520 --> 00:21:22,910
+تتجسم إلى ثمانية مناطق، مصبور؟ هذه هي أول شي تطلع على
+
+269
+00:21:22,910 --> 00:21:26,770
+اللي فوق اللي فوق اللي هو هيتجسم لأربعة أجزاء هي
+
+270
+00:21:26,770 --> 00:21:31,950
+جزء هي جزءين هي ثلاثة هي أربعة وفي أربعة تحتين هي
+
+271
+00:21:31,950 --> 00:21:35,770
+واحد هي اثنين هي ثلاثة هي أربعة وفي أربعة تحتين
+
+272
+00:21:35,770 --> 00:21:42,650
+بيصير كمان اللي هي أجزاء يعني eight octants، eight
+
+273
+00:21:42,650 --> 00:21:46,680
+octants هذه اللي هي الأجزاء، لأن الآن اللي كله
+
+274
+00:21:46,680 --> 00:21:49,780
+positive هو بيسميه first octant، اللي بقى جي مش
+
+275
+00:21:49,780 --> 00:21:53,760
+كتير احنا ما .. ماحكيش حاكيين عن standard تسميات
+
+276
+00:21:53,760 --> 00:21:56,720
+زي اللي هي كانت قبل الربع الأول، الربع الثاني،
+
+277
+00:21:56,720 --> 00:21:59,800
+الربع الثالث، لأ احنا بس مهتمين في اننا سمينا هذا
+
+278
+00:21:59,800 --> 00:22:03,920
+هو عبارة عن الربع اللي هو الـ .. الـ .. الـ first
+
+279
+00:22:03,920 --> 00:22:07,940
+octant اللي هو الـ X موجبة و الـ Y موجبة و الـ Z
+
+280
+00:22:07,940 --> 00:22:11,620
+موجبة، هاي هذه المنطقة، هاي الربح هذا، شايف؟ الثمن
+
+281
+00:22:11,620 --> 00:22:15,080
+هذا، هاي المنطقة هذه، طيب يا جماعة صلوا على النبي
+
+282
+00:22:15,080 --> 00:22:21,860
+عليه الصلاة والسلام، نيجي الآن لو اجينا قلنا زي ما
+
+283
+00:22:21,860 --> 00:22:27,120
+حكينا قبل شوية، اطلعولي لـ z بتساوي خمسة، z بتساوي
+
+284
+00:22:27,120 --> 00:22:31,960
+خمسة معناته هي الـ z axis، هاجي عند الـ z بتساوي خمسة
+
+285
+00:22:32,630 --> 00:22:36,170
+أو برسم plane لأن اتفاقنا z بيساوي خمسة معناته X
+
+286
+00:22:36,170 --> 00:22:39,550
+مطلقة و Y مطلقة يعني X بتاخد كل النقاط و Y كل
+
+287
+00:22:39,550 --> 00:22:45,430
+النقاط و Z ثابت، إذا كطبيعة هترسملي هذا الـ plane
+
+288
+00:22:45,430 --> 00:22:52,510
+اللي بقطع الـ Z Axis عند النقطة خمسة، طيب هاي المستوى
+
+289
+00:22:52,510 --> 00:22:56,190
+ده اللي بالأزرق، Z بالساوية خمسة، طيب لو اجينا
+
+290
+00:22:56,190 --> 00:23:02,190
+للمستوى هذا اللي هو Y بتساوي ثلاثة يعني Y ثابتة
+
+291
+00:23:02,190 --> 00:23:07,090
+مدام Y ثابتة، Y بتساوي ثلاثة إذا هتعمل لي plane زي
+
+292
+00:23:07,090 --> 00:23:11,510
+ما قلنا قبل شوية، موازي للـ X Y plane عند مين؟
+
+293
+00:23:11,510 --> 00:23:17,840
+النقطة ثلاثة، هاي ثلاثة قيمة Y ثلاثة، و برسم موازية Z
+
+294
+00:23:17,840 --> 00:23:23,000
+X Plane هذا اللي هو اللي بالبردجاني، الآن اللي
+
+295
+00:23:23,000 --> 00:23:29,060
+بالبردجاني هذا و اللي بالأزرق هذا لأن الاثنين مع
+
+296
+00:23:29,060 --> 00:23:35,260
+بعض تقاطعن، إيش هيكون تقاطع مستويين؟ عبارة عن خط هي
+
+297
+00:23:35,260 --> 00:23:39,500
+المستوى الأول وهي المستوى الثاني هيتقاطع بخط هذا
+
+298
+00:23:39,500 --> 00:23:45,450
+الخط هذا إيش معادلته؟ اللي هو Y بتساوي ثلاثة وفي نفس
+
+299
+00:23:45,450 --> 00:23:49,270
+الوقت Z بتساوي خمسة، لأن Y بتساوي ثلاثة هو عبارة عن
+
+300
+00:23:49,270 --> 00:23:53,370
+بلين و Z بتساوي خمسة عبارة عن بلين، المشترك
+
+301
+00:23:53,370 --> 00:23:58,190
+بينهما أو تقاطعهما هو عبارة عن خط، إذن Y بتساوي
+
+302
+00:23:58,190 --> 00:24:02,190
+ثلاثة و Z بتساوي خمسة تعني خط التقاطع بين هدول
+
+303
+00:24:02,190 --> 00:24:06,370
+المستويين، وهيوه ماشي، طيب صلوا على النبي عليه
+
+304
+00:24:06,370 --> 00:24:13,400
+الصلاة والسلام، نيجي الآن، أخدنا z بتساوي اللي هو
+
+305
+00:24:13,400 --> 00:24:19,300
+خمسة طلع بلين موازي الـ X Y، Y بتساوي ثلاثة هو
+
+306
+00:24:19,300 --> 00:24:26,460
+بلين موازي الـ Z X، الآن بتاخد X بتساوي اثنين، X
+
+307
+00:24:26,460 --> 00:24:31,580
+بتساوي 2 يعني 2 الثابتة، هي الـ X بتساوي 2 إذا
+
+308
+00:24:31,580 --> 00:24:36,940
+موازي للـ Y Z بلين، هذا اللي بالأخضر هو عبارة
+
+309
+00:24:36,940 --> 00:24:42,420
+عن X بتساوي 2، X بتساوي 2 هيتقطع مع الأزرق في خط
+
+310
+00:24:42,420 --> 00:24:48,800
+الأزرق مين هو؟ Y بتساوي 3، إذا هذا الخط عبارة عن X
+
+311
+00:24:48,800 --> 00:24:57,130
+بتساوي 2 اللي هو الأخضر مع Z بتساوي خمسة اللي هو
+
+312
+00:24:57,130 --> 00:25:03,690
+عبارة عن الأزرق، هاي خط التقاطع بينهم، طيب الآن
+
+313
+00:25:03,690 --> 00:25:13,710
+تقاطع الثلاثة مستويات مع بعض اللي هو Z بخمسة، Z بخمسة
+
+314
+00:25:13,710 --> 00:25:19,510
+مع اللي هو Y بثلاثة
+
+315
+00:25:20,500 --> 00:25:27,960
+مع X بتساوي 2، هيتقاطع اللي هو في النقطة هذه لأن هذا
+
+316
+00:25:27,960 --> 00:25:32,720
+الأول تقاطع الاثنين الأولانيات المستويين وهذا تقاطع
+
+317
+00:25:32,720 --> 00:25:37,240
+الاثنين الثانيات، لما نوجد النقاط المشتركة بين
+
+318
+00:25:37,240 --> 00:25:41,180
+الثلاثة هيكون تقاطع الحد مع تقاطع الحد اللي هي يمين
+
+319
+00:25:41,180 --> 00:25:45,750
+النقطة هذه، إذا الثلاثة مستويات هذا المستوى اللي هو
+
+320
+00:25:45,750 --> 00:25:50,390
+البردجاني مع الأخضر مع الأزرق هيتقاطع في النقطة
+
+321
+00:25:50,390 --> 00:25:56,390
+هذه اللي احنا هنسميها نقطة التقاطع اللي هي اثنين و
+
+322
+00:25:56,390 --> 00:26:04,790
+ثلاثة و خمسة، اثنين و ثلاثة و خمسة لو دي نطلعها، هذه
+
+323
+00:26:04,790 --> 00:26:09,930
+النقطة اللي هي في عند اللي هي z بتساوي خمسة اللي
+
+324
+00:26:09,930 --> 00:26:15,250
+هي على الـ z axis هتكون صفر و صفر و خمسة، النقطة
+
+325
+00:26:15,250 --> 00:26:18,690
+اللي هي في المستوى هذا على الـ x axis اللي هي اثنين
+
+326
+00:26:18,690 --> 00:26:22,590
+صفر صفر، إحداثيات مجرد إحداثيات النقطة على المستوى
+
+327
+00:26:22,590 --> 00:26:26,450
+y بتساوي ثلاثة اللي على الـ y axis اللي هي صفر و
+
+328
+00:26:26,450 --> 00:26:32,070
+ثلاثة وصفر، هذه طبعا الآن خط تقاطع اللي هو
+
+329
+00:26:32,070 --> 00:26:37,070
+المستوى x بتساوي اثنين مع المستوى y بتساوي ثلاثة هو
+
+330
+00:26:37,070 --> 00:26:40,950
+الخط x بتساوي اثنين و y بتساوي ثلاثة، هيه إذا هذا خط
+
+331
+00:26:40,950 --> 00:26:45,150
+تقاطع المستويين، مستويين و هذول الاثنين الثانيات و
+
+332
+00:26:45,150 --> 00:26:50,530
+هذول الاثنين الثالثات طبعا تقاطع الثلاثة خطوط اللي هي
+
+333
+00:26:50,530 --> 00:26:55,710
+تقاطع المستويات بالترتيب هو عبارة عن النقطة اللي
+
+334
+00:26:55,710 --> 00:27:00,510
+مذكورة، هذه هي قراءة أخرى لعملية التقاطع الثلاثة
+
+335
+00:27:00,510 --> 00:27:05,010
+مستويات عن طريق معرفة التقاطع كل مستويين مع بعض
+
+336
+00:27:05,010 --> 00:27:09,570
+بخطهم، تقاطع الثلاثة خطوط في النقطة اللي بتتقاطع فيها
+
+337
+00:27:09,570 --> 00:27:15,270
+الثلاثة مستويات وهي نقطة اثنين وثلاثة وخمسة، المفروض
+
+338
+00:27:15,270 --> 00:27:21,840
+هيك يعني الأمور واضحة أو واضحة، طيب الآن we
+
+339
+00:27:21,840 --> 00:27:25,300
+interpret these equations and inequalities
+
+340
+00:27:25,300 --> 00:27:28,700
+geometrically، الآن بدنا نشوف بعض اللي هي الـ
+
+341
+00:27:28,700 --> 00:27:33,760
+equations أو الـ inequalities، نشوف إيش بنعني فيها
+
+342
+00:27:34,360 --> 00:27:39,580
+الآن لما نقول z أكبر أو يساوي صفر معناته أنا بدي
+
+343
+00:27:39,580 --> 00:27:44,620
+المنطقة كلها اللي بتكون الـ z فيها أكبر أو يساوي
+
+344
+00:27:44,620 --> 00:27:49,580
+صفر، خليني أرجع لكم هنا، z أكبر أو يساوي صفر فهي كل
+
+345
+00:27:49,580 --> 00:27:54,040
+المنطقة اللي فوق هذه، المنطقة اللي فوق كلها هذه، من
+
+346
+00:27:54,040 --> 00:27:59,600
+الأرضية هذه الـ XY واطلعوا فوق، هذه عبارة عن Z أكبر
+
+347
+00:27:59,600 --> 00:28:04,380
+أو يساوي صفر، يعني بمعنى آخر Z أكبر أو يساوي صفر هي
+
+348
+00:28:04,380 --> 00:28:09,100
+عبارة عن مين يا جماعة؟ عن الـ half space
+
+349
+00:28:09,100 --> 00:28:13,500
+consisting of the points on and above الـ XY plane
+
+350
+00:28:13,500 --> 00:28:18,000
+إذن هي عبارة عن الـ اللي هي نفس الأرضية الـ XY
+
+351
+00:28:18,000 --> 00:28:22,580
+plane مع القيم الأعلى، يعني عبارة عن نص الـ space
+
+352
+00:28:22,580 --> 00:28:27,560
+بالإضافة للأرضية، إذا X بتساوي ناقص ثلاثة، اتفقنا X
+
+353
+00:28:27,560 --> 00:28:30,760
+بتساوي ثلاثة قبل شوية أو اثنين أو أربعة عبارة عن
+
+354
+00:28:30,760 --> 00:28:39,040
+إيش؟ عن بلين الموازي لمين؟ للـ YZ بلين بس الآن
+
+355
+00:28:39,040 --> 00:28:44,260
+ما لتهت تحت X بتساوي ناقص ثلاثة يعني تحت في المنطقة
+
+356
+00:28:44,260 --> 00:28:48,420
+هذه تحت، المقصود فيها يعني هي بتيجي هنا ناقص ثلاثة
+
+357
+00:28:48,420 --> 00:28:53,640
+و بترسم مستوى موازي الـ ZY plane يعني زي هذا
+
+358
+00:28:53,640 --> 00:28:56,720
+المستوى اللي بالأزرق بس على الجهة الثانية على
+
+359
+00:28:56,720 --> 00:29:01,960
+الجهة اللي هناك، بيكون X بتساوي ناقص ثلاثة، إذا X
+
+360
+00:29:01,960 --> 00:29:06,260
+بتساوي ناقص ثلاثة يا جماعة هو عبارة عن الـ plane
+
+361
+00:29:06,260 --> 00:29:10,700
+العمودي، perpendicular to the x-axis عند الـ x
+
+362
+00:29:10,700 --> 00:29:15,140
+بتساوي ناقص 2، this plane lies parallel to the yz
+
+363
+00:29:15,140 --> 00:29:18,980
+plane and three units behind it، المفهوم يعني
+
+364
+00:29:18,980 --> 00:29:24,240
+هو عبارة عن المستوى الموازي للـ yz plane وخلف يعني
+
+365
+00:29:24,240 --> 00:29:30,080
+خلف المنطقة السالبة three units بعيد عن نقطة الأصل
+
+366
+00:29:30,080 --> 00:29:34,290
+اللي هنا، يعني وراء اللي هو في المنطقة السالبة
+
+367
+00:29:34,290 --> 00:29:39,990
+بالمقدار ثلاثة، طيب نيجي لـ الآن Z بتساوي صفر، اتفقنا Z
+
+368
+00:29:39,990 --> 00:29:44,830
+بتساوي صفر هو عبارة عن الـ X Y بلين، لما أقول X أصغر
+
+369
+00:29:44,830 --> 00:29:49,670
+أو يساوي صفر و Y أكبر أو يساوي صفر اللي هو أول
+
+370
+00:29:49,670 --> 00:29:53,310
+حاجة، Z بتساوي صفر معناته جاعدنا على الأرض، ماشي إيش؟
+
+371
+00:29:53,310 --> 00:29:56,540
+جاعدنا على الأرض؟ جاعدنا على الأرض، يهدي بعدنا على
+
+372
+00:29:56,540 --> 00:30:00,620
+الأرض يهدي، لما يقول Z بتساوي صفر، لما يقولك يحددك
+
+373
+00:30:00,620 --> 00:30:05,000
+كمان X أصغر يساوي صفر ويقولك أنا بدي المنطقة اللي
+
+374
+00:30:05,000 --> 00:30:08,680
+هان للـ Xات اللي أصغر أو يساوي صفر، يعني هالجيت لما
+
+375
+00:30:08,680 --> 00:30:12,410
+جالك Z بتساوي صفر و أصغر يساوي صفر، ده نحطك هنا، طيب
+
+376
+00:30:12,410 --> 00:30:16,130
+ما لسه ما خلصش تحديد بيقول لي y أكبر وأصغر وصفر
+
+377
+00:30:16,130 --> 00:30:19,270
+مدام Y أكبر وأصغر وصفر و X أصغر وأصغر وصفر صار
+
+378
+00:30:19,270 --> 00:30:24,150
+هذا، شايفين هذه المنطقة، هذا الربع تبع زمان اللي كنا
+
+379
+00:30:24,150 --> 00:30:33,910
+نسميه الربع اللي هو الـ X سالبة و Y موجبة يعني
+
+380
+00:30:33,910 --> 00:30:41,310
+الربع الثاني، هنا اللي هو عبارة عن الرابع اللي موجود
+
+381
+00:30:41,310 --> 00:30:46,150
+هان اللي بنسميه the second quadrant of the XY Plane
+
+382
+00:30:46,150 --> 00:30:51,430
+إن شاء الله تكونوا فهمتوا عليها، X أكبر أو يساوي صفر
+
+383
+00:30:51,430 --> 00:30:54,830
+مدام X أكبر أو يساوي صفر إذا جالك في المنطقة هذه، أنا
+
+384
+00:30:54,830 --> 00:30:58,430
+في العالم هذا، العالم الكبير هذا في الجهة اللي اللي
+
+385
+00:30:58,430 --> 00:31:02,910
+جاي هذه، X أكبر أو يساوي صفر، لما جالك برضه Y أكبر أو
+
+386
+00:31:02,910 --> 00:31:06,570
+يساوي صفر برضه جالك، أنا في الجهة هذه، جالك سيبك من
+
+387
+00:31:06,570 --> 00:31:09,690
+المنطقة هذه، هي الـ Y أكبر أو يساوي صفر و X أكبر أو
+
+388
+00:31:09,690 --> 00:31:14,270
+يساوي صفر، صرنا احنا هذا في المنطقة هذا
+
+445
+00:35:27,780 --> 00:35:32,820
+ليست هذه يكتبها أيه دون ما يكتب Z، معناته دوائر فوق
+
+446
+00:35:32,820 --> 00:35:38,140
+بعض فوق بعض فوق بعض، هتعمل لي cylinder، هتعمل لي عبارة
+
+447
+00:35:38,140 --> 00:35:41,640
+عن أسطوانة. إذا X تربيع زائد Y تربيع بيساوي أربعة، و
+
+448
+00:35:41,640 --> 00:35:48,360
+يساوي، هي عبارة عن أسطوانة، اللي هي دوائرها قاطعها
+
+449
+00:35:48,360 --> 00:35:52,680
+عبارة عن، أو المقاطع التابعة لها عبارة عن دوائر موازية
+
+450
+00:35:52,680 --> 00:35:56,720
+للـ X Y plane. لكن لما Z بتساوي ثلاثة، و كده
+
+451
+00:35:56,720 --> 00:36:01,780
+بيقول لي اسمها: أنا بدي الدائرة X تربيع زائد Y تربيع
+
+452
+00:36:01,780 --> 00:36:05,220
+بيساوي أربعة، اللي في المستوى Z بيساوي ثلاثة، لأن
+
+453
+00:36:05,220 --> 00:36:09,320
+هذه كلها لو أسطوانة بتمثل X تربيع زائد Y تربيع بيساوي
+
+454
+00:36:09,320 --> 00:36:12,920
+أربعة. لكن لما أقول لك Z بيساوي ثلاثة، غصب عنك أنا
+
+455
+00:36:12,920 --> 00:36:16,980
+بدي عند Z بيساوي ثلاثة، وأرسم لك هذه، بيصير هذه
+
+456
+00:36:16,980 --> 00:36:20,800
+المستوى Z بيساوي ثلاثة، وهي اللي بيحقق X تربيع زائد Y
+
+457
+00:36:20,800 --> 00:36:27,300
+تربيع بيساوي أربعة. إذا هذه الرسمة دائرة، دائرة في
+
+458
+00:36:27,300 --> 00:36:34,460
+المستوى Z بيساوي ثلاثة، بمركز صفر، صفر، وطبعا الـ
+
+459
+00:36:34,460 --> 00:36:38,640
+Z التابعة لها ثلاثة، ونصف قطرها اثنين. يعني هذه
+
+460
+00:36:38,640 --> 00:36:47,080
+النقطة المركز عبارة عن صفر، صفر، وثلاثة، وإيش نصف
+
+461
+00:36:47,080 --> 00:36:50,060
+قطرها اثنين، إذا ده circle: X
+
+462
+00:36:50,060 --> 00:36:53,780
+تربيع زائد Y تربيع بيساوي أربعة، in the plane Z بيساوي
+
+463
+00:36:53,780 --> 00:37:01,130
+ثلاثة، وهي اللي هو الحديث عن الحل اللي حكيته. نجي
+
+464
+00:37:01,130 --> 00:37:03,990
+الآن يا جماعة، حاجة أنتو بتعرفوها، حاجة اسمها
+
+465
+00:37:03,990 --> 00:37:11,150
+distance and sphere in space، اللي هي بدنا نشوف إيش
+
+466
+00:37:11,150 --> 00:37:16,770
+المسافة بين نقطتين في الفراغ، وإيش هي معادلة مين
+
+467
+00:37:16,770 --> 00:37:20,690
+الكرة في الفراغ. هتلاقي هذه حاجة سهلة، وحاجة أنتو
+
+468
+00:37:20,690 --> 00:37:24,590
+من زمان بتعرفوها. الآن الـ formula
+
+469
+00:37:24,590 --> 00:37:27,430
+for distance between two points in the xy plane
+
+470
+00:37:27,430 --> 00:37:30,370
+extends to points in space. يعني احنا بدل ما عملنا
+
+471
+00:37:30,370 --> 00:37:34,530
+formula في الـ xy plane، بتمدها لزمان لما كانت النقطة
+
+472
+00:37:34,530 --> 00:37:40,470
+P1 (x1, y1)، و P2 (x2, y2)، كنا نقول إيش المسافة بين P1 و
+
+473
+00:37:40,470 --> 00:37:44,810
+بين الاثنين هي عبارة عن الجذر التربيعي لـ (x1 - x2) تربيع
+
+474
+00:37:44,810 --> 00:37:49,740
+زائد (y1 - y2) تربيع. إلا لما يكون
+
+475
+00:37:49,740 --> 00:37:54,080
+النقطتين في الفراغ، فبكون المسافة بين النقطتين P1 و
+
+476
+00:37:54,080 --> 00:38:00,540
+P2 عبارة عن الجذر التربيعي لـ (X1 - X2) تربيع زائد (Y1 -
+
+477
+00:38:00,540 --> 00:38:05,240
+Y2) تربيع زائد (Z1 - Z2) تربيع، بنجمعهم
+
+478
+00:38:05,240 --> 00:38:09,900
+وبناخد الجذر التربيعي، بتطلع المسافة بين النقطتين
+
+479
+00:38:09,900 --> 00:38:16,790
+P1 و P2. وشوف هذا بتتضح هنا أكثر لما نحكي هذا. فرضنا
+
+480
+00:38:16,790 --> 00:38:19,950
+أنه هي عند نقطة، هذا، انسوا أن أنا بحكي عن معادلة
+
+481
+00:38:19,950 --> 00:38:23,750
+الدائرة. الـ G، لو فرضنا عند نقطتين في الفراغ (X naught،
+
+482
+00:38:23,750 --> 00:38:28,070
+Y naught، Z naught)، وهذا X، Y، Z هي نقطتين، هذول
+
+483
+00:38:28,070 --> 00:38:31,990
+النقطتين، المسافة بين النقطتين هذول هي عبارة عن
+
+484
+00:38:31,990 --> 00:38:38,760
+الجذر التربيعي لـ (X naught - X) تربيع زائد (Y naught - Y) تربيع
+
+485
+00:38:38,760 --> 00:38:43,440
+زائد (Z naught - Z) تربيع. زائد (Y naught - Y) تربيع
+
+486
+00:38:43,440 --> 00:38:47,260
+زائد (Z naught - Z) تربيع. زائد (Y naught - Y) تربيع
+
+487
+00:38:47,260 --> 00:38:51,620
+زائد (Z naught - Z) تربيع. زائد (Y naught - Y) تربيع
+
+488
+00:38:51,620 --> 00:38:53,440
+زائد (Z naught - Z) تربيع. زائد (Z naught - Z) تربيع
+
+489
+00:38:53,440 --> 00:38:54,280
+زائد (Z naught - Z) تربيع. زائد (Z naught - Z) تربيع
+
+490
+00:38:54,280 --> 00:38:55,020
+زائد (Z naught - Z) تربيع. زائد (Z naught - Z) تربيع
+
+491
+00:38:55,020 --> 00:39:00,630
+زائد (Z naught - Z) تربيع. و...و...و...ونصف قطرها
+
+492
+00:39:00,630 --> 00:39:06,290
+بيساوي أيه؟ يعني بدي كل الـ...بدي الآن هوية كل
+
+493
+00:39:06,290 --> 00:39:09,790
+النقاط اللي بعد...اللي ببعد عن (x naught، y naught،
+
+494
+00:39:09,790 --> 00:39:15,690
+z naught) ببعد عنه بمقدار اللي هو، جداش، بمقدار بيساوي
+
+495
+00:39:15,690 --> 00:39:20,870
+أيه؟ هذا اللي بدي، اللي هي عبارة عن، اللي هي بدي
+
+496
+00:39:20,870 --> 00:39:27,520
+أشوف هوية، أو معادلة، اللي هي اللي بتعبّر عن كل
+
+497
+00:39:27,520 --> 00:39:32,400
+النقاط التي تبعد بعد ثابت عن أيه، يعني بمعنى آخر
+
+498
+00:39:32,400 --> 00:39:36,860
+بدي معادلة الكرة، إيش معادلة الكرة؟ اللي هي اللي
+
+499
+00:39:36,860 --> 00:39:41,020
+المسافة، قلنا إيش المسافة بين هذه وهذه؟ نفرض أن
+
+500
+00:39:41,020 --> 00:39:43,640
+النقطة اللي بتكون على الكرة اللي بتبعد بعد ثابت
+
+501
+00:39:43,640 --> 00:39:47,580
+على مقدار أيه اسمها (x، y، z)، فبيصير عندي الجذر
+
+502
+00:39:47,580 --> 00:39:51,220
+التربيعي لـ (x - x naught) تربيع زائد (y - y naught)
+
+503
+00:39:51,220 --> 00:39:56,470
+تربيع زائد (z - z naught) تربيع بيساوي إيش اللي
+
+504
+00:39:56,470 --> 00:39:59,770
+هو a، هذا الجذر التربيعي، يعني المسافة بين النقطتين
+
+505
+00:39:59,770 --> 00:40:04,250
+اللي هي a، بيساوي الجذر التربيعي لهذا، مربع الجهتين،
+
+506
+00:40:04,250 --> 00:40:10,450
+بتطلع عبارة عن المسافة اللي هي، المسافة بتساوي
+
+507
+00:40:10,450 --> 00:40:14,510
+المربع، اللي مربعه بتطلع مربع المسافة بيساوي الـ a
+
+508
+00:40:14,510 --> 00:40:20,700
+تربيع، اللي هو البعد هذا. وهذا هيعبر لي عن كل النقاط
+
+509
+00:40:20,700 --> 00:40:26,460
+التي تبعد بعد ثابت عن مين؟ اللي هي x naught، و y
+
+510
+00:40:26,460 --> 00:40:31,060
+naught، و z naught، بمقدار البعد، إيش بيساوي a؟ إذا
+
+511
+00:40:31,060 --> 00:40:37,080
+صارت معادلة الكرة هي عبارة عن (X - X naught) تربيع زائد (Y -
+
+512
+00:40:37,080 --> 00:40:39,180
+Y naught) تربيع زائد (Z - Z naught) تربيع
+
+513
+00:40:39,180 --> 00:40:40,480
+(Z - Z naught) تربيع (Z - Z naught) تربيع (Z - Z naught) تربيع
+
+514
+00:40:40,480 --> 00:40:42,240
+تربيع (Z - Z naught) تربيع (Z - Z naught) تربيع (Z -
+
+515
+00:40:42,240 --> 00:40:45,760
+Z naught) تربيع (Z - Z naught) تربيع (Z - Z naught) تربيع
+
+516
+00:40:45,760 --> 00:40:50,420
+(Z - Z naught) تربيع (Z - Z naught) تربيع (Z - Z naught) تربيع
+
+517
+00:40:50,420 --> 00:40:54,980
+(Z - Z naught) بيساوي a تربيع، عن x0، y0، z0، اجت من الجذر
+
+518
+00:40:54,980 --> 00:40:58,800
+التربيعي اللي هو قيمة البعد، بيساوي a، وبعدين ربعت
+
+519
+00:40:58,800 --> 00:41:02,740
+الجهتين عشان تكون المعادلة أنظف، أو تكون أسهل في
+
+520
+00:41:02,740 --> 00:41:06,360
+التعامل. إذا هذه الـ sphere of radius a centered at
+
+521
+00:41:06,360 --> 00:41:13,100
+the point x0، y0. طيب، نيجي ناخد مثال، بقول: find the
+
+522
+00:41:13,100 --> 00:41:19,040
+center and radius of the sphere، أوجد اللي هو مركز
+
+523
+00:41:19,040 --> 00:41:22,460
+ونصف قطر للسفير: X تربيع زائد Y تربيع زائد Z تربيع
+
+524
+00:41:22,460 --> 00:41:25,920
+زائد 3X - 4Z + 1، إيش بيساوي؟ بيساوي
+
+525
+00:41:25,920 --> 00:41:31,560
+صفر. الآن بدي أخلي هذه شكلها على شكل اللي قبل بشوية،
+
+526
+00:41:31,560 --> 00:41:37,060
+اللي هي، بدي أكتبها على الشكل هذا، هيك، هيك، هيك، هيك.
+
+527
+00:41:37,060 --> 00:41:43,100
+ده نشوف كيف اللي بحكيه، بنجي، نلمّ المتغيرات التابعة للـ
+
+528
+00:41:43,100 --> 00:41:48,640
+X مع بعض، هي X تربيع زائد 3X مع بعض، والـ Y تربيع
+
+529
+00:41:49,720 --> 00:41:53,280
+ماعليهاش، مافيش Y تربيع، أيه، مافيش، وياها زيها، الحالة الـ
+
+530
+00:41:53,280 --> 00:41:56,460
+Z تربيع مع الـ 4Z، هيها، الـ 4Z، تسيبوا منها
+
+531
+00:41:56,460 --> 00:42:01,740
+دول، جدعان، بنجمعهم، مش قصتهم، الآن الآن زائد واحد، هذا
+
+532
+00:42:01,740 --> 00:42:04,980
+بنحوله على الجهتين، بيصير ناقص واحد. إذا أنا
+
+533
+00:42:04,980 --> 00:42:09,060
+حاطط الـ X تربيع، و الـ 3X، هدول مع بعض، و الـ Y
+
+534
+00:42:09,060 --> 00:42:12,220
+تربيع، هذه الحالة، لأنها مافيش، والـ Z تربيع مع
+
+535
+00:42:12,220 --> 00:42:16,260
+اللي هي اللي فيها الـ Zات مع بعض. الآن، والواحد لحاله
+
+536
+00:42:16,260 --> 00:42:18,900
+المطلق هذا، بدي أنقله على الجهة الثانية، صار ناقص
+
+537
+00:42:18,900 --> 00:42:23,560
+واحد. لأن إيش سويت هنا أنا؟ الآن بدي أحولها لمين؟
+
+538
+00:42:23,560 --> 00:42:26,700
+لِلصيغة اللي قبل بشوية، يعني بدي أعمل هذا على صورة
+
+539
+00:42:26,700 --> 00:42:29,860
+مربع كامل، وهذا مربع كامل، وهذا مربع كامل، هذه
+
+540
+00:42:29,860 --> 00:42:35,000
+جاهزة مربع كامل، دول، طب إيش بدي أسوي؟ بدي أكمل
+
+541
+00:42:35,000 --> 00:42:41,640
+المربع، يعني بمعنى آخر بدي أحط الحد، هذا، هذا، اللي هو
+
+542
+00:42:41,640 --> 00:42:45,200
+اللي بيخليه هذا عبارة عن أيه؟ عشان عندي X تربيع زائد
+
+543
+00:42:45,200 --> 00:42:52,570
+3X، ماشي. الآن بدي أخلي هذا بحيث أنه يصير (X + 3/2) تربيع، زائد
+
+544
+00:42:52,570 --> 00:42:55,670
+اللي هو طبعاً، زائد 3/2 لكل تربيع زائد - 
+
+545
+00:42:55,670 --> 00:42:58,230
+2 لكل تربيع، زائد Y تربيع بيساوي عشان... عشان
+
+546
+00:42:58,230 --> 00:42:59,110
+عشان... عشان... عشان... عشان... عشان... عشان... عشان... عشان
+
+547
+00:42:59,110 --> 00:43:01,410
+عشان... عشان... عشان... عشان... عشان... عشان... عشان... عشان
+
+548
+00:43:01,410 --> 00:43:02,010
+عشان... عشان... عشان... عشان... عشان... عشان... عشان... عشان
+
+549
+00:43:02,010 --> 00:43:02,570
+عشان... عشان... عشان... عشان... عشان... عشان... عشان... عشان
+
+550
+00:43:02,570 --> 00:43:06,590
+عشان... عشان... عشان... عشان...
+
+551
+00:43:18,280 --> 00:43:24,260
+بأخذ اللي هو الحد المعامل لـ X، وبقسمه على 2: 3
+
+552
+00:43:24,260 --> 00:43:28,300
+على 2، وبربعه، وبضيفه، صار عبارة عن 3/2، بيصير 9/4. نفس الشيء، الحد هذا اللي
+
+553
+00:43:28,300 --> 00:43:33,420
+هو عبارة عن إيش؟ اللي هو -4، بقسمه على 2،
+
+554
+00:43:33,420 --> 00:43:38,490
+بيصير -2، بربعه، بيصير 4، صار هذا
+
+555
+00:43:38,490 --> 00:43:46,970
+مربع كامل، وهذا مربع كامل، وهذا مربع كامل. اللي ضفته
+
+556
+00:43:46,970 --> 00:43:50,030
+هنا الـ 9/4، بضيفه على الطرف الثاني، عشان
+
+557
+00:43:50,030 --> 00:43:55,470
+تظلّ الميزان واحد، الـ 4 ضفتها، ده، ضيفها ده، ده، ده
+
+558
+00:43:55,470 --> 00:43:55,610
+ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده
+
+559
+00:43:55,610 --> 00:43:59,990
+ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده
+
+560
+00:43:59,990 --> 00:44:01,990
+ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده
+
+561
+00:44:01,990 --> 00:44:02,890
+ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده
+
+562
+00:44:02,890 --> 00:44:03,430
+ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده
+
+563
+00:44:03,430 --> 00:44:11,670
+ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده، ده
+
+564
+00:44:11,670 --> 00:44:20,200
+ده، ده، هذا مربع كامل، وجمعت هذول، ربعها، طلعت 21/4
+
+565
+00:44:20,200 --> 00:44:24,720
+الآن جاهزة المعادلة، أن أقول: hence the center of
+
+566
+00:44:24,720 --> 00:44:28,900
+the sphere is، إيش الـ center؟ اللي هو -3/2،
+
+567
+00:44:28,900 --> 00:44:32,780
+لأن هي الصورة يا جماعة، هي الصورة اللي هو أيه؟
+
+568
+00:44:32,780 --> 00:44:36,760
+هاد الـ center، ها X naught، لما كانت -Y naught، Z naught،
+
+569
+00:44:36,760 --> 00:44:41,320
+وهي نصف القطر، إذا هنا صارت هذا عبارة عن -، -
+
+570
+00:44:41,320 --> 00:44:46,250
+يعني -3/2، وهذه -2، جاهزة، عبارة عن
+
+571
+00:44:46,250 --> 00:44:51,030
+2، وهي قيمة Z طبعاً، وليس Y، و Y - 0 لكل تربيع،
+
+572
+00:44:51,030 --> 00:44:56,330
+يعني 0، صارت هذه الـ Center، وهذه جذرها اللي
+
+573
+00:44:56,330 --> 00:45:00,270
+هو عبارة عن الجذر الـ 21/4، هو عبارة عن الـ
+
+574
+00:45:00,270 --> 00:45:05,450
+Radius لمين؟ للسفير اللي عندي الآن.
+
+576
+00:45:08,170 --> 00:45:13,310
+بظل عندي اللي هو، اللي هي، مثال آخر: here are some
+
+577
+00:45:13,310 --> 00:45:15,570
+geometric interpretations of inequalities and
+
+578
+00:45:15,570 --> 00:45:21,030
+equations involving spheres. الآن نشوف اللي هي بعض
+
+579
+00:45:21,030 --> 00:45:26,890
+اللي هي المعادلات والـ inequalities اللي تتضمن
+
+580
+00:45:26,890 --> 00:45:31,880
+spheres في داخلها، أو في مضمونها، نشوف كيف. عند X
+
+581
+00:45:31,880 --> 00:45:36,160
+تربيع زائد Y تربيع زائد Z تربيع أصغر من 4. احنا بنعرف X
+
+582
+00:45:36,160 --> 00:45:41,140
+تربيع زائد Y تربيع زائد Z تربيع بتساوي 4، هي عبارة عن
+
+583
+00:45:41,140 --> 00:45:45,140
+الاسفير، زي اللي قبل بشوية، اللي مركزها (0، 0، 0)، لأن X
+
+584
+00:45:45,140 --> 00:45:50,380
+تربيع زائد Y تربيع زائد Z تربيع بيساوي 4، عبارة عن
+
+585
+00:45:50,380 --> 00:45:56,390
+كرة مركزها (0، 0، 0)، ونصف قطرها 2. لما أقول strictly
+
+586
+00:45:56,390 --> 00:46:01,850
+أصغر من أربعة، يعني اللي داخل الكرة بدون جُدرتها،
+
+587
+00:46:01,850 --> 00:46:05,890
+يعني the interior of the sphere، X تربيع زائد Y تربيع
+
+588
+00:46:05,890 --> 00:46:10,310
+زائد Z تربيع بيساوي أربعة. لما أقول هذا أصغر أو يساوي،
+
+589
+00:46:10,310 --> 00:46:15,130
+معناته أنا قُلت الـ interior، جوا الكرة مع جُدرتها، ده
+
+590
+00:46:15,130 --> 00:46:20,730
+solid، اللي هو المجسم، ball bounded، مجسم الكرة، أو الـ
+
+591
+00:46:20,730 --> 00:46:24,970
+ball، الـ bounded by the سفير، اللي هي جُدرتها، الـ X
+
+592
+00:46:24,970 --> 00:46:29,990
+تربيع زائد Y تربيع زائد Z تربيع = 4، بالإضافة إلى داخلها،
+
+593
+00:46:29,990 --> 00:46:35,190
+يعني الـ solid المتكون من السفير ومن بداخل السفير،
+
+594
+00:46:35,190 --> 00:46:40,250
+معناته اللي هي، بمعنى آخر، alternatively the sphere
+
+595
+00:46:40,250 --> 00:46:43,290
+X تربيع زائد Y تربيع زائد Z تربيع = 4،
+
+596
+00:46:45,410 --> 00:46:51,030
+together with its interior، مع اللي هي إيش؟ كلها
+
+597
+00:46:51,030 --> 00:46:54,730
+لما نجي لـ X تربيع زائد Y تربيع زائد Z تربيع أكبر، أكبر
+
+598
+00:46:54,730 --> 00:46:57,830
+من أربعة، كلكم عارفين، قلت، هذي مادام هذي اللي جوا، أو
+
+599
+00:46:57,830 --> 00:47:01,310
+هذي اللي جوا مع الجُدرة، هذا هو عبارة عن اللي برا،
+
+600
+00:47:01,310 --> 00:47:04,290
+بدون الجُدرة، the exterior of the sphere، X تربيع زائد
+
+601
+00:47:04,290 --> 00:47:08,920
+Y تربيع زائد Z تربيع بيساوي أربعة، له خارج الكرة، إلى
+
+602
+00:47:08,920 --> 00:47:12,340
+ما لا نهاية. بيظلّوا، إلا لما أقول X تربيع زائد Y تربيع
+
+603
+00:47:12,340 --> 00:47:15,520
+زائد Z تربيع بيساوي أربعة، هي السفير،
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/B5_wN_rcAT0_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/B5_wN_rcAT0_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..7568fb437f0226adc80fe70e03a5129c3b95e2f9
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/B5_wN_rcAT0_postprocess.srt
@@ -0,0 +1,2492 @@
+1
+00:00:01,080 --> 00:00:05,600
+بسم الله الرحمن الرحيم هذه هي المحاضرة الأولى
+
+2
+00:00:05,600 --> 00:00:10,080
+لمساق تفاضل و تكامل جيم لطلاب و طالبات الجامعة
+
+3
+00:00:10,080 --> 00:00:15,020
+الإسلامية الفصل
+
+4
+00:00:15,020 --> 00:00:25,500
+الصيفي 2020 الذي اعتمد فيه التعليم الإلكتروني وهي
+
+5
+00:00:25,500 --> 00:00:32,560
+المحاضرة الأولىالتي سنبدأ بها بإذن الله الحديث
+
+6
+00:00:32,560 --> 00:00:39,300
+بداية على .. على ال course outline لهذا الفصل
+
+7
+00:00:39,300 --> 00:00:47,570
+فاهلا و سهلا بجميع الطلاب والطالبات في هذا الفصلال
+
+8
+00:00:47,570 --> 00:00:52,870
+course outline هو كالعادة كما هو في الفصول الماضية
+
+9
+00:00:52,870 --> 00:00:56,210
+من نفس الكتاب اللي هو textbook اللي هو calculus
+
+10
+00:00:56,210 --> 00:01:02,870
+Thomas اللي هي الطبعة 12 اللي هي اللي احنا بنكمل
+
+11
+00:01:02,870 --> 00:01:08,250
+فيها اللي أخدنا في تفاضل الف وتفاضل ب الفصول
+
+12
+00:01:08,250 --> 00:01:15,570
+الماضيةالشباتر المطلوبة اللي هو هنبدأ بشبطر 12 ثم
+
+13
+00:01:15,570 --> 00:01:23,170
+شبطر 13 ثم 14 ثم 15 شبطر 12 اللي هو عبارة عن اللي
+
+14
+00:01:23,170 --> 00:01:27,290
+هو هنتحدث فيه على three dimensional coordinate
+
+15
+00:01:27,290 --> 00:01:32,940
+systemsأحنا طبعا نتحدثنا في تفاضل ألف و تفاضل باء
+
+16
+00:01:32,940 --> 00:01:37,780
+في ال two dimensions اللي هو في لبنان كنا أخدنا كل
+
+17
+00:01:37,780 --> 00:01:44,580
+متعلقات التفاضل والتكامل في اللي هي لبنان أو ال
+
+18
+00:01:44,580 --> 00:01:50,160
+two dimensionsالان هنبدأ الحديث على اللي هو في
+
+19
+00:01:50,160 --> 00:01:54,360
+تفاضل جيم على ال three-dimensional coordinate
+
+20
+00:01:54,360 --> 00:01:58,820
+systems ونبني على هذا الكلام هنحكي برضه عن ال
+
+21
+00:01:58,820 --> 00:02:05,580
+vectors and dot و ال cross product و هنيجي ان شاء
+
+22
+00:02:05,580 --> 00:02:10,060
+الله بالتفصيل لكل هذه الأمور lines and planes in
+
+23
+00:02:10,060 --> 00:02:14,580
+space اللي هي معادلة الخط مستقيم طبعا أخدناها في
+
+24
+00:02:14,580 --> 00:02:14,820
+ال
+
+25
+00:02:17,640 --> 00:02:21,020
+معادلة الخط المستقيم في الـ space في الفراغ
+
+26
+00:02:21,020 --> 00:02:24,880
+ومعادلة المستوى في الفراغ وان شاء الله هنشوف
+
+27
+00:02:24,880 --> 00:02:30,820
+الكلام هذا وان شاء الله بكون كلام سهل الان ايضا ال
+
+28
+00:02:30,820 --> 00:02:35,580
+section اللي هو الأخير في ال chapter 12 هو
+
+29
+00:02:35,580 --> 00:02:40,910
+cylinders and quadratic surfacesاللي هي حاجة اسمها
+
+30
+00:02:40,910 --> 00:02:45,530
+تعميم للسيلندر العادية واللي هي ال quadratic
+
+31
+00:02:45,530 --> 00:02:51,430
+surfaces أو السطوح الناجمة عن اللي هي المعادلات
+
+32
+00:02:51,430 --> 00:02:55,750
+التربيعية وحنشوف هذا الكلام أيضا بالتفصيلبعد هيك
+
+33
+00:02:55,750 --> 00:02:59,350
+هنبدأ في شبتر 13 اللي هو curves in space and their
+
+34
+00:02:59,350 --> 00:03:03,630
+tangent ال curves in space and their tangent اللي
+
+35
+00:03:03,630 --> 00:03:08,010
+هو هنتحدث عنها ان شاء الله و هنتحدث عن integrals
+
+36
+00:03:08,010 --> 00:03:13,410
+of vector functions اللي هي الدوال اللي .. اللي هي
+
+37
+00:03:13,410 --> 00:03:20,850
+قيمتها عبارة عن vectorsهنتحدث عنها أيضا هنتحدث عن
+
+38
+00:03:20,850 --> 00:03:25,870
+ال arc length in space بعد هيك هنكون احنا وصلنا
+
+39
+00:03:25,870 --> 00:03:29,770
+لمرحلة اللي هو حاجة اسمها ال first exam طبعا زي ما
+
+40
+00:03:29,770 --> 00:03:33,550
+انتوا عارفين اللي هو الفصل الصيفي هذا هيكون عندي
+
+41
+00:03:33,550 --> 00:03:38,770
+اللي هو ال first exam و ال second exam و ال final
+
+42
+00:03:38,770 --> 00:03:44,350
+exam كلهم هيكون عبارة عن امتحانات اللي هوعبر الـ
+
+43
+00:03:44,350 --> 00:03:49,190
+model و هتكون إلكترونية أو في البيت ان شاء الله
+
+44
+00:03:49,190 --> 00:03:53,550
+الآن ال chapter اللي .. اللي .. اللي هنكمل في
+
+45
+00:03:53,550 --> 00:03:56,970
+chapter 13 هنحكي عن ال curve vectors and normal
+
+46
+00:03:56,970 --> 00:04:02,330
+vectors of curveهنحكي عن الـ tangential and normal
+
+47
+00:04:02,330 --> 00:04:07,310
+component of acceleration وبنكون هيك خلصنا شبطر 13
+
+48
+00:04:07,310 --> 00:04:11,990
+بنبدأ في شبطر 14 اللي هو functions of several
+
+49
+00:04:11,990 --> 00:04:16,710
+variables اللي هو الدوال فيه عدد من المجاهيل طبعا
+
+50
+00:04:16,710 --> 00:04:19,630
+احنا في ال ال two dimensions وفي calculus alpha
+
+51
+00:04:19,630 --> 00:04:22,370
+وcalculus beta حكينا عن function of one variable
+
+52
+00:04:22,370 --> 00:04:26,030
+اللي هنحكي عن functions of two variables three
+
+53
+00:04:26,030 --> 00:04:31,660
+variables and so onهنحكي ما يتعلق بها من ال limits
+
+54
+00:04:31,660 --> 00:04:34,920
+و ال continuity in higher dimensions نفس اللي هو
+
+55
+00:04:34,920 --> 00:04:40,540
+الأسلوب اللي كان في تفاضل ألف و با و لكن بما يتعلق
+
+56
+00:04:40,540 --> 00:04:44,820
+بالدوال الجديد الآن اللي هنحكينا برضه في تفاضل ألف
+
+57
+00:04:44,820 --> 00:04:48,060
+و تفاضل با عن اللي هو ال derivatives الآن هنحكي عن
+
+58
+00:04:48,060 --> 00:04:51,340
+ال partial derivatives لأنه صار في عندي أكتر من
+
+59
+00:04:51,340 --> 00:04:55,260
+مجهول فهيصير عندي اللي هو حاجة اسمها التفاضل
+
+60
+00:04:55,260 --> 00:05:00,770
+الجزئيو هناخد برضه الـ chain rule في هذه الحالات
+
+61
+00:05:00,770 --> 00:05:04,670
+زي ما أخدنا chain rule سابقا في الحالة اللي هي ال
+
+62
+00:05:04,670 --> 00:05:08,050
+function of any variable بعدين هنحكي عن حاجة اسمها
+
+63
+00:05:08,050 --> 00:05:11,490
+directional derivatives and gradient vectors و
+
+64
+00:05:11,490 --> 00:05:15,170
+هنجيل كل هذا الكلام بالتفصيل ان شاء الله tangent
+
+65
+00:05:15,170 --> 00:05:18,210
+planes and differentials أيضا اللي هو زي ما كان
+
+66
+00:05:18,210 --> 00:05:24,100
+فيه ممساتفي الحالة ال curves العادية في عندى اللي
+
+67
+00:05:24,100 --> 00:05:30,300
+هي مستويات تمثل اللي هي الأجسام اللي عندنا أو
+
+68
+00:05:30,300 --> 00:05:34,540
+الرسمات اللي عندنا الرسمات اللي عندنا أي ان كانت و
+
+69
+00:05:34,540 --> 00:05:37,920
+هنيجي نحكي عن شو اللي هو معادلة ال tangent planes
+
+70
+00:05:37,920 --> 00:05:40,400
+و كيف نوجدها و هنحكي عن حاجة اسمها برضه
+
+71
+00:05:40,400 --> 00:05:44,300
+differentials و برضه ممكن نسمع عنها بس في حالة
+
+72
+00:05:44,300 --> 00:05:46,800
+اللي هو ال two dimensions او function of one
+
+73
+00:05:46,800 --> 00:05:50,760
+variableالآن هنحكي أيضًا عن extreme values and
+
+74
+00:05:50,760 --> 00:05:54,400
+saddle points وبرضه حكينا عن مقابل الهن في اللي هو
+
+75
+00:05:54,400 --> 00:05:59,100
+تفاضل ب أو تفاضل ألف اللي هو اللي هي النقاط العظمى
+
+76
+00:05:59,100 --> 00:06:03,220
+و النقاط الصغرى وقولها نحد اسمها ال saddle point و
+
+77
+00:06:03,220 --> 00:06:08,010
+هنيجي إليها ان شاء الله لاحقًاعند هنا في نهاية
+
+78
+00:06:08,010 --> 00:06:12,070
+شبتر هذا شبتر طول 14 بيجي عندنا اللي هو second
+
+79
+00:06:12,070 --> 00:06:14,910
+exam او الناس اللي بيقول second exam بيقول second
+
+80
+00:06:14,910 --> 00:06:19,290
+quiz اللي هو برضه هيكون زي ما اتفجنا على اللي هو
+
+81
+00:06:19,290 --> 00:06:23,700
+ال modelهذا اللي بحكي فيه بحكي في ال course
+
+82
+00:06:23,700 --> 00:06:30,140
+outline لسه لهذا المساق نجي الى اللي هو الجزء
+
+83
+00:06:30,140 --> 00:06:32,800
+الثالث من المادة اللي هو عبارة عن ال chapter
+
+84
+00:06:32,800 --> 00:06:36,900
+الأخير اللي هو chapter 15 اللي هو ما يتعلق بال
+
+85
+00:06:36,900 --> 00:06:40,360
+integration هنجي على نحكي عن ال double and
+
+86
+00:06:40,360 --> 00:06:44,080
+iterated integrals over rectangles اللي هي ال
+
+87
+00:06:44,080 --> 00:06:47,080
+double integral اللي هو ال integration اللي هي two
+
+88
+00:06:47,080 --> 00:06:53,000
+integrations اللي هنانشوف ايش معناه integral
+
+89
+00:06:53,000 --> 00:06:57,220
+integral و بعدين I treated integrals و بعدين اللي
+
+90
+00:06:57,220 --> 00:07:02,080
+هو هنبدأ نحكي عن ال double integrals over الان
+
+91
+00:07:02,080 --> 00:07:06,780
+general region هتلاقوا نفس المنهجية اللي تحدثنا
+
+92
+00:07:06,780 --> 00:07:11,630
+فيهافي تفاضل الف عن ال integration هنتحدث فيها في
+
+93
+00:07:11,630 --> 00:07:16,210
+اللي هو ال integration او اللي هو integration of
+
+94
+00:07:16,210 --> 00:07:18,910
+function of two variables او three variables حسب
+
+95
+00:07:18,910 --> 00:07:21,910
+اللي بدناها و هيكون عندي double integral او triple
+
+96
+00:07:21,910 --> 00:07:25,370
+integral حسب عدد ال variables كله هنجيب و هتفهم
+
+97
+00:07:25,370 --> 00:07:29,570
+عليا ايش اللي بقوله area by double integration
+
+98
+00:07:29,570 --> 00:07:33,780
+هنجيب ال areaاللي هي تحت منحنة by double
+
+99
+00:07:33,780 --> 00:07:36,880
+integration كنا نجيبها عادي بالقبل ال integration
+
+100
+00:07:36,880 --> 00:07:39,520
+العادي الآن هنجيبها by double integration و هنشوف
+
+101
+00:07:39,520 --> 00:07:42,860
+كيف و هنجي الكلام سهل لا تقلقوا double
+
+102
+00:07:42,860 --> 00:07:46,380
+integrations in polar forms هنستخدم ال double
+
+103
+00:07:46,380 --> 00:07:50,280
+integrations وسط البولر بتحللنا بعض المشاكل بعض
+
+104
+00:07:50,280 --> 00:07:54,280
+الدول بتكون صعبة التكامل بالعادي بنحولها للبولر
+
+105
+00:07:54,280 --> 00:07:57,880
+ولا تقلقوا برضه في التحويل للبولر ان شاء الله كله
+
+106
+00:07:57,880 --> 00:08:01,460
+مجدور عليهلأن triple integrals in rectangular
+
+107
+00:08:01,460 --> 00:08:05,080
+coordinates هيصير عندنا بدل ما هو double integral
+
+108
+00:08:05,080 --> 00:08:09,660
+انتجرشين اتنين واحد واحد هنشوف ايش بقصد فيه بيصير
+
+109
+00:08:09,660 --> 00:08:13,460
+عندى تلاتة integration وش شو معناته وإيش كيف بدنا
+
+110
+00:08:13,460 --> 00:08:16,780
+نشتغل عليه ده بيبقى triple integral in rectangular
+
+111
+00:08:16,780 --> 00:08:23,000
+coordinatesالجزء اللي قبل الأخير Terrible integral
+
+112
+00:08:23,000 --> 00:08:27,140
+in cylindrical and spherical يعني حق نيجي اللي هو
+
+113
+00:08:27,140 --> 00:08:32,160
+باستخدام cylindrical و spherical coordinates اتسهل
+
+114
+00:08:32,160 --> 00:08:36,180
+علينا أحيانا بعض ال integrations بواسطة ال
+
+115
+00:08:36,180 --> 00:08:39,480
+rectangular coordinates وحنا عارف ايش معناته اللي
+
+116
+00:08:39,480 --> 00:08:43,060
+هو بتكون أسهل لنا في استخدام cylindrical و
+
+117
+00:08:43,060 --> 00:08:46,610
+sphericalوبعدين هنحكي عن ال substitution in
+
+118
+00:08:46,610 --> 00:08:51,330
+multiple integrals ومنكون هيك احنا انهينا اللي هو
+
+119
+00:08:51,330 --> 00:08:56,430
+ال course مع اللي هو .. اللي هو chapter 15 و
+
+120
+00:08:56,430 --> 00:09:00,230
+هيجينا ال final exam ال final exam هيكون في كل
+
+121
+00:09:00,230 --> 00:09:08,270
+المادة و هيكون اللي هو ال final exam برضه إلكتروني
+
+122
+00:09:08,270 --> 00:09:12,160
+و كله عبر الموديليعني التلت تقييمات اللي عندنا في
+
+123
+00:09:12,160 --> 00:09:15,140
+الكورس عشان نكون واضحين هكون اللي هو عندي تلت
+
+124
+00:09:15,140 --> 00:09:19,580
+شغلات الكورس لكوز الأول أو ال first exam لكوز
+
+125
+00:09:19,580 --> 00:09:23,320
+التاني اللي هو ال second exam و لكوز ال final exam
+
+126
+00:09:23,320 --> 00:09:31,900
+الآن ال .. ال .. هيكون عند التقييم كما يلي التقييم
+
+127
+00:09:31,900 --> 00:09:37,560
+هيكون عندي اللي هو ال .. ال term exams هيكون عندي
+
+128
+00:09:37,560 --> 00:09:42,590
+أربعين في الميةيعني الأول اللي هو عشرين والتاني
+
+129
+00:09:42,590 --> 00:09:46,550
+عشرين أو حسب ما نتفق يعني ان شاء الله مبدئيا
+
+130
+00:09:46,550 --> 00:09:50,170
+اربعين في المية اللي هو عطوهم ال term exams اللي
+
+131
+00:09:50,170 --> 00:09:54,330
+هو كل واحد عشرين في المية وممكن نحكي في غير هيك ان
+
+132
+00:09:54,330 --> 00:10:00,720
+شاء اللهالنهائي هيكون ستين في المية الان يا جماعة
+
+133
+00:10:00,720 --> 00:10:06,640
+الطالب طبعا حسب قرار الجامعة مخير بده bus يعني بده
+
+134
+00:10:06,640 --> 00:10:10,980
+bus bus بدهوش علامة bus او fail بده bus او fail
+
+135
+00:10:10,980 --> 00:10:17,380
+بدخل اللي هو ال first وال second وال final كلهين
+
+136
+00:10:17,380 --> 00:10:21,240
+إلكتروني وفي البيت مش مشكلة في البيت أو وين ما بده
+
+137
+00:10:21,670 --> 00:10:25,710
+لأن لو الطالب بده اللي هو علامة في الصيف برضه
+
+138
+00:10:25,710 --> 00:10:29,870
+هنعطيه علامة كيف هنعطيه علامة؟ بدخل طبعا عادي اللي
+
+139
+00:10:29,870 --> 00:10:33,750
+هو ال first و second عادي بدخل في البيت اللي هو
+
+140
+00:10:33,750 --> 00:10:38,090
+بياخد الامتحان او ال quiz في البيت على ال model
+
+141
+00:10:38,090 --> 00:10:45,890
+لكن ال final اللي هو بيكون في الجامعة او عبر ال
+
+142
+00:10:45,890 --> 00:10:49,580
+modelأو عبر اللي هو الوجه هي حسب ما نتفج مش مشكلة
+
+143
+00:10:49,580 --> 00:10:53,260
+بس انهم بيكونوا في ال final في الجامعة طبعا و كيف
+
+144
+00:10:53,260 --> 00:10:58,500
+بيحصل على هذه العلامة بياخد غير مكتمل قبل ما نعمل
+
+145
+00:10:58,500 --> 00:11:02,490
+اللي هو لباقي الطلاب اللي بدهم ال busيعني الآن
+
+146
+00:11:02,490 --> 00:11:07,050
+خيار العلامة تاخد غير مكتمل بتكون مقدّم ال first
+
+147
+00:11:07,050 --> 00:11:12,090
+ومقدّم ال second ومحسبالك اللي هو من ال 40% اللي
+
+148
+00:11:12,090 --> 00:11:18,050
+بيظل 60% ال final بتقدمه في داخل الجامعة بال model
+
+149
+00:11:18,050 --> 00:11:23,330
+أو بال .. اللي هي الوجهة هي حسب ما نتفق وغالبا
+
+150
+00:11:23,330 --> 00:11:27,650
+احنا هنعمل ان شاء الله بال model لو حتى كان في
+
+151
+00:11:27,650 --> 00:11:32,570
+عندي اللي هو علامة لكن في الجامعةهذا اللي هو ال
+
+152
+00:11:32,570 --> 00:11:39,890
+course outline بالنسبة للمثاب اللي عندنا نيجي الآن
+
+153
+00:11:39,890 --> 00:11:44,810
+نبدأ في اللي هو chapter 12 chapter 12 اللي هو
+
+154
+00:11:44,810 --> 00:11:50,050
+بالظبط اسمه vectors and the geometry of spaceالتي
+
+155
+00:11:50,050 --> 00:11:54,190
+هي المتجهات والهندسة اللي هي طبعا المقصودة عن
+
+156
+00:11:54,190 --> 00:11:57,370
+الهندسة التحليلية أو ما يقابل الهندسة التحليلية في
+
+157
+00:11:57,370 --> 00:12:00,930
+الفراغ يعني الآن بدل ما نحكي عن اللي هي اللي
+
+158
+00:12:00,930 --> 00:12:05,530
+حكيناه في calculus ألف و calculus باك كنا نحكي عن
+
+159
+00:12:05,530 --> 00:12:10,420
+ال geometry in a planeالان بنحكي عن Geometry in
+
+160
+00:12:10,420 --> 00:12:13,940
+Space ماحدش يروح في كل هندسة فراغية لأ لأ احنا مش
+
+161
+00:12:13,940 --> 00:12:16,880
+هنحكي عن الهندسة الاقليدية الفراغية لأ احنا هنحكي
+
+162
+00:12:16,880 --> 00:12:21,120
+عن ما يتعلق بامور الدوال والتفاضل والتكامل ووالاخر
+
+163
+00:12:21,120 --> 00:12:25,100
+يعني حاجة اللي كنا بنسميها او تسمى بالهندسة
+
+164
+00:12:25,100 --> 00:12:31,050
+التحليليةنجي الآن لـ 12 واحد ال section الأول 12
+
+165
+00:12:31,050 --> 00:12:35,010
+واحد اللي هو عبارة عن اسمه three dimensional
+
+166
+00:12:35,010 --> 00:12:39,250
+coordinate system لأن three dimensional coordinate
+
+167
+00:12:39,250 --> 00:12:43,970
+system اللي هو خلينا نتطلع نشوف بالظبط ايش اللي
+
+168
+00:12:43,970 --> 00:12:49,330
+كنا احنا كنا هنا ايش اللي مقصود فيه لأن زمان كنا
+
+169
+00:12:49,330 --> 00:12:53,290
+نحكي عن اللي هو لبنان اذا في كان عندي X وY لإن كنا
+
+170
+00:12:53,290 --> 00:12:58,490
+على العرض نشتغلالان هنشتغل في الفراغ فطبيعي هيكون
+
+171
+00:12:58,490 --> 00:13:04,250
+عندى X وY وفيه coordinate ثالث ايه شماله اسمه Z
+
+172
+00:13:04,250 --> 00:13:11,600
+Axis بالظبط هي عندي هي X وهي Y وهي Zالان كيف بدنا
+
+173
+00:13:11,600 --> 00:13:16,160
+نحدد النقاط اللى هى فى الفرام قبلها طبعا ال
+
+174
+00:13:16,160 --> 00:13:19,240
+coordinates ال coordinates معناته فى positive
+
+175
+00:13:19,240 --> 00:13:23,480
+coordinate وعالجهة الثانية فى ال negative اللى هى
+
+176
+00:13:23,480 --> 00:13:28,220
+ال x axis الان هى ال positive y axis وعالجهة
+
+177
+00:13:28,220 --> 00:13:33,880
+الثانية فى ال negative y axis الان هى ال positive
+
+178
+00:13:33,880 --> 00:13:39,790
+z axis وهى اللى تحت اللى هو ال negative z axisهيك
+
+179
+00:13:39,790 --> 00:13:47,670
+بيكون غطينا الفرار كله بواسطة X Y Z coordinates
+
+180
+00:13:47,670 --> 00:13:52,990
+يعني أي نقطة في الفرار هنصلها من خلال اللي هي ال
+
+181
+00:13:52,990 --> 00:13:57,280
+coordinates الثلاثة اللي موجودةالان ايش الايد هذه
+
+182
+00:13:57,280 --> 00:14:01,640
+اللي بنسميها قاعدة اليد اليمنى قاعدة اليد اليمنى
+
+183
+00:14:01,640 --> 00:14:06,360
+بتيجي بترفع ايدك اليمين هيها او بتيجي اللي بترفع
+
+184
+00:14:06,360 --> 00:14:11,980
+أصبعك هذا اللي هان بكون موجه باتجاه ال positive z
+
+185
+00:14:11,980 --> 00:14:18,740
+axis الان بتضغط بايدك بتضغط اللي هي بال X على ال Y
+
+186
+00:14:18,740 --> 00:14:21,960
+فالمنطقة اللي انت بتضغط فيها في اتجاه ال X هذه
+
+187
+00:14:21,960 --> 00:14:26,750
+بتكون الموجه بقى وهذه بتكون ايشاللي هي الموجبة للـ
+
+188
+00:14:26,750 --> 00:14:31,790
+Y والأعلى اللي أصبح هذا بتدل على المنطقة الموجبة
+
+189
+00:14:31,790 --> 00:14:38,910
+إذا هيك Z positive هيك Y positive هيك X positive
+
+190
+00:14:38,910 --> 00:14:43,270
+الآن عكسهم تماما في الجهة الأخرى هي مين الـ
+
+191
+00:14:43,270 --> 00:14:48,570
+Negative خلينا نشوف بعض اللي هي الشغلاتاللي هي
+
+192
+00:14:48,570 --> 00:14:52,730
+تحديد النقاط يعني ده نشوف خلينا مع بعض لو اجينا
+
+193
+00:14:52,730 --> 00:14:58,080
+للنقطة هي نقطة امشينا على ال X axis هنا بمقدار Xو
+
+194
+00:14:58,080 --> 00:15:01,960
+بعدين ماشينا هنا موازي ال Y اكسس زي ما كنا نعمل في
+
+195
+00:15:01,960 --> 00:15:06,440
+العادى و نوشي موازي ال Y اكسس بمقدار Y يعني جطعنا
+
+196
+00:15:06,440 --> 00:15:10,860
+من ال Y Y و بعدين طلعنا لفوق بمقدار Z النقطة اللي
+
+197
+00:15:10,860 --> 00:15:14,020
+وصلناها هذه طبعا كلها أعمد هذي زي ما انتوا عارفين
+
+198
+00:15:14,020 --> 00:15:20,180
+هذا الآن هذا النقطة هي عبارة عن ال coordinates لها
+
+199
+00:15:20,180 --> 00:15:26,770
+او إحداثيتها عبارة عن X و Y و Zالان X اللي هو هيه
+
+200
+00:15:26,770 --> 00:15:33,350
+البعد من هان لهان X هيه اللي
+
+201
+00:15:33,350 --> 00:15:37,990
+أخدنا هيه من هان لهان هيه ال X وهذا البعد من هان
+
+202
+00:15:37,990 --> 00:15:42,950
+لهان اللي هو Z اللي ارتفعنا ارتفاع فيه إلى أعلى
+
+203
+00:15:43,280 --> 00:15:48,040
+والبعد من هان لعند الهان اللي أخدناها هنا اللي هو
+
+204
+00:15:48,040 --> 00:15:53,680
+Y يعني أتينا أحنا أول شي مشينا X و بعدين مشينا Y و
+
+205
+00:15:53,680 --> 00:15:58,960
+بعدين مشينا Z ووصلنا لنقطة X و Y و Z طيب الآن لو
+
+206
+00:15:58,960 --> 00:16:05,520
+أتينا تخيلنا أن أنا أتيت ثبتت ال X هي ال X هذه إيه
+
+207
+00:16:05,520 --> 00:16:13,000
+شمالها مقدار ثابتإذا صارت قيمة أي ال X ثابت أي هذا
+
+208
+00:16:13,000 --> 00:16:19,500
+ثابت الآن صارت اللي هي ال Y بتاخد هنا X ثابت هي من
+
+209
+00:16:19,500 --> 00:16:23,300
+هنا إلى هنا هي ثابت صارت هذه قيمة ال Y تمشي على ال
+
+210
+00:16:23,300 --> 00:16:27,220
+X الثابت يعني ضلت ماشية هي قيمة Y يعني ما أخد
+
+211
+00:16:27,220 --> 00:16:32,090
+حريتهاوالـ Z بأخد حريتها إلى أعلى فالآن الـ Z و
+
+212
+00:16:32,090 --> 00:16:37,750
+الـ Y لما يكون المتغيرات التنتين مع ثبوت ال X بعمل
+
+213
+00:16:37,750 --> 00:16:42,590
+لي Blame هذا هاي الـ Blame هذا طبعا بمتد إلى أعلى
+
+214
+00:16:42,590 --> 00:16:45,010
+المال الأهية و إلى أسفل المال الأهية و من هنا
+
+215
+00:16:45,010 --> 00:16:48,890
+المال الأهية و من هنا المال الأهية فهذا X بتساوي
+
+216
+00:16:48,890 --> 00:16:58,050
+constant هو عبارة عن Blameموازنة من لل Y زد بلين
+
+217
+00:16:58,050 --> 00:17:04,590
+كله ماشي الحال طيب الان لو Y ثابتة معناته احنا
+
+218
+00:17:04,590 --> 00:17:10,370
+ثبتنا Y و ال X و ال Z انطلجت و قيمة ال X انطلجت
+
+219
+00:17:10,370 --> 00:17:13,930
+هذي بتنطج و بتاخد اللي بتهيأ و هذي ال Z بتاخد اللي
+
+220
+00:17:13,930 --> 00:17:19,170
+بتهيأ فبتصنعني لبلين هذا إذا Y بيساوي ثابت هو
+
+221
+00:17:19,170 --> 00:17:27,690
+عبارة عن مستوىموازي لل Z X بلين يعني ال Y ثابت
+
+222
+00:17:27,690 --> 00:17:32,790
+موازي لل Z X بلين لو ال Y سفر بنكون وين جعدنا
+
+223
+00:17:32,790 --> 00:17:37,090
+جعدنا في ال X Z بلين سيرنا وين جاعدين في ال X Z
+
+224
+00:17:37,090 --> 00:17:42,410
+بلين يعني بيصير ال X متغيرة وال Z متغيرة لكن فش Y
+
+225
+00:17:42,410 --> 00:17:47,490
+فكادنا جاعدين وين هنا طيب الان لو اجينا وقلنا
+
+226
+00:17:47,490 --> 00:17:55,300
+والله خد لأ انا بدي اخد الثابت Zالان اخدت ثابت z
+
+227
+00:17:55,300 --> 00:18:00,380
+هي y و شايفين يا جماعة هي ثابت z صارت ال y هي
+
+228
+00:18:00,380 --> 00:18:05,160
+بتاخد حريتها و ال x بتاخد حريتها فبتعملها plane
+
+229
+00:18:05,160 --> 00:18:10,200
+هذا بنسميه z بالثاوض ثابت هيكون موازلة ال x y
+
+230
+00:18:10,200 --> 00:18:16,300
+plane الأرض هذه اذا z ثابت عبارة عن مستوى موازلة
+
+231
+00:18:16,300 --> 00:18:22,180
+ال x y planeY ثابت معناته مستوى موازي للـ X وZ
+
+232
+00:18:22,180 --> 00:18:28,280
+بلين اللي هي X ثابت معناته مستوى موازي للـ Y Z
+
+233
+00:18:28,280 --> 00:18:32,660
+بلين و هكذا طب لو أخدنا Z بصفر يعني لو قلنا Z
+
+234
+00:18:32,660 --> 00:18:37,730
+بساوة صفرهذه تعني انه صار عند كلنا على الأرضية تبع
+
+235
+00:18:37,730 --> 00:18:42,130
+الزمان اللي هو ال X Y plane تبع الزمان إذا Z
+
+236
+00:18:42,130 --> 00:18:45,790
+بالساوية سفر هو عبارة عن X Y plane اللي هو Y
+
+237
+00:18:45,790 --> 00:18:49,770
+بالساوية سفر عبارة عن X Z plane X بالساوية سفر
+
+238
+00:18:49,770 --> 00:18:56,240
+عبارة عن Z Y plane و هكذاطب لو الآن اللي هو عندي
+
+239
+00:18:56,240 --> 00:19:01,920
+النقطة x و y و 0 بنصير هذا قيمة x وهذا قيمة y
+
+240
+00:19:01,920 --> 00:19:04,520
+ومافي ز ز على الأرض احنا قاعدين على الأرض فبصير
+
+241
+00:19:04,520 --> 00:19:09,000
+سفر و هكذا طب لو كانت ال .. ال .. ال y سفر بنصير
+
+242
+00:19:09,000 --> 00:19:13,060
+قاعدين في وين في ال xz اللي عندنا نقطة x و 0 و z
+
+243
+00:19:13,060 --> 00:19:18,780
+هيهاو هكذا تستطيع تكتب كل النقاط اللى بدك هيها
+
+244
+00:19:18,780 --> 00:19:22,500
+وتحصلها على اللى هو في ال three dimensional space
+
+245
+00:19:22,500 --> 00:19:31,240
+تصلى اي نقطة في ال spaceنأتي الآن لبعض المعادلات
+
+246
+00:19:31,240 --> 00:19:37,600
+اللي ناخدها عنفلة بمعنى آخر هاي ال space هاي ال
+
+247
+00:19:37,600 --> 00:19:40,240
+coordinates هدا ال x coordinate هاي المنطقة
+
+248
+00:19:40,240 --> 00:19:44,460
+الموجبة وهي المنطقة السالبة وهذا ال y coordinate
+
+249
+00:19:44,460 --> 00:19:49,540
+وهي ال y المنطقة الموجبة وهي السالبة وهي ال z
+
+250
+00:19:49,540 --> 00:19:53,360
+coordinate الموجبة وهي ال z coordinate اللي هي
+
+251
+00:19:53,360 --> 00:19:58,860
+axis اللي هي السالبماشى يا جماعة طيب الآن زى ما
+
+252
+00:19:58,860 --> 00:20:03,000
+قلنا قبل بشوية هذا النقطة الان ال origin اللى
+
+253
+00:20:03,000 --> 00:20:07,940
+عبارة عن سفر سفر سفر قبل بشوية قلنا X بتساوي سفر
+
+254
+00:20:07,940 --> 00:20:13,540
+إيش هتكون عبارة عن ال X ال Y زد بلين قبل بشوية
+
+255
+00:20:13,540 --> 00:20:17,930
+ذكرتها أنا في الرسم اللى فاتتلأن لو أخدت y بتساوي
+
+256
+00:20:17,930 --> 00:20:23,110
+سفر بيصير x متغيرة و z متغيرة بكون قعدت في ال xz
+
+257
+00:20:23,110 --> 00:20:30,130
+plane هاي ال xz plane هذا البردجاني الان لو أخدت و
+
+258
+00:20:30,130 --> 00:20:34,990
+z بتساوي سفر هاي أخدت هذا المستوى هاي هذا هذا هذا
+
+259
+00:20:35,330 --> 00:20:40,770
+ايو هيك هيك هذا z بتساوي 0 الان هدول ال planes لو
+
+260
+00:20:40,770 --> 00:20:45,290
+جينا قولنا ال planes التلاتة z بتساوي 0 و y بتساوي
+
+261
+00:20:45,290 --> 00:20:49,490
+0 و x بتساوي 0 في إيش هيتقاطعن؟ هيتقاطعن في نقطة
+
+262
+00:20:49,490 --> 00:20:54,270
+واحدة اللي هي مين هيتقاطعن في نقطة اللي هي ال 000
+
+263
+00:20:54,270 --> 00:20:58,290
+لما أقول z بتساوي 0 و y بتساوي 0 و x بتساوي 0 في
+
+264
+00:20:58,290 --> 00:21:03,500
+نفس الوقت بس عند ال origin هذهالان لاحظوا زمان
+
+265
+00:21:03,500 --> 00:21:09,020
+كانت ال X Y plane تتجسم إلى أربع ايش امناطق اللي
+
+266
+00:21:09,020 --> 00:21:11,800
+هو الربع الأول وربع التاني وربع التالت وربع الرابع
+
+267
+00:21:11,800 --> 00:21:16,520
+الاخر هنا لأ احنا عندنا الان لما نيجي نجسم هي
+
+268
+00:21:16,520 --> 00:21:22,910
+تتجسم إلى تمام مناطقمصبور ؟ هذه هي اول شي تطلع على
+
+269
+00:21:22,910 --> 00:21:26,770
+اللي فوق اللي فوق اللي هو هيتجسم لأربع أجزاء هي
+
+270
+00:21:26,770 --> 00:21:31,950
+جزء هي جزءين هي تلاتة هي أربعة وفي أربعة تحتين هي
+
+271
+00:21:31,950 --> 00:21:35,770
+واحد هي اتنين هي تلاتة هي أربعة وفي أربعة تحتين
+
+272
+00:21:35,770 --> 00:21:42,650
+بيصير كمان اللي هي أجزاء يعني eight octants eight
+
+273
+00:21:42,650 --> 00:21:46,680
+octants هذه اللي هي الأجزاء لانالان اللي كله
+
+274
+00:21:46,680 --> 00:21:49,780
+positive هو بيسميه first octant اللي بقى جي مش
+
+275
+00:21:49,780 --> 00:21:53,760
+كتير احنا ما .. ماحكيش حاكيين عن standard تسميات
+
+276
+00:21:53,760 --> 00:21:56,720
+زي اللي هي كانت قبل الربع الأول، الربع التاني،
+
+277
+00:21:56,720 --> 00:21:59,800
+الربع التالت، لأ احنا بس مهتمين في اننا سمينا هذا
+
+278
+00:21:59,800 --> 00:22:03,920
+هو عبارة عن الربع اللي هو ال .. ال .. ال first
+
+279
+00:22:03,920 --> 00:22:07,940
+octant اللي هو ال X موجبة و ال Y موجبة و ال Z
+
+280
+00:22:07,940 --> 00:22:11,620
+موجبة، هاي هذه المنطقة، هاي الربح هذا، شايف؟ التمن
+
+281
+00:22:11,620 --> 00:22:15,080
+هذا، هاي المنطقة هذهطيب يا جماعة صلوا على النبي
+
+282
+00:22:15,080 --> 00:22:21,860
+عليه الصلاة والسلام نيجي الآن لو اجينا قولنا زي ما
+
+283
+00:22:21,860 --> 00:22:27,120
+حكينا قبل بشوية اطلعولي ل z بتساوي خمسة z بتساوي
+
+284
+00:22:27,120 --> 00:22:31,960
+خمسة معناته هي ال z axis هاجي عند ال z بتساوي خمسة
+
+285
+00:22:32,630 --> 00:22:36,170
+أو برسم plane لأن اتفاقنا ز بيساوي خمسة معناته X
+
+286
+00:22:36,170 --> 00:22:39,550
+مطلوقة و Y مطلوقة يعني X بتاخد كل النقاط و Y كل
+
+287
+00:22:39,550 --> 00:22:45,430
+النقاط و Z ثابت اذا كطبيعة هترسملي هذا ال plane
+
+288
+00:22:45,430 --> 00:22:52,510
+اللي بقطع ال Z Axis عند النقطة خمسةطيب هاي المستوى
+
+289
+00:22:52,510 --> 00:22:56,190
+ده اللي بالأزرق Z بالساوية خمسة طيب لو أجينا
+
+290
+00:22:56,190 --> 00:23:02,190
+للمستوى هذا اللي هو Y بتساوية تلاتة يعني Y ثابتة
+
+291
+00:23:02,190 --> 00:23:07,090
+مدام Y ثابتة Y بتساوية تلاتة إذا هتعمللي plane زي
+
+292
+00:23:07,090 --> 00:23:11,510
+ما قولنا قبل شوية موازي لل X Y plane عند مين
+
+293
+00:23:11,510 --> 00:23:17,840
+النقطة تلاتة هاي تلاتة قيمة Y تلاتةو برسم موازلة Z
+
+294
+00:23:17,840 --> 00:23:23,000
+X Plane هذا اللي هو اللي بالبردجاني الان اللي
+
+295
+00:23:23,000 --> 00:23:29,060
+بالبردجاني هذا و اللي بالأزراج هذا لأن التنين مع
+
+296
+00:23:29,060 --> 00:23:35,260
+بعض تقاطحن ايش هيكون تقاطع مستويين عبارة عن خط هي
+
+297
+00:23:35,260 --> 00:23:39,500
+المستوى الأول وهي المستوى الثاني هيتقاطع بخط هذا
+
+298
+00:23:39,500 --> 00:23:45,450
+الخط هذا ايش معدلتهاللي هو Y بتساوي تلاتة وفي نفس
+
+299
+00:23:45,450 --> 00:23:49,270
+الوقت Z بتساوي خمسة لأن Y بتساوي تلاتة هو عبارة عن
+
+300
+00:23:49,270 --> 00:23:53,370
+إبلين و Z بتساوي خمسة عبارة عن إبلين المشترك
+
+301
+00:23:53,370 --> 00:23:58,190
+بينهما أو تقاطعهما هو عبارة عن خط إذن Y بتساوي
+
+302
+00:23:58,190 --> 00:24:02,190
+تلاتة و Z بتساوي خمسة تعني خط التقاطع بين هدول
+
+303
+00:24:02,190 --> 00:24:06,370
+المستويين وهيوه ماشي طيب صلوا على النبي عليه
+
+304
+00:24:06,370 --> 00:24:13,400
+الصلاة والسلام نيدي الآنأخدنا z بتساوي اللي هو
+
+305
+00:24:13,400 --> 00:24:19,300
+خمسة طلع ابلين موازن ال X Y Y بتساوي تلاتة هو
+
+306
+00:24:19,300 --> 00:24:26,460
+ابلين موازن ال Z X الآن بتاخد X بتساوي اتنينX
+
+307
+00:24:26,460 --> 00:24:31,580
+بتساوى 2 يعني 2 الثابتة هي الـ X بتساوى 2 إذا
+
+308
+00:24:31,580 --> 00:24:36,940
+موازي للـ Y زي الدبلين هذا اللي بالأخضر هو عبارة
+
+309
+00:24:36,940 --> 00:24:42,420
+عن X بتساوى 2 X بتساوى 2 هيتقطع مع الأزرق في خط
+
+310
+00:24:42,420 --> 00:24:48,800
+الأزرق مين هو؟ Y بتساوى 3 إذا هذا الخط عبارة عن X
+
+311
+00:24:48,800 --> 00:24:57,130
+بتساوى 2اللي هو الأخضر مع Z بتساوي خمسة اللي هو
+
+312
+00:24:57,130 --> 00:25:03,690
+عبارة عن الأزرق هاي خط التقاطع بينهم طيب الآن
+
+313
+00:25:03,690 --> 00:25:13,710
+تقاطع التلت مستويات مع بعض اللي هو Z بخمسة Z بخمسة
+
+314
+00:25:13,710 --> 00:25:19,510
+مع اللي هو Y بتلاتة
+
+315
+00:25:20,500 --> 00:25:27,960
+مع X بتساو 2 هيتقاطع اللي هو في النقطة هذه لأن هذا
+
+316
+00:25:27,960 --> 00:25:32,720
+الأول تقاطع التنين الأولانيات المستويين وهذا تقاطع
+
+317
+00:25:32,720 --> 00:25:37,240
+التنين التانيات لما نبدأوجد النقاط المشتركة بين
+
+318
+00:25:37,240 --> 00:25:41,180
+التلاتة هيكون تقاطع حد مع تقاطع حد اللي هي يمين
+
+319
+00:25:41,180 --> 00:25:45,750
+النقطة هذه إذا التلت مستويات هذا المستوىاللي هو
+
+320
+00:25:45,750 --> 00:25:50,390
+البردجاني مع الأخضر مع الأزرق هيتقاطع في النقطة
+
+321
+00:25:50,390 --> 00:25:56,390
+هذه اللي احنا هنسميها نقطة التقاطع اللي هي اتنين و
+
+322
+00:25:56,390 --> 00:26:04,790
+تلاتة و خمسة اتنين و تلاتة و خمسة لو دي نطلعناهذه
+
+323
+00:26:04,790 --> 00:26:09,930
+النقطة اللي هي في العند اللي هي z بتساوي خمسة اللي
+
+324
+00:26:09,930 --> 00:26:15,250
+هي على ال z axis هتكون سفر و سفر و خمسة النقطة
+
+325
+00:26:15,250 --> 00:26:18,690
+اللي هي في المستوى هذا على ال x axis اللي هي اتنين
+
+326
+00:26:18,690 --> 00:26:22,590
+سفر سفر احداتيات مجرد احداتيات النقطة على المستوى
+
+327
+00:26:22,590 --> 00:26:26,450
+y بتساوي تلاتة اللي على ال y axis اللي هي سفر و
+
+328
+00:26:26,450 --> 00:26:32,070
+تلاتة او زيروهذه طبعا الآن خط تقاطع اللي هو
+
+329
+00:26:32,070 --> 00:26:37,070
+المستوى x بتساوي اتنين مع المستوى y بتساوي تلتة هو
+
+330
+00:26:37,070 --> 00:26:40,950
+الخط x بتساوي اتنين و y بتساوي تلتة هيه إذا هذا خط
+
+331
+00:26:40,950 --> 00:26:45,150
+تقاطع المستويين مستويين و هذولة التنين التانيات و
+
+332
+00:26:45,150 --> 00:26:50,530
+هذولة التنين التلتات طبعا تقاطع التلت خطوط اللي هي
+
+333
+00:26:50,530 --> 00:26:55,710
+تقاطع المستويات بالترتيب هو عبارة عن النقطة اللي
+
+334
+00:26:55,710 --> 00:27:00,510
+مذكورة هذه هي قراءة أخرىلعملية التقاطع التلت
+
+335
+00:27:00,510 --> 00:27:05,010
+مستويات عن طريق معرفة التقاطع كل مستويين مع بعض
+
+336
+00:27:05,010 --> 00:27:09,570
+بخطهم تقاطع التلت خطوط في النقطة اللي بتتقاطع فيها
+
+337
+00:27:09,570 --> 00:27:15,270
+التلت مستويات وهي نقطة اتنين وتلاتة وخمسة المفروض
+
+338
+00:27:15,270 --> 00:27:21,840
+هيك يعني الأمور واضحة أو واضحةطيب الآن we
+
+339
+00:27:21,840 --> 00:27:25,300
+interpret these equations and inequalities
+
+340
+00:27:25,300 --> 00:27:28,700
+geometrically الآن بدنا نشوف بعض اللي هي ال
+
+341
+00:27:28,700 --> 00:27:33,760
+equations أو ال inequalities نشوف إيش بنعني فيها
+
+342
+00:27:34,360 --> 00:27:39,580
+الآن لما نقول z أكبر أو يساوي سفر معناته أنا بدي
+
+343
+00:27:39,580 --> 00:27:44,620
+المنطقة كلها اللي بتكون ال z فيها أكبر أو يساوي
+
+344
+00:27:44,620 --> 00:27:49,580
+سفر خليني أرجعلكم هنا z أكبر أو يساوي سفر فهي كل
+
+345
+00:27:49,580 --> 00:27:54,040
+المنطقة اللي فوق هذه المنطقة اللي فوق كلها هذهمن
+
+346
+00:27:54,040 --> 00:27:59,600
+الأرضية هذه الـ XY واطلعوا فوق هذه عبارة عن Z أكبر
+
+347
+00:27:59,600 --> 00:28:04,380
+أو يساوي سفر يعني بمعنى آخر Z أكبر أو يساوي سفر هي
+
+348
+00:28:04,380 --> 00:28:09,100
+عبارة عن مين يا جماعة؟ عن الـ half space
+
+349
+00:28:09,100 --> 00:28:13,500
+consisting of the points on and above الـ XY plane
+
+350
+00:28:13,500 --> 00:28:18,000
+إذن هي عبارة عن أق اللي هي نفس الأرضية الـ XY
+
+351
+00:28:18,000 --> 00:28:22,580
+plane معالقيم الأعلى يعني عبارة عن نص ال space
+
+352
+00:28:22,580 --> 00:28:27,560
+بالإضافة للأرضية إذا X بتساوي ناقص تلاتة اتفجنا X
+
+353
+00:28:27,560 --> 00:28:30,760
+بتساوي تلاتة قبل شوية او اتنين او اربع عبارة عن
+
+354
+00:28:30,760 --> 00:28:39,040
+إيش عن لبلين الموازي لمين لل YZ لبلين بس الأنوين
+
+355
+00:28:39,040 --> 00:28:44,260
+ما لتهتتحت X بتساوي ناقص ثلاثة يعني تحت في المنطقه
+
+356
+00:28:44,260 --> 00:28:48,420
+هذه تحت المقصود فيها يعني هي بتيجي هنا ناقص ثلاثة
+
+357
+00:28:48,420 --> 00:28:53,640
+و بترسم مستوى موازي ال ZY plane يعني زي هذا
+
+358
+00:28:53,640 --> 00:28:56,720
+المستوى اللي بالأزرق بس على الجهه الثانيه على
+
+359
+00:28:56,720 --> 00:29:01,960
+الجهه اللي هناك بيكون X بتساوي ناقص ثلاثة إذا X
+
+360
+00:29:01,960 --> 00:29:06,260
+بتساوي ناقص ثلاثة يا جماعةهو عبارة عن الـ plane
+
+361
+00:29:06,260 --> 00:29:10,700
+العمودي perpendicular to the x-axis عند الـ x
+
+362
+00:29:10,700 --> 00:29:15,140
+بتساوي نقص 2 this plane lies parallel to the yz
+
+363
+00:29:15,140 --> 00:29:18,980
+plane and the three units behind it المفهوم يعني
+
+364
+00:29:18,980 --> 00:29:24,240
+هو عبارة عن المستوى الموازي للـ yz plane وخلف يعني
+
+365
+00:29:24,240 --> 00:29:30,080
+خلف المنطقة السالبة three unit بعيد عن نقطة الأصل
+
+366
+00:29:30,080 --> 00:29:34,290
+اللي هنايعني ورا اللي هو في المنطقة السالبة
+
+367
+00:29:34,290 --> 00:29:39,990
+بالمقدار تلاتة طيب نيجي لان Z بتساوي سفر اتفاجنا Z
+
+368
+00:29:39,990 --> 00:29:44,830
+بتساوي سفر هو عبارة عن ال X Y بلين لما أقول X أصغر
+
+369
+00:29:44,830 --> 00:29:49,670
+أوي يساوي سفر و Y أكبر أوي يساوي سفر اللي هو أول
+
+370
+00:29:49,670 --> 00:29:53,310
+حاجة Z بتساوي سفر معناته جاعدنا على الأرض ماشي إيش
+
+371
+00:29:53,310 --> 00:29:56,540
+جاعدنا على الأرض؟ جاعدنا على الأرض يهديبعدنا على
+
+372
+00:29:56,540 --> 00:30:00,620
+الأرض يهدي لما يقول Z بتساوي سفر لما يقولك يحددك
+
+373
+00:30:00,620 --> 00:30:05,000
+كمان X أصغر يساوي سفر ويقولك أنا بدي المنطقة اللي
+
+374
+00:30:05,000 --> 00:30:08,680
+هان لل Xات اللي أصغر أوي ساوي سفر يعني هالجيت لما
+
+375
+00:30:08,680 --> 00:30:12,410
+جالك Z بتساوي سفر و أصغر ساوي سفر ده نحطك هناطيب
+
+376
+00:30:12,410 --> 00:30:16,130
+ما لسه ماخلصش تحديد بيقولي y أكبر و أصغر و سفر
+
+377
+00:30:16,130 --> 00:30:19,270
+مدام I أكبر و أصغر و سفر و X أصغر و أصغر و سفر صار
+
+378
+00:30:19,270 --> 00:30:24,150
+هذا شايفين هذا المنطقة هذا الربع تبع زمان اللي كنا
+
+379
+00:30:24,150 --> 00:30:33,910
+نسميه الربع اللي هو ال X سالبة و Y مودبة يعني
+
+380
+00:30:33,910 --> 00:30:41,310
+الربع التانيهنا اللي هو عبارة عن الرابع اللي موجود
+
+381
+00:30:41,310 --> 00:30:46,150
+هان اللي بنسمي the second quadrant of the XY Plane
+
+382
+00:30:46,150 --> 00:30:51,430
+ان شاء الله تكونوا فهمتوا عليهاX أكبر أو سوى سفر
+
+383
+00:30:51,430 --> 00:30:54,830
+مدام X أكبر أو سوى سفر إذا جالك في المنطقة هذه أنا
+
+384
+00:30:54,830 --> 00:30:58,430
+في العالم هذا العالم الكبير هذا في الجهة اللي اللي
+
+385
+00:30:58,430 --> 00:31:02,910
+جاي هذه X أكبر أو سوى سفر لما جالك برضه Y أكبر أو
+
+386
+00:31:02,910 --> 00:31:06,570
+سوى سفر برضه جالك أنا في الجهة هذه جالك سيبك من
+
+387
+00:31:06,570 --> 00:31:09,690
+المنطقة هذه هي ال Y أكبر أو سوى سفر و X أكبر أو
+
+388
+00:31:09,690 --> 00:31:14,270
+سوى سفرصرنا احنا هذا في المنطقة هذا في ال octant
+
+389
+00:31:14,270 --> 00:31:18,330
+هذا اللي فوق و اللي تحت ماخلصش تحديد قال زي أكبر
+
+390
+00:31:18,330 --> 00:31:22,210
+أو ساوى سفر إذا مدام X أكبر أو ساوى سفر و Y أكبر
+
+391
+00:31:22,210 --> 00:31:25,370
+أو ساوى سفر و Z أكبر أو ساوى سفر إذا نحط حاله في
+
+392
+00:31:25,370 --> 00:31:30,330
+اللي سمناه ال first octant اللي هو الثمن الأول
+
+393
+00:31:30,330 --> 00:31:37,280
+نيجي لهذه يا جماعةالان Y أكبر يسوى سالب واحد طبعا
+
+394
+00:31:37,280 --> 00:31:41,880
+احنا بنعرف Y أصغر سوى واحد احنا .. طبعا X .. X و Z
+
+395
+00:31:41,880 --> 00:31:47,040
+ماحكيش عنهم معناته X و Z بطلقات طيب ماحنا بنعرف Y
+
+396
+00:31:47,040 --> 00:31:51,480
+بتسوى واحد عبارة عن plain Y بتسوى واحد plain وين Y
+
+397
+00:31:51,480 --> 00:31:55,000
+بتسوى واحد؟ هاي Y بتسوى واحد مثلا عند المنطقى هذا
+
+398
+00:31:55,000 --> 00:31:59,260
+هذا plain هايوهيك بلين هيك وهيك وهيك وهيك وهيك
+
+399
+00:31:59,260 --> 00:32:07,180
+وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك
+
+400
+00:32:07,180 --> 00:32:07,240
+وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك
+
+401
+00:32:07,240 --> 00:32:07,340
+وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك
+
+402
+00:32:07,340 --> 00:32:08,000
+وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك
+
+403
+00:32:08,000 --> 00:32:08,240
+وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيك
+
+404
+00:32:08,240 --> 00:32:12,810
+وهيك وهيك وهيك وهيك وهيك وهيك وهيك وهيكو Y بتساوي
+
+405
+00:32:12,810 --> 00:32:18,410
+سالب واحد هيك و المنطقة كلها اللي بين التنين هذه
+
+406
+00:32:18,410 --> 00:32:22,770
+هي Y أكبر ساول سالب واحد و أصغر ساول واحد اللي هي
+
+407
+00:32:22,770 --> 00:32:28,390
+بتعمل لي slab هاي واحد هاي رسمها و هاي هنا و ال
+
+408
+00:32:28,390 --> 00:32:32,360
+slab اللي طويل هذا اللي بيضلوا إلى بالنهايةهو
+
+409
+00:32:32,360 --> 00:32:37,080
+الاسلاب اللى بتويل لـ Blanes Y بيساوى نقص واحد و Y
+
+410
+00:32:37,080 --> 00:32:41,280
+بتساوي واحد هذا الاسلاب اللى هو Y أكبر يساوي سالب
+
+411
+00:32:41,280 --> 00:32:47,440
+واحد اصغر او يساوي واحد طيب نيجي الان Y بيساوى نقص
+
+412
+00:32:47,440 --> 00:32:50,640
+اتنين و Z بيساوى اتنين طبعا خلاص حفظناها عن غيباد
+
+413
+00:32:50,640 --> 00:32:58,100
+Y بيساوى نقص اتنين عبارة عن مستوى يوازن X Z بلينZ
+
+414
+00:32:58,100 --> 00:33:04,900
+بيساوى 2 مستوى يوازي الـ Y X بلين عنده نقطة 2 وهذا
+
+415
+00:33:04,900 --> 00:33:09,600
+عنده نقطة نقص 2 تقاطع مستويين هذا معناته تقاطع
+
+416
+00:33:09,600 --> 00:33:12,560
+مستويين يعني هيطلع عبارة عن خط زي ما عرفنا قبل
+
+417
+00:33:12,560 --> 00:33:17,540
+شوية إذا هو عبارة عن الخط اللي بمر بالنقطة نقص 2
+
+418
+00:33:17,540 --> 00:33:23,260
+و2 وبتكون هي ال X المتغيرة فبكون موازي لل X Axis
+
+419
+00:33:23,580 --> 00:33:26,340
+لأنه هو اللي بدأ تكون متغيرة إذا y بساوة نقص اتنين
+
+420
+00:33:26,340 --> 00:33:32,260
+و z بساوة اتنين هو عبارة عن خط بمر بالنقطة اللي هي
+
+421
+00:33:32,260 --> 00:33:37,220
+z بساوة اتنين مثلا هي اتنين و y بساوة سلب اتنين
+
+422
+00:33:37,220 --> 00:33:43,480
+بكون هذا الخط اللي زي هيك موازي بمر بالنقطة هذه y
+
+423
+00:33:43,480 --> 00:33:47,700
+وازي ال x axis إذا ال line in which he blends y
+
+424
+00:33:47,700 --> 00:33:49,780
+بساوة نقص اتنين و z بساوة اتنين intersect
+
+425
+00:33:50,640 --> 00:33:54,760
+Alternatively, the line through the point كذا بمر
+
+426
+00:33:54,760 --> 00:34:00,040
+بالنقطة اللي هي 0 ونقص 2 و2 وكمان 1 ونقص 2 و2 و3
+
+427
+00:34:00,040 --> 00:34:05,340
+ونقص 2 و2 لأنه بمر بكل النقاط main اللي هي ال Xات
+
+428
+00:34:05,340 --> 00:34:11,600
+يعني بمر بالنقطة يعني الخط بمثل X نقص 2 و2 حيث X
+
+429
+00:34:11,600 --> 00:34:18,180
+متغير هذا يعني برسم لخط موازي لل X axis طيب ماشي
+
+430
+00:34:18,180 --> 00:34:22,400
+الحاجة صلّع النبي يا جماعةعليه الصلاة والسلام لما
+
+431
+00:34:22,400 --> 00:34:27,720
+بقول لي what points bxy satisfy the equations إيش
+
+432
+00:34:27,720 --> 00:34:33,460
+اللي هي هذا النقطة إيش بتحبط التالي إيش بتمثل هذه
+
+433
+00:34:33,460 --> 00:34:38,440
+x تربيع زي ضي تربيع بساوة أربعة و z بتساوى كده
+
+434
+00:34:38,440 --> 00:34:43,760
+تلاتة لما يقول لي x تربيع زي ضي تربيع بساوة أربعة
+
+435
+00:34:43,760 --> 00:34:52,710
+و يسكت الآن أنا بعرف إن هذه معادلة دائرةصح في ال X
+
+436
+00:34:52,710 --> 00:34:57,070
+Y plane هي معدلة الدائرة اللي هو اللي هي مركزها
+
+437
+00:34:57,070 --> 00:35:01,150
+سفر و سفر و نص قطرها اتنين هي سفر و سفر و نص قطرها
+
+438
+00:35:01,150 --> 00:35:06,930
+اتنين بس انا رسمت هنا وين و كأنه ز بسفر طب ما هي ز
+
+439
+00:35:06,930 --> 00:35:10,010
+مش بسفر انا سيبوكوا من هذا انا بحكيش عن هذا الان
+
+440
+00:35:10,010 --> 00:35:13,870
+انا بحكي X تربية ز وY تربية بساوة واربعة الان X
+
+441
+00:35:13,870 --> 00:35:17,030
+تربية ز وY تربية بساوة واربعة جيت رسمت هنا دائرة
+
+442
+00:35:17,030 --> 00:35:19,530
+في ال .. عند ز بالساوة و سفر بس هو قال ليه ز
+
+443
+00:35:19,530 --> 00:35:23,700
+بالساوة و سفرلأن لو كانت Z بساوية واحد هيرسم كمان
+
+444
+00:35:23,700 --> 00:35:27,780
+دائرة Z بساوية واحد و نص كمان دائرة إذا لما يطلق
+
+445
+00:35:27,780 --> 00:35:32,820
+لياه هذه يكتبها إيه دون ما يكتب Z معناته دوائر فوق
+
+446
+00:35:32,820 --> 00:35:38,140
+بعض فوق بعض فوق بعض هتعمللي cylinderهتعمل لي عبارة
+
+447
+00:35:38,140 --> 00:35:41,640
+عن أسطوانة إذا X تربيه زاد و Y تربيه بساوء أربعة و
+
+448
+00:35:41,640 --> 00:35:48,360
+يسكت هي عبارة عن أسطوانة اللي هي دوائرها قاطعها
+
+449
+00:35:48,360 --> 00:35:52,680
+عبارة عن أو المقاطعة تبعتها عبارة عن دوائر موازية
+
+450
+00:35:52,680 --> 00:35:56,720
+لل X Y plane لكن لما جلزت بتساوي تلاتة و كده
+
+451
+00:35:56,720 --> 00:36:01,780
+بيقولي اسمهاأنا بدي الدائرة X تربيع Z Y تربيع
+
+452
+00:36:01,780 --> 00:36:05,220
+بساوية أربعة اللي في المستوى Z بتساوية تلاتة لأن
+
+453
+00:36:05,220 --> 00:36:09,320
+هذه كلها لو سطوانة بتمثل X تربيع Z Y تربيع بساوية
+
+454
+00:36:09,320 --> 00:36:12,920
+أربعة لكن لما أقول لك Z بتساوية تلاتة غصب عنك أنا
+
+455
+00:36:12,920 --> 00:36:16,980
+بدي عند Z بتساوية تلاتة و ارسم ليه هذه بيصير هذه
+
+456
+00:36:16,980 --> 00:36:20,800
+المستوى Z بتساوية تلاتة وهي اللي بيحقق X تربيع Z Y
+
+457
+00:36:20,800 --> 00:36:27,300
+تربيع بالساوية أربعةإذا هذه الرسمة دائرة دائرة في
+
+458
+00:36:27,300 --> 00:36:34,460
+المستوى z بتساوي تلاتة بمركز سفر و سفر و طبعا الـ
+
+459
+00:36:34,460 --> 00:36:38,640
+z تبعتها تلاتة و نصف قطرها إيش اتنين يعني هذه
+
+460
+00:36:38,640 --> 00:36:47,080
+النقطة المركز عبارة عن سفر و سفر و تلاتةو ايش نصف
+
+461
+00:36:47,080 --> 00:36:50,060
+قطرها منها اللي هي ابقى عن اتنين إذا ده circle X
+
+462
+00:36:50,060 --> 00:36:53,780
+تربيه زي Y تربيه بتسوء أربعة in the plane Z بتسوء
+
+463
+00:36:53,780 --> 00:37:01,130
+تلاتة وهي اللي هو الحديث عن الحل اللي حكيتهنجي
+
+464
+00:37:01,130 --> 00:37:03,990
+الآن يا جماعة حاجة انتوا بتعرفوها حاجة اسمها
+
+465
+00:37:03,990 --> 00:37:11,150
+distance and sphere in space اللي هي بدنا نشوف إيش
+
+466
+00:37:11,150 --> 00:37:16,770
+المسافة بين نقطتين في الفراغ و إيش هي معادلة مين
+
+467
+00:37:16,770 --> 00:37:20,690
+الكرة في الفراغ هتلاقي هذه حاجة سهلة و حاجة انتوا
+
+468
+00:37:20,690 --> 00:37:24,590
+من زي الحاجات اللي بتعرفوها زمانالأن الـ formula
+
+469
+00:37:24,590 --> 00:37:27,430
+for distance between two points in the xy plane
+
+470
+00:37:27,430 --> 00:37:30,370
+extends to points in space يعني احنا بدل ما عملنا
+
+471
+00:37:30,370 --> 00:37:34,530
+formula في الـ xy plane تبعد زمن لما كانت النقطة
+
+472
+00:37:34,530 --> 00:37:40,470
+b1 x1 y1 و b2 x2 y2 كنا نقول ايش المسافة بين b1 و
+
+473
+00:37:40,470 --> 00:37:44,810
+بين الاتنين هي عبارة عن جدر التربيع إلى x1 نقص x2
+
+474
+00:37:44,810 --> 00:37:49,740
+لكل تربيع زاد y1 نقص y2 لكل تربيعإلا لما يكون
+
+475
+00:37:49,740 --> 00:37:54,080
+النقطتين في الفراغ فبكون المسافة بين النقطتين P1 و
+
+476
+00:37:54,080 --> 00:38:00,540
+P2 عبارة عن الجدر التربيعي ل X1 نقص X2 تربيع Y1
+
+477
+00:38:00,540 --> 00:38:05,240
+نقص Y2 كل تربيع Z1 نقص Z2 كل تربيع بنجمحهم
+
+478
+00:38:05,240 --> 00:38:09,900
+وبناخدهم الجدر التربيعي بتطلع المسافة بين النقطتين
+
+479
+00:38:09,900 --> 00:38:16,790
+P1 و P2 وشوف هذا بتتضح هنا أكتر لما نحكي هذافرضنا
+
+480
+00:38:16,790 --> 00:38:19,950
+إنه هي عند نقطة هذه انسوا إن أنا بحكي عن معادل
+
+481
+00:38:19,950 --> 00:38:23,750
+الدائرة الـ G لو فرضنا عند نقطين في الفرار X نوت و
+
+482
+00:38:23,750 --> 00:38:28,070
+Y نوت و Z نوت وهذه X و Y و Z هي نقطين هذولة
+
+483
+00:38:28,070 --> 00:38:31,990
+النقطين المسافة بين النقطين هذولة هي عبارة عن
+
+484
+00:38:31,990 --> 00:38:38,760
+الجدر التربيعي ل X نوت هي المسافةنقص X لكل تربيع
+
+485
+00:38:38,760 --> 00:38:43,440
+ونقص Y لكل تربيع ونقص Z لكل تربيع ونقص Y لكل تربيع
+
+486
+00:38:43,440 --> 00:38:47,260
+ونقص Z لكل تربيع ونقص Y لكل تربيع ونقص Z لكل تربيع
+
+487
+00:38:47,260 --> 00:38:51,620
+ونقص Y لكل تربيع ونقص Z لكل تربيع ونقص Y لكل تربيع
+
+488
+00:38:51,620 --> 00:38:53,440
+ونقص Z لكل تربيع ونقص Z لكل تربيع ونقص Z لكل تربيع
+
+489
+00:38:53,440 --> 00:38:54,280
+ونقص Z لكل تربيع ونقص Z لكل تربيع ونقص Z لكل تربيع
+
+490
+00:38:54,280 --> 00:38:55,020
+ونقص Z لكل تربيع ونقص Z لكل تربيع ونقص Z لكل تربيع
+
+491
+00:38:55,020 --> 00:39:00,630
+ونقص Z لكل تربيع ونقص Zو .. و .. و .. و نصف قطرها
+
+492
+00:39:00,630 --> 00:39:06,290
+بساوي ايه؟ يعني بدي كل ال .. بدي الآن هوية كل
+
+493
+00:39:06,290 --> 00:39:09,790
+النقاط اللي بعد .. اللي ببعد عن x note و y note و
+
+494
+00:39:09,790 --> 00:39:15,690
+z note ببعد عنه بمقدار اللي هو جداش بمقدار بساوي
+
+495
+00:39:15,690 --> 00:39:20,870
+ايه؟ هذا اللي بدي اللي هي عبارة عن اللي هي بدي
+
+496
+00:39:20,870 --> 00:39:27,520
+اشوف هوية او معادلةاللي هي اللي بتعبّر عن كل
+
+497
+00:39:27,520 --> 00:39:32,400
+النقاط اللتي تبعد بعد ثابت عن إيه يعني بمعنى أخر
+
+498
+00:39:32,400 --> 00:39:36,860
+بدي معادلة الكورة إيش معادلة الكورة اللي هي اللي
+
+499
+00:39:36,860 --> 00:39:41,020
+المسافة قلنا إيش المسافة بين هذه وهذه نفرض أن
+
+500
+00:39:41,020 --> 00:39:43,640
+النقطة اللي بتكون على الكورة اللي بتبعد بعد ثابت
+
+501
+00:39:43,640 --> 00:39:47,580
+على مقدار إيه اسمها x y و z فبصير عند الجدر
+
+502
+00:39:47,580 --> 00:39:51,220
+التربيع إلى x نقص x نقطة تربية زي y minus y نقطة
+
+503
+00:39:51,220 --> 00:39:56,470
+كل تربية زي z minus z نقطة كل تربيةبساوي أيش اللي
+
+504
+00:39:56,470 --> 00:39:59,770
+هو a هذا الجدر التربيعي يعني المسافة بين النقطين
+
+505
+00:39:59,770 --> 00:40:04,250
+اللي هي a بساوي الجدر التربيعي لهذا مربع الجهتين
+
+506
+00:40:04,250 --> 00:40:10,450
+بتطلع عبارة عن المسافة اللي هي المسافة بتساوي
+
+507
+00:40:10,450 --> 00:40:14,510
+المربع اللي مربحها بتطلع مربع المسافة بساوي ال a
+
+508
+00:40:14,510 --> 00:40:20,700
+تربيع اللي هو البعد هذاوهذا هيعبّرلي عن كل النقاط
+
+509
+00:40:20,700 --> 00:40:26,460
+التي تبعد بعضا ثابتا عن مين اللي هي x naught و y
+
+510
+00:40:26,460 --> 00:40:31,060
+naught و z naught بمقدار البعض إيش بساوي إيه إذا
+
+511
+00:40:31,060 --> 00:40:37,080
+صارت معادلة الكرة هي عبارة عنX نقص X نقطة تربيع Y
+
+512
+00:40:37,080 --> 00:40:39,180
+نقطة تربيع Y نقطة تربيع Y نقطة تربيع Z نقطة تربيع
+
+513
+00:40:39,180 --> 00:40:40,480
+Z نقطة تربيع Z نقطة تربيع Z نقطة تربيع Z نقطة
+
+514
+00:40:40,480 --> 00:40:42,240
+تربيع Z نقطة تربيع Z نقطة تربيع Z نقطة تربيع Z
+
+515
+00:40:42,240 --> 00:40:45,760
+نقطة تربيع Z نقطة تربيع Z نقطة تربيع Z نقطة تربيع
+
+516
+00:40:45,760 --> 00:40:50,420
+نقطة تربيع Z نقطة تربيع Z نقطة تربيع Z نقطة تربيع
+
+517
+00:40:50,420 --> 00:40:54,980
+Z نقطةبمقدار a عن x0 و y0 و z0 إجت من الجذر
+
+518
+00:40:54,980 --> 00:40:58,800
+التربيعي اللي هو قيمة البعد بساول a وبعدين ربعت
+
+519
+00:40:58,800 --> 00:41:02,740
+الجهتين عشان تكون المعادلة ألطف أو تكون أسهل في
+
+520
+00:41:02,740 --> 00:41:06,360
+التعامل إذا هذه الـ sphere of radius a centered at
+
+521
+00:41:06,360 --> 00:41:13,100
+the point x0 و y0 طيب نيجي ناخد مثال بقول find the
+
+522
+00:41:13,100 --> 00:41:19,040
+center and radius of the sphere اوجد اللي هو مركزو
+
+523
+00:41:19,040 --> 00:41:22,460
+نصف قطر لسفير X تربية ز و Y تربية ز و Z تربية ز
+
+524
+00:41:22,460 --> 00:41:25,920
+ثلاثة X نقص اربعة ز زائد واحد ايش بيساوي بيساوي
+
+525
+00:41:25,920 --> 00:41:31,560
+سفر الان بده اخلي هذه شكلها ع شكل اللي جابل بشوية
+
+526
+00:41:31,560 --> 00:41:37,060
+اللي هي بده اكتبها على الشكل هذا هيك هيك هيك هيك
+
+527
+00:41:37,060 --> 00:41:43,100
+ده نشوف كيف اللي بحكيه بنجيالملموني اللي تبعات ال
+
+528
+00:41:43,100 --> 00:41:48,640
+X مع بعض هي X تربيع زي التلتة X مع بعض وال Y تربيع
+
+529
+00:41:49,720 --> 00:41:53,280
+ماعلق فيش y تربيع ايه فيش ويات هيها زيها الحالة ال
+
+530
+00:41:53,280 --> 00:41:56,460
+z تربيع مع الأربعة z هيها الأربعة z تسيبوا منها
+
+531
+00:41:56,460 --> 00:42:01,740
+دولة جد بجمعكم مش قصتهم الآن الآن زائد واحد هذا
+
+532
+00:42:01,740 --> 00:42:04,980
+انجلوليه على الجهتان يصيب يصير نقص واحد اذا انا
+
+533
+00:42:04,980 --> 00:42:09,060
+حاطيت ال x تربيع و التلاتة x هدولة مع بعض و ال y
+
+534
+00:42:09,060 --> 00:42:12,220
+تربيع هذه الحالة لأنها فيش ويات و ال z تربيع مع
+
+535
+00:42:12,220 --> 00:42:16,260
+اللي هي اللي فيها الزدات مع بعضالان و الواحد لحد
+
+536
+00:42:16,260 --> 00:42:18,900
+المطلق هذا بدي ارنجله على الجهة التانية صار ناقص
+
+537
+00:42:18,900 --> 00:42:23,560
+واحد لان ايش سويت هنا انا الان بدي احولها لمين
+
+538
+00:42:23,560 --> 00:42:26,700
+للصيغة اللي قبل بشوية يعني بدي اعمل هذا على صورة
+
+539
+00:42:26,700 --> 00:42:29,860
+مربع كامل و هذا مربع كامل و هذا مربع كامل هذه
+
+540
+00:42:29,860 --> 00:42:35,000
+جاهزة مربع كامل دل هدولة طب ايش بدي اسوي بدي اكمل
+
+541
+00:42:35,000 --> 00:42:41,640
+المربع يعني بمعنى اخر بدي احط الحد هذا هذااللي هو
+
+542
+00:42:41,640 --> 00:42:45,200
+اللي بيخليه لهذا عبارة عن إيه عشان عندي X تربيع زي
+
+543
+00:42:45,200 --> 00:42:52,570
+تلاتة X ماشي الان بدي أخلي هذا بحيث أنه يصير Xاللي
+
+544
+00:42:52,570 --> 00:42:55,670
+هو طبعا زائد تلاتة على اتنين لكل تربيع زائد minus
+
+545
+00:42:55,670 --> 00:42:58,230
+اتنين لكل تربيع زياد واي تربيع بيساوي عشان عشان
+
+546
+00:42:58,230 --> 00:42:59,110
+عشان عشان عشان عشان عشان عشان عشان عشان عشان عشان
+
+547
+00:42:59,110 --> 00:43:01,410
+عشان عشان عشان عشان عشان عشان عشان عشان عشان عشان
+
+548
+00:43:01,410 --> 00:43:02,010
+عشان عشان عشان عشان عشان عشان عشان عشان عشان عشان
+
+549
+00:43:02,010 --> 00:43:02,570
+عشان عشان عشان عشان عشان عشان عشان عشان عشان عشان
+
+550
+00:43:02,570 --> 00:43:06,590
+عشان عشان عشان عشان عشان
+
+551
+00:43:18,280 --> 00:43:24,260
+باخد اللي هو الحد المعامل x و بجسمه على 2 ثلاثة
+
+552
+00:43:24,260 --> 00:43:28,300
+على اتنين و بربعه و باضيفه صار عبارة عن تلاتة على
+
+553
+00:43:28,300 --> 00:43:33,420
+اتنين بيصير تسعة على اربعة نفس الاشي الحد هذا اللي
+
+554
+00:43:33,420 --> 00:43:38,490
+هو عبارة عن ايشاللي هو ناقص أربعة بجسمه على اتنين
+
+555
+00:43:38,490 --> 00:43:42,450
+بيصير ناقص اتنين بربعه بيصير جداش أربعة صار هذا
+
+556
+00:43:42,450 --> 00:43:46,970
+مربع كامل وهذا مربع كامل وهذا مربع كامل اللي ضفته
+
+557
+00:43:46,970 --> 00:43:50,030
+هنا التسعة على أربعة بضيفه على الطرف الثاني عشان
+
+558
+00:43:50,030 --> 00:43:55,470
+تظل الميزان واحدالاربعة ضيفتها ده ضيفها ده ده ده
+
+559
+00:43:55,470 --> 00:43:55,610
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+560
+00:43:55,610 --> 00:43:59,990
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+561
+00:43:59,990 --> 00:44:01,990
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+562
+00:44:01,990 --> 00:44:02,890
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+563
+00:44:02,890 --> 00:44:03,430
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+564
+00:44:03,430 --> 00:44:11,670
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+565
+00:44:11,670 --> 00:44:20,200
+ده دوهذا مربع كامل وجمعت هذول ربعها طلعت 21 على 4
+
+566
+00:44:20,200 --> 00:44:24,720
+الان جاهزة المعادلة ان اقول hence the center of
+
+567
+00:44:24,720 --> 00:44:28,900
+the sphere is ايش ال center اللي هو ناقص تلاتة على
+
+568
+00:44:28,900 --> 00:44:32,780
+اتنين لان هي الصورة يا جماعة هي الصورة اللي هو ايه
+
+569
+00:44:32,780 --> 00:44:36,760
+هاد ال center ها X نوت لما كانت ناقص Y نوت Z نوت
+
+570
+00:44:36,760 --> 00:44:41,320
+وهي نصف القطر اذا هنا صارت هذا عبارة عن ناقص ناقص
+
+571
+00:44:41,320 --> 00:44:46,250
+يعني ناقص تلاتة على اتنينوهذه ناقص جاهزة عبارة عن
+
+572
+00:44:46,250 --> 00:44:51,030
+2 وهي قيمة z طبعا وليس Y وY ناقص صفر لكل تربيع
+
+573
+00:44:51,030 --> 00:44:56,330
+يعني إيه صفر صارت هذه الـ Center وهذه جدرها اللي
+
+574
+00:44:56,330 --> 00:45:00,270
+هو عبارة عن الجدر ال 21 على 2 هو عبارة عن ال
+
+575
+00:45:00,270 --> 00:45:05,450
+Radius لمين لإسفل اللي عندي الآن
+
+576
+00:45:08,170 --> 00:45:13,310
+بظل عندى اللى هو اللى هى مثال اخر here are some
+
+577
+00:45:13,310 --> 00:45:15,570
+geometric interpretations of inequalities and
+
+578
+00:45:15,570 --> 00:45:21,030
+equations involving spheres الآن نشوف اللى هى بعض
+
+579
+00:45:21,030 --> 00:45:26,890
+اللى هى المعادلات والinqualities اللى تتضمن
+
+580
+00:45:26,890 --> 00:45:31,880
+spheres فى داخلها او فى مضمونها نشوف كيفعند X
+
+581
+00:45:31,880 --> 00:45:36,160
+تربيع ZY تربيع ZZ تربيع أصغر من 4 احنا بنعرف X
+
+582
+00:45:36,160 --> 00:45:41,140
+تربيع ZY تربيع ZZ تربيع بتساوي 4 هي عبارة عن
+
+583
+00:45:41,140 --> 00:45:45,140
+الاسفير زي اللي جابل بشوية اللي مركزها 0000 لأن X
+
+584
+00:45:45,140 --> 00:45:50,380
+تربيع ZZ تربيع ZY تربيع ZZ تربيع بساوي 4 عبارة عن
+
+585
+00:45:50,380 --> 00:45:56,390
+كرة مركزها 0000 و نصف قطرها 2لما أقول strictly
+
+586
+00:45:56,390 --> 00:46:01,850
+أصغر من أربعة يعني نابي داخل الكرة بدون جشرتها
+
+587
+00:46:01,850 --> 00:46:05,890
+يعني the interior of the sphere X تربيع زي Y تربيع
+
+588
+00:46:05,890 --> 00:46:10,310
+زي Z تربيع بيساوي أربعة لما أقول هذا أصغر أو يساوي
+
+589
+00:46:10,310 --> 00:46:15,130
+معناته أنا قولت ال interior جوا الكرة مع الجشرة ده
+
+590
+00:46:15,130 --> 00:46:20,730
+solidاللي هو المجسم ball bounded مجسم الكرة أو الـ
+
+591
+00:46:20,730 --> 00:46:24,970
+ball الـ bounded by the سفير اللي هي الجشرة الـ X
+
+592
+00:46:24,970 --> 00:46:29,990
+تربيع Z واطر بزت تربيع أربعة بالإضافة إلى داخلها
+
+593
+00:46:29,990 --> 00:46:35,190
+يعني ال solid المتكون من السفير ومن بداخل السفير
+
+594
+00:46:35,190 --> 00:46:40,250
+معناته اللي هي بمعنى آخر alternatively the sphere
+
+595
+00:46:40,250 --> 00:46:43,290
+X تربيع Z واطر بزت تربيع أربعة
+
+596
+00:46:45,410 --> 00:46:51,030
+together with its interior مع اللي هي إيش دا كلها
+
+597
+00:46:51,030 --> 00:46:54,730
+لانيجي ل X تربية زي Y تربية زي Z تربية أكبر أكبر
+
+598
+00:46:54,730 --> 00:46:57,830
+من أربعة كلكم عارف قال هذي مادام هذك اللي جوا أو
+
+599
+00:46:57,830 --> 00:47:01,310
+هذك اللي جوا مع الجشرة هذا هو عبارة عن اللي برا
+
+600
+00:47:01,310 --> 00:47:04,290
+بدون الجشرة the exterior of the sphere X تربية زي
+
+601
+00:47:04,290 --> 00:47:08,920
+Y تربية زي Z تربية بساوية أربعة له خارج الكرةإلى
+
+602
+00:47:08,920 --> 00:47:12,340
+ما لنهايه بيظلوا إلا لما أقول X تربيع زي Y تربيع
+
+603
+00:47:12,340 --> 00:47:15,520
+زي Z تربيع يساوي أربعة هي السفير هذه الجشرة يعني
+
+604
+00:47:15,520 --> 00:47:20,360
+نفسها أه لما أقول Z أصغر يساوي سفر معناته أنا بدي
+
+605
+00:47:20,360 --> 00:47:24,480
+الجشرة بس اللي أيش اللي أصغر أو يساوي سفر يعني
+
+606
+00:47:24,480 --> 00:47:28,620
+الجزء من الكرة اللي في المنطقة السالبة the lower
+
+607
+00:47:28,620 --> 00:47:33,000
+hemisphere يعني نص الكرة cut from the sphere X
+
+608
+00:47:33,000 --> 00:47:36,540
+تربيع زي Y تربيع زي Z تربيع يساوي أربعة by the XY
+
+609
+00:47:36,540 --> 00:47:41,350
+plane يعني عبارة عنهذا تقاطع هذه الـ sphere مع
+
+610
+00:47:41,350 --> 00:47:45,090
+اللي اللي هو نص العالم تبعنا اللي هو زي الأصغر
+
+611
+00:47:45,090 --> 00:47:49,370
+السفر اللي تحت معناته صار عبارة عن نص الكرة اللي
+
+612
+00:47:49,370 --> 00:47:55,850
+تحت ماشي الحال هذه هي بتكون اللي هيالبثال الأخير
+
+613
+00:47:55,850 --> 00:48:00,590
+بكون عندكم معاكم exercise جماعة ياريت ان انتوا هذه
+
+614
+00:48:00,590 --> 00:48:05,090
+ال exercise هي لحد الأدنى بفضل تحلوها هذه كلها ال
+
+615
+00:48:05,090 --> 00:48:09,630
+exercises عندك ايش هذا بتمثل X بساوة 2 و Y بساوة 3
+
+616
+00:48:09,630 --> 00:48:14,570
+قلناها هي عندك اللي هو دائرةأو كورة الكورة هي
+
+617
+00:48:14,570 --> 00:48:18,670
+بتعرف مركزها وهي Z بساوية 0 لأن الكورة في ال X Y
+
+618
+00:48:18,670 --> 00:48:23,110
+plane هذه الاخيرة ربما بتحل تشوف ايش النقص ده وهي
+
+619
+00:48:23,110 --> 00:48:27,270
+عندك كمان اللي هي ال homework اللي عندنا وهيك
+
+620
+00:48:27,270 --> 00:48:31,290
+بتكون تمت بحمد الله والسلام عليكم ورحمة الله وإلى
+
+621
+00:48:31,290 --> 00:48:35,910
+لقاء آخر سامحوني على طول المحاضرة لإن كان في حديث
+
+622
+00:48:35,910 --> 00:48:40,570
+في البداية عن اللي هو ال course online والسلام
+
+623
+00:48:40,570 --> 00:48:40,830
+عليكم
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/C_FiCc2vxyg.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/C_FiCc2vxyg.srt
new file mode 100644
index 0000000000000000000000000000000000000000..ee76cdade2373142bb418cea7693a5ea603d29ad
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/C_FiCc2vxyg.srt
@@ -0,0 +1,2103 @@
+1
+00:00:00,710 --> 00:00:04,030
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,030 --> 00:00:09,030
+وبركاته هذه هي المحاضرة الثامنة مساق تفاضل وتكامل
+
+3
+00:00:09,030 --> 00:00:15,630
+جيم وهي المحاضرة الأولى في Chapter ثلاثة عشر تحت عنوان
+
+4
+00:00:15,630 --> 00:00:20,510
+vector valued functions and motion in space أحنا
+
+5
+00:00:20,510 --> 00:00:24,250
+يعني كنا اتعلمنا شو معنى الـ .. الـ functions في الـ
+
+6
+00:00:24,250 --> 00:00:28,310
+.. سواء في الإعدادية أو حتى مرت معنا في تفاضل ألف
+
+7
+00:00:28,310 --> 00:00:32,210
+لكن كنا نحكي عن real valued function لما نقول real
+
+8
+00:00:32,210 --> 00:00:37,330
+valued function معناته هي من domain  اللي عندنا real
+
+9
+00:00:37,330 --> 00:00:41,790
+valued إلى R لأنه نقول عنها real valued بينما لما
+
+10
+00:00:41,790 --> 00:00:45,470
+نقول vector valued function الـ vector valued
+
+11
+00:00:45,470 --> 00:00:50,170
+function يعني قيمة الدالة بدها تكون وين؟ في الـ
+
+12
+00:00:50,170 --> 00:00:52,790
+vector عبارة عن vector لما نقول vector valued
+
+13
+00:00:52,790 --> 00:00:58,700
+function يعني اللي هو range  بيكون vector أو set
+
+14
+00:00:58,700 --> 00:01:03,620
+of vectors بينما اللي هو الـ domain طبعا حسب ما
+
+15
+00:01:03,620 --> 00:01:07,840
+اللي هو ابن حكي أو اللي هو ابن عمله هنشوف إيش
+
+16
+00:01:07,840 --> 00:01:10,560
+معنى الكلام هذا إذا أنا ما أقول complex valued
+
+17
+00:01:10,560 --> 00:01:14,060
+function يعني معناته قيمة الدالة complex real
+
+18
+00:01:14,060 --> 00:01:17,820
+valued معناته قيمة الدالة real vector valued
+
+19
+00:01:17,820 --> 00:01:21,920
+معناته قيمة الدالة الدالة عبارة عن vector and
+
+20
+00:01:21,920 --> 00:01:26,380
+motion in space اللي هي الحركة في الفراغ، نشوف
+
+21
+00:01:26,380 --> 00:01:30,100
+احنا اللي هو نبدأ بالـ section الأول في ثلاثة عشر
+
+22
+00:01:30,100 --> 00:01:33,420
+اللي هو ثلاثة عشر واحد اللي اليوم هنحكي فيه في هذا
+
+23
+00:01:33,420 --> 00:01:37,700
+المحاضرة ثلاثة عشر واحد اللي هو curves in space and
+
+24
+00:01:37,700 --> 00:01:43,780
+their tangents اللي هي المنحنيات في الفراغ ومماسات
+
+25
+00:01:43,780 --> 00:01:48,040
+المنحنيات، نشوف إيش معنى المنحنى في الفراغ وإيش
+
+26
+00:01:48,040 --> 00:01:52,950
+معنى اللي هو مماس المنحنى، الآن خلّينا يا جماعة بس
+
+27
+00:01:52,950 --> 00:01:57,390
+خلّينا نروح للرسمة، خلّينا نتخيل أنه في جزيء بيسير
+
+28
+00:01:57,390 --> 00:02:01,910
+في الفراغ عبر اللي هو إيش؟ curve معين، وليكن هذا الـ
+
+29
+00:02:01,910 --> 00:02:06,970
+curve، الآن نفترض أنه الجزيء بدأ من النقطة اللي هنا
+
+30
+00:02:07,500 --> 00:02:12,320
+بدأ عند نقطة وهي عند وقت محدد اسمه T note يعني
+
+31
+00:02:12,320 --> 00:02:15,580
+عند T note اللي هو بدأ من هنا، هذه الآن في
+
+32
+00:02:15,580 --> 00:02:20,860
+الإحداثيات في الـ X و في الـ Y و في الـ Z، الآن لما
+
+33
+00:02:20,860 --> 00:02:25,480
+يتحرك مع مرور الوقت، مر الوقت الآن لما يتحرك مع
+
+34
+00:02:25,480 --> 00:02:30,950
+مرور الوقت الجزيء هذا وصل هنا مثلا، إذا الآن مع مرور
+
+35
+00:02:30,950 --> 00:02:37,310
+الوقت يتحرك الجزيء وتختلف إحداثياته، يعني بيصير الـ
+
+36
+00:02:37,310 --> 00:02:41,550
+X والـ Y والـ Z، هنا هي الإحداثيات الـ X، وهي إحداثيات
+
+37
+00:02:41,550 --> 00:02:46,370
+الـ Y، وفوق اللي هي إحداثيات من الـ Z، ماشي كمان شوية
+
+38
+00:02:46,370 --> 00:02:51,370
+انتقل هنا برضه اتغيرت إحداثياته، ماشي لعند الـ hand
+
+39
+00:02:51,370 --> 00:02:57,350
+اتغيرت إحداثياته وصارت النقطة هذه، لاحظوا إنه تتغير
+
+40
+00:02:57,350 --> 00:03:03,030
+الإحداثيات قيمة X وقيمة Y وقيمة Z تتغير بمرور
+
+41
+00:03:03,030 --> 00:03:08,850
+الوقت، يعني بمعنى آخر اللي هي X تبعته عبارة عن
+
+42
+00:03:08,850 --> 00:03:14,310
+function of T و Y تبعته function of T و Z تبعته
+
+43
+00:03:14,310 --> 00:03:19,020
+function of T يعني الإحداثيات صارت X اللي هي عبارة
+
+44
+00:03:19,020 --> 00:03:24,500
+بتساوي F of T والـ Y عبارة عن G of T والـ Z عبارة
+
+45
+00:03:24,500 --> 00:03:29,360
+عن H of T، ليش كتبناها H of T؟ لأن زي ما شوفنا أنه
+
+46
+00:03:29,360 --> 00:03:33,500
+يتغير اللي هو مكان الـ particle بتغير الزمن، بمعنى
+
+47
+00:03:33,500 --> 00:03:36,440
+آخر تتغير الإحداثيات اللي هي الـ X والـ Y والـ Z
+
+48
+00:03:36,440 --> 00:03:40,880
+بتغير الزمن، إذا طبيعي ومنطقي أنه نعبر عن الـ X، F
+
+49
+00:03:40,880 --> 00:03:45,940
+of T والـ Y، G of T والـ Z، H of T، الآن وصل الـ
+
+50
+00:03:45,940 --> 00:03:50,920
+particle هنا، بهي اللي عندي اللي هي الإحداثيات
+
+51
+00:03:50,920 --> 00:03:55,820
+تبعته، الآن بنقدر نجيب الـ position vector له يعني
+
+52
+00:03:55,820 --> 00:03:59,780
+الآن position vector بعد مرور T من الزمن للـ
+
+53
+00:03:59,780 --> 00:04:04,140
+particle اللي هو هذا الـ position vector الـ O P هذا
+
+54
+00:04:04,140 --> 00:04:09,020
+الـ R هو عبارة عن R of T، هيصير إيش ما قاله ده الـ
+
+55
+00:04:09,020 --> 00:04:11,960
+R of T لأن الـ position vector كان هنا وبعدين كان
+
+56
+00:04:11,960 --> 00:04:15,190
+هنا، حسب الإحداثيات، ما عرفناش الـ position vector
+
+57
+00:04:15,190 --> 00:04:19,770
+اللي كان هنا، الآن الـ position vector R نفسه صار
+
+58
+00:04:19,770 --> 00:04:23,590
+اللي هو يعتمد على T في التغير، والـ T عبارة عن إيه؟
+
+59
+00:04:23,590 --> 00:04:28,030
+يا جماعة، عبارة عن عدد، عشان هيك بنقول عنه الـ R of
+
+60
+00:04:28,030 --> 00:04:31,550
+T، هذا الـ vector عبارة عن vector valued function
+
+61
+00:04:31,550 --> 00:04:37,600
+يتغير بمرور الزمن T، يعني اللي هو domain اللي هو
+
+62
+00:04:37,600 --> 00:04:41,600
+اللي بيعتمد عليه اللي هي الـ T، عبارة عن R و range
+
+63
+00:04:41,600 --> 00:04:45,360
+عبارة عن vector، عشان هيك بنسميه هذا عبارة عن
+
+64
+00:04:45,360 --> 00:04:50,700
+vector valued function، وخلّينا نرجع للتعريف اللي
+
+65
+00:04:50,700 --> 00:04:53,380
+احنا بدنا نبدأ فيه أو الحديث اللي بدنا نبدأ فيه
+
+66
+00:04:53,380 --> 00:04:56,860
+اللي في الواقع أنا حكيته على الرسمة، الآن بنقول when
+
+67
+00:04:56,860 --> 00:04:59,440
+a particle moves هذا اللي حكيته بس بشكل أساسي يكون في
+
+68
+00:04:59,440 --> 00:05:02,840
+الصورة والكاملة when a particle moves through a
+
+69
+00:05:02,840 --> 00:05:07,320
+space space during a time interval I, we think of a
+
+70
+00:05:07,320 --> 00:05:10,600
+particle’s coordinates as a function defined on I
+
+71
+00:05:10,600 --> 00:05:15,060
+يعني لما يتحرك الـ particle بتصير مواقعه إحداثية X
+
+72
+00:05:15,060 --> 00:05:17,960
+function في T، F of T و Y بيساوي G of T و Z بتساوي
+
+73
+00:05:17,960 --> 00:05:21,360
+H of T و T عبارة عن element in I اللي هي interval
+
+74
+00:05:21,360 --> 00:05:24,920
+من الزمن، يعني فترة من الزمن، عشر دقائق، نص ساعة،
+
+75
+00:05:24,920 --> 00:05:30,100
+ساعتين، ثلاثة، كلها أعداد، أعداد حقيقية، الآن
+
+76
+00:05:30,100 --> 00:05:33,520
+الـ points X و Y و Z بتساوي F of T و G of T و H of
+
+77
+00:05:33,520 --> 00:05:39,680
+T makes up the curve in space that we call اللي هو
+
+78
+00:05:39,680 --> 00:05:45,890
+path of the particle الآن زي ما قلنا قبل شوية أنه
+
+79
+00:05:45,890 --> 00:05:50,990
+مع مرور الزمن، مع مرور الزمن بيجي عندي بيسير لأن
+
+80
+00:05:50,990 --> 00:05:54,450
+كان الـ particle هنا وصار هنا، يعني رسم في طريقه
+
+81
+00:05:54,450 --> 00:06:02,210
+curve، هذا الـ curve بنسميه the path of the particle هو
+
+82
+00:06:02,210 --> 00:06:06,270
+عبارة عن طريق الـ particle اللي عبر الـ curve، لأن الـ
+
+83
+00:06:06,270 --> 00:06:09,210
+equations هذه الـ x والـ y والـ z بنسميها
+
+84
+00:06:09,210 --> 00:06:13,290
+parametric equations يعني عملنا parameterization
+
+85
+00:06:13,290 --> 00:06:20,190
+للـ curve بواسطة أن هذه تعبر عن الـ curve عن طريق
+
+86
+00:06:20,190 --> 00:06:24,490
+براميتر الـ T اللي في هذه الحالة اسمه الزمن، الآن a
+
+87
+00:06:24,490 --> 00:06:28,070
+curve in space can be represented in a vector form
+
+88
+00:06:28,070 --> 00:06:31,530
+زي ما شوفنا قبل شوية الـ R of T كيف عملنا عبارة عن
+
+89
+00:06:31,530 --> 00:06:36,230
+O B اللي هو F of T في اتجاه I، زي G of T في اتجاه J، زي H
+
+90
+00:06:36,230 --> 00:06:39,510
+of T في اتجاه K، هذا اللي هو اللي بنسميه
+
+91
+00:06:39,510 --> 00:06:45,970
+الـ position vector in time T اللي هو بعبر عن مسار
+
+92
+00:06:45,970 --> 00:06:50,430
+الـ particle، الـ position vector، الآن الـ functions
+
+93
+00:06:50,430 --> 00:06:54,250
+هذه الـ F و G و H هي عبارة عن الـ component
+
+94
+00:06:54,250 --> 00:06:59,490
+functions اللي هي عبارة عن دوال حقيقية وقيمتها F
+
+95
+00:06:59,490 --> 00:07:02,450
+of T عبارة عن عدد، و G of T عبارة عن عدد، و H of T
+
+96
+00:07:02,450 --> 00:07:05,970
+عبارة عن عدد، لكن الدالة النهائية اللي هي هذه عبارة
+
+97
+00:07:05,970 --> 00:07:09,690
+عن vector valued function اللي هي دالة قيمتها
+
+98
+00:07:09,690 --> 00:07:12,610
+vector اللي هو اللي بعبر عنه الـ position vector
+
+99
+00:07:13,110 --> 00:07:19,830
+الآن نفكر في طريقة الـ particle الذي قبل، نسميه
+
+100
+00:07:19,830 --> 00:07:22,930
+الـ particle as the curve traced by R يعني اللي هو
+
+101
+00:07:22,930 --> 00:07:29,150
+المنحنى اللي يرسم بواسطة مين أو يشق بطريقة، بواسطة
+
+102
+00:07:29,150 --> 00:07:34,010
+مين؟ الـ vector، الـ position vector R هذه اللي هي
+
+103
+00:07:34,010 --> 00:07:39,560
+الشيء اللي عندي، اللي مثال على اللي هو curve in
+
+104
+00:07:39,560 --> 00:07:43,300
+space، الآن يا جماعة الـ curve in space من الشغلات
+
+105
+00:07:43,300 --> 00:07:48,290
+اللي مش سهل أن نرسمها باليد، يعني احنا ممكن ناخد
+
+106
+00:07:48,290 --> 00:07:52,310
+مثال مجدول عليه وهذا مثال من أبسط الأمثلة عشان
+
+107
+00:07:52,310 --> 00:07:56,010
+نرسمه ونوضح كيف نرسمه باليد، لكن اللي هو in
+
+108
+00:07:56,010 --> 00:08:00,310
+general، الرسمات اللي هي بواسطة الحاسوب، خلّيني ناخد
+
+109
+00:08:00,310 --> 00:08:03,970
+مثال، هاي عندي مثال على vector valued function R of
+
+110
+00:08:03,970 --> 00:08:08,840
+T بتساوي sin ثلاثة T و cos T في اتجاه I، sin ثلاثة T
+
+111
+00:08:08,840 --> 00:08:12,400
+sin T في اتجاه الـ J زائد T في اتجاه الـ K، هذا
+
+112
+00:08:12,400 --> 00:08:16,120
+الآن الـ function أو الـ vector valued function أو
+
+113
+00:08:16,120 --> 00:08:20,900
+الـ position vector اللي بيسير بواسطة الـ particle
+
+114
+00:08:20,900 --> 00:08:25,660
+هي عند نقطة صفر T أو صفر K هنا بعدين انتقل بعدين
+
+115
+00:08:25,660 --> 00:08:29,200
+انتقل بعدين انتقل ومشي لما نوصل عند النقطة هذه
+
+116
+00:08:29,200 --> 00:08:35,590
+وهكذا، هذه هي عبارة عن رسمة من الـ position أو اللي
+
+117
+00:08:35,590 --> 00:08:39,630
+هو الـ particle اللي بيسير عبر الـ curve اللي أمامي
+
+118
+00:08:39,630 --> 00:08:44,470
+هذا، هذا اللي بده يعمل إيه؟ الـ curve اللي أمامنا لما
+
+119
+00:08:44,470 --> 00:08:49,950
+ناخد T من صفر إلى عدد معين، Similarly بنفس الطريقة
+
+120
+00:08:49,950 --> 00:08:53,910
+مثال آخر R of T بيساوي Cos T في جهة I، Sine T في
+
+121
+00:08:53,910 --> 00:08:58,630
+جهة J، زي Sine 2T في جهة K، برضه هذا لو أجينا رسمناه
+
+122
+00:08:58,630 --> 00:09:03,810
+بالكمبيوتر، اللي هو غيّرنا T بأرقام حقيقية هنصل ..
+
+123
+00:09:03,810 --> 00:09:07,970
+هنشوف على فترة معينة زي ما بدك بتيجي بتاخد بتلاقي
+
+124
+00:09:07,970 --> 00:09:11,810
+اللي هو هاي الـ curve اللي عندك اللي بيصير، الآن هاي
+
+125
+00:09:11,810 --> 00:09:15,290
+الـ curve أكثر تعقيدا، R of T، R of T بيساوي أربعة زي sin
+
+126
+00:09:15,290 --> 00:09:21,970
+عشرين T cos T في جهة I زائد أربعة زي sin عشرين T في sin T في
+
+127
+00:09:21,970 --> 00:09:28,690
+جهة J زائد cos عشرين T في جهة K، هذا مثال، رسمته هي، طبعا
+
+128
+00:09:28,690 --> 00:09:31,950
+تقول لي ارسمها أنت أو نرسمها باليد، طبعا لا يمكن أن
+
+129
+00:09:31,950 --> 00:09:34,890
+احنا نعرف نرسمها باليد أو صعب جدا، ليش نقول لا
+
+130
+00:09:34,890 --> 00:09:39,050
+يمكن، يمكن في ناس عندها قدرات أو يعني أمور فنية في
+
+131
+00:09:39,050 --> 00:09:43,470
+الرسم تقدر ترسم جزء منها، لكن in general هذه
+
+132
+00:09:43,470 --> 00:09:47,950
+الرسمات بواسطة الحاسوب، الآن
+
+133
+00:09:50,880 --> 00:09:54,800
+اللي حكينا عنها الـ R of T، زي ما قلنا الـ equation
+
+134
+00:09:54,800 --> 00:09:57,220
+اثنين اللي حكينا عنها قبل شوية defines R as a
+
+135
+00:09:57,220 --> 00:10:00,400
+vector function of real variable T on the interval
+
+136
+00:10:00,400 --> 00:10:03,200
+I، اللي هي سميناها R of T بتساوي F of T في اتجاه I
+
+137
+00:10:03,200 --> 00:10:07,300
+زائد G of T في اتجاه J زائد H of T في اتجاه K، هذا
+
+138
+00:10:07,300 --> 00:10:11,500
+بتسميها زي ما قلت vector valued function of a real
+
+139
+00:10:11,500 --> 00:10:16,920
+variable يعني قيمة الـ R قيمة الدالة vector والـ
+
+140
+00:10:16,920 --> 00:10:21,820
+domain تبعها اللي هو T، قيمة T عبارة عن real number
+
+141
+00:10:22,370 --> 00:10:25,290
+More generally, a vector valued function or a
+
+142
+00:10:25,290 --> 00:10:28,130
+vector function with domain D is a rule that
+
+143
+00:10:28,130 --> 00:10:30,670
+assigns a vector in space to each element in D
+
+144
+00:10:30,670 --> 00:10:33,370
+يعني اللي بقوله هنا أن الـ vector valued function
+
+145
+00:10:33,370 --> 00:10:38,410
+هذه بتاخد كل عنصر في domain  D و بتودي لمين؟ لـ
+
+146
+00:10:38,410 --> 00:10:41,750
+vector in space، زي ما قلنا اللي هو vector valued
+
+147
+00:10:41,750 --> 00:10:45,950
+function، الآن زي ما حكيت في الأول أنا real valued
+
+148
+00:10:45,950 --> 00:10:50,430
+function اللي هي real valued functions معناته إنه
+
+149
+00:10:50,430 --> 00:10:54,830
+قيمتها real، domain هي إيش؟ هو حسب ما بدي أنا، الآن
+
+150
+00:10:54,830 --> 00:10:57,650
+هذه بنسميها الـ Real Valued Function، عادة بنسميها
+
+151
+00:10:57,650 --> 00:11:02,370
+الـ Scalar Function عشان نميزها عن الـ Vector
+
+152
+00:11:02,370 --> 00:11:05,990
+Function، الآن الـ Components اللي حكيناه قبل شوية لما
+
+153
+00:11:05,990 --> 00:11:09,010
+قلنا R of T بتساوي F of T في اتجاه الـ I زائد
+
+154
+00:11:09,010 --> 00:11:12,190
+G of T في اتجاه الـ J زائد H of T في اتجاه الـ K
+
+155
+00:11:12,190 --> 00:11:16,670
+الـ F of T والـ H of T والـ G of T هذه عبارة عن
+
+156
+00:11:16,670 --> 00:11:19,550
+أمثلة اللي هي الـ Components للـ R، أمثلة على
+
+157
+00:11:19,550 --> 00:11:25,770
+Scalar Functions. The domain of a vector-valued
+
+158
+00:11:25,770 --> 00:11:29,250
+function is the common domain of its component، إيش
+
+159
+00:11:29,250 --> 00:11:33,750
+هذه يعني؟ بقول لك لما نقول اللي هو R of T مثلا
+
+160
+00:11:33,750 --> 00:11:38,610
+بتساوي F of T في اتجاه I، G of T في اتجاه J، H of
+
+161
+00:11:38,610 --> 00:11:43,330
+T في اتجاه K، domain  هذه الـ function هو الـ common
+
+162
+00:11:43,330 --> 00:11:46,210
+domain بين التلاتة، يعني الـ domain المشترك بين
+
+163
+00:11:46,210 --> 00:11:51,080
+التلاتة، الـ domain المشترك بين domain F of T و H of T
+
+164
+00:11:51,080 --> 00:11:58,420
+ومين؟ و G of T، و H of T، يعني الآن لو أجينا
+
+165
+00:11:58,420 --> 00:12:01,520
+للـ function، طبعا هذا المثال يقول graph the vector
+
+166
+00:12:01,520 --> 00:12:04,120
+function R of T بتساوي Cosine T في اتجاه I زائد Cosine
+
+167
+00:12:04,120 --> 00:12:07,720
+T في اتجاه J زائد T في اتجاه K، لو شفنا الـ vector
+
+168
+00:12:07,720 --> 00:12:10,820
+valued function هذه وقال لي إيش الـ domain؟ domain  ها
+
+169
+00:12:10,820 --> 00:12:15,280
+هو الـ domain المشترك لهذه وهذه وهذه، سهلة الـ domain
+
+170
+00:12:15,280 --> 00:12:18,990
+الـ cosine، كل R، الـ cosine تشتغل على كل الـ R، الـ
+
+171
+00:12:18,990 --> 00:12:21,410
+cosine الـ T بتشتغل على كل الـ R، والـ T بتشتغل
+
+172
+00:12:21,410 --> 00:12:23,590
+على كل الـ R، إذا الـ domain هو الـ domain المشترك
+
+173
+00:12:23,590 --> 00:12:26,370
+بين إيه؟ الـ domain المشترك بين هذين؟ اللي هو
+
+174
+00:12:26,370 --> 00:12:30,770
+عبارة عن R كله، عشان إيه؟ كما نقول domain R of T
+
+1
+
+223
+00:16:10,940 --> 00:16:15,660
+بتمشي على اللي هي الأسطوانة يعني الآن بتكون عند
+
+224
+00:16:15,660 --> 00:16:19,540
+النقطة هذه إحداثياتها عبارة عن  واحد، صفر، صفر
+
+225
+00:16:19,540 --> 00:16:22,240
+على اتنين، صفر، صفر على اتنين، واحد يعني
+
+226
+00:16:22,240 --> 00:16:27,000
+عبارة عن صفر ونصف، صفر،  نصف هذه النقطة لاحظ
+
+227
+00:16:27,000 --> 00:16:30,480
+أنه مع مرور الزمن هتدلها تمشي دائما تمشي على
+
+228
+00:16:30,480 --> 00:16:33,780
+الأسطوانة لأن العلاقة ثابتة  x بتساوي y على
+
+229
+00:16:33,780 --> 00:16:34,200
+اتنين
+
+230
+00:16:37,260 --> 00:16:43,200
+قيمة الـ X التي هي عبارة عن كوساين
+
+231
+00:16:43,200 --> 00:16:47,400
+الـ π التي هي عبارة عن  ناقص واحد، و Y ساين التي هي
+
+232
+00:16:47,400 --> 00:16:52,060
+عبارة عن صفر، بنرجع التي هي ساين الـ π، و Z بتساوي
+
+233
+00:16:52,060 --> 00:16:55,620
+التي هي كذا التي هي بتساوي π، بنرتفع π
+
+234
+00:16:55,620 --> 00:17:00,630
+بالإحداثية اللي ذكرناها برضه بتظهر على أسطوانة الآن
+
+235
+00:17:00,630 --> 00:17:04,810
+بس لاحظ ملاحظة التالية: x بتساوي π على اتنين
+
+236
+00:17:04,810 --> 00:17:08,770
+بترجع اللي هي لأنها بتدور، بتكون اكتملت لل cosine و
+
+237
+00:17:08,770 --> 00:17:12,090
+ال sine بتصير cosine π على اتنين اللي هي عبارة
+
+238
+00:17:12,090 --> 00:17:16,190
+عن واحد، و sine π على اتنين، 2π بتساوي صفر
+
+239
+00:17:16,190 --> 00:17:20,630
+cosine 2π واحد، سالب واحد آسف، وال sine اللي
+
+240
+00:17:20,630 --> 00:17:24,220
+هي 2π صفر، عند t بتساوي π على اتنين يعني هذه
+
+241
+00:17:24,220 --> 00:17:28,780
+بتسير على جبال هذه بالظبط، وبتبدأ دورة جديدة يعني
+
+242
+00:17:28,780 --> 00:17:34,620
+الآن بعد كل 2π بتسير اللي هي بتكمل حالها
+
+243
+00:17:34,620 --> 00:17:39,400
+عشان هيك بنسميها هذه عبارة عن helix، الأسطوانة اللي
+
+244
+00:17:39,400 --> 00:17:42,760
+هي رسمة اللي هو اللي عندي، فبتكون ال helix هي
+
+245
+00:17:42,760 --> 00:17:48,080
+بالشكل اللي أمامي، هذه اللي هي رسمة cos t، sin t،
+
+246
+00:17:48,080 --> 00:17:54,960
+t، هذه هي الرسمة اللي اتغلبنا عليها، وهي بسيطة
+
+247
+00:17:54,960 --> 00:18:03,100
+الآن بس شغلة سريعة، لو اجينا يعني الآن بدي أضغط
+
+248
+00:18:03,100 --> 00:18:06,660
+السلك، أو أمد السلك لفوق، يعني يا إما بدي أضغط السلك،
+
+249
+00:18:06,660 --> 00:18:11,040
+أخليه مضغوط، أو أضغطه، كيف بدك تضغطه، وكيف بدك تفرده؟
+
+250
+00:18:11,040 --> 00:18:14,860
+شوف كيف بدي أضغطه، بدي أضغطه بشغلتين، خليني أقول الشغلة
+
+251
+00:18:14,860 --> 00:18:19,850
+الأولى: لو اجينا ضربنا، هدفنا كسر الآن اللي هي معامل z
+
+252
+00:18:19,850 --> 00:18:23,330
+في الكسم، الآن ال sine وال cosine دورتهم كل 2
+
+253
+00:18:23,330 --> 00:18:27,930
+π، الآن بعد 2π، لما t بتساوي 2π قيمة
+
+254
+00:18:27,930 --> 00:18:30,510
+ال z اللي هي طول اللي ... اللي ... اللي ... اللي
+
+255
+00:18:30,510 --> 00:18:34,250
+بعدها بتصير اللي ... اللي هي عندها عند t بتساوي
+
+256
+00:18:34,250 --> 00:18:39,490
+2π، بنبدأ دورة جديدة، طب وقت إيش بنبنها على
+
+257
+00:18:39,490 --> 00:18:44,270
+الـ Z في هذه الحالة بيصير 2π مضروبة في  نصف
+
+258
+00:18:44,270 --> 00:18:47,090
+خليني أسميها هذا نصف، ممكن أسماها، نقول هذا نصف
+
+259
+00:18:47,090 --> 00:18:50,910
+بتصير عند لما
+
+260
+00:18:50,910 --> 00:18:55,610
+صارت 2π بتكون اكتملت الدورة، بيصير قيمة t π يعني
+
+261
+00:18:55,610 --> 00:19:00,070
+الآن بدل ما بعد من هان لهان، ارتفع الـ π بتكون
+
+262
+00:19:00,070 --> 00:19:04,580
+اكتملت الدورة، لأن هذه بيصير 2π، هذه لأنها مضروبة في
+
+263
+00:19:04,580 --> 00:19:10,040
+نصف، إذا صارت الدورة بعد π، تكتمل بعد π تانية كمان
+
+264
+00:19:10,040 --> 00:19:14,640
+مرة، بعد π تالتة كمان مرة، ولكن لو لاحظ، لو ضربنا هذه
+
+265
+00:19:14,640 --> 00:19:20,540
+مش في كسر، في عدد، لو قلنا في 2π، هذه دورتها عبارة عن
+
+266
+00:19:20,540 --> 00:19:26,850
+إيش؟ 2π، الآن في 2π لما بتصير ال t طبعا ال t
+
+267
+00:19:26,850 --> 00:19:30,950
+في هذه الحالة بتصير 4π، فبتصير قيمة ال t
+
+268
+00:19:30,950 --> 00:19:34,630
+من هنا لعند الأربعة بتطلع، إذا هذه ... لما نصل
+
+269
+00:19:34,630 --> 00:19:37,770
+للاربعة بتكون اكتملت اللي هي 2π هنا لأن هذه
+
+270
+00:19:37,770 --> 00:19:42,570
+بتمشي بسرعة، فبتصير الدورة بعد ايش؟ بتصير الدورة هذه
+
+271
+00:19:42,570 --> 00:19:45,510
+اكتملت لما صارت 2π، هذه صارت إيش؟ 4π
+
+272
+00:19:45,510 --> 00:19:49,310
+يعني بيصير الارتفاع تبعها أربعة، وكأننا مددنا السلك
+
+273
+00:19:49,310 --> 00:19:53,500
+لفوق، وسحبناه لفوق، صار اللي هو الدورة بدل ما تيجي من
+
+274
+00:19:53,500 --> 00:19:55,520
+هنا إلى 2π، بيصير تمشي من هنا إلى عند
+
+275
+00:19:55,520 --> 00:19:59,980
+الـ 4π، فبكون السلك زي اللي هو إيش؟ فرضناه الآن
+
+276
+00:19:59,980 --> 00:20:04,740
+هنا العكس، لو اجينا إيش؟ سوينا؟ بدنا نضغط السلك، عشان
+
+277
+00:20:04,740 --> 00:20:09,220
+نضغط السلك، بنضرب معامل ال sign ومعامل ال cosine
+
+278
+00:20:09,220 --> 00:20:14,000
+بالعدد، لو ضربناها هنا في اتنين، فبيصير لما ال t وصلت
+
+279
+00:20:14,000 --> 00:20:18,720
+π، هذه بيصير وصلت إيش؟ 2π، لأنه لما ضربنا
+
+280
+00:20:18,720 --> 00:20:23,710
+هذا في اتنين، بيصير cosine 2t، sin 2t، الآن ال
+
+281
+00:20:23,710 --> 00:20:27,270
+cosine وال cosine دورتهم بعد ما وصلت اللي هي 2
+
+282
+00:20:27,270 --> 00:20:34,510
+π، هذه كانت t، وهذه اتنين، وهذه اتنين، الآن لما t
+
+283
+00:20:34,510 --> 00:20:40,150
+تصير π، اتنين تصير 2π، وهذا يصير 2π
+
+284
+00:20:40,150 --> 00:20:44,330
+يعني اكتملت الدورة، اكتملت الدورة عند من عند t
+
+285
+00:20:44,330 --> 00:20:48,410
+بتساوي π، يعني انضغط السلك، يعني بيصير بعد ارتفاع
+
+286
+00:20:48,410 --> 00:20:53,510
+π، اللي هو اكتملت الدورة، لكن لو جينا ضربنا هذه في
+
+287
+00:20:53,510 --> 00:20:59,010
+كسر، في نصف مثلا، بيصير عشان تصل هذه 2π، بدها
+
+288
+00:20:59,010 --> 00:21:02,310
+تكون ال t لازم تكون 4π، يعني بمطّ هذا و
+
+289
+00:21:02,310 --> 00:21:08,110
+بطلع لفوق وهكذا، طيب إذا اللي حكيته اللي هو الضرب
+
+290
+00:21:08,110 --> 00:21:16,730
+هنا في عدد، في ضرب في عدد، يعني اللي هو صحيح، بيزيد
+
+291
+00:21:16,730 --> 00:21:20,490
+طبعا عدد موجة، بصحيح، بيزيد اللي هي ب ... بضغط السلك،
+
+292
+00:21:20,490 --> 00:21:25,050
+ضربه في كسر، ب ... ب ... ب ... بمط السلك، ب ... بفرد
+
+293
+00:21:25,050 --> 00:21:30,650
+السلك، يعني لأن ضرب هذه في كسر، ب ... ب ... بضغط السلك،
+
+294
+00:21:30,650 --> 00:21:36,950
+ضربه في اللي هو عدد، ب ... بفرد اللي هو السلك وهكذا
+
+295
+00:21:36,950 --> 00:21:40,370
+هذه اللي هي ال ... الكسر اللي أمامنا الآن اللي هو
+
+296
+00:21:40,370 --> 00:21:46,640
+Limits and continuity، definition، تعريف، خليني أعرف شو
+
+297
+00:21:46,640 --> 00:21:50,000
+معناه ال limits، ال epsilon delta definition، لنمر
+
+298
+00:21:50,000 --> 00:21:54,520
+عليها مرور سريع، let R of T بتساوي F of T في اتجاه
+
+299
+00:21:54,520 --> 00:21:56,940
+i، زي الـ G of T في اتجاه الـ j، زي الـ H of T في
+
+300
+00:21:56,940 --> 00:22:00,500
+اتجاه الـ k، هذا عبارة عن vector valued function، زي
+
+301
+00:22:00,500 --> 00:22:04,840
+ما قلنا، domain، هنا افترض أن D يعني قيمة t بتتاخد
+
+302
+00:22:04,840 --> 00:22:09,500
+من D، و L vector، و L عبارة عن إيش؟ vector، لو
+
+303
+00:22:09,500 --> 00:22:14,240
+فرضنا، بنقول أنه we say that R has a limit L، بنقول
+
+304
+00:22:14,240 --> 00:22:19,200
+limit R of T، لما T تروح لـ T naught، اللي هي T<sub>0</sub>، بتقول
+
+305
+00:22:19,200 --> 00:22:23,570
+إلى T<sub>0</sub> بتساوي L، إذا فرضنا إن هذه الـ limit R of
+
+306
+00:22:23,570 --> 00:22:27,830
+T موجودة، بتساوي L، هذا معناته كلما صغرت المسافة
+
+307
+00:22:27,830 --> 00:22:32,690
+بين T و T<sub>0</sub>، تصغر المسافة بين R of T و L، هذا الكلام
+
+308
+00:22:32,690 --> 00:22:35,450
+نعبر عنه بالـ epsilon delta definition كما يلي:
+
+309
+00:22:35,450 --> 00:22:40,230
+أنه مهما كانت epsilon صغيرة، يعني اللي تعبر عن قصر
+
+310
+00:22:40,230 --> 00:22:44,510
+المسافة بين R of T و L، باللاجة delta، اللي هي تعبر
+
+311
+00:22:44,510 --> 00:22:51,270
+عن صغر المسافة بين T و T<sub>0</sub>، بحيث أنه لما T تروح لـ T<sub>0</sub>،
+
+312
+00:22:51,270 --> 00:22:55,010
+يعني بمعنى آخر، تصغر المسافة بين T و T<sub>0</sub> وتصير أصغر
+
+313
+00:22:55,010 --> 00:23:01,410
+من delta، وأكبر من 0، معناته هذا يؤدي إلى المسافة
+
+314
+00:23:01,410 --> 00:23:05,250
+بين R و T، أقل تكون أصغر من مين؟ من أي epsilon في
+
+315
+00:23:05,250 --> 00:23:09,290
+الدنيا، رقم صغير، يعني بمعنى آخر، عندما تصغر المسافة
+
+316
+00:23:09,290 --> 00:23:14,690
+بين T و T<sub>0</sub>، تصير أصغر من delta، بضمن لك أن المسافة بين
+
+317
+00:23:14,690 --> 00:23:19,870
+R و T، أقل تكون أصغر من أي رقم في الدنيا، يعني بمعنى
+
+318
+00:23:19,870 --> 00:23:25,520
+آخر، عندما تقول T لـ T<sub>0</sub>، بتقول R of T، نعدلها بس
+
+319
+00:23:25,520 --> 00:23:29,780
+هذا اللي بدي إياكم تعرفوه في اللي هو تعريف اللي هو ال
+
+320
+00:23:29,780 --> 00:23:33,500
+limit بواسطة اللي هو ال epsilon delta definition
+
+321
+00:23:33,500 --> 00:23:37,440
+الآن الشيء المهم، الشيء، الشغل الممكن اللي بدنا
+
+322
+00:23:37,440 --> 00:23:41,620
+نستخدمه احنا، واللي هنستخدمه اللي هو كما يلي: لو
+
+323
+00:23:41,620 --> 00:23:45,200
+الآن عندي ... عندي R of T عبارة عن vector valued
+
+324
+00:23:45,200 --> 00:23:49,400
+function، بتساوي F of T في اتجاه i، زائد G في اتجاه
+
+325
+00:23:49,400 --> 00:23:53,300
+j، زائد H of T في اتجاه k، زي اللي هي اللي فوق، لأن
+
+326
+00:23:53,300 --> 00:23:57,500
+لو كانت ... بقول إن ال limit هذه موجودة، limit ال R
+
+327
+00:23:57,500 --> 00:24:02,500
+of T موجودة، إذا كانت limit كل component لحالها
+
+328
+00:24:02,500 --> 00:24:06,720
+موجودة، يعني عشان أحسب ال limit هذه، باجي باخد ال
+
+329
+00:24:06,720 --> 00:24:10,470
+limit للـ F of T، و limit g of t، و limit h of t، لو
+
+330
+00:24:10,470 --> 00:24:14,710
+كان إن هذول التلاتة موجودات، بتصير ال limit كلها
+
+331
+00:24:14,710 --> 00:24:18,330
+للـ R of T موجودة، وتساوي limit R of T بتساوي limit
+
+332
+00:24:18,330 --> 00:24:20,670
+الأول في اتجاه الـ i، زائد limit الثاني في اتجاه الـ
+
+333
+00:24:20,670 --> 00:24:24,470
+j، زائد limit الثالث في اتجاه الـ k، نشوف هذا عملياً
+
+334
+00:24:24,470 --> 00:24:28,850
+عبر مثال، المثال الثاني مثلاً، يعني أخدنا R of T
+
+335
+00:24:28,850 --> 00:24:32,750
+بتساوي cos T في اتجاه الـ i، و sin T في اتجاه الـ j، و
+
+336
+00:24:32,750 --> 00:24:36,940
+T في اتجاه الـ k، then احسب لي بالله limit of R of T
+
+337
+00:24:36,940 --> 00:24:41,360
+لما T تروح لمين؟ لـ π على أربعة، الحساب سهل يا
+
+338
+00:24:41,360 --> 00:24:45,120
+جماعة، بيجي بنحسب الأول limitه في اتجاه الـ i، و
+
+339
+00:24:45,120 --> 00:24:47,420
+limit الثاني في اتجاه الـ j، و limit الثالث في
+
+340
+00:24:47,420 --> 00:24:51,920
+اتجاه الـ k،  ده فهم شغل ما كان سهل، هذا هذه الآن على
+
+341
+00:24:51,920 --> 00:24:54,900
+طول  تعويض مباشرة، لأن ده أصلاً متصل زي ما بقى بنعرف
+
+342
+00:24:54,900 --> 00:24:58,300
+احنا زمان في calculus ألف، limit cosine T لما T
+
+343
+00:24:58,300 --> 00:25:01,420
+تروح لـ π على أربعة، بتصير cosine الـ π على أربعة
+
+344
+00:25:01,420 --> 00:25:05,020
+اللي هي جذر الاثنين على اثنين، في اتجاه الـ i، sin T هو
+
+345
+00:25:05,020 --> 00:25:08,720
+sine π على 4، sine π على 4، جذر الاثنين على 2، تعويض
+
+346
+00:25:08,720 --> 00:25:13,380
+مباشر لأنه متصلة، زائد هذه برضه T هي π على 4
+
+347
+00:25:13,380 --> 00:25:18,100
+فبتكون limit R of T لها دي بتساوي اللي هو اللي
+
+348
+00:25:18,100 --> 00:25:21,920
+أمامنا، طيب الجماعة، صلوا على النبي، الله الصلاة والسلام
+
+349
+00:25:29,870 --> 00:25:36,430
+تعريف الاتصال، زي ... زي الاتصال العادي عند
+
+350
+00:25:36,430 --> 00:25:40,890
+نقطة T، بيساوي R of T is continuous in its domain
+
+351
+00:25:40,890 --> 00:25:45,130
+إذا كان limit R of T لما T تروح لـ T<sub>0</sub>، بيساوي R of
+
+352
+00:25:45,130 --> 00:25:48,690
+T<sub>0</sub>، زي ما كنا نقول زمان، limit F of T لما T تروح لـ
+
+353
+00:25:48,690 --> 00:25:52,820
+T<sub>0</sub> بيساوي F of T<sub>0</sub>، هذا اللي هو معنى الاتصال برضه للـ
+
+354
+00:25:52,820 --> 00:25:56,800
+vector-valued function، بنفس المعنى اللي كنا ناخده
+
+355
+00:25:56,800 --> 00:26:00,580
+زمان في الدوال العادية اللي هي الـ real-valued
+
+356
+00:26:00,580 --> 00:26:04,540
+function، لما نقول الـ function is continuous
+
+357
+00:26:04,540 --> 00:26:09,500
+معناته بنقصد فيها إنها continuous عند كل نقطة من
+
+358
+00:26:09,500 --> 00:26:12,760
+نقاط الدومين، الـ function is continuous if it is
+
+359
+00:26:12,760 --> 00:26:16,520
+continuous at every point in its domain، ناخد مثلاً
+
+360
+00:26:16,750 --> 00:26:20,830
+الآن، all the space curves shown in figure 13-2
+
+361
+00:26:20,830 --> 00:26:23,610
+و 13-4 are continuous because their component
+
+362
+00:26:23,610 --> 00:26:27,330
+functions are continuous at every point of T، إيش
+
+363
+00:26:27,330 --> 00:26:30,670
+بيقللنا هذا؟ بيقللنا ما يلي ... خلينا نرجع لـ 13-2
+
+364
+00:26:30,670 --> 00:26:32,450
+و 13-3
+
+365
+00:26:39,110 --> 00:26:42,690
+هذه الدوال مثلاً، على سبيل المثال، هو R of T بيساوي
+
+366
+00:26:42,690 --> 00:26:46,870
+cos T، sin T، T، بما أن ال components هذه اللي هي
+
+367
+00:26:46,870 --> 00:26:50,310
+cos T و sin T و T كلهم continuous، إذا ده continuous
+
+368
+00:26:50,310 --> 00:26:56,410
+ال vector valued function، بما أن هذه
+
+369
+00:26:56,410 --> 00:26:58,890
+continuous، وهذه continuous، وهذه continuous، إذا الـ
+
+370
+00:26:58,890 --> 00:27:02,650
+R of T برضه إيش مالها؟ continuous، والتالتة زيها، إذا
+
+371
+00:27:02,650 --> 00:27:05,650
+ال continuity لل vector valued function اللي بتحكم
+
+372
+00:27:05,650 --> 00:27:09,850
+عليها أن يكون كلهم إيش مالهم؟ continuous، وإن بيكون
+
+373
+00:27:09,850 --> 00:27:12,650
+كلهم continuous، بنقول عنهم اللي هي ال vector
+
+374
+00:27:12,650 --> 00:27:16,630
+valued continuous، وهي مثال اللي هو اللي هي مثال
+
+375
+00:27:16,630 --> 00:27:19,550
+جابته برضه هذه، إذا كلهم أمثلة على continuous
+
+376
+00:27:19,550 --> 00:27:23,760
+functions، الآن نجينا ال function g of t بتساوي cos
+
+377
+00:27:23,760 --> 00:27:27,940
+t، زائد sin t، زائد اللي هو ال greatest integer، أو ال
+
+378
+00:27:27,940 --> 00:27:33,860
+least integer، اللي هو ال t، الآن ال floor integer
+
+379
+00:27:33,860 --> 00:27:39,240
+احنا بنعرف إن هذه continuous، وين مكان ما عدا عند
+
+380
+00:27:39,240 --> 00:27:44,700
+وين يا جماعة؟ ما عدا عند اللي هي ال integers، عند ال
+
+381
+00:27:44,700 --> 00:27:48,920
+integers، مش continuous، عشان هيك بنقدرش نقول عن ال
+
+382
+00:27:48,920 --> 00:27:52,340
+g of t إنها continuous عند ال integers، رغم إنه هذه
+
+383
+00:27:52,340 --> 00:27:56,260
+continuous، وهذه continuous، لكن بنقول إن continuous
+
+384
+00:27:56,260 --> 00:27:59,800
+هدول وين ما كان، لأن هدول وين ما كان، وهذه وين ما كان
+
+385
+00:27:59,800 --> 00:28:02,920
+وهذه وين ما كان، ما عدا عند مين؟ ال integers، عشان هيك
+
+386
+00:28:02,920 --> 00:28:07,770
+بنقول continuous وين ما كان ما عدا عند ال integers
+
+387
+00:28:07,770 --> 00:28:11,270
+إذا ال function الكلية، ال vector valued function
+
+388
+00:28:11,270 --> 00:28:17,410
+g of t، continuous وين ما كان ما عدا عند مين؟ عند ال
+
+389
+00:28:17,
+
+445
+00:32:03,690 --> 00:32:05,930
+إن اللي هو معنى الـ derivative اللي أخدناه في الـ
+
+446
+00:32:05,930 --> 00:32:08,770
+calculus اللي هو هاي الـ position vector R of T
+
+447
+00:32:08,770 --> 00:32:12,530
+وهذا الـ position R of T زي الـ Delta T الآن هذا
+
+448
+00:32:12,530 --> 00:32:16,460
+اللي هو عبارة عن اللي بيسكر هو عبارة عن القاطع اللي
+
+449
+00:32:16,460 --> 00:32:21,400
+هو بين P و Q هو في الواقع بيصير هذا زائد هذا بيساوي
+
+450
+00:32:21,400 --> 00:32:26,330
+هذا يعني بمعنى آخر اللي هو هذا اللي هو R of T زي
+
+451
+00:32:26,330 --> 00:32:31,550
+الـ Delta T ناقص R of T هو في الواقع هذا ناقص هذا
+
+452
+00:32:31,550 --> 00:32:36,330
+لأن بيصير لما تقول R of T زائد هذا برجح هذا طيب
+
+453
+00:32:36,330 --> 00:32:39,230
+إذا بالأصفر هذا نتجنا من هذا ومن هذا بصورة طبعة
+
+454
+00:32:39,230 --> 00:32:42,090
+المتجهة اللي أنتم عارفينها هذا اللي هو القاطع
+
+455
+00:32:42,090 --> 00:32:45,370
+الآن ميل القاطع هذا اللي هو عبارة عن إيش يا جماعة؟
+
+456
+00:32:45,370 --> 00:32:49,890
+اللي هو R of T زائد Delta T ناقص R of T على Delta T
+
+457
+00:32:49,890 --> 00:32:55,400
+Delta T اللي هو قيمة الـ Y هنا ناقص قيمة الـ y هنا
+
+458
+00:32:55,400 --> 00:33:02,440
+اللي هي هذه ناقص هذه اللي هي بتطلع هذه على اللي هو
+
+459
+00:33:02,440 --> 00:33:05,500
+هذه اللي هي قيمة الـ X لأنه بنزل عمود هيك وقيمة الـ
+
+460
+00:33:05,500 --> 00:33:09,700
+X هذه اللي هي عبارة عن Delta T لو كانت T وصارت
+
+461
+00:33:09,700 --> 00:33:12,580
+اللي هو R زائد Delta T يعني حصل الطرح اللي بتطلع
+
+462
+00:33:12,580 --> 00:33:16,880
+اللي هي Delta T يعني ميل المماس .. ميل القاطع هذا
+
+463
+00:33:16,880 --> 00:33:19,940
+هو عبارة عن R of T زائد Delta T ناقص R of T على
+
+464
+00:33:19,940 --> 00:33:22,600
+Delta T الآن القاطع هذا
+
+465
+00:33:25,390 --> 00:33:29,730
+لما تجرب هذه النقطة على الـ Q وتنطبق عليها ميل
+
+466
+00:33:29,730 --> 00:33:33,570
+القاطع إيش؟
+
+467
+00:33:33,570 --> 00:33:37,130
+بيصير ميل القاطع بيصل هنا بيصل هنا لما يصير الـ Q
+
+468
+00:33:37,130 --> 00:33:40,490
+تروح للـ P الـ Delta T تروح للصفر يعني لما الـ Q
+
+469
+00:33:40,490 --> 00:33:45,250
+تروح للـ P هذا بيصير مماس القاطع ف الآن limit هذا
+
+470
+00:33:45,250 --> 00:33:51,230
+المقدار هو عبارة عن مين يا جماعة؟ هو عبارة عن R' of
+
+471
+00:33:51,230 --> 00:33:55,550
+T يعني limit من القاطع بيطلع مين؟ من المماس وهذا هو
+
+472
+00:33:55,550 --> 00:34:00,910
+tangent للـ vector اللي هو R الـ tangent vector لمين؟
+
+473
+00:34:01,520 --> 00:34:05,400
+لـ curve اللي عندي عند نقطتي هذا الـ
+
+474
+00:34:05,400 --> 00:34:11,000
+jump اللي هو ميل المماس
+
+475
+00:34:11,000 --> 00:34:15,400
+الصحيح اللي هو عبارة عن الـ derivative للـ R of T
+
+476
+00:34:15,400 --> 00:34:21,620
+يساوي اللي أمامنا هذا اللي هو اللي عندنا طيب
+
+477
+00:34:21,620 --> 00:34:27,610
+الجماعة بنيجي لحاجة بيسميها piecewise smooth piecewise
+
+478
+00:34:27,610 --> 00:34:32,550
+smooth بيسميها
+
+479
+00:34:32,550 --> 00:34:37,790
+piecewise smooth بيسميها piecewise smooth بيسميها
+
+480
+00:34:37,790 --> 00:34:37,970
+بيسميها بيسميها بيسميها بيسميها بيسميها بيسميها
+
+481
+00:34:37,970 --> 00:34:38,090
+بيسميها بيسميها بيسميها بيسميها بيسميها بيسميها
+
+482
+00:34:38,090 --> 00:34:41,010
+بيسميها بيسميها بيسميها بيسميها بيسميها بيسميها
+
+483
+00:34:41,010 --> 00:34:46,220
+بيسميها  لما نكون كلها متصلات مع بعض اتصالًا يعني
+
+484
+00:34:46,220 --> 00:34:48,520
+الدالة منها متصلة والدالة منها متصلة والدالة
+
+485
+00:34:48,520 --> 00:34:49,460
+منها متصلة والدالة منها متصلة والدالة منها
+
+486
+00:34:49,460 --> 00:34:50,380
+متصلة والدالة منها متصلة والدالة منها
+
+487
+00:34:50,380 --> 00:34:51,460
+متصلة والدالة منها متصلة والدالة منها
+
+488
+00:34:51,460 --> 00:34:52,240
+متصلة والدالة منها متصلة والدالة منها
+
+489
+00:34:52,240 --> 00:34:54,680
+متصلة والدالة منها متصلة والدالة منها
+
+490
+00:34:54,680 --> 00:34:57,180
+متصلة والدالة منها متصلة والدالة منها
+
+491
+00:34:57,180 --> 00:34:58,680
+متصلة والدالة منها متصلة والدالة منها
+
+492
+00:34:58,680 --> 00:35:00,400
+متصلة والدالة منها متصلة والدالة منها
+
+493
+00:35:00,400 --> 00:35:03,280
+متصلة والدالة منها متصلة والدالة منها الآن هذه
+
+494
+00:35:03,280 --> 00:35:07,020
+مقاومة مقاومة وليست موجودة وشرط لـ smoothness أن
+
+495
+00:35:07,020 --> 00:35:10,300
+تكون الـ derivative موجودة لكن هذا قطع كاملة
+
+496
+00:35:10,300 --> 00:35:13,640
+smooth وهذا قطع كاملة smooth مرتبطات مع بعض عشان
+
+497
+00:35:13,640 --> 00:35:19,740
+هيك بنسميهم piecewise smooth يعني smooth قطعة قطعة
+
+498
+00:35:19,740 --> 00:35:24,400
+طيب الآن definition if R is the position vector of
+
+499
+00:35:24,400 --> 00:35:27,540
+a particle moving along the smooth curve in space
+
+500
+00:35:27,540 --> 00:35:32,610
+then لأن لو فرضنا R عبارة عن position vector لـ
+
+501
+00:35:32,610 --> 00:35:36,710
+particle يتحرك على curve على smooth curve في الـ
+
+502
+00:35:36,710 --> 00:35:42,690
+space فبتكون V of T سرعة الـ particle على اللي هو
+
+503
+00:35:42,690 --> 00:35:47,530
+إيش؟ سرعة الـ particle بتساوي DR by DT يعني الـ
+
+504
+00:35:47,530 --> 00:35:52,740
+position vector لما نأخذه اللي هو DR by DT هو معدل
+
+505
+00:35:52,740 --> 00:35:57,220
+تغير المسافة بالنسبة للزمن أو الإزاحة بالنسبة للزمن
+
+506
+00:35:57,220 --> 00:36:00,840
+وهو عبارة عن الـ velocity is the particle velocity
+
+507
+00:36:00,840 --> 00:36:06,000
+vector particle velocity vector tangent to the
+
+508
+00:36:06,000 --> 00:36:11,220
+curve زي ما قلنا الآن anytime T the direction of V
+
+509
+00:36:11,220 --> 00:36:15,360
+اللي هو اتجاه الـ V هو is the direction of motion
+
+510
+00:36:15,360 --> 00:36:20,500
+إذا اتجاه الـ V هو اللي بنسميه اتجاه حركة الـ
+
+511
+00:36:20,500 --> 00:36:20,940
+particle
+
+512
+00:36:35,200 --> 00:36:38,300
+طيب the magnitude of v طبعًا كله أخذته في الـ
+
+513
+00:36:38,300 --> 00:36:40,880
+physics عشان أفكر مثلًا أنا the magnitude of v is
+
+514
+00:36:40,880 --> 00:36:44,980
+the particle's speed مظبوط اللي هي الـ magnitude للـ
+
+515
+00:36:44,980 --> 00:36:48,340
+V الـ magnitude للـ velocity يعني القيمة العددية
+
+516
+00:36:48,340 --> 00:36:53,160
+الموجبة للـ velocity بنسميها إيش؟ الـ speed and the
+
+517
+00:36:53,160 --> 00:36:56,400
+derivative dV by dt هي اللي بنسميها الـ
+
+518
+00:36:56,400 --> 00:37:02,220
+acceleration أو العجلة لمين؟ للـ vector in summary
+
+519
+00:37:02,220 --> 00:37:05,460
+velocity is the derivative of the position vector
+
+520
+00:37:10,540 --> 00:37:17,490
+Dv by Dt بيساوي D بقرار R بقرار D الآن الـ unit
+
+521
+00:37:17,490 --> 00:37:20,150
+vector v على الـ magnitude لـ v هو عبارة عن the
+
+522
+00:37:20,150 --> 00:37:23,470
+direction of the motion at the time t الـ direction
+
+523
+00:37:23,470 --> 00:37:27,050
+للحركة هو الـ direction لـ الـ v إيش الـ direction لـ
+
+524
+00:37:27,050 --> 00:37:29,910
+الـ v اللي هو الـ v على الـ magnitude لـ الـ v يعني اللي
+
+525
+00:37:29,910 --> 00:37:34,390
+هو متجه الوحدة في اتجاه الـ velocity هو اللي
+
+526
+00:37:34,390 --> 00:37:41,110
+بنسميه the direction of the motion نأخذ مثلًا Find
+
+527
+00:37:41,110 --> 00:37:44,590
+the velocity, speed and acceleration of a particle
+
+528
+00:37:44,590 --> 00:37:47,610
+whose motion in space is given by the position
+
+529
+00:37:47,610 --> 00:37:52,310
+vector R of T بتساوي كذا الآن بيقول لي أو دي الـ
+
+530
+00:37:52,310 --> 00:37:58,520
+velocity و الـ speed و الـ acceleration لجزئ حركته
+
+531
+00:37:58,520 --> 00:38:02,080
+مرتبطة بالـ Position Vector أو ممثلة أو يمثلها الـ
+
+532
+00:38:02,080 --> 00:38:06,060
+Position Vector R of T اللي أمامنا وبعدين بيقول الـ
+
+533
+00:38:06,060 --> 00:38:11,320
+sketch ده Vector Velocity Vector V 7 باي على أربعة
+
+534
+00:38:11,320 --> 00:38:14,780
+طبعًا نرسم زي ما قلنا نرسم ما نعرفش نرسمه إلا
+
+535
+00:38:14,780 --> 00:38:17,660
+بواسطة الكمبيوتر أو بعض الرسومات السهلة زي الـ helix
+
+536
+00:38:17,660 --> 00:38:21,040
+اللي عملناها قبل شوية لأن الـ Velocity and
+
+537
+00:38:21,040 --> 00:38:24,140
+acceleration vector at a time t هو R الـ Velocity
+
+538
+00:38:24,140 --> 00:38:28,680
+سهلة التي هي dR by dt هي R بنفاضل هذه بالنسبة لـ t
+
+539
+00:38:28,680 --> 00:38:35,360
+تصبح نفضلها ناقص اثنين sin t زائد اثنين cos t ناقص
+
+540
+00:38:35,360 --> 00:38:39,240
+اثنين في خمسة التي هي عشرة cos t في sin t تعرفوا
+
+541
+00:38:39,240 --> 00:38:44,880
+كيف نفاضل الآن في اتجاه الـ K الآن بنبسطها
+
+542
+00:38:45,440 --> 00:38:49,760
+التي هي ناقص اثنين sin T في جهة I زائد اثنين cos T
+
+543
+00:38:49,760 --> 00:38:53,920
+في جهة J وهذا عبارة عن اثنين cos T sin T التي
+
+544
+00:38:53,920 --> 00:38:55,120
+عبارة عن sin 2T
+
+545
+00:38:58,860 --> 00:39:02,080
+الآن الـ Acceleration بس عشان أسهل إحنا التفاضل
+
+546
+00:39:02,080 --> 00:39:05,660
+كمان مرة، مش ممكن نفاضل ونتغلبها الآن الـ
+
+547
+00:39:05,660 --> 00:39:08,220
+Acceleration اللي طالبها الـ Acceleration اللي هي
+
+548
+00:39:08,220 --> 00:39:14,260
+عجلة Dv by Dt أو R'' of T اللي هي بالسبب نفاضل
+
+549
+00:39:14,260 --> 00:39:17,620
+هذه كمان مرة اللي هي بيصير ناقص اثنين Cos T في
+
+550
+00:39:17,620 --> 00:39:20,660
+اتجاه الـ I ناقص اثنين Sin T في اتجاه الـ J
+
+551
+00:39:20,660 --> 00:39:26,870
+ناقص اللي هي بيصير زائد عشرة Cos اثنين T ناقص عشرة
+
+552
+00:39:26,870 --> 00:39:29,950
+Cos اثنين T في اتجاه الـ K هذه اللي هي الـ mean الـ
+
+553
+00:39:29,950 --> 00:39:34,170
+acceleration الـ speed اللي طالبها كلها تطبيقات
+
+554
+00:39:34,170 --> 00:39:37,710
+سهلة هي الـ magnitude لـ V الـ magnitude لـ V of T هي
+
+555
+00:39:37,710 --> 00:39:40,150
+عبارة عن جذر التربيع لهذه تربيع زائد هذه تربيع
+
+556
+00:39:40,150 --> 00:39:43,730
+زائد هذه تربيع هي هذه تربيع وهي هذه تربيع وهي هذه
+
+557
+00:39:43,730 --> 00:39:47,460
+تربيع هذه تربيع مع هذه تربيع اللي هي عبارة عن جذر
+
+558
+00:39:47,460 --> 00:39:52,000
+أربعة sin تربيع زائد أربعة cos تربيع اللي هي أربعة
+
+559
+00:39:52,000 --> 00:39:54,700
+أربعة sin تربيع زائد أربعة cos تربيع اللي هي عبارة
+
+560
+00:39:54,700 --> 00:39:57,300
+عن أربعة لأن sin تربيع زائد cos تربيع اللي
+
+561
+00:39:57,300 --> 00:39:59,480
+بناخذه الأربعة منها عامل مشترك و sin تربيع زائد
+
+562
+00:39:59,480 --> 00:40:02,420
+cos تربيع بيساوي واحد زي ما أنتم عارفين بيصير
+
+563
+00:40:02,420 --> 00:40:06,700
+أربعة وهذه تنزل خمسة وعشرين sin تربيع اثنين T وهذه
+
+564
+00:40:06,700 --> 00:40:10,770
+عبارة عن مين؟ الـ velocity لو عوضنا عن T بيساوي 7π على
+
+565
+00:40:10,770 --> 00:40:16,970
+4 بتوجد V of 7π على 4 هنا بتعوض مكان هنا بتوجد
+
+566
+00:40:16,970 --> 00:40:20,030
+قيمتها بتطلع عندك القيمة جذر الاثنين في اتجاه I زي جذر
+
+567
+00:40:20,030 --> 00:40:24,510
+الاثنين في اتجاه K زي 5K عوض عن الـ acceleration هنا
+
+568
+00:40:24,510 --> 00:40:27,930
+عند 7π على 4 عوض في هدول بتطلع القيمة اللي أمامنا
+
+569
+00:40:27,930 --> 00:40:33,330
+هنا وعوض عن اللي هي .. خذ الـ .. الـ .. آسف مش تعوض
+
+570
+00:40:33,540 --> 00:40:38,600
+خذ الـ magnitude لهذه جذر التربيع اللي هو 2،2،4،5
+
+571
+00:40:38,600 --> 00:40:41,900
+اللي هي الجذر التاسع والعشرين بسرعة لأنه هدول
+
+572
+00:40:41,900 --> 00:40:45,700
+السهلات هذه اللي هي اللي هو حل السؤال أو المثال
+
+573
+00:40:45,700 --> 00:40:52,340
+وهي عندي اللي هو عند اللي هي مثلًا نقطة عند سبعة
+
+574
+00:40:52,340 --> 00:40:55,400
+باي على أربعة عند سبعة باي على أربعة T بتساوي
+
+575
+00:40:55,400 --> 00:41:00,380
+سبعة باي على أربعة بتحسب قيمة اللي هو X و Y و Z X
+
+576
+00:41:00,380 --> 00:41:02,180
+Y Z هاي X
+
+577
+00:41:08,360 --> 00:41:09,760
+YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
+
+578
+00:41:18,160 --> 00:41:23,000
+الآن عند اللي هي 7π على 4 هنا بالظبط عند اللي هي
+
+579
+00:41:23,000 --> 00:41:28,120
+اللي بتكون R الـ X هي قيمتها بتعود لـ 7π على 4 و
+
+580
+00:41:28,120 --> 00:41:32,800
+أنا 7π على 4 و أنا 7π على 4 فبتوجد الـ X تبعتها
+
+581
+00:41:32,800 --> 00:41:37,360
+الـ X هنا تأخذ لها اللي هو المماس بيكون هو هذا R
+
+582
+00:41:37,400 --> 00:41:44,800
+نعوض عن T بالسبعة
+
+583
+00:41:44,800 --> 00:41:48,300
+و سبعة باي على أربعة و باي على أربعة و باي على
+
+584
+00:41:48,300 --> 00:41:50,780
+أربعة و باي على أربعة و باي على أربعة و باي على
+
+585
+00:41:50,780 --> 00:41:50,880
+أربعة و باي على أربعة و باي على أربعة و باي على
+
+586
+00:41:50,880 --> 00:41:51,160
+أربعة و باي على أربعة و باي على أربعة و باي على
+
+587
+00:41:51,160 --> 00:41:51,360
+أربعة و باي على أربعة و باي على أربعة و باي على
+
+588
+00:41:51,360 --> 00:41:54,640
+أربعة و باي على أربعة و باي على أربعة و باي على
+
+589
+00:41:54,640 --> 00:42:00,800
+أربعة و باي على أربعة و باي على أربعة و باي الآن قلت
+
+590
+00:42:00,800 --> 00:42:04,700
+هذه وعملناها قبل هكذا remark we can express the
+
+591
+00:42:04,700 --> 00:42:08,560
+velocity of a moving particle يعني بنقدر نعبر عن
+
+592
+00:42:08,560 --> 00:42:13,500
+الـ velocity لـ particle يتحرك as the product of its
+
+593
+00:42:13,500 --> 00:42:17,080
+speed and its direction يعني دائمًا الـ velocity
+
+594
+00:42:18,090 --> 00:42:22,610
+بتساوي الـ magnitude للـ V اللي هو S هو الـ speed
+
+595
+00:42:22,610 --> 00:42:26,530
+مضروب في الـ direction للـ velocity إيش الـ direction
+
+596
+00:42:26,530 --> 00:42:29,970
+الـ V على الـ magnitude للـ V إذا الـ velocity دائمًا
+
+597
+00:42:29,970 --> 00:42:32,870
+بيساوي الـ speed ضرب الـ direction هذا الـ direction
+
+598
+00:42:32,870 --> 00:42:36,060
+وهذا الـ speed زي ما أنتم عارفين Differentiation
+
+599
+00:42:36,060 --> 00:42:38,200
+rules for vector functions الـ differentiation
+
+600
+00:42:38,200 --> 00:42:42,080
+rules هي الـ differentiation rules للعادية بالضبط
+
+601
+00:42:42,080 --> 00:42:46,360
+let u, v be differentiable vector functions of D وC
+
+602
+00:42:46,360 --> 00:42:50,860
+a constant vector وC أي scalar و F أي
+
+603
+00:42:50,860 --> 00:42:54,890
+differentiable scalar function الآن كله قابل
+
+604
+00:42:54,890 --> 00:42:58,970
+الاشتقاق علشان هيك بدنا نشتق بسرعة هي اللي هي دالة
+
+605
+00:42:58,970 --> 00:43:02,210
+ثابتة اللي هي مثلًا C1 في اتجاه I زائد C2 في اتجاه
+
+606
+00:43:02,210 --> 00:43:06,670
+J زائد C3 في اتجاه K بحيث يساوي الـ Zero vector
+
+607
+00:43:06,670 --> 00:43:11,200
+تفاضل الدالة الـ vector valued function الثابتة وهو
+
+608
+00:43:11,200 --> 00:43:13,160
+عبارة عن الـ Zero vector إيش الـ Zero vector اللي
+
+609
+00:43:13,160 --> 00:43:15,340
+هو صفر في اتجاه الـ I زي الـ صفر في اتجاه الـ J
+
+610
+00:43:15,340 --> 00:43:20,120
+زي الـ صفر في اتجاه الـ K الآن D by DT لـ C في U
+
+611
+00:43:20,120 --> 00:43:23,580
+of T يعني هذا ثابت بتطلع الثابت برا بتفاضل من U
+
+612
+00:43:23,580 --> 00:43:27,800
+prime of T تفاضل حاصل ضرب دالتين هذا scalar وهذا
+
+613
+00:43:27,800 --> 00:43:33,560
+vector عادي F prime في U زائد F في U prime تفاضل
+
+614
+00:43:33,560 --> 00:43:37,660
+مجموع دالتين vectors عادي زي العادي u prime زائد v
+
+615
+00:43:37,660 --> 00:43:41,180
+prime حاصل طرح u prime ناقص v prime عادي زي ما كنا
+
+616
+00:43:41,180 --> 00:43:45,360
+تعلمنا قبل u dot v اللي هو الـ dot product هذا يا
+
+617
+00:43:45,360 --> 00:43:49,520
+جماعة اللي هو برضه مشابه للـ vector العادي u prime
+
+618
+00:43:49,520 --> 00:43:55,780
+dot v زائد u dot v prime حتى الـ cross product
+
+619
+00:43:55,780 -->
+
+667
+00:47:33,440 --> 00:47:39,830
+Dot و Product بينهما صفر، إذا كان R Orthogonal
+
+668
+00:47:39,830 --> 00:47:45,510
+صار الـ R'(T) والـ R(T)  Orthogonal، يعني if R
+
+669
+00:47:45,510 --> 00:47:49,690
+is a differentiable function of T of constant length
+
+670
+00:47:49,690 --> 00:47:55,650
+يعني طول الـ R ثابت، then الـ R.dR/dT
+
+671
+00:47:55,650 --> 00:48:00,930
+بيساوي صفر، أو بمعنى آخر، الـ R (اللي هو الـ position
+
+672
+00:48:00,930 --> 00:48:06,470
+vector) والمماس للـ curve اللي بيمشي فيه الـ
+
+673
+00:48:06,470 --> 00:48:11,890
+position vector (اللي هو dr/dt) دائماً متعامدين، زي
+
+674
+00:48:11,890 --> 00:48:16,610
+ما شفنا هنا، اللي هو dr/dt والـ R(T) الاثنين
+
+675
+00:48:16,610 --> 00:48:21,230
+متعامدين.  هيك، وتكون انتهت المحاضرة والسلام عليكم
+
+676
+00:48:21,230 --> 00:48:23,270
+ورحمة الله وبركاته
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/EFnvj3yPdag.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/EFnvj3yPdag.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b6ef5f9e9c3719ea400b6c93f8369bdd505839bf
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/EFnvj3yPdag.srt
@@ -0,0 +1,1275 @@
+1
+00:00:00,490 --> 00:00:04,210
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله هذه
+
+2
+00:00:04,210 --> 00:00:09,590
+هي المحاضرة رقم 7 لمادة أو مساق تفاضل وتكامل
+
+3
+00:00:09,590 --> 00:00:14,150
+الـGene وهي الجزء الثاني من الـ section الأخير في
+
+4
+00:00:14,150 --> 00:00:21,090
+شبطر 12، الـ 12.6، كنا حكينا عن اللي هو الـ cylinders
+
+5
+00:00:21,090 --> 00:00:25,210
+and quadratic surfaces، وحكينا عن ثلاثة من الـ
+
+6
+00:00:25,210 --> 00:00:28,230
+quadratic surfaces، ده بتتذكروا حكينا عن الـ
+
+7
+00:00:28,230 --> 00:00:34,030
+ellipsoid والـ hyperbolic paraboloid وعن اللي هو
+
+8
+00:00:34,030 --> 00:00:38,620
+elliptical cone. ضل عندنا ثلاثة اللي هو أنواع جديدة
+
+9
+00:00:38,620 --> 00:00:43,200
+هنحكي عنها، اللي هي من الـ quadratic surfaces اللي
+
+10
+00:00:43,200 --> 00:00:49,940
+الـ surfaces أو اللي هي السطوح أو المجسمات اللي هي
+
+11
+00:00:49,940 --> 00:00:54,890
+رسم لمعادلات تربيعية. الآن نجي لـ example خمسة
+
+12
+00:00:54,890 --> 00:00:57,830
+example خمسة اللي هو هيكون عندنا حاجة اسمها
+
+13
+00:00:57,830 --> 00:01:01,170
+elliptical paraboloid، لنشوف ايش elliptical
+
+14
+00:01:01,170 --> 00:01:04,690
+paraboloid، زي ما قلنا قبل هيك في المحاضرة السابقة
+
+15
+00:01:04,690 --> 00:01:11,750
+أنّه هيكون اللي هو أكيد الاسم له علاقة بالجسم أو
+
+16
+00:01:11,750 --> 00:01:17,420
+الشكل اللي هيطلع. الجراف ده اللي هو X تربيع زائد Y
+
+17
+00:01:17,420 --> 00:01:22,780
+تربيع على B تربيع بيساوي Z على C. بدنا الآن
+
+18
+00:01:22,780 --> 00:01:27,500
+اللي هو نشوف هذا اللي هو الجراف، أي هذا المعادلة
+
+19
+00:01:27,500 --> 00:01:32,980
+بتمثل جراف ماذا؟ قلنا احنا في الأسئلة من هالنوع
+
+20
+00:01:32,980 --> 00:01:39,320
+أول ما بنحكي، والشغلة الأساسية في تحديد الشكل هي
+
+21
+00:01:39,320 --> 00:01:44,820
+المقاطع، اللي هي الشكل. الآن أول حاجة بناخد X بيساوي
+
+22
+00:01:44,820 --> 00:01:49,600
+0. لما X بيساوي 0 بيصير عندي Y تربيع على B تربيع
+
+23
+00:01:49,600 --> 00:01:54,100
+بيساوي Z على C، أو بمعنى آخر تطلع عندي Z بيساوي C على
+
+24
+00:01:54,100 --> 00:02:00,100
+B تربيع في Y تربيع. وهذا واضح أن هي عبارة عن معادلة
+
+25
+00:02:00,100 --> 00:02:06,400
+parabola. إذًا اللي هو في الـ ZY plane هيكون في عندنا
+
+26
+00:02:06,400 --> 00:02:11,910
+parabola، هي اللي هي parabola اللي عندنا في الـ ZY plane.
+
+27
+00:02:11,910 --> 00:02:16,030
+نأخذ الآن X بيساوي 0، أو أخدنا المقطع اللي هو اللي
+
+28
+00:02:16,030 --> 00:02:20,790
+هي الـ plane، قطع هذا اللي هي الشكل فيه اللي هو
+
+29
+00:02:20,790 --> 00:02:25,470
+الـ parabola أمامنا. إذا صارت عندي Z بيساوي C على B
+
+30
+00:02:25,470 --> 00:02:30,740
+تربيع Y، هي عبارة عن parabola بين Z وبين مين؟ الـ Y
+
+31
+00:02:30,740 --> 00:02:34,880
+هي بين الـ Z وبين الـ Y، هي parabola اللي أمامنا
+
+32
+00:02:34,880 --> 00:02:38,960
+اللي هي Z بيساوي C على A تربيع في X تربيع، يعني ثابت
+
+33
+00:02:38,960 --> 00:02:42,140
+في X تربيع. الأعلى لإنه هذا موجب وهذا موجب زي ما
+
+34
+00:02:42,140 --> 00:02:45,990
+احنا عارفين C أكبر من 0، ماخدينها. اللي هناخده المقطع
+
+35
+00:02:45,990 --> 00:02:49,530
+الآخر اللي هو Y بيساوي 0. لما ناخد Y بيساوي 0 برضه
+
+36
+00:02:49,530 --> 00:02:52,850
+هيطلع عندي X تربيع على A تربيع بيساوي Z على C، أو
+
+37
+00:02:52,850 --> 00:02:57,370
+بمعنى آخر Z بتساوي C على A تربيع في X تربيع برضه
+
+38
+00:02:57,370 --> 00:03:00,530
+هذه معادلة parabola زي ما أنتم عارفين، بس بين مين
+
+39
+00:03:00,530 --> 00:03:04,950
+بين الـ Z والـ X، وهي parabola بين الـ Z وبين الـ X
+
+40
+00:03:04,950 --> 00:03:09,270
+الآن بنجمع هذه parabola الأولى وهي parabola
+
+41
+00:03:09,270 --> 00:03:13,050
+الثانية. إذا صار مقطعين، كل واحدة منهم parabola. نيجي
+
+42
+00:03:13,050 --> 00:03:16,610
+للمقطع الثالث. المقطع الثالث لو جينا أخدنا اللي هو
+
+43
+00:03:16,610 --> 00:03:20,870
+Z بصفر، هتطلع اللي هي X وY بصفر، لأنه بيصير X تربيع
+
+44
+00:03:20,870 --> 00:03:25,250
+على A تربيع، وY تربيع على B تربيع بصفر، يعني كلمتين
+
+45
+00:03:25,250 --> 00:03:28,030
+مربعتين مجموعهم صفر، يعني كل واحدة فيهن صفر، يعني X
+
+46
+00:03:28,030 --> 00:03:32,430
+بصفر وY بصفر وZ بصفر، يعني بمعنى آخر بتمر بنقطة من
+
+47
+00:03:32,430 --> 00:03:38,340
+الأصل. اللي هي الـ graph اللي عندي الآن. عشان هيك أن
+
+48
+00:03:38,340 --> 00:03:41,740
+عندي الـ C أكبر من 0. تمام، الـ C أكبر من 0. خذ الآن
+
+49
+00:03:41,740 --> 00:03:46,700
+قيمة Z مثلًا، خذها بيساوي C، يعني بدي مقطع Z بيساوي C
+
+50
+00:03:46,700 --> 00:03:53,480
+اللي ايش بعمل؟ اللي يعني بوازي الـ XY plane، اللي هو
+
+51
+00:03:53,480 --> 00:04:01,600
+أشوف ايش لما أقطع اللي هو الشكل بمستوى يوازي الـ X
+
+52
+00:04:01,600 --> 00:04:04,700
+Y plane. عند المستوى Z بيساوي 0 أشوف ايش اللي بتطلع
+
+53
+00:04:04,700 --> 00:04:08,140
+عندي. Z بتساوي C، بيصير X تربيع على A تربيع، زائد Y
+
+54
+00:04:08,140 --> 00:04:09,220
+تربيع على B تربيع
+
+55
+00:04:09,220 --> 00:04:10,120
+على B تربيع على B تربيع على B
+
+56
+00:04:10,120 --> 00:04:10,480
+تربيع على B تربيع على B تربيع على B
+
+57
+00:04:10,480 --> 00:04:11,420
+تربيع على B تربيع على B تربيع على B تربيع على B
+
+58
+00:04:11,420 --> 00:04:11,460
+تربيع على B تربيع على B تربيع على B تربيع على B
+
+59
+00:04:11,460 --> 00:04:16,740
+تربيع على B تربيع
+
+60
+00:04:16,770 --> 00:04:21,690
+اللي هو موازية الـ xy plane، طلع عندها ellipse، ولو
+
+61
+00:04:21,690 --> 00:04:26,310
+اجينا أخدنا أي مقطع آخر برضه هتطلع كمية ثابتة. هذه
+
+62
+00:04:26,310 --> 00:04:30,410
+هنجسم عليها، هيطلع واحد، هتطلع ellipse، وأي قيمة لـ Z
+
+63
+00:04:30,410 --> 00:04:33,870
+موجبة هنلاقيها برضه بتطلع ellipse. لكن لو أحد قال
+
+64
+00:04:33,870 --> 00:04:37,710
+ليش؟ طب فش رسم تحت؟ خذ Z بيساوي ناقص C، لو طلعت ناقص C
+
+65
+00:04:37,710 --> 00:04:40,930
+على C، بطلع ناقص واحد. مستحيل الـ X تربيع على إيه
+
+66
+00:04:40,930 --> 00:04:43,630
+تربيع، والـ Y تربيع على B تربيع تساوي ناقص واحد.
+
+67
+00:04:43,630 --> 00:04:47,610
+مستحيل تكون كمية سالبة، عشان هيك فش رسم وين؟ أسفل.
+
+68
+00:04:47,610 --> 00:04:52,670
+إذًا الرسم أعلى. عرفنا أن المقطع هذا parabola، والمقطع
+
+69
+00:04:52,670 --> 00:04:57,310
+الثاني parabola، قلنا في الـ XZ وYZ، وبينما في الـ XY
+
+70
+00:04:57,310 --> 00:05:00,930
+plane، الموازي للـ XY plane عند Z بيساوي C، طلع عند
+
+71
+00:05:00,930 --> 00:05:04,010
+ellipse، وأي مقطع آخر بيكون ellipse، عشان هيك بتطلع
+
+72
+00:05:04,010 --> 00:05:09,210
+عندي الشكل هذا، اللي هي بنسميها elliptical، اللي هي
+
+73
+00:05:09,210 --> 00:05:11,790
+الوصف elliptical، لإنه مقطع واحد اسمه ellipse
+
+74
+00:05:11,790 --> 00:05:16,050
+paraboloid. قلنا الـ paraboloid يعني صار في عندي
+
+75
+00:05:16,050 --> 00:05:19,470
+more than one parabola، اللي هي عندها نتو parabolas
+
+76
+00:05:19,470 --> 00:05:23,790
+صار عندي اللي هو اسمه elliptical paraboloid، يعني
+
+77
+00:05:23,790 --> 00:05:27,970
+في عندنا واحدة ellipse، واتنتين parabola. هذا اللي هو
+
+78
+00:05:27,970 --> 00:05:34,110
+الشكل، المعادلة التربيعية اللي أمامنا. نيجي الآن ناخذ
+
+79
+00:05:34,110 --> 00:05:38,150
+مثال آخر. example 3b: What is the graph of the
+
+80
+00:05:38,150 --> 00:05:43,090
+quadratic equation؟ X تربيع على 36، زائد Z تربيع على
+
+81
+00:05:43,090 --> 00:05:48,210
+25، بيساوي Y على 2. لاحظوا أن هذولا التنتين المربعات
+
+82
+00:05:48,210 --> 00:05:53,550
+وهذه بدون تربيع، اللي هو واضح أن هذه هي نفس اللي
+
+83
+00:05:53,550 --> 00:06:00,690
+جابلها، بس مبدلين الـ Z بمين؟ بـ Y. لاحظ هنا كان الـ X
+
+84
+00:06:00,690 --> 00:06:06,030
+آسف على اللي قبلها. الآن لما كانت الـ Z هي المش
+
+85
+00:06:06,030 --> 00:06:10,110
+تربيعي، كانت اللي هي الـ axis تبع الـ elliptical
+
+86
+00:06:10,110 --> 00:06:13,750
+paraboloid هو الـ Z، والـ X والـ Y هما اللي فيهم
+
+87
+00:06:13,750 --> 00:06:19,410
+تربيع، فكان الشكل اللي أمامنا. الآن لو عملنا Y مكان
+
+88
+00:06:19,410 --> 00:06:22,750
+Z زي ما في السؤال اللي بعد هذه، بيصير عندي X تربيع
+
+89
+00:06:22,750 --> 00:06:27,010
+على A تربيع، زائد Z تربيع على B تربيع، بيساوي Y على C.
+
+90
+00:06:27,010 --> 00:06:32,970
+بيصير الآن اللي هو الـ axis للشكل اللي عندي اللي هو
+
+91
+00:06:32,970 --> 00:06:37,850
+الـ Y، يعني بتصير هذه مجلوبة بس وجهها وين؟ للـ Y هيك، و
+
+92
+00:06:37,850 --> 00:06:43,710
+بتكون نفس الرسمة، لكن على الـ y-axis، اللي بيكون الـ
+
+93
+00:06:43,710 --> 00:06:48,730
+axis تبع الـ parabola. إذًا الـ X تربيعها 36، Z تربيعها
+
+94
+00:06:48,730 --> 00:06:54,710
+25، Y على 2، هي عبارة عن elliptical paraboloid، نفس
+
+95
+00:06:54,710 --> 00:06:59,490
+الأولاني، but ايش its axis؟ اللي هو مين صار الآن؟ الـ
+
+96
+00:06:59,490 --> 00:07:04,930
+Y بدل الـ Z. It is y-axis instead of the z-axis. Its
+
+97
+00:07:04,930 --> 00:07:09,410
+parabolas are in the yz plane. لو الآن على السريع
+
+98
+00:07:09,410 --> 00:07:14,290
+لو حطيت هذه Z بـ 0، هتطلع parabola بين الـ y، x. حط الـ
+
+99
+00:07:14,290 --> 00:07:17,390
+X بـ 0، parabola بين مين؟ الـ Z والـ Y. إذا صارت الـ
+
+100
+00:07:17,390 --> 00:07:21,790
+parabolas الآن، الـ two parabolas، واحدة اللي هي في
+
+101
+00:07:21,790 --> 00:07:26,850
+الـ yz plane، والثانية في الـ xz plane. في الـ yz
+
+102
+00:07:26,850 --> 00:07:31,230
+plane parabola، وفي الـ xz plane parabola، بينما الـ cross
+
+103
+00:07:31,230 --> 00:07:35,110
+section اللي هو الـ ellipse، الـ Y بتساوي عند Y
+
+104
+00:07:35,110 --> 00:07:38,090
+بتساوي C، أو Y بتساوي 2 في هذه الحالة، لأن الـ C
+
+105
+00:07:38,090 --> 00:07:41,830
+مأخذينها ايش؟ واحنا اتنين. Y بتساوي 2 بيصير عندي
+
+106
+00:07:41,830 --> 00:07:46,800
+اللي هو هذا عبارة عن ايش؟ عن ellipse، أو طبعًا لو أخدت
+
+107
+00:07:46,800 --> 00:07:50,380
+أي C أكبر من 0 برضه بيطلع عبارة عن ellipse زي ما
+
+108
+00:07:50,380 --> 00:07:53,600
+قلنا. إذًا هذا هو الشكل الأولاني بس نايم على الـ Y
+
+109
+00:07:53,600 --> 00:07:57,700
+-axis، بالاتجاه المدبب للـ z-axis. نيجي ناخذ المعادلة
+
+110
+00:07:57,700 --> 00:08:02,100
+الثانية. X تربيع ناقص 4X، زائد 2Y تربيع، زائد 10Y، ناقص 13Z
+
+111
+00:08:02,100 --> 00:08:06,270
+زائد 29، بيساوي 0. هذا المثال درّينا عليه. على طول بنعمل
+
+112
+00:08:06,270 --> 00:08:09,550
+بين هدول إكمال مربع، وبين هدول إكمال مربع، وهذه
+
+113
+00:08:09,550 --> 00:08:13,610
+المعادلة
+
+114
+00:08:13,610 --> 00:08:18,110
+خطية لا تلزم إكمال مربع ولا غيره. فبنكمل المربع و
+
+115
+00:08:18,110 --> 00:08:20,590
+بنزيد أربعة، وعارفين كيف بنزيد أربعة وعملناها قبل
+
+116
+00:08:20,590 --> 00:08:24,510
+هيك، وهذه بنزيد ايش؟ 25، لأنه بيصير هادم
+
+117
+00:08:24,510 --> 00:08:27,770
+مع بعض التنتين. اللي هو ناقص 4 على 2 بيطلع
+
+118
+00:08:27,770 --> 00:08:32,290
+ناقص 2، بتربيعها بتطلع 4، والـ 10 على 2
+
+119
+00:08:32,290 --> 00:08:35,270
+معامل Y بتطلع 5، بتربيعها بتطلع 25. بيصير
+
+120
+00:08:35,270 --> 00:08:38,690
+هذا مربع كامل، وهذا مربع كامل. ضفت أنا 29،
+
+121
+00:08:38,690 --> 00:08:41,810
+لازم تشيل 29، هي شلناها الـ 29،
+
+122
+00:08:41,810 --> 00:08:46,930
+وطلعت اللي هي المعادلة زي اللي أمامنا. أصارعت لأنه
+
+123
+00:08:46,930 --> 00:08:50,670
+وكلام معاد هذا. الآن هذه مربع كامل، طلعت X ناقص 2
+
+124
+00:08:50,670 --> 00:08:53,050
+تربيع على 36، بعد ما قسمنا الكل على 36،
+
+125
+00:08:53,050 --> 00:08:55,970
+و36، وهذه نفس الشيء، قسمنا على 36، صارت
+
+126
+00:08:55,970 --> 00:08:59,750
+بيساوي هذه. الآن نفس الأولى، بس صار لها shift على الـ
+
+127
+00:08:59,750 --> 00:09:04,970
+X axis باتنين، وعلى الـ Y axis بناقص خمسة. إذًا it is an
+
+128
+00:09:04,970 --> 00:09:10,200
+elliptical paraboloid. هذه مربعين، عندي الـ X تربيع و
+
+129
+00:09:10,200 --> 00:09:15,580
+الـ Y تربيع مع shift، وهنا مربع، وهنا بدون مربع، إذًا
+
+130
+00:09:15,580 --> 00:09:19,500
+صارت عندي عبارة عن elliptical paraboloid. مش مصدق؟
+
+131
+00:09:19,500 --> 00:09:23,340
+حط Y بيساوي ناقص 5، بيطلع عندي هذه parabola، وهذه
+
+132
+00:09:23,340 --> 00:09:26,160
+بحط X بيساوي 2 بيصير هذه برضه parabola ثانية، إذًا
+
+133
+00:09:26,160 --> 00:09:29,680
+two parabola. زي ده، parabola Y والـ Z، بحطها اللي هو
+
+134
+00:09:29,680 --> 00:09:33,300
+بتساوي 1، بيصير عبارة عن ايش؟ عن ellipse، يعني فعلا
+
+135
+00:09:33,300 --> 00:09:37,260
+elliptical paraboloid، زي اللي قبل. Similar to that
+
+136
+00:09:37,260 --> 00:09:41,960
+example خمسة، but it is shifted two in the positive
+
+137
+00:09:41,960 --> 00:09:47,010
+x-axis وخمسة units in the negative y-axis. عشان هيك
+
+138
+00:09:47,010 --> 00:09:51,630
+هتلاقيها بتمر بمين؟ الآن كانت تتمر في الأول بـ 0 و0
+
+139
+00:09:51,630 --> 00:09:58,030
+و0، هالجيل صارت تتمر بـ 2 و-5 و0. In a state of 00
+
+140
+00:09:58,030 --> 00:10:04,410
+and its axis. لو كانت هذه Y تربيع وهذه X تربيع،
+
+141
+00:10:04,410 --> 00:10:09,530
+بيصير الـ axis اللي هو Z-axis، لكن اللي هو صار shift
+
+142
+00:10:09,530 --> 00:10:15,530
+لكل graph بمقدار 2 على الـ X وناقص 5 على الـ Y، عشان
+
+143
+00:10:15,530 --> 00:10:21,410
+هيك اللي هو الـ axis تبعها مش هيكون الـ Z-axis نفسه،
+
+144
+00:10:21,410 --> 00:10:26,190
+لأ، هيكون line parallel to the Z-axis، بس بمر بمين؟
+
+145
+00:10:26,190 --> 00:10:33,150
+بنقطة 2 وناقص 5، وأي نقطة Z. الآن لأنه shift 2 على
+
+146
+00:10:33,150 --> 00:10:38,530
+الـ X-axis و5 على الـ Y-axis، فبيصير اللي هو الـ axis
+
+147
+00:10:38,530 --> 00:10:42,490
+تبعها بدل ما كان الـ Z، صار اللي هو الـ axis الموازي
+
+148
+00:10:42,490 --> 00:10:47,630
+للـ Z-axis، و بيمر بـ 2 وناقص 5. طبعًا عمودي عليهم،
+
+149
+00:10:47,630 --> 00:10:52,290
+موازي للـ Z-axis، والـ Z بتأخذ أي قيمة عندي، والـ A
+
+150
+00:10:52,290 --> 00:10:57,850
+طبعًا 6، والـ B 6، والـ C بتساوي 1 زي ما أنتم عارفين.
+
+151
+00:10:58,140 --> 00:11:03,780
+إذًا هذا اللي عندي اللي هو مثال على اللي هي الـ
+
+152
+00:11:03,780 --> 00:11:10,340
+elliptical paraboloid. نيجي الآن للنوع الثاني، أو
+
+153
+00:11:10,340 --> 00:11:13,800
+الخامس، يعني احنا أخذنا الآن صار عندنا الـ
+
+154
+00:11:13,800 --> 00:11:18,460
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+155
+00:11:18,460 --> 00:11:19,080
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+156
+00:11:19,080 --> 00:11:19,740
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+157
+00:11:19,740 --> 00:11:20,020
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+158
+00:11:20,020 --> 00:11:21,220
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+159
+00:11:21,220 --> 00:11:22,920
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+160
+00:11:22,920 --> 00:11:24,820
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+161
+00:11:24,820 --> 00:11:25,680
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+162
+00:11:25,680 --> 00:11:25,700
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+163
+00:11:34,510 --> 00:11:39,970
+الآن بدنا نحكي عن نوع خامس، اللي هو قبل الأخير
+
+223
+00:16:11,800 --> 00:16:15,560
+المربع له بيصير إكمال المربع، هنا أكملنا المربع
+
+224
+00:16:15,560 --> 00:16:20,580
+ماشي الحال، ضفنا أربعة، وضفنا خمسة وعشرين، الآن بيصير
+
+225
+00:16:20,580 --> 00:16:23,420
+ضفنا تسعة وعشرين، وفي عندنا عشرين، يعني بيكون ضفنا
+
+226
+00:16:23,420 --> 00:16:26,880
+تسعة، إذا بنضيف على... شلنا العشرين، أقصد من
+
+227
+00:16:26,880 --> 00:16:30,000
+هنا بيظل إيش؟ تسعة، لازم نضيفها على الطرف الثاني
+
+228
+00:16:30,000 --> 00:16:33,930
+بنكسب كلّه على تسعة، وبنكمل المربع، بتطلع عندي
+
+229
+00:16:33,930 --> 00:16:37,630
+المعادلة اللي أمامي هذه، هذه المعادلة اللي أمامي
+
+230
+00:16:37,630 --> 00:16:42,510
+بالضبط، هي تربيع، وهي تربيع، موجب، وهذه السالب، السالب
+
+231
+00:16:42,510 --> 00:16:49,450
+هو اللي هيكون مالَه؟ اللي هو هو، أو ما يوازي الـ axis
+
+232
+00:16:49,450 --> 00:16:54,790
+تبع الـ hyperboloid of one sheet، هو إذا كان ما فيش
+
+233
+00:16:54,790 --> 00:16:59,620
+أي shift، يعني فيش shift، كل مرة بيكون هو، لو في shift
+
+234
+00:16:59,620 --> 00:17:03,600
+زي هنا بيصير اللي هو الـ axis، اللي هو الـ z-axis
+
+235
+00:17:03,600 --> 00:17:08,560
+انزاح إثنين باتجاه الـ positive x-axis، ونقص خمسة
+
+236
+00:17:08,560 --> 00:17:13,980
+باتجاه الـ negative y-axis، فبيصير اللي هو الـ axis
+
+237
+00:17:13,980 --> 00:17:19,820
+للـ hyperboloid of one sheet، هنا هو عبارة عن المحور
+
+238
+00:17:19,820 --> 00:17:27,460
+الموازي لمحور Z والذي يمر بالنقطة 2 وناقص 5، يعني
+
+239
+00:17:27,460 --> 00:17:30,740
+بيصير عندنا الخط اللي بيمر بـ 2 ناقص 5 و Z اللي هي
+
+240
+00:17:30,740 --> 00:17:34,880
+المتغيرة، بيكون هذا الخط الموازي اللي هو الـ Z-axis
+
+241
+00:17:34,880 --> 00:17:40,320
+هو عبارة عن الـ axis للـ hyperboloid of one sheet
+
+242
+00:17:40,460 --> 00:17:43,740
+عشان هيك هذا بيمثل hyperboloid of one sheet
+
+243
+00:17:43,740 --> 00:17:47,280
+similar to that in example ستة اللي قلنا عنه قبل
+
+244
+00:17:47,280 --> 00:17:51,560
+وشوية، but it is shifted two units in the positive
+
+245
+00:17:51,560 --> 00:17:56,720
+x-axis، وخمسة units in the negative y-axis، so its
+
+246
+00:17:56,720 --> 00:18:00,140
+axis is the line that is parallel to the z-axis plane
+
+247
+00:18:00,140 --> 00:18:03,600
+and passes through إثنين وناقص خمسة، أو Z أيًّا
+
+248
+00:18:03,600 --> 00:18:14,190
+كانت B3 وB3 وC1، نجي الآن للنوع الأخير، أو
+
+249
+00:18:14,190 --> 00:18:18,010
+الصنف الأخير اللي هو من الـ quadratic surfaces
+
+250
+00:18:18,010 --> 00:18:22,850
+اللي هندرسه، اللي هو اللي بنسميه hyperboloid of two
+
+251
+00:18:22,850 --> 00:18:29,780
+sheets، الآن هنا بدل واحدة صار اثنتين، عندنا فئتين
+
+252
+00:18:29,780 --> 00:18:34,900
+سالب، الآن اللي ما لهوش سالب، اللي موجب، الغريب يعني
+
+253
+00:18:34,900 --> 00:18:38,060
+هناك كان الغريب اللي هو اللي الـ Z اللي له سالب
+
+254
+00:18:38,060 --> 00:18:42,860
+واحد، الآن اللي الغريب هنا اللي موجب، الغريب هو اللي
+
+255
+00:18:42,860 --> 00:18:49,730
+بيحدد اللي هو الـ axis، لمَن؟ للشكل اللي عندنا، شوف
+
+256
+00:18:49,730 --> 00:18:56,090
+كيف نتبع نفس النهج، ونفس الأسلوب، بدنا نعرف مين يا
+
+257
+00:18:56,090 --> 00:19:02,730
+جماعة المقاطع، خد X بصفر، بيصير عندنا Z تربيع على C
+
+258
+00:19:02,730 --> 00:19:05,530
+تربيع ناقص Y تربيع على B تربيع، بيساوي واحد
+
+259
+00:19:05,530 --> 00:19:10,230
+وكلكم عارف إن هذه معادلة Hyperbola، في الـ ... في الـ
+
+260
+00:19:10,230 --> 00:19:17,960
+YZ-plane، Hyperbola في الـ YZ-plane، هي اللي هو بين الـ
+
+261
+00:19:17,960 --> 00:19:23,180
+Y والـ Z، هي واحدة بين الـ Y والـ Z، وهي الثانية بين
+
+262
+00:19:23,180 --> 00:19:26,670
+الـ Y والـ Z على الجهة الثانية، عملت للـ hyperbola
+
+263
+00:19:26,670 --> 00:19:31,870
+شوفها، الـ gate الآن، خد Z بصفر... لأ...
+
+264
+00:19:31,870 --> 00:19:35,670
+أجل... Y بصفر، بيصير الـ hyperbola Z تربيع على C تربيع
+
+265
+00:19:35,670 --> 00:19:39,250
+ناقص X تربيع على A تربيع، بيساوي إيش؟ واحد برضه
+
+266
+00:19:39,250 --> 00:19:44,110
+هي دي إيش إسمها؟ hyperbola، بس بين الـ X والـ Z، هي
+
+267
+00:19:44,110 --> 00:19:48,990
+الـ hyperbola بين الـ X هيّا والـ Z، وهي الـ
+
+268
+00:19:48,990 --> 00:19:56,190
+hyperbola،  عجبالها بين الـ X والـ Z، هيّا هي هذه
+
+269
+00:19:56,190 --> 00:20:06,540
+وهذه، والأولى هذه بين الـ Y والـ Z، ماشي، طيب نيجي إذا
+
+270
+00:20:06,540 --> 00:20:11,240
+صار عندنا المقطعين hyperbola بالشكل اللي أمامنا، طيب
+
+271
+00:20:11,240 --> 00:20:14,100
+خد Z  بـالساوية صفر، عشان نقولنا إيش؟ طب أنتَ تركت لنا
+
+272
+00:20:14,100 --> 00:20:17,100
+الفرار ده ليش؟ الفرار هذا، هنشوف، تعالوا نشوف، نأخذ
+
+273
+00:20:17,100 --> 00:20:21,040
+الكلام اللي بأحكي عنه، اللي هنأخذ Z بـالساوية صفر، لما Z
+
+274
+00:20:21,040 --> 00:20:24,760
+بـالساوية صفر، بيصير X تربيع على A تربيع زائد Y تربيع على B تربيع
+
+275
+00:20:24,760 --> 00:20:28,880
+بيساوي سالب واحد، لأنه بيصير ناقص ناقص وواحد، فبيصير
+
+276
+00:20:28,880 --> 00:20:31,880
+وجهتين في ناقص وبيطلع موجب تربيع تربيع تربيع، بيه
+
+277
+00:20:31,880 --> 00:20:35,040
+تربيع يساوي ناقص واحد، ده كلام مستحيل يتحقق، هدول
+
+278
+00:20:35,040 --> 00:20:38,720
+موجبات، وهذه سالبة، إذًا فش رسم عند Z بتساوي صفر
+
+279
+00:20:38,720 --> 00:20:46,420
+رأيك أصلًا، خد أي قيمة Z من هذه الـ C لعند ناقص C من
+
+280
+00:20:46,420 --> 00:20:50,000
+هنا لها، أي Z، يعني خد أي Z، الـ absolute value للـ Z
+
+281
+00:20:50,000 --> 00:20:56,890
+أصغر من C، بيصير عندك الآن، لما تأخذ قيمة أصغر من الـ
+
+282
+00:20:56,890 --> 00:21:00,050
+C، بيصير اللي فوق أصغر من اللي تحت، بيصير هذا كلّه
+
+283
+00:21:00,050 --> 00:21:04,230
+أصغر من واحد، لما أصغر من واحد وأجيبه هان، بيصير
+
+284
+00:21:04,230 --> 00:21:07,530
+واحد ناقص شيء أصغر من واحد، بيطلع كمية موجبة، اللي
+
+285
+00:21:07,530 --> 00:21:10,890
+هان بيصير الآن، أضرب الكل في سالب، بيصير X تربيع على
+
+286
+00:21:10,890 --> 00:21:14,190
+A تربيع زائد Y تربيع على B تربيع، بيساوي كمية سالبة
+
+287
+00:21:14,190 --> 00:21:19,550
+بعد ما ضربت بالسالب، إذًا مستحيل يتحقق، يعني أي Z في
+
+288
+00:21:19,550 --> 00:21:24,130
+المنطقة هذه بين C وبين ناقص C، هتعمل لي اللي هو
+
+289
+00:21:24,130 --> 00:21:28,030
+كمية مجبرة بيساوي سالبة، يعني لن تتحقق هذه
+
+290
+00:21:28,030 --> 00:21:32,550
+المعادلة، مدام المعادلة هذه لن تتحقق، معناته فيش رسم
+
+291
+00:21:32,550 --> 00:21:36,930
+في المنطقة هذه، إذًا الرسم هيكون من عند C وطالع، ومن
+
+292
+00:21:36,930 --> 00:21:41,990
+عند ناقص C ونازل، لأن لو حطيت الآن عند Z بتساوي C
+
+293
+00:21:41,990 --> 00:21:46,590
+بتطلع هذه واحد، أو ناقص C، بتنقل هان، بيصير صفر، بيصير X
+
+294
+00:21:46,590 --> 00:21:49,490
+تربيع على A تربيع زائد Y تربيع على B تربيع، بيساوي
+
+295
+00:21:49,490 --> 00:21:53,930
+صفر، يعني X صفر، وY صفر، وZ بتساوي C، يعني بتمر
+
+296
+00:21:53,930 --> 00:21:58,510
+بالنقطة هذه صفر وصفر وC، وبتمر بالنقطة صفر وصفر
+
+297
+00:21:58,510 --> 00:22:03,470
+و ناقص C، الآن الرسمة هتكون هنا وهنا، وأكدنا عليه
+
+298
+00:22:03,470 --> 00:22:06,470
+بالـ hyperbola هذه والـ hyperbola هذه، أقول لكم بدي أؤكد
+
+299
+00:22:06,470 --> 00:22:09,930
+عليه بطريقة ثانية، خد Z  ده إيش بتساوي؟ 2C، يعني اطلع
+
+300
+00:22:09,930 --> 00:22:14,290
+على فوقها 2C مثلًا، أو سالب 2C، مش فارقة، اللي بدك إياه
+
+301
+00:22:14,510 --> 00:22:18,810
+أو الاثنين مع بعض، الآن بيصير عندنا X تربيع، هذه 2C
+
+302
+00:22:18,810 --> 00:22:22,030
+لما أربعها بيصير 4C تربيع، 4C تربيع، C
+
+303
+00:22:22,030 --> 00:22:24,890
+تربيع يعني أربعة، أربعة، أجيبها على الجهة الثانية
+
+304
+00:22:24,890 --> 00:22:28,030
+بيصير ناقص أربعة، واحد ناقص أربعة، جدّيش؟ ناقص ثلاثة
+
+305
+00:22:28,030 --> 00:22:31,640
+ناقص ثلاثة، وهنا ناقص وهنا ناقص، أضرب الكل في ناقص
+
+306
+00:22:31,640 --> 00:22:34,660
+بيصير X تربيع على A تربيع زائد Y تربيع على B تربيع، بيساوي ثلاثة
+
+307
+00:22:34,660 --> 00:22:39,140
+وهذه عبارة عن مين؟ عن ellipse، إذًا المقطع هنا عند
+
+308
+00:22:39,140 --> 00:22:44,060
+2C أو قبلها شوية، وقبلها شوية، وهنا وهنا، واتحد
+
+309
+00:22:44,060 --> 00:22:48,480
+كلهن بيطلع عن مين؟ ellipses، عشان هيك بنقول عنها
+
+310
+00:22:48,850 --> 00:22:53,410
+اللي هو hyperbola دي، يعني في عندنا مقطعين hyperbola
+
+311
+00:22:53,410 --> 00:22:57,310
+بس مش of one sheet زي اللي قبله شوية، of two
+
+312
+00:22:57,310 --> 00:23:03,600
+sheets، sheet فوق وsheet تحت، طبعًا هذه زي ما
+
+313
+00:23:03,600 --> 00:23:09,160
+نسميها الـ vertex 0,0,C وهذه الـ vertex 0,0,-C اللي
+
+314
+00:23:09,160 --> 00:23:14,160
+بتمر فيها الـ hyperboloid
+
+315
+00:23:14,160 --> 00:23:18,240
+of one sheet، of two sheets، وإذا بدك تسميها
+
+316
+00:23:18,240 --> 00:23:25,520
+elliptical hyperboloid، تنفع، لكن الاسم الشائع اللي هو
+
+317
+00:23:25,520 --> 00:23:33,250
+hyperboloid، الأولى of one sheet
+
+318
+00:23:33,250 --> 00:23:36,350
+of one sheet، والثانية of two sheets، دي أخذ المثال
+
+319
+00:23:36,350 --> 00:23:40,070
+الأخير، الـ example، what is the graph of
+
+320
+00:23:40,070 --> 00:23:44,160
+the quadratic equation؟ هذا الآن عملنا نفس اللي
+
+321
+00:23:44,160 --> 00:23:48,060
+عملناه قبل، نفس النهج، الآن بدل الـ Z حطيت الـ Y، يعني
+
+322
+00:23:48,060 --> 00:23:52,000
+هيصير الآن الـ axis الغريب، من الغريب الآن اللي هو
+
+323
+00:23:52,000 --> 00:23:55,780
+الـ Y تربيع، الـ Y الغريب، ليش؟ لأن هذا شرط سالب، هذا شرط
+
+324
+00:23:55,780 --> 00:23:59,740
+سالب، وزيّه، وهذا المختلف عنه، إذًا هو الـ axis تبع الـ
+
+325
+00:23:59,740 --> 00:24:03,760
+hyperboloid of one of two sheets، of
+
+326
+00:24:03,760 --> 00:24:07,420
+two sheets، لإنه عندنا two سالب's، الآن في الأولاني
+
+327
+00:24:07,420 --> 00:24:11,420
+of one sheet اللي عنده فيه كان عندنا one سالب، الآن
+
+328
+00:24:11,420 --> 00:24:13,480
+هذه هي عبارة عن... عبارة عن... عبارة عن... عبارة عن... عبارة
+
+329
+00:24:13,480 --> 00:24:14,660
+عن... عبارة عن... عبارة عن... عبارة عن... عبارة عن... عبارة
+
+330
+00:24:14,660 --> 00:24:16,000
+عن... عبارة عن... عبارة عن... عبارة عن... عبارة عن... عبارة
+
+331
+00:24:16,000 --> 00:24:18,160
+عن... عبارة عن... عبارة عن... عبارة عن... عبارة عن... عبارة
+
+332
+00:24:18,160 --> 00:24:22,780
+عن... عبارة عن... عبارة عن... عبارة عن... عبارة عن... عبارة
+
+333
+00:24:22,780 --> 00:24:23,040
+عن... عبارة عن... عبارة عن... عبارة عن... عبارة عن... عبارة
+
+334
+00:24:23,040 --> 00:24:23,100
+عن... عبارة عن... عبارة عن... عبارة عن... عبارة عن... عبارة
+
+335
+00:24:23,100 --> 00:24:27,760
+عن... عبارة عن... عبارة عن... عبارة عن... عبارة عن... عبارة
+
+336
+00:24:27,760 --> 00:24:30,740
+عن
+
+337
+00:24:40,280 --> 00:24:44,200
+المعادلة الأخيرة هذه، نفس الشيء، سنعملها إكمال مربع
+
+338
+00:24:44,200 --> 00:24:48,480
+عملنا لها إكمال مربع، صارت Z تربيع، أخدت السالب
+
+339
+00:24:48,480 --> 00:24:52,660
+برا، بصارت Z تربيع ناقص أربعة، ناقص... ناقص Y تربيع ناقص
+
+340
+00:24:52,660 --> 00:24:52,800
+ناقص، وY تربيع ناقص... ناقص، وY تربيع ناقص... ناقص، وY
+
+341
+00:24:52,800 --> 00:24:53,060
+تربيع ناقص... وY تربيع ناقص... وY تربيع ناقص... وY
+
+342
+00:24:53,060 --> 00:24:56,160
+تربيع ناقص... وY تربيع ناقص... وY تربيع ناقص... وY
+
+343
+00:24:56,160 --> 00:25:02,000
+تربيع ناقص... وY تربيع ناقص... وY تربيع ناقص... وY
+
+344
+00:25:02,000 --> 00:25:06,910
+تربيع ناقص... وY تربيع، اللي هو سالب أربعة، وضفت سالب
+
+345
+00:25:06,910 --> 00:25:10,190
+خمسة وعشرين، يعني ضفت سالب تسعة وعشرين، بضيف على
+
+346
+00:25:10,190 --> 00:25:13,230
+الطرف الثالث سالب تسعة وعشرين، وكان
+
+347
+00:25:13,230 --> 00:25:18,870
+ثلاثين، بيصير إيش؟ واحد، لأن هذه جاهزة بجسمها اللي هو
+
+348
+00:25:20,670 --> 00:25:27,150
+لأ، اسمحولي بس أغير هنا، خلي هذه الآن بنضيف كمان
+
+349
+00:25:27,150 --> 00:25:30,590
+ثمانية، يعني خليها ثمانية وثلاثين، هذه أساس أنها
+
+350
+00:25:30,590 --> 00:25:34,530
+تصير هذه تسعة، ولما أجسم على تسعة بيطلع إيش؟ واحد
+
+351
+00:25:34,530 --> 00:25:37,530
+فاهمين عليه؟ يعني هذا في السؤال، هذه مش ثلاثين، هذه
+
+352
+00:25:37,530 --> 00:25:41,070
+بيساوي ثمانية وثلاثين، ثمانية وثلاثين، وتسعة وعشرين، و
+
+353
+00:25:41,070 --> 00:25:49,810
+تسعة... تسعة... تسعة... تسعة
+
+354
+00:25:49,810 --> 00:25:50,370
+وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة...
+
+355
+00:25:50,370 --> 00:25:50,490
+وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة...
+
+356
+00:25:50,490 --> 00:25:50,770
+وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة...
+
+357
+00:25:50,770 --> 00:25:51,450
+وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة...
+
+358
+00:25:51,450 --> 00:25:52,030
+وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة... وتسعة...
+
+359
+00:25:52,030 --> 00:26:00,110
+وتسعة... وتسعة... وتسعة... وتسعة
+
+360
+00:26:00,110 --> 00:26:08,510
+وتسعة،  لو لم يكن هناك shift في المرة
+
+361
+00:26:08,510 --> 00:26:12,410
+فسيكون الـ Z-axis، لكن مادام صار فيه shift، صار الـ
+
+362
+00:26:12,410 --> 00:26:19,370
+Z-axis اللي هو المحور، هيه، بس إذا سحبنا اللي هو 2
+
+363
+00:26:19,370 --> 00:26:24,210
+على الـ X، وسالب 5 على الـ Y، فبيصير عبارة عن الـ axis
+
+364
+00:26:24,210 --> 00:26:30,250
+لهذه الـ hyperboloid of two sheets، عبارة عن الخط اللي
+
+365
+00:26:30,250 --> 00:26:35,690
+بوازي الـ Z-axis، ويمر بالنقطة 2 وناقص 5، وأي
+
+366
+00:26:35,690 --> 00:26:39,970
+نقطة Z، هذا اللي هو محور الـ hyperboloid of two
+
+367
+00:26:39,970 --> 00:26:45,030
+sheets، الآن هذه اللي هي نفس الأولى اللي حكينا عنها
+
+368
+00:26:45,030 --> 00:26:51,450
+قبل هيك، اللي هي هذه، بس  إضافتها عبارة عن shift على الـ
+
+369
+00:26:51,450 --> 00:26:56,230
+X-axis بمقدار 2، وهذه ستتكرر، وعلى الـ Y-axis بمقدار 5
+
+370
+00:26:56,230 --> 00:27:01,150
+بس بالاتجاه السالب، الحالة هي، كبسيرة، تمر اللي هي في 2
+
+371
+00:27:01,150 --> 00:27:06,890
+و5، وZ بتساوي 1 زائد أو ناقص 1، ليه؟ قيمة C هذه
+
+372
+00:27:06,890 --> 00:27:12,250
+instead of 0,0، وزائد C، والـ X-axis زي ما قلنا is
+
+373
+00:27:12,250 --> 00:27:15,930
+the line that is parallel to the Z-axis، and  بصفر
+
+374
+00:27:15,930 --> 00:27:20,230
+إثنين وناقص خمسة وZ الـ... وبعدين الـ a تبعتها
+
+375
+00:27:20,230 --> 00:27:24,710
+بتساوي... اللي هي ثلاثة، والـ b ثلاثة، والـ c بتساوي
+
+376
+00:27:24,710 --> 00:27:29,570
+واحد، وهيك بيكون إحنا خلصنا اللي هو كل اثنين وستة
+
+377
+00:27:29,570 --> 00:27:34,490
+وهي الـ exercises اللي داعمة، وبارك الله فيكم، وإلى
+
+378
+00:27:34,490 --> 00:27:35,670
+لقاء آخر
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/EFnvj3yPdag_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/EFnvj3yPdag_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..3aa45d584bc7a1f57ba3ca35fd150d05cf962a23
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/EFnvj3yPdag_postprocess.srt
@@ -0,0 +1,1512 @@
+1
+00:00:00,490 --> 00:00:04,210
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله هذه
+
+2
+00:00:04,210 --> 00:00:09,590
+هي المحاضرة رقم 7 لمادة أو مساق تفاضل وتكامل
+
+3
+00:00:09,590 --> 00:00:14,150
+الـGene وهي الجزء الثاني من ال section الأخير في
+
+4
+00:00:14,150 --> 00:00:21,090
+شبطر 12 الـ12.6كننا حكينا عن اللي هو ال cylinders
+
+5
+00:00:21,090 --> 00:00:25,210
+and quadratic surfaces و حكينا عن ثلاثة من ال
+
+6
+00:00:25,210 --> 00:00:28,230
+quadratic surfaces ده بتتذكروا حكينا عن ال
+
+7
+00:00:28,230 --> 00:00:34,030
+ellipsoid و ال hyperbolic paraboloid و عن اللي هو
+
+8
+00:00:34,030 --> 00:00:38,620
+elliptical coneضال عندنا تلت اللي هو أنواع جديدة
+
+9
+00:00:38,620 --> 00:00:43,200
+هنحكي عنها اللي هي من ال quadratic surfaces اللي
+
+10
+00:00:43,200 --> 00:00:49,940
+ال surfaces أو اللي هي السطوح أو المجسمات اللي هي
+
+11
+00:00:49,940 --> 00:00:54,890
+رسم لمعادلات تربيعيةالان نجي ال example خمسة
+
+12
+00:00:54,890 --> 00:00:57,830
+example خمسة اللي هو هيكون عندنا حاجة اسمها
+
+13
+00:00:57,830 --> 00:01:01,170
+elliptical paraboloid لتنشوف ايش elliptical
+
+14
+00:01:01,170 --> 00:01:04,690
+paraboloid زي ما قلنا قبل هيك في المحاضرة السابقة
+
+15
+00:01:04,690 --> 00:01:11,750
+انه هيكون اللي هو اكيد الاسم له علاقة بالجسم او
+
+16
+00:01:11,750 --> 00:01:17,420
+الشكل اللي هيطلعالجراف ده اللي هو X تربيع زائد Y
+
+17
+00:01:17,420 --> 00:01:22,780
+تربيع على B تربيع بساوي Z عالميا على C بدنا الآن
+
+18
+00:01:22,780 --> 00:01:27,500
+اللي هو نشوف هذا اللي هو الجراف اي هذا المعادلة
+
+19
+00:01:27,500 --> 00:01:32,980
+بتمثل جراف ماذا قولنا احنا في الأسئلة من هالنوع
+
+20
+00:01:32,980 --> 00:01:39,320
+اول ما بنحكي والشغلة الأساسية في تحديد الشكل هي
+
+21
+00:01:39,320 --> 00:01:44,820
+المقاطعة اللي هي الشكلفالان اول حاجة بناخد X بساوة
+
+22
+00:01:44,820 --> 00:01:49,600
+0 لما X بساوة 0 بيصير عندى Y تربيع على B تربيع
+
+23
+00:01:49,600 --> 00:01:54,100
+بساوة Z على C او بمعنى اخر تطلع عندى Z بساوة C على
+
+24
+00:01:54,100 --> 00:02:00,100
+B تربيع في Y تربيع وهذا واضح ان هي عبارة عن معادلة
+
+25
+00:02:00,100 --> 00:02:06,400
+برابولة اذا اللى هو في ال ZY plane هيكون في عندنا
+
+26
+00:02:06,400 --> 00:02:11,910
+برابولة هي اللى هي البرابولة اللى عندنان ي ز بلين
+
+27
+00:02:11,910 --> 00:02:16,030
+نما ناخدنا X بساوة 0 أو أخدنا المقطع اللي هو اللي
+
+28
+00:02:16,030 --> 00:02:20,790
+هي ال plane قطع هذا اللي هي الشكل فيه اللي هو
+
+29
+00:02:20,790 --> 00:02:25,470
+البرابولة الأمامنا إذا صارت عندي Z بساوة C على B
+
+30
+00:02:25,470 --> 00:02:30,740
+تربيع Y هي عبارة عن برابولة بين Zو بين مين؟ الـ Y
+
+31
+00:02:30,740 --> 00:02:34,880
+هي بين الـ Z و بين الـ Y هي البرابولة اللى أمامنا
+
+32
+00:02:34,880 --> 00:02:38,960
+اللى هي Z بساوة C على A تربيع في X تربيع يعني ثابت
+
+33
+00:02:38,960 --> 00:02:42,140
+في X تربيع الأعلى لإنه هذا موجب و هذا موجب زي ما
+
+34
+00:02:42,140 --> 00:02:45,990
+احنا عارفين C أكبر من 0 ماخدينهااللي هناخده المقطع
+
+35
+00:02:45,990 --> 00:02:49,530
+الاخر اللي هو Y بيساوي 0 لما ناخد Y بيساوي 0 برضه
+
+36
+00:02:49,530 --> 00:02:52,850
+هيطلع عند X تربي على A تربيع بيساوي Z على C أو
+
+37
+00:02:52,850 --> 00:02:57,370
+بمعنى أخر Z بتساوي C على A تربيع في X تربيع برضه
+
+38
+00:02:57,370 --> 00:03:00,530
+هذه معادلة برابولة زي ما انتوا عارفين بس بين مين
+
+39
+00:03:00,530 --> 00:03:04,950
+بين ال Z وال X وهي البرابولة بين ال Zو بين الـ X
+
+40
+00:03:04,950 --> 00:03:09,270
+الآن بنجمع هذه البرابولة الأولى وهي البرابولة
+
+41
+00:03:09,270 --> 00:03:13,050
+الثانية اذا صار مقطعين كل واحدة منهم برابولة نيجي
+
+42
+00:03:13,050 --> 00:03:16,610
+للمقطع الثالث المقطع الثالث لو جينا أخدنا اللي هو
+
+43
+00:03:16,610 --> 00:03:20,870
+Z بصفر هتطلع اللي هي X وY بصفر لأنه بيصير X تربيع
+
+44
+00:03:20,870 --> 00:03:25,250
+على A تربيع Z وY تربيع بتربيع بصفر يعني كلمتين
+
+45
+00:03:25,250 --> 00:03:28,030
+مربعتين مجموحين صفر يعني كل واحدة فين صفر يعني X
+
+46
+00:03:28,030 --> 00:03:32,430
+بصفر وY بصفر وZ بصفر يعني بمعنى أخر بتمر بنقطة من
+
+47
+00:03:32,430 --> 00:03:38,340
+الأصلاللي هي ال graph اللي عندي الان عشان هيك ان
+
+48
+00:03:38,340 --> 00:03:41,740
+عندي ال C أكبر من 0 مزام ال C أكبر من 0 خد الآن
+
+49
+00:03:41,740 --> 00:03:46,700
+قيمة Z مثلا خدها بساوي C يعني بدي مقطع Z بساوي C
+
+50
+00:03:46,700 --> 00:03:53,480
+اللي ايش بعمل اللي يعني بوازي ال X Y plane اللي هو
+
+51
+00:03:53,480 --> 00:04:01,600
+اشوف ايه لما اقطعاللي هو الشكل بمستوى يوازي الـ X
+
+52
+00:04:01,600 --> 00:04:04,700
+Y plane عند المستوى Z بساوة 0 اشوف ايش اللي بتطلع
+
+53
+00:04:04,700 --> 00:04:08,140
+عندى Z بتساوى C بيصير X تربيه على A تربيه على Z Y
+
+54
+00:04:08,140 --> 00:04:09,220
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+55
+00:04:09,220 --> 00:04:10,120
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+56
+00:04:10,120 --> 00:04:10,480
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+57
+00:04:10,480 --> 00:04:11,420
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+58
+00:04:11,420 --> 00:04:11,460
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+59
+00:04:11,460 --> 00:04:16,740
+تربيه على B تربيه
+
+60
+00:04:16,770 --> 00:04:21,690
+اللي هو موازلة الـ xy plane طلع عندها ellipse و لو
+
+61
+00:04:21,690 --> 00:04:26,310
+اجينا اخدنا اي مقطع اخر برضه هتطلع كمية ثابتة هذه
+
+62
+00:04:26,310 --> 00:04:30,410
+هنجسم عليها هيطلع واحد هتطلع ellipse و اي قيمة لزد
+
+63
+00:04:30,410 --> 00:04:33,870
+موجبة هنلاقيهاش برضه بتطلع ellipseلكن لو أحد قال
+
+64
+00:04:33,870 --> 00:04:37,710
+لي طب فش رسم تحت خد Z بساوة ناقص C لو طلعت ناقص C
+
+65
+00:04:37,710 --> 00:04:40,930
+على C بطلع ناقص واحد مستحيل ال X تربيع على إيه
+
+66
+00:04:40,930 --> 00:04:43,630
+تربيع Z وY تربيع على بيت تربيع تساوية ناقص واحد
+
+67
+00:04:43,630 --> 00:04:47,610
+مستحيل تكون كمية سالبة عشان هيك فش رسم وين أسفل
+
+68
+00:04:47,610 --> 00:04:52,670
+إذا الرسم أعلى عرفنا أن المقطع هذا بربولة و المقطع
+
+69
+00:04:52,670 --> 00:04:57,310
+الثاني بربولة قلنا في ال XZ وYZ و بينما في ال XY
+
+70
+00:04:57,310 --> 00:05:00,930
+plane الموازي لل XY plane عند Z بساوة C طلع عند
+
+71
+00:05:00,930 --> 00:05:04,010
+Ellipseو أي مقطع اخر بيسموه ellipse عشان هيك بتطلع
+
+72
+00:05:04,010 --> 00:05:09,210
+عندى الشكل هذه اللى هى بنسميها elliptical اللى هى
+
+73
+00:05:09,210 --> 00:05:11,790
+الوصف elliptical لإنه مقطع واحد اسمه ellipse
+
+74
+00:05:11,790 --> 00:05:16,050
+paraboloid قلنا ال paraboloid يعني صار في عندى
+
+75
+00:05:16,050 --> 00:05:19,470
+more than one parabola اللى هى عندها نتو parabolas
+
+76
+00:05:19,470 --> 00:05:23,790
+فصار عندى اللى هو اسمه elliptical paraboloid يعني
+
+77
+00:05:23,790 --> 00:05:27,970
+في عندنا واحدة ellipse وتنتين parabola هذا اللى هو
+
+78
+00:05:27,970 --> 00:05:34,110
+الشكل المعادلةالتربيعية اللي أمامنا نيجي الأن ناخد
+
+79
+00:05:34,110 --> 00:05:38,150
+مثال آخر example 3b what is the graph of the
+
+80
+00:05:38,150 --> 00:05:43,090
+quadratic equations X تربيع على 36 زي Z تربيع على
+
+81
+00:05:43,090 --> 00:05:48,210
+25 بساوي Y على 2 لاحظوا أن هذولا التنتين المربعات
+
+82
+00:05:48,210 --> 00:05:53,550
+وهذه دون تربيع اللي هو واضح أن هذه هي نفس اللي
+
+83
+00:05:53,550 --> 00:06:00,690
+جابلها بس مبدلين ال Z بمين of Yلاحظ هنا كان الـ X
+
+84
+00:06:00,690 --> 00:06:06,030
+أسف على اللي قبلها الآن لما كانت الـ Z هي المش
+
+85
+00:06:06,030 --> 00:06:10,110
+تربيعي كانت اللي هي ال axis تبع ال elliptical
+
+86
+00:06:10,110 --> 00:06:13,750
+paraboloid هو الـ Zوالـ X و الـ Y هما اللي فيهم
+
+87
+00:06:13,750 --> 00:06:19,410
+تربيع فكان الشكل اللي أمامنا الان لو اعملنا Y مكان
+
+88
+00:06:19,410 --> 00:06:22,750
+Z زي ما في السؤال اللي بعد هذي بيصير عند X تربيع
+
+89
+00:06:22,750 --> 00:06:27,010
+على A تربيع زائد Z تربيع على B تربيع بساوي Y على C
+
+90
+00:06:27,010 --> 00:06:32,970
+بيصير الان اللي هو ال axis للشكل اللي عندى اللي هو
+
+91
+00:06:32,970 --> 00:06:37,850
+ال Y يعني بتصير هذه مجلوبة بس وجهها وين لل Y هيك و
+
+92
+00:06:37,850 --> 00:06:43,710
+بتكون نفس الرسمةلكن على الـ y-axis اللي بيكون الـ
+
+93
+00:06:43,710 --> 00:06:48,730
+axis تبع البرابولة إذا الـ x تربيعها 36, z تربيعها
+
+94
+00:06:48,730 --> 00:06:54,710
+25, y على 2 هي عبارة عن elliptical paraboloidنفس
+
+95
+00:06:54,710 --> 00:06:59,490
+الأولاني but إيش its axis اللي هو مين صار الآن ال
+
+96
+00:06:59,490 --> 00:07:04,930
+y بدل ال z is y axis instead of the z axis its
+
+97
+00:07:04,930 --> 00:07:09,410
+parabolas are in the yz plane لو الآن على السريع
+
+98
+00:07:09,410 --> 00:07:14,290
+لو حطيت هذه z ب 0 هتطلع parabola بين ال y x حط ال
+
+99
+00:07:14,290 --> 00:07:17,390
+x ب 0 parabola بين مين ال z و ال y إذا صارت ال
+
+100
+00:07:17,390 --> 00:07:21,790
+parabolas الآن ال two parabolas واحدةاللي هي في
+
+101
+00:07:21,790 --> 00:07:26,850
+الـ yz plane و التاني في ال xz plane في ال yz
+
+102
+00:07:26,850 --> 00:07:31,230
+plane بربلا و في ال xz plane بربلا بينما ال cross
+
+103
+00:07:31,230 --> 00:07:35,110
+section اللي هو ال ellipse ال y بتساوي عند y
+
+104
+00:07:35,110 --> 00:07:38,090
+بتساوي c او y بتساوي 2 في هذه الحالة لإن ال c
+
+105
+00:07:38,090 --> 00:07:41,830
+ماخدينا ايش و احنا اتنين y بتساوي اتنين بيصير عند
+
+106
+00:07:41,830 --> 00:07:46,800
+d اللي هو هذا عبارة عن ايشعن إليبس أو طبعا لو أخدت
+
+107
+00:07:46,800 --> 00:07:50,380
+أي c أكبر من 0 برضه بيطلع عبارة عن إليبس زي ما
+
+108
+00:07:50,380 --> 00:07:53,600
+قلنا إذا هذا هو الشكل الأولاني بس نايم على ال y
+
+109
+00:07:53,600 --> 00:07:57,700
+-axis بالاتجاه المدبّد للz-axis نيجي ناخد المعادلة
+
+110
+00:07:57,700 --> 00:08:02,100
+التانية X تربية نقص 4X زي 2ي تربية زي 10Y نقص 13Z
+
+111
+00:08:02,100 --> 00:08:06,270
+زي 29 بساوي 0 هذا ال مثال درين عليهعلى طول بنعمل
+
+112
+00:08:06,270 --> 00:08:09,550
+بين هدول اكمال مربع و بين هدول اكمال مربع وهذه
+
+113
+00:08:09,550 --> 00:08:13,610
+المعادلة
+
+114
+00:08:13,610 --> 00:08:18,110
+خطية لا تلزم اكمال مربع ولا غيره فبنكمل المربع و
+
+115
+00:08:18,110 --> 00:08:20,590
+بنزيد اربعة وعارفين كيف بنزيد اربعة وعملناها قبل
+
+116
+00:08:20,590 --> 00:08:24,510
+هيك وهذه بنزيد ايه اش خمسة وعشرين لانه بيصير هادم
+
+117
+00:08:24,510 --> 00:08:27,770
+مع بعض التنتيناللي هو ناقص اربعة عال اتنين بيطلع
+
+118
+00:08:27,770 --> 00:08:32,290
+ناقص اتنين بتربحها بتطلع اربعة والعشرة على اتنين
+
+119
+00:08:32,290 --> 00:08:35,270
+معامل Y بتطلع خمسة بتربحها بتطلع خمسة وعشرين بيصير
+
+120
+00:08:35,270 --> 00:08:38,690
+هذا مربع كامل وهذا مربع كامل ضفت انت تسعة وعشرين
+
+121
+00:08:38,690 --> 00:08:41,810
+لازم تشيل تسعة وعشرين هي نشلناها التسعة وعشرين
+
+122
+00:08:41,810 --> 00:08:46,930
+وطلعت اللي هي المعادلة زي اللي أمامنا أصارعت لأنه
+
+123
+00:08:46,930 --> 00:08:50,670
+وكلام معاد هذاالان هذه مربع كامل طلعت X نقص اتنين
+
+124
+00:08:50,670 --> 00:08:53,050
+V على ستة وتلاتين بعد ما جسمنا الكل على ستة
+
+125
+00:08:53,050 --> 00:08:55,970
+وتلاتين وهذه نفس الشيء جسمنا على ستة تلاتين صارت
+
+126
+00:08:55,970 --> 00:08:59,750
+بساوين هذه الان نفس الأولى بس صار لها shift على ال
+
+127
+00:08:59,750 --> 00:09:04,970
+X axis بتنين وعلى ال Y axis بنقص خمسة اذا it is an
+
+128
+00:09:04,970 --> 00:09:10,200
+elliptical paraboloid هذه مربعينعندي ال X تربيع و
+
+129
+00:09:10,200 --> 00:09:15,580
+ال Y تربيع مع shift و هنا مربع و هنا بدون مربع إذا
+
+130
+00:09:15,580 --> 00:09:19,500
+صارت عندي عبارة عن elliptical paraboloid مش مصدق
+
+131
+00:09:19,500 --> 00:09:23,340
+حط Y بسوا نقص خمسة بطلع عندي هذه parabola و هذه
+
+132
+00:09:23,340 --> 00:09:26,160
+بحط X بسوا اتنين بيصير هذه برضه parabola تانية إذا
+
+133
+00:09:26,160 --> 00:09:29,680
+two parabola زي ده parabola Yوالـ z بحطها اللي هو
+
+134
+00:09:29,680 --> 00:09:33,300
+بتساوي واحد بصير عبارة عن إيش عن ellipse يعني فعلا
+
+135
+00:09:33,300 --> 00:09:37,260
+elliptical paraboloid زي اللي قبل similar to that
+
+136
+00:09:37,260 --> 00:09:41,960
+example خمسة but it is shifted two in the positive
+
+137
+00:09:41,960 --> 00:09:47,010
+x axis وخمسة units in the negative y axisعشان هيك
+
+138
+00:09:47,010 --> 00:09:51,630
+هتلاقيها بتمر بمين الآن كانت تتمر في الأول بـ0 و0
+
+139
+00:09:51,630 --> 00:09:58,030
+و0 هالجيل صارت تتمر بـ2 و-5 و0 in a state of 00
+
+140
+00:09:58,030 --> 00:10:04,410
+and its axis لو كانت هذه Y تربيع وهذه X تربيع
+
+141
+00:10:04,410 --> 00:10:09,530
+بيصير ال axis اللي هو Z axis لكن اللي هو صار shift
+
+142
+00:10:09,530 --> 00:10:15,530
+لكل graphبمقدار 2 على ال X و نقص 5 على ال Y عشان
+
+143
+00:10:15,530 --> 00:10:21,410
+هيك اللي هو ال axis تبعها مش هيكون ال Z axis نفسه
+
+144
+00:10:21,410 --> 00:10:26,190
+لأ هيكون line parallel to the Z axis بس بمر بمين
+
+145
+00:10:26,190 --> 00:10:33,150
+بنقطة 2 و نقص 5 و أي نقطة Zالان لانه shift 2 على
+
+146
+00:10:33,150 --> 00:10:38,530
+ال X axis و 5 على ال Y axis فبصير اللي هو ال axis
+
+147
+00:10:38,530 --> 00:10:42,490
+تبعها بدل ما كان ال Z صار اللي هو ال axis الموازي
+
+148
+00:10:42,490 --> 00:10:47,630
+لل Z axis و بمره ب2 و ناقص 5 طبعا عمودي عليهم
+
+149
+00:10:47,630 --> 00:10:52,290
+موازيا لل Z axis و ال Z بتاخد أي قيمة عندي و ال A
+
+150
+00:10:52,290 --> 00:10:57,850
+طبعا 6 و ال B 6 و ال C بتساوي 1 زي ما انتوا عارفين
+
+151
+00:10:58,140 --> 00:11:03,780
+إذا هذا اللي عندي اللي هو مثال على اللي هي ال
+
+152
+00:11:03,780 --> 00:11:10,340
+elliptical paraboloid نيجي الآنللنوع الثاني او
+
+153
+00:11:10,340 --> 00:11:13,800
+الصينفي الخامس يعني احنا اخدنا الان صار عندنا ال
+
+154
+00:11:13,800 --> 00:11:18,460
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+155
+00:11:18,460 --> 00:11:19,080
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+156
+00:11:19,080 --> 00:11:19,740
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+157
+00:11:19,740 --> 00:11:20,020
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+158
+00:11:20,020 --> 00:11:21,220
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+159
+00:11:21,220 --> 00:11:22,920
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+160
+00:11:22,920 --> 00:11:24,820
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+161
+00:11:24,820 --> 00:11:25,680
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+162
+00:11:25,680 --> 00:11:25,700
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+163
+00:11:34,510 --> 00:11:39,970
+الان بدنا نحكي عن نوع خامس اللي هو قبل الأخير من
+
+164
+00:11:39,970 --> 00:11:43,750
+ال graphs اللي معناه graph x تربيع على a تربيع و y
+
+165
+00:11:43,750 --> 00:11:46,370
+تربيع على b تربيع نقص z تربيع على c تربيع ايش
+
+166
+00:11:46,370 --> 00:11:51,970
+بيساوي بيساوي واحد الان هذه بتختلف عن ال ellipsoid
+
+167
+00:11:51,970 --> 00:11:57,770
+بس بالناقص اول بقى معادلة اذا اختلفت واحدة بالناقص
+
+168
+00:11:57,770 --> 00:12:05,640
+اختلف الآن اللي هي الشكلنشوف عشان نعرف الشكل بناخد
+
+169
+00:12:05,640 --> 00:12:10,040
+المقاطع Z بصفر هيطلع X تربية على أي تربية و Y
+
+170
+00:12:10,040 --> 00:12:12,880
+تربية على بيه تربية بصفر واحد اللي هي ellipse اللي
+
+171
+00:12:12,880 --> 00:12:18,060
+هي X تربية على أي تربية و Y تربية على بيه تربية
+
+172
+00:12:18,060 --> 00:12:21,000
+بصفر واحد اللي هي ellipse وين موجودة في ال X Y
+
+173
+00:12:21,000 --> 00:12:25,910
+plane هذا لما أخدنا Z بصفرلو أخدت X بصفر هيطلع Y
+
+174
+00:12:25,910 --> 00:12:30,050
+تربيع بي تربيع نقص Z تربيع على C تربيع بسواحة طبعا
+
+175
+00:12:30,050 --> 00:12:34,610
+هذه اشمالها Hyperbola Hyperbola وين؟ في ال Y زي
+
+176
+00:12:34,610 --> 00:12:41,350
+الدبلين طيب خد الآن Y بصفر Y بصفر بتطلع X تربيع
+
+177
+00:12:41,350 --> 00:12:44,530
+على A تربيع نقص Z تربيع على C تربيع بسواحة العبارة
+
+178
+00:12:44,530 --> 00:12:49,170
+عن مين يا جماعة؟ عن Hyperbola برضه في ال X والZ ده
+
+179
+00:12:49,170 --> 00:12:53,350
+اتصار عندى Hyperbolaهذه في الـ xz plane هي
+
+180
+00:12:53,350 --> 00:12:58,270
+الهيبربول الأولى وفي ال yz plane هي الهيبربول
+
+181
+00:12:58,270 --> 00:13:04,870
+الثانية والمقطع الموازي اللي هو في ال xy plane
+
+182
+00:13:04,870 --> 00:13:09,450
+اللي هو المقطع اعتبر عندي إليبس فهذه اللي هي بتطلع
+
+183
+00:13:09,450 --> 00:13:15,390
+عندى اللي هي بنسميها hyperboloid of one sheet
+
+184
+00:13:15,720 --> 00:13:19,800
+هيبربولايد وفانشيت طبعا هو يمكن الواحد يقول طب ليش
+
+185
+00:13:19,800 --> 00:13:25,040
+احنا ماسمناهاش اللي هو elliptical hyperboloidهي في
+
+186
+00:13:25,040 --> 00:13:28,760
+الواقع elliptical hyperboloid بس of one sheet لأن
+
+187
+00:13:28,760 --> 00:13:32,160
+بعد شوية هنشوف واحدة تانية of two sheets لكن عساس
+
+188
+00:13:32,160 --> 00:13:36,180
+ان نفرجهم عن بعض كتبناهم اسمناهم hyperboloid of
+
+189
+00:13:36,180 --> 00:13:39,920
+one sheet اختصارا والتانية هنسميها hyperboloid of
+
+190
+00:13:39,920 --> 00:13:44,740
+two sheets اذا هذا الشكل اسمه hyperboloid of one
+
+191
+00:13:44,740 --> 00:13:48,680
+sheet اللي هي بالظبط زي الطابلة يا جماعة اللي هي
+
+192
+00:13:48,680 --> 00:13:51,980
+مفتوحة من الجهتين واضدها طبعا إلى مالة نهايةإذا
+
+193
+00:13:51,980 --> 00:13:57,460
+مقاطعها مقطعين hyperbola و مقطع ellipse بنسميها
+
+194
+00:13:57,460 --> 00:14:03,640
+hyperboloid of one sheet لاحظ الآن المعامل السالب
+
+195
+00:14:03,640 --> 00:14:10,280
+هو اللي عمل ليه ال axis لمين لل hyperboloid of one
+
+196
+00:14:10,280 --> 00:14:15,650
+sheetيعني الآن لما يكون معامل الـ Z أو لو صار
+
+197
+00:14:15,650 --> 00:14:20,930
+روشيفت بيكون اللي هو السالم هو بيكون الـ Z axis أو
+
+198
+00:14:20,930 --> 00:14:25,990
+ما يوازيه هو الـ Axis لو صار اللي هو هادي بدلها X
+
+199
+00:14:25,990 --> 00:14:29,570
+تربيع على A تربيع ناقص Y تربيع بيه تربيع زائد Z
+
+200
+00:14:29,570 --> 00:14:33,410
+تربيع C تربيع يعني صار ناقص للـ Y بيصير اللي هي
+
+201
+00:14:33,410 --> 00:14:38,840
+هادي نفسها بس بتنطعج وين بتنحطعلى اللي هو ال y
+
+202
+00:14:38,840 --> 00:14:42,940
+axis يعني بتصير الطابلة بس محطوطة اللي هو ال axis
+
+203
+00:14:42,940 --> 00:14:47,040
+تبعها ال y و لو كان إشارة ال x تربيحية سالبة بيصير
+
+204
+00:14:47,040 --> 00:14:51,360
+اللي هو ال axis هو من ال x axis إذا الإشارة
+
+205
+00:14:51,360 --> 00:14:56,260
+السالبة هي اللتي تحدد أنه من اللي هو ال axis ل
+
+206
+00:14:56,260 --> 00:14:59,180
+اللي هو الشكل اللي عندي بس الشكل اللي عنده هو
+
+207
+00:14:59,180 --> 00:15:04,580
+الشكل هذا ل اللي هو ال hyperboloid بنسميه of one
+
+208
+00:15:04,580 --> 00:15:11,870
+sheetوهي اللي هو هنا تفاصيل عند اللي هي الرسمة
+
+209
+00:15:11,870 --> 00:15:15,730
+اللي حكيتها فوق طيب يا جماعة صلى النبي عليه الصلاة
+
+210
+00:15:15,730 --> 00:15:20,130
+والسلام ناخد كالعادة مثال داعن للمثال الأولان اللي
+
+211
+00:15:20,130 --> 00:15:23,870
+جابله example 6b what is the graph of the
+
+212
+00:15:23,870 --> 00:15:26,090
+quadratic equations ايش ال graph of the quadratic
+
+213
+00:15:26,090 --> 00:15:30,010
+equations اللي أمامي هذه اللي سويته أنا بس حطيت
+
+214
+00:15:30,010 --> 00:15:33,980
+لمين لل y تربيع الإشارة السالبةوحطيت لهنا أرقام
+
+215
+00:15:33,980 --> 00:15:38,000
+إذا هذه عبارة عن الطابلة اللي جتوها بالظبط اللي هي
+
+216
+00:15:38,000 --> 00:15:43,220
+ال hyperboloid of one sheet بس ال axis تبع حمين ال
+
+217
+00:15:43,220 --> 00:15:47,420
+y-axis but its axis is the y-axis instead of the x
+
+218
+00:15:47,420 --> 00:15:52,200
+-axis وكمان اللي هي ال hyperbolas اللي هتكون ال
+
+219
+00:15:52,200 --> 00:15:56,440
+hyperbolas بين ال x و ال y و بين ال z و ال y بينما
+
+220
+00:15:56,440 --> 00:16:01,750
+ال ellipse هتكون بين ال x و ال zالان نجينا الجزء
+
+221
+00:16:01,750 --> 00:16:05,890
+الثاني X تربيع نقص اربع اكزاد Y تربيع اكزاد عشر Y
+
+222
+00:16:05,890 --> 00:16:11,800
+نقص اكزاد تربيع اكزاد عشرين اكزاد عشرينبنعمل اكمال
+
+223
+00:16:11,800 --> 00:16:15,560
+المربع له بيصير اكمال المربع هنا اكملنا المربع
+
+224
+00:16:15,560 --> 00:16:20,580
+ماشي الحال ضفنا اربعة وضفنا خمسة وعشرين الان بيصير
+
+225
+00:16:20,580 --> 00:16:23,420
+ضفنا تسعة وعشرين وفي عند عشرين يعني بيكون ضفنا
+
+226
+00:16:23,420 --> 00:16:26,880
+تسعة اذا بنضيف على .. شيلنا العشرين على قصدي من
+
+227
+00:16:26,880 --> 00:16:30,000
+هنا بيظل ايش تسعة لازم نضيفها على الطرف الثاني
+
+228
+00:16:30,000 --> 00:16:33,930
+بنكسب كله على تسعةوبنكمل المربع بتطلع عندي
+
+229
+00:16:33,930 --> 00:16:37,630
+المعادلة اللي أمامي هذه هذه المعادلة اللي أمامي
+
+230
+00:16:37,630 --> 00:16:42,510
+بالظبط هي تربيع وهي تربيع مجبات وهذه السالب السالب
+
+231
+00:16:42,510 --> 00:16:49,450
+هو اللي هيكون ماله اللي هو هو أو ما يوزي ال axis
+
+232
+00:16:49,450 --> 00:16:54,790
+تبع ال hyperboloid of one sheet هو إذا كان مافيش
+
+233
+00:16:54,790 --> 00:16:59,620
+أي shiftيعني فيش shift كل مرة بيكون هو لو في shift
+
+234
+00:16:59,620 --> 00:17:03,600
+زي هنا بيصير اللي هو ال axis اللي هو ال z-axis
+
+235
+00:17:03,600 --> 00:17:08,560
+انزاح واتنين باتجاه ال positive x-axis ونقص خمسة
+
+236
+00:17:08,560 --> 00:17:13,980
+باتجاه ال negative y-axis فبيصير اللي هو ال axis
+
+237
+00:17:13,980 --> 00:17:19,820
+للhyperboloid of one sheet هناهو عبارة عن المحور
+
+238
+00:17:19,820 --> 00:17:27,460
+الموازي لمحور Z والذي يمر بالنقطة 2 وناقص 5 يعني
+
+239
+00:17:27,460 --> 00:17:30,740
+بيصير عند الخط اللي بيمر 2 ناقص 5 و Z اللي هي
+
+240
+00:17:30,740 --> 00:17:34,880
+المتغيرة بيكون هذا الخط الموازي اللي هو الـ Z-axis
+
+241
+00:17:34,880 --> 00:17:40,320
+هو عبارة عن الـ axis للhyperboloid of one sheet
+
+242
+00:17:40,460 --> 00:17:43,740
+عشان هيك هذا بتمثل hyperboloid of one sheet
+
+243
+00:17:43,740 --> 00:17:47,280
+similar to that in example ستة اللي قولنا عنه قبل
+
+244
+00:17:47,280 --> 00:17:51,560
+وشوية but it is shifted two units in the positive
+
+245
+00:17:51,560 --> 00:17:56,720
+x-axis وخمسة units in the negative y-axis so its
+
+246
+00:17:56,720 --> 00:18:00,140
+axis is the line that parallel to the z-axis plane
+
+247
+00:18:00,140 --> 00:18:03,600
+and passes through اتنين ونقص خمسة او زد اي ان
+
+248
+00:18:03,600 --> 00:18:14,190
+كانت الذنبB3 وB3 وC1 نجي الأن للنوع الأخير أو
+
+249
+00:18:14,190 --> 00:18:18,010
+الصنف الأخير اللي هو من الـ quadratic surfaces
+
+250
+00:18:18,010 --> 00:18:22,850
+اللي هندرسه اللي هو اللي بنسمي hyperboloid of two
+
+251
+00:18:22,850 --> 00:18:29,780
+sheetsالان هنا بدل موحدة صار انت انتين عند فئل
+
+252
+00:18:29,780 --> 00:18:34,900
+سالب الان اللي مالوش سالب اللي موجب الغريب يعني
+
+253
+00:18:34,900 --> 00:18:38,060
+هناك كان الغريب اللي هو اللي الزد اللي له سالب
+
+254
+00:18:38,060 --> 00:18:42,860
+واحد الان اللي الغريب هنا اللي موجب الغريب هو اللي
+
+255
+00:18:42,860 --> 00:18:49,730
+بحيعمله اللي هو ال axisلمن ؟ لشكل اللي عندى تشوف
+
+256
+00:18:49,730 --> 00:18:56,090
+كيف نتبع نفس النهج و نفس الأسلوب بدنا نعرف مين يا
+
+257
+00:18:56,090 --> 00:19:02,730
+جماعة المقاطع خد X بصفر بصير عند Z تربية على C
+
+258
+00:19:02,730 --> 00:19:05,530
+تربية نقص Y تربية على B تربية عشان بساوية واحد
+
+259
+00:19:05,530 --> 00:19:10,230
+وكلكم عرف ان هذه معادلة Hyperbola في الوين؟ في ال
+
+260
+00:19:10,230 --> 00:19:17,960
+YZ دبلين Hyperbola في ال YZ دبلينهي اللي هو بين ال
+
+261
+00:19:17,960 --> 00:19:23,180
+y و ال z هي واحدة بين ال y و ال z و هي التانية بين
+
+262
+00:19:23,180 --> 00:19:26,670
+ال y و ال z على الجهة الثانيةعملت للـhyperbola
+
+263
+00:19:26,670 --> 00:19:31,870
+شوفها الـ gate الآن خد z بصفر خد z بصفر .. لأ ..
+
+264
+00:19:31,870 --> 00:19:35,670
+جبل .. y بصفر بصير الhyperbola z تربيع على c تربيع
+
+265
+00:19:35,670 --> 00:19:39,250
+نقص x تربيع على a تربيع بساوي ايه؟ شوية واحد برضه
+
+266
+00:19:39,250 --> 00:19:44,110
+هي دي ايه اشمالها؟ hyperbola بس بين ال x و ال z هي
+
+267
+00:19:44,110 --> 00:19:48,990
+ال hyperbola بين ال x هيها و ال z و هي ال
+
+268
+00:19:48,990 --> 00:19:56,190
+hyperbola عجبالها بين ال xالـ X والـ Z هيها هي هذه
+
+269
+00:19:56,190 --> 00:20:06,540
+وهي هذه والأول هذي بين ال Y والـ Zماشي طيب نحي إذا
+
+270
+00:20:06,540 --> 00:20:11,240
+صار عندي المقطعين hyperbolat بالشكل اللي أمامي طيب
+
+271
+00:20:11,240 --> 00:20:14,100
+خد z بالساوة و سفر عشان نقولنا إيه طب انت تركتلنا
+
+272
+00:20:14,100 --> 00:20:17,100
+الفرار ده ليش الفرار هذا هنشوف تعالوا نشوفوا نكد
+
+273
+00:20:17,100 --> 00:20:21,040
+الكلام اللي بحكيه اللي هنخد z بالساوة و سفر لما z
+
+274
+00:20:21,040 --> 00:20:24,760
+بالساوة و سفر بصير x تربع a تربع z y تربع b تربع
+
+275
+00:20:24,760 --> 00:20:28,880
+بساوة سالب واحدلأنه بيصير ناقص ناقص و واحد فدربت
+
+276
+00:20:28,880 --> 00:20:31,880
+وجهتين في ناقص و بطلع لوكس تربيعي تربيع تربيع بيه
+
+277
+00:20:31,880 --> 00:20:35,040
+تربيع سمه ناقص واحد ده كلام مستحيل يتحقق هدوله
+
+278
+00:20:35,040 --> 00:20:38,720
+موجبات و هده سالبة إذا فش رسم عند Z بتساوي سفر
+
+279
+00:20:38,720 --> 00:20:46,420
+رأيك أصلا خد أي قيمة Z من هذه ال C لعند ناقص C من
+
+280
+00:20:46,420 --> 00:20:50,000
+هنا لها أي Z يعني خد أي Z ال absolute value لل Z
+
+281
+00:20:50,000 --> 00:20:56,890
+أصغر من C بيصير عندك الآنلما تأخذ قيمة أصغر من الـ
+
+282
+00:20:56,890 --> 00:21:00,050
+C بيصير اللي فوق أصغر من اللي تحت بيصير هذا كله
+
+283
+00:21:00,050 --> 00:21:04,230
+أصغر من واحد لما أصغر من واحد واتجيبه هان بيصير
+
+284
+00:21:04,230 --> 00:21:07,530
+واحد ناقص أشي أصغر من واحد بيطلع كمية موجبة اللي
+
+285
+00:21:07,530 --> 00:21:10,890
+هان بيصير الأن أضرب الكل في سالب بيصير X تربيع على
+
+286
+00:21:10,890 --> 00:21:14,190
+أي تربيع زائد Y تربيع بيه تربيع بيساوي كمية سالبة
+
+287
+00:21:14,190 --> 00:21:19,550
+بعد ما ضربت بالسالب إذا مستحيل يتحققيعني أي Z في
+
+288
+00:21:19,550 --> 00:21:24,130
+المنطقة هذه بين C وبين ناقص C هتعمل لي اللي هو
+
+289
+00:21:24,130 --> 00:21:28,030
+كمية مجبرة بالساوي سالبة يعني لن تتحقق هذه
+
+290
+00:21:28,030 --> 00:21:32,550
+المعادلة مدام المعادلة هذه لن تتحقق معناته فيش رسم
+
+291
+00:21:32,550 --> 00:21:36,930
+في المنطقة هذه إذن الرسم هيكون من عند C وطالع ومن
+
+292
+00:21:36,930 --> 00:21:41,990
+عند ناقص C ونازل لأن لو حطيت الآن عند Z بتساوي C
+
+293
+00:21:41,990 --> 00:21:46,590
+بتطلع حد واحد أو ناقص Cبتنقل و هان بيصير 0 بيصير X
+
+294
+00:21:46,590 --> 00:21:49,490
+تربيع على Y تربيع زيادة Y تربيع على B تربيع بيساوي
+
+295
+00:21:49,490 --> 00:21:53,930
+0 يعني X صفر و Y صفر و Z بتساوي C يعني بتمر
+
+296
+00:21:53,930 --> 00:21:58,510
+بالنقطة هذه صفر و صفر و C و بتمر بالنقطة صفر و صفر
+
+297
+00:21:58,510 --> 00:22:03,470
+و ناقص C الآن الرسمة هتكون هنا و هنا وأكدنا عليه
+
+298
+00:22:03,470 --> 00:22:06,470
+بالhyperbola هذي و الhyperbola هذي أقولك بدي أكد
+
+299
+00:22:06,470 --> 00:22:09,930
+عليه بطريقة تانية خد Z ده إيش بتساوي 2C يعني اطلع
+
+300
+00:22:09,930 --> 00:22:14,290
+على فوقها 2C مثلا أو شالب 2C مش فارقة اللي بدك إيه
+
+301
+00:22:14,510 --> 00:22:18,810
+أو تنتين مع بعض الان بيصير عند x ترقلهده اتنين C
+
+302
+00:22:18,810 --> 00:22:22,030
+لما اربحها بيصير اربعة C تربيع اربعة C تربيع C
+
+303
+00:22:22,030 --> 00:22:24,890
+تربيع يعني اربعة اربعة جيبها على الجهة الثانية
+
+304
+00:22:24,890 --> 00:22:28,030
+بيصير ناقص اربعة واحد ناقص اربعة جديش ناقص تلاتة
+
+305
+00:22:28,030 --> 00:22:31,640
+ناقص تلاتة وهنا ناقص وهنا ناقصأضرب الكل في نقص
+
+306
+00:22:31,640 --> 00:22:34,660
+بصير X تربيع A تربيع زي Y تربيع B تربيع بسوء تلاتة
+
+307
+00:22:34,660 --> 00:22:39,140
+وهذه عبارة عن مين عن ellipse إذا المقطع هنا X عند
+
+308
+00:22:39,140 --> 00:22:44,060
+2C أو قبلها شوية و قبلها شوية و هنا و هنا و اتحد
+
+309
+00:22:44,060 --> 00:22:48,480
+كلهين بيطلع عن مين ellipses عشان هيك بنقول عنها
+
+310
+00:22:48,850 --> 00:22:53,410
+اللي هو hyperbola دي يعني في عندي مقطعين hyperbola
+
+311
+00:22:53,410 --> 00:22:57,310
+بس مش of one sheet زي اللي قبله شوية of two
+
+312
+00:22:57,310 --> 00:23:03,600
+sheets، say sheet فوق و sheet تحتطبعا هذه زي ما
+
+313
+00:23:03,600 --> 00:23:09,160
+نسميها ال vertex 00c و هذه ال vertex 00-c اللي
+
+314
+00:23:09,160 --> 00:23:14,160
+بتمر فيها ال hyperboloid
+
+315
+00:23:14,160 --> 00:23:18,240
+of one sheet of two sheets و إذا بدك تسميها
+
+316
+00:23:18,240 --> 00:23:25,520
+elliptical hyperboloid تنفع لكن اسم الشائع اللي هو
+
+317
+00:23:25,520 --> 00:23:33,250
+hyperboloidالأولى of one sheet
+
+318
+00:23:33,250 --> 00:23:36,350
+of one sheet والتانية of two sheets دي أخد المثال
+
+319
+00:23:36,350 --> 00:23:40,070
+الأخير الداعيم المثال الداعيم what is the graph of
+
+320
+00:23:40,070 --> 00:23:44,160
+the quadratic equationsهذا الآن عملت نفس اللي
+
+321
+00:23:44,160 --> 00:23:48,060
+عملته قبل نفس النهج الآن بدل ال Z حطيت ال Y يعني
+
+322
+00:23:48,060 --> 00:23:52,000
+هيصير الآن ال axis الغريب من الغريب الآن اللي هو
+
+323
+00:23:52,000 --> 00:23:55,780
+ال Y تربي ال Y الغريب ليش؟ لأن هذا شرط سلب هذا شرط
+
+324
+00:23:55,780 --> 00:23:59,740
+سلب زيه وهذا المختلف عنه إذن هو ال axis تبع ال
+
+325
+00:23:59,740 --> 00:24:03,760
+hyperbola ال hyperboloid of one of two sheets of
+
+326
+00:24:03,760 --> 00:24:07,420
+two sheets لإنه عندي two سالب's الآن في الأولاني
+
+327
+00:24:07,420 --> 00:24:11,420
+of one sheet اللي عنده فيه كان عندي one سالبالان
+
+328
+00:24:11,420 --> 00:24:13,480
+هذه هي عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+329
+00:24:13,480 --> 00:24:14,660
+عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن
+
+330
+00:24:14,660 --> 00:24:16,000
+عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+331
+00:24:16,000 --> 00:24:18,160
+عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن
+
+332
+00:24:18,160 --> 00:24:22,780
+عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+333
+00:24:22,780 --> 00:24:23,040
+عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن
+
+334
+00:24:23,040 --> 00:24:23,100
+عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+335
+00:24:23,100 --> 00:24:27,760
+عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن
+
+336
+00:24:27,760 --> 00:24:30,740
+عبارة عن
+
+337
+00:24:40,280 --> 00:24:44,200
+المعادلة الأخيرة هذه نفس الشيء سنعملها إكمال مربع
+
+338
+00:24:44,200 --> 00:24:48,480
+عملنا لها إكمال مربع صارت تزعزت تاربيع أخدت السالب
+
+339
+00:24:48,480 --> 00:24:52,660
+برا بصيرت تاربيع ناقص أربع ناقص ناقص Y تاربيع ناقص
+
+340
+00:24:52,660 --> 00:24:52,800
+ناقص و Y تاربيع ناقص ناقص و Y تاربيع ناقص ناقص و Y
+
+341
+00:24:52,800 --> 00:24:53,060
+تاربيع ناقص و Y تاربيع ناقص و Y تاربيع ناقص و Y
+
+342
+00:24:53,060 --> 00:24:56,160
+تاربيع ناقص و Y تاربيع ناقص و Y تاربيع ناقص و Y
+
+343
+00:24:56,160 --> 00:25:02,000
+تاربيع ناقص و Y تاربيع ناقص و Y تاربيع ناقص و Y
+
+344
+00:25:02,000 --> 00:25:06,910
+تاربيع ناقص و Y تاراللي هو سالب أربعة وضفت سالب
+
+345
+00:25:06,910 --> 00:25:10,190
+خمسة وعشرين يعني ضفت سالب تسعة وعشرين بضيف على
+
+346
+00:25:10,190 --> 00:25:13,230
+الطرف التالت سالب التالت سالب تسعة وعشرين وكان
+
+347
+00:25:13,230 --> 00:25:18,870
+تلاتين بيصير إيش واحد لأن هذه جاهزة بجسمها اللي هو
+
+348
+00:25:20,670 --> 00:25:27,150
+لأ اسمحولي بس أغير هنا خلّي هذه الآن بتضيف كمان
+
+349
+00:25:27,150 --> 00:25:30,590
+تمانية يعني خلّيها تمانية و تلاتين هذه عساس أنها
+
+350
+00:25:30,590 --> 00:25:34,530
+تصير هذه تسعة و لما أجسم على تسعة بطلع إيش واحد
+
+351
+00:25:34,530 --> 00:25:37,530
+فاهمين عليه يعني هذا في السؤال هذه مش تلاتين هذه
+
+352
+00:25:37,530 --> 00:25:41,070
+بيساوي تمانية و تلاتينثمانية و تلاتين و تسعة و
+
+353
+00:25:41,070 --> 00:25:49,810
+عشرين و تسعة و تسعة و تسعة و تسعة
+
+354
+00:25:49,810 --> 00:25:50,370
+و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+355
+00:25:50,370 --> 00:25:50,490
+تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+356
+00:25:50,490 --> 00:25:50,770
+تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+357
+00:25:50,770 --> 00:25:51,450
+تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+358
+00:25:51,450 --> 00:25:52,030
+تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+359
+00:25:52,030 --> 00:26:00,110
+تسعة و تسعة و تسعة و تسعة
+
+360
+00:26:00,110 --> 00:26:08,510
+و تسعة و تسلأن لو لم يكن هناك shift في المرة
+
+361
+00:26:08,510 --> 00:26:12,410
+فسيكون الـ Z-axis لكن مادام صار فيه shift صار الـ
+
+362
+00:26:12,410 --> 00:26:19,370
+Z-axis اللي هو المحور هيه بس إذا سحبنا اللي هو 2
+
+363
+00:26:19,370 --> 00:26:24,210
+على ال X و سالب 5 على ال Y فبصير عبارة عن ال axis
+
+364
+00:26:24,210 --> 00:26:30,250
+لهذه ال hyperbola of two sheets عبارة عن الخط اللي
+
+365
+00:26:30,250 --> 00:26:35,690
+بوازي ال Z-axis و يمر في النقطة 2 و ناقص 5و أي
+
+366
+00:26:35,690 --> 00:26:39,970
+نقطة زد هذا اللي هو محور الـ hyperboloid of two
+
+367
+00:26:39,970 --> 00:26:45,030
+sheets الان هذه اللي هي نفس الأولى اللي حكينا عنها
+
+368
+00:26:45,030 --> 00:26:51,450
+قبل هيكاللي هي هذه بس اشترلها عبارة عن shift ع ال
+
+369
+00:26:51,450 --> 00:26:56,230
+X-axis بمقدار 2 وهذه ستتكرر وع ال Y-axis بمقدار 5
+
+370
+00:26:56,230 --> 00:27:01,150
+بس بالاتجاه السالب حالة هي كبسيرة تمر اللي هي في 2
+
+371
+00:27:01,150 --> 00:27:06,890
+و 5 و Z بتساوي 1 زائد أو ماقص 1 ليه قيمة C هذه
+
+372
+00:27:06,890 --> 00:27:12,250
+instead of 00 وزائد C and X-axis زي ما قولنا is
+
+373
+00:27:12,250 --> 00:27:15,930
+the line that parallel to the Z-axisand بص صفره
+
+374
+00:27:15,930 --> 00:27:20,230
+اتنين ونقص خمسة وزد ال .. و بعدين ال a تبعتها
+
+375
+00:27:20,230 --> 00:27:24,710
+بتساوي .. اللي هي تلاتة و ال b تلاتة و ال c بتساوي
+
+376
+00:27:24,710 --> 00:27:29,570
+واحد و هيك بكون احنا خلصنا اللي هو كل اتناش ستة
+
+377
+00:27:29,570 --> 00:27:34,490
+وهي ال exercises اللي داعمة و بارك الله فيكم و إلى
+
+378
+00:27:34,490 --> 00:27:35,670
+لقاء اخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/EFnvj3yPdag_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/EFnvj3yPdag_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..94e0c5236b7e459e98b0bac06ddd394e81689946
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/EFnvj3yPdag_raw.srt
@@ -0,0 +1,1532 @@
+1
+00:00:00,490 --> 00:00:04,210
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله هذه
+
+2
+00:00:04,210 --> 00:00:09,590
+هي المحاضرة رقم 7 لمادة أو مساق تفاضل وتكامل
+
+3
+00:00:09,590 --> 00:00:14,150
+الـGene وهي الجزء الثاني من ال section الأخير في
+
+4
+00:00:14,150 --> 00:00:21,090
+شبطر 12 الـ12.6كننا حكينا عن اللي هو ال cylinders
+
+5
+00:00:21,090 --> 00:00:25,210
+and quadratic surfaces و حكينا عن ثلاثة من ال
+
+6
+00:00:25,210 --> 00:00:28,230
+quadratic surfaces ده بتتذكروا حكينا عن ال
+
+7
+00:00:28,230 --> 00:00:34,030
+ellipsoid و ال hyperbolic paraboloid و عن اللي هو
+
+8
+00:00:34,030 --> 00:00:38,620
+elliptical coneضال عندنا تلت اللي هو أنواع جديدة
+
+9
+00:00:38,620 --> 00:00:43,200
+هنحكي عنها اللي هي من ال quadratic surfaces اللي
+
+10
+00:00:43,200 --> 00:00:49,940
+ال surfaces أو اللي هي السطوح أو المجسمات اللي هي
+
+11
+00:00:49,940 --> 00:00:54,890
+رسم لمعادلات تربيعيةالان نجي ال example خمسة
+
+12
+00:00:54,890 --> 00:00:57,830
+example خمسة اللي هو هيكون عندنا حاجة اسمها
+
+13
+00:00:57,830 --> 00:01:01,170
+elliptical paraboloid لتنشوف ايش elliptical
+
+14
+00:01:01,170 --> 00:01:04,690
+paraboloid زي ما قلنا قبل هيك في المحاضرة السابقة
+
+15
+00:01:04,690 --> 00:01:11,750
+انه هيكون اللي هو اكيد الاسم له علاقة بالجسم او
+
+16
+00:01:11,750 --> 00:01:17,420
+الشكل اللي هيطلعالجراف ده اللي هو X تربيع زائد Y
+
+17
+00:01:17,420 --> 00:01:22,780
+تربيع على B تربيع بساوي Z عالميا على C بدنا الآن
+
+18
+00:01:22,780 --> 00:01:27,500
+اللي هو نشوف هذا اللي هو الجراف اي هذا المعادلة
+
+19
+00:01:27,500 --> 00:01:32,980
+بتمثل جراف ماذا قولنا احنا في الأسئلة من هالنوع
+
+20
+00:01:32,980 --> 00:01:39,320
+اول ما بنحكي والشغلة الأساسية في تحديد الشكل هي
+
+21
+00:01:39,320 --> 00:01:44,820
+المقاطعة اللي هي الشكلفالان اول حاجة بناخد X بساوة
+
+22
+00:01:44,820 --> 00:01:49,600
+0 لما X بساوة 0 بيصير عندى Y تربيع على B تربيع
+
+23
+00:01:49,600 --> 00:01:54,100
+بساوة Z على C او بمعنى اخر تطلع عندى Z بساوة C على
+
+24
+00:01:54,100 --> 00:02:00,100
+B تربيع في Y تربيع وهذا واضح ان هي عبارة عن معادلة
+
+25
+00:02:00,100 --> 00:02:06,400
+برابولة اذا اللى هو في ال ZY plane هيكون في عندنا
+
+26
+00:02:06,400 --> 00:02:11,910
+برابولة هي اللى هي البرابولة اللى عندنان ي ز بلين
+
+27
+00:02:11,910 --> 00:02:16,030
+نما ناخدنا X بساوة 0 أو أخدنا المقطع اللي هو اللي
+
+28
+00:02:16,030 --> 00:02:20,790
+هي ال plane قطع هذا اللي هي الشكل فيه اللي هو
+
+29
+00:02:20,790 --> 00:02:25,470
+البرابولة الأمامنا إذا صارت عندي Z بساوة C على B
+
+30
+00:02:25,470 --> 00:02:30,740
+تربيع Y هي عبارة عن برابولة بين Zو بين مين؟ الـ Y
+
+31
+00:02:30,740 --> 00:02:34,880
+هي بين الـ Z و بين الـ Y هي البرابولة اللى أمامنا
+
+32
+00:02:34,880 --> 00:02:38,960
+اللى هي Z بساوة C على A تربيع في X تربيع يعني ثابت
+
+33
+00:02:38,960 --> 00:02:42,140
+في X تربيع الأعلى لإنه هذا موجب و هذا موجب زي ما
+
+34
+00:02:42,140 --> 00:02:45,990
+احنا عارفين C أكبر من 0 ماخدينهااللي هناخده المقطع
+
+35
+00:02:45,990 --> 00:02:49,530
+الاخر اللي هو Y بيساوي 0 لما ناخد Y بيساوي 0 برضه
+
+36
+00:02:49,530 --> 00:02:52,850
+هيطلع عند X تربي على A تربيع بيساوي Z على C أو
+
+37
+00:02:52,850 --> 00:02:57,370
+بمعنى أخر Z بتساوي C على A تربيع في X تربيع برضه
+
+38
+00:02:57,370 --> 00:03:00,530
+هذه معادلة برابولة زي ما انتوا عارفين بس بين مين
+
+39
+00:03:00,530 --> 00:03:04,950
+بين ال Z وال X وهي البرابولة بين ال Zو بين الـ X
+
+40
+00:03:04,950 --> 00:03:09,270
+الآن بنجمع هذه البرابولة الأولى وهي البرابولة
+
+41
+00:03:09,270 --> 00:03:13,050
+الثانية اذا صار مقطعين كل واحدة منهم برابولة نيجي
+
+42
+00:03:13,050 --> 00:03:16,610
+للمقطع الثالث المقطع الثالث لو جينا أخدنا اللي هو
+
+43
+00:03:16,610 --> 00:03:20,870
+Z بصفر هتطلع اللي هي X وY بصفر لأنه بيصير X تربيع
+
+44
+00:03:20,870 --> 00:03:25,250
+على A تربيع Z وY تربيع بتربيع بصفر يعني كلمتين
+
+45
+00:03:25,250 --> 00:03:28,030
+مربعتين مجموحين صفر يعني كل واحدة فين صفر يعني X
+
+46
+00:03:28,030 --> 00:03:32,430
+بصفر وY بصفر وZ بصفر يعني بمعنى أخر بتمر بنقطة من
+
+47
+00:03:32,430 --> 00:03:38,340
+الأصلاللي هي ال graph اللي عندي الان عشان هيك ان
+
+48
+00:03:38,340 --> 00:03:41,740
+عندي ال C أكبر من 0 مزام ال C أكبر من 0 خد الآن
+
+49
+00:03:41,740 --> 00:03:46,700
+قيمة Z مثلا خدها بساوي C يعني بدي مقطع Z بساوي C
+
+50
+00:03:46,700 --> 00:03:53,480
+اللي ايش بعمل اللي يعني بوازي ال X Y plane اللي هو
+
+51
+00:03:53,480 --> 00:04:01,600
+اشوف ايه لما اقطعاللي هو الشكل بمستوى يوازي الـ X
+
+52
+00:04:01,600 --> 00:04:04,700
+Y plane عند المستوى Z بساوة 0 اشوف ايش اللي بتطلع
+
+53
+00:04:04,700 --> 00:04:08,140
+عندى Z بتساوى C بيصير X تربيه على A تربيه على Z Y
+
+54
+00:04:08,140 --> 00:04:09,220
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+55
+00:04:09,220 --> 00:04:10,120
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+56
+00:04:10,120 --> 00:04:10,480
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+57
+00:04:10,480 --> 00:04:11,420
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+58
+00:04:11,420 --> 00:04:11,460
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+59
+00:04:11,460 --> 00:04:11,460
+تربيه على B تربيه على B تربيه على B تربيه على B
+
+60
+00:04:11,460 --> 00:04:16,740
+تربيه على B تربيه
+
+61
+00:04:16,770 --> 00:04:21,690
+اللي هو موازلة الـ xy plane طلع عندها ellipse و لو
+
+62
+00:04:21,690 --> 00:04:26,310
+اجينا اخدنا اي مقطع اخر برضه هتطلع كمية ثابتة هذه
+
+63
+00:04:26,310 --> 00:04:30,410
+هنجسم عليها هيطلع واحد هتطلع ellipse و اي قيمة لزد
+
+64
+00:04:30,410 --> 00:04:33,870
+موجبة هنلاقيهاش برضه بتطلع ellipseلكن لو أحد قال
+
+65
+00:04:33,870 --> 00:04:37,710
+لي طب فش رسم تحت خد Z بساوة ناقص C لو طلعت ناقص C
+
+66
+00:04:37,710 --> 00:04:40,930
+على C بطلع ناقص واحد مستحيل ال X تربيع على إيه
+
+67
+00:04:40,930 --> 00:04:43,630
+تربيع Z وY تربيع على بيت تربيع تساوية ناقص واحد
+
+68
+00:04:43,630 --> 00:04:47,610
+مستحيل تكون كمية سالبة عشان هيك فش رسم وين أسفل
+
+69
+00:04:47,610 --> 00:04:52,670
+إذا الرسم أعلى عرفنا أن المقطع هذا بربولة و المقطع
+
+70
+00:04:52,670 --> 00:04:57,310
+الثاني بربولة قلنا في ال XZ وYZ و بينما في ال XY
+
+71
+00:04:57,310 --> 00:05:00,930
+plane الموازي لل XY plane عند Z بساوة C طلع عند
+
+72
+00:05:00,930 --> 00:05:04,010
+Ellipseو أي مقطع اخر بيسموه ellipse عشان هيك بتطلع
+
+73
+00:05:04,010 --> 00:05:09,210
+عندى الشكل هذه اللى هى بنسميها elliptical اللى هى
+
+74
+00:05:09,210 --> 00:05:11,790
+الوصف elliptical لإنه مقطع واحد اسمه ellipse
+
+75
+00:05:11,790 --> 00:05:16,050
+paraboloid قلنا ال paraboloid يعني صار في عندى
+
+76
+00:05:16,050 --> 00:05:19,470
+more than one parabola اللى هى عندها نتو parabolas
+
+77
+00:05:19,470 --> 00:05:23,790
+فصار عندى اللى هو اسمه elliptical paraboloid يعني
+
+78
+00:05:23,790 --> 00:05:27,970
+في عندنا واحدة ellipse وتنتين parabola هذا اللى هو
+
+79
+00:05:27,970 --> 00:05:34,110
+الشكل المعادلةالتربيعية اللي أمامنا نيجي الأن ناخد
+
+80
+00:05:34,110 --> 00:05:38,150
+مثال آخر example 3b what is the graph of the
+
+81
+00:05:38,150 --> 00:05:43,090
+quadratic equations X تربيع على 36 زي Z تربيع على
+
+82
+00:05:43,090 --> 00:05:48,210
+25 بساوي Y على 2 لاحظوا أن هذولا التنتين المربعات
+
+83
+00:05:48,210 --> 00:05:53,550
+وهذه دون تربيع اللي هو واضح أن هذه هي نفس اللي
+
+84
+00:05:53,550 --> 00:06:00,690
+جابلها بس مبدلين ال Z بمين of Yلاحظ هنا كان الـ X
+
+85
+00:06:00,690 --> 00:06:06,030
+أسف على اللي قبلها الآن لما كانت الـ Z هي المش
+
+86
+00:06:06,030 --> 00:06:10,110
+تربيعي كانت اللي هي ال axis تبع ال elliptical
+
+87
+00:06:10,110 --> 00:06:13,750
+paraboloid هو الـ Zوالـ X و الـ Y هما اللي فيهم
+
+88
+00:06:13,750 --> 00:06:19,410
+تربيع فكان الشكل اللي أمامنا الان لو اعملنا Y مكان
+
+89
+00:06:19,410 --> 00:06:22,750
+Z زي ما في السؤال اللي بعد هذي بيصير عند X تربيع
+
+90
+00:06:22,750 --> 00:06:27,010
+على A تربيع زائد Z تربيع على B تربيع بساوي Y على C
+
+91
+00:06:27,010 --> 00:06:32,970
+بيصير الان اللي هو ال axis للشكل اللي عندى اللي هو
+
+92
+00:06:32,970 --> 00:06:37,850
+ال Y يعني بتصير هذه مجلوبة بس وجهها وين لل Y هيك و
+
+93
+00:06:37,850 --> 00:06:43,710
+بتكون نفس الرسمةلكن على الـ y-axis اللي بيكون الـ
+
+94
+00:06:43,710 --> 00:06:48,730
+axis تبع البرابولة إذا الـ x تربيعها 36, z تربيعها
+
+95
+00:06:48,730 --> 00:06:54,710
+25, y على 2 هي عبارة عن elliptical paraboloidنفس
+
+96
+00:06:54,710 --> 00:06:59,490
+الأولاني but إيش its axis اللي هو مين صار الآن ال
+
+97
+00:06:59,490 --> 00:07:04,930
+y بدل ال z is y axis instead of the z axis its
+
+98
+00:07:04,930 --> 00:07:09,410
+parabolas are in the yz plane لو الآن على السريع
+
+99
+00:07:09,410 --> 00:07:14,290
+لو حطيت هذه z ب 0 هتطلع parabola بين ال y x حط ال
+
+100
+00:07:14,290 --> 00:07:17,390
+x ب 0 parabola بين مين ال z و ال y إذا صارت ال
+
+101
+00:07:17,390 --> 00:07:21,790
+parabolas الآن ال two parabolas واحدةاللي هي في
+
+102
+00:07:21,790 --> 00:07:26,850
+الـ yz plane و التاني في ال xz plane في ال yz
+
+103
+00:07:26,850 --> 00:07:31,230
+plane بربلا و في ال xz plane بربلا بينما ال cross
+
+104
+00:07:31,230 --> 00:07:35,110
+section اللي هو ال ellipse ال y بتساوي عند y
+
+105
+00:07:35,110 --> 00:07:38,090
+بتساوي c او y بتساوي 2 في هذه الحالة لإن ال c
+
+106
+00:07:38,090 --> 00:07:41,830
+ماخدينا ايش و احنا اتنين y بتساوي اتنين بيصير عند
+
+107
+00:07:41,830 --> 00:07:46,800
+d اللي هو هذا عبارة عن ايشعن إليبس أو طبعا لو أخدت
+
+108
+00:07:46,800 --> 00:07:50,380
+أي c أكبر من 0 برضه بيطلع عبارة عن إليبس زي ما
+
+109
+00:07:50,380 --> 00:07:53,600
+قلنا إذا هذا هو الشكل الأولاني بس نايم على ال y
+
+110
+00:07:53,600 --> 00:07:57,700
+-axis بالاتجاه المدبّد للz-axis نيجي ناخد المعادلة
+
+111
+00:07:57,700 --> 00:08:02,100
+التانية X تربية نقص 4X زي 2ي تربية زي 10Y نقص 13Z
+
+112
+00:08:02,100 --> 00:08:06,270
+زي 29 بساوي 0 هذا ال مثال درين عليهعلى طول بنعمل
+
+113
+00:08:06,270 --> 00:08:09,550
+بين هدول اكمال مربع و بين هدول اكمال مربع وهذه
+
+114
+00:08:09,550 --> 00:08:13,610
+المعادلة
+
+115
+00:08:13,610 --> 00:08:18,110
+خطية لا تلزم اكمال مربع ولا غيره فبنكمل المربع و
+
+116
+00:08:18,110 --> 00:08:20,590
+بنزيد اربعة وعارفين كيف بنزيد اربعة وعملناها قبل
+
+117
+00:08:20,590 --> 00:08:24,510
+هيك وهذه بنزيد ايه اش خمسة وعشرين لانه بيصير هادم
+
+118
+00:08:24,510 --> 00:08:27,770
+مع بعض التنتيناللي هو ناقص اربعة عال اتنين بيطلع
+
+119
+00:08:27,770 --> 00:08:32,290
+ناقص اتنين بتربحها بتطلع اربعة والعشرة على اتنين
+
+120
+00:08:32,290 --> 00:08:35,270
+معامل Y بتطلع خمسة بتربحها بتطلع خمسة وعشرين بيصير
+
+121
+00:08:35,270 --> 00:08:38,690
+هذا مربع كامل وهذا مربع كامل ضفت انت تسعة وعشرين
+
+122
+00:08:38,690 --> 00:08:41,810
+لازم تشيل تسعة وعشرين هي نشلناها التسعة وعشرين
+
+123
+00:08:41,810 --> 00:08:46,930
+وطلعت اللي هي المعادلة زي اللي أمامنا أصارعت لأنه
+
+124
+00:08:46,930 --> 00:08:50,670
+وكلام معاد هذاالان هذه مربع كامل طلعت X نقص اتنين
+
+125
+00:08:50,670 --> 00:08:53,050
+V على ستة وتلاتين بعد ما جسمنا الكل على ستة
+
+126
+00:08:53,050 --> 00:08:55,970
+وتلاتين وهذه نفس الشيء جسمنا على ستة تلاتين صارت
+
+127
+00:08:55,970 --> 00:08:59,750
+بساوين هذه الان نفس الأولى بس صار لها shift على ال
+
+128
+00:08:59,750 --> 00:09:04,970
+X axis بتنين وعلى ال Y axis بنقص خمسة اذا it is an
+
+129
+00:09:04,970 --> 00:09:10,200
+elliptical paraboloid هذه مربعينعندي ال X تربيع و
+
+130
+00:09:10,200 --> 00:09:15,580
+ال Y تربيع مع shift و هنا مربع و هنا بدون مربع إذا
+
+131
+00:09:15,580 --> 00:09:19,500
+صارت عندي عبارة عن elliptical paraboloid مش مصدق
+
+132
+00:09:19,500 --> 00:09:23,340
+حط Y بسوا نقص خمسة بطلع عندي هذه parabola و هذه
+
+133
+00:09:23,340 --> 00:09:26,160
+بحط X بسوا اتنين بيصير هذه برضه parabola تانية إذا
+
+134
+00:09:26,160 --> 00:09:29,680
+two parabola زي ده parabola Yوالـ z بحطها اللي هو
+
+135
+00:09:29,680 --> 00:09:33,300
+بتساوي واحد بصير عبارة عن إيش عن ellipse يعني فعلا
+
+136
+00:09:33,300 --> 00:09:37,260
+elliptical paraboloid زي اللي قبل similar to that
+
+137
+00:09:37,260 --> 00:09:41,960
+example خمسة but it is shifted two in the positive
+
+138
+00:09:41,960 --> 00:09:47,010
+x axis وخمسة units in the negative y axisعشان هيك
+
+139
+00:09:47,010 --> 00:09:51,630
+هتلاقيها بتمر بمين الآن كانت تتمر في الأول بـ0 و0
+
+140
+00:09:51,630 --> 00:09:58,030
+و0 هالجيل صارت تتمر بـ2 و-5 و0 in a state of 00
+
+141
+00:09:58,030 --> 00:10:04,410
+and its axis لو كانت هذه Y تربيع وهذه X تربيع
+
+142
+00:10:04,410 --> 00:10:09,530
+بيصير ال axis اللي هو Z axis لكن اللي هو صار shift
+
+143
+00:10:09,530 --> 00:10:15,530
+لكل graphبمقدار 2 على ال X و نقص 5 على ال Y عشان
+
+144
+00:10:15,530 --> 00:10:21,410
+هيك اللي هو ال axis تبعها مش هيكون ال Z axis نفسه
+
+145
+00:10:21,410 --> 00:10:26,190
+لأ هيكون line parallel to the Z axis بس بمر بمين
+
+146
+00:10:26,190 --> 00:10:33,150
+بنقطة 2 و نقص 5 و أي نقطة Zالان لانه shift 2 على
+
+147
+00:10:33,150 --> 00:10:38,530
+ال X axis و 5 على ال Y axis فبصير اللي هو ال axis
+
+148
+00:10:38,530 --> 00:10:42,490
+تبعها بدل ما كان ال Z صار اللي هو ال axis الموازي
+
+149
+00:10:42,490 --> 00:10:47,630
+لل Z axis و بمره ب2 و ناقص 5 طبعا عمودي عليهم
+
+150
+00:10:47,630 --> 00:10:52,290
+موازيا لل Z axis و ال Z بتاخد أي قيمة عندي و ال A
+
+151
+00:10:52,290 --> 00:10:57,850
+طبعا 6 و ال B 6 و ال C بتساوي 1 زي ما انتوا عارفين
+
+152
+00:10:58,140 --> 00:11:03,780
+إذا هذا اللي عندي اللي هو مثال على اللي هي ال
+
+153
+00:11:03,780 --> 00:11:10,340
+elliptical paraboloid نيجي الآنللنوع الثاني او
+
+154
+00:11:10,340 --> 00:11:13,800
+الصينفي الخامس يعني احنا اخدنا الان صار عندنا ال
+
+155
+00:11:13,800 --> 00:11:18,460
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+156
+00:11:18,460 --> 00:11:19,080
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+157
+00:11:19,080 --> 00:11:19,740
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+158
+00:11:19,740 --> 00:11:20,020
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+159
+00:11:20,020 --> 00:11:21,220
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+160
+00:11:21,220 --> 00:11:22,920
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+161
+00:11:22,920 --> 00:11:24,820
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+162
+00:11:24,820 --> 00:11:25,680
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+163
+00:11:25,680 --> 00:11:25,680
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+164
+00:11:25,680 --> 00:11:25,680
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+165
+00:11:25,680 --> 00:11:25,700
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+166
+00:11:25,700 --> 00:11:25,700
+ال ال ال ال ال
+
+167
+00:11:34,510 --> 00:11:39,970
+الان بدنا نحكي عن نوع خامس اللي هو قبل الأخير من
+
+168
+00:11:39,970 --> 00:11:43,750
+ال graphs اللي معناه graph x تربيع على a تربيع و y
+
+169
+00:11:43,750 --> 00:11:46,370
+تربيع على b تربيع نقص z تربيع على c تربيع ايش
+
+170
+00:11:46,370 --> 00:11:51,970
+بيساوي بيساوي واحد الان هذه بتختلف عن ال ellipsoid
+
+171
+00:11:51,970 --> 00:11:57,770
+بس بالناقص اول بقى معادلة اذا اختلفت واحدة بالناقص
+
+172
+00:11:57,770 --> 00:12:05,640
+اختلف الآن اللي هي الشكلنشوف عشان نعرف الشكل بناخد
+
+173
+00:12:05,640 --> 00:12:10,040
+المقاطع Z بصفر هيطلع X تربية على أي تربية و Y
+
+174
+00:12:10,040 --> 00:12:12,880
+تربية على بيه تربية بصفر واحد اللي هي ellipse اللي
+
+175
+00:12:12,880 --> 00:12:18,060
+هي X تربية على أي تربية و Y تربية على بيه تربية
+
+176
+00:12:18,060 --> 00:12:21,000
+بصفر واحد اللي هي ellipse وين موجودة في ال X Y
+
+177
+00:12:21,000 --> 00:12:25,910
+plane هذا لما أخدنا Z بصفرلو أخدت X بصفر هيطلع Y
+
+178
+00:12:25,910 --> 00:12:30,050
+تربيع بي تربيع نقص Z تربيع على C تربيع بسواحة طبعا
+
+179
+00:12:30,050 --> 00:12:34,610
+هذه اشمالها Hyperbola Hyperbola وين؟ في ال Y زي
+
+180
+00:12:34,610 --> 00:12:41,350
+الدبلين طيب خد الآن Y بصفر Y بصفر بتطلع X تربيع
+
+181
+00:12:41,350 --> 00:12:44,530
+على A تربيع نقص Z تربيع على C تربيع بسواحة العبارة
+
+182
+00:12:44,530 --> 00:12:49,170
+عن مين يا جماعة؟ عن Hyperbola برضه في ال X والZ ده
+
+183
+00:12:49,170 --> 00:12:53,350
+اتصار عندى Hyperbolaهذه في الـ xz plane هي
+
+184
+00:12:53,350 --> 00:12:58,270
+الهيبربول الأولى وفي ال yz plane هي الهيبربول
+
+185
+00:12:58,270 --> 00:13:04,870
+الثانية والمقطع الموازي اللي هو في ال xy plane
+
+186
+00:13:04,870 --> 00:13:09,450
+اللي هو المقطع اعتبر عندي إليبس فهذه اللي هي بتطلع
+
+187
+00:13:09,450 --> 00:13:15,390
+عندى اللي هي بنسميها hyperboloid of one sheet
+
+188
+00:13:15,720 --> 00:13:19,800
+هيبربولايد وفانشيت طبعا هو يمكن الواحد يقول طب ليش
+
+189
+00:13:19,800 --> 00:13:25,040
+احنا ماسمناهاش اللي هو elliptical hyperboloidهي في
+
+190
+00:13:25,040 --> 00:13:28,760
+الواقع elliptical hyperboloid بس of one sheet لأن
+
+191
+00:13:28,760 --> 00:13:32,160
+بعد شوية هنشوف واحدة تانية of two sheets لكن عساس
+
+192
+00:13:32,160 --> 00:13:36,180
+ان نفرجهم عن بعض كتبناهم اسمناهم hyperboloid of
+
+193
+00:13:36,180 --> 00:13:39,920
+one sheet اختصارا والتانية هنسميها hyperboloid of
+
+194
+00:13:39,920 --> 00:13:44,740
+two sheets اذا هذا الشكل اسمه hyperboloid of one
+
+195
+00:13:44,740 --> 00:13:48,680
+sheet اللي هي بالظبط زي الطابلة يا جماعة اللي هي
+
+196
+00:13:48,680 --> 00:13:51,980
+مفتوحة من الجهتين واضدها طبعا إلى مالة نهايةإذا
+
+197
+00:13:51,980 --> 00:13:57,460
+مقاطعها مقطعين hyperbola و مقطع ellipse بنسميها
+
+198
+00:13:57,460 --> 00:14:03,640
+hyperboloid of one sheet لاحظ الآن المعامل السالب
+
+199
+00:14:03,640 --> 00:14:10,280
+هو اللي عمل ليه ال axis لمين لل hyperboloid of one
+
+200
+00:14:10,280 --> 00:14:15,650
+sheetيعني الآن لما يكون معامل الـ Z أو لو صار
+
+201
+00:14:15,650 --> 00:14:20,930
+روشيفت بيكون اللي هو السالم هو بيكون الـ Z axis أو
+
+202
+00:14:20,930 --> 00:14:25,990
+ما يوازيه هو الـ Axis لو صار اللي هو هادي بدلها X
+
+203
+00:14:25,990 --> 00:14:29,570
+تربيع على A تربيع ناقص Y تربيع بيه تربيع زائد Z
+
+204
+00:14:29,570 --> 00:14:33,410
+تربيع C تربيع يعني صار ناقص للـ Y بيصير اللي هي
+
+205
+00:14:33,410 --> 00:14:38,840
+هادي نفسها بس بتنطعج وين بتنحطعلى اللي هو ال y
+
+206
+00:14:38,840 --> 00:14:42,940
+axis يعني بتصير الطابلة بس محطوطة اللي هو ال axis
+
+207
+00:14:42,940 --> 00:14:47,040
+تبعها ال y و لو كان إشارة ال x تربيحية سالبة بيصير
+
+208
+00:14:47,040 --> 00:14:51,360
+اللي هو ال axis هو من ال x axis إذا الإشارة
+
+209
+00:14:51,360 --> 00:14:56,260
+السالبة هي اللتي تحدد أنه من اللي هو ال axis ل
+
+210
+00:14:56,260 --> 00:14:59,180
+اللي هو الشكل اللي عندي بس الشكل اللي عنده هو
+
+211
+00:14:59,180 --> 00:15:04,580
+الشكل هذا ل اللي هو ال hyperboloid بنسميه of one
+
+212
+00:15:04,580 --> 00:15:11,870
+sheetوهي اللي هو هنا تفاصيل عند اللي هي الرسمة
+
+213
+00:15:11,870 --> 00:15:15,730
+اللي حكيتها فوق طيب يا جماعة صلى النبي عليه الصلاة
+
+214
+00:15:15,730 --> 00:15:20,130
+والسلام ناخد كالعادة مثال داعن للمثال الأولان اللي
+
+215
+00:15:20,130 --> 00:15:23,870
+جابله example 6b what is the graph of the
+
+216
+00:15:23,870 --> 00:15:26,090
+quadratic equations ايش ال graph of the quadratic
+
+217
+00:15:26,090 --> 00:15:30,010
+equations اللي أمامي هذه اللي سويته أنا بس حطيت
+
+218
+00:15:30,010 --> 00:15:33,980
+لمين لل y تربيع الإشارة السالبةوحطيت لهنا أرقام
+
+219
+00:15:33,980 --> 00:15:38,000
+إذا هذه عبارة عن الطابلة اللي جتوها بالظبط اللي هي
+
+220
+00:15:38,000 --> 00:15:43,220
+ال hyperboloid of one sheet بس ال axis تبع حمين ال
+
+221
+00:15:43,220 --> 00:15:47,420
+y-axis but its axis is the y-axis instead of the x
+
+222
+00:15:47,420 --> 00:15:52,200
+-axis وكمان اللي هي ال hyperbolas اللي هتكون ال
+
+223
+00:15:52,200 --> 00:15:56,440
+hyperbolas بين ال x و ال y و بين ال z و ال y بينما
+
+224
+00:15:56,440 --> 00:16:01,750
+ال ellipse هتكون بين ال x و ال zالان نجينا الجزء
+
+225
+00:16:01,750 --> 00:16:05,890
+الثاني X تربيع نقص اربع اكزاد Y تربيع اكزاد عشر Y
+
+226
+00:16:05,890 --> 00:16:11,800
+نقص اكزاد تربيع اكزاد عشرين اكزاد عشرينبنعمل اكمال
+
+227
+00:16:11,800 --> 00:16:15,560
+المربع له بيصير اكمال المربع هنا اكملنا المربع
+
+228
+00:16:15,560 --> 00:16:20,580
+ماشي الحال ضفنا اربعة وضفنا خمسة وعشرين الان بيصير
+
+229
+00:16:20,580 --> 00:16:23,420
+ضفنا تسعة وعشرين وفي عند عشرين يعني بيكون ضفنا
+
+230
+00:16:23,420 --> 00:16:26,880
+تسعة اذا بنضيف على .. شيلنا العشرين على قصدي من
+
+231
+00:16:26,880 --> 00:16:30,000
+هنا بيظل ايش تسعة لازم نضيفها على الطرف الثاني
+
+232
+00:16:30,000 --> 00:16:33,930
+بنكسب كله على تسعةوبنكمل المربع بتطلع عندي
+
+233
+00:16:33,930 --> 00:16:37,630
+المعادلة اللي أمامي هذه هذه المعادلة اللي أمامي
+
+234
+00:16:37,630 --> 00:16:42,510
+بالظبط هي تربيع وهي تربيع مجبات وهذه السالب السالب
+
+235
+00:16:42,510 --> 00:16:49,450
+هو اللي هيكون ماله اللي هو هو أو ما يوزي ال axis
+
+236
+00:16:49,450 --> 00:16:54,790
+تبع ال hyperboloid of one sheet هو إذا كان مافيش
+
+237
+00:16:54,790 --> 00:16:59,620
+أي shiftيعني فيش shift كل مرة بيكون هو لو في shift
+
+238
+00:16:59,620 --> 00:17:03,600
+زي هنا بيصير اللي هو ال axis اللي هو ال z-axis
+
+239
+00:17:03,600 --> 00:17:08,560
+انزاح واتنين باتجاه ال positive x-axis ونقص خمسة
+
+240
+00:17:08,560 --> 00:17:13,980
+باتجاه ال negative y-axis فبيصير اللي هو ال axis
+
+241
+00:17:13,980 --> 00:17:19,820
+للhyperboloid of one sheet هناهو عبارة عن المحور
+
+242
+00:17:19,820 --> 00:17:27,460
+الموازي لمحور Z والذي يمر بالنقطة 2 وناقص 5 يعني
+
+243
+00:17:27,460 --> 00:17:30,740
+بيصير عند الخط اللي بيمر 2 ناقص 5 و Z اللي هي
+
+244
+00:17:30,740 --> 00:17:34,880
+المتغيرة بيكون هذا الخط الموازي اللي هو الـ Z-axis
+
+245
+00:17:34,880 --> 00:17:40,320
+هو عبارة عن الـ axis للhyperboloid of one sheet
+
+246
+00:17:40,460 --> 00:17:43,740
+عشان هيك هذا بتمثل hyperboloid of one sheet
+
+247
+00:17:43,740 --> 00:17:47,280
+similar to that in example ستة اللي قولنا عنه قبل
+
+248
+00:17:47,280 --> 00:17:51,560
+وشوية but it is shifted two units in the positive
+
+249
+00:17:51,560 --> 00:17:56,720
+x-axis وخمسة units in the negative y-axis so its
+
+250
+00:17:56,720 --> 00:18:00,140
+axis is the line that parallel to the z-axis plane
+
+251
+00:18:00,140 --> 00:18:03,600
+and passes through اتنين ونقص خمسة او زد اي ان
+
+252
+00:18:03,600 --> 00:18:14,190
+كانت الذنبB3 وB3 وC1 نجي الأن للنوع الأخير أو
+
+253
+00:18:14,190 --> 00:18:18,010
+الصنف الأخير اللي هو من الـ quadratic surfaces
+
+254
+00:18:18,010 --> 00:18:22,850
+اللي هندرسه اللي هو اللي بنسمي hyperboloid of two
+
+255
+00:18:22,850 --> 00:18:29,780
+sheetsالان هنا بدل موحدة صار انت انتين عند فئل
+
+256
+00:18:29,780 --> 00:18:34,900
+سالب الان اللي مالوش سالب اللي موجب الغريب يعني
+
+257
+00:18:34,900 --> 00:18:38,060
+هناك كان الغريب اللي هو اللي الزد اللي له سالب
+
+258
+00:18:38,060 --> 00:18:42,860
+واحد الان اللي الغريب هنا اللي موجب الغريب هو اللي
+
+259
+00:18:42,860 --> 00:18:49,730
+بحيعمله اللي هو ال axisلمن ؟ لشكل اللي عندى تشوف
+
+260
+00:18:49,730 --> 00:18:56,090
+كيف نتبع نفس النهج و نفس الأسلوب بدنا نعرف مين يا
+
+261
+00:18:56,090 --> 00:19:02,730
+جماعة المقاطع خد X بصفر بصير عند Z تربية على C
+
+262
+00:19:02,730 --> 00:19:05,530
+تربية نقص Y تربية على B تربية عشان بساوية واحد
+
+263
+00:19:05,530 --> 00:19:10,230
+وكلكم عرف ان هذه معادلة Hyperbola في الوين؟ في ال
+
+264
+00:19:10,230 --> 00:19:17,960
+YZ دبلين Hyperbola في ال YZ دبلينهي اللي هو بين ال
+
+265
+00:19:17,960 --> 00:19:23,180
+y و ال z هي واحدة بين ال y و ال z و هي التانية بين
+
+266
+00:19:23,180 --> 00:19:26,670
+ال y و ال z على الجهة الثانيةعملت للـhyperbola
+
+267
+00:19:26,670 --> 00:19:31,870
+شوفها الـ gate الآن خد z بصفر خد z بصفر .. لأ ..
+
+268
+00:19:31,870 --> 00:19:35,670
+جبل .. y بصفر بصير الhyperbola z تربيع على c تربيع
+
+269
+00:19:35,670 --> 00:19:39,250
+نقص x تربيع على a تربيع بساوي ايه؟ شوية واحد برضه
+
+270
+00:19:39,250 --> 00:19:44,110
+هي دي ايه اشمالها؟ hyperbola بس بين ال x و ال z هي
+
+271
+00:19:44,110 --> 00:19:48,990
+ال hyperbola بين ال x هيها و ال z و هي ال
+
+272
+00:19:48,990 --> 00:19:56,190
+hyperbola عجبالها بين ال xالـ X والـ Z هيها هي هذه
+
+273
+00:19:56,190 --> 00:20:06,540
+وهي هذه والأول هذي بين ال Y والـ Zماشي طيب نحي إذا
+
+274
+00:20:06,540 --> 00:20:11,240
+صار عندي المقطعين hyperbolat بالشكل اللي أمامي طيب
+
+275
+00:20:11,240 --> 00:20:14,100
+خد z بالساوة و سفر عشان نقولنا إيه طب انت تركتلنا
+
+276
+00:20:14,100 --> 00:20:17,100
+الفرار ده ليش الفرار هذا هنشوف تعالوا نشوفوا نكد
+
+277
+00:20:17,100 --> 00:20:21,040
+الكلام اللي بحكيه اللي هنخد z بالساوة و سفر لما z
+
+278
+00:20:21,040 --> 00:20:24,760
+بالساوة و سفر بصير x تربع a تربع z y تربع b تربع
+
+279
+00:20:24,760 --> 00:20:28,880
+بساوة سالب واحدلأنه بيصير ناقص ناقص و واحد فدربت
+
+280
+00:20:28,880 --> 00:20:31,880
+وجهتين في ناقص و بطلع لوكس تربيعي تربيع تربيع بيه
+
+281
+00:20:31,880 --> 00:20:35,040
+تربيع سمه ناقص واحد ده كلام مستحيل يتحقق هدوله
+
+282
+00:20:35,040 --> 00:20:38,720
+موجبات و هده سالبة إذا فش رسم عند Z بتساوي سفر
+
+283
+00:20:38,720 --> 00:20:46,420
+رأيك أصلا خد أي قيمة Z من هذه ال C لعند ناقص C من
+
+284
+00:20:46,420 --> 00:20:50,000
+هنا لها أي Z يعني خد أي Z ال absolute value لل Z
+
+285
+00:20:50,000 --> 00:20:56,890
+أصغر من C بيصير عندك الآنلما تأخذ قيمة أصغر من الـ
+
+286
+00:20:56,890 --> 00:21:00,050
+C بيصير اللي فوق أصغر من اللي تحت بيصير هذا كله
+
+287
+00:21:00,050 --> 00:21:04,230
+أصغر من واحد لما أصغر من واحد واتجيبه هان بيصير
+
+288
+00:21:04,230 --> 00:21:07,530
+واحد ناقص أشي أصغر من واحد بيطلع كمية موجبة اللي
+
+289
+00:21:07,530 --> 00:21:10,890
+هان بيصير الأن أضرب الكل في سالب بيصير X تربيع على
+
+290
+00:21:10,890 --> 00:21:14,190
+أي تربيع زائد Y تربيع بيه تربيع بيساوي كمية سالبة
+
+291
+00:21:14,190 --> 00:21:19,550
+بعد ما ضربت بالسالب إذا مستحيل يتحققيعني أي Z في
+
+292
+00:21:19,550 --> 00:21:24,130
+المنطقة هذه بين C وبين ناقص C هتعمل لي اللي هو
+
+293
+00:21:24,130 --> 00:21:28,030
+كمية مجبرة بالساوي سالبة يعني لن تتحقق هذه
+
+294
+00:21:28,030 --> 00:21:32,550
+المعادلة مدام المعادلة هذه لن تتحقق معناته فيش رسم
+
+295
+00:21:32,550 --> 00:21:36,930
+في المنطقة هذه إذن الرسم هيكون من عند C وطالع ومن
+
+296
+00:21:36,930 --> 00:21:41,990
+عند ناقص C ونازل لأن لو حطيت الآن عند Z بتساوي C
+
+297
+00:21:41,990 --> 00:21:46,590
+بتطلع حد واحد أو ناقص Cبتنقل و هان بيصير 0 بيصير X
+
+298
+00:21:46,590 --> 00:21:49,490
+تربيع على Y تربيع زيادة Y تربيع على B تربيع بيساوي
+
+299
+00:21:49,490 --> 00:21:53,930
+0 يعني X صفر و Y صفر و Z بتساوي C يعني بتمر
+
+300
+00:21:53,930 --> 00:21:58,510
+بالنقطة هذه صفر و صفر و C و بتمر بالنقطة صفر و صفر
+
+301
+00:21:58,510 --> 00:22:03,470
+و ناقص C الآن الرسمة هتكون هنا و هنا وأكدنا عليه
+
+302
+00:22:03,470 --> 00:22:06,470
+بالhyperbola هذي و الhyperbola هذي أقولك بدي أكد
+
+303
+00:22:06,470 --> 00:22:09,930
+عليه بطريقة تانية خد Z ده إيش بتساوي 2C يعني اطلع
+
+304
+00:22:09,930 --> 00:22:14,290
+على فوقها 2C مثلا أو شالب 2C مش فارقة اللي بدك إيه
+
+305
+00:22:14,510 --> 00:22:18,810
+أو تنتين مع بعض الان بيصير عند x ترقلهده اتنين C
+
+306
+00:22:18,810 --> 00:22:22,030
+لما اربحها بيصير اربعة C تربيع اربعة C تربيع C
+
+307
+00:22:22,030 --> 00:22:24,890
+تربيع يعني اربعة اربعة جيبها على الجهة الثانية
+
+308
+00:22:24,890 --> 00:22:28,030
+بيصير ناقص اربعة واحد ناقص اربعة جديش ناقص تلاتة
+
+309
+00:22:28,030 --> 00:22:31,640
+ناقص تلاتة وهنا ناقص وهنا ناقصأضرب الكل في نقص
+
+310
+00:22:31,640 --> 00:22:34,660
+بصير X تربيع A تربيع زي Y تربيع B تربيع بسوء تلاتة
+
+311
+00:22:34,660 --> 00:22:39,140
+وهذه عبارة عن مين عن ellipse إذا المقطع هنا X عند
+
+312
+00:22:39,140 --> 00:22:44,060
+2C أو قبلها شوية و قبلها شوية و هنا و هنا و اتحد
+
+313
+00:22:44,060 --> 00:22:48,480
+كلهين بيطلع عن مين ellipses عشان هيك بنقول عنها
+
+314
+00:22:48,850 --> 00:22:53,410
+اللي هو hyperbola دي يعني في عندي مقطعين hyperbola
+
+315
+00:22:53,410 --> 00:22:57,310
+بس مش of one sheet زي اللي قبله شوية of two
+
+316
+00:22:57,310 --> 00:23:03,600
+sheets، say sheet فوق و sheet تحتطبعا هذه زي ما
+
+317
+00:23:03,600 --> 00:23:09,160
+نسميها ال vertex 00c و هذه ال vertex 00-c اللي
+
+318
+00:23:09,160 --> 00:23:14,160
+بتمر فيها ال hyperboloid
+
+319
+00:23:14,160 --> 00:23:18,240
+of one sheet of two sheets و إذا بدك تسميها
+
+320
+00:23:18,240 --> 00:23:25,520
+elliptical hyperboloid تنفع لكن اسم الشائع اللي هو
+
+321
+00:23:25,520 --> 00:23:33,250
+hyperboloidالأولى of one sheet
+
+322
+00:23:33,250 --> 00:23:36,350
+of one sheet والتانية of two sheets دي أخد المثال
+
+323
+00:23:36,350 --> 00:23:40,070
+الأخير الداعيم المثال الداعيم what is the graph of
+
+324
+00:23:40,070 --> 00:23:44,160
+the quadratic equationsهذا الآن عملت نفس اللي
+
+325
+00:23:44,160 --> 00:23:48,060
+عملته قبل نفس النهج الآن بدل ال Z حطيت ال Y يعني
+
+326
+00:23:48,060 --> 00:23:52,000
+هيصير الآن ال axis الغريب من الغريب الآن اللي هو
+
+327
+00:23:52,000 --> 00:23:55,780
+ال Y تربي ال Y الغريب ليش؟ لأن هذا شرط سلب هذا شرط
+
+328
+00:23:55,780 --> 00:23:59,740
+سلب زيه وهذا المختلف عنه إذن هو ال axis تبع ال
+
+329
+00:23:59,740 --> 00:24:03,760
+hyperbola ال hyperboloid of one of two sheets of
+
+330
+00:24:03,760 --> 00:24:07,420
+two sheets لإنه عندي two سالب's الآن في الأولاني
+
+331
+00:24:07,420 --> 00:24:11,420
+of one sheet اللي عنده فيه كان عندي one سالبالان
+
+332
+00:24:11,420 --> 00:24:13,480
+هذه هي عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+333
+00:24:13,480 --> 00:24:14,660
+عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن
+
+334
+00:24:14,660 --> 00:24:16,000
+عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+335
+00:24:16,000 --> 00:24:18,160
+عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن
+
+336
+00:24:18,160 --> 00:24:22,780
+عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+337
+00:24:22,780 --> 00:24:23,040
+عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن
+
+338
+00:24:23,040 --> 00:24:23,100
+عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+339
+00:24:23,100 --> 00:24:27,760
+عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن
+
+340
+00:24:27,760 --> 00:24:30,740
+عبارة عن
+
+341
+00:24:40,280 --> 00:24:44,200
+المعادلة الأخيرة هذه نفس الشيء سنعملها إكمال مربع
+
+342
+00:24:44,200 --> 00:24:48,480
+عملنا لها إكمال مربع صارت تزعزت تاربيع أخدت السالب
+
+343
+00:24:48,480 --> 00:24:52,660
+برا بصيرت تاربيع ناقص أربع ناقص ناقص Y تاربيع ناقص
+
+344
+00:24:52,660 --> 00:24:52,800
+ناقص و Y تاربيع ناقص ناقص و Y تاربيع ناقص ناقص و Y
+
+345
+00:24:52,800 --> 00:24:53,060
+تاربيع ناقص و Y تاربيع ناقص و Y تاربيع ناقص و Y
+
+346
+00:24:53,060 --> 00:24:56,160
+تاربيع ناقص و Y تاربيع ناقص و Y تاربيع ناقص و Y
+
+347
+00:24:56,160 --> 00:25:02,000
+تاربيع ناقص و Y تاربيع ناقص و Y تاربيع ناقص و Y
+
+348
+00:25:02,000 --> 00:25:06,910
+تاربيع ناقص و Y تاراللي هو سالب أربعة وضفت سالب
+
+349
+00:25:06,910 --> 00:25:10,190
+خمسة وعشرين يعني ضفت سالب تسعة وعشرين بضيف على
+
+350
+00:25:10,190 --> 00:25:13,230
+الطرف التالت سالب التالت سالب تسعة وعشرين وكان
+
+351
+00:25:13,230 --> 00:25:18,870
+تلاتين بيصير إيش واحد لأن هذه جاهزة بجسمها اللي هو
+
+352
+00:25:20,670 --> 00:25:27,150
+لأ اسمحولي بس أغير هنا خلّي هذه الآن بتضيف كمان
+
+353
+00:25:27,150 --> 00:25:30,590
+تمانية يعني خلّيها تمانية و تلاتين هذه عساس أنها
+
+354
+00:25:30,590 --> 00:25:34,530
+تصير هذه تسعة و لما أجسم على تسعة بطلع إيش واحد
+
+355
+00:25:34,530 --> 00:25:37,530
+فاهمين عليه يعني هذا في السؤال هذه مش تلاتين هذه
+
+356
+00:25:37,530 --> 00:25:41,070
+بيساوي تمانية و تلاتينثمانية و تلاتين و تسعة و
+
+357
+00:25:41,070 --> 00:25:49,810
+عشرين و تسعة و تسعة و تسعة و تسعة
+
+358
+00:25:49,810 --> 00:25:50,370
+و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+359
+00:25:50,370 --> 00:25:50,490
+تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+360
+00:25:50,490 --> 00:25:50,770
+تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+361
+00:25:50,770 --> 00:25:51,450
+تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+362
+00:25:51,450 --> 00:25:52,030
+تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+363
+00:25:52,030 --> 00:25:52,030
+تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و تسعة و
+
+364
+00:25:52,030 --> 00:26:00,110
+تسعة و تسعة و تسعة و تسعة
+
+365
+00:26:00,110 --> 00:26:08,510
+و تسعة و تسلأن لو لم يكن هناك shift في المرة
+
+366
+00:26:08,510 --> 00:26:12,410
+فسيكون الـ Z-axis لكن مادام صار فيه shift صار الـ
+
+367
+00:26:12,410 --> 00:26:19,370
+Z-axis اللي هو المحور هيه بس إذا سحبنا اللي هو 2
+
+368
+00:26:19,370 --> 00:26:24,210
+على ال X و سالب 5 على ال Y فبصير عبارة عن ال axis
+
+369
+00:26:24,210 --> 00:26:30,250
+لهذه ال hyperbola of two sheets عبارة عن الخط اللي
+
+370
+00:26:30,250 --> 00:26:35,690
+بوازي ال Z-axis و يمر في النقطة 2 و ناقص 5و أي
+
+371
+00:26:35,690 --> 00:26:39,970
+نقطة زد هذا اللي هو محور الـ hyperboloid of two
+
+372
+00:26:39,970 --> 00:26:45,030
+sheets الان هذه اللي هي نفس الأولى اللي حكينا عنها
+
+373
+00:26:45,030 --> 00:26:51,450
+قبل هيكاللي هي هذه بس اشترلها عبارة عن shift ع ال
+
+374
+00:26:51,450 --> 00:26:56,230
+X-axis بمقدار 2 وهذه ستتكرر وع ال Y-axis بمقدار 5
+
+375
+00:26:56,230 --> 00:27:01,150
+بس بالاتجاه السالب حالة هي كبسيرة تمر اللي هي في 2
+
+376
+00:27:01,150 --> 00:27:06,890
+و 5 و Z بتساوي 1 زائد أو ماقص 1 ليه قيمة C هذه
+
+377
+00:27:06,890 --> 00:27:12,250
+instead of 00 وزائد C and X-axis زي ما قولنا is
+
+378
+00:27:12,250 --> 00:27:15,930
+the line that parallel to the Z-axisand بص صفره
+
+379
+00:27:15,930 --> 00:27:20,230
+اتنين ونقص خمسة وزد ال .. و بعدين ال a تبعتها
+
+380
+00:27:20,230 --> 00:27:24,710
+بتساوي .. اللي هي تلاتة و ال b تلاتة و ال c بتساوي
+
+381
+00:27:24,710 --> 00:27:29,570
+واحد و هيك بكون احنا خلصنا اللي هو كل اتناش ستة
+
+382
+00:27:29,570 --> 00:27:34,490
+وهي ال exercises اللي داعمة و بارك الله فيكم و إلى
+
+383
+00:27:34,490 --> 00:27:35,670
+لقاء اخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/GtNEO-ZOqgE_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/GtNEO-ZOqgE_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0c50b309f9aacd85a04f08e2f606b552d589a30b
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/GtNEO-ZOqgE_postprocess.srt
@@ -0,0 +1,2224 @@
+1
+00:00:00,620 --> 00:00:04,700
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,700 --> 00:00:10,940
+وبركاته هذه هي المحاضرة رقم 21 تفاضل وتكامل جيم
+
+3
+00:00:10,940 --> 00:00:14,840
+وهي المحاضرة الثانية في الشبطر الأخير في المادة
+
+4
+00:00:14,840 --> 00:00:20,870
+اللي هو 15سنبدأ اليوم في 15.2 Double integrals
+
+5
+00:00:20,870 --> 00:00:24,990
+over general regions اللي حكينا عنه المرة الماضية
+
+6
+00:00:24,990 --> 00:00:29,570
+Double integrals over rectangles وكانت تحديد حدود
+
+7
+00:00:29,570 --> 00:00:33,130
+ال integration في حينه سهلة جدا اللي هي عبارة عن
+
+8
+00:00:33,130 --> 00:00:39,650
+حدود ال rectangle من A لB لC لD وليس فيها يعني
+
+9
+00:00:39,650 --> 00:00:44,330
+اللي هو غلبة على طول بس تعود عنها هذه اللي هي
+
+10
+00:00:44,330 --> 00:00:48,310
+الحدودالان بدنا نجي الى double integral over
+
+11
+00:00:48,310 --> 00:00:51,890
+bounded non rectangular region يعني الان هذه ال
+
+12
+00:00:51,890 --> 00:00:56,070
+region عندى region arbitrary region اللي هي في
+
+13
+00:00:56,070 --> 00:00:59,770
+المنطقة المحدودة طبعا اللي هي هذه المنطقة الذركة
+
+14
+00:00:59,770 --> 00:01:02,990
+هي ال region اللي عليها بدنا نعمل integration وهي
+
+15
+00:01:02,990 --> 00:01:08,070
+طبعا بتكون جزء من domain الدالة المراد اللي هي عمل
+
+16
+00:01:08,070 --> 00:01:12,670
+عليها او اللي هو إيجاد ال double integral لها على
+
+17
+00:01:12,670 --> 00:01:18,370
+المنطقة المذكورةنفس الفكرة السابقة بنجسم برضه
+
+18
+00:01:18,370 --> 00:01:26,710
+بناخد أول حد و آخر حد في المنطقة وبناخد x0, x1, x2
+
+19
+00:01:26,710 --> 00:01:33,450
+إلى آخره أو بنجزء الموازي للـ x-axis وبالمثل بنجزء
+
+20
+00:01:33,450 --> 00:01:36,350
+الموازي للـ y-axis للـ x-axis
+
+21
+00:01:42,330 --> 00:01:48,770
+وهو الموازيق لهم Y0 Y1 Y2 لعند YN وطبعا الفترة
+
+22
+00:01:48,770 --> 00:01:52,670
+النموذجية اللي بنشتغل عليها زي ما كنا ليها XK-1
+
+23
+00:01:52,670 --> 00:01:57,670
+لعند XK على X وYK-1 لعند YK على Y فبتطلع عندي اللي
+
+24
+00:01:57,670 --> 00:02:03,090
+هو المستطيل هذا او اللي هو المربع هذا اللي هو طوله
+
+25
+00:02:03,090 --> 00:02:09,540
+Delta XK وطوله Delta YKالآن زي ما أخدنا جمل بناخد
+
+26
+00:02:09,540 --> 00:02:16,700
+نقطة عشوائية هنا وهي xk و yk فوبنجمع اللي هي على
+
+27
+00:02:16,700 --> 00:02:19,900
+كل المنطقة اللي عملناها اللي هي بتطلع عند ال
+
+28
+00:02:19,900 --> 00:02:24,480
+summation طبعا عارفين إيش بنجمع نجمع f of xk و yk
+
+29
+00:02:24,480 --> 00:02:30,220
+قيمة الدالة مضروبة في delta xk اللي هي المنطقة
+
+30
+00:02:30,220 --> 00:02:33,620
+اللي هي نموذجية هذه اللي هو المستطيل اللي عندنا
+
+31
+00:02:33,620 --> 00:02:38,220
+مساحتهفبطلع الـ summation هذا هو remand sum لو كان
+
+32
+00:02:38,220 --> 00:02:41,300
+الـ remand sum هذا limit as N as N goes to
+
+33
+00:02:41,300 --> 00:02:45,480
+infinity اللي هو exist بنسميه اللي هو الـ double
+
+34
+00:02:45,480 --> 00:02:50,060
+integral over the region R زي ما عملنا قبل بالظبط
+
+35
+00:02:50,060 --> 00:02:53,320
+الـ N as N goes to infinity طبعا كل هنا دولة بروح
+
+36
+00:02:53,320 --> 00:02:57,680
+بيسير هذه المنطقة كلها تقطة المنطقة المحدودة عندنا
+
+37
+00:02:58,060 --> 00:03:02,100
+الان هنا كاتب لنا norm لل بي بيروح للسفر norm لل
+
+38
+00:03:02,100 --> 00:03:07,240
+بي ال بي هنا هو عبارة عن الان باجي بقى بشوف كل
+
+39
+00:03:07,240 --> 00:03:11,600
+الفترات اللي زي هذه الان باجي باخد delta xk و
+
+40
+00:03:11,600 --> 00:03:16,260
+delta yk فيهم كلهم أكبر واحدة في كل الموجودات من
+
+41
+00:03:16,260 --> 00:03:20,160
+ال delta xk أو من ال delta yk هي اللي بسميها ايش
+
+42
+00:03:20,160 --> 00:03:23,540
+ال norm لل بي فالان ال norm لل بي لما يروح للسفر
+
+43
+00:03:23,540 --> 00:03:27,670
+يعني هذه بمعنى آخر لو كانت هي أكبر فترةأو أكبر وهو
+
+44
+00:03:27,670 --> 00:03:31,350
+طول الفترة Delta XK لما يذهب للسفر معناته كل اللي
+
+45
+00:03:31,350 --> 00:03:34,130
+أصغر منها هيذهب للسفر و نفس الشيء كل اللي أصغر
+
+46
+00:03:34,130 --> 00:03:38,010
+منها من هنا يذهب للسفرمعناته أنه حصلت عندما الـ
+
+47
+00:03:38,010 --> 00:03:40,830
+Normal بيذهب إلى Zero أصبحت هذه كما قلنا قبل هي
+
+48
+00:03:40,830 --> 00:03:44,210
+عبارة عن خطوط أو بمعنى آخر الـ N هتذهب إلى مالة
+
+49
+00:03:44,210 --> 00:03:49,990
+نهاية فالتي هي هذه تعبير آخر على بدل الـ N تذهب
+
+50
+00:03:49,990 --> 00:03:54,030
+إلى مالة نهاية الـ Normal بيذهب لمين؟ للصفر وهذا
+
+51
+00:03:54,030 --> 00:03:59,420
+اللي هو الـ Double Integral الآن في سيكشن هذااللي
+
+52
+00:03:59,420 --> 00:04:04,180
+هو الشطارة عنا ان كيف نحدد حدود ال integration R
+
+53
+00:04:04,180 --> 00:04:07,360
+زمان كانت ال rectangle بكل سهولة من A ل B و من C ل
+
+54
+00:04:07,360 --> 00:04:11,460
+D لأ الآن زي ما شايفين احنا لا نقدرش نقول من A ل B
+
+55
+00:04:11,460 --> 00:04:15,880
+ل C ل D لأن هذه محكومة اللي هي ب curve و اللي فوق
+
+56
+00:04:15,880 --> 00:04:19,580
+ب curve او اذا بدأنا بالجهة الثانية ب curve و
+
+57
+00:04:19,580 --> 00:04:22,020
+الجهة الثالثة ب curve فبعدين كيف نغطي هذه المنطقة
+
+58
+00:04:22,020 --> 00:04:30,650
+ككل الآن نشوف كيف نعملنأتي الآن للـ Volume نحكي عن
+
+59
+00:04:30,650 --> 00:04:32,610
+الـ Volumes اللي هو الحجم زي ما انتوا عارفين ده
+
+60
+00:04:32,610 --> 00:04:36,410
+كانت اللي هي اللي جابل بشوية هذه اللي هي F of X وY
+
+61
+00:04:36,410 --> 00:04:41,650
+وK ده كانت موجبة بيصير يمثل هذا اللي هو ال limit
+
+62
+00:04:41,650 --> 00:04:44,790
+لهذا ال summation لأنه هذا بيصير تقدير للحجم في
+
+63
+00:04:44,790 --> 00:04:48,150
+فقدان أو في زيادة حسب اللي هو ال partitions اللي
+
+64
+00:04:48,150 --> 00:04:51,970
+بناخده الآن لما نقنط روح لما ننهي و نذهب للـ Zero
+
+65
+00:04:51,970 --> 00:04:54,810
+بيصير بالظبط هذا حجم الجسم اللي هو الـ Double
+
+66
+00:04:54,810 --> 00:05:00,200
+Integralالان نأتي ونشوف نظرية فبونيز تيوريم
+
+67
+00:05:00,200 --> 00:05:03,000
+الـStronger Form التي كانت المرة الماضية عن
+
+68
+00:05:03,000 --> 00:05:07,320
+الـRectangles والان على الـCurves أو الـRegion
+
+69
+00:05:07,320 --> 00:05:11,520
+التي ليست ركتانجل لت F of XY be continuous on a
+
+70
+00:05:11,520 --> 00:05:15,300
+region R لأ لذلك المطلوب من النظرية المعطيتها أن F
+
+71
+00:05:15,300 --> 00:05:20,100
+of XY يكون continuous على R لأن if R is defined on
+
+72
+00:05:20,100 --> 00:05:26,380
+أو byX من A لعند B وY من G1 لعند G2 يعني الآن
+
+73
+00:05:26,380 --> 00:05:32,540
+المنطقة R محددة محددة بواسطة اللي هو Y ال curve
+
+74
+00:05:32,540 --> 00:05:36,920
+بروح من G1 of X ل G2 of X والـ X كلها بتروح من A
+
+75
+00:05:36,920 --> 00:05:41,360
+لعند B والـ G1 و G2 التنتين continuous إذن ال
+
+76
+00:05:41,360 --> 00:05:45,680
+double integration لل F of X و Y over R بتساوي
+
+77
+00:05:46,430 --> 00:05:50,110
+الانتجريشن الان dy بنكمل بالنسبة لمين لـ dy لأن y
+
+78
+00:05:50,110 --> 00:05:53,630
+ربطناها بمتغير ده ان هي اللي بنكملها في الأول dy و
+
+79
+00:05:53,630 --> 00:05:57,890
+هلشوف هذا الكلام ايش معناته بعد شوية dy اللي هي من
+
+80
+00:05:57,890 --> 00:06:02,870
+g1 of x ال y لعند g2 of x الآن بعدين بنكمل ل x من
+
+81
+00:06:02,870 --> 00:06:06,730
+وين من عند a لعند b فبطلع عندى اللي هان عبارة عن
+
+82
+00:06:06,730 --> 00:06:11,010
+اللي هو رقم في النهايةالأن إذا كاملة الـ R نفسها
+
+83
+00:06:11,010 --> 00:06:15,910
+هذه الـ R لكن معرفة عن طريق مين؟ عن طريق أن الـ L
+
+84
+00:06:15,910 --> 00:06:18,830
+ننكامل أول شي بالنسبة لـ X يعني الـ X محصورة بين
+
+85
+00:06:18,830 --> 00:06:23,950
+كيرفين H1 of Y وH2 of Y و الـ Y في هذه الحالة
+
+86
+00:06:23,950 --> 00:06:27,670
+محصورة بين الـ C و بين الـ D في هذه الحالة بيكون
+
+87
+00:06:27,670 --> 00:06:31,490
+الـ Double Integration F of XY DA بساوية الـ
+
+88
+00:06:31,490 --> 00:06:35,630
+Integration أول شي بنكامل بالنسبة لـ XH1 of Y H2
+
+89
+00:06:35,630 --> 00:06:41,130
+of Y C لعند الهو HD اللي هانشطرتنا انه احنا نجيب
+
+90
+00:06:41,130 --> 00:06:44,170
+اللي هي حدود ال integration بعد شوية كيف نستخدم في
+
+91
+00:06:44,170 --> 00:06:47,230
+بونيز فيرم فإننا نودد حدود ال integration سواء
+
+92
+00:06:47,230 --> 00:06:53,090
+نكامل DX DY أو نكامل DY DX وحنلاقي انه اللي هو
+
+93
+00:06:53,090 --> 00:06:57,190
+الجهتين بيعطي هذه او هذه بيعطي نفس القيمة اللي هو
+
+94
+00:06:57,190 --> 00:07:01,820
+الحجم المطلوب او ال integration اللي مطلوبهذا اللي
+
+95
+00:07:01,820 --> 00:07:06,400
+حكيته عنه قبل بشوية دلتا A K وهي F of XK وYK
+
+96
+00:07:06,400 --> 00:07:12,560
+الارتفاع حجمه هذا اللي هو حاصل ضرب دلتا A K في F
+
+97
+00:07:12,560 --> 00:07:17,180
+of XK لما نجمحن K من عند واحد لعند N وناخد N تروح
+
+98
+00:07:17,180 --> 00:07:20,000
+لما لنهاية بنكون اللي هو عمل integration للجسم
+
+99
+00:07:20,000 --> 00:07:23,400
+وجدنا الحق بيصير حجم جنب بعض جنب بعض جنب بعض
+
+100
+00:07:23,400 --> 00:07:28,400
+بتتكوّر لحجمين الجسم اللي أمامي نشوف الآن مثلًا
+
+101
+00:07:29,180 --> 00:07:33,300
+الأن بقول find the volume of the prism whose base
+
+102
+00:07:33,300 --> 00:07:36,740
+is the triangle in the xy plane bounded by the x
+
+103
+00:07:36,740 --> 00:07:41,240
+-axis and the lines y بيساوي x and x بيساوي واحد
+
+104
+00:07:41,240 --> 00:07:44,940
+and whose top lies in the plane اللي هو z بيساوي
+
+105
+00:07:44,940 --> 00:07:50,940
+تلاتة ناقص x ناقص y عندي الآن منشور قاعدته اللي هو
+
+106
+00:07:50,940 --> 00:07:54,700
+عبارة عن الشكل اللي أمامي هذا اللي هو إيش y بتساوي
+
+107
+00:07:54,700 --> 00:07:59,790
+x هي y بتساوي xأو X بتساوي واحد و X بتساوي واحد
+
+108
+00:07:59,790 --> 00:08:04,170
+هيها والـ X axis المنطقة المحصورة هذه هي عبارة عن
+
+109
+00:08:04,170 --> 00:08:08,590
+قاعدة المنشور لكن سقف المنشور وينه؟ سقف المنشور
+
+110
+00:08:08,590 --> 00:08:12,190
+طبعا هذا الآن في البلين لأن سقفه أعلى اللي هو
+
+111
+00:08:12,190 --> 00:08:16,650
+شماله Z بتساوي تلاتة ناقص X ناقص Y يعني سقفه عبارة
+
+112
+00:08:16,650 --> 00:08:21,760
+عن البلين هذا الآن بلزمنيش أرسمبالظبط، يجب أن أرسم
+
+113
+00:08:21,760 --> 00:08:26,400
+المنطقة التي هي عبارة عن قاعدة الجسم الذي أريد أن
+
+114
+00:08:26,400 --> 00:08:33,720
+أجد وهو من حجمه وأكون عارف بس الدالة التي هي ثقف
+
+115
+00:08:33,720 --> 00:08:42,070
+من ثقف المجسم الذي قاعدته هذابصير عندي اللي هو الـ
+
+116
+00:08:42,070 --> 00:08:46,210
+Volume الـ Volume الـ Volume اللي هو عبارة عن اللي
+
+117
+00:08:46,210 --> 00:08:49,430
+أنا بدأ فاضل الـ dy بيدي ال dx في الأول ماشي الحال
+
+118
+00:08:49,430 --> 00:08:52,330
+شوفوا عليها ال volume هو عبارة عن ال double
+
+119
+00:08:52,330 --> 00:08:56,770
+integration لأن في عند جثثتي اللي هي الحدود صبروا
+
+120
+00:08:56,770 --> 00:09:00,170
+عليها شوية V بيساوي ال double integral لمين للدالة
+
+121
+00:09:00,170 --> 00:09:04,180
+اللي بدأ فاضلها اللي هي ارتفاع المنشوراللي هو dy
+
+122
+00:09:04,180 --> 00:09:08,540
+dx اللي هي dy by dx مع حد اللي هتمثل roughly هذا
+
+123
+00:09:08,540 --> 00:09:12,400
+الكلام مساحة هذا فبصير المساحة في الارتفاع اللي هو
+
+124
+00:09:12,400 --> 00:09:15,940
+الحجم هذا الكلام roughly زي ما قلنا by Fubini's
+
+125
+00:09:15,940 --> 00:09:19,680
+theorem إذا نضع بال integration لدالة dy dx هو
+
+126
+00:09:19,680 --> 00:09:24,200
+هيطلع من ال volume أنا الآن شوفوا عليّ، بدأ فاضل
+
+127
+00:09:24,200 --> 00:09:29,980
+بالنسبة ل Y مدام بدأ فاضل أول إشي بالنسبة ل Yأيش
+
+128
+00:09:29,980 --> 00:09:34,720
+بيجيباج على المنطقة باخد
+
+129
+00:09:34,720 --> 00:09:39,500
+خطوط موازية لل Y axis في داخل المنطقة شايفين
+
+130
+00:09:39,500 --> 00:09:44,020
+المنطقة باخد خطوط موازية لل Y axis إذا لجيتها كلها
+
+131
+00:09:44,020 --> 00:09:49,240
+على نفس المنوال إذا لجيتها كلها على نفس المنوال
+
+132
+00:09:50,090 --> 00:09:53,010
+معنى نفس المنوال يعني جاية من مكان و رايحة لأ
+
+133
+00:09:53,010 --> 00:09:55,690
+للمكان نفسه جاية من المكان و رايحة للمكان نفسه
+
+134
+00:09:55,690 --> 00:10:01,410
+لاحظوا الخط الطولي الآن هذا داخل من وين المنطقة من
+
+135
+00:10:01,410 --> 00:10:05,770
+عند اللي هو Y بتساوي سفر هاي على ال X axis و طالع
+
+136
+00:10:05,770 --> 00:10:10,990
+وين طالع من الخط هذا الخط هذا ايش قيمة Y؟ Y بتساوي
+
+137
+00:10:10,990 --> 00:10:15,270
+X عشان هيك لو اجيت اخدت اي خط تاني جنبه اي خط تاني
+
+138
+00:10:15,270 --> 00:10:18,630
+جنبه هتلاقي انه بتطلع من السفر و تروح لمن؟ للخط Y
+
+139
+00:10:18,630 --> 00:10:24,800
+بتساوي Xفعلاً كل على نفس المنوالفيش داعي لأني أجزئ
+
+140
+00:10:24,800 --> 00:10:28,500
+المنطقة لأن كلهم في نفس المنوال وبجيكوا بعدين شو
+
+141
+00:10:28,500 --> 00:10:33,000
+معناه تجزئ المنطقة الآن إذا قيمة Y كاملة بالنسبة
+
+142
+00:10:33,000 --> 00:10:38,440
+لـ Y أول اشي أخدنا خط موازي للـ Y Axis في داخل
+
+143
+00:10:38,440 --> 00:10:41,940
+المنطقة من أين دخل و من أين خرج هي حدود ال
+
+144
+00:10:41,940 --> 00:10:45,960
+integration اتفجنا؟ طيب إذا صار عندي Y بتساوي سفر
+
+145
+00:10:45,960 --> 00:10:53,820
+وY لما تطلع تكون إيه شمالها؟ Y بتساوي Xهذا حد مين
+
+146
+00:10:53,820 --> 00:11:00,860
+حد اللي هي Y لأننا نجي لـ X تشوفوا كيف نجي لـ X X
+
+147
+00:11:00,860 --> 00:11:04,460
+بالداجة أشوف الخطوط هذه اللي في داخل المنطقة
+
+148
+00:11:04,460 --> 00:11:08,780
+الطولية من وين بدأت و من وين انتهت تلاحظوا الخط
+
+149
+00:11:08,780 --> 00:11:11,340
+الطولي الأول عند الـ Zero بعدين بيطلع بعدها بعدها
+
+150
+00:11:11,340 --> 00:11:15,620
+بعدها كل الخطوط الطولية لما نقصلها نتخيل إن عندك
+
+151
+00:11:15,620 --> 00:11:19,460
+هذا عبارة عن إش بينضيق بيمسح في المنطقة بيمشي هيك
+
+152
+00:11:19,460 --> 00:11:23,270
+و بيمشي هيك لما مسح المنطقة كلهاالأن لما يمسح
+
+153
+00:11:23,270 --> 00:11:27,650
+المنطقة ببدأ من عند X بـ 0 و بنتهيأ لعند X بـ 1
+
+154
+00:11:27,650 --> 00:11:32,110
+بنكون هيك احنا هذا الخط المتغير لما يتحرك من هنا
+
+155
+00:11:32,110 --> 00:11:36,750
+لهنا و أخدنا X اللي هي من 0 لعند 1 و مسحنا كل
+
+156
+00:11:36,750 --> 00:11:40,830
+المنطقة يعني حدود التكامل بالنسبة لـ Xهي من صفر
+
+157
+00:11:40,830 --> 00:11:44,570
+لعند مين لعند واحد لأنه بداية المنطقة بصفر وبتنتهي
+
+158
+00:11:44,570 --> 00:11:48,930
+بواحد اللي هي الخطوط الطولية هذه بتبدأ من عند Zero
+
+159
+00:11:48,930 --> 00:11:52,550
+وتنتهي لعند واحد ومن مساحة المنطقة كلها الآن بدنا
+
+160
+00:11:52,550 --> 00:11:55,130
+نبدأ الكامل التكامل سهل يعني ماعني نضيع وقت
+
+161
+00:11:55,130 --> 00:11:59,290
+التكامل صفر لعند واحد تكامل التلاتة بالنسبة ل Y
+
+162
+00:11:59,290 --> 00:12:04,730
+تلاتة Y وال X XY وY يصبح Y تربيع على 2 بنعوض عنهم
+
+163
+00:12:04,730 --> 00:12:08,450
+من وين لوين حدود ال integration من صفر لعند Xبنعوض
+
+164
+00:12:08,450 --> 00:12:16,090
+عن هذه من 0 لعند X بطلع عندي اللي هو مكان ال Y ب X
+
+165
+00:12:16,090 --> 00:12:19,430
+بيصير 3X ومكان ال Y ب X بيصير نقص X تربيعه ومكان
+
+166
+00:12:19,430 --> 00:12:23,110
+ال Y ب X بيصير نقص X تربيعه على 2 بنبسطها إيه
+
+167
+00:12:23,110 --> 00:12:26,650
+بتطلع ليه نقص X تربيعه نقص X تربيعه على 2 بتطلع 3X
+
+168
+00:12:26,650 --> 00:12:31,920
+نقص 3X تربية على 2هذا الان المقدار انا بدي اكمله
+
+169
+00:12:31,920 --> 00:12:34,400
+بالنسبة لمين لا اكثر انه خلصنا اللي هو ال
+
+170
+00:12:34,400 --> 00:12:37,460
+integration الأول نجينا ال integration التاني لان
+
+171
+00:12:37,460 --> 00:12:39,620
+احنا بنحكي عن double integral او iterated
+
+172
+00:12:39,620 --> 00:12:42,940
+integrals ورا بعض التانين الان من 0 ل عند واحد
+
+173
+00:12:42,940 --> 00:12:45,560
+نفاض الكامل بالنسبة لمين لا اكثر هذا كلام جديد
+
+174
+00:12:45,560 --> 00:12:48,780
+يعنيتلاتة X بيصير تلاتة X تربيع ع تنين نقص تلاتة X
+
+175
+00:12:48,780 --> 00:12:52,420
+تربيع بيصير نقص تلاتة X تكعيب على تلاتة في اتنين
+
+176
+00:12:52,420 --> 00:12:55,920
+يعني اللي هو X تكعيب على اتنين الان من و إلى وين
+
+177
+00:12:55,920 --> 00:12:58,560
+حدود ال integration عند سفر اللي عند واحد بنعوض
+
+178
+00:12:58,560 --> 00:13:02,200
+عنها اللي هي بواحد اول اشي اللي هي و السفر طبعا
+
+179
+00:13:02,200 --> 00:13:05,160
+هيطلع سفر بطلع تلاتة ع اتنين نقص نص اللي هو ايش
+
+180
+00:13:05,160 --> 00:13:08,840
+بطلع واحدواحد قال انا اسمعه انا مانا مانا مانا
+
+181
+00:13:08,840 --> 00:13:09,680
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+182
+00:13:09,680 --> 00:13:10,800
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+183
+00:13:10,800 --> 00:13:13,760
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+184
+00:13:13,760 --> 00:13:14,040
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+185
+00:13:14,040 --> 00:13:14,180
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+186
+00:13:14,180 --> 00:13:15,140
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+187
+00:13:15,140 --> 00:13:16,420
+مانا مانا مانا مانا مانا مانا مانا مانا
+
+188
+00:13:35,840 --> 00:13:39,620
+الان هنا بتاخد موازي لمين للـ X axis في المنطقة
+
+189
+00:13:39,620 --> 00:13:42,900
+مادام بدأ أكامل بالنسبة للـ X الآن بدأ أشوف كل
+
+190
+00:13:42,900 --> 00:13:46,720
+الخطوط اللي هي العرضية هذه اللي بالشكل هذه كلها
+
+191
+00:13:46,720 --> 00:13:50,560
+على نفس المنوال جاملة عند هذا الخط و رايحة عند
+
+192
+00:13:50,560 --> 00:13:55,760
+الخط هذاالان شوفوا قيمة X علي هذه المنطقة هي هذه
+
+193
+00:13:55,760 --> 00:13:59,400
+الخطوط وهي الخط الأحمر النموذجي لها اللي هو بيجي
+
+194
+00:13:59,400 --> 00:14:03,820
+من X بتساوي Y لعند X ايش بتساوي واحد اذا حدود
+
+195
+00:14:03,820 --> 00:14:08,770
+التكامل ل X هي من عند Y لعند ايش واحدنأتي الآن
+
+196
+00:14:08,770 --> 00:14:13,630
+لقيمة مين بعد أن نكمل لـ X نكمل لـ Y يعني الآن كيف
+
+197
+00:14:13,630 --> 00:14:18,050
+نحدد Y في الخطوط العرضية هذه كلها كلها كلها كلها
+
+198
+00:14:18,050 --> 00:14:20,370
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+199
+00:14:20,370 --> 00:14:20,750
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+200
+00:14:20,750 --> 00:14:20,810
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+201
+00:14:20,810 --> 00:14:21,250
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+202
+00:14:21,250 --> 00:14:22,710
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+203
+00:14:22,710 --> 00:14:28,310
+كلها كلها كلها كلها
+
+204
+00:14:28,310 --> 00:14:32,160
+كلها كلهااللي هو Y بالساوية سفر وبتنتهي لعند Y
+
+205
+00:14:32,160 --> 00:14:35,780
+بالساوية أحد بيكون اللي هو الضوء اللي أخدنا لو
+
+206
+00:14:35,780 --> 00:14:38,420
+ماشينا من هنا ومضلنا نمشي فيه نمشي فيه حسب اللي هو
+
+207
+00:14:38,420 --> 00:14:43,000
+هان هيكون مسح المنطقة هذه كلها ففعلا بيمسحها من
+
+208
+00:14:43,000 --> 00:14:46,340
+عند Y بالساوية سفر لعند Y بالساوية أحد عشان هيك
+
+209
+00:14:46,530 --> 00:14:49,190
+اللي هو حدود الـ integration من عند 0 لعند 1 و
+
+210
+00:14:49,190 --> 00:14:51,990
+بنكاملها دي زمان كمان اللي قبل شوية اللي هي
+
+211
+00:14:51,990 --> 00:14:55,670
+بالنسبة ل X بيصير ثالثة X نقص X تربية على 2 نقص XY
+
+212
+00:14:55,670 --> 00:15:02,980
+من Y لعند 1 و بنعوض عند X بتساوي 1X بالساوية واحد
+
+213
+00:15:02,980 --> 00:15:07,560
+بيصير تلاتة ناقص نص ناقص Y ناقص اللي هو عند X
+
+214
+00:15:07,560 --> 00:15:10,420
+بالساوية Y بيصير تلاتة Y ناقص Y تربيع على اتنين
+
+215
+00:15:10,420 --> 00:15:14,780
+ناقص Y تربيع اللي هي هاي هادى مع هادى بنختصرها
+
+216
+00:15:14,780 --> 00:15:17,880
+بتطلع عندى اللي هو المقدار هذا خمس على اتنين ناقص
+
+217
+00:15:17,880 --> 00:15:21,700
+كده كده الاخرى هادى نكملها بالنسبة ل 100 بالنسبة ل
+
+218
+00:15:21,700 --> 00:15:27,220
+Y وأكيد كلكم بعرفي كامل بالنسبة ل Y بكل سهولة اللي
+
+219
+00:15:27,220 --> 00:15:31,940
+كملناها بيصير خمس على اتنين Yنقص 4Y³ على 2 يعني
+
+220
+00:15:31,940 --> 00:15:37,060
+نقص 2Y³ التي هي زائد تصبح ثلاثة Y تكعيب على ثلاثة
+
+221
+00:15:37,060 --> 00:15:40,440
+في اتنين يعني Y تكعيب على ثلاثة على اتنين من سفر
+
+222
+00:15:40,440 --> 00:15:43,780
+لعند واحد بنعوض بنلاقي ان القيمة فعلا طلعت عندنا
+
+223
+00:15:43,780 --> 00:15:49,040
+ايش بتساوي بتساوي اللي هي عبارة عن اللي هي واحد
+
+224
+00:15:49,040 --> 00:15:53,960
+نفس القيمة الأولى اجربها لحالك الحسابة تدل تحت انا
+
+225
+00:15:53,960 --> 00:15:57,160
+حسبت كمالت لكن تبقى الحسابات بسيطة تعملها
+
+226
+00:15:59,560 --> 00:16:04,440
+سنذهب الآن إلى مثال تاني مثال تاني يقول أنك ليس
+
+227
+00:16:04,440 --> 00:16:08,880
+دائما حظك أن تذهب وتكمل بالنسبة ل X بالنسبة ل Y أو
+
+228
+00:16:08,880 --> 00:16:11,960
+تكمل بالنسبة ل Y بالنسبة ل X أحيانا بحكمك الـ
+
+229
+00:16:11,960 --> 00:16:15,940
+Integrant اللي هو الكمية اللي بدك تكملهااللي هي
+
+230
+00:16:15,940 --> 00:16:18,880
+مثلا عندنا hand sign X على X بقول لي calculate ال
+
+231
+00:16:18,880 --> 00:16:22,980
+integration هذا over mean over R R إيش هي R اللي
+
+232
+00:16:22,980 --> 00:16:27,320
+ماخدها بقول لي R where R is the triangle in the X
+
+233
+00:16:27,320 --> 00:16:30,440
+Y بلين عبارة عن triangle in the X Y بلين bounded
+
+234
+00:16:30,440 --> 00:16:33,920
+by the X axis زي اللي توا بالظبط ب X axis و ال
+
+235
+00:16:33,920 --> 00:16:38,720
+line Y بيساوي X و الخط X بتساوي 1 بقول لي كملي
+
+236
+00:16:38,720 --> 00:16:43,020
+هادي على المنطقة هذهالان الان واحد قال اللي بدأ
+
+237
+00:16:43,020 --> 00:16:46,840
+كامل dx dy اول شي بدأ كامل بالنسبة ل x لما بدك
+
+238
+00:16:46,840 --> 00:16:49,740
+تكامل بالنسبة ل x في الأول لما تيجي تكامل sin
+
+239
+00:16:49,740 --> 00:16:52,620
+الجبل حتى ما نحط اللي هو حدد حدود ال integration
+
+240
+00:16:52,620 --> 00:16:56,560
+حدود ال integration بنحددها مافي مشكلة لكن الان لو
+
+241
+00:16:56,560 --> 00:17:00,840
+بدك تحدد هنا تيجي تكامل sin x على x بالنسبة ل x
+
+242
+00:17:00,840 --> 00:17:04,440
+بتجف منعرفش نكملها by elementary اللي هي
+
+243
+00:17:04,440 --> 00:17:09,400
+integrationيعني ايه بدك اللي هو الان مابتقدرش
+
+244
+00:17:09,400 --> 00:17:12,820
+اتكاملها بالشكل هذا اذا فيه بيصير تحايل ان انا
+
+245
+00:17:12,820 --> 00:17:16,900
+اكمل dy بdx هذا عشان نعرف جداش في وبونيز theorem
+
+246
+00:17:16,900 --> 00:17:19,900
+مهمة يعني ال double integration اللي احنا مش هنعرف
+
+247
+00:17:19,900 --> 00:17:23,380
+انه نكامله بالنسبة ل x بقول سهلا هو اكيد التكامل
+
+248
+00:17:23,380 --> 00:17:26,760
+حسب في وبونيز theorem هيساوي اللي هو التكامل نفسه
+
+249
+00:17:26,760 --> 00:17:31,460
+لما نكامل بالنسبة ل x ل y او ل y ل x عشان هيك بروح
+
+250
+00:17:31,460 --> 00:17:36,020
+بشرد وبكامل بالنسبة لمين ل y عشان هيكV بدّى كاملها
+
+251
+00:17:36,020 --> 00:17:38,880
+هو الاشي بالنسبة لـY وبعدين بالنسبة لـX لـSin X
+
+252
+00:17:38,880 --> 00:17:41,800
+على X الآن بدّى أحدد حدود ال integration حدود ال
+
+253
+00:17:41,800 --> 00:17:44,960
+integration زى ما عملنا قبله بشوية بالظبط مدام لـY
+
+254
+00:17:44,960 --> 00:17:49,540
+إذا بدي أخد موازي لـY Axis الموازي لـY Axis عملنا
+
+255
+00:17:49,540 --> 00:17:52,930
+قبله شويةالخط الموازية الـ y axis اللي هو بيبدأ من
+
+256
+00:17:52,930 --> 00:17:57,050
+عند y بتساوي سفر و بنتهي للخط y بتساوي x فبتكون
+
+257
+00:17:57,050 --> 00:18:00,450
+حدود ال integration لل y من سفر لعند x عملتها قبل
+
+258
+00:18:00,450 --> 00:18:03,930
+هيك لأن كل اللي هي الموازية لل y axis هذه لو
+
+259
+00:18:03,930 --> 00:18:07,390
+مسحناها هنلاقيها بتبدأ عشان تمسح المنطقة كلها
+
+260
+00:18:07,390 --> 00:18:12,690
+بتبدأ من سفر لعند مين لعند الواحد فبكون هو هذا ال
+
+261
+00:18:12,690 --> 00:18:13,530
+volume
+
+262
+00:18:15,760 --> 00:18:19,860
+الان كامل sin x على x بالنسبة ل y صار ثابت هذا
+
+263
+00:18:19,860 --> 00:18:25,120
+المقدار فبصير تفاض و sin x على x في y و بده كامله
+
+264
+00:18:25,120 --> 00:18:30,420
+من أين؟ بده حدود التكامل من 0 ل x فبعوض من 0 ل x
+
+265
+00:18:30,420 --> 00:18:36,140
+طبعا الفعلة y ب0 تطلع حد 0 إذن بتظهر مين حالة x
+
+266
+00:18:36,140 --> 00:18:40,590
+sin x على x في xبتروح الـ x مع الـ x بتظهر صين الـ
+
+267
+00:18:40,590 --> 00:18:44,010
+x من صفر ل واحد و جيت لا تكامل يا محلا و كلنا
+
+268
+00:18:44,010 --> 00:18:47,390
+بنعرف ان كامله اللي هو بساوي اللي هو ناقص cosine
+
+269
+00:18:47,390 --> 00:18:52,570
+الـ x من صفر ل عند واحد اللي هي عبارة عن اللي عند
+
+270
+00:18:52,570 --> 00:18:55,250
+واحد اللي هي ناقص cosine واحد هيها و عند الصفر
+
+271
+00:18:55,250 --> 00:18:58,910
+ناقص ناقص بصير زائد cosine ال zero اللي هي عبارة
+
+272
+00:18:58,910 --> 00:19:02,980
+عن واحدإذن الـ integration بالنسبة لـ y قبل و
+
+273
+00:19:02,980 --> 00:19:06,820
+بعدين x هو الحل في هذه الحالة و زي ما قلنا when
+
+274
+00:19:06,820 --> 00:19:09,720
+the order of integration is reversed the integral
+
+275
+00:19:09,720 --> 00:19:14,740
+is hard يعني لما نأتي و ناخد دي x by دي y بنكون و
+
+276
+00:19:14,740 --> 00:19:18,560
+ده يعني مشكلةنجي الآن نلخص اللي حكيناه Finding
+
+277
+00:19:18,560 --> 00:19:21,460
+limits of integration هذا الكلام اللي حكيته قبله
+
+278
+00:19:21,460 --> 00:19:25,100
+شوية و احنا حكيناه تمام التمام الان عشان توجد اللي
+
+279
+00:19:25,100 --> 00:19:29,000
+هو limits of integration للمنطقة R أول حاجة فيها
+
+280
+00:19:29,000 --> 00:19:31,900
+اندي طريقتين يا ال cross-section يا ال vertical
+
+281
+00:19:31,900 --> 00:19:34,820
+-section ال cross-section يعني بده كامل بالنسبة
+
+282
+00:19:34,820 --> 00:19:40,120
+موازي لمن؟ ال cross-section اللي هو بده ياخد موازي
+
+283
+00:19:40,120 --> 00:19:45,410
+ال Y-axisاللي هو بده ياخد اسف using vertical cross
+
+284
+00:19:45,410 --> 00:19:48,890
+section اه vertical cross section الان هذا using
+
+285
+00:19:48,890 --> 00:19:51,770
+ال vertical مدام ال vertical يعني موازية ال y axis
+
+286
+00:19:51,770 --> 00:19:55,010
+يعني بده كامل اول اش بالنسبة لمن بالنسبة ل Y اذا
+
+287
+00:19:55,010 --> 00:19:58,610
+هذه الطريقة بالنسبة ل Yهذا الكلام شرحته بس خلّيني
+
+288
+00:19:58,610 --> 00:20:03,010
+أشرحه لكم كمان مرة لأنه فرضه لحاله هو شرحته من
+
+289
+00:20:03,010 --> 00:20:07,770
+خلال المثالين أول حاجة sketch the region and label
+
+290
+00:20:07,770 --> 00:20:13,130
+the bounding curves يعني الأول حاجة بدك تعملها في
+
+291
+00:20:13,130 --> 00:20:15,490
+تحديد limits of integration لازم المنطقة تكون
+
+292
+00:20:15,490 --> 00:20:19,350
+مرسومة هي المنطقة مرسومة فرضنا أن معطيني هذه
+
+293
+00:20:19,350 --> 00:20:24,730
+المنطقة هي ال region R اللي عليها بدك تعملهاأول
+
+294
+00:20:24,730 --> 00:20:29,030
+اشي برسم من المنطقة الرسم ليها X تربيع زي Y تربيع
+
+295
+00:20:29,030 --> 00:20:32,930
+بسوء واحد و X زي Y بسوء واحد هذا طبعا X تربيع زي Y
+
+296
+00:20:32,930 --> 00:20:39,170
+تربيع بسوء واحد جزء من من الدائرة هيها مع X زي Y
+
+297
+00:20:39,170 --> 00:20:43,690
+بسوء واحد هيها هذه المنطقة المحصورة بين الجهتين هي
+
+298
+00:20:43,690 --> 00:20:46,430
+لو انت كملت الدائرة وكملت الخطين هي المنطقة
+
+299
+00:20:46,430 --> 00:20:50,770
+المحصورةإذا هذه المنطقة R هي المنطقة اللي بديها هو
+
+300
+00:20:50,770 --> 00:20:55,830
+نوجد التكامل عليها إذا أول شيء عشان أوجد حدود
+
+301
+00:20:55,830 --> 00:20:59,790
+التكامل برسم المنطقة الان find the Y limit of
+
+302
+00:20:59,790 --> 00:21:05,330
+integration يعني الآن بتوجد حدود ال YImagine a
+
+303
+00:21:05,330 --> 00:21:09,790
+vertical line cutting R in the direction of
+
+304
+00:21:09,790 --> 00:21:14,330
+increasing Y نحن نأخد خط موازن منين للـ Y-axis
+
+305
+00:21:14,330 --> 00:21:19,810
+بتزايد يعني من أين يدخل المنطقة و من أين يخرج منها
+
+306
+00:21:19,810 --> 00:21:22,750
+هاي الـ Vertical-Y لاحظوا الـ Vertical-Line هاي و
+
+307
+00:21:22,750 --> 00:21:26,610
+كلهم على نفس المنوال كلهم بيجي من الخط لمينلأ
+
+308
+00:21:26,610 --> 00:21:31,210
+القوس هذا الان ببدأ من عند اللي هو قيمة y هنا إيش
+
+309
+00:21:31,210 --> 00:21:34,690
+عند x زي y بسعة واحد معادلة إتاة إذا قيمة y هنا
+
+310
+00:21:34,690 --> 00:21:38,050
+اللي هي واحد ناقص x إذا حد ال integration اللي تحت
+
+311
+00:21:38,050 --> 00:21:42,790
+واحد ناقص x الان من وين خرج من وين خرج اللي هو من
+
+312
+00:21:42,790 --> 00:21:45,830
+الدائرة هذه الدائرة هذه اللي هي x تربيع زي y تربيع
+
+313
+00:21:45,830 --> 00:21:50,730
+بسعة واحدفي المنطقة Y موجبة يعني Y تربيع بيساوي 1
+
+314
+00:21:50,730 --> 00:21:54,790
+ناقص X تربيع يعني Y بيساوي الجدر الموجب للواحد
+
+315
+00:21:54,790 --> 00:21:59,350
+ناقص X تربيع لأنه في المنطقة الموجبة فبنخرج من
+
+316
+00:21:59,350 --> 00:22:03,390
+الار وين عند Y بيساوي الجدر التربيع لواحد ناقص X
+
+317
+00:22:03,390 --> 00:22:08,100
+تربيع فبتكون هذا حدود مينحدود الـ Y نجي الآن نجي
+
+318
+00:22:08,100 --> 00:22:11,900
+نشوف هدول الخطوط ال vertical من وين لوين بمسحن ال
+
+319
+00:22:11,900 --> 00:22:16,820
+X بمسحن المنطقة بقيمة X بمسح المنطقة هي و هي و هي
+
+320
+00:22:16,820 --> 00:22:18,300
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+321
+00:22:18,300 --> 00:22:20,240
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+322
+00:22:20,240 --> 00:22:22,220
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+323
+00:22:22,220 --> 00:22:23,440
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+324
+00:22:23,440 --> 00:22:31,860
+و هي و هي و هي و هي و هي و هي و هي و
+
+325
+00:22:31,860 --> 00:22:39,460
+هي و هيوشغّلنا العداد LX من صفر لعند واحد عشان هيك
+
+326
+00:22:39,460 --> 00:22:43,760
+حدود ال integration من صفر لعند واحدنأتي الآن لـ
+
+327
+00:22:43,760 --> 00:22:46,640
+Using Horizontal Cross Section زي ما حكيته برضه
+
+328
+00:22:46,640 --> 00:22:51,260
+قبل قليل شرحته اللي هو نفس الاشي باخد اللي هو اللي
+
+329
+00:22:51,260 --> 00:22:54,040
+في حالة اللي هو بتكامل بالنسبة لـ X قبل أو ال
+
+330
+00:22:54,040 --> 00:22:58,040
+cross section بتاخد موازية لـ X Axis موازية لـ X
+
+331
+00:22:58,040 --> 00:23:00,900
+Axis من وين دخل المنطقة و من وين غادرها من وين دخل
+
+332
+00:23:00,900 --> 00:23:04,040
+المنطقة قيمة X بالساوية واحد ناقص Y هذه الحد الأول
+
+333
+00:23:04,040 --> 00:23:07,000
+من وين خرج المنطقة عند X بالساوية جدر التربيع إلى
+
+334
+00:23:07,000 --> 00:23:10,060
+واحد ناقص Y تربيع عملنا زي هذا قبل قليل لأن أصل X
+
+335
+00:23:10,060 --> 00:23:18,540
+تربيع زي Y تربيع بساوية واحدكل الخطوط التي بدأت
+
+336
+00:23:18,540 --> 00:23:21,860
+هذه الخطوط هي الـ Horizontal اللي بتمسح المنطقة
+
+337
+00:23:21,860 --> 00:23:26,000
+بتفتغر المنطقة المنطقة بتبدأ من عند Y بيتساوي و أ
+
+338
+00:23:26,000 --> 00:23:30,620
+سفر لعند Y أش بتساوي واحد أخر أش في المنطقة بيصير
+
+339
+00:23:30,620 --> 00:23:36,180
+من سفر لواحد بيكون هذا هو قيمة ال integrationبناخد
+
+340
+00:23:36,180 --> 00:23:39,800
+مثال بيقولي sketch the region of integration for
+
+341
+00:23:39,800 --> 00:23:44,040
+the integral الان بيقولي اول اشي هاي اعطيتك
+
+342
+00:23:44,040 --> 00:23:48,920
+integration انت تشاطر و ارسملي ايش المنطقة هذا اول
+
+343
+00:23:48,920 --> 00:23:52,480
+حاجة وبعدين بيقولك and write an equivalent
+
+344
+00:23:52,480 --> 00:23:55,600
+integral with the order of integration reversed
+
+345
+00:23:55,600 --> 00:24:00,880
+وكمل integration اللي هو مكافئله بس ايش معكوس يعني
+
+346
+00:24:00,880 --> 00:24:07,070
+dx by dy قولنا له زينأول حاجة لما بادي بطلع بدا
+
+347
+00:24:07,070 --> 00:24:10,590
+كامل بالنسبة ل Y إذا مدان بدا كامل بالنسبة ل Y في
+
+348
+00:24:10,590 --> 00:24:15,870
+الأول ما هو هنا إذا هذه حدود Y يعني Y في الأول X
+
+349
+00:24:15,870 --> 00:24:20,630
+تربيع و Y في الآخر 2X يعني Y بتساوي X تربيع بادي
+
+350
+00:24:20,630 --> 00:24:24,390
+برسم اللي هي الرسمة Y بتساوي X تربيع حيث Y بتساوي
+
+351
+00:24:24,390 --> 00:24:28,990
+X تربيع تمشي هك و تمشي هك ماشي الآن اللي عند Y
+
+352
+00:24:28,990 --> 00:24:33,490
+بتساوي 2X Y بتساوي 2X بتعرف ترسمه Y بتساوي 2Xبتاخد
+
+353
+00:24:33,490 --> 00:24:38,910
+X بصفر هيها لأنها خط و X بصفر واحد او اتنين بتطلع
+
+354
+00:24:38,910 --> 00:24:43,530
+هيها بتجي بترسم الخط هذا على استقامة و بتمده طبعا
+
+355
+00:24:43,530 --> 00:24:47,290
+واضح ان المنطقة المحصورة بين الاتنين بس هذه ماشي
+
+356
+00:24:47,290 --> 00:24:51,810
+الان هذه المنطقة اذا هي هذه المنطقة المذكورةالان
+
+357
+00:24:51,810 --> 00:24:57,770
+انا عشان بالزمن اللي هو خطوط التقاطع اللي هي تقاطع
+
+358
+00:24:57,770 --> 00:25:01,430
+الخطين خطوة ال curve و تقاطع الخطوة ال curve بدي
+
+359
+00:25:01,430 --> 00:25:04,590
+بحل هدولة مع بعض التنتين y بتساوي x تربيع و y
+
+360
+00:25:04,590 --> 00:25:08,770
+بتساوي 2x بقول 2x بتساوي x تربيع إذا x تربيع ناقص
+
+361
+00:25:08,770 --> 00:25:13,670
+2x بتساوي 0 يعني x اللي هو في x minus 2 بتساوي 0
+
+362
+00:25:13,670 --> 00:25:18,090
+يعني x بتساوي 0هي X بتساوى 2 في حالة X بتساوى 0
+
+363
+00:25:18,090 --> 00:25:21,290
+تطلع Y بتساوى 0 هي النقطة الأولى وفي حالة X بتساوى
+
+364
+00:25:21,290 --> 00:25:25,510
+2 اللى هو اكيد ال Y لو عوضت هنا ال X أو هنا بتطلع
+
+365
+00:25:25,510 --> 00:25:30,410
+4 أربعة فتطلع ال Y أربعة يعني نقطتي التقاطع اللى
+
+366
+00:25:30,410 --> 00:25:34,590
+عبارة عن 2 و 4 و 0 و 0 إذن هي المنطقة إيجينا
+
+367
+00:25:34,590 --> 00:25:41,440
+حصيناهاأحصى انتهاما الان اغير حدود ال integration
+
+368
+00:25:41,440 --> 00:25:48,100
+دي اكس دي واي او ال reverse ال integration دي اكس
+
+369
+00:25:48,100 --> 00:25:51,740
+دي واي مادام دي اكس دي واي اذا بدي اشوف قيمة X كيف
+
+370
+00:25:51,740 --> 00:25:58,020
+باخد X موازي في المنطقة بدخل موازي من ال X axisكل
+
+371
+00:25:58,020 --> 00:26:01,940
+الموازيات للـ x-axis بتيجي من عند الخط هذا و
+
+372
+00:26:01,940 --> 00:26:05,560
+بتنتهي لعند ال curve إذا كلها على نفس المنوال إذا
+
+373
+00:26:05,560 --> 00:26:09,980
+أنا مرتاح integration واحد و بدأ أكمل الآن قيمة x
+
+374
+00:26:09,980 --> 00:26:13,620
+هنا احنا y بيساوي 2x إذا x أش تكون y على 2 إذا
+
+375
+00:26:13,620 --> 00:26:18,740
+حدود ال integration y على 2 إلىالان قيمة X هنا Y
+
+376
+00:26:18,740 --> 00:26:22,420
+بيستوي X تربيع و المنطقة هذه لـ X موجبة إذا X
+
+377
+00:26:22,420 --> 00:26:26,920
+بيستوي زرع من مقص جدر Y لأ هي جدر Y لأن المنطقة مش
+
+378
+00:26:26,920 --> 00:26:32,040
+موجبة إذا اللي هي دخلت X من عند Y على 2 و انتهت
+
+379
+00:26:32,040 --> 00:26:35,540
+لعين جدر Y نيجي الآن الخطوط هذه اللي هي ال cross
+
+380
+00:26:35,540 --> 00:26:39,180
+sections ال horizontals كلهم بيبدأ من وين لوين
+
+381
+00:26:39,180 --> 00:26:44,100
+بيبدأ من عند السفراللي هي قيمة Y من عند السفر وبظل
+
+382
+00:26:44,100 --> 00:26:48,880
+بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع
+
+383
+00:26:48,880 --> 00:26:51,900
+.. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع ..
+
+384
+00:26:51,900 --> 00:26:52,520
+بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع
+
+385
+00:26:52,520 --> 00:26:52,580
+.. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع ..
+
+386
+00:26:52,580 --> 00:26:55,920
+بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع
+
+387
+00:26:55,920 --> 00:26:58,800
+.. بتطلع .. بتطلع .. بتطل
+
+388
+00:27:01,950 --> 00:27:05,450
+بتصور هيك الأمور ان شاء الله تكون انفهمت منيح في
+
+389
+00:27:05,450 --> 00:27:09,390
+موضوع إيجاد حدود ال integration الان بعض خواص
+
+390
+00:27:09,390 --> 00:27:12,870
+properties of double integrals بعض خواص ال double
+
+391
+00:27:12,870 --> 00:27:15,770
+integrals هي مشابه للخواص اللي هي ال integration
+
+392
+00:27:15,770 --> 00:27:19,950
+العادى لو كانت f of x y و d of x y are continuous
+
+393
+00:27:19,950 --> 00:27:22,750
+and abound in the region R then the following
+
+394
+00:27:22,750 --> 00:27:25,940
+properties holds أول حاجة constant multiplesالـ
+
+395
+00:27:25,940 --> 00:27:28,860
+Double Integral C F of X, Y, D, A يظهر الـ
+
+396
+00:27:28,860 --> 00:27:32,120
+Constant برة مثل الـ Integration العادى العادي for
+
+397
+00:27:32,120 --> 00:27:36,780
+any number C Some Undifference التي هي تكامل مجموع
+
+398
+00:27:36,780 --> 00:27:40,580
+دالتين الـ Double Integration طبعا لمجموع دالتين
+
+399
+00:27:40,580 --> 00:27:44,900
+أو حاصل طرحين بسوء مجموع أو حاصل طرح التكاملين زي
+
+400
+00:27:44,900 --> 00:27:48,260
+ما كنا ناخد في التكامل العادى Domination لو كان
+
+401
+00:27:48,260 --> 00:27:53,220
+الـ Integration F of X, Y, D, A هو أكبر أو يساوي
+
+402
+00:27:53,220 --> 00:27:57,910
+سفرأسف، لو كان الـ F of XY أكبر أو يساوي سفر على
+
+403
+00:27:57,910 --> 00:28:01,910
+الـ R أكيد الـ Integration تبعها أكبر أو يساوي سفر
+
+404
+00:28:01,910 --> 00:28:04,490
+لأنه أصلا بيصير Volume، زي ما قلت قبل شوية هذا
+
+405
+00:28:04,490 --> 00:28:07,810
+بيصير الارتفاع، اللي هي عبارة عن مساحة، بيصير
+
+406
+00:28:07,810 --> 00:28:10,230
+المساحة في الارتفاع بيطلع Volume فعلا، إن أما يكون
+
+407
+00:28:10,230 --> 00:28:12,390
+F of XY أكبر أو يساوي سفر، بيكون الـ Double
+
+408
+00:28:12,390 --> 00:28:17,640
+Integral أكبر أو يساوي سفر، ومن ثم in generalلو
+
+409
+00:28:17,640 --> 00:28:22,020
+كان عندي Fxy أكبر أو يساوي Dxy على المنطقة R اللي
+
+410
+00:28:22,020 --> 00:28:24,560
+بنحكي عليها بيكون الـ Double Integration للأولى
+
+411
+00:28:24,560 --> 00:28:28,280
+أكبر أو يساوي الـ Double Integration للثانية إلى
+
+412
+00:28:28,280 --> 00:28:31,900
+الأربعة اللي هي الـ Additivity اللي هي Double
+
+413
+00:28:31,900 --> 00:28:36,730
+Integra لـ Fxy DAالأن لو الـ R مجسم لمنطقتين
+
+414
+00:28:36,730 --> 00:28:40,870
+disjoint يعني R بتساوي R واحد تقاطع اتحاد R اتنين
+
+415
+00:28:40,870 --> 00:28:44,850
+حيث R واحد تقاطع R اتنين اشمالين اللي هي non
+
+416
+00:28:44,850 --> 00:28:50,170
+overlapping يعني مش متداخلة مش متقاطعات اللي هو
+
+417
+00:28:50,170 --> 00:28:54,500
+تقاطحين اللي هو Phi بيصير عند ال double integralما
+
+418
+00:28:54,500 --> 00:28:59,040
+هو التكامل على المنطقة الأولى زائد التكامل على
+
+419
+00:28:59,040 --> 00:29:03,280
+المنطقة التانية زي زمان كانت من A ل B و كانت C بين
+
+420
+00:29:03,280 --> 00:29:08,040
+ال A و ال B و كنا نقول من A ل C زائد من C ل B إذا
+
+421
+00:29:08,040 --> 00:29:14,280
+هو مشابه للقديم نأخد مثال و نقول find the volume
+
+422
+00:29:14,280 --> 00:29:18,500
+of the wood you likeSolid that lies beneath the
+
+423
+00:29:18,500 --> 00:29:22,180
+surface Z بالساوية 16 نقص X تربيع نقص Y تربيع and
+
+424
+00:29:22,180 --> 00:29:24,800
+above the region R bounded by the curve Y بيساوية
+
+425
+00:29:24,800 --> 00:29:29,180
+2 جدرى X and the line Y بيساوية 4X نقص 2 and the X
+
+426
+00:29:29,180 --> 00:29:36,060
+axis الكلام واضح بقول او جدلي حجم الجسم اللي اللي
+
+427
+00:29:36,060 --> 00:29:41,400
+هو الجزء العلوي له اللي هو عبارة عن إيش ال surface
+
+428
+00:29:41,400 --> 00:29:46,770
+Z بالساوية 16 نقص X تربيع نقص Y تربيعوقاعته مين هي
+
+429
+00:29:46,770 --> 00:29:52,310
+المنطقة المحصورة بين Y بساوة 2 جدر ال X هاي Y
+
+430
+00:29:52,310 --> 00:29:59,210
+بساوة 2 جدر ال X هاي الكرفة ده و ال line Y بساوة 4
+
+431
+00:29:59,210 --> 00:30:05,810
+X نقص 2 و ال line 4 X minus 2 و X أكسيزي محدد لك
+
+432
+00:30:05,810 --> 00:30:11,800
+يا تماما بقولي أوجد حجم الجسم الجسم هذا الليقاعدته
+
+433
+00:30:11,800 --> 00:30:16,100
+هيهة والتي هي سقفه من فوق الدالة الهادئة ماشي
+
+434
+00:30:16,100 --> 00:30:19,760
+الحال اذا اكيد بدنا نكامل هذه الدالة على المنطقة
+
+435
+00:30:19,760 --> 00:30:23,040
+هذه بطلع ال volume ال volume is V بالساوية ال
+
+436
+00:30:23,040 --> 00:30:29,160
+double integral الان DX DY لأ الدالة هذه 16 نقص X
+
+437
+00:30:29,160 --> 00:30:33,860
+تربيع نقص Y تربيع اللي هو الارتفاع DX DY وبدنا
+
+438
+00:30:33,860 --> 00:30:38,660
+نحدد منطقة التكامل اللي هي حدود التكامللاحظ الآن
+
+439
+00:30:38,660 --> 00:30:42,420
+هو كامل بالنسبة لـ X أول شيء و بعدين بالنسبة لمين؟
+
+440
+00:30:42,420 --> 00:30:45,880
+لـ Y طبعا هو قال لك أنا بدا كامل بالـ X قبله و
+
+441
+00:30:45,880 --> 00:30:50,260
+بعدين لـ Y لأنه أسهل ليه؟ و خلّينا نشوف ليش أسهله
+
+442
+00:30:50,260 --> 00:30:53,640
+مادان بدا كامل لـ X في الأول إذا بتاخد إيه شماله؟
+
+443
+00:30:53,640 --> 00:30:56,780
+Horizontal Line أو موازي لمين؟ لل X Axis هي
+
+444
+00:30:56,780 --> 00:31:00,020
+المنطقة هذه، لو جيت أخدت أي موازي لل X Axis؟ هيك،
+
+445
+00:31:00,020 --> 00:31:04,470
+هيك، هيك، هيك،ستجدهم جميعًا على نفس المنوال ما هو
+
+446
+00:31:04,470 --> 00:31:08,690
+نفس المنوال؟ يعني يدخل المنطقة من هذا الـ Curve
+
+447
+00:31:08,690 --> 00:31:13,070
+ويخرج من مين؟ من هذه الخطة طبعًا هذه المنطقة هي
+
+448
+00:31:13,070 --> 00:31:17,650
+هذه اللي توّى اللي هو الخط الأول Y بساوية 2 جذر
+
+449
+00:31:17,650 --> 00:31:22,830
+الـ X اللي هو الـ Curve جذده والخط الثاني يسمى Y
+
+450
+00:31:22,830 --> 00:31:26,990
+بساوية 4 نقص X نقص 2واللي هو الـ X axis هذه
+
+451
+00:31:26,990 --> 00:31:30,050
+المنطقة هي المنطقة اللي عليها التكوين للـ R هنا بس
+
+452
+00:31:30,050 --> 00:31:34,650
+نجلنا لبطة وضيح طيب الان أخدنا اللي هو ال
+
+453
+00:31:34,650 --> 00:31:38,170
+horizontal line من وين دخل هيه دخل من عند هذا جداش
+
+454
+00:31:38,170 --> 00:31:42,710
+قيمة X هنا X بتساوي من هنا بنربع الجهتين بيصير Y
+
+455
+00:31:42,710 --> 00:31:46,510
+تربيع بيساوي 2 اللي هو بيصير 4X بيصير X بيساوي Y
+
+456
+00:31:46,510 --> 00:31:50,450
+تربيع على 4 إذا قيمة X هنا اللي هي Y تربيع على 4
+
+457
+00:31:50,450 --> 00:31:53,450
+ما تخافوش من التربيع لأن قيمة X موجبة وحضّلها إيش
+
+458
+00:31:53,450 --> 00:31:58,390
+موجبةالان X بساوي صار لحد السفل يكون Y تربيع على 4
+
+459
+00:31:58,390 --> 00:32:02,430
+و لعند من يخرج من هنا؟ يخرج من هنا، ما هي قيمة X؟
+
+460
+00:32:02,430 --> 00:32:06,490
+عند I بساوية 4X minus 2 جيب Y هنا بيصير 2 هنا
+
+461
+00:32:06,490 --> 00:32:10,790
+بيصير 4X بساوية Y زائد 2 يعني X بساوية Y زائد 2
+
+462
+00:32:10,790 --> 00:32:15,050
+على 4 يعني قيمة X هنا مالها؟ Y زائد 2 على 4 في حد
+
+463
+00:32:15,050 --> 00:32:19,000
+ال integration لل X التاني Y زائد 2على أربعة نأتي
+
+464
+00:32:19,000 --> 00:32:22,260
+الآن كل الخطوط هذه الـ Horizontal لو جينا مسحنا
+
+465
+00:32:22,260 --> 00:32:26,260
+مسحنا مسحنا ستبدأ من عند ال Y بـ 0 وتنتهي إلى ال Y
+
+466
+00:32:26,260 --> 00:32:29,740
+أيش بتساوي 2 عشان هنا حدود ال integration من 0
+
+467
+00:32:29,740 --> 00:32:34,360
+لعند 2 وبنأتي من كامل هذه الكملة بغلب شوية يعني
+
+468
+00:32:34,360 --> 00:32:38,580
+خلّيني أقول مش بغلب هو التكامل سهل بس بده حسابات
+
+469
+00:32:38,580 --> 00:32:41,400
+طويلة خلّيني أعلى السريع لأن الفكرة في أن أوجد
+
+470
+00:32:41,400 --> 00:32:43,800
+حدود ال integration والتكامل اللي أنتوا بتعملوه
+
+471
+00:32:43,800 --> 00:32:48,590
+لحالكم هذه بيصير 16X ناقص X تكعيب على 3نقص y تربيع
+
+472
+00:32:48,590 --> 00:32:52,010
+في x من أين لو ينمن y تربيع على أربعة على y زائد
+
+473
+00:32:52,010 --> 00:32:55,330
+اتنين على اياش على أربعة الآن بدنا نعوض اللي هو
+
+474
+00:32:55,330 --> 00:33:00,710
+أول اشي x باللي فوق ونقص ال x باللي تحت الان x
+
+475
+00:33:00,710 --> 00:33:04,610
+باللي فوق هيبصير عوضنا على y زائد اتنين على أربعة
+
+476
+00:33:04,610 --> 00:33:11,060
+ف16 تطلع 4 في y زايد اتنينناقص Y2 كل تكعيب على
+
+477
+00:33:11,060 --> 00:33:16,380
+تلاتة في 64 ناقص
+
+478
+00:33:16,380 --> 00:33:21,920
+X في Y تربيع X هو Y زي 2 على 4 في Y تربيع ناقص Y
+
+479
+00:33:21,920 --> 00:33:27,470
+تربيع على 4بصير المكان الـ XY تربيع على 4 بصير 4Y
+
+480
+00:33:27,470 --> 00:33:32,970
+تربيع اللي قاعد ناقص، ناقص بصير زائد اللي هي Y أُس
+
+481
+00:33:32,970 --> 00:33:37,070
+اللي هي تربيع المكان الـ X بصير Y أُس 6 على 3 في
+
+482
+00:33:37,070 --> 00:33:41,850
+مين في 64 لأنه هذه لما تتكعب بصير 64 زي ما انتوا
+
+483
+00:33:41,850 --> 00:33:49,300
+عارفينزائد ناطص ناطص يصبح زائد X يصبح Y زائد 2 على
+
+484
+00:33:49,300 --> 00:33:54,960
+4 Y
+
+485
+00:33:54,960 --> 00:34:01,580
+تربيع على 2 يصبح Y أقصى 4 على 4الكل إيه شماله الـ
+
+486
+00:34:01,580 --> 00:34:04,420
+DY هذا بدي أعمله كمان مرة بالنسبالي لمين لـ DY
+
+487
+00:34:04,420 --> 00:34:07,900
+بغلبكم بس انتوا فكوه اللي حالكم هدولة اللي .. اللي
+
+488
+00:34:07,900 --> 00:34:11,340
+حاكينا عنها هدولة فكوه فكوا هدولة بغلبنا شوية بس
+
+489
+00:34:11,340 --> 00:34:13,660
+في البيت احنا لما بجيب في الامتحان بجيب الشغلة
+
+490
+00:34:13,660 --> 00:34:16,600
+اللي بتغلب في الحسابات زي هيك اللي هي تضيع الفكرة
+
+491
+00:34:16,600 --> 00:34:19,640
+اللي جيبنا منها ولكن هذا في البيت بتقدر هذا تكعبه
+
+492
+00:34:19,640 --> 00:34:24,340
+وتوجده تفرده تاخد ال Y زي بعض و ال Y تربيع زي بعض
+
+493
+00:34:24,340 --> 00:34:26,360
+و ال Y تكعيق زي بعض و ال Y أقصى أربعة زي بعض
+
+494
+00:34:26,530 --> 00:34:30,890
+وبتكامله هيطلع عندك اللي هو المقدار اللي أمامي هذا
+
+495
+00:34:30,890 --> 00:34:36,750
+هو تكامل هذا المقدار مثلا Y6 هي عبارة عن Y7 على 7
+
+496
+00:34:36,750 --> 00:34:41,870
+في هذا المقدار اللي هو Y مثلا Y4 هي Y5 على 5 في 24
+
+497
+00:34:41,870 --> 00:34:46,210
+وهكذا البجيات بنفس الإسلوبالان باعوض عنه من صفر
+
+498
+00:34:46,210 --> 00:34:49,950
+الاتنين بطلع عندي القيبة اللي عندي الان خلّيني
+
+499
+00:34:49,950 --> 00:34:53,490
+أنتبه للنقطة الأخيرة دي مهمة اللي هي another
+
+500
+00:34:53,490 --> 00:34:57,250
+method using dy dx الان واحد قاللي انا اسمع انا
+
+501
+00:34:57,250 --> 00:35:01,970
+مديش ارد عليكم و مدي اشتغل dy by dx dy by dx قلتله
+
+502
+00:35:01,970 --> 00:35:05,600
+ماشي بس هتغلب انت انت حرالان دي واي باي دي اكس
+
+503
+00:35:05,600 --> 00:35:09,040
+باجي الان باخد اللي هو مدام دي واي او الاشي اذا
+
+504
+00:35:09,040 --> 00:35:11,780
+بدي اخد vertical line يعني موازية من ال Y-axis
+
+505
+00:35:11,780 --> 00:35:15,420
+لاحظوا ال vertical lines هنا من هنا لهنا من هنا
+
+506
+00:35:15,420 --> 00:35:19,300
+لعندي لهنا مافيش عندي مشكلة كلها بتيجي من عند اللي
+
+507
+00:35:19,300 --> 00:35:23,020
+هو Y بصفر و بتيجي الى ال curve هذا لكن لما اطلع
+
+508
+00:35:23,020 --> 00:35:27,120
+فوق و بدي اكمل باقي المنطقة بصير اللي هو ال
+
+509
+00:35:27,120 --> 00:35:31,100
+vertical line ع منواه الاخرالان المنوال هذا بيسير
+
+510
+00:35:31,100 --> 00:35:35,420
+من جاي من عند Y اللي هي على الخط هذا اللي هي 4X-2
+
+511
+00:35:35,420 --> 00:35:39,160
+و بروح لل Y اللي هنا هي 2 جدر ال X إذا المنطقتين
+
+512
+00:35:39,160 --> 00:35:42,700
+غصب عني هذه المنطقة بدي أجزئها لمنطقتين مدام بدي
+
+513
+00:35:42,700 --> 00:35:46,220
+أكمل بالنسبة ل Y في الأول فباجي على خط الحد هذا و
+
+514
+00:35:46,220 --> 00:35:51,330
+بحط الحدماجي باخد الان vertical line في المنطقة
+
+515
+00:35:51,330 --> 00:35:54,690
+الأولى الـ R1 اللي هو بلاقيه في الـ R1 هذه المنطقة
+
+516
+00:35:54,690 --> 00:35:58,310
+R1 هي قصدي هي في الـ R1 هلاقيها كلها على نفس
+
+517
+00:35:58,310 --> 00:36:02,450
+المنوال بتيجي من عند Y بتساوي 0 و بتروح ل Y إيه
+
+518
+00:36:02,450 --> 00:36:07,210
+شمالها بساوي 2 جدر ال X إذا من 0 ل 2 جدر ال X الآن
+
+519
+00:36:07,510 --> 00:36:17,190
+الان المنطقة هذه بتنمسح من عند الـ 0 لعند من؟ لعند
+
+520
+00:36:17,190 --> 00:36:21,330
+النقطة هذه النقطة هذه جديش عبارة عن نص كيف وجدناها
+
+521
+00:36:21,330 --> 00:36:23,770
+نص كمان شويه بقولكم كيف وجدناها نص لعند النص إذا
+
+522
+00:36:23,770 --> 00:36:28,070
+من عند 0 لعند نص اللي هي حدود من ال X بكون خلصنا
+
+523
+00:36:28,070 --> 00:36:31,630
+المنطقة الأولى نيجي على من؟ المنطقة الثانية زائد
+
+524
+00:36:31,630 --> 00:36:34,710
+ال integration للمنطقة الثانية لأنه أخد مع المنطقة
+
+525
+00:36:34,710 --> 00:36:38,640
+الأولى لسه المنطقة الثانيةبتجي من عند كلها على نفس
+
+526
+00:36:38,640 --> 00:36:44,720
+المنوال من عند الخط اللي هي Y بتساوي 4X-2 4X-2
+
+527
+00:36:44,720 --> 00:36:49,160
+وبتروح على الخط اللي فجأة هو 2GX لأن هذه المنطقة
+
+528
+00:36:49,160 --> 00:36:53,320
+الخطوط الطولية هذه الموازية للـ Y Axis بتبدأ من
+
+529
+00:36:53,320 --> 00:36:58,060
+عند X بتساوي نصفوبتنتهي في آخر المنطقة لـ x أيش
+
+530
+00:36:58,060 --> 00:37:01,360
+بتساوي واحد فبصير حدول integration من نص لعند واحد
+
+531
+00:37:01,360 --> 00:37:06,940
+فبكون ال volume هذا اللي هو بطريقة إيجاد dy by dx
+
+532
+00:37:06,940 --> 00:37:10,220
+إذن اضطرينا لتجزئة المنطقة منطقتين عشان هيك قلنا
+
+533
+00:37:10,220 --> 00:37:13,360
+احنا المنطقة هذه في منطقة واحدة أسهلنا و أريحنا و
+
+534
+00:37:13,360 --> 00:37:16,820
+خلّينا نشتغل فيها لإن هذه هتغلبنا كمان بكرا لما
+
+535
+00:37:16,820 --> 00:37:20,340
+أنا كامل و ادعوض و مجدور و مجدور هتغلب انسى
+
+536
+00:37:22,920 --> 00:37:25,840
+طبعاً في الاخر كما عملنا قبل قليل sign ال X على X
+
+537
+00:37:25,840 --> 00:37:29,120
+يضطر لكي نكمل ال Y قبل ال X إذا كان هناك عقول و
+
+538
+00:37:29,120 --> 00:37:34,200
+نستخدمها لتلافي التجزئة لمنطقتين إلا إذا كان الأمر
+
+539
+00:37:34,200 --> 00:37:37,760
+في كل الحوار بالجزة أو لتلافي ال integration إذا
+
+540
+00:37:37,760 --> 00:37:40,550
+لم نكن نعرف كيف نكملهبنعمل reversing لل
+
+541
+00:37:40,550 --> 00:37:44,150
+integration الان هذا كيف وجدناها اللي هو عندكم
+
+542
+00:37:44,150 --> 00:37:48,890
+اللي هو اللي هو الخط y بسوء 4x نقص 2 ماشي الحال
+
+543
+00:37:48,890 --> 00:37:54,970
+الان هذا اللي هو خد اللي هو y بزفر لأن هذا فقط و
+
+544
+00:37:54,970 --> 00:38:00,350
+مع مين؟ مع ال zero يعني هذا على الخط هذا مظبوط خد
+
+545
+00:38:00,350 --> 00:38:06,090
+اللي هيY بصير على الخط هذا 4X
+
+546
+00:38:06,090 --> 00:38:11,670
+بيساوي 2 يعني X بيساوي 2 على 4 يعني نص الان هذه
+
+547
+00:38:11,670 --> 00:38:16,080
+النقطة هي عبارة عن تقاطع هذا الخط مع هذا الخطمع
+
+548
+00:38:16,080 --> 00:38:20,240
+هذا، كيف نجد التقاطع؟ كما عملنا قبل قليل Y بساوي 4
+
+549
+00:38:20,240 --> 00:38:25,880
+ناقص X Y بساوي 4 X minus 2 و Y بساوي 2 جدر ال X
+
+550
+00:38:25,880 --> 00:38:30,400
+بأخد 2 جدر ال X و بسويها بهذه و بأوجد الحلول مع
+
+551
+00:38:30,400 --> 00:38:34,380
+بعض الحلول بوجد قيمة X و بعدها بوجد Y بكون أوجد
+
+552
+00:38:34,380 --> 00:38:38,480
+نقاط التقاطع ستجدها 1 و 2 و ستجدها 0 و 0 إذن خطوط
+
+553
+00:38:38,480 --> 00:38:42,180
+نقاط التقاطع بتعرفوا كيف وجدوها من حل هذا الخط ال
+
+554
+00:38:42,180 --> 00:38:47,750
+curve مع هذا الخطبتطلع عندي اللي هي ضغاط التقاطع و
+
+555
+00:38:47,750 --> 00:38:50,350
+هيك بيكون خلصنا المحاضرة و برك الله فيكم و السلام
+
+556
+00:38:50,350 --> 00:38:52,730
+عليكم و رحمة الله و بركاته و إلى لقاء آخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/GtNEO-ZOqgE_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/GtNEO-ZOqgE_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..886a3eb5175f28a87392a6fdfe5b14f9ca9e4966
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/GtNEO-ZOqgE_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 1562, "start": 0.62, "end": 15.62, "text": "بسم الله الرحمن الرحيم السلام عليكم ورحمة الله وبركاته هذه هي المحاضرة رقم 21 تفاضل وتكامل جيم وهي المحاضرة الثانية في الشبطر الأخير في المادة اللي هو 15", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 29538, 39896, 9673, 5016, 46958, 25720, 12602, 4587, 2304, 5080, 6055, 5172, 46958, 1211, 34683, 4117, 10943, 1211, 10874, 32640, 37037, 1829, 9673, 5016, 46958, 25720, 6024, 104, 7649, 10632, 8978, 25124, 3555, 9566, 2288, 16247, 9778, 13546, 8978, 9673, 18513, 3660, 13672, 1829, 31439, 2119], "avg_logprob": -0.1359375006386212, "compression_ratio": 1.8211920529801324, "no_speech_prob": 9.238719940185547e-06, "words": [{"start": 0.62, "end": 0.9, "word": "بسم", "probability": 0.870361328125}, {"start": 0.9, "end": 1.16, "word": " الله", "probability": 0.94775390625}, {"start": 1.16, "end": 1.62, 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السابقة بنجسم برضه بناخد أول حد و آخر حد في المنطقة وبناخد x0, x1, x2 إلى آخره أو بنجزء الموازي للـ x-axis وبالمثل بنجزء الموازي للـ y-axis للـ x-axis", "tokens": [1863, 36178, 27188, 4117, 25720, 21136, 16758, 28671, 44945, 7435, 38251, 4724, 2288, 11242, 3224, 44945, 47283, 3215, 5551, 12610, 11331, 3215, 4032, 19753, 34740, 11331, 3215, 8978, 9673, 1863, 9566, 28671, 46599, 1863, 47283, 3215, 2031, 15, 11, 2031, 16, 11, 2031, 17, 30731, 19753, 34740, 3224, 34051, 44945, 7435, 11622, 38207, 9673, 2407, 31377, 1829, 24976, 39184, 2031, 12, 24633, 4032, 3555, 45340, 12984, 1211, 44945, 7435, 11622, 38207, 9673, 2407, 31377, 1829, 24976, 39184, 288, 12, 24633, 24976, 39184, 2031, 12, 24633], "avg_logprob": -0.33648254912953046, "compression_ratio": 1.7272727272727273, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 74.47, "end": 75.11, "word": "نفس", "probability": 0.7374267578125}, {"start": 75.11, "end": 75.59, "word": " الفكرة", "probability": 0.9869791666666666}, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/GtNEO-ZOqgE_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/GtNEO-ZOqgE_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b4679389d1c8217f8b0660127e8c5a0b1742a9d7
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/GtNEO-ZOqgE_raw.srt
@@ -0,0 +1,2256 @@
+1
+00:00:00,620 --> 00:00:04,700
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,700 --> 00:00:10,940
+وبركاته هذه هي المحاضرة رقم 21 تفاضل وتكامل جيم
+
+3
+00:00:10,940 --> 00:00:14,840
+وهي المحاضرة الثانية في الشبطر الأخير في المادة
+
+4
+00:00:14,840 --> 00:00:20,870
+اللي هو 15سنبدأ اليوم في 15.2 Double integrals
+
+5
+00:00:20,870 --> 00:00:24,990
+over general regions اللي حكينا عنه المرة الماضية
+
+6
+00:00:24,990 --> 00:00:29,570
+Double integrals over rectangles وكانت تحديد حدود
+
+7
+00:00:29,570 --> 00:00:33,130
+ال integration في حينه سهلة جدا اللي هي عبارة عن
+
+8
+00:00:33,130 --> 00:00:39,650
+حدود ال rectangle من A لB لC لD وليس فيها يعني
+
+9
+00:00:39,650 --> 00:00:44,330
+اللي هو غلبة على طول بس تعود عنها هذه اللي هي
+
+10
+00:00:44,330 --> 00:00:48,310
+الحدودالان بدنا نجي الى double integral over
+
+11
+00:00:48,310 --> 00:00:51,890
+bounded non rectangular region يعني الان هذه ال
+
+12
+00:00:51,890 --> 00:00:56,070
+region عندى region arbitrary region اللي هي في
+
+13
+00:00:56,070 --> 00:00:59,770
+المنطقة المحدودة طبعا اللي هي هذه المنطقة الذركة
+
+14
+00:00:59,770 --> 00:01:02,990
+هي ال region اللي عليها بدنا نعمل integration وهي
+
+15
+00:01:02,990 --> 00:01:08,070
+طبعا بتكون جزء من domain الدالة المراد اللي هي عمل
+
+16
+00:01:08,070 --> 00:01:12,670
+عليها او اللي هو إيجاد ال double integral لها على
+
+17
+00:01:12,670 --> 00:01:18,370
+المنطقة المذكورةنفس الفكرة السابقة بنجسم برضه
+
+18
+00:01:18,370 --> 00:01:26,710
+بناخد أول حد و آخر حد في المنطقة وبناخد x0, x1, x2
+
+19
+00:01:26,710 --> 00:01:33,450
+إلى آخره أو بنجزء الموازي للـ x-axis وبالمثل بنجزء
+
+20
+00:01:33,450 --> 00:01:36,350
+الموازي للـ y-axis للـ x-axis
+
+21
+00:01:42,330 --> 00:01:48,770
+وهو الموازيق لهم Y0 Y1 Y2 لعند YN وطبعا الفترة
+
+22
+00:01:48,770 --> 00:01:52,670
+النموذجية اللي بنشتغل عليها زي ما كنا ليها XK-1
+
+23
+00:01:52,670 --> 00:01:57,670
+لعند XK على X وYK-1 لعند YK على Y فبتطلع عندي اللي
+
+24
+00:01:57,670 --> 00:02:03,090
+هو المستطيل هذا او اللي هو المربع هذا اللي هو طوله
+
+25
+00:02:03,090 --> 00:02:09,540
+Delta XK وطوله Delta YKالآن زي ما أخدنا جمل بناخد
+
+26
+00:02:09,540 --> 00:02:16,700
+نقطة عشوائية هنا وهي xk و yk فوبنجمع اللي هي على
+
+27
+00:02:16,700 --> 00:02:19,900
+كل المنطقة اللي عملناها اللي هي بتطلع عند ال
+
+28
+00:02:19,900 --> 00:02:24,480
+summation طبعا عارفين إيش بنجمع نجمع f of xk و yk
+
+29
+00:02:24,480 --> 00:02:30,220
+قيمة الدالة مضروبة في delta xk اللي هي المنطقة
+
+30
+00:02:30,220 --> 00:02:33,620
+اللي هي نموذجية هذه اللي هو المستطيل اللي عندنا
+
+31
+00:02:33,620 --> 00:02:38,220
+مساحتهفبطلع الـ summation هذا هو remand sum لو كان
+
+32
+00:02:38,220 --> 00:02:41,300
+الـ remand sum هذا limit as N as N goes to
+
+33
+00:02:41,300 --> 00:02:45,480
+infinity اللي هو exist بنسميه اللي هو الـ double
+
+34
+00:02:45,480 --> 00:02:50,060
+integral over the region R زي ما عملنا قبل بالظبط
+
+35
+00:02:50,060 --> 00:02:53,320
+الـ N as N goes to infinity طبعا كل هنا دولة بروح
+
+36
+00:02:53,320 --> 00:02:57,680
+بيسير هذه المنطقة كلها تقطة المنطقة المحدودة عندنا
+
+37
+00:02:58,060 --> 00:03:02,100
+الان هنا كاتب لنا norm لل بي بيروح للسفر norm لل
+
+38
+00:03:02,100 --> 00:03:07,240
+بي ال بي هنا هو عبارة عن الان باجي بقى بشوف كل
+
+39
+00:03:07,240 --> 00:03:11,600
+الفترات اللي زي هذه الان باجي باخد delta xk و
+
+40
+00:03:11,600 --> 00:03:16,260
+delta yk فيهم كلهم أكبر واحدة في كل الموجودات من
+
+41
+00:03:16,260 --> 00:03:20,160
+ال delta xk أو من ال delta yk هي اللي بسميها ايش
+
+42
+00:03:20,160 --> 00:03:23,540
+ال norm لل بي فالان ال norm لل بي لما يروح للسفر
+
+43
+00:03:23,540 --> 00:03:27,670
+يعني هذه بمعنى آخر لو كانت هي أكبر فترةأو أكبر وهو
+
+44
+00:03:27,670 --> 00:03:31,350
+طول الفترة Delta XK لما يذهب للسفر معناته كل اللي
+
+45
+00:03:31,350 --> 00:03:34,130
+أصغر منها هيذهب للسفر و نفس الشيء كل اللي أصغر
+
+46
+00:03:34,130 --> 00:03:38,010
+منها من هنا يذهب للسفرمعناته أنه حصلت عندما الـ
+
+47
+00:03:38,010 --> 00:03:40,830
+Normal بيذهب إلى Zero أصبحت هذه كما قلنا قبل هي
+
+48
+00:03:40,830 --> 00:03:44,210
+عبارة عن خطوط أو بمعنى آخر الـ N هتذهب إلى مالة
+
+49
+00:03:44,210 --> 00:03:49,990
+نهاية فالتي هي هذه تعبير آخر على بدل الـ N تذهب
+
+50
+00:03:49,990 --> 00:03:54,030
+إلى مالة نهاية الـ Normal بيذهب لمين؟ للصفر وهذا
+
+51
+00:03:54,030 --> 00:03:59,420
+اللي هو الـ Double Integral الآن في سيكشن هذااللي
+
+52
+00:03:59,420 --> 00:04:04,180
+هو الشطارة عنا ان كيف نحدد حدود ال integration R
+
+53
+00:04:04,180 --> 00:04:07,360
+زمان كانت ال rectangle بكل سهولة من A ل B و من C ل
+
+54
+00:04:07,360 --> 00:04:11,460
+D لأ الآن زي ما شايفين احنا لا نقدرش نقول من A ل B
+
+55
+00:04:11,460 --> 00:04:15,880
+ل C ل D لأن هذه محكومة اللي هي ب curve و اللي فوق
+
+56
+00:04:15,880 --> 00:04:19,580
+ب curve او اذا بدأنا بالجهة الثانية ب curve و
+
+57
+00:04:19,580 --> 00:04:22,020
+الجهة الثالثة ب curve فبعدين كيف نغطي هذه المنطقة
+
+58
+00:04:22,020 --> 00:04:30,650
+ككل الآن نشوف كيف نعملنأتي الآن للـ Volume نحكي عن
+
+59
+00:04:30,650 --> 00:04:32,610
+الـ Volumes اللي هو الحجم زي ما انتوا عارفين ده
+
+60
+00:04:32,610 --> 00:04:36,410
+كانت اللي هي اللي جابل بشوية هذه اللي هي F of X وY
+
+61
+00:04:36,410 --> 00:04:41,650
+وK ده كانت موجبة بيصير يمثل هذا اللي هو ال limit
+
+62
+00:04:41,650 --> 00:04:44,790
+لهذا ال summation لأنه هذا بيصير تقدير للحجم في
+
+63
+00:04:44,790 --> 00:04:48,150
+فقدان أو في زيادة حسب اللي هو ال partitions اللي
+
+64
+00:04:48,150 --> 00:04:51,970
+بناخده الآن لما نقنط روح لما ننهي و نذهب للـ Zero
+
+65
+00:04:51,970 --> 00:04:54,810
+بيصير بالظبط هذا حجم الجسم اللي هو الـ Double
+
+66
+00:04:54,810 --> 00:05:00,200
+Integralالان نأتي ونشوف نظرية فبونيز تيوريم
+
+67
+00:05:00,200 --> 00:05:03,000
+الـStronger Form التي كانت المرة الماضية عن
+
+68
+00:05:03,000 --> 00:05:07,320
+الـRectangles والان على الـCurves أو الـRegion
+
+69
+00:05:07,320 --> 00:05:11,520
+التي ليست ركتانجل لت F of XY be continuous on a
+
+70
+00:05:11,520 --> 00:05:15,300
+region R لأ لذلك المطلوب من النظرية المعطيتها أن F
+
+71
+00:05:15,300 --> 00:05:20,100
+of XY يكون continuous على R لأن if R is defined on
+
+72
+00:05:20,100 --> 00:05:26,380
+أو byX من A لعند B وY من G1 لعند G2 يعني الآن
+
+73
+00:05:26,380 --> 00:05:32,540
+المنطقة R محددة محددة بواسطة اللي هو Y ال curve
+
+74
+00:05:32,540 --> 00:05:36,920
+بروح من G1 of X ل G2 of X والـ X كلها بتروح من A
+
+75
+00:05:36,920 --> 00:05:41,360
+لعند B والـ G1 و G2 التنتين continuous إذن ال
+
+76
+00:05:41,360 --> 00:05:45,680
+double integration لل F of X و Y over R بتساوي
+
+77
+00:05:46,430 --> 00:05:50,110
+الانتجريشن الان dy بنكمل بالنسبة لمين لـ dy لأن y
+
+78
+00:05:50,110 --> 00:05:53,630
+ربطناها بمتغير ده ان هي اللي بنكملها في الأول dy و
+
+79
+00:05:53,630 --> 00:05:57,890
+هلشوف هذا الكلام ايش معناته بعد شوية dy اللي هي من
+
+80
+00:05:57,890 --> 00:06:02,870
+g1 of x ال y لعند g2 of x الآن بعدين بنكمل ل x من
+
+81
+00:06:02,870 --> 00:06:06,730
+وين من عند a لعند b فبطلع عندى اللي هان عبارة عن
+
+82
+00:06:06,730 --> 00:06:11,010
+اللي هو رقم في النهايةالأن إذا كاملة الـ R نفسها
+
+83
+00:06:11,010 --> 00:06:15,910
+هذه الـ R لكن معرفة عن طريق مين؟ عن طريق أن الـ L
+
+84
+00:06:15,910 --> 00:06:18,830
+ننكامل أول شي بالنسبة لـ X يعني الـ X محصورة بين
+
+85
+00:06:18,830 --> 00:06:23,950
+كيرفين H1 of Y وH2 of Y و الـ Y في هذه الحالة
+
+86
+00:06:23,950 --> 00:06:27,670
+محصورة بين الـ C و بين الـ D في هذه الحالة بيكون
+
+87
+00:06:27,670 --> 00:06:31,490
+الـ Double Integration F of XY DA بساوية الـ
+
+88
+00:06:31,490 --> 00:06:35,630
+Integration أول شي بنكامل بالنسبة لـ XH1 of Y H2
+
+89
+00:06:35,630 --> 00:06:41,130
+of Y C لعند الهو HD اللي هانشطرتنا انه احنا نجيب
+
+90
+00:06:41,130 --> 00:06:44,170
+اللي هي حدود ال integration بعد شوية كيف نستخدم في
+
+91
+00:06:44,170 --> 00:06:47,230
+بونيز فيرم فإننا نودد حدود ال integration سواء
+
+92
+00:06:47,230 --> 00:06:53,090
+نكامل DX DY أو نكامل DY DX وحنلاقي انه اللي هو
+
+93
+00:06:53,090 --> 00:06:57,190
+الجهتين بيعطي هذه او هذه بيعطي نفس القيمة اللي هو
+
+94
+00:06:57,190 --> 00:07:01,820
+الحجم المطلوب او ال integration اللي مطلوبهذا اللي
+
+95
+00:07:01,820 --> 00:07:06,400
+حكيته عنه قبل بشوية دلتا A K وهي F of XK وYK
+
+96
+00:07:06,400 --> 00:07:12,560
+الارتفاع حجمه هذا اللي هو حاصل ضرب دلتا A K في F
+
+97
+00:07:12,560 --> 00:07:17,180
+of XK لما نجمحن K من عند واحد لعند N وناخد N تروح
+
+98
+00:07:17,180 --> 00:07:20,000
+لما لنهاية بنكون اللي هو عمل integration للجسم
+
+99
+00:07:20,000 --> 00:07:23,400
+وجدنا الحق بيصير حجم جنب بعض جنب بعض جنب بعض
+
+100
+00:07:23,400 --> 00:07:28,400
+بتتكوّر لحجمين الجسم اللي أمامي نشوف الآن مثلًا
+
+101
+00:07:29,180 --> 00:07:33,300
+الأن بقول find the volume of the prism whose base
+
+102
+00:07:33,300 --> 00:07:36,740
+is the triangle in the xy plane bounded by the x
+
+103
+00:07:36,740 --> 00:07:41,240
+-axis and the lines y بيساوي x and x بيساوي واحد
+
+104
+00:07:41,240 --> 00:07:44,940
+and whose top lies in the plane اللي هو z بيساوي
+
+105
+00:07:44,940 --> 00:07:50,940
+تلاتة ناقص x ناقص y عندي الآن منشور قاعدته اللي هو
+
+106
+00:07:50,940 --> 00:07:54,700
+عبارة عن الشكل اللي أمامي هذا اللي هو إيش y بتساوي
+
+107
+00:07:54,700 --> 00:07:59,790
+x هي y بتساوي xأو X بتساوي واحد و X بتساوي واحد
+
+108
+00:07:59,790 --> 00:08:04,170
+هيها والـ X axis المنطقة المحصورة هذه هي عبارة عن
+
+109
+00:08:04,170 --> 00:08:08,590
+قاعدة المنشور لكن سقف المنشور وينه؟ سقف المنشور
+
+110
+00:08:08,590 --> 00:08:12,190
+طبعا هذا الآن في البلين لأن سقفه أعلى اللي هو
+
+111
+00:08:12,190 --> 00:08:16,650
+شماله Z بتساوي تلاتة ناقص X ناقص Y يعني سقفه عبارة
+
+112
+00:08:16,650 --> 00:08:21,760
+عن البلين هذا الآن بلزمنيش أرسمبالظبط، يجب أن أرسم
+
+113
+00:08:21,760 --> 00:08:26,400
+المنطقة التي هي عبارة عن قاعدة الجسم الذي أريد أن
+
+114
+00:08:26,400 --> 00:08:33,720
+أجد وهو من حجمه وأكون عارف بس الدالة التي هي ثقف
+
+115
+00:08:33,720 --> 00:08:42,070
+من ثقف المجسم الذي قاعدته هذابصير عندي اللي هو الـ
+
+116
+00:08:42,070 --> 00:08:46,210
+Volume الـ Volume الـ Volume اللي هو عبارة عن اللي
+
+117
+00:08:46,210 --> 00:08:49,430
+أنا بدأ فاضل الـ dy بيدي ال dx في الأول ماشي الحال
+
+118
+00:08:49,430 --> 00:08:52,330
+شوفوا عليها ال volume هو عبارة عن ال double
+
+119
+00:08:52,330 --> 00:08:56,770
+integration لأن في عند جثثتي اللي هي الحدود صبروا
+
+120
+00:08:56,770 --> 00:09:00,170
+عليها شوية V بيساوي ال double integral لمين للدالة
+
+121
+00:09:00,170 --> 00:09:04,180
+اللي بدأ فاضلها اللي هي ارتفاع المنشوراللي هو dy
+
+122
+00:09:04,180 --> 00:09:08,540
+dx اللي هي dy by dx مع حد اللي هتمثل roughly هذا
+
+123
+00:09:08,540 --> 00:09:12,400
+الكلام مساحة هذا فبصير المساحة في الارتفاع اللي هو
+
+124
+00:09:12,400 --> 00:09:15,940
+الحجم هذا الكلام roughly زي ما قلنا by Fubini's
+
+125
+00:09:15,940 --> 00:09:19,680
+theorem إذا نضع بال integration لدالة dy dx هو
+
+126
+00:09:19,680 --> 00:09:24,200
+هيطلع من ال volume أنا الآن شوفوا عليّ، بدأ فاضل
+
+127
+00:09:24,200 --> 00:09:29,980
+بالنسبة ل Y مدام بدأ فاضل أول إشي بالنسبة ل Yأيش
+
+128
+00:09:29,980 --> 00:09:34,720
+بيجيباج على المنطقة باخد
+
+129
+00:09:34,720 --> 00:09:39,500
+خطوط موازية لل Y axis في داخل المنطقة شايفين
+
+130
+00:09:39,500 --> 00:09:44,020
+المنطقة باخد خطوط موازية لل Y axis إذا لجيتها كلها
+
+131
+00:09:44,020 --> 00:09:49,240
+على نفس المنوال إذا لجيتها كلها على نفس المنوال
+
+132
+00:09:50,090 --> 00:09:53,010
+معنى نفس المنوال يعني جاية من مكان و رايحة لأ
+
+133
+00:09:53,010 --> 00:09:55,690
+للمكان نفسه جاية من المكان و رايحة للمكان نفسه
+
+134
+00:09:55,690 --> 00:10:01,410
+لاحظوا الخط الطولي الآن هذا داخل من وين المنطقة من
+
+135
+00:10:01,410 --> 00:10:05,770
+عند اللي هو Y بتساوي سفر هاي على ال X axis و طالع
+
+136
+00:10:05,770 --> 00:10:10,990
+وين طالع من الخط هذا الخط هذا ايش قيمة Y؟ Y بتساوي
+
+137
+00:10:10,990 --> 00:10:15,270
+X عشان هيك لو اجيت اخدت اي خط تاني جنبه اي خط تاني
+
+138
+00:10:15,270 --> 00:10:18,630
+جنبه هتلاقي انه بتطلع من السفر و تروح لمن؟ للخط Y
+
+139
+00:10:18,630 --> 00:10:24,800
+بتساوي Xفعلاً كل على نفس المنوالفيش داعي لأني أجزئ
+
+140
+00:10:24,800 --> 00:10:28,500
+المنطقة لأن كلهم في نفس المنوال وبجيكوا بعدين شو
+
+141
+00:10:28,500 --> 00:10:33,000
+معناه تجزئ المنطقة الآن إذا قيمة Y كاملة بالنسبة
+
+142
+00:10:33,000 --> 00:10:38,440
+لـ Y أول اشي أخدنا خط موازي للـ Y Axis في داخل
+
+143
+00:10:38,440 --> 00:10:41,940
+المنطقة من أين دخل و من أين خرج هي حدود ال
+
+144
+00:10:41,940 --> 00:10:45,960
+integration اتفجنا؟ طيب إذا صار عندي Y بتساوي سفر
+
+145
+00:10:45,960 --> 00:10:53,820
+وY لما تطلع تكون إيه شمالها؟ Y بتساوي Xهذا حد مين
+
+146
+00:10:53,820 --> 00:11:00,860
+حد اللي هي Y لأننا نجي لـ X تشوفوا كيف نجي لـ X X
+
+147
+00:11:00,860 --> 00:11:04,460
+بالداجة أشوف الخطوط هذه اللي في داخل المنطقة
+
+148
+00:11:04,460 --> 00:11:08,780
+الطولية من وين بدأت و من وين انتهت تلاحظوا الخط
+
+149
+00:11:08,780 --> 00:11:11,340
+الطولي الأول عند الـ Zero بعدين بيطلع بعدها بعدها
+
+150
+00:11:11,340 --> 00:11:15,620
+بعدها كل الخطوط الطولية لما نقصلها نتخيل إن عندك
+
+151
+00:11:15,620 --> 00:11:19,460
+هذا عبارة عن إش بينضيق بيمسح في المنطقة بيمشي هيك
+
+152
+00:11:19,460 --> 00:11:23,270
+و بيمشي هيك لما مسح المنطقة كلهاالأن لما يمسح
+
+153
+00:11:23,270 --> 00:11:27,650
+المنطقة ببدأ من عند X بـ 0 و بنتهيأ لعند X بـ 1
+
+154
+00:11:27,650 --> 00:11:32,110
+بنكون هيك احنا هذا الخط المتغير لما يتحرك من هنا
+
+155
+00:11:32,110 --> 00:11:36,750
+لهنا و أخدنا X اللي هي من 0 لعند 1 و مسحنا كل
+
+156
+00:11:36,750 --> 00:11:40,830
+المنطقة يعني حدود التكامل بالنسبة لـ Xهي من صفر
+
+157
+00:11:40,830 --> 00:11:44,570
+لعند مين لعند واحد لأنه بداية المنطقة بصفر وبتنتهي
+
+158
+00:11:44,570 --> 00:11:48,930
+بواحد اللي هي الخطوط الطولية هذه بتبدأ من عند Zero
+
+159
+00:11:48,930 --> 00:11:52,550
+وتنتهي لعند واحد ومن مساحة المنطقة كلها الآن بدنا
+
+160
+00:11:52,550 --> 00:11:55,130
+نبدأ الكامل التكامل سهل يعني ماعني نضيع وقت
+
+161
+00:11:55,130 --> 00:11:59,290
+التكامل صفر لعند واحد تكامل التلاتة بالنسبة ل Y
+
+162
+00:11:59,290 --> 00:12:04,730
+تلاتة Y وال X XY وY يصبح Y تربيع على 2 بنعوض عنهم
+
+163
+00:12:04,730 --> 00:12:08,450
+من وين لوين حدود ال integration من صفر لعند Xبنعوض
+
+164
+00:12:08,450 --> 00:12:16,090
+عن هذه من 0 لعند X بطلع عندي اللي هو مكان ال Y ب X
+
+165
+00:12:16,090 --> 00:12:19,430
+بيصير 3X ومكان ال Y ب X بيصير نقص X تربيعه ومكان
+
+166
+00:12:19,430 --> 00:12:23,110
+ال Y ب X بيصير نقص X تربيعه على 2 بنبسطها إيه
+
+167
+00:12:23,110 --> 00:12:26,650
+بتطلع ليه نقص X تربيعه نقص X تربيعه على 2 بتطلع 3X
+
+168
+00:12:26,650 --> 00:12:31,920
+نقص 3X تربية على 2هذا الان المقدار انا بدي اكمله
+
+169
+00:12:31,920 --> 00:12:34,400
+بالنسبة لمين لا اكثر انه خلصنا اللي هو ال
+
+170
+00:12:34,400 --> 00:12:37,460
+integration الأول نجينا ال integration التاني لان
+
+171
+00:12:37,460 --> 00:12:39,620
+احنا بنحكي عن double integral او iterated
+
+172
+00:12:39,620 --> 00:12:42,940
+integrals ورا بعض التانين الان من 0 ل عند واحد
+
+173
+00:12:42,940 --> 00:12:45,560
+نفاض الكامل بالنسبة لمين لا اكثر هذا كلام جديد
+
+174
+00:12:45,560 --> 00:12:48,780
+يعنيتلاتة X بيصير تلاتة X تربيع ع تنين نقص تلاتة X
+
+175
+00:12:48,780 --> 00:12:52,420
+تربيع بيصير نقص تلاتة X تكعيب على تلاتة في اتنين
+
+176
+00:12:52,420 --> 00:12:55,920
+يعني اللي هو X تكعيب على اتنين الان من و إلى وين
+
+177
+00:12:55,920 --> 00:12:58,560
+حدود ال integration عند سفر اللي عند واحد بنعوض
+
+178
+00:12:58,560 --> 00:13:02,200
+عنها اللي هي بواحد اول اشي اللي هي و السفر طبعا
+
+179
+00:13:02,200 --> 00:13:05,160
+هيطلع سفر بطلع تلاتة ع اتنين نقص نص اللي هو ايش
+
+180
+00:13:05,160 --> 00:13:08,840
+بطلع واحدواحد قال انا اسمعه انا مانا مانا مانا
+
+181
+00:13:08,840 --> 00:13:09,680
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+182
+00:13:09,680 --> 00:13:10,800
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+183
+00:13:10,800 --> 00:13:13,760
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+184
+00:13:13,760 --> 00:13:14,040
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+185
+00:13:14,040 --> 00:13:14,180
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+186
+00:13:14,180 --> 00:13:15,140
+مانا مانا مانا مانا مانا مانا مانا مانا مانا مانا
+
+187
+00:13:15,140 --> 00:13:16,420
+مانا مانا مانا مانا مانا مانا مانا مانا
+
+188
+00:13:35,840 --> 00:13:39,620
+الان هنا بتاخد موازي لمين للـ X axis في المنطقة
+
+189
+00:13:39,620 --> 00:13:42,900
+مادام بدأ أكامل بالنسبة للـ X الآن بدأ أشوف كل
+
+190
+00:13:42,900 --> 00:13:46,720
+الخطوط اللي هي العرضية هذه اللي بالشكل هذه كلها
+
+191
+00:13:46,720 --> 00:13:50,560
+على نفس المنوال جاملة عند هذا الخط و رايحة عند
+
+192
+00:13:50,560 --> 00:13:55,760
+الخط هذاالان شوفوا قيمة X علي هذه المنطقة هي هذه
+
+193
+00:13:55,760 --> 00:13:59,400
+الخطوط وهي الخط الأحمر النموذجي لها اللي هو بيجي
+
+194
+00:13:59,400 --> 00:14:03,820
+من X بتساوي Y لعند X ايش بتساوي واحد اذا حدود
+
+195
+00:14:03,820 --> 00:14:08,770
+التكامل ل X هي من عند Y لعند ايش واحدنأتي الآن
+
+196
+00:14:08,770 --> 00:14:13,630
+لقيمة مين بعد أن نكمل لـ X نكمل لـ Y يعني الآن كيف
+
+197
+00:14:13,630 --> 00:14:18,050
+نحدد Y في الخطوط العرضية هذه كلها كلها كلها كلها
+
+198
+00:14:18,050 --> 00:14:20,370
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+199
+00:14:20,370 --> 00:14:20,750
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+200
+00:14:20,750 --> 00:14:20,810
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+201
+00:14:20,810 --> 00:14:20,810
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+202
+00:14:20,810 --> 00:14:20,810
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+203
+00:14:20,810 --> 00:14:21,250
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+204
+00:14:21,250 --> 00:14:22,710
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+205
+00:14:22,710 --> 00:14:22,710
+كلها كلها كلها كلها كلها كلها كلها كلها كلها كلها
+
+206
+00:14:22,710 --> 00:14:28,310
+كلها كلها كلها كلها
+
+207
+00:14:28,310 --> 00:14:32,160
+كلها كلهااللي هو Y بالساوية سفر وبتنتهي لعند Y
+
+208
+00:14:32,160 --> 00:14:35,780
+بالساوية أحد بيكون اللي هو الضوء اللي أخدنا لو
+
+209
+00:14:35,780 --> 00:14:38,420
+ماشينا من هنا ومضلنا نمشي فيه نمشي فيه حسب اللي هو
+
+210
+00:14:38,420 --> 00:14:43,000
+هان هيكون مسح المنطقة هذه كلها ففعلا بيمسحها من
+
+211
+00:14:43,000 --> 00:14:46,340
+عند Y بالساوية سفر لعند Y بالساوية أحد عشان هيك
+
+212
+00:14:46,530 --> 00:14:49,190
+اللي هو حدود الـ integration من عند 0 لعند 1 و
+
+213
+00:14:49,190 --> 00:14:51,990
+بنكاملها دي زمان كمان اللي قبل شوية اللي هي
+
+214
+00:14:51,990 --> 00:14:55,670
+بالنسبة ل X بيصير ثالثة X نقص X تربية على 2 نقص XY
+
+215
+00:14:55,670 --> 00:15:02,980
+من Y لعند 1 و بنعوض عند X بتساوي 1X بالساوية واحد
+
+216
+00:15:02,980 --> 00:15:07,560
+بيصير تلاتة ناقص نص ناقص Y ناقص اللي هو عند X
+
+217
+00:15:07,560 --> 00:15:10,420
+بالساوية Y بيصير تلاتة Y ناقص Y تربيع على اتنين
+
+218
+00:15:10,420 --> 00:15:14,780
+ناقص Y تربيع اللي هي هاي هادى مع هادى بنختصرها
+
+219
+00:15:14,780 --> 00:15:17,880
+بتطلع عندى اللي هو المقدار هذا خمس على اتنين ناقص
+
+220
+00:15:17,880 --> 00:15:21,700
+كده كده الاخرى هادى نكملها بالنسبة ل 100 بالنسبة ل
+
+221
+00:15:21,700 --> 00:15:27,220
+Y وأكيد كلكم بعرفي كامل بالنسبة ل Y بكل سهولة اللي
+
+222
+00:15:27,220 --> 00:15:31,940
+كملناها بيصير خمس على اتنين Yنقص 4Y³ على 2 يعني
+
+223
+00:15:31,940 --> 00:15:37,060
+نقص 2Y³ التي هي زائد تصبح ثلاثة Y تكعيب على ثلاثة
+
+224
+00:15:37,060 --> 00:15:40,440
+في اتنين يعني Y تكعيب على ثلاثة على اتنين من سفر
+
+225
+00:15:40,440 --> 00:15:43,780
+لعند واحد بنعوض بنلاقي ان القيمة فعلا طلعت عندنا
+
+226
+00:15:43,780 --> 00:15:49,040
+ايش بتساوي بتساوي اللي هي عبارة عن اللي هي واحد
+
+227
+00:15:49,040 --> 00:15:53,960
+نفس القيمة الأولى اجربها لحالك الحسابة تدل تحت انا
+
+228
+00:15:53,960 --> 00:15:57,160
+حسبت كمالت لكن تبقى الحسابات بسيطة تعملها
+
+229
+00:15:59,560 --> 00:16:04,440
+سنذهب الآن إلى مثال تاني مثال تاني يقول أنك ليس
+
+230
+00:16:04,440 --> 00:16:08,880
+دائما حظك أن تذهب وتكمل بالنسبة ل X بالنسبة ل Y أو
+
+231
+00:16:08,880 --> 00:16:11,960
+تكمل بالنسبة ل Y بالنسبة ل X أحيانا بحكمك الـ
+
+232
+00:16:11,960 --> 00:16:15,940
+Integrant اللي هو الكمية اللي بدك تكملهااللي هي
+
+233
+00:16:15,940 --> 00:16:18,880
+مثلا عندنا hand sign X على X بقول لي calculate ال
+
+234
+00:16:18,880 --> 00:16:22,980
+integration هذا over mean over R R إيش هي R اللي
+
+235
+00:16:22,980 --> 00:16:27,320
+ماخدها بقول لي R where R is the triangle in the X
+
+236
+00:16:27,320 --> 00:16:30,440
+Y بلين عبارة عن triangle in the X Y بلين bounded
+
+237
+00:16:30,440 --> 00:16:33,920
+by the X axis زي اللي توا بالظبط ب X axis و ال
+
+238
+00:16:33,920 --> 00:16:38,720
+line Y بيساوي X و الخط X بتساوي 1 بقول لي كملي
+
+239
+00:16:38,720 --> 00:16:43,020
+هادي على المنطقة هذهالان الان واحد قال اللي بدأ
+
+240
+00:16:43,020 --> 00:16:46,840
+كامل dx dy اول شي بدأ كامل بالنسبة ل x لما بدك
+
+241
+00:16:46,840 --> 00:16:49,740
+تكامل بالنسبة ل x في الأول لما تيجي تكامل sin
+
+242
+00:16:49,740 --> 00:16:52,620
+الجبل حتى ما نحط اللي هو حدد حدود ال integration
+
+243
+00:16:52,620 --> 00:16:56,560
+حدود ال integration بنحددها مافي مشكلة لكن الان لو
+
+244
+00:16:56,560 --> 00:17:00,840
+بدك تحدد هنا تيجي تكامل sin x على x بالنسبة ل x
+
+245
+00:17:00,840 --> 00:17:04,440
+بتجف منعرفش نكملها by elementary اللي هي
+
+246
+00:17:04,440 --> 00:17:09,400
+integrationيعني ايه بدك اللي هو الان مابتقدرش
+
+247
+00:17:09,400 --> 00:17:12,820
+اتكاملها بالشكل هذا اذا فيه بيصير تحايل ان انا
+
+248
+00:17:12,820 --> 00:17:16,900
+اكمل dy بdx هذا عشان نعرف جداش في وبونيز theorem
+
+249
+00:17:16,900 --> 00:17:19,900
+مهمة يعني ال double integration اللي احنا مش هنعرف
+
+250
+00:17:19,900 --> 00:17:23,380
+انه نكامله بالنسبة ل x بقول سهلا هو اكيد التكامل
+
+251
+00:17:23,380 --> 00:17:26,760
+حسب في وبونيز theorem هيساوي اللي هو التكامل نفسه
+
+252
+00:17:26,760 --> 00:17:31,460
+لما نكامل بالنسبة ل x ل y او ل y ل x عشان هيك بروح
+
+253
+00:17:31,460 --> 00:17:36,020
+بشرد وبكامل بالنسبة لمين ل y عشان هيكV بدّى كاملها
+
+254
+00:17:36,020 --> 00:17:38,880
+هو الاشي بالنسبة لـY وبعدين بالنسبة لـX لـSin X
+
+255
+00:17:38,880 --> 00:17:41,800
+على X الآن بدّى أحدد حدود ال integration حدود ال
+
+256
+00:17:41,800 --> 00:17:44,960
+integration زى ما عملنا قبله بشوية بالظبط مدام لـY
+
+257
+00:17:44,960 --> 00:17:49,540
+إذا بدي أخد موازي لـY Axis الموازي لـY Axis عملنا
+
+258
+00:17:49,540 --> 00:17:52,930
+قبله شويةالخط الموازية الـ y axis اللي هو بيبدأ من
+
+259
+00:17:52,930 --> 00:17:57,050
+عند y بتساوي سفر و بنتهي للخط y بتساوي x فبتكون
+
+260
+00:17:57,050 --> 00:18:00,450
+حدود ال integration لل y من سفر لعند x عملتها قبل
+
+261
+00:18:00,450 --> 00:18:03,930
+هيك لأن كل اللي هي الموازية لل y axis هذه لو
+
+262
+00:18:03,930 --> 00:18:07,390
+مسحناها هنلاقيها بتبدأ عشان تمسح المنطقة كلها
+
+263
+00:18:07,390 --> 00:18:12,690
+بتبدأ من سفر لعند مين لعند الواحد فبكون هو هذا ال
+
+264
+00:18:12,690 --> 00:18:13,530
+volume
+
+265
+00:18:15,760 --> 00:18:19,860
+الان كامل sin x على x بالنسبة ل y صار ثابت هذا
+
+266
+00:18:19,860 --> 00:18:25,120
+المقدار فبصير تفاض و sin x على x في y و بده كامله
+
+267
+00:18:25,120 --> 00:18:30,420
+من أين؟ بده حدود التكامل من 0 ل x فبعوض من 0 ل x
+
+268
+00:18:30,420 --> 00:18:36,140
+طبعا الفعلة y ب0 تطلع حد 0 إذن بتظهر مين حالة x
+
+269
+00:18:36,140 --> 00:18:40,590
+sin x على x في xبتروح الـ x مع الـ x بتظهر صين الـ
+
+270
+00:18:40,590 --> 00:18:44,010
+x من صفر ل واحد و جيت لا تكامل يا محلا و كلنا
+
+271
+00:18:44,010 --> 00:18:47,390
+بنعرف ان كامله اللي هو بساوي اللي هو ناقص cosine
+
+272
+00:18:47,390 --> 00:18:52,570
+الـ x من صفر ل عند واحد اللي هي عبارة عن اللي عند
+
+273
+00:18:52,570 --> 00:18:55,250
+واحد اللي هي ناقص cosine واحد هيها و عند الصفر
+
+274
+00:18:55,250 --> 00:18:58,910
+ناقص ناقص بصير زائد cosine ال zero اللي هي عبارة
+
+275
+00:18:58,910 --> 00:19:02,980
+عن واحدإذن الـ integration بالنسبة لـ y قبل و
+
+276
+00:19:02,980 --> 00:19:06,820
+بعدين x هو الحل في هذه الحالة و زي ما قلنا when
+
+277
+00:19:06,820 --> 00:19:09,720
+the order of integration is reversed the integral
+
+278
+00:19:09,720 --> 00:19:14,740
+is hard يعني لما نأتي و ناخد دي x by دي y بنكون و
+
+279
+00:19:14,740 --> 00:19:18,560
+ده يعني مشكلةنجي الآن نلخص اللي حكيناه Finding
+
+280
+00:19:18,560 --> 00:19:21,460
+limits of integration هذا الكلام اللي حكيته قبله
+
+281
+00:19:21,460 --> 00:19:25,100
+شوية و احنا حكيناه تمام التمام الان عشان توجد اللي
+
+282
+00:19:25,100 --> 00:19:29,000
+هو limits of integration للمنطقة R أول حاجة فيها
+
+283
+00:19:29,000 --> 00:19:31,900
+اندي طريقتين يا ال cross-section يا ال vertical
+
+284
+00:19:31,900 --> 00:19:34,820
+-section ال cross-section يعني بده كامل بالنسبة
+
+285
+00:19:34,820 --> 00:19:40,120
+موازي لمن؟ ال cross-section اللي هو بده ياخد موازي
+
+286
+00:19:40,120 --> 00:19:45,410
+ال Y-axisاللي هو بده ياخد اسف using vertical cross
+
+287
+00:19:45,410 --> 00:19:48,890
+section اه vertical cross section الان هذا using
+
+288
+00:19:48,890 --> 00:19:51,770
+ال vertical مدام ال vertical يعني موازية ال y axis
+
+289
+00:19:51,770 --> 00:19:55,010
+يعني بده كامل اول اش بالنسبة لمن بالنسبة ل Y اذا
+
+290
+00:19:55,010 --> 00:19:58,610
+هذه الطريقة بالنسبة ل Yهذا الكلام شرحته بس خلّيني
+
+291
+00:19:58,610 --> 00:20:03,010
+أشرحه لكم كمان مرة لأنه فرضه لحاله هو شرحته من
+
+292
+00:20:03,010 --> 00:20:07,770
+خلال المثالين أول حاجة sketch the region and label
+
+293
+00:20:07,770 --> 00:20:13,130
+the bounding curves يعني الأول حاجة بدك تعملها في
+
+294
+00:20:13,130 --> 00:20:15,490
+تحديد limits of integration لازم المنطقة تكون
+
+295
+00:20:15,490 --> 00:20:19,350
+مرسومة هي المنطقة مرسومة فرضنا أن معطيني هذه
+
+296
+00:20:19,350 --> 00:20:24,730
+المنطقة هي ال region R اللي عليها بدك تعملهاأول
+
+297
+00:20:24,730 --> 00:20:29,030
+اشي برسم من المنطقة الرسم ليها X تربيع زي Y تربيع
+
+298
+00:20:29,030 --> 00:20:32,930
+بسوء واحد و X زي Y بسوء واحد هذا طبعا X تربيع زي Y
+
+299
+00:20:32,930 --> 00:20:39,170
+تربيع بسوء واحد جزء من من الدائرة هيها مع X زي Y
+
+300
+00:20:39,170 --> 00:20:43,690
+بسوء واحد هيها هذه المنطقة المحصورة بين الجهتين هي
+
+301
+00:20:43,690 --> 00:20:46,430
+لو انت كملت الدائرة وكملت الخطين هي المنطقة
+
+302
+00:20:46,430 --> 00:20:50,770
+المحصورةإذا هذه المنطقة R هي المنطقة اللي بديها هو
+
+303
+00:20:50,770 --> 00:20:55,830
+نوجد التكامل عليها إذا أول شيء عشان أوجد حدود
+
+304
+00:20:55,830 --> 00:20:59,790
+التكامل برسم المنطقة الان find the Y limit of
+
+305
+00:20:59,790 --> 00:21:05,330
+integration يعني الآن بتوجد حدود ال YImagine a
+
+306
+00:21:05,330 --> 00:21:09,790
+vertical line cutting R in the direction of
+
+307
+00:21:09,790 --> 00:21:14,330
+increasing Y نحن نأخد خط موازن منين للـ Y-axis
+
+308
+00:21:14,330 --> 00:21:19,810
+بتزايد يعني من أين يدخل المنطقة و من أين يخرج منها
+
+309
+00:21:19,810 --> 00:21:22,750
+هاي الـ Vertical-Y لاحظوا الـ Vertical-Line هاي و
+
+310
+00:21:22,750 --> 00:21:26,610
+كلهم على نفس المنوال كلهم بيجي من الخط لمينلأ
+
+311
+00:21:26,610 --> 00:21:31,210
+القوس هذا الان ببدأ من عند اللي هو قيمة y هنا إيش
+
+312
+00:21:31,210 --> 00:21:34,690
+عند x زي y بسعة واحد معادلة إتاة إذا قيمة y هنا
+
+313
+00:21:34,690 --> 00:21:38,050
+اللي هي واحد ناقص x إذا حد ال integration اللي تحت
+
+314
+00:21:38,050 --> 00:21:42,790
+واحد ناقص x الان من وين خرج من وين خرج اللي هو من
+
+315
+00:21:42,790 --> 00:21:45,830
+الدائرة هذه الدائرة هذه اللي هي x تربيع زي y تربيع
+
+316
+00:21:45,830 --> 00:21:50,730
+بسعة واحدفي المنطقة Y موجبة يعني Y تربيع بيساوي 1
+
+317
+00:21:50,730 --> 00:21:54,790
+ناقص X تربيع يعني Y بيساوي الجدر الموجب للواحد
+
+318
+00:21:54,790 --> 00:21:59,350
+ناقص X تربيع لأنه في المنطقة الموجبة فبنخرج من
+
+319
+00:21:59,350 --> 00:22:03,390
+الار وين عند Y بيساوي الجدر التربيع لواحد ناقص X
+
+320
+00:22:03,390 --> 00:22:08,100
+تربيع فبتكون هذا حدود مينحدود الـ Y نجي الآن نجي
+
+321
+00:22:08,100 --> 00:22:11,900
+نشوف هدول الخطوط ال vertical من وين لوين بمسحن ال
+
+322
+00:22:11,900 --> 00:22:16,820
+X بمسحن المنطقة بقيمة X بمسح المنطقة هي و هي و هي
+
+323
+00:22:16,820 --> 00:22:18,300
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+324
+00:22:18,300 --> 00:22:20,240
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+325
+00:22:20,240 --> 00:22:22,220
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+326
+00:22:22,220 --> 00:22:23,440
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+327
+00:22:23,440 --> 00:22:23,440
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+328
+00:22:23,440 --> 00:22:23,440
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+329
+00:22:23,440 --> 00:22:23,440
+و هي و هي و هي و هي و هي و هي و هي و هي و هي و هي
+
+330
+00:22:23,440 --> 00:22:31,860
+و هي و هي و هي و هي و هي و هي و هي و
+
+331
+00:22:31,860 --> 00:22:39,460
+هي و هيوشغّلنا العداد LX من صفر لعند واحد عشان هيك
+
+332
+00:22:39,460 --> 00:22:43,760
+حدود ال integration من صفر لعند واحدنأتي الآن لـ
+
+333
+00:22:43,760 --> 00:22:46,640
+Using Horizontal Cross Section زي ما حكيته برضه
+
+334
+00:22:46,640 --> 00:22:51,260
+قبل قليل شرحته اللي هو نفس الاشي باخد اللي هو اللي
+
+335
+00:22:51,260 --> 00:22:54,040
+في حالة اللي هو بتكامل بالنسبة لـ X قبل أو ال
+
+336
+00:22:54,040 --> 00:22:58,040
+cross section بتاخد موازية لـ X Axis موازية لـ X
+
+337
+00:22:58,040 --> 00:23:00,900
+Axis من وين دخل المنطقة و من وين غادرها من وين دخل
+
+338
+00:23:00,900 --> 00:23:04,040
+المنطقة قيمة X بالساوية واحد ناقص Y هذه الحد الأول
+
+339
+00:23:04,040 --> 00:23:07,000
+من وين خرج المنطقة عند X بالساوية جدر التربيع إلى
+
+340
+00:23:07,000 --> 00:23:10,060
+واحد ناقص Y تربيع عملنا زي هذا قبل قليل لأن أصل X
+
+341
+00:23:10,060 --> 00:23:18,540
+تربيع زي Y تربيع بساوية واحدكل الخطوط التي بدأت
+
+342
+00:23:18,540 --> 00:23:21,860
+هذه الخطوط هي الـ Horizontal اللي بتمسح المنطقة
+
+343
+00:23:21,860 --> 00:23:26,000
+بتفتغر المنطقة المنطقة بتبدأ من عند Y بيتساوي و أ
+
+344
+00:23:26,000 --> 00:23:30,620
+سفر لعند Y أش بتساوي واحد أخر أش في المنطقة بيصير
+
+345
+00:23:30,620 --> 00:23:36,180
+من سفر لواحد بيكون هذا هو قيمة ال integrationبناخد
+
+346
+00:23:36,180 --> 00:23:39,800
+مثال بيقولي sketch the region of integration for
+
+347
+00:23:39,800 --> 00:23:44,040
+the integral الان بيقولي اول اشي هاي اعطيتك
+
+348
+00:23:44,040 --> 00:23:48,920
+integration انت تشاطر و ارسملي ايش المنطقة هذا اول
+
+349
+00:23:48,920 --> 00:23:52,480
+حاجة وبعدين بيقولك and write an equivalent
+
+350
+00:23:52,480 --> 00:23:55,600
+integral with the order of integration reversed
+
+351
+00:23:55,600 --> 00:24:00,880
+وكمل integration اللي هو مكافئله بس ايش معكوس يعني
+
+352
+00:24:00,880 --> 00:24:07,070
+dx by dy قولنا له زينأول حاجة لما بادي بطلع بدا
+
+353
+00:24:07,070 --> 00:24:10,590
+كامل بالنسبة ل Y إذا مدان بدا كامل بالنسبة ل Y في
+
+354
+00:24:10,590 --> 00:24:15,870
+الأول ما هو هنا إذا هذه حدود Y يعني Y في الأول X
+
+355
+00:24:15,870 --> 00:24:20,630
+تربيع و Y في الآخر 2X يعني Y بتساوي X تربيع بادي
+
+356
+00:24:20,630 --> 00:24:24,390
+برسم اللي هي الرسمة Y بتساوي X تربيع حيث Y بتساوي
+
+357
+00:24:24,390 --> 00:24:28,990
+X تربيع تمشي هك و تمشي هك ماشي الآن اللي عند Y
+
+358
+00:24:28,990 --> 00:24:33,490
+بتساوي 2X Y بتساوي 2X بتعرف ترسمه Y بتساوي 2Xبتاخد
+
+359
+00:24:33,490 --> 00:24:38,910
+X بصفر هيها لأنها خط و X بصفر واحد او اتنين بتطلع
+
+360
+00:24:38,910 --> 00:24:43,530
+هيها بتجي بترسم الخط هذا على استقامة و بتمده طبعا
+
+361
+00:24:43,530 --> 00:24:47,290
+واضح ان المنطقة المحصورة بين الاتنين بس هذه ماشي
+
+362
+00:24:47,290 --> 00:24:51,810
+الان هذه المنطقة اذا هي هذه المنطقة المذكورةالان
+
+363
+00:24:51,810 --> 00:24:57,770
+انا عشان بالزمن اللي هو خطوط التقاطع اللي هي تقاطع
+
+364
+00:24:57,770 --> 00:25:01,430
+الخطين خطوة ال curve و تقاطع الخطوة ال curve بدي
+
+365
+00:25:01,430 --> 00:25:04,590
+بحل هدولة مع بعض التنتين y بتساوي x تربيع و y
+
+366
+00:25:04,590 --> 00:25:08,770
+بتساوي 2x بقول 2x بتساوي x تربيع إذا x تربيع ناقص
+
+367
+00:25:08,770 --> 00:25:13,670
+2x بتساوي 0 يعني x اللي هو في x minus 2 بتساوي 0
+
+368
+00:25:13,670 --> 00:25:18,090
+يعني x بتساوي 0هي X بتساوى 2 في حالة X بتساوى 0
+
+369
+00:25:18,090 --> 00:25:21,290
+تطلع Y بتساوى 0 هي النقطة الأولى وفي حالة X بتساوى
+
+370
+00:25:21,290 --> 00:25:25,510
+2 اللى هو اكيد ال Y لو عوضت هنا ال X أو هنا بتطلع
+
+371
+00:25:25,510 --> 00:25:30,410
+4 أربعة فتطلع ال Y أربعة يعني نقطتي التقاطع اللى
+
+372
+00:25:30,410 --> 00:25:34,590
+عبارة عن 2 و 4 و 0 و 0 إذن هي المنطقة إيجينا
+
+373
+00:25:34,590 --> 00:25:41,440
+حصيناهاأحصى انتهاما الان اغير حدود ال integration
+
+374
+00:25:41,440 --> 00:25:48,100
+دي اكس دي واي او ال reverse ال integration دي اكس
+
+375
+00:25:48,100 --> 00:25:51,740
+دي واي مادام دي اكس دي واي اذا بدي اشوف قيمة X كيف
+
+376
+00:25:51,740 --> 00:25:58,020
+باخد X موازي في المنطقة بدخل موازي من ال X axisكل
+
+377
+00:25:58,020 --> 00:26:01,940
+الموازيات للـ x-axis بتيجي من عند الخط هذا و
+
+378
+00:26:01,940 --> 00:26:05,560
+بتنتهي لعند ال curve إذا كلها على نفس المنوال إذا
+
+379
+00:26:05,560 --> 00:26:09,980
+أنا مرتاح integration واحد و بدأ أكمل الآن قيمة x
+
+380
+00:26:09,980 --> 00:26:13,620
+هنا احنا y بيساوي 2x إذا x أش تكون y على 2 إذا
+
+381
+00:26:13,620 --> 00:26:18,740
+حدود ال integration y على 2 إلىالان قيمة X هنا Y
+
+382
+00:26:18,740 --> 00:26:22,420
+بيستوي X تربيع و المنطقة هذه لـ X موجبة إذا X
+
+383
+00:26:22,420 --> 00:26:26,920
+بيستوي زرع من مقص جدر Y لأ هي جدر Y لأن المنطقة مش
+
+384
+00:26:26,920 --> 00:26:32,040
+موجبة إذا اللي هي دخلت X من عند Y على 2 و انتهت
+
+385
+00:26:32,040 --> 00:26:35,540
+لعين جدر Y نيجي الآن الخطوط هذه اللي هي ال cross
+
+386
+00:26:35,540 --> 00:26:39,180
+sections ال horizontals كلهم بيبدأ من وين لوين
+
+387
+00:26:39,180 --> 00:26:44,100
+بيبدأ من عند السفراللي هي قيمة Y من عند السفر وبظل
+
+388
+00:26:44,100 --> 00:26:48,880
+بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع
+
+389
+00:26:48,880 --> 00:26:51,900
+.. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع ..
+
+390
+00:26:51,900 --> 00:26:52,520
+بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع
+
+391
+00:26:52,520 --> 00:26:52,580
+.. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع ..
+
+392
+00:26:52,580 --> 00:26:52,580
+بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع
+
+393
+00:26:52,580 --> 00:26:52,580
+.. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع ..
+
+394
+00:26:52,580 --> 00:26:55,920
+بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع .. بتطلع
+
+395
+00:26:55,920 --> 00:26:58,800
+.. بتطلع .. بتطلع .. بتطل
+
+396
+00:27:01,950 --> 00:27:05,450
+بتصور هيك الأمور ان شاء الله تكون انفهمت منيح في
+
+397
+00:27:05,450 --> 00:27:09,390
+موضوع إيجاد حدود ال integration الان بعض خواص
+
+398
+00:27:09,390 --> 00:27:12,870
+properties of double integrals بعض خواص ال double
+
+399
+00:27:12,870 --> 00:27:15,770
+integrals هي مشابه للخواص اللي هي ال integration
+
+400
+00:27:15,770 --> 00:27:19,950
+العادى لو كانت f of x y و d of x y are continuous
+
+401
+00:27:19,950 --> 00:27:22,750
+and abound in the region R then the following
+
+402
+00:27:22,750 --> 00:27:25,940
+properties holds أول حاجة constant multiplesالـ
+
+403
+00:27:25,940 --> 00:27:28,860
+Double Integral C F of X, Y, D, A يظهر الـ
+
+404
+00:27:28,860 --> 00:27:32,120
+Constant برة مثل الـ Integration العادى العادي for
+
+405
+00:27:32,120 --> 00:27:36,780
+any number C Some Undifference التي هي تكامل مجموع
+
+406
+00:27:36,780 --> 00:27:40,580
+دالتين الـ Double Integration طبعا لمجموع دالتين
+
+407
+00:27:40,580 --> 00:27:44,900
+أو حاصل طرحين بسوء مجموع أو حاصل طرح التكاملين زي
+
+408
+00:27:44,900 --> 00:27:48,260
+ما كنا ناخد في التكامل العادى Domination لو كان
+
+409
+00:27:48,260 --> 00:27:53,220
+الـ Integration F of X, Y, D, A هو أكبر أو يساوي
+
+410
+00:27:53,220 --> 00:27:57,910
+سفرأسف، لو كان الـ F of XY أكبر أو يساوي سفر على
+
+411
+00:27:57,910 --> 00:28:01,910
+الـ R أكيد الـ Integration تبعها أكبر أو يساوي سفر
+
+412
+00:28:01,910 --> 00:28:04,490
+لأنه أصلا بيصير Volume، زي ما قلت قبل شوية هذا
+
+413
+00:28:04,490 --> 00:28:07,810
+بيصير الارتفاع، اللي هي عبارة عن مساحة، بيصير
+
+414
+00:28:07,810 --> 00:28:10,230
+المساحة في الارتفاع بيطلع Volume فعلا، إن أما يكون
+
+415
+00:28:10,230 --> 00:28:12,390
+F of XY أكبر أو يساوي سفر، بيكون الـ Double
+
+416
+00:28:12,390 --> 00:28:17,640
+Integral أكبر أو يساوي سفر، ومن ثم in generalلو
+
+417
+00:28:17,640 --> 00:28:22,020
+كان عندي Fxy أكبر أو يساوي Dxy على المنطقة R اللي
+
+418
+00:28:22,020 --> 00:28:24,560
+بنحكي عليها بيكون الـ Double Integration للأولى
+
+419
+00:28:24,560 --> 00:28:28,280
+أكبر أو يساوي الـ Double Integration للثانية إلى
+
+420
+00:28:28,280 --> 00:28:31,900
+الأربعة اللي هي الـ Additivity اللي هي Double
+
+421
+00:28:31,900 --> 00:28:36,730
+Integra لـ Fxy DAالأن لو الـ R مجسم لمنطقتين
+
+422
+00:28:36,730 --> 00:28:40,870
+disjoint يعني R بتساوي R واحد تقاطع اتحاد R اتنين
+
+423
+00:28:40,870 --> 00:28:44,850
+حيث R واحد تقاطع R اتنين اشمالين اللي هي non
+
+424
+00:28:44,850 --> 00:28:50,170
+overlapping يعني مش متداخلة مش متقاطعات اللي هو
+
+425
+00:28:50,170 --> 00:28:54,500
+تقاطحين اللي هو Phi بيصير عند ال double integralما
+
+426
+00:28:54,500 --> 00:28:59,040
+هو التكامل على المنطقة الأولى زائد التكامل على
+
+427
+00:28:59,040 --> 00:29:03,280
+المنطقة التانية زي زمان كانت من A ل B و كانت C بين
+
+428
+00:29:03,280 --> 00:29:08,040
+ال A و ال B و كنا نقول من A ل C زائد من C ل B إذا
+
+429
+00:29:08,040 --> 00:29:14,280
+هو مشابه للقديم نأخد مثال و نقول find the volume
+
+430
+00:29:14,280 --> 00:29:18,500
+of the wood you likeSolid that lies beneath the
+
+431
+00:29:18,500 --> 00:29:22,180
+surface Z بالساوية 16 نقص X تربيع نقص Y تربيع and
+
+432
+00:29:22,180 --> 00:29:24,800
+above the region R bounded by the curve Y بيساوية
+
+433
+00:29:24,800 --> 00:29:29,180
+2 جدرى X and the line Y بيساوية 4X نقص 2 and the X
+
+434
+00:29:29,180 --> 00:29:36,060
+axis الكلام واضح بقول او جدلي حجم الجسم اللي اللي
+
+435
+00:29:36,060 --> 00:29:41,400
+هو الجزء العلوي له اللي هو عبارة عن إيش ال surface
+
+436
+00:29:41,400 --> 00:29:46,770
+Z بالساوية 16 نقص X تربيع نقص Y تربيعوقاعته مين هي
+
+437
+00:29:46,770 --> 00:29:52,310
+المنطقة المحصورة بين Y بساوة 2 جدر ال X هاي Y
+
+438
+00:29:52,310 --> 00:29:59,210
+بساوة 2 جدر ال X هاي الكرفة ده و ال line Y بساوة 4
+
+439
+00:29:59,210 --> 00:30:05,810
+X نقص 2 و ال line 4 X minus 2 و X أكسيزي محدد لك
+
+440
+00:30:05,810 --> 00:30:11,800
+يا تماما بقولي أوجد حجم الجسم الجسم هذا الليقاعدته
+
+441
+00:30:11,800 --> 00:30:16,100
+هيهة والتي هي سقفه من فوق الدالة الهادئة ماشي
+
+442
+00:30:16,100 --> 00:30:19,760
+الحال اذا اكيد بدنا نكامل هذه الدالة على المنطقة
+
+443
+00:30:19,760 --> 00:30:23,040
+هذه بطلع ال volume ال volume is V بالساوية ال
+
+444
+00:30:23,040 --> 00:30:29,160
+double integral الان DX DY لأ الدالة هذه 16 نقص X
+
+445
+00:30:29,160 --> 00:30:33,860
+تربيع نقص Y تربيع اللي هو الارتفاع DX DY وبدنا
+
+446
+00:30:33,860 --> 00:30:38,660
+نحدد منطقة التكامل اللي هي حدود التكامللاحظ الآن
+
+447
+00:30:38,660 --> 00:30:42,420
+هو كامل بالنسبة لـ X أول شيء و بعدين بالنسبة لمين؟
+
+448
+00:30:42,420 --> 00:30:45,880
+لـ Y طبعا هو قال لك أنا بدا كامل بالـ X قبله و
+
+449
+00:30:45,880 --> 00:30:50,260
+بعدين لـ Y لأنه أسهل ليه؟ و خلّينا نشوف ليش أسهله
+
+450
+00:30:50,260 --> 00:30:53,640
+مادان بدا كامل لـ X في الأول إذا بتاخد إيه شماله؟
+
+451
+00:30:53,640 --> 00:30:56,780
+Horizontal Line أو موازي لمين؟ لل X Axis هي
+
+452
+00:30:56,780 --> 00:31:00,020
+المنطقة هذه، لو جيت أخدت أي موازي لل X Axis؟ هيك،
+
+453
+00:31:00,020 --> 00:31:04,470
+هيك، هيك، هيك،ستجدهم جميعًا على نفس المنوال ما هو
+
+454
+00:31:04,470 --> 00:31:08,690
+نفس المنوال؟ يعني يدخل المنطقة من هذا الـ Curve
+
+455
+00:31:08,690 --> 00:31:13,070
+ويخرج من مين؟ من هذه الخطة طبعًا هذه المنطقة هي
+
+456
+00:31:13,070 --> 00:31:17,650
+هذه اللي توّى اللي هو الخط الأول Y بساوية 2 جذر
+
+457
+00:31:17,650 --> 00:31:22,830
+الـ X اللي هو الـ Curve جذده والخط الثاني يسمى Y
+
+458
+00:31:22,830 --> 00:31:26,990
+بساوية 4 نقص X نقص 2واللي هو الـ X axis هذه
+
+459
+00:31:26,990 --> 00:31:30,050
+المنطقة هي المنطقة اللي عليها التكوين للـ R هنا بس
+
+460
+00:31:30,050 --> 00:31:34,650
+نجلنا لبطة وضيح طيب الان أخدنا اللي هو ال
+
+461
+00:31:34,650 --> 00:31:38,170
+horizontal line من وين دخل هيه دخل من عند هذا جداش
+
+462
+00:31:38,170 --> 00:31:42,710
+قيمة X هنا X بتساوي من هنا بنربع الجهتين بيصير Y
+
+463
+00:31:42,710 --> 00:31:46,510
+تربيع بيساوي 2 اللي هو بيصير 4X بيصير X بيساوي Y
+
+464
+00:31:46,510 --> 00:31:50,450
+تربيع على 4 إذا قيمة X هنا اللي هي Y تربيع على 4
+
+465
+00:31:50,450 --> 00:31:53,450
+ما تخافوش من التربيع لأن قيمة X موجبة وحضّلها إيش
+
+466
+00:31:53,450 --> 00:31:58,390
+موجبةالان X بساوي صار لحد السفل يكون Y تربيع على 4
+
+467
+00:31:58,390 --> 00:32:02,430
+و لعند من يخرج من هنا؟ يخرج من هنا، ما هي قيمة X؟
+
+468
+00:32:02,430 --> 00:32:06,490
+عند I بساوية 4X minus 2 جيب Y هنا بيصير 2 هنا
+
+469
+00:32:06,490 --> 00:32:10,790
+بيصير 4X بساوية Y زائد 2 يعني X بساوية Y زائد 2
+
+470
+00:32:10,790 --> 00:32:15,050
+على 4 يعني قيمة X هنا مالها؟ Y زائد 2 على 4 في حد
+
+471
+00:32:15,050 --> 00:32:19,000
+ال integration لل X التاني Y زائد 2على أربعة نأتي
+
+472
+00:32:19,000 --> 00:32:22,260
+الآن كل الخطوط هذه الـ Horizontal لو جينا مسحنا
+
+473
+00:32:22,260 --> 00:32:26,260
+مسحنا مسحنا ستبدأ من عند ال Y بـ 0 وتنتهي إلى ال Y
+
+474
+00:32:26,260 --> 00:32:29,740
+أيش بتساوي 2 عشان هنا حدود ال integration من 0
+
+475
+00:32:29,740 --> 00:32:34,360
+لعند 2 وبنأتي من كامل هذه الكملة بغلب شوية يعني
+
+476
+00:32:34,360 --> 00:32:38,580
+خلّيني أقول مش بغلب هو التكامل سهل بس بده حسابات
+
+477
+00:32:38,580 --> 00:32:41,400
+طويلة خلّيني أعلى السريع لأن الفكرة في أن أوجد
+
+478
+00:32:41,400 --> 00:32:43,800
+حدود ال integration والتكامل اللي أنتوا بتعملوه
+
+479
+00:32:43,800 --> 00:32:48,590
+لحالكم هذه بيصير 16X ناقص X تكعيب على 3نقص y تربيع
+
+480
+00:32:48,590 --> 00:32:52,010
+في x من أين لو ينمن y تربيع على أربعة على y زائد
+
+481
+00:32:52,010 --> 00:32:55,330
+اتنين على اياش على أربعة الآن بدنا نعوض اللي هو
+
+482
+00:32:55,330 --> 00:33:00,710
+أول اشي x باللي فوق ونقص ال x باللي تحت الان x
+
+483
+00:33:00,710 --> 00:33:04,610
+باللي فوق هيبصير عوضنا على y زائد اتنين على أربعة
+
+484
+00:33:04,610 --> 00:33:11,060
+ف16 تطلع 4 في y زايد اتنينناقص Y2 كل تكعيب على
+
+485
+00:33:11,060 --> 00:33:16,380
+تلاتة في 64 ناقص
+
+486
+00:33:16,380 --> 00:33:21,920
+X في Y تربيع X هو Y زي 2 على 4 في Y تربيع ناقص Y
+
+487
+00:33:21,920 --> 00:33:27,470
+تربيع على 4بصير المكان الـ XY تربيع على 4 بصير 4Y
+
+488
+00:33:27,470 --> 00:33:32,970
+تربيع اللي قاعد ناقص، ناقص بصير زائد اللي هي Y أُس
+
+489
+00:33:32,970 --> 00:33:37,070
+اللي هي تربيع المكان الـ X بصير Y أُس 6 على 3 في
+
+490
+00:33:37,070 --> 00:33:41,850
+مين في 64 لأنه هذه لما تتكعب بصير 64 زي ما انتوا
+
+491
+00:33:41,850 --> 00:33:49,300
+عارفينزائد ناطص ناطص يصبح زائد X يصبح Y زائد 2 على
+
+492
+00:33:49,300 --> 00:33:54,960
+4 Y
+
+493
+00:33:54,960 --> 00:34:01,580
+تربيع على 2 يصبح Y أقصى 4 على 4الكل إيه شماله الـ
+
+494
+00:34:01,580 --> 00:34:04,420
+DY هذا بدي أعمله كمان مرة بالنسبالي لمين لـ DY
+
+495
+00:34:04,420 --> 00:34:07,900
+بغلبكم بس انتوا فكوه اللي حالكم هدولة اللي .. اللي
+
+496
+00:34:07,900 --> 00:34:11,340
+حاكينا عنها هدولة فكوه فكوا هدولة بغلبنا شوية بس
+
+497
+00:34:11,340 --> 00:34:13,660
+في البيت احنا لما بجيب في الامتحان بجيب الشغلة
+
+498
+00:34:13,660 --> 00:34:16,600
+اللي بتغلب في الحسابات زي هيك اللي هي تضيع الفكرة
+
+499
+00:34:16,600 --> 00:34:19,640
+اللي جيبنا منها ولكن هذا في البيت بتقدر هذا تكعبه
+
+500
+00:34:19,640 --> 00:34:24,340
+وتوجده تفرده تاخد ال Y زي بعض و ال Y تربيع زي بعض
+
+501
+00:34:24,340 --> 00:34:26,360
+و ال Y تكعيق زي بعض و ال Y أقصى أربعة زي بعض
+
+502
+00:34:26,530 --> 00:34:30,890
+وبتكامله هيطلع عندك اللي هو المقدار اللي أمامي هذا
+
+503
+00:34:30,890 --> 00:34:36,750
+هو تكامل هذا المقدار مثلا Y6 هي عبارة عن Y7 على 7
+
+504
+00:34:36,750 --> 00:34:41,870
+في هذا المقدار اللي هو Y مثلا Y4 هي Y5 على 5 في 24
+
+505
+00:34:41,870 --> 00:34:46,210
+وهكذا البجيات بنفس الإسلوبالان باعوض عنه من صفر
+
+506
+00:34:46,210 --> 00:34:49,950
+الاتنين بطلع عندي القيبة اللي عندي الان خلّيني
+
+507
+00:34:49,950 --> 00:34:53,490
+أنتبه للنقطة الأخيرة دي مهمة اللي هي another
+
+508
+00:34:53,490 --> 00:34:57,250
+method using dy dx الان واحد قاللي انا اسمع انا
+
+509
+00:34:57,250 --> 00:35:01,970
+مديش ارد عليكم و مدي اشتغل dy by dx dy by dx قلتله
+
+510
+00:35:01,970 --> 00:35:05,600
+ماشي بس هتغلب انت انت حرالان دي واي باي دي اكس
+
+511
+00:35:05,600 --> 00:35:09,040
+باجي الان باخد اللي هو مدام دي واي او الاشي اذا
+
+512
+00:35:09,040 --> 00:35:11,780
+بدي اخد vertical line يعني موازية من ال Y-axis
+
+513
+00:35:11,780 --> 00:35:15,420
+لاحظوا ال vertical lines هنا من هنا لهنا من هنا
+
+514
+00:35:15,420 --> 00:35:19,300
+لعندي لهنا مافيش عندي مشكلة كلها بتيجي من عند اللي
+
+515
+00:35:19,300 --> 00:35:23,020
+هو Y بصفر و بتيجي الى ال curve هذا لكن لما اطلع
+
+516
+00:35:23,020 --> 00:35:27,120
+فوق و بدي اكمل باقي المنطقة بصير اللي هو ال
+
+517
+00:35:27,120 --> 00:35:31,100
+vertical line ع منواه الاخرالان المنوال هذا بيسير
+
+518
+00:35:31,100 --> 00:35:35,420
+من جاي من عند Y اللي هي على الخط هذا اللي هي 4X-2
+
+519
+00:35:35,420 --> 00:35:39,160
+و بروح لل Y اللي هنا هي 2 جدر ال X إذا المنطقتين
+
+520
+00:35:39,160 --> 00:35:42,700
+غصب عني هذه المنطقة بدي أجزئها لمنطقتين مدام بدي
+
+521
+00:35:42,700 --> 00:35:46,220
+أكمل بالنسبة ل Y في الأول فباجي على خط الحد هذا و
+
+522
+00:35:46,220 --> 00:35:51,330
+بحط الحدماجي باخد الان vertical line في المنطقة
+
+523
+00:35:51,330 --> 00:35:54,690
+الأولى الـ R1 اللي هو بلاقيه في الـ R1 هذه المنطقة
+
+524
+00:35:54,690 --> 00:35:58,310
+R1 هي قصدي هي في الـ R1 هلاقيها كلها على نفس
+
+525
+00:35:58,310 --> 00:36:02,450
+المنوال بتيجي من عند Y بتساوي 0 و بتروح ل Y إيه
+
+526
+00:36:02,450 --> 00:36:07,210
+شمالها بساوي 2 جدر ال X إذا من 0 ل 2 جدر ال X الآن
+
+527
+00:36:07,510 --> 00:36:17,190
+الان المنطقة هذه بتنمسح من عند الـ 0 لعند من؟ لعند
+
+528
+00:36:17,190 --> 00:36:21,330
+النقطة هذه النقطة هذه جديش عبارة عن نص كيف وجدناها
+
+529
+00:36:21,330 --> 00:36:23,770
+نص كمان شويه بقولكم كيف وجدناها نص لعند النص إذا
+
+530
+00:36:23,770 --> 00:36:28,070
+من عند 0 لعند نص اللي هي حدود من ال X بكون خلصنا
+
+531
+00:36:28,070 --> 00:36:31,630
+المنطقة الأولى نيجي على من؟ المنطقة الثانية زائد
+
+532
+00:36:31,630 --> 00:36:34,710
+ال integration للمنطقة الثانية لأنه أخد مع المنطقة
+
+533
+00:36:34,710 --> 00:36:38,640
+الأولى لسه المنطقة الثانيةبتجي من عند كلها على نفس
+
+534
+00:36:38,640 --> 00:36:44,720
+المنوال من عند الخط اللي هي Y بتساوي 4X-2 4X-2
+
+535
+00:36:44,720 --> 00:36:49,160
+وبتروح على الخط اللي فجأة هو 2GX لأن هذه المنطقة
+
+536
+00:36:49,160 --> 00:36:53,320
+الخطوط الطولية هذه الموازية للـ Y Axis بتبدأ من
+
+537
+00:36:53,320 --> 00:36:58,060
+عند X بتساوي نصفوبتنتهي في آخر المنطقة لـ x أيش
+
+538
+00:36:58,060 --> 00:37:01,360
+بتساوي واحد فبصير حدول integration من نص لعند واحد
+
+539
+00:37:01,360 --> 00:37:06,940
+فبكون ال volume هذا اللي هو بطريقة إيجاد dy by dx
+
+540
+00:37:06,940 --> 00:37:10,220
+إذن اضطرينا لتجزئة المنطقة منطقتين عشان هيك قلنا
+
+541
+00:37:10,220 --> 00:37:13,360
+احنا المنطقة هذه في منطقة واحدة أسهلنا و أريحنا و
+
+542
+00:37:13,360 --> 00:37:16,820
+خلّينا نشتغل فيها لإن هذه هتغلبنا كمان بكرا لما
+
+543
+00:37:16,820 --> 00:37:20,340
+أنا كامل و ادعوض و مجدور و مجدور هتغلب انسى
+
+544
+00:37:22,920 --> 00:37:25,840
+طبعاً في الاخر كما عملنا قبل قليل sign ال X على X
+
+545
+00:37:25,840 --> 00:37:29,120
+يضطر لكي نكمل ال Y قبل ال X إذا كان هناك عقول و
+
+546
+00:37:29,120 --> 00:37:34,200
+نستخدمها لتلافي التجزئة لمنطقتين إلا إذا كان الأمر
+
+547
+00:37:34,200 --> 00:37:37,760
+في كل الحوار بالجزة أو لتلافي ال integration إذا
+
+548
+00:37:37,760 --> 00:37:40,550
+لم نكن نعرف كيف نكملهبنعمل reversing لل
+
+549
+00:37:40,550 --> 00:37:44,150
+integration الان هذا كيف وجدناها اللي هو عندكم
+
+550
+00:37:44,150 --> 00:37:48,890
+اللي هو اللي هو الخط y بسوء 4x نقص 2 ماشي الحال
+
+551
+00:37:48,890 --> 00:37:54,970
+الان هذا اللي هو خد اللي هو y بزفر لأن هذا فقط و
+
+552
+00:37:54,970 --> 00:38:00,350
+مع مين؟ مع ال zero يعني هذا على الخط هذا مظبوط خد
+
+553
+00:38:00,350 --> 00:38:06,090
+اللي هيY بصير على الخط هذا 4X
+
+554
+00:38:06,090 --> 00:38:11,670
+بيساوي 2 يعني X بيساوي 2 على 4 يعني نص الان هذه
+
+555
+00:38:11,670 --> 00:38:16,080
+النقطة هي عبارة عن تقاطع هذا الخط مع هذا الخطمع
+
+556
+00:38:16,080 --> 00:38:20,240
+هذا، كيف نجد التقاطع؟ كما عملنا قبل قليل Y بساوي 4
+
+557
+00:38:20,240 --> 00:38:25,880
+ناقص X Y بساوي 4 X minus 2 و Y بساوي 2 جدر ال X
+
+558
+00:38:25,880 --> 00:38:30,400
+بأخد 2 جدر ال X و بسويها بهذه و بأوجد الحلول مع
+
+559
+00:38:30,400 --> 00:38:34,380
+بعض الحلول بوجد قيمة X و بعدها بوجد Y بكون أوجد
+
+560
+00:38:34,380 --> 00:38:38,480
+نقاط التقاطع ستجدها 1 و 2 و ستجدها 0 و 0 إذن خطوط
+
+561
+00:38:38,480 --> 00:38:42,180
+نقاط التقاطع بتعرفوا كيف وجدوها من حل هذا الخط ال
+
+562
+00:38:42,180 --> 00:38:47,750
+curve مع هذا الخطبتطلع عندي اللي هي ضغاط التقاطع و
+
+563
+00:38:47,750 --> 00:38:50,350
+هيك بيكون خلصنا المحاضرة و برك الله فيكم و السلام
+
+564
+00:38:50,350 --> 00:38:52,730
+عليكم و رحمة الله و بركاته و إلى لقاء آخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/HMa43OLUsAQ_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/HMa43OLUsAQ_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6f1d0ccf8ee8a5c35472cec3fccd24b048904095
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/HMa43OLUsAQ_postprocess.srt
@@ -0,0 +1,1672 @@
+1
+00:00:00,440 --> 00:00:03,720
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:03,720 --> 00:00:10,840
+وبركاته هذه المحاضرة رقم 12 نساق تفاضل وتكامل جيم
+
+3
+00:00:10,840 --> 00:00:16,500
+وهي المحاضرة الخامسة في chapter 13 وهي الأخيرة في
+
+4
+00:00:16,500 --> 00:00:21,640
+chapter 13 وهي تحت عنوان tangential and normal
+
+5
+00:00:21,640 --> 00:00:25,500
+component of accelerationيعني بدنا اللي هو العجلة
+
+6
+00:00:25,500 --> 00:00:32,100
+ان نعملها على او نحليلها على two components واحد
+
+7
+00:00:32,100 --> 00:00:35,720
+في اتجاه اللي هو ال unit tangent vector واحد في
+
+8
+00:00:35,720 --> 00:00:40,500
+اتجاه ال principle unit normal vector كمان سنراه
+
+9
+00:00:40,500 --> 00:00:45,740
+لاحقا الان بدنا نعرف احنا بنعرف قولنا لو كان لنا
+
+10
+00:00:45,740 --> 00:00:53,370
+particle بيسيراللي هو عبر curve بال position
+
+11
+00:00:53,370 --> 00:00:58,970
+vector R of T بتساوي كده اللي هو برسم النخط الآن
+
+12
+00:00:58,970 --> 00:01:03,210
+في أثناء حركته اللي هو اللي بقيس اتجاه الحركة اللي
+
+13
+00:01:03,210 --> 00:01:07,260
+هو ال unit tangent vector اللي قلنا عنه Tواللي
+
+14
+00:01:07,260 --> 00:01:13,100
+بيجيس اللى هو مدى التواؤه او اللى هو ال turning لل
+
+15
+00:01:13,100 --> 00:01:18,100
+curve او للمصار اللى هو اللى بيجيسه اللى هو ال
+
+16
+00:01:18,100 --> 00:01:23,680
+normal او ال principle unit normal vector اللى هو
+
+17
+00:01:23,680 --> 00:01:29,700
+دائما باتجاه باتجاه اللى هو مركز اللى هو ال circle
+
+18
+00:01:29,700 --> 00:01:33,470
+of curvature اللى حكينا عنها المرة الماضيةالان في
+
+19
+00:01:33,470 --> 00:01:37,310
+عنا vector ثالث ايضا بوس في حركة ال particle او
+
+20
+00:01:37,310 --> 00:01:42,060
+مهم لحركة ال market ال particleاللي هو ال vector
+
+21
+00:01:42,060 --> 00:01:47,260
+اللي بيكون عمودي على اللي هو ال plane تبع ال T و
+
+22
+00:01:47,260 --> 00:01:50,500
+ال N اللي هو عبارة عن ال T cross N اللي هو اللي
+
+23
+00:01:50,500 --> 00:01:55,940
+بنسميه احنا او هنسميه ال by normal vector اذا الأن
+
+24
+00:01:55,940 --> 00:01:59,360
+بنعرف حاجة اسمها ال by normal vector of a curve in
+
+25
+00:01:59,360 --> 00:02:04,080
+space SB بيستوي T cross N وهذا عبارة عن unit
+
+26
+00:02:04,080 --> 00:02:09,720
+vector orthogonal to both T and Nالعنوان الـ three
+
+27
+00:02:09,720 --> 00:02:14,580
+vectors هدول ال T و ال N و ال B في الواقع هم بعرف
+
+28
+00:02:14,580 --> 00:02:17,440
+حركة تدفع حركة تتحرك على حركة تتحرك على حركة تتحرك
+
+29
+00:02:17,440 --> 00:02:18,040
+على حركة تتحرك على حركة تتحرك على حركة تتحرك على
+
+30
+00:02:18,040 --> 00:02:19,420
+حركة تتحرك على حركة تتحرك على حركة تتحرك على حركة
+
+31
+00:02:19,420 --> 00:02:22,700
+تتحرك على حركة تتحرك على حركة تتحرك على حركة تتحرك
+
+32
+00:02:22,700 --> 00:02:23,020
+على حركة تتحرك على حركة تتحرك على حركة تتحرك على
+
+33
+00:02:23,020 --> 00:02:23,280
+حركة تتحرك على حركة تتحرك على حركة تتحرك على حركة
+
+34
+00:02:23,280 --> 00:02:26,340
+تتحرك على حركة تتحرك على حركة تتحرك على حركة تتحرك
+
+35
+00:02:26,340 --> 00:02:30,070
+على حركة تيعني الآن أثناء سير اللي هو ال vector في
+
+36
+00:02:30,070 --> 00:02:33,630
+ال space .. الجزء في ال space أو الجسم في ال space
+
+37
+00:02:33,630 --> 00:02:37,830
+هذا عندي مجسم صار المجسم هذا بيسير على اللي هو ال
+
+38
+00:02:37,830 --> 00:02:41,370
+curve هذا في اللي هو ال T اللي هو ال unit tangent
+
+39
+00:02:41,370 --> 00:02:47,400
+vector بجيس اللي هو اتجاه الحركةالان الان بيجيسلي
+
+40
+00:02:47,400 --> 00:02:52,280
+مدى التواق اللى هو او ال turning تبع مين تبع اللى
+
+41
+00:02:52,280 --> 00:02:57,640
+هو مصار ال particle جداش ال turning بيكون من خلال
+
+42
+00:02:57,640 --> 00:03:01,620
+ال normal اللى هو دايما بيكون باتجاه اللى هو ال
+
+43
+00:03:01,620 --> 00:03:06,120
+center لل circle of curvature عند لحظة ما لا اللى
+
+44
+00:03:06,120 --> 00:03:13,040
+هو ال particleالان الجسم هذا اثناء سيرو الان ال
+
+45
+00:03:13,040 --> 00:03:16,540
+normal هذا بجيس لل twisting تبع اللي هو لو اتخيلنا
+
+46
+00:03:16,540 --> 00:03:22,200
+جطار ماشي جطار ماشي هى هذا بشوف كيف حركته هذا بشوف
+
+47
+00:03:22,200 --> 00:03:26,260
+كيف التواؤل اللي هو الان الان هذا اللي هو ال
+
+48
+00:03:26,260 --> 00:03:30,800
+normal عليهم بجيس لل twisting تبع اللي هو كجسم
+
+49
+00:03:30,800 --> 00:03:34,670
+كامل الان في الفراغ بجيساللي هو ايش .. كيف ال
+
+50
+00:03:34,670 --> 00:03:39,290
+twisting بيصير ل .. اللي هو القطر او ال particle
+
+51
+00:03:39,290 --> 00:03:44,510
+اللي بيصير اذا هذه ال T و ال N و ال B هي فعلا ال
+
+52
+00:03:44,510 --> 00:03:48,950
+frame بوصف حركة اللي هو الجسم او ال particle اللي
+
+53
+00:03:48,950 --> 00:03:55,910
+بيكون سائر على طريق ما بيدالة معينةالان الـ Normal
+
+54
+00:03:55,910 --> 00:03:59,350
+طبعا احنا عرفناه اللي هو عبارة عن واحدة ركابة في
+
+55
+00:03:59,350 --> 00:04:04,250
+DT by DS و الـ T اللي هو عبارة عن DR by DS او
+
+56
+00:04:04,250 --> 00:04:09,530
+بواسطة تلية الـ T معناته V ع المجنتيوت للـ V و هذا
+
+57
+00:04:09,530 --> 00:04:14,310
+الـ DT by DT ع المجنتيوت تبعها اللي هنشوفه .. اللي
+
+58
+00:04:14,310 --> 00:04:23,280
+شوفنا سابقا عندي الآن نشوف كيف اللي هونوجد حاجة
+
+59
+00:04:23,280 --> 00:04:27,240
+مهمة او جزء ثاني من الدراسة اللي هو حاجة اسمة
+
+60
+00:04:27,240 --> 00:04:30,920
+tangential and normal components of acceleration
+
+61
+00:04:30,920 --> 00:04:34,460
+tangential and normal components of acceleration
+
+62
+00:04:34,460 --> 00:04:39,000
+ايش اللي بنقصد فيه الان عندي اللي هو ال
+
+63
+00:04:39,000 --> 00:04:44,400
+acceleration العجلة العجلة بلزمني و هيكون مفيد ان
+
+64
+00:04:44,400 --> 00:04:49,800
+احللها الى متجهينمتعمدين متجهة في اتجاه او
+
+65
+00:04:49,800 --> 00:04:52,960
+component في اتجاه اللي هو ال unit tangent vector
+
+66
+00:04:52,960 --> 00:04:56,660
+و component اخرى في اتجاه عمودي عليها اللي هو ال
+
+67
+00:04:56,660 --> 00:04:59,800
+unit ال principle ال unit normal vector و خلينا
+
+68
+00:04:59,800 --> 00:05:04,040
+نشوف ايش اللي بنقوله احنا بنعرف ان ال velocity هي
+
+69
+00:05:04,040 --> 00:05:08,750
+عبارة عن dr by dtهو يساوي اللي هو لو استخدمنا ال
+
+70
+00:05:08,750 --> 00:05:12,930
+chain rule هي عبارة عن DR by DS في DS by DT اللي
+
+71
+00:05:12,930 --> 00:05:16,530
+هي بالساوي DR by DS زي ما عرفنا ان هي عبارة عن من؟
+
+72
+00:05:16,530 --> 00:05:23,390
+عن ال unit tangent vector D في DS by DT الان اللي
+
+73
+00:05:23,390 --> 00:05:28,340
+هي S اللي هي عبارة عن ال longالـ T هي عبارة عن
+
+74
+00:05:28,340 --> 00:05:34,900
+الزمن ثم
+
+75
+00:05:34,900 --> 00:05:41,840
+و يدفرنشيش both sides اللي هو انفاض
+
+76
+00:05:41,840 --> 00:05:47,040
+الفنسبة التي بيصير عندي Aلأن إذا فضلنا العجلة V
+
+77
+00:05:47,040 --> 00:05:51,460
+السرعة بتطلع من الـ acceleration بساوة DV بي DT
+
+78
+00:05:51,460 --> 00:05:55,360
+اللي هي بتفاضل هذه هذه بتساوي هذه بتفاضل الجهتين
+
+79
+00:05:55,360 --> 00:06:00,020
+بيصير D بي DT لت في DS بي DT حاصل ضغط دالتين اللي
+
+80
+00:06:00,020 --> 00:06:03,860
+هي التفاضل الأولى هي Fث ثانية زي التفاضل الثانية
+
+81
+00:06:03,860 --> 00:06:07,000
+في الأولى زي ما انتوا عارفين فبصير هذا D تربيع S
+
+82
+00:06:07,000 --> 00:06:08,440
+بي DT تربية
+
+83
+00:06:14,800 --> 00:06:21,420
+و يساوي وين رايح انا رايح اعملكم اللي هو two
+
+84
+00:06:21,420 --> 00:06:24,760
+components واحدة في اتجاه ال T واحدة في اتجاه ال N
+
+85
+00:06:24,760 --> 00:06:29,880
+و دعنا نشوف كيف تظهرالأن D تربيع S by DT تربيع في
+
+86
+00:06:29,880 --> 00:06:35,800
+اتجاه الـ T هذه ظهرت خلصنا منها زائد الان DS by DT
+
+87
+00:06:35,800 --> 00:06:39,900
+في DT by DT هذه بتستخدم اللي هو ال .. ال .. ال ..
+
+88
+00:06:39,900 --> 00:06:47,880
+ال chain rule بيصير DT by DS في DS by DT الان Y
+
+89
+00:06:47,880 --> 00:06:53,300
+ساوي هذه زي ما هي زائد DS by DT في كده الان هذه
+
+90
+00:06:53,300 --> 00:06:58,290
+اللي هي DT by DSالان هي بيساوي كابا فعلا ليش لأن
+
+91
+00:06:58,290 --> 00:07:04,850
+ان بتساوي dt by ds على كابا مظبوط على كابا رجعوا
+
+92
+00:07:04,850 --> 00:07:08,590
+لها لاش تكون خربطنا قبل هل جيت فعلا ان هي الان
+
+93
+00:07:08,590 --> 00:07:12,750
+بتساوي dt by ds عالميا على كابا واضح ايه بترجع
+
+94
+00:07:12,750 --> 00:07:16,270
+للقانون هذا و بتحفظوه ان شاء الله عساس ان هو تسهل
+
+95
+00:07:16,270 --> 00:07:20,950
+عليكم الأمور في كل الحساباتالان هذه زي ما هي بتنزل
+
+96
+00:07:20,950 --> 00:07:26,910
+لان هي انا غرضي هيك DS by DT الان في كابا في DS by
+
+97
+00:07:26,910 --> 00:07:31,770
+DT بتصير ايش هي DS by DT الكل تربيع في كابا في مين
+
+98
+00:07:31,770 --> 00:07:36,690
+في ال and اللي انا اصلا بتعمل ال a اللي هي عبارة
+
+99
+00:07:36,690 --> 00:07:40,210
+عن two components واحدة في اتجاه ال T واحدة في
+
+100
+00:07:40,210 --> 00:07:45,480
+اتجاه مين ال and هذه يا جماعة هذههي اللي بتسميها
+
+101
+00:07:45,480 --> 00:07:50,900
+AT اللي هي الـ tangential يعني ال tangential
+
+102
+00:07:50,900 --> 00:07:55,080
+acceleration او ال component في اتجاه T تبع ال
+
+103
+00:07:55,080 --> 00:07:58,480
+acceleration وهذا ع بعضها اللي بتسميها اللي هي ايش
+
+104
+00:07:58,480 --> 00:08:04,620
+اللي هي ال AN اللي هي ال normal component of
+
+105
+00:08:04,620 --> 00:08:09,780
+acceleration انشوف هذا الكلام if تحت ال N صرنا
+
+106
+00:08:09,780 --> 00:08:14,540
+مؤهلين ان نعرفإذا كتبت أكسلراشن فيكتور باسم a
+
+107
+00:08:14,540 --> 00:08:18,620
+بتساوي at في اتجاه الان زائد an في اتجاه الان
+
+108
+00:08:18,620 --> 00:08:23,020
+بيصير يعني كتبنا ال acceleration عبارة عن عدد في t
+
+109
+00:08:23,020 --> 00:08:29,440
+اللي هو ال unit tangent vector زائد an عدد في n
+
+110
+00:08:29,440 --> 00:08:33,500
+اللي هو ال principle unit normal vector صارت عند
+
+111
+00:08:33,500 --> 00:08:38,520
+ال a مكتوبة على صورةاللي هو اللي هو two components
+
+112
+00:08:38,520 --> 00:08:41,440
+واحدة في اتجاه ال T واحدة في اتجاه ال N هذا ال a
+
+113
+00:08:41,440 --> 00:08:45,220
+tangential من الحكي اللي قبل بشوية هيطلع اللي هو
+
+114
+00:08:45,220 --> 00:08:50,200
+لإنه قبل بشوية قولنا أنه أثبتنا أنه a هتكون هدف T
+
+115
+00:08:50,200 --> 00:08:56,030
+و هدف N أي إش بنكتب بساوي a عدد في T زائدعدد في N
+
+116
+00:08:56,030 --> 00:08:59,370
+معناته هذا هو ال .. ال .. ال .. العدد اللي حكينا
+
+117
+00:08:59,370 --> 00:09:03,090
+عنه اللي سمناه قبل شوية A T و هذا اللي سمناه A N
+
+118
+00:09:03,090 --> 00:09:07,690
+معناته انه صار عند ال A T بيساوي D تربيع أصبعي D
+
+119
+00:09:07,690 --> 00:09:12,310
+تربيع أو A N بيساوي كابا في هذا المقدار طيب احنا
+
+120
+00:09:12,310 --> 00:09:16,430
+كل أمورنا سهلة بتسير لما نحولها بدل من بال DS ب ..
+
+121
+00:09:16,430 --> 00:09:24,190
+ب .. اللي هي اللي نحولها ل ال V أو ال Tأو الـ R أو
+
+122
+00:09:24,190 --> 00:09:29,470
+الـ Acceleration فاهمين جصدي الـ AT أسهلنا إن الآن
+
+123
+00:09:29,470 --> 00:09:35,990
+نقول هي عبارة عن طبعا DS by DT DS by DT اللي هي ال
+
+124
+00:09:35,990 --> 00:09:40,070
+magnitude لمين؟ لـ V لما نقفض لهذه بتصير عبارة عن
+
+125
+00:09:40,070 --> 00:09:48,200
+D by DT للمagnitude لـ Vالـ AN هي عبارة عن كيف DS
+
+126
+00:09:48,200 --> 00:09:53,120
+by DT لكل تربيع برضه DS by DT هي مين ؟ هي عبارة عن
+
+127
+00:09:53,120 --> 00:09:58,450
+اللي هي الـ Speedلأن الـ R دي R by DT اللي هي الـ
+
+128
+00:09:58,450 --> 00:10:01,770
+R direction الـ S اللي هي الـ Length بعتمد على T
+
+129
+00:10:01,770 --> 00:10:06,050
+يعني هو كأنه هذه طول المسافة أما اللي هي إيش هذه
+
+130
+00:10:06,050 --> 00:10:11,350
+الـ V الـ R الإزاحة فلما أفضلت المسافة DS by DT
+
+131
+00:10:11,350 --> 00:10:15,210
+اللي هي المسافة بدلال الـ T فتطلع عندي اللي هو
+
+132
+00:10:15,210 --> 00:10:18,570
+Speedيعني اللي هو بيصيح ال magnitude ال V لكل
+
+133
+00:10:18,570 --> 00:10:23,930
+تربيع أو هذا DS by DT صارت ال AN هيقيمتها بوجدها
+
+134
+00:10:23,930 --> 00:10:28,530
+بالشكل هذا أو هاي AT هيقيمتها بوجدها بالشكل هذا
+
+135
+00:10:28,530 --> 00:10:32,570
+هذه الأولى بتسميها tangential والثانية بتسميها
+
+136
+00:10:32,570 --> 00:10:33,610
+normal scalar
+
+137
+00:10:36,880 --> 00:10:41,880
+Tangential Scalar Component of Acceleration الان
+
+138
+00:10:41,880 --> 00:10:48,620
+هذا اختصارا توضيح للي بنحكيه هى عندى ال a هو اتجاه
+
+139
+00:10:48,620 --> 00:10:52,620
+العجلة طبعا هى ال vertical ماشي هنا هى ال particle
+
+140
+00:10:52,620 --> 00:10:57,240
+هنا هى ال acceleration تبعته هى ال unit normal
+
+141
+00:10:57,240 --> 00:11:00,880
+vectorهذا unit tangent vector وهذا ال principal
+
+142
+00:11:00,880 --> 00:11:06,160
+unit normal vector الأكسلاريشين هذا بدي أحللها
+
+143
+00:11:06,160 --> 00:11:09,940
+واحدة هنا واحدة هنا التحليل اللي هنا هيطلع عند ال
+
+144
+00:11:09,940 --> 00:11:13,980
+AT التي في اتجاه ال T هو D تربيع أسوأ DT تربيع
+
+145
+00:11:13,980 --> 00:11:18,430
+اللي حكيناه هنا و اللي في اتجاه ال normal هذاالتي
+
+146
+00:11:18,430 --> 00:11:23,290
+حكيناها كابع في DS by DT تربيع هذا اللي هو ال A N
+
+147
+00:11:23,290 --> 00:11:27,390
+في اتجاه ال N و ال A T في اتجاه ال T بتصير ال A
+
+148
+00:11:27,390 --> 00:11:32,670
+بتساوي A T في اتجاه ال T مضروبة في ال T كمتجه و A
+
+149
+00:11:32,670 --> 00:11:36,710
+N في اتجاه زائد A N في اتجاه ال N دي مضروبة في N
+
+150
+00:11:36,710 --> 00:11:37,650
+كمتجه
+
+151
+00:11:39,970 --> 00:11:43,570
+Formula for calculating the normal components of
+
+152
+00:11:43,570 --> 00:11:49,590
+acceleration الان أسهلنا لو رجعنا شوية ال AT سهل
+
+153
+00:11:49,590 --> 00:11:55,410
+إيجادها ليش ديتها مش معطينا ال R بوجد الان DR by
+
+154
+00:11:55,410 --> 00:11:59,530
+DT بطلع من ال velocity باخده ال magnitude بتطلع ال
+
+155
+00:11:59,530 --> 00:12:04,070
+speed بقى فاضله بالنسبة ليه بطلع ال AT اذا سهل
+
+156
+00:12:04,070 --> 00:12:07,970
+المأمر ضال هذه اللي بتخلق شوية في الحسابات بنجيبها
+
+157
+00:12:07,970 --> 00:12:12,620
+بطريقة ثانيةالتي هي الطريقة كيف أننا نكون هنا على
+
+158
+00:12:12,620 --> 00:12:15,860
+طول بوجد الأودادات الات زي قبل بشهية بالطريقة
+
+159
+00:12:15,860 --> 00:12:19,940
+السهلة بوجد الان عبارة عن الجدر التربيعي للمجنتيود
+
+160
+00:12:19,940 --> 00:12:25,480
+للات تربيع ناقص ات لكل تربيع لإنه عند الأي
+
+161
+00:12:25,480 --> 00:12:29,620
+بالساوية هذه زائد هذه فالمجنتيود لهذه هو عبارة عن
+
+162
+00:12:29,620 --> 00:12:33,580
+الجدر التربيعي لهذه تربيع زائد هذه تربيع ومن ثم
+
+163
+00:12:33,580 --> 00:12:40,040
+الانبساوي الجدر التربية الى الات تربية ناقص الات
+
+164
+00:12:40,040 --> 00:12:45,360
+تربيةالان لو جينا في ال circle ع ال circle لو كانت
+
+165
+00:12:45,360 --> 00:12:48,860
+الحركة هي عندي اللي هو ال .. ال .. ال .. ال
+
+166
+00:12:48,860 --> 00:12:53,300
+particle بيمشي على circle هذه الآن في اتجاه ال T
+
+167
+00:12:53,300 --> 00:12:57,500
+وهذه في اتجاه مين الان حللنا مين ال acceleration
+
+168
+00:12:57,500 --> 00:13:02,680
+لما نحلل ال acceleration بتتحلل ل D تربيع أسوأ و D
+
+169
+00:13:02,680 --> 00:13:07,420
+تربيع هادى اللي هي في اتجاه اللي هو ال .. ال unit
+
+170
+00:13:07,420 --> 00:13:12,980
+tangent vectorالتي نسميها AT التي هي D absolute
+
+171
+00:13:12,980 --> 00:13:18,300
+value of V by DT او D ال speed by DT وهذه اللي هي
+
+172
+00:13:18,300 --> 00:13:21,020
+عبارة عن كابة في absolute value of V تربيع في D
+
+173
+00:13:21,020 --> 00:13:26,720
+جهة ال N او ال magnitude of V تربيع على ال role
+
+174
+00:13:26,720 --> 00:13:32,060
+اللي هي عبارة عن واحد على كابة في Nالان هي اللي هي
+
+175
+00:13:32,060 --> 00:13:37,160
+ال components ال a tangential و ال a n اللي هي هذه
+
+176
+00:13:37,160 --> 00:13:42,140
+ال a n و هذه ال a tangential بالنسبة لل particle
+
+177
+00:13:42,140 --> 00:13:51,460
+اللي عجلته a على دائرة طيب لو كان سرعت اللي هي ال
+
+178
+00:13:51,460 --> 00:13:58,400
+particle ثابتة لو سرعته ثابتة a على الدائرة سرعته
+
+179
+00:13:58,400 --> 00:14:02,140
+ثابتةاللي هو ال magnitude لسرعته A و ال magnitude
+
+180
+00:14:02,140 --> 00:14:05,740
+ل V اللي هو A شماله ثابت يعني هو غير اتجاهه طبعا
+
+181
+00:14:05,740 --> 00:14:10,360
+اللي هو اتجاه السرعة بتغير لكن لو كان Zero على
+
+182
+00:14:10,360 --> 00:14:15,430
+اللي هو الدائرة ب speed ثابتعلى طول مدام بسبيد
+
+183
+00:14:15,430 --> 00:14:20,350
+ثابت إذا حيطلع عندي D تربيع أسبوع DT تربيع أو هذه
+
+184
+00:14:20,350 --> 00:14:24,690
+اللي هي عبارة عن D لسبيد by DT هيطلع سفر إذا مش
+
+185
+00:14:24,690 --> 00:14:30,150
+هيكون في عندي two tangential AT هيكون كل اتجاه
+
+186
+00:14:30,150 --> 00:14:35,650
+العجل أشماله اتجاه المركزلأن كل شيء سيكون عادى لأن
+
+187
+00:14:35,650 --> 00:14:39,750
+هذه سيكون صفر و سيكون الـ acceleration اتجاهها فقط
+
+188
+00:14:39,750 --> 00:14:44,230
+هنا إذا في حالة اللي هو الـ speed constant على
+
+189
+00:14:44,230 --> 00:14:50,330
+الدائرة بيكون اتجاه العجلة دائما في اتجاه المين
+
+190
+00:14:50,330 --> 00:14:54,770
+المركز يعني يوم ما تشوف يوم ما تشوف الان particle
+
+191
+00:14:54,770 --> 00:15:01,030
+عجلته اللي هو مش في اتجاه المركز عقول هتقول هذا
+
+192
+00:15:01,030 --> 00:15:06,120
+ممشيش بسرعة ب speed ثابتلأ سبيده متغير لأن لو
+
+193
+00:15:06,120 --> 00:15:12,520
+سبيده ثابت بيكون اتجاه عجلت وين بس باتجاه المركز
+
+194
+00:15:12,520 --> 00:15:17,520
+او باتجاه ال normal يعني اي particle ال a له
+
+195
+00:15:17,520 --> 00:15:22,880
+اتجاهه مش في اتجاه المركز معناته ال speed is not
+
+196
+00:15:22,880 --> 00:15:30,300
+constant خليكوا اللي هو مذكري طيبالان ناخد مثال
+
+197
+00:15:30,300 --> 00:15:34,100
+without finding T and N write the acceleration of
+
+198
+00:15:34,100 --> 00:15:39,260
+the motion R of T بساوية كذا in the form A تنجنتيا
+
+199
+00:15:39,260 --> 00:15:42,740
+في اتجاه T و A N في اتجاه الـ N وقول يعني بدون ما
+
+200
+00:15:42,740 --> 00:15:46,960
+تحسب T و N اقعد تحسبهم لان انا اصلا بدي ابدالتهم
+
+201
+00:15:46,960 --> 00:15:51,600
+اصلا انابدي ال a تنكتب ال aT في اتجاه ال T زي ال
+
+202
+00:15:51,600 --> 00:15:56,100
+aN في اتجاه ال N يعني بدي اكتبها ال acceleration
+
+203
+00:15:56,100 --> 00:16:02,180
+بهذه لا تحسب ال T ولا تحسب ال N بقوله منيح ال aT
+
+204
+00:16:02,180 --> 00:16:05,780
+سهلة بجيب لك ال aT و انا بعرف العجلة اللي انا
+
+205
+00:16:05,780 --> 00:16:10,040
+بعرفها لأنه اصلا لازم اوجدها باخد ال magnitude لها
+
+206
+00:16:10,040 --> 00:16:13,720
+و بربحها و بنقصها لتربيحها منها و باخد لها الجذر
+
+207
+00:16:13,720 --> 00:16:17,300
+تطلع aN بكون كل أمورنا جاهزة ايش اللي بتقولوا يلا
+
+208
+00:16:17,300 --> 00:16:22,340
+معاياالأن هاي الـ R of T جيب الـ V V بتساوي ناقص
+
+209
+00:16:22,340 --> 00:16:27,750
+Sin Tزائد التفاضل اللي حصل ضرب دلتين sin t زائد t
+
+210
+00:16:27,750 --> 00:16:31,930
+cos t كله في اتجاه ال I زائد تفاضلها cos t و
+
+211
+00:16:31,930 --> 00:16:36,710
+تفاضلها اللي حصل ضرب دلتين تي واحد يعني بيصير cos
+
+212
+00:16:36,710 --> 00:16:42,310
+t زائد اللي هو سالب sin t في T بيصير عندى هذا اللي
+
+213
+00:16:42,310 --> 00:16:46,190
+هو ال velocity لاحظ هنا اللي هو السادة بتروح مع
+
+214
+00:16:46,190 --> 00:16:51,300
+هذه تطلع T cos T وهذه بتروح مع هذه تطلع T sin Tخد
+
+215
+00:16:51,300 --> 00:16:54,500
+المجنتيو دي لها المجنتيو دي عبارة عن جدر التربيع
+
+216
+00:16:54,500 --> 00:16:57,800
+إلى هذه تربيع زائد هذه تربيع طبعا cos تربيع زائد
+
+217
+00:16:57,800 --> 00:17:01,120
+sin تربيع لما نطلع ال T تربيع برا بيصير واحد في T
+
+218
+00:17:01,120 --> 00:17:03,960
+تربيع يعني اللي هو جدر ال T تربيع جدر ال T تربيع
+
+219
+00:17:03,960 --> 00:17:07,340
+علشان ال value لل T لكن ال T عندنا اصلا احنا مفترض
+
+220
+00:17:07,340 --> 00:17:11,000
+انها اكبر من 0 اذا بتساوي ايش؟ بتساوي T اذا حصلنا
+
+221
+00:17:11,000 --> 00:17:14,480
+.. طب ليش وجدت .. وجدت هلول؟ لإن احنا عارفين ال a
+
+222
+00:17:14,480 --> 00:17:20,010
+tangential ايش بتساوي D ل speed by dtو يساوي ايش
+
+223
+00:17:20,010 --> 00:17:23,450
+بتساوي تفضلها واحد اذا الـ a تنجنش الا وجدناها
+
+224
+00:17:23,450 --> 00:17:29,290
+واحد الان ال acceleration الان سهل الان خلينا نجد
+
+225
+00:17:29,290 --> 00:17:32,230
+اول اشي مين ال acceleration ال acceleration هي ال
+
+226
+00:17:32,230 --> 00:17:37,030
+V عندنا بده فاضل هذه بيصير تفضلها عبارة عن cos T
+
+227
+00:17:37,030 --> 00:17:42,430
+لما حصل ضرب تلتين ناقص T sin T في اتجاه I زائد هذا
+
+228
+00:17:42,430 --> 00:17:47,830
+حصل ضرب دلتين sin T زائد T في cos T في اتجاه Jالان
+
+229
+00:17:47,830 --> 00:17:51,770
+هادى بدى اوجدها هادى ال acceleration هيها هادى بدى
+
+230
+00:17:51,770 --> 00:17:55,590
+اكتبها على الصورة هادى هيها اللى أمامى طيب كيف بدى
+
+231
+00:17:55,590 --> 00:17:59,510
+اكتبها؟ شوفوا معايا كيف خد ال magnitude لمين الآن
+
+232
+00:17:59,510 --> 00:18:04,150
+لل a ليش ال magnitude لل a عشان اوجد ال a انت زى
+
+233
+00:18:04,150 --> 00:18:06,490
+ما قلنا قبل بشوية ال magnitude لل a هو عبارة عن
+
+234
+00:18:06,490 --> 00:18:11,610
+جدر التربيعي لهادى تربيع هيها زائد هادى تربيع هيها
+
+235
+00:18:11,610 --> 00:18:16,080
+بالربع الجهتين هادى بيصير cosine تربيعوهنا بيصير
+
+236
+00:18:16,080 --> 00:18:19,220
+sin تربيع كوسين تربيع زاد sin تربيع اش مالها واحد
+
+237
+00:18:19,220 --> 00:18:24,660
+خلصنا الهد التفاضل تربيع حياش زاد T تربيع sin
+
+238
+00:18:24,660 --> 00:18:28,640
+تربيع وهد T تربيع كوسين تربيع بتطلع T تربيع برا
+
+239
+00:18:28,640 --> 00:18:31,480
+بيصير T تربيع في sin تربيع زاد كوسين تربيع يعني T
+
+240
+00:18:31,480 --> 00:18:35,140
+تربيع لأن sin تربيع زاد كوسين تربيع واحد الحد اللي
+
+241
+00:18:35,140 --> 00:18:40,230
+في النصتنين كوساين في sin في T بالسادة هنا تنين
+
+242
+00:18:40,230 --> 00:18:43,410
+sin في كوساين في T بالموجة بروح مع بعض إذا بضل
+
+243
+00:18:43,410 --> 00:18:46,770
+عندي بس المقدار فعلا جدر التربيع إلى واحد زائد T
+
+244
+00:18:46,770 --> 00:18:50,710
+تربيع لأن ال AN عبارة عن زي ما قلنا جدر التربيع
+
+245
+00:18:50,710 --> 00:18:56,410
+إلى ال magnitude ال A تربيع ناقص ال AT تربيعالـ A
+
+246
+00:18:56,410 --> 00:18:59,630
+T تربيع أوجدناه ليه سالب واحد وهذا أوجدناه ليه
+
+247
+00:18:59,630 --> 00:19:02,990
+واحد زائد T تربيع بيصير الجدر التربيع واحد زائد T
+
+248
+00:19:02,990 --> 00:19:05,850
+تربيع نقصه اللي هو الجدر التربيع إلى T تربيع اللي
+
+249
+00:19:05,850 --> 00:19:08,170
+هو absolute value للـ T بيساوي الـ T لإن الـ T
+
+250
+00:19:08,170 --> 00:19:13,930
+أكبر من سفر إذاً A أنجهزة .. A أنجهزة يا جماعة أو
+
+251
+00:19:13,930 --> 00:19:17,930
+A T جاهزة هو طالب الـ A تكتب على صورة هذه في هذه
+
+252
+00:19:17,930 --> 00:19:21,370
+إذا ما خلاص جهزة A .. A إيش هتساوي الـ A
+
+253
+00:19:21,370 --> 00:19:25,640
+tangentialفي اتجاه الـ T زائد A N في اتجاه الـ N A
+
+254
+00:19:25,640 --> 00:19:29,940
+tangential لو وجدناها واحد اذا بصير T A N ايش هي T
+
+255
+00:19:29,940 --> 00:19:34,700
+فN فصارت اللي هي الان كاتبين ال A على صورة اللي هو
+
+256
+00:19:34,700 --> 00:19:40,040
+T زائد T في اتجاه ال N اللي هو tangential و normal
+
+257
+00:19:40,040 --> 00:19:47,140
+components يعني هذه هي عبارة عن هذه العجلة لكن
+
+258
+00:19:47,140 --> 00:19:51,160
+احنا كتبناها بالصورة اللي هو طلبها عننا
+
+259
+00:19:53,250 --> 00:19:56,830
+شوفوا الآن في حاجة اسمها الـ torsion الـ torsion
+
+260
+00:19:56,830 --> 00:20:01,210
+هي عملية حسابات هنلاقيها بقى قليل شوية بس قبل ..
+
+261
+00:20:01,210 --> 00:20:03,970
+بس قبل ما نروح هذا .. هذا اللي هو ال curve لمين
+
+262
+00:20:03,970 --> 00:20:06,810
+اللي توح لناه السؤال اللي توح لناه ال curve إيه
+
+263
+00:20:06,810 --> 00:20:10,370
+له؟ هاي ال particle اللي .. اللي بمشي هنا هاي اللي
+
+264
+00:20:10,370 --> 00:20:13,710
+هو ال .. ال acceleration أوجدناها و أوجدنا ان ال
+
+265
+00:20:13,710 --> 00:20:18,110
+acceleration ال AT في اتجاه ال T و ال AN في اتجاه
+
+266
+00:20:18,110 --> 00:20:23,460
+اللي هي مين ال Tالـ A T طلعت واحد هي بصير T وهذه
+
+267
+00:20:23,460 --> 00:20:27,740
+طلعت اللي هو T فبصير T في N هذه اللي هي T في N
+
+268
+00:20:27,740 --> 00:20:32,760
+اللي هي تحليل ال A في اتجاه ال N وهذه تحليل ال A
+
+269
+00:20:32,760 --> 00:20:38,080
+في اتجاه من ال T اللي هي واحد طيب الآن بدنا نعرف
+
+270
+00:20:38,080 --> 00:20:41,840
+حاجة اسمها torsion أو العزم let B بتساوي T cross N
+
+271
+00:20:42,340 --> 00:20:47,020
+هذا الـ binomial vector اللي حكينا عنه قبل شوية
+
+272
+00:20:47,020 --> 00:20:51,840
+لأن the torsion function هذا تعريف of a smooth
+
+273
+00:20:51,840 --> 00:20:55,180
+curve is طبعا فيه له مجرات وفيه له حكي لتعريف مش
+
+274
+00:20:55,180 --> 00:21:01,080
+مجالنا لأن الـ tau هذه عبارة عن ناقص db by ds في
+
+275
+00:21:01,080 --> 00:21:06,380
+ضطن طبعا هذه ناقص اللي هو استلاحة عشان نحكي كلنا
+
+276
+00:21:06,380 --> 00:21:10,610
+في لغة واحدةطبعا الان الـ Tau بتساوي في الآخر عدد
+
+277
+00:21:10,610 --> 00:21:14,570
+عدد بتحط اللي ناقص بتطلع مين اللي هو التورجين وفيه
+
+278
+00:21:14,570 --> 00:21:18,150
+طبعا تفسيرات فيزيائية مش مجالها الآن الـ Tau
+
+279
+00:21:18,150 --> 00:21:22,330
+بتساوي ناقص DB by DS فعلا طبعا هذا الكلام احنا
+
+280
+00:21:22,330 --> 00:21:27,290
+مابنحبوش راحظوا ان احنا نبعد عن الحسابات اللي هو
+
+281
+00:21:27,290 --> 00:21:31,510
+التفاضل بالنسبة لـ S بنروح للحسابات بالنسبة لمين
+
+282
+00:21:31,510 --> 00:21:35,800
+بالنسبة التي أسهلناعشان هي هو ما أعطينا القانون
+
+283
+00:21:35,800 --> 00:21:41,660
+اللي هو بعد شوية قانون لحساب التورجين اللي هي
+
+284
+00:21:41,660 --> 00:21:44,800
+computational formula هو يقول لك the most widely
+
+285
+00:21:44,800 --> 00:21:48,580
+used formula for torgen derived in more advanced
+
+286
+00:21:48,580 --> 00:21:52,400
+text يعني مش مجالنا اللي هو اشتقاقها التورجين وهذا
+
+287
+00:21:52,400 --> 00:21:55,020
+اللي هنستخدمه في حساباتنا هذا اللي هنستخدمه في
+
+288
+00:21:55,020 --> 00:22:00,960
+حساباتنا مش هذهالتي هي تورجين X dot Y dot Z dot
+
+289
+00:22:00,960 --> 00:22:05,000
+اهل الفيزياء لايعرفون معنى هذا على مجموعه V cross
+
+290
+00:22:05,000 --> 00:22:10,340
+A كل تربيع V cross A ال V معروفة تنجب بـDR by DT و
+
+291
+00:22:10,340 --> 00:22:14,320
+A تكون DV by DT نعمل cross product ناخد المجموعه و
+
+292
+00:22:14,320 --> 00:22:19,200
+بالربعه تطلع الكمية هذه نجي للفوق اللي فوق ما هي X
+
+293
+00:22:19,200 --> 00:22:25,040
+dot هذه معناتها DX by DT احنا عارفين ان R of Tالـ
+
+294
+00:22:25,040 --> 00:22:28,760
+position vector أو اللي بتعمل ال .. ال .. ال .. ال
+
+295
+00:22:28,760 --> 00:22:33,760
+.. ال curve R of T بساوي X of T في اتجاه I زائد Y
+
+296
+00:22:33,760 --> 00:22:38,680
+of T في اتجاه G زائد Z of T في اتجاه الـ K لأن X
+
+297
+00:22:38,680 --> 00:22:44,850
+ضد هذه معناته DX by DT في الـ Rوهذه dy by dt التي
+
+298
+00:22:44,850 --> 00:22:50,430
+في الار وهذه zz by dt التي في الار وهذه معامل k
+
+299
+00:22:50,430 --> 00:22:56,270
+وهذه معامل i وهذه معامل j وهذه معامل i الآن بفضلها
+
+300
+00:22:56,270 --> 00:23:00,170
+كمان مرة دي تربيع x by dt تربيع تطلع هذه و بفضل
+
+301
+00:23:00,170 --> 00:23:03,450
+هذه كمان مرة تطلع هذه و بفضل هذه كمان مرة تطلع هذه
+
+302
+00:23:03,880 --> 00:23:07,360
+الان اللي بقعده بفاضل هذه كمان مرة تطلع هذه بفاضل
+
+303
+00:23:07,360 --> 00:23:09,780
+هذه كمان مرة تطلع هذه وهذه بفاضلها كمان مرة تطلع
+
+304
+00:23:09,780 --> 00:23:15,560
+هذه يعني هذه عبارة عن D تكعيب X by DT تكعيب يعني
+
+305
+00:23:15,560 --> 00:23:20,000
+التفاضل معامل I تلت مرات هذه التفاضل معامل I مرتين
+
+306
+00:23:20,000 --> 00:23:23,960
+هذا التفاضل بوجد ال determinant هذا بجسمه على ال
+
+307
+00:23:23,960 --> 00:23:26,860
+determinant أعلى اللي هو القيمة هذه ايه بتطلع مين
+
+308
+00:23:26,860 --> 00:23:31,610
+ال torsion وهذه طريقة الحل اللي احنا هنستخدمهاماشي
+
+309
+00:23:31,610 --> 00:23:35,470
+الحال اذا الان هذه الان في اسئلة عندى في ال
+
+310
+00:23:35,470 --> 00:23:41,230
+homework هيها طالب مين طالب ال tau مثلا هي عندى
+
+311
+00:23:41,230 --> 00:23:45,110
+معطيك هذا ال position vector هيه و طلبلك ال tau
+
+312
+00:23:45,110 --> 00:23:49,530
+عشان اوجد ال tau بدى اوجد ال determinant اللى فاضل
+
+313
+00:23:49,530 --> 00:23:53,210
+هذه اول مرة و بعدين تانى مرة و بعدين تلت مرة بحطن
+
+314
+00:23:53,210 --> 00:23:56,650
+فى العمود الاولوهذا اول مرة ثاني مرة ثالث مرة في
+
+315
+00:23:56,650 --> 00:24:00,070
+العمود التاني وهذا اول مرة ثاني مرة ثالث مرة ثالث
+
+316
+00:24:00,070 --> 00:24:00,830
+مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة
+
+317
+00:24:00,830 --> 00:24:00,970
+ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث
+
+318
+00:24:00,970 --> 00:24:01,470
+مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة
+
+319
+00:24:01,470 --> 00:24:02,810
+ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث
+
+320
+00:24:02,810 --> 00:24:02,930
+مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة
+
+321
+00:24:02,930 --> 00:24:10,110
+ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث
+
+322
+00:24:10,110 --> 00:24:17,970
+مرة ثالث مرة ثالث مرة
+
+323
+00:24:17,970 --> 00:24:20,810
+ثالث
+
+324
+00:24:21,640 --> 00:24:26,380
+اللي هو تطبق القانون هذا بتطلع عند مين التاو إذا
+
+325
+00:24:26,380 --> 00:24:32,620
+هذه الطريقة لإيجاد التاو زي ما شبهناالان في شغله
+
+326
+00:24:32,620 --> 00:24:37,720
+ثانيه قبل ما نروح للشغله الثانيه خليني نكمل عندي
+
+327
+00:24:37,720 --> 00:24:42,060
+computational formula for curves in space ماشي هذه
+
+328
+00:24:42,060 --> 00:24:46,380
+الان الملخص للي حكيناه كله يعني ال unit tangent
+
+329
+00:24:46,380 --> 00:24:52,020
+vector T V على ال magnitude ل V كله تفاضل من R R
+
+330
+00:24:52,020 --> 00:24:55,560
+of T تفاضل بالنسبة ل T هذه الان principal unit
+
+331
+00:24:55,560 --> 00:24:58,200
+normal vector NDT
+
+332
+00:25:05,330 --> 00:25:12,720
+بعد ذلك نفضل الار ونجيب الانالان ال B T cross N
+
+333
+00:25:12,720 --> 00:25:16,200
+اللي أوددناها خلصناها هذا ال by normal T cross N
+
+334
+00:25:16,200 --> 00:25:21,380
+اللي هي بتكون عمودية عالميا على اللي هي ال T و ال
+
+335
+00:25:21,380 --> 00:25:25,900
+N أو على ال plane اللي بيحملها قابة اللي هي DT by
+
+336
+00:25:25,900 --> 00:25:29,120
+DS magnitude وهذا مش هنستخدمها في الحسابات هنستخدم
+
+337
+00:25:29,120 --> 00:25:34,920
+اللي هو اللي هي قبل اللي حكينا عنها القانون اللي
+
+338
+00:25:34,920 --> 00:25:39,440
+حكينا عنه سابقا اللي هو بدلالة T أو اللي هو هي فيه
+
+339
+00:25:39,440 --> 00:25:44,310
+قانون تانياللي برضه ممكن لو انت طلب منك اللي هو
+
+340
+00:25:44,310 --> 00:25:50,230
+توجد لسؤال واحد اللي هو تاو أقابعمانت موجد الـ V و
+
+341
+00:25:50,230 --> 00:25:53,650
+الـ A و الـ cross product لهن فأسهلك تستخدم القابة
+
+342
+00:25:53,650 --> 00:25:57,930
+بالقانون هذا V cross A magnitude على magnitude V
+
+343
+00:25:57,930 --> 00:26:02,430
+تكعيد و جرب حللي السؤال على قابة بهذا القانون أو
+
+344
+00:26:02,430 --> 00:26:06,430
+بالقانون السابق حلليه أو شوف الجوابين بطلع زي بعض
+
+345
+00:26:06,430 --> 00:26:10,290
+الأسئلة الهمور كل بعد شوية ال torsion tau هيها مش
+
+346
+00:26:10,290 --> 00:26:14,410
+هنستخدم هذا زي ما قلنا هنستخدمين هذا اللي عندنا
+
+347
+00:26:14,410 --> 00:26:18,410
+إلا إذا كان في معطينة بيه بدلالة Sمباشرة وهذا طبعا
+
+348
+00:26:18,410 --> 00:26:21,890
+نادرا ما يحدث اما هو اي قانون اللي احنا الاكتر
+
+349
+00:26:21,890 --> 00:26:25,290
+شيوعا في الاستخدام الان ال tangential and normal
+
+350
+00:26:25,290 --> 00:26:27,730
+scalar components of acceleration a بيساوي a
+
+351
+00:26:27,730 --> 00:26:30,170
+tangential في اتجاه ال T زي ال a n في اتجاه ال N
+
+352
+00:26:30,170 --> 00:26:36,630
+ال a tangential هي DV by DT DT بالفاضل اللي هي ال
+
+353
+00:26:36,630 --> 00:26:40,150
+speed وال a n اللي هي قابة في ال speed التربيع
+
+354
+00:26:40,150 --> 00:26:44,010
+اللي هي هتساوي الجذر التربيعي طبعا هذه بنوجدها من
+
+355
+00:26:44,010 --> 00:26:49,020
+هنا أسلنا من وجهة دهانلأن هذه إيجادة قد يتطلب هذا
+
+356
+00:26:49,020 --> 00:26:52,740
+و هذا بيغلب أو يتطلب السابق برضه بيغلب لكن هذا
+
+357
+00:26:52,740 --> 00:26:55,140
+أسألك على طول جدر التربيع لـ Magnitude لـ A تربيع
+
+358
+00:26:55,140 --> 00:26:59,420
+نقص A T تربيع التربيع اللي احنا حكينا عنه سابقاً
+
+359
+00:26:59,420 --> 00:27:03,620
+طيب نرجع ل .. لسيبنا هنا دولة اللي هن ال .. ال ..
+
+360
+00:27:03,620 --> 00:27:07,900
+ال .. ال .. ال blends ال blends اللي .. اللي ..
+
+361
+00:27:07,900 --> 00:27:13,170
+اللي يصنع من مين من ال T و ال N و ال Dالان هى ال
+
+362
+00:27:13,170 --> 00:27:17,050
+particle عندى او القطار اللى ماشى القطار اللى ماشى
+
+363
+00:27:17,050 --> 00:27:21,350
+او ال particle هذا زى ما قلنا فى له اللى هو نقطة
+
+364
+00:27:21,350 --> 00:27:26,090
+unit tangent vector وفى عندى ال principle normal
+
+365
+00:27:26,090 --> 00:27:32,630
+vector ال plane اللى بيحملهن اللى بيحمل ال T والان
+
+366
+00:27:32,630 --> 00:27:36,650
+بنسميه escalating plane عينكم الله بتكونوا تحفظوها
+
+367
+00:27:38,420 --> 00:27:42,780
+الان كيف نوجد معدلته هذا إيجاد معدلته انه طبعا مين
+
+368
+00:27:42,780 --> 00:27:46,880
+بيبقى معاطيك اللي هو R of T اللي هو R اللي هو اللي
+
+369
+00:27:46,880 --> 00:27:50,100
+برسم لنا ال curve اللي بيمشي عليه ال particle من
+
+370
+00:27:50,100 --> 00:27:53,440
+ال R بوجد زي ما بتعرف طبعا زي ما مشينا قبله شوية
+
+371
+00:27:53,440 --> 00:27:57,200
+بتوجدوا ال T و بتوجدوا ال N أكيد بتعرفوا توجدوه و
+
+372
+00:27:57,200 --> 00:28:00,900
+من ال R برضه بقدر أوجد نقطة اللي هو بيبقى بعطيك
+
+373
+00:28:00,900 --> 00:28:05,320
+أنه هذا بدي معدلتهاللي معادلة الـ Plane الـ
+
+374
+00:28:05,320 --> 00:28:08,380
+Scalating in Plane اللي بمر بالنقطة كذا هذه بيبقى
+
+375
+00:28:08,380 --> 00:28:12,620
+معاطيك نقطة بس ضال عليك توجد مين اللي هو Normal
+
+376
+00:28:12,620 --> 00:28:16,200
+على الـ Plane مين ال Normal على الـ Plane اللي هو
+
+377
+00:28:16,200 --> 00:28:20,440
+T cross N اللي هو ال By Normal اللي بناه وجده إذا
+
+378
+00:28:20,440 --> 00:28:24,080
+ال Scalating in Plane معادلته سهل إيجادها وهذا
+
+379
+00:28:24,080 --> 00:28:27,140
+اللي هو معلومات سابقة يعنيليس داعي ناخد أمثلة
+
+380
+00:28:27,140 --> 00:28:31,120
+عليها لأنه بتعرفوا توجدوها اللي هو بتعرفوا هي نقطة
+
+381
+00:28:31,120 --> 00:28:34,420
+موجودة و ال normal اللي على ال escalating اللي هو
+
+382
+00:28:34,420 --> 00:28:39,580
+ال B بتعرفوا توجدوه فبتوجدوه عدلته هذا الآن ال
+
+383
+00:28:39,580 --> 00:28:45,000
+plane المكوّن من ال N و ال B المكوّن من binormal و
+
+384
+00:28:45,000 --> 00:28:49,180
+normalاسمه Normal لبنان يعني هنا في كلمة Normal
+
+385
+00:28:49,180 --> 00:28:53,140
+وهنا كلمة Normal إذا اسمه Normal لبنان ال Normal
+
+386
+00:28:53,140 --> 00:28:56,420
+لبنان هي النقطة المعروفة و ال unit tangent vector
+
+387
+00:28:56,420 --> 00:29:00,860
+معروف فمن العمودي على اللي هو ال Normal لبنان اللي
+
+388
+00:29:00,860 --> 00:29:05,120
+هو ال unit tangent vector هو النقطة تعرفه معادلة
+
+389
+00:29:05,120 --> 00:29:09,800
+لبنان زي ما أخدنا سابقاالان هذا اللي بين الـ unit
+
+390
+00:29:09,800 --> 00:29:13,180
+tangent vector و الـ binormal بنسميه rectifying
+
+391
+00:29:13,180 --> 00:29:18,580
+plane هذا rectifying plane معدلته سهلة هذه النقطة
+
+392
+00:29:18,580 --> 00:29:21,840
+معروفة وهي ال normal على ال plane المعروف و
+
+393
+00:29:21,840 --> 00:29:25,120
+بتعرفوا معدلته
+
+394
+00:29:25,560 --> 00:29:28,120
+الان قبل بشوية انا حكيته هذا الكلام اللي هو
+
+395
+00:29:28,120 --> 00:29:31,140
+التعليق عليها اذا هذا في عندكم اسئلة على ال
+
+396
+00:29:31,140 --> 00:29:35,360
+escalating وعلى ال normal وعلى ال rectifying بدكم
+
+397
+00:29:35,360 --> 00:29:40,580
+تحلوها باللي هي معرفة نقطة ومؤيجاد اللي هو هذا او
+
+398
+00:29:40,580 --> 00:29:44,650
+هذا او هذاوكل هى هدولة اوددناها وعرفنا كيف نوددها
+
+399
+00:29:44,650 --> 00:29:50,470
+ومعادلة ال plane بمعرفة نقطة وعمودي عليه اعرفناها
+
+400
+00:29:50,470 --> 00:29:54,430
+نيجى هدرس اعيد اللى حكيته قبل بشوية في اول تقريبا
+
+401
+00:29:54,430 --> 00:29:59,130
+المحاضرة قولنا ال particle او اللى هو القطار ماشى
+
+402
+00:29:59,130 --> 00:30:04,370
+هى ماشى هو ماشى اللى هو في الواقع ماشى ب frame ايش
+
+403
+00:30:04,370 --> 00:30:09,850
+ال frame اللى ماشى فيه اللى هو ال T و ال N و ال B
+
+404
+00:30:10,090 --> 00:30:16,070
+كل واحد فيهم كل واحد فيهم له معنى للحركة قولنا هذا
+
+405
+00:30:16,070 --> 00:30:22,890
+بيبين اتجاه الحركة اللي هو اتجاه ال velocity V على
+
+406
+00:30:22,890 --> 00:30:26,490
+ال magnitude لل V ال unit tangent vector هذا الان
+
+407
+00:30:26,490 --> 00:30:34,510
+هو اللي بيحدد اللي هوالتواء أو الـ Turning لمن؟
+
+408
+00:30:34,510 --> 00:30:39,430
+للبس أو للقطار أو للبارتيكل عبر الطريق اللي موجود
+
+409
+00:30:39,430 --> 00:30:45,770
+دائما هذا التواء بيكون باتجاه مركز الـ Circle of
+
+410
+00:30:45,770 --> 00:30:46,410
+curvature
+
+411
+00:30:50,650 --> 00:30:55,130
+الانحناء او ال curvature عند النقطة المذكورة والان
+
+412
+00:30:55,130 --> 00:31:00,130
+هذا ال B هو اللي بيحدد لي ال twisting لمين لل
+
+413
+00:31:00,130 --> 00:31:04,510
+particle او للقطار لما يتحرك بيصيرله طبعا حركة
+
+414
+00:31:04,510 --> 00:31:09,390
+حركة الآن بعمل اللي هو مش عارف ده انا جاعد بحرك
+
+415
+00:31:09,390 --> 00:31:13,310
+وانتوا مش شايفيني اللي هو بعمل اللي هو ال twisting
+
+416
+00:31:13,310 --> 00:31:17,230
+للقطار القطار هي و هيك بعمل هيك هيك هيك هذا ال
+
+417
+00:31:17,230 --> 00:31:21,710
+twisting اللي بجيسه مين اللي هو ال Bأو هكذا ننهي
+
+418
+00:31:21,710 --> 00:31:25,670
+المحاضرة و بارك الله فيكم و إلى لقاء آخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/HMa43OLUsAQ_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/HMa43OLUsAQ_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..87098880b3e35a5a6e6989ad09814972941b2827
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/HMa43OLUsAQ_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2332, "start": 0.44, "end": 23.32, "text": "بسم الله الرحمن الرحيم السلام عليكم ورحمة الله وبركاته هذه المحاضرة رقم 12 نساق تفاضل وتكامل جيم وهي المحاضرة الخامسة في chapter 13 وهي الأخيرة في chapter 13 وهي تحت عنوان tangential and normal component of acceleration", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 29538, 9673, 5016, 46958, 25720, 12602, 4587, 2304, 2272, 8717, 3794, 995, 4587, 6055, 5172, 46958, 1211, 34683, 4117, 10943, 1211, 10874, 32640, 37037, 1829, 9673, 5016, 46958, 25720, 33962, 10943, 3794, 3660, 8978, 7187, 3705, 37037, 1829, 16247, 9778, 48923, 8978, 7187, 3705, 37037, 1829, 6055, 33753, 18871, 2407, 7649, 10266, 2549, 293, 2710, 6542, 295, 17162], "avg_logprob": -0.14624618557168217, "compression_ratio": 1.6666666666666667, "no_speech_prob": 4.708766937255859e-06, "words": [{"start": 0.44, "end": 0.82, "word": 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9.4, "word": " نساق", "probability": 0.70819091796875}, {"start": 9.4, "end": 9.88, "word": " تفاضل", "probability": 0.9466552734375}, {"start": 9.88, "end": 10.46, "word": " وتكامل", "probability": 0.9010009765625}, {"start": 10.46, "end": 10.84, "word": " جيم", "probability": 0.720703125}, {"start": 10.84, "end": 11.82, "word": " وهي", "probability": 0.6273193359375}, {"start": 11.82, "end": 12.48, "word": " المحاضرة", "probability": 0.9912109375}, {"start": 12.48, "end": 13.78, "word": " الخامسة", "probability": 0.9727783203125}, {"start": 13.78, "end": 14.08, "word": " في", "probability": 0.94384765625}, {"start": 14.08, "end": 14.52, "word": " chapter", "probability": 0.556640625}, {"start": 14.52, "end": 15.74, "word": " 13", "probability": 0.904296875}, {"start": 15.74, "end": 15.98, "word": " وهي", "probability": 0.811279296875}, {"start": 15.98, "end": 16.38, "word": " الأخيرة", "probability": 0.9422200520833334}, {"start": 16.38, "end": 16.5, "word": " في", "probability": 0.9423828125}, {"start": 16.5, "end": 16.72, "word": " chapter", "probability": 0.80810546875}, {"start": 16.72, "end": 17.46, "word": " 13", "probability": 0.97021484375}, {"start": 17.46, "end": 18.38, "word": " وهي", "probability": 0.8125}, {"start": 18.38, "end": 18.7, "word": " تحت", "probability": 0.996337890625}, {"start": 18.7, "end": 19.26, "word": " عنوان", "probability": 0.98193359375}, {"start": 19.26, "end": 20.14, "word": " tangential", "probability": 0.71044921875}, {"start": 20.14, "end": 20.5, "word": " and", "probability": 0.9443359375}, {"start": 20.5, "end": 21.64, "word": " normal", "probability": 0.8408203125}, {"start": 21.64, "end": 22.38, "word": " component", "probability": 0.57275390625}, {"start": 22.38, "end": 22.66, "word": " of", "probability": 0.9677734375}, {"start": 22.66, "end": 23.32, "word": " acceleration", "probability": 0.81591796875}], "temperature": 1.0}, {"id": 2, "seek": 4722, "start": 23.74, "end": 47.22, "text": "يعني بدنا اللي هو العجلة ان نعملها على او نحليلها على two components واحد في اتجاه اللي هو ال unit tangent vector واحد في اتجاه ال principle unit normal vector كمان سنراه لاحقا الان بدنا نعرف احنا بنعرف قولنا لو كان لنا particle بيسير", "tokens": [40228, 22653, 47525, 8315, 13672, 1829, 31439, 18863, 7435, 37977, 16472, 8717, 25957, 1211, 11296, 15844, 1975, 2407, 8717, 5016, 20292, 1211, 11296, 15844, 732, 6677, 36764, 24401, 8978, 1975, 2655, 7435, 40294, 13672, 1829, 31439, 2423, 4985, 27747, 8062, 36764, 24401, 8978, 1975, 2655, 7435, 40294, 2423, 8665, 4985, 2710, 8062, 9122, 2304, 7649, 8608, 1863, 23557, 3224, 20193, 5016, 4587, 995, 2423, 7649, 47525, 8315, 8717, 3615, 28480, 1975, 5016, 8315, 44945, 3615, 28480, 12174, 12610, 8315, 45164, 25961, 5296, 8315, 12359, 4724, 1829, 3794, 13546], "avg_logprob": -0.23929072497935777, "compression_ratio": 1.7832512315270936, "no_speech_prob": 0.0, "words": [{"start": 23.74, "end": 24.18, "word": "يعني", "probability": 0.6605224609375}, {"start": 24.18, "end": 24.56, "word": " بدنا", "probability": 0.804931640625}, {"start": 24.56, "end": 24.76, "word": " اللي", "probability": 0.52783203125}, {"start": 24.76, "end": 24.94, "word": " هو", "probability": 0.94482421875}, {"start": 24.94, "end": 25.5, "word": " العجلة", "probability": 0.9879557291666666}, {"start": 25.5, "end": 26.44, "word": " ان", "probability": 0.220458984375}, {"start": 26.44, "end": 28.5, "word": " نعملها", "probability": 0.767333984375}, {"start": 28.5, "end": 29.12, "word": " على", "probability": 0.61962890625}, {"start": 29.12, "end": 29.76, "word": " او", "probability": 0.84765625}, {"start": 29.76, "end": 30.24, "word": " نحليلها", "probability": 0.75947265625}, {"start": 30.24, "end": 30.38, "word": " على", "probability": 0.3818359375}, {"start": 30.38, "end": 30.56, "word": " two", "probability": 0.7265625}, {"start": 30.56, "end": 31.34, "word": " components", "probability": 0.91943359375}, {"start": 31.34, "end": 32.1, "word": " واحد", "probability": 0.955322265625}, {"start": 32.1, "end": 32.24, "word": " في", "probability": 0.90576171875}, {"start": 32.24, "end": 32.64, "word": " اتجاه", "probability": 0.9805908203125}, {"start": 32.64, "end": 32.82, "word": " اللي", "probability": 0.911865234375}, {"start": 32.82, "end": 33.02, "word": " هو", "probability": 0.9833984375}, {"start": 33.02, "end": 33.66, "word": " ال", "probability": 0.5087890625}, {"start": 33.66, "end": 33.82, "word": " unit", "probability": 0.966796875}, {"start": 33.82, "end": 34.24, "word": " tangent", "probability": 0.90283203125}, {"start": 34.24, "end": 34.76, "word": " vector", "probability": 0.966796875}, {"start": 34.76, "end": 35.6, "word": " واحد", "probability": 0.896728515625}, {"start": 35.6, "end": 35.72, "word": " في", "probability": 0.95556640625}, {"start": 35.72, "end": 36.26, "word": " اتجاه", "probability": 0.9864501953125}, {"start": 36.26, "end": 36.72, "word": " ال", "probability": 0.92431640625}, {"start": 36.72, "end": 37.14, "word": " principle", "probability": 0.4365234375}, {"start": 37.14, "end": 37.7, "word": " unit", "probability": 0.96875}, {"start": 37.7, "end": 38.32, "word": " normal", "probability": 0.77734375}, {"start": 38.32, "end": 39.18, "word": " vector", "probability": 0.97900390625}, {"start": 39.18, "end": 39.84, "word": " كمان", "probability": 0.64794921875}, {"start": 39.84, "end": 40.5, "word": " سنراه", "probability": 0.8575439453125}, {"start": 40.5, "end": 41.14, "word": " لاحقا", "probability": 0.939697265625}, {"start": 41.14, "end": 42.52, "word": " الان", "probability": 0.65771484375}, {"start": 42.52, "end": 42.8, "word": " بدنا", "probability": 0.94873046875}, {"start": 42.8, "end": 43.4, "word": " نعرف", "probability": 0.9715169270833334}, {"start": 43.4, "end": 43.8, "word": " احنا", "probability": 0.9065755208333334}, {"start": 43.8, "end": 44.4, "word": " بنعرف", "probability": 0.9285481770833334}, {"start": 44.4, "end": 44.82, "word": " قولنا", "probability": 0.597412109375}, {"start": 44.82, "end": 45.0, "word": " لو", "probability": 0.9326171875}, {"start": 45.0, "end": 45.24, "word": " كان", "probability": 0.90966796875}, {"start": 45.24, "end": 45.74, "word": " لنا", "probability": 0.5909423828125}, {"start": 45.74, "end": 46.24, "word": " particle", "probability": 0.80615234375}, {"start": 46.24, "end": 47.22, "word": " بيسير", "probability": 0.77386474609375}], "temperature": 1.0}, {"id": 3, "seek": 6583, "start": 47.67, "end": 65.83, "text": "اللي هو عبر curve بال position vector R of T بتساوي كده اللي هو برسم النخط الآن في أثناء حركته اللي هو اللي بقيس اتجاه الحركة اللي هو ال unit tangent vector اللي قلنا عنه T", "tokens": [6027, 20292, 31439, 6225, 26890, 7605, 20666, 2535, 8062, 497, 295, 314, 39894, 3794, 995, 45865, 9122, 3215, 3224, 13672, 1829, 31439, 4724, 2288, 38251, 28239, 9778, 9566, 6024, 48506, 8978, 5551, 12984, 1863, 16606, 11331, 31747, 47395, 13672, 1829, 31439, 13672, 1829, 4724, 38436, 3794, 1975, 2655, 7435, 40294, 21542, 31747, 3660, 13672, 1829, 31439, 2423, 4985, 27747, 8062, 13672, 1829, 12174, 1211, 8315, 18871, 3224, 314], "avg_logprob": -0.1704030788463095, "compression_ratio": 1.6047904191616766, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 47.67, "end": 48.01, "word": "اللي", "probability": 0.2889404296875}, {"start": 48.01, "end": 48.31, "word": " هو", "probability": 0.9189453125}, {"start": 48.31, "end": 49.05, "word": " عبر", "probability": 0.961669921875}, {"start": 49.05, "end": 49.87, "word": " curve", "probability": 0.51953125}, {"start": 49.87, "end": 51.87, "word": " بال", "probability": 0.65576171875}, {"start": 51.87, "end": 53.37, "word": " position", "probability": 0.6220703125}, {"start": 53.37, "end": 53.81, "word": " vector", "probability": 0.96337890625}, {"start": 53.81, "end": 54.03, "word": " R", "probability": 0.311767578125}, {"start": 54.03, "end": 54.21, "word": " of", "probability": 0.31103515625}, {"start": 54.21, "end": 54.39, "word": " T", "probability": 0.9013671875}, {"start": 54.39, "end": 54.95, "word": " بتساوي", "probability": 0.8779296875}, {"start": 54.95, "end": 55.77, "word": " كده", "probability": 0.8204752604166666}, {"start": 55.77, "end": 56.27, "word": " اللي", "probability": 0.901611328125}, {"start": 56.27, "end": 56.61, "word": " هو", "probability": 0.94482421875}, {"start": 56.61, "end": 57.29, "word": " برسم", "probability": 0.9742838541666666}, {"start": 57.29, "end": 58.15, "word": " النخط", "probability": 0.8349609375}, {"start": 58.15, "end": 58.97, "word": " الآن", "probability": 0.6588134765625}, {"start": 58.97, "end": 59.15, "word": " في", "probability": 0.748046875}, {"start": 59.15, "end": 59.55, "word": " أثناء", "probability": 0.9234619140625}, {"start": 59.55, "end": 60.19, "word": " حركته", "probability": 0.9817708333333334}, {"start": 60.19, "end": 60.93, "word": " اللي", "probability": 0.982666015625}, {"start": 60.93, "end": 61.17, "word": " هو", "probability": 0.93359375}, {"start": 61.17, "end": 61.65, "word": " اللي", "probability": 0.931640625}, {"start": 61.65, "end": 62.07, "word": " بقيس", "probability": 0.9274088541666666}, {"start": 62.07, "end": 62.47, "word": " اتجاه", "probability": 0.9566650390625}, {"start": 62.47, "end": 63.03, "word": " الحركة", "probability": 0.9853515625}, {"start": 63.03, "end": 63.21, "word": " اللي", "probability": 0.953369140625}, {"start": 63.21, "end": 63.77, "word": " هو", "probability": 0.98095703125}, {"start": 63.77, "end": 63.91, "word": " ال", "probability": 0.9169921875}, {"start": 63.91, "end": 64.13, "word": " unit", "probability": 0.92724609375}, {"start": 64.13, "end": 64.53, "word": " tangent", "probability": 0.96533203125}, {"start": 64.53, "end": 64.91, "word": " vector", "probability": 0.97265625}, {"start": 64.91, "end": 65.05, "word": " اللي", "probability": 0.966796875}, {"start": 65.05, "end": 65.29, "word": " قلنا", "probability": 0.8289388020833334}, {"start": 65.29, "end": 65.57, "word": " عنه", "probability": 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0.83837890625}, {"start": 72.42, "end": 73.1, "word": " لل", "probability": 0.5625}, {"start": 73.1, "end": 74.16, "word": " curve", "probability": 0.91455078125}, {"start": 74.16, "end": 74.54, "word": " او", "probability": 0.87841796875}, {"start": 74.54, "end": 76.16, "word": " للمصار", "probability": 0.8524169921875}, {"start": 76.16, "end": 76.42, "word": " اللى", "probability": 0.976806640625}, {"start": 76.42, "end": 76.54, "word": " هو", "probability": 0.5673828125}, {"start": 76.54, "end": 76.68, "word": " اللى", "probability": 0.98681640625}, {"start": 76.68, "end": 77.24, "word": " بيجيسه", "probability": 0.9898274739583334}, {"start": 77.24, "end": 77.5, "word": " اللى", "probability": 0.929931640625}, {"start": 77.5, "end": 77.8, "word": " هو", "probability": 0.966796875}, {"start": 77.8, "end": 78.1, "word": " ال", "probability": 0.8974609375}, {"start": 78.1, "end": 78.72, "word": " normal", "probability": 0.5830078125}, {"start": 78.72, "end": 79.42, "word": " او", "probability": 0.943603515625}, {"start": 79.42, "end": 79.56, "word": " ال", "probability": 0.94677734375}, {"start": 79.56, "end": 80.0, "word": " principle", "probability": 0.4482421875}, {"start": 80.0, "end": 81.1, "word": " unit", "probability": 0.92578125}, {"start": 81.1, "end": 81.54, "word": " normal", "probability": 0.7529296875}, {"start": 81.54, "end": 82.1, "word": " vector", "probability": 0.95556640625}, {"start": 82.1, "end": 83.22, "word": " اللى", "probability": 0.984130859375}, {"start": 83.22, "end": 83.68, "word": " هو", "probability": 0.95166015625}, {"start": 83.68, "end": 85.26, "word": " دائما", "probability": 0.8942057291666666}, {"start": 85.26, "end": 86.08, "word": " باتجاه", "probability": 0.8199462890625}, {"start": 86.08, "end": 86.8, "word": " باتجاه", "probability": 0.843994140625}, {"start": 86.8, "end": 87.4, "word": " اللى", "probability": 0.98486328125}, {"start": 87.4, "end": 87.7, "word": " هو", "probability": 0.982421875}, {"start": 87.7, "end": 88.66, "word": " مركز", "probability": 0.9793294270833334}, {"start": 88.66, "end": 89.04, "word": " اللى", "probability": 0.9736328125}, {"start": 89.04, "end": 89.22, "word": " هو", "probability": 0.958984375}, {"start": 89.22, "end": 89.36, "word": " ال", "probability": 0.55810546875}, {"start": 89.36, "end": 89.7, "word": " circle", "probability": 0.904296875}, {"start": 89.7, "end": 89.96, "word": " of", "probability": 0.9384765625}, {"start": 89.96, "end": 90.42, "word": " curvature", "probability": 0.9814453125}, {"start": 90.42, "end": 90.7, "word": " اللى", "probability": 0.992431640625}, {"start": 90.7, "end": 91.1, "word": " حكينا", "probability": 0.9627685546875}, {"start": 91.1, "end": 91.48, "word": " عنها", "probability": 0.942626953125}, {"start": 91.48, "end": 91.94, "word": " المرة", "probability": 0.96337890625}, {"start": 91.94, "end": 92.4, "word": " الماضية", "probability": 0.9825846354166666}], "temperature": 1.0}, {"id": 5, "seek": 9961, "start": 92.73, "end": 99.61, "text": "الان في عنا vector ثالث ايضا بوس في حركة ال particle او مهم لحركة ال market ال particle", "tokens": [6027, 7649, 8978, 6225, 8315, 8062, 38637, 6027, 12984, 1975, 1829, 11242, 995, 4724, 41779, 8978, 11331, 31747, 3660, 2423, 12359, 1975, 2407, 3714, 16095, 5296, 5016, 31747, 3660, 2423, 2142, 2423, 12359], "avg_logprob": -0.2637867734712713, "compression_ratio": 1.29, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 92.73, "end": 93.31, "word": "الان", "probability": 0.7734375}, {"start": 93.31, "end": 93.47, "word": " في", "probability": 0.578125}, {"start": 93.47, "end": 93.73, "word": " عنا", "probability": 0.56597900390625}, {"start": 93.73, "end": 94.07, "word": " vector", "probability": 0.580078125}, {"start": 94.07, "end": 94.69, "word": " ثالث", "probability": 0.94384765625}, {"start": 94.69, "end": 95.05, "word": " ايضا", "probability": 0.8875732421875}, {"start": 95.05, "end": 95.37, "word": " بوس", "probability": 0.583251953125}, {"start": 95.37, "end": 95.51, "word": " في", "probability": 0.595703125}, {"start": 95.51, "end": 95.93, "word": " حركة", "probability": 0.9689127604166666}, {"start": 95.93, "end": 96.07, "word": " ال", "probability": 0.64453125}, {"start": 96.07, "end": 96.67, "word": " particle", "probability": 0.7421875}, {"start": 96.67, "end": 97.31, "word": " او", "probability": 0.83251953125}, {"start": 97.31, "end": 97.59, "word": " مهم", "probability": 0.944091796875}, {"start": 97.59, "end": 98.11, "word": " لحركة", "probability": 0.8887939453125}, {"start": 98.11, "end": 98.23, "word": " ال", "probability": 0.80810546875}, {"start": 98.23, "end": 98.61, "word": " market", "probability": 0.90673828125}, {"start": 98.61, "end": 99.15, "word": " ال", "probability": 0.4619140625}, {"start": 99.15, "end": 99.61, "word": " particle", "probability": 0.93359375}], "temperature": 1.0}, {"id": 6, "seek": 12722, "start": 100.32, "end": 127.22, "text": "اللي هو ال vector اللي بيكون عمودي على اللي هو ال plane تبع ال T و ال N اللي هو عبارة عن ال T cross N اللي هو اللي بنسميه احنا او هنسميه ال by normal vector اذا الأن بنعرف حاجة اسمها ال by normal vector of a curve in space SB بيستوي T cross N وهذا عبارة عن unit vector orthogonal to both T and N", "tokens": [6027, 20292, 31439, 2423, 8062, 13672, 1829, 4724, 1829, 30544, 6225, 2304, 2407, 16254, 15844, 13672, 1829, 31439, 2423, 5720, 6055, 3555, 3615, 2423, 314, 4032, 2423, 426, 13672, 1829, 31439, 6225, 3555, 9640, 3660, 18871, 2423, 314, 3278, 426, 13672, 1829, 31439, 13672, 1829, 44945, 38251, 1829, 3224, 1975, 5016, 8315, 1975, 2407, 8032, 1863, 38251, 1829, 3224, 2423, 538, 2710, 8062, 1975, 15730, 16247, 1863, 44945, 3615, 28480, 11331, 26108, 3660, 24525, 2304, 11296, 2423, 538, 2710, 8062, 295, 257, 7605, 294, 1901, 26944, 4724, 1829, 14851, 2407, 1829, 314, 3278, 426, 37037, 15730, 6225, 3555, 9640, 3660, 18871, 4985, 8062, 41488, 281, 1293, 314, 293, 426], "avg_logprob": -0.21278409050269562, "compression_ratio": 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مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث", "tokens": [2407, 3224, 15730, 1975, 12610, 3714, 25720, 38637, 7649, 1829, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 8978, 18863, 2304, 23328, 16712, 7649, 1829, 4032, 3224, 15730, 1975, 12610, 3714, 25720, 38637, 7649, 1829, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 3714, 25720, 38637, 6027, 12984, 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"probability": 0.9853515625}, {"start": 1745.54, "end": 1745.76, "word": " زي", "probability": 0.949951171875}, {"start": 1745.76, "end": 1745.8, "word": " ما", "probability": 0.9736328125}, {"start": 1745.8, "end": 1746.3, "word": " أخدنا", "probability": 0.864013671875}, {"start": 1746.3, "end": 1747.18, "word": " سابقا", "probability": 0.97509765625}], "temperature": 1.0}, {"id": 78, "seek": 176512, "start": 1747.4, "end": 1765.12, "text": "الان هذا اللي بين الـ unit tangent vector و الـ binormal بنسميه rectifying plane هذا rectifying plane معدلته سهلة هذه النقطة معروفة وهي ال normal على ال plane المعروف و بتعرفوا معدلته", "tokens": [6027, 7649, 23758, 13672, 1829, 49374, 2423, 39184, 4985, 27747, 8062, 4032, 2423, 39184, 5171, 24440, 44945, 38251, 1829, 3224, 11048, 5489, 5720, 23758, 11048, 5489, 5720, 20449, 3215, 1211, 47395, 8608, 3224, 37977, 29538, 28239, 47432, 3660, 20449, 32887, 5172, 3660, 37037, 1829, 2423, 2710, 15844, 2423, 5720, 9673, 3615, 32887, 5172, 4032, 39894, 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"end": 1845.49, "word": " Circle", "probability": 0.5234375}, {"start": 1845.49, "end": 1845.77, "word": " of", "probability": 0.908203125}, {"start": 1845.77, "end": 1846.41, "word": " curvature", "probability": 0.64013671875}], "temperature": 1.0}, {"id": 83, "seek": 187969, "start": 1850.65, "end": 1879.69, "text": "الانحناء او ال curvature عند النقطة المذكورة والان هذا ال B هو اللي بيحدد لي ال twisting لمين لل particle او للقطار لما يتحرك بيصيرله طبعا حركة حركة الآن بعمل اللي هو مش عارف ده انا جاعد بحرك وانتوا مش شايفيني اللي هو بعمل اللي هو ال twisting للقطار القطار هي و هيك بعمل هيك هيك هيك هذا ال twisting اللي بجيسه مين اللي هو ال B", "tokens": [6027, 7649, 5016, 1863, 16606, 1975, 2407, 2423, 37638, 43242, 28239, 47432, 3660, 9673, 8848, 4117, 13063, 3660, 16070, 7649, 23758, 2423, 363, 31439, 13672, 1829, 4724, 1829, 24401, 3215, 32239, 2423, 34491, 32767, 9957, 24976, 12359, 1975, 2407, 24976, 47432, 9640, 5296, 15042, 7251, 2655, 5016, 31747, 4724, 1829, 9381, 13546, 43761, 23032, 3555, 3615, 995, 11331, 31747, 3660, 11331, 31747, 3660, 6024, 48506, 4724, 25957, 1211, 13672, 1829, 31439, 37893, 6225, 9640, 5172, 11778, 3224, 1975, 8315, 10874, 995, 22488, 4724, 5016, 31747, 4032, 7649, 2655, 14407, 37893, 13412, 995, 33911, 9957, 1829, 13672, 1829, 31439, 4724, 25957, 1211, 13672, 1829, 31439, 2423, 34491, 24976, 47432, 9640, 25062, 9566, 9640, 39896, 4032, 39896, 4117, 4724, 25957, 1211, 39896, 4117, 39896, 4117, 39896, 4117, 23758, 2423, 34491, 13672, 1829, 4724, 7435, 1829, 3794, 3224, 3714, 9957, 13672, 1829, 31439, 2423, 363], "avg_logprob": -0.27469405886176584, "compression_ratio": 2.0884615384615386, "no_speech_prob": 0.00018775463104248047, "words": [{"start": 1850.65, "end": 1851.45, "word": "الانحناء", "probability": 0.702978515625}, {"start": 1851.45, "end": 1851.63, "word": " او", "probability": 0.6402587890625}, {"start": 1851.63, "end": 1851.83, "word": " ال", "probability": 0.45751953125}, {"start": 1851.83, "end": 1852.25, "word": " curvature", "probability": 0.302490234375}, {"start": 1852.25, "end": 1853.21, "word": " عند", "probability": 0.857421875}, {"start": 1853.21, "end": 1853.65, "word": " النقطة", "probability": 0.9269205729166666}, {"start": 1853.65, "end": 1854.25, "word": " المذكورة", "probability": 0.97548828125}, {"start": 1854.25, "end": 1855.13, "word": " والان", "probability": 0.613037109375}, {"start": 1855.13, "end": 1855.55, "word": " هذا", "probability": 0.7587890625}, {"start": 1855.55, "end": 1855.77, "word": " ال", "probability": 0.919921875}, {"start": 1855.77, "end": 1856.01, "word": " B", "probability": 0.251953125}, {"start": 1856.01, "end": 1856.63, "word": " هو", "probability": 0.94921875}, {"start": 1856.63, "end": 1857.01, "word": " اللي", "probability": 0.830810546875}, {"start": 1857.01, "end": 1857.87, "word": " بيحدد", "probability": 0.91943359375}, {"start": 1857.87, "end": 1858.11, "word": " لي", "probability": 0.378173828125}, {"start": 1858.11, "end": 1858.35, "word": " ال", "probability": 0.340087890625}, {"start": 1858.35, "end": 1858.89, "word": " twisting", "probability": 0.9072265625}, {"start": 1858.89, "end": 1859.83, "word": " لمين", "probability": 0.6072998046875}, {"start": 1859.83, "end": 1860.13, "word": " لل", "probability": 0.52880859375}, {"start": 1860.13, "end": 1860.63, "word": " particle", "probability": 0.70361328125}, {"start": 1860.63, "end": 1861.53, "word": " او", "probability": 0.823486328125}, {"start": 1861.53, "end": 1862.17, "word": " للقطار", "probability": 0.8523763020833334}, {"start": 1862.17, "end": 1862.49, "word": " لما", "probability": 0.90185546875}, {"start": 1862.49, "end": 1863.15, "word": " يتحرك", "probability": 0.89306640625}, {"start": 1863.15, "end": 1863.65, "word": " بيصيرله", "probability": 0.68662109375}, {"start": 1863.65, "end": 1863.97, "word": " طبعا", "probability": 0.920654296875}, {"start": 1863.97, "end": 1864.51, "word": " حركة", "probability": 0.9674479166666666}, {"start": 1864.51, "end": 1865.27, "word": " حركة", "probability": 0.8948567708333334}, {"start": 1865.27, "end": 1865.65, "word": " الآن", "probability": 0.6494140625}, {"start": 1865.65, "end": 1866.57, "word": " بعمل", "probability": 0.8650716145833334}, {"start": 1866.57, "end": 1867.07, "word": " اللي", "probability": 0.81787109375}, {"start": 1867.07, "end": 1867.47, "word": " هو", "probability": 0.7646484375}, {"start": 1867.47, "end": 1868.01, "word": " مش", "probability": 0.77392578125}, {"start": 1868.01, "end": 1868.29, "word": " عارف", "probability": 0.9931640625}, {"start": 1868.29, "end": 1868.69, "word": " ده", "probability": 0.4019775390625}, {"start": 1868.69, "end": 1868.83, "word": " انا", "probability": 0.73486328125}, {"start": 1868.83, "end": 1869.07, "word": " جاعد", "probability": 0.5205891927083334}, {"start": 1869.07, "end": 1869.39, "word": " بحرك", "probability": 0.970703125}, {"start": 1869.39, "end": 1869.67, "word": " وانتوا", "probability": 0.65631103515625}, {"start": 1869.67, "end": 1869.77, "word": " مش", "probability": 0.966796875}, {"start": 1869.77, "end": 1870.35, "word": " شايفيني", "probability": 0.8986328125}, {"start": 1870.35, "end": 1870.87, "word": " اللي", "probability": 0.96630859375}, {"start": 1870.87, "end": 1871.01, "word": " هو", "probability": 0.98779296875}, {"start": 1871.01, "end": 1871.53, "word": " بعمل", "probability": 0.8876953125}, {"start": 1871.53, "end": 1872.17, "word": " اللي", "probability": 0.6806640625}, {"start": 1872.17, "end": 1872.49, "word": " هو", "probability": 0.9912109375}, {"start": 1872.49, "end": 1872.97, "word": " ال", "probability": 0.60498046875}, {"start": 1872.97, "end": 1873.31, "word": " twisting", "probability": 0.9814453125}, {"start": 1873.31, "end": 1874.19, "word": " للقطار", "probability": 0.96826171875}, {"start": 1874.19, "end": 1874.93, "word": " القطار", "probability": 0.9539388020833334}, {"start": 1874.93, "end": 1875.13, "word": " هي", "probability": 0.9013671875}, {"start": 1875.13, "end": 1875.23, "word": " و", "probability": 0.1259765625}, {"start": 1875.23, "end": 1875.49, "word": " هيك", "probability": 0.939208984375}, {"start": 1875.49, "end": 1875.81, "word": " بعمل", "probability": 0.8650716145833334}, {"start": 1875.81, "end": 1876.11, "word": " هيك", "probability": 0.9404296875}, {"start": 1876.11, "end": 1876.39, "word": " هيك", "probability": 0.9140625}, {"start": 1876.39, "end": 1876.63, "word": " هيك", "probability": 0.865478515625}, {"start": 1876.63, "end": 1876.83, "word": " هذا", "probability": 0.185791015625}, {"start": 1876.83, "end": 1877.23, "word": " ال", "probability": 0.413818359375}, {"start": 1877.23, "end": 1877.63, "word": " twisting", "probability": 0.96337890625}, {"start": 1877.63, "end": 1877.91, "word": " اللي", "probability": 0.9638671875}, {"start": 1877.91, "end": 1878.39, "word": " بجيسه", "probability": 0.78505859375}, {"start": 1878.39, "end": 1878.75, "word": " مين", "probability": 0.9208984375}, {"start": 1878.75, "end": 1879.15, "word": " اللي", "probability": 0.96240234375}, {"start": 1879.15, "end": 1879.39, "word": " هو", "probability": 0.9833984375}, {"start": 1879.39, "end": 1879.55, "word": " ال", "probability": 0.95458984375}, {"start": 1879.55, "end": 1879.69, "word": " B", "probability": 0.89404296875}], "temperature": 1.0}, {"id": 84, "seek": 188567, "start": 1879.97, "end": 1885.67, "text": "أو هكذا ننهي المحاضرة و بارك الله فيكم و إلى لقاء آخر", "tokens": [10721, 2407, 8032, 4117, 15730, 8717, 1863, 3224, 1829, 9673, 5016, 46958, 25720, 4032, 4724, 9640, 4117, 21984, 8978, 24793, 4032, 30731, 5296, 4587, 16606, 19753, 34740], "avg_logprob": -0.35825894134385244, "compression_ratio": 1.1728395061728396, "no_speech_prob": 0.0, "words": [{"start": 1879.97, "end": 1880.35, "word": "أو", "probability": 0.717041015625}, {"start": 1880.35, "end": 1880.97, "word": " هكذا", "probability": 0.5624186197916666}, {"start": 1880.97, "end": 1881.71, "word": " ننهي", "probability": 0.68402099609375}, {"start": 1881.71, "end": 1882.49, "word": " المحاضرة", "probability": 0.9478759765625}, {"start": 1882.49, "end": 1882.91, "word": " و", "probability": 0.6787109375}, {"start": 1882.91, "end": 1883.39, "word": " بارك", "probability": 0.741455078125}, {"start": 1883.39, "end": 1883.61, "word": " الله", "probability": 0.94580078125}, {"start": 1883.61, "end": 1884.15, "word": " فيكم", "probability": 0.904296875}, {"start": 1884.15, "end": 1884.33, "word": " و", "probability": 0.74169921875}, {"start": 1884.33, "end": 1884.61, "word": " إلى", "probability": 0.86962890625}, {"start": 1884.61, "end": 1885.19, "word": " لقاء", "probability": 0.8015950520833334}, {"start": 1885.19, "end": 1885.67, "word": " آخر", "probability": 0.81689453125}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 1890.1971875, "duration_after_vad": 1875.2031249999982}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/HMa43OLUsAQ_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/HMa43OLUsAQ_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6f1d0ccf8ee8a5c35472cec3fccd24b048904095
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/HMa43OLUsAQ_raw.srt
@@ -0,0 +1,1672 @@
+1
+00:00:00,440 --> 00:00:03,720
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:03,720 --> 00:00:10,840
+وبركاته هذه المحاضرة رقم 12 نساق تفاضل وتكامل جيم
+
+3
+00:00:10,840 --> 00:00:16,500
+وهي المحاضرة الخامسة في chapter 13 وهي الأخيرة في
+
+4
+00:00:16,500 --> 00:00:21,640
+chapter 13 وهي تحت عنوان tangential and normal
+
+5
+00:00:21,640 --> 00:00:25,500
+component of accelerationيعني بدنا اللي هو العجلة
+
+6
+00:00:25,500 --> 00:00:32,100
+ان نعملها على او نحليلها على two components واحد
+
+7
+00:00:32,100 --> 00:00:35,720
+في اتجاه اللي هو ال unit tangent vector واحد في
+
+8
+00:00:35,720 --> 00:00:40,500
+اتجاه ال principle unit normal vector كمان سنراه
+
+9
+00:00:40,500 --> 00:00:45,740
+لاحقا الان بدنا نعرف احنا بنعرف قولنا لو كان لنا
+
+10
+00:00:45,740 --> 00:00:53,370
+particle بيسيراللي هو عبر curve بال position
+
+11
+00:00:53,370 --> 00:00:58,970
+vector R of T بتساوي كده اللي هو برسم النخط الآن
+
+12
+00:00:58,970 --> 00:01:03,210
+في أثناء حركته اللي هو اللي بقيس اتجاه الحركة اللي
+
+13
+00:01:03,210 --> 00:01:07,260
+هو ال unit tangent vector اللي قلنا عنه Tواللي
+
+14
+00:01:07,260 --> 00:01:13,100
+بيجيس اللى هو مدى التواؤه او اللى هو ال turning لل
+
+15
+00:01:13,100 --> 00:01:18,100
+curve او للمصار اللى هو اللى بيجيسه اللى هو ال
+
+16
+00:01:18,100 --> 00:01:23,680
+normal او ال principle unit normal vector اللى هو
+
+17
+00:01:23,680 --> 00:01:29,700
+دائما باتجاه باتجاه اللى هو مركز اللى هو ال circle
+
+18
+00:01:29,700 --> 00:01:33,470
+of curvature اللى حكينا عنها المرة الماضيةالان في
+
+19
+00:01:33,470 --> 00:01:37,310
+عنا vector ثالث ايضا بوس في حركة ال particle او
+
+20
+00:01:37,310 --> 00:01:42,060
+مهم لحركة ال market ال particleاللي هو ال vector
+
+21
+00:01:42,060 --> 00:01:47,260
+اللي بيكون عمودي على اللي هو ال plane تبع ال T و
+
+22
+00:01:47,260 --> 00:01:50,500
+ال N اللي هو عبارة عن ال T cross N اللي هو اللي
+
+23
+00:01:50,500 --> 00:01:55,940
+بنسميه احنا او هنسميه ال by normal vector اذا الأن
+
+24
+00:01:55,940 --> 00:01:59,360
+بنعرف حاجة اسمها ال by normal vector of a curve in
+
+25
+00:01:59,360 --> 00:02:04,080
+space SB بيستوي T cross N وهذا عبارة عن unit
+
+26
+00:02:04,080 --> 00:02:09,720
+vector orthogonal to both T and Nالعنوان الـ three
+
+27
+00:02:09,720 --> 00:02:14,580
+vectors هدول ال T و ال N و ال B في الواقع هم بعرف
+
+28
+00:02:14,580 --> 00:02:17,440
+حركة تدفع حركة تتحرك على حركة تتحرك على حركة تتحرك
+
+29
+00:02:17,440 --> 00:02:18,040
+على حركة تتحرك على حركة تتحرك على حركة تتحرك على
+
+30
+00:02:18,040 --> 00:02:19,420
+حركة تتحرك على حركة تتحرك على حركة تتحرك على حركة
+
+31
+00:02:19,420 --> 00:02:22,700
+تتحرك على حركة تتحرك على حركة تتحرك على حركة تتحرك
+
+32
+00:02:22,700 --> 00:02:23,020
+على حركة تتحرك على حركة تتحرك على حركة تتحرك على
+
+33
+00:02:23,020 --> 00:02:23,280
+حركة تتحرك على حركة تتحرك على حركة تتحرك على حركة
+
+34
+00:02:23,280 --> 00:02:26,340
+تتحرك على حركة تتحرك على حركة تتحرك على حركة تتحرك
+
+35
+00:02:26,340 --> 00:02:30,070
+على حركة تيعني الآن أثناء سير اللي هو ال vector في
+
+36
+00:02:30,070 --> 00:02:33,630
+ال space .. الجزء في ال space أو الجسم في ال space
+
+37
+00:02:33,630 --> 00:02:37,830
+هذا عندي مجسم صار المجسم هذا بيسير على اللي هو ال
+
+38
+00:02:37,830 --> 00:02:41,370
+curve هذا في اللي هو ال T اللي هو ال unit tangent
+
+39
+00:02:41,370 --> 00:02:47,400
+vector بجيس اللي هو اتجاه الحركةالان الان بيجيسلي
+
+40
+00:02:47,400 --> 00:02:52,280
+مدى التواق اللى هو او ال turning تبع مين تبع اللى
+
+41
+00:02:52,280 --> 00:02:57,640
+هو مصار ال particle جداش ال turning بيكون من خلال
+
+42
+00:02:57,640 --> 00:03:01,620
+ال normal اللى هو دايما بيكون باتجاه اللى هو ال
+
+43
+00:03:01,620 --> 00:03:06,120
+center لل circle of curvature عند لحظة ما لا اللى
+
+44
+00:03:06,120 --> 00:03:13,040
+هو ال particleالان الجسم هذا اثناء سيرو الان ال
+
+45
+00:03:13,040 --> 00:03:16,540
+normal هذا بجيس لل twisting تبع اللي هو لو اتخيلنا
+
+46
+00:03:16,540 --> 00:03:22,200
+جطار ماشي جطار ماشي هى هذا بشوف كيف حركته هذا بشوف
+
+47
+00:03:22,200 --> 00:03:26,260
+كيف التواؤل اللي هو الان الان هذا اللي هو ال
+
+48
+00:03:26,260 --> 00:03:30,800
+normal عليهم بجيس لل twisting تبع اللي هو كجسم
+
+49
+00:03:30,800 --> 00:03:34,670
+كامل الان في الفراغ بجيساللي هو ايش .. كيف ال
+
+50
+00:03:34,670 --> 00:03:39,290
+twisting بيصير ل .. اللي هو القطر او ال particle
+
+51
+00:03:39,290 --> 00:03:44,510
+اللي بيصير اذا هذه ال T و ال N و ال B هي فعلا ال
+
+52
+00:03:44,510 --> 00:03:48,950
+frame بوصف حركة اللي هو الجسم او ال particle اللي
+
+53
+00:03:48,950 --> 00:03:55,910
+بيكون سائر على طريق ما بيدالة معينةالان الـ Normal
+
+54
+00:03:55,910 --> 00:03:59,350
+طبعا احنا عرفناه اللي هو عبارة عن واحدة ركابة في
+
+55
+00:03:59,350 --> 00:04:04,250
+DT by DS و الـ T اللي هو عبارة عن DR by DS او
+
+56
+00:04:04,250 --> 00:04:09,530
+بواسطة تلية الـ T معناته V ع المجنتيوت للـ V و هذا
+
+57
+00:04:09,530 --> 00:04:14,310
+الـ DT by DT ع المجنتيوت تبعها اللي هنشوفه .. اللي
+
+58
+00:04:14,310 --> 00:04:23,280
+شوفنا سابقا عندي الآن نشوف كيف اللي هونوجد حاجة
+
+59
+00:04:23,280 --> 00:04:27,240
+مهمة او جزء ثاني من الدراسة اللي هو حاجة اسمة
+
+60
+00:04:27,240 --> 00:04:30,920
+tangential and normal components of acceleration
+
+61
+00:04:30,920 --> 00:04:34,460
+tangential and normal components of acceleration
+
+62
+00:04:34,460 --> 00:04:39,000
+ايش اللي بنقصد فيه الان عندي اللي هو ال
+
+63
+00:04:39,000 --> 00:04:44,400
+acceleration العجلة العجلة بلزمني و هيكون مفيد ان
+
+64
+00:04:44,400 --> 00:04:49,800
+احللها الى متجهينمتعمدين متجهة في اتجاه او
+
+65
+00:04:49,800 --> 00:04:52,960
+component في اتجاه اللي هو ال unit tangent vector
+
+66
+00:04:52,960 --> 00:04:56,660
+و component اخرى في اتجاه عمودي عليها اللي هو ال
+
+67
+00:04:56,660 --> 00:04:59,800
+unit ال principle ال unit normal vector و خلينا
+
+68
+00:04:59,800 --> 00:05:04,040
+نشوف ايش اللي بنقوله احنا بنعرف ان ال velocity هي
+
+69
+00:05:04,040 --> 00:05:08,750
+عبارة عن dr by dtهو يساوي اللي هو لو استخدمنا ال
+
+70
+00:05:08,750 --> 00:05:12,930
+chain rule هي عبارة عن DR by DS في DS by DT اللي
+
+71
+00:05:12,930 --> 00:05:16,530
+هي بالساوي DR by DS زي ما عرفنا ان هي عبارة عن من؟
+
+72
+00:05:16,530 --> 00:05:23,390
+عن ال unit tangent vector D في DS by DT الان اللي
+
+73
+00:05:23,390 --> 00:05:28,340
+هي S اللي هي عبارة عن ال longالـ T هي عبارة عن
+
+74
+00:05:28,340 --> 00:05:34,900
+الزمن ثم
+
+75
+00:05:34,900 --> 00:05:41,840
+و يدفرنشيش both sides اللي هو انفاض
+
+76
+00:05:41,840 --> 00:05:47,040
+الفنسبة التي بيصير عندي Aلأن إذا فضلنا العجلة V
+
+77
+00:05:47,040 --> 00:05:51,460
+السرعة بتطلع من الـ acceleration بساوة DV بي DT
+
+78
+00:05:51,460 --> 00:05:55,360
+اللي هي بتفاضل هذه هذه بتساوي هذه بتفاضل الجهتين
+
+79
+00:05:55,360 --> 00:06:00,020
+بيصير D بي DT لت في DS بي DT حاصل ضغط دالتين اللي
+
+80
+00:06:00,020 --> 00:06:03,860
+هي التفاضل الأولى هي Fث ثانية زي التفاضل الثانية
+
+81
+00:06:03,860 --> 00:06:07,000
+في الأولى زي ما انتوا عارفين فبصير هذا D تربيع S
+
+82
+00:06:07,000 --> 00:06:08,440
+بي DT تربية
+
+83
+00:06:14,800 --> 00:06:21,420
+و يساوي وين رايح انا رايح اعملكم اللي هو two
+
+84
+00:06:21,420 --> 00:06:24,760
+components واحدة في اتجاه ال T واحدة في اتجاه ال N
+
+85
+00:06:24,760 --> 00:06:29,880
+و دعنا نشوف كيف تظهرالأن D تربيع S by DT تربيع في
+
+86
+00:06:29,880 --> 00:06:35,800
+اتجاه الـ T هذه ظهرت خلصنا منها زائد الان DS by DT
+
+87
+00:06:35,800 --> 00:06:39,900
+في DT by DT هذه بتستخدم اللي هو ال .. ال .. ال ..
+
+88
+00:06:39,900 --> 00:06:47,880
+ال chain rule بيصير DT by DS في DS by DT الان Y
+
+89
+00:06:47,880 --> 00:06:53,300
+ساوي هذه زي ما هي زائد DS by DT في كده الان هذه
+
+90
+00:06:53,300 --> 00:06:58,290
+اللي هي DT by DSالان هي بيساوي كابا فعلا ليش لأن
+
+91
+00:06:58,290 --> 00:07:04,850
+ان بتساوي dt by ds على كابا مظبوط على كابا رجعوا
+
+92
+00:07:04,850 --> 00:07:08,590
+لها لاش تكون خربطنا قبل هل جيت فعلا ان هي الان
+
+93
+00:07:08,590 --> 00:07:12,750
+بتساوي dt by ds عالميا على كابا واضح ايه بترجع
+
+94
+00:07:12,750 --> 00:07:16,270
+للقانون هذا و بتحفظوه ان شاء الله عساس ان هو تسهل
+
+95
+00:07:16,270 --> 00:07:20,950
+عليكم الأمور في كل الحساباتالان هذه زي ما هي بتنزل
+
+96
+00:07:20,950 --> 00:07:26,910
+لان هي انا غرضي هيك DS by DT الان في كابا في DS by
+
+97
+00:07:26,910 --> 00:07:31,770
+DT بتصير ايش هي DS by DT الكل تربيع في كابا في مين
+
+98
+00:07:31,770 --> 00:07:36,690
+في ال and اللي انا اصلا بتعمل ال a اللي هي عبارة
+
+99
+00:07:36,690 --> 00:07:40,210
+عن two components واحدة في اتجاه ال T واحدة في
+
+100
+00:07:40,210 --> 00:07:45,480
+اتجاه مين ال and هذه يا جماعة هذههي اللي بتسميها
+
+101
+00:07:45,480 --> 00:07:50,900
+AT اللي هي الـ tangential يعني ال tangential
+
+102
+00:07:50,900 --> 00:07:55,080
+acceleration او ال component في اتجاه T تبع ال
+
+103
+00:07:55,080 --> 00:07:58,480
+acceleration وهذا ع بعضها اللي بتسميها اللي هي ايش
+
+104
+00:07:58,480 --> 00:08:04,620
+اللي هي ال AN اللي هي ال normal component of
+
+105
+00:08:04,620 --> 00:08:09,780
+acceleration انشوف هذا الكلام if تحت ال N صرنا
+
+106
+00:08:09,780 --> 00:08:14,540
+مؤهلين ان نعرفإذا كتبت أكسلراشن فيكتور باسم a
+
+107
+00:08:14,540 --> 00:08:18,620
+بتساوي at في اتجاه الان زائد an في اتجاه الان
+
+108
+00:08:18,620 --> 00:08:23,020
+بيصير يعني كتبنا ال acceleration عبارة عن عدد في t
+
+109
+00:08:23,020 --> 00:08:29,440
+اللي هو ال unit tangent vector زائد an عدد في n
+
+110
+00:08:29,440 --> 00:08:33,500
+اللي هو ال principle unit normal vector صارت عند
+
+111
+00:08:33,500 --> 00:08:38,520
+ال a مكتوبة على صورةاللي هو اللي هو two components
+
+112
+00:08:38,520 --> 00:08:41,440
+واحدة في اتجاه ال T واحدة في اتجاه ال N هذا ال a
+
+113
+00:08:41,440 --> 00:08:45,220
+tangential من الحكي اللي قبل بشوية هيطلع اللي هو
+
+114
+00:08:45,220 --> 00:08:50,200
+لإنه قبل بشوية قولنا أنه أثبتنا أنه a هتكون هدف T
+
+115
+00:08:50,200 --> 00:08:56,030
+و هدف N أي إش بنكتب بساوي a عدد في T زائدعدد في N
+
+116
+00:08:56,030 --> 00:08:59,370
+معناته هذا هو ال .. ال .. ال .. العدد اللي حكينا
+
+117
+00:08:59,370 --> 00:09:03,090
+عنه اللي سمناه قبل شوية A T و هذا اللي سمناه A N
+
+118
+00:09:03,090 --> 00:09:07,690
+معناته انه صار عند ال A T بيساوي D تربيع أصبعي D
+
+119
+00:09:07,690 --> 00:09:12,310
+تربيع أو A N بيساوي كابا في هذا المقدار طيب احنا
+
+120
+00:09:12,310 --> 00:09:16,430
+كل أمورنا سهلة بتسير لما نحولها بدل من بال DS ب ..
+
+121
+00:09:16,430 --> 00:09:24,190
+ب .. اللي هي اللي نحولها ل ال V أو ال Tأو الـ R أو
+
+122
+00:09:24,190 --> 00:09:29,470
+الـ Acceleration فاهمين جصدي الـ AT أسهلنا إن الآن
+
+123
+00:09:29,470 --> 00:09:35,990
+نقول هي عبارة عن طبعا DS by DT DS by DT اللي هي ال
+
+124
+00:09:35,990 --> 00:09:40,070
+magnitude لمين؟ لـ V لما نقفض لهذه بتصير عبارة عن
+
+125
+00:09:40,070 --> 00:09:48,200
+D by DT للمagnitude لـ Vالـ AN هي عبارة عن كيف DS
+
+126
+00:09:48,200 --> 00:09:53,120
+by DT لكل تربيع برضه DS by DT هي مين ؟ هي عبارة عن
+
+127
+00:09:53,120 --> 00:09:58,450
+اللي هي الـ Speedلأن الـ R دي R by DT اللي هي الـ
+
+128
+00:09:58,450 --> 00:10:01,770
+R direction الـ S اللي هي الـ Length بعتمد على T
+
+129
+00:10:01,770 --> 00:10:06,050
+يعني هو كأنه هذه طول المسافة أما اللي هي إيش هذه
+
+130
+00:10:06,050 --> 00:10:11,350
+الـ V الـ R الإزاحة فلما أفضلت المسافة DS by DT
+
+131
+00:10:11,350 --> 00:10:15,210
+اللي هي المسافة بدلال الـ T فتطلع عندي اللي هو
+
+132
+00:10:15,210 --> 00:10:18,570
+Speedيعني اللي هو بيصيح ال magnitude ال V لكل
+
+133
+00:10:18,570 --> 00:10:23,930
+تربيع أو هذا DS by DT صارت ال AN هيقيمتها بوجدها
+
+134
+00:10:23,930 --> 00:10:28,530
+بالشكل هذا أو هاي AT هيقيمتها بوجدها بالشكل هذا
+
+135
+00:10:28,530 --> 00:10:32,570
+هذه الأولى بتسميها tangential والثانية بتسميها
+
+136
+00:10:32,570 --> 00:10:33,610
+normal scalar
+
+137
+00:10:36,880 --> 00:10:41,880
+Tangential Scalar Component of Acceleration الان
+
+138
+00:10:41,880 --> 00:10:48,620
+هذا اختصارا توضيح للي بنحكيه هى عندى ال a هو اتجاه
+
+139
+00:10:48,620 --> 00:10:52,620
+العجلة طبعا هى ال vertical ماشي هنا هى ال particle
+
+140
+00:10:52,620 --> 00:10:57,240
+هنا هى ال acceleration تبعته هى ال unit normal
+
+141
+00:10:57,240 --> 00:11:00,880
+vectorهذا unit tangent vector وهذا ال principal
+
+142
+00:11:00,880 --> 00:11:06,160
+unit normal vector الأكسلاريشين هذا بدي أحللها
+
+143
+00:11:06,160 --> 00:11:09,940
+واحدة هنا واحدة هنا التحليل اللي هنا هيطلع عند ال
+
+144
+00:11:09,940 --> 00:11:13,980
+AT التي في اتجاه ال T هو D تربيع أسوأ DT تربيع
+
+145
+00:11:13,980 --> 00:11:18,430
+اللي حكيناه هنا و اللي في اتجاه ال normal هذاالتي
+
+146
+00:11:18,430 --> 00:11:23,290
+حكيناها كابع في DS by DT تربيع هذا اللي هو ال A N
+
+147
+00:11:23,290 --> 00:11:27,390
+في اتجاه ال N و ال A T في اتجاه ال T بتصير ال A
+
+148
+00:11:27,390 --> 00:11:32,670
+بتساوي A T في اتجاه ال T مضروبة في ال T كمتجه و A
+
+149
+00:11:32,670 --> 00:11:36,710
+N في اتجاه زائد A N في اتجاه ال N دي مضروبة في N
+
+150
+00:11:36,710 --> 00:11:37,650
+كمتجه
+
+151
+00:11:39,970 --> 00:11:43,570
+Formula for calculating the normal components of
+
+152
+00:11:43,570 --> 00:11:49,590
+acceleration الان أسهلنا لو رجعنا شوية ال AT سهل
+
+153
+00:11:49,590 --> 00:11:55,410
+إيجادها ليش ديتها مش معطينا ال R بوجد الان DR by
+
+154
+00:11:55,410 --> 00:11:59,530
+DT بطلع من ال velocity باخده ال magnitude بتطلع ال
+
+155
+00:11:59,530 --> 00:12:04,070
+speed بقى فاضله بالنسبة ليه بطلع ال AT اذا سهل
+
+156
+00:12:04,070 --> 00:12:07,970
+المأمر ضال هذه اللي بتخلق شوية في الحسابات بنجيبها
+
+157
+00:12:07,970 --> 00:12:12,620
+بطريقة ثانيةالتي هي الطريقة كيف أننا نكون هنا على
+
+158
+00:12:12,620 --> 00:12:15,860
+طول بوجد الأودادات الات زي قبل بشهية بالطريقة
+
+159
+00:12:15,860 --> 00:12:19,940
+السهلة بوجد الان عبارة عن الجدر التربيعي للمجنتيود
+
+160
+00:12:19,940 --> 00:12:25,480
+للات تربيع ناقص ات لكل تربيع لإنه عند الأي
+
+161
+00:12:25,480 --> 00:12:29,620
+بالساوية هذه زائد هذه فالمجنتيود لهذه هو عبارة عن
+
+162
+00:12:29,620 --> 00:12:33,580
+الجدر التربيعي لهذه تربيع زائد هذه تربيع ومن ثم
+
+163
+00:12:33,580 --> 00:12:40,040
+الانبساوي الجدر التربية الى الات تربية ناقص الات
+
+164
+00:12:40,040 --> 00:12:45,360
+تربيةالان لو جينا في ال circle ع ال circle لو كانت
+
+165
+00:12:45,360 --> 00:12:48,860
+الحركة هي عندي اللي هو ال .. ال .. ال .. ال
+
+166
+00:12:48,860 --> 00:12:53,300
+particle بيمشي على circle هذه الآن في اتجاه ال T
+
+167
+00:12:53,300 --> 00:12:57,500
+وهذه في اتجاه مين الان حللنا مين ال acceleration
+
+168
+00:12:57,500 --> 00:13:02,680
+لما نحلل ال acceleration بتتحلل ل D تربيع أسوأ و D
+
+169
+00:13:02,680 --> 00:13:07,420
+تربيع هادى اللي هي في اتجاه اللي هو ال .. ال unit
+
+170
+00:13:07,420 --> 00:13:12,980
+tangent vectorالتي نسميها AT التي هي D absolute
+
+171
+00:13:12,980 --> 00:13:18,300
+value of V by DT او D ال speed by DT وهذه اللي هي
+
+172
+00:13:18,300 --> 00:13:21,020
+عبارة عن كابة في absolute value of V تربيع في D
+
+173
+00:13:21,020 --> 00:13:26,720
+جهة ال N او ال magnitude of V تربيع على ال role
+
+174
+00:13:26,720 --> 00:13:32,060
+اللي هي عبارة عن واحد على كابة في Nالان هي اللي هي
+
+175
+00:13:32,060 --> 00:13:37,160
+ال components ال a tangential و ال a n اللي هي هذه
+
+176
+00:13:37,160 --> 00:13:42,140
+ال a n و هذه ال a tangential بالنسبة لل particle
+
+177
+00:13:42,140 --> 00:13:51,460
+اللي عجلته a على دائرة طيب لو كان سرعت اللي هي ال
+
+178
+00:13:51,460 --> 00:13:58,400
+particle ثابتة لو سرعته ثابتة a على الدائرة سرعته
+
+179
+00:13:58,400 --> 00:14:02,140
+ثابتةاللي هو ال magnitude لسرعته A و ال magnitude
+
+180
+00:14:02,140 --> 00:14:05,740
+ل V اللي هو A شماله ثابت يعني هو غير اتجاهه طبعا
+
+181
+00:14:05,740 --> 00:14:10,360
+اللي هو اتجاه السرعة بتغير لكن لو كان Zero على
+
+182
+00:14:10,360 --> 00:14:15,430
+اللي هو الدائرة ب speed ثابتعلى طول مدام بسبيد
+
+183
+00:14:15,430 --> 00:14:20,350
+ثابت إذا حيطلع عندي D تربيع أسبوع DT تربيع أو هذه
+
+184
+00:14:20,350 --> 00:14:24,690
+اللي هي عبارة عن D لسبيد by DT هيطلع سفر إذا مش
+
+185
+00:14:24,690 --> 00:14:30,150
+هيكون في عندي two tangential AT هيكون كل اتجاه
+
+186
+00:14:30,150 --> 00:14:35,650
+العجل أشماله اتجاه المركزلأن كل شيء سيكون عادى لأن
+
+187
+00:14:35,650 --> 00:14:39,750
+هذه سيكون صفر و سيكون الـ acceleration اتجاهها فقط
+
+188
+00:14:39,750 --> 00:14:44,230
+هنا إذا في حالة اللي هو الـ speed constant على
+
+189
+00:14:44,230 --> 00:14:50,330
+الدائرة بيكون اتجاه العجلة دائما في اتجاه المين
+
+190
+00:14:50,330 --> 00:14:54,770
+المركز يعني يوم ما تشوف يوم ما تشوف الان particle
+
+191
+00:14:54,770 --> 00:15:01,030
+عجلته اللي هو مش في اتجاه المركز عقول هتقول هذا
+
+192
+00:15:01,030 --> 00:15:06,120
+ممشيش بسرعة ب speed ثابتلأ سبيده متغير لأن لو
+
+193
+00:15:06,120 --> 00:15:12,520
+سبيده ثابت بيكون اتجاه عجلت وين بس باتجاه المركز
+
+194
+00:15:12,520 --> 00:15:17,520
+او باتجاه ال normal يعني اي particle ال a له
+
+195
+00:15:17,520 --> 00:15:22,880
+اتجاهه مش في اتجاه المركز معناته ال speed is not
+
+196
+00:15:22,880 --> 00:15:30,300
+constant خليكوا اللي هو مذكري طيبالان ناخد مثال
+
+197
+00:15:30,300 --> 00:15:34,100
+without finding T and N write the acceleration of
+
+198
+00:15:34,100 --> 00:15:39,260
+the motion R of T بساوية كذا in the form A تنجنتيا
+
+199
+00:15:39,260 --> 00:15:42,740
+في اتجاه T و A N في اتجاه الـ N وقول يعني بدون ما
+
+200
+00:15:42,740 --> 00:15:46,960
+تحسب T و N اقعد تحسبهم لان انا اصلا بدي ابدالتهم
+
+201
+00:15:46,960 --> 00:15:51,600
+اصلا انابدي ال a تنكتب ال aT في اتجاه ال T زي ال
+
+202
+00:15:51,600 --> 00:15:56,100
+aN في اتجاه ال N يعني بدي اكتبها ال acceleration
+
+203
+00:15:56,100 --> 00:16:02,180
+بهذه لا تحسب ال T ولا تحسب ال N بقوله منيح ال aT
+
+204
+00:16:02,180 --> 00:16:05,780
+سهلة بجيب لك ال aT و انا بعرف العجلة اللي انا
+
+205
+00:16:05,780 --> 00:16:10,040
+بعرفها لأنه اصلا لازم اوجدها باخد ال magnitude لها
+
+206
+00:16:10,040 --> 00:16:13,720
+و بربحها و بنقصها لتربيحها منها و باخد لها الجذر
+
+207
+00:16:13,720 --> 00:16:17,300
+تطلع aN بكون كل أمورنا جاهزة ايش اللي بتقولوا يلا
+
+208
+00:16:17,300 --> 00:16:22,340
+معاياالأن هاي الـ R of T جيب الـ V V بتساوي ناقص
+
+209
+00:16:22,340 --> 00:16:27,750
+Sin Tزائد التفاضل اللي حصل ضرب دلتين sin t زائد t
+
+210
+00:16:27,750 --> 00:16:31,930
+cos t كله في اتجاه ال I زائد تفاضلها cos t و
+
+211
+00:16:31,930 --> 00:16:36,710
+تفاضلها اللي حصل ضرب دلتين تي واحد يعني بيصير cos
+
+212
+00:16:36,710 --> 00:16:42,310
+t زائد اللي هو سالب sin t في T بيصير عندى هذا اللي
+
+213
+00:16:42,310 --> 00:16:46,190
+هو ال velocity لاحظ هنا اللي هو السادة بتروح مع
+
+214
+00:16:46,190 --> 00:16:51,300
+هذه تطلع T cos T وهذه بتروح مع هذه تطلع T sin Tخد
+
+215
+00:16:51,300 --> 00:16:54,500
+المجنتيو دي لها المجنتيو دي عبارة عن جدر التربيع
+
+216
+00:16:54,500 --> 00:16:57,800
+إلى هذه تربيع زائد هذه تربيع طبعا cos تربيع زائد
+
+217
+00:16:57,800 --> 00:17:01,120
+sin تربيع لما نطلع ال T تربيع برا بيصير واحد في T
+
+218
+00:17:01,120 --> 00:17:03,960
+تربيع يعني اللي هو جدر ال T تربيع جدر ال T تربيع
+
+219
+00:17:03,960 --> 00:17:07,340
+علشان ال value لل T لكن ال T عندنا اصلا احنا مفترض
+
+220
+00:17:07,340 --> 00:17:11,000
+انها اكبر من 0 اذا بتساوي ايش؟ بتساوي T اذا حصلنا
+
+221
+00:17:11,000 --> 00:17:14,480
+.. طب ليش وجدت .. وجدت هلول؟ لإن احنا عارفين ال a
+
+222
+00:17:14,480 --> 00:17:20,010
+tangential ايش بتساوي D ل speed by dtو يساوي ايش
+
+223
+00:17:20,010 --> 00:17:23,450
+بتساوي تفضلها واحد اذا الـ a تنجنش الا وجدناها
+
+224
+00:17:23,450 --> 00:17:29,290
+واحد الان ال acceleration الان سهل الان خلينا نجد
+
+225
+00:17:29,290 --> 00:17:32,230
+اول اشي مين ال acceleration ال acceleration هي ال
+
+226
+00:17:32,230 --> 00:17:37,030
+V عندنا بده فاضل هذه بيصير تفضلها عبارة عن cos T
+
+227
+00:17:37,030 --> 00:17:42,430
+لما حصل ضرب تلتين ناقص T sin T في اتجاه I زائد هذا
+
+228
+00:17:42,430 --> 00:17:47,830
+حصل ضرب دلتين sin T زائد T في cos T في اتجاه Jالان
+
+229
+00:17:47,830 --> 00:17:51,770
+هادى بدى اوجدها هادى ال acceleration هيها هادى بدى
+
+230
+00:17:51,770 --> 00:17:55,590
+اكتبها على الصورة هادى هيها اللى أمامى طيب كيف بدى
+
+231
+00:17:55,590 --> 00:17:59,510
+اكتبها؟ شوفوا معايا كيف خد ال magnitude لمين الآن
+
+232
+00:17:59,510 --> 00:18:04,150
+لل a ليش ال magnitude لل a عشان اوجد ال a انت زى
+
+233
+00:18:04,150 --> 00:18:06,490
+ما قلنا قبل بشوية ال magnitude لل a هو عبارة عن
+
+234
+00:18:06,490 --> 00:18:11,610
+جدر التربيعي لهادى تربيع هيها زائد هادى تربيع هيها
+
+235
+00:18:11,610 --> 00:18:16,080
+بالربع الجهتين هادى بيصير cosine تربيعوهنا بيصير
+
+236
+00:18:16,080 --> 00:18:19,220
+sin تربيع كوسين تربيع زاد sin تربيع اش مالها واحد
+
+237
+00:18:19,220 --> 00:18:24,660
+خلصنا الهد التفاضل تربيع حياش زاد T تربيع sin
+
+238
+00:18:24,660 --> 00:18:28,640
+تربيع وهد T تربيع كوسين تربيع بتطلع T تربيع برا
+
+239
+00:18:28,640 --> 00:18:31,480
+بيصير T تربيع في sin تربيع زاد كوسين تربيع يعني T
+
+240
+00:18:31,480 --> 00:18:35,140
+تربيع لأن sin تربيع زاد كوسين تربيع واحد الحد اللي
+
+241
+00:18:35,140 --> 00:18:40,230
+في النصتنين كوساين في sin في T بالسادة هنا تنين
+
+242
+00:18:40,230 --> 00:18:43,410
+sin في كوساين في T بالموجة بروح مع بعض إذا بضل
+
+243
+00:18:43,410 --> 00:18:46,770
+عندي بس المقدار فعلا جدر التربيع إلى واحد زائد T
+
+244
+00:18:46,770 --> 00:18:50,710
+تربيع لأن ال AN عبارة عن زي ما قلنا جدر التربيع
+
+245
+00:18:50,710 --> 00:18:56,410
+إلى ال magnitude ال A تربيع ناقص ال AT تربيعالـ A
+
+246
+00:18:56,410 --> 00:18:59,630
+T تربيع أوجدناه ليه سالب واحد وهذا أوجدناه ليه
+
+247
+00:18:59,630 --> 00:19:02,990
+واحد زائد T تربيع بيصير الجدر التربيع واحد زائد T
+
+248
+00:19:02,990 --> 00:19:05,850
+تربيع نقصه اللي هو الجدر التربيع إلى T تربيع اللي
+
+249
+00:19:05,850 --> 00:19:08,170
+هو absolute value للـ T بيساوي الـ T لإن الـ T
+
+250
+00:19:08,170 --> 00:19:13,930
+أكبر من سفر إذاً A أنجهزة .. A أنجهزة يا جماعة أو
+
+251
+00:19:13,930 --> 00:19:17,930
+A T جاهزة هو طالب الـ A تكتب على صورة هذه في هذه
+
+252
+00:19:17,930 --> 00:19:21,370
+إذا ما خلاص جهزة A .. A إيش هتساوي الـ A
+
+253
+00:19:21,370 --> 00:19:25,640
+tangentialفي اتجاه الـ T زائد A N في اتجاه الـ N A
+
+254
+00:19:25,640 --> 00:19:29,940
+tangential لو وجدناها واحد اذا بصير T A N ايش هي T
+
+255
+00:19:29,940 --> 00:19:34,700
+فN فصارت اللي هي الان كاتبين ال A على صورة اللي هو
+
+256
+00:19:34,700 --> 00:19:40,040
+T زائد T في اتجاه ال N اللي هو tangential و normal
+
+257
+00:19:40,040 --> 00:19:47,140
+components يعني هذه هي عبارة عن هذه العجلة لكن
+
+258
+00:19:47,140 --> 00:19:51,160
+احنا كتبناها بالصورة اللي هو طلبها عننا
+
+259
+00:19:53,250 --> 00:19:56,830
+شوفوا الآن في حاجة اسمها الـ torsion الـ torsion
+
+260
+00:19:56,830 --> 00:20:01,210
+هي عملية حسابات هنلاقيها بقى قليل شوية بس قبل ..
+
+261
+00:20:01,210 --> 00:20:03,970
+بس قبل ما نروح هذا .. هذا اللي هو ال curve لمين
+
+262
+00:20:03,970 --> 00:20:06,810
+اللي توح لناه السؤال اللي توح لناه ال curve إيه
+
+263
+00:20:06,810 --> 00:20:10,370
+له؟ هاي ال particle اللي .. اللي بمشي هنا هاي اللي
+
+264
+00:20:10,370 --> 00:20:13,710
+هو ال .. ال acceleration أوجدناها و أوجدنا ان ال
+
+265
+00:20:13,710 --> 00:20:18,110
+acceleration ال AT في اتجاه ال T و ال AN في اتجاه
+
+266
+00:20:18,110 --> 00:20:23,460
+اللي هي مين ال Tالـ A T طلعت واحد هي بصير T وهذه
+
+267
+00:20:23,460 --> 00:20:27,740
+طلعت اللي هو T فبصير T في N هذه اللي هي T في N
+
+268
+00:20:27,740 --> 00:20:32,760
+اللي هي تحليل ال A في اتجاه ال N وهذه تحليل ال A
+
+269
+00:20:32,760 --> 00:20:38,080
+في اتجاه من ال T اللي هي واحد طيب الآن بدنا نعرف
+
+270
+00:20:38,080 --> 00:20:41,840
+حاجة اسمها torsion أو العزم let B بتساوي T cross N
+
+271
+00:20:42,340 --> 00:20:47,020
+هذا الـ binomial vector اللي حكينا عنه قبل شوية
+
+272
+00:20:47,020 --> 00:20:51,840
+لأن the torsion function هذا تعريف of a smooth
+
+273
+00:20:51,840 --> 00:20:55,180
+curve is طبعا فيه له مجرات وفيه له حكي لتعريف مش
+
+274
+00:20:55,180 --> 00:21:01,080
+مجالنا لأن الـ tau هذه عبارة عن ناقص db by ds في
+
+275
+00:21:01,080 --> 00:21:06,380
+ضطن طبعا هذه ناقص اللي هو استلاحة عشان نحكي كلنا
+
+276
+00:21:06,380 --> 00:21:10,610
+في لغة واحدةطبعا الان الـ Tau بتساوي في الآخر عدد
+
+277
+00:21:10,610 --> 00:21:14,570
+عدد بتحط اللي ناقص بتطلع مين اللي هو التورجين وفيه
+
+278
+00:21:14,570 --> 00:21:18,150
+طبعا تفسيرات فيزيائية مش مجالها الآن الـ Tau
+
+279
+00:21:18,150 --> 00:21:22,330
+بتساوي ناقص DB by DS فعلا طبعا هذا الكلام احنا
+
+280
+00:21:22,330 --> 00:21:27,290
+مابنحبوش راحظوا ان احنا نبعد عن الحسابات اللي هو
+
+281
+00:21:27,290 --> 00:21:31,510
+التفاضل بالنسبة لـ S بنروح للحسابات بالنسبة لمين
+
+282
+00:21:31,510 --> 00:21:35,800
+بالنسبة التي أسهلناعشان هي هو ما أعطينا القانون
+
+283
+00:21:35,800 --> 00:21:41,660
+اللي هو بعد شوية قانون لحساب التورجين اللي هي
+
+284
+00:21:41,660 --> 00:21:44,800
+computational formula هو يقول لك the most widely
+
+285
+00:21:44,800 --> 00:21:48,580
+used formula for torgen derived in more advanced
+
+286
+00:21:48,580 --> 00:21:52,400
+text يعني مش مجالنا اللي هو اشتقاقها التورجين وهذا
+
+287
+00:21:52,400 --> 00:21:55,020
+اللي هنستخدمه في حساباتنا هذا اللي هنستخدمه في
+
+288
+00:21:55,020 --> 00:22:00,960
+حساباتنا مش هذهالتي هي تورجين X dot Y dot Z dot
+
+289
+00:22:00,960 --> 00:22:05,000
+اهل الفيزياء لايعرفون معنى هذا على مجموعه V cross
+
+290
+00:22:05,000 --> 00:22:10,340
+A كل تربيع V cross A ال V معروفة تنجب بـDR by DT و
+
+291
+00:22:10,340 --> 00:22:14,320
+A تكون DV by DT نعمل cross product ناخد المجموعه و
+
+292
+00:22:14,320 --> 00:22:19,200
+بالربعه تطلع الكمية هذه نجي للفوق اللي فوق ما هي X
+
+293
+00:22:19,200 --> 00:22:25,040
+dot هذه معناتها DX by DT احنا عارفين ان R of Tالـ
+
+294
+00:22:25,040 --> 00:22:28,760
+position vector أو اللي بتعمل ال .. ال .. ال .. ال
+
+295
+00:22:28,760 --> 00:22:33,760
+.. ال curve R of T بساوي X of T في اتجاه I زائد Y
+
+296
+00:22:33,760 --> 00:22:38,680
+of T في اتجاه G زائد Z of T في اتجاه الـ K لأن X
+
+297
+00:22:38,680 --> 00:22:44,850
+ضد هذه معناته DX by DT في الـ Rوهذه dy by dt التي
+
+298
+00:22:44,850 --> 00:22:50,430
+في الار وهذه zz by dt التي في الار وهذه معامل k
+
+299
+00:22:50,430 --> 00:22:56,270
+وهذه معامل i وهذه معامل j وهذه معامل i الآن بفضلها
+
+300
+00:22:56,270 --> 00:23:00,170
+كمان مرة دي تربيع x by dt تربيع تطلع هذه و بفضل
+
+301
+00:23:00,170 --> 00:23:03,450
+هذه كمان مرة تطلع هذه و بفضل هذه كمان مرة تطلع هذه
+
+302
+00:23:03,880 --> 00:23:07,360
+الان اللي بقعده بفاضل هذه كمان مرة تطلع هذه بفاضل
+
+303
+00:23:07,360 --> 00:23:09,780
+هذه كمان مرة تطلع هذه وهذه بفاضلها كمان مرة تطلع
+
+304
+00:23:09,780 --> 00:23:15,560
+هذه يعني هذه عبارة عن D تكعيب X by DT تكعيب يعني
+
+305
+00:23:15,560 --> 00:23:20,000
+التفاضل معامل I تلت مرات هذه التفاضل معامل I مرتين
+
+306
+00:23:20,000 --> 00:23:23,960
+هذا التفاضل بوجد ال determinant هذا بجسمه على ال
+
+307
+00:23:23,960 --> 00:23:26,860
+determinant أعلى اللي هو القيمة هذه ايه بتطلع مين
+
+308
+00:23:26,860 --> 00:23:31,610
+ال torsion وهذه طريقة الحل اللي احنا هنستخدمهاماشي
+
+309
+00:23:31,610 --> 00:23:35,470
+الحال اذا الان هذه الان في اسئلة عندى في ال
+
+310
+00:23:35,470 --> 00:23:41,230
+homework هيها طالب مين طالب ال tau مثلا هي عندى
+
+311
+00:23:41,230 --> 00:23:45,110
+معطيك هذا ال position vector هيه و طلبلك ال tau
+
+312
+00:23:45,110 --> 00:23:49,530
+عشان اوجد ال tau بدى اوجد ال determinant اللى فاضل
+
+313
+00:23:49,530 --> 00:23:53,210
+هذه اول مرة و بعدين تانى مرة و بعدين تلت مرة بحطن
+
+314
+00:23:53,210 --> 00:23:56,650
+فى العمود الاولوهذا اول مرة ثاني مرة ثالث مرة في
+
+315
+00:23:56,650 --> 00:24:00,070
+العمود التاني وهذا اول مرة ثاني مرة ثالث مرة ثالث
+
+316
+00:24:00,070 --> 00:24:00,830
+مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة
+
+317
+00:24:00,830 --> 00:24:00,970
+ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث
+
+318
+00:24:00,970 --> 00:24:01,470
+مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة
+
+319
+00:24:01,470 --> 00:24:02,810
+ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث
+
+320
+00:24:02,810 --> 00:24:02,930
+مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة
+
+321
+00:24:02,930 --> 00:24:10,110
+ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث مرة ثالث
+
+322
+00:24:10,110 --> 00:24:17,970
+مرة ثالث مرة ثالث مرة
+
+323
+00:24:17,970 --> 00:24:20,810
+ثالث
+
+324
+00:24:21,640 --> 00:24:26,380
+اللي هو تطبق القانون هذا بتطلع عند مين التاو إذا
+
+325
+00:24:26,380 --> 00:24:32,620
+هذه الطريقة لإيجاد التاو زي ما شبهناالان في شغله
+
+326
+00:24:32,620 --> 00:24:37,720
+ثانيه قبل ما نروح للشغله الثانيه خليني نكمل عندي
+
+327
+00:24:37,720 --> 00:24:42,060
+computational formula for curves in space ماشي هذه
+
+328
+00:24:42,060 --> 00:24:46,380
+الان الملخص للي حكيناه كله يعني ال unit tangent
+
+329
+00:24:46,380 --> 00:24:52,020
+vector T V على ال magnitude ل V كله تفاضل من R R
+
+330
+00:24:52,020 --> 00:24:55,560
+of T تفاضل بالنسبة ل T هذه الان principal unit
+
+331
+00:24:55,560 --> 00:24:58,200
+normal vector NDT
+
+332
+00:25:05,330 --> 00:25:12,720
+بعد ذلك نفضل الار ونجيب الانالان ال B T cross N
+
+333
+00:25:12,720 --> 00:25:16,200
+اللي أوددناها خلصناها هذا ال by normal T cross N
+
+334
+00:25:16,200 --> 00:25:21,380
+اللي هي بتكون عمودية عالميا على اللي هي ال T و ال
+
+335
+00:25:21,380 --> 00:25:25,900
+N أو على ال plane اللي بيحملها قابة اللي هي DT by
+
+336
+00:25:25,900 --> 00:25:29,120
+DS magnitude وهذا مش هنستخدمها في الحسابات هنستخدم
+
+337
+00:25:29,120 --> 00:25:34,920
+اللي هو اللي هي قبل اللي حكينا عنها القانون اللي
+
+338
+00:25:34,920 --> 00:25:39,440
+حكينا عنه سابقا اللي هو بدلالة T أو اللي هو هي فيه
+
+339
+00:25:39,440 --> 00:25:44,310
+قانون تانياللي برضه ممكن لو انت طلب منك اللي هو
+
+340
+00:25:44,310 --> 00:25:50,230
+توجد لسؤال واحد اللي هو تاو أقابعمانت موجد الـ V و
+
+341
+00:25:50,230 --> 00:25:53,650
+الـ A و الـ cross product لهن فأسهلك تستخدم القابة
+
+342
+00:25:53,650 --> 00:25:57,930
+بالقانون هذا V cross A magnitude على magnitude V
+
+343
+00:25:57,930 --> 00:26:02,430
+تكعيد و جرب حللي السؤال على قابة بهذا القانون أو
+
+344
+00:26:02,430 --> 00:26:06,430
+بالقانون السابق حلليه أو شوف الجوابين بطلع زي بعض
+
+345
+00:26:06,430 --> 00:26:10,290
+الأسئلة الهمور كل بعد شوية ال torsion tau هيها مش
+
+346
+00:26:10,290 --> 00:26:14,410
+هنستخدم هذا زي ما قلنا هنستخدمين هذا اللي عندنا
+
+347
+00:26:14,410 --> 00:26:18,410
+إلا إذا كان في معطينة بيه بدلالة Sمباشرة وهذا طبعا
+
+348
+00:26:18,410 --> 00:26:21,890
+نادرا ما يحدث اما هو اي قانون اللي احنا الاكتر
+
+349
+00:26:21,890 --> 00:26:25,290
+شيوعا في الاستخدام الان ال tangential and normal
+
+350
+00:26:25,290 --> 00:26:27,730
+scalar components of acceleration a بيساوي a
+
+351
+00:26:27,730 --> 00:26:30,170
+tangential في اتجاه ال T زي ال a n في اتجاه ال N
+
+352
+00:26:30,170 --> 00:26:36,630
+ال a tangential هي DV by DT DT بالفاضل اللي هي ال
+
+353
+00:26:36,630 --> 00:26:40,150
+speed وال a n اللي هي قابة في ال speed التربيع
+
+354
+00:26:40,150 --> 00:26:44,010
+اللي هي هتساوي الجذر التربيعي طبعا هذه بنوجدها من
+
+355
+00:26:44,010 --> 00:26:49,020
+هنا أسلنا من وجهة دهانلأن هذه إيجادة قد يتطلب هذا
+
+356
+00:26:49,020 --> 00:26:52,740
+و هذا بيغلب أو يتطلب السابق برضه بيغلب لكن هذا
+
+357
+00:26:52,740 --> 00:26:55,140
+أسألك على طول جدر التربيع لـ Magnitude لـ A تربيع
+
+358
+00:26:55,140 --> 00:26:59,420
+نقص A T تربيع التربيع اللي احنا حكينا عنه سابقاً
+
+359
+00:26:59,420 --> 00:27:03,620
+طيب نرجع ل .. لسيبنا هنا دولة اللي هن ال .. ال ..
+
+360
+00:27:03,620 --> 00:27:07,900
+ال .. ال .. ال blends ال blends اللي .. اللي ..
+
+361
+00:27:07,900 --> 00:27:13,170
+اللي يصنع من مين من ال T و ال N و ال Dالان هى ال
+
+362
+00:27:13,170 --> 00:27:17,050
+particle عندى او القطار اللى ماشى القطار اللى ماشى
+
+363
+00:27:17,050 --> 00:27:21,350
+او ال particle هذا زى ما قلنا فى له اللى هو نقطة
+
+364
+00:27:21,350 --> 00:27:26,090
+unit tangent vector وفى عندى ال principle normal
+
+365
+00:27:26,090 --> 00:27:32,630
+vector ال plane اللى بيحملهن اللى بيحمل ال T والان
+
+366
+00:27:32,630 --> 00:27:36,650
+بنسميه escalating plane عينكم الله بتكونوا تحفظوها
+
+367
+00:27:38,420 --> 00:27:42,780
+الان كيف نوجد معدلته هذا إيجاد معدلته انه طبعا مين
+
+368
+00:27:42,780 --> 00:27:46,880
+بيبقى معاطيك اللي هو R of T اللي هو R اللي هو اللي
+
+369
+00:27:46,880 --> 00:27:50,100
+برسم لنا ال curve اللي بيمشي عليه ال particle من
+
+370
+00:27:50,100 --> 00:27:53,440
+ال R بوجد زي ما بتعرف طبعا زي ما مشينا قبله شوية
+
+371
+00:27:53,440 --> 00:27:57,200
+بتوجدوا ال T و بتوجدوا ال N أكيد بتعرفوا توجدوه و
+
+372
+00:27:57,200 --> 00:28:00,900
+من ال R برضه بقدر أوجد نقطة اللي هو بيبقى بعطيك
+
+373
+00:28:00,900 --> 00:28:05,320
+أنه هذا بدي معدلتهاللي معادلة الـ Plane الـ
+
+374
+00:28:05,320 --> 00:28:08,380
+Scalating in Plane اللي بمر بالنقطة كذا هذه بيبقى
+
+375
+00:28:08,380 --> 00:28:12,620
+معاطيك نقطة بس ضال عليك توجد مين اللي هو Normal
+
+376
+00:28:12,620 --> 00:28:16,200
+على الـ Plane مين ال Normal على الـ Plane اللي هو
+
+377
+00:28:16,200 --> 00:28:20,440
+T cross N اللي هو ال By Normal اللي بناه وجده إذا
+
+378
+00:28:20,440 --> 00:28:24,080
+ال Scalating in Plane معادلته سهل إيجادها وهذا
+
+379
+00:28:24,080 --> 00:28:27,140
+اللي هو معلومات سابقة يعنيليس داعي ناخد أمثلة
+
+380
+00:28:27,140 --> 00:28:31,120
+عليها لأنه بتعرفوا توجدوها اللي هو بتعرفوا هي نقطة
+
+381
+00:28:31,120 --> 00:28:34,420
+موجودة و ال normal اللي على ال escalating اللي هو
+
+382
+00:28:34,420 --> 00:28:39,580
+ال B بتعرفوا توجدوه فبتوجدوه عدلته هذا الآن ال
+
+383
+00:28:39,580 --> 00:28:45,000
+plane المكوّن من ال N و ال B المكوّن من binormal و
+
+384
+00:28:45,000 --> 00:28:49,180
+normalاسمه Normal لبنان يعني هنا في كلمة Normal
+
+385
+00:28:49,180 --> 00:28:53,140
+وهنا كلمة Normal إذا اسمه Normal لبنان ال Normal
+
+386
+00:28:53,140 --> 00:28:56,420
+لبنان هي النقطة المعروفة و ال unit tangent vector
+
+387
+00:28:56,420 --> 00:29:00,860
+معروف فمن العمودي على اللي هو ال Normal لبنان اللي
+
+388
+00:29:00,860 --> 00:29:05,120
+هو ال unit tangent vector هو النقطة تعرفه معادلة
+
+389
+00:29:05,120 --> 00:29:09,800
+لبنان زي ما أخدنا سابقاالان هذا اللي بين الـ unit
+
+390
+00:29:09,800 --> 00:29:13,180
+tangent vector و الـ binormal بنسميه rectifying
+
+391
+00:29:13,180 --> 00:29:18,580
+plane هذا rectifying plane معدلته سهلة هذه النقطة
+
+392
+00:29:18,580 --> 00:29:21,840
+معروفة وهي ال normal على ال plane المعروف و
+
+393
+00:29:21,840 --> 00:29:25,120
+بتعرفوا معدلته
+
+394
+00:29:25,560 --> 00:29:28,120
+الان قبل بشوية انا حكيته هذا الكلام اللي هو
+
+395
+00:29:28,120 --> 00:29:31,140
+التعليق عليها اذا هذا في عندكم اسئلة على ال
+
+396
+00:29:31,140 --> 00:29:35,360
+escalating وعلى ال normal وعلى ال rectifying بدكم
+
+397
+00:29:35,360 --> 00:29:40,580
+تحلوها باللي هي معرفة نقطة ومؤيجاد اللي هو هذا او
+
+398
+00:29:40,580 --> 00:29:44,650
+هذا او هذاوكل هى هدولة اوددناها وعرفنا كيف نوددها
+
+399
+00:29:44,650 --> 00:29:50,470
+ومعادلة ال plane بمعرفة نقطة وعمودي عليه اعرفناها
+
+400
+00:29:50,470 --> 00:29:54,430
+نيجى هدرس اعيد اللى حكيته قبل بشوية في اول تقريبا
+
+401
+00:29:54,430 --> 00:29:59,130
+المحاضرة قولنا ال particle او اللى هو القطار ماشى
+
+402
+00:29:59,130 --> 00:30:04,370
+هى ماشى هو ماشى اللى هو في الواقع ماشى ب frame ايش
+
+403
+00:30:04,370 --> 00:30:09,850
+ال frame اللى ماشى فيه اللى هو ال T و ال N و ال B
+
+404
+00:30:10,090 --> 00:30:16,070
+كل واحد فيهم كل واحد فيهم له معنى للحركة قولنا هذا
+
+405
+00:30:16,070 --> 00:30:22,890
+بيبين اتجاه الحركة اللي هو اتجاه ال velocity V على
+
+406
+00:30:22,890 --> 00:30:26,490
+ال magnitude لل V ال unit tangent vector هذا الان
+
+407
+00:30:26,490 --> 00:30:34,510
+هو اللي بيحدد اللي هوالتواء أو الـ Turning لمن؟
+
+408
+00:30:34,510 --> 00:30:39,430
+للبس أو للقطار أو للبارتيكل عبر الطريق اللي موجود
+
+409
+00:30:39,430 --> 00:30:45,770
+دائما هذا التواء بيكون باتجاه مركز الـ Circle of
+
+410
+00:30:45,770 --> 00:30:46,410
+curvature
+
+411
+00:30:50,650 --> 00:30:55,130
+الانحناء او ال curvature عند النقطة المذكورة والان
+
+412
+00:30:55,130 --> 00:31:00,130
+هذا ال B هو اللي بيحدد لي ال twisting لمين لل
+
+413
+00:31:00,130 --> 00:31:04,510
+particle او للقطار لما يتحرك بيصيرله طبعا حركة
+
+414
+00:31:04,510 --> 00:31:09,390
+حركة الآن بعمل اللي هو مش عارف ده انا جاعد بحرك
+
+415
+00:31:09,390 --> 00:31:13,310
+وانتوا مش شايفيني اللي هو بعمل اللي هو ال twisting
+
+416
+00:31:13,310 --> 00:31:17,230
+للقطار القطار هي و هيك بعمل هيك هيك هيك هذا ال
+
+417
+00:31:17,230 --> 00:31:21,710
+twisting اللي بجيسه مين اللي هو ال Bأو هكذا ننهي
+
+418
+00:31:21,710 --> 00:31:25,670
+المحاضرة و بارك الله فيكم و إلى لقاء آخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/LGA6DA-unSY.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/LGA6DA-unSY.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a128cf6537c39663107139fbb984d84171e37114
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/LGA6DA-unSY.srt
@@ -0,0 +1,1996 @@
+1
+00:00:00,710 --> 00:00:04,290
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,290 --> 00:00:09,590
+وبركاته هذه المحاضرة السادسة في مساق تفاضل وتكامل جيم
+
+3
+00:00:09,590 --> 00:00:14,330
+المحاضرة بعنوان Cylinders and Quadratic Surfaces
+
+4
+00:00:14,330 --> 00:00:17,750
+وهو القسم الأخير في هذا الفصل ولكن
+
+5
+00:00:17,750 --> 00:00:22,030
+هذه المحاضرة هي عبارة عن الجزء الأول من 12-6
+
+6
+00:00:22,030 --> 00:00:25,590
+والجزء الثاني إن شاء الله سيكون في المحاضرة السابعة
+
+7
+00:00:26,040 --> 00:00:29,700
+الآن، العنوان Cylinders and Quadratic
+
+8
+00:00:29,700 --> 00:00:33,100
+Surfaces. سنتحدث أولًا عن الاسطوانات، وبعدين
+
+9
+00:00:33,100 --> 00:00:36,120
+سنتحدث عن ثلاثة من الأسطح التربيعية في هذه
+
+10
+00:00:36,120 --> 00:00:39,200
+المحاضرة، وإن شاء الله نكمل الثلاثة
+
+11
+00:00:39,200 --> 00:00:43,340
+الأخريات. الآن، نأتي لنحدث عن الاسطوانات، نعرف الآن
+
+12
+00:00:43,340 --> 00:00:46,920
+الاسطوانات. الآن تعريف الاسطوانات، الاسطوانة
+
+13
+00:00:46,920 --> 00:00:50,280
+عندنا هنا هو تعميم لما نعرفه عن الاسطوانة.
+
+14
+00:00:50,550 --> 00:00:54,690
+ركزوا معي، فلنرى كيف نعرف الاسطوانة الآن.
+
+15
+00:00:54,690 --> 00:01:02,030
+نجيب منحنى في الفراغ، هذا المنحنى سنسميه
+
+16
+00:01:02,030 --> 00:01:07,830
+"generating curve". نجيب مثلًا هذا المنحنى في مستوى yz
+
+17
+00:01:07,830 --> 00:01:11,090
+في الفراغ،  لكنه سيكون في المستوى، ليس
+
+18
+00:01:11,090 --> 00:01:13,790
+هو الذي سيكون في الفراغ، سيكون شكله
+
+19
+00:01:13,790 --> 00:01:17,490
+كالاسطوانة الكلاسيكية في الفراغ. أما المنحنى العام الذي
+
+20
+00:01:17,490 --> 00:01:20,930
+عندنا هنا، سنأخذه في مستوى xy على سبيل المثال.
+
+21
+00:01:20,930 --> 00:01:25,970
+الآن، نريد أن نعرف الاسطوانة، نأخذ الخط، الخط
+
+22
+00:01:25,970 --> 00:01:30,670
+هذا يكون موازيًا لمحور x
+
+23
+00:01:30,670 --> 00:01:36,090
+الآن، محور x، آخذ موازي له، وهو خط
+
+24
+00:01:36,090 --> 00:01:40,070
+يمر به على طول، تمرير
+
+25
+00:01:43,720 --> 00:01:48,780
+الشكل الذي يظهر عندي هذا، هذا نسميه اسطوانة، إذا
+
+26
+00:01:48,780 --> 00:01:51,480
+هذه الاسطوانة هي عبارة عن تعميم للاسطوانة
+
+27
+00:01:51,480 --> 00:01:55,580
+العامة، المنحنى الذي عندي هو عبارة عن
+
+28
+00:01:55,580 --> 00:02:00,920
+المنحنى الذي أمامي هذا. لأن اسطوانة
+
+29
+00:02:00,920 --> 00:02:03,940
+مرة أخرى، هي سطح ينتج من تحريك
+
+30
+00:02:03,940 --> 00:02:07,380
+خط مستقيم، هذا الخط المستقيم على طول منحنى مستو
+
+31
+00:02:07,380 --> 00:02:10,420
+هذا المنحنى المستوي، وهو منحنى في
+
+32
+00:02:10,420 --> 00:02:14,040
+المستوي، مع إبقاء الخط موازيًا لخط ثابت
+
+33
+00:02:14,040 --> 00:02:18,320
+وهو، يعني، يكون هذا الخط
+
+34
+00:02:18,320 --> 00:02:22,260
+موازيًا لخط ما. نحن أخذنا محور x
+
+35
+00:02:22,260 --> 00:02:25,730
+-axis، ممكن نأخذ أي خط آخر، لكن محور x-Axis أسهل لأنها
+
+36
+00:02:25,730 --> 00:02:32,830
+تطلع اسطوانة من مستوى zy على محور
+
+37
+00:02:32,830 --> 00:02:37,430
+x، المقطع كلها نفس الشيء لأن هذا يطلع
+
+38
+00:02:37,430 --> 00:02:40,810
+عندي ما نسميه اسطوانة. في الهندسة الفراغية،
+
+39
+00:02:40,810 --> 00:02:44,230
+حيث أن الاسطوانات تعني اسطوانات دائرية، سابقًا
+
+40
+00:02:44,230 --> 00:02:47,810
+كنا نتحدث فيها في الهندسة الفراغية، كانت الاسطوانة
+
+41
+00:02:47,810 --> 00:02:50,930
+لدينا اسطوانة دائرية، يعني يكون ال
+
+42
+00:02:50,930 --> 00:02:55,530
+منحنى الذي يولدها هو عبارة عن دائرة،
+
+43
+00:02:55,530 --> 00:02:59,710
+والمنحنى الذي يصنع الاسطوانة أو الخط الذي يصنع
+
+44
+00:02:59,710 --> 00:03:03,170
+الاسطوانة هو عبارة عن خط موازي لمحور x أو z
+
+45
+00:03:03,170 --> 00:03:06,610
+-axis أو y-axis حسب مكان الاسطوانة
+
+46
+00:03:06,610 --> 00:03:12,580
+سوف نرى ما الذي نتحدث عنه، نأخذ مثالًا آخر يوضح
+
+47
+00:03:12,580 --> 00:03:17,400
+الأمر. يقول: أوجد معادلة للاسطوانة المصنوعة
+
+48
+00:03:17,400 --> 00:03:21,960
+من الخطوط الموازية لمحور z التي تمر
+
+49
+00:03:21,960 --> 00:03:24,560
+من خلال القطع المكافئ y = x² و z
+
+50
+00:03:24,560 --> 00:03:28,440
+= 0. لدينا شيئان يجب تحديدهما،
+
+51
+00:03:28,440 --> 00:03:33,530
+الاسطوانة أولًا، وهو المنحنى المستوي، أو
+
+52
+00:03:33,530 --> 00:03:37,570
+المنحنى المولد، في هذه الحالة عبارة عن قطع مكافئ
+
+53
+00:03:37,570 --> 00:03:41,830
+بين y و x، وفي مستوى z = 0،
+
+54
+00:03:41,830 --> 00:03:45,810
+يعني في مستوى xy. هذا القطع المكافئ الذي عندي، وهو
+
+55
+00:03:45,810 --> 00:03:49,540
+المنحنى المولد، الذي أقول: أوجد ال
+
+56
+00:03:49,540 --> 00:03:55,920
+اسطوانة الدائرية، آسف، الاسطوانة التي تنشأ من
+
+57
+00:03:55,920 --> 00:04:00,340
+الخط الذي... الذي... الذي يمر، الخط الموازي لـ z
+
+58
+00:04:00,340 --> 00:04:04,320
+-axis، خط موازٍ لـ z-axis، هذا خط موازٍ لـ z-axis
+
+59
+00:04:04,320 --> 00:04:09,380
+يمر موازيًا لـ z-axis  على القطع المكافئ الذي
+
+60
+00:04:09,380 --> 00:04:12,830
+أمامي، وهو المنحنى المولد. واضح أنه عندما
+
+61
+00:04:12,830 --> 00:04:16,690
+أرسم خطًا هنا، وخطًا هنا، وجانبه خطًا آخر، وجانبه خطًا آخر،
+
+62
+00:04:16,690 --> 00:04:20,830
+يظل ماشيًا، ماشيًا، ماشيًا، الخط إلى مالا نهاية، فيصنع لي
+
+63
+00:04:20,830 --> 00:04:24,690
+الشكل الذي أمامي، هذا الشكل الذي أمامي هو
+
+64
+00:04:24,690 --> 00:04:30,760
+عبارة عن اسطوانة. المنحنى المولد لها y = x² و z = 0، يعني في مستوى xy
+
+65
+00:04:30,760 --> 00:04:34,800
+و الخط الموازي للمحور الرئيسي الذي
+
+66
+00:04:34,800 --> 00:04:38,440
+نريد أن نصنعه حول الاسطوانة، وهو محور z-axis
+
+67
+00:04:38,440 --> 00:04:40,940
+بالتمرير، وتظهر عندي الاسطوانة التي أمامي الآن.
+
+68
+00:04:44,380 --> 00:04:48,280
+نريد أن نرى ما معادلة هذه الاسطوانة التي تحقق
+
+69
+00:04:48,280 --> 00:04:52,410
+هذه الاسطوانة، التي نشأت، ما الذي يحققها؟ لاحظوا لو
+
+70
+00:04:52,410 --> 00:04:55,970
+جئنا وأخذنا هذه النقطة على الاسطوانة، أكيد هذه النقطة
+
+71
+00:04:55,970 --> 00:04:59,230
+إحداثياتها x-note و x-note تربيع، لأنها أين؟ على
+
+72
+00:04:59,230 --> 00:05:04,170
+الاسطوانة، على القطع المكافئ y = x تربيع. لو كانت هذه
+
+73
+00:05:04,170 --> 00:05:08,050
+x-note، ستكون y لها ماذا؟ x-note تربيع، و
+
+74
+00:05:08,050 --> 00:05:11,450
+طبعًا z ماذا أخذناها؟ صفر.  طيب، لو طلعنا لأعلى قليلًا،
+
+75
+00:05:11,450 --> 00:05:15,830
+هنا، سيظل x-note و x-note تربيع، ومَن؟ و
+
+76
+00:05:15,830 --> 00:05:20,010
+z تتغير، لأن لو جئنا هنا، ستكون النقطة التي
+
+77
+00:05:20,010 --> 00:05:24,730
+هنا عبارة عن مَن؟ مثلًا x1 و x1 تربيع و صفر. التي
+
+78
+00:05:24,730 --> 00:05:31,910
+فوق ستكون z و x1 و x1 تربيع و صفر. يعني أي نقطة على
+
+79
+00:05:31,910 --> 00:05:38,450
+الاسطوانة التي أمامي ستكون إحداثياتها عبارة عن
+
+80
+00:05:38,450 --> 00:05:46,380
+x1 مثلًا، و x1 تربيع، و z المتغيرة هذه. إذًا z
+
+81
+00:05:46,380 --> 00:05:50,900
+تأخذ أي قيمة، لكن هناك علاقة ثابتة بين أي نقطة، وهو
+
+82
+00:05:50,900 --> 00:05:57,240
+نقطة x ونقطة y، وهي العلاقة y = x
+
+83
+00:05:57,240 --> 00:06:00,920
+تربيع. إذًا الآن، لأعبر عن هذه الاسطوانة
+
+84
+00:06:00,920 --> 00:06:06,480
+كاملة، أقول y = x تربيع، وأسكت. معناته z تأخذ
+
+85
+00:06:06,480 --> 00:06:09,440
+صفر، و z تأخذ واحد، و z تأخذ اثنين، و z تأخذ
+
+86
+00:06:09,440 --> 00:06:12,400
+ثلاثة، يعني تطلع وتنزل، لكنها تطلع أين؟ على
+
+87
+00:06:12,400 --> 00:06:18,000
+هذا، يعني صارت الآن y = x تربيع هي عبارة
+
+88
+00:06:18,000 --> 00:06:23,520
+عن المعادلة المعبرة عن الاسطوانة التي
+
+89
+00:06:23,520 --> 00:06:27,690
+أمامي، التي منحناها المولد y
+
+90
+00:06:27,690 --> 00:06:32,690
+= x تربيع و z = 0. أما عندما أقول y =
+
+91
+00:06:32,690 --> 00:06:36,330
+x تربيع، وأسكت، معناته z تأخذ أي قيمة، معناته ستكون
+
+92
+00:06:36,330 --> 00:06:40,090
+التي على الاسطوانة. إذًا، معادلة الاسطوانة التي أريد
+
+93
+00:06:40,090 --> 00:06:43,890
+أن أجد معادلة الاسطوانة هي y = x
+
+94
+00:06:43,890 --> 00:06:47,580
+تربيع. طيب، يا جماعة، الآن لاحظوا في المثال الذي...
+
+95
+00:06:47,580 --> 00:06:53,440
+الذي تحدثنا فيه، عندي y = x تربيع، وهو ما ظهر
+
+96
+00:06:53,440 --> 00:06:57,320
+عندي، وهو اسطوانة، صحيح. الاسطوانة، لو جئنا
+
+97
+00:06:57,320 --> 00:07:01,810
+بـ y و x تربيعها، يصبح y - x تربيع = 0. y
+
+98
+00:07:01,810 --> 00:07:05,490
+- x تربيع = 0. يعني وكأني أقول، لو قلنا f
+
+99
+00:07:05,490 --> 00:07:09,990
+of (x, y) = 0، يعني صارت f of (x, y) تمثل y =
+
+100
+00:07:09,990 --> 00:07:14,150
+y و x تربيع، والصفر هو الثابت، وكأنها
+
+101
+00:07:14,150 --> 00:07:18,090
+تعطينا الملاحظة التالية، وسنفهمها إن شاء الله
+
+102
+00:07:18,460 --> 00:07:22,780
+على سبيل المثال، يظهر أن أي معادلة f(x,y) = c
+
+103
+00:07:22,780 --> 00:07:27,940
+في متغيرات x,y تعرّف عن اسطوانة حول محور z-axis
+
+104
+00:07:27,940 --> 00:07:32,420
+والتي تُعرّف أيضًا f(x,y) = c التي قلنا...
+
+105
+00:07:32,420 --> 00:07:35,240
+التي، كما قلنا قبل قليل، f(x,y) قلنا عنها
+
+106
+00:07:35,240 --> 00:07:40,700
+عبارة عن y تربيع... x - y تربيع = c، يعني
+
+107
+00:07:40,700 --> 00:07:49,720
+هذه تصبح x - y تربيع، y - x تربيع، y - x تربيع، y
+
+108
+00:07:49,720 --> 00:07:55,780
+- x تربيع، y - x تربيع، y - x تربيع، y - x
+
+109
+00:07:55,780 --> 00:07:56,720
+- x تربيع، y - x تربيع، y - x تربيع، y - x
+
+110
+00:07:56,720 --> 00:07:57,200
+تربيع، y - x تربيع، y - x تربيع، y - x تربيع، y
+
+111
+00:07:57,200 --> 00:07:59,440
+- x تربيع، y - x تربيع، y - x تربيع، y - x تربيع،
+
+112
+00:07:59,440 --> 00:08:04,980
+y - x تربيع، y -
+
+113
+00:08:06,220 --> 00:08:12,340
+وهو في مستوى معين، لكن عندما أترك z بدون ذكر،
+
+114
+00:08:12,340 --> 00:08:17,360
+معناته أنه في c، f(x, y) = c تعني اسطوانة
+
+115
+00:08:17,360 --> 00:08:22,000
+التي منحناها المستوي أو المولد،
+
+116
+00:08:22,000 --> 00:08:27,560
+في مستوى xy ومحورها موازي ل
+
+117
+00:08:27,560 --> 00:08:30,860
+المحور أو الخط الذي نرسمه موازي للمحور،
+
+118
+00:08:30,860 --> 00:08:34,140
+لذلك، المحور، سأتحدث عن أمثلة، كي تفهموا أكثر.
+
+119
+00:08:34,140 --> 00:08:37,910
+على سبيل المثال، المعادلة x تربيع + y تربيع
+
+120
+00:08:37,910 --> 00:08:41,130
+= 1. دعونا نسميها، دعونا ننسى الكلام السابق
+
+121
+00:08:41,130 --> 00:08:46,770
+هنا قليلًا، x تربيع + y تربيع، نسميها f(x, y)، والثابت
+
+122
+00:08:46,770 --> 00:08:50,670
+c نسميها واحدًا.  يعني هذه تمثل f(x, y)
+
+123
+00:08:50,670 --> 00:08:57,110
+= c. هذه f(x, y)، وهذه من c، الآن f(x, y)
+
+124
+00:08:57,110 --> 00:08:59,610
+التي هي عبارة عن x تربيع + y تربيع، وتساوي c
+
+125
+00:08:59,610 --> 00:09:05,910
+وتساوي 1. هذه تصبح دائرة، حسنًا. الآن، الدائرة
+
+126
+00:09:05,910 --> 00:09:10,680
+هذه فقط بـ z مطلقة. إذًا وكأنه المنحنى المستوي
+
+127
+00:09:10,680 --> 00:09:16,240
+التابع لها في مستوى xy، والمحور التابع لها أين؟
+
+128
+00:09:16,240 --> 00:09:19,480
+على محور z أو موازٍ لمحور z، فتصبح
+
+129
+00:09:19,480 --> 00:09:22,660
+الدائرة، عندما نأخذ خطوطًا وخطوطًا وخطوطًا وخطوطًا،
+
+130
+00:09:22,660 --> 00:09:27,020
+موازية لمحور z، سيظهر اسطوانة دائرية، التي
+
+131
+00:09:27,020 --> 00:09:31,810
+نحن نعرفها. هذا البرنامج يسمح بإنشاء اسطوانة دائرية
+
+132
+00:09:31,810 --> 00:09:34,790
+مصنوعة من الخطوط الموازية لمحور z.
+
+133
+00:09:48,920 --> 00:09:54,300
+اسطوانة مصنوعة من الخطوط التي تمر من خلال
+
+134
+00:09:54,300 --> 00:09:57,760
+الخطوط في الدائرة x تربيع + y تربيع = 1،
+
+135
+00:09:57,760 --> 00:10:03,840
+في مستوى xy. بنفس الطريقة، لو جئت بـ x و
+
+136
+00:10:03,840 --> 00:10:10,180
+z = c، وطرحنا y بدون تحديد، يعني y متغيرة، لكن
+
+137
+00:10:10,180 --> 00:10:16,380
+العلاقة بين x و z، هناك علاقة بين x و z، هذه ستصنع لي ما
+
+138
+00:10:16,380 --> 00:10:22,730
+هو الآن الاسطوانة،  المنحنى المولد لها
+
+139
+00:10:22,730 --> 00:10:29,770
+العلاقة بين x و z، والخط الموازي الآن
+
+140
+00:10:29,770 --> 00:10:35,190
+لمحور y الذي غير موجود، إذًا x و z = c هي
+
+141
+00:10:35,190 --> 00:10:41,740
+عبارة عن هذا المنحنى يصنع اسطوانة، وهو
+
+142
+00:10:41,740 --> 00:10:46,980
+موازي للمحور، لمحور y، معادلته
+
+143
+00:10:46,980 --> 00:10:54,320
+g(x, z) = c. الآن، بنفس الطريقة، المنحنى h(y, z)
+
+144
+00:10:54,320 --> 00:11:00,820
+= c، يُعرّف لي اسطوانة، محورها غير موجود هنا،
+
+145
+00:11:00,820 --> 00:11:07,180
+من هو؟ محور x،  ومعادلته تصبح
+
+146
+00:11:07,180 --> 00:11:11,480
+h(y, z) = c، و x يكون حرًا، فيصنع لي
+
+147
+00:11:11,480 --> 00:11:13,940
+هذا المنحنى، كما فعلنا قبل قليل، نرسم المنحنى
+
+148
+00:11:13,940 --> 00:11:18,560
+هذا، ونرسم الخطوط الموازية لمحور z على
+
+149
+00:11:18,560 --> 00:11:22,920
+المنحنى، فيصنع لي الاسطوانة حسب طبيعة المنحنى المولد
+
+150
+00:11:22,920 --> 00:11:27,900
+وعامودية على محور x، بينما هذا عامودية على
+
+151
+00:11:27,900 --> 00:11:31,900
+محور y، والأولى، التابعة لـ f عامودية على محور z،
+
+152
+00:11:31,900 --> 00:11:35,920
+وهكذا، هذا هو الاسطوانة، وهناك في التمارين
+
+153
+00:11:35,920 --> 00:11:40,940
+هناك أسئلة على الموضوع، توضح الأمر بشكل
+
+154
+00:11:40,940 --> 00:11:43,820
+أكبر، وإن شاء الله تكونون قد فهمتم ما نتحدث عنه. الآن
+
+155
+00:11:43,820 --> 00:11:47,580
+نأتي لما هو موضوعنا الأهم، وهو
+
+156
+00:11:47,580 --> 00:11:53,120
+الأسطح التربيعية، أو السطوح الناجمة عن
+
+157
+00:11:53,120 --> 00:11:57,840
+المعادلات من الدرجة الثانية. طبعًا نحن أخذنا في
+
+158
+00:11:57,840 --> 00:12:00,960
+في حساب التفاضل والتكامل 1000، في حساب التفاضل والتكامل 1000، وبقي
+
+159
+00:12:00,960 --> 00:12:07,550
+المعادلات... المعادلات من الدرجة الثانية، و
+
+160
+00:12:07,550 --> 00:12:12,030
+رسوماتها. يعني مثلًا أخذنا y = x تربيع، وهي
+
+161
+00:12:12,030 --> 00:12:15,850
+قطع مكافئ، وحتى لو كانت منقولة على
+
+162
+00:12:15,850 --> 00:12:19,550
+x أو على y، تبقى قطع مكافئ. الآن،
+
+163
+00:12:19,550 --> 00:12:23,790
+المعادلة x تربيع / a تربيع + y تربيع
+
+164
+00:12:23,790 --> 00:12:29,930
+/ b تربيع = 1، أو تساوي قيمة ما، هذه
+
+165
+00:12:29,930 --> 00:12:33,550
+ماذا سميناها؟ عبارة عن قطع ناقص، معادلة تربيعية، ممكن
+
+166
+00:12:33,550 --> 00:12:36,610
+المعادلة التربيعية تصنع قطعًا زائدا x تربيع
+
+167
+00:12:36,610 --> 00:12:39,150
+/ a تربيع - y تربيع / b تربيع = 1،
+
+168
+00:12:39,150 --> 00:12:44,050
+أو ما شابه. الآن، برضه تصنع قطعًا زائدا، أو ممكن
+
+169
+00:12:44,050 --> 00:12:49,150
+تصنع x تربيع = y تربيع، وهو ما
+
+170
+00:12:49,150 --> 00:12:52,390
+يصنع خطين، y = +x أو -x. إذًا المعادلات
+
+171
+00:12:52,390 --> 
+
+223
+00:16:50,620 --> 00:16:54,740
+فيكم جالس بهِ هذا برضه إليبس إليبس بس وين معناه في
+
+224
+00:16:54,740 --> 00:17:00,100
+الـ Y زي الدبلين فبنيدي بنرسم إليبس في وين في الـ Y
+
+225
+00:17:00,100 --> 00:17:05,340
+زي الدبلين هي الـ Y زي الدبلين بنرسم الـ إليبس
+
+226
+00:17:05,340 --> 00:17:12,160
+تبعتنا هي الـ Y وهي الانف تلحيك وهاي Z بلين وهي الـ
+
+227
+00:17:12,160 --> 00:17:16,640
+C وهاي الـ B هاي كمان المقطع الثاني A H إليبس إذا
+
+228
+00:17:16,640 --> 00:17:20,080
+برضه في الـ Y Z بلين طلع عندي إليبس واحد سبجني وقال
+
+229
+00:17:20,080 --> 00:17:23,740
+طب هو اصلا انت لو أخدت Y بصفر هيطلع X تربيع على A
+
+230
+00:17:23,740 --> 00:17:27,220
+تربيع زي Z تربيع على C تربيع بساوي واحد برضه طلعت
+
+231
+00:17:27,220 --> 00:17:32,100
+الناقش إليبس في الـ X Z إبلين يعني وكأنك بتقولنا أن
+
+232
+00:17:32,100 --> 00:17:38,320
+كل المقاطع تبعت نقاش مالها إليبس إذا هو كأنه كل
+
+233
+00:17:38,320 --> 00:17:41,780
+المقطع في الـ XY و في الـ XZ و في الـ YX كلها
+
+234
+00:17:41,780 --> 00:17:46,080
+Ellipses عشان هيك اللي هي بتطلع عند الشكل اللي هو
+
+235
+00:17:46,080 --> 00:17:50,100
+الشكل اللي هو البيضاوي زي كرة الـ Baseball هذه تبعت
+
+236
+00:17:50,100 --> 00:17:54,520
+اللي .. اللي بلعبوها الأمريكان اللي هي هذه الـ ..
+
+237
+00:17:54,520 --> 00:17:58,080
+الـ .. الـ .. أو يعني مش تبعت الـ basket بس مش مطورة
+
+238
+00:17:58,080 --> 00:18:02,490
+يعني مطورة شوية لأ زي تبعت الـ Baseball هذه الآن
+
+239
+00:18:02,490 --> 00:18:07,190
+اللي هي إيش يا جماعة؟ هي الـ ellipse اللي عندنا اللي
+
+240
+00:18:07,190 --> 00:18:11,010
+مقطعها كلها إيه؟ شمالها ellipses عشان هيك بنسميها
+
+241
+00:18:11,010 --> 00:18:16,070
+ellipsoid ellipsoid و بس ellipsoid لأن مافيش مقطع
+
+242
+00:18:16,070 --> 00:18:21,260
+يعني اسمها من مقاطعها مقاطعها كلها ellipses علشان
+
+243
+00:18:21,260 --> 00:18:25,420
+هيك بنسميها ellipsoid الآن و هذا الشكل اللي هي
+
+244
+00:18:25,420 --> 00:18:30,140
+ellipsoid الآن لاحظوا كلها معادلة أربعية و في
+
+245
+00:18:30,140 --> 00:18:33,420
+الآخر كلهم مجموعات لبعض في طرف و في الطرف الثاني
+
+246
+00:18:33,420 --> 00:18:37,140
+يشماله ثابت في هذه الحالة واحد أن لو كان ثابت الـ
+
+247
+00:18:37,140 --> 00:18:41,260
+ellipse معالج اللي هنا مضل اللي هو إيش؟ واحد الآن
+
+248
+00:18:41,260 --> 00:18:45,430
+نيجي مثال عددي بس قبلها خلّيني أقول ليه لو الـ .. الـ
+
+249
+00:18:45,430 --> 00:18:47,530
+.. الـ curve cut from the surface by the plane z
+
+250
+00:18:47,530 --> 00:18:51,490
+بيساوي z note يعني لو أنت أخدت مقطع لفوق شوية لفوق
+
+251
+00:18:51,490 --> 00:18:54,250
+شوية طب هو هيطلع برضه إيش؟ إيليبسات لو خدنا لفوق
+
+252
+00:18:54,250 --> 00:18:58,190
+هنا أخدنا عند z z نوت بساوي نوت أو لتحت شوية أو
+
+253
+00:18:58,190 --> 00:19:01,330
+على الجهة هنا عند الـ Y أو عند الجهة ده كله هيطلع
+
+254
+00:19:01,330 --> 00:19:07,890
+في الآخر إيش؟ المقاطع إليبسات وهي المقطع ابدأ اجرب
+
+255
+00:19:07,890 --> 00:19:11,570
+أن تحط z بساوي z نوت واختصر المعادلة وهمدك تحط z
+
+256
+00:19:11,570 --> 00:19:15,570
+بساوي z نوت و تحطها أنا z بساوي z نوت و بتحلها
+
+257
+00:19:15,570 --> 00:19:21,660
+بتطلع ليها لو كانت هذه C أو الـ Z نوت بالساوية أو Z
+
+258
+00:19:21,660 --> 00:19:22,720
+نوت بالساوية أو Z نوت بالساوية أو Z نوت بالساوية
+
+259
+00:19:22,720 --> 00:19:23,440
+أو Z نوت بالساوية أو Z نوت بالساوية أو Z نوت
+
+260
+00:19:23,440 --> 00:19:25,540
+بالساوية أو Z نوت بالساوية أو Z نوت بالساوية أو Z
+
+261
+00:19:25,540 --> 00:19:27,060
+نوت بالساوية أو Z نوت بالساوية أو Z نوت بالساوية
+
+262
+00:19:27,060 --> 00:19:27,760
+أو Z نوت بالساوية أو Z نوت بالساوية أو Z نوت
+
+263
+00:19:27,760 --> 00:19:27,920
+بالساوية أو Z نوت بالساوية أو Z نوت بالساوية أو Z
+
+264
+00:19:27,920 --> 00:19:28,140
+نوت بالساوية أو Z نوت بالساوية أو Z نوت بالساوية
+
+265
+00:19:28,140 --> 00:19:28,340
+أو Z نوت بالساوية أو Z نوت بالساوية أو Z نوت
+
+266
+00:19:28,340 --> 00:19:31,700
+بالساوية أو Z نوت بالساوية أو Z نوت بالطبع لو
+
+267
+00:19:31,700 --> 00:19:36,640
+أخدت صفر تطلع نقطة اللي هي المنطقة هذه عند C و
+
+268
+00:19:36,640 --> 00:19:40,760
+المنطقة هذه عند ناقص C شايفينها؟ اه فبناخدها اللي
+
+269
+00:19:40,760 --> 00:19:44,540
+هي Z نقطة اللي بنعوض عليها يا بتكون لتحت شوية ده
+
+270
+00:19:44,540 --> 00:19:48,540
+تحت شوية لو أحد أخد لفوق مش هيطلع رسم هيطلع حادة
+
+271
+00:19:48,540 --> 00:19:52,380
+بساوي كمية سالبة يعني فش رسم فأكيد آخر حد اللي عند
+
+272
+00:19:52,380 --> 00:19:58,220
+Z اللي طولها مايتعداش الـ C و من تحت مايتعداش الـ C
+
+273
+00:19:58,220 --> 00:20:04,270
+اللي بدك تاخدها كمقطع عرضي طيب إذا هو أخد z بساوي z
+
+274
+00:20:04,270 --> 00:20:07,210
+note ل z note أصغر من c عشان يضمن فيه رسم لو أخدها
+
+275
+00:20:07,210 --> 00:20:10,870
+أكبر من c بيصير اللي هي برا رسم هتطلع معادلة فيها
+
+276
+00:20:10,870 --> 00:20:14,730
+سالب وجرب لحالك فـ .. فبتكون مش متحقق كمية موجبة
+
+277
+00:20:14,730 --> 00:20:18,670
+بتساوي سالب فمش متحقق فأكيد يكون فيش رسم عند z
+
+278
+00:20:18,670 --> 00:20:22,550
+بتسا.. z note بتساوي بالضبط c أو ناقص c هيطلع
+
+279
+00:20:22,550 --> 00:20:26,510
+عبارة عن اللي هو نقطة هيطلع اللي هي معادلة 0 و c
+
+280
+00:20:26,510 --> 00:20:29,950
+اللي هي النقطة اللي ورّجناكم إياها لأن لما أخدها z
+
+281
+00:20:29,950 --> 00:20:33,310
+note بحيث الـ magnitude لـ z note أو قيمة الـ
+
+282
+00:20:33,310 --> 00:20:35,370
+absolute value ل z note أصغر من c يعني جوات
+
+283
+00:20:35,370 --> 00:20:39,210
+المنطقة اللي هي قراصة أشرت لكم عليها و عوضت في
+
+284
+00:20:39,210 --> 00:20:41,710
+المعادلة حيطلع عند x note على هذا المقدر و y note
+
+285
+00:20:41,710 --> 00:20:46,330
+على هذا المقدر بساوي برضه هتطلع شمال عبارة عن
+
+286
+00:20:46,330 --> 00:20:52,320
+ellipse عشان هيك البعض بيقول هذه ellipse of
+
+287
+00:20:52,320 --> 00:20:56,480
+revolution يعني و كأنه إيش؟ شمالها اللي هي ناتجة
+
+288
+00:20:56,480 --> 00:21:02,140
+كلها مقطعها ellipses أو اللي هو من دوران ellipse
+
+289
+00:21:10,340 --> 00:21:14,680
+معادلة أعطانا x تربيع ناقص أربعة x زائد Y تربيع
+
+290
+00:21:14,680 --> 00:21:20,540
+زائد عشرة Y زائد ستة و ثلاثين Z التربيع بساوي سبعة
+
+291
+00:21:20,540 --> 00:21:25,100
+أعطانا معادلة ثانية x تربيع ناقص اثنين Y ناقص واحد
+
+292
+00:21:25,100 --> 00:21:28,540
+لكل تربيع زائد Z زائد أربعة لكل تربيع بساوي إيش؟
+
+293
+00:21:28,540 --> 00:21:37,790
+جماله بساوي خمسة و عشرين نشوف الآن لو جينا نتطلع
+
+294
+00:21:37,790 --> 00:21:41,710
+لهذه خلينا نتطلع لهذه بعد إذا نكون جابل اه عشان
+
+295
+00:21:41,710 --> 00:21:47,330
+مافيش فيها شغل كثير هذه عبارة عن اللي هي X تربيع
+
+296
+00:21:47,330 --> 00:21:50,370
+ناقص اثنين كتربيع Y ناقص واحد كتربيع زي Z أربعة
+
+297
+00:21:50,370 --> 00:21:55,430
+كتربيع على خمسة و عشرين هذا واضح أن جسمه على 25 بيصير
+
+298
+00:21:55,430 --> 00:22:00,650
+هنا الـ a على 25 وهنا الـ b على 25 وهنا الـ c على 25
+
+299
+00:22:00,650 --> 00:22:07,970
+بس ما حدث إيش؟ اللي هي شفت يعني الآن الـ ellipse اللي
+
+300
+00:22:07,970 --> 00:22:13,770
+جاب اللي بشوية اللي رسمناها هذه الـ a هنا بيصير خمسة
+
+301
+00:22:13,770 --> 00:22:19,890
+و هنا خمسة و هنا خمسة بس الآن النقطة هذه و كأنه x
+
+302
+00:22:19,890 --> 00:22:22,230
+ناقص اثنين لكل تربيع و كأنه الـ x axis صار له shift
+
+303
+00:22:22,230 --> 00:22:25,670
+اثنين و هنا اللي برضه صار لها shift و هنا صار لها
+
+304
+00:22:25,670 --> 00:22:34,110
+shift إذا الرسم اللي عندي هنا هذه هي معادلة على 25
+
+305
+00:22:34,110 --> 00:22:38,990
+عبارة عن معادلة ellipse بس لاحظوا أن الـ a و الـ b و
+
+306
+00:22:38,990 --> 00:22:43,070
+الـ c كلها متساويات يعني لو جيت أخدت أي مقطع عرضي
+
+307
+00:22:43,070 --> 00:22:47,810
+لها لو أخدت x بساوي 2 فبيصير هذه المعادلة اللي دلّت
+
+308
+00:22:47,810 --> 00:22:53,770
+هي عبارة عن معادلة إيش؟ دائرة مركزها واحد و ناقص
+
+309
+00:22:53,770 --> 00:22:57,690
+أربعة في الـ Y و Z ولو أخدت هذه اللي هي z
+
+310
+00:22:57,690 --> 00:23:00,750
+بالساعة و ناقص أربعة يعني عشان أصفّرها يا جماعة
+
+311
+00:23:00,750 --> 00:23:05,090
+بأخد هيك بتظل المعادلة إيش برضه؟ عبارة عن دائرة
+
+312
+00:23:05,090 --> 00:23:09,930
+يعني كل مقطعها دوائر يعني في الواقع هي هذه عبارة
+
+313
+00:23:09,930 --> 00:23:14,790
+عن كرة لأن لما تكون أريحكم لما تكون الـ a و الـ b و
+
+314
+00:23:14,790 --> 00:23:20,670
+الـ c زي بعض بتطلع عندي كل المقاطع اللي هي Circles
+
+315
+00:23:20,670 --> 00:23:25,950
+يعني بتطلع إيش مالها؟ Ellipsoid مدورة يعني سُفير
+
+316
+00:23:25,950 --> 00:23:31,110
+يعني إيش؟ مركزها بدل ما كان مركزها 000 صار مركزها
+
+317
+00:23:31,110 --> 00:23:35,370
+إيش؟ اللي في الأول كان مركزها 000 لاحظوا اللي إيش؟
+
+318
+00:23:35,370 --> 00:23:39,390
+اللي بقصده قبل قليل من الوقت لما قمنا باختيار هذه
+
+319
+00:23:39,390 --> 00:23:46,370
+الاسم كانت هذه الاسم X ناقص Zero كل تربيع Y ناقص
+
+320
+00:23:46,370 --> 00:23:49,610
+Zero كل تربيع Z ناقص Zero كل تربيع الآن الجديدة
+
+321
+00:23:49,610 --> 00:23:53,170
+تبعتنا هذه الاسم X ناقص واحد كل تربيع ناقص اثنين كل
+
+322
+00:23:53,170 --> 00:23:58,160
+تربيع ناقص واحد كل تربيع زائد أربعة كل تربيع اللي هي
+
+323
+00:23:58,160 --> 00:24:02,080
+with center إذا صح التعبير هو عبارة عن اثنين واحد
+
+324
+00:24:02,080 --> 00:24:07,160
+و مين و سالب أربعة بدل اللي كان في المثال الأولان
+
+325
+00:24:07,160 --> 00:24:12,400
+اللي شرحنا عليه اللي هو zero zero و zero طيب نيجي
+
+326
+00:24:12,400 --> 00:24:17,380
+الآن للي هي المعادلة هذه المعادلة هذه اللي عندنا
+
+327
+00:24:17,380 --> 00:24:22,520
+شكلها بده شغل هذه اللي بالشكل هذا بده عملها على
+
+328
+00:24:22,520 --> 00:24:27,380
+صورة تربيعية زي هيك أو تربيعية على إشي على إشي على
+
+329
+00:24:27,380 --> 00:24:35,640
+إشي على إشي على إشي على إشي على إشي على إشي على
+
+330
+00:24:35,640 --> 00:24:36,040
+إشي على إشي على إشي على إشي على إشي على إشي على
+
+331
+00:24:36,040 --> 00:24:37,520
+إشي على إشي على إشي على إشي على إشي على إشي على
+
+332
+00:24:37,520 --> 00:24:39,920
+إشي على إشي على إشي على إشي على إشي على إشي على
+
+333
+00:24:39,920 --> 00:24:40,080
+إشي على إشي على إشي على إشي على إشي على إشي على
+
+334
+00:24:40,080 --> 00:24:40,800
+إشي على إشي على إشي على إشي على إشي على إشي على
+
+335
+00:24:40,800 --> 00:24:44,510
+إشي على إشي على إشي على إشي على و 36 Z التربيع هذه
+
+336
+00:24:44,510 --> 00:24:49,810
+معمولة و خلاص مافيش معاملة Z صفر معاملة Z عشان هيك
+
+337
+00:24:49,810 --> 00:24:52,990
+الإكمال المربع لها دول طب أني ما جبت اللي الأربعة
+
+338
+00:24:52,990 --> 00:24:57,870
+هذه الآن x تربيع ناقص أربعة x اللي هي الحد الثاني
+
+339
+00:24:57,870 --> 00:25:04,590
+الحد الثاني على اثنين بطلع جدّيش؟ ناقص اثنين اللي هو بربعه
+
+340
+00:25:04,590 --> 00:25:08,810
+بطلع عندي أربعة الحد الثالث اللي بده يعمل هذا مربع
+
+341
+00:25:08,810 --> 00:25:12,690
+كامل للي ناسي كيف بنكمل المربع وهي تربيع زائد عشرة Y
+
+342
+00:25:12,690 --> 00:25:16,930
+عشرة على اثنين بيطلع خمسة المعامل Y يعني على اثنين
+
+343
+00:25:16,930 --> 00:25:21,670
+بيطلع خمسة بنربعه بيطلع كده؟ خمسة و عشرين فضفت هنا
+
+344
+00:25:21,670 --> 00:25:24,570
+أربعة وضفت هنا خمسة و عشرين هذا عشان أعمله مربع
+
+345
+00:25:24,570 --> 00:25:27,810
+كامل و هنعمله شغال مربع كامل اللي أنا اللي أضفته
+
+346
+00:25:27,810 --> 00:25:30,370
+على الجهة دي بدي أضيفه على الجهة الثانية فبيصير
+
+347
+00:25:30,370 --> 00:25:34,570
+زائد أربعة زائد خمسة و عشرين و السبعة الأصلية اللي
+
+348
+00:25:34,570 --> 00:25:38,190
+في المعادلة فصار لما نجمع إن هدولة بيصير نجدّيش؟ ستة
+
+349
+00:25:38,190 --> 00:25:41,910
+و ثلاثين لأن نيجي نعملنا دولة نعمل مربع كامل هذه
+
+350
+00:25:41,910 --> 00:25:45,410
+عبارة عن X ناقص اثنين لكل تربيع وهذه Y زائد خمسة
+
+351
+00:25:45,410 --> 00:25:49,850
+لكل تربيع وجسمت كل عالمين على ستة و ثلاثين فصارت
+
+352
+00:25:49,850 --> 00:25:53,310
+عبارة
+
+353
+00:25:53,310 --> 00:25:56,310
+عن X ناقص اثنين كل تربيع على ستة و ثلاثين Y زائد
+
+354
+00:25:56,310 --> 00:25:59,710
+الخمسة كل تربيع على ستة و ثلاثين Z التربيع بساوي
+
+355
+00:25:59,710 --> 00:26:04,170
+واحد لاحظ أن هذه هي عبارة عن معادلة ellipsoid هي
+
+356
+00:26:04,170 --> 00:26:10,350
+هذه مربعة وهذه مربعة وهذه مربعة بخصوص الـ X مع الـ
+
+357
+00:26:10,350 --> 00:26:15,570
+2 وكأنه متغير و X ناقص 2 أو Y زائد خمسة و Z نفسها
+
+358
+00:26:15,570 --> 00:26:19,670
+كل هدولة مربعات مدام مربعاته بالشكل هذا هذه
+
+359
+00:26:19,670 --> 00:26:24,450
+برضه هتكون عبارة عن ellipsoid اللي هي الـ A تبعتها
+
+360
+00:26:24,450 --> 00:26:30,610
+6 والـ B 6 والـ C 1 الـ A والـ B 6 والـ C 1 والـ
+
+361
+00:26:30,610 --> 00:26:37,770
+Center الآن بدل ما كان 000 صار 2 و 5 و 0 2 و 5 و 0
+
+362
+00:26:37,770 --> 00:26:40,730
+برضه إذا هذا الشكل عبارة عن إيش يا جماعة؟ ellipsoid
+
+363
+00:26:40,730 --> 00:26:45,230
+إذا حفظنا الشكل هذا ellipsoid و هذا ellipsoid لكن
+
+364
+00:26:45,230 --> 00:26:48,910
+كل المتساويات المعاملات عشان هيقول لنا عبارة عن إيش؟
+
+365
+00:26:48,910 --> 00:26:53,320
+يا جماعة؟ عبارة عن سفير لو أحد قال لي طب إيش هتقرأ؟
+
+366
+00:26:53,320 --> 00:26:56,120
+أخد z بصفر بتطلع ليه المعادلة هذه معادلة دائرة
+
+367
+00:26:56,120 --> 00:26:59,940
+مظبوط أما يفي هتلاقي مقطع دائرة و مقطعين بعملين
+
+368
+00:26:59,940 --> 00:27:04,860
+ellipses طيب إذا هذا تطبيق على اللي أولاني نيجي
+
+369
+00:27:04,860 --> 00:27:10,620
+الآن نشوف اللي هو المثال الثاني اللي هو أو المثال
+
+370
+00:27:10,620 --> 00:27:14,180
+الثالث في الـ chapter اللي هو the hyperbolic
+
+371
+00:27:14,180 --> 00:27:18,220
+paraboloid وقال لي هاي hyperbolik و parabolaik فإيش
+
+372
+00:27:18,220 --> 00:27:23,600
+سماها هيك أكيد من مقطعها سماها هيك قلنا الآن عشان
+
+373
+00:27:23,600 --> 00:27:28,020
+إيش هي؟ اللي بدنا إياها نرسمها Y تربيع على B تربيع ناقص
+
+374
+00:27:28,020 --> 00:27:31,820
+X تربيع على A تربيع بساوي Z على مين؟ على C حيثُ
+
+375
+00:27:31,820 --> 00:27:36,140
+الـ C شمالها أكبر من 0 ماشي الآن عشان الـ C أكبر من
+
+376
+00:27:36,140 --> 00:27:39,900
+0 عشان أن لو أخد الـ C أصغر من 0 بيصير المقام X
+
+377
+00:27:39,900 --> 00:27:43,120
+تربيع على A تربيع ناقص Y تربيع على B تربيع بساوي
+
+378
+00:27:43,120 --> 00:27:48,500
+كمية اللي هي الـ C أكبر من 0 عشان نحدد الكلام كلنا
+
+379
+00:27:48,500 --> 00:27:52,880
+زي بعض خلينا نقول C أكبر من 0 وبصير الشكل هذا بيقول
+
+380
+00:27:52,880 --> 00:27:56,500
+لي اللي هو وين الـ hyper parabola أو وين الـ parabola
+
+381
+00:27:56,500 --> 00:28:01,980
+متجهها الـ get it فإيش؟ بقول طيب أي تربيع بيه تربيع
+
+382
+00:28:01,980 --> 00:28:04,060
+ناقص six تربيع أي تربيع بيه تربيع بساوي ده ناسي
+
+383
+00:28:04,060 --> 00:28:10,510
+اللي يحدد الشكل هو مقاطعه مقاطع الشكل و بنختار
+
+384
+00:28:10,510 --> 00:28:15,470
+مقاطع تكون سهلة خد الآن طبعاً بيقول لي في symmetry
+
+385
+00:28:15,470 --> 00:28:20,070
+with respect to the plane X بتساوي صفر أو Y بتساوي
+
+386
+00:28:20,070 --> 00:28:25,750
+صفر هنلاقي اللي هو يعني لو أخذنا الآن X بساوي صفر
+
+387
+00:28
+
+445
+00:32:56,840 --> 00:32:56,960
+وانا وانا وانا وانا وانا وانا وانا وانا وانا وانا
+
+446
+00:32:56,960 --> 00:32:58,900
+وانا وانا وانا وانا وانا وانا وانا وانا وانا وانا
+
+447
+00:32:58,900 --> 00:32:59,160
+وانا وانا وانا وانا وانا وانا وانا وانا وانا وانا
+
+448
+00:32:59,160 --> 00:32:59,960
+وانا وانا وانا وانا وانا وانا وانا وانا وانا وانا
+
+449
+00:32:59,960 --> 00:33:07,080
+وانا وانا
+
+450
+00:33:07,080 --> 00:33:16,590
+وانا والبارابولا بين الـ Y و الـ Z، الـ Y و الـ Z
+
+451
+00:33:16,590 --> 00:33:22,020
+الأعلى، هذه النقطة تمثل لها minimum بينما في
+
+452
+00:33:22,020 --> 00:33:26,880
+المنطقة اللي هي البارابولا اللي بالأخضر، هذه هيها
+
+453
+00:33:26,880 --> 00:33:33,600
+اللي هي بين الـ X و الـ Z plane، البارابولا إلى أسفل، عشان
+
+454
+00:33:33,600 --> 00:33:37,380
+هيك، هذه النقطة تمثلها نقطة الأصل، اللي هي نقطة
+
+455
+00:33:37,380 --> 00:33:41,440
+maximum، عشان هيك لو جيت أخدت أي جوار حواليها، هتلاقي
+
+456
+00:33:41,440 --> 00:33:47,520
+مرة ممكن تكون أكبر ومرة بتكون أسفل، عشان هيك
+
+457
+00:33:47,520 --> 00:33:51,210
+هنُفصِّل لها لـ maximum و لـ minimum هنا، وهذا طبعاً لما
+
+458
+00:33:51,210 --> 00:33:54,990
+نحكي عن maximum و minimum، نسمي هذا عبارة عن saddle
+
+459
+00:33:54,990 --> 00:33:59,430
+point أو نقطة سرج الحصان، إذا الـ saddle point هنا
+
+460
+00:33:59,430 --> 00:34:04,250
+هي عبارة عن 0,0,0، هذا اللي بدنا نعرفه عن اللي هي
+
+461
+00:34:04,250 --> 00:34:08,290
+hyperbolic paraboloid، نيجي الآن ناخذ مثال، what is
+
+462
+00:34:08,290 --> 00:34:10,310
+the graph of each of the following quadratic
+
+463
+00:34:10,310 --> 00:34:13,820
+equations؟ بدنا نعرف الـ graph لها، الدولة الآن Y تربيع
+
+464
+00:34:13,820 --> 00:34:19,780
+على 6 + 3 - Z تربيع على 4 = X على 5، بس
+
+465
+00:34:19,780 --> 00:34:22,960
+خليني أعطيكم ملاحظة عشان... على اللي قبل... لاحظوا
+
+466
+00:34:22,960 --> 00:34:27,160
+هنا اللي هي اثنتين تربيعيات وبينهم سالب، والجهة
+
+467
+00:34:27,160 --> 00:34:31,640
+الأخرى اللي هي طبعاً موجبة، و...و...و...والمتغير
+
+468
+00:34:31,640 --> 00:34:36,520
+الثالث Z بدون تربيع، يعني في الـEllipses كان كلهم
+
+469
+00:34:36,520 --> 00:34:40,740
+موجبات وفي الآخر constant، وكلهم اللي هم مربعات
+
+470
+00:34:41,120 --> 00:34:44,400
+بحكي عن اللي هي المتغيرات الـ X والـ Y والـ Z
+
+471
+00:34:44,400 --> 00:34:49,140
+هنا المتغيرين تربيعين بين إشارة سالب، و
+
+472
+00:34:49,140 --> 00:34:52,820
+المتغير الآخر على الطرف الآخر، واللي هو إيه إشارته
+
+473
+00:34:52,820 --> 00:34:57,160
+إشارته موجب، هذا طبعاً بيمثل لي hyperbolic
+
+474
+00:34:57,160 --> 00:35:01,160
+paraboloid، أي شكل أي معادلة زي هيك بتكون برضه
+
+475
+00:35:01,160 --> 00:35:05,980
+hyperbolic paraboloid، وهي في عندي الآن نطلع المثال
+
+476
+00:35:05,980 --> 00:35:11,200
+الثالثة، بهاللي هنتطلع Y تربيع على 36 - Z تربيع
+
+477
+00:35:11,200 --> 00:35:15,600
+على 4 = X على 5، هاي تربيع وهي تربيع بين إشارة سالب،
+
+478
+00:35:15,600 --> 00:35:20,460
+وهذه لحالها بدون تربيع، Linear يعني، وإشارتها موجبة
+
+479
+00:35:20,460 --> 00:35:24,810
+في هذه الحالة، هذه عبارة عن hyperboloid paraboloid
+
+480
+00:35:24,810 --> 00:35:29,730
+يعني حط Z بصفر، تطلع hyperbola بين Y و X، وحط هنا Y
+
+481
+00:35:29,730 --> 00:35:34,110
+بصفر، تطلع hyperbola بين X و Z، وحط هذه عبارة عن عدد
+
+482
+00:35:34,110 --> 00:35:37,610
+بتطلع عبارة عن hyperbola، بارابولا، اثنتين آسف
+
+483
+00:35:37,610 --> 00:35:41,050
+بارابولا بارابولا وبعدين hyperbola، الشكل اللي قلت
+
+484
+00:35:41,050 --> 00:35:45,390
+hyperbola في الأول، البارابولا بين الـ Y و الـ X، hyperbola
+
+485
+00:35:45,390 --> 00:35:51,150
+بين الـ X و الـ Z، وهي hyperbola بين الـ Y و الـ X، يعني
+
+486
+00:35:51,150 --> 00:35:56,670
+مُشابهة للجبل بالظبط، بس اختلفت مين اللي هو
+
+487
+00:35:56,670 --> 00:36:03,020
+البارابولات والـ hyperbola بين مين ومين، اللي قبل
+
+488
+00:36:03,020 --> 00:36:08,220
+كانت كيف واللي حالياً صارت كيف، إذا هذه التفصيل is a
+
+489
+00:36:08,220 --> 00:36:11,960
+hyperbolic paraboloid similar to that in example
+
+490
+00:36:11,960 --> 00:36:16,740
+ثلاثة، but the parabolas are in the XY plane, ZX plane
+
+491
+00:36:16,740 --> 00:36:25,550
+الآن، in the XY plane, ZX plane عن الـ hyperbola اللي
+
+492
+00:36:25,550 --> 00:36:29,450
+هو وين؟ في الـ plane اللي هو x بتساوي k، أكبر من 0، زي
+
+493
+00:36:29,450 --> 00:36:34,230
+ما قلنا قبل شوية، يعني يعني الـ الـ الـ نفس الحكي
+
+494
+00:36:34,230 --> 00:36:38,570
+الأول، يعني بس بدل الـ x إيش؟ بدل z الـ x في اللي قبل
+
+495
+00:36:38,570 --> 00:36:42,930
+طيب، هنشوف
+
+496
+00:36:42,930 --> 00:36:46,150
+اللي بعدها، طبعاً سيدي الـ point إيش؟ هتكون برضه
+
+497
+00:36:46,150 --> 00:36:49,530
+0,0,0، ما فيها مشكلة، عن طريق الـ Equation
+
+498
+00:36:49,530 --> 00:36:54,830
+التي قمنا بالتحقق
+
+499
+00:36:54,830 --> 00:36:59,530
+عليها، الآن X تربيع - 8X + Y تربيع - 2Y + Z
+
+500
+00:36:59,530 --> 00:37:03,710
+= 15، نعملها على صورة تربيعيات، واللي بيظل
+
+501
+00:37:03,710 --> 00:37:07,330
+لحاله يظل لحاله، وبنشوف بيطلع إيش الشكل اللي عندك
+
+502
+00:37:07,330 --> 00:37:12,150
+the equation becomes، الآن بدي أعمل إكمال مربع، - X
+
+503
+00:37:12,150 --> 00:37:15,750
+تربيع - 8X مع بعض، ضربت في -، هو الأشياء بعد
+
+504
+00:37:15,750 --> 00:37:20,950
+إذنكم + Y تربيع - 2Y - Z = -15، ضربت
+
+505
+00:37:20,950 --> 00:37:26,070
+كل هذا في -، عشان أكمل المربع، الآن عندي سحبت
+
+506
+00:37:26,070 --> 00:37:29,910
+الناقص منها، أنا بصير X تربيع - 8X، الآن بدي
+
+507
+00:37:29,910 --> 00:37:33,170
+أخلي مربع كامل، هذا بقسم هذا على اثنين، بيطلع أربعة
+
+508
+00:37:33,170 --> 00:37:36,490
+بربعه بيصير ستة عشر، إذا انضفت ستة عشر، فالواقع -
+
+509
+00:37:36,490 --> 00:37:41,090
+ستة عشر، الآن +، الآن الثاني Y تربيع - 2Y
+
+510
+00:37:41,090 --> 00:37:44,970
++ 1، الآن عارفين كيف جبته، هذا قسمته هذا على
+
+511
+00:37:44,970 --> 00:37:48,640
+اثنين بيطلع واحد، ربعته بيطلع واحد، ده انضفت واحد، إذا
+
+512
+00:37:48,640 --> 00:37:51,520
+أضفت واحد وطرحت هنا ستة عشر، لأن زيتها ستة عشر
+
+513
+00:37:51,520 --> 00:37:56,240
+وناقصي طرحت ستة عشر - Z زي ما هي = -
+
+514
+00:37:56,240 --> 00:37:59,200
+خمسة عشر زي ما هي، وحطيت ناقص ستة عشر اللي عملت على
+
+515
+00:37:59,200 --> 00:38:02,600
+الجهة اللي هذه، وواحد اللي عملت على الجهة الثانية
+
+516
+00:38:02,600 --> 00:38:05,800
+إلى أن أجمع -16 و -15 - 1
+
+517
+00:38:05,800 --> 00:38:09,160
+و = -32، إذا صار عندي that
+
+518
+00:38:09,160 --> 00:38:16,000
+is اللي هو هذا (Y - 1)² صح؟ معايا يا
+
+519
+00:38:16,000 --> 00:38:20,820
+جماعة، (Y - 1)² - (X - 4)
+
+520
+00:38:20,820 --> 00:38:25,220
+)²، هذين اللي هو = الـ Z، بنقلها على الطرف
+
+521
+00:38:25,220 --> 00:38:29,420
+الثاني، بصير Z - 32، إذا هاي صارت الشكل
+
+522
+00:38:29,420 --> 00:38:35,440
+اللي هو هاي مربع - مربع + مربع، وهذا إيش اللي هو
+
+523
+00:38:35,440 --> 00:38:39,250
+من الدرجة الأولى، إذا هذه عبارة عن hyperbolic
+
+524
+00:38:39,250 --> 00:38:45,650
+paraboloid، زي الأولى بالظبط، بس بتختلف أنه عندي
+
+525
+00:38:45,650 --> 00:38:49,970
+اللي هو صار shift على الـ Y axis 1 وعلى الـ X axis 4
+
+526
+00:38:49,970 --> 00:38:55,310
+وعلى الـ Z axis 30، يعني بيصير الآن اللي هي الـ
+
+527
+00:38:55,310 --> 00:38:59,910
+saddle point، بدل ما كانت 0,0,0، بيصير الـ saddle point
+
+528
+00:38:59,910 --> 00:39:05,980
+عبارة عن 4، 1، 30، والـ a والـ b والـ
+
+529
+00:39:05,980 --> 00:39:09,300
+c، طبعاً ناخذهم إيش بيساوي واحد، لأنه هنا على واحد
+
+530
+00:39:09,300 --> 00:39:13,540
+هنا على واحد، هنا على إيش؟ واحد، واضح الكلام، هاي عندي
+
+531
+00:39:13,540 --> 00:39:19,840
+مثالين، الآن نيجي لأخر مثال أو أخر مثالين أو النوع
+
+532
+00:39:19,840 --> 00:39:22,620
+الثالث اللي بناخذه في هذه المحاضرة، اللي هو حاجة
+
+533
+00:39:22,620 --> 00:39:26,640
+بنسميها elliptical cone، elliptical cone يا جماعة
+
+534
+00:39:26,640 --> 00:39:32,700
+من اسمها، عندي شكل مقطعه اسمه ellipse و cone، بعمل
+
+535
+00:39:32,700 --> 00:39:38,290
+لكون، إيش الكون؟ بنشوف هو مخروط، طيب الـ Graph، X تربيع
+
+536
+00:39:38,290 --> 00:39:40,510
+على A تربيع + Y تربيع على B تربيع = Z تربيع
+
+537
+00:39:40,510 --> 00:39:45,790
+على C تربيع، لاحظوا ده كله تربيع، كله تربيع، لكن بيختلف
+
+538
+00:39:45,790 --> 00:39:50,170
+عن مين؟ عن الـ Ellipse، أن هذا مش زائد، بيساوي ثابت، لأ
+
+539
+00:39:50,170 --> 00:39:53,310
+X تربيع على A تربيع + Y تربيع على B تربيع =
+
+540
+00:39:53,310 --> 00:39:57,430
+Z تربيع على C تربيع، بيساوي إيش؟ تربيع، ومش فيه توابط
+
+541
+00:39:57,430 --> 00:40:01,770
+أه، هذه اللي هي هتطلع عندي حاجة اسمها elliptical cone
+
+542
+00:40:01,770 --> 00:40:07,160
+وزي ما قلنا بنعرف الشكل من مين؟ من المقاطع، المقطع
+
+543
+00:40:07,160 --> 00:40:11,360
+الأول، خذ X بصفر، هنشتغل في الـ yz plane، لما X بصفر
+
+544
+00:40:11,360 --> 00:40:15,120
+...لما...آسف، لما X بصفر، إيش بتصير عندي؟ اللي هو Y
+
+545
+00:40:15,120 --> 00:40:18,100
+تربيع على B تربيع = Z تربيع على C تربيع، يعني
+
+546
+00:40:18,100 --> 00:40:23,640
+مكانها Z = ± C/B في Y، يعني خطين
+
+547
+00:40:23,640 --> 00:40:30,500
+خطين، هيك بين الـ X و...بين الـ Z والـ Y، هاي الـ Z و
+
+548
+00:40:30,500 --> 00:40:36,400
+هاي الـ Z والـ Y، ماشي؟ هاي خط، خطين، إذن هدول خطين، أو
+
+549
+00:40:36,400 --> 00:40:41,540
+ميّالين هيك متعاكسين، اللي هو لما ناخذ X بصفر، إذا
+
+550
+00:40:41,540 --> 00:40:50,660
+عمل لي خطين بين Z وY، طيب خذ Y بصفر، هيطلع عندي
+
+551
+00:40:50,660 --> 00:40:54,200
+شبيه، اللي هو X تربيع على A تربيع = Z تربيع على C تربيع
+
+552
+00:40:54,200 --> 00:40:58,420
+حلّها بيطلع Z = ± C/A في X، برضه two lines،
+
+553
+00:40:58,420 --> 00:41:02,000
+لإن مش في الـ Y، زي دي blade في
+
+554
+00:41:02,000 --> 00:41:07,440
+الـ X زي دي blade، بين الـ X، هي الـ X وهي الـ X على
+
+555
+00:41:07,440 --> 00:41:11,370
+الجهة الثانية، فبيطلع اللي إنهم خطين، الآن هذه
+
+556
+00:41:11,370 --> 00:41:17,310
+المقاطع بتعمل خطوط، خطين، خطين، الآن نشوف المقطع
+
+557
+00:41:17,310 --> 00:41:22,170
+العرضي، لو أخذنا Z بصفر، هنلاقي إيش؟ 0, 0, 0
+
+558
+00:41:22,170 --> 00:41:26,470
+هيصير كمية مربعة = 0 + 0، X تربيع على A
+
+559
+00:41:26,470 --> 00:41:28,430
+تربيع + Y تربيع على B تربيع، مستوى = 0، معناه
+
+560
+00:41:28,430 --> 00:41:35,380
+توازي = 0، معناه بيمر بمين؟ بنقطة الأصل، إذا
+
+561
+00:41:35,380 --> 00:41:38,840
+الآن ناخذ طب مقطع لفوق شوية أو لتحت شوية، زي مثلاً
+
+562
+00:41:38,840 --> 00:41:42,780
+بتساوي C أو -C، زي ما بدك، اللي هو C عدد لايفسي
+
+563
+00:41:42,780 --> 00:41:46,120
+أو سفر، لما بيصير عندي X تربيع على A تربيع + Y
+
+564
+00:41:46,120 --> 00:41:49,900
+تربيع على B تربيع = 1، هذا المقطع
+
+565
+00:41:49,900 --> 00:41:54,360
+إيش ماله؟ Ellipse، يعني هو كأنه المقطع الموازي لـ
+
+566
+00:41:54,360 --> 00:41:59,700
+XY plane على بعد C، على بعد -C، عبارة عن Ellipse
+
+567
+00:41:59,700 --> 00:42:04,240
+Ellipse، إذا مقاطعه ellipses، عشان إيه نكملها بتطلع
+
+568
+00:42:04,240 --> 00:42:09,100
+عبارة عن الكون الأمامي، كل المقطع العرضية ellipses
+
+569
+00:42:09,100 --> 00:42:13,880
+وهذول الخطوط اللي ذكرناهم، فبيصير عندي اللي هو الـ X
+
+570
+00:42:13,880 --> 00:42:17,380
+تربيع على A تربيع + Y تربيع على B تربيع = Z
+
+571
+00:42:17,380 --> 00:42:21,720
+تربيع على C تربيع، هو كون سمينه، elliptical cone، عشان
+
+572
+00:42:21,720 --> 00:42:26,720
+نقول اللي هو مقطع العرضي هو عبارة عن ellipse، لو الـ
+
+573
+00:42:26,720 --> 00:42:31,570
+A = الـ B، بيصير المقطع العرضي عبارة عن اللي هو
+
+574
+00:42:31,570 --> 00:42:35,430
+Circle، فبيصير Circle الكون، والـ Circle طبعاً حالة
+
+575
+00:42:35,430 --> 00:42:38,770
+خاصة من الـ Ellipse، الـ Ellipse، إذن Elliptical cone
+
+576
+00:42:38,770 --> 00:42:44,770
+زي ما شفنا الآن، لاحظوا أنه مين محوره الآن أو هي
+
+577
+00:42:44,770 --> 00:42:47,890
+وين مقطعه العرضي اللي هو ذاك الموازي للـ XY Plane
+
+578
+00:42:47,890 --> 00:42:52,110
+يعني الـ Z، وكأنه تجاوز محوره مين؟ الـ Z Axis، يعني
+
+579
+00:42:52,110 --> 00:42:57,500
+كون على مين؟ على الـ Z Axis، لأن لو بدّلنا الـ X مكان
+
+580
+00:42:57,500 --> 00:43:01,740
+الـ Z هنا، هيصير...لو قلنا X تربيع على A تربيع + Z
+
+581
+00:43:01,740 --> 00:43:05,120
+تربيع على B تربيع = X تربيع على A تربيع، بيصير
+
+582
+00:43:05,120 --> 00:43:09,880
+الكون هذا، X هذه صارت مين؟ الـ Y، الـ Y، معناته هيكون
+
+583
+00:43:09,880 --> 00:43:12,700
+على وين؟ على الـ Y Axis، هيصير الـ A هذا على الـ Y
+
+584
+00:43:12,700 --> 00:43:17,280
+Axis، لو كانت هذه بدل الـ Z X، هيصير الكون مين؟ على
+
+585
+00:43:17,280 --> 00:43:20,740
+الـ X Axis، مائل على الـ X Axis، إن شاء الله واضحة
+
+586
+00:43:20,740 --> 00:43:24,180
+الصورة، نيجي ناخذ المثال الأخير، What is the graph of
+
+587
+00:43:24,180 --> 00:43:27,400
+each of the following quadratic equations؟ X تربيع
+
+588
+00:43:27,400 --> 00:43:32,820
+على 36 + Z تربيع على 25 + Y تربيع على 4، إيش اللي
+
+589
+00:43:32,820 --> 00:43:37,720
+عملته؟ زي ما قلت توجه قبله شوية، بدلت الـ Z بدل الـ Y
+
+590
+00:43:37,720 --> 00:43:41,840
+صار الكون على مين؟ على الـ Z، يعني المقطع العرضي
+
+591
+00:43:41,840 --> 00:43:46,650
+موازي لمين؟ للـ XZ Plane، وهو كمعنى الـ elliptical cone
+
+592
+00:43:46,650 --> 00:43:53,570
+برضه، لكن أكسيها هي Y-axis، هي اللي هنا بدل من Z
+
+593
+00:43:53,570 --> 00:43:57,250
+-axis، ومقاطع الـ X ومقاطع الـ Y، أكسيها هي الـ X
+
+594
+00:43:57,250 --> 00:44:00,190
+-axis، ومقاطع الـ Y أكسيها هي الـ X-axis، ومقاطع الـ
+
+595
+00:44:00,190 --> 00:44:01,890
+Y أكسيها هي الـ X-axis، ومقاطع الـ Y أكسيها هي الـ
+
+596
+00:44:01,890 --> 00:44:03,730
+X-axis، ومقاطع الـ Y أكسيها هي الـ X-axis، ومقاطع
+
+597
+00:44:03,730 --> 00:44:06,190
+الـ Y أكسيها هي الـ X-axis، ومقاطع الـ Y أكسيها هي
+
+598
+00:44:06,190 --> 00:44:06,670
+الـ X-axis، ومقاطع الـ Y أكسيها هي الـ X-axis،
+
+599
+00:44:06,670 --> 00:44:09,890
+ومقاطع الـ Y أكسيها هي الـ X-axis، ومقاطع الـ X
+
+600
+00:44:17,990 --> 00:44:20,910
+هذا مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً
+
+601
+00:44:20,910 --> 00:44:30,530
+مثلاً مثلاً
+
+602
+00:44:30,530 --> 00:44:33,630
+مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً
+
+603
+00:44:33,630 --> 00:44:35,330
+مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً
+
+604
+00:44:35,330 --> 00:44:35,970
+مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً =
+
+605
+00:44:35,970 --> 00:44:44,970
+على 25 و 4 أو 29 أو بلغة أخرى 29، قسمتها لـ 4 أو 25
+
+606
+00:44:44,970 --> 00:44:48,250
+طالع هذا إيش بيساوي؟ بيساوي 0، طبعاً عملتها هيك عشان
+
+607
+00:44:48,250 --> 00:44:52,790
+أظبطها، تطلع اللي هي ellipsoid as elliptical cone
+
+608
+00:44:52,
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/LGA6DA-unSY_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/LGA6DA-unSY_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..23262751d947b103935ec39cc52b8ce4762524d5
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/LGA6DA-unSY_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2559, "start": 0.71, "end": 25.59, "text": "بسم الله الرحمن الرحيم السلام عليكم ورحمة الله وبركاته هذه المحاضرة السادسة مساق تفاضل وتكامل جيم المحاضرة بعنوان cylinders and quadratic surfaces وهو ال section الأخير في هذا ال chapter ولكن المحاضرة هذه اللي هي عبارة عن الجزء الأول من 12-6 والجزء الثاني ان شاء الله بكون هي المحاضرة السابعة", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 29538, 9673, 5016, 46958, 25720, 21136, 18513, 3794, 3660, 47524, 995, 4587, 6055, 5172, 46958, 1211, 34683, 4117, 10943, 1211, 10874, 32640, 9673, 5016, 46958, 25720, 45030, 1863, 2407, 7649, 42166, 293, 37262, 16130, 37037, 2407, 2423, 3541, 16247, 9778, 13546, 8978, 23758, 2423, 7187, 35525, 19452, 9673, 5016, 46958, 25720, 29538, 13672, 1829, 39896, 6225, 3555, 9640, 3660, 18871, 25724, 11622, 38207, 16247, 12610, 9154, 2272, 12, 21, 16070, 7435, 11622, 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" هي", "probability": 0.9423828125}, {"start": 19.43, "end": 20.07, "word": " عبارة", "probability": 0.962158203125}, {"start": 20.07, "end": 20.23, "word": " عن", "probability": 0.99755859375}, {"start": 20.23, "end": 20.67, "word": " الجزء", "probability": 0.9837239583333334}, {"start": 20.67, "end": 21.13, "word": " الأول", "probability": 0.96826171875}, {"start": 21.13, "end": 21.33, "word": " من", "probability": 0.984375}, {"start": 21.33, "end": 21.59, "word": " 12", "probability": 0.53125}, {"start": 21.59, "end": 22.03, "word": "-6", "probability": 0.6640625}, {"start": 22.03, "end": 22.95, "word": " والجزء", "probability": 0.887451171875}, {"start": 22.95, "end": 23.33, "word": " الثاني", "probability": 0.947509765625}, {"start": 23.33, "end": 23.45, "word": " ان", "probability": 0.619140625}, {"start": 23.45, "end": 23.65, "word": " شاء", "probability": 0.95263671875}, {"start": 23.65, "end": 23.65, "word": " الله", "probability": 0.94873046875}, {"start": 23.65, "end": 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"word": " X", "probability": 0.958984375}, {"start": 92.71, "end": 93.11, "word": " Axis", "probability": 0.970947265625}, {"start": 93.11, "end": 93.37, "word": " عندي", "probability": 0.803955078125}, {"start": 93.37, "end": 93.69, "word": " باخد", "probability": 0.98291015625}, {"start": 93.69, "end": 94.33, "word": " موازي", "probability": 0.949951171875}, {"start": 94.33, "end": 94.79, "word": " له", "probability": 0.7138671875}, {"start": 94.79, "end": 95.43, "word": " اللي", "probability": 0.97509765625}, {"start": 95.43, "end": 95.65, "word": " هو", "probability": 0.9912109375}, {"start": 95.65, "end": 96.09, "word": " خط", "probability": 0.99462890625}, {"start": 96.09, "end": 96.35, "word": " على", "probability": 0.94189453125}, {"start": 96.35, "end": 97.25, "word": " طول", "probability": 0.99560546875}, {"start": 97.25, "end": 98.11, "word": " الاش", "probability": 0.4337158203125}, {"start": 98.11, "end": 98.81, "word": " بمرره", "probability": 0.8450927734375}, {"start": 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paraboloid زي الأولى بالظبط بس بتختلف انه عندى اللى هو صار shift على ال y axis 1 وعلى ال x axis 4 وعلى ال z axis 30 يعني بيصير الآن اللى هى ال saddle point بدل ما كانت 000 بيصير ال saddle point عبارة عن 4", "tokens": [28814, 15730, 29538, 6225, 3555, 9640, 3660, 18871, 9848, 65, 7940, 45729, 7902, 327, 30767, 1829, 16247, 12610, 7578, 20666, 19913, 3555, 9566, 4724, 3794, 39894, 46456, 46538, 16472, 3224, 43242, 7578, 13672, 7578, 31439, 20328, 9640, 5513, 15844, 2423, 288, 10298, 502, 4032, 3615, 23942, 2423, 2031, 10298, 1017, 4032, 3615, 23942, 2423, 710, 10298, 2217, 37495, 22653, 4724, 1829, 9381, 13546, 6024, 48506, 13672, 7578, 8032, 7578, 2423, 30459, 935, 47525, 1211, 19446, 25961, 2655, 13711, 4724, 1829, 9381, 13546, 2423, 30459, 935, 6225, 3555, 9640, 3660, 18871, 1017], "avg_logprob": -0.187330163853324, "compression_ratio": 1.702970297029703, "no_speech_prob": 1.1920928955078125e-06, "words": [{"start": 2316.75, "end": 2317.25, "word": "إذا", "probability": 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new file mode 100644
index 0000000000000000000000000000000000000000..7c92b027f889ec1dd6513e14b296577927752c07
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/MDfHgpCP1Ww.srt
@@ -0,0 +1,2050 @@
+1
+00:00:00,000 --> 00:00:04,160
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,160 --> 00:00:12,080
+وبركاته هذه هي المحاضرة رقم 26 مساق  فاضل وتكامل
+
+3
+00:00:12,080 --> 00:00:19,180
+جيم وهي المحاضرة الأخيرة وهي المحاضرة السابعة وليست
+
+4
+00:00:19,180 --> 00:00:22,520
+في الثامنة السابعة في chapter 15 المحاضرة السابقة
+
+5
+00:00:22,520 --> 00:00:26,260
+اللي قلت عنها السابعة، لأ هي كانت السادسة وهي
+
+6
+00:00:26,260 --> 00:00:30,420
+بعنوان substitutions in multiple integrals اللي هي
+
+7
+00:00:30,420 --> 00:00:34,580
+عملية التعويض في multiple integrals أو حتى ال
+
+8
+00:00:34,580 --> 00:00:38,480
+double. طبعًا الآن احنا كنا نعمل substitutions في
+
+9
+00:00:38,480 --> 00:00:43,500
+تفاضل الف أو تفاضل باء ونعوض تعويضًا مناسبًا عشان نسهل
+
+10
+00:00:43,500 --> 00:00:47,620
+علينا عملية ال integration ونكون التكامل بيكون صعبًا
+
+11
+00:00:47,620 --> 00:00:53,460
+بنسهّل علينا التكامل. الآن في شيء مشابه فيه اللي هو
+
+12
+00:00:53,460 --> 00:00:57,800
+ال double integrals و multiple integrals أنه بنعوض
+
+13
+00:00:57,800 --> 00:01:04,040
+عن ال X وعن ال Y بشيء اسمه U و V بدلالة X وY، طبعًا
+
+14
+00:01:04,040 --> 00:01:08,850
+بيكون هذا التعويض اللي هو بيسهّل علينا عملية
+
+15
+00:01:08,850 --> 00:01:12,750
+التكامل. ولكن تبعًا لهذا التعويض بيصير عندنا اللي
+
+16
+00:01:12,750 --> 00:01:17,410
+هو تابعات معينة بدنا نأخذها بعين الاعتبار لما نعوض
+
+17
+00:01:17,410 --> 00:01:20,930
+خلينا نشوف، نبدأ الآن. Substitution in double
+
+18
+00:01:20,930 --> 00:01:23,370
+integrals. بدنا نبدأ في Substitution in double
+
+19
+00:01:23,370 --> 00:01:27,190
+integrals. طبعًا عندنا احنا ال integration لل F(x,y)
+
+20
+00:01:27,190 --> 00:01:31,870
+dx dy هي اللي بدنا اللي هو نكاملها over R، over ال
+
+21
+00:01:31,870 --> 00:01:35,610
+region R، هي ال region R اللي عندي اللي بدنا نكامل
+
+22
+00:01:35,610 --> 00:01:39,390
+عليها. طبعًا هذه اللي هو ممكن تكون اللي هو الدالة
+
+23
+00:01:39,390 --> 00:01:44,590
+صعبة التكامل، فبدنا نحول ال F(x,y) نحولها بدلالة
+
+24
+00:01:44,590 --> 00:01:49,690
+اللي هو U و V بطريقة معينة. فعشان هيك اللي هو
+
+25
+00:01:49,690 --> 00:01:53,980
+بنيجي بناخد حاجة اسمها اللي هي transformation
+
+26
+00:01:53,980 --> 00:02:01,180
+بنحول من اللي هو ال X و Y عبر دالة X بتساوي G(U,V)
+
+27
+00:02:01,180 --> 00:02:07,500
+و V بنعوض عن كل X بـG(U,V) و Y بنعوضها بـH(U,V) و
+
+28
+00:02:07,500 --> 00:02:12,450
+V. فاللي هي المنطقة اللي اللي اللي اللي اللي اللي
+
+29
+00:02:12,450 --> 00:02:12,750
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+30
+00:02:12,750 --> 00:02:13,110
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+31
+00:02:13,110 --> 00:02:13,230
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+32
+00:02:13,230 --> 00:02:13,250
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+33
+00:02:13,250 --> 00:02:13,310
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+34
+00:02:13,310 --> 00:02:15,990
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+35
+00:02:15,990 --> 00:02:16,070
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+36
+00:02:16,070 --> 00:02:16,250
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+37
+00:02:16,250 --> 00:02:16,790
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+38
+00:02:16,790 --> 00:02:16,870
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+39
+00:02:16,870 --> 00:02:28,950
+اللي
+
+40
+00:02:28,950 --> 00:02:32,530
+اللي بيصير عندنا F(G(U,V), H(U,V)) بعد ما عوضنا عن X
+
+41
+00:02:32,530 --> 00:02:38,350
+و H(U,V) بعد ما عوضنا عن Y. الآن الـ dx dy هذه
+
+42
+00:02:38,350 --> 00:02:42,810
+الآن إيش هتساوي؟ حاجة اسمها الـ Jacobian لـ
+
+43
+00:02:42,810 --> 00:02:46,710
+|Jacobian(U,V)| du dv. طبعًا هذا
+
+44
+00:02:46,710 --> 00:02:49,750
+اللي هو قانون فيه الاشتقاقات، و فيه اللي و كده، مش
+
+45
+00:02:49,750 --> 00:02:54,290
+موضوعنا الآن. إذا F(x,y)، بنعوض عن X بدلالتها
+
+46
+00:02:54,290 --> 00:03:00,050
+اللي هي بدلالة U و V، وY بدلالتها بدلالة U, V، و dx dy
+
+47
+00:03:00,050 --> 00:03:05,600
+بتصير تستبدل باللي هي الـ Jacobian في du dv. الآن
+
+48
+00:03:05,600 --> 00:03:10,180
+ال region R طبعًا هتتحول إلى region إيش اسمها؟ D. إذا
+
+49
+00:03:10,180 --> 00:03:13,600
+الآن عندي في عملية التحويل في ثلاث أمور اللي هو
+
+50
+00:03:13,600 --> 00:03:17,920
+مكان اللي هي ال F(x,y) هنعوض، dx dy في إلى
+
+51
+00:03:17,920 --> 00:03:22,100
+تعويض هي اللي هو اللي قلنا عنه، و ال region اللي
+
+52
+00:03:22,100 --> 00:03:25,200
+هو بيصير region جديدة، وهنشوف كيف نتعامل في كل
+
+53
+00:03:25,200 --> 00:03:28,590
+واحدة من الثلاث اللي عندنا. الآن شو الـ Jacobian؟
+
+54
+00:03:28,590 --> 00:03:31,550
+هذا اللي بتحكي عنه ال Jacobian بنعرفه كما يلي:
+
+55
+00:03:31,550 --> 00:03:35,630
+Definition: the Jacobian determinant or Jacobian of
+
+56
+00:03:35,630 --> 00:03:39,070
+the coordinates Transformation: X = G(U,V) و Y
+
+57
+00:03:39,070 --> 00:03:42,730
+= H(U,V). يعني ال Jacobian لهذا ال
+
+58
+00:03:42,730 --> 00:03:46,190
+Transformation اللي لما عوضنا شلنا X وعوضنا
+
+59
+00:03:46,190 --> 00:03:49,510
+مكانها بدلالة U و V، وشلنا Y وعوضنا مكانها بدلالة
+
+60
+00:03:49,510 --> 00:03:54,430
+U و V، ال Jacobian هو كما يلي: بنعرفه هو عبارة عن:
+
+61
+00:03:54,430 --> 00:03:55,070
+∂x/∂u    ∂x/∂v
+.
+
+62
+00:04:03,680 --> 00:04:07,200
+والسطر الثاني اللي هو العمود الثاني في السطر
+
+63
+00:04:07,200 --> 00:04:11,610
+الثاني اللي هو ∂y/∂u  ∂y/∂v اختصارًا
+
+64
+00:04:11,610 --> 00:04:15,190
+طبعًا هذا ∂x/∂u في ∂y/∂v بيصير ضرب في هذا، بتطلع هذا
+
+65
+00:04:15,190 --> 00:04:18,830
+المقدار ناقص هذا المقدار في هذا المقدار فبطلع هذا
+
+66
+00:04:18,830 --> 00:04:23,170
+مين هو يا جماعة؟ الجاكوبيان. إذا الجاكوبيان اللي هو
+
+67
+00:04:23,170 --> 00:04:28,230
+∂x/∂u الأولى، ∂x/∂v الثانية، وبنعمل
+
+68
+00:04:28,230 --> 00:04:33,990
+السطر الأول، بعدين بنيجي ∂y/∂u، وبعدين ∂y
+
+69
+00:04:33,990 --> 00:04:37,450
+/∂v، وبنجد ال determinant، بنحط له absolute
+
+70
+00:04:37,450 --> 00:04:42,270
+value وبنضربه في هذا، وبيصير du dv. بيكون حولنا هذا
+
+71
+00:04:42,270 --> 00:04:46,830
+الـ integral. نضل عليها اللي هي الـ G، أحصلها من مين؟ من R، و
+
+72
+00:04:46,830 --> 00:04:52,570
+خلينا اللي هو نشوف المثال التالي اللي حاجة احنا
+
+73
+00:04:52,570 --> 00:04:57,040
+عارفينها، نشوف إيش بيقول. بيقول مثال واحد: find the
+
+74
+00:04:57,040 --> 00:04:59,920
+Jacobian for the polar coordinate transformation.
+
+75
+00:04:59,920 --> 00:05:04,600
+أنتم متذكرين اللي هو العلاقة بين اللي هو ال X و
+
+76
+00:05:04,600 --> 00:05:09,100
+ال R و θ، و ال Y و ال R و θ؟ هذا الآن هو ال
+
+77
+00:05:09,100 --> 00:05:12,000
+transformation اللي بدنا نشتغل عليه. المثال هيك
+
+78
+00:05:12,000 --> 00:05:14,580
+بتقول: find the Jacobian for the polar coordinate
+
+79
+00:05:14,580 --> 00:05:18,620
+transformation: X = R cos θ، و Y = R sin θ.
+
+80
+00:05:18,620 --> 00:05:21,240
+و Use equation one اللي قبله شوية to write the
+
+81
+00:05:21,240 --> 00:05:25,580
+Cartesian integral as a polar integral. إذا الآن
+
+82
+00:05:25,580 --> 00:05:30,960
+هذه المنطقة، هذا ال integration بدي أعوض عن X بقيمة
+
+83
+00:05:30,960 --> 00:05:35,940
+R cos θ، و Y = R sin θ، وأوجد اللي هو ال
+
+84
+00:05:35,940 --> 00:05:40,140
+Jacobian، و dx dy أستبدلها بال Jacobian في اللي هو
+
+85
+00:05:40,140 --> 00:05:45,560
+dR dθ، وهنشوف أن هذا هيشتغلنا القانون اللي احنا
+
+86
+00:05:45,560 --> 00:05:50,150
+عارفينه زمان، هتشوف كيف. الآن أول شيء اللي هو ال
+
+87
+00:05:50,150 --> 00:05:53,010
+region، آخذ اللي هو ال .. ال .. ال .. ال .. ال ..
+
+88
+00:05:53,010 --> 00:05:58,510
+المنطقة اللي هي اللي بديها على R، هذا R وهذا إيش
+
+89
+00:05:58,510 --> 00:06:02,950
+θ؟ أنا آخذهم θ في ال Cartesian اللي هو R مع
+
+90
+00:06:02,950 --> 00:06:08,070
+مين؟ مع θ. ماشي، الآن قيم R اللي هي بديها من من
+
+91
+00:06:08,070 --> 00:06:11,750
+صفر لعند واحد. هذه المنطقة ال region اللي بديها
+
+92
+00:06:11,750 --> 00:06:15,810
+هذا ال region. الآن نزيله عند واحد ونزيله عند π
+
+93
+00:06:15,810 --> 00:06:20,010
+على اثنين. هذه هتطلع ال region G في R، بس الآن كيف
+
+94
+00:06:20,010 --> 00:06:23,970
+هتطلع؟ هتطلع بعد ما نشوف اللي هو ال region اللي
+
+95
+00:06:23,970 --> 00:06:28,210
+معطيني إياها هو، اللي هي R هذه معطاها. الآن ال R
+
+96
+00:06:28,210 --> 00:06:34,250
+أشملها، ال R معطيني إياها اللي هو ال X عندي بتساوي
+
+97
+00:06:34,250 --> 00:06:41,670
+R Cos θ، هي عندي اللي هو ال .. ال .. ال θ،  آخذها
+
+98
+00:06:41,670 --> 00:06:45,030
+عندي اللي بتأخذ المنطقة هذه، بس أنا بأخذ المنطقة
+
+99
+00:06:45,030 --> 00:06:49,310
+المنطقة هذه اللي هي عبارة عن ال .. هذا اللي هو
+
+100
+00:06:49,310 --> 00:06:55,150
+طبعًا Y = 0، وهذا X = 0، وهذا جزء من X²
+
+101
+00:06:55,150 --> 00:07:00,210
++ Y² = 1. أو بالـ Polar اللي هي R
+
+102
+00:07:00,210 --> 00:07:05,930
+بتساوي إيش؟ واحد. و θ لأنه ماخذها في الربع الأول،
+
+103
+00:07:05,930 --> 00:07:09,470
+θ من صفر لعند مين؟ لعند π على اثنين. إذا
+
+104
+00:07:09,470 --> 00:07:14,870
+المنطقة هذه اللي هي المرسومة دائرة هذه لما بتأخذ
+
+105
+00:07:14,870 --> 00:07:19,030
+قيمها، طبعًا قيمها ب R هنا اللي هي R بـ directed
+
+106
+00:07:19,030 --> 00:07:23,970
+distance. ماشي الحال، اللي هي عبارة عن R اللي عندي
+
+107
+00:07:23,970 --> 00:07:29,710
+اللي هي من صفر لعند واحد. ماشي الحال؟ أو θ من صفر
+
+108
+00:07:29,710 --> 00:07:33,150
+لعند مين؟ لعند π على اثنين. عشان هيك لو بدأت ارسم
+
+109
+00:07:33,150 --> 00:07:37,630
+الـ R و θ كـ Cartesian زي ما رسمت x و y كـ Cartesian
+
+110
+00:07:37,630 --> 00:07:42,450
+بدأت ارسم R و θ كـ Cartesian. الآن أنا بتعامل معهم كـ
+
+111
+00:07:42,450 --> 00:07:45,370
+transformations، يعني x = R cos θ، و y =
+
+112
+00:07:45,370 --> 00:07:49,150
+R sin θ. العلاقة بين هنا لما برسم هنا على مكان
+
+113
+00:07:49,150 --> 00:07:53,570
+معين بتصير اللي هي هذه اللي هي ال .. ال .. ال ..
+
+114
+00:07:53,570 --> 00:07:56,850
+ال .. الـ directed distance هي الـ R بالشكل هذا، و
+
+115
+00:07:56,850 --> 00:07:59,790
+الـ θ اللي هي مع الـ positive x axis، لكن لما
+
+116
+00:07:59,790 --> 00:08:04,370
+برسمها هنا، هذا إحداثي R، وهذا إحداثي θ. هذا بشتغل
+
+117
+00:08:04,370 --> 00:08:08,050
+فيه عادي، اللي هي R اللي كانت هنا من 0 لـ 1، بيصير
+
+118
+00:08:08,050 --> 00:08:14,110
+هاي R عندي، هاي R طبعًا هي الـ R، هاي عندي 0. الآن
+
+119
+00:08:14,110 --> 00:08:18,310
+ماشي الحال، الـ R = 0، as if هي هنا، R = 0
+
+120
+00:08:18,310 --> 00:08:24,420
+هيها، وهي اللي هو R = 1. الآن θ من وين لوين؟ θ
+
+121
+00:08:24,420 --> 00:08:32,760
+من عند 0 لعند π على 2. فهذه هي المنطقة اللي ..
+
+122
+00:08:32,760 --> 00:08:37,120
+اللي هذي R هنا صارت G هنا، أنا برسمها في الـ
+
+123
+00:08:37,120 --> 00:08:41,380
+Cartesian يا جماعة. R و θ بأعتبرها عادي زي متغيرات
+
+124
+00:08:41,380 --> 00:08:46,050
+عادية زي ال X و ال Y، تحت. لكن العلاقة بين X و Y و R
+
+125
+00:08:46,050 --> 00:08:50,270
+و θ على نفس الرسمة هذه القديمة، هي العلاقة بينهم
+
+126
+00:08:50,270 --> 00:08:53,830
+اللي هي θ بتكون اللي هو الزاوية اللي بيصنعها مع الـ
+
+127
+00:08:53,830 --> 00:08:57,250
+positive x axis اللي هو X هذا، وبتعمل اللي هو
+
+128
+00:08:57,250 --> 00:09:01,590
+directed distance هذا اللي هو بمثل LR. ماشي، الآن الـ
+
+129
+00:09:01,590 --> 00:09:06,970
+G هي صارت R. إذاً هي الـ R صارت إيش؟ G. الآن إيش بدنا
+
+130
+00:09:06,970 --> 00:09:11,270
+نسوي؟ F(x,y), dx, dy. إيش بدنا نعملها؟ بنشيل ال X
+
+131
+00:09:11,270 --> 00:09:15,710
+بيصير R cos θ، وال Y R sin θ حسب اللي معطيني إياه هو
+
+132
+00:09:15,710 --> 00:09:19,870
+هيه معطيني إياها وينها؟ عند X = R cos θ، و Y = R
+
+133
+00:09:19,870 --> 00:09:22,910
+sin θ، زي ما أعطانا ال transformation اللي بدنا
+
+134
+00:09:22,910 --> 00:09:27,130
+نستخدمه.  بس بال Integral over mean over G اللي
+
+135
+00:09:27,130 --> 00:09:32,130
+عندي هذه اللي أمامي. بيصير F(R cos θ, R sin θ) في
+
+136
+00:09:32,130 --> 00:09:36,500
+ال Jacobian dR dθ، اللي كانت du dv.
+
+137
+00:09:36,500 --> 00:09:40,700
+ال Jacobian ل R و θ، إيش ال Jacobian ل R و θ؟
+
+138
+00:09:40,700 --> 00:09:43,840
+خلينا نشوف حسب القانون اللي قبل شوية. ∂x/∂r
+
+139
+00:09:43,840 --> 00:09:48,620
+∂x/∂θ، ∂y/∂r، ∂y/∂θ. نوجد ∂x/∂r، ∂x/∂r بنعرف
+
+140
+00:09:48,620 --> 00:09:52,400
+نوجدها هي اللي هو ال X، ∂x/∂r اللي هي
+
+141
+00:09:52,400 --> 00:09:56,080
+cos θ، صح؟ طيب ∂x/∂θ، -R sin
+
+142
+00:09:56,080 --> 00:10:01,600
+θ، مضبوط؟ طيب ∂y/∂r، ∂y/∂r اللي هي
+
+143
+00:10:06,530 --> 00:10:13,110
+عبارة عن sin θ، ∂y/∂θ اللي هو R cos θ، هي R
+
+144
+00:10:13,110 --> 00:10:19,110
+cos θ. أضرب هذه في هذه بيصير R cos²θ، لأن ناقص
+
+145
+00:10:19,110 --> 00:10:23,630
+ناقص هذه بيصير زائد R sin²θ. نأخذ ال R برا
+
+146
+00:10:23,630 --> 00:10:26,630
+بيصير R في cos²θ + sin²θ، هذه عبارة عن
+
+147
+00:10:26,630 --> 00:10:32,670
+واحد، بتطلع ال R. الآن احنا مفترضين فوق عندنا R
+
+148
+00:10:32,670 --> 00:10:36,250
+إيش مالها؟ أكبر أو يساوي صفر، هي الـ R ≥ 0 و
+
+149
+00:10:36,250 --> 00:10:40,590
+أصغر أو يساوي إيش؟ واحد. بناء عليه الآن لما نأخذ
+
+150
+00:10:40,590 --> 00:10:43,450
+القانون الـ double integration عوضنا زي ما قلنا
+
+151
+00:10:43,450 --> 00:10:47,250
+هنا، بدنا نعوض مكان الـ dx dy، Jacobian |
+
+152
+00:10:47,250 --> 00:10:52,630
+Jacobian| في dR dθ. الجاكوبيان الآن طلع إيش؟ = R، لـ
+
+153
+00:10:52,630 --> 00:10:55,290
+|Jacobian|، الـ |Jacobian| = R،
+
+154
+00:10:55,290 --> 00:10:58,890
+لكن R ≥ 0، ماخذينها. إذا إيش بتساوي؟ R. ففعلاً
+
+155
+00:10:58,890 --> 00:11:03,770
+عندي الـ double integral ل R(x,y) dx dy، زي زمان
+
+156
+00:11:03,770 --> 00:11:08,890
+اللي هي R dR dθ. يعني وكأننا بنقول الـ Polar
+
+157
+00:11:08,890 --> 00:11:13,330
+Transformation هو عبارة، أو الـ polar اللي هي ال ..
+
+158
+00:11:13,330 --> 00:11:15,410
+ال .. ال .. ال double integral in polar
+
+159
+00:11:15,410 --> 00:11:20,410
+coordinates هي عبارة عن حالة خاصة من اللي هو الـ
+
+160
+00:11:20,410 --> 00:11:24,590
+general transformation اللي بنحكي عنها، أنه اللي هو
+
+161
+00:11:24,590 --> 00:11:29,290
+X = G(U,V)، و H = U و V. هذه الآن U و V
+
+162
+00:11:29,290 --> 00:11:32,110
+هي ال R و θ اللي قبل بشوية. إذا صارت اللي هي عبارة
+
+163
+00:11:32,110 --> 00:11:36,470
+عن حالة خاصة من هذا التعويض. يعني ما هي إلا اللي هي
+
+164
+00:11:36,470 --> 00:11:40,300
+ال double integral عبارة عن issue التعويض اللي هو
+
+165
+00:11:40,300 --> 00:11:43,940
+لـ R و θ، وفق القانون اللي حكيناه قبل بشوية. نيجي
+
+166
+00:11:43,940 --> 00:11:48,840
+أن نثبت المعلومة أكثر. جالي example: evaluate ال
+
+167
+00:11:48,840 --> 00:11:53,720
+double integral لـ (2x - y)/2 dx dy، x =
+
+168
+00:11:53,720 --> 00:12:00,090
+y/2 + 1، x = y/2، و من 0 لـ 4. الآن يا
+
+169
+00:12:00,090 --> 00:12:03,790
+جماعة، by the following transformation، يعني بيقول
+
+170
+00:12:03,790 --> 00:12:07,570
+evaluate هذا بواسطة اللي هو الـ Transformation
+
+171
+00:12:07,570 --> 00:12:13,350
+التالي، يعني أعوض عن X و Y وفق القوانين التالية:
+
+172
+00:12:13,350 --> 00:12
+
+223
+00:16:11,110 --> 00:16:14,710
+كده لعند قيمة x بتساوي كده هي على الخطين دول
+
+224
+00:16:14,710 --> 00:16:18,110
+لاحظوا أنه لو جينا نأخذ عبر هذول الخطوط هيك
+
+225
+00:16:18,110 --> 00:16:22,970
+بموازية فعلاً بنلاقي اللي هو الـ x بتيجي من اثنين من
+
+226
+00:16:22,970 --> 00:16:27,210
+الـ x بتطلع من y على اثنين لعند x بتساوي اللي هي
+
+227
+00:16:27,210 --> 00:16:31,270
+اللي هي عبارة عن y على اثنين زائد واحد، فاهمين؟
+
+228
+00:16:31,270 --> 00:16:34,650
+هذيك وهذيك زي ما كنا نعمل زيها، اللي هان بده أحدد
+
+229
+00:16:34,650 --> 00:16:38,150
+المنطقة، أكملها مين اللي بيكملها؟ اللي هو y، دي y، y
+
+230
+00:16:38,150 --> 00:16:41,410
+بتقول من صفر لعند مين؟ لعند أربعة، يعني y بدأنا بصفر
+
+231
+00:16:41,410 --> 00:16:45,970
+هيها بصفر وانتهينا بعند مين؟ بعند أربعة، بنرسم الخط
+
+232
+00:16:45,970 --> 00:16:50,770
+بتبقى هذه هي المنطقة اللي قاعد بكمل عليها، اللي هي
+
+233
+00:16:50,770 --> 00:16:57,050
+اسمها إيش؟ اسمها r، بدي أحولها لمنطقة g، طيب اللي
+
+234
+00:16:57,050 --> 00:17:03,340
+قلنا التحويل الكامل اللي هو أنه بتيجي الـ خط الأول x
+
+235
+00:17:03,340 --> 00:17:06,760
+بيساوي y على 2، اللي قلنا عنه إيش بده يصير في الـ ..
+
+236
+00:17:06,760 --> 00:17:10,540
+اللي هو .. بده يصير في الـ .. في الـ u والـ v؟ بنعوض
+
+237
+00:17:10,540 --> 00:17:15,020
+عن x بقيمتها اللي عبارة عن u زائد v، وy على 2
+
+238
+00:17:15,020 --> 00:17:19,660
+بقيمتها اللي هي v، بيصير u زائد v بيساوي v، يعني v مع
+
+239
+00:17:19,660 --> 00:17:23,620
+v، يعني u بيساوي 0، إذًا اتحول الخط المائل هذا لـ u
+
+240
+00:17:23,620 --> 00:17:27,060
+بيساوي 0، يعني الخط هي محلّه، اللي هو زي اللي بده
+
+241
+00:17:27,060 --> 00:17:31,080
+يعملني rectangle كمان شوية، x بيصير y على 2 زائد 1
+
+242
+00:17:31,080 --> 00:17:35,020
+بشيل الـ x بحط u زائد v، والـ y على 2 اللي هي عبارة
+
+243
+00:17:35,020 --> 00:17:39,360
+عن v، وهذا الـ 1، فبيصير u زائد v بيساوي v زائد 1، الـ v
+
+244
+00:17:39,360 --> 00:17:43,340
+مع الـ v، بيصير الـ u إيش بتساوي؟ 1، إذًا اللي هو الـ du
+
+245
+00:17:43,340 --> 00:17:47,800
+أو الـ u صارت عبارة عن إيش؟ من 0 إلى 1، يعني مكان
+
+246
+00:17:47,800 --> 00:17:52,730
+اللي بشتغل عليه rectangular region، y بتساوي 0، اللي
+
+247
+00:17:52,730 --> 00:17:57,750
+هي y إيش؟ 2v، 2v، بسيط، الصفر، إذًا v بتساوي 0
+
+248
+00:17:57,750 --> 00:18:02,010
+هيحدد اللي هو الـ v، الخط الثاني y بيساوي 4، اللي هو 2v
+
+249
+00:18:02,010 --> 00:18:05,970
+بيساوي 4، يعني الـ v بتساوي 2، إذًا هذه الآن اللي هي
+
+250
+00:18:05,970 --> 00:18:10,310
+حدود المنطقة u و v، u بتساوي 0، احفظوها، u بتساوي 0 و
+
+251
+00:18:10,310 --> 00:18:14,510
+u بتساوي 1، u بتساوي 0 و u بتساوي 1، u بتساوي 0، هي u
+
+252
+00:18:14,510 --> 00:18:21,820
+بتساوي 0، ماشي الحال، هيها u بتساوي 0 و u بتساوي 1، و
+
+253
+00:18:21,820 --> 00:18:25,060
+بعدين v بيساوي صفر و v بيساوي اثنين، فعملت لي الـ
+
+254
+00:18:25,060 --> 00:18:29,160
+region هذه الحلوة، سهلة التكامل، يعني بيصير على طول
+
+255
+00:18:29,160 --> 00:18:32,940
+حدود الـ integration الآن، بدل ما هي كانت اللي هي
+
+256
+00:18:32,940 --> 00:18:36,420
+بالشكل هذا، على طول بيصير إيش؟ هيها حدود
+
+257
+00:18:36,420 --> 00:18:40,820
+الـ integration، u اللي بده يتكامل لـ u، حدود u من صفر
+
+258
+00:18:40,820 --> 00:18:43,860
+لعند الواحد، لأنه rectangle من صفر لعند الواحد، و v
+
+259
+00:18:43,860 --> 00:18:48,430
+من عند صفر لعند من عند الاثنين، الآن خلصت من اللي هي
+
+260
+00:18:48,430 --> 00:18:52,410
+المنطقة اللي بتغلبنا، الآن ضل عليّ أوجد من اثنين x
+
+261
+00:18:52,410 --> 00:18:55,730
+ناقص y على اثنين، هذا ما عوضنا، ما كان اسمها u،
+
+262
+00:18:55,730 --> 00:18:59,930
+ضل الآن dx dy، إيش؟ jacobian، الـ absolute value في
+
+263
+00:18:59,930 --> 00:19:04,450
+du dv، الآن كيف بدي أوجد الـ jacobian؟ هذا الـ jacobian
+
+264
+00:19:04,450 --> 00:19:09,530
+زي ما قلنا بيساوي del x by del u، del x by del v، del y by
+
+265
+00:19:09,530 --> 00:19:14,730
+del u، del y by del v، del x by del u اللي هو عبارة عن
+
+266
+00:19:14,730 --> 00:19:20,090
+واحد، صح ولا لا؟ أيوه واحد، del x by del v، del x by del v
+
+267
+00:19:20,090 --> 00:19:26,990
+برضه واحد، del y by del u، del y by del u صفر، del y by del
+
+268
+00:19:26,990 --> 00:19:30,840
+v اثنين، إذًا هذول حددنا اللي هي على طول
+
+269
+00:19:30,840 --> 00:19:33,180
+الـ jacobian، يعني هو واحد في اثنين ناقص صفر،
+
+270
+00:19:33,180 --> 00:19:37,080
+اللي بطلع إيش؟ اثنين، إذًا بشيل اللي هو الـ jacobian و
+
+271
+00:19:37,080 --> 00:19:42,480
+بحط مكانه اثنين، وهاي u، المكان دي دي u دي v، لأن
+
+272
+00:19:42,480 --> 00:19:45,740
+absolute value اثنين هو نفسه اثنين، بنكمل هذه u
+
+273
+00:19:45,740 --> 00:19:49,220
+اثنين u، بدنا نكامل بالنسبة لـ u، بيصير اثنين u تربيع
+
+274
+00:19:49,220 --> 00:19:52,480
+على اثنين، يعني u تربيع من صفر لواحد، بتطلع عبارة عن
+
+275
+00:19:52,480 --> 00:19:56,490
+واحد، الآن كله دي v بيطلع عبارة عن v من صفر لـ 2،
+
+276
+00:19:56,490 --> 00:20:00,410
+بيطلع عندي 2، وبيكون هي قيمة الـ integration، وكان
+
+277
+00:20:00,410 --> 00:20:05,170
+سهل اللي هو الـ integration بعد ما حولناه، نأخذ مثال
+
+278
+00:20:05,170 --> 00:20:08,250
+آخر، evaluate الـ double integral من 0 لـ 1 ناقص x، و
+
+279
+00:20:08,250 --> 00:20:13,910
+من 0 لعند 1، الجذر التربيعي لـ x زائد y ناقص 2x
+
+280
+00:20:13,910 --> 00:20:18,830
+كله تربيع dy dx، طبعاً الآن اللي هي المنطقة اللي هي
+
+281
+00:20:18,830 --> 00:20:22,230
+بدنا نعمل evaluation لهذه، بدنا نحدد المنطقة و
+
+282
+00:20:22,230 --> 00:20:27,390
+بعدين نحدد u والـ v، وبعدين نشيل هذول ونحط مكان
+
+283
+00:20:27,390 --> 00:20:31,550
+مدلهم u و v، ونشيل الـ dy و dx ونحط مكانهم jacobian
+
+284
+00:20:31,550 --> 00:20:35,850
+في du و dv، طيب، ندور الأشياء على المنطقة اللي هو
+
+285
+00:20:35,850 --> 00:20:40,730
+عندي الـ y بتساوي صفر لعند y بيساوي 1 ناقص x، y
+
+286
+00:20:40,730 --> 00:20:45,830
+بتساوي صفر لعند 1، يعني لعند y بيساوي 1 ناقص x، يعني
+
+287
+00:20:45,830 --> 00:20:51,540
+x زائد y بيساوي 1، هذه، لأن y بتساوي صفر، هيلأن من
+
+288
+00:20:51,540 --> 00:20:56,160
+فوق إيش بيكون؟ x زي y بيساوي 1، x زي y بيساوي 1، هي x
+
+289
+00:20:56,160 --> 00:21:03,020
+بصفر، y بصفر، x بواحد، x بصفر، y بواحد، فبيصير هذا الخط
+
+290
+00:21:03,020 --> 00:21:06,340
+المستقيم اللي عندي، فبيصير y فعلاً، y جاي من عند الـ
+
+291
+00:21:06,340 --> 00:21:11,530
+zero لعند 1 ناقص x، نأتي الآن لمنطقة هذه، جاي نلاقي
+
+292
+00:21:11,530 --> 00:21:14,150
+إيش الـ x عندها؟ من صفر لعند واحد، بدأ من صفر لعند
+
+293
+00:21:14,150 --> 00:21:17,310
+واحد، لأن هذا بيظل ممتد هيك وهيك، طيب إيش حدوده اللي
+
+294
+00:21:17,310 --> 00:21:21,070
+بدنا نحددها؟ اللي هو قيمة x، x بيقول لي من عند صفر
+
+295
+00:21:21,070 --> 00:21:24,230
+إذًا نحط الخط هذا اللي عند واحد، وأحط الخط هذا
+
+296
+00:21:24,230 --> 00:21:28,810
+فبتطلع المنطقة اللي أمامي هذه، لأن هذه المنطقة بدنا
+
+297
+00:21:28,810 --> 00:21:34,150
+نحولها لمين؟ لمنطقة g اللي في الـ u والـ v، الآن لو
+
+298
+00:21:34,150 --> 00:21:39,000
+تطلعنا على الـ integration اللي أمامنا، الآن أمامنا
+
+299
+00:21:39,000 --> 00:21:43,920
+خيارات إن نأخذ هذا كله u وهذا كله v، لو أخذته هيك
+
+300
+00:21:43,920 --> 00:21:47,220
+هتلاقي اللي هو، جرب لحالك، هتلاقي الأمور اللي هو
+
+301
+00:21:47,220 --> 00:21:52,140
+بدأت تتعدد، حيصير عندك اللي هو الوضع صعب، الآن اللي
+
+302
+00:21:52,140 --> 00:21:56,620
+جوا هذا بأخذه اللي هو u، بعينيّ، بقدر أكمل جذر الـ
+
+303
+00:21:56,620 --> 00:22:01,140
+u مش صعب، وهذا v، وبقدر أكمل الـ v تربيع ومش صعب، خذ
+
+304
+00:22:01,140 --> 00:22:05,500
+الـ u بيساوي x زائد y، والـ v بيساوي y ناقص 2x، الآن
+
+305
+00:22:05,500 --> 00:22:10,150
+هذول نحلهم مع بعض عشان نجيب x و y، الآن لو اجمعت ضربت
+
+306
+00:22:10,150 --> 00:22:14,950
+اللي هي هذه بـ 2، بيصير 2u زائد 2x زائد
+
+307
+00:22:14,950 --> 00:22:20,370
+2y، مضبوط، واجمعهم لبعض، بيصير 2u زائد اللي
+
+308
+00:22:20,370 --> 00:22:26,150
+هو v، 2u زائد v بيساوي اللي هو 2y زائد y،
+
+309
+00:22:26,150 --> 00:22:31,530
+اللي هي 3y، بيصير اللي هو 2u زائد v، عن طريق
+
+310
+00:22:31,530 --> 00:22:41,290
+عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+311
+00:22:41,290 --> 00:22:49,150
+عن اللي هو 2 .. 2u بيساوي اللي هو 2
+
+312
+00:22:49,150 --> 00:22:53,990
+طبعاً زائد v، نسينا، 2u زائد v بيساوي 2y
+
+313
+00:22:53,990 --> 00:22:58,610
+زائد y، 3y، الآن بدي قيمة الـ y، بيصير y بتساوي
+
+314
+00:22:58,610 --> 00:23:05,140
+أقسم هذول على اللي هو 3، وهذا على 3، فبيصير
+
+315
+00:23:05,140 --> 00:23:09,240
+عندك y بيساوي 2u على 3 زائد v على 3،
+
+316
+00:23:09,240 --> 00:23:14,300
+الآن بدي أحصل على قيمة x، يعني بدي أطير
+
+317
+00:23:14,300 --> 00:23:19,210
+قيمة y، اللي هو اضرب هذا في ناقص، كله في ناقص، أو اضرب
+
+318
+00:23:19,210 --> 00:23:22,450
+هذه في ناقص، زي ما بدك، مش مشكلة كثير، لو ضربت هذه
+
+319
+00:23:22,450 --> 00:23:26,930
+كلها في ناقص بيصير ناقص v بيساوي ناقص y زائد 2
+
+320
+00:23:26,930 --> 00:23:30,830
+x، وأجمعهم مع بعض، بعد الضرب بيصير u
+
+321
+00:23:30,830 --> 00:23:36,910
+ناقص v بيساوي 3x، u ناقص v بيساوي 3x، x مع
+
+322
+00:23:36,910 --> 00:23:40,190
+3، بيصير x بيساوي u على 3 ناقص v على 3،
+
+323
+00:23:40,190 --> 00:23:45,270
+و بتوجد اللي هي x و y، وهذا اللي هو الحل، أنهينا،
+
+324
+00:23:45,270 --> 00:23:50,610
+سهلة زي ما شفنا، الآن جبنا x و y بدلالة الـ u والـ v،
+
+325
+00:23:50,610 --> 00:23:57,550
+لأن بيصير، لأن طبعاً عملتها هيك، ليش؟ عشان اللي هو أسهل
+
+326
+00:23:57,550 --> 00:24:01,090
+على حالي إيجاد المنطقة، وأسهل على حالي إيجاد
+
+327
+00:24:01,090 --> 00:24:05,240
+الـ jacobian، طيب خلينا نشوف أول شيء اللي هي الخطوط
+
+328
+00:24:05,240 --> 00:24:08,400
+هذه إيش؟ بدي أشوف عندها، أول شيء x زائد y بيساوي
+
+329
+00:24:08,400 --> 00:24:12,920
+واحد، x زائد y بيساوي واحد، عوض عن x بقيمتها u على
+
+330
+00:24:12,920 --> 00:24:17,360
+3 ناقص v على 3، زائد y اللي هي 2u على 3 زائد
+
+331
+00:24:17,360 --> 00:24:21,500
+v على 3، اللي هي هذه بتروح مع هذه، بيصير اللي هو
+
+332
+00:24:21,500 --> 00:24:26,820
+2u، u، 3u على 3، يعني u بيساوي واحد، خلصنا منها،
+
+333
+00:24:27,210 --> 00:24:30,090
+لأن x بيساوي صفر، لأن الخط هذا صار u
+
+334
+00:24:30,090 --> 00:24:33,910
+بيساوي إيش؟ واحد، ما حلّه، ما يسأل، x بتساوي صفر،
+
+335
+00:24:33,910 --> 00:24:37,990
+بيصير اللي هو u على 3 ناقص v على 3 بتساوي
+
+336
+00:24:37,990 --> 00:24:41,490
+صفر، أنقل هذا على الجهة الثانية بيبقى 3
+
+337
+00:24:41,490 --> 00:24:47,890
+بيصير v بتساوي u، v بتساوي u، سهل، هاي v بتساوي إيش؟
+
+338
+00:24:47,890 --> 00:24:53,500
+u، ماشي الحال، الآن y بيساوي صفر، إيش بيساوي؟ y بيساوي صفر
+
+339
+00:24:53,500 --> 00:24:58,080
+اللي هو عبارة عن اللي هو 2u على 3 زائد v على 3
+
+340
+00:24:58,080 --> 00:25:01,700
+بيساوي صفر، أو اللي أنا أقسم على 3، أو أنقل، بيصير u
+
+341
+00:25:01,700 --> 00:25:07,120
+زائد v بيساوي ناقص 2u، v بيساوي ناقص 2u، هي اللي هي v
+
+342
+00:25:07,120 --> 00:25:11,700
+بيساوي ناقص 2u، لو عوضت من هون، واحدة بصفر، وتانية
+
+343
+00:25:11,700 --> 00:25:16,330
+هتطلع لك اللي هو هذه، وهذا، الآن u إيش بيساوي؟ u بيساوي
+
+344
+00:25:16,330 --> 00:25:19,850
+واحد، اللي طلع عندي، ماشي الحال، إذًا المنطقة المحصورة
+
+345
+00:25:19,850 --> 00:25:23,830
+هيها، المنطقة هذه، v بيساوي u و v بيساوي ناقص 2u،
+
+346
+00:25:23,830 --> 00:25:27,990
+طبعاً بتعرفوا ترسموها هذه، أكيد، اللي هو u بواحد،
+
+347
+00:25:27,990 --> 00:25:32,830
+بيصير v بيساوي ناقص 2، هيها، الآن وهذه u واحد و
+
+348
+00:25:32,830 --> 00:25:36,670
+v واحد، أنزلوا لبعض، وهذه zero zero، إذًا هي المنطقة
+
+349
+00:25:36,670 --> 00:25:40,720
+المذكورة، الآن إذًا أنا أريد أكامل بالنسبة لـ v قبل
+
+350
+00:25:40,720 --> 00:25:44,940
+مثلاً، فبيصير v جاي من وين لوين؟ بطلع v بيساوي ناقص
+
+351
+00:25:44,940 --> 00:25:49,240
+2u لعند u، الآن طيب الـ u كلها من وين بتيجي؟ الـ u
+
+352
+00:25:49,240 --> 00:25:52,920
+بتاخذ من عند 0 لعند 1، تحديد حدود الـ integration
+
+353
+00:25:52,920 --> 00:25:57,260
+سهل الآن، طيب الآن قبل نوجد مين يا جماعة؟ الـ jacobian،
+
+354
+00:25:57,260 --> 00:26:02,610
+هاي الـ x، وهي الـ u، دي الـ x by دي الـ u، ثلث، del x by
+
+355
+00:26:02,610 --> 00:26:08,210
+del v ناقص ثلث، هاي السطر الأول، ثلث وناقص ثلث، نيجي
+
+356
+00:26:08,210 --> 00:26:12,190
+لـ del y by del u، del y by del u، del y by del u
+
+357
+00:26:12,190 --> 00:26:17,310
+ثلثين، و del y by del v ثلث، إذًا اللي هو بيصير عندي
+
+358
+00:26:17,310 --> 00:26:22,270
+ثلثين، ثلثين، وثلث، هذا الـ Jacobian، بوجد قيمته تسعة
+
+359
+00:26:22,270 --> 00:26:27,360
+ناقص اللي هي، بيصير زائد، تسعة، هو 3 على 9، يعني
+
+360
+00:26:27,360 --> 00:26:30,440
+ثلث، إذًا هذا الـ jacobian، إذًا صار الـ integration هذا
+
+361
+00:26:30,440 --> 00:26:37,280
+الآن سهل المنال، هذه dy dx بأعوضها dv du مع مين؟ مع
+
+362
+00:26:37,280 --> 00:26:40,820
+الـ jacobian، الـ jacobian قديش قلنا؟ ثلث، إذًا بتصير ثلث
+
+363
+00:26:40,820 --> 00:26:45,600
+dv du، الآن الـ .. الـ .. هذه اللي هي زي ما عوضنا هذه
+
+364
+00:26:45,600 --> 00:26:48,860
+اللي هي عبارة عن u أُس نص .. u هذه، فبيصير general u
+
+365
+00:26:48,860 --> 00:26:52,900
+يعني u أُس نص، وهذه قلنا v تربيع، هذه v، واللي برا
+
+366
+00:26:52,900 --> 00:26:55,700
+اثنين، بيصي .. بيقص اثنين، بيصير v تربيع، لأن هذا
+
+367
+00:26:55,700 --> 00:26:59,320
+تكامله سهل، بس قبل ما نكامله، موجود ثلث في u أُس نص
+
+368
+00:26:59,320 --> 00:27:02,260
+في v تربيع، اللي هي حدود الـ integration زي ما قلنا
+
+369
+00:27:02,530 --> 00:27:07,590
+الـ v بتيجي من عند ناقص 2u لعند الـ u، والـ u بتروح
+
+370
+00:27:07,590 --> 00:27:11,070
+من 0 لعند 1، زي ما حكينا قبل شوية، اللي هم بنكامل
+
+371
+00:27:11,070 --> 00:27:16,110
+بالنسبة لـ v، بيصير اللي هو v تكعيب على اللي هو من
+
+372
+00:27:16,110 --> 00:27:20,830
+3، يعني، وفي ثلث برا طلعناها، بيصير v تكعيب على 3 من
+
+373
+00:27:20,830 --> 00:27:25,380
+v من ناقص 2u لعند الـ u، بنعوض عنها بتطلع عندي عبارة
+
+374
+00:27:25,380 --> 00:27:29,700
+عن اللي هو u أُس نص في u زي ما هي في u تكعيب زائد
+
+375
+00:27:29,700 --> 00:27:35,580
+ثمانية u تكعيب du، هذه اللي هو الآن بنضربها جوا،
+
+376
+00:27:35,580 --> 00:27:42,120
+الآن بيصير اللي هي عندي المقدار، عندي u أُس نص في u
+
+377
+00:27:42,120 --> 00:27:48,440
+تكعيب، تسعة u تكعيب في u، تس
+
+445
+00:32:27,090 --> 00:32:31,590
+V بتروح مع الـ U فبصير 2U E to the U DU DV وهذا
+
+446
+00:32:31,590 --> 00:32:36,030
+مشهور بالـ integration by parts بتحلو اللي هو وبنجده
+
+447
+00:32:36,030 --> 00:32:39,730
+خلّينا نشوف وصلنا هذه بتساوي هذه الآن ضلّ علينا نحدد
+
+448
+00:32:39,730 --> 00:32:43,790
+مين حدود الـ integration هذه؟ R هي هذه الآن بدي أجي
+
+449
+00:32:43,790 --> 00:32:47,790
+اللي هي هذه. لنشوف كيف بنحصلها أكيد كلكم عارفين كيف
+
+450
+00:32:47,790 --> 00:32:52,580
+ايش بتساوي؟ طبعًا التحصيل الآن اللي هو مش هعمل جدول
+
+451
+00:32:52,580 --> 00:32:55,580
+خلينا نقولها شفهيًا، هيك مش هعمل جدول بس
+
+452
+00:32:55,580 --> 00:32:59,180
+احنا بنعمل جدول لحظة يا جماعة الآن y بيساوي اثنين
+
+453
+00:32:59,180 --> 00:33:02,020
+y بيساوي اثنين، ايش حيصير؟ y بيساوي اثنين اللي هو
+
+454
+00:33:02,020 --> 00:33:04,840
+بيصير y بيساوي اثنين، بيصير u في v بيساوي اثنين
+
+455
+00:33:04,840 --> 00:33:09,900
+يعني uv بيساوي اثنين، اللي هو u في v بيساوي اثنين
+
+456
+00:33:09,900 --> 00:33:13,840
+اللي هو برضه ايش زي مثل x y بيساوي واحد، و u v
+
+457
+00:33:13,840 --> 00:33:16,940
+بيساوي اثنين، هي والجهة الثانية بس احنا ايش منطقة
+
+458
+00:33:16,940 --> 00:33:21,120
+المنطقة الـ UV بيساوي ايش؟ بيساوي اثنين، هاي U هذا الخط
+
+459
+00:33:21,120 --> 00:33:25,760
+الأصفر يتحول لهذا الخط الأصفر اللي هان الآن X في Y
+
+460
+00:33:25,760 --> 00:33:31,540
+بيساوي واحد. خذ الـ X في Y اللي هو U على V في U في V
+
+461
+00:33:31,540 --> 00:33:34,760
+الـ V بتروح مع الـ V بيصير U تربيع بيساوي واحد، يعني
+
+462
+00:33:34,760 --> 00:33:40,340
+U بيساوي واحد، U بتساوي واحد اللي هي عبارة عن هاي U
+
+463
+00:33:40,340 --> 00:33:43,560
+بيساوي واحد الأحمر بدل اللي هو الخط هذا المائل
+
+464
+00:33:43,560 --> 00:33:46,950
+اللي هو الأحمر. بنشوف الأزرق هذا y بيساوي 2 و y
+
+465
+00:33:46,950 --> 00:33:52,010
+بيساوي 2 اللي هي بيصير u، أنا ايش هذا الخط الأصفر؟
+
+466
+00:33:52,010 --> 00:33:55,130
+لأ، عملنا الخط الأصفر الأزرق هذا، الأزرق y بيساوي x
+
+467
+00:33:55,130 --> 00:33:59,550
+y بيساوي x، y بيساوي x، هذه ساوي هذه uv بيساوي u على
+
+468
+00:33:59,550 --> 00:34:04,150
+v اللي هو بيصير عندك اللي هو الـ u مع الـ u بتروح
+
+469
+00:34:04,150 --> 00:34:09,250
+بيصير v تربيع بيساوي اللي هو 2، يعني اللي هو v
+
+470
+00:34:09,250 --> 00:34:13,390
+تربيع بيساوي 2، ولا قد ايش؟ بيصير احنا قلنا ايش هذا y
+
+471
+00:34:13,390 --> 00:34:17,390
+بيساوي x، y بيساوي x، بنساويهم لبعض، y بصير u على v
+
+472
+00:34:17,390 --> 00:34:21,390
+بيساوي u في v، لأ واحد، u مع الـ u بتروح بصير اللي هو
+
+473
+00:34:21,390 --> 00:34:24,730
+واحد على v بيساوي v، يعني v تربيع بيساوي واحد، يعني v
+
+474
+00:34:24,730 --> 00:34:27,950
+بتساوي واحد، فاهمين عليّ؟ ان شاء الله v بتساوي واحد
+
+475
+00:34:27,950 --> 00:34:31,510
+اللي هو ايش هذا الخط الأحمر، إذن هي المنطقة اللي ما
+
+476
+00:34:31,510 --> 00:34:35,220
+اللي مطلوبة. الآن بدنا نكامل بالنسبة لـ U قبل، إذا كنا
+
+477
+00:34:35,220 --> 00:34:39,040
+نكامل بالنسبة لـ U، ناخذ موازي، فبيصير U جاي من عند
+
+478
+00:34:39,040 --> 00:34:45,580
+اللي هو واحد لعند اللي هو اثنين على V، اه، والـ V من
+
+479
+00:34:45,580 --> 00:34:49,520
+عند الواحد لعند اخر ايش؟ اثنين، واذا بدنا نكامل
+
+480
+00:34:49,520 --> 00:34:53,380
+العكس بناخد اللي هو اول اش باتجاه هذا اللي بتصير V
+
+481
+00:34:53,380 --> 00:34:58,220
+من عند واحد لعند اثنين على V، اي منهم بصير الان هو
+
+482
+00:34:58,220 --> 00:35:03,700
+مكامل بالنسبة لمين قبل؟ هي اعوض اثنين U E to the U
+
+483
+00:35:03,700 --> 00:35:08,540
+لأن بدنا اللي هو نوجد حدود الـ integration الآن، DV
+
+484
+00:35:08,540 --> 00:35:13,480
+V من واحد لعندي اثنين على U، الخط العمودي هذا من
+
+485
+00:35:13,480 --> 00:35:16,900
+واحد ل اثنين على U، U من واحد لعندي اثنين لعندي
+
+486
+00:35:16,900 --> 00:35:20,860
+اثنين، U من واحد لعندي اثنين، ارجع لكم، U من عند واحد
+
+487
+00:35:20,860 --> 00:35:26,140
+هي U من عند اللي هي بتبدأ الادة طبعًا V بتمشي هيك
+
+488
+00:35:26,140 --> 00:35:30,060
+تيجي من عند الواحد وبيصير لعند U، V، V بيساوي اثنين
+
+489
+00:35:30,060 --> 00:35:34,260
+يعني V بيساوي اثنين على U، اذا V من عند واحد ل اثنين
+
+490
+00:35:34,260 --> 00:35:37,540
+على U، وكل هنا دول الموازيات اللي اخذناها هنا
+
+491
+00:35:37,540 --> 00:35:43,080
+هيكون البدن من عند واحد لعند ايش؟ اثنين، من واحد لعند
+
+492
+00:35:43,080 --> 00:35:47,860
+اثنين اللي هنكاملها، تصبح اثنين U تكاملها بالنسبة لـ V
+
+493
+00:35:47,860 --> 00:35:52,940
+V U E to the U من اثنين على U لعند الواحد، بنعوض عن
+
+494
+00:35:52,940 --> 00:35:58,520
+V اللي هي بتصبح اثنين E to the U ناقص اللي هي لما
+
+495
+00:35:58,520 --> 00:36:01,080
+نعوض عنها اثنين على U لما بتروح اليوم على U تصبح
+
+496
+00:36:01,080 --> 00:36:06,820
+اثنين E to the U ناقص اللي هو اللي هنا V بواحد التي
+
+497
+00:36:06,820 --> 00:36:10,960
+هي u في e to the u du هو يساوي اثنين ناقص u to the
+
+498
+00:36:10,960 --> 00:36:13,600
+u هذا integration by parts بيطلع عندي اللي هو
+
+499
+00:36:13,600 --> 00:36:17,640
+المقدار هذا، احنا مش هنطلب منكم شغلات صعبة زي كذا أو مش
+
+500
+00:36:17,640 --> 00:36:20,340
+مستذكرينها، لكن الـ integration by parts يعني ممكن
+
+501
+00:36:20,340 --> 00:36:23,940
+واحد يصادفها بسرعة، لكن نحاول لما نعجبكم مش يعني
+
+502
+00:36:24,250 --> 00:36:27,950
+لأنه يعني بيبقى، أقصد أنه أنه بنحاول أنه مش في
+
+503
+00:36:27,950 --> 00:36:30,510
+اختبركم في integration by parts، لأ، بدي أختبركم في
+
+504
+00:36:30,510 --> 00:36:34,350
+الـ transformation، ويساوي 2 اللي هو زي ما قلنا بعد
+
+505
+00:36:34,350 --> 00:36:37,450
+ما عوضنا، عملنا integration by parts لحد التكامل
+
+506
+00:36:37,450 --> 00:36:41,930
+عوضنا μ 2 لعند واحد لعند اثنين، عوضنا عن قيمة
+
+507
+00:36:41,930 --> 00:36:46,010
+طلعت القيمة اللي أمامكم، والتعويض سهل وبتعرفوا تعملهم
+
+508
+00:36:46,910 --> 00:36:51,430
+طيب، نجي الآن، substitution in triple integrals
+
+509
+00:36:51,430 --> 00:36:58,610
+اللي هو نفس روح الـ double integral، الآن f of x و
+
+510
+00:36:58,610 --> 00:37:03,970
+y و z، بس في dx dy dz، هذه بيصير بدلها بدنا نعوض بـ u
+
+511
+00:37:03,970 --> 00:37:09,970
+و v و w، بيصير جاكوبيان، ننتبه له مضروب في مين؟ في du dv
+
+512
+00:37:09,970 --> 00:37:14,750
+dw، اذا نفس الشيء بس شلنا الـ dz جيبنا dz زيادة و
+
+513
+00:37:14,750 --> 00:37:20,730
+حطينا dw بعد ما عوضنا عن ما هو X و Y و Z بما يلاقيه
+
+514
+00:37:20,730 --> 00:37:24,970
+منهم U و V و W، الـ X بتساوي G و V و V و W، والـ Y
+
+515
+00:37:24,970 --> 00:37:28,650
+بتساوي H و V و V و W، والـ Z بتساوي K و V و V و W
+
+516
+00:37:28,650 --> 00:37:31,550
+هاي اللي هي الـ region اللي هي الـ in space اللي هي
+
+517
+00:37:31,550 --> 00:37:37,450
+G صارت in space تبع الاسف اللي كانت X و Y و Z بعد
+
+518
+00:37:37,450 --> 00:37:41,030
+ما عوضنا عنها اللي هي بدلت U و V و كده و W صارت
+
+519
+00:37:41,030 --> 00:37:43,350
+الـ preimage هذا Transformation عشان هيك من hand ل
+
+520
+00:37:43,350 --> 00:37:46,940
+hand، لكن اما هو اللي هي نبدأ بهذه والـ preimage
+
+521
+00:37:46,940 --> 00:37:50,780
+اللي هي هذه، طيب هذه الـ G يجي للـ Jacobian اللي
+
+522
+00:37:50,780 --> 00:37:54,200
+عندي، ايش تعريفه؟ Jacobian نفس الأولاني بس الآن زادت
+
+523
+00:37:54,200 --> 00:37:59,620
+اللي هي Z و W، ده الـ X by ده الـ U، ده الـ X by ده الـ
+
+524
+00:37:59,620 --> 00:38:03,750
+V، ده الـ X by ده الـ W، هي السطر الأول، الثاني، واختصارًا
+
+525
+00:38:03,750 --> 00:38:13,090
+عشان اسلوب للحفظ، واختصارًا
+
+526
+00:38:13,090 --> 00:38:15,630
+عشان اسلوب للحفظ واختصارًا، عشان اسلوب للحفظ واختصارًا
+
+527
+00:38:15,630 --> 00:38:21,310
+عشان اسلوب للحفظ واختصارًا
+
+528
+00:38:21,310 --> 00:38:26,190
+عشان اسلوب للحفظ واختصارًا، عشان اسلوب للحفظ
+
+529
+00:38:33,900 --> 00:38:37,240
+واللي طبعًا ممكن اللي اولانيه، جاكوبيان اللي اولانيه
+
+530
+00:38:37,240 --> 00:38:41,380
+تبع زمان، دي x، دي الـ x bar، و y على دي الـ u، و دي الـ
+
+531
+00:38:41,380 --> 00:38:47,680
+v، والاستذكار زي ما قلنا كيف تستعملوها، الآن نأتي
+
+532
+00:38:47,680 --> 00:38:50,320
+للـ cylindrical coordinates اللي اخذناه اللي هو
+
+533
+00:38:50,320 --> 00:38:55,280
+العلاقات بين R, θ و Z، وبدنا الآن نحوّلها اللي هي
+
+534
+00:38:55,280 --> 00:39:00,640
+من الـ Cartesian لـ cylindrical، مين اللي بيحوّل؟ X
+
+535
+00:39:00,640 --> 00:39:04,440
+بيساوي R Cos θ، و Y بيساوي R Sin θ، و Z بيساوي Z
+
+536
+00:39:04,440 --> 00:39:08,440
+بدنا نثبت لكم أن هذا التحويل اللي كنا نحوّل فيه
+
+537
+00:39:08,440 --> 00:39:13,420
+يتلاقم مع اللي بنحكي فيه في هذا الـ Transformation
+
+538
+00:39:13,810 --> 00:39:20,890
+التغيير الجاكوبياني هو جاكوبيان R ثتا وZ، دل X by
+
+539
+00:39:20,890 --> 00:39:27,770
+دل R، دل X by دل ثتا، دل X by دل Z، دل Y by دل ثتا R
+
+540
+00:39:27,770 --> 00:39:32,190
+دل Y by دل ثتا، دل Y دل Z، الآخر، دل Z by دل R، دل Z
+
+541
+00:39:32,190 --> 00:39:36,030
+by دل ثتا، دل Z by دل Z، طبعًا الـ Z في الجهتين نفس
+
+542
+00:39:36,030 --> 00:39:38,270
+الشيء في الـ ... في الـ ... في الـ ... في الـ ... في الـ ...
+
+543
+00:39:38,270 --> 00:39:42,860
+في الـ ... في الـ ... في الـ ... في الـ rectangular، بتحول
+
+544
+00:39:42,860 --> 00:39:48,340
+من rectangular لـ cylindrical، دل X by دل R، ايش يا
+
+545
+00:39:48,340 --> 00:39:54,120
+جماعة؟ دل X by دل R اللي هي cos θ، دل X by دل θ اللي
+
+546
+00:39:54,120 --> 00:39:59,140
+هي ناقص R sin θ، دل X by دل Z صفر، نفس الأشياء
+
+547
+00:39:59,140 --> 00:40:05,460
+دل Y by دل R اللي هو sin θ، دل Y by دل θ اللي هو R
+
+548
+00:40:05,460 --> 00:40:14,040
+cos θ، دل Y by دل Z صفر، نأتي للأخيرة DEL Z by DEL θ
+
+549
+00:40:14,040 --> 00:40:18,780
+صفر، DEL Z by DEL R صفر، DEL Z by DEL R صفر
+
+550
+00:40:18,780 --> 00:40:20,060
+DEL Z by DEL R صفر، DEL Z by DEL R صفر
+
+551
+00:40:20,060 --> 00:40:20,500
+DEL Z by DEL R صفر، DEL Z by DEL R صفر
+
+552
+00:40:20,500 --> 00:40:20,840
+DEL Z by DEL R صفر، DEL Z by DEL R صفر
+
+553
+00:40:20,840 --> 00:40:22,940
+DEL Z by DEL R صفر، DEL Z by DEL R صفر
+
+554
+00:40:22,940 --> 00:40:25,020
+DEL Z by DEL R صفر، DEL Z by DEL R صفر
+
+555
+00:40:25,020 --> 00:40:29,120
+DEL Z by DEL R صفر، DEL Z by DEL R صفر
+
+556
+00:40:29,120 --> 00:40:34,760
+DEL Z by DEL R صفر، DEL Z by DEL R صفر
+
+557
+00:40:34,760 --> 00:40:37,910
+زائد Rsin تربيع اللي هو في النهاية ايش بيطلع؟
+
+558
+00:40:37,910 --> 00:40:42,150
+بيطلع R، DR، اذا F of X Y و Z DX DY DZ هاي الـ D
+
+559
+00:40:42,150 --> 00:40:46,830
+صارت G، الآن الـ F of X Y و Z صارت H of R و θ و Z او
+
+560
+00:40:46,830 --> 00:40:51,590
+زي ما كنا نعمل قبل، F of R Cos θ، F of R Sin θ و Z
+
+561
+00:40:51,590 --> 00:40:56,870
+ماشي، الآن DX DY DZ بدأت تعمل بالجاكوبيان، الجاكوبيان
+
+562
+00:40:56,870 --> 00:41:00,610
+مين هو؟ الـ absolute value of R DR D θ DZ وهو اللي
+
+563
+00:41:00,610 --> 00:41:06,310
+هي القانون اللي احنا اشتغلنا فيه، قبل نجي الآن نجي
+
+564
+00:41:06,310 --> 00:41:09,110
+نعمل مثال الـ spherical، الـ spherical نفس الأشياء
+
+565
+00:41:09,110 --> 00:41:13,150
+عندي اللي هو فيه transformations، بحوله عشان اقول لكم
+
+566
+00:41:13,150 --> 00:41:17,930
+أن هذا اللي هي الـ transformations هذا تعميم لتحويل
+
+567
+00:41:17,930 --> 00:41:22,510
+من الـ rectangular الى cylindrical او from
+
+568
+00:41:22,510 --> 00:41:29,350
+rectangular الى اللي هو spherical، اللي عندي X
+
+569
+00:41:29,350 --> 00:41:33,310
+بيساوي رو Sin φ Cos θ، و Y بيساوي رو Sin φ Sin θ، و Z بيساوي رو Cos φ
+
+570
+00:41:33,310 --> 00:41:37,150
+هذه اللي هو بنحول من X و Y و Z الى رو و φ و θ، الآن الـ Jacobian
+
+571
+00:41:37,150 --> 00:41:42,280
+بتصير اللي هو عبارة عن دي الـ X by دي الـ Rho، دي
+
+572
+00:41:42,280 --> 00:41:45,480
+الـ X by دي الـ Phi، دي الـ X by دي الـ Theta، ونفس
+
+573
+00:41:45,480 --> 00:41:48,200
+الشيء دي الـ Y، ونفس الشيء دي الـ Z، الآن هذا الـ
+
+574
+00:41:48,200 --> 00:41:51,460
+Jacobian بتوجدوه اللي هو هذه، دي الـ X by دي الـ
+
+575
+00:41:51,460 --> 00:41:55,320
+Rho بتعرفوا توجدوه أكيد اللي هي Sin φ Cos θ
+
+576
+00:41:55,320 --> 00:41:58,800
+و الثانية اللي هي دي الـ X by دي الـ Phi اللي هي
+
+577
+00:41:58,800 --> 00:42:01,640
+عبارة عن Rho Cos φ Cos θ، و دي الـ X by دي
+
+578
+00:42:01,640 --> 00:42:06,460
+الـ Theta، Rho Sin φ اللي هو ناقص Sin θ بتطلع برا
+
+579
+00:42:06,460 --> 00:42:12,260
+الناقص طبعًا وبيصير تحطها هنا، بتعوض عن هذه وبتعوض
+
+580
+00:42:12,260 --> 00:42:15,420
+عن هدول بعد ما تفضل اللي هو للـ Y، وبعد ما تفضل للـ
+
+581
+00:42:15,420 --> 00:42:18,720
+Z بتعوضهم وبتوجد الـ determinant طبعًا بغلب ايجاده
+
+582
+00:42:18,720 --> 00:42:22,160
+يعني او خلينا نقول بطول، بس عادي يعني بتستخدم أي سطر
+
+583
+00:42:22,160 --> 00:42:25,420
+أو أي عمود بتطلع لك في النهاية عبر علاقات معينة
+
+584
+00:42:25,420 --> 00:42:28,420
+رو تربيع sin θ في، وهذا هو exercise 19، طيب الآن
+
+585
+00:42:34,580 --> 00:42:37,640
+بدنا نجي نعوض عن ... مافيش داعي للتفصيل، ما بدي أفصل
+
+586
+00:42:37,640 --> 00:42:43,380
+لايجاد determinant، f of x y و z اللي هو بيصير h و
+
+587
+00:42:43,380 --> 00:42:47,080
+rho و phi و theta لما نعوّض مكان x y و z بقيمها
+
+588
+00:42:47,080 --> 00:42:51,800
+dx dy dz اللي بيصير اللي هو عبارة عن جاكوبيان اللي
+
+589
+00:42:51,800 --> 00:42:55,060
+هو absolute value of sin تربيع phi d rho d phi و
+
+590
+00:42:55,060 --> 00:42:57,820
+d θ، هذه اللي هي طبعًا هذه موجبة طبيعي وهذه موجبة في
+
+591
+00:42:58,110 --> 00:43:02,550
+نطاق الهدا فبتشيل absolute value، مش حتفرق، بتطلع
+
+592
+00:43:02,550 --> 00:43:05,710
+نفس القيم اللي عملناها قبل، طبعًا، والـ region D
+
+593
+00:43:05,710 --> 00:43:10,570
+بتتحول الـ region G، اذا اللي هي الـ spherical، وده
+
+594
+00:43:10,570 --> 00:43:13,730
+كمية اللي عبارة عن حلقة خاصة من اللي هو اللي بنحكي
+
+595
+00:43:13,730 --> 00:43:16,670
+عنه in general transformations، نيجي بقى اللي هو
+
+596
+00:43:16,670 --> 00:43:19,810
+example خمسة، الـ Example الخامس بيقول ايش؟ هذا طبعًا
+
+597
+00:43:19,810 --> 00:43:22,570
+عالميا على الـ triple ... على جزء ... اللي هو الـ
+
+598
+00:43:22,570 --> 00:43:26,870
+triple integrals على x, y و z اللي هو 2x ناقص y
+
+599
+00:43:26,870 --> 00:43:31,790
+على 2, z على 3, dx, dy, dz، وحدود الـ integration
+
+600
+00:43:31,790 --> 00:43:35,540
+هي أمامنا، بقول لك by applying the transformation
+
+601
+00:43:35,540 --> 00:43:40,000
+اللي هو U بيساوي 2X ناقص U على 2 اللي هي هذه V
+
+602
+00:43:40,000 --> 00:43:43,880
+بيساوي Y على 2 اللي هي استعاني منها بهذه، و W
+
+603
+00:43:43,880 --> 00:43:47,960
+بيساوي Z على 3، طبعًا هو عارف انه احنا يعني هنتغلب
+
+604
+00:43:47,960 --> 00:43:50,760
+لو بدنا نوجد اللي هو transformation، لكن ممكن
+
+605
+00:43:50,760 --> 00:43:54,880
+الواحد اللي هو U يستنتج من هذا و W من هنا ويقول
+
+606
+00:43:54,880 --> 00:43:59,340
+لك V هذه اللي هو هي هنا، UY على 2 وهذا كأنه بناحط
+
+607
+00:43:59,340 --> 00:44:04,240
+V بيساوي على 2 بتتناسق مع حد وبتتناسق مع حد فيمكن
+
+608
+00:44:04,240 -->
+
+667
+00:48:05,330 --> 00:48:06,370
+وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد
+
+668
+00:48:06,370 --> 00:48:07,750
+وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد
+
+669
+00:48:07,750 --> 00:48:11,550
+وحد وحد وحد وحد
+
+670
+00:48:20,390 --> 00:48:26,050
+دل z by دل u ودل z by دل v صفر ودل z by دل w اللي
+
+671
+00:48:26,050 --> 00:48:29,430
+هو بيساوي تلاتة فبيصير صفر صفر تلاتة بنوجد ال
+
+672
+00:48:29,430 --> 00:48:32,810
+determinant هذا بواسطه السطر هذا مثلا بيصير تلاتة
+
+673
+00:48:32,810 --> 00:48:36,410
+في هذا الـ determinant اللي هو اتنين نقل صفر يعني
+
+674
+00:48:36,410 --> 00:48:40,590
+اتنين في تلاتة بتطلع ستة الان بنيجي بنعوض عن هذا
+
+675
+00:48:40,590 --> 00:48:45,010
+المقدار عوضنا عن هذه اللي هي u وهذا w وهذا
+
+676
+00:48:45,010 --> 00:48:48,230
+الجاكوبيان عن حدود الـ integration زي ما قلنا U من
+
+677
+00:48:48,230 --> 00:48:52,370
+صفر لواحد كله rectangles و DV من صفر ل اتنين و W
+
+678
+00:48:52,370 --> 00:48:56,430
+من صفر ل عند واحد الان بنعوض مكانها دي بصير ستة
+
+679
+00:48:56,430 --> 00:49:00,130
+بنطلع الستة برا و بنكامل U زائد V بالنسبة لـ U بصير
+
+680
+00:49:00,130 --> 00:49:04,570
+U تربيع على اتنين UV من صفر ل عند واحد بنعوض عنها
+
+681
+00:49:04,570 --> 00:49:08,390
+بواحد بصير نص  الـ W وهذه كما تصبح صفر دي
+
+682
+00:49:08,390 --> 00:49:15,170
+VDW الـ V من فاضل بصير من كامل V على اتنين زي VW من
+
+683
+00:49:15,170 --> 00:49:19,810
+صفر لعند اتنين بنعوض الآن عن V بقيمتها بصير عبارة
+
+684
+00:49:19,810 --> 00:49:24,460
+عن واحد زي اتنين W بالكامل هذه بيصير واحد زي الـ W
+
+685
+00:49:24,460 --> 00:49:28,360
+تربيع من صفر لواحد والستة البره بيصير ستة في اتنين
+
+686
+00:49:28,360 --> 00:49:32,100
+واحد على اتنين ساوى اتناشر و هيك بيكون احنا انهينا
+
+687
+00:49:32,100 --> 00:49:36,680
+الـ محاضرة والسلام عليكم ورحمة الله وبركاته وكل
+
+688
+00:49:36,680 --> 00:49:41,520
+عام ولا كل فصل وانتم بخير ربنا ينجحكم جميعا
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/MDfHgpCP1Ww_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/MDfHgpCP1Ww_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..381bdcbfe3b210835346c90390d6e75acbf7914a
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/MDfHgpCP1Ww_postprocess.srt
@@ -0,0 +1,2752 @@
+1
+00:00:00,000 --> 00:00:04,160
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,160 --> 00:00:12,080
+وبركاته هذه هي المحاضرة رقم 26 مساقطة فاضل وتكامل
+
+3
+00:00:12,080 --> 00:00:19,180
+جيم وهي المحاضرة الأخيرة وهي المحاضرة السابعةوليس
+
+4
+00:00:19,180 --> 00:00:22,520
+في ثامنة السابعة في chapter 15 المحاضرة السابقة
+
+5
+00:00:22,520 --> 00:00:26,260
+اللي قلت عنها السابعة لأ هي كانت السادسة وهي
+
+6
+00:00:26,260 --> 00:00:30,420
+بعنوان substitutions in multiple integrals اللي هي
+
+7
+00:00:30,420 --> 00:00:34,580
+عملية التعويض في multiple integrals أو حتى ال
+
+8
+00:00:34,580 --> 00:00:38,480
+double طبعا الان احنا كنا نعمل substitutions في
+
+9
+00:00:38,480 --> 00:00:43,500
+تفاضل الفأو تفاضل باه ونعوض تعويض مناسب عشان نسهل
+
+10
+00:00:43,500 --> 00:00:47,620
+علينا عملية ال integration ونكون التكامل بيكون صعب
+
+11
+00:00:47,620 --> 00:00:53,460
+بنسهّل علينا التكامل الآن في شيء مشابه فيه اللي هو
+
+12
+00:00:53,460 --> 00:00:57,800
+ال double integrals و multiple integrals انه بنعوض
+
+13
+00:00:57,800 --> 00:01:04,040
+عن ال X وعن ال Y بشيء اسمه U V بدلالة X وY طبعا
+
+14
+00:01:04,040 --> 00:01:08,850
+بيكونهذا التعويض اللي هو بيسهّل علينا عالميا عملية
+
+15
+00:01:08,850 --> 00:01:12,750
+التكامل ولكن تبعاً لهذا التعويض بيصير عندنا اللي
+
+16
+00:01:12,750 --> 00:01:17,410
+هو تابعات معينة بدنا ناخدها بعين الاعتبار لما نعوض
+
+17
+00:01:17,410 --> 00:01:20,930
+خلينا نشوف نبدأ الآن Substitution in double
+
+18
+00:01:20,930 --> 00:01:23,370
+integrals بدنا نبدأ في Substitution in double
+
+19
+00:01:23,370 --> 00:01:27,190
+integrals طبعا عندنا احنا ال integration لل Fxy
+
+20
+00:01:27,190 --> 00:01:31,870
+dxdy هي اللي بدنا اللي هو نكملها over R over ال
+
+21
+00:01:31,870 --> 00:01:35,610
+region R هي ال region R اللي عندىاللي بدنا نكمل
+
+22
+00:01:35,610 --> 00:01:39,390
+عليها طبعا هذه اللي هو ممكن تكون اللي هو الدالة
+
+23
+00:01:39,390 --> 00:01:44,590
+صعبة التكامل فبدنا نحول ال F of XY نحولها بدلالة
+
+24
+00:01:44,590 --> 00:01:49,690
+اللي هو U of V بطريقة معينة فعشان هيك اللي هو
+
+25
+00:01:49,690 --> 00:01:53,980
+بنيجي بناخد حاجة اسمهااللي هي transformation
+
+26
+00:01:53,980 --> 00:02:01,180
+بنحوّل من اللي هو ال X و Y عبر دالة X بتساوي G و V
+
+27
+00:02:01,180 --> 00:02:07,500
+و V بنعاود عن كل X بـG و V و V و Y شمالها H و V و
+
+28
+00:02:07,500 --> 00:02:12,450
+Vفاللي هي المنطقة اللي اللي اللي اللي اللي اللي
+
+29
+00:02:12,450 --> 00:02:12,750
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+30
+00:02:12,750 --> 00:02:13,110
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+31
+00:02:13,110 --> 00:02:13,230
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+32
+00:02:13,230 --> 00:02:13,250
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+33
+00:02:13,250 --> 00:02:13,310
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+34
+00:02:13,310 --> 00:02:15,990
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+35
+00:02:15,990 --> 00:02:16,070
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+36
+00:02:16,070 --> 00:02:16,250
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+37
+00:02:16,250 --> 00:02:16,790
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+38
+00:02:16,790 --> 00:02:16,870
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+39
+00:02:16,870 --> 00:02:28,950
+اللي
+
+40
+00:02:28,950 --> 00:02:32,530
+الليفبصير عندنا F of G of U of V بعد ما عوضنا عن X
+
+41
+00:02:32,530 --> 00:02:38,350
+وH of U of V بعد ما عوضنا عن Y الآن الـ DXDY هذه
+
+42
+00:02:38,350 --> 00:02:42,810
+الآن إيش هتساوي حاجة اسمها الـ Jacobian لـ
+
+43
+00:02:42,810 --> 00:02:46,710
+Absolute value of Jacobian UV دي و دي V طبعا هذا
+
+44
+00:02:46,710 --> 00:02:49,750
+اللي هو قانون فيه الاشتقاخات و فيه اللي و كده مش
+
+45
+00:02:49,750 --> 00:02:54,290
+موضوعنا الآن إذا F of X و Y بنعوض عن X بدلالتها
+
+46
+00:02:54,290 --> 00:03:00,050
+اللي هي بدلالة U و VوY دلالتها بدلالة UV وDXDY
+
+47
+00:03:00,050 --> 00:03:05,600
+بتصير تستبدل بالـ اللي هي الـ Jacobian في DUDVالان
+
+48
+00:03:05,600 --> 00:03:10,180
+ال region R طبعا هتتحول الى region ايش اسمها D اذا
+
+49
+00:03:10,180 --> 00:03:13,600
+الان عندى فى عملية التحويل فى تلت أمور اللى هو
+
+50
+00:03:13,600 --> 00:03:17,920
+مكان اللى هى ال F of X و Y هنعوض ال D X و I فى الى
+
+51
+00:03:17,920 --> 00:03:22,100
+تعويض هيه اللى هو اللى قلنا عنه و ال region اللى
+
+52
+00:03:22,100 --> 00:03:25,200
+هو بيصير region جديدة و هنشوف كيف نتعامل فى كل
+
+53
+00:03:25,200 --> 00:03:28,590
+واحدة من التلاتة اللى عندناالان شو الـ Jacobian
+
+54
+00:03:28,590 --> 00:03:31,550
+هذا اللي بتحكي عنه ال Jacobian بنعرفه كما يلي
+
+55
+00:03:31,550 --> 00:03:35,630
+Definition the Jacobian determinant or Jacobian of
+
+56
+00:03:35,630 --> 00:03:39,070
+the coordinates Transformation X بيساوي G of U و Y
+
+57
+00:03:39,070 --> 00:03:42,730
+بيساوي H of U و V يعني ال Jacobian لهذا ال
+
+58
+00:03:42,730 --> 00:03:46,190
+Transformation اللي لما نعوضنا شيلنا X وعوضنا
+
+59
+00:03:46,190 --> 00:03:49,510
+مكانها بدلالة U و V وشيلنا Y وعوضنا مكانها بدلالة
+
+60
+00:03:49,510 --> 00:03:54,430
+U و V ال Jacobian هو كما يلي بنعرفه هو عبارة عن دل
+
+61
+00:03:54,430 --> 00:03:55,070
+X
+
+62
+00:04:03,680 --> 00:04:07,200
+والسطر التاني اللي هو العمود التاني في السطر
+
+63
+00:04:07,200 --> 00:04:11,610
+التاني اللي هو del u by del y by del vاختصارًا
+
+64
+00:04:11,610 --> 00:04:15,190
+طبعًا هذا دل X في دل U بيصير دروب في هذا بتطلع هذا
+
+65
+00:04:15,190 --> 00:04:18,830
+المقدار ناقص هذا المقدار في هذا المقدار فبطلع هذا
+
+66
+00:04:18,830 --> 00:04:23,170
+مين هو يا جماعة الجاكوبيان اذا الجاكوبيان اللي هو
+
+67
+00:04:23,170 --> 00:04:28,230
+دل X by دل U الأولى دل X by دل V الثانية وبنعمل
+
+68
+00:04:28,230 --> 00:04:33,990
+السطر الأول بعدين بنيجي دل Y by دل Uو بعدين del y
+
+69
+00:04:33,990 --> 00:04:37,450
+by del v و بنودد ال determinant بنحطله absolute
+
+70
+00:04:37,450 --> 00:04:42,270
+value و بنضربه في هذا و بيصير du dv بكون حوضنا هذا
+
+71
+00:04:42,270 --> 00:04:46,830
+درجة نضل عليها اللي هي ال G أحصلها من مين من R و
+
+72
+00:04:46,830 --> 00:04:52,570
+خلينا اللي هو نشوف المثال التالي اللي حاجة احنا
+
+73
+00:04:52,570 --> 00:04:57,040
+عارفينها نشوف ايش بقولبقول مثال واحد find the
+
+74
+00:04:57,040 --> 00:04:59,920
+jacobian for the polar coordinate transformation
+
+75
+00:04:59,920 --> 00:05:04,600
+انتوا متذكرين اللي هو العلاقة بين اللي هو ال X و
+
+76
+00:05:04,600 --> 00:05:09,100
+ال R و الثتا و ال Y و ال R و الثتا هذا الآن هو ال
+
+77
+00:05:09,100 --> 00:05:12,000
+transformation اللي بدنا نشتغل عليه المثال هيك
+
+78
+00:05:12,000 --> 00:05:14,580
+بتقول find the jacobian for the polar coordinate
+
+79
+00:05:14,580 --> 00:05:18,620
+transformation X بالسواء R cos θوY بسهولة R sin θ
+
+80
+00:05:18,620 --> 00:05:21,240
+وUse equation one اللي قبله شوية to write the
+
+81
+00:05:21,240 --> 00:05:25,580
+cartesian integral as a polar integral إذا الآن
+
+82
+00:05:25,580 --> 00:05:30,960
+هذه المنطقة هذا ال integration بدي أعوض عن X بقيمة
+
+83
+00:05:30,960 --> 00:05:35,940
+R cos θ وY بسهولة R sin θ وأوددكم اللي هو ال
+
+84
+00:05:35,940 --> 00:05:40,140
+Jacobian وDX في DY أستبدلها بالjacobian في اللي هو
+
+85
+00:05:40,140 --> 00:05:45,560
+DR Dθ وحنشوف أن هذا هيشتقلنا القانون اللي احنا
+
+86
+00:05:45,560 --> 00:05:50,150
+عارفينه زمان هتشوف كيفالان اول اشي اللى هو ال
+
+87
+00:05:50,150 --> 00:05:53,010
+region اخد اللى هو ال .. ال .. ال .. ال .. ال ..
+
+88
+00:05:53,010 --> 00:05:58,510
+المنطقة اللى هي اللى بديها على R هذا R و هذا ايش
+
+89
+00:05:58,510 --> 00:06:02,950
+فتا انا باخدهن فتا اش في ال Cartesian اللى هو R مع
+
+90
+00:06:02,950 --> 00:06:08,070
+مين مع ثتا ماشي الان قيم R اللى هي بديها من من
+
+91
+00:06:08,070 --> 00:06:11,750
+Zero لعند واحد هذه المنطقة ال region اللى بديها
+
+92
+00:06:11,750 --> 00:06:15,810
+هذا ال regionالان نزيله عند واحد ونزيله عند باي
+
+93
+00:06:15,810 --> 00:06:20,010
+على اتنين هذه هتطلع ال region G في R بس الان كيف
+
+94
+00:06:20,010 --> 00:06:23,970
+هتطلع .. هتطلع بعد ما نشوف اللي هو ال region اللي
+
+95
+00:06:23,970 --> 00:06:28,210
+معطيني إياها هو اللي هي R هذه معطاها الان ال R
+
+96
+00:06:28,210 --> 00:06:34,250
+أشملها ال R معطيني إياها اللي هوالـ X عندى بتساوي
+
+97
+00:06:34,250 --> 00:06:41,670
+R Cos θ هي عندى اللى هو ال .. ال .. ال θ هاخدها
+
+98
+00:06:41,670 --> 00:06:45,030
+عندى اللى بتاخدوا المنطقة هذه بس أنا بتاخدوا
+
+99
+00:06:45,030 --> 00:06:49,310
+المنطقة هذه اللى هى عبارة عن ال .. هذا اللى هو
+
+100
+00:06:49,310 --> 00:06:55,150
+طبعا Y بساوى 0 وهذا X بساوى 0 وهذا ده جزء من X
+
+101
+00:06:55,150 --> 00:07:00,210
+تربيع زاد Y تربيع بساوى 1أو بالـ Polar اللي هي R
+
+102
+00:07:00,210 --> 00:07:05,930
+بتساوي إيش؟ واحد و Theta لإنه ماخدها في الربح حدى
+
+103
+00:07:05,930 --> 00:07:09,470
+Theta من سفر لعند مين لعند Pi على اتنين إذا
+
+104
+00:07:09,470 --> 00:07:14,870
+المنطقة هذه اللي هي المرسومة دائرة هذه لما بتاخد
+
+105
+00:07:14,870 --> 00:07:19,030
+قيمها طبعا قيمها بR هنا اللي هي R ب Direct
+
+106
+00:07:19,030 --> 00:07:23,970
+Distance ماشي الحال اللي هي عبارة عن R اللي عندي
+
+107
+00:07:23,970 --> 00:07:29,710
+اللي هي من سفر لعند واحدماشي الحال او θ من صفر
+
+108
+00:07:29,710 --> 00:07:33,150
+لعند مين لعند باي على اتنين عشان هيك لو بدأ ارسم
+
+109
+00:07:33,150 --> 00:07:37,630
+الار والثتا ككارتيزيان زي ما رسمت x و y ككارتيزيان
+
+110
+00:07:37,630 --> 00:07:42,450
+بدأ ارسم ار وثتا ككارتيزيان الان انا بتعامل محن ك
+
+111
+00:07:42,450 --> 00:07:45,370
+transformations يعني x بيسوأ ار كصين ثتا و y بيسوأ
+
+112
+00:07:45,370 --> 00:07:49,150
+ار صين ثتا العلاقة بين هنا لما بدرسم هنا على مكان
+
+113
+00:07:49,150 --> 00:07:53,570
+معين بتصير اللي هي هذه اللي هي ال .. ال .. ال ..
+
+114
+00:07:53,570 --> 00:07:56,850
+ال ..الـ Directed Distance هي الـ R بالشكل هذا و
+
+115
+00:07:56,850 --> 00:07:59,790
+الـ θة اللي هي مع الـ Positive X Axis لكن لما
+
+116
+00:07:59,790 --> 00:08:04,370
+بدرسمها هان هذا أحداثي R و هذا أحداثي θة هذا بشتغل
+
+117
+00:08:04,370 --> 00:08:08,050
+فيه عادي اللي هي R اللي كانت هنا من 0 ل 1 بيصير
+
+118
+00:08:08,050 --> 00:08:14,110
+هاي R عندي هاي R طبعا هي الـ R هاي عندي 0 الآن
+
+119
+00:08:14,110 --> 00:08:18,310
+ماشي الحال الـ R بتساوي 0 as if هي هان R بتساوي 0
+
+120
+00:08:18,310 --> 00:08:24,420
+هيها و هي اللي هو R بتساوي 1الان θ من وين لوين؟ θ
+
+121
+00:08:24,420 --> 00:08:32,760
+من عند 0 لعند باي على 2 فهذه هي المنطقة اللي ..
+
+122
+00:08:32,760 --> 00:08:37,120
+اللي هذي R هنا صارت G هنا أنا برسمها في الـ
+
+123
+00:08:37,120 --> 00:08:41,380
+Cartesian يا جماعة R و θ باعتبرها عادي زي متغيرات
+
+124
+00:08:41,380 --> 00:08:46,050
+عادية زي ال X و ال Y تحتلكن العلاقة بين X و Y و R
+
+125
+00:08:46,050 --> 00:08:50,270
+و θ على نفس الرسمة هذه القديمة هي العلاقة بينهن
+
+126
+00:08:50,270 --> 00:08:53,830
+اللي هي θ بتكون اللي هو الزاوية اللي بيصنحها مع ال
+
+127
+00:08:53,830 --> 00:08:57,250
+positive x axis اللي هو X هذا و بتعمل اللي هو
+
+128
+00:08:57,250 --> 00:09:01,590
+directed distance هذا اللي هو بمثل LRماشي الآن الـ
+
+129
+00:09:01,590 --> 00:09:06,970
+G هي صارت R إذاً هي الـ R صارت إيش G الآن إيش بدنا
+
+130
+00:09:06,970 --> 00:09:11,270
+نسوي F of X,Y,D,X,Z,Y إيش بدنا نعملها بنشيل ال X
+
+131
+00:09:11,270 --> 00:09:15,710
+بيصير R Cos θ و ال Y R Sin θ حسب اللي معطينيه هو
+
+132
+00:09:15,710 --> 00:09:19,870
+هيه معطينيها وينها عند X بيسوي R Cos θ و Y بيسوي R
+
+133
+00:09:19,870 --> 00:09:22,910
+Sin θ زي ما أعطانا ال transformation اللي بدنا
+
+134
+00:09:22,910 --> 00:09:27,130
+نستعمله بسوء بال Integral over mean over G اللي
+
+135
+00:09:27,130 --> 00:09:32,130
+عندي هذه اللي أماميلم ينلف F of R cos θ R sin θ في
+
+136
+00:09:32,130 --> 00:09:36,500
+الجاكوبياندر دي ثيتا اللي كانت دي يو دي ذي
+
+137
+00:09:36,500 --> 00:09:40,700
+الجاكوبيا للار وثيتا ايش الجاكوبيا للار وثيتا
+
+138
+00:09:40,700 --> 00:09:43,840
+خلينا نشوف حسب القانون اللي قبل شوية دل اكس
+
+139
+00:09:43,840 --> 00:09:48,620
+بايدلار دل اكس بايدلثيتا دل واي بايدلار دل واي
+
+140
+00:09:48,620 --> 00:09:52,400
+بايدلثيتا نوجد دل اكس بايدلار دل اكس بايدلار بنعرف
+
+141
+00:09:52,400 --> 00:09:56,080
+نوجدها هي اللي هو ال X دل اكس بايدلار اللي هي
+
+142
+00:09:56,080 --> 00:10:01,600
+cosine ثيتا صح طيب دل اكس بايدلثيتا ناقص R sin
+
+143
+00:10:01,600 --> 00:10:06,530
+ثيتامظبوط؟ طيب دل Y by دل R دل Y by دل R اللي هي
+
+144
+00:10:06,530 --> 00:10:13,110
+عبارة عن sin θ دل Y by دل θ اللي هو R cos θ هي R
+
+145
+00:10:13,110 --> 00:10:19,110
+cos θ أضرب هذه في هذه بيصير R cos تربيع لأن ناقص
+
+146
+00:10:19,110 --> 00:10:23,630
+ناقص هذه بيصير زائد R sin تربيع ناخد ال R برا
+
+147
+00:10:23,630 --> 00:10:26,630
+بيصير R في cos تربيع زي sin تربيع هذه عبارة عن
+
+148
+00:10:26,630 --> 00:10:32,670
+واحد بتطلع من ال Rالان احنا مفترضين فوق عندنا R
+
+149
+00:10:32,670 --> 00:10:36,250
+ايش مالها اكبر يساوي سفر هي الـ R أكبر يساوي سفر و
+
+150
+00:10:36,250 --> 00:10:40,590
+اصغر او يساوي ايش واحد بناء عليه الان لما ناخد
+
+151
+00:10:40,590 --> 00:10:43,450
+القانون الـ double integration عوضنا زي ما قلنا
+
+152
+00:10:43,450 --> 00:10:47,250
+هنا دلنا نعوض مكان الـ dx dy جاكوبين absolute
+
+153
+00:10:47,250 --> 00:10:52,630
+value له في DRDθ الجاكوبين الان طلع ايش بساوي R لـ
+
+154
+00:10:52,630 --> 00:10:55,290
+absolute value الجاكوبينالـ Absolute Value لـ R
+
+155
+00:10:55,290 --> 00:10:58,890
+لكن R أكبر سواء صفر مخدينة إذا إيش بتساوي R ففعلا
+
+156
+00:10:58,890 --> 00:11:03,770
+عندي الـ Double Integral لـ R of XY DX DY زي زمان
+
+157
+00:11:03,770 --> 00:11:08,890
+اللي هي RDR Dθ يعني و كأننا بنقول الـ Polar
+
+158
+00:11:08,890 --> 00:11:13,330
+Transformation هو عبارةأو الـ polar اللي هي ال ..
+
+159
+00:11:13,330 --> 00:11:15,410
+ال .. ال .. ال double integral in polar
+
+160
+00:11:15,410 --> 00:11:20,410
+coordinates هي عبارة عن حالة خاصة من اللي هو الـ
+
+161
+00:11:20,410 --> 00:11:24,590
+general transformation اللي بنحكي عنها انه اللي هو
+
+162
+00:11:24,590 --> 00:11:29,290
+X بيساوي D و V و V و H بيساوي U و V هذي الآن U و V
+
+163
+00:11:29,290 --> 00:11:32,110
+هي الاروتيتر اللي قبل بشوية اذا صارت اللي هي عبارة
+
+164
+00:11:32,110 --> 00:11:36,470
+عن حالة خاصة من هذا التعويض يعني ما هي الا اللي هي
+
+165
+00:11:36,470 --> 00:11:40,300
+ال double integral عبارة عن issue التعويضاللي هو
+
+166
+00:11:40,300 --> 00:11:43,940
+لأ روتيتها وفق القانون اللي حكيناها قبل بشوية نيجي
+
+167
+00:11:43,940 --> 00:11:48,840
+أن نثبت المعلومة أكتر جالي example evaluate ال
+
+168
+00:11:48,840 --> 00:11:53,720
+double integral ل 2x ناقص y على 2 dx dy x بيساوي
+
+169
+00:11:53,720 --> 00:12:00,090
+على 2 زائد 1 و x بيساوي y على 2 و من 0 ل 4الان يا
+
+170
+00:12:00,090 --> 00:12:03,790
+جماعة by the following transformation يعني بيقول
+
+171
+00:12:03,790 --> 00:12:07,570
+evaluate هذا بواسطة اللي هو الـ Transformation
+
+172
+00:12:07,570 --> 00:12:13,350
+القاتى يعني اتعود عن X و Y وفق القوانين التالية
+
+173
+00:12:13,350 --> 00:12:18,620
+ايش ماخد الـ U؟ ماخد الـ U بسوء 2 X نقص Y على 2هو
+
+174
+00:12:18,620 --> 00:12:21,980
+الآن طبعا احنا ماعناش خبرة للتعويض لكن هو الشغلات
+
+175
+00:12:21,980 --> 00:12:25,480
+اللي سهلة ممكن احنا نعوض لحالنا لكن الصعبة اللي هي
+
+176
+00:12:25,480 --> 00:12:28,540
+طبعا و في امتحان و في كده أكيد بنكون اللي هو
+
+177
+00:12:28,540 --> 00:12:33,020
+بنعطيكم إياها ال U و ال V عشان تعوضوا أسهل يعني
+
+178
+00:12:33,020 --> 00:12:36,860
+لأن ممكن ما تقدروش توجدوها في امتحان لكن في بيت في
+
+179
+00:12:36,860 --> 00:12:40,720
+كده ممكن بعد شوية مع الميران و محل التمارين يصير
+
+180
+00:12:40,720 --> 00:12:45,900
+في عندكم إمكانية لإيش بدك تعوض عن U و عن V اللي هي
+
+181
+00:12:45,900 --> 00:12:50,080
+تستبدلها بمادةإذن هو استبدل الـ Integrand هذا كله
+
+182
+00:12:50,080 --> 00:12:54,740
+بـ 2x-y على 2 و استبدل الـ V بالـ y على 2 و الـ y
+
+183
+00:12:54,740 --> 00:12:58,140
+على 2 اللي عندي هنا و بيقول لي and integrate over
+
+184
+00:12:58,140 --> 00:13:01,000
+appropriate region in the UV plane يعني الآن
+
+185
+00:13:01,000 --> 00:13:06,000
+بيقوللي بالتعويض عن U عن x و y بدلالة اللي هو الـ
+
+186
+00:13:06,000 --> 00:13:10,860
+U و V اللي معطية عبر العلاقة التالية أو ددلة
+
+187
+00:13:10,860 --> 00:13:16,520
+double integral هذا in the UV planeقلنا أن هناك
+
+188
+00:13:16,520 --> 00:13:22,240
+تلاتة أشياء تتغير تتغير الـ Integrant و تتغير DX و
+
+189
+00:13:22,240 --> 00:13:27,180
+DY بالسير الجاكوبين في DU و DV وهذه الأشياء تتغير
+
+190
+00:13:27,180 --> 00:13:33,720
+الحدود الانتيجريشن من region في الـ XY إلى region
+
+191
+00:13:33,720 --> 00:13:39,440
+في الـ U و V و دعنا نشوف كل واحدة على حدةنأتي الآن
+
+192
+00:13:39,440 --> 00:13:45,240
+قبل أن نرسم الرسمة هذه نشوف الآن كيف بدنا اللي هو
+
+193
+00:13:45,240 --> 00:13:49,240
+N عوّض لأنه الرسمة بعد شوية هنستنتجها من كده بدنا
+
+194
+00:13:49,240 --> 00:13:53,780
+نعمل يا جماعة اللي هو الجدول التالي بس قبل أن نعمل
+
+195
+00:13:53,780 --> 00:13:58,460
+الجدول التالي خلّينا نجيب اللي هو X بدلالة U أو V
+
+196
+00:13:58,460 --> 00:14:04,100
+وY بدلالة U أو V هنا Y جاهزة Y بيساو 2V ناشي Y
+
+197
+00:14:04,100 --> 00:14:08,730
+بيساو 2V شايفينها Y بيساو 2Vإلى أن صارت Y بيساوي
+
+198
+00:14:08,730 --> 00:14:15,510
+2V عوض هنا بيصير 2V على 2 و 2X نقص 2V على 2
+
+199
+00:14:15,510 --> 00:14:19,250
+التنينات بروح مع بعض بيصير X نقص V بيساوي U يعني X
+
+200
+00:14:19,250 --> 00:14:24,030
+بيساوي U زائد V إذا عندي الآن حليت التنتين طلعت
+
+201
+00:14:24,030 --> 00:14:30,010
+عندي X بيساوي U زائد V و Y بيساوي إيش 2Vالان يا
+
+202
+00:14:30,010 --> 00:14:35,290
+جماعة اللي بده سويه كما يليه اللي بده سويه بده ادي
+
+203
+00:14:35,290 --> 00:14:39,190
+اللي هو اوددلك اللي هو حدود ال integration هذه
+
+204
+00:14:39,190 --> 00:14:44,090
+اشوفها احولها من ال X و ال Y الى ال U و ال V عبر
+
+205
+00:14:44,090 --> 00:14:47,650
+الجدول التالي و بعدين بتلاقي اني automatic حولتلك
+
+206
+00:14:47,650 --> 00:14:53,500
+اللي هي المنطقة R الى المنطقة اللي عنديأول حاجة لو
+
+207
+00:14:53,500 --> 00:14:58,120
+نجي على السريع X بيساوي Y على 2 هذا حدود ال X اللي
+
+208
+00:14:58,120 --> 00:15:01,660
+عند X بيساوي Y على 2 زي 1 X بيساوي Y على 2 يعني Y
+
+209
+00:15:01,660 --> 00:15:09,560
+بيساوي 2X Y بيساوي 2X هي Y بصفر و X مثلا ب2 بيصير
+
+210
+00:15:09,560 --> 00:15:15,920
+Y ب4 يعني نوصل بينهم هذا بيكون خط Y بيساوي 2X
+
+211
+00:15:19,320 --> 00:15:24,260
+الخط التاني X بيساوي Y زائد 2 على 2 زائد 1 يعني
+
+212
+00:15:24,260 --> 00:15:28,740
+نفس الخط هذا بس مزيوح بمقدار واحد فبصير اللي هو
+
+213
+00:15:28,740 --> 00:15:32,700
+عبارة عن لو جينا رسمناه بيصير Y على 2 بيساوي X
+
+214
+00:15:32,700 --> 00:15:36,940
+ناقص واحد يعني Y بيساوي 2 X ناقص 2 رسمليها بنفس
+
+215
+00:15:36,940 --> 00:15:42,300
+الطريقة خد اللي هو مثلاعندي اللي هو X عندي اللي هو
+
+216
+00:15:42,300 --> 00:15:46,480
+Y بصفر بتطلع X اتنين X بيساوي اتنين يعني X بيساوي
+
+217
+00:15:46,480 --> 00:15:50,600
+واحد بتيجي هنا النقطة خد الآن اللي هي Y مثلا
+
+218
+00:15:50,600 --> 00:15:54,800
+بتساوي اربعة بيصير اتنين X ناقص اتنين بيساوي اللي
+
+219
+00:15:54,800 --> 00:15:58,360
+هو اربعة يعني اتنين X بتساوي ستة يعني X بتساوي إيش
+
+220
+00:15:58,360 --> 00:16:01,700
+تلاتة يعني هذه النقطة اللي هي عند تلاتة و بنرسم
+
+221
+00:16:01,700 --> 00:16:05,960
+الخط هذا بنعرف نرسمه الآن هذه اللي هو الخط اللي
+
+222
+00:16:05,960 --> 00:16:11,110
+بيحدد لمينالخيام اللي هيش اللي هي قيمة x بتساوي
+
+223
+00:16:11,110 --> 00:16:14,710
+كده لعند قيمة x بتساوي كده هي على الخطين دول
+
+224
+00:16:14,710 --> 00:16:18,110
+اللاحظوا انه لو جينا ناخد عبر هدول الخطوط هيك
+
+225
+00:16:18,110 --> 00:16:22,970
+بموازية فعلا بنلاقي اللي هو ال x بتيجي من اتنين من
+
+226
+00:16:22,970 --> 00:16:27,210
+ال x بتطلع من y على اتنين لعند x بتساوي اللي هي
+
+227
+00:16:27,210 --> 00:16:31,270
+اللي هي عبارة عن y على اتنين زائد واحد فاهمين عليه
+
+228
+00:16:31,270 --> 00:16:34,650
+هذيك و هذيك زي ما كنا نعمل زيهااللي هان بده أحدد
+
+229
+00:16:34,650 --> 00:16:38,150
+المنطقة أكملها مين اللي بيكمل ليه اللي هو Y دي Y Y
+
+230
+00:16:38,150 --> 00:16:41,410
+بتقول من سفر لعند مين لعند أربع يعني Y بدينا بسفر
+
+231
+00:16:41,410 --> 00:16:45,970
+هيها بسفر وانتهينا بعند مين بعند أربع بنرسم الخط
+
+232
+00:16:45,970 --> 00:16:50,770
+بتبقى هذه هي المنطقة اللي قاعد بكمل عليها اللي هي
+
+233
+00:16:50,770 --> 00:16:57,050
+اسمها إيش اسمها R بده أحولها لمنطقة جي طيب اللي
+
+234
+00:16:57,050 --> 00:17:03,340
+قلنا التحويل كمالي اللي هو انه بتيجيالخط الأول X
+
+235
+00:17:03,340 --> 00:17:06,760
+بيسوا Y على 2 اللي قلنا عنه أيش بده يحصل في ال ..
+
+236
+00:17:06,760 --> 00:17:10,540
+اللي هو .. بده يحصل في ال .. في ال U و ال V؟ بنعوض
+
+237
+00:17:10,540 --> 00:17:15,020
+عن X بقيمتها اللي عبارة عن U زائد V و Y على 2
+
+238
+00:17:15,020 --> 00:17:19,660
+بقيمتها اللي هي V بيصير U زائد V بساوة V يعني V مع
+
+239
+00:17:19,660 --> 00:17:23,620
+V يعني U بساوة 0 إذا اتحول الخط المائل هذا ل U
+
+240
+00:17:23,620 --> 00:17:27,060
+بساوة 0 يعني الخط هي محلهااللي هو زي اللي بده
+
+241
+00:17:27,060 --> 00:17:31,080
+يعملني rectangle كمان شوية X بيصير Y على 2 زائد 1
+
+242
+00:17:31,080 --> 00:17:35,020
+بشيل ال X بحط U زائد V و ال Y على 2 اللي هي عبارة
+
+243
+00:17:35,020 --> 00:17:39,360
+عن V و هذا ال 1 فبصير U زائد V بيصير V زائد 1 ال V
+
+244
+00:17:39,360 --> 00:17:43,340
+مع ال V بيصير ال U أش بتساوي 1 إذا اللي هو ال DU
+
+245
+00:17:43,340 --> 00:17:47,800
+أو ال U صارت عبارة عن إيش؟ من 0 إلى 1 يعني مكان
+
+246
+00:17:47,800 --> 00:17:52,730
+اللي بشتغل على rectangular region Y بتساوي 0اللي
+
+247
+00:17:52,730 --> 00:17:57,750
+هي Y أشق منتها 2V 2V بسيط السفر إذا V بتساوى 0
+
+248
+00:17:57,750 --> 00:18:02,010
+هيحدد اللي هو ال V الخط التان Y بساوى 4 اللي هو 2V
+
+249
+00:18:02,010 --> 00:18:05,970
+بساوى 4 يعني ال V بتساوى 2 إذا هذه الآن اللي هي
+
+250
+00:18:05,970 --> 00:18:10,310
+حدود المنطقة U و V U بتساوى 0 احفظوها U بتساوى 0 و
+
+251
+00:18:10,310 --> 00:18:14,510
+U بتساوى 1 U بتساوى 0 و U بتساوى 1 U بتساوى 0 هي U
+
+252
+00:18:14,510 --> 00:18:21,820
+بتساوى 0 ماشي الحال هيها U بتساوى 0و U سوا واحد و
+
+253
+00:18:21,820 --> 00:18:25,060
+بعدين V بالساوة سفر و V بتساوة اتنين فعملتلي ال
+
+254
+00:18:25,060 --> 00:18:29,160
+region هذه الحلوة سهلة التكامل يعني بيصير على طول
+
+255
+00:18:29,160 --> 00:18:32,940
+حدود ال integration الآن بدل ما هي كانت اللي هي
+
+256
+00:18:32,940 --> 00:18:36,420
+بالشكل هذا على طول بيصير اشملها جبل اللي لدك حدود
+
+257
+00:18:36,420 --> 00:18:40,820
+ال integration U اللي بده كامل ل U جبل U من سفر
+
+258
+00:18:40,820 --> 00:18:43,860
+لعند الواحد لأنه rectangle من سفر لعند الواحد و V
+
+259
+00:18:43,860 --> 00:18:48,430
+من عند سفر لعند من عند الاتنينالان خلصت من اللي هي
+
+260
+00:18:48,430 --> 00:18:52,410
+المنطقة اللي بتغلب الان ضال علي اوجد من اتنين X
+
+261
+00:18:52,410 --> 00:18:55,730
+minus Y على اتنين هذا ما عوضنا ما كان الاسم منها U
+
+262
+00:18:55,730 --> 00:18:59,930
+ضال الان DX DY اشمل جاكوبيا ال absolute value في
+
+263
+00:18:59,930 --> 00:19:04,450
+DU DVالان كيف بدأ اوجد الجاكوبيان هذا الجاكوبيان
+
+264
+00:19:04,450 --> 00:19:09,530
+زي ما قلنا بساوي دل x by دل u دل x by دل v دل y by
+
+265
+00:19:09,530 --> 00:19:14,730
+دل u دل y by دل v دل x by دل u اللي هو عبارة عن
+
+266
+00:19:14,730 --> 00:19:20,090
+واحد صح ولا لا ايه واحد دل x by دل v دل x by دل v
+
+267
+00:19:20,090 --> 00:19:26,990
+برضه واحد دل y by دل u دل y by دل u سفر دل y by دل
+
+268
+00:19:26,990 --> 00:19:30,840
+v اتنينإذا هاي هدولة حددين اللي هي على طول
+
+269
+00:19:30,840 --> 00:19:33,180
+الجاكوبيا يعني هو واحد في اتنين باتنين نقص صفر
+
+270
+00:19:33,180 --> 00:19:37,080
+اللي بطلع إيه إتنين إذا بشيل اللي هو الجاكوبيا و
+
+271
+00:19:37,080 --> 00:19:42,480
+بحط مكانه اتنين و هاي U المكان دي دي U دي V لأن
+
+272
+00:19:42,480 --> 00:19:45,740
+absolute value اتنين هو نفسه اتنين بنكمل هذه U
+
+273
+00:19:45,740 --> 00:19:49,220
+اتنين U بدنا نكمل بالنسبة ل U بيصير اتنين U تربيع
+
+274
+00:19:49,220 --> 00:19:52,480
+على اتنين يعني U تربيع من صفر لواحد بتطلع عبارة عن
+
+275
+00:19:52,480 --> 00:19:56,490
+واحدالان كله دي V بيطلع عبارة عن V من صفر ل 2
+
+276
+00:19:56,490 --> 00:20:00,410
+بيطلع عند 2 و بيكون هي قيمة ال integration و كان
+
+277
+00:20:00,410 --> 00:20:05,170
+سهل اللي هو ال integration بعد ما حولناهناخد مثال
+
+278
+00:20:05,170 --> 00:20:08,250
+آخر evaluate الـ double integral من 0 ل 1 نقص x و
+
+279
+00:20:08,250 --> 00:20:13,910
+من 0 ل عند 1 الجدر التربيع إلى x زائد u y نقص 2x
+
+280
+00:20:13,910 --> 00:20:18,830
+لكل تربيع dy dx طبعا الآن اللي هي المنطقة اللي هي
+
+281
+00:20:18,830 --> 00:20:22,230
+بدنا نعمل evaluation لهذه بدنا نحدد المنطقة و
+
+282
+00:20:22,230 --> 00:20:27,390
+بعدين نحدد u و ال V و بعدين نشيل هدولة و نحط مكان
+
+283
+00:20:27,390 --> 00:20:31,550
+مدلة u و v و نشيل ال dy و y dx و نحط مكان جاكوبيا
+
+284
+00:20:31,550 --> 00:20:35,850
+في du و dvطيب ندور الاشياء على المنطقة اللي هو
+
+285
+00:20:35,850 --> 00:20:40,730
+عندي ال Y بتساوي سفر لعند Y بيساوي 1 نقص X Y
+
+286
+00:20:40,730 --> 00:20:45,830
+بتساوي سفر لعند 1 يعني لعند Y بيساوي 1 نقص X يعني
+
+287
+00:20:45,830 --> 00:20:51,540
+X زائد Y بيساوي 1 هذي لأن Y بتساوي سفر هيلأن من
+
+288
+00:20:51,540 --> 00:20:56,160
+فوق ايش بيكون X زي Y بيسوا 1 X زي Y بيسوا 1 هي X
+
+289
+00:20:56,160 --> 00:21:03,020
+بسفر Y بسفر X بواحد X بسفر Y بواحد فبصير هذا الخط
+
+290
+00:21:03,020 --> 00:21:06,340
+المستقيم اللي عندي فبصير Y فعلا Y جاي من عند ال
+
+291
+00:21:06,340 --> 00:21:11,530
+zero لعند واحد نقص Xنأتي الآن لمنطقة هذه جاي نلاقي
+
+292
+00:21:11,530 --> 00:21:14,150
+إيش ال X عنده من سفر لعند واحد بدأ من سفر لعند
+
+293
+00:21:14,150 --> 00:21:17,310
+واحد لأن هذا بيظل ممتد هيك و هيك طب إيش حدوده اللي
+
+294
+00:21:17,310 --> 00:21:21,070
+بدوا حددلناها اللي هو قيمة X X بيقولي من عند سفر
+
+295
+00:21:21,070 --> 00:21:24,230
+إذا نحط الخط هذا اللي عند واحد و أحط الخط هذا
+
+296
+00:21:24,230 --> 00:21:28,810
+فبتطلع المنطقة اللي أمامي هذه لأن هذه المنطقة بدنا
+
+297
+00:21:28,810 --> 00:21:34,150
+نحولها لمين لمنطقة جي اللي في ال U و ال V الان لو
+
+298
+00:21:34,150 --> 00:21:39,000
+تطلعنا على ال integration اللي أمامناالان امامنا
+
+299
+00:21:39,000 --> 00:21:43,920
+خيارات ان ناخد هذا كل يوموهذا كله V لو أخدته هك
+
+300
+00:21:43,920 --> 00:21:47,220
+هتلاقي اللي هو بجرب لحالك هتلاقي الأمور اللي هو
+
+301
+00:21:47,220 --> 00:21:52,140
+بدت تتعدد وحيصير عندك اللي هو الوضع صعب فالأن اللي
+
+302
+00:21:52,140 --> 00:21:56,620
+جوا هذا باخده اللي هو U بعين الله بقدر أكمل جدر ال
+
+303
+00:21:56,620 --> 00:22:01,140
+U مش صعب وهذا V و بقدر أكمل ال V تربيع ومش صعب خد
+
+304
+00:22:01,140 --> 00:22:05,500
+ال U بيساوي X زي Y و ال V بيساوي Y نقص 2X الان
+
+305
+00:22:05,500 --> 00:22:10,150
+هدول حلهم مع بعض عشان تجيب X وUالان لو اجهد ضربت
+
+306
+00:22:10,150 --> 00:22:14,950
+اللي هي هادف اتنين بيصير اتنين U زائد اتنين X زائد
+
+307
+00:22:14,950 --> 00:22:20,370
+اتنين Y مظبوط واجمح لبعض بيصير اتنين U زائد اللي
+
+308
+00:22:20,370 --> 00:22:26,150
+هو V اتنين U زائد V بساوي اللي هو اتنين Y زائد Y
+
+309
+00:22:26,150 --> 00:22:31,530
+اللي هي تلتة Y بيصير اللي هو اتنين U زائد Vعن طريق
+
+310
+00:22:31,530 --> 00:22:41,290
+عبارة عن عبارة عن عبارة عن عبارة عن عبارة عن عبارة
+
+311
+00:22:41,290 --> 00:22:49,150
+عناللي هو اتنين .. اتنين U بساوي اللي هو اتنين
+
+312
+00:22:49,150 --> 00:22:53,990
+طبعا زائد V نسينا اتنين U زائد V بساوية اتنين Y
+
+313
+00:22:53,990 --> 00:22:58,610
+زائد ال Y تلاتة Y الان بدي قيمة ال Y بصير Y بتساوي
+
+314
+00:22:58,610 --> 00:23:05,140
+اجسم هدولة على اللي هوتلاتة و هذا على تلاتة فبصير
+
+315
+00:23:05,140 --> 00:23:09,240
+عندك y بيساوي اتنين u على تلاتة زائد v على تلاتة
+
+316
+00:23:09,240 --> 00:23:14,300
+الان بدي احصل على mean على قيمة x يعني بدي اطيّر
+
+317
+00:23:14,300 --> 00:23:19,210
+mean y اللي هو اضرب هذا في ناقص كله في ناقصأو اضرب
+
+318
+00:23:19,210 --> 00:23:22,450
+هذه في ناقص زي ما بدك مش مشكلة كثير لو ضربت هذه
+
+319
+00:23:22,450 --> 00:23:26,930
+كلها في ناقص بيصير ناقص V بيساوي ناقص Y زاد اتنين
+
+320
+00:23:26,930 --> 00:23:30,830
+X و اترحى او اجمحوا مع بعض دولة بعد الضرب بيصير U
+
+321
+00:23:30,830 --> 00:23:36,910
+ناقص V بساوي تلاتة X U ناقص V بيساوي تلاتة X X مع
+
+322
+00:23:36,910 --> 00:23:40,190
+تلاتة بيصير X بيساوي U على تلاتة ناقص V على تلاتة
+
+323
+00:23:40,190 --> 00:23:45,270
+و بتيوجد اللي هي X و Y وهذا اللي هو الحل انهين
+
+324
+00:23:45,270 --> 00:23:50,610
+سهلات زي ما شفنالان جيبنا x و y بدلالة ال u و ال v
+
+325
+00:23:50,610 --> 00:23:57,550
+لان بيصير لان طبعا عملته هيك ليش عشان اللي هو أسهل
+
+326
+00:23:57,550 --> 00:24:01,090
+على حالي إيجاد المنطقة و أسهل على حالي إيجاد
+
+327
+00:24:01,090 --> 00:24:05,240
+الجاكوبينطيب خلّينا نشوف أول إشي اللي هي الخطوط
+
+328
+00:24:05,240 --> 00:24:08,400
+هذه إيش بدي تشير عندها أول إشي X زائد Y بيساوي
+
+329
+00:24:08,400 --> 00:24:12,920
+واحد X زائد Y بيساوي واحد عوض عن X بقيمتها U ع
+
+330
+00:24:12,920 --> 00:24:17,360
+تلاتة نقص V ع تلاتة زائد Y اللي هي 2U ع تلاتة زائد
+
+331
+00:24:17,360 --> 00:24:21,500
+V ع تلاتة اللي هي هذه بتروح مع هذه بيصير اللي هو
+
+332
+00:24:21,500 --> 00:24:26,820
+2U U تلاتة U ع تلاتة يعني U بساوي واحد خلصنا منها
+
+333
+00:24:27,210 --> 00:24:30,090
+لأن A التي هي X بيساوي سفر لأن الخط هذا صار U
+
+334
+00:24:30,090 --> 00:24:33,910
+بيساوي إيش واحد ماحله و ما يسأله X بتساوي سفر
+
+335
+00:24:33,910 --> 00:24:37,990
+بيصير اللي هو U على ثلاثة نقص V على ثلاثة بتساوي
+
+336
+00:24:37,990 --> 00:24:41,490
+سفر انجل هذا على الجهة التانية بيبقوا ضروف ثلاثة
+
+337
+00:24:41,490 --> 00:24:47,890
+بيصير V بتساوي U و V بتساوي U سهل هاي V بتساوي إيش
+
+338
+00:24:47,890 --> 00:24:53,500
+U ماشيالان Y بيساوي سفر ايش بيساوي Y بيساوي سفر
+
+339
+00:24:53,500 --> 00:24:58,080
+اللي هو عبارة عن اللي هو 2U على 3 زي ال V على 3
+
+340
+00:24:58,080 --> 00:25:01,700
+بيساوي سفر او اللي انا اجسم على 3 او ورنجل بيصير U
+
+341
+00:25:01,700 --> 00:25:07,120
+V بيساوي ناقص 2U V بيساوي ناقص 2U هي اللي هي V
+
+342
+00:25:07,120 --> 00:25:11,700
+بيساوي ناقص 2U لو عوضت منه وتين واحدة بسفر وتانية
+
+343
+00:25:11,700 --> 00:25:16,330
+هتطلعلك اللي هو هذهوهذا الان U ايش بيساوي U بيساوي
+
+344
+00:25:16,330 --> 00:25:19,850
+واحد اللي طلع عندي ماشي الحل اذا المنطقة المحصورة
+
+345
+00:25:19,850 --> 00:25:23,830
+هيها المنطقة هذه V بيساوي U و V بيساوي نقص اتنين U
+
+346
+00:25:23,830 --> 00:25:27,990
+طبعا بتعرفوا ترسموها هذه اكيد اللي هو U بواحد
+
+347
+00:25:27,990 --> 00:25:32,830
+بيصير V بيساوي نقص اتنين هيها الان و هذي U واحد و
+
+348
+00:25:32,830 --> 00:25:36,670
+V واحد انزلوا بعض و هذي Zero Zero اذا هي المنطقة
+
+349
+00:25:36,670 --> 00:25:40,720
+المذكورةالان اذا انا اريد كامل بالنسبة ل V قبل
+
+350
+00:25:40,720 --> 00:25:44,940
+مثلا فبصير V جاي من وين لوين بطلع V بالساوية نقص
+
+351
+00:25:44,940 --> 00:25:49,240
+2U لعند U الان طب ال U كلها من وين بتيجي ال U
+
+352
+00:25:49,240 --> 00:25:52,920
+بتاخد من عند 0 لعند 1 تحديد حدود ال integration
+
+353
+00:25:52,920 --> 00:25:57,260
+سهل الآن طيب الان قبل نوجد مين يا جماعة الجاكوبين
+
+354
+00:25:57,260 --> 00:26:02,610
+هاي ال X وهي ال U دي ال X by دي ال U تلتو del x by
+
+355
+00:26:02,610 --> 00:26:08,210
+del v ناقص تلت هاي السطر الأول تلت و ناقص تلت نيجي
+
+356
+00:26:08,210 --> 00:26:12,190
+ل del y by del u del y by del u del y by del u
+
+357
+00:26:12,190 --> 00:26:17,310
+تلتين و del y by del v تلت اذا اللي هو بيصير عندى
+
+358
+00:26:17,310 --> 00:26:22,270
+تلتين تلتين و تلت هذا ال Jacobian بوجد قيمته تسعة
+
+359
+00:26:22,270 --> 00:26:27,360
+ناقص اللي هي بيصير زائدتسعين هو تلاتة ع تسعة يعني
+
+360
+00:26:27,360 --> 00:26:30,440
+تلت إذا هذا الجاكوبيان إذا صار ال integration هذا
+
+361
+00:26:30,440 --> 00:26:37,280
+الآن سهل المنال هذه dy dx باعوضها dv du مع مين مع
+
+362
+00:26:37,280 --> 00:26:40,820
+الجاكوبيان الجاكوبيان جداش قلنا تلت إذا تصير تلت
+
+363
+00:26:40,820 --> 00:26:45,600
+dv duالان ال .. ال .. هذه اللي هي زي ما عوضنا هذه
+
+364
+00:26:45,600 --> 00:26:48,860
+اللي هي عبارة عن U أص نص .. U هذه فبصير general U
+
+365
+00:26:48,860 --> 00:26:52,900
+يعني U أص نص و هذه قولنا V تربيع هذه V و اللي برا
+
+366
+00:26:52,900 --> 00:26:55,700
+اتنين بيصي .. بيقص اتنين بيصي V تربيع لأن هذا
+
+367
+00:26:55,700 --> 00:26:59,320
+تكامله سهل بس قبل ما نكامله ممعود تلت في U أص نص
+
+368
+00:26:59,320 --> 00:27:02,260
+في V تربيع اللي هي حدود ال integration زي ما قولنا
+
+369
+00:27:02,530 --> 00:27:07,590
+الـ V بتيجي من عند نقص 2U لعند الـ U و الـ U بتروح
+
+370
+00:27:07,590 --> 00:27:11,070
+من 0 لعند 1 زي ما حكينا قبل شوية اللي هم بنكامل
+
+371
+00:27:11,070 --> 00:27:16,110
+بالنسبة لـ V بيصير اللي هو V تكعيب على اللي هو من
+
+372
+00:27:16,110 --> 00:27:20,830
+3 يعني و في تلت برة طلعناها بيصير V تكعيب على 3 من
+
+373
+00:27:20,830 --> 00:27:25,380
+V من نقص 2U لعند الـ Uبنعوض عنها بتطلع عندي عبارة
+
+374
+00:27:25,380 --> 00:27:29,700
+عن اللي هو U أص نص في U زي ما هي في U تكعيب زائد
+
+375
+00:27:29,700 --> 00:27:35,580
+تمانية U تكعيب DU هذه اللي هو الآن بنضربها جوا
+
+376
+00:27:35,580 --> 00:27:42,120
+الآن بيصير اللي هي عندي المقدارعندي U نص في U
+
+377
+00:27:42,120 --> 00:27:48,440
+تكعيب تسعة U تكعيب في U تسعة وفي عندي اللي هو تسعة
+
+378
+00:27:48,440 --> 00:27:51,260
+برا بروحهم مع بعض تسعة فوق و تسعة أخر بروحهم مع
+
+379
+00:27:51,260 --> 00:27:55,200
+بعض بيصير U تكعيب في نص بيصير U سبعة على اتنين من
+
+380
+00:27:55,200 --> 00:27:58,080
+سفر اللي عنده واحد دي U بنكملها بيصير سبعة على
+
+381
+00:27:58,080 --> 00:28:00,940
+اتنين زائد واحد اللي هي يعني تسعة على اتنين U تسعة
+
+382
+00:28:00,940 --> 00:28:03,720
+اتنين اللي هو مجسومة على تسعة على اتنين يعني
+
+383
+00:28:03,720 --> 00:28:07,180
+مضروبة في تسعين فاهمين من سفر اللي عنده واحد اللي
+
+384
+00:28:07,180 --> 00:28:13,660
+هو بساوي تسعينو هيكون أو جدنية اللي هي القيمة نيجي
+
+385
+00:28:13,660 --> 00:28:17,860
+ناخد value of the integral مثال آخر الجدر التربيعي
+
+386
+00:28:17,860 --> 00:28:22,600
+ل Y على X يتعدى جدر X في Y دي X دي Y من واحد على Y
+
+387
+00:28:22,600 --> 00:28:27,140
+ل Y من واحد ل مين ل نص طبعا الجثة انفهمت اللي هو
+
+388
+00:28:27,140 --> 00:28:35,290
+بدنا اللي هو نعوض عن U Vو بدنا اللي هو ان نعوذ
+
+389
+00:28:35,290 --> 00:28:39,450
+عن DX DY بقى Jakobian في D و DZ و بدنا نعوذ عن
+
+390
+00:28:39,450 --> 00:28:45,860
+المنطقة هذه من R إلى Gالان اول اشي بنحدد المنطقة
+
+391
+00:28:45,860 --> 00:28:49,600
+هذه خلينا نحددها مش مشكلة لو اول اشي عوضت من فعل
+
+392
+00:28:49,600 --> 00:28:53,220
+او حددت المنطقة هذه خلينا نحدد المنطقة هذه X بساوة
+
+393
+00:28:53,220 --> 00:28:57,380
+واحد على Y يعني X في Y بتساوة واحد X بساوة Y في Y
+
+394
+00:28:57,380 --> 00:29:00,880
+هي بتساوة واحد هي هذه رسم التعارفين على الجهة ده
+
+395
+00:29:00,880 --> 00:29:03,260
+وعلى الجهة الثانية بس على المنطقة الموجودة اللي
+
+396
+00:29:03,260 --> 00:29:08,210
+عندناهلقيت ما اقول لك ليش؟ الان عندي Y X بتساوي Y
+
+397
+00:29:08,210 --> 00:29:12,950
+X بتساوي Y هايه X بتساوي Y شايفينه؟ هاي اللي هو
+
+398
+00:29:12,950 --> 00:29:17,830
+الجهة الثانية الان هذه بتمتد هيكة و بتمتد هيكة و
+
+399
+00:29:17,830 --> 00:29:21,090
+عالجهة الثانية برضه نفس الشيطان من من هنا جلي اللي
+
+400
+00:29:21,090 --> 00:29:25,230
+هو Y من واحديعني y ابدأ من عند واحد لأنه نقطة
+
+401
+00:29:25,230 --> 00:29:28,430
+التقاطع هنا دولة واحد وسهل انك تعرفها واحد لأن y
+
+402
+00:29:28,430 --> 00:29:32,170
+بيساوي x بيصير اللي هو او اضمحها بيصير y تربع
+
+403
+00:29:32,170 --> 00:29:34,130
+بيساوي واحد يعني y بيساوي زادة ونقص واحد يعني y
+
+404
+00:29:34,130 --> 00:29:36,830
+بيساوي واحد هنا وx بيساوي واحد إذا هذا نقطة
+
+405
+00:29:36,830 --> 00:29:39,570
+التقاطع إذا لما تقول y بتساوي واحد هيها من نقطة
+
+406
+00:29:39,570 --> 00:29:43,150
+التقاطع لعند y بتساوي اتنين هي المستقيم إذا هي
+
+407
+00:29:43,150 --> 00:29:48,710
+المنطقة اللي بينهين ماشي الحال طيبلأن هذه المنطقة
+
+408
+00:29:48,710 --> 00:29:51,830
+اللي بينهم اللي هي منطقتنا بدنا نعملها
+
+409
+00:29:51,830 --> 00:29:53,690
+transformation و بعد شوية بنشوف نعملها
+
+410
+00:29:53,690 --> 00:29:57,030
+transformation بس بقى نعرف عن إيش بنعوض الآن لو
+
+411
+00:29:57,030 --> 00:30:00,430
+واحد جالي زي ما عملنا قبل شوية بتعوض X في Y لجوا
+
+412
+00:30:00,430 --> 00:30:05,690
+أس في V بيصير هذه E to the V أس نص أيش اللي كملها
+
+413
+00:30:06,210 --> 00:30:10,710
+أه نقدرش نكون عجبناها اكتر اذا منطقي ان انا اخد كل
+
+414
+00:30:10,710 --> 00:30:16,770
+الجدر ال X Y ايش ماله بساوي اللي هو U الان و هذا Y
+
+415
+00:30:16,770 --> 00:30:21,950
+على X واخدوا الجدر ايش بساوي اللي هو HV فبصير V E
+
+416
+00:30:21,950 --> 00:30:25,450
+to the U هذا ثابت V و E to the U سهل التكامل و هذا
+
+417
+00:30:25,450 --> 00:30:29,110
+سهل التكامل فبشجعني على ان انا اخد التعويض اللي
+
+418
+00:30:29,110 --> 00:30:31,910
+اخدته خليني اشوف ان انا اخد التعويض اللي اخدته لان
+
+419
+00:30:31,910 --> 00:30:36,560
+ده اقعد اشتغل فادي هيك مش عارف اكملهاالان هذا الان
+
+420
+00:30:36,560 --> 00:30:40,000
+بدي أجيب قيمة x و y ربّع أجهتين بيصير u تربيع
+
+421
+00:30:40,000 --> 00:30:44,560
+بيساوي xy و هذيك ربحها بيصير v تربيع y على x الان
+
+422
+00:30:44,560 --> 00:30:48,020
+اضربلي هذه في هذه بيصير u تربيع v تربيع بيساوي y
+
+423
+00:30:48,020 --> 00:30:52,920
+تربيع يعني y بتساوي u في v الان هدول أجسمها على
+
+424
+00:30:52,920 --> 00:30:56,660
+بعض هذه أجسمها على هذه بيصير u تربيع على v تربيع
+
+425
+00:30:56,660 --> 00:31:00,020
+بتساوي x تربيع لإن ال y هتروح مع مين مع ال y و
+
+426
+00:31:00,020 --> 00:31:02,700
+تطلع هذه ال x فوق و تضرب في ال x التانية بيصير x
+
+427
+00:31:02,700 --> 00:31:08,280
+تربيعوهذه بتعطيني x ايش بتساوي u على v x بتساوي u
+
+428
+00:31:08,280 --> 00:31:11,860
+على v إذن هاي ال transformations اللي أنا بتعود
+
+429
+00:31:11,860 --> 00:31:17,840
+بواسطتهم بتعود عن x ب u على vأه و Y بنعوض و U في V
+
+430
+00:31:17,840 --> 00:31:22,500
+حسب اللي عوضناها انا من الاستقناء منها بنكون اللي
+
+431
+00:31:22,500 --> 00:31:25,740
+هو بنحصل على اللي بدناها نوجد الجاكوبيا لإن هو
+
+432
+00:31:25,740 --> 00:31:30,020
+يوجد ده هي لزمنا الجاكوبيا ان هو دل X by دل U دل X
+
+433
+00:31:30,020 --> 00:31:34,420
+by دل U دل X by دل V دل Y by دل U دل Y by دل V زي
+
+434
+00:31:34,420 --> 00:31:38,350
+ما احفظناها الآندل X by دل U هذه عبارة عن واحد على
+
+435
+00:31:38,350 --> 00:31:43,810
+V دل X by دل V ناقص U لأنها ثابت على V تربيع دل Y
+
+436
+00:31:43,810 --> 00:31:48,550
+by دل U اللي هو عبارة عن V دل Y by دل V اللي هي
+
+437
+00:31:48,550 --> 00:31:55,180
+عبارة عن Uلنضرب طرفين اللي هو U على V ناقص اللي
+
+438
+00:31:55,180 --> 00:32:00,380
+بيصير زائد U على V لإن مع ال V تربية بتروح بيظل U
+
+439
+00:32:00,380 --> 00:32:02,960
+على V ناقص و في عندنا ناقص بيصير زائد و بعدها
+
+440
+00:32:02,960 --> 00:32:07,610
+بيصير تنتين U على Vإذاً هذا قيمة من الـ Jacobian.
+
+441
+00:32:07,750 --> 00:32:11,590
+إذاً أمورنا الآن صارت سهلة اللي هي الـ R و هي الـ
+
+442
+00:32:11,590 --> 00:32:16,530
+G ال region بعد الـ Transformation. لأن جدر ال Y
+
+443
+00:32:16,530 --> 00:32:20,950
+على X هو E to the جدر X Y D X D Y. الآن شيل ال Y
+
+444
+00:32:20,950 --> 00:32:27,090
+على X اللي سمنها شحن V وهو E to the U و2U على V ال
+
+445
+00:32:27,090 --> 00:32:31,590
+V بتروح مع ال U فبصير 2U E to the U DU DV وهذا
+
+446
+00:32:31,590 --> 00:32:36,030
+مشهور بال integration by parts بتحلو اللي هووبنجده
+
+447
+00:32:36,030 --> 00:32:39,730
+خلّينا نشوف وصلنا هذه بتساوي هذه الآن ضلنا احدد
+
+448
+00:32:39,730 --> 00:32:43,790
+مين حدود ال integration هذه R هي هذه الآن بدي جي
+
+449
+00:32:43,790 --> 00:32:47,790
+اللي هي هذه لنشوف كيف بنحصلها أكيد كلكم عارف كيف
+
+450
+00:32:47,790 --> 00:32:52,580
+ايش بتساوي طبعا التحصيل الآناللي هو مش هعمل جدول
+
+451
+00:32:52,580 --> 00:32:55,580
+خليني احنا نقولها شفا و مزام هيك مش هعمل جدول بس
+
+452
+00:32:55,580 --> 00:32:59,180
+احنا بنعمل جدول لحظة يا جماعة الان y بيساوة اتنين
+
+453
+00:32:59,180 --> 00:33:02,020
+y بيساوة اتنين ايش هيصير y بيساوة اتنين اللي هو
+
+454
+00:33:02,020 --> 00:33:04,840
+بيصير y بيساوة اتنين بيصير u في v بيساوة اتنين
+
+455
+00:33:04,840 --> 00:33:09,900
+يعني uv بيساوة اتنين اللي هو u في v بيساوة اتنين
+
+456
+00:33:09,900 --> 00:33:13,840
+اللي هو برضه ايش زي ghost زي x y بيساوة واحد و u v
+
+457
+00:33:13,840 --> 00:33:16,940
+بيساوة اتنين هيه وعالجهة التانية بس احنا ايش منطقة
+
+458
+00:33:16,940 --> 00:33:21,120
+الوضUV بيساوي ايش بيساوي اتنين هاي U هذا الخط
+
+459
+00:33:21,120 --> 00:33:25,760
+الأصفر يتحول لهذا الخط الأصفر اللي هان الان X في Y
+
+460
+00:33:25,760 --> 00:33:31,540
+بيساوي واحد خد ال X في Y اللي هو U على V في U في V
+
+461
+00:33:31,540 --> 00:33:34,760
+ال V بتروح مع ال V بيصير U تربيع بيساوي واحد يعني
+
+462
+00:33:34,760 --> 00:33:40,340
+U بيساوي واحد U بتساوي واحد اللي هي عبارة عن هاي U
+
+463
+00:33:40,340 --> 00:33:43,560
+بيساوي واحد الأحمر بدل اللي هو الخط هذا المائل
+
+464
+00:33:43,560 --> 00:33:46,950
+اللي هو الأحمربنشوف الأزرق هذا y بيساوي 2 و y
+
+465
+00:33:46,950 --> 00:33:52,010
+بيساوي 2 اللي هي بيصير u انا ايهش هذا الخط الأصفر
+
+466
+00:33:52,010 --> 00:33:55,130
+لأ عملنا الخط الأصفر الأزرق هذا الأزرق y بيساوي x
+
+467
+00:33:55,130 --> 00:33:59,550
+y بيساوي x y بيساوي x هذي ساوي هذي uv بيساوي u على
+
+468
+00:33:59,550 --> 00:34:04,150
+v اللي هو بيصير عندك اللي هو ال u مع ال u بتروح
+
+469
+00:34:04,150 --> 00:34:09,250
+بيصير v تربيع بالساوي اللي هو 2 يعني اللي هو v
+
+470
+00:34:09,250 --> 00:34:13,390
+تربيع بيساوي 2 ولا قداش بيصيرأحنا قلنا ايش هذا y
+
+471
+00:34:13,390 --> 00:34:17,390
+بساوة x y بساوة x بنسويهم البعض y بصير u على v
+
+472
+00:34:17,390 --> 00:34:21,390
+بساوة u في v لأ واحد u مع ال u بتروح بصير اللي هو
+
+473
+00:34:21,390 --> 00:34:24,730
+واحد على v بساوة v يعني v تربيع بساوة واحد يعني v
+
+474
+00:34:24,730 --> 00:34:27,950
+بتساوة واحد فاهمين علي ان شاء الله v بتساوة واحد
+
+475
+00:34:27,950 --> 00:34:31,510
+اللي هو ايش هذا الخط الأحمر إذن هي المنطقة اللي ما
+
+476
+00:34:31,510 --> 00:34:35,220
+اللي مطلوبةالان بدنا نكامل بالنسبة ل U قبل اذا كنا
+
+477
+00:34:35,220 --> 00:34:39,040
+نكامل بالنسبة ل U ناخد موازي فبصير U جا من عند
+
+478
+00:34:39,040 --> 00:34:45,580
+اللي هو واحد لعند اللي هو اتنين على V اه و ال V من
+
+479
+00:34:45,580 --> 00:34:49,520
+عند الواحد لعند اخر اش ايش اتنين و اذا بدنا نكامل
+
+480
+00:34:49,520 --> 00:34:53,380
+العكس بناخد اللي هو اول اش باتجاه هذا اللي بتصير V
+
+481
+00:34:53,380 --> 00:34:58,220
+من عند واحد لعند اتنين على V اي منهم بصير الان هو
+
+482
+00:34:58,220 --> 00:35:03,700
+مكامل بالنسبة ل مين قبلهي اعوض اتنين U E to the U
+
+483
+00:35:03,700 --> 00:35:08,540
+لان بدنا اللي هو نوجد حدود ال integration الان DV
+
+484
+00:35:08,540 --> 00:35:13,480
+V من واحد لعندي اتنين على U الخطة العمودي هذا من
+
+485
+00:35:13,480 --> 00:35:16,900
+واحد لاتنين على U U من واحد لعندي اتنين لعندي
+
+486
+00:35:16,900 --> 00:35:20,860
+اتنين U من واحد لعندي اتنين ارجعلكم U من عند واحد
+
+487
+00:35:20,860 --> 00:35:26,140
+هي U من عنداللي هي بتبدأ الادة طبعا V بتمشي هيك
+
+488
+00:35:26,140 --> 00:35:30,060
+تيجي من عند الواحد و بيصير لعند U V V بيسوا اتنين
+
+489
+00:35:30,060 --> 00:35:34,260
+يعني V بيسوا اتنين على U اذا V من عند واحد لاتنين
+
+490
+00:35:34,260 --> 00:35:37,540
+على U و كل هنا دول الموازيات اللي أخدناها هنا
+
+491
+00:35:37,540 --> 00:35:43,080
+هيكون البدن من عند واحد لعند ايش اتنينمن واحد عند
+
+492
+00:35:43,080 --> 00:35:47,860
+اتنين اللي هبنكمل تصبح اتنين U تكملها بالنسبة ل V
+
+493
+00:35:47,860 --> 00:35:52,940
+V U E to the U من اتنين علي U لعند الواحد بنعوض عن
+
+494
+00:35:52,940 --> 00:35:58,520
+V اللي هي بتصبح اتنين E to the U ناقص اللي هي لما
+
+495
+00:35:58,520 --> 00:36:01,080
+نعوض عليها اتنين علي U لما بتروح اليوم علي U تصبح
+
+496
+00:36:01,080 --> 00:36:06,820
+اتنين E to the U ناقص اللي هو اللي هنا V بواحدالتي
+
+497
+00:36:06,820 --> 00:36:10,960
+هي u في e to the u du هو يساوي اتنين نقص u to the
+
+498
+00:36:10,960 --> 00:36:13,600
+u هذا integration by parts بيطلع عندى اللى هو
+
+499
+00:36:13,600 --> 00:36:17,640
+المقدار هذا احنا مش هنطلبكم شغلات صعبة زيك او مش
+
+500
+00:36:17,640 --> 00:36:20,340
+مستذكرينها لكن ال integration by parts يعني ممكن
+
+501
+00:36:20,340 --> 00:36:23,940
+واحد يصادقها بسرعة لكن نحاول لما نعجبكم مش يعني
+
+502
+00:36:24,250 --> 00:36:27,950
+لأنه يعني بيبقى أقصد أنه أنه بنحاول أنه مش في
+
+503
+00:36:27,950 --> 00:36:30,510
+أختبركم في integration by parts لأ بدي أختبركم في
+
+504
+00:36:30,510 --> 00:36:34,350
+ال transformation ويساوى 2 اللي هو زي ما قلنا بعد
+
+505
+00:36:34,350 --> 00:36:37,450
+ما عوضنا عملنا integration by parts لحد التكامل
+
+506
+00:36:37,450 --> 00:36:41,930
+عوضنا ميو 2 لعند واحد لعند اتنين عوضنا عن قيمة
+
+507
+00:36:41,930 --> 00:36:46,010
+طلعة القيمة اللي أمامكم والتعويض سهل و تعرف تعملهم
+
+508
+00:36:46,910 --> 00:36:51,430
+طيب نجي الآن substitution is in triple integrals
+
+509
+00:36:51,430 --> 00:36:58,610
+اللي هو نفس روح الـ double integral الان f of x و
+
+510
+00:36:58,610 --> 00:37:03,970
+y و z بس في dx dy dz هذه بيصير بدلها بدنا نعوض ب u
+
+511
+00:37:03,970 --> 00:37:09,970
+و v و w بيصير جاكوب ينتبههم مضروب في مين في du dv
+
+512
+00:37:09,970 --> 00:37:14,750
+dw إذا نفس الأشي بس شيلنا ال dz جيبنا dz زيادة و
+
+513
+00:37:14,750 --> 00:37:20,730
+حطينا dw بعد ماعوضنا عن ما هو X و Y و Z بما يلاقيه
+
+514
+00:37:20,730 --> 00:37:24,970
+منهم U و V و W الـ X بتساوي G و V و V و W و الـ Y
+
+515
+00:37:24,970 --> 00:37:28,650
+بتساوي H و V و V و W و الـ Z بتساوي K و V و V و W
+
+516
+00:37:28,650 --> 00:37:31,550
+هاي اللي هي ال region اللي هي ال in space اللي هي
+
+517
+00:37:31,550 --> 00:37:37,450
+G صارت in space تبع الاسف اللي كانت X و Y و Z بعد
+
+518
+00:37:37,450 --> 00:37:41,030
+ما عوضنا عنها اللي هي بدلت U و V و كده و W صارت
+
+519
+00:37:41,030 --> 00:37:43,350
+الـ Preimage هذا Transformation عشان هيك من hand ل
+
+520
+00:37:43,350 --> 00:37:46,940
+handلكن اما هو اللى هى نبدأ بهذه و الـ Preimage
+
+521
+00:37:46,940 --> 00:37:50,780
+اللى هى هذه طيب هذه الـ G يجى لل Jakobian اللى
+
+522
+00:37:50,780 --> 00:37:54,200
+عندى اش تعريفه Jakobian نفس الأولانى بس الآن زادت
+
+523
+00:37:54,200 --> 00:37:59,620
+اللى هى Z و W ده ال X by ده ال U ده ال X by ده ال
+
+524
+00:37:59,620 --> 00:38:03,750
+V ده ال X by ده ال W هى السطر الأول التانىواختصارا
+
+525
+00:38:03,750 --> 00:38:13,090
+عشان اسلب لحفظ واختصارا
+
+526
+00:38:13,090 --> 00:38:15,630
+عشان اسلب لحفظ واختصارا عشان اسلب لحفظ واختصارا
+
+527
+00:38:15,630 --> 00:38:21,310
+عشان اسلب لحفظ واختصارا
+
+528
+00:38:21,310 --> 00:38:26,190
+عشان اسلب لحفظ واختصارا عشان اسلب لحفظ
+
+529
+00:38:33,900 --> 00:38:37,240
+واللي طبعا ممكن اللي اولانيه تجاكبوا اللي اولانيه
+
+530
+00:38:37,240 --> 00:38:41,380
+تبع زمان دي x دي ال x bar و y على دي ال u و دي ال
+
+531
+00:38:41,380 --> 00:38:47,680
+v و الاستذكار زي ما قلنا كيف تستعملوهاالان نأتي
+
+532
+00:38:47,680 --> 00:38:50,320
+للـ cylindrical coordinates اللي أخدناه اللي هو
+
+533
+00:38:50,320 --> 00:38:55,280
+العلاقات بين R, θ و Z و بدنا الآن نحوّلها اللي هي
+
+534
+00:38:55,280 --> 00:39:00,640
+من الـ Cartesian لـ cylindrical مين اللي بيحوّل؟ X
+
+535
+00:39:00,640 --> 00:39:04,440
+بيساوي R Cos θ و Y بيساوي R Sin θ و Z بيساوي Ehz
+
+536
+00:39:04,440 --> 00:39:08,440
+بدنا نثبت لكم أن هذا التحويل اللي كنا نحوّل فيه
+
+537
+00:39:08,440 --> 00:39:13,420
+يتلاقم مع اللي بنحكي فيه في هذا الـ Transformation
+
+538
+00:39:13,810 --> 00:39:20,890
+التغيير الجاكوبياني هو جاكوبيان R ثتا وZ دل X by
+
+539
+00:39:20,890 --> 00:39:27,770
+دل R دل X by دل ثتا دل X by دل Z دل Y by دل ثتا R
+
+540
+00:39:27,770 --> 00:39:32,190
+دل Y by دل ثتا دل Y دل Z الاخر دل Z by دل R دل Z
+
+541
+00:39:32,190 --> 00:39:36,030
+by دل ثتا دل Z by دل Z طبعا ال Z في الجهتين نفس
+
+542
+00:39:36,030 --> 00:39:38,270
+الاشي في ال .. في ال .. في ال .. في ال .. في ال ..
+
+543
+00:39:38,270 --> 00:39:42,860
+في ال .. في ال .. في ال .. في ال rectangularبتحول
+
+544
+00:39:42,860 --> 00:39:48,340
+من rectangular لـ cylindrical دل X by دل R ايش يا
+
+545
+00:39:48,340 --> 00:39:54,120
+جماعة دل X by دل R اللي هي cos θ دل X by دل θ اللي
+
+546
+00:39:54,120 --> 00:39:59,140
+هي ناقص R sin θ دل X by دل Z سفر إلى نفس الاشياء
+
+547
+00:39:59,140 --> 00:40:05,460
+دل Y by دل R اللي هو sin θ دل Y by دل θ اللي هو R
+
+548
+00:40:05,460 --> 00:40:14,040
+cos θ دل Y by دل Z سفرنأتي للأخيرة DEL Z by DEL θ
+
+549
+00:40:14,040 --> 00:40:18,780
+سفر DEL Z by DEL أرض سفر DEL DEL Z by DEL أرض سفر
+
+550
+00:40:18,780 --> 00:40:20,060
+DEL DEL Z by DEL أرض سفر DEL DEL Z by DEL أرض سفر
+
+551
+00:40:20,060 --> 00:40:20,500
+DEL DEL Z by DEL أرض سفر DEL DEL Z by DEL أرض سفر
+
+552
+00:40:20,500 --> 00:40:20,840
+DEL DEL Z by DEL أرض سفر DEL DEL Z by DEL أرض سفر
+
+553
+00:40:20,840 --> 00:40:22,940
+DEL DEL Z by DEL أرض سفر DEL DEL Z by DEL أرض سفر
+
+554
+00:40:22,940 --> 00:40:25,020
+DEL DEL Z by DEL أرض سفر DEL DEL Z by DEL أرض سفر
+
+555
+00:40:25,020 --> 00:40:29,120
+DEL DEL Z by DEL أرض سفر DEL DEL Z by DEL أرض سفر
+
+556
+00:40:29,120 --> 00:40:34,760
+DEL DEL Z by DEL أرض سفر DEL DEL Z by DEL أرض سفر
+
+557
+00:40:34,760 --> 00:40:37,910
+Dزائد Rsin تربيع اللي هو في النهاية ايش بيطلع
+
+558
+00:40:37,910 --> 00:40:42,150
+بيطلع عن DR اذا F of X Y و Z DX DY DZ هاي ال D
+
+559
+00:40:42,150 --> 00:40:46,830
+صارت G الان ال F of X Y و Z صارت H of R و θ و Z او
+
+560
+00:40:46,830 --> 00:40:51,590
+زي ما كنا نعمل قبل F of R Cos θ F of R Sin θ و Z
+
+561
+00:40:51,590 --> 00:40:56,870
+ماشي الان DX DY DZ بدأت تعمل بالجاكوبيا الجاكوبيا
+
+562
+00:40:56,870 --> 00:41:00,610
+المين هو ال absolute value of R DR D θ DZ وهو اللي
+
+563
+00:41:00,610 --> 00:41:06,310
+هي القانون اللي احنا اشتغلنا فيقبلك نجي لان نجي
+
+564
+00:41:06,310 --> 00:41:09,110
+نعمل ميلاي الـ spherical الـ spherical نفس الأشياء
+
+565
+00:41:09,110 --> 00:41:13,150
+عندى اللى هو فيه transformations بحوله عشان أقولكم
+
+566
+00:41:13,150 --> 00:41:17,930
+أن هذا اللي هي ال transformations هذا تعميم لتحويل
+
+567
+00:41:17,930 --> 00:41:22,510
+من ال rectangularالى cylindrical او from
+
+568
+00:41:22,510 --> 00:41:29,350
+rectangular الى اللى هو spherical اللى عندي X
+
+569
+00:41:29,350 --> 00:41:33,310
+بيسوي رو صين فاى كوسين ثتا و Y بيسوي رو صين فاى
+
+570
+00:41:33,310 --> 00:41:37,150
+صين ثتا و Z بيسوي رو كوسين فاى هذه اللى هو بنحول
+
+571
+00:41:37,150 --> 00:41:42,280
+من X و Y و Z الى رو و فاى و ثتاالان الـ Jacobian
+
+572
+00:41:42,280 --> 00:41:45,480
+تبحين اللي هو عبارة عن دي الـ X by دي الـ Rho دي
+
+573
+00:41:45,480 --> 00:41:48,200
+الـ X by دي الـ Phi دي الـ X by دي الـ Theta و نفس
+
+574
+00:41:48,200 --> 00:41:51,460
+الاشي دي الـ Y و نفس الاشي دي الـ Z الان هذا الـ
+
+575
+00:41:51,460 --> 00:41:55,320
+Jacobian بتوجدوا اللي هو هذه دي الـ X by دي الـ
+
+576
+00:41:55,320 --> 00:41:58,800
+Rho بتعرفوا توجدوها أكيد اللي هي Sin Phi Cos Theta
+
+577
+00:41:58,800 --> 00:42:01,640
+و التانية اللي هي دي الـ X by دي الـ Phi اللي هي
+
+578
+00:42:01,640 --> 00:42:06,460
+عبارة عن Rho Cos Phi Cos Theta و دي الـ X by دي
+
+579
+00:42:06,460 --> 00:42:12,260
+الـ Theta Rho Sin Phiاللي هو ناقص sin θ بتطلع برا
+
+580
+00:42:12,260 --> 00:42:15,420
+الناقص طبعا و بيصير تحطها هنا بتعوض عن هذه و بتعوض
+
+581
+00:42:15,420 --> 00:42:18,720
+عن هدول بعد ما تفضل اللي هو لل Y و بعد ما تفضل لل
+
+582
+00:42:18,720 --> 00:42:22,160
+Z بتعوضهم و بتوجد ال determinant طبعا بغلب إيجاده
+
+583
+00:42:22,160 --> 00:42:25,420
+يعني او خلنا نقول بطول بس عادي يعني بتستخدم أي سطر
+
+584
+00:42:25,420 --> 00:42:28,420
+أو أي عمود بتطلعلك في النهاية عبر علاقات معينة
+
+585
+00:42:28,420 --> 00:42:34,580
+روتر بيع sin θ في و هذا هو exercise 19 طيبالان
+
+586
+00:42:34,580 --> 00:42:37,640
+بيدناجي نعوض عن .. مافيش داعي للتفصيل ماديش أفصل
+
+587
+00:42:37,640 --> 00:42:43,380
+لإيجاد determinant f of x y و z اللي هو بيصير h و
+
+588
+00:42:43,380 --> 00:42:47,080
+rho و phi و theta لما نعوروا مكان x y و z بقيامها
+
+589
+00:42:47,080 --> 00:42:51,800
+dx dy dz اللي بيصير اللي هو عبارة عن جاكوبين اللي
+
+590
+00:42:51,800 --> 00:42:55,060
+هو absolute value of sin تربيع phi d rho d phi و
+
+591
+00:42:55,060 --> 00:42:57,820
+هذه اللي هي طبعا هذه موجبة طبيعي و هذه موجبة في
+
+592
+00:42:58,110 --> 00:43:02,550
+نطاق الهداك فبتشيلها absolute value مش هتفرج بتطلع
+
+593
+00:43:02,550 --> 00:43:05,710
+نفس القيم اللي عملناها قبل طبعا وال region D
+
+594
+00:43:05,710 --> 00:43:10,570
+بتتحول ال region G إذا اللي هي ال spherical وده
+
+595
+00:43:10,570 --> 00:43:13,730
+كمية اللي عبارة عن حلقة خاصة من اللي هو اللي بنحكي
+
+596
+00:43:13,730 --> 00:43:16,670
+عنه in general transformations نيجي بقى اللي هو
+
+597
+00:43:16,670 --> 00:43:19,810
+example خمسةالـ Example الخامس بيقول ليه؟ هذا طبعا
+
+598
+00:43:19,810 --> 00:43:22,570
+عالميا على ال triple .. على جزء .. اللي هو ال
+
+599
+00:43:22,570 --> 00:43:26,870
+triple integrals على x, y و z اللي هو 2x ناقص y
+
+600
+00:43:26,870 --> 00:43:31,790
+على 2, z على 3, dx, dy, dz و الحدود ال integration
+
+601
+00:43:31,790 --> 00:43:35,540
+هي أمامنابقول لك by applying the transformation
+
+602
+00:43:35,540 --> 00:43:40,000
+اللي هو U بالساوية 2 X ناقص U على 2 اللي هي هذه V
+
+603
+00:43:40,000 --> 00:43:43,880
+بالساوية Y على 2 اللي هي استعاني منها بهذه و W
+
+604
+00:43:43,880 --> 00:43:47,960
+بالساوية Z على 3 طبعا هو عارف انه احنا يعني هنتغلب
+
+605
+00:43:47,960 --> 00:43:50,760
+لو بدنا نوجد اللي هو transformation لكن ممكن
+
+606
+00:43:50,760 --> 00:43:54,880
+الواحد اللي هو U استنتج من هذا و W من هنا و يقول
+
+607
+00:43:54,880 --> 00:43:59,340
+لك V هذه اللي هو هي هنا UY على 2 و هذا كأنه بناحط
+
+608
+00:43:59,340 --> 00:44:04,240
+V بساوية على 2بتتناسق مع حد و بتتناسق مع حد فيمكن
+
+609
+00:44:04,240 --> 00:44:09,640
+كده و بيجرب و بيطلع اللي هو عنده او طبعا يعني واحد
+
+610
+00:44:09,640 --> 00:44:13,420
+جاء خطر ع باله فكرة تانية في التعويض هيلاجي ممكن
+
+611
+00:44:13,420 --> 00:44:16,640
+تظبط بعض الشغلات و بعض الشغلات ماتظبطش لكن الحاجات
+
+612
+00:44:16,640 --> 00:44:18,880
+اللي مش مقدور عليها و مانعرفاش لسه احنا خبرتنا
+
+613
+00:44:18,880 --> 00:44:22,800
+جليلة بيعطينا اللي هو ال transformations نيجي الآن
+
+614
+00:44:22,800 --> 00:44:26,850
+تلات مرات بتتغير هي واحد اتنين وهي تلاتةالى الـ
+
+615
+00:44:26,850 --> 00:44:28,810
+Nigella اللى هي ال region اللى عليها من الكامل
+
+616
+00:44:28,810 --> 00:44:33,170
+عندى X بالساوية Y على 2 لعند Y على 2 زائد 1 ماشي
+
+617
+00:44:33,170 --> 00:44:38,950
+الحال عندى اللى هو الخط اللى هو هيه المائل X
+
+618
+00:44:38,950 --> 00:44:43,110
+بالساوية Y على 2 و هذا الخط المائل التاني X
+
+619
+00:44:43,110 --> 00:44:45,430
+بالساوية Y على 2 زائد 1 رسمناه قبل شوية هذا
+
+620
+00:44:45,430 --> 00:44:50,720
+المنطقةواللي هي قيمة Y بتبدأ من عند سفر و بتنتهي
+
+621
+00:44:50,720 --> 00:44:54,880
+لعند مين لعند أربعة هي من سفر لعند أربعة و Z بتبدأ
+
+622
+00:44:54,880 --> 00:44:59,700
+من عند سفر و بتنتهي لعند مين لعند تلاتة فوق ماشي
+
+623
+00:44:59,700 --> 00:45:06,310
+فبيصير عندي اللي هوالشكل اللي عندي عبارة عن الشكل
+
+624
+00:45:06,310 --> 00:45:10,350
+اللي أمامي طبعا هو ما إلا جاي لإنه حسب ميول اللي
+
+625
+00:45:10,350 --> 00:45:14,530
+هو X بسواية على اتنين هي هذه و X بسواية على اتنين
+
+626
+00:45:14,530 --> 00:45:18,770
+زائد واحد فبطلع عندي الصندوق اللي أمامي هذا هو ال
+
+627
+00:45:18,770 --> 00:45:21,590
+region تبعتي لإن هذه ال region بدي أحولها و أصلا
+
+628
+00:45:21,590 --> 00:45:24,370
+مش ضروري أرسمها لإن أنا بدي أحول منها على طول بحول
+
+629
+00:45:24,370 --> 00:45:27,610
+من هنا و هنا عبر التحويل اللي هو هذا لا بلزم أرسم
+
+630
+00:45:27,610 --> 00:45:32,080
+هذه ولا بلزم أرسم هذه أصلا فنشوف كيفالان نجي
+
+631
+00:45:32,080 --> 00:45:38,580
+لانوجد X و Y لو جينا هنا الـ Z بتسيي .. ده نوجد X
+
+632
+00:45:38,580 --> 00:45:43,820
+و Y و Z بدلالة U و V و W عند و W بتساوي Z ع دلاتة
+
+633
+00:45:43,820 --> 00:45:49,250
+يعني اللي هي Z بتساوي تلاتة Wوهنا برضه واضحة Y
+
+634
+00:45:49,250 --> 00:45:55,350
+بيساوي 2V Y بيساوي 2V حط هنا 2V بيصير 2X نقص 2V
+
+635
+00:45:55,350 --> 00:46:00,150
+على 2 يعني X نقص V بتنجل ال V هنا بيصير X بيساوي U
+
+636
+00:46:00,150 --> 00:46:06,000
+زائد Vنأتي الآن يا جماعة اللي هو إيش بدنا نسوي
+
+637
+00:46:06,000 --> 00:46:11,760
+نأتي الآن بدنا اللي هو نوجد الـ Jacobian أو نوجد
+
+638
+00:46:11,760 --> 00:46:16,260
+اللي هي المنطقة إذا بدنا نحوّل كل خطوط هذولة إلى
+
+639
+00:46:16,260 --> 00:46:20,340
+خطوط في الـ U و الـ V هاي X بيساوي على 2 الأولاني
+
+640
+00:46:20,340 --> 00:46:26,020
+و X بيساوي على 2 زائد 1 و التاني 0 ل 4 و التالت 0
+
+641
+00:46:26,020 --> 00:46:29,460
+ل 3Z نشوف كيف الخطوط هذي بدأت تتحول في الـ U و الـ
+
+642
+00:46:29,460 --> 00:46:33,950
+V زي ما عملنا X بيساوي U زائد V هيهاو ال Y على 2
+
+643
+00:46:33,950 --> 00:46:37,610
+اللي هي عبارة عن V ال V مع ال V بيصير U بساوة 0
+
+644
+00:46:37,610 --> 00:46:40,990
+عملناها قبل شوية هذه اللي هي عبارة عن U زائد V
+
+645
+00:46:40,990 --> 00:46:45,490
+مكان ال X بساوة V زائد 1 هي ال V و هي زائد 1 ال V
+
+646
+00:46:45,490 --> 00:46:48,510
+مع ال V بيصير U بساوة 1 ده U بساوة 0 ده U بساوة 1
+
+647
+00:46:48,510 --> 00:46:52,650
+كله حلو و rectangle Y بساوة 0 اللي هي 2V بساوة 0
+
+648
+00:46:52,650 --> 00:46:58,370
+يعني V بساوة 0 و Y بساوة 4 2V بساوة 4 يعني V بساوة
+
+649
+00:46:58,370 --> 00:47:03,040
+2 إذن V من 0 إلى 2 كله محلاة كله rectanglesو z
+
+650
+00:47:03,040 --> 00:47:08,100
+بالساوة 0 بيصير 3w بالساوة 0 يعني w بالساوة 0 z
+
+651
+00:47:08,100 --> 00:47:12,020
+بالساوة 3 يعني 3w بالساوة 3 يعني w بتساوة 1 إذا
+
+652
+00:47:12,020 --> 00:47:16,860
+حدود ال integration حدود سهلة خالص كلها عبارة عن
+
+653
+00:47:16,860 --> 00:47:20,400
+rectangle وهي رسمة اللي هو ال rectangle اللي عندي
+
+654
+00:47:20,400 --> 00:47:26,640
+هي U من 0 ل1 وهي اللي عندي V طبعا هو كتبها هي V من
+
+655
+00:47:26,640 --> 00:47:31,900
+0 ل2أو Z أو W اللي هو من سفر اللي عند واحد طلع عند
+
+656
+00:47:31,900 --> 00:47:36,580
+الصندوق أو اللي هو المتوازي اللي أمامنا هذا أو
+
+657
+00:47:36,580 --> 00:47:43,300
+اللي هو المتوازي المستطيلات طيب نيجي الآن عند the
+
+658
+00:47:43,300 --> 00:47:46,580
+Jacobian of the transformation again from ذاك ال
+
+659
+00:47:46,580 --> 00:47:50,020
+Jacobian دل X بي دل U دل X بي دل V دل X بي دل W
+
+660
+00:47:50,020 --> 00:47:53,880
+طبعا ال Jacobian سهل اللي عندنا دل X بي دل U واحد
+
+661
+00:47:53,880 --> 00:47:55,220
+دل X بي دل V
+
+662
+00:47:59,970 --> 00:48:02,390
+وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد
+
+663
+00:48:02,390 --> 00:48:04,110
+وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد
+
+664
+00:48:04,110 --> 00:48:04,910
+وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد
+
+665
+00:48:04,910 --> 00:48:05,190
+وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد
+
+666
+00:48:05,190 --> 00:48:05,330
+وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد
+
+667
+00:48:05,330 --> 00:48:06,370
+وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد
+
+668
+00:48:06,370 --> 00:48:07,750
+وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد وحد
+
+669
+00:48:07,750 --> 00:48:11,550
+وحد وحد وحد وحد
+
+670
+00:48:20,390 --> 00:48:26,050
+دل z by دل u ودل z by دل v صفر ودل z by دل w اللي
+
+671
+00:48:26,050 --> 00:48:29,430
+هو بيساوي تلاتة فبيصير صفر صفر تلاتة بنوجد ال
+
+672
+00:48:29,430 --> 00:48:32,810
+determinant هذا بواسط السطر هذا مثلا بيصير تلاتة
+
+673
+00:48:32,810 --> 00:48:36,410
+في هذا ال determinant اللي هو اتنين نقل صفر يعني
+
+674
+00:48:36,410 --> 00:48:40,590
+اتنين في تلاتة بتطلع ستة الان بنيجي بنعوض عن هذا
+
+675
+00:48:40,590 --> 00:48:45,010
+المقدار عوضنا عن هذه اللي هي u وهذا w وهذا
+
+676
+00:48:45,010 --> 00:48:48,230
+الجاكوبينالعن حدود ال integration زي ما قلنا U من
+
+677
+00:48:48,230 --> 00:48:52,370
+صفر ل واحد كله rectangles و DV من صفر ل اتنين و W
+
+678
+00:48:52,370 --> 00:48:56,430
+من صفر ل عند واحد الان بنعوض مكانها دي بصير ستة
+
+679
+00:48:56,430 --> 00:49:00,130
+بنطلع الستة برا و بنكامل U زائد V بالنسبة ل U بصير
+
+680
+00:49:00,130 --> 00:49:04,570
+U تربيع على اتنين UV من صفر ل عند واحد بنعوض عنها
+
+681
+00:49:04,570 --> 00:49:08,390
+بواحدبصير نص الدقيقة زي ال W وهذه كما تصبح صفر دي
+
+682
+00:49:08,390 --> 00:49:15,170
+VDW ال V من فاضل بصير من كامل V على اتنين زي VW من
+
+683
+00:49:15,170 --> 00:49:19,810
+صفر لعند اتنين بنعوض الآن عن V بقيمتها بصير عبارة
+
+684
+00:49:19,810 --> 00:49:24,460
+عن واحد زي اتنين Wبالكامل هذه بيصير واحد زي ال W
+
+685
+00:49:24,460 --> 00:49:28,360
+تربيع من سفر ل واحد والست البره بيصير ستة في اتنين
+
+686
+00:49:28,360 --> 00:49:32,100
+واحد على اتنين ساوى اتناشر و هيك بيكون احنا انهينا
+
+687
+00:49:32,100 --> 00:49:36,680
+الاخر محاضرة والسلام عليكم ورحمة الله وبركاته وكل
+
+688
+00:49:36,680 --> 00:49:41,520
+عام ولا كل فصل وانتم بخير ربنا ينجحكم جميعا
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/MDfHgpCP1Ww_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/MDfHgpCP1Ww_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..2659c891eb994b13d932afdd94a47049eac820f1
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/MDfHgpCP1Ww_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 1788, "start": 0.0, "end": 17.88, "text": "بسم الله الرحمن الرحيم السلام عليكم ورحمة الله وبركاته هذه هي المحاضرة رقم 26 مساقطة فاضل وتكامل جيم وهي المحاضرة الأخيرة وهي المحاضرة السابعة", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 29538, 39896, 9673, 5016, 46958, 25720, 12602, 4587, 2304, 7551, 47524, 995, 47432, 3660, 6156, 46958, 1211, 34683, 4117, 10943, 1211, 10874, 32640, 37037, 1829, 9673, 5016, 46958, 25720, 16247, 9778, 48923, 37037, 1829, 9673, 5016, 46958, 25720, 21136, 16758, 27884], "avg_logprob": -0.09194711538461539, "compression_ratio": 1.8633093525179856, "no_speech_prob": 0.00032448768615722656, "words": [{"start": 0.0, "end": 1.02, "word": "بسم", "probability": 0.8037109375}, {"start": 1.02, "end": 1.24, "word": " الله", "probability": 0.94970703125}, {"start": 1.24, "end": 1.7, "word": " الرحمن", "probability": 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"avg_logprob": -0.5032894783898404, "compression_ratio": 1.2127659574468086, "no_speech_prob": 0.0, "words": [{"start": 176.37, "end": 176.95, "word": "وY", "probability": 0.328826904296875}, {"start": 176.95, "end": 177.41, "word": " دلالتها", "probability": 0.72646484375}, {"start": 177.41, "end": 177.95, "word": " بدلالة", "probability": 0.747216796875}, {"start": 177.95, "end": 178.43, "word": " UV", "probability": 0.86474609375}, {"start": 178.43, "end": 180.05, "word": " وDXDY", "probability": 0.5404052734375}, {"start": 180.05, "end": 180.47, "word": " بتصير", "probability": 0.6241048177083334}, {"start": 180.47, "end": 181.21, "word": " تستبدل", "probability": 0.9710693359375}, {"start": 181.21, "end": 181.57, "word": " بالـ", "probability": 0.424072265625}, {"start": 181.57, "end": 181.81, "word": " اللي", "probability": 0.54290771484375}, {"start": 181.81, "end": 182.11, "word": " هي", "probability": 0.8935546875}, {"start": 182.11, "end": 182.45, "word": " الـ", 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the Jacobian determinant or Jacobian of the coordinates Transformation X بيساوي G of U و Y بيساوي H of U و V يعني ال Jacobian لهذا ال Transformation اللي لما نعوضنا شيلنا X وعوضنا مكانها بدلالة U و V وشيلنا Y وعوضنا مكانها بدلالة U و V ال Jacobian هو كما يلي بنعرفه هو عبارة عن دل X", "tokens": [6027, 7649, 13412, 2407, 2423, 39184, 14117, 952, 23758, 13672, 1829, 39894, 5016, 4117, 1829, 18871, 3224, 2423, 14117, 952, 44945, 3615, 28480, 3224, 9122, 15042, 7251, 20292, 46245, 849, 264, 14117, 952, 41296, 420, 14117, 952, 295, 264, 21056, 6531, 8663, 1783, 4724, 1829, 3794, 995, 45865, 460, 295, 624, 4032, 398, 4724, 1829, 3794, 995, 45865, 389, 295, 624, 4032, 691, 37495, 22653, 2423, 14117, 952, 46740, 15730, 2423, 6531, 8663, 13672, 1829, 5296, 15042, 8717, 3615, 2407, 11242, 8315, 13412, 26895, 8315, 1783, 4032, 3615, 2407, 11242, 8315, 3714, 41361, 11296, 47525, 1211, 6027, 3660, 624, 4032, 691, 4032, 8592, 26895, 8315, 398, 4032, 3615, 2407, 11242, 8315, 3714, 41361, 11296, 47525, 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"probability": 0.5859375}, {"start": 215.83, "end": 216.49, "word": " coordinates", "probability": 0.77734375}, {"start": 216.49, "end": 217.49, "word": " Transformation", "probability": 0.85302734375}, {"start": 217.49, "end": 217.79, "word": " X", "probability": 0.740234375}, {"start": 217.79, "end": 218.15, "word": " بيساوي", "probability": 0.647412109375}, {"start": 218.15, "end": 218.35, "word": " G", "probability": 0.716796875}, {"start": 218.35, "end": 218.53, "word": " of", "probability": 0.69482421875}, {"start": 218.53, "end": 218.73, "word": " U", "probability": 0.9814453125}, {"start": 218.73, "end": 218.93, "word": " و", "probability": 0.96142578125}, {"start": 218.93, "end": 219.07, "word": " Y", "probability": 0.65283203125}, {"start": 219.07, "end": 219.41, "word": " بيساوي", "probability": 0.9763671875}, {"start": 219.41, "end": 219.67, "word": " H", "probability": 0.96337890625}, {"start": 219.67, "end": 219.91, "word": " of", "probability": 0.919921875}, {"start": 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شيلنا", "probability": 0.9475911458333334}, {"start": 225.49, "end": 225.73, "word": " X", "probability": 0.97265625}, {"start": 225.73, "end": 226.19, "word": " وعوضنا", "probability": 0.8962890625}, {"start": 226.19, "end": 226.61, "word": " مكانها", "probability": 0.9822591145833334}, {"start": 226.61, "end": 227.11, "word": " بدلالة", "probability": 0.890869140625}, {"start": 227.11, "end": 227.31, "word": " U", "probability": 0.96533203125}, {"start": 227.31, "end": 227.39, "word": " و", "probability": 0.96484375}, {"start": 227.39, "end": 227.53, "word": " V", "probability": 0.82470703125}, {"start": 227.53, "end": 228.03, "word": " وشيلنا", "probability": 0.9130859375}, {"start": 228.03, "end": 228.19, "word": " Y", "probability": 0.984375}, {"start": 228.19, "end": 228.69, "word": " وعوضنا", "probability": 0.97548828125}, {"start": 228.69, "end": 229.07, "word": " مكانها", "probability": 0.9923502604166666}, {"start": 229.07, "end": 229.51, "word": " بدلالة", "probability": 0.98828125}, {"start": 229.51, "end": 229.73, "word": " U", "probability": 0.9599609375}, {"start": 229.73, "end": 229.83, "word": " و", "probability": 0.94921875}, {"start": 229.83, "end": 230.03, "word": " V", "probability": 0.8857421875}, {"start": 230.03, "end": 230.53, "word": " ال", "probability": 0.80908203125}, {"start": 230.53, "end": 231.05, "word": " Jacobian", "probability": 0.9794921875}, {"start": 231.05, "end": 231.39, "word": " هو", "probability": 0.97998046875}, {"start": 231.39, "end": 231.85, "word": " كما", "probability": 0.979248046875}, {"start": 231.85, "end": 232.09, "word": " يلي", "probability": 0.9814453125}, {"start": 232.09, "end": 232.71, "word": " بنعرفه", "probability": 0.9600830078125}, {"start": 232.71, "end": 232.91, "word": " هو", "probability": 0.7119140625}, {"start": 232.91, "end": 233.29, "word": " عبارة", "probability": 0.9957275390625}, {"start": 233.29, "end": 233.61, "word": " عن", "probability": 0.98974609375}, {"start": 233.61, "end": 234.43, "word": " دل", "probability": 0.59808349609375}, {"start": 234.43, "end": 235.07, "word": " X", "probability": 0.966796875}], "temperature": 1.0}, {"id": 12, "seek": 25026, "start": 243.68, "end": 250.26, "text": "والسطر التاني اللي هو العمود التاني في السطر التاني اللي هو del u by del y by del v", "tokens": [2407, 6027, 3794, 9566, 2288, 16712, 7649, 1829, 13672, 1829, 31439, 18863, 2304, 23328, 16712, 7649, 1829, 8978, 21136, 9566, 2288, 16712, 7649, 1829, 13672, 1829, 31439, 1103, 344, 538, 1103, 288, 538, 1103, 371], "avg_logprob": -0.3357204919060071, "compression_ratio": 1.8591549295774648, "no_speech_prob": 0.00016224384307861328, "words": [{"start": 243.68, "end": 244.36, "word": "والسطر", "probability": 0.528076171875}, {"start": 244.36, "end": 244.78, "word": " التاني", "probability": 0.8955078125}, {"start": 244.78, "end": 245.38, "word": " اللي", "probability": 0.58056640625}, {"start": 245.38, "end": 245.44, "word": " هو", "probability": 0.97021484375}, {"start": 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{"start": 257.49, "end": 257.69, "word": " هذا", "probability": 0.97802734375}, {"start": 257.69, "end": 258.15, "word": " المقدار", "probability": 0.9949544270833334}, {"start": 258.15, "end": 258.63, "word": " فبطلع", "probability": 0.87919921875}, {"start": 258.63, "end": 258.83, "word": " هذا", "probability": 0.88232421875}, {"start": 258.83, "end": 259.09, "word": " مين", "probability": 0.91552734375}, {"start": 259.09, "end": 259.31, "word": " هو", "probability": 0.9775390625}, {"start": 259.31, "end": 259.43, "word": " يا", "probability": 0.9453125}, {"start": 259.43, "end": 259.95, "word": " جماعة", "probability": 0.9954427083333334}, {"start": 259.95, "end": 260.87, "word": " الجاكوبيان", "probability": 0.6873372395833334}, {"start": 260.87, "end": 261.55, "word": " اذا", "probability": 0.5732421875}, {"start": 261.55, "end": 262.37, "word": " الجاكوبيان", "probability": 0.90234375}, {"start": 262.37, "end": 262.93, "word": " اللي", "probability": 0.923095703125}, {"start": 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"probability": 0.904541015625}, {"start": 267.91, "end": 268.23, "word": " وبنعمل", "probability": 0.7482421875}, {"start": 268.23, "end": 268.53, "word": " السطر", "probability": 0.8670247395833334}, {"start": 268.53, "end": 268.97, "word": " الأول", "probability": 0.989013671875}, {"start": 268.97, "end": 269.71, "word": " بعدين", "probability": 0.817138671875}, {"start": 269.71, "end": 270.11, "word": " بنيجي", "probability": 0.858154296875}, {"start": 270.11, "end": 270.43, "word": " دل", "probability": 0.97412109375}, {"start": 270.43, "end": 270.95, "word": " Y", "probability": 0.9892578125}, {"start": 270.95, "end": 271.43, "word": " by", "probability": 0.97216796875}, {"start": 271.43, "end": 271.71, "word": " دل", "probability": 0.98876953125}, {"start": 271.71, "end": 271.99, "word": " U", "probability": 0.99365234375}], "temperature": 1.0}, {"id": 14, "seek": 29417, "start": 272.55, "end": 294.17, "text": "و بعدين del y by del v و بنودد ال determinant بنحطله absolute value و بنضربه في هذا و بيصير du dv بكون حوضنا هذا درجة نضل عليها اللي هي ال G أحصلها من مين من R و خلينا اللي هو نشوف المثال التالي اللي حاجة احنا عارفينها نشوف ايش بقول", "tokens": [2407, 39182, 9957, 1103, 288, 538, 1103, 371, 4032, 44945, 23328, 3215, 2423, 41296, 44945, 5016, 9566, 43761, 8236, 2158, 4032, 44945, 11242, 25513, 3224, 8978, 23758, 4032, 4724, 1829, 9381, 13546, 1581, 274, 85, 4724, 30544, 11331, 2407, 11242, 8315, 23758, 11778, 47341, 3660, 8717, 11242, 1211, 25894, 11296, 13672, 1829, 39896, 2423, 460, 5551, 5016, 36520, 11296, 9154, 3714, 9957, 9154, 497, 4032, 16490, 20292, 8315, 13672, 1829, 31439, 8717, 8592, 38688, 9673, 12984, 6027, 16712, 6027, 1829, 13672, 1829, 11331, 26108, 3660, 1975, 5016, 8315, 6225, 9640, 5172, 9957, 11296, 8717, 8592, 38688, 1975, 1829, 8592, 4724, 39648], "avg_logprob": -0.29335171218011896, "compression_ratio": 1.6331877729257642, "no_speech_prob": 1.4722347259521484e-05, "words": [{"start": 272.55, "end": 272.81, "word": "و", 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71870, "start": 699.72, "end": 718.7, "text": "اللي هو لأ روتيتها وفق القانون اللي حكيناها قبل بشوية نيجي أن نثبت المعلومة أكتر جالي example evaluate ال double integral ل 2x ناقص y على 2 dx dy x بيساوي على 2 زائد 1 و x بيساوي y على 2 و من 0 ل 4", "tokens": [6027, 20292, 31439, 5296, 10721, 12602, 35473, 36081, 11296, 4032, 5172, 4587, 25062, 7649, 11536, 13672, 1829, 11331, 4117, 1829, 8315, 11296, 12174, 36150, 4724, 8592, 2407, 10632, 8717, 1829, 7435, 1829, 14739, 8717, 12984, 3555, 2655, 9673, 30241, 20498, 3660, 5551, 4117, 2655, 2288, 10874, 6027, 1829, 1365, 13059, 2423, 3834, 11573, 5296, 568, 87, 8717, 995, 4587, 9381, 288, 15844, 568, 30017, 14584, 2031, 4724, 1829, 3794, 995, 45865, 15844, 568, 30767, 16373, 3215, 502, 4032, 2031, 4724, 1829, 3794, 995, 45865, 288, 15844, 568, 4032, 9154, 1958, 5296, 1017], "avg_logprob": -0.2684811943320818, "compression_ratio": 1.4757281553398058, "no_speech_prob": 1.6033649444580078e-05, "words": [{"start": 699.72, "end": 700.08, "word": 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"end": 794.44, "text": "إذن هو استبدل الـ Integrand هذا كله بـ 2x-y على 2 و استبدل الـ V بالـ y على 2 و الـ y على 2 اللي عندي هنا و بيقول لي and integrate over appropriate region in the UV plane يعني الآن بيقوللي بالتعويض عن U عن x و y بدلالة اللي هو الـ U و V اللي معطية عبر العلاقة التالية أو ددلة double integral هذا in the UV plane", "tokens": [28814, 8848, 1863, 31439, 44713, 44510, 1211, 2423, 39184, 5681, 1146, 3699, 23758, 28242, 3224, 4724, 39184, 568, 87, 12, 88, 15844, 568, 4032, 44713, 44510, 1211, 2423, 39184, 691, 20666, 39184, 288, 15844, 568, 4032, 2423, 39184, 288, 15844, 568, 13672, 1829, 18871, 16254, 34105, 4032, 4724, 1829, 39648, 32239, 293, 13365, 670, 6854, 4458, 294, 264, 17887, 5720, 37495, 22653, 6024, 48506, 4724, 1829, 39648, 20292, 20666, 2655, 3615, 45865, 11242, 18871, 624, 18871, 2031, 4032, 288, 47525, 1211, 6027, 3660, 13672, 1829, 31439, 2423, 39184, 624, 4032, 691, 13672, 1829, 20449, 9566, 10632, 6225, 26890, 18863, 15040, 28671, 16712, 6027, 10632, 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114, "seek": 257782, "start": 2554.16, "end": 2577.82, "text": "الان بيدناجي نعوض عن .. مافيش داعي للتفصيل ماديش أفصل لإيجاد determinant f of x y و z اللي هو بيصير h و rho و phi و theta لما نعوروا مكان x y و z بقيامها dx dy dz اللي بيصير اللي هو عبارة عن جاكوبين اللي هو absolute value of sin تربيع phi d rho d phi و هذه اللي هي طبعا هذه موجبة طبيعي و هذه موجبة في", "tokens": [6027, 7649, 4724, 25708, 1863, 26108, 1829, 8717, 3615, 2407, 11242, 18871, 4386, 19446, 41185, 8592, 11778, 45761, 1829, 24976, 2655, 5172, 9381, 26895, 19446, 16254, 8592, 5551, 5172, 36520, 5296, 28814, 1829, 7435, 18513, 41296, 283, 295, 2031, 288, 4032, 710, 13672, 1829, 31439, 4724, 1829, 9381, 13546, 276, 4032, 20293, 4032, 13107, 4032, 9725, 5296, 15042, 8717, 3615, 13063, 14407, 3714, 41361, 2031, 288, 4032, 710, 4724, 38436, 10943, 11296, 30017, 14584, 9758, 13672, 1829, 4724, 1829, 9381, 13546, 13672, 1829, 31439, 6225, 3555, 9640, 3660, 18871, 10874, 995, 4117, 37746, 9957, 13672, 1829, 31439, 8236, 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"probability": 0.9851888020833334}, {"start": 2592.49, "end": 2593.01, "word": " من", "probability": 0.9580078125}, {"start": 2593.01, "end": 2593.21, "word": " اللي", "probability": 0.980712890625}, {"start": 2593.21, "end": 2593.31, "word": " هو", "probability": 0.7001953125}, {"start": 2593.31, "end": 2593.41, "word": " اللي", "probability": 0.983154296875}, {"start": 2593.41, "end": 2593.73, "word": " بنحكي", "probability": 0.86688232421875}, {"start": 2593.73, "end": 2593.99, "word": " عنه", "probability": 0.986572265625}, {"start": 2593.99, "end": 2594.17, "word": " in", "probability": 0.12115478515625}, {"start": 2594.17, "end": 2594.43, "word": " general", "probability": 0.84765625}, {"start": 2594.43, "end": 2595.19, "word": " transformations", "probability": 0.95947265625}, {"start": 2595.19, "end": 2596.03, "word": " نيجي", "probability": 0.789306640625}, {"start": 2596.03, "end": 2596.21, "word": " بقى", "probability": 0.7918294270833334}, {"start": 2596.21, "end": 2596.45, 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" ال", "probability": 0.52880859375}, {"start": 2600.91, "end": 2601.23, "word": " triple", "probability": 0.42236328125}, {"start": 2601.23, "end": 2601.35, "word": " ..", "probability": 0.198974609375}, {"start": 2601.35, "end": 2601.65, "word": " على", "probability": 0.56689453125}, {"start": 2601.65, "end": 2602.05, "word": " جزء", "probability": 0.62939453125}, {"start": 2602.05, "end": 2602.15, "word": " ..", "probability": 0.5849609375}, {"start": 2602.15, "end": 2602.39, "word": " اللي", "probability": 0.9833984375}, {"start": 2602.39, "end": 2602.51, "word": " هو", "probability": 0.984375}, {"start": 2602.51, "end": 2602.57, "word": " ال", "probability": 0.47509765625}, {"start": 2602.57, "end": 2602.75, "word": " triple", "probability": 0.71044921875}, {"start": 2602.75, "end": 2603.37, "word": " integrals", "probability": 0.940185546875}, {"start": 2603.37, "end": 2603.91, "word": " على", "probability": 0.71533203125}, {"start": 2603.91, "end": 2604.13, "word": " x,", 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عند", "probability": 0.97412109375}, {"start": 2942.65, "end": 2943.11, "word": " واحد", "probability": 0.993896484375}, {"start": 2943.11, "end": 2944.13, "word": " بنعوض", "probability": 0.87158203125}, {"start": 2944.13, "end": 2944.57, "word": " عنها", "probability": 0.984130859375}, {"start": 2944.57, "end": 2945.13, "word": " بواحد", "probability": 0.9886067708333334}], "temperature": 1.0}, {"id": 131, "seek": 296121, "start": 2945.33, "end": 2961.21, "text": "بصير نص الدقيقة زي ال W وهذه كما تصبح صفر دي VDW ال V من فاضل بصير من كامل V على اتنين زي VW من صفر لعند اتنين بنعوض الآن عن V بقيمتها بصير عبارة عن واحد زي اتنين W", "tokens": [3555, 9381, 13546, 8717, 9381, 32748, 38436, 28671, 30767, 1829, 2423, 343, 37037, 24192, 9122, 15042, 6055, 9381, 49628, 20328, 5172, 2288, 11778, 1829, 691, 35, 54, 2423, 691, 9154, 6156, 46958, 1211, 4724, 9381, 13546, 9154, 9122, 10943, 1211, 691, 15844, 1975, 2655, 1863, 9957, 30767, 1829, 691, 54, 9154, 20328, 5172, 2288, 5296, 3615, 41260, 1975, 2655, 1863, 9957, 44945, 3615, 2407, 11242, 6024, 48506, 18871, 691, 4724, 4587, 32640, 2655, 11296, 4724, 9381, 13546, 6225, 3555, 9640, 3660, 18871, 36764, 24401, 30767, 1829, 1975, 2655, 1863, 9957, 343], "avg_logprob": -0.34425950601049093, "compression_ratio": 1.701219512195122, "no_speech_prob": 5.799531936645508e-05, "words": [{"start": 2945.33, "end": 2945.69, "word": "بصير", "probability": 0.554443359375}, {"start": 2945.69, "end": 2945.93, "word": " نص", "probability": 0.723876953125}, {"start": 2945.93, "end": 2946.29, "word": " الدقيقة", "probability": 0.740234375}, {"start": 2946.29, "end": 2946.41, "word": " زي", "probability": 0.509521484375}, {"start": 2946.41, "end": 2946.51, "word": " ال", "probability": 0.5859375}, {"start": 2946.51, "end": 2946.79, "word": " W", "probability": 0.173583984375}, {"start": 2946.79, "end": 2947.03, "word": " وهذه", "probability": 0.3914794921875}, {"start": 2947.03, "end": 2947.13, "word": " كما", "probability": 0.26220703125}, {"start": 2947.13, "end": 2947.37, "word": " تصبح", "probability": 0.635009765625}, {"start": 2947.37, "end": 2947.73, "word": " صفر", "probability": 0.6690266927083334}, {"start": 2947.73, "end": 2948.39, "word": " دي", "probability": 0.5936279296875}, {"start": 2948.39, "end": 2949.17, "word": " VDW", "probability": 0.66650390625}, {"start": 2949.17, "end": 2949.57, "word": " ال", "probability": 0.447265625}, {"start": 2949.57, "end": 2949.79, "word": " V", "probability": 0.94384765625}, {"start": 2949.79, "end": 2950.03, "word": " من", "probability": 0.732421875}, {"start": 2950.03, "end": 2951.81, "word": " فاضل", "probability": 0.9142252604166666}, {"start": 2951.81, "end": 2952.37, "word": " بصير", "probability": 0.5516764322916666}, {"start": 2952.37, "end": 2952.61, "word": " من", "probability": 0.3486328125}, {"start": 2952.61, "end": 2952.97, "word": " كامل", "probability": 0.9754231770833334}, {"start": 2952.97, "end": 2953.15, "word": " V", "probability": 0.916015625}, {"start": 2953.15, "end": 2953.33, "word": " على", "probability": 0.59619140625}, {"start": 2953.33, "end": 2953.79, "word": " اتنين", "probability": 0.8193359375}, {"start": 2953.79, "end": 2954.31, "word": " زي", "probability": 0.85693359375}, {"start": 2954.31, "end": 2954.93, "word": " VW", "probability": 0.757568359375}, {"start": 2954.93, "end": 2955.17, "word": " من", "probability": 0.95947265625}, {"start": 2955.17, "end": 2955.47, "word": " صفر", "probability": 0.8367513020833334}, {"start": 2955.47, "end": 2955.73, "word": " لعند", "probability": 0.7259114583333334}, {"start": 2955.73, "end": 2956.21, "word": " اتنين", "probability": 0.9398193359375}, {"start": 2956.21, "end": 2957.13, "word": " بنعوض", "probability": 0.832275390625}, {"start": 2957.13, "end": 2957.45, "word": " الآن", "probability": 0.60223388671875}, {"start": 2957.45, "end": 2957.69, "word": " عن", "probability": 0.9150390625}, {"start": 2957.69, "end": 2957.93, "word": " V", "probability": 0.97021484375}, {"start": 2957.93, "end": 2958.61, "word": " بقيمتها", "probability": 0.8642578125}, {"start": 2958.61, "end": 2959.43, "word": " بصير", "probability": 0.8790690104166666}, {"start": 2959.43, "end": 2959.81, "word": " عبارة", "probability": 0.9881591796875}, {"start": 2959.81, "end": 2959.95, "word": " عن", "probability": 0.9736328125}, {"start": 2959.95, "end": 2960.33, "word": " واحد", "probability": 0.94482421875}, {"start": 2960.33, "end": 2960.53, "word": " زي", "probability": 0.917724609375}, {"start": 2960.53, "end": 2960.81, "word": " اتنين", "probability": 0.962646484375}, {"start": 2960.81, "end": 2961.21, "word": " W", "probability": 0.5126953125}], "temperature": 1.0}, {"id": 132, "seek": 298152, "start": 2961.44, "end": 2981.52, "text": "بالكامل هذه بيصير واحد زي ال W تربيع من سفر ل واحد والست البره بيصير ستة في اتنين واحد على اتنين ساوى اتناشر و هيك بيكون احنا انهينا الاخر محاضرة والسلام عليكم ورحمة الله وبركاته وكل عام ولا كل فصل وانتم بخير ربنا ينجحكم جميعا", "tokens": 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2962.28, "end": 2962.64, "word": " هذه", "probability": 0.51025390625}, {"start": 2962.64, "end": 2963.12, "word": " بيصير", "probability": 0.77813720703125}, {"start": 2963.12, "end": 2963.6, "word": " واحد", "probability": 0.849609375}, {"start": 2963.6, "end": 2963.88, "word": " زي", "probability": 0.815673828125}, {"start": 2963.88, "end": 2964.02, "word": " ال", "probability": 0.8271484375}, {"start": 2964.02, "end": 2964.46, "word": " W", "probability": 0.474853515625}, {"start": 2964.46, "end": 2965.26, "word": " تربيع", "probability": 0.9124755859375}, {"start": 2965.26, "end": 2965.64, "word": " من", "probability": 0.978515625}, {"start": 2965.64, "end": 2965.96, "word": " سفر", "probability": 0.7937825520833334}, {"start": 2965.96, "end": 2966.08, "word": " ل", "probability": 0.6826171875}, {"start": 2966.08, "end": 2966.46, "word": " واحد", "probability": 0.788818359375}, {"start": 2966.46, "end": 2966.8, "word": " والست", "probability": 0.64599609375}, {"start": 2966.8, "end": 2967.28, "word": " البره", "probability": 0.5673828125}, {"start": 2967.28, "end": 2967.54, "word": " بيصير", "probability": 0.94921875}, {"start": 2967.54, "end": 2967.86, "word": " ستة", "probability": 0.8525390625}, {"start": 2967.86, "end": 2967.94, "word": " في", "probability": 0.496826171875}, {"start": 2967.94, "end": 2968.36, "word": " اتنين", "probability": 0.9605712890625}, {"start": 2968.36, "end": 2968.98, "word": " واحد", "probability": 0.908935546875}, {"start": 2968.98, "end": 2969.08, "word": " على", "probability": 0.1817626953125}, {"start": 2969.08, "end": 2969.42, "word": " اتنين", "probability": 0.9798583984375}, {"start": 2969.42, "end": 2969.82, "word": " ساوى", "probability": 0.47711181640625}, {"start": 2969.82, "end": 2970.36, "word": " اتناشر", "probability": 0.81259765625}, {"start": 2970.36, "end": 2970.76, "word": " و", "probability": 0.5751953125}, {"start": 2970.76, "end": 2970.96, "word": " هيك", "probability": 0.795166015625}, {"start": 2970.96, "end": 2971.4, "word": " بيكون", "probability": 0.9466145833333334}, {"start": 2971.4, "end": 2971.74, "word": " احنا", "probability": 0.9645182291666666}, {"start": 2971.74, "end": 2972.1, "word": " انهينا", "probability": 0.8638916015625}, {"start": 2972.1, "end": 2972.54, "word": " الاخر", "probability": 0.7030029296875}, {"start": 2972.54, "end": 2973.12, "word": " محاضرة", "probability": 0.98876953125}, {"start": 2973.12, "end": 2973.78, "word": " والسلام", "probability": 0.88232421875}, {"start": 2973.78, "end": 2974.12, "word": " عليكم", "probability": 0.94677734375}, {"start": 2974.12, "end": 2974.46, "word": " ورحمة", "probability": 0.908203125}, {"start": 2974.46, "end": 2974.64, "word": " الله", "probability": 0.96142578125}, {"start": 2974.64, "end": 2975.58, "word": " وبركاته", "probability": 0.990234375}, {"start": 2975.58, "end": 2976.68, "word": " وكل", "probability": 0.79345703125}, {"start": 2976.68, "end": 2977.44, "word": " عام", "probability": 0.984619140625}, {"start": 2977.44, "end": 2977.64, "word": " ولا", "probability": 0.40380859375}, {"start": 2977.64, "end": 2977.92, "word": " كل", "probability": 0.9619140625}, {"start": 2977.92, "end": 2978.34, "word": " فصل", "probability": 0.6856282552083334}, {"start": 2978.34, "end": 2978.94, "word": " وانتم", "probability": 0.7096354166666666}, {"start": 2978.94, "end": 2979.84, "word": " بخير", "probability": 0.99755859375}, {"start": 2979.84, "end": 2980.26, "word": " ربنا", "probability": 0.8478190104166666}, {"start": 2980.26, "end": 2980.78, "word": " ينجحكم", "probability": 0.9869140625}, {"start": 2980.78, "end": 2981.52, "word": " جميعا", "probability": 0.9918212890625}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2981.97625, "duration_after_vad": 2966.753468749996}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/O9Voa-iueFc_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/O9Voa-iueFc_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..5d20134968806475ce7b197fea015e9095a99927
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/O9Voa-iueFc_postprocess.srt
@@ -0,0 +1,840 @@
+1
+00:00:00,640 --> 00:00:03,860
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:03,860 --> 00:00:09,400
+وبركاته هذه المحاضرة رقم 22 مساق تفاضل وتكامل جيم
+
+3
+00:00:09,400 --> 00:00:16,200
+وهي المحاضرة التالتة في شفتر 15 سنتناول فيها اللي
+
+4
+00:00:16,200 --> 00:00:19,580
+هو 15 3 اللي هو تحت عنوان area by double
+
+5
+00:00:19,580 --> 00:00:25,200
+integration وهذه المحاضرة سريعة ووقتها قليل تتناول
+
+6
+00:00:25,200 --> 00:00:30,290
+فكرة واحدة وهي جزء من الفكرة اللي في 15 2الان
+
+7
+00:00:30,290 --> 00:00:35,450
+تعريف احنا في المحاضرة الماضية عرفنا الـ Volume هو
+
+8
+00:00:35,450 --> 00:00:40,450
+عبارة عن الـ Double Integration لاللي هي الـ Da في
+
+9
+00:00:40,450 --> 00:00:49,580
+F of X Y حيث F of X Y هي اللي هي سطح الجسماللي
+
+10
+00:00:49,580 --> 00:00:53,060
+قاعدته
+
+11
+00:00:53,060 --> 00:00:58,640
+اللي هي R وسطحه اللي هو F of XY وZ فكان ال volume
+
+12
+00:00:58,640 --> 00:01:02,540
+بيساوي ال double integral F of XY DA over R على
+
+13
+00:01:02,540 --> 00:01:08,450
+الشتراط أن F of XY أكبر أو يساوي سفرالآن لو كانت
+
+14
+00:01:08,450 --> 00:01:13,910
+الـ function f of x,y بتساوي واحد بيصير عندي وكأني
+
+15
+00:01:13,910 --> 00:01:17,530
+أنا الآن اللي هو اللي .. اللي .. اللي .. اللي عندي
+
+16
+00:01:17,530 --> 00:01:22,070
+عبارة عن area مضروبة في ارتفاع واحد يعني عبارة عن
+
+17
+00:01:22,070 --> 00:01:26,910
+الـ Areaأذا منطقي أن أقول دفينيشن الـ Area of a
+
+18
+00:01:26,910 --> 00:01:30,490
+Closed Bounded Region R يعني الآن لو بدنا مساحة
+
+19
+00:01:30,490 --> 00:01:35,270
+الـ Closed Bounded Region R على طول بس بعمل Double
+
+20
+00:01:35,270 --> 00:01:39,950
+Integration over R دي أيه؟ دي إكس دي واي أو دي واي
+
+21
+00:01:39,950 --> 00:01:44,050
+دي إكس هتطلع عندي على طول من الـ Areaأذا الآن
+
+22
+00:01:44,050 --> 00:01:47,590
+خلّيني أخد مثال على هذا الكلام Find the area of
+
+23
+00:01:47,590 --> 00:01:52,030
+the region R bounded by Y بتساوي X and Y بتساوي X
+
+24
+00:01:52,030 --> 00:01:56,250
+تربيع in the first quadrant بقولي أوجد مساحة
+
+25
+00:01:56,250 --> 00:02:02,110
+المنطقة R المحصورة بين Y بتساوي X وY بتساوي X
+
+26
+00:02:02,110 --> 00:02:05,590
+تربيع وفي ال first quadrant فهيجي برسم Y بتساوي X
+
+27
+00:02:05,590 --> 00:02:10,090
+تربيع هي Y بتساوي X تربيع الآن Y بتساوي X هي Y
+
+28
+00:02:10,090 --> 00:02:14,930
+بتساوي Xهيتقطع في نقطتين كيف بعرف التقاطع زي ما
+
+29
+00:02:14,930 --> 00:02:18,550
+قلنا في المحاضرة الماضية اللي هي بساوي اللي هو ال
+
+30
+00:02:18,550 --> 00:02:21,730
+Y هنا بال Y هنا بيصير X بالساوية X تربيعي يعني X
+
+31
+00:02:21,730 --> 00:02:25,230
+في X minus واحد بيساوي سفر يعني يا X بتساوي سفر يا
+
+32
+00:02:25,230 --> 00:02:28,610
+X بتساوي واحد X بتساوي سفر بتعطيني Y بتساوي سفر
+
+33
+00:02:28,610 --> 00:02:32,210
+فتطلع النقطة هذه سفر سفر X بتساوي واحد بتعطيني Y
+
+34
+00:02:32,210 --> 00:02:36,030
+بتساوي واحد بيصير النقطة هذه واحد واحد هي نقطتي
+
+35
+00:02:36,030 --> 00:02:41,710
+التقاطع وهي المنطقة الموجودة المطلوب إيجاد مسحتها
+
+36
+00:02:41,850 --> 00:02:48,950
+حسب إيجاد المساحة بالـ Single
+
+37
+00:02:48,950 --> 00:02:51,470
+Integration الان الـ Y Double Integration الأمور
+
+38
+00:02:51,470 --> 00:02:55,950
+سهلة كمان الـ Area بالسهولة Double Integral الـ
+
+39
+00:02:55,950 --> 00:03:00,130
+Area بالسهولة Double Integral يفضل بالنسبالي دي Y
+
+40
+00:03:00,130 --> 00:03:04,310
+بي دي X دي Y بي دي X مدام دي Y جامل إذا باخد في
+
+41
+00:03:04,310 --> 00:03:08,810
+المحاضرة الماضية موازي للـ Y Axis في المنطقة بشوف
+
+42
+00:03:08,810 --> 00:03:13,090
+اللي هووين بدخل المنطقة وين بخرج المنطقة وهل هي
+
+43
+00:03:13,090 --> 00:03:16,990
+على نفس المنوال أو لا فعلا هي بدخل المنطقة من عند
+
+44
+00:03:16,990 --> 00:03:20,450
+Y بساوي X تربيع و بخرج من عندها اللي هو من عند Y
+
+45
+00:03:20,450 --> 00:03:23,490
+بساوي X يعني بدخل من عند ال curve و بطلع من عند
+
+46
+00:03:23,490 --> 00:03:28,290
+الخط و كلهم في نفس المنوال عشان هي كقيمة Y هتكون
+
+47
+00:03:28,290 --> 00:03:34,250
+من عند X تربيع لعند مين لعند Xلإيجي لكل الخطوط
+
+48
+00:03:34,250 --> 00:03:36,930
+اللي على نفس المنوال زي ما قلنا المحاضرة الماضية
+
+49
+00:03:36,930 --> 00:03:41,410
+لو اتخيلنا اللي هو اشي مضيق هيك هدا بنزل هيك و
+
+50
+00:03:41,410 --> 00:03:44,930
+بنزل هيك بشكل عموق يعني عمودي موازي للـY Axis
+
+51
+00:03:44,930 --> 00:03:49,250
+فبتلاقيه بمسح كل المنطقة هذه اللي عنده يعني لو
+
+52
+00:03:49,250 --> 00:03:54,310
+اتخيلنا ان الماجو هو هذا المضيق و بمشي هيفدأ يمشي
+
+53
+00:03:54,310 --> 00:03:57,510
+من عند الـ0 و ينتهي عند الواحد و يكون مسح المنطقة
+
+54
+00:03:57,510 --> 00:04:02,630
+كلها فبصير حد X اللي هو من 0 لعند 1وبنكمل الآن
+
+55
+00:04:02,630 --> 00:04:07,890
+التكامل سهل اللي هو على طول تكامل الـ DIY من وين
+
+56
+00:04:07,890 --> 00:04:11,590
+من X تربيع لعند X إلى أن بنجد قيمتها بيصير ال
+
+57
+00:04:11,590 --> 00:04:16,530
+integration من 0 ل1 ل X ناقص X تربيع لكل DX بفاضل
+
+58
+00:04:16,530 --> 00:04:19,970
+.. بكامل هذه لجوا بيصير ناقص X تربية على 2 ناقص X
+
+59
+00:04:19,970 --> 00:04:25,320
+تقريبة على 3 من 0 لعند 1 بعوض الآنبصير نص ناقص طول
+
+60
+00:04:25,320 --> 00:04:29,580
+يساوي سدوس هذه اللي هي مساحة المنطقة اللي عندنا لو
+
+61
+00:04:29,580 --> 00:04:32,920
+واحد قال ليا بدي أعمل DX by DY بقوله مافيش مشكلة
+
+62
+00:04:32,920 --> 00:04:36,700
+أعمل DX by DY بصير الآن تاخد اللي هو الـHorizontal
+
+63
+00:04:36,700 --> 00:04:39,300
+Line اللي هو موازير الـX Axis في هذه الحالة
+
+64
+00:04:39,300 --> 00:04:42,880
+الـHorizontal Line بيكون كله على نفس المنوال و جاي
+
+65
+00:04:42,880 --> 00:04:47,180
+من عند X بتساوي Y لعند X بيساوي جدر الـY الموجب
+
+66
+00:04:47,180 --> 00:04:50,220
+فبصير عنده اللي هي حدود الـIntegration من Y لعند
+
+67
+00:04:50,220 --> 00:04:56,290
+جدر الـYوالتي هي حدود الـ Y في هذه الحالة من 0 إلى
+
+68
+00:04:56,290 --> 00:05:00,950
+1 وتمسح كل الـ Horizontal Lines برضه من 0 إلى 1
+
+69
+00:05:00,950 --> 00:05:05,330
+ومن كامل بنفس الأسلوب سنجد جواب هو نفسه عبارة عن 6
+
+70
+00:05:05,330 --> 00:05:10,830
+وجربهطيب، الآن نأتي ناخد مثال آخر شكله مختلف Find
+
+71
+00:05:10,830 --> 00:05:13,830
+the area of the region R enclosed by the parabola
+
+72
+00:05:13,830 --> 00:05:16,810
+Y بساوية X تربيع and the line Y بساوية X زائد 2
+
+73
+00:05:16,810 --> 00:05:21,370
+بقول لأوجد اللي هي مساحة المنطقة R المحصورة بين Y
+
+74
+00:05:21,370 --> 00:05:25,670
+بساوية X تربيع أو المغلقة بواسط Y بساوية X تربيع و
+
+75
+00:05:25,670 --> 00:05:29,130
+Line Y بساوية X زائد 2 هاي Y بساوية X تربيع، هاي
+
+76
+00:05:29,130 --> 00:05:34,940
+نرسمناها كاملالأن Y بيساوي X زايد اتنين وهي Y
+
+77
+00:05:34,940 --> 00:05:38,080
+بيساوي X زايد اتنين رسمناها بتعرفوا ترسموها كلكم
+
+78
+00:05:38,080 --> 00:05:42,200
+الخط مستقيم بيطلع عندي اللي هو الشكل الأمامي بدي
+
+79
+00:05:42,200 --> 00:05:46,850
+أودد ال area اللي في المنطقة اللي أمامي هذهالان
+
+80
+00:05:46,850 --> 00:05:50,070
+اول اشي بوجد مين يا جماعة خطوط التقاطع وين بتقاطع
+
+81
+00:05:50,070 --> 00:05:54,070
+ان المنحنيين هي المنحنى والمستقيم هي هذه و هذه كيف
+
+82
+00:05:54,070 --> 00:05:58,030
+وجدناها زي ما قلنا المرة الماضية y بيساوي x زي 2 و
+
+83
+00:05:58,030 --> 00:06:01,750
+y بيساوي x تربيع بتيجي تاخد x تربيع بتساوي x زي 2
+
+84
+00:06:01,750 --> 00:06:06,290
+و بتحلل هذه هتلاقي انه اللي هو قيمة x يا 2 يا ناقص
+
+85
+00:06:06,290 --> 00:06:11,430
+1 بتعود عن قيمتها هنا بتطلع عندك 1 و 4 فبتطلع
+
+86
+00:06:11,430 --> 00:06:15,230
+النقطة الأولى ناقص 1 و 1 و النقطة التانية 2 و 4
+
+87
+00:06:15,600 --> 00:06:18,700
+الأن أمورنا جاهزة لكي نجد الـ Area الـ Area ماذا
+
+88
+00:06:18,700 --> 00:06:23,300
+بتساوي؟ بتساوي اللي هو الـ Integration بده كامل
+
+89
+00:06:23,300 --> 00:06:26,900
+مثلا بالنسبة لمين؟ بالنسبة هو جال .. خلينا نكمل
+
+90
+00:06:26,900 --> 00:06:30,780
+بالنسبة ل X جاب الـ Y الآن مدام DX ده مضي أخد
+
+91
+00:06:30,780 --> 00:06:35,820
+موازي لمين في المنطقة؟ لل X-Axis يعني أنا ماشي، هي
+
+92
+00:06:35,820 --> 00:06:38,820
+موازي لل X-Axis هيو .. هيو .. هيو .. هيو .. هيو ..
+
+93
+00:06:38,820 --> 00:06:43,550
+هيو ..هذه المنطقة المنوال تبعها بيجي من هنا لعند
+
+94
+00:06:43,550 --> 00:06:47,350
+هنا بينما اللي فوق منوال آخر بيجي من خط هذا لعند
+
+95
+00:06:47,350 --> 00:06:51,250
+هذا عشان أيه؟ كمضطر أن أقسم المنطقة لمنطقتين فبصير
+
+96
+00:06:51,250 --> 00:06:54,530
+التكامل الـ Area بسوء على المنطقة الأولى R1 زائد
+
+97
+00:06:54,530 --> 00:06:58,930
+على المنطقة التانية R2خلّينا نشوف الآن منطقة R1
+
+98
+00:06:58,930 --> 00:07:03,590
+أول شيء ناخد موازلة الـ X axis فبصير قيمة X هنا
+
+99
+00:07:03,590 --> 00:07:08,010
+جاي من هنا لعندها هذه وين على Y بتساوي X تربيع
+
+100
+00:07:08,010 --> 00:07:12,710
+يعني قيمة الـ X هنا بساوي جدر زائد أو ناقص جدر الـ
+
+101
+00:07:12,710 --> 00:07:15,450
+Y لأن Y بتساوي X تربيع قيمة X بتساوي زائد أو ناقص
+
+102
+00:07:15,450 --> 00:07:19,000
+جدر الـ Yهذه المنطقة طبيعية X بالساوية ناقص جذر Y
+
+103
+00:07:19,000 --> 00:07:23,620
+لأنها المنطقة السالمة في X وهذه المنطقة جذر Y لذلك
+
+104
+00:07:23,620 --> 00:07:28,060
+X أو الـ Horizontal Line جاء من عند X بالساوية
+
+105
+00:07:28,060 --> 00:07:34,520
+ناقص جذر Y وهي ناقص جذر Y ورايح لمن؟ لجذر Y نرى
+
+106
+00:07:34,520 --> 00:07:37,970
+كيف هذه المنطقة تنمسحهي قيمة Y يعني اللي هدول الـ
+
+107
+00:07:37,970 --> 00:07:40,710
+Horizontal Lines بيظل مشيها تيك تيك تيك تيك لما
+
+108
+00:07:40,710 --> 00:07:44,630
+يصل هناك يعني بيصل من عند الصفر لعند المنطقة هذه
+
+109
+00:07:44,630 --> 00:07:49,210
+اللي هي Y برضه بتساوي واحد إذا حدود الـ Y هتكون من
+
+110
+00:07:49,210 --> 00:07:53,390
+صفر لعند واحد زائد التكامل على المنطقة الثانية لما
+
+111
+00:07:53,390 --> 00:07:56,150
+نتكامل على المنطقة الثانية ناخد وين Horizontal
+
+112
+00:07:56,150 --> 00:08:00,330
+Lines كل على نفس المنوال هي بالأحمر الواضح اللي هي
+
+113
+00:08:00,330 --> 00:08:04,870
+من اللي هو الخط لعند اللي هو إيش ال curveالآن في
+
+114
+00:08:04,870 --> 00:08:11,930
+منطقة هذا الخط ما هي قيمة X عليه؟ Y-2 لأن Y زاد X
+
+115
+00:08:11,930 --> 00:08:15,490
+بالساوية 2 يعني X بالساوية Y-2 على الخط هذا قيمة X
+
+116
+00:08:15,490 --> 00:08:20,350
+هي Y-2 لعند هذا هدودنا قيمته اللي هو عبارة عن جذر
+
+117
+00:08:20,350 --> 00:08:24,460
+الـ Yهنا نشوف الـ Horizontal Lines هذه من أين إلى
+
+118
+00:08:24,460 --> 00:08:27,080
+أين تبدأ ومن أين إلى أين تبدأ هي تبدأ من عند الـ
+
+119
+00:08:27,080 --> 00:08:29,540
+Hand اللي كلها على نفسه من ويه اللي عند يعني ويه
+
+120
+00:08:29,540 --> 00:08:34,120
+بتساوي واحد وبتبتسح المنطقة لما تصل إلى ويه بساوي
+
+121
+00:08:34,120 --> 00:08:39,540
+أربعة عشان أيه لحدوث التكامل بالنسبة لويه من واحد
+
+122
+00:08:39,540 --> 00:08:44,160
+لعند أربعة هذه هي ال area و بتحسبها و بتوجد قيمتها
+
+123
+00:08:44,470 --> 00:08:48,210
+الآن لو أحد قال لأ اسمعوا أنا بديش أشتغل زيكم بدي
+
+124
+00:08:48,210 --> 00:08:51,030
+أشتغل أو أقول شي دي واي باي دي إكس ونقوله نوع على
+
+125
+00:08:51,030 --> 00:08:54,250
+نوع و أحسنلك كمان، ليش أحسنلك؟ لأن كلها جاية من
+
+126
+00:08:54,250 --> 00:08:59,010
+أيش؟ على نفس المنوال اللي هي جاية من عند الـ curve
+
+127
+00:08:59,010 --> 00:09:03,200
+Y بالساوي X تربيعY بيساوي X تربيع لعند Y أيش
+
+128
+00:09:03,200 --> 00:09:06,580
+بتساوي X زائد 2؟ واحد يقول لي لأ ما هو هذا مختلف
+
+129
+00:09:06,580 --> 00:09:10,460
+عن هذا لأ أنا قيمة Y هنا و هنا زي بعض لأنه أنا
+
+130
+00:09:10,460 --> 00:09:13,420
+الآن مازال باخد موازي للـ Y axis لإنه بدأ كامل
+
+131
+00:09:13,420 --> 00:09:17,100
+بالنسبة للـ Y في الأول إذا قيمة Y هنا بيساوي X
+
+132
+00:09:17,100 --> 00:09:20,860
+تربيع وقيمة Y هنا برضه بيساوي X تربيع كلها جاي من
+
+133
+00:09:20,860 --> 00:09:25,060
+عند Y بيساوي X تربيع لعند Y أيش بيساوي X زائد 2؟
+
+134
+00:09:25,120 --> 00:09:30,640
+اللي هي ده من X تربيع لعند X زائد اتنين لأن قيمة Y
+
+135
+00:09:30,640 --> 00:09:34,160
+على هذا الجزء وعلى هذا الجزء زي بعض بخلاف X اللي
+
+136
+00:09:34,160 --> 00:09:38,160
+كانت هنا X بيساوي نقل جذر ال Y وهنا X بيساوي جذر
+
+137
+00:09:38,160 --> 00:09:42,040
+ال Y لإن هو منطقة X سالب أو منطقة X موجبة بينما Y
+
+138
+00:09:42,040 --> 00:09:45,780
+كلها بتساوي X تربيع على هذا الكيرت سواء من المنطقة
+
+139
+00:09:45,780 --> 00:09:50,540
+هذه أو من المنطقة هذهطيب، الان واضح انه اللي هو
+
+140
+00:09:50,540 --> 00:09:53,880
+كله على نفس المنوال بيجي من هنا الموازيات للـY
+
+141
+00:09:53,880 --> 00:09:57,940
+Axis وبروح للي فوق اللي هو الخط الان طب هدولة اللي
+
+142
+00:09:57,940 --> 00:10:01,580
+مشيت هي الشكل هذا اللي هو الـ Vertical من وين
+
+143
+00:10:01,580 --> 00:10:07,340
+ببدن؟ ببدن من عند اللي هي X-1وبتدرك تمشي كتابة
+
+144
+00:10:07,340 --> 00:10:08,160
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+145
+00:10:08,160 --> 00:10:09,400
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+146
+00:10:09,400 --> 00:10:10,160
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+147
+00:10:10,160 --> 00:10:13,940
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+148
+00:10:13,940 --> 00:10:14,340
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+149
+00:10:14,340 --> 00:10:21,080
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+150
+00:10:21,080 --> 00:10:22,700
+كتابة كتابة كتابة كتابة كتابةاللي هو من X تربيع
+
+151
+00:10:22,700 --> 00:10:26,060
+إلى X زائد 2 بتعوض بيصير X زائد 2 ناقص X تربيع
+
+152
+00:10:26,060 --> 00:10:29,180
+بتكامل كمان مرة بالنسبة ل X أو بتكامل بالنسبة ل X
+
+153
+00:10:29,180 --> 00:10:33,280
+بيصير X تربيع على 2 زائد 2 X ناقص X تكعيب على 3
+
+154
+00:10:33,280 --> 00:10:38,700
+بناقص 1 لعند 2 بنعوض بيصير اللي هي عبارة عن 4 على
+
+155
+00:10:38,700 --> 00:10:44,080
+2 اللي هي 2 زائد 4 ناقص اللي هي 8 على 3 بتطلع 9
+
+156
+00:10:44,080 --> 00:10:49,910
+على 2 وهيك بيكون حلنا السؤالديجي لحاجة اسم الـ
+
+157
+00:10:49,910 --> 00:10:54,290
+Average Value of F over R الـ Average Value للـ F
+
+158
+00:10:54,290 --> 00:10:58,590
+over R اللي هي زمان كنا نحكي عن الـ Average Value
+
+159
+00:10:58,590 --> 00:11:02,790
+العادية مجموع القراءات على عددها أو لو كان على خط
+
+160
+00:11:02,790 --> 00:11:07,730
+مستقيم لو كلهم كانت متصلات بيكون اللي هو عبارة عن
+
+161
+00:11:07,730 --> 00:11:11,210
+اللي هي عندي الـ Average Value مجموع اللي هي القيم
+
+162
+00:11:11,210 --> 00:11:15,400
+اللي عليها على طول اللي هي الفترة ماشي الحالهنا
+
+163
+00:11:15,400 --> 00:11:18,860
+الأن المنطقي بيقول لك الـ Average Value of F over
+
+164
+00:11:18,860 --> 00:11:24,440
+R قيمة F التي بتصنحها الـ Double Integral FDA على
+
+165
+00:11:24,440 --> 00:11:29,960
+المساحة الكلية بتطلع اللي هي و كأنه صار حجم على
+
+166
+00:11:29,960 --> 00:11:33,700
+مساحة بتطلع اللي هي مين بتطلع و بتظلها الـ F لأن
+
+167
+00:11:33,700 --> 00:11:36,960
+هذه عبارة عن .. خلينا نقول Roughly عبارة عن F
+
+168
+00:11:36,960 --> 00:11:40,740
+مساحة على مساحة بتطلع الـ F لأن هذه بتعمل مساحة و
+
+169
+00:11:40,740 --> 00:11:43,580
+هذه بتعمل مساحة هذا الكلام Roughly طبعا يعني لكن
+
+170
+00:11:43,580 --> 00:11:48,030
+الـ Meaningاللي هو الـ Double Integral over R FDA
+
+171
+00:11:48,030 --> 00:11:51,090
+على الـ Area للـ R التي تظهر وهي الـ Average Value
+
+172
+00:11:51,090 --> 00:11:55,270
+لمين الـ F over R فمنطق التعريف هذا الـ Average
+
+173
+00:11:55,270 --> 00:11:59,030
+Value of F over R بسوء واحد على الـ Area of R في
+
+174
+00:11:59,030 --> 00:12:03,550
+الـ Double Integral للـ FDA over R نيجي ناخد مثال،
+
+175
+00:12:03,550 --> 00:12:07,870
+ناخدلكم مثال بسيط هو الآن احنا تقريبا بالمحاضرة
+
+176
+00:12:07,870 --> 00:12:11,200
+هذه والمحاضرة اللي فاتتغطينا كل الإمكانيات كيف
+
+177
+00:12:11,200 --> 00:12:14,020
+نودد اللي هو الـ Double Integration على أي منطقة
+
+178
+00:12:14,020 --> 00:12:17,440
+حتى لو كانت صعبة و بتتجذب Find the average value
+
+179
+00:12:17,440 --> 00:12:22,380
+of f of x,y بساوية x cos x,y over the rectangle x
+
+180
+00:12:22,380 --> 00:12:25,480
+بين السفر و بين الـπي و y بين السفر و بين الواحد
+
+181
+00:12:25,620 --> 00:12:29,080
+واضح أن الـ Integration على ركتانجل إذا وانا مغمض
+
+182
+00:12:29,080 --> 00:12:35,400
+عينيا اللي هي DX X من سفر لعند Y و DY من عند سفر Y
+
+183
+00:12:35,400 --> 00:12:37,620
+من عند سفر لعند واحد على الـ Rectangle لكن لو
+
+184
+00:12:37,620 --> 00:12:40,880
+Region بدنا نضطر نعمل زي ما عملنا قبل شوية في حدود
+
+185
+00:12:40,880 --> 00:12:44,280
+الـ Integration الان الـ Value of the integral F
+
+186
+00:12:44,280 --> 00:12:47,960
+over R نوجد أول شيء هذا و بعدين نوجد الـ Area و
+
+187
+00:12:47,960 --> 00:12:51,300
+بنكسب هذا على هذا بتطلع الـ Average Valueالان نأخد
+
+188
+00:12:51,300 --> 00:12:55,580
+الـ integration الـ integration لمن؟ لـ X cos XY
+
+189
+00:12:55,580 --> 00:13:01,780
+DY DX من 0 لعند 1 و من 0 لعند مين؟ لعند Y طبعاً
+
+190
+00:13:01,780 --> 00:13:05,240
+اعملنا DY by DX لأنه DX .. DY .. هنعجّد الأمور فDY
+
+191
+00:13:05,240 --> 00:13:10,940
+by DX على طول بتطلع هذه هي عبارة عن اللي هي تفاضل
+
+192
+00:13:10,940 --> 00:13:15,780
+اللي هي sin X في Y عشان هي هذي تكملة اللي هي sin
+
+193
+00:13:15,780 --> 00:13:22,250
+XYمن عند اللي هو Zero لعند الواحد اللي كله DX Sine
+
+194
+00:13:22,250 --> 00:13:26,850
+XY اللي هي مايمن Zero لعند واحد إيش بتساوي اللي هي
+
+195
+00:13:26,850 --> 00:13:32,270
+Sine الواحد ال X ناقص لإنه Y بنعوض ناقص Sine Zero
+
+196
+00:13:32,270 --> 00:13:35,470
+اللي هو Zero اللي بيصير Sine X من سفر لضاي Sine X
+
+197
+00:13:35,470 --> 00:13:39,130
+من سفر لضاي اللي عبارة عن ناقص Cos X من سفر لضاي
+
+198
+00:13:39,130 --> 00:13:43,980
+اللي هو من سفر لضاي بنعوض اللي هيعبارة عن ناقص cos
+
+199
+00:13:43,980 --> 00:13:48,040
+πي هي ناقص ناقص واحد يعني واحد ناقص ناقص اللي هي
+
+200
+00:13:48,040 --> 00:13:50,360
+cosine الـ zero يعني زائد cosine الـ zero اللي هو
+
+201
+00:13:50,360 --> 00:13:53,060
+واحد يعني بيطلع اللي هو ال average value لل
+
+202
+00:13:53,060 --> 00:13:56,880
+function هذه over المنطقة هذه بساوية لأ آسف لأن
+
+203
+00:13:56,880 --> 00:13:59,240
+قيمة ال integration بساوية اتنين أما ال average
+
+204
+00:13:59,240 --> 00:14:03,380
+value احنا بدنا نجيب مساحة لمستطيل مستطيل طوله باي
+
+205
+00:14:03,510 --> 00:14:09,330
+وعرضه 1 مثلا بيصير مساحته 1 في باي يعني باي بيصير
+
+206
+00:14:09,330 --> 00:14:12,190
+1 على باي مضروب في الاتنين اللي هي قيمة ال
+
+207
+00:14:12,190 --> 00:14:15,110
+integration اللي وجدناها بتطلع ال average value of
+
+208
+00:14:15,110 --> 00:14:19,350
+F over R اللي عبارة عن 2 على المساحة اللي هي باي
+
+209
+00:14:19,350 --> 00:14:23,550
+وهو المطلوب ورأيت ممكن خلصنا المحاضرة وإلى لقاء
+
+210
+00:14:23,550 --> 00:14:24,670
+آخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/O9Voa-iueFc_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/O9Voa-iueFc_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..2d92c34a0cb1e97a793e43e1e9f0fa2c410ebfab
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/O9Voa-iueFc_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2924, "start": 0.64, "end": 29.24, "text": "بسم الله الرحمن الرحيم السلام عليكم ورحمة الله وبركاته هذه المحاضرة رقم 22 مساق تفاضل وتكامل جيم وهي المحاضرة التالتة في شفتر 15 سنتناول فيها اللي هو 15 3 اللي هو تحت عنوان area by double integration وهذه المحاضرة سريعة ووقتها قليل تتناول فكرة واحدة وهي جزء من الفكرة اللي في 15 2", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 29538, 9673, 5016, 46958, 25720, 12602, 4587, 2304, 5853, 47524, 995, 4587, 6055, 5172, 46958, 1211, 34683, 4117, 10943, 1211, 10874, 32640, 37037, 1829, 9673, 5016, 46958, 25720, 16712, 6027, 2655, 3660, 8978, 13412, 5172, 2655, 2288, 2119, 8608, 29399, 8315, 12610, 8978, 11296, 13672, 1829, 31439, 2119, 805, 13672, 1829, 31439, 6055, 33753, 18871, 2407, 7649, 1859, 538, 3834, 10980, 37037, 24192, 9673, 5016, 46958, 25720, 8608, 16572, 27884, 4032, 30543, 2655, 11296, 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"compression_ratio": 1.4639639639639639, "no_speech_prob": 5.960464477539062e-07, "words": [{"start": 729.22, "end": 729.88, "word": "غطينا", "probability": 0.871826171875}, {"start": 729.88, "end": 730.14, "word": " كل", "probability": 0.92041015625}, {"start": 730.14, "end": 730.96, "word": " الإمكانيات", "probability": 0.89599609375}, {"start": 730.96, "end": 731.2, "word": " كيف", "probability": 0.765380859375}, {"start": 731.2, "end": 731.58, "word": " نودد", "probability": 0.6002604166666666}, {"start": 731.58, "end": 731.7, "word": " اللي", "probability": 0.6630859375}, {"start": 731.7, "end": 731.92, "word": " هو", "probability": 0.91748046875}, {"start": 731.92, "end": 732.2, "word": " الـ", "probability": 0.927490234375}, {"start": 732.2, "end": 732.3, "word": " Double", "probability": 0.5048828125}, {"start": 732.3, "end": 732.94, "word": " Integration", "probability": 0.68603515625}, {"start": 732.94, "end": 733.26, "word": " على", "probability": 0.77783203125}, {"start": 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" of", "probability": 0.95166015625}, {"start": 737.7, "end": 738.26, "word": " f", "probability": 0.370849609375}, {"start": 738.26, "end": 738.48, "word": " of", "probability": 0.08544921875}, {"start": 738.48, "end": 738.66, "word": " x", "probability": 0.662109375}, {"start": 738.66, "end": 738.96, "word": ",y", "probability": 0.6634521484375}, {"start": 738.96, "end": 739.42, "word": " بساوية", "probability": 0.617578125}, {"start": 739.42, "end": 739.66, "word": " x", "probability": 0.8291015625}, {"start": 739.66, "end": 739.9, "word": " cos", "probability": 0.6220703125}, {"start": 739.9, "end": 740.32, "word": " x", "probability": 0.54541015625}, {"start": 740.32, "end": 740.7, "word": ",y", "probability": 0.87890625}, {"start": 740.7, "end": 741.28, "word": " over", "probability": 0.81884765625}, {"start": 741.28, "end": 741.5, "word": " the", "probability": 0.873046875}, {"start": 741.5, "end": 742.0, "word": " rectangle", "probability": 0.921875}, {"start": 742.0, "end": 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"probability": 0.9627278645833334}], "temperature": 1.0}, {"id": 34, "seek": 77002, "start": 745.62, "end": 770.02, "text": "واضح أن الـ Integration على ركتانجل إذا وانا مغمض عينيا اللي هي DX X من سفر لعند Y و DY من عند سفر Y من عند سفر لعند واحد على الـ Rectangle لكن لو Region بدنا نضطر نعمل زي ما عملنا قبل شوية في حدود الـ Integration الان الـ Value of the integral F over R نوجد أول شيء هذا و بعدين نوجد الـ Area و بنكسب هذا على هذا بتطلع الـ Average Value", "tokens": [2407, 46958, 5016, 14739, 2423, 39184, 47713, 15844, 12602, 4117, 2655, 7649, 7435, 1211, 11933, 15730, 4032, 7649, 995, 3714, 17082, 2304, 11242, 6225, 9957, 25528, 13672, 1829, 39896, 48817, 1783, 9154, 8608, 5172, 2288, 5296, 3615, 41260, 398, 4032, 48427, 9154, 43242, 8608, 5172, 2288, 398, 9154, 43242, 8608, 5172, 2288, 5296, 3615, 41260, 36764, 24401, 15844, 2423, 39184, 497, 557, 7846, 44381, 45164, 25121, 47525, 8315, 8717, 11242, 9566, 2288, 8717, 25957, 1211, 30767, 1829, 19446, 6225, 42213, 8315, 12174, 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3714, 43020, 16490, 1211, 9381, 8315, 9673, 5016, 46958, 25720, 4032, 28814, 23942, 5296, 4587, 16606, 19753, 34740], "avg_logprob": -0.23831644305237779, "compression_ratio": 1.7565217391304349, "no_speech_prob": 0.00012445449829101562, "words": [{"start": 843.51, "end": 844.15, "word": "وعرضه", "probability": 0.579803466796875}, {"start": 844.15, "end": 844.39, "word": " 1", "probability": 0.66796875}, {"start": 844.39, "end": 844.85, "word": " مثلا", "probability": 0.714599609375}, {"start": 844.85, "end": 845.75, "word": " بيصير", "probability": 0.68817138671875}, {"start": 845.75, "end": 846.83, "word": " مساحته", "probability": 0.76123046875}, {"start": 846.83, "end": 847.37, "word": " 1", "probability": 0.67724609375}, {"start": 847.37, "end": 847.65, "word": " في", "probability": 0.7314453125}, {"start": 847.65, "end": 847.85, "word": " باي", "probability": 0.488250732421875}, {"start": 847.85, "end": 848.21, "word": " يعني", "probability": 0.904296875}, {"start": 848.21, "end": 848.61, "word": " باي", "probability": 0.885986328125}, {"start": 848.61, "end": 849.33, "word": " بيصير", "probability": 0.9156494140625}, {"start": 849.33, "end": 849.71, "word": " 1", "probability": 0.78466796875}, {"start": 849.71, "end": 850.01, "word": " على", "probability": 0.810546875}, {"start": 850.01, "end": 850.39, "word": " باي", "probability": 0.85009765625}, {"start": 850.39, "end": 850.89, "word": " مضروب", "probability": 0.9581298828125}, {"start": 850.89, "end": 851.03, "word": " في", "probability": 0.8671875}, {"start": 851.03, "end": 851.49, "word": " الاتنين", "probability": 0.76580810546875}, {"start": 851.49, "end": 851.65, "word": " اللي", "probability": 0.75244140625}, {"start": 851.65, "end": 851.77, "word": " هي", "probability": 0.9267578125}, {"start": 851.77, "end": 852.09, "word": " قيمة", "probability": 0.9930013020833334}, {"start": 852.09, "end": 852.19, "word": " ال", "probability": 0.9814453125}, {"start": 852.19, "end": 852.59, "word": " integration", "probability": 0.75}, {"start": 852.59, "end": 852.81, "word": " اللي", "probability": 0.968505859375}, {"start": 852.81, "end": 853.43, "word": " وجدناها", "probability": 0.8680419921875}, {"start": 853.43, "end": 854.19, "word": " بتطلع", "probability": 0.848876953125}, {"start": 854.19, "end": 854.29, "word": " ال", "probability": 0.88671875}, {"start": 854.29, "end": 854.55, "word": " average", "probability": 0.7529296875}, {"start": 854.55, "end": 854.95, "word": " value", "probability": 0.98095703125}, {"start": 854.95, "end": 855.11, "word": " of", "probability": 0.37939453125}, {"start": 855.11, "end": 855.27, "word": " F", "probability": 0.59423828125}, {"start": 855.27, "end": 855.59, "word": " over", "probability": 0.83837890625}, {"start": 855.59, "end": 855.83, "word": " R", "probability": 0.9501953125}, {"start": 855.83, "end": 855.95, "word": " اللي", "probability": 0.6077880859375}, {"start": 855.95, "end": 856.33, "word": " عبارة", "probability": 0.94921875}, {"start": 856.33, "end": 856.55, "word": " عن", "probability": 0.986328125}, {"start": 856.55, "end": 857.07, "word": " 2", "probability": 0.69287109375}, {"start": 857.07, "end": 857.95, "word": " على", "probability": 0.82373046875}, {"start": 857.95, "end": 858.63, "word": " المساحة", "probability": 0.953369140625}, {"start": 858.63, "end": 858.75, "word": " اللي", "probability": 0.931640625}, {"start": 858.75, "end": 858.97, "word": " هي", "probability": 0.9091796875}, {"start": 858.97, "end": 859.35, "word": " باي", "probability": 0.824462890625}, {"start": 859.35, "end": 859.97, "word": " وهو", "probability": 0.833984375}, {"start": 859.97, "end": 860.61, "word": " المطلوب", "probability": 0.9830322265625}, {"start": 860.61, "end": 861.21, "word": " ورأيت", "probability": 0.54290771484375}, {"start": 861.21, "end": 861.43, "word": " ممكن", "probability": 0.75439453125}, {"start": 861.43, "end": 861.95, "word": " خلصنا", "probability": 0.8875732421875}, {"start": 861.95, "end": 862.45, "word": " المحاضرة", "probability": 0.9617919921875}, {"start": 862.45, "end": 862.91, "word": " وإلى", "probability": 0.84375}, {"start": 862.91, "end": 863.55, "word": " لقاء", "probability": 0.9612630208333334}, {"start": 863.55, "end": 864.67, "word": " آخر", "probability": 0.81396484375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 865.872125, "duration_after_vad": 860.6143750000002}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/O9Voa-iueFc_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/O9Voa-iueFc_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..5d20134968806475ce7b197fea015e9095a99927
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/O9Voa-iueFc_raw.srt
@@ -0,0 +1,840 @@
+1
+00:00:00,640 --> 00:00:03,860
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:03,860 --> 00:00:09,400
+وبركاته هذه المحاضرة رقم 22 مساق تفاضل وتكامل جيم
+
+3
+00:00:09,400 --> 00:00:16,200
+وهي المحاضرة التالتة في شفتر 15 سنتناول فيها اللي
+
+4
+00:00:16,200 --> 00:00:19,580
+هو 15 3 اللي هو تحت عنوان area by double
+
+5
+00:00:19,580 --> 00:00:25,200
+integration وهذه المحاضرة سريعة ووقتها قليل تتناول
+
+6
+00:00:25,200 --> 00:00:30,290
+فكرة واحدة وهي جزء من الفكرة اللي في 15 2الان
+
+7
+00:00:30,290 --> 00:00:35,450
+تعريف احنا في المحاضرة الماضية عرفنا الـ Volume هو
+
+8
+00:00:35,450 --> 00:00:40,450
+عبارة عن الـ Double Integration لاللي هي الـ Da في
+
+9
+00:00:40,450 --> 00:00:49,580
+F of X Y حيث F of X Y هي اللي هي سطح الجسماللي
+
+10
+00:00:49,580 --> 00:00:53,060
+قاعدته
+
+11
+00:00:53,060 --> 00:00:58,640
+اللي هي R وسطحه اللي هو F of XY وZ فكان ال volume
+
+12
+00:00:58,640 --> 00:01:02,540
+بيساوي ال double integral F of XY DA over R على
+
+13
+00:01:02,540 --> 00:01:08,450
+الشتراط أن F of XY أكبر أو يساوي سفرالآن لو كانت
+
+14
+00:01:08,450 --> 00:01:13,910
+الـ function f of x,y بتساوي واحد بيصير عندي وكأني
+
+15
+00:01:13,910 --> 00:01:17,530
+أنا الآن اللي هو اللي .. اللي .. اللي .. اللي عندي
+
+16
+00:01:17,530 --> 00:01:22,070
+عبارة عن area مضروبة في ارتفاع واحد يعني عبارة عن
+
+17
+00:01:22,070 --> 00:01:26,910
+الـ Areaأذا منطقي أن أقول دفينيشن الـ Area of a
+
+18
+00:01:26,910 --> 00:01:30,490
+Closed Bounded Region R يعني الآن لو بدنا مساحة
+
+19
+00:01:30,490 --> 00:01:35,270
+الـ Closed Bounded Region R على طول بس بعمل Double
+
+20
+00:01:35,270 --> 00:01:39,950
+Integration over R دي أيه؟ دي إكس دي واي أو دي واي
+
+21
+00:01:39,950 --> 00:01:44,050
+دي إكس هتطلع عندي على طول من الـ Areaأذا الآن
+
+22
+00:01:44,050 --> 00:01:47,590
+خلّيني أخد مثال على هذا الكلام Find the area of
+
+23
+00:01:47,590 --> 00:01:52,030
+the region R bounded by Y بتساوي X and Y بتساوي X
+
+24
+00:01:52,030 --> 00:01:56,250
+تربيع in the first quadrant بقولي أوجد مساحة
+
+25
+00:01:56,250 --> 00:02:02,110
+المنطقة R المحصورة بين Y بتساوي X وY بتساوي X
+
+26
+00:02:02,110 --> 00:02:05,590
+تربيع وفي ال first quadrant فهيجي برسم Y بتساوي X
+
+27
+00:02:05,590 --> 00:02:10,090
+تربيع هي Y بتساوي X تربيع الآن Y بتساوي X هي Y
+
+28
+00:02:10,090 --> 00:02:14,930
+بتساوي Xهيتقطع في نقطتين كيف بعرف التقاطع زي ما
+
+29
+00:02:14,930 --> 00:02:18,550
+قلنا في المحاضرة الماضية اللي هي بساوي اللي هو ال
+
+30
+00:02:18,550 --> 00:02:21,730
+Y هنا بال Y هنا بيصير X بالساوية X تربيعي يعني X
+
+31
+00:02:21,730 --> 00:02:25,230
+في X minus واحد بيساوي سفر يعني يا X بتساوي سفر يا
+
+32
+00:02:25,230 --> 00:02:28,610
+X بتساوي واحد X بتساوي سفر بتعطيني Y بتساوي سفر
+
+33
+00:02:28,610 --> 00:02:32,210
+فتطلع النقطة هذه سفر سفر X بتساوي واحد بتعطيني Y
+
+34
+00:02:32,210 --> 00:02:36,030
+بتساوي واحد بيصير النقطة هذه واحد واحد هي نقطتي
+
+35
+00:02:36,030 --> 00:02:41,710
+التقاطع وهي المنطقة الموجودة المطلوب إيجاد مسحتها
+
+36
+00:02:41,850 --> 00:02:48,950
+حسب إيجاد المساحة بالـ Single
+
+37
+00:02:48,950 --> 00:02:51,470
+Integration الان الـ Y Double Integration الأمور
+
+38
+00:02:51,470 --> 00:02:55,950
+سهلة كمان الـ Area بالسهولة Double Integral الـ
+
+39
+00:02:55,950 --> 00:03:00,130
+Area بالسهولة Double Integral يفضل بالنسبالي دي Y
+
+40
+00:03:00,130 --> 00:03:04,310
+بي دي X دي Y بي دي X مدام دي Y جامل إذا باخد في
+
+41
+00:03:04,310 --> 00:03:08,810
+المحاضرة الماضية موازي للـ Y Axis في المنطقة بشوف
+
+42
+00:03:08,810 --> 00:03:13,090
+اللي هووين بدخل المنطقة وين بخرج المنطقة وهل هي
+
+43
+00:03:13,090 --> 00:03:16,990
+على نفس المنوال أو لا فعلا هي بدخل المنطقة من عند
+
+44
+00:03:16,990 --> 00:03:20,450
+Y بساوي X تربيع و بخرج من عندها اللي هو من عند Y
+
+45
+00:03:20,450 --> 00:03:23,490
+بساوي X يعني بدخل من عند ال curve و بطلع من عند
+
+46
+00:03:23,490 --> 00:03:28,290
+الخط و كلهم في نفس المنوال عشان هي كقيمة Y هتكون
+
+47
+00:03:28,290 --> 00:03:34,250
+من عند X تربيع لعند مين لعند Xلإيجي لكل الخطوط
+
+48
+00:03:34,250 --> 00:03:36,930
+اللي على نفس المنوال زي ما قلنا المحاضرة الماضية
+
+49
+00:03:36,930 --> 00:03:41,410
+لو اتخيلنا اللي هو اشي مضيق هيك هدا بنزل هيك و
+
+50
+00:03:41,410 --> 00:03:44,930
+بنزل هيك بشكل عموق يعني عمودي موازي للـY Axis
+
+51
+00:03:44,930 --> 00:03:49,250
+فبتلاقيه بمسح كل المنطقة هذه اللي عنده يعني لو
+
+52
+00:03:49,250 --> 00:03:54,310
+اتخيلنا ان الماجو هو هذا المضيق و بمشي هيفدأ يمشي
+
+53
+00:03:54,310 --> 00:03:57,510
+من عند الـ0 و ينتهي عند الواحد و يكون مسح المنطقة
+
+54
+00:03:57,510 --> 00:04:02,630
+كلها فبصير حد X اللي هو من 0 لعند 1وبنكمل الآن
+
+55
+00:04:02,630 --> 00:04:07,890
+التكامل سهل اللي هو على طول تكامل الـ DIY من وين
+
+56
+00:04:07,890 --> 00:04:11,590
+من X تربيع لعند X إلى أن بنجد قيمتها بيصير ال
+
+57
+00:04:11,590 --> 00:04:16,530
+integration من 0 ل1 ل X ناقص X تربيع لكل DX بفاضل
+
+58
+00:04:16,530 --> 00:04:19,970
+.. بكامل هذه لجوا بيصير ناقص X تربية على 2 ناقص X
+
+59
+00:04:19,970 --> 00:04:25,320
+تقريبة على 3 من 0 لعند 1 بعوض الآنبصير نص ناقص طول
+
+60
+00:04:25,320 --> 00:04:29,580
+يساوي سدوس هذه اللي هي مساحة المنطقة اللي عندنا لو
+
+61
+00:04:29,580 --> 00:04:32,920
+واحد قال ليا بدي أعمل DX by DY بقوله مافيش مشكلة
+
+62
+00:04:32,920 --> 00:04:36,700
+أعمل DX by DY بصير الآن تاخد اللي هو الـHorizontal
+
+63
+00:04:36,700 --> 00:04:39,300
+Line اللي هو موازير الـX Axis في هذه الحالة
+
+64
+00:04:39,300 --> 00:04:42,880
+الـHorizontal Line بيكون كله على نفس المنوال و جاي
+
+65
+00:04:42,880 --> 00:04:47,180
+من عند X بتساوي Y لعند X بيساوي جدر الـY الموجب
+
+66
+00:04:47,180 --> 00:04:50,220
+فبصير عنده اللي هي حدود الـIntegration من Y لعند
+
+67
+00:04:50,220 --> 00:04:56,290
+جدر الـYوالتي هي حدود الـ Y في هذه الحالة من 0 إلى
+
+68
+00:04:56,290 --> 00:05:00,950
+1 وتمسح كل الـ Horizontal Lines برضه من 0 إلى 1
+
+69
+00:05:00,950 --> 00:05:05,330
+ومن كامل بنفس الأسلوب سنجد جواب هو نفسه عبارة عن 6
+
+70
+00:05:05,330 --> 00:05:10,830
+وجربهطيب، الآن نأتي ناخد مثال آخر شكله مختلف Find
+
+71
+00:05:10,830 --> 00:05:13,830
+the area of the region R enclosed by the parabola
+
+72
+00:05:13,830 --> 00:05:16,810
+Y بساوية X تربيع and the line Y بساوية X زائد 2
+
+73
+00:05:16,810 --> 00:05:21,370
+بقول لأوجد اللي هي مساحة المنطقة R المحصورة بين Y
+
+74
+00:05:21,370 --> 00:05:25,670
+بساوية X تربيع أو المغلقة بواسط Y بساوية X تربيع و
+
+75
+00:05:25,670 --> 00:05:29,130
+Line Y بساوية X زائد 2 هاي Y بساوية X تربيع، هاي
+
+76
+00:05:29,130 --> 00:05:34,940
+نرسمناها كاملالأن Y بيساوي X زايد اتنين وهي Y
+
+77
+00:05:34,940 --> 00:05:38,080
+بيساوي X زايد اتنين رسمناها بتعرفوا ترسموها كلكم
+
+78
+00:05:38,080 --> 00:05:42,200
+الخط مستقيم بيطلع عندي اللي هو الشكل الأمامي بدي
+
+79
+00:05:42,200 --> 00:05:46,850
+أودد ال area اللي في المنطقة اللي أمامي هذهالان
+
+80
+00:05:46,850 --> 00:05:50,070
+اول اشي بوجد مين يا جماعة خطوط التقاطع وين بتقاطع
+
+81
+00:05:50,070 --> 00:05:54,070
+ان المنحنيين هي المنحنى والمستقيم هي هذه و هذه كيف
+
+82
+00:05:54,070 --> 00:05:58,030
+وجدناها زي ما قلنا المرة الماضية y بيساوي x زي 2 و
+
+83
+00:05:58,030 --> 00:06:01,750
+y بيساوي x تربيع بتيجي تاخد x تربيع بتساوي x زي 2
+
+84
+00:06:01,750 --> 00:06:06,290
+و بتحلل هذه هتلاقي انه اللي هو قيمة x يا 2 يا ناقص
+
+85
+00:06:06,290 --> 00:06:11,430
+1 بتعود عن قيمتها هنا بتطلع عندك 1 و 4 فبتطلع
+
+86
+00:06:11,430 --> 00:06:15,230
+النقطة الأولى ناقص 1 و 1 و النقطة التانية 2 و 4
+
+87
+00:06:15,600 --> 00:06:18,700
+الأن أمورنا جاهزة لكي نجد الـ Area الـ Area ماذا
+
+88
+00:06:18,700 --> 00:06:23,300
+بتساوي؟ بتساوي اللي هو الـ Integration بده كامل
+
+89
+00:06:23,300 --> 00:06:26,900
+مثلا بالنسبة لمين؟ بالنسبة هو جال .. خلينا نكمل
+
+90
+00:06:26,900 --> 00:06:30,780
+بالنسبة ل X جاب الـ Y الآن مدام DX ده مضي أخد
+
+91
+00:06:30,780 --> 00:06:35,820
+موازي لمين في المنطقة؟ لل X-Axis يعني أنا ماشي، هي
+
+92
+00:06:35,820 --> 00:06:38,820
+موازي لل X-Axis هيو .. هيو .. هيو .. هيو .. هيو ..
+
+93
+00:06:38,820 --> 00:06:43,550
+هيو ..هذه المنطقة المنوال تبعها بيجي من هنا لعند
+
+94
+00:06:43,550 --> 00:06:47,350
+هنا بينما اللي فوق منوال آخر بيجي من خط هذا لعند
+
+95
+00:06:47,350 --> 00:06:51,250
+هذا عشان أيه؟ كمضطر أن أقسم المنطقة لمنطقتين فبصير
+
+96
+00:06:51,250 --> 00:06:54,530
+التكامل الـ Area بسوء على المنطقة الأولى R1 زائد
+
+97
+00:06:54,530 --> 00:06:58,930
+على المنطقة التانية R2خلّينا نشوف الآن منطقة R1
+
+98
+00:06:58,930 --> 00:07:03,590
+أول شيء ناخد موازلة الـ X axis فبصير قيمة X هنا
+
+99
+00:07:03,590 --> 00:07:08,010
+جاي من هنا لعندها هذه وين على Y بتساوي X تربيع
+
+100
+00:07:08,010 --> 00:07:12,710
+يعني قيمة الـ X هنا بساوي جدر زائد أو ناقص جدر الـ
+
+101
+00:07:12,710 --> 00:07:15,450
+Y لأن Y بتساوي X تربيع قيمة X بتساوي زائد أو ناقص
+
+102
+00:07:15,450 --> 00:07:19,000
+جدر الـ Yهذه المنطقة طبيعية X بالساوية ناقص جذر Y
+
+103
+00:07:19,000 --> 00:07:23,620
+لأنها المنطقة السالمة في X وهذه المنطقة جذر Y لذلك
+
+104
+00:07:23,620 --> 00:07:28,060
+X أو الـ Horizontal Line جاء من عند X بالساوية
+
+105
+00:07:28,060 --> 00:07:34,520
+ناقص جذر Y وهي ناقص جذر Y ورايح لمن؟ لجذر Y نرى
+
+106
+00:07:34,520 --> 00:07:37,970
+كيف هذه المنطقة تنمسحهي قيمة Y يعني اللي هدول الـ
+
+107
+00:07:37,970 --> 00:07:40,710
+Horizontal Lines بيظل مشيها تيك تيك تيك تيك لما
+
+108
+00:07:40,710 --> 00:07:44,630
+يصل هناك يعني بيصل من عند الصفر لعند المنطقة هذه
+
+109
+00:07:44,630 --> 00:07:49,210
+اللي هي Y برضه بتساوي واحد إذا حدود الـ Y هتكون من
+
+110
+00:07:49,210 --> 00:07:53,390
+صفر لعند واحد زائد التكامل على المنطقة الثانية لما
+
+111
+00:07:53,390 --> 00:07:56,150
+نتكامل على المنطقة الثانية ناخد وين Horizontal
+
+112
+00:07:56,150 --> 00:08:00,330
+Lines كل على نفس المنوال هي بالأحمر الواضح اللي هي
+
+113
+00:08:00,330 --> 00:08:04,870
+من اللي هو الخط لعند اللي هو إيش ال curveالآن في
+
+114
+00:08:04,870 --> 00:08:11,930
+منطقة هذا الخط ما هي قيمة X عليه؟ Y-2 لأن Y زاد X
+
+115
+00:08:11,930 --> 00:08:15,490
+بالساوية 2 يعني X بالساوية Y-2 على الخط هذا قيمة X
+
+116
+00:08:15,490 --> 00:08:20,350
+هي Y-2 لعند هذا هدودنا قيمته اللي هو عبارة عن جذر
+
+117
+00:08:20,350 --> 00:08:24,460
+الـ Yهنا نشوف الـ Horizontal Lines هذه من أين إلى
+
+118
+00:08:24,460 --> 00:08:27,080
+أين تبدأ ومن أين إلى أين تبدأ هي تبدأ من عند الـ
+
+119
+00:08:27,080 --> 00:08:29,540
+Hand اللي كلها على نفسه من ويه اللي عند يعني ويه
+
+120
+00:08:29,540 --> 00:08:34,120
+بتساوي واحد وبتبتسح المنطقة لما تصل إلى ويه بساوي
+
+121
+00:08:34,120 --> 00:08:39,540
+أربعة عشان أيه لحدوث التكامل بالنسبة لويه من واحد
+
+122
+00:08:39,540 --> 00:08:44,160
+لعند أربعة هذه هي ال area و بتحسبها و بتوجد قيمتها
+
+123
+00:08:44,470 --> 00:08:48,210
+الآن لو أحد قال لأ اسمعوا أنا بديش أشتغل زيكم بدي
+
+124
+00:08:48,210 --> 00:08:51,030
+أشتغل أو أقول شي دي واي باي دي إكس ونقوله نوع على
+
+125
+00:08:51,030 --> 00:08:54,250
+نوع و أحسنلك كمان، ليش أحسنلك؟ لأن كلها جاية من
+
+126
+00:08:54,250 --> 00:08:59,010
+أيش؟ على نفس المنوال اللي هي جاية من عند الـ curve
+
+127
+00:08:59,010 --> 00:09:03,200
+Y بالساوي X تربيعY بيساوي X تربيع لعند Y أيش
+
+128
+00:09:03,200 --> 00:09:06,580
+بتساوي X زائد 2؟ واحد يقول لي لأ ما هو هذا مختلف
+
+129
+00:09:06,580 --> 00:09:10,460
+عن هذا لأ أنا قيمة Y هنا و هنا زي بعض لأنه أنا
+
+130
+00:09:10,460 --> 00:09:13,420
+الآن مازال باخد موازي للـ Y axis لإنه بدأ كامل
+
+131
+00:09:13,420 --> 00:09:17,100
+بالنسبة للـ Y في الأول إذا قيمة Y هنا بيساوي X
+
+132
+00:09:17,100 --> 00:09:20,860
+تربيع وقيمة Y هنا برضه بيساوي X تربيع كلها جاي من
+
+133
+00:09:20,860 --> 00:09:25,060
+عند Y بيساوي X تربيع لعند Y أيش بيساوي X زائد 2؟
+
+134
+00:09:25,120 --> 00:09:30,640
+اللي هي ده من X تربيع لعند X زائد اتنين لأن قيمة Y
+
+135
+00:09:30,640 --> 00:09:34,160
+على هذا الجزء وعلى هذا الجزء زي بعض بخلاف X اللي
+
+136
+00:09:34,160 --> 00:09:38,160
+كانت هنا X بيساوي نقل جذر ال Y وهنا X بيساوي جذر
+
+137
+00:09:38,160 --> 00:09:42,040
+ال Y لإن هو منطقة X سالب أو منطقة X موجبة بينما Y
+
+138
+00:09:42,040 --> 00:09:45,780
+كلها بتساوي X تربيع على هذا الكيرت سواء من المنطقة
+
+139
+00:09:45,780 --> 00:09:50,540
+هذه أو من المنطقة هذهطيب، الان واضح انه اللي هو
+
+140
+00:09:50,540 --> 00:09:53,880
+كله على نفس المنوال بيجي من هنا الموازيات للـY
+
+141
+00:09:53,880 --> 00:09:57,940
+Axis وبروح للي فوق اللي هو الخط الان طب هدولة اللي
+
+142
+00:09:57,940 --> 00:10:01,580
+مشيت هي الشكل هذا اللي هو الـ Vertical من وين
+
+143
+00:10:01,580 --> 00:10:07,340
+ببدن؟ ببدن من عند اللي هي X-1وبتدرك تمشي كتابة
+
+144
+00:10:07,340 --> 00:10:08,160
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+145
+00:10:08,160 --> 00:10:09,400
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+146
+00:10:09,400 --> 00:10:10,160
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+147
+00:10:10,160 --> 00:10:13,940
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+148
+00:10:13,940 --> 00:10:14,340
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+149
+00:10:14,340 --> 00:10:21,080
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+150
+00:10:21,080 --> 00:10:22,700
+كتابة كتابة كتابة كتابة كتابةاللي هو من X تربيع
+
+151
+00:10:22,700 --> 00:10:26,060
+إلى X زائد 2 بتعوض بيصير X زائد 2 ناقص X تربيع
+
+152
+00:10:26,060 --> 00:10:29,180
+بتكامل كمان مرة بالنسبة ل X أو بتكامل بالنسبة ل X
+
+153
+00:10:29,180 --> 00:10:33,280
+بيصير X تربيع على 2 زائد 2 X ناقص X تكعيب على 3
+
+154
+00:10:33,280 --> 00:10:38,700
+بناقص 1 لعند 2 بنعوض بيصير اللي هي عبارة عن 4 على
+
+155
+00:10:38,700 --> 00:10:44,080
+2 اللي هي 2 زائد 4 ناقص اللي هي 8 على 3 بتطلع 9
+
+156
+00:10:44,080 --> 00:10:49,910
+على 2 وهيك بيكون حلنا السؤالديجي لحاجة اسم الـ
+
+157
+00:10:49,910 --> 00:10:54,290
+Average Value of F over R الـ Average Value للـ F
+
+158
+00:10:54,290 --> 00:10:58,590
+over R اللي هي زمان كنا نحكي عن الـ Average Value
+
+159
+00:10:58,590 --> 00:11:02,790
+العادية مجموع القراءات على عددها أو لو كان على خط
+
+160
+00:11:02,790 --> 00:11:07,730
+مستقيم لو كلهم كانت متصلات بيكون اللي هو عبارة عن
+
+161
+00:11:07,730 --> 00:11:11,210
+اللي هي عندي الـ Average Value مجموع اللي هي القيم
+
+162
+00:11:11,210 --> 00:11:15,400
+اللي عليها على طول اللي هي الفترة ماشي الحالهنا
+
+163
+00:11:15,400 --> 00:11:18,860
+الأن المنطقي بيقول لك الـ Average Value of F over
+
+164
+00:11:18,860 --> 00:11:24,440
+R قيمة F التي بتصنحها الـ Double Integral FDA على
+
+165
+00:11:24,440 --> 00:11:29,960
+المساحة الكلية بتطلع اللي هي و كأنه صار حجم على
+
+166
+00:11:29,960 --> 00:11:33,700
+مساحة بتطلع اللي هي مين بتطلع و بتظلها الـ F لأن
+
+167
+00:11:33,700 --> 00:11:36,960
+هذه عبارة عن .. خلينا نقول Roughly عبارة عن F
+
+168
+00:11:36,960 --> 00:11:40,740
+مساحة على مساحة بتطلع الـ F لأن هذه بتعمل مساحة و
+
+169
+00:11:40,740 --> 00:11:43,580
+هذه بتعمل مساحة هذا الكلام Roughly طبعا يعني لكن
+
+170
+00:11:43,580 --> 00:11:48,030
+الـ Meaningاللي هو الـ Double Integral over R FDA
+
+171
+00:11:48,030 --> 00:11:51,090
+على الـ Area للـ R التي تظهر وهي الـ Average Value
+
+172
+00:11:51,090 --> 00:11:55,270
+لمين الـ F over R فمنطق التعريف هذا الـ Average
+
+173
+00:11:55,270 --> 00:11:59,030
+Value of F over R بسوء واحد على الـ Area of R في
+
+174
+00:11:59,030 --> 00:12:03,550
+الـ Double Integral للـ FDA over R نيجي ناخد مثال،
+
+175
+00:12:03,550 --> 00:12:07,870
+ناخدلكم مثال بسيط هو الآن احنا تقريبا بالمحاضرة
+
+176
+00:12:07,870 --> 00:12:11,200
+هذه والمحاضرة اللي فاتتغطينا كل الإمكانيات كيف
+
+177
+00:12:11,200 --> 00:12:14,020
+نودد اللي هو الـ Double Integration على أي منطقة
+
+178
+00:12:14,020 --> 00:12:17,440
+حتى لو كانت صعبة و بتتجذب Find the average value
+
+179
+00:12:17,440 --> 00:12:22,380
+of f of x,y بساوية x cos x,y over the rectangle x
+
+180
+00:12:22,380 --> 00:12:25,480
+بين السفر و بين الـπي و y بين السفر و بين الواحد
+
+181
+00:12:25,620 --> 00:12:29,080
+واضح أن الـ Integration على ركتانجل إذا وانا مغمض
+
+182
+00:12:29,080 --> 00:12:35,400
+عينيا اللي هي DX X من سفر لعند Y و DY من عند سفر Y
+
+183
+00:12:35,400 --> 00:12:37,620
+من عند سفر لعند واحد على الـ Rectangle لكن لو
+
+184
+00:12:37,620 --> 00:12:40,880
+Region بدنا نضطر نعمل زي ما عملنا قبل شوية في حدود
+
+185
+00:12:40,880 --> 00:12:44,280
+الـ Integration الان الـ Value of the integral F
+
+186
+00:12:44,280 --> 00:12:47,960
+over R نوجد أول شيء هذا و بعدين نوجد الـ Area و
+
+187
+00:12:47,960 --> 00:12:51,300
+بنكسب هذا على هذا بتطلع الـ Average Valueالان نأخد
+
+188
+00:12:51,300 --> 00:12:55,580
+الـ integration الـ integration لمن؟ لـ X cos XY
+
+189
+00:12:55,580 --> 00:13:01,780
+DY DX من 0 لعند 1 و من 0 لعند مين؟ لعند Y طبعاً
+
+190
+00:13:01,780 --> 00:13:05,240
+اعملنا DY by DX لأنه DX .. DY .. هنعجّد الأمور فDY
+
+191
+00:13:05,240 --> 00:13:10,940
+by DX على طول بتطلع هذه هي عبارة عن اللي هي تفاضل
+
+192
+00:13:10,940 --> 00:13:15,780
+اللي هي sin X في Y عشان هي هذي تكملة اللي هي sin
+
+193
+00:13:15,780 --> 00:13:22,250
+XYمن عند اللي هو Zero لعند الواحد اللي كله DX Sine
+
+194
+00:13:22,250 --> 00:13:26,850
+XY اللي هي مايمن Zero لعند واحد إيش بتساوي اللي هي
+
+195
+00:13:26,850 --> 00:13:32,270
+Sine الواحد ال X ناقص لإنه Y بنعوض ناقص Sine Zero
+
+196
+00:13:32,270 --> 00:13:35,470
+اللي هو Zero اللي بيصير Sine X من سفر لضاي Sine X
+
+197
+00:13:35,470 --> 00:13:39,130
+من سفر لضاي اللي عبارة عن ناقص Cos X من سفر لضاي
+
+198
+00:13:39,130 --> 00:13:43,980
+اللي هو من سفر لضاي بنعوض اللي هيعبارة عن ناقص cos
+
+199
+00:13:43,980 --> 00:13:48,040
+πي هي ناقص ناقص واحد يعني واحد ناقص ناقص اللي هي
+
+200
+00:13:48,040 --> 00:13:50,360
+cosine الـ zero يعني زائد cosine الـ zero اللي هو
+
+201
+00:13:50,360 --> 00:13:53,060
+واحد يعني بيطلع اللي هو ال average value لل
+
+202
+00:13:53,060 --> 00:13:56,880
+function هذه over المنطقة هذه بساوية لأ آسف لأن
+
+203
+00:13:56,880 --> 00:13:59,240
+قيمة ال integration بساوية اتنين أما ال average
+
+204
+00:13:59,240 --> 00:14:03,380
+value احنا بدنا نجيب مساحة لمستطيل مستطيل طوله باي
+
+205
+00:14:03,510 --> 00:14:09,330
+وعرضه 1 مثلا بيصير مساحته 1 في باي يعني باي بيصير
+
+206
+00:14:09,330 --> 00:14:12,190
+1 على باي مضروب في الاتنين اللي هي قيمة ال
+
+207
+00:14:12,190 --> 00:14:15,110
+integration اللي وجدناها بتطلع ال average value of
+
+208
+00:14:15,110 --> 00:14:19,350
+F over R اللي عبارة عن 2 على المساحة اللي هي باي
+
+209
+00:14:19,350 --> 00:14:23,550
+وهو المطلوب ورأيت ممكن خلصنا المحاضرة وإلى لقاء
+
+210
+00:14:23,550 --> 00:14:24,670
+آخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4.srt
new file mode 100644
index 0000000000000000000000000000000000000000..cbb7ea69010363593a4b76594769b337b09bc673
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4.srt
@@ -0,0 +1,2185 @@
+1
+00:00:00,670 --> 00:00:04,150
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,150 --> 00:00:10,210
+وبركاته هذه المحاضرة رقم 18 لمساق  فاضل وتكامل
+
+3
+00:00:10,210 --> 00:00:18,090
+الجين وهي المحاضرة السادسة، Chapter 14 حيث سنتناول
+
+4
+00:00:18,090 --> 00:00:24,910
+الحديث عن 14.6 وهو تحت عنوان tangent planes and
+
+5
+00:00:24,910 --> 00:00:30,200
+differentials. سنرى كيف نجد معادلة tangent planes و
+
+6
+00:00:30,200 --> 00:00:33,920
+كيف نحكي عن الـ differentials. طبعاً الـ tangent
+
+7
+00:00:33,920 --> 00:00:38,500
+planes فكرتها، حتى إيجاد معادلتها، بنفس فكرة tangent
+
+8
+00:00:38,500 --> 00:00:42,680
+line في calculus 1.  والـ differentials هي تعميم
+
+9
+00:00:42,680 --> 00:00:48,090
+للـ differentials التي أخذناها في تفاضل ألف، وكذلك
+
+10
+00:00:48,090 --> 00:00:51,570
+سنحكي عن الـ Linearization،  الذي هو  أفدنا به لـ
+
+11
+00:00:51,570 --> 00:00:54,470
+Function of one variable. هناخذه لـ Function of
+
+12
+00:00:54,470 --> 00:00:58,030
+two variables أو  Equivariables. نيجي الآن للـ
+
+13
+00:00:58,030 --> 00:01:01,670
+Tangent planes and normal lines، التي هي التي
+
+14
+00:01:01,670 --> 00:01:06,490
+سنتحدث عنها اليوم. لو كان عندي R  بساوي G of T في
+
+15
+00:01:06,490 --> 00:01:09,770
+اتجاه I و H of T في اتجاه J، و K of T في اتجاه
+
+16
+00:01:09,770 --> 00:01:13,630
+الـK، is a smooth curve on a level surface F of XY
+
+17
+00:01:13,630 --> 00:01:19,290
+وZ  بساوي C. يعني زي ما حكينا قبل هيك، اللي هو
+
+18
+00:01:25,330 --> 00:01:29,130
+عند الـ Smooth Curve R  بساوي U of T،  زائد U of T
+
+19
+00:01:29,130 --> 00:01:34,450
+زائد K of T، نعوضه في داخل الـ F of X, Y, وZ بعملية
+
+20
+00:01:34,450 --> 00:01:38,990
+constant، يعني بعملية level curve. الآن زي ما عملنا
+
+21
+00:01:38,990 --> 00:01:43,750
+في المحاضرة التي فاتت، أخذنا f of x, f of g, و h, h
+
+22
+00:01:43,750 --> 00:01:46,310
+of t, k of t. كانت two variables، أن أجد
+
+23
+00:01:46,310 --> 00:01:50,790
+variables  بساوي c. هذه  فاضلة تنتين بالنسبة لها
+
+24
+00:01:50,790 --> 00:01:55,450
+ثابتة بالنسبة لها. إذاً أنا أكيد أُفاضلها صفراً. هذه
+
+25
+00:01:55,450 --> 00:01:59,330
+التي هي أُفاضلها by Chain Rule، التي هي Del F by Del
+
+26
+00:01:59,330 --> 00:02:05,690
+X في Del X by Del T. الـ X  هنا G، وDel F by Del
+
+27
+00:02:05,690 --> 00:02:10,430
+Y في Del Y by Del T، التي هي H في هذه الحالة، وDel F
+
+28
+00:02:10,430 --> 00:02:15,410
+by Del Z في Del K by Del T، التي هي Z في هذه الحالة.
+
+29
+00:02:15,410 --> 00:02:19,470
+كله هذا يساوي صفراً. هذه مثلًا عملناها في المرة الماضية.
+
+30
+00:02:32,260 --> 00:02:36,800
+يعني الآن أقول لكم أن الـ Gradient للـ F عمودي
+
+31
+00:02:36,800 --> 00:02:44,840
+على الـ Velocity لأي Curve يمر بالنقطة P0.
+
+32
+00:02:44,840 --> 00:02:48,920
+يعني لو كان عندي هذا الـ Curve يمر بالنقطة P0، التي
+
+33
+00:02:48,920 --> 00:02:53,260
+هو مماسها اللي هو V1،  الذي يكون ال Gradient للـ
+
+34
+00:02:53,260 --> 00:02:57,580
+F عمودي عليه. وأيضاً لو كان هذا عندي كمان واحد ثاني
+
+35
+00:02:57,740 --> 00:03:02,740
+أو هذا واحد ثاني، فبكون مماسّه اللي هو V2، التي هي ال
+
+36
+00:03:02,740 --> 00:03:06,360
+velocity، اللي هي DR ب DT،  أيضاً عمودي عليهما على ال
+
+37
+00:03:06,360 --> 00:03:09,760
+gradient. يعني لو كان عندي أيضاً Curve آخر يمرّ به
+
+38
+00:03:09,760 --> 00:03:13,180
+نقطة، اللي هو الـ curve هذا، الذي يخلق اللي هي هذه
+
+39
+00:03:13,180 --> 00:03:18,270
+بساوي C، على الـ surface، فيخلي الـ Gradient للـ F
+
+40
+00:03:18,270 --> 00:03:23,510
+عمودي على مماسّه الـ dR/dT. يعني الـ Gradient للـ F
+
+41
+00:03:23,510 --> 00:03:28,730
+سيكون عمودي على كل، اللي هي على الـ Plane التي
+
+42
+00:03:28,730 --> 00:03:35,050
+تكون من الـ V1 و V2،  ويُحدد  ما هو  الـ Tangent
+
+43
+00:03:35,050 --> 00:03:41,590
+لـ curve يمر بالنقطة P0، ويخلق هذا
+
+44
+00:03:41,590 --> 00:03:47,290
+الذي يساوي Level Surface. طيب إذاً هذا الذي حكينا
+
+45
+00:03:47,290 --> 00:03:51,130
+عنه هو الـ tangent plane at the point P0، هو ال
+
+46
+00:03:51,130 --> 00:03:54,310
+tangent plane الذي حكينا عنه on the level surface
+
+47
+00:03:54,310 --> 00:03:57,790
+F of X, Y, Z بساوي c، of a differentiable
+
+48
+00:03:57,790 --> 00:04:02,480
+function is the plane through P0.  الآن، normal هو Normal
+
+49
+00:04:02,480 --> 00:04:07,580
+على Gradient. إذاً صار عندي Gradient Normal على
+
+50
+00:04:07,580 --> 00:04:12,660
+Tangent Plane. هذا هو الـ Tangent Plane. صار الـ
+
+51
+00:04:12,660 --> 00:04:16,360
+Gradient عمودي عليه. إذاً الـ Tangent Plane هو عبارة
+
+52
+00:04:16,360 --> 00:04:22,560
+عن الـ Plane الذي يحمل الـ V1 و V2 و V3،
+
+53
+00:04:25,790 --> 00:04:30,650
+المماسات للمنحنيات التي تمرّ بالنقطة P0، التي
+
+54
+00:04:30,650 --> 00:04:34,710
+يجعل اللي هو F of X, Y, و Z بساوي C is a level
+
+55
+00:04:34,710 --> 00:04:38,210
+surface. هذا هو الـ Tangent Plane، وعرفنا العمودي
+
+56
+00:04:38,210 --> 00:04:41,610
+عليه، اللي هو الـ Gradient، وعارفين نقطة. إذاً بمعرفة
+
+57
+00:04:41,610 --> 00:04:44,970
+نقطة وعمودي، على طول بنعرف معادلة الـ Plane، زي ما
+
+58
+00:04:44,970 --> 00:04:49,230
+أنتم عارفين، وبذلك  مكون معادلة الـ Plane سهلة.
+
+59
+00:04:49,420 --> 00:04:55,040
+التانجنت بلين لـ F(x,y,z) بساوي C at P0 (X0, Y0, Z0) هي
+
+60
+00:04:55,040 --> 00:04:59,560
+عبارة عن اللي هو Fx، اللي هي مكونات الـ Gradient، لأن
+
+61
+00:04:59,560 --> 00:05:03,180
+Gradient بساوي Fx في اتجاه I، زي ما هي Gradient زي Fy
+
+62
+00:05:03,180 --> 00:05:06,460
+في اتجاه J، زي Fz في اتجاه K. هذا اللي هي ال
+
+63
+00:05:06,460 --> 00:05:10,900
+Gradient. ومن ثم ال Gradient، اللي هو العمودي على ال
+
+64
+00:05:10,900 --> 00:05:15,570
+surface، ومعرفة نقطة على الـ Surface، بنعرف معادلة
+
+65
+00:05:15,570 --> 00:05:20,410
+مين الـ Plane، زي ما احنا عارفين، بكون Fx في X
+
+66
+00:05:20,410 --> 00:05:24,230
+minus X0، Fy في Y minus Y0، Fz في
+
+67
+00:05:24,230 --> 00:05:27,970
+Z minus Z0، حيث الـ Fx و الـ Fy و الـ Fz هما
+
+68
+00:05:27,970 --> 00:05:31,410
+الذين يُكوّنون الـ Gradient، اللي هو العمودي على الـ
+
+69
+00:05:31,410 --> 00:05:33,730
+Surface الذي نحن عارفينه، و الـ X0 و الـ Y
+
+70
+00:05:33,730 --> 00:05:36,810
+0 و الـ Z0 هي عبارة عن النقاط التي على الـ
+
+71
+00:05:36,810 --> 00:05:39,740
+Tangent Plane. بما عرفنا الـ Gradient، الذي هو
+
+72
+00:05:39,740 --> 00:05:42,560
+عمودي على الـ Tangent Plane، وبما عرفنا النقطة X0
+
+73
+00:05:42,560 --> 00:05:46,500
+وY0، عرفنا معادلة الـ Tangent Plane. هل نستطيع أن
+
+74
+00:05:46,500 --> 00:05:51,200
+نعرف الـ Normal Line؟ الـ Normal Line على الـ Level
+
+75
+00:05:51,200 --> 00:05:55,490
+Surface F(X,Y,Z)؟ أجل، لأننا وجدنا أن ال
+
+76
+00:05:55,490 --> 00:05:59,250
+gradient عمودي عليه، اللي هو ال level، على ال
+
+77
+00:05:59,250 --> 00:06:02,970
+tangent plane، الذي هو عمودي عليه، اللي هو هذه
+
+78
+00:06:02,970 --> 00:06:06,670
+الـ surface التي عندي. مادام عمودي عليه، إذاً كل خط
+
+79
+00:06:06,670 --> 00:06:10,790
+عمودي عليه يكون موازياً للـ gradient. ما دام عرفنا أنه
+
+80
+00:06:10,790 --> 00:06:17,270
+موازي للـ Line، اتجاهه، يعني بمعنى آخر، وعرفنا نقطة
+
+81
+00:06:17,270 --> 00:06:22,410
+يعني معادلة الخط الموازي للـ Gradient، اللي هو الـ
+
+82
+00:06:22,410 --> 00:06:25,690
+Normal Line على اللي هو الـ Tangent Plane، هو عبارة
+
+83
+00:06:25,690 --> 00:06:30,010
+عن X بساوي X0 زائد Fx في T. هذه تبعت  مكونات الـ
+
+84
+00:06:30,010 --> 00:06:33,530
+Gradient. Y بساوي Y0 زائد Fy تبعت الـ Gradient في
+
+85
+00:06:33,530 --> 00:06:41,870
+T. و Z بساوي Z0 زائد Fz في T. الآن بمعرفة موازي للخط
+
+86
+00:06:41,870 --> 00:06:47,050
+الـ Gradient، ومعرفة نقطة على الخط X0, Y0, Z0، عرفنا
+
+87
+00:06:47,050 --> 00:06:51,970
+أننا وجدنا معادلة الـ Line Normal على الـ Tangent
+
+88
+00:06:51,970 --> 00:06:55,490
+Plane أو Normal على الـ Surface التي قبل قليل
+
+89
+00:06:55,490 --> 00:07:02,760
+شاهدناها. الخط الموازي لهذا نقول عمودي على الـ Surface
+
+90
+00:07:02,760 --> 00:07:06,600
+أو التي عندي عمودي على الـ Tangent Plane. استطعنا
+
+91
+00:07:06,600 --> 00:07:14,060
+نعرف منه ونعرف معادلته من معرفة اتجاهه ومعرفة نقطة
+
+92
+00:07:14,060 --> 00:07:19,880
+عليه، زي ما كنا سابقاً في Chapter 13.  ناخذ مثال
+
+93
+00:07:19,880 --> 00:07:22,980
+عددي. Find the Tangent Plane and Normal Line to the
+
+94
+00:07:22,980 --> 00:07:29,040
+Surface. ومعطين الـ Surface عن طريق الـ X²Y²Z - 9 = 0
+
+95
+00:07:29,040 --> 00:07:34,480
+يعني أعطيني level surface. عند نقطة (1, 2, 4)
+
+96
+00:07:34,480 --> 00:07:43,980
+هذه عندي اللي هو ال level surface X²Y²Z - 9 = 0. بدي
+
+97
+00:07:43,980 --> 00:07:48,210
+معادلة tangent plane  عنده. عند نقطة (1, 2, 4)،
+
+98
+00:07:48,210 --> 00:07:52,330
+ومعادلة مين؟ الـ Normal line الذي عليه.
+
+99
+00:07:52,330 --> 00:07:55,870
+لأننا عارفين الـ Plane، نعرف معادلته لو
+
+100
+00:07:55,870 --> 00:07:59,010
+عرفنا نقطة وعمودي عليه. طيب ما نحن عارفين أن الـ
+
+101
+00:07:59,010 --> 00:08:03,910
+Gradient لـ Level Surface عمودي عليه، على اللي
+
+102
+00:08:03,910 --> 00:08:09,570
+هو هذا الـ Plane. ما دام عارفين الـ Gradient عمودي
+
+103
+00:08:09,570 --> 00:08:13,130
+وعارفين النقطة، بنعرف معادلة الـ Tangent. إذاً الـ
+
+104
+00:08:13,130 --> 00:08:16,750
+Tangent Plane عند النقطة (1, 2, 4) عمودي
+
+105
+00:08:16,750 --> 00:08:20,010
+perpendicular to the gradient of F عند النقطة هذه.
+
+106
+00:08:21,050 --> 00:08:25,010
+من الحديث الذي ذكرناه قبل قليل. والـ Gradient لـ F
+
+107
+00:08:25,010 --> 00:08:32,030
+يساوي 2X في اتجاه I زائد 2Y في اتجاه J زائد K
+
+108
+00:08:32,030 --> 00:08:39,270
+عارفين ايش بيساوي؟ Fx هيها، و Fy هيها 2Y، و Fz هي فقط
+
+109
+00:08:39,270 --> 00:08:43,300
+1. للـ Gradient الـ F صار 2X في تجاه الـ I زائد 2Y في
+
+110
+00:08:43,300 --> 00:08:48,720
+تجاه الـ J زائد K، التي هي FX, FY, FZ، لما تحسبها الـ
+
+111
+00:08:48,720 --> 00:08:52,260
+Gradient عند مين؟ عند النقطة المذكورة، التي هي (1,
+
+112
+00:08:52,260 --> 00:08:56,220
+2, 4)، ويساوي 2 في اتجاه الـ I زائد اللي هي مكان الـ
+
+113
+00:08:56,220 --> 00:08:59,800
+Y 2 يصير 4 في اتجاه الـ J زائد الـ K. هذا الـ
+
+114
+00:08:59,800 --> 00:09:02,460
+Gradient، اللي هو العمودي على الـ Plane الذي نريده.
+
+115
+00:09:02,460 --> 00:09:07,640
+عرفنا نقطة على الـ Plane. إذاً جهّزنا معادلتنا
+
+116
+00:09:07,640 --> 00:09:12,440
+التي هي 2، التي كنا نسميها A، في X minus
+
+117
+00:09:12,440 --> 00:09:16,660
+1، زائد B، التي هي 4، Y minus 2،
+
+118
+00:09:16,660 --> 00:09:21,060
+عارفين ليش Y minus 2؟ Y ناقص 2، Z ناقص 4،
+
+119
+00:09:21,060 --> 00:09:24,920
+و الـ X ناقص 1، النقطة التي عرفناها في، اللي هي C
+
+120
+00:09:24,920 --> 00:09:28,600
+هنا ايش بتساوي؟ اللي هي معامل K، الذي هو 1. إذاً
+
+121
+00:09:28,600 --> 00:09:34,470
+صارت هذه معادلة Tangent Plane، التي سنبسطها. 2X -2 و
+
+122
+00:09:34,470 --> 00:09:40,620
+كذا بتطلع عندي 2X زائد 4Y زائد Z = 14. الآن بدنا نعرف الـ
+
+123
+00:09:40,620 --> 00:09:43,740
+Normal Line. الـ Normal Line على الـ Surface هو
+
+124
+00:09:43,740 --> 00:09:45,880
+موازٍ للـ Gradient، والـ Gradient عمودي على الـ
+
+125
+00:09:45,880 --> 00:09:50,260
+Surface. إذاً الـ Normal Line، عارفين موازٍ لإيش؟ اللي
+
+126
+00:09:50,260 --> 00:09:55,140
+هو عبارة عن 2I زائد 4J زائد K. وعارفين النقطة عليه؟ إذاً
+
+127
+00:09:55,140 --> 00:10:00,420
+على طول معادلته X بتساوي 1، اللي هي النقطة، زائد 2T،
+
+128
+00:10:00,420 --> 00:10:07,500
+اللي هي 2 من مكونات T. Y بساوي 2، اللي هي قيمة النقطة Y
+
+129
+00:10:07,800 --> 00:10:12,760
+زائد اللي هي الـ... الـ... الـ... الـ... الـ... التي في
+
+130
+00:10:12,760 --> 00:10:18,200
+الـ line الموازي، اللي هي 4T. و هذه الأخيرة T زائد 4
+
+131
+00:10:18,200 --> 00:10:20,560
+فبكون، و T طبعاً parameter زي ما أنتم عارفين، الـ
+
+132
+00:10:20,560 --> 00:10:24,320
+element in R. فبكون الـ normal line هي معادلته، وطبعاً
+
+133
+00:10:24,320 --> 00:10:28,240
+هذه شغلات قديمة لكن نحن نعملها لنستغل معلوماتنا
+
+134
+00:10:28,240 --> 00:10:34,810
+الجديدة في إيجادها. طيب، الآن لو الصيغة التي عندنا
+
+135
+00:10:34,810 --> 00:10:39,990
+كانت F(x, y, z) بساوي ثابت. طيب لو كان غير معروض
+
+136
+00:10:39,990 --> 00:10:43,670
+علينا بالشكل هذا، لو معروض علينا اللي هو معادلة
+
+137
+00:10:43,670 --> 00:10:48,030
+بالشكل الثاني، يعني معروض علينا surface الآن عند الذي
+
+138
+00:10:48,030 --> 00:10:55,340
+عرضناها قبل، اللي هي Z التي عندي، F(X,Y)
+
+139
+00:10:55,340 --> 00:10:59,700
+function،  التي هي،  وتساوي C.  صار  IH level surface
+
+140
+00:10:59,700 --> 00:11:03,700
+لو الموضوع غير معروض بالشكل هذا، لو معروض بشكل آخر.
+
+141
+00:11:03,700 --> 00:11:07,340
+خليني أشوف ايش المقصود. لو عندي to find an equation
+
+142
+00:11:07,340 --> 00:11:10,140
+for the plane tangent to a smooth curve. لو بدنا
+
+143
+00:11:10,140 --> 00:11:13,900
+نعرف اللي هو معادلة الـ tangent plane على smooth
+
+144
+00:11:13,900 --> 00:11:16,960
+curve. الآن Z بساوي F(X,Y)، غير مكتوب على صورة level
+
+145
+00:11:16,960 --> 00:11:21,770
+surface. At the point (x0, y0, z0) where z0 بساوي f
+
+146
+00:11:21,770 --> 00:11:25,590
+of x0 و y0. بدنا نعرف اللي هو الـ Tangent Plane
+
+147
+00:11:25,590 --> 00:11:29,770
+لهذا الـ Surface. Z بساوي F(x, y). الآن بدنا نعمل
+
+148
+00:11:29,770 --> 00:11:33,770
+التالي. لاحظوا الآن هذه Z بساوي F(x, y)  بساوي
+
+149
+00:11:33,770 --> 00:11:38,460
+F(x, y). ننقل Z.  بتصير F(x,y) - Z بتساوي 0. الآن الـ Surface Z بساوي
+
+150
+00:11:38,460 --> 00:11:43,120
+F(X, Y). اللي هو هذا يُعبّر عنه لو سمينا هذه كلها
+
+151
+00:11:43,120 --> 00:11:46,280
+على بعض، Function F كبير  of X, Y, Z، زي تابعنا
+
+152
+00:11:46,280 --> 00:11:52,280
+سابقا، هتساوي F(X, Y) ناقص Z. صار هذا هذا عبارة عن
+
+153
+00:11:52,280 --> 00:11:56,380
+Level Surface، ماشي.  بساوي Level Surface. الآن
+
+154
+00:11:56,380 --> 00:11:59,320
+نتعامل معاه زي قبل قليل. الآن الـ partial
+
+155
+00:11:59,320 --> 00:12:01,540
+derivatives of F. لو جينا أخذنا الـ partial
+
+156
+00:12:01,540 --> 00:12:05,690
+derivatives،  لِنُطبّق عليه  مين يا جماعة؟ اللي هي
+
+157
+00:12:05,690 --> 00:12:09,930
+التي قبل قليل، المعادلة الـ Fx لها بساوي Del By
+
+158
+00:12:09,930 --> 00:12:14,670
+Del X لهذا المقدار. هذه بالنسبة لـ X بس، Fx
+
+159
+00:12:14,670 --> 00:12:19,730
+لأن بالنسبة لـ Y صفر، بالنسبة لـ Z صفر. إذاً Fx ناقص
+
+160
+00:12:19,730 --> 00:12:25,120
+صفر، وساوي Fx صغير. هنا Fy بيصير Del فضلتها Del
+
+161
+00:12:25,120 --> 00:12:27,760
+بالنسبة ل Y جزئياً. هذه فضلتها بالنسبة ل Y كلها
+
+162
+00:12:27,760 --> 00:12:31,620
+بيصير Del by Del y F(x, y) ناقص Z. نفس الشيء،
+
+163
+00:12:31,620 --> 00:12:36,120
+بيصير Fy ناقص، اللي هي بالنسبة ل Y صفر، ويساوي Fy
+
+164
+00:12:36,120 --> 00:12:39,900
+of z، التي هي Del هذا المقدار كله بالنسبة ل Z. هذا
+
+165
+00:12:39,900 --> 00:12:43,780
+بالنسبة ل Z صفر. ماشي الحال. الآن صار ال X و ال Y
+
+166
+00:12:43,780 --> 00:12:47,180
+و ال Z هنا، في هذه الحالة independent. ثلاثة X
+
+167
+00:12:47,180 -->
+
+223
+00:16:27,780 --> 00:16:33,530
+الـ tangent plane to the surface شايفين مثلها؟ الآن
+
+224
+00:16:33,530 --> 00:16:37,990
+مثلاً آخر الـ surface f of x y و z بساوية x تربيع
+
+225
+00:16:37,990 --> 00:16:41,550
+z ناقص y تربيع ناقص اتنين بساوية صفر هذا عبارة عن الـ
+
+226
+00:16:41,550 --> 00:16:45,090
+level surface هذا بيعمل ليهاش cylinder x تربيع زائد y تربيع
+
+227
+00:16:45,090 --> 00:16:48,650
+ناقص اتنين بساوية صفر الـ z ثابتة  متغيرة
+
+228
+00:16:48,650 --> 00:16:53,110
+مطلوقة زي ما نقول وهذا دائرة فبتعمل ليهاش الـ
+
+229
+00:16:53,110 --> 00:16:58,950
+circular الآن x زائد y زائد z ناقص أربعة بيساوي صفر، جي
+
+230
+00:16:58,950 --> 00:17:02,330
+و x و y و z هذا الـ level surface بيعمل لي إيش
+
+231
+00:17:02,330 --> 00:17:06,830
+إبلين، إيش اللي طلبنيها؟ بقول لي ده surface هاي و
+
+232
+00:17:06,830 --> 00:17:10,110
+هذا اللي هو الـ cylinder و الـ surface هذا الإبلين
+
+233
+00:17:10,110 --> 00:17:13,250
+بقول لي meet in an ellipse، اتقاطعن في إيش؟ اتقاطعن
+
+234
+00:17:13,250 --> 00:17:16,150
+في ellipse، هي الـ ellipse اللي اتقاطعن فيها، ماشي
+
+235
+00:17:16,150 --> 00:17:20,010
+الحال؟ بقول find parametric equations for the line
+
+236
+00:17:20,010 --> 00:17:25,650
+tangent to E at point 1 و 3 و 3 بقول لي أوجد معادلة
+
+237
+00:17:25,650 --> 00:17:31,850
+الـ parametric equation  لـ معادلة اللي هي مين الخط
+
+238
+00:17:31,850 --> 00:17:36,810
+المماس اللي بيمس هذه الـ ellipse عند النقطة
+
+239
+00:17:36,810 --> 00:17:41,280
+المذكورة. أنا عشان أجد معادلة المماس أو معادلة خط
+
+240
+00:17:41,280 --> 00:17:45,760
+بالزمن شغلتين بالزمن اللي هو vector موازي له لخط
+
+241
+00:17:45,760 --> 00:17:49,900
+الماس وبالزمن نقطة النقطة المعطينيها. طب الـ vector
+
+242
+00:17:49,900 --> 00:17:54,830
+الموازي شوفوا كيف نجد الـ vector الموازي. أول حاجة لو
+
+243
+00:17:54,830 --> 00:18:00,050
+ألحظنا على هذه اللي هي الـ ellipse الـ ellipse هي
+
+244
+00:18:00,050 --> 00:18:03,790
+عبارة موجودة طبعاً هي عبارة عن تفاطع الـ plane و
+
+245
+00:18:03,790 --> 00:18:09,890
+الـ cylinder. الآن الـ ellipse هذه موجودة في
+
+246
+00:18:09,890 --> 00:18:12,810
+الجهتين يعني موجودة في الـ cylinder و موجودة في
+
+247
+00:18:12,810 --> 00:18:17,650
+الـ plane. الآن الـ gradient للـ F عمودي على مين؟
+
+248
+00:18:17,650 --> 00:18:21,310
+على اللي هو الـ cylinder. إذا عمودي على اللي هو أي
+
+249
+00:18:21,310 --> 00:18:25,990
+منحنة فيها من ضمنها اللي هي الـ ellipse. إذا
+
+250
+00:18:25,990 --> 00:18:28,950
+الـ gradient للـ F عمودي على الـ ellipse. هاي واحد،
+
+251
+00:18:28,950 --> 00:18:32,570
+خلوه في الذاكرة. إتنين، الـ gradient للـ G عمودي على
+
+252
+00:18:32,570 --> 00:18:37,370
+مين؟ على الـ surface هذا اللي هو G بساوية صفر. مدام
+
+253
+00:18:37,370 --> 00:18:41,430
+عمودي عليه إذا عمودي على كل كرفي من ضمنه المين؟
+
+254
+00:18:41,540 --> 00:18:46,020
+اللي هي الـ ellipse. إذا صار الـ gradient للـ G
+
+255
+00:18:46,020 --> 00:18:48,980
+عمودي على الـ ellipse عند نقطة هذه. و الـ gradient
+
+256
+00:18:48,980 --> 00:18:52,680
+للـ F عمودي على الـ ellipse عند النقطة هذه. إذا أكيد
+
+257
+00:18:52,680 --> 00:18:57,440
+الـ cross product بينهم هيكون ماله؟ عمودي على
+
+258
+00:18:57,440 --> 00:19:03,080
+الاثنين، مظبوط؟ يعني بمعنى آخر موازي للمماس. لأن
+
+259
+00:19:03,080 --> 00:19:06,660
+المماس اللي هو هنا الـ gradient للألف عمودي عليه
+
+260
+00:19:06,660 --> 00:19:09,240
+و الـ gradient للـ G برضه عمودي على المماس اللي
+
+261
+00:19:09,240 --> 00:19:13,360
+بدنا إياه. لأن المماس عنده النقطة هذه اللي هو الـ
+
+262
+00:19:13,360 --> 00:19:15,420
+gradient عمودي على الـ curve اللي هو عمودي على
+
+263
+00:19:15,420 --> 00:19:18,120
+المماس و الـ gradient للـ F عمودي على الـ curve
+
+264
+00:19:18,120 --> 00:19:21,260
+اللي هو عمودي على المماس. إذن الـ gradient للـ F
+
+265
+00:19:21,260 --> 00:19:24,540
+و الـ gradient للـ G عمودي على المماس. إذن اللي هو
+
+266
+00:19:24,540 --> 00:19:30,780
+الـ vector الموازي اللي هو للمماس اللي هو العمودي
+
+267
+00:19:30,780 --> 00:19:34,600
+على اللي هو الـ gradient للـ F و الـ gradient للـ G
+
+268
+00:19:34,600 --> 00:19:39,740
+زي مين؟ زي المماس، إذا صار المماس معروف الـ vector
+
+269
+00:19:39,740 --> 00:19:42,340
+موازي له. الـ gradient للـ F كروس الـ gradient للـ G
+
+270
+00:19:42,340 --> 00:19:46,060
+ومعروف نقطة، إذا بنقدر نعرف اللي هو معادلته، خلّيني
+
+271
+00:19:46,060 --> 00:19:49,000
+نعرف الـ gradient للـ F كروس الـ gradient للـ G الـ
+
+272
+00:19:49,000 --> 00:19:51,740
+tangent line is orthogonal to both الـ gradient
+
+273
+00:19:51,740 --> 00:19:56,020
+للـ F and الـ gradient للـ G at 1 و 1 و 3 زي ما
+
+274
+00:19:56,020 --> 00:20:00,510
+ذكرت قبل شوية. وهكذا بدأت بالمعنى الـ vector الـ
+
+275
+00:20:00,510 --> 00:20:05,150
+gradient للـ F كروس الـ gradient للـ G. إذا نجد
+
+276
+00:20:05,150 --> 00:20:07,730
+الـ gradient للـ F عنده نقطة واحد و تلاتة و تلاتة
+
+277
+00:20:07,730 --> 00:20:11,510
+الـ gradient للـ F هو اتنين x في اتجاه الـ i زائد
+
+278
+00:20:11,510 --> 00:20:14,570
+اتنين y في اتجاه الـ j احسبها عند الواحد والواحد
+
+279
+00:20:14,570 --> 00:20:19,970
+والتلاتة بتطلع اتنين i زائد اتنين j هذا الـ
+
+280
+00:20:19,970 --> 00:20:23,290
+gradient للـ F. الـ gradient للـ G is similar للإن
+
+281
+00:20:23,290 --> 00:20:29,510
+الـ تفاضل بالنسبة للـ x هو i بتفضل بالنسبة لـ y صفر
+
+282
+00:20:29,510 --> 00:20:34,370
+بالنسبة لـ z بساوي واحد يعني k الـ  gradient للـ G
+
+283
+00:20:34,370 --> 00:20:39,170
+بساوي i زائد k. لأن صار معروف هذا صار معروف هذا لأن
+
+284
+00:20:39,170 --> 00:20:43,190
+توجد الـ cross product بينهم بينهم الـ gradient انتلاف
+
+285
+00:20:43,190 --> 00:20:47,190
+في cross الـ gradient للـ G بتعرفوا كيف بحط i, j, k بعدين
+
+286
+00:20:47,190 --> 00:20:50,990
+عناصر الأول اتنين، اتنين، صفر هي اتنين، اتنين، صفر
+
+287
+00:20:50,990 --> 00:20:52,690
+وهذه 1، 0، 1
+
+288
+00:20:56,990 --> 00:21:00,730
+بنعمل مفبوك الـ determinant شطب هذا و شطب هذا
+
+289
+00:21:00,730 --> 00:21:05,670
+بيصير 2 في اتجاه الـ i. هي شطب هذا و شطب هذا ناقص
+
+290
+00:21:05,670 --> 00:21:11,030
+اللي هو 2 في اتجاه الـ j لأن هذا صفر اللي بتظلها و
+
+291
+00:21:11,030 --> 00:21:15,670
+شطب هذا و شطب هذا بيصير 2 اللي هي صفر يعني ناقص 2
+
+292
+00:21:15,670 --> 00:21:19,690
+في اتجاه الـ k. وعندي هذا اللي هو الـ gradient للـ
+
+293
+00:21:19,690 --> 00:21:23,520
+F كروس الـ gradient للـ G. لأن عرفت اللي هو الـ vector
+
+294
+00:21:23,520 --> 00:21:26,660
+موازي للخط وعرفت اللي هي نقطة على الخط إذا بعرف
+
+295
+00:21:26,660 --> 00:21:31,040
+معادلة الخط. x بتساوي اللي هي واحد زائد اتنين في t
+
+296
+00:21:31,040 --> 00:21:37,140
+اللي هو y بتساوي واحد ناقص اتنين في t و z
+
+297
+00:21:37,140 --> 00:21:41,800
+بتساوي اللي هو تلاتة ناقص اتنين في t
+
+298
+00:21:41,800 --> 00:21:46,060
+بيكون هو ده معادلة المماس اللي هي الـ ellipse
+
+299
+00:21:46,060 --> 00:21:51,460
+التقاطع للـ surface. الآن نجي لـ Estimating Exchange
+
+300
+00:21:51,460 --> 00:21:55,340
+in a specific direction اللي هو احنا زمان كنا نحكي
+
+301
+00:21:55,340 --> 00:22:01,640
+عن Estimating Exchange in general في الكلكولة
+
+302
+00:22:01,640 --> 00:22:05,480
+الصالفة كانت اللي هي سهلة الـ df كنا نقول بساوي
+
+303
+00:22:05,480 --> 00:22:12,240
+اللي هي f' في ds. الآن لأ احنا صار عندنا او دي اف
+
+304
+00:22:12,240 --> 00:22:15,520
+بساوي f برايم في دي واي زي ما قلنا او دي اكس حسب
+
+305
+00:22:15,520 --> 00:22:19,280
+اللي عندنا. الآن احنا صار function of two variables
+
+306
+00:22:19,280 --> 00:22:23,300
+فهيكون الـ estimating the change in F in the
+
+307
+00:22:23,300 --> 00:22:26,940
+direction of unit vector U شوف كيف to estimate the
+
+308
+00:22:26,940 --> 00:22:29,620
+change in the value of a differentiable function F
+
+309
+00:22:29,620 --> 00:22:33,120
+when we move a small distance ds from point
+
+310
+00:22:33,120 --> 00:22:36,980
+P0 in a particular direction U use the formula  اتة
+
+311
+00:22:36,980 --> 00:22:45,170
+دي الان. بنقرب التغيير اللي بيصير اللي هو في الدالة
+
+312
+00:22:45,170 --> 00:22:50,130
+of a differentiable function F لما نتحرك بمقدار
+
+313
+00:22:50,130 --> 00:22:56,490
+مسافة ds من نقطة P0 في الـ direction لـ U. يعني الآن
+
+314
+00:22:56,490 --> 00:23:00,350
+عندي اللي هو لما اتحركت اللي هي من عند الـ P0
+
+315
+00:23:03,310 --> 00:23:07,610
+لنقطة أخرى في اتجاه مين؟ على اتجاه الـ U. صار في هذه
+
+316
+00:23:07,610 --> 00:23:11,850
+النقطة دي مشيناها صار هذا اللي هو فيه تغيير في
+
+317
+00:23:11,850 --> 00:23:16,090
+مين؟ في الـ F. التغيير في الـ F اللي هو تقريبي دلتا
+
+318
+00:23:16,090 --> 00:23:20,480
+df. طبعاً اللي هو بالضبط الـ ΔF بساوي اللي هو بقاله
+
+319
+00:23:20,480 --> 00:23:23,360
+علاقة Δx و Δy و مش عارف إيش و كده و كده. زمان
+
+320
+00:23:23,360 --> 00:23:26,340
+كانت لما كانت بس function of one variable دلتا F
+
+321
+00:23:26,340 --> 00:23:32,080
+اللي بساوي اللي هو y ناقص y نوت أو دلتا y بساوي y
+
+322
+00:23:32,080 --> 00:23:34,960
+ناقص y نوت و دلتا x بساوي x ناقص x نوت. لكن الآن
+
+323
+00:23:34,960 --> 00:23:37,800
+صار عندي لأ في direction معين فبصير الآن الـ
+
+324
+00:23:37,800 --> 00:23:41,900
+differential دي F تقريباً تقريباً بساوي هذا المقدار
+
+325
+00:23:41,900 --> 00:23:46,610
+اللي هو directional derivative مضروب في مين؟ في الـ
+
+326
+00:23:46,610 --> 00:23:51,510
+ds الـ distance increment اللي اتحركناها فبكون هذا
+
+327
+00:23:51,510 --> 00:23:55,670
+مقدار التغير تقريباً. هذا مقدار التغير طبعاً فيه اللي
+
+328
+00:23:55,670 --> 00:23:59,890
+هو خطأ نحسبها بعد شوية هذا قانونه المهم لأن
+
+329
+00:23:59,890 --> 00:24:05,070
+قانون اللي هو لو اتحركنا دالة اللي هو اتحركنا في
+
+330
+00:24:05,070 --> 00:24:09,610
+اتجاه الـ U بمقدار ds صار فيه تغيير في الـ F مقدار
+
+331
+00:24:09,610 --> 00:24:12,570
+التغيير في الـ F تقريباً بساوية directional
+
+332
+00:24:12,570 --> 00:24:16,750
+derivative في هذا الاتجاه مضروبة في مقدار اللي هي
+
+333
+00:24:16,750 --> 00:24:21,530
+المسافة اللي مشيناها ds على اللي هو الـ U فبكون هو
+
+334
+00:24:21,530 --> 00:24:26,130
+مقدار التغير في الـ F وخلينا نشوف المثال التالي
+
+335
+00:24:27,340 --> 00:24:31,160
+نشوف التطبيق الأول على اللي هو الـ Example 4
+
+336
+00:24:31,160 --> 00:24:35,440
+Estimate how much the value of f of x, y, z بساوي
+
+337
+00:24:35,440 --> 00:24:40,400
+sin x زائد 2y, z اللي هي will change if the point P
+
+338
+00:24:40,400 --> 00:24:45,960
+moves 0.1 units from P0 straight toward اللي
+
+339
+00:24:45,960 --> 00:24:51,520
+هو P1 اللي هو 2 و 2 و ناقص 2 يعني بدنا نشوف .. نشوف
+
+340
+00:24:51,520 --> 00:24:59,060
+.. نقرب قد إيش التغير في F التي تعمل تغيير الـ
+
+341
+00:24:59,060 --> 00:25:05,140
+estimation هو الـ ΔF لما النقطة
+
+342
+00:25:06,600 --> 00:25:11,260
+تتغير وتتحرك بمقدار 0.1 من أين إلى أين تتحرك
+
+343
+00:25:11,260 --> 00:25:15,060
+من النقطة P0 في اتجاه مباشرة straightforward
+
+344
+00:25:15,060 --> 00:25:21,700
+في اتجاه P1 يعني P1 هو الـ direction اللي بدنا
+
+345
+00:25:21,700 --> 00:25:25,580
+نتحرك فيه ومقدار التحرك اللي هو الـ ds في هذه
+
+346
+00:25:25,580 --> 00:25:30,080
+الحالة بساوي 0.1. إذن عشان أقدر أوجد أعمل
+
+347
+00:25:30,080 --> 00:25:34,200
+estimation للتغير في F اللي هو الـ df بدا أوجد الـ
+
+348
+00:25:34,200 --> 00:25:37,680
+gradient و بدا أوجد الـ U و بدا أوجد الـ ds. ds هي
+
+349
+00:25:37,680 --> 00:25:42,600
+موجودة و الـ U هو عبارة عن الـ direction اللي هو
+
+350
+00:25:42,600 --> 00:25:46,640
+من P0 لـ P1 اللي هو بمعنى آخر الـ unit vector في
+
+351
+00:25:46,640 --> 00:25:53,810
+اتجاه P0 P1. الآن اللي هي الـ gradient اللي نجيبه هو
+
+352
+00:25:53,810 --> 00:26:01,130
+الـ direction of this vector from 0, 1 و 0 إلى 2 و
+
+353
+00:26:01,130 --> 00:26:05,930
+2 و ناقص 2. أول شيء اللي هو الـ vector بين هذه وهذه
+
+354
+00:26:05,930 --> 00:26:09,980
+اللي هو عبارة عن 2i, 2j, -2k
+
+355
+00:26:21,150 --> 00:26:24,290
+الآن بدنا نوجد اللي هو الـ direction له. إذا على
+
+356
+00:26:24,290 --> 00:26:28,030
+مجموغه مجموغه هو الجذر التربيعي للأربعة زائد
+
+357
+00:26:28,030 --> 00:26:32,050
+الواحد زائد الأربعة اللي هي عبارة عن بيصير 2i
+
+358
+00:26:32,050 --> 00:26:35,690
+زائد j ناقص 2k على 3 هذا اللي هو الـ
+
+359
+00:26:35,690 --> 00:26:40,270
+direction U اللي احنا بدنا التغيير يصير في اتجاهه
+
+360
+00:26:40,640 --> 00:26:43,460
+الآن من الـ gradient للـ F نيجي من جديد من الـ
+
+361
+00:26:43,460 --> 00:26:47,720
+gradient للـ F نوجد الدالة اللي هو Fx Fx اللي هي
+
+362
+00:26:47,720 --> 00:26:51,000
+هاد بالنسبة لـ Fx بيصير y cos x في اتجاه الـ i
+
+363
+00:26:51,000 --> 00:26:56,340
+زائد Fy Fy إيش Fy في هذه الحالة اللي هي sin x لأن
+
+364
+00:26:56,340 --> 00:27:00,620
+تفاضل الـ y واحد sin x زائد اللي هو تفاضل الـ 2yz
+
+365
+00:27:00,620 --> 00:27:05,060
+بالنسبة لـ y اللي هو 2z كله في اتجاه من الـ j زائد
+
+366
+00:27:05,060 --> 00:27:09,400
+اللي هي Fz Fz هذا صفر بالنسبة لـ z وهذا بالنسبة لـ z
+
+367
+00:27:09,400 --> 00:27:13,760
+اللي هي 2y فبتنزل 2y k. هذه الـ gradient الـ
+
+368
+00:27:13,760 --> 00:27:17,460
+gradient بدنا إياه عند مين؟ احنا عند اللي هو طبعاً
+
+369
+00:27:17,460 --> 00:27:22,800
+التحرك من عند النقطة P0 ,1 ,0 إذا عند نقطة مين؟ P0
+
+370
+00:27:22,800 --> 00:27:27,000
+,1 ,0 هي النقطة اللي من عندها صار التحرك اللي عندها
+
+371
+00:27:27,000 --> 00:27:32,050
+بنجيب الـ gradient اللي هي بتساوي عوض عن x بـ 0
+
+372
+00:27:32,050 --> 00:27:40,430
+بيصير الـ y بـ 1 يعني i زائد sin 0 صفر زائد 2z
+
+373
+00:27:40,430 --> 00:27:44,650
+اللي هي 2 في صفر يعني بيصير في الـ j لأن هذا صفر و
+
+374
+00:27:44,650 --> 00:27:49,790
+هذا صفر زائد 2yk y بـ 1 بيصير زائد 2k. إذا صار عند
+
+375
+00:27:49,790 --> 00:27:54,280
+الـ gradient للـ F عند النقطة (0, 1, 0). لو ضربنا dot
+
+376
+00:27:54,280 --> 00:27:58,540
+product مع U بيصير عبارة عن ناقص 2/3 فاهمين ليش؟ لأنه
+
+377
+00:27:58,540 --> 00:28:02,280
+احنا بدنا نجد اللي هو df نسميه الـ gradient dot
+
+378
+00:28:02,280 --> 00:28:06,220
+الـ U مضروب في مين؟ في الـ ds ده صار عدد مضروب في
+
+379
+00:28:06,220 --> 00:28:08,960
+عدد. او ليش نجيب الـ dot product هذا؟ هذا هو الـ
+
+380
+00:28:08,960 --> 00:28:13,660
+gradient لـ F و dot لـ U. gradient لـ F i زائد 2k
+
+381
+00:28:13,660 --> 00:28:17,580
+و dot لـ U. دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+382
+00:28:17,580 --> 00:28:19,080
+نجدها فوق دعونا نجدها فوق دعونا
+
+383
+00:28:19,080 --> 00:28:19,420
+نجدها فوق دعونا نجدها فوق دعونا
+
+384
+00:28:19,420 --> 00:28:19,980
+نجدها فوق دعونا نجدها فوق دعونا
+
+385
+00:28:19,980 --> 00:28:21,100
+نجدها فوق دعونا نجدها فوق دعونا
+
+386
+00:28:21,100 --> 00:28:25,860
+نجدها فوق دعونا نجدها فوق دعونا
+
+387
+00:28:25,860 --> 00:28:32,560
+نجدها فوق دعونا نجدها فوق. الآن الـ «تغيير df في F
+
+388
+00:28:32,560 --> 00:28:38,020
+التي تنتج من المتحرك ds» يتساوي 0.1 معطيني unit
+
+389
+00:28:38,020 --> 00:28:41,920
+away from (0, 1, 0) in the direction of U is
+
+390
+00:28:41,920 --> 00:28:45,520
+approximately df بتساوي حسب القانون اللي فوق دلتا
+
+391
+00:28:45,520 --> 00
+
+445
+00:32:46,590 --> 00:32:51,350
+قلنا قبل شوية هي قانونها L بتساوي F نحسبها عندنا
+
+446
+00:32:51,350 --> 00:32:54,470
+النقطة المطلوبة Fx عندنا النقطة المطلوبة في X نقص
+
+447
+00:32:54,470 --> 00:32:57,710
+X not of Y برضه عندنا النقطة المطلوبة في Y minus Y
+
+448
+00:32:57,710 --> 00:33:01,310
+not هذا هو Linearization خلينا الآن نطبق على مثالنا
+
+449
+00:33:01,830 --> 00:33:06,590
+عند الـ Fx إيش بيساوي 2x ناقص Y فاهمين عليها Fy
+
+450
+00:33:06,590 --> 00:33:10,370
+إيش بيساوي اللي هو ده صفر اللي هي بيصير ناقص X
+
+451
+00:33:10,370 --> 00:33:14,570
+معايا زائد 2 في نص اللي هي بيصير اللي هو زائد Y
+
+452
+00:33:14,570 --> 00:33:18,790
+هذا Fy لما تحسب هذا عند النقطة 3 و 2 وهذا 3 و 2
+
+453
+00:33:18,790 --> 00:33:22,530
+وهذا عند النقطة 3 و 2 الأولى بتصير اللي هي 2 في 3
+
+454
+00:33:22,530 --> 00:33:27,330
+ب6 ناقص 2 اللي هي 4  إذن Fy عند الـ 3 و 2 اللي
+
+455
+00:33:27,330 --> 00:33:33,780
+هي ناقص 3 مكان X، مكان نقص X وY مكانها 2 بيصير نقص 3
+
+456
+00:33:33,780 --> 00:33:37,340
+زائد 2 بيساوي نقص 1 الآن أيضًا ماذا نلزمنا في
+
+457
+00:33:37,340 --> 00:33:43,420
+القانون F عند 3 و 2 F عند 3 و 2 بيصير 9 ناقص 3 في 2
+
+458
+00:33:43,420 --> 00:33:50,070
+ب6 زائد نص اللي هو في 2 في 2 ب4 لأن Y ب2اللي بيصير
+
+459
+00:33:50,070 --> 00:33:55,210
+عنده 2 زائد 3 والمجموع هذا بيساوي 8 إذا حصلنا
+
+460
+00:33:55,210 --> 00:33:59,830
+Fx و Fy و F عند نقطة 3 و 2 إذا نعوض على L و X Y
+
+461
+00:33:59,830 --> 00:34:03,770
+بيساوي F في 3 و 2 زائد Fx في X minus 3 زائد Fy في
+
+462
+00:34:03,770 --> 00:34:08,610
+Y minus 2 و Y يساوي اللي هو التالي F في 3 و 2
+
+463
+00:34:08,610 --> 00:34:13,990
+ثمانية هذه Fx في 3 و 2 اللي هي 4 مضروبة في X ناقص
+
+464
+00:34:13,990 --> 00:34:15,670
+اللي هي X note اللي هي 3
+
+465
+00:34:23,600 --> 00:34:31,680
+Y-Y-Y-Y-Y-Y-Y-Y-Y-Y-Y
+
+466
+00:34:31,890 --> 00:34:35,630
+وهي نقطة
+
+467
+00:34:35,630 --> 00:34:41,750
+3 و2 في الـ X Y بيساوي 4 X ناقص Y ناقص 2 يمكن أن
+
+468
+00:34:41,750 --> 00:34:45,270
+يكون هكذا حلنا أو جدنا الـ Linearization زي ما
+
+469
+00:34:45,270 --> 00:34:49,390
+قلنا أن الـ Linearization في الآخر L of X Y ليس
+
+470
+00:34:49,390 --> 00:34:53,430
+بالضبط بيساوي F of X Y بيساوي تقريباً 4 points
+
+471
+00:34:53,430 --> 00:34:59,110
+close to X Y  ليه 3 X نقطة Y نقطة 3 و2 طب في خطأ
+
+472
+00:34:59,110 --> 00:35:04,000
+مقدار الخطأ نحسبه بالنظرية التالية، الخطأ اللي نقرأه
+
+473
+00:35:04,000 --> 00:35:06,900
+الـ Approximation هو F of XY بيساوي تقريباً L of
+
+474
+00:35:06,900 --> 00:35:12,320
+XY بحسب التالي، نشوف الخطأ فيه يعني الخطأ اللي
+
+475
+00:35:12,320 --> 00:35:17,520
+نسميه إيه هو عبارة عن الفرق بين اللي هو F of XY و
+
+476
+00:35:17,520 --> 00:35:22,200
+L of XY الفرق بينهم اللي هو عبارة عن أكبر فرق
+
+477
+00:35:22,200 --> 00:35:26,500
+له  شيء اسمه الـ Error و هنشوف قديش قيمته نشوف
+
+478
+00:35:26,500 --> 00:35:27,360
+أيش النظرية بتقول
+
+479
+00:35:34,050 --> 00:35:39,370
+لو فرضنا أن F و Fxx و Fyy و Fx و Fy كلها  متصلة
+
+480
+00:35:39,370 --> 00:35:42,430
+اللي هو الـ first و second partial derivatives
+
+481
+00:35:42,430 --> 00:35:46,830
+المفروض أن تكون متصلة عند نقطة x0 و y0 في
+
+482
+00:35:46,830 --> 00:35:51,370
+open set  حول نقطة x0 و y0يعني فرضنا أن
+
+483
+00:35:51,370 --> 00:35:55,810
+الـ F و الـ partial derivatives كلها متصلة في
+
+484
+00:35:55,810 --> 00:36:00,870
+rectangle حول الـ X0 و Y0 ومنفترض أن M إيش هي
+
+485
+00:36:00,870 --> 00:36:05,230
+is any upper bound for the values of يعني هي عبارة
+
+486
+00:36:05,230 --> 00:36:11,370
+عن حد أعلى لقيمة الـ Fxx والـ Fyy والـ Fxy  على
+
+487
+00:36:11,370 --> 00:36:15,230
+ال rectangle R اللي بنحكي عليه then ال error
+
+488
+00:36:16,660 --> 00:36:20,280
+Incured in replacing f of x,y on R by its
+
+489
+00:36:20,280 --> 00:36:25,540
+linearization f of x,y يعني بنقول f of x,y و f of
+
+490
+00:36:25,540 --> 00:36:29,620
+x,y الفرق بينهم في داخل الـ rectangle هذا اللي
+
+491
+00:36:29,620 --> 00:36:35,960
+حكينا عليه هيكون الفرق بينهم لن يتعدى لن يتعدى الـ
+
+492
+00:36:35,960 --> 00:36:40,520
+absolute value لـ x,y أصغر أو يساوي نص الـ M في x
+
+493
+00:36:40,520 --> 00:36:45,780
+minus x0 زائد y minus y0 الكل تربيع إذا عملية
+
+494
+00:36:45,780 --> 00:36:51,820
+تقريب L of X Y بتساوي
+
+495
+00:36:51,820 --> 00:36:59,130
+تقريباً F of X Y في المنطقة R، بتكون تقريباً بتساويها
+
+496
+00:36:59,130 --> 00:37:06,410
+لكن في Error الـ L of X Y ناقص F of X Y لن يتعدى
+
+497
+00:37:06,410 --> 00:37:10,270
+إشارة سالبة أو موجبة بهذه القيمة اللي هي قيمة الـ Error
+
+498
+00:37:10,270 --> 00:37:14,710
+أصغر أو يساوي نص الـ M الـ M يعني الـ Upper Bound
+
+499
+00:37:14,710 --> 00:37:22,650
+لهذه الـ FXX و FYY و FXY في X – X¹ زائد Y – Y¹ الكل
+
+500
+00:37:22,650 --> 00:37:26,110
+تربيع وحتى تتضح هذه الصورة أكثر خلينا ناخد
+
+501
+00:37:26,110 --> 00:37:31,270
+example example find an upper bound for the error
+
+502
+00:37:31,270 --> 00:37:34,530
+in the approximation of X Y بيساوي ألف X Y in
+
+503
+00:37:34,530 --> 00:37:38,750
+example خمسة خمسة over the rectangle  كده بقول لي
+
+504
+00:37:38,750 --> 00:37:41,330
+أنت اللي جابلي شوية ال example الخمسة اللي جابلي
+
+505
+00:37:41,330 --> 00:37:44,210
+بشوية هذا أو قيمته ال linearization هي
+
+506
+00:37:44,210 --> 00:37:47,270
+linearization إلا أن هذا الـ Linearization اللي
+
+507
+00:37:47,270 --> 00:37:54,390
+أوجدنا تقريباً  هذه بتساوي هذه الدالة وبين في
+
+508
+00:37:54,390 --> 00:37:57,650
+حول التلاتة واتنين كل ما بعدنا عن التلاتة
+
+509
+00:37:57,650 --> 00:38:01,860
+واتنين بيصير اللي هو الفرق بين هذا وهذا كبير الآن
+
+510
+00:38:01,860 --> 00:38:06,160
+بدأ أشوف لك قديش اللي هو الـ Error بين ال .. الحساب
+
+511
+00:38:06,160 --> 00:38:09,960
+أنه .. أنه XR بيبقى نقص XY زائد نص Y تربيع زائد
+
+512
+00:38:09,960 --> 00:38:15,380
+تلاتة بيساوي 4X ناقص Y ناقص 2 تقريباً في المنطقة
+
+513
+00:38:15,380 --> 00:38:19,040
+حول التلاتة اتنين تلاتة و اتنين حول النقطة
+
+514
+00:38:19,040 --> 00:38:21,700
+تلاتة و اتنين بدأ أشوف لك الآن الفرق بين هذه وهذه
+
+515
+00:38:21,700 --> 00:38:27,440
+يعني عملية التقريب هذه بهذه قديش تقريباً الـ Error
+
+516
+00:38:27,440 --> 00:38:32,520
+فيها اللي بده يقول لنا إياها التي هي المثال التالي
+
+517
+00:38:32,520 --> 00:38:36,580
+وخلينا نستذكر هذه لو أردنا fx فخلينا نستخدم
+
+518
+00:38:36,580 --> 00:38:41,600
+fxx إيش هتساوي في هذه الحالة؟ اتنين fyy إيش هتساوي
+
+519
+00:38:41,600 --> 00:38:48,920
+واحد،  مظبوط fxy إيش هتساوي؟ ناقص واحد، ناشي خلينا
+
+520
+00:38:48,920 --> 00:38:51,920
+نحفظهم و نستخدمهم بعد شوية لأن بدنا نلزمنا في
+
+521
+00:38:51,920 --> 00:38:56,280
+السؤال الآن، الآن بيقول لي find an upper bound for
+
+522
+00:38:56,280 --> 00:38:59,420
+the error in the approximation of of xy بيساوي
+
+523
+00:38:59,420 --> 00:39:03,040
+تقريباً الـf of x y في خطأ، الخطأ قلنا طبعاً بده
+
+524
+00:39:03,040 --> 00:39:07,620
+يشمل على ال rectangle R X minus ثلاثة أصغر من
+
+525
+00:39:07,620 --> 00:39:10,480
+point one يعني ما نبعدش عن التلاتة ب point one هي
+
+526
+00:39:10,480 --> 00:39:13,190
+نقطة تلاتة و اتنين اللي إحنا حكينا عنها اللي
+
+527
+00:39:13,190 --> 00:39:16,710
+حولها بدنا نعمل إيه؟ عشان نعمل اللي هي دي يعني في
+
+528
+00:39:16,710 --> 00:39:20,970
+داخل المستطيل هذا اللي هو X ناقص تلاتة من X هيك
+
+529
+00:39:20,970 --> 00:39:25,630
+point one يعني من اتنين تسعة من عشرة لتلاتة واحد
+
+530
+00:39:25,630 --> 00:39:31,650
+من عشرة وتحت من اللي هي Y اللي هي واحد وتسعة من
+
+531
+00:39:31,650 --> 00:39:34,470
+عشرة وتحت اتنين واحد من عشرة في داخل ال rectangle
+
+532
+00:39:34,470 --> 00:39:40,450
+هذا اللي بتكون من هذه وهذه بدنا نجد اللي هو الخطأ
+
+533
+00:39:40,450 --> 00:39:43,750
+في الحساب في الـ approximation اللي هو f of x y
+
+534
+00:39:43,750 --> 00:39:47,390
+بيساوي قيمة f of x y  إيش طالب لي express the upper
+
+535
+00:39:47,390 --> 00:39:51,770
+bound as a percentage of f of تلاتة و اتنين بقول
+
+536
+00:39:51,770 --> 00:39:56,110
+لي اللي هو وجد لي نسبة الخطأ بالنسبة لقيم اتنين
+
+537
+00:39:56,110 --> 00:39:59,750
+الدالة عند تلاتة و اتنين يعني بدك تحسب الخطأ أول
+
+538
+00:39:59,750 --> 00:40:04,870
+شيء اللي هو بدك توجد الـ E اللي قبل شوية و تحسب ال
+
+539
+00:40:04,870 --> 00:40:08,330
+percentage لل error بالنسبة لل F of 3 و 2 The
+
+540
+00:40:08,330 --> 00:40:11,670
+value of F centered at the rectangle يعني F 3 و 2
+
+541
+00:40:11,670 --> 00:40:17,170
+هي قيمة F اللي هو مركز ال rectangle اللي بتكون منه
+
+542
+00:40:17,170 --> 00:40:21,060
+هذا اللي هو ال rectangle اللي حكيته قبل شوية الآن
+
+543
+00:40:21,060 --> 00:40:26,640
+نجد الـ Error نستخدم الـ Inquality  A of XY أصغر من
+
+544
+00:40:26,640 --> 00:40:31,360
+النص M في هذا المقدار هذا القانون الآن M إيش هي؟ M
+
+545
+00:40:31,360 --> 00:40:35,220
+هي عبارة عن .. قلنا إيه؟ هي Upper Bound لمن؟ لـ
+
+546
+00:40:35,220 --> 00:40:40,800
+FXX و FXY و FYY من القانون هذا الآن نجد هذول FXX و
+
+547
+00:40:40,800 --> 00:40:45,480
+FXY  أوجدناهم يا إنت قبل شوية FXX طلعت قديش؟ 2 لـ
+
+548
+00:40:45,480 --> 00:40:48,240
+Absolute Value لها 2 FXY طلعت ناقص 1 لـ Absolute
+
+549
+00:40:48,240 --> 00:40:51,960
+Value لها 1، FYY وجدناها واحد طلعت الـ
+
+550
+00:40:51,960 --> 00:40:55,300
+Absolute لها واحد إذا الآن الـ Upper bound لها دول
+
+551
+00:40:55,300 --> 00:40:58,940
+يعني أكبر واحد يعني الحد الأعلى اللي هنا  بين
+
+552
+00:40:58,940 --> 00:41:02,140
+الاتنين والواحد والواحد أكيد اتنين هو اتنين إذا نأخذ
+
+553
+00:41:02,140 --> 00:41:05,960
+الـ M واحنا مرتاحين بتساوي اتنين إذا صار عندي اللي
+
+554
+00:41:05,960 --> 00:41:09,440
+هو الـ Error هذا أصغر أو يساوي نص في اتنين اللي هي
+
+555
+00:41:09,440 --> 00:41:13,600
+الـ M في X minus ثلاثة اللي هي X0 زائد Y minus
+
+556
+00:41:13,600 --> 00:41:16,740
+اتنين اللي هي Y0 هذا الكل تربيع هذا القانون طبعاً
+
+557
+00:41:17,210 --> 00:41:20,670
+الآن X-3 معطيني إياها دائماً إيه؟ أصغر أو يساوي
+
+558
+00:41:20,670 --> 00:41:25,670
+0.1 داخل المستطيل تبعنا اللي هي زائد Y-Y2 اللي هي
+
+559
+00:41:25,670 --> 00:41:28,730
+أصغر من 0.1 اللي هو داخل المستطيل تبعنا إذا هذه
+
+560
+00:41:28,730 --> 00:41:31,970
+نستبدلها بأصغر أو يساوي اللي هو بيصير ال error
+
+561
+00:41:31,970 --> 00:41:36,590
+أصغر أو يساوي نصف 2 اللي هي بواحد ماشي اللي هي ده
+
+562
+00:41:36,590 --> 00:41:41,330
+صار  اللي هي point one الكل زائد point one الكل
+
+563
+00:41:41,330 --> 00:41:43,850
+إيش؟ تربيع لأن هو أصغر أو يساوي point one أو
+
+564
+00:41:43,850 --> 00:41:46,490
+أصغر أو يساوي point one لذا صار أصغر أو يساوي point one
+
+565
+00:41:46,490 --> 00:41:51,190
+تربيع اللي هو point 04 لذا صار ال error لا يتعدى
+
+566
+00:41:51,190 --> 00:41:57,970
+point 04 الآن الـ Percentage of F of 3 و 2 بيساوي
+
+567
+00:41:57,970 --> 00:42:03,150
+طبعاً اللي هي نسبة الخطأ بالنسبة لمين؟ F of 3 و 2 F
+
+568
+00:42:03,150 --> 00:42:05,990
+of 3 و 2 إيش طلعت معانا قبل شوية؟ أو جدناها في
+
+569
+00:42:05,990 --> 00:42:09,250
+المثال اللي قبل ثمانية The error is no greater
+
+570
+00:42:09,250 --> 00:42:12,770
+than يعني هيكون ال error نسبة ال percentage في ال
+
+571
+00:42:12,770 --> 00:42:20,460
+error لن تتعدى هتكون أصغر أو يساوي .04 على ثمانية
+
+572
+00:42:20,460 --> 00:42:27,220
+مضروبة 100% بيطلع 0.5 من مائة إذا هذه نسبة
+
+573
+00:42:27,220 --> 00:42:34,200
+الخطأ هذا هو الخطأ الفرق بين f of x,y و f of x,y
+
+574
+00:42:34,200 --> 00:42:37,920
+إن شاء الله تكون الأمور واضحة الآن اللي حكيناه عن
+
+575
+00:42:37,920 --> 00:42:40,560
+الـ functions of two variables ده نحكي عن الـ
+
+576
+00:42:40,560 --> 00:42:44,280
+functions of more than two variables يعني of three
+
+577
+00:42:44,280 --> 00:42:47,780
+variables أو غيره Analogous result hold for
+
+578
+00:42:47,780 --> 00:42:50,760
+differentiable functions for more than two
+
+579
+00:42:50,760 --> 00:42:54,200
+variables يعني الآن في عندنا اللي هي شغلات مشابهة
+
+580
+00:42:54,200 --> 00:42:58,860
+هتحدث لـ function of three variables خلينا نشوف
+
+581
+00:42:58,860 --> 00:43:01,520
+اللي هو linearization الـ linearization of f of x,
+
+582
+00:43:01,640 --> 00:43:07,520
+y, z at a point x0, y0, z0 is يعني الآن الدالة
+
+583
+00:43:07,520 --> 00:43:12,460
+اللي عندي F of X وY وZ بدي أقربها بدالة خطية اسمها
+
+584
+00:43:12,460 --> 00:43:16,960
+L of X وY وZ طبعاً أقربها بدالة خطية حول النقطة X0
+
+585
+00:43:16,960 --> 00:43:20,720
+و Y0 و Z0 طبعاً في تقريبات أخرى  اللي يمكن
+
+586
+00:43:20,720 --> 00:43:23,340
+تكون .. بعض الدوال اللي تكون تقريب بدالة خطية
+
+587
+00:43:23,340 --> 00:43:27,600
+صعبة، يمكن أن تكون الـ Quadratic Approximation أسهل
+
+588
+00:43:27,600 --> 00:43:31,020
+و كذا في طبعاً طرق في الـ .. الـ .. الـ .. اللي هي
+
+589
+00:43:31,020 --> 00:43:34,140
+numerical analysis بتاخدوها كتير بس إحنا بناخد هنا
+
+590
+00:43:34,140 --> 00:43:37,160
+اللي هي linearization تقريب اللي هو الدالة بدالة
+
+591
+00:43:37,160 --> 00:43:41,120
+خطية حول النقطة X Note و Y Note و Z Note اللي قاله
+
+592
+00:43:41,120 --> 00:43:44,060
+في X و Y و Z زي ما جاب بالضبط قيمة الـ F عند
+
+593
+00:43:44,060 --> 00:43:48,070
+النقطة المذكورة زائد fx عند النقطة المذكورة في x
+
+594
+00:43:48,070 --> 00:43:52,550
+-x0 زائد fy عند النقطة المذكورة في y-y0 زائد fz
+
+595
+00:43:52,550 --> 00:43:56,670
+عند النقطة المذكورة في z-z0 زي الأول بالضبط بس  إيه
+
+596
+00:43:56,670 --> 00:44:01,570
+إضافته بس هذا على اللي هو ال function of three
+
+597
+00:44:01,570 --> 00:44:07,430
+variable باقي الحديث هيكون بنفس الأسلوب نفس اللي
+
+598
+00:44:07,430 --> 00:44:11,950
+حكينا على الـ function of two variables نفترض أن R
+
+599
+00:44:11,950 --> 00:44:15,150
+is a closed rectangle solid centered at P0 and
+
+600
+00:44:15,150 --> 00:44:18,010
+lying in an open region on which the second
+
+601
+00:44:18,010 --> 00:44:21,270
+partial derivatives of R continuous suppose also
+
+602
+00:44:21,270 --> 00:44:26,890
+that هدول كلهم Fxx, Fyy, Fzz صاروا في متغير ثالث
+
+603
+00:44:26,890 --> 00:44:32,270
+Z, X, Y, Xz and Fyz are all less than or equal to
+
+604
+00:44:32,270 --> 00:44:35,370
+M يعني الآن نأخذ ال upper bound هدول through R
+
+605
+00:44:35,600 --> 00:44:40,780
+والخطأ هو كما قلنا الفرق بين هذه وهذه أن الـ
+
+606
+00:44:40,780 --> 00:44:44,180
+approximation of F by L is bounded through R by
+
+607
+00:44:44,180 --> 00:44:47,940
+the inequality يعني الآن الخطأ لن يتعدى اللي هو
+
+608
+00:44:47,940 --> 00:44:51,160
+سيكون أصغر أو يساوي نص ال M في ال X minus X node زائد
+
+609
+00:44:51,160 --> 00:44:55,180
+هذه نفسها نفس القانون بس ضفنا لهم Z minus Z node
+
+610
+00:44:55,180 --> 00:45:00,560
+طبعاً كل تربيع ده اللي هو L of X Y و Z بيساوي F of X
+
+611
+00:45:00,560 --> 00:45:03,860
+Y و Z في داخل ال rectangle طبعاً ال rectangle صارت
+
+612
+00:45:03,860 --> 00:45:08,610
+الآن عبارة عن صندوق اللي هو الـ X minus X naught
+
+613
+00:45:08,610 --> 00:45:11,230
+أص
+
+667
+00:49:15,990 --> 00:49:22,090
+Error؟ كالتالي قلنا من نحتاج Fxx،  هذا Fxx عنده
+
+668
+00:49:22,090 --> 00:49:30,990
+هذا Fxx ، واثنين Fyy هنا، على طول Fxy ، Fxy  اللي هي
+
+669
+00:49:30,990 --> 00:49:40,300
+ناقص واحد، ΔX وهو صفر، و ΔY هو صفر لأن التفاضل
+
+670
+00:49:40,300 --> 00:49:43,440
+بالنسبة لـ X أو Y هو صفر، لذلك ΔY و ΔX كلهما
+
+671
+00:49:43,440 --> 00:49:48,500
+صفر. لكن ΔZ هي ناقص ثلاثة Sin Z، إذن الآن
+
+672
+00:49:48,500 --> 00:49:51,800
+أريد أن أجلب Upper Bound لهذه المجموعات، أين؟ على
+
+673
+00:49:51,800 --> 00:49:54,880
+المنطقة هذه. نحن دائماً كان يطلع عند الـ Upper
+
+674
+00:49:54,880 --> 00:49:58,340
+Bound اللي هو يطلع أن هذه أعداد جاهزة، ما عدا هذه
+
+675
+00:49:58,340 --> 00:50:01,860
+ليست عدد، إذا هذه التي نريد... التي نريد أن نشتغل عليها
+
+676
+00:50:01,860 --> 00:50:04,800
+لأن الأعداد هذه جاهزة: 2، الـ Absolute Value لها 2
+
+677
+00:50:04,800 --> 00:50:07,900
+وهذه جاهزة 0، وهذه الـ Absolute Value  لها
+
+678
+00:50:07,900 --> 00:50:11,300
+1، وهذه، وهذه الـ Absolute Value لها 1
+
+679
+00:50:11,300 --> 00:50:14,500
+لما نشوف الـ Absolute Value of ZZ هذه، ماذا يساوي؟
+
+680
+00:50:14,500 --> 00:50:17,590
+الـ Absolute Value لهذه؟ الـ Absolute Value لهذه
+
+681
+00:50:17,590 --> 00:50:20,650
+الآن، لا أحد يقول لي أصغر أو أشهر وثلاثة، لا أنا
+
+682
+00:50:20,650 --> 00:50:24,110
+أريدها أين؟ في المنطقة التي Z أصغر من أو تساوي 0.1
+
+683
+00:50:24,110 --> 00:50:30,270
+في المنطقة الصغيرة هذه. الآن قيمة Z، قيمة Z في
+
+684
+00:50:30,270 --> 00:50:35,150
+المنطقة التي تكون... التي هي قريبة من... التي هي قريبة
+
+685
+00:50:35,150 --> 00:50:38,650
+من الصفر، يعني Z بتكون قيمتها
+
+686
+00:50:38,650 --> 00:50:42,590
+Z تقريباً تقريباً بتساوي 0.1،  يعني الـ Absolute
+
+687
+00:50:42,590 --> 00:50:47,730
+Value لقيمة Z أصغر من ثلاثة،  sign من الـ point
+
+688
+00:50:47,730 --> 00:50:52,330
+one، عندي ماشي، sign Z بيكون أصغر من أو يساوي
+
+689
+00:50:52,330 --> 00:50:56,590
+اللي هي sign الـ point one، حتى ما نحتاج أن نقول أن الـ
+
+690
+00:50:56,590 --> 00:51:01,050
+sign يساوي تقريباً القيمة، لأن sign Z أصغر من أو
+
+691
+00:51:01,050 --> 00:51:05,170
+يساوي ثلاثة sign الـ point one في الفترة،  أين؟ حول
+
+692
+00:51:05,170 --> 00:51:10,620
+هذه المنطقة. طيب، الآن في المنطقة هذه يا جماعة
+
+693
+00:51:10,620 --> 00:51:16,060
+عندي قيمة Z أصغر من أو تساوي قيمة 0.01، هذه قيمة
+
+694
+00:51:16,060 --> 00:51:18,760
+الـ 0.01، احسبيها على الآلة، واضربيها في الثلاثة،
+
+695
+00:51:18,760 --> 00:51:24,180
+ستجدينها تقريباً بـ 0.3، بكل الأحوال الآن هذه الـ
+
+696
+00:51:24,180 --> 00:51:29,800
+FZZ صغيرة 0.3، الآن الـ Upper Bound من يأخذه في
+
+697
+00:51:29,800 --> 00:51:32,900
+هذه الحالة؟ اللي هو أكبر واحد، اللي هو H2 as a bound
+
+698
+00:51:32,900 --> 00:51:36,500
+in the second partial derivatives. ماشي، إذا الـ
+
+699
+00:51:36,500 --> 00:51:39,960
+Upper bound في هذه الحالة اللي هو ماذا؟ Fxx يساوي
+
+700
+00:51:39,960 --> 00:51:43,320
+اثنين، الآن نعوض، hence the error included by
+
+701
+00:51:43,320 --> 00:51:48,940
+replacing F by L on R satisfies E، نصف الكلام هذا، الـ
+
+702
+00:51:48,940 --> 00:51:52,680
+M هي اثنين، نصف في اثنين في هذه 0.1، هي من
+
+703
+00:51:52,680 --> 00:51:55,300
+فوق، أصغر من أو تساوي 0.1، وهذا أصغر من 0.2
+
+704
+00:51:55,300 --> 00:51:59,780
+والأخيرة التي هي أصغر من أو تساوي 0.01. الآن كلنا
+
+705
+00:51:59,780 --> 00:52:03,960
+هنا نجمع مع بعض، ونضرب، فهذا المقدار الذي
+
+706
+00:52:03,960 --> 00:52:07,840
+عندي، وبهذا يكون أننا نحن الذي هو أوجدنا الخطأ في
+
+707
+00:52:07,840 --> 00:52:11,340
+التقريب، وبهذا نكون قد أنهينا المحاضرة، وإلى
+
+708
+00:52:11,340 --> 00:52:14,220
+لقاء آخر. السلام عليكم ورحمة الله وبركاته
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8f176904717eeffe922bfe8cd1be6486ff8c7ca5
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4_postprocess.srt
@@ -0,0 +1,2832 @@
+1
+00:00:00,670 --> 00:00:04,150
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,150 --> 00:00:10,210
+وبركاته هذه المحاضرة رقم 18 لمساقطة فاضل وتكامل
+
+3
+00:00:10,210 --> 00:00:18,090
+الجين وهي المحاضرة السادسة chapter 14 حيث سنتناول
+
+4
+00:00:18,090 --> 00:00:24,910
+الحديث عن 146 وهو تحت منوع tangent planes and
+
+5
+00:00:24,910 --> 00:00:30,200
+differentialsبنشوف كيف نجد معادلة tangent planes و
+
+6
+00:00:30,200 --> 00:00:33,920
+كيف نحكي عن الـ differentials طبعا الـ tangent
+
+7
+00:00:33,920 --> 00:00:38,500
+planes فكرتها حتى إيجاد معادلتها بنفس فكرة tangent
+
+8
+00:00:38,500 --> 00:00:42,680
+line في calculus 1000 و الـ differentials هي تعنيم
+
+9
+00:00:42,680 --> 00:00:48,090
+للـ differentials اللي أخدناهافي تفاضل ألف وكذلك
+
+10
+00:00:48,090 --> 00:00:51,570
+هنحكي عن الـ Linearization اللي هو أفدنا زيه لـ
+
+11
+00:00:51,570 --> 00:00:54,470
+Function of one variable هناخدها لـ Function of
+
+12
+00:00:54,470 --> 00:00:58,030
+two variables أو Equivariables نيجي الآن للـ
+
+13
+00:00:58,030 --> 00:01:01,670
+Tangent planes and normal lines اللي هى اللي
+
+14
+00:01:01,670 --> 00:01:06,490
+هنتحدث عنها اليوملو كان عندي R بساوية G of T في
+
+15
+00:01:06,490 --> 00:01:09,770
+تجاه I and H of T في تجاه J زي الـK وT في تجاه
+
+16
+00:01:09,770 --> 00:01:13,630
+الـK is a smooth curve on a level surface F of XY
+
+17
+00:01:13,630 --> 00:01:19,290
+وZ بساوية C يعني زي ما حكينا قبل هيك اللي هو
+
+18
+00:01:25,330 --> 00:01:29,130
+عند الـ Smooth Curve R بسوء U of T زادة U of T
+
+19
+00:01:29,130 --> 00:01:34,450
+زادة K of T عوضه في داخل الـ F of X Y وZ بعملية
+
+20
+00:01:34,450 --> 00:01:38,990
+constant يعني بعملية level curveالان زي ما عملنا
+
+21
+00:01:38,990 --> 00:01:43,750
+في المحاضرة اللي فاتت اخدنا f of x, f of g و h, h
+
+22
+00:01:43,750 --> 00:01:46,310
+of t, k of t كانت two variables ان انا تلقي
+
+23
+00:01:46,310 --> 00:01:50,790
+variables بساوة c هذه فاضلة تنتين بالنسبة ليها
+
+24
+00:01:50,790 --> 00:01:55,450
+ثابتة بالنسبة ليها اذا انا اكيد تفضلها سفروهذه
+
+25
+00:01:55,450 --> 00:01:59,330
+اللي هي تفضلها by Chain Rule اللي هي Del F by Del
+
+26
+00:01:59,330 --> 00:02:05,690
+X في Del X by Del T الـ X اللي هنا G وDel F by Del
+
+27
+00:02:05,690 --> 00:02:10,430
+Y في Del Y by Del T اللي هي H في هذه الحالة وDel F
+
+28
+00:02:10,430 --> 00:02:15,410
+by Del Z في Del K by Del T اللي هي Z في هذه الحالة
+
+29
+00:02:15,410 --> 00:02:19,470
+كله هي ساوي 7هذه مثلًا عملناها في المرة الماضية
+
+30
+00:02:32,260 --> 00:02:36,800
+يعني الآن بقول لي اللي هو الـ Gradient للـ F عمودي
+
+31
+00:02:36,800 --> 00:02:44,840
+على الـ Velocity لأي اللي هو Curve بمرق النقطة P0
+
+32
+00:02:44,840 --> 00:02:48,920
+يعني لو كان عندي هذا الـ Curve بمرق النقطة P0 اللي
+
+33
+00:02:48,920 --> 00:02:53,260
+هو مماسق اللي هو V1 اللي هو بيكون ال Gradient للـ
+
+34
+00:02:53,260 --> 00:02:57,580
+F عمودي عليه وأيضا لو كان هذه عندي كمان واحد تاني
+
+35
+00:02:57,740 --> 00:03:02,740
+أو هذا واحد تاني فبكون مماسه اللي هو V2 اللي هي ال
+
+36
+00:03:02,740 --> 00:03:06,360
+velocity اللي هي DR بDT برضه عمودي عالميين على ال
+
+37
+00:03:06,360 --> 00:03:09,760
+gradient يعني لو كان عندي كمان برضه curve موربل بي
+
+38
+00:03:09,760 --> 00:03:13,180
+نوت اللي هو ال curve هذا اللي بيخلق اللي هي هذه
+
+39
+00:03:13,180 --> 00:03:18,270
+ساوة C لبس ال surfaceبخلّي الـ Gradient للـ F
+
+40
+00:03:18,270 --> 00:03:23,510
+عمودي على مماسه الـ DRIDT يعني الـ Gradient للـ F
+
+41
+00:03:23,510 --> 00:03:28,730
+هيكون عمودي على كل اللي هي على الـ Plane اللي
+
+42
+00:03:28,730 --> 00:03:35,050
+بتكون من الـ V1 و V2 و بحمل أيه اللي هو Tangent
+
+43
+00:03:35,050 --> 00:03:41,590
+لأيه اللي هو Curve بمر بالنقطة D Note و بخلّي هذا
+
+44
+00:03:41,590 --> 00:03:47,290
+اللي هو بساوي Level Surfaceطيب إذا هذا اللي حكينا
+
+45
+00:03:47,290 --> 00:03:51,130
+عنه هو الـ tangent plane at the point B0 هو ال
+
+46
+00:03:51,130 --> 00:03:54,310
+tangent plane اللي حكينا عنه on the level surface
+
+47
+00:03:54,310 --> 00:03:57,790
+F of XYZ بالساوية سفق C of a differentiable
+
+48
+00:03:57,790 --> 00:04:02,480
+function is the plane through B0الأنمال هو Normal
+
+49
+00:04:02,480 --> 00:04:07,580
+على Gradient إذا صار عندي Gradient Normal على
+
+50
+00:04:07,580 --> 00:04:12,660
+Tangent Plane هذا هو الـ Tangent Plane صار الـ
+
+51
+00:04:12,660 --> 00:04:16,360
+Gradient عمودي عليه إذا الـ Tangent Plane هو عبارة
+
+52
+00:04:16,360 --> 00:04:22,560
+عن الـ Plane اللي بيحمل الـ V1 و V2 و V3
+
+53
+00:04:25,790 --> 00:04:30,650
+المماسات للمنحنيات اللي بمرن بالنقطة بـ Note اللي
+
+54
+00:04:30,650 --> 00:04:34,710
+بعمل ان اللي هو F of X Y و Z بساوة C is a level
+
+55
+00:04:34,710 --> 00:04:38,210
+surface هذا هو الـ Tangent Plane وعرفنا العمودي
+
+56
+00:04:38,210 --> 00:04:41,610
+عليه اللي هو الـ Gradient وعارفين نقطة إذا بمعرفة
+
+57
+00:04:41,610 --> 00:04:44,970
+نقطة وعمودي على طول بنعرف معادلة الـ Plane زي ما
+
+58
+00:04:44,970 --> 00:04:49,230
+انتوا عارفين وبذلك مكون معادلة الـ Plane سهلة
+
+59
+00:04:49,420 --> 00:04:55,040
+التانجنت بلينتو Fxy بيساوي C at P0 وX0 وY0 وZ0 هي
+
+60
+00:04:55,040 --> 00:04:59,560
+عبارة عن اللي هو Fx اللي هي تبعات الـ Gradient لأن
+
+61
+00:04:59,560 --> 00:05:03,180
+Gradient بيساوي Fx في اتجاه I زي دي Gradient زي Fy
+
+62
+00:05:03,180 --> 00:05:06,460
+في اتجاه G زي Fz في اتجاه K هذا اللي هي ال
+
+63
+00:05:06,460 --> 00:05:10,900
+Gradient ومن ثم ال Gradient اللي هو العمودي على ال
+
+64
+00:05:10,900 --> 00:05:15,570
+surfaceومعرفة نقطة على الـ Surface بنعرف معادلة
+
+65
+00:05:15,570 --> 00:05:20,410
+مين الـ Plane زي ما احنا عارفين بكون Fx في اتجاه X
+
+66
+00:05:20,410 --> 00:05:24,230
+في مضروف X minus X knot Fy في Y minus Y knot Fz في
+
+67
+00:05:24,230 --> 00:05:27,970
+Z minus Z knot حيث الـ Fx و الـ Fy و الـ Fz هما
+
+68
+00:05:27,970 --> 00:05:31,410
+اللي بيكوّن الـ Gradient اللي هو العمودي على الـ
+
+69
+00:05:31,410 --> 00:05:33,730
+Surface اللي احنا عارفينه و الـ X knot و الـ Y
+
+70
+00:05:33,730 --> 00:05:36,810
+knot و الـ Z knot هي عبارة عن النقاط اللي على الـ
+
+71
+00:05:36,810 --> 00:05:39,740
+Tangent الـ Planeبما عرفت الـ Gradient اللي هو
+
+72
+00:05:39,740 --> 00:05:42,560
+عمودي على الـ Tangent Plane وبما عرفت النقطة X0
+
+73
+00:05:42,560 --> 00:05:46,500
+وY0 عرفنا معادلة الـ Tangent Plane هل نستطيع أن
+
+74
+00:05:46,500 --> 00:05:51,200
+نعرف الـ Normal Line الـ Normal Line على الـ Level
+
+75
+00:05:51,200 --> 00:05:55,490
+Surface F of X Y وZ؟أه لإن احنا نجينا ان ال
+
+76
+00:05:55,490 --> 00:05:59,250
+gradient عمودي عالميا على اللي هو ال level على ال
+
+77
+00:05:59,250 --> 00:06:02,970
+tangent plane اللي هو عمودي عالميا على اللي هو هذه
+
+78
+00:06:02,970 --> 00:06:06,670
+ال surface اللي عندي مدام عمودي عليه اذا كل خط
+
+79
+00:06:06,670 --> 00:06:10,790
+عمودي عليه مكون موازية ال gradientما دام عرفت أنه
+
+80
+00:06:10,790 --> 00:06:17,270
+موازي للـ Line اتجاهه يعني بمعنى آخر وعرفت نقطة
+
+81
+00:06:17,270 --> 00:06:22,410
+يعني معادلة الخط الموازية للـ Gradient اللي هو الـ
+
+82
+00:06:22,410 --> 00:06:25,690
+Normal Line على اللي هو الـ Tangent Plane هو عبارة
+
+83
+00:06:25,690 --> 00:06:30,010
+عن X بساوي X Note زاد FX في T هذه تبعت Mean لـ
+
+84
+00:06:30,010 --> 00:06:33,530
+Gradient Y بساوي Y Note زاد FY تبعت لـ Gradient في
+
+85
+00:06:33,530 --> 00:06:41,870
+T زاد Z بساوي Z Note FZ في Tالان بمعرفة موازي للخط
+
+86
+00:06:41,870 --> 00:06:47,050
+الـ Gradient ومعرفة نقطة على الخط X0 وY0 وZ0 اعرفت
+
+87
+00:06:47,050 --> 00:06:51,970
+ان انا معادلة الـ Line Normal على الـ Tangent
+
+88
+00:06:51,970 --> 00:06:55,490
+Plane أو Normal على الـ Surface اللي قبل بشوية
+
+89
+00:06:55,490 --> 00:07:02,760
+شفناهالخط الموازي لهذا بقول عمودي على الـ Surface
+
+90
+00:07:02,760 --> 00:07:06,600
+أو اللي عندي عمودي على الـ Tangent Plane اقدرنا
+
+91
+00:07:06,600 --> 00:07:14,060
+نعرف مين ونعرف معادلته من معرفة اتجاهه ومعرفة نقطة
+
+92
+00:07:14,060 --> 00:07:19,880
+عليه زي ما كنا سابقا في شهر تلت معاشر ناخد مثال
+
+93
+00:07:19,880 --> 00:07:22,980
+عددي Find the Tangent Plane and Normal Line to the
+
+94
+00:07:22,980 --> 00:07:29,040
+Surface ومعطين الـ Surfaceعن طريق الـ X²Y²Z-9Y0
+
+95
+00:07:29,040 --> 00:07:34,480
+يعني نعطيني level surface أكتر نقطة من 1 و 2 و 4
+
+96
+00:07:34,480 --> 00:07:43,980
+هاي عندي اللي هو ال level surface X²Y²Z-9Y0 بدي
+
+97
+00:07:43,980 --> 00:07:48,210
+معادلة tangent plane لهعنده نقطة مين ليه واحد
+
+98
+00:07:48,210 --> 00:07:52,330
+اتنين اربعة ومعادلة مين الـ Normal line اللي عليه
+
+99
+00:07:52,330 --> 00:07:55,870
+لأن احنا عارفين الـ Plane اللي بنعرف معادلته لو
+
+100
+00:07:55,870 --> 00:07:59,010
+عرفنا نقطة عمودي عليه طب ما احنا عارفين انه الـ
+
+101
+00:07:59,010 --> 00:08:03,910
+Gradient لـ Level Surface عمودي عالميين على اللي
+
+102
+00:08:03,910 --> 00:08:09,570
+هو هذا الـ Planeما دام عارفين الـ G العمودي
+
+103
+00:08:09,570 --> 00:08:13,130
+وعارفين النقطة بنعرف عن عادلة الـ Tangent إذا الـ
+
+104
+00:08:13,130 --> 00:08:16,750
+Tangent بلينة ثلوج نقطة واحد واثنين واربعة عمودي
+
+105
+00:08:16,750 --> 00:08:20,010
+perpendicular to the gradient of F عند النقطة هذه
+
+106
+00:08:21,050 --> 00:08:25,010
+من الحديث اللي حدثناه قبل قليل والـ Gradient لـ F
+
+107
+00:08:25,010 --> 00:08:32,030
+يساوي 2X في اتجاه I زي 2Y في اتجاه J زائد المين K
+
+108
+00:08:32,030 --> 00:08:39,270
+عارفين ايش بسوي؟ Fx هيها و Fy هيها 2Y و Fz هي فقط
+
+109
+00:08:39,270 --> 00:08:43,300
+1لـ Gradient الـ F صار 2X في تجاه الـ I زائد 2Y في
+
+110
+00:08:43,300 --> 00:08:48,720
+تجاه الـ J زائد K اللي هي FX FY FZ لما تحسبها الـ
+
+111
+00:08:48,720 --> 00:08:52,260
+Gradient عند مين عنده النقطة المذكورة اللي هي 1 و
+
+112
+00:08:52,260 --> 00:08:56,220
+2 و 4 و سوى 2 في تجاه الـ I زائد اللي هي مكان الـ
+
+113
+00:08:56,220 --> 00:08:59,800
+Y2 يصير 4 في تجاه الـ J زائد الـ K هذا الـ
+
+114
+00:08:59,800 --> 00:09:02,460
+Gradient اللي هو العمودي على الـ Plane اللي بدنا
+
+115
+00:09:02,460 --> 00:09:07,640
+إياهعرفين نقطة على الـ Plane إذا جهزة معادلتنا
+
+116
+00:09:07,640 --> 00:09:12,440
+اللي هي اتنين اللي زمنا كنا نسميها A في X minus
+
+117
+00:09:12,440 --> 00:09:16,660
+واحد اللي هي زائد B اللي هي أربعة Y minus اتنين
+
+118
+00:09:16,660 --> 00:09:21,060
+عارفين ليش Y minus اتنين Y نقص اتنين Z نقص أربعة
+
+119
+00:09:21,060 --> 00:09:24,920
+والـ X نقص واحد النقطة اللي عارفينها في اللي هي C
+
+120
+00:09:24,920 --> 00:09:28,600
+هنا إيش بتساوي اللي هي معامل K اللي هو واحد إذا
+
+121
+00:09:28,600 --> 00:09:34,470
+صارت هذه معادلة Tangent Planeاللي هنبسطها 2x-2 وان
+
+122
+00:09:34,470 --> 00:09:40,620
+كده بتطلع عندى 2x زي 4y زي 14الآن بدنا نعرف الـ
+
+123
+00:09:40,620 --> 00:09:43,740
+Normal Line الـ Normal Line على الـ Surface هو
+
+124
+00:09:43,740 --> 00:09:45,880
+موازي من الـ Gradient والـ Gradient عمودي على الـ
+
+125
+00:09:45,880 --> 00:09:50,260
+Surface إذا الـ Normal Line عارفين موازي إيه؟ اللي
+
+126
+00:09:50,260 --> 00:09:55,140
+هو عبارة من 2I زي 4G زي K وعارفين النقطة عليه؟ إذا
+
+127
+00:09:55,140 --> 00:10:00,420
+على طول معدلته X بتساوي 1 اللي هي النقطة زي 2T
+
+128
+00:10:00,420 --> 00:10:07,500
+اللي هي 2 منها T Y بيساوي 2 اللي هي قيمة النقطة Y
+
+129
+00:10:07,800 --> 00:10:12,760
+زائد اللي هي ال .. ال .. ال .. ال .. ال .. اللي في
+
+130
+00:10:12,760 --> 00:10:18,200
+ال line الموازي اللي هي A4T و هذه الأخيرة T زائد 4
+
+131
+00:10:18,200 --> 00:10:20,560
+فبكون و T طبعا parameter زي ما انتوا عارفين ال
+
+132
+00:10:20,560 --> 00:10:24,320
+element in R فبكون ال normal line هي معدلة و طبعا
+
+133
+00:10:24,320 --> 00:10:28,240
+هذه شغلات قديمة لكن احنا بنعملها نستغل معلوماتنا
+
+134
+00:10:28,240 --> 00:10:34,810
+الجديدة في إيجادها طيبالان لو الصيغة اللى عندنا
+
+135
+00:10:34,810 --> 00:10:39,990
+كانت f of x y وz بساول ثابت طب لو كان مش معروض
+
+136
+00:10:39,990 --> 00:10:43,670
+علينا بالشكل هذا لو معروض علينا اللى هو معادلة
+
+137
+00:10:43,670 --> 00:10:48,030
+بالشكل ثاني يعني معروض علينا surface الان عند اللى
+
+138
+00:10:48,030 --> 00:10:55,340
+أعرضناها قبلاللي هي Z اللي هي عندي F of XY و Z
+
+139
+00:10:55,340 --> 00:10:59,700
+function اللي هو و بتساوي C سارة IH level surface
+
+140
+00:10:59,700 --> 00:11:03,700
+لو الموضوع مش معروض بالشكل هذا لو معروض بشكل آخر
+
+141
+00:11:03,700 --> 00:11:07,340
+خليني أشوف إيش مقصود لو عندي to find an equation
+
+142
+00:11:07,340 --> 00:11:10,140
+for the plane tangent to a smooth curve لو بدنا
+
+143
+00:11:10,140 --> 00:11:13,900
+نعرف اللي هو معادلة ال tangent plane على smooth
+
+144
+00:11:13,900 --> 00:11:16,960
+curve الآن Z بسوء F of XY مش مكتوب على صورة level
+
+145
+00:11:16,960 --> 00:11:21,770
+surfaceAt the point x0, y0 و z0 where z0 بيساوي f
+
+146
+00:11:21,770 --> 00:11:25,590
+of x0 و y0 بدنا نعرف اللي هو الـ Tangent Plane
+
+147
+00:11:25,590 --> 00:11:29,770
+لهذا الـ Surface z بيساوي f of x, y الآن بدنا نعمل
+
+148
+00:11:29,770 --> 00:11:33,770
+مياليه، لاحظوا الآن هذه z بيساوي f of x, y بيساوي
+
+149
+00:11:33,770 --> 00:11:38,460
+f of x, y نقل zH بتساوي 0الأن الـ Surface Z ساوية
+
+150
+00:11:38,460 --> 00:11:43,120
+F of X Y اللي هو هذا بعبّر عنه لو سمينا هذه كلها
+
+151
+00:11:43,120 --> 00:11:46,280
+على بعض Function F Capital of X Y و Z زي تابعت
+
+152
+00:11:46,280 --> 00:11:52,280
+زمان هتساوية F of X Y ناقص Z صار هذا هذا عبارة عن
+
+153
+00:11:52,280 --> 00:11:56,380
+Level Surface ماشي بساوية Level Surface الآن
+
+154
+00:11:56,380 --> 00:11:59,320
+بنتعامل معاه زي قبل بشوية الان ال partial
+
+155
+00:11:59,320 --> 00:12:01,540
+derivatives of F لو جينا أخدنا ال partial
+
+156
+00:12:01,540 --> 00:12:05,690
+derivatives عشان نطبّق عليه مين يا جماعةاللي هي
+
+157
+00:12:05,690 --> 00:12:09,930
+اللي جابل بشوية المعادلة الـ Fx إلها سوى Deal By
+
+158
+00:12:09,930 --> 00:12:14,670
+Deal X لمين لهذا المقدار هذه بالنسبة لـ X بس Fx
+
+159
+00:12:14,670 --> 00:12:19,730
+لأنه بالنسبة لـ Y صفر بالنسبة لـ Z صفر إذا Fx نقص
+
+160
+00:12:19,730 --> 00:12:25,120
+صفر وسوى Fx Smallفهنا fy بيصير deal فضلتها دي
+
+161
+00:12:25,120 --> 00:12:27,760
+بالنسبة ل y جزيياً هذه بتفضلها بالنسبة ل y كلها
+
+162
+00:12:27,760 --> 00:12:31,620
+بيصير deal by deal y f of x و y ناقص z نفس الأشي
+
+163
+00:12:31,620 --> 00:12:36,120
+بيصير fy ناقص اللي هي بالنسبة ل y سفر و يساوي fy
+
+164
+00:12:36,120 --> 00:12:39,900
+of z اللي هي deal هذا المقدار كله بالنسبة ل z هذا
+
+165
+00:12:39,900 --> 00:12:43,780
+بالنسبة ل z سفر ماشي الحال الآن نصارة ال x و ال y
+
+166
+00:12:43,780 --> 00:12:47,180
+و ال z عنده هنا في هذه الحالة independent تلاتة x
+
+167
+00:12:47,180 --> 00:12:51,190
+و y و z dependentأسف independent فمدار ساعة
+
+168
+00:12:51,190 --> 00:12:55,570
+independent إذن تفضل بالنسبة لبعض سفر إذن del هذا
+
+169
+00:12:55,570 --> 00:13:00,090
+المقدار by del z بيساوي سفر ناقص del by del z by
+
+170
+00:13:00,090 --> 00:13:03,670
+del z بيساوي إيش واحد وفعلا ناقص بس ناقص واحد لأن
+
+171
+00:13:03,670 --> 00:13:06,830
+هذه بتدعوضها في المعادلة اللي جاب المشوية اللي
+
+172
+00:13:06,830 --> 00:13:11,190
+حكينا عنها إما بس كانت F capital مثلا كنا بسميناها
+
+173
+00:13:11,190 --> 00:13:14,450
+F small اللي جاب المشوية يا جماعة بتدعوضها في هذه
+
+174
+00:13:14,450 --> 00:13:17,850
+اللي هي Fx بس هذه F capital بالسين وهذه F capital
+
+175
+00:13:17,850 --> 00:13:21,040
+وهذه F capitalأف كابيتال هي أف صغيرة اللي قبل شوية
+
+176
+00:13:21,040 --> 00:13:24,680
+و أف كابيتال هي أف صغيرة هذا لكن أف كابيتال للـ z
+
+177
+00:13:24,680 --> 00:13:28,240
+طلعت إيه؟ بتساوي سالب واحد فبتصير هذه المعادلة هي
+
+178
+00:13:28,240 --> 00:13:32,240
+بس بسالب واحد خليني أشوف المعادلة الجديدة بصير
+
+179
+00:13:32,240 --> 00:13:38,560
+عندي اللي هي عوضنا Fxy هي Fy هي Fz لما نعوض مع
+
+180
+00:13:38,560 --> 00:13:42,710
+مكانهم اللي هي ال formula اللي قبل شويةهذه Fx فهي
+
+181
+00:13:42,710 --> 00:13:46,550
+Fx صغيرة وهذه Fy فهي Fy صغيرة وهي من فوق ما
+
+182
+00:13:46,550 --> 00:13:51,710
+اخترناها والـ Fz هذه اشتلعت سالب واحد فبصير هذه هي
+
+183
+00:13:51,710 --> 00:13:59,140
+معادلة الـ tangent plane لل surface Z بتساوي Fof X
+
+184
+00:13:59,140 --> 00:14:03,540
+و Y يعني الآن هذه اللي هي كانت Bridge أو جسر اللي
+
+185
+00:14:03,540 --> 00:14:06,720
+هي هذه F Capital للوصول للمعادلة اللي أمامنا هذه
+
+186
+00:14:06,720 --> 00:14:09,700
+خلاص الآن لما تنعرض علينا Z بسيط و أقفف X و Y على
+
+187
+00:14:09,700 --> 00:14:13,520
+طول بنقدر نجد تنجين تـBlend لها بس بنجيب F X و F Y
+
+188
+00:14:13,520 --> 00:14:18,480
+وهذا بيصير Z minus Z not طيب نشوف إذا Blended
+
+189
+00:14:18,480 --> 00:14:21,260
+tangent to the surface Z بسيط و أقفف X Y at X not
+
+190
+00:14:21,260 --> 00:14:25,040
+و Y not و أقفف X not و Y notالتي هي the plane
+
+191
+00:14:25,040 --> 00:14:27,420
+tangent to the surface z بساوية f of x, y of a
+
+192
+00:14:27,420 --> 00:14:30,840
+differentiable function f at the point x0 و y0 و
+
+193
+00:14:30,840 --> 00:14:35,300
+z0 بساوية x0 و y0 و f of x0 و f of y و y0 is
+
+194
+00:14:35,300 --> 00:14:39,920
+لتوّه، قلت عشان أنا حكيت لتوّه، fx في x minus x0،
+
+195
+00:14:39,920 --> 00:14:44,300
+fy في y و y minus y0، ماقع الـ z minus z0 هي
+
+196
+00:14:44,300 --> 00:14:47,980
+معدلتنا وهي الـ tangent plane إلها أو surfaceنا
+
+197
+00:14:47,980 --> 00:14:50,800
+وهي الـ tangent plane إلها خلّيني أخد مثال عددي
+
+198
+00:14:50,800 --> 00:14:54,110
+جماعةFind her at a plane tangent to the surface z
+
+199
+00:14:54,110 --> 00:14:57,330
+بالساوية x cos y نقص y e to the x عند مين عنده
+
+200
+00:14:57,330 --> 00:15:01,530
+نقطة 00 الموضوع سهل هي عند z بالساوية كذا يعني
+
+201
+00:15:01,530 --> 00:15:06,230
+هيصير هذه عبارة عن تمثل لمين F of x y F small إذا
+
+202
+00:15:06,230 --> 00:15:09,050
+صارت عند z بالساوية F of x y يعني F of x y
+
+203
+00:15:09,050 --> 00:15:12,770
+بالساوية x cos y نقص y e to the x وبنطبق هذه
+
+204
+00:15:12,770 --> 00:15:18,770
+المعادلة ال Fx في هذه الحالة اللي هي عبارة عنCos y
+
+205
+00:15:18,770 --> 00:15:23,190
+ناقص y e to the x Cos y ناقص y e to the x تحسبها
+
+206
+00:15:23,190 --> 00:15:28,050
+عند من؟ عند الـ 0 و الـ 0 و الـ 0 تحسبها عند الـ 0
+
+207
+00:15:28,050 --> 00:15:29,450
+و الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+208
+00:15:29,450 --> 00:15:29,910
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+209
+00:15:29,910 --> 00:15:30,630
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+210
+00:15:30,630 --> 00:15:30,790
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+211
+00:15:30,790 --> 00:15:31,350
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+212
+00:15:31,350 --> 00:15:36,230
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+213
+00:15:36,230 --> 00:15:41,730
+عند الـ 0 وصلى الله عليه وسلم عن نبي عليه الصلاة
+
+214
+00:15:41,730 --> 00:15:50,350
+والسلام And fy ليها سالب x sin y سالب x sin y
+
+215
+00:15:50,350 --> 00:15:55,530
+تفاضلها بالاسم y e to the x بالسالي كان عند 0 0 0
+
+216
+00:15:55,530 --> 00:15:59,470
+هذا يصبح 0 وهذا يصبح e to the 0 يعني واحد يعني
+
+217
+00:15:59,470 --> 00:16:04,940
+سالب واحد من عين المارس اللي قدامهاإذا جهزة Fx و
+
+218
+00:16:04,940 --> 00:16:10,800
+Fy إذا الـ tangent plane is واحد هي مكان Fx في X
+
+219
+00:16:10,800 --> 00:16:13,360
+ناقص X نوت طبعاً Zero نحن نحسبها Zero Zero Zero
+
+220
+00:16:13,360 --> 00:16:18,700
+فناقص واحد اللي هي تبع ال Y هذه Fy مضروب Fy minus
+
+221
+00:16:18,700 --> 00:16:22,420
+Zero ناقص Z minus Zero اللي هو حسب القانون يسوى
+
+222
+00:16:22,420 --> 00:16:27,780
+Zero بيصير X ناقص Y ناقص Z بيسوى Zero هذه اللي هي
+
+223
+00:16:27,780 --> 00:16:33,530
+ال tangent plane to the surface شايفين مثلها؟الان
+
+224
+00:16:33,530 --> 00:16:37,990
+مثلًا آخر الـ Surface f of x y و z بساوية x تربيع
+
+225
+00:16:37,990 --> 00:16:41,550
+z y تربيعه نقص اتنين بساوية سفر هذا عبارة عن الـ
+
+226
+00:16:41,550 --> 00:16:45,090
+Level Surface هذا بيعمل ليهاش Cylinder x تربيع z y
+
+227
+00:16:45,090 --> 00:16:48,650
+تربيعه نقص اتنين بساوية سفر الـ z ثابتة متغيرة
+
+228
+00:16:48,650 --> 00:16:53,110
+مطلوقة زي ما نقول وهذا دائرة فبتعمل ليهاش الـ
+
+229
+00:16:53,110 --> 00:16:58,950
+Circularالآن X زائد Y زائد Z نقص أربعة بيساوي 0 جي
+
+230
+00:16:58,950 --> 00:17:02,330
+و X و Y و Z هذا ال level surface بيعمل لي إيش
+
+231
+00:17:02,330 --> 00:17:06,830
+إبلاين، إيش اللي طلبنيها؟ بقوللي ده surface هاي و
+
+232
+00:17:06,830 --> 00:17:10,110
+هذا اللي هو الـ cylinder و ال surface هذا الإبلاين
+
+233
+00:17:10,110 --> 00:17:13,250
+بقوللي meet in an ellipse، اتقاطعن في إيش؟ اتقاطعن
+
+234
+00:17:13,250 --> 00:17:16,150
+في ellipse، هي ال ellipse اللي اتقاطعن فيها، ماشي
+
+235
+00:17:16,150 --> 00:17:20,010
+الحال؟بقول find parametric equations for the line
+
+236
+00:17:20,010 --> 00:17:25,650
+tangent to E at point 1 و3 و3 بقوللي أوجد معادلة
+
+237
+00:17:25,650 --> 00:17:31,850
+الـ Parametric equation لأ معادلة اللي هي مين الخط
+
+238
+00:17:31,850 --> 00:17:36,810
+المماس اللي بيمس هذه الـ Ellipse عنده النقطة
+
+239
+00:17:36,810 --> 00:17:41,280
+المذكورةأنا عشان أجد معادلة المماس أو معادلة خط
+
+240
+00:17:41,280 --> 00:17:45,760
+بالزمن شغلتين بالزمن اللي هو vector موازي له لخط
+
+241
+00:17:45,760 --> 00:17:49,900
+التماس وبالزمن نقطة النقطة المعطينيها طب ال vector
+
+242
+00:17:49,900 --> 00:17:54,830
+الموازي شوفوا كيف نجد ال vector الموازي أول حاجةلو
+
+243
+00:17:54,830 --> 00:18:00,050
+ألحظنا على هذه اللي هي الـ Ellipse الـ Ellipse هي
+
+244
+00:18:00,050 --> 00:18:03,790
+عبارة موجودة طبعاً هي عبارة عن تفاطع الـ Plane و
+
+245
+00:18:03,790 --> 00:18:09,890
+الـ Cylinder الآن الـ Ellipse هذه موجودة في
+
+246
+00:18:09,890 --> 00:18:12,810
+الجهتين يعني موجودة في الـ Cylinder و موجودة في
+
+247
+00:18:12,810 --> 00:18:17,650
+الـ Plane الآن الـ Gradient للـ Fعمودي على مين؟
+
+248
+00:18:17,650 --> 00:18:21,310
+على اللي هو الـCylinder إذا عمودي على اللي هو أي
+
+249
+00:18:21,310 --> 00:18:25,990
+منحنة فيها من ضمنها اللي هي الـEllipse إذا
+
+250
+00:18:25,990 --> 00:18:28,950
+الـGradient للـF عمودي على الـEllipse هاي واحد،
+
+251
+00:18:28,950 --> 00:18:32,570
+خلوه في الذاكرة إتنين، الـGradient للـG عمودي على
+
+252
+00:18:32,570 --> 00:18:37,370
+مين؟ على الـSurface هذا اللي هو G بالساوية 0 مدام
+
+253
+00:18:37,370 --> 00:18:41,430
+عمودي عليه إذا عمودي على كل كرفي من ضمنه المين؟
+
+254
+00:18:41,540 --> 00:18:46,020
+اللي هي الـ Ellipse إذا صار الـ Gradient للـ G
+
+255
+00:18:46,020 --> 00:18:48,980
+عمودي على الـ Ellipse عنده نقطة هذه والـ Gradient
+
+256
+00:18:48,980 --> 00:18:52,680
+لل F عمودي على الـ Ellipse عند النقطة هذه إذا أكيد
+
+257
+00:18:52,680 --> 00:18:57,440
+الـ Cross Product بينهم هيكون ماله عمودي على
+
+258
+00:18:57,440 --> 00:19:03,080
+التنين، مظبوط؟ يعني بمعنى آخر موازي للمماسلأن
+
+259
+00:19:03,080 --> 00:19:06,660
+المماس اللي هو هنا الـ Gradient للألف عمودي عليه
+
+260
+00:19:06,660 --> 00:19:09,240
+والـ Gradient للـ D برضه عمودي على المماس اللي
+
+261
+00:19:09,240 --> 00:19:13,360
+بدناها لأن المماس عنده النقطة هذه اللي هو الـ
+
+262
+00:19:13,360 --> 00:19:15,420
+Gradient عمودي على الـ Curve اللي هو عمودي على
+
+263
+00:19:15,420 --> 00:19:18,120
+المماس والـ Gradient للألف عمودي على الـ Curve
+
+264
+00:19:18,120 --> 00:19:21,260
+اللي هو عمودي على المماس إذن الـ Gradient للأف
+
+265
+00:19:21,260 --> 00:19:24,540
+والـ Gradient للـ D عمودي على المماس إذن اللي هو
+
+266
+00:19:24,540 --> 00:19:30,780
+الـ Vector الموازياللي هو للمماس اللي هو العمودي
+
+267
+00:19:30,780 --> 00:19:34,600
+على اللي هو الـ gradient للـ F والـ gradient للـ G
+
+268
+00:19:34,600 --> 00:19:39,740
+زي مين؟ زي المماس، إذا صار المماس معروف vector
+
+269
+00:19:39,740 --> 00:19:42,340
+موازيئله الـ gradient للـ F كرس الـ gradient للـ G
+
+270
+00:19:42,340 --> 00:19:46,060
+ومعروف نقطة، إذا بنقدر نعرف اللي هو معدلته، خلّيني
+
+271
+00:19:46,060 --> 00:19:49,000
+نعرف الـ gradient للـ F كرس الـ gradient للـ G الـ
+
+272
+00:19:49,000 --> 00:19:51,740
+tangent line is orthogonal to both الـ gradient
+
+273
+00:19:51,740 --> 00:19:56,020
+للـ F and الـ gradient للـ G at 1 و 1 و 3 زي ما
+
+274
+00:19:56,020 --> 00:20:00,510
+ذكرت قبل شويةوهكذا بدأت بالمعنى الـ vector الـ
+
+275
+00:20:00,510 --> 00:20:05,150
+gradient للـ F across الـ gradient للـ G إذا نجد
+
+276
+00:20:05,150 --> 00:20:07,730
+الـ gradient للـ F عنده نقطة واحد و تلاتة و تلاتة
+
+277
+00:20:07,730 --> 00:20:11,510
+الـ gradient للـ F هو اتنين X اتجاه الـ I زاد
+
+278
+00:20:11,510 --> 00:20:14,570
+اتنين Y في اتجاه الـ G احسبها عند الواحد والواحد
+
+279
+00:20:14,570 --> 00:20:19,970
+والتلاتة بتطلع اتنين I زاد اتنين G هذا الـ
+
+280
+00:20:19,970 --> 00:20:23,290
+gradient للـ F الـ gradient للـ G is similar للإن
+
+281
+00:20:23,290 --> 00:20:29,510
+التفاضل بالنسبة للـ X هو Iبتفضل بالنسبة لـ Y صفر
+
+282
+00:20:29,510 --> 00:20:34,370
+بالنسبة لـ Z بسوء واحد يعني K الـ C لـ Grid لـ G
+
+283
+00:20:34,370 --> 00:20:39,170
+بسوء I زائد K لأن صار معروف هذا صار معروف هذا لأن
+
+284
+00:20:39,170 --> 00:20:43,190
+توجد الـ cross product بينهم بينهم الـ Grid انتلاف
+
+285
+00:20:43,190 --> 00:20:47,190
+في cross ال Grid لـ G بتعرفوا كيف بحط IGK بعدين
+
+286
+00:20:47,190 --> 00:20:50,990
+عناصر الأول اتنين اتنين صفر هي اتنين اتنين صفر
+
+287
+00:20:50,990 --> 00:20:52,690
+وهذه 101
+
+288
+00:20:56,990 --> 00:21:00,730
+بنعمل مفبوك الـ Determinant شطب هذا و شطب هذا
+
+289
+00:21:00,730 --> 00:21:05,670
+بيصير 2 في اتجاه الـ I هي شطب هذا و شطب هذا ناقص
+
+290
+00:21:05,670 --> 00:21:11,030
+اللي هو 2 في اتجاه الـ J لأن هذا سفر اللي بتظلها و
+
+291
+00:21:11,030 --> 00:21:15,670
+شطب هذا و شطب هذا بيصير 2 اللي هي سفر يعني ناقص 2
+
+292
+00:21:15,670 --> 00:21:19,690
+في اتجاه الـ K وعندي هذا اللي هو الـ Gradient للـ
+
+293
+00:21:19,690 --> 00:21:23,520
+F cross ال Gradient للـ Gلأن عرفت اللي هو vector
+
+294
+00:21:23,520 --> 00:21:26,660
+موازي للخط وعرفت اللي هي نقطة على الخط إذا بعرف
+
+295
+00:21:26,660 --> 00:21:31,040
+الميال الخط X بتساوي اللي هي واحد زاد اتنين في T
+
+296
+00:21:31,040 --> 00:21:37,140
+اللي هو Y بتساوي واحد اللي هو ناقص اتنين في T و Z
+
+297
+00:21:37,140 --> 00:21:41,800
+بتساوي اللي هو تلاتة هي تلاتة ناقص اتنين في T
+
+298
+00:21:41,800 --> 00:21:46,060
+بيكون هو ده معادلة المماس اللي هي ال ellipse
+
+299
+00:21:46,060 --> 00:21:51,460
+التقاطع لل surfaceالان نجي لـ Estimating Exchange
+
+300
+00:21:51,460 --> 00:21:55,340
+in a Specific Direction اللي هو احنا زمان كنا نحكي
+
+301
+00:21:55,340 --> 00:22:01,640
+عن Estimating Exchange in general في الكلكولة
+
+302
+00:22:01,640 --> 00:22:05,480
+الصالف كانت اللي هي سهلة الـ DF كنا نقول بساوي
+
+303
+00:22:05,480 --> 00:22:12,240
+اللي هي F' في DSالان لأ احنا صار عنده او ا دي اف
+
+304
+00:22:12,240 --> 00:22:15,520
+باسامة اف برايم في دي واي زي ما قلنا او دي اكس حسب
+
+305
+00:22:15,520 --> 00:22:19,280
+اللي عندنا الان احنا صار function of two variables
+
+306
+00:22:19,280 --> 00:22:23,300
+فهيكون ال estimating the change in F in the
+
+307
+00:22:23,300 --> 00:22:26,940
+direction of mean of U شوف كيف to estimate the
+
+308
+00:22:26,940 --> 00:22:29,620
+change in the value of a differentiable function F
+
+309
+00:22:29,620 --> 00:22:33,120
+when we move at the small distance dS from point
+
+310
+00:22:33,120 --> 00:22:36,980
+B0 in a particular direction U use the formula اتة
+
+311
+00:22:36,980 --> 00:22:45,170
+دي الانبنقرب التغيير اللي بيصير اللي هو في الدالة
+
+312
+00:22:45,170 --> 00:22:50,130
+of a differentiable function F لما نتحرك بمقدار
+
+313
+00:22:50,130 --> 00:22:56,490
+مسافة DS من نقطة P0 في الـ direction لـU يعني الآن
+
+314
+00:22:56,490 --> 00:23:00,350
+عندي اللي هو لما اتحركت اللي هي من عند الـ P0
+
+315
+00:23:03,310 --> 00:23:07,610
+لنقطة أخرى في اتجاه مين على اتجاه الـ U صار في هذه
+
+316
+00:23:07,610 --> 00:23:11,850
+النقطة دي السمشينا صار هذا اللي هو فيه تغيير في
+
+317
+00:23:11,850 --> 00:23:16,090
+مين في الـ F التغيير في الـ F اللي هو تقريبي دلت
+
+318
+00:23:16,090 --> 00:23:20,480
+DFطبعاً اللي هو بالظبط الـ Δ F بساوي اللي هو بقاله
+
+319
+00:23:20,480 --> 00:23:23,360
+علاقة Δ X و Δ Y و مش عارف إيش و كده و كده زمان
+
+320
+00:23:23,360 --> 00:23:26,340
+كانت لما كانت بس function of one variable دلتة F
+
+321
+00:23:26,340 --> 00:23:32,080
+اللي بساوي اللي هو Y ناقص Y نوت أو دلتة Y بساوي Y
+
+322
+00:23:32,080 --> 00:23:34,960
+ماينس Y نوت و دلتة X بساوي X ماينس X نوت لكن الآن
+
+323
+00:23:34,960 --> 00:23:37,800
+صار عندي لأ في direction معين فبصير الآن الـ
+
+324
+00:23:37,800 --> 00:23:41,900
+differential دي F تقريبا تقريبا بساوي هذا المقدار
+
+325
+00:23:41,900 --> 00:23:46,610
+اللي هو directional derivativeمضروب في مين في الـ
+
+326
+00:23:46,610 --> 00:23:51,510
+DS الـ distance increment اللي اتحركناها فبكون هذا
+
+327
+00:23:51,510 --> 00:23:55,670
+مقدار التغير تقريبا هذا مقدار التغير طبعا فيه اللي
+
+328
+00:23:55,670 --> 00:23:59,890
+هو خطأ بحان عسبه بعد بشوية هذا قانونه المهم لأن
+
+329
+00:23:59,890 --> 00:24:05,070
+قانون اللي هو لو اتحركنادالة اللي هو اتحركنا في
+
+330
+00:24:05,070 --> 00:24:09,610
+اتجاه الـ U بمقدار DS صار فيه تغيير في الـ F مقدار
+
+331
+00:24:09,610 --> 00:24:12,570
+التغيير في الـ F تقريبا بساوية Directional
+
+332
+00:24:12,570 --> 00:24:16,750
+Derivative في هذا الاتجاه مضروبة في مقدار اللي هي
+
+333
+00:24:16,750 --> 00:24:21,530
+المسافة اللي مشيناها DS على اللي هو الـ U فبكون هو
+
+334
+00:24:21,530 --> 00:24:26,130
+مقدار التغير في الـ F وخلينا نشوف المثال التالي
+
+335
+00:24:27,340 --> 00:24:31,160
+نشوف التطبيق الأول على اللي هو الـ Example 4
+
+336
+00:24:31,160 --> 00:24:35,440
+Estimate how much the value of f of x, y, z بسواي
+
+337
+00:24:35,440 --> 00:24:40,400
+sin x زاد 2y, z اللي هي will change if the point B
+
+338
+00:24:40,400 --> 00:24:45,960
+moves point 1 units from B0 straight toward اللي
+
+339
+00:24:45,960 --> 00:24:51,520
+هو B1 اللي هو 2 و 2 و نقص 2 يعني بدنا نشوف .. نشوف
+
+340
+00:24:51,520 --> 00:24:59,060
+.. نقرب قداشة التغير في Fالتي تعمل تغيير الـ
+
+341
+00:24:59,060 --> 00:25:05,140
+Estimation هو الـ ADF لما النقطة
+
+342
+00:25:06,600 --> 00:25:11,260
+تتغير وتتحرك بمقدار point one من أين إلى أين تتحرك
+
+343
+00:25:11,260 --> 00:25:15,060
+من النقطة B Note في اتجاه مباشرة straightforward
+
+344
+00:25:15,060 --> 00:25:21,700
+في اتجاه B1 يعني B1 هو الـ direction اللي بدنا
+
+345
+00:25:21,700 --> 00:25:25,580
+نتحرك فيه ومقدار التحرك اللي هو الـ DS في هذه
+
+346
+00:25:25,580 --> 00:25:30,080
+الحالة بساوية point one إذن عشان أقدر أوجد أعمل
+
+347
+00:25:30,080 --> 00:25:34,200
+estimation للتغير في F اللي هو الـ DFبداوجد الـ
+
+348
+00:25:34,200 --> 00:25:37,680
+Gradient و بداوجد الـ U و بداوجد الـ DS DS هي
+
+349
+00:25:37,680 --> 00:25:42,600
+موجودة و الـ U هو عبارة عن الـ Direction اللي هو
+
+350
+00:25:42,600 --> 00:25:46,640
+من P0 ل P1 اللي هو بمعنى أخر الـ Unit Vector في
+
+351
+00:25:46,640 --> 00:25:53,810
+اتجاه P0 P1الآن اللي هي الـ Gradient اللي نجيبه هو
+
+352
+00:25:53,810 --> 00:26:01,130
+الـ Direction of this vector from 0, 1 و 0 إلى 2 و
+
+353
+00:26:01,130 --> 00:26:05,930
+2 و نقص 2 أول شيء اللي هو الـ vector بين هذه وهذه
+
+354
+00:26:05,930 --> 00:26:09,980
+اللي هو عبارة عن 2I2
+
+355
+00:26:21,150 --> 00:26:24,290
+الأن بدنا نوجد اللي هو الـ Direction له إذا على
+
+356
+00:26:24,290 --> 00:26:28,030
+مجموعته مجموعته هو الجدر التربيعي للأربعة زاد
+
+357
+00:26:28,030 --> 00:26:32,050
+الواحد زاد الأربعة اللي هي عبارة عن بيصير اتنين I
+
+358
+00:26:32,050 --> 00:26:35,690
+زاد J نقص اتنين K على تلاتة هذا اللي هو الـ
+
+359
+00:26:35,690 --> 00:26:40,270
+Direction U اللي احنا بدنا التغيير يصير في اتجاهه
+
+360
+00:26:40,640 --> 00:26:43,460
+الآن من الـ Gradient للـ F نيجي من جديد من الـ
+
+361
+00:26:43,460 --> 00:26:47,720
+Gradient للـ F نفعظ الدلالة اللي هو Fx Fx اللي هي
+
+362
+00:26:47,720 --> 00:26:51,000
+هاد بالنسبة لـ Fx بيصير Y Cos X في اتجاه الـ I
+
+363
+00:26:51,000 --> 00:26:56,340
+زائد Fy Fy ايش Fy في هذه الحالة اللي هي Sin X لأن
+
+364
+00:26:56,340 --> 00:27:00,620
+اتفاض الـ Y واحد Sin X زائد اللي هو اتفاضه الـ 2YZ
+
+365
+00:27:00,620 --> 00:27:05,060
+بالنسبة لـ Y اللي هو 2Zكله في اتجاه من الـ J زائد
+
+366
+00:27:05,060 --> 00:27:09,400
+اللي هي FZ FZ هذا صفر بنسبة لـ Z وهذا بالنسبة لـ Z
+
+367
+00:27:09,400 --> 00:27:13,760
+اللي هي 2Y فبتنزل 2YK هذه الـ Gradient الـ
+
+368
+00:27:13,760 --> 00:27:17,460
+Gradient بدنا يهيأ عند مين احنا عند اللي هو طبعا
+
+369
+00:27:17,460 --> 00:27:22,800
+التحرك من عند النقطة P0,1,0 إذا عنده نقطة مين؟ P0
+
+370
+00:27:22,800 --> 00:27:27,000
+,1,0 هي النقطة اللي من عندها صار التحرك اللي عندها
+
+371
+00:27:27,000 --> 00:27:32,050
+بنجيب الـ Gradientاللي هي بتساوي عوضه عن X بـ 0
+
+372
+00:27:32,050 --> 00:27:40,430
+بيصير الـ Y بـ 1 يعني I زائد صين الـ 00 زائد 2 زد
+
+373
+00:27:40,430 --> 00:27:44,650
+اللي هي 2 في صفر يعني بيصير في الـ J لأن هذا صفر و
+
+374
+00:27:44,650 --> 00:27:49,790
+هذا صفر زائد 2YK Y بـ 1 بيصير زائد 2K إذا صار عند
+
+375
+00:27:49,790 --> 00:27:54,280
+الـ Gradient للـ F عند النقطة 100لو ضربنا dot
+
+376
+00:27:54,280 --> 00:27:58,540
+product مع U بيصير عبارة عن I فاهمين ليش؟ لأنه
+
+377
+00:27:58,540 --> 00:28:02,280
+احنا بدنا نجد اللي هو DF نسميه الـ Gradient Dot
+
+378
+00:28:02,280 --> 00:28:06,220
+الـ U مضروف في مين؟ في الـ S ده صار عدد مضروف في
+
+379
+00:28:06,220 --> 00:28:08,960
+عدد او ليش نجيب الـ dot product هذا؟هذا هو الـ
+
+380
+00:28:08,960 --> 00:28:13,660
+Gradient لـ F وDot لـ U Gradient لـ F I زيادة 2K
+
+381
+00:28:13,660 --> 00:28:17,580
+وDot لـ U دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+382
+00:28:17,580 --> 00:28:19,080
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+383
+00:28:19,080 --> 00:28:19,420
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+384
+00:28:19,420 --> 00:28:19,980
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+385
+00:28:19,980 --> 00:28:21,100
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+386
+00:28:21,100 --> 00:28:25,860
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+387
+00:28:25,860 --> 00:28:32,560
+نجدها فوق دعونا نجدها فوقالأن الـ «تغيير DF في F
+
+388
+00:28:32,560 --> 00:28:38,020
+التي تنتج من المتحرك DS» يتساوي 0.1 معطينيا unit
+
+389
+00:28:38,020 --> 00:28:41,920
+away from 0 و 1 و 0 in the direction of U is
+
+390
+00:28:41,920 --> 00:28:45,520
+approximately DF بتساوي حسب القانون اللي فوق Delta
+
+391
+00:28:45,520 --> 00:28:49,460
+F ضد الـ U مضروبة في الـ DS هي هذا أوددناها عبارة
+
+392
+00:28:49,460 --> 00:28:53,380
+عن ناقص تلتين وهذا الـ DS عبارة عن 0.1 اتتين مع
+
+393
+00:28:53,380 --> 00:28:57,920
+بعض بيطلع اللي هو هذا المقداروهذا اللي هو الـ DF
+
+394
+00:28:57,920 --> 00:29:02,740
+اللي هو تقريب التغيير فيه أو التغيير اللي حدث في F
+
+395
+00:29:02,740 --> 00:29:09,580
+عندما تحركنا Point 1 في اتجاه من النقطة B Note إلى
+
+396
+00:29:09,580 --> 00:29:14,150
+النقطة B1الآن بدنا نجي على حاجة اسمها
+
+397
+00:29:14,150 --> 00:29:18,310
+Linearization Linearization طبعا اللي هو أو تقريب
+
+398
+00:29:18,310 --> 00:29:23,670
+الدالة اللي هو معادلة خطية أو حاجة بنسميها
+
+399
+00:29:23,670 --> 00:29:26,630
+Linearization اللي أخدناها في calculus الف هذه
+
+400
+00:29:26,630 --> 00:29:32,580
+تعميم اللي بس في calculus دالة متغيريةHow to
+
+401
+00:29:32,580 --> 00:29:36,520
+Linearize a Function of Two Variables كيف بدنا أن
+
+402
+00:29:36,520 --> 00:29:42,260
+نقرّب الدالة بالـ Line اللي جنبها كنا زمان نقرّب
+
+403
+00:29:42,260 --> 00:29:47,180
+الدالة بالمماس حول نقطة التماس لأن حول نقطة التماس
+
+404
+00:29:47,180 --> 00:29:52,300
+الفرق بين اللي هو الخط و بين ال curve جليل جدا
+
+405
+00:29:52,300 --> 00:29:55,900
+لإنه حول المماس هذا خدنا فكرة كلصة لافيش دي عينا
+
+406
+00:29:55,900 --> 00:29:58,750
+نعيدهإلا أن هذه الفكرة بنعملها function of two
+
+407
+00:29:58,750 --> 00:30:02,450
+variables عند لو .. لو جينا بدنا نعرف
+
+408
+00:30:02,450 --> 00:30:06,430
+linearization of a function f of x,y at a point x0
+
+409
+00:30:06,430 --> 00:30:10,210
+,y0 بدنا نعرف ال linearization لل function f of x
+
+410
+00:30:10,210 --> 00:30:14,650
+,y عنده النقطة x0,y0 متى؟ طبعا للدلة اللي بتكون
+
+411
+00:30:14,650 --> 00:30:18,570
+أفضل الـ differentialهو الـ function L of X Y
+
+412
+00:30:18,570 --> 00:30:23,110
+بتساوي F of X Note Y Note زاد F X of X Note Y Note
+
+413
+00:30:23,110 --> 00:30:27,430
+في X Minus X Note زاد F Y of X Note Y Note في Y
+
+414
+00:30:27,430 --> 00:30:31,330
+Minus Y Note الآن هذه الـ linearization
+
+415
+00:30:33,670 --> 00:30:37,150
+اللي في X و Y يعني هي عند النقطة X knot و Y knot
+
+416
+00:30:37,150 --> 00:30:42,150
+وهي النقطة X Y اللي أصار تغير فيها انتقلت لهان لأن
+
+417
+00:30:42,150 --> 00:30:44,810
+هذا المقدار هو عبارة عن Delta X اللي هو X minus X
+
+418
+00:30:44,810 --> 00:30:49,030
+knot وهذا اللي هو Delta Y Y minus Y knot الان اللي
+
+419
+00:30:49,030 --> 00:30:53,290
+هي ال function is differential لهان فهذه النقطة ال
+
+420
+00:30:53,290 --> 00:30:57,790
+X و Y حوالين النقطة X knot و Y knotكل ما اقتربت
+
+421
+00:30:57,790 --> 00:31:02,510
+الـ X من الـ X0 وY0 كل ما كان فيه إمكانية أن نقول
+
+422
+00:31:02,510 --> 00:31:07,010
+الـ Linearization هذا اللي عرفناها تقريبا بساومين
+
+423
+00:31:07,010 --> 00:31:11,790
+F of X أو Y لأنه لما نقترب منه الـ X minus X0
+
+424
+00:31:11,790 --> 00:31:16,410
+بتصغر والـ Y minus Y0 بتصغر لما نقترب كتير بسير
+
+425
+00:31:16,410 --> 00:31:21,050
+حوالين النقطة هذهالخط اللي هو الـ Linearization
+
+426
+00:31:21,050 --> 00:31:25,950
+هذا والـ Function الأصلية بيسير تقريباً جات بعض
+
+427
+00:31:25,950 --> 00:31:30,330
+يعني ألّف XY بنقدر اللي هو الـ Function الدالة
+
+428
+00:31:30,330 --> 00:31:36,080
+اللي ممكن تكون معقدة انقربها بواسطة خطالخط اللي هو
+
+429
+00:31:36,080 --> 00:31:40,020
+عبارة عن هي دالة خطية اللي هي في X بس أس واحد و Y
+
+430
+00:31:40,020 --> 00:31:44,940
+أس واحد فصارت اللي هي جربنا الدالة الـ F of X knot
+
+431
+00:31:44,940 --> 00:31:50,100
+و Y knot أو F of XY تقريبا بساوة L of XY for
+
+432
+00:31:50,100 --> 00:31:57,180
+points XY that are nearbyأو very close to x0 و y0
+
+433
+00:31:57,180 --> 00:32:01,260
+لأنه بيصير هذه صغيرة جدا وهذا صغير جدا فبتصير أقله
+
+434
+00:32:01,260 --> 00:32:05,440
+في x و y تقريبا بتساوي f of x و y في حالة اللي هي
+
+435
+00:32:05,440 --> 00:32:11,160
+x و y جريبة جدا من x0 و y0 هذا بنسمي is the
+
+436
+00:32:11,160 --> 00:32:15,600
+standard linear approximation of f at x0 و y0 طبعا
+
+437
+00:32:15,600 --> 00:32:19,800
+أكيد في التقريب هذا في خطأ في هذا الخطأ احنا بدنا
+
+438
+00:32:19,800 --> 00:32:23,790
+نحاول نحسبه بعد شويةأولا نشوف كيف يوجد الـ
+
+439
+00:32:23,790 --> 00:32:28,110
+Linearization Find the linearization of f of xy
+
+440
+00:32:28,110 --> 00:32:32,470
+بساوي x تربيع ناقص xy زائد نص y تربيع زائد تلاتة
+
+441
+00:32:32,470 --> 00:32:36,470
+at the point من تلاتة و اتنين سنشوف ان وجد
+
+442
+00:32:36,470 --> 00:32:39,850
+Linearization للـ Function اللي أمامناما هو
+
+443
+00:32:39,850 --> 00:32:43,330
+التطبيق على القانون اللي قبل بشوية الـ Fx ايش
+
+444
+00:32:43,330 --> 00:32:46,590
+هتساوي الـ Fx لأن الـ Linearization هي قانونها
+
+445
+00:32:46,590 --> 00:32:51,350
+قلنا قبل بشوية هي قانونها L بتساوي F نحسبها عندنا
+
+446
+00:32:51,350 --> 00:32:54,470
+النقطة المطلوبة Fx عندنا النقطة المطلوبة في X نقص
+
+447
+00:32:54,470 --> 00:32:57,710
+X not of Y برضه عندنا النقطة المطلوبة في Y minus Y
+
+448
+00:32:57,710 --> 00:33:01,310
+not هذا هو Linearization خلينا الآن نطبق على مثلنا
+
+449
+00:33:01,830 --> 00:33:06,590
+عند الـ Fx إيش بيساوي 2x ناقص Y فاهمين عليها Fy
+
+450
+00:33:06,590 --> 00:33:10,370
+إيش بيساوي اللي هو ده سفر اللي هي بيصير ناقص X
+
+451
+00:33:10,370 --> 00:33:14,570
+معايا زائد 2 في نص اللي هي بيصير اللي هو زائد Y
+
+452
+00:33:14,570 --> 00:33:18,790
+هذا Fy لما تحسب هذا عند النقطة 3 ع 2 وهذا 3 و 2
+
+453
+00:33:18,790 --> 00:33:22,530
+وهذا عند النقطة 3 و 2 الأولى بتصير اللي هي 2 في 3
+
+454
+00:33:22,530 --> 00:33:27,330
+ب6 ناقص 2 اللي هي 4 التامي Fy عند الـ 3 و 2 اللي
+
+455
+00:33:27,330 --> 00:33:33,780
+هي ناقص 3 مكان Xمكان نقص X وY مكانها 2 بصير نقص 3
+
+456
+00:33:33,780 --> 00:33:37,340
+زي 2 بساوي نقص 1 الآن أيضًا ماذا بنلزمنا في
+
+457
+00:33:37,340 --> 00:33:43,420
+القانون F عند 3 و 2 F عند 3 و 2 بصير 9 نقص 3 في 2
+
+458
+00:33:43,420 --> 00:33:50,070
+ب6 زي نص اللي هو في 2 في 2 ب4 لأن Y ب2اللي بيصير
+
+459
+00:33:50,070 --> 00:33:55,210
+عنده 2 التي زائد 3 والمجموع هذا بيساوي 8 إذا حصلنا
+
+460
+00:33:55,210 --> 00:33:59,830
+Fx و Fy و F عنده نقطة 3 و 2 إذا بنعوض على L و X Y
+
+461
+00:33:59,830 --> 00:34:03,770
+بيساوي F في 3 ع 2 زادة Fx في X minus 3 زادة Fy في
+
+462
+00:34:03,770 --> 00:34:08,610
+Y minus 2 و Y ساوي اللي هو التالي F في 3 ع 2
+
+463
+00:34:08,610 --> 00:34:13,990
+تمانية هذه Fx في 3 ع 2 اللي هي 4 مضروبة في X ناقص
+
+464
+00:34:13,990 --> 00:34:15,670
+اللي هي X note اللي هي 3
+
+465
+00:34:23,600 --> 00:34:31,680
+Y-Y-Y-Y-Y-Y-Y-Y-Y-Y-Y
+
+466
+00:34:31,890 --> 00:34:35,630
+وهي نقطة
+
+467
+00:34:35,630 --> 00:34:41,750
+3 و2 في الـ X Y بساوية 4 X ناقص Y ناقص 2 يمكن أن
+
+468
+00:34:41,750 --> 00:34:45,270
+يكون هكذا حلنا أو جدنا الـ Linearization زي ما
+
+469
+00:34:45,270 --> 00:34:49,390
+قلنا ان الـ Linearization في الآخر L of X Y ليس
+
+470
+00:34:49,390 --> 00:34:53,430
+بالظبط بساوية F of X Y بساوية تقريبا 4 points
+
+471
+00:34:53,430 --> 00:34:59,110
+close to X Y ليه 3 X نقطة Y نقطة 3 و2 طب في خطأ
+
+472
+00:34:59,110 --> 00:35:04,000
+مقدار الخطأ بتحسبه نظرية التاليةالخطأ اللي نقرأه
+
+473
+00:35:04,000 --> 00:35:06,900
+الـ Approximation هو F of XY بيساوي تقريبًا L of
+
+474
+00:35:06,900 --> 00:35:12,320
+XY بحسب التالي نشوف الخطأ فيه يعني الخطأ اللي
+
+475
+00:35:12,320 --> 00:35:17,520
+نسميه إيه هو عبارة عن الفرق بين اللي هو F of XY و
+
+476
+00:35:17,520 --> 00:35:22,200
+L of XY الفرق بينهم اللي هو عبارة عن أكتر أكتر فرق
+
+477
+00:35:22,200 --> 00:35:26,500
+له حاجة اسمها الـ Error و هنشوف قداش قيمته نشوف
+
+478
+00:35:26,500 --> 00:35:27,360
+أيش النظرية بتقول
+
+479
+00:35:34,050 --> 00:35:39,370
+لو فرضنا أن F و Fxx و Fyy و Fx و Fy كل هن دولة
+
+480
+00:35:39,370 --> 00:35:42,430
+اللي هو الـ first و second partial derivatives
+
+481
+00:35:42,430 --> 00:35:46,830
+المظبوط أن يكون in continuous عنده نقطة x0 و y0 في
+
+482
+00:35:46,830 --> 00:35:51,370
+open set as if حوالين نقطة x0 و y0يعني فرضنا إن
+
+483
+00:35:51,370 --> 00:35:55,810
+الـ F و الـ partial derivatives كلهم continuous في
+
+484
+00:35:55,810 --> 00:36:00,870
+rectangle حوالين الـ X0 و Y0 ومنفترض أن M إيش هي
+
+485
+00:36:00,870 --> 00:36:05,230
+is any upper bound for the values of يعني هي عبارة
+
+486
+00:36:05,230 --> 00:36:11,370
+عن حد أعلى لقيمة الـ Fxx والـ Fyy والـ Fxy وين على
+
+487
+00:36:11,370 --> 00:36:15,230
+ال rectangle R اللي بنحكي عليه then ال error
+
+488
+00:36:16,660 --> 00:36:20,280
+Incured in replacing f of x,y on R by its
+
+489
+00:36:20,280 --> 00:36:25,540
+linearization f of x,y يعني بنقول f of x,y و f of
+
+490
+00:36:25,540 --> 00:36:29,620
+x,y الفرق بينهم في داخل الـ rectangle هذا اللي
+
+491
+00:36:29,620 --> 00:36:35,960
+حكينا عليه هيكون الفرق بينهم لن يتعدى لن يتعدى الـ
+
+492
+00:36:35,960 --> 00:36:40,520
+absolute value لـ x,y أصغر أو سوى نص الـ M في x
+
+493
+00:36:40,520 --> 00:36:45,780
+minus x0 زي y minus y0 الكل إيش تربيةأذا عملية
+
+494
+00:36:45,780 --> 00:36:51,820
+تقريب L of X Y بتساوي
+
+495
+00:36:51,820 --> 00:36:59,130
+تقريبًا F of X Y في المنطقة Rبتكون تقريباً بتسويها
+
+496
+00:36:59,130 --> 00:37:06,410
+لكن في Error الـ L of X Y ناقص F of X Y لن يتعدى
+
+497
+00:37:06,410 --> 00:37:10,270
+لوسالب أو موجة بهذه القيمة اللي هي أياش الـ Error
+
+498
+00:37:10,270 --> 00:37:14,710
+أصغر أو يساوي نص الـ M الـ M يعني الـ Upper Bound
+
+499
+00:37:14,710 --> 00:37:22,650
+لهذه الـ FXX و FYY و FXYفي X – X¹ زي Y – Y¹ كل
+
+500
+00:37:22,650 --> 00:37:26,110
+تربيع وحتى تتضح هذه الصورة أكتر خلّينا ناخد
+
+501
+00:37:26,110 --> 00:37:31,270
+example example find an upper bound for the error
+
+502
+00:37:31,270 --> 00:37:34,530
+in the approximation of X Y بساوي ألف X Y in
+
+503
+00:37:34,530 --> 00:37:38,750
+example خمسة خمسة over the rectangle كده بقوللي
+
+504
+00:37:38,750 --> 00:37:41,330
+أنت اللي جابل بشوية ال example الخمسة اللي جابل
+
+505
+00:37:41,330 --> 00:37:44,210
+بشوية هذا او جنانه ال linearization هي
+
+506
+00:37:44,210 --> 00:37:47,270
+linearizationإلا أن هذا الـ Linearization اللي
+
+507
+00:37:47,270 --> 00:37:54,390
+أوجدنا تقريبًا هذا هذا بتساوي هذه الدالة وبين في
+
+508
+00:37:54,390 --> 00:37:57,650
+حوالين التلاتة واتنين كل ما بعدنا عن التلاتة
+
+509
+00:37:57,650 --> 00:38:01,860
+واتنين بيصير اللي هو الفرق بين هذا وهذا كبيرالان
+
+510
+00:38:01,860 --> 00:38:06,160
+بدأ أشوف لك جديش اللي هو الـ Error بين ال .. الحس
+
+511
+00:38:06,160 --> 00:38:09,960
+انه .. انه XR بيبقى نقص XY زيادة نص Y تربيع زيادة
+
+512
+00:38:09,960 --> 00:38:15,380
+تلاتة بتساوي 4X نقص Y نقص 2 تقريبا في المنطقة
+
+513
+00:38:15,380 --> 00:38:19,040
+حوالين التلاتة اتنين تلاتة و اتنين حوالين النقطة
+
+514
+00:38:19,040 --> 00:38:21,700
+تلاتة و اتنين بدأ أشوف لك الآن الفرق بين هذه وهذه
+
+515
+00:38:21,700 --> 00:38:27,440
+يعني عملية التقريب هذه بهذه جديش تقريبا الـ Error
+
+516
+00:38:27,440 --> 00:38:32,520
+فيها اللي بده يقوليلنا إياهاالتي هي المثال التالي
+
+517
+00:38:32,520 --> 00:38:36,580
+وخلّينا نستذكر هذه لو اوددنا fx فسخلّيني استخدم
+
+518
+00:38:36,580 --> 00:38:41,600
+fxx ايش هتساوي في هذه الحالة اتنين fyy ايش هتساوي
+
+519
+00:38:41,600 --> 00:38:48,920
+واحدمظبوط فكس واي ايش هتساوي ناقص واحد ناشي خلينا
+
+520
+00:38:48,920 --> 00:38:51,920
+نحفظهم و نستخدمهم بعد شوية لان بدنا نلزمنا في
+
+521
+00:38:51,920 --> 00:38:56,280
+السؤال الان الان بيقولي find an upper bound for
+
+522
+00:38:56,280 --> 00:38:59,420
+the error in the approximation of of xy بيساوي
+
+523
+00:38:59,420 --> 00:39:03,040
+تقريبا الوفيكس واي في خطأ الخطأ قلنا طبعا بده
+
+524
+00:39:03,040 --> 00:39:07,620
+ياشمله على ال rectangle R X minus ثلاثة أصغر من
+
+525
+00:39:07,620 --> 00:39:10,480
+point one يعني ما نبعلش عن التلاتة ب point one هي
+
+526
+00:39:10,480 --> 00:39:13,190
+نقطة تلاتة و اتنين اللي احنا حكينا عنهااللي
+
+527
+00:39:13,190 --> 00:39:16,710
+حواليها بدنا نقل إيه؟ عشان نعمل اللي هي دي يعني في
+
+528
+00:39:16,710 --> 00:39:20,970
+داخل المستطيل هذا اللي هو X نقص تلاتة من X هيك
+
+529
+00:39:20,970 --> 00:39:25,630
+point one يعني من اتنين تسعة من عشرة لتلاتة واحد
+
+530
+00:39:25,630 --> 00:39:31,650
+من عشرة وتحت من اللي هي Y اللي هي واحد وتسعة من
+
+531
+00:39:31,650 --> 00:39:34,470
+عشرة وتحت اتنين واحد من عشرة في داخل ال rectangle
+
+532
+00:39:34,470 --> 00:39:40,450
+هذا اللي بتكون من هذه وهذهبدنا نجد اللي هو الخطأ
+
+533
+00:39:40,450 --> 00:39:43,750
+في الحساب في الـ approximation اللي هو f of x y
+
+534
+00:39:43,750 --> 00:39:47,390
+بسواقله f of x y إيش طالب لي express the upper
+
+535
+00:39:47,390 --> 00:39:51,770
+bound as a percentage of f of تلاتة ع اتنين بقول
+
+536
+00:39:51,770 --> 00:39:56,110
+لي اللي هو وجد لي نسبة الخطأ بالنسبة لقيم اتنين
+
+537
+00:39:56,110 --> 00:39:59,750
+اتدالة عن تلاتة و اتنين يعني بدك تحسب الخطأ أول
+
+538
+00:39:59,750 --> 00:40:04,870
+إشياللي هو بدك توجد الـ E اللي قبل بشوية و تحسب ال
+
+539
+00:40:04,870 --> 00:40:08,330
+percentage لل error بالنسبة لل F of 3 ع 2 The
+
+540
+00:40:08,330 --> 00:40:11,670
+value of F centered at the rectangle يعني F 3 ع 2
+
+541
+00:40:11,670 --> 00:40:17,170
+هي قيمة F اللي هو مركز ال rectangle اللي بتكون منه
+
+542
+00:40:17,170 --> 00:40:21,060
+هذا اللي هو ال rectangle اللي حكيته قبل بشويةالأن
+
+543
+00:40:21,060 --> 00:40:26,640
+نجد الـ Error نستخدم الـ Inquality A of XY أزرع من
+
+544
+00:40:26,640 --> 00:40:31,360
+النص M في هذا المقدار هذا القانون الان M إيش هي؟ M
+
+545
+00:40:31,360 --> 00:40:35,220
+هي عبارة عن .. قولنا إيه؟ هي Upper Bound لمن؟ لـ
+
+546
+00:40:35,220 --> 00:40:40,800
+FXX و FXY و FYY من القانون هذا الان نجد هذول FXX و
+
+547
+00:40:40,800 --> 00:40:45,480
+FXY أو جدلكم يا انت قبل شوية FXX طلعت قداش 2 لـ
+
+548
+00:40:45,480 --> 00:40:48,240
+Absolute Value لها 2 FXY طلعت ناقص 1 لـ Absolute
+
+549
+00:40:48,240 --> 00:40:51,960
+Value لها 1أفواي واي وجدناها واحد طلعت الـ
+
+550
+00:40:51,960 --> 00:40:55,300
+Absolute لها واحد إذا الأن الـ Upper bound لها دول
+
+551
+00:40:55,300 --> 00:40:58,940
+يعني أكبر واحد يعني الأكبر حد اللي هنا دولة بين
+
+552
+00:40:58,940 --> 00:41:02,140
+الاتنين والواحد والواحد أكيد نين هو اتنين إذا ناخد
+
+553
+00:41:02,140 --> 00:41:05,960
+الـ M واحنا مرتحين بتساوي اتنين إذا صار عندي اللي
+
+554
+00:41:05,960 --> 00:41:09,440
+هو الـ Error هذا أظهر وساوي نص في اتنين اللي هي
+
+555
+00:41:09,440 --> 00:41:13,600
+الـ M في X minus ثلاثة اللي هي X0 زائد Y minus
+
+556
+00:41:13,600 --> 00:41:16,740
+اتنين اللي هي Y0 هذا الكل تربيح هذا القانون طبعا
+
+557
+00:41:17,210 --> 00:41:20,670
+الان X-3 ماعطينيها دايما إيه شمالها أصغر أو يساوي
+
+558
+00:41:20,670 --> 00:41:25,670
+0.1 داخل المستطيل تبعنا اللي هي زائد Y-Y2 اللي هي
+
+559
+00:41:25,670 --> 00:41:28,730
+أصغر من 0.1 اللي هو داخل المستطيل تبعنا إذا هذه
+
+560
+00:41:28,730 --> 00:41:31,970
+بنستبدلها أصغر أو يساوي اللي هو بيصير ال error
+
+561
+00:41:31,970 --> 00:41:36,590
+أصغر أو يساوي نصف 2 اللي هي بواحدماشي اللي هي ده
+
+562
+00:41:36,590 --> 00:41:41,330
+صارت ا اللي هي point one الكل زائد point one الكل
+
+563
+00:41:41,330 --> 00:41:43,850
+ايه اشماله تربيع لإن هو اصغر يساوي point one او
+
+564
+00:41:43,850 --> 00:41:46,490
+اصغر يساوي point one لذا صار اصغر يساوي point one
+
+565
+00:41:46,490 --> 00:41:51,190
+تربيع اللي هو point four لذا صار ال error لا يتعدل
+
+566
+00:41:51,190 --> 00:41:57,970
+point fourالأن الـ Percentage of F of 3 ع 2 بساوي
+
+567
+00:41:57,970 --> 00:42:03,150
+طبعاً اللي هي نسبة الخطأ بالنسبة لمين F of 3 ع 2 F
+
+568
+00:42:03,150 --> 00:42:05,990
+of 3 ع 2 إيش حطلعت معانا قبل بشوية أو جدناها في
+
+569
+00:42:05,990 --> 00:42:09,250
+السوية اللي قبل تمانية The error is no greater
+
+570
+00:42:09,250 --> 00:42:12,770
+than يعني هيكون ال error نسبة ال percentage في ال
+
+571
+00:42:12,770 --> 00:42:20,460
+error لن تتعدى هتكون أصغر أو ساوي.04 على تمانية
+
+572
+00:42:20,460 --> 00:42:27,220
+مضروبة 100% بيطلع بوينت خمسة من مائة إذا هذه نسبة
+
+573
+00:42:27,220 --> 00:42:34,200
+الخطأ هذا هو الخطأ الفرق بين f of x,y و f of x,y
+
+574
+00:42:34,200 --> 00:42:37,920
+ان شاء الله تكون متاحة الأمورالآن اللي حكيناه عن
+
+575
+00:42:37,920 --> 00:42:40,560
+الـ functions of two variables ده نحكي عن الـ
+
+576
+00:42:40,560 --> 00:42:44,280
+functions of more than two variables يعني of three
+
+577
+00:42:44,280 --> 00:42:47,780
+variables أو غيره Analogous result hold for
+
+578
+00:42:47,780 --> 00:42:50,760
+differentiable functions for more than two
+
+579
+00:42:50,760 --> 00:42:54,200
+variables يعني الآن في عندنا اللي هي شغلات مشابهة
+
+580
+00:42:54,200 --> 00:42:58,860
+هتحدث لـ function of three variables خليني نشوف
+
+581
+00:42:58,860 --> 00:43:01,520
+اللي هو linearization الـ linearization of f of x,
+
+582
+00:43:01,640 --> 00:43:07,520
+y, z at a point b0, x0, y0, z0 isيعني الآن الدالة
+
+583
+00:43:07,520 --> 00:43:12,460
+اللي عندي F of X وY وZ بدي أقربها بدالة خطية اسمها
+
+584
+00:43:12,460 --> 00:43:16,960
+L of X وY وZ طبعاً أقربها بدالة خطية حول النقطة X0
+
+585
+00:43:16,960 --> 00:43:20,720
+أو Y0 أو Z0 طبعاً في تقريبات أخرى بدواي اللي يمكن
+
+586
+00:43:20,720 --> 00:43:23,340
+تكون .. بعض الدواي اللي تكون تقريبة بدالة خطية
+
+587
+00:43:23,340 --> 00:43:27,600
+صعبةيمكن أن تكون الـ Quadratic Approximation أسهل
+
+588
+00:43:27,600 --> 00:43:31,020
+و كذا في طبعا طرق في ال .. ال .. ال .. اللي هي
+
+589
+00:43:31,020 --> 00:43:34,140
+numerical analysis بتاخدوها كتير بس احنا بناخد هنا
+
+590
+00:43:34,140 --> 00:43:37,160
+اللي هي linearization تقريب اللي هو الدالة بدالة
+
+591
+00:43:37,160 --> 00:43:41,120
+خطية حول النقطة X Note و Y Note و Z Note اللي قاله
+
+592
+00:43:41,120 --> 00:43:44,060
+في X و Y و Z زي اللي جابل بالظبط قيمة الـ F عند
+
+593
+00:43:44,060 --> 00:43:48,070
+النقطة المذكورةزائد fx عنده النقطة المذكورة في x
+
+594
+00:43:48,070 --> 00:43:52,550
+-x0 زائد fy عنده النقطة المذكورة في y-y0 زائد fz
+
+595
+00:43:52,550 --> 00:43:56,670
+عنده النقطة المذكورة في z-z0 زي الأول بالظبط بس اي
+
+596
+00:43:56,670 --> 00:44:01,570
+شماله بس هذا على اللي هو ال function of three
+
+597
+00:44:01,570 --> 00:44:07,430
+variable باقي الحديث هيكون بنفس الأسلوبالنفس اللي
+
+598
+00:44:07,430 --> 00:44:11,950
+حكينا على الـ function of two variables نفترض أن R
+
+599
+00:44:11,950 --> 00:44:15,150
+is a closed rectangle solid centered at P0 and
+
+600
+00:44:15,150 --> 00:44:18,010
+lying in an open region on which the second
+
+601
+00:44:18,010 --> 00:44:21,270
+partial derivatives of R continuous suppose also
+
+602
+00:44:21,270 --> 00:44:26,890
+that هدولة كلهم Fxx, Fyy, Fzz صاروا في متغير تالت
+
+603
+00:44:26,890 --> 00:44:32,270
+Z, X, Y, Xz and Fyz are all less than or equal to
+
+604
+00:44:32,270 --> 00:44:35,370
+M يعني الآن بناخد ال upper bound هدول through R
+
+605
+00:44:35,600 --> 00:44:40,780
+والخطأ هو كما قلنا الفرق بين هذه وهذه ان الـ
+
+606
+00:44:40,780 --> 00:44:44,180
+approximation of F by L is bounded through R by
+
+607
+00:44:44,180 --> 00:44:47,940
+the inequality يعني الآن الخطأ لن يتعدى اللي هو
+
+608
+00:44:47,940 --> 00:44:51,160
+سيكون أزرع بسوء نص ال M في ال X minus X node زي
+
+609
+00:44:51,160 --> 00:44:55,180
+هذه نفسها نفس القانون بس ضفنا لهمين Z minus Z node
+
+610
+00:44:55,180 --> 00:45:00,560
+طبعا كل تربيحة ده اللي هو L of X Y و Z بسوء F of X
+
+611
+00:45:00,560 --> 00:45:03,860
+Y و Z في داخل ال rectangle طبعا ال rectangle صارها
+
+612
+00:45:03,860 --> 00:45:08,610
+الآن عبارة عن صندوقاللي هو الـ X minus X naught
+
+613
+00:45:08,610 --> 00:45:11,230
+أصغر من something Y minus Y naught أصغر من
+
+614
+00:45:11,230 --> 00:45:14,070
+something Z minus Z naught أصغر من something بحيث
+
+615
+00:45:14,070 --> 00:45:18,570
+أن الـ M الـ Upper bound تبعه بين هذه الـ M هي نص
+
+616
+00:45:18,570 --> 00:45:24,690
+M حوالين داخل هذه المنطقة بتكون تقريب الـ F بالـ L
+
+617
+00:45:24,690 --> 00:45:29,570
+لن يكون أكثر من هذا المقدار اللي هو الـ Error
+
+618
+00:45:30,050 --> 00:45:32,310
+العنوان if the second partial derivatives of f are
+
+619
+00:45:32,310 --> 00:45:36,150
+continuous and if x y و z it change from x note y
+
+620
+00:45:36,150 --> 00:45:40,070
+note by small amount dx dy dz يعني لو كانت اللي هي
+
+621
+00:45:40,070 --> 00:45:46,170
+x تغيرها dx و y تغيرت بمقدار dy و z بمقدار dz
+
+622
+00:45:46,170 --> 00:45:48,490
+بنقول عن حاجة اسمها الـ differential total
+
+623
+00:45:48,490 --> 00:45:51,970
+differential اللي مش هنحكي عنها كتير بس للمعرفة df
+
+624
+00:45:51,970 --> 00:45:58,360
+بتساوي fx في dx زي fy في dy زي fz في dzGives a
+
+625
+00:45:58,360 --> 00:46:01,200
+good approximation of the resulting change enough
+
+626
+00:46:01,200 --> 00:46:06,060
+هذا اللي بتقرّب اللي هو مين التغير في مين فقف احنا
+
+627
+00:46:06,060 --> 00:46:09,880
+قبل شوية أخدنا التغير فقف لما نكون وين اللي هو
+
+628
+00:46:09,880 --> 00:46:13,700
+اتجاه التغير في اتجاه معين الآن لو كان اللي هو بس
+
+629
+00:46:13,700 --> 00:46:19,250
+تغير في X و X naught و Y Y naught و Z naughtاللي
+
+630
+00:46:19,250 --> 00:46:23,730
+هو بمقدار DX, DY, DZ يعني مش تغير في الـ X والـ Y
+
+631
+00:46:23,730 --> 00:46:28,110
+والـ Z معروف تغير كل واحدة بيكون الـ DF بهذا
+
+632
+00:46:28,110 --> 00:46:34,310
+الأسلوب طيب الان نيجي الان لمثال عملي لهو حديثنا
+
+633
+00:46:34,310 --> 00:46:38,230
+هذا انشوف اللي هو المثال عشرة find a linearization
+
+634
+00:46:38,230 --> 00:46:43,750
+L of XYZ of اللي هو المقدار هذا at the point 2 1 0
+
+635
+00:46:43,750 --> 00:46:47,640
+و بعدين بيقولي find an upper boundبالنسبة للخطأ
+
+636
+00:46:47,640 --> 00:46:52,380
+المطلوب في تغيير الـ F by L لما نقرب الـ F بالـ
+
+637
+00:46:52,380 --> 00:46:56,160
+Linearization تبعنا اللي هو على الـ Rectangle هذه
+
+638
+00:46:56,160 --> 00:47:00,560
+نشوف قداشي الـ Error في هذه الحالة خلّينا نشوف أول
+
+639
+00:47:00,560 --> 00:47:07,080
+شيء نطبق القانون أكيد بالزمنة Fx اللي هي 2X-Y Fy
+
+640
+00:47:07,080 --> 00:47:12,100
+هي عبارة عن ناقص X Fz هي دولة طبعا سفر بتبقى عبارة
+
+641
+00:47:12,100 --> 00:47:19,220
+عن تلاتة Cos Z إذا صار عندي Fx عند النقطة 2 1 0
+
+642
+00:47:19,220 --> 00:47:24,780
+بنحسبها Fx عند النقطة 2 1 0 اللي هي بساوي تلاتة
+
+643
+00:47:24,780 --> 00:47:30,950
+لأنه 4 ناقص 1 تلاتة Fy عند النقطة 2 1 0 عوضاللي هي
+
+644
+00:47:30,950 --> 00:47:35,650
+مكان الـ X بالـ 2 بيصير نقص 2 FZ اللي هي عبارة عن
+
+645
+00:47:35,650 --> 00:47:41,370
+3 لأن Cos Zero عبارة عن Zero إذا تلاتة و F عند 2 و
+
+646
+00:47:41,370 --> 00:47:47,130
+1 و 0 F عند 2 و 1 و 0 هذا أربعة ناقص 2 بيطلع 2
+
+647
+00:47:47,130 --> 00:47:53,820
+زائد 0 اللي هي بيطلع جديش اتنينالأن هاي أوجدنا
+
+648
+00:47:53,820 --> 00:47:57,920
+اللي هي كل المتطلبات Fx و Fy و Fz عند النقاط
+
+649
+00:47:57,920 --> 00:48:01,280
+المذكورة والـ F كمان إذن الـ Linearization حسب
+
+650
+00:48:01,280 --> 00:48:05,480
+القانون قالوا في X و Y و Z بساوي قيمة الـ F Fx في
+
+651
+00:48:05,480 --> 00:48:10,400
+X minus X note اللي هي اتنين اللي هي Fy عنده نقطة
+
+652
+00:48:10,400 --> 00:48:15,300
+هادة في Y minus Y note واحد زائد Fz في Z minus Z
+
+653
+00:48:15,300 --> 00:48:18,640
+note اللي هي سفر هاي عند هدول بنعوض القيامة اللي
+
+654
+00:48:18,640 --> 00:48:22,870
+أوجدناهااللي هي F الأولى عبارة عن اتنين هيها زائد
+
+655
+00:48:22,870 --> 00:48:26,750
+F X اللي هي تلاتة في X minus اتنين و F Y نقص اتنين
+
+656
+00:48:26,750 --> 00:48:31,630
+و F Z تلاتة نضربهم و نبسطهم بيطلع عندى هذا اللي هو
+
+657
+00:48:31,630 --> 00:48:34,610
+الـ Linearization اللي هون ده Linearization of F
+
+658
+00:48:34,610 --> 00:48:38,810
+at اتنين و واحد و Zero is بالظبط هذا المقدار اللي
+
+659
+00:48:38,810 --> 00:48:42,670
+طلع أمامنا اللي هون هذا المقدار احنا بنقول بساوي
+
+660
+00:48:42,670 --> 00:48:46,550
+تقريبا هذا المقدار بساوي تقريبا هذا المقدار في
+
+661
+00:48:46,550 --> 00:48:51,950
+المنطقة المذكورة اللي عندهانلكن هناك خطأ نحسب نسبة
+
+662
+00:48:51,950 --> 00:48:57,270
+الخطأ بتقريب الـ F إيه بهذه انتوا جربوا اعطوا قيم
+
+663
+00:48:57,270 --> 00:49:02,650
+لـ X و Y و Z هنا في هذه المنطقة في الـ X و Y و Z
+
+664
+00:49:02,650 --> 00:49:09,590
+اللي بتحقق هذهأعطوها أوجدوا قيمة الـ F عندها وقيمة
+
+665
+00:49:09,590 --> 00:49:12,750
+الـ L عندها هتلاقوا الفرق بين الجهتين لن يتعدى الـ
+
+666
+00:49:12,750 --> 00:49:15,990
+Error اللي بدنا نوجده بعد شوية كيف بدنا نوجد الـ
+
+667
+00:49:15,990 --> 00:49:22,090
+Error؟ كالتالي قلنا مين بلزمنا Fxx هاي Fxx عنده
+
+668
+00:49:22,090 --> 00:49:30,990
+هاي Fxx اتنين Fyy هنا على طول سفر Fxy Fxy اللي هي
+
+669
+00:49:30,990 --> 00:49:40,300
+نقص واحدأفزد X وهو سفر و أفزد Y هو سفر لأن التفاضل
+
+670
+00:49:40,300 --> 00:49:43,440
+بالنسبة ل X أو Y هو سفر لذلك أفزد Y و أفزد X كله
+
+671
+00:49:43,440 --> 00:49:48,500
+مياشي أصفر لكن أفزد Z هي ضاقص تلاتة Sin Z إذن الآن
+
+672
+00:49:48,500 --> 00:49:51,800
+أريد أن أجلب Upper Bound لهذه المجموعات وين؟ على
+
+673
+00:49:51,800 --> 00:49:54,880
+المنطقة هذهأحنا دايماً كان يطلع عند الـ Upper
+
+674
+00:49:54,880 --> 00:49:58,340
+Bound اللي هو يطلع ان هدول أعداد جهزات ما عدا هذه
+
+675
+00:49:58,340 --> 00:50:01,860
+مش عدد، إذا هذه اللي بدنا .. اللي بدنا نشتغل عليها
+
+676
+00:50:01,860 --> 00:50:04,800
+لأن الأعداد هذه جاهزة 2، الـ Absolute Value لها 2
+
+677
+00:50:04,800 --> 00:50:07,900
+وهذه جاهزة 0 وهذه الـ Absolute Value الجاهزة إليها
+
+678
+00:50:07,900 --> 00:50:11,300
+1 وهذه وهذه الـ Absolute Value جاهزة إليها ياشتفر
+
+679
+00:50:11,300 --> 00:50:14,500
+لما نشوف الـ Absolute Value of ZZ هذه إيش بيسوي
+
+680
+00:50:14,500 --> 00:50:17,590
+الـ Absolute Value لهذه؟الـ Absolute Value لهذه
+
+681
+00:50:17,590 --> 00:50:20,650
+الآن ماحدش يقول لي أصغر أو أشهر و تلاتة لأ أنا
+
+682
+00:50:20,650 --> 00:50:24,110
+بديها وين في المنطقة اللي هي Z أصغر أو أشهر و 0.1
+
+683
+00:50:24,110 --> 00:50:30,270
+في المنطقة الصغيرة هذه الآن صين الـ Z صين الـ Z في
+
+684
+00:50:30,270 --> 00:50:35,150
+المنطقة اللي بتكون اللي هي جريبة من اللي هي الكاسر
+
+685
+00:50:35,150 --> 00:50:38,650
+هذا جريبة من الـ Z الصفر يعني الـ Z بتكون الصين
+
+686
+00:50:38,650 --> 00:50:42,590
+الـ Z تقريبا تقريبا بتساوي 100 Z يعني الـ Absolute
+
+687
+00:50:42,590 --> 00:50:47,730
+Value للصين الـ Zأصغر من تلاتة sign من الـ point
+
+688
+00:50:47,730 --> 00:50:52,330
+one عندي ماشي sign الـ z بيكون أصغر أو ساوي اللي
+
+689
+00:50:52,330 --> 00:50:56,590
+هي sign الـ point one حتى ما لازمتيش أقول أن الـ
+
+690
+00:50:56,590 --> 00:51:01,050
+sign بيساوي تقريبا القيمة لأن sign الـ z أصغر أو
+
+691
+00:51:01,050 --> 00:51:05,170
+ساوي تلاتة sign الـ point one في الفترة وين حوالين
+
+692
+00:51:05,170 --> 00:51:10,620
+هذه المنطقة أه طيب الآنفي المنطقة هذه يا جماعة
+
+693
+00:51:10,620 --> 00:51:16,060
+عندي صين الـ Z أصغر أو ساوي صين الـ 0.01 هذه صينة
+
+694
+00:51:16,060 --> 00:51:18,760
+الـ 0.01 احسبها على الألة وضربها في التلاتة
+
+695
+00:51:18,760 --> 00:51:24,180
+هتلاقيها تقريبا بـ 0.3 بكل الأحوال الآن هذه الـ
+
+696
+00:51:24,180 --> 00:51:29,800
+FZZ صغيرة 0.3 الآن الـ Upper Bound مين هياخده في
+
+697
+00:51:29,800 --> 00:51:32,900
+هذه الحالة اللي هو أكبر واحد اللي هو H2 as a bound
+
+698
+00:51:32,900 --> 00:51:36,500
+in the second partial derivativesماشي إذا الـ
+
+699
+00:51:36,500 --> 00:51:39,960
+Upper bound في هذه الحالة اللي هو ايش Fxx بساوي
+
+700
+00:51:39,960 --> 00:51:43,320
+اتنين الان بنعوض hence the error included by
+
+701
+00:51:43,320 --> 00:51:48,940
+replacing F by L on R satisfies E نص الكلام هذا ال
+
+702
+00:51:48,940 --> 00:51:52,680
+M هيها اتنين نص في اتنين في هذه point one هي من
+
+703
+00:51:52,680 --> 00:51:55,300
+فوق أصغر و سوى point one وهذا أصغر من point two
+
+704
+00:51:55,300 --> 00:51:59,780
+والأخيرة اللي هي أصغر و سوى point O oneالان كلنا
+
+705
+00:51:59,780 --> 00:52:03,960
+هنا نجمح مع بعض و نربح و نضرب فهذا المقدار اللي
+
+706
+00:52:03,960 --> 00:52:07,840
+عندي و بهيك يكون اننا احنا اللي هو اوجدنا الارو في
+
+707
+00:52:07,840 --> 00:52:11,340
+ال approximation وهيك يكون خلصنا المحاضرة و إلى
+
+708
+00:52:11,340 --> 00:52:14,220
+لقاء آخر السلام عليكم و رحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..b846237506c993a1ae61fdad7ba43792765c89fb
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4_raw.json
@@ -0,0 +1 @@
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مثلًا عملناها في المرة الماضية", "tokens": [3224, 24192, 50113, 1211, 14111, 995, 6225, 42213, 8315, 11296, 8978, 9673, 25720, 9673, 46958, 10632], "avg_logprob": -0.6819852941176471, "compression_ratio": 1.0161290322580645, "no_speech_prob": 7.152557373046875e-07, "words": [{"start": 137.21, "end": 137.61, "word": "هذه", "probability": 0.718994140625}, {"start": 137.61, "end": 137.91, "word": " مثلًا", "probability": 0.6858978271484375}, {"start": 137.91, "end": 138.57, "word": " عملناها", "probability": 0.727081298828125}, {"start": 138.57, "end": 138.63, "word": " في", "probability": 0.25732421875}, {"start": 138.63, "end": 138.85, "word": " المرة", "probability": 0.646484375}, {"start": 138.85, "end": 139.47, "word": " الماضية", "probability": 0.8455403645833334}], "temperature": 1.0}, {"id": 9, "seek": 17758, "start": 152.26, "end": 177.58, "text": "يعني الآن بقول لي اللي هو الـ Gradient للـ F عمودي على الـ Velocity لأي اللي هو Curve بمرق النقطة P0 يعني لو كان عندي هذا الـ Curve 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معرفة اتجاهه ومعرفة نقطة عليه زي ما كنا سابقا في شهر تلت معاشر ناخد مثال عددي Find the Tangent Plane and Normal Line to the Surface ومعطين الـ Surface", "tokens": [6027, 9778, 9566, 9673, 2407, 31377, 1829, 46740, 15730, 4724, 39648, 6225, 2304, 2407, 16254, 15844, 2423, 39184, 36052, 34051, 13672, 1829, 18871, 16254, 6225, 2304, 2407, 16254, 15844, 2423, 39184, 22063, 317, 2149, 1929, 1975, 28543, 2288, 8315, 8717, 3615, 28480, 3714, 9957, 4032, 1863, 3615, 28480, 20449, 18513, 1211, 47395, 9154, 20449, 28480, 3660, 1975, 2655, 7435, 40294, 3224, 4032, 2304, 3615, 28480, 3660, 8717, 47432, 3660, 47356, 30767, 1829, 19446, 9122, 8315, 8608, 16758, 4587, 995, 8978, 13412, 3224, 2288, 6055, 1211, 2655, 20449, 33599, 2288, 8717, 47283, 3215, 50113, 6027, 6225, 3215, 16254, 11809, 264, 22063, 317, 2149, 1929, 293, 21277, 14670, 281, 264, 36052, 4032, 2304, 3615, 9566, 9957, 2423, 39184, 36052], "avg_logprob": -0.295021172549765, "compression_ratio": 1.7416666666666667, "no_speech_prob": 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"probability": 0.9580078125}, {"start": 430.88, "end": 433.68, "word": " ومعرفة", "probability": 0.80390625}, {"start": 433.68, "end": 434.06, "word": " نقطة", "probability": 0.7599283854166666}, {"start": 434.06, "end": 434.28, "word": " عليه", "probability": 0.9658203125}, {"start": 434.28, "end": 434.54, "word": " زي", "probability": 0.827880859375}, {"start": 434.54, "end": 434.64, "word": " ما", "probability": 0.94873046875}, {"start": 434.64, "end": 435.04, "word": " كنا", "probability": 0.9638671875}, {"start": 435.04, "end": 436.28, "word": " سابقا", "probability": 0.97314453125}, {"start": 436.28, "end": 436.48, "word": " في", "probability": 0.884765625}, {"start": 436.48, "end": 436.84, "word": " شهر", "probability": 0.6313883463541666}, {"start": 436.84, "end": 437.32, "word": " تلت", "probability": 0.482666015625}, {"start": 437.32, "end": 437.74, "word": " معاشر", "probability": 0.7462565104166666}, {"start": 437.74, "end": 439.64, "word": " ناخد", "probability": 0.7535807291666666}, {"start": 439.64, "end": 439.88, "word": " مثال", "probability": 0.902099609375}, {"start": 439.88, "end": 440.26, "word": " عددي", "probability": 0.6951497395833334}, {"start": 440.26, "end": 441.0, "word": " Find", "probability": 0.369140625}, {"start": 441.0, "end": 441.18, "word": " the", "probability": 0.5615234375}, {"start": 441.18, "end": 441.5, "word": " Tangent", "probability": 0.822265625}, {"start": 441.5, "end": 441.84, "word": " Plane", "probability": 0.933837890625}, {"start": 441.84, "end": 442.04, "word": " and", "probability": 0.94091796875}, {"start": 442.04, "end": 442.36, "word": " Normal", "probability": 0.69921875}, {"start": 442.36, "end": 442.66, "word": " Line", "probability": 0.75537109375}, {"start": 442.66, "end": 442.82, "word": " to", "probability": 0.875}, {"start": 442.82, "end": 442.98, "word": " the", "probability": 0.65869140625}, {"start": 442.98, "end": 443.48, "word": " Surface", "probability": 0.9130859375}, {"start": 443.48, "end": 443.98, "word": " ومعطين", "probability": 0.876171875}, {"start": 443.98, "end": 444.18, "word": " الـ", "probability": 0.4681396484375}, {"start": 444.18, "end": 444.58, "word": " Surface", "probability": 0.92919921875}], "temperature": 1.0}, {"id": 22, "seek": 46570, "start": 445.38, "end": 465.7, "text": "عن طريق الـ X²Y²Z-9Y0 يعني نعطيني level surface أكتر نقطة من 1 و 2 و 4 هاي عندي اللي هو ال level surface X²Y²Z-9Y0 بدي معادلة tangent plane له", "tokens": [3615, 1863, 23032, 16572, 4587, 2423, 39184, 1783, 27643, 56, 27643, 57, 12, 24, 56, 15, 37495, 22653, 8717, 3615, 9566, 9957, 1829, 1496, 3753, 5551, 4117, 2655, 2288, 8717, 47432, 3660, 9154, 502, 4032, 568, 4032, 1017, 8032, 47302, 18871, 16254, 13672, 1829, 31439, 2423, 1496, 3753, 1783, 27643, 56, 27643, 57, 12, 24, 56, 15, 4724, 16254, 20449, 18513, 37977, 27747, 5720, 46740], "avg_logprob": -0.5549242460366451, "compression_ratio": 1.2929936305732483, "no_speech_prob": 0.0008893013000488281, "words": [{"start": 445.38, "end": 445.9, "word": "عن", "probability": 0.287872314453125}, {"start": 445.9, "end": 445.9, "word": " طريق", "probability": 0.8977864583333334}, {"start": 445.9, "end": 445.94, "word": " الـ", "probability": 0.38201904296875}, {"start": 445.94, "end": 447.64, "word": " X²Y²Z", "probability": 0.324261474609375}, {"start": 447.64, "end": 449.04, "word": "-9Y0", "probability": 0.5063638687133789}, {"start": 449.04, "end": 449.26, "word": " يعني", "probability": 0.6107177734375}, {"start": 449.26, "end": 449.78, "word": " نعطيني", "probability": 0.5622802734375}, {"start": 449.78, "end": 450.08, "word": " level", "probability": 0.5654296875}, {"start": 450.08, "end": 450.68, "word": " surface", "probability": 0.79736328125}, {"start": 450.68, "end": 452.08, "word": " أكتر", "probability": 0.71392822265625}, {"start": 452.08, "end": 452.44, "word": " نقطة", "probability": 0.7491048177083334}, {"start": 452.44, "end": 452.6, "word": " من", "probability": 0.94970703125}, {"start": 452.6, "end": 453.42, "word": " 1", "probability": 0.5498046875}, {"start": 453.42, "end": 453.6, "word": " و", "probability": 0.378173828125}, {"start": 453.6, "end": 453.84, "word": " 2", "probability": 0.52197265625}, {"start": 453.84, "end": 454.16, "word": " و", "probability": 0.8515625}, {"start": 454.16, "end": 454.48, "word": " 4", "probability": 0.875}, {"start": 454.48, "end": 455.16, "word": " هاي", "probability": 0.437408447265625}, {"start": 455.16, "end": 455.56, "word": " عندي", "probability": 0.685791015625}, {"start": 455.56, "end": 455.84, "word": " اللي", "probability": 0.84912109375}, {"start": 455.84, "end": 456.22, "word": " هو", "probability": 0.92529296875}, {"start": 456.22, "end": 457.18, "word": " ال", "probability": 0.54736328125}, {"start": 457.18, "end": 459.9, "word": " level", "probability": 0.564453125}, {"start": 459.9, "end": 460.4, "word": " surface", "probability": 0.88134765625}, {"start": 460.4, "end": 461.72, "word": " X²Y²Z", "probability": 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واحد", "probability": 0.60491943359375}, {"start": 468.21, "end": 468.51, "word": " اتنين", "probability": 0.7904052734375}, {"start": 468.51, "end": 468.95, "word": " اربعة", "probability": 0.8240559895833334}, {"start": 468.95, "end": 469.97, "word": " ومعادلة", "probability": 0.88857421875}, {"start": 469.97, "end": 470.53, "word": " مين", "probability": 0.85693359375}, {"start": 470.53, "end": 471.21, "word": " الـ", "probability": 0.5986328125}, {"start": 471.21, "end": 471.43, "word": " Normal", "probability": 0.5126953125}, {"start": 471.43, "end": 471.79, "word": " line", "probability": 0.5673828125}, {"start": 471.79, "end": 471.97, "word": " اللي", "probability": 0.880126953125}, {"start": 471.97, "end": 472.33, "word": " عليه", "probability": 0.8388671875}, {"start": 472.33, "end": 473.03, "word": " لأن", "probability": 0.802490234375}, {"start": 473.03, "end": 473.27, "word": " احنا", "probability": 0.8304036458333334}, {"start": 473.27, "end": 473.83, "word": " عارفين", 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{"start": 477.89, "end": 478.05, "word": " احنا", "probability": 0.9383138020833334}, {"start": 478.05, "end": 478.55, "word": " عارفين", "probability": 0.9935302734375}, {"start": 478.55, "end": 478.81, "word": " انه", "probability": 0.850830078125}, {"start": 478.81, "end": 479.01, "word": " الـ", "probability": 0.56787109375}, {"start": 479.01, "end": 479.79, "word": " Gradient", "probability": 0.86962890625}, {"start": 479.79, "end": 480.73, "word": " لـ", "probability": 0.67333984375}, {"start": 480.73, "end": 481.03, "word": " Level", "probability": 0.80224609375}, {"start": 481.03, "end": 481.63, "word": " Surface", "probability": 0.8740234375}, {"start": 481.63, "end": 482.11, "word": " عمودي", "probability": 0.9932861328125}, {"start": 482.11, "end": 482.93, "word": " عالميين", "probability": 0.6246337890625}, {"start": 482.93, "end": 483.49, "word": " على", "probability": 0.84375}, {"start": 483.49, "end": 483.91, "word": " اللي", "probability": 0.9775390625}, {"start": 483.91, "end": 484.17, "word": " هو", "probability": 0.9775390625}, {"start": 484.17, "end": 484.49, "word": " هذا", "probability": 0.791015625}, {"start": 484.49, "end": 484.95, "word": " الـ", "probability": 0.920654296875}, {"start": 484.95, "end": 486.15, "word": " Plane", "probability": 0.926513671875}], "temperature": 1.0}, {"id": 24, "seek": 50001, "start": 486.71, "end": 500.01, "text": "ما دام عارفين الـ G العمودي وعارفين النقطة بنعرف عن عادلة الـ Tangent إذا الـ Tangent بلينة ثلوج نقطة واحد واثنين واربعة عمودي perpendicular to the gradient of F عند النقطة هذه", "tokens": [15042, 11778, 10943, 6225, 9640, 5172, 9957, 2423, 39184, 460, 18863, 2304, 2407, 16254, 4032, 3615, 9640, 5172, 9957, 28239, 47432, 3660, 44945, 3615, 28480, 18871, 6225, 18513, 37977, 2423, 39184, 22063, 317, 11933, 15730, 2423, 39184, 22063, 317, 4724, 1211, 9957, 3660, 38637, 1211, 29245, 8717, 47432, 3660, 36764, 24401, 4032, 5718, 104, 1863, 9957, 4032, 9640, 3555, 27884, 6225, 2304, 2407, 16254, 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{"start": 3103.1, "end": 3103.32, "word": " by", "probability": 0.6083984375}, {"start": 3103.32, "end": 3104.04, "word": " replacing", "probability": 0.904296875}, {"start": 3104.04, "end": 3104.4, "word": " F", "probability": 0.80517578125}, {"start": 3104.4, "end": 3105.0, "word": " by", "probability": 0.7314453125}, {"start": 3105.0, "end": 3105.26, "word": " L", "probability": 0.9736328125}, {"start": 3105.26, "end": 3105.44, "word": " on", "probability": 0.81201171875}, {"start": 3105.44, "end": 3105.64, "word": " R", "probability": 0.99365234375}, {"start": 3105.64, "end": 3106.18, "word": " satisfies", "probability": 0.78662109375}, {"start": 3106.18, "end": 3106.88, "word": " E", "probability": 0.5224609375}, {"start": 3106.88, "end": 3107.52, "word": " نص", "probability": 0.889404296875}, {"start": 3107.52, "end": 3108.18, "word": " الكلام", "probability": 0.82763671875}, {"start": 3108.18, "end": 3108.48, "word": " هذا", "probability": 0.75927734375}, {"start": 3108.48, "end": 3108.94, "word": " ال", "probability": 0.94140625}, {"start": 3108.94, "end": 3109.18, "word": " M", "probability": 0.7197265625}, {"start": 3109.18, "end": 3109.58, "word": " هيها", "probability": 0.659912109375}, {"start": 3109.58, "end": 3109.9, "word": " اتنين", "probability": 0.8148193359375}, {"start": 3109.9, "end": 3110.12, "word": " نص", "probability": 0.931640625}, {"start": 3110.12, "end": 3110.24, "word": " في", "probability": 0.394775390625}, {"start": 3110.24, "end": 3110.54, "word": " اتنين", "probability": 0.97607421875}, {"start": 3110.54, "end": 3110.74, "word": " في", "probability": 0.923828125}, {"start": 3110.74, "end": 3111.52, "word": " هذه", "probability": 0.251708984375}, {"start": 3111.52, "end": 3111.94, "word": " point", "probability": 0.80908203125}, {"start": 3111.94, "end": 3112.28, "word": " one", "probability": 0.80078125}, {"start": 3112.28, "end": 3112.5, "word": " هي", "probability": 0.609375}, {"start": 3112.5, "end": 3112.68, "word": " من", "probability": 0.97802734375}, {"start": 3112.68, "end": 3113.18, "word": " فوق", "probability": 0.939453125}, {"start": 3113.18, "end": 3113.58, "word": " أصغر", "probability": 0.9027099609375}, {"start": 3113.58, "end": 3113.7, "word": " و", "probability": 0.78466796875}, {"start": 3113.7, "end": 3113.72, "word": " سوى", "probability": 0.5076497395833334}, {"start": 3113.72, "end": 3113.96, "word": " point", "probability": 0.9482421875}, {"start": 3113.96, "end": 3114.16, "word": " one", "probability": 0.91455078125}, {"start": 3114.16, "end": 3114.38, "word": " وهذا", "probability": 0.5130615234375}, {"start": 3114.38, "end": 3114.6, "word": " أصغر", "probability": 0.9317626953125}, {"start": 3114.6, "end": 3114.72, "word": " من", "probability": 0.951171875}, {"start": 3114.72, "end": 3114.96, "word": " point", "probability": 0.90478515625}, {"start": 3114.96, "end": 3115.3, "word": " two", "probability": 0.90283203125}, {"start": 3115.3, "end": 3116.38, "word": " والأخيرة", "probability": 0.76953125}, {"start": 3116.38, "end": 3116.72, "word": " اللي", "probability": 0.823486328125}, {"start": 3116.72, "end": 3116.88, "word": " هي", "probability": 0.8408203125}, {"start": 3116.88, "end": 3117.16, "word": " أصغر", "probability": 0.9615478515625}, {"start": 3117.16, "end": 3117.28, "word": " و", "probability": 0.9658203125}, {"start": 3117.28, "end": 3117.48, "word": " سوى", "probability": 0.9191080729166666}, {"start": 3117.48, "end": 3117.86, "word": " point", "probability": 0.9736328125}, {"start": 3117.86, "end": 3118.2, "word": " O", "probability": 0.267333984375}, {"start": 3118.2, "end": 3118.5, "word": " one", "probability": 0.712890625}], "temperature": 1.0}, {"id": 139, "seek": 313422, "start": 3118.92, "end": 3134.22, "text": "الان كلنا هنا نجمح مع بعض و نربح و نضرب فهذا المقدار اللي عندي و بهيك يكون اننا احنا اللي هو اوجدنا الارو في ال approximation وهيك يكون خلصنا المحاضرة و إلى لقاء آخر السلام عليكم و رحمة الله وبركاته", "tokens": [6027, 7649, 28242, 8315, 34105, 8717, 7435, 2304, 5016, 20449, 45030, 11242, 4032, 8717, 25513, 5016, 4032, 8717, 11242, 25513, 6156, 3224, 15730, 9673, 28543, 9640, 13672, 1829, 18871, 16254, 4032, 39627, 1829, 4117, 7251, 30544, 16472, 8315, 1975, 5016, 8315, 13672, 1829, 31439, 1975, 29245, 3215, 8315, 2423, 9640, 2407, 8978, 2423, 28023, 37037, 1829, 4117, 7251, 30544, 16490, 1211, 9381, 8315, 9673, 5016, 46958, 25720, 4032, 30731, 5296, 4587, 16606, 19753, 34740, 21136, 37440, 25894, 24793, 4032, 12602, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224], "avg_logprob": -0.35252809256650086, "compression_ratio": 1.7114427860696517, "no_speech_prob": 4.410743713378906e-06, "words": [{"start": 3118.92, "end": 3119.36, "word": "الان", "probability": 0.706298828125}, {"start": 3119.36, "end": 3119.78, "word": " كلنا", "probability": 0.653564453125}, {"start": 3119.78, "end": 3119.9, "word": " هنا", "probability": 0.1807861328125}, {"start": 3119.9, "end": 3120.18, "word": " نجمح", "probability": 0.8516845703125}, {"start": 3120.18, "end": 3120.36, "word": " مع", "probability": 0.6162109375}, {"start": 3120.36, "end": 3120.66, "word": " بعض", "probability": 0.99609375}, {"start": 3120.66, "end": 3120.74, "word": " و", "probability": 0.8203125}, {"start": 3120.74, "end": 3121.1, "word": " نربح", "probability": 0.720703125}, {"start": 3121.1, "end": 3121.3, "word": " و", "probability": 0.79638671875}, {"start": 3121.3, "end": 3121.56, "word": " نضرب", "probability": 0.9305013020833334}, {"start": 3121.56, "end": 3121.9, "word": " فهذا", "probability": 0.5082194010416666}, {"start": 3121.9, "end": 3123.66, "word": " المقدار", "probability": 0.6192220052083334}, {"start": 3123.66, "end": 3123.96, "word": " اللي", "probability": 0.59716796875}, {"start": 3123.96, "end": 3124.36, "word": " عندي", "probability": 0.642822265625}, {"start": 3124.36, "end": 3124.78, "word": " و", "probability": 0.1622314453125}, {"start": 3124.78, "end": 3125.08, "word": " بهيك", "probability": 0.484375}, {"start": 3125.08, "end": 3125.38, "word": " يكون", "probability": 0.6768798828125}, {"start": 3125.38, "end": 3125.74, "word": " اننا", "probability": 0.3775634765625}, {"start": 3125.74, "end": 3126.42, "word": " احنا", "probability": 0.8317057291666666}, {"start": 3126.42, "end": 3126.7, "word": " اللي", "probability": 0.7783203125}, {"start": 3126.7, "end": 3126.94, "word": " هو", "probability": 0.93505859375}, {"start": 3126.94, "end": 3127.44, "word": " اوجدنا", "probability": 0.808837890625}, {"start": 3127.44, "end": 3127.76, "word": " الارو", "probability": 0.4352620442708333}, {"start": 3127.76, "end": 3127.84, "word": " في", "probability": 0.6650390625}, {"start": 3127.84, "end": 3127.92, "word": " ال", "probability": 0.896484375}, {"start": 3127.92, "end": 3128.54, "word": " approximation", "probability": 0.75}, {"start": 3128.54, "end": 3129.22, "word": " وهيك", "probability": 0.6660970052083334}, {"start": 3129.22, "end": 3129.4, "word": " يكون", "probability": 0.927978515625}, {"start": 3129.4, "end": 3129.86, "word": " خلصنا", "probability": 0.94580078125}, {"start": 3129.86, "end": 3130.54, "word": " المحاضرة", "probability": 0.9874267578125}, {"start": 3130.54, "end": 3130.98, "word": " و", "probability": 0.728515625}, {"start": 3130.98, "end": 3131.34, "word": " إلى", "probability": 0.62939453125}, {"start": 3131.34, "end": 3132.02, "word": " لقاء", "probability": 0.9191080729166666}, {"start": 3132.02, "end": 3132.64, "word": " آخر", "probability": 0.813232421875}, {"start": 3132.64, "end": 3133.02, "word": " السلام", "probability": 0.842041015625}, {"start": 3133.02, "end": 3133.32, "word": " عليكم", "probability": 0.947509765625}, {"start": 3133.32, "end": 3133.46, "word": " و", "probability": 0.97607421875}, {"start": 3133.46, "end": 3133.56, "word": " رحمة", "probability": 0.7942708333333334}, {"start": 3133.56, "end": 3133.7, "word": " الله", "probability": 0.9677734375}, {"start": 3133.7, "end": 3134.22, "word": " وبركاته", 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..ace8c2266f39c841d6962497945bb6d4cb9e36cf
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P0WvQniRYz4_raw.srt
@@ -0,0 +1,2836 @@
+1
+00:00:00,670 --> 00:00:04,150
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,150 --> 00:00:10,210
+وبركاته هذه المحاضرة رقم 18 لمساقطة فاضل وتكامل
+
+3
+00:00:10,210 --> 00:00:18,090
+الجين وهي المحاضرة السادسة chapter 14 حيث سنتناول
+
+4
+00:00:18,090 --> 00:00:24,910
+الحديث عن 146 وهو تحت منوع tangent planes and
+
+5
+00:00:24,910 --> 00:00:30,200
+differentialsبنشوف كيف نجد معادلة tangent planes و
+
+6
+00:00:30,200 --> 00:00:33,920
+كيف نحكي عن الـ differentials طبعا الـ tangent
+
+7
+00:00:33,920 --> 00:00:38,500
+planes فكرتها حتى إيجاد معادلتها بنفس فكرة tangent
+
+8
+00:00:38,500 --> 00:00:42,680
+line في calculus 1000 و الـ differentials هي تعنيم
+
+9
+00:00:42,680 --> 00:00:48,090
+للـ differentials اللي أخدناهافي تفاضل ألف وكذلك
+
+10
+00:00:48,090 --> 00:00:51,570
+هنحكي عن الـ Linearization اللي هو أفدنا زيه لـ
+
+11
+00:00:51,570 --> 00:00:54,470
+Function of one variable هناخدها لـ Function of
+
+12
+00:00:54,470 --> 00:00:58,030
+two variables أو Equivariables نيجي الآن للـ
+
+13
+00:00:58,030 --> 00:01:01,670
+Tangent planes and normal lines اللي هى اللي
+
+14
+00:01:01,670 --> 00:01:06,490
+هنتحدث عنها اليوملو كان عندي R بساوية G of T في
+
+15
+00:01:06,490 --> 00:01:09,770
+تجاه I and H of T في تجاه J زي الـK وT في تجاه
+
+16
+00:01:09,770 --> 00:01:13,630
+الـK is a smooth curve on a level surface F of XY
+
+17
+00:01:13,630 --> 00:01:19,290
+وZ بساوية C يعني زي ما حكينا قبل هيك اللي هو
+
+18
+00:01:25,330 --> 00:01:29,130
+عند الـ Smooth Curve R بسوء U of T زادة U of T
+
+19
+00:01:29,130 --> 00:01:34,450
+زادة K of T عوضه في داخل الـ F of X Y وZ بعملية
+
+20
+00:01:34,450 --> 00:01:38,990
+constant يعني بعملية level curveالان زي ما عملنا
+
+21
+00:01:38,990 --> 00:01:43,750
+في المحاضرة اللي فاتت اخدنا f of x, f of g و h, h
+
+22
+00:01:43,750 --> 00:01:46,310
+of t, k of t كانت two variables ان انا تلقي
+
+23
+00:01:46,310 --> 00:01:50,790
+variables بساوة c هذه فاضلة تنتين بالنسبة ليها
+
+24
+00:01:50,790 --> 00:01:55,450
+ثابتة بالنسبة ليها اذا انا اكيد تفضلها سفروهذه
+
+25
+00:01:55,450 --> 00:01:59,330
+اللي هي تفضلها by Chain Rule اللي هي Del F by Del
+
+26
+00:01:59,330 --> 00:02:05,690
+X في Del X by Del T الـ X اللي هنا G وDel F by Del
+
+27
+00:02:05,690 --> 00:02:10,430
+Y في Del Y by Del T اللي هي H في هذه الحالة وDel F
+
+28
+00:02:10,430 --> 00:02:15,410
+by Del Z في Del K by Del T اللي هي Z في هذه الحالة
+
+29
+00:02:15,410 --> 00:02:19,470
+كله هي ساوي 7هذه مثلًا عملناها في المرة الماضية
+
+30
+00:02:32,260 --> 00:02:36,800
+يعني الآن بقول لي اللي هو الـ Gradient للـ F عمودي
+
+31
+00:02:36,800 --> 00:02:44,840
+على الـ Velocity لأي اللي هو Curve بمرق النقطة P0
+
+32
+00:02:44,840 --> 00:02:48,920
+يعني لو كان عندي هذا الـ Curve بمرق النقطة P0 اللي
+
+33
+00:02:48,920 --> 00:02:53,260
+هو مماسق اللي هو V1 اللي هو بيكون ال Gradient للـ
+
+34
+00:02:53,260 --> 00:02:57,580
+F عمودي عليه وأيضا لو كان هذه عندي كمان واحد تاني
+
+35
+00:02:57,740 --> 00:03:02,740
+أو هذا واحد تاني فبكون مماسه اللي هو V2 اللي هي ال
+
+36
+00:03:02,740 --> 00:03:06,360
+velocity اللي هي DR بDT برضه عمودي عالميين على ال
+
+37
+00:03:06,360 --> 00:03:09,760
+gradient يعني لو كان عندي كمان برضه curve موربل بي
+
+38
+00:03:09,760 --> 00:03:13,180
+نوت اللي هو ال curve هذا اللي بيخلق اللي هي هذه
+
+39
+00:03:13,180 --> 00:03:18,270
+ساوة C لبس ال surfaceبخلّي الـ Gradient للـ F
+
+40
+00:03:18,270 --> 00:03:23,510
+عمودي على مماسه الـ DRIDT يعني الـ Gradient للـ F
+
+41
+00:03:23,510 --> 00:03:28,730
+هيكون عمودي على كل اللي هي على الـ Plane اللي
+
+42
+00:03:28,730 --> 00:03:35,050
+بتكون من الـ V1 و V2 و بحمل أيه اللي هو Tangent
+
+43
+00:03:35,050 --> 00:03:41,590
+لأيه اللي هو Curve بمر بالنقطة D Note و بخلّي هذا
+
+44
+00:03:41,590 --> 00:03:47,290
+اللي هو بساوي Level Surfaceطيب إذا هذا اللي حكينا
+
+45
+00:03:47,290 --> 00:03:51,130
+عنه هو الـ tangent plane at the point B0 هو ال
+
+46
+00:03:51,130 --> 00:03:54,310
+tangent plane اللي حكينا عنه on the level surface
+
+47
+00:03:54,310 --> 00:03:57,790
+F of XYZ بالساوية سفق C of a differentiable
+
+48
+00:03:57,790 --> 00:04:02,480
+function is the plane through B0الأنمال هو Normal
+
+49
+00:04:02,480 --> 00:04:07,580
+على Gradient إذا صار عندي Gradient Normal على
+
+50
+00:04:07,580 --> 00:04:12,660
+Tangent Plane هذا هو الـ Tangent Plane صار الـ
+
+51
+00:04:12,660 --> 00:04:16,360
+Gradient عمودي عليه إذا الـ Tangent Plane هو عبارة
+
+52
+00:04:16,360 --> 00:04:22,560
+عن الـ Plane اللي بيحمل الـ V1 و V2 و V3
+
+53
+00:04:25,790 --> 00:04:30,650
+المماسات للمنحنيات اللي بمرن بالنقطة بـ Note اللي
+
+54
+00:04:30,650 --> 00:04:34,710
+بعمل ان اللي هو F of X Y و Z بساوة C is a level
+
+55
+00:04:34,710 --> 00:04:38,210
+surface هذا هو الـ Tangent Plane وعرفنا العمودي
+
+56
+00:04:38,210 --> 00:04:41,610
+عليه اللي هو الـ Gradient وعارفين نقطة إذا بمعرفة
+
+57
+00:04:41,610 --> 00:04:44,970
+نقطة وعمودي على طول بنعرف معادلة الـ Plane زي ما
+
+58
+00:04:44,970 --> 00:04:49,230
+انتوا عارفين وبذلك مكون معادلة الـ Plane سهلة
+
+59
+00:04:49,420 --> 00:04:55,040
+التانجنت بلينتو Fxy بيساوي C at P0 وX0 وY0 وZ0 هي
+
+60
+00:04:55,040 --> 00:04:59,560
+عبارة عن اللي هو Fx اللي هي تبعات الـ Gradient لأن
+
+61
+00:04:59,560 --> 00:05:03,180
+Gradient بيساوي Fx في اتجاه I زي دي Gradient زي Fy
+
+62
+00:05:03,180 --> 00:05:06,460
+في اتجاه G زي Fz في اتجاه K هذا اللي هي ال
+
+63
+00:05:06,460 --> 00:05:10,900
+Gradient ومن ثم ال Gradient اللي هو العمودي على ال
+
+64
+00:05:10,900 --> 00:05:15,570
+surfaceومعرفة نقطة على الـ Surface بنعرف معادلة
+
+65
+00:05:15,570 --> 00:05:20,410
+مين الـ Plane زي ما احنا عارفين بكون Fx في اتجاه X
+
+66
+00:05:20,410 --> 00:05:24,230
+في مضروف X minus X knot Fy في Y minus Y knot Fz في
+
+67
+00:05:24,230 --> 00:05:27,970
+Z minus Z knot حيث الـ Fx و الـ Fy و الـ Fz هما
+
+68
+00:05:27,970 --> 00:05:31,410
+اللي بيكوّن الـ Gradient اللي هو العمودي على الـ
+
+69
+00:05:31,410 --> 00:05:33,730
+Surface اللي احنا عارفينه و الـ X knot و الـ Y
+
+70
+00:05:33,730 --> 00:05:36,810
+knot و الـ Z knot هي عبارة عن النقاط اللي على الـ
+
+71
+00:05:36,810 --> 00:05:39,740
+Tangent الـ Planeبما عرفت الـ Gradient اللي هو
+
+72
+00:05:39,740 --> 00:05:42,560
+عمودي على الـ Tangent Plane وبما عرفت النقطة X0
+
+73
+00:05:42,560 --> 00:05:46,500
+وY0 عرفنا معادلة الـ Tangent Plane هل نستطيع أن
+
+74
+00:05:46,500 --> 00:05:51,200
+نعرف الـ Normal Line الـ Normal Line على الـ Level
+
+75
+00:05:51,200 --> 00:05:55,490
+Surface F of X Y وZ؟أه لإن احنا نجينا ان ال
+
+76
+00:05:55,490 --> 00:05:59,250
+gradient عمودي عالميا على اللي هو ال level على ال
+
+77
+00:05:59,250 --> 00:06:02,970
+tangent plane اللي هو عمودي عالميا على اللي هو هذه
+
+78
+00:06:02,970 --> 00:06:06,670
+ال surface اللي عندي مدام عمودي عليه اذا كل خط
+
+79
+00:06:06,670 --> 00:06:10,790
+عمودي عليه مكون موازية ال gradientما دام عرفت أنه
+
+80
+00:06:10,790 --> 00:06:17,270
+موازي للـ Line اتجاهه يعني بمعنى آخر وعرفت نقطة
+
+81
+00:06:17,270 --> 00:06:22,410
+يعني معادلة الخط الموازية للـ Gradient اللي هو الـ
+
+82
+00:06:22,410 --> 00:06:25,690
+Normal Line على اللي هو الـ Tangent Plane هو عبارة
+
+83
+00:06:25,690 --> 00:06:30,010
+عن X بساوي X Note زاد FX في T هذه تبعت Mean لـ
+
+84
+00:06:30,010 --> 00:06:33,530
+Gradient Y بساوي Y Note زاد FY تبعت لـ Gradient في
+
+85
+00:06:33,530 --> 00:06:41,870
+T زاد Z بساوي Z Note FZ في Tالان بمعرفة موازي للخط
+
+86
+00:06:41,870 --> 00:06:47,050
+الـ Gradient ومعرفة نقطة على الخط X0 وY0 وZ0 اعرفت
+
+87
+00:06:47,050 --> 00:06:51,970
+ان انا معادلة الـ Line Normal على الـ Tangent
+
+88
+00:06:51,970 --> 00:06:55,490
+Plane أو Normal على الـ Surface اللي قبل بشوية
+
+89
+00:06:55,490 --> 00:07:02,760
+شفناهالخط الموازي لهذا بقول عمودي على الـ Surface
+
+90
+00:07:02,760 --> 00:07:06,600
+أو اللي عندي عمودي على الـ Tangent Plane اقدرنا
+
+91
+00:07:06,600 --> 00:07:14,060
+نعرف مين ونعرف معادلته من معرفة اتجاهه ومعرفة نقطة
+
+92
+00:07:14,060 --> 00:07:19,880
+عليه زي ما كنا سابقا في شهر تلت معاشر ناخد مثال
+
+93
+00:07:19,880 --> 00:07:22,980
+عددي Find the Tangent Plane and Normal Line to the
+
+94
+00:07:22,980 --> 00:07:29,040
+Surface ومعطين الـ Surfaceعن طريق الـ X²Y²Z-9Y0
+
+95
+00:07:29,040 --> 00:07:34,480
+يعني نعطيني level surface أكتر نقطة من 1 و 2 و 4
+
+96
+00:07:34,480 --> 00:07:43,980
+هاي عندي اللي هو ال level surface X²Y²Z-9Y0 بدي
+
+97
+00:07:43,980 --> 00:07:48,210
+معادلة tangent plane لهعنده نقطة مين ليه واحد
+
+98
+00:07:48,210 --> 00:07:52,330
+اتنين اربعة ومعادلة مين الـ Normal line اللي عليه
+
+99
+00:07:52,330 --> 00:07:55,870
+لأن احنا عارفين الـ Plane اللي بنعرف معادلته لو
+
+100
+00:07:55,870 --> 00:07:59,010
+عرفنا نقطة عمودي عليه طب ما احنا عارفين انه الـ
+
+101
+00:07:59,010 --> 00:08:03,910
+Gradient لـ Level Surface عمودي عالميين على اللي
+
+102
+00:08:03,910 --> 00:08:09,570
+هو هذا الـ Planeما دام عارفين الـ G العمودي
+
+103
+00:08:09,570 --> 00:08:13,130
+وعارفين النقطة بنعرف عن عادلة الـ Tangent إذا الـ
+
+104
+00:08:13,130 --> 00:08:16,750
+Tangent بلينة ثلوج نقطة واحد واثنين واربعة عمودي
+
+105
+00:08:16,750 --> 00:08:20,010
+perpendicular to the gradient of F عند النقطة هذه
+
+106
+00:08:21,050 --> 00:08:25,010
+من الحديث اللي حدثناه قبل قليل والـ Gradient لـ F
+
+107
+00:08:25,010 --> 00:08:32,030
+يساوي 2X في اتجاه I زي 2Y في اتجاه J زائد المين K
+
+108
+00:08:32,030 --> 00:08:39,270
+عارفين ايش بسوي؟ Fx هيها و Fy هيها 2Y و Fz هي فقط
+
+109
+00:08:39,270 --> 00:08:43,300
+1لـ Gradient الـ F صار 2X في تجاه الـ I زائد 2Y في
+
+110
+00:08:43,300 --> 00:08:48,720
+تجاه الـ J زائد K اللي هي FX FY FZ لما تحسبها الـ
+
+111
+00:08:48,720 --> 00:08:52,260
+Gradient عند مين عنده النقطة المذكورة اللي هي 1 و
+
+112
+00:08:52,260 --> 00:08:56,220
+2 و 4 و سوى 2 في تجاه الـ I زائد اللي هي مكان الـ
+
+113
+00:08:56,220 --> 00:08:59,800
+Y2 يصير 4 في تجاه الـ J زائد الـ K هذا الـ
+
+114
+00:08:59,800 --> 00:09:02,460
+Gradient اللي هو العمودي على الـ Plane اللي بدنا
+
+115
+00:09:02,460 --> 00:09:07,640
+إياهعرفين نقطة على الـ Plane إذا جهزة معادلتنا
+
+116
+00:09:07,640 --> 00:09:12,440
+اللي هي اتنين اللي زمنا كنا نسميها A في X minus
+
+117
+00:09:12,440 --> 00:09:16,660
+واحد اللي هي زائد B اللي هي أربعة Y minus اتنين
+
+118
+00:09:16,660 --> 00:09:21,060
+عارفين ليش Y minus اتنين Y نقص اتنين Z نقص أربعة
+
+119
+00:09:21,060 --> 00:09:24,920
+والـ X نقص واحد النقطة اللي عارفينها في اللي هي C
+
+120
+00:09:24,920 --> 00:09:28,600
+هنا إيش بتساوي اللي هي معامل K اللي هو واحد إذا
+
+121
+00:09:28,600 --> 00:09:34,470
+صارت هذه معادلة Tangent Planeاللي هنبسطها 2x-2 وان
+
+122
+00:09:34,470 --> 00:09:40,620
+كده بتطلع عندى 2x زي 4y زي 14الآن بدنا نعرف الـ
+
+123
+00:09:40,620 --> 00:09:43,740
+Normal Line الـ Normal Line على الـ Surface هو
+
+124
+00:09:43,740 --> 00:09:45,880
+موازي من الـ Gradient والـ Gradient عمودي على الـ
+
+125
+00:09:45,880 --> 00:09:50,260
+Surface إذا الـ Normal Line عارفين موازي إيه؟ اللي
+
+126
+00:09:50,260 --> 00:09:55,140
+هو عبارة من 2I زي 4G زي K وعارفين النقطة عليه؟ إذا
+
+127
+00:09:55,140 --> 00:10:00,420
+على طول معدلته X بتساوي 1 اللي هي النقطة زي 2T
+
+128
+00:10:00,420 --> 00:10:07,500
+اللي هي 2 منها T Y بيساوي 2 اللي هي قيمة النقطة Y
+
+129
+00:10:07,800 --> 00:10:12,760
+زائد اللي هي ال .. ال .. ال .. ال .. ال .. اللي في
+
+130
+00:10:12,760 --> 00:10:18,200
+ال line الموازي اللي هي A4T و هذه الأخيرة T زائد 4
+
+131
+00:10:18,200 --> 00:10:20,560
+فبكون و T طبعا parameter زي ما انتوا عارفين ال
+
+132
+00:10:20,560 --> 00:10:24,320
+element in R فبكون ال normal line هي معدلة و طبعا
+
+133
+00:10:24,320 --> 00:10:28,240
+هذه شغلات قديمة لكن احنا بنعملها نستغل معلوماتنا
+
+134
+00:10:28,240 --> 00:10:34,810
+الجديدة في إيجادها طيبالان لو الصيغة اللى عندنا
+
+135
+00:10:34,810 --> 00:10:39,990
+كانت f of x y وz بساول ثابت طب لو كان مش معروض
+
+136
+00:10:39,990 --> 00:10:43,670
+علينا بالشكل هذا لو معروض علينا اللى هو معادلة
+
+137
+00:10:43,670 --> 00:10:48,030
+بالشكل ثاني يعني معروض علينا surface الان عند اللى
+
+138
+00:10:48,030 --> 00:10:55,340
+أعرضناها قبلاللي هي Z اللي هي عندي F of XY و Z
+
+139
+00:10:55,340 --> 00:10:59,700
+function اللي هو و بتساوي C سارة IH level surface
+
+140
+00:10:59,700 --> 00:11:03,700
+لو الموضوع مش معروض بالشكل هذا لو معروض بشكل آخر
+
+141
+00:11:03,700 --> 00:11:07,340
+خليني أشوف إيش مقصود لو عندي to find an equation
+
+142
+00:11:07,340 --> 00:11:10,140
+for the plane tangent to a smooth curve لو بدنا
+
+143
+00:11:10,140 --> 00:11:13,900
+نعرف اللي هو معادلة ال tangent plane على smooth
+
+144
+00:11:13,900 --> 00:11:16,960
+curve الآن Z بسوء F of XY مش مكتوب على صورة level
+
+145
+00:11:16,960 --> 00:11:21,770
+surfaceAt the point x0, y0 و z0 where z0 بيساوي f
+
+146
+00:11:21,770 --> 00:11:25,590
+of x0 و y0 بدنا نعرف اللي هو الـ Tangent Plane
+
+147
+00:11:25,590 --> 00:11:29,770
+لهذا الـ Surface z بيساوي f of x, y الآن بدنا نعمل
+
+148
+00:11:29,770 --> 00:11:33,770
+مياليه، لاحظوا الآن هذه z بيساوي f of x, y بيساوي
+
+149
+00:11:33,770 --> 00:11:38,460
+f of x, y نقل zH بتساوي 0الأن الـ Surface Z ساوية
+
+150
+00:11:38,460 --> 00:11:43,120
+F of X Y اللي هو هذا بعبّر عنه لو سمينا هذه كلها
+
+151
+00:11:43,120 --> 00:11:46,280
+على بعض Function F Capital of X Y و Z زي تابعت
+
+152
+00:11:46,280 --> 00:11:52,280
+زمان هتساوية F of X Y ناقص Z صار هذا هذا عبارة عن
+
+153
+00:11:52,280 --> 00:11:56,380
+Level Surface ماشي بساوية Level Surface الآن
+
+154
+00:11:56,380 --> 00:11:59,320
+بنتعامل معاه زي قبل بشوية الان ال partial
+
+155
+00:11:59,320 --> 00:12:01,540
+derivatives of F لو جينا أخدنا ال partial
+
+156
+00:12:01,540 --> 00:12:05,690
+derivatives عشان نطبّق عليه مين يا جماعةاللي هي
+
+157
+00:12:05,690 --> 00:12:09,930
+اللي جابل بشوية المعادلة الـ Fx إلها سوى Deal By
+
+158
+00:12:09,930 --> 00:12:14,670
+Deal X لمين لهذا المقدار هذه بالنسبة لـ X بس Fx
+
+159
+00:12:14,670 --> 00:12:19,730
+لأنه بالنسبة لـ Y صفر بالنسبة لـ Z صفر إذا Fx نقص
+
+160
+00:12:19,730 --> 00:12:25,120
+صفر وسوى Fx Smallفهنا fy بيصير deal فضلتها دي
+
+161
+00:12:25,120 --> 00:12:27,760
+بالنسبة ل y جزيياً هذه بتفضلها بالنسبة ل y كلها
+
+162
+00:12:27,760 --> 00:12:31,620
+بيصير deal by deal y f of x و y ناقص z نفس الأشي
+
+163
+00:12:31,620 --> 00:12:36,120
+بيصير fy ناقص اللي هي بالنسبة ل y سفر و يساوي fy
+
+164
+00:12:36,120 --> 00:12:39,900
+of z اللي هي deal هذا المقدار كله بالنسبة ل z هذا
+
+165
+00:12:39,900 --> 00:12:43,780
+بالنسبة ل z سفر ماشي الحال الآن نصارة ال x و ال y
+
+166
+00:12:43,780 --> 00:12:47,180
+و ال z عنده هنا في هذه الحالة independent تلاتة x
+
+167
+00:12:47,180 --> 00:12:51,190
+و y و z dependentأسف independent فمدار ساعة
+
+168
+00:12:51,190 --> 00:12:55,570
+independent إذن تفضل بالنسبة لبعض سفر إذن del هذا
+
+169
+00:12:55,570 --> 00:13:00,090
+المقدار by del z بيساوي سفر ناقص del by del z by
+
+170
+00:13:00,090 --> 00:13:03,670
+del z بيساوي إيش واحد وفعلا ناقص بس ناقص واحد لأن
+
+171
+00:13:03,670 --> 00:13:06,830
+هذه بتدعوضها في المعادلة اللي جاب المشوية اللي
+
+172
+00:13:06,830 --> 00:13:11,190
+حكينا عنها إما بس كانت F capital مثلا كنا بسميناها
+
+173
+00:13:11,190 --> 00:13:14,450
+F small اللي جاب المشوية يا جماعة بتدعوضها في هذه
+
+174
+00:13:14,450 --> 00:13:17,850
+اللي هي Fx بس هذه F capital بالسين وهذه F capital
+
+175
+00:13:17,850 --> 00:13:21,040
+وهذه F capitalأف كابيتال هي أف صغيرة اللي قبل شوية
+
+176
+00:13:21,040 --> 00:13:24,680
+و أف كابيتال هي أف صغيرة هذا لكن أف كابيتال للـ z
+
+177
+00:13:24,680 --> 00:13:28,240
+طلعت إيه؟ بتساوي سالب واحد فبتصير هذه المعادلة هي
+
+178
+00:13:28,240 --> 00:13:32,240
+بس بسالب واحد خليني أشوف المعادلة الجديدة بصير
+
+179
+00:13:32,240 --> 00:13:38,560
+عندي اللي هي عوضنا Fxy هي Fy هي Fz لما نعوض مع
+
+180
+00:13:38,560 --> 00:13:42,710
+مكانهم اللي هي ال formula اللي قبل شويةهذه Fx فهي
+
+181
+00:13:42,710 --> 00:13:46,550
+Fx صغيرة وهذه Fy فهي Fy صغيرة وهي من فوق ما
+
+182
+00:13:46,550 --> 00:13:51,710
+اخترناها والـ Fz هذه اشتلعت سالب واحد فبصير هذه هي
+
+183
+00:13:51,710 --> 00:13:59,140
+معادلة الـ tangent plane لل surface Z بتساوي Fof X
+
+184
+00:13:59,140 --> 00:14:03,540
+و Y يعني الآن هذه اللي هي كانت Bridge أو جسر اللي
+
+185
+00:14:03,540 --> 00:14:06,720
+هي هذه F Capital للوصول للمعادلة اللي أمامنا هذه
+
+186
+00:14:06,720 --> 00:14:09,700
+خلاص الآن لما تنعرض علينا Z بسيط و أقفف X و Y على
+
+187
+00:14:09,700 --> 00:14:13,520
+طول بنقدر نجد تنجين تـBlend لها بس بنجيب F X و F Y
+
+188
+00:14:13,520 --> 00:14:18,480
+وهذا بيصير Z minus Z not طيب نشوف إذا Blended
+
+189
+00:14:18,480 --> 00:14:21,260
+tangent to the surface Z بسيط و أقفف X Y at X not
+
+190
+00:14:21,260 --> 00:14:25,040
+و Y not و أقفف X not و Y notالتي هي the plane
+
+191
+00:14:25,040 --> 00:14:27,420
+tangent to the surface z بساوية f of x, y of a
+
+192
+00:14:27,420 --> 00:14:30,840
+differentiable function f at the point x0 و y0 و
+
+193
+00:14:30,840 --> 00:14:35,300
+z0 بساوية x0 و y0 و f of x0 و f of y و y0 is
+
+194
+00:14:35,300 --> 00:14:39,920
+لتوّه، قلت عشان أنا حكيت لتوّه، fx في x minus x0،
+
+195
+00:14:39,920 --> 00:14:44,300
+fy في y و y minus y0، ماقع الـ z minus z0 هي
+
+196
+00:14:44,300 --> 00:14:47,980
+معدلتنا وهي الـ tangent plane إلها أو surfaceنا
+
+197
+00:14:47,980 --> 00:14:50,800
+وهي الـ tangent plane إلها خلّيني أخد مثال عددي
+
+198
+00:14:50,800 --> 00:14:54,110
+جماعةFind her at a plane tangent to the surface z
+
+199
+00:14:54,110 --> 00:14:57,330
+بالساوية x cos y نقص y e to the x عند مين عنده
+
+200
+00:14:57,330 --> 00:15:01,530
+نقطة 00 الموضوع سهل هي عند z بالساوية كذا يعني
+
+201
+00:15:01,530 --> 00:15:06,230
+هيصير هذه عبارة عن تمثل لمين F of x y F small إذا
+
+202
+00:15:06,230 --> 00:15:09,050
+صارت عند z بالساوية F of x y يعني F of x y
+
+203
+00:15:09,050 --> 00:15:12,770
+بالساوية x cos y نقص y e to the x وبنطبق هذه
+
+204
+00:15:12,770 --> 00:15:18,770
+المعادلة ال Fx في هذه الحالة اللي هي عبارة عنCos y
+
+205
+00:15:18,770 --> 00:15:23,190
+ناقص y e to the x Cos y ناقص y e to the x تحسبها
+
+206
+00:15:23,190 --> 00:15:28,050
+عند من؟ عند الـ 0 و الـ 0 و الـ 0 تحسبها عند الـ 0
+
+207
+00:15:28,050 --> 00:15:29,450
+و الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+208
+00:15:29,450 --> 00:15:29,450
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+209
+00:15:29,450 --> 00:15:29,910
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+210
+00:15:29,910 --> 00:15:30,630
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+211
+00:15:30,630 --> 00:15:30,790
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+212
+00:15:30,790 --> 00:15:31,350
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+213
+00:15:31,350 --> 00:15:36,230
+عند الـ 0 و الـ 0 تحسبها عند الـ 0 و الـ 0 تحسبها
+
+214
+00:15:36,230 --> 00:15:41,730
+عند الـ 0 وصلى الله عليه وسلم عن نبي عليه الصلاة
+
+215
+00:15:41,730 --> 00:15:50,350
+والسلام And fy ليها سالب x sin y سالب x sin y
+
+216
+00:15:50,350 --> 00:15:55,530
+تفاضلها بالاسم y e to the x بالسالي كان عند 0 0 0
+
+217
+00:15:55,530 --> 00:15:59,470
+هذا يصبح 0 وهذا يصبح e to the 0 يعني واحد يعني
+
+218
+00:15:59,470 --> 00:16:04,940
+سالب واحد من عين المارس اللي قدامهاإذا جهزة Fx و
+
+219
+00:16:04,940 --> 00:16:10,800
+Fy إذا الـ tangent plane is واحد هي مكان Fx في X
+
+220
+00:16:10,800 --> 00:16:13,360
+ناقص X نوت طبعاً Zero نحن نحسبها Zero Zero Zero
+
+221
+00:16:13,360 --> 00:16:18,700
+فناقص واحد اللي هي تبع ال Y هذه Fy مضروب Fy minus
+
+222
+00:16:18,700 --> 00:16:22,420
+Zero ناقص Z minus Zero اللي هو حسب القانون يسوى
+
+223
+00:16:22,420 --> 00:16:27,780
+Zero بيصير X ناقص Y ناقص Z بيسوى Zero هذه اللي هي
+
+224
+00:16:27,780 --> 00:16:33,530
+ال tangent plane to the surface شايفين مثلها؟الان
+
+225
+00:16:33,530 --> 00:16:37,990
+مثلًا آخر الـ Surface f of x y و z بساوية x تربيع
+
+226
+00:16:37,990 --> 00:16:41,550
+z y تربيعه نقص اتنين بساوية سفر هذا عبارة عن الـ
+
+227
+00:16:41,550 --> 00:16:45,090
+Level Surface هذا بيعمل ليهاش Cylinder x تربيع z y
+
+228
+00:16:45,090 --> 00:16:48,650
+تربيعه نقص اتنين بساوية سفر الـ z ثابتة متغيرة
+
+229
+00:16:48,650 --> 00:16:53,110
+مطلوقة زي ما نقول وهذا دائرة فبتعمل ليهاش الـ
+
+230
+00:16:53,110 --> 00:16:58,950
+Circularالآن X زائد Y زائد Z نقص أربعة بيساوي 0 جي
+
+231
+00:16:58,950 --> 00:17:02,330
+و X و Y و Z هذا ال level surface بيعمل لي إيش
+
+232
+00:17:02,330 --> 00:17:06,830
+إبلاين، إيش اللي طلبنيها؟ بقوللي ده surface هاي و
+
+233
+00:17:06,830 --> 00:17:10,110
+هذا اللي هو الـ cylinder و ال surface هذا الإبلاين
+
+234
+00:17:10,110 --> 00:17:13,250
+بقوللي meet in an ellipse، اتقاطعن في إيش؟ اتقاطعن
+
+235
+00:17:13,250 --> 00:17:16,150
+في ellipse، هي ال ellipse اللي اتقاطعن فيها، ماشي
+
+236
+00:17:16,150 --> 00:17:20,010
+الحال؟بقول find parametric equations for the line
+
+237
+00:17:20,010 --> 00:17:25,650
+tangent to E at point 1 و3 و3 بقوللي أوجد معادلة
+
+238
+00:17:25,650 --> 00:17:31,850
+الـ Parametric equation لأ معادلة اللي هي مين الخط
+
+239
+00:17:31,850 --> 00:17:36,810
+المماس اللي بيمس هذه الـ Ellipse عنده النقطة
+
+240
+00:17:36,810 --> 00:17:41,280
+المذكورةأنا عشان أجد معادلة المماس أو معادلة خط
+
+241
+00:17:41,280 --> 00:17:45,760
+بالزمن شغلتين بالزمن اللي هو vector موازي له لخط
+
+242
+00:17:45,760 --> 00:17:49,900
+التماس وبالزمن نقطة النقطة المعطينيها طب ال vector
+
+243
+00:17:49,900 --> 00:17:54,830
+الموازي شوفوا كيف نجد ال vector الموازي أول حاجةلو
+
+244
+00:17:54,830 --> 00:18:00,050
+ألحظنا على هذه اللي هي الـ Ellipse الـ Ellipse هي
+
+245
+00:18:00,050 --> 00:18:03,790
+عبارة موجودة طبعاً هي عبارة عن تفاطع الـ Plane و
+
+246
+00:18:03,790 --> 00:18:09,890
+الـ Cylinder الآن الـ Ellipse هذه موجودة في
+
+247
+00:18:09,890 --> 00:18:12,810
+الجهتين يعني موجودة في الـ Cylinder و موجودة في
+
+248
+00:18:12,810 --> 00:18:17,650
+الـ Plane الآن الـ Gradient للـ Fعمودي على مين؟
+
+249
+00:18:17,650 --> 00:18:21,310
+على اللي هو الـCylinder إذا عمودي على اللي هو أي
+
+250
+00:18:21,310 --> 00:18:25,990
+منحنة فيها من ضمنها اللي هي الـEllipse إذا
+
+251
+00:18:25,990 --> 00:18:28,950
+الـGradient للـF عمودي على الـEllipse هاي واحد،
+
+252
+00:18:28,950 --> 00:18:32,570
+خلوه في الذاكرة إتنين، الـGradient للـG عمودي على
+
+253
+00:18:32,570 --> 00:18:37,370
+مين؟ على الـSurface هذا اللي هو G بالساوية 0 مدام
+
+254
+00:18:37,370 --> 00:18:41,430
+عمودي عليه إذا عمودي على كل كرفي من ضمنه المين؟
+
+255
+00:18:41,540 --> 00:18:46,020
+اللي هي الـ Ellipse إذا صار الـ Gradient للـ G
+
+256
+00:18:46,020 --> 00:18:48,980
+عمودي على الـ Ellipse عنده نقطة هذه والـ Gradient
+
+257
+00:18:48,980 --> 00:18:52,680
+لل F عمودي على الـ Ellipse عند النقطة هذه إذا أكيد
+
+258
+00:18:52,680 --> 00:18:57,440
+الـ Cross Product بينهم هيكون ماله عمودي على
+
+259
+00:18:57,440 --> 00:19:03,080
+التنين، مظبوط؟ يعني بمعنى آخر موازي للمماسلأن
+
+260
+00:19:03,080 --> 00:19:06,660
+المماس اللي هو هنا الـ Gradient للألف عمودي عليه
+
+261
+00:19:06,660 --> 00:19:09,240
+والـ Gradient للـ D برضه عمودي على المماس اللي
+
+262
+00:19:09,240 --> 00:19:13,360
+بدناها لأن المماس عنده النقطة هذه اللي هو الـ
+
+263
+00:19:13,360 --> 00:19:15,420
+Gradient عمودي على الـ Curve اللي هو عمودي على
+
+264
+00:19:15,420 --> 00:19:18,120
+المماس والـ Gradient للألف عمودي على الـ Curve
+
+265
+00:19:18,120 --> 00:19:21,260
+اللي هو عمودي على المماس إذن الـ Gradient للأف
+
+266
+00:19:21,260 --> 00:19:24,540
+والـ Gradient للـ D عمودي على المماس إذن اللي هو
+
+267
+00:19:24,540 --> 00:19:30,780
+الـ Vector الموازياللي هو للمماس اللي هو العمودي
+
+268
+00:19:30,780 --> 00:19:34,600
+على اللي هو الـ gradient للـ F والـ gradient للـ G
+
+269
+00:19:34,600 --> 00:19:39,740
+زي مين؟ زي المماس، إذا صار المماس معروف vector
+
+270
+00:19:39,740 --> 00:19:42,340
+موازيئله الـ gradient للـ F كرس الـ gradient للـ G
+
+271
+00:19:42,340 --> 00:19:46,060
+ومعروف نقطة، إذا بنقدر نعرف اللي هو معدلته، خلّيني
+
+272
+00:19:46,060 --> 00:19:49,000
+نعرف الـ gradient للـ F كرس الـ gradient للـ G الـ
+
+273
+00:19:49,000 --> 00:19:51,740
+tangent line is orthogonal to both الـ gradient
+
+274
+00:19:51,740 --> 00:19:56,020
+للـ F and الـ gradient للـ G at 1 و 1 و 3 زي ما
+
+275
+00:19:56,020 --> 00:20:00,510
+ذكرت قبل شويةوهكذا بدأت بالمعنى الـ vector الـ
+
+276
+00:20:00,510 --> 00:20:05,150
+gradient للـ F across الـ gradient للـ G إذا نجد
+
+277
+00:20:05,150 --> 00:20:07,730
+الـ gradient للـ F عنده نقطة واحد و تلاتة و تلاتة
+
+278
+00:20:07,730 --> 00:20:11,510
+الـ gradient للـ F هو اتنين X اتجاه الـ I زاد
+
+279
+00:20:11,510 --> 00:20:14,570
+اتنين Y في اتجاه الـ G احسبها عند الواحد والواحد
+
+280
+00:20:14,570 --> 00:20:19,970
+والتلاتة بتطلع اتنين I زاد اتنين G هذا الـ
+
+281
+00:20:19,970 --> 00:20:23,290
+gradient للـ F الـ gradient للـ G is similar للإن
+
+282
+00:20:23,290 --> 00:20:29,510
+التفاضل بالنسبة للـ X هو Iبتفضل بالنسبة لـ Y صفر
+
+283
+00:20:29,510 --> 00:20:34,370
+بالنسبة لـ Z بسوء واحد يعني K الـ C لـ Grid لـ G
+
+284
+00:20:34,370 --> 00:20:39,170
+بسوء I زائد K لأن صار معروف هذا صار معروف هذا لأن
+
+285
+00:20:39,170 --> 00:20:43,190
+توجد الـ cross product بينهم بينهم الـ Grid انتلاف
+
+286
+00:20:43,190 --> 00:20:47,190
+في cross ال Grid لـ G بتعرفوا كيف بحط IGK بعدين
+
+287
+00:20:47,190 --> 00:20:50,990
+عناصر الأول اتنين اتنين صفر هي اتنين اتنين صفر
+
+288
+00:20:50,990 --> 00:20:52,690
+وهذه 101
+
+289
+00:20:56,990 --> 00:21:00,730
+بنعمل مفبوك الـ Determinant شطب هذا و شطب هذا
+
+290
+00:21:00,730 --> 00:21:05,670
+بيصير 2 في اتجاه الـ I هي شطب هذا و شطب هذا ناقص
+
+291
+00:21:05,670 --> 00:21:11,030
+اللي هو 2 في اتجاه الـ J لأن هذا سفر اللي بتظلها و
+
+292
+00:21:11,030 --> 00:21:15,670
+شطب هذا و شطب هذا بيصير 2 اللي هي سفر يعني ناقص 2
+
+293
+00:21:15,670 --> 00:21:19,690
+في اتجاه الـ K وعندي هذا اللي هو الـ Gradient للـ
+
+294
+00:21:19,690 --> 00:21:23,520
+F cross ال Gradient للـ Gلأن عرفت اللي هو vector
+
+295
+00:21:23,520 --> 00:21:26,660
+موازي للخط وعرفت اللي هي نقطة على الخط إذا بعرف
+
+296
+00:21:26,660 --> 00:21:31,040
+الميال الخط X بتساوي اللي هي واحد زاد اتنين في T
+
+297
+00:21:31,040 --> 00:21:37,140
+اللي هو Y بتساوي واحد اللي هو ناقص اتنين في T و Z
+
+298
+00:21:37,140 --> 00:21:41,800
+بتساوي اللي هو تلاتة هي تلاتة ناقص اتنين في T
+
+299
+00:21:41,800 --> 00:21:46,060
+بيكون هو ده معادلة المماس اللي هي ال ellipse
+
+300
+00:21:46,060 --> 00:21:51,460
+التقاطع لل surfaceالان نجي لـ Estimating Exchange
+
+301
+00:21:51,460 --> 00:21:55,340
+in a Specific Direction اللي هو احنا زمان كنا نحكي
+
+302
+00:21:55,340 --> 00:22:01,640
+عن Estimating Exchange in general في الكلكولة
+
+303
+00:22:01,640 --> 00:22:05,480
+الصالف كانت اللي هي سهلة الـ DF كنا نقول بساوي
+
+304
+00:22:05,480 --> 00:22:12,240
+اللي هي F' في DSالان لأ احنا صار عنده او ا دي اف
+
+305
+00:22:12,240 --> 00:22:15,520
+باسامة اف برايم في دي واي زي ما قلنا او دي اكس حسب
+
+306
+00:22:15,520 --> 00:22:19,280
+اللي عندنا الان احنا صار function of two variables
+
+307
+00:22:19,280 --> 00:22:23,300
+فهيكون ال estimating the change in F in the
+
+308
+00:22:23,300 --> 00:22:26,940
+direction of mean of U شوف كيف to estimate the
+
+309
+00:22:26,940 --> 00:22:29,620
+change in the value of a differentiable function F
+
+310
+00:22:29,620 --> 00:22:33,120
+when we move at the small distance dS from point
+
+311
+00:22:33,120 --> 00:22:36,980
+B0 in a particular direction U use the formula اتة
+
+312
+00:22:36,980 --> 00:22:45,170
+دي الانبنقرب التغيير اللي بيصير اللي هو في الدالة
+
+313
+00:22:45,170 --> 00:22:50,130
+of a differentiable function F لما نتحرك بمقدار
+
+314
+00:22:50,130 --> 00:22:56,490
+مسافة DS من نقطة P0 في الـ direction لـU يعني الآن
+
+315
+00:22:56,490 --> 00:23:00,350
+عندي اللي هو لما اتحركت اللي هي من عند الـ P0
+
+316
+00:23:03,310 --> 00:23:07,610
+لنقطة أخرى في اتجاه مين على اتجاه الـ U صار في هذه
+
+317
+00:23:07,610 --> 00:23:11,850
+النقطة دي السمشينا صار هذا اللي هو فيه تغيير في
+
+318
+00:23:11,850 --> 00:23:16,090
+مين في الـ F التغيير في الـ F اللي هو تقريبي دلت
+
+319
+00:23:16,090 --> 00:23:20,480
+DFطبعاً اللي هو بالظبط الـ Δ F بساوي اللي هو بقاله
+
+320
+00:23:20,480 --> 00:23:23,360
+علاقة Δ X و Δ Y و مش عارف إيش و كده و كده زمان
+
+321
+00:23:23,360 --> 00:23:26,340
+كانت لما كانت بس function of one variable دلتة F
+
+322
+00:23:26,340 --> 00:23:32,080
+اللي بساوي اللي هو Y ناقص Y نوت أو دلتة Y بساوي Y
+
+323
+00:23:32,080 --> 00:23:34,960
+ماينس Y نوت و دلتة X بساوي X ماينس X نوت لكن الآن
+
+324
+00:23:34,960 --> 00:23:37,800
+صار عندي لأ في direction معين فبصير الآن الـ
+
+325
+00:23:37,800 --> 00:23:41,900
+differential دي F تقريبا تقريبا بساوي هذا المقدار
+
+326
+00:23:41,900 --> 00:23:46,610
+اللي هو directional derivativeمضروب في مين في الـ
+
+327
+00:23:46,610 --> 00:23:51,510
+DS الـ distance increment اللي اتحركناها فبكون هذا
+
+328
+00:23:51,510 --> 00:23:55,670
+مقدار التغير تقريبا هذا مقدار التغير طبعا فيه اللي
+
+329
+00:23:55,670 --> 00:23:59,890
+هو خطأ بحان عسبه بعد بشوية هذا قانونه المهم لأن
+
+330
+00:23:59,890 --> 00:24:05,070
+قانون اللي هو لو اتحركنادالة اللي هو اتحركنا في
+
+331
+00:24:05,070 --> 00:24:09,610
+اتجاه الـ U بمقدار DS صار فيه تغيير في الـ F مقدار
+
+332
+00:24:09,610 --> 00:24:12,570
+التغيير في الـ F تقريبا بساوية Directional
+
+333
+00:24:12,570 --> 00:24:16,750
+Derivative في هذا الاتجاه مضروبة في مقدار اللي هي
+
+334
+00:24:16,750 --> 00:24:21,530
+المسافة اللي مشيناها DS على اللي هو الـ U فبكون هو
+
+335
+00:24:21,530 --> 00:24:26,130
+مقدار التغير في الـ F وخلينا نشوف المثال التالي
+
+336
+00:24:27,340 --> 00:24:31,160
+نشوف التطبيق الأول على اللي هو الـ Example 4
+
+337
+00:24:31,160 --> 00:24:35,440
+Estimate how much the value of f of x, y, z بسواي
+
+338
+00:24:35,440 --> 00:24:40,400
+sin x زاد 2y, z اللي هي will change if the point B
+
+339
+00:24:40,400 --> 00:24:45,960
+moves point 1 units from B0 straight toward اللي
+
+340
+00:24:45,960 --> 00:24:51,520
+هو B1 اللي هو 2 و 2 و نقص 2 يعني بدنا نشوف .. نشوف
+
+341
+00:24:51,520 --> 00:24:59,060
+.. نقرب قداشة التغير في Fالتي تعمل تغيير الـ
+
+342
+00:24:59,060 --> 00:25:05,140
+Estimation هو الـ ADF لما النقطة
+
+343
+00:25:06,600 --> 00:25:11,260
+تتغير وتتحرك بمقدار point one من أين إلى أين تتحرك
+
+344
+00:25:11,260 --> 00:25:15,060
+من النقطة B Note في اتجاه مباشرة straightforward
+
+345
+00:25:15,060 --> 00:25:21,700
+في اتجاه B1 يعني B1 هو الـ direction اللي بدنا
+
+346
+00:25:21,700 --> 00:25:25,580
+نتحرك فيه ومقدار التحرك اللي هو الـ DS في هذه
+
+347
+00:25:25,580 --> 00:25:30,080
+الحالة بساوية point one إذن عشان أقدر أوجد أعمل
+
+348
+00:25:30,080 --> 00:25:34,200
+estimation للتغير في F اللي هو الـ DFبداوجد الـ
+
+349
+00:25:34,200 --> 00:25:37,680
+Gradient و بداوجد الـ U و بداوجد الـ DS DS هي
+
+350
+00:25:37,680 --> 00:25:42,600
+موجودة و الـ U هو عبارة عن الـ Direction اللي هو
+
+351
+00:25:42,600 --> 00:25:46,640
+من P0 ل P1 اللي هو بمعنى أخر الـ Unit Vector في
+
+352
+00:25:46,640 --> 00:25:53,810
+اتجاه P0 P1الآن اللي هي الـ Gradient اللي نجيبه هو
+
+353
+00:25:53,810 --> 00:26:01,130
+الـ Direction of this vector from 0, 1 و 0 إلى 2 و
+
+354
+00:26:01,130 --> 00:26:05,930
+2 و نقص 2 أول شيء اللي هو الـ vector بين هذه وهذه
+
+355
+00:26:05,930 --> 00:26:09,980
+اللي هو عبارة عن 2I2
+
+356
+00:26:21,150 --> 00:26:24,290
+الأن بدنا نوجد اللي هو الـ Direction له إذا على
+
+357
+00:26:24,290 --> 00:26:28,030
+مجموعته مجموعته هو الجدر التربيعي للأربعة زاد
+
+358
+00:26:28,030 --> 00:26:32,050
+الواحد زاد الأربعة اللي هي عبارة عن بيصير اتنين I
+
+359
+00:26:32,050 --> 00:26:35,690
+زاد J نقص اتنين K على تلاتة هذا اللي هو الـ
+
+360
+00:26:35,690 --> 00:26:40,270
+Direction U اللي احنا بدنا التغيير يصير في اتجاهه
+
+361
+00:26:40,640 --> 00:26:43,460
+الآن من الـ Gradient للـ F نيجي من جديد من الـ
+
+362
+00:26:43,460 --> 00:26:47,720
+Gradient للـ F نفعظ الدلالة اللي هو Fx Fx اللي هي
+
+363
+00:26:47,720 --> 00:26:51,000
+هاد بالنسبة لـ Fx بيصير Y Cos X في اتجاه الـ I
+
+364
+00:26:51,000 --> 00:26:56,340
+زائد Fy Fy ايش Fy في هذه الحالة اللي هي Sin X لأن
+
+365
+00:26:56,340 --> 00:27:00,620
+اتفاض الـ Y واحد Sin X زائد اللي هو اتفاضه الـ 2YZ
+
+366
+00:27:00,620 --> 00:27:05,060
+بالنسبة لـ Y اللي هو 2Zكله في اتجاه من الـ J زائد
+
+367
+00:27:05,060 --> 00:27:09,400
+اللي هي FZ FZ هذا صفر بنسبة لـ Z وهذا بالنسبة لـ Z
+
+368
+00:27:09,400 --> 00:27:13,760
+اللي هي 2Y فبتنزل 2YK هذه الـ Gradient الـ
+
+369
+00:27:13,760 --> 00:27:17,460
+Gradient بدنا يهيأ عند مين احنا عند اللي هو طبعا
+
+370
+00:27:17,460 --> 00:27:22,800
+التحرك من عند النقطة P0,1,0 إذا عنده نقطة مين؟ P0
+
+371
+00:27:22,800 --> 00:27:27,000
+,1,0 هي النقطة اللي من عندها صار التحرك اللي عندها
+
+372
+00:27:27,000 --> 00:27:32,050
+بنجيب الـ Gradientاللي هي بتساوي عوضه عن X بـ 0
+
+373
+00:27:32,050 --> 00:27:40,430
+بيصير الـ Y بـ 1 يعني I زائد صين الـ 00 زائد 2 زد
+
+374
+00:27:40,430 --> 00:27:44,650
+اللي هي 2 في صفر يعني بيصير في الـ J لأن هذا صفر و
+
+375
+00:27:44,650 --> 00:27:49,790
+هذا صفر زائد 2YK Y بـ 1 بيصير زائد 2K إذا صار عند
+
+376
+00:27:49,790 --> 00:27:54,280
+الـ Gradient للـ F عند النقطة 100لو ضربنا dot
+
+377
+00:27:54,280 --> 00:27:58,540
+product مع U بيصير عبارة عن I فاهمين ليش؟ لأنه
+
+378
+00:27:58,540 --> 00:28:02,280
+احنا بدنا نجد اللي هو DF نسميه الـ Gradient Dot
+
+379
+00:28:02,280 --> 00:28:06,220
+الـ U مضروف في مين؟ في الـ S ده صار عدد مضروف في
+
+380
+00:28:06,220 --> 00:28:08,960
+عدد او ليش نجيب الـ dot product هذا؟هذا هو الـ
+
+381
+00:28:08,960 --> 00:28:13,660
+Gradient لـ F وDot لـ U Gradient لـ F I زيادة 2K
+
+382
+00:28:13,660 --> 00:28:17,580
+وDot لـ U دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+383
+00:28:17,580 --> 00:28:19,080
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+384
+00:28:19,080 --> 00:28:19,420
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+385
+00:28:19,420 --> 00:28:19,980
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+386
+00:28:19,980 --> 00:28:21,100
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+387
+00:28:21,100 --> 00:28:25,860
+نجدها فوق دعونا نجدها فوق دعونا نجدها فوق دعونا
+
+388
+00:28:25,860 --> 00:28:32,560
+نجدها فوق دعونا نجدها فوقالأن الـ «تغيير DF في F
+
+389
+00:28:32,560 --> 00:28:38,020
+التي تنتج من المتحرك DS» يتساوي 0.1 معطينيا unit
+
+390
+00:28:38,020 --> 00:28:41,920
+away from 0 و 1 و 0 in the direction of U is
+
+391
+00:28:41,920 --> 00:28:45,520
+approximately DF بتساوي حسب القانون اللي فوق Delta
+
+392
+00:28:45,520 --> 00:28:49,460
+F ضد الـ U مضروبة في الـ DS هي هذا أوددناها عبارة
+
+393
+00:28:49,460 --> 00:28:53,380
+عن ناقص تلتين وهذا الـ DS عبارة عن 0.1 اتتين مع
+
+394
+00:28:53,380 --> 00:28:57,920
+بعض بيطلع اللي هو هذا المقداروهذا اللي هو الـ DF
+
+395
+00:28:57,920 --> 00:29:02,740
+اللي هو تقريب التغيير فيه أو التغيير اللي حدث في F
+
+396
+00:29:02,740 --> 00:29:09,580
+عندما تحركنا Point 1 في اتجاه من النقطة B Note إلى
+
+397
+00:29:09,580 --> 00:29:14,150
+النقطة B1الآن بدنا نجي على حاجة اسمها
+
+398
+00:29:14,150 --> 00:29:18,310
+Linearization Linearization طبعا اللي هو أو تقريب
+
+399
+00:29:18,310 --> 00:29:23,670
+الدالة اللي هو معادلة خطية أو حاجة بنسميها
+
+400
+00:29:23,670 --> 00:29:26,630
+Linearization اللي أخدناها في calculus الف هذه
+
+401
+00:29:26,630 --> 00:29:32,580
+تعميم اللي بس في calculus دالة متغيريةHow to
+
+402
+00:29:32,580 --> 00:29:36,520
+Linearize a Function of Two Variables كيف بدنا أن
+
+403
+00:29:36,520 --> 00:29:42,260
+نقرّب الدالة بالـ Line اللي جنبها كنا زمان نقرّب
+
+404
+00:29:42,260 --> 00:29:47,180
+الدالة بالمماس حول نقطة التماس لأن حول نقطة التماس
+
+405
+00:29:47,180 --> 00:29:52,300
+الفرق بين اللي هو الخط و بين ال curve جليل جدا
+
+406
+00:29:52,300 --> 00:29:55,900
+لإنه حول المماس هذا خدنا فكرة كلصة لافيش دي عينا
+
+407
+00:29:55,900 --> 00:29:58,750
+نعيدهإلا أن هذه الفكرة بنعملها function of two
+
+408
+00:29:58,750 --> 00:30:02,450
+variables عند لو .. لو جينا بدنا نعرف
+
+409
+00:30:02,450 --> 00:30:06,430
+linearization of a function f of x,y at a point x0
+
+410
+00:30:06,430 --> 00:30:10,210
+,y0 بدنا نعرف ال linearization لل function f of x
+
+411
+00:30:10,210 --> 00:30:14,650
+,y عنده النقطة x0,y0 متى؟ طبعا للدلة اللي بتكون
+
+412
+00:30:14,650 --> 00:30:18,570
+أفضل الـ differentialهو الـ function L of X Y
+
+413
+00:30:18,570 --> 00:30:23,110
+بتساوي F of X Note Y Note زاد F X of X Note Y Note
+
+414
+00:30:23,110 --> 00:30:27,430
+في X Minus X Note زاد F Y of X Note Y Note في Y
+
+415
+00:30:27,430 --> 00:30:31,330
+Minus Y Note الآن هذه الـ linearization
+
+416
+00:30:33,670 --> 00:30:37,150
+اللي في X و Y يعني هي عند النقطة X knot و Y knot
+
+417
+00:30:37,150 --> 00:30:42,150
+وهي النقطة X Y اللي أصار تغير فيها انتقلت لهان لأن
+
+418
+00:30:42,150 --> 00:30:44,810
+هذا المقدار هو عبارة عن Delta X اللي هو X minus X
+
+419
+00:30:44,810 --> 00:30:49,030
+knot وهذا اللي هو Delta Y Y minus Y knot الان اللي
+
+420
+00:30:49,030 --> 00:30:53,290
+هي ال function is differential لهان فهذه النقطة ال
+
+421
+00:30:53,290 --> 00:30:57,790
+X و Y حوالين النقطة X knot و Y knotكل ما اقتربت
+
+422
+00:30:57,790 --> 00:31:02,510
+الـ X من الـ X0 وY0 كل ما كان فيه إمكانية أن نقول
+
+423
+00:31:02,510 --> 00:31:07,010
+الـ Linearization هذا اللي عرفناها تقريبا بساومين
+
+424
+00:31:07,010 --> 00:31:11,790
+F of X أو Y لأنه لما نقترب منه الـ X minus X0
+
+425
+00:31:11,790 --> 00:31:16,410
+بتصغر والـ Y minus Y0 بتصغر لما نقترب كتير بسير
+
+426
+00:31:16,410 --> 00:31:21,050
+حوالين النقطة هذهالخط اللي هو الـ Linearization
+
+427
+00:31:21,050 --> 00:31:25,950
+هذا والـ Function الأصلية بيسير تقريباً جات بعض
+
+428
+00:31:25,950 --> 00:31:30,330
+يعني ألّف XY بنقدر اللي هو الـ Function الدالة
+
+429
+00:31:30,330 --> 00:31:36,080
+اللي ممكن تكون معقدة انقربها بواسطة خطالخط اللي هو
+
+430
+00:31:36,080 --> 00:31:40,020
+عبارة عن هي دالة خطية اللي هي في X بس أس واحد و Y
+
+431
+00:31:40,020 --> 00:31:44,940
+أس واحد فصارت اللي هي جربنا الدالة الـ F of X knot
+
+432
+00:31:44,940 --> 00:31:50,100
+و Y knot أو F of XY تقريبا بساوة L of XY for
+
+433
+00:31:50,100 --> 00:31:57,180
+points XY that are nearbyأو very close to x0 و y0
+
+434
+00:31:57,180 --> 00:32:01,260
+لأنه بيصير هذه صغيرة جدا وهذا صغير جدا فبتصير أقله
+
+435
+00:32:01,260 --> 00:32:05,440
+في x و y تقريبا بتساوي f of x و y في حالة اللي هي
+
+436
+00:32:05,440 --> 00:32:11,160
+x و y جريبة جدا من x0 و y0 هذا بنسمي is the
+
+437
+00:32:11,160 --> 00:32:15,600
+standard linear approximation of f at x0 و y0 طبعا
+
+438
+00:32:15,600 --> 00:32:19,800
+أكيد في التقريب هذا في خطأ في هذا الخطأ احنا بدنا
+
+439
+00:32:19,800 --> 00:32:23,790
+نحاول نحسبه بعد شويةأولا نشوف كيف يوجد الـ
+
+440
+00:32:23,790 --> 00:32:28,110
+Linearization Find the linearization of f of xy
+
+441
+00:32:28,110 --> 00:32:32,470
+بساوي x تربيع ناقص xy زائد نص y تربيع زائد تلاتة
+
+442
+00:32:32,470 --> 00:32:36,470
+at the point من تلاتة و اتنين سنشوف ان وجد
+
+443
+00:32:36,470 --> 00:32:39,850
+Linearization للـ Function اللي أمامناما هو
+
+444
+00:32:39,850 --> 00:32:43,330
+التطبيق على القانون اللي قبل بشوية الـ Fx ايش
+
+445
+00:32:43,330 --> 00:32:46,590
+هتساوي الـ Fx لأن الـ Linearization هي قانونها
+
+446
+00:32:46,590 --> 00:32:51,350
+قلنا قبل بشوية هي قانونها L بتساوي F نحسبها عندنا
+
+447
+00:32:51,350 --> 00:32:54,470
+النقطة المطلوبة Fx عندنا النقطة المطلوبة في X نقص
+
+448
+00:32:54,470 --> 00:32:57,710
+X not of Y برضه عندنا النقطة المطلوبة في Y minus Y
+
+449
+00:32:57,710 --> 00:33:01,310
+not هذا هو Linearization خلينا الآن نطبق على مثلنا
+
+450
+00:33:01,830 --> 00:33:06,590
+عند الـ Fx إيش بيساوي 2x ناقص Y فاهمين عليها Fy
+
+451
+00:33:06,590 --> 00:33:10,370
+إيش بيساوي اللي هو ده سفر اللي هي بيصير ناقص X
+
+452
+00:33:10,370 --> 00:33:14,570
+معايا زائد 2 في نص اللي هي بيصير اللي هو زائد Y
+
+453
+00:33:14,570 --> 00:33:18,790
+هذا Fy لما تحسب هذا عند النقطة 3 ع 2 وهذا 3 و 2
+
+454
+00:33:18,790 --> 00:33:22,530
+وهذا عند النقطة 3 و 2 الأولى بتصير اللي هي 2 في 3
+
+455
+00:33:22,530 --> 00:33:27,330
+ب6 ناقص 2 اللي هي 4 التامي Fy عند الـ 3 و 2 اللي
+
+456
+00:33:27,330 --> 00:33:33,780
+هي ناقص 3 مكان Xمكان نقص X وY مكانها 2 بصير نقص 3
+
+457
+00:33:33,780 --> 00:33:37,340
+زي 2 بساوي نقص 1 الآن أيضًا ماذا بنلزمنا في
+
+458
+00:33:37,340 --> 00:33:43,420
+القانون F عند 3 و 2 F عند 3 و 2 بصير 9 نقص 3 في 2
+
+459
+00:33:43,420 --> 00:33:50,070
+ب6 زي نص اللي هو في 2 في 2 ب4 لأن Y ب2اللي بيصير
+
+460
+00:33:50,070 --> 00:33:55,210
+عنده 2 التي زائد 3 والمجموع هذا بيساوي 8 إذا حصلنا
+
+461
+00:33:55,210 --> 00:33:59,830
+Fx و Fy و F عنده نقطة 3 و 2 إذا بنعوض على L و X Y
+
+462
+00:33:59,830 --> 00:34:03,770
+بيساوي F في 3 ع 2 زادة Fx في X minus 3 زادة Fy في
+
+463
+00:34:03,770 --> 00:34:08,610
+Y minus 2 و Y ساوي اللي هو التالي F في 3 ع 2
+
+464
+00:34:08,610 --> 00:34:13,990
+تمانية هذه Fx في 3 ع 2 اللي هي 4 مضروبة في X ناقص
+
+465
+00:34:13,990 --> 00:34:15,670
+اللي هي X note اللي هي 3
+
+466
+00:34:23,600 --> 00:34:31,680
+Y-Y-Y-Y-Y-Y-Y-Y-Y-Y-Y
+
+467
+00:34:31,890 --> 00:34:35,630
+وهي نقطة
+
+468
+00:34:35,630 --> 00:34:41,750
+3 و2 في الـ X Y بساوية 4 X ناقص Y ناقص 2 يمكن أن
+
+469
+00:34:41,750 --> 00:34:45,270
+يكون هكذا حلنا أو جدنا الـ Linearization زي ما
+
+470
+00:34:45,270 --> 00:34:49,390
+قلنا ان الـ Linearization في الآخر L of X Y ليس
+
+471
+00:34:49,390 --> 00:34:53,430
+بالظبط بساوية F of X Y بساوية تقريبا 4 points
+
+472
+00:34:53,430 --> 00:34:59,110
+close to X Y ليه 3 X نقطة Y نقطة 3 و2 طب في خطأ
+
+473
+00:34:59,110 --> 00:35:04,000
+مقدار الخطأ بتحسبه نظرية التاليةالخطأ اللي نقرأه
+
+474
+00:35:04,000 --> 00:35:06,900
+الـ Approximation هو F of XY بيساوي تقريبًا L of
+
+475
+00:35:06,900 --> 00:35:12,320
+XY بحسب التالي نشوف الخطأ فيه يعني الخطأ اللي
+
+476
+00:35:12,320 --> 00:35:17,520
+نسميه إيه هو عبارة عن الفرق بين اللي هو F of XY و
+
+477
+00:35:17,520 --> 00:35:22,200
+L of XY الفرق بينهم اللي هو عبارة عن أكتر أكتر فرق
+
+478
+00:35:22,200 --> 00:35:26,500
+له حاجة اسمها الـ Error و هنشوف قداش قيمته نشوف
+
+479
+00:35:26,500 --> 00:35:27,360
+أيش النظرية بتقول
+
+480
+00:35:34,050 --> 00:35:39,370
+لو فرضنا أن F و Fxx و Fyy و Fx و Fy كل هن دولة
+
+481
+00:35:39,370 --> 00:35:42,430
+اللي هو الـ first و second partial derivatives
+
+482
+00:35:42,430 --> 00:35:46,830
+المظبوط أن يكون in continuous عنده نقطة x0 و y0 في
+
+483
+00:35:46,830 --> 00:35:51,370
+open set as if حوالين نقطة x0 و y0يعني فرضنا إن
+
+484
+00:35:51,370 --> 00:35:55,810
+الـ F و الـ partial derivatives كلهم continuous في
+
+485
+00:35:55,810 --> 00:36:00,870
+rectangle حوالين الـ X0 و Y0 ومنفترض أن M إيش هي
+
+486
+00:36:00,870 --> 00:36:05,230
+is any upper bound for the values of يعني هي عبارة
+
+487
+00:36:05,230 --> 00:36:11,370
+عن حد أعلى لقيمة الـ Fxx والـ Fyy والـ Fxy وين على
+
+488
+00:36:11,370 --> 00:36:15,230
+ال rectangle R اللي بنحكي عليه then ال error
+
+489
+00:36:16,660 --> 00:36:20,280
+Incured in replacing f of x,y on R by its
+
+490
+00:36:20,280 --> 00:36:25,540
+linearization f of x,y يعني بنقول f of x,y و f of
+
+491
+00:36:25,540 --> 00:36:29,620
+x,y الفرق بينهم في داخل الـ rectangle هذا اللي
+
+492
+00:36:29,620 --> 00:36:35,960
+حكينا عليه هيكون الفرق بينهم لن يتعدى لن يتعدى الـ
+
+493
+00:36:35,960 --> 00:36:40,520
+absolute value لـ x,y أصغر أو سوى نص الـ M في x
+
+494
+00:36:40,520 --> 00:36:45,780
+minus x0 زي y minus y0 الكل إيش تربيةأذا عملية
+
+495
+00:36:45,780 --> 00:36:51,820
+تقريب L of X Y بتساوي
+
+496
+00:36:51,820 --> 00:36:59,130
+تقريبًا F of X Y في المنطقة Rبتكون تقريباً بتسويها
+
+497
+00:36:59,130 --> 00:37:06,410
+لكن في Error الـ L of X Y ناقص F of X Y لن يتعدى
+
+498
+00:37:06,410 --> 00:37:10,270
+لوسالب أو موجة بهذه القيمة اللي هي أياش الـ Error
+
+499
+00:37:10,270 --> 00:37:14,710
+أصغر أو يساوي نص الـ M الـ M يعني الـ Upper Bound
+
+500
+00:37:14,710 --> 00:37:22,650
+لهذه الـ FXX و FYY و FXYفي X – X¹ زي Y – Y¹ كل
+
+501
+00:37:22,650 --> 00:37:26,110
+تربيع وحتى تتضح هذه الصورة أكتر خلّينا ناخد
+
+502
+00:37:26,110 --> 00:37:31,270
+example example find an upper bound for the error
+
+503
+00:37:31,270 --> 00:37:34,530
+in the approximation of X Y بساوي ألف X Y in
+
+504
+00:37:34,530 --> 00:37:38,750
+example خمسة خمسة over the rectangle كده بقوللي
+
+505
+00:37:38,750 --> 00:37:41,330
+أنت اللي جابل بشوية ال example الخمسة اللي جابل
+
+506
+00:37:41,330 --> 00:37:44,210
+بشوية هذا او جنانه ال linearization هي
+
+507
+00:37:44,210 --> 00:37:47,270
+linearizationإلا أن هذا الـ Linearization اللي
+
+508
+00:37:47,270 --> 00:37:54,390
+أوجدنا تقريبًا هذا هذا بتساوي هذه الدالة وبين في
+
+509
+00:37:54,390 --> 00:37:57,650
+حوالين التلاتة واتنين كل ما بعدنا عن التلاتة
+
+510
+00:37:57,650 --> 00:38:01,860
+واتنين بيصير اللي هو الفرق بين هذا وهذا كبيرالان
+
+511
+00:38:01,860 --> 00:38:06,160
+بدأ أشوف لك جديش اللي هو الـ Error بين ال .. الحس
+
+512
+00:38:06,160 --> 00:38:09,960
+انه .. انه XR بيبقى نقص XY زيادة نص Y تربيع زيادة
+
+513
+00:38:09,960 --> 00:38:15,380
+تلاتة بتساوي 4X نقص Y نقص 2 تقريبا في المنطقة
+
+514
+00:38:15,380 --> 00:38:19,040
+حوالين التلاتة اتنين تلاتة و اتنين حوالين النقطة
+
+515
+00:38:19,040 --> 00:38:21,700
+تلاتة و اتنين بدأ أشوف لك الآن الفرق بين هذه وهذه
+
+516
+00:38:21,700 --> 00:38:27,440
+يعني عملية التقريب هذه بهذه جديش تقريبا الـ Error
+
+517
+00:38:27,440 --> 00:38:32,520
+فيها اللي بده يقوليلنا إياهاالتي هي المثال التالي
+
+518
+00:38:32,520 --> 00:38:36,580
+وخلّينا نستذكر هذه لو اوددنا fx فسخلّيني استخدم
+
+519
+00:38:36,580 --> 00:38:41,600
+fxx ايش هتساوي في هذه الحالة اتنين fyy ايش هتساوي
+
+520
+00:38:41,600 --> 00:38:48,920
+واحدمظبوط فكس واي ايش هتساوي ناقص واحد ناشي خلينا
+
+521
+00:38:48,920 --> 00:38:51,920
+نحفظهم و نستخدمهم بعد شوية لان بدنا نلزمنا في
+
+522
+00:38:51,920 --> 00:38:56,280
+السؤال الان الان بيقولي find an upper bound for
+
+523
+00:38:56,280 --> 00:38:59,420
+the error in the approximation of of xy بيساوي
+
+524
+00:38:59,420 --> 00:39:03,040
+تقريبا الوفيكس واي في خطأ الخطأ قلنا طبعا بده
+
+525
+00:39:03,040 --> 00:39:07,620
+ياشمله على ال rectangle R X minus ثلاثة أصغر من
+
+526
+00:39:07,620 --> 00:39:10,480
+point one يعني ما نبعلش عن التلاتة ب point one هي
+
+527
+00:39:10,480 --> 00:39:13,190
+نقطة تلاتة و اتنين اللي احنا حكينا عنهااللي
+
+528
+00:39:13,190 --> 00:39:16,710
+حواليها بدنا نقل إيه؟ عشان نعمل اللي هي دي يعني في
+
+529
+00:39:16,710 --> 00:39:20,970
+داخل المستطيل هذا اللي هو X نقص تلاتة من X هيك
+
+530
+00:39:20,970 --> 00:39:25,630
+point one يعني من اتنين تسعة من عشرة لتلاتة واحد
+
+531
+00:39:25,630 --> 00:39:31,650
+من عشرة وتحت من اللي هي Y اللي هي واحد وتسعة من
+
+532
+00:39:31,650 --> 00:39:34,470
+عشرة وتحت اتنين واحد من عشرة في داخل ال rectangle
+
+533
+00:39:34,470 --> 00:39:40,450
+هذا اللي بتكون من هذه وهذهبدنا نجد اللي هو الخطأ
+
+534
+00:39:40,450 --> 00:39:43,750
+في الحساب في الـ approximation اللي هو f of x y
+
+535
+00:39:43,750 --> 00:39:47,390
+بسواقله f of x y إيش طالب لي express the upper
+
+536
+00:39:47,390 --> 00:39:51,770
+bound as a percentage of f of تلاتة ع اتنين بقول
+
+537
+00:39:51,770 --> 00:39:56,110
+لي اللي هو وجد لي نسبة الخطأ بالنسبة لقيم اتنين
+
+538
+00:39:56,110 --> 00:39:59,750
+اتدالة عن تلاتة و اتنين يعني بدك تحسب الخطأ أول
+
+539
+00:39:59,750 --> 00:40:04,870
+إشياللي هو بدك توجد الـ E اللي قبل بشوية و تحسب ال
+
+540
+00:40:04,870 --> 00:40:08,330
+percentage لل error بالنسبة لل F of 3 ع 2 The
+
+541
+00:40:08,330 --> 00:40:11,670
+value of F centered at the rectangle يعني F 3 ع 2
+
+542
+00:40:11,670 --> 00:40:17,170
+هي قيمة F اللي هو مركز ال rectangle اللي بتكون منه
+
+543
+00:40:17,170 --> 00:40:21,060
+هذا اللي هو ال rectangle اللي حكيته قبل بشويةالأن
+
+544
+00:40:21,060 --> 00:40:26,640
+نجد الـ Error نستخدم الـ Inquality A of XY أزرع من
+
+545
+00:40:26,640 --> 00:40:31,360
+النص M في هذا المقدار هذا القانون الان M إيش هي؟ M
+
+546
+00:40:31,360 --> 00:40:35,220
+هي عبارة عن .. قولنا إيه؟ هي Upper Bound لمن؟ لـ
+
+547
+00:40:35,220 --> 00:40:40,800
+FXX و FXY و FYY من القانون هذا الان نجد هذول FXX و
+
+548
+00:40:40,800 --> 00:40:45,480
+FXY أو جدلكم يا انت قبل شوية FXX طلعت قداش 2 لـ
+
+549
+00:40:45,480 --> 00:40:48,240
+Absolute Value لها 2 FXY طلعت ناقص 1 لـ Absolute
+
+550
+00:40:48,240 --> 00:40:51,960
+Value لها 1أفواي واي وجدناها واحد طلعت الـ
+
+551
+00:40:51,960 --> 00:40:55,300
+Absolute لها واحد إذا الأن الـ Upper bound لها دول
+
+552
+00:40:55,300 --> 00:40:58,940
+يعني أكبر واحد يعني الأكبر حد اللي هنا دولة بين
+
+553
+00:40:58,940 --> 00:41:02,140
+الاتنين والواحد والواحد أكيد نين هو اتنين إذا ناخد
+
+554
+00:41:02,140 --> 00:41:05,960
+الـ M واحنا مرتحين بتساوي اتنين إذا صار عندي اللي
+
+555
+00:41:05,960 --> 00:41:09,440
+هو الـ Error هذا أظهر وساوي نص في اتنين اللي هي
+
+556
+00:41:09,440 --> 00:41:13,600
+الـ M في X minus ثلاثة اللي هي X0 زائد Y minus
+
+557
+00:41:13,600 --> 00:41:16,740
+اتنين اللي هي Y0 هذا الكل تربيح هذا القانون طبعا
+
+558
+00:41:17,210 --> 00:41:20,670
+الان X-3 ماعطينيها دايما إيه شمالها أصغر أو يساوي
+
+559
+00:41:20,670 --> 00:41:25,670
+0.1 داخل المستطيل تبعنا اللي هي زائد Y-Y2 اللي هي
+
+560
+00:41:25,670 --> 00:41:28,730
+أصغر من 0.1 اللي هو داخل المستطيل تبعنا إذا هذه
+
+561
+00:41:28,730 --> 00:41:31,970
+بنستبدلها أصغر أو يساوي اللي هو بيصير ال error
+
+562
+00:41:31,970 --> 00:41:36,590
+أصغر أو يساوي نصف 2 اللي هي بواحدماشي اللي هي ده
+
+563
+00:41:36,590 --> 00:41:41,330
+صارت ا اللي هي point one الكل زائد point one الكل
+
+564
+00:41:41,330 --> 00:41:43,850
+ايه اشماله تربيع لإن هو اصغر يساوي point one او
+
+565
+00:41:43,850 --> 00:41:46,490
+اصغر يساوي point one لذا صار اصغر يساوي point one
+
+566
+00:41:46,490 --> 00:41:51,190
+تربيع اللي هو point four لذا صار ال error لا يتعدل
+
+567
+00:41:51,190 --> 00:41:57,970
+point fourالأن الـ Percentage of F of 3 ع 2 بساوي
+
+568
+00:41:57,970 --> 00:42:03,150
+طبعاً اللي هي نسبة الخطأ بالنسبة لمين F of 3 ع 2 F
+
+569
+00:42:03,150 --> 00:42:05,990
+of 3 ع 2 إيش حطلعت معانا قبل بشوية أو جدناها في
+
+570
+00:42:05,990 --> 00:42:09,250
+السوية اللي قبل تمانية The error is no greater
+
+571
+00:42:09,250 --> 00:42:12,770
+than يعني هيكون ال error نسبة ال percentage في ال
+
+572
+00:42:12,770 --> 00:42:20,460
+error لن تتعدى هتكون أصغر أو ساوي.04 على تمانية
+
+573
+00:42:20,460 --> 00:42:27,220
+مضروبة 100% بيطلع بوينت خمسة من مائة إذا هذه نسبة
+
+574
+00:42:27,220 --> 00:42:34,200
+الخطأ هذا هو الخطأ الفرق بين f of x,y و f of x,y
+
+575
+00:42:34,200 --> 00:42:37,920
+ان شاء الله تكون متاحة الأمورالآن اللي حكيناه عن
+
+576
+00:42:37,920 --> 00:42:40,560
+الـ functions of two variables ده نحكي عن الـ
+
+577
+00:42:40,560 --> 00:42:44,280
+functions of more than two variables يعني of three
+
+578
+00:42:44,280 --> 00:42:47,780
+variables أو غيره Analogous result hold for
+
+579
+00:42:47,780 --> 00:42:50,760
+differentiable functions for more than two
+
+580
+00:42:50,760 --> 00:42:54,200
+variables يعني الآن في عندنا اللي هي شغلات مشابهة
+
+581
+00:42:54,200 --> 00:42:58,860
+هتحدث لـ function of three variables خليني نشوف
+
+582
+00:42:58,860 --> 00:43:01,520
+اللي هو linearization الـ linearization of f of x,
+
+583
+00:43:01,640 --> 00:43:07,520
+y, z at a point b0, x0, y0, z0 isيعني الآن الدالة
+
+584
+00:43:07,520 --> 00:43:12,460
+اللي عندي F of X وY وZ بدي أقربها بدالة خطية اسمها
+
+585
+00:43:12,460 --> 00:43:16,960
+L of X وY وZ طبعاً أقربها بدالة خطية حول النقطة X0
+
+586
+00:43:16,960 --> 00:43:20,720
+أو Y0 أو Z0 طبعاً في تقريبات أخرى بدواي اللي يمكن
+
+587
+00:43:20,720 --> 00:43:23,340
+تكون .. بعض الدواي اللي تكون تقريبة بدالة خطية
+
+588
+00:43:23,340 --> 00:43:27,600
+صعبةيمكن أن تكون الـ Quadratic Approximation أسهل
+
+589
+00:43:27,600 --> 00:43:31,020
+و كذا في طبعا طرق في ال .. ال .. ال .. اللي هي
+
+590
+00:43:31,020 --> 00:43:34,140
+numerical analysis بتاخدوها كتير بس احنا بناخد هنا
+
+591
+00:43:34,140 --> 00:43:37,160
+اللي هي linearization تقريب اللي هو الدالة بدالة
+
+592
+00:43:37,160 --> 00:43:41,120
+خطية حول النقطة X Note و Y Note و Z Note اللي قاله
+
+593
+00:43:41,120 --> 00:43:44,060
+في X و Y و Z زي اللي جابل بالظبط قيمة الـ F عند
+
+594
+00:43:44,060 --> 00:43:48,070
+النقطة المذكورةزائد fx عنده النقطة المذكورة في x
+
+595
+00:43:48,070 --> 00:43:52,550
+-x0 زائد fy عنده النقطة المذكورة في y-y0 زائد fz
+
+596
+00:43:52,550 --> 00:43:56,670
+عنده النقطة المذكورة في z-z0 زي الأول بالظبط بس اي
+
+597
+00:43:56,670 --> 00:44:01,570
+شماله بس هذا على اللي هو ال function of three
+
+598
+00:44:01,570 --> 00:44:07,430
+variable باقي الحديث هيكون بنفس الأسلوبالنفس اللي
+
+599
+00:44:07,430 --> 00:44:11,950
+حكينا على الـ function of two variables نفترض أن R
+
+600
+00:44:11,950 --> 00:44:15,150
+is a closed rectangle solid centered at P0 and
+
+601
+00:44:15,150 --> 00:44:18,010
+lying in an open region on which the second
+
+602
+00:44:18,010 --> 00:44:21,270
+partial derivatives of R continuous suppose also
+
+603
+00:44:21,270 --> 00:44:26,890
+that هدولة كلهم Fxx, Fyy, Fzz صاروا في متغير تالت
+
+604
+00:44:26,890 --> 00:44:32,270
+Z, X, Y, Xz and Fyz are all less than or equal to
+
+605
+00:44:32,270 --> 00:44:35,370
+M يعني الآن بناخد ال upper bound هدول through R
+
+606
+00:44:35,600 --> 00:44:40,780
+والخطأ هو كما قلنا الفرق بين هذه وهذه ان الـ
+
+607
+00:44:40,780 --> 00:44:44,180
+approximation of F by L is bounded through R by
+
+608
+00:44:44,180 --> 00:44:47,940
+the inequality يعني الآن الخطأ لن يتعدى اللي هو
+
+609
+00:44:47,940 --> 00:44:51,160
+سيكون أزرع بسوء نص ال M في ال X minus X node زي
+
+610
+00:44:51,160 --> 00:44:55,180
+هذه نفسها نفس القانون بس ضفنا لهمين Z minus Z node
+
+611
+00:44:55,180 --> 00:45:00,560
+طبعا كل تربيحة ده اللي هو L of X Y و Z بسوء F of X
+
+612
+00:45:00,560 --> 00:45:03,860
+Y و Z في داخل ال rectangle طبعا ال rectangle صارها
+
+613
+00:45:03,860 --> 00:45:08,610
+الآن عبارة عن صندوقاللي هو الـ X minus X naught
+
+614
+00:45:08,610 --> 00:45:11,230
+أصغر من something Y minus Y naught أصغر من
+
+615
+00:45:11,230 --> 00:45:14,070
+something Z minus Z naught أصغر من something بحيث
+
+616
+00:45:14,070 --> 00:45:18,570
+أن الـ M الـ Upper bound تبعه بين هذه الـ M هي نص
+
+617
+00:45:18,570 --> 00:45:24,690
+M حوالين داخل هذه المنطقة بتكون تقريب الـ F بالـ L
+
+618
+00:45:24,690 --> 00:45:29,570
+لن يكون أكثر من هذا المقدار اللي هو الـ Error
+
+619
+00:45:30,050 --> 00:45:32,310
+العنوان if the second partial derivatives of f are
+
+620
+00:45:32,310 --> 00:45:36,150
+continuous and if x y و z it change from x note y
+
+621
+00:45:36,150 --> 00:45:40,070
+note by small amount dx dy dz يعني لو كانت اللي هي
+
+622
+00:45:40,070 --> 00:45:46,170
+x تغيرها dx و y تغيرت بمقدار dy و z بمقدار dz
+
+623
+00:45:46,170 --> 00:45:48,490
+بنقول عن حاجة اسمها الـ differential total
+
+624
+00:45:48,490 --> 00:45:51,970
+differential اللي مش هنحكي عنها كتير بس للمعرفة df
+
+625
+00:45:51,970 --> 00:45:58,360
+بتساوي fx في dx زي fy في dy زي fz في dzGives a
+
+626
+00:45:58,360 --> 00:46:01,200
+good approximation of the resulting change enough
+
+627
+00:46:01,200 --> 00:46:06,060
+هذا اللي بتقرّب اللي هو مين التغير في مين فقف احنا
+
+628
+00:46:06,060 --> 00:46:09,880
+قبل شوية أخدنا التغير فقف لما نكون وين اللي هو
+
+629
+00:46:09,880 --> 00:46:13,700
+اتجاه التغير في اتجاه معين الآن لو كان اللي هو بس
+
+630
+00:46:13,700 --> 00:46:19,250
+تغير في X و X naught و Y Y naught و Z naughtاللي
+
+631
+00:46:19,250 --> 00:46:23,730
+هو بمقدار DX, DY, DZ يعني مش تغير في الـ X والـ Y
+
+632
+00:46:23,730 --> 00:46:28,110
+والـ Z معروف تغير كل واحدة بيكون الـ DF بهذا
+
+633
+00:46:28,110 --> 00:46:34,310
+الأسلوب طيب الان نيجي الان لمثال عملي لهو حديثنا
+
+634
+00:46:34,310 --> 00:46:38,230
+هذا انشوف اللي هو المثال عشرة find a linearization
+
+635
+00:46:38,230 --> 00:46:43,750
+L of XYZ of اللي هو المقدار هذا at the point 2 1 0
+
+636
+00:46:43,750 --> 00:46:47,640
+و بعدين بيقولي find an upper boundبالنسبة للخطأ
+
+637
+00:46:47,640 --> 00:46:52,380
+المطلوب في تغيير الـ F by L لما نقرب الـ F بالـ
+
+638
+00:46:52,380 --> 00:46:56,160
+Linearization تبعنا اللي هو على الـ Rectangle هذه
+
+639
+00:46:56,160 --> 00:47:00,560
+نشوف قداشي الـ Error في هذه الحالة خلّينا نشوف أول
+
+640
+00:47:00,560 --> 00:47:07,080
+شيء نطبق القانون أكيد بالزمنة Fx اللي هي 2X-Y Fy
+
+641
+00:47:07,080 --> 00:47:12,100
+هي عبارة عن ناقص X Fz هي دولة طبعا سفر بتبقى عبارة
+
+642
+00:47:12,100 --> 00:47:19,220
+عن تلاتة Cos Z إذا صار عندي Fx عند النقطة 2 1 0
+
+643
+00:47:19,220 --> 00:47:24,780
+بنحسبها Fx عند النقطة 2 1 0 اللي هي بساوي تلاتة
+
+644
+00:47:24,780 --> 00:47:30,950
+لأنه 4 ناقص 1 تلاتة Fy عند النقطة 2 1 0 عوضاللي هي
+
+645
+00:47:30,950 --> 00:47:35,650
+مكان الـ X بالـ 2 بيصير نقص 2 FZ اللي هي عبارة عن
+
+646
+00:47:35,650 --> 00:47:41,370
+3 لأن Cos Zero عبارة عن Zero إذا تلاتة و F عند 2 و
+
+647
+00:47:41,370 --> 00:47:47,130
+1 و 0 F عند 2 و 1 و 0 هذا أربعة ناقص 2 بيطلع 2
+
+648
+00:47:47,130 --> 00:47:53,820
+زائد 0 اللي هي بيطلع جديش اتنينالأن هاي أوجدنا
+
+649
+00:47:53,820 --> 00:47:57,920
+اللي هي كل المتطلبات Fx و Fy و Fz عند النقاط
+
+650
+00:47:57,920 --> 00:48:01,280
+المذكورة والـ F كمان إذن الـ Linearization حسب
+
+651
+00:48:01,280 --> 00:48:05,480
+القانون قالوا في X و Y و Z بساوي قيمة الـ F Fx في
+
+652
+00:48:05,480 --> 00:48:10,400
+X minus X note اللي هي اتنين اللي هي Fy عنده نقطة
+
+653
+00:48:10,400 --> 00:48:15,300
+هادة في Y minus Y note واحد زائد Fz في Z minus Z
+
+654
+00:48:15,300 --> 00:48:18,640
+note اللي هي سفر هاي عند هدول بنعوض القيامة اللي
+
+655
+00:48:18,640 --> 00:48:22,870
+أوجدناهااللي هي F الأولى عبارة عن اتنين هيها زائد
+
+656
+00:48:22,870 --> 00:48:26,750
+F X اللي هي تلاتة في X minus اتنين و F Y نقص اتنين
+
+657
+00:48:26,750 --> 00:48:31,630
+و F Z تلاتة نضربهم و نبسطهم بيطلع عندى هذا اللي هو
+
+658
+00:48:31,630 --> 00:48:34,610
+الـ Linearization اللي هون ده Linearization of F
+
+659
+00:48:34,610 --> 00:48:38,810
+at اتنين و واحد و Zero is بالظبط هذا المقدار اللي
+
+660
+00:48:38,810 --> 00:48:42,670
+طلع أمامنا اللي هون هذا المقدار احنا بنقول بساوي
+
+661
+00:48:42,670 --> 00:48:46,550
+تقريبا هذا المقدار بساوي تقريبا هذا المقدار في
+
+662
+00:48:46,550 --> 00:48:51,950
+المنطقة المذكورة اللي عندهانلكن هناك خطأ نحسب نسبة
+
+663
+00:48:51,950 --> 00:48:57,270
+الخطأ بتقريب الـ F إيه بهذه انتوا جربوا اعطوا قيم
+
+664
+00:48:57,270 --> 00:49:02,650
+لـ X و Y و Z هنا في هذه المنطقة في الـ X و Y و Z
+
+665
+00:49:02,650 --> 00:49:09,590
+اللي بتحقق هذهأعطوها أوجدوا قيمة الـ F عندها وقيمة
+
+666
+00:49:09,590 --> 00:49:12,750
+الـ L عندها هتلاقوا الفرق بين الجهتين لن يتعدى الـ
+
+667
+00:49:12,750 --> 00:49:15,990
+Error اللي بدنا نوجده بعد شوية كيف بدنا نوجد الـ
+
+668
+00:49:15,990 --> 00:49:22,090
+Error؟ كالتالي قلنا مين بلزمنا Fxx هاي Fxx عنده
+
+669
+00:49:22,090 --> 00:49:30,990
+هاي Fxx اتنين Fyy هنا على طول سفر Fxy Fxy اللي هي
+
+670
+00:49:30,990 --> 00:49:40,300
+نقص واحدأفزد X وهو سفر و أفزد Y هو سفر لأن التفاضل
+
+671
+00:49:40,300 --> 00:49:43,440
+بالنسبة ل X أو Y هو سفر لذلك أفزد Y و أفزد X كله
+
+672
+00:49:43,440 --> 00:49:48,500
+مياشي أصفر لكن أفزد Z هي ضاقص تلاتة Sin Z إذن الآن
+
+673
+00:49:48,500 --> 00:49:51,800
+أريد أن أجلب Upper Bound لهذه المجموعات وين؟ على
+
+674
+00:49:51,800 --> 00:49:54,880
+المنطقة هذهأحنا دايماً كان يطلع عند الـ Upper
+
+675
+00:49:54,880 --> 00:49:58,340
+Bound اللي هو يطلع ان هدول أعداد جهزات ما عدا هذه
+
+676
+00:49:58,340 --> 00:50:01,860
+مش عدد، إذا هذه اللي بدنا .. اللي بدنا نشتغل عليها
+
+677
+00:50:01,860 --> 00:50:04,800
+لأن الأعداد هذه جاهزة 2، الـ Absolute Value لها 2
+
+678
+00:50:04,800 --> 00:50:07,900
+وهذه جاهزة 0 وهذه الـ Absolute Value الجاهزة إليها
+
+679
+00:50:07,900 --> 00:50:11,300
+1 وهذه وهذه الـ Absolute Value جاهزة إليها ياشتفر
+
+680
+00:50:11,300 --> 00:50:14,500
+لما نشوف الـ Absolute Value of ZZ هذه إيش بيسوي
+
+681
+00:50:14,500 --> 00:50:17,590
+الـ Absolute Value لهذه؟الـ Absolute Value لهذه
+
+682
+00:50:17,590 --> 00:50:20,650
+الآن ماحدش يقول لي أصغر أو أشهر و تلاتة لأ أنا
+
+683
+00:50:20,650 --> 00:50:24,110
+بديها وين في المنطقة اللي هي Z أصغر أو أشهر و 0.1
+
+684
+00:50:24,110 --> 00:50:30,270
+في المنطقة الصغيرة هذه الآن صين الـ Z صين الـ Z في
+
+685
+00:50:30,270 --> 00:50:35,150
+المنطقة اللي بتكون اللي هي جريبة من اللي هي الكاسر
+
+686
+00:50:35,150 --> 00:50:38,650
+هذا جريبة من الـ Z الصفر يعني الـ Z بتكون الصين
+
+687
+00:50:38,650 --> 00:50:42,590
+الـ Z تقريبا تقريبا بتساوي 100 Z يعني الـ Absolute
+
+688
+00:50:42,590 --> 00:50:47,730
+Value للصين الـ Zأصغر من تلاتة sign من الـ point
+
+689
+00:50:47,730 --> 00:50:52,330
+one عندي ماشي sign الـ z بيكون أصغر أو ساوي اللي
+
+690
+00:50:52,330 --> 00:50:56,590
+هي sign الـ point one حتى ما لازمتيش أقول أن الـ
+
+691
+00:50:56,590 --> 00:51:01,050
+sign بيساوي تقريبا القيمة لأن sign الـ z أصغر أو
+
+692
+00:51:01,050 --> 00:51:05,170
+ساوي تلاتة sign الـ point one في الفترة وين حوالين
+
+693
+00:51:05,170 --> 00:51:10,620
+هذه المنطقة أه طيب الآنفي المنطقة هذه يا جماعة
+
+694
+00:51:10,620 --> 00:51:16,060
+عندي صين الـ Z أصغر أو ساوي صين الـ 0.01 هذه صينة
+
+695
+00:51:16,060 --> 00:51:18,760
+الـ 0.01 احسبها على الألة وضربها في التلاتة
+
+696
+00:51:18,760 --> 00:51:24,180
+هتلاقيها تقريبا بـ 0.3 بكل الأحوال الآن هذه الـ
+
+697
+00:51:24,180 --> 00:51:29,800
+FZZ صغيرة 0.3 الآن الـ Upper Bound مين هياخده في
+
+698
+00:51:29,800 --> 00:51:32,900
+هذه الحالة اللي هو أكبر واحد اللي هو H2 as a bound
+
+699
+00:51:32,900 --> 00:51:36,500
+in the second partial derivativesماشي إذا الـ
+
+700
+00:51:36,500 --> 00:51:39,960
+Upper bound في هذه الحالة اللي هو ايش Fxx بساوي
+
+701
+00:51:39,960 --> 00:51:43,320
+اتنين الان بنعوض hence the error included by
+
+702
+00:51:43,320 --> 00:51:48,940
+replacing F by L on R satisfies E نص الكلام هذا ال
+
+703
+00:51:48,940 --> 00:51:52,680
+M هيها اتنين نص في اتنين في هذه point one هي من
+
+704
+00:51:52,680 --> 00:51:55,300
+فوق أصغر و سوى point one وهذا أصغر من point two
+
+705
+00:51:55,300 --> 00:51:59,780
+والأخيرة اللي هي أصغر و سوى point O oneالان كلنا
+
+706
+00:51:59,780 --> 00:52:03,960
+هنا نجمح مع بعض و نربح و نضرب فهذا المقدار اللي
+
+707
+00:52:03,960 --> 00:52:07,840
+عندي و بهيك يكون اننا احنا اللي هو اوجدنا الارو في
+
+708
+00:52:07,840 --> 00:52:11,340
+ال approximation وهيك يكون خلصنا المحاضرة و إلى
+
+709
+00:52:11,340 --> 00:52:14,220
+لقاء آخر السلام عليكم و رحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P92tIJL5vI0.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P92tIJL5vI0.srt
new file mode 100644
index 0000000000000000000000000000000000000000..d50d07122a48a7ac32e768bd0859f7cdcd473d5d
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P92tIJL5vI0.srt
@@ -0,0 +1,1246 @@
+1
+00:00:01,200 --> 00:00:04,160
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,160 --> 00:00:10,120
+وبركاته هذه هي المحاضرة العاشرة تفاضل وتكامل جيم
+
+3
+00:00:10,120 --> 00:00:16,060
+وهي المحاضرة الثالثة في chapter 13, section اليوم
+
+4
+00:00:16,060 --> 00:00:22,480
+اللي هو 13.3, R length in space اللي هو طول القوس
+
+5
+00:00:22,480 --> 00:00:30,750
+أو طول اللي هو المنحنى في الفراغ. الآن احنا إذا
+
+6
+00:00:30,750 --> 00:00:37,410
+بتتذكروا في تفاضل ألف، في آخر chapter في تفاضل ألف
+
+7
+00:00:37,410 --> 00:00:41,130
+أخذنا كيف نوجد ال length لل curve لو كان عندنا
+
+8
+00:00:41,130 --> 00:00:45,590
+اللي هو arc أو curve عندنا في اللي هو كان بس مش في
+
+9
+00:00:45,590 --> 00:00:50,290
+الفراغ. كان عندنا إذا بتتذكروا ال arc أو ال curve
+
+10
+00:00:50,290 --> 00:00:55,310
+كان موجود في ال .. في .. في  فلنفترض كنا نقول ال ..
+
+11
+00:00:55,310 --> 00:00:58,990
+ال .. ال .. ال .. ال ال length اللي هو عبارة عن ال
+
+12
+00:00:58,990 --> 00:01:02,810
+integration من a ل b للجذر التربيعي لـ (dx/dt)²
+
+13
+00:01:02,810 --> 00:01:07,130
+زائد (dy/dt)²  dt
+
+14
+00:01:07,130 --> 00:01:11,990
+لأنّه كان عندنا اللي هو الشغل بس في المستوى لكن
+
+15
+00:01:11,990 --> 00:01:16,090
+الآن حديثنا احنا اللي هو عن اللي هو ال Arc أو ال
+
+16
+00:01:16,090 --> 00:01:21,310
+Curve في ال Space لو كان في عندنا اللي هو Arc في
+
+17
+00:01:21,310 --> 00:01:25,590
+ال Space بدنا نشوف كيف نوجد اللي هو طول ال Arc
+
+18
+00:01:25,590 --> 00:01:32,320
+اللي عندنا. ال Arc طوله. الآن نفس الأسلوب اللي حلّينا
+
+19
+00:01:32,320 --> 00:01:37,000
+فيه اللي هو في التفاضل ألف، ولكن دخل عندنا اللي هو
+
+20
+00:01:37,000 --> 00:01:42,100
+متغير جديد اللي هو Z(t) أو اللي هو ال coordinate
+
+21
+00:01:42,100 --> 00:01:47,430
+لل Z فصار عندي الآن اللي هو ال .. ال .. ال .. طول
+
+22
+00:01:47,430 --> 00:01:54,050
+ال curve اللي هو بدلالة ال X و ال Y و ال Z بدل
+
+23
+00:01:54,050 --> 00:01:56,830
+المستوى اللي هو بدلالة ال X و ال Y. خلينا نكون
+
+24
+00:01:56,830 --> 00:02:00,690
+أكثر تحديدا. تعالوا نعرف اللي هو تعريفنا، definition
+
+25
+00:02:00,690 --> 00:02:04,410
+the length of a smooth curve. طبعا احنا R(t)
+
+26
+00:02:04,410 --> 00:02:07,810
+بيساوي X(t) في اتجاه i زائد Y(t) في اتجاه j زائد Z(t)
+
+27
+00:02:07,810 --> 00:02:11,650
+في اتجاه k. قلنا إن هذا اللي هو برسم لنا curve زي ما
+
+28
+00:02:11,650 --> 00:02:19,950
+اتفقنا. هي حركة ال Particle وفق ال Position Vector
+
+29
+00:02:19,950 --> 00:02:24,830
+اللي أمامنا. ببدأ عند نقطة t₀ و بستمر و كل ما
+
+30
+00:02:24,830 --> 00:02:28,590
+اتحرك بتتحرك اللي هي نقاط ال X و Y و Z و بتعملنا
+
+31
+00:02:28,590 --> 00:02:30,130
+Curve
+
+32
+00:02:32,090 --> 00:02:36,990
+من عند t من عند A لعند B. يعني الآن t تتحرك من A
+
+33
+00:02:36,990 --> 00:02:40,590
+لعند B. بناء عليه، ال R(t) برسم لنا اللي هو ال
+
+34
+00:02:40,590 --> 00:02:43,350
+position vector، برسم لنا ال curve. لنشوف كيف اللي
+
+35
+00:02:43,350 --> 00:02:48,470
+هو جدّ طول هذا ال curve اللي برسمه R(t) لما t
+
+36
+00:02:48,470 --> 00:02:52,370
+تبدأ من عند ال A و تنتهي عند مين؟ عند ال B. التعريف
+
+37
+00:02:52,370 --> 00:02:56,130
+بالضبط اللي هو that is traced exactly once as t
+
+38
+00:02:56,130 --> 00:02:59,250
+increases from t = A to t = B. يعني بقولي
+
+39
+00:02:59,530 --> 00:03:04,290
+اللي هو R(t) اللي هو عند كل نقطة t بترسم بمرّة
+
+40
+00:03:04,290 --> 00:03:07,910
+واحدة. t تغيّرت، t بترسم كما هي النقطة اللي
+
+41
+00:03:07,910 --> 00:03:10,850
+بعدها. هو أكيد لما ترسم على ال curve جدّاش بالضبط
+
+42
+00:03:10,850 --> 00:03:12,770
+طول ال curve. زي ما قلنا، ال length يساوي ال
+
+43
+00:03:12,770 --> 00:03:15,790
+integration من A لـ B للجذر التربيعي لـ (dx/dt)² زائد
+
+44
+00:03:15,790 --> 00:03:20,210
+(dy/dt)² زائد (dz/dt)²
+
+45
+00:03:20,210 --> 00:03:24,810
+هذا كله تحت الجذر. نكمله بالنسبة
+
+46
+00:03:24,810 --> 00:03:29,820
+لمين؟ للـ t، اللي هي dt. الآن لو جئنا طلعنا لهذا
+
+47
+00:03:29,820 --> 00:03:34,520
+المقدار. احنا ما نعرف ايش هو. لو جئنا أخذنا dr/dt
+
+48
+00:03:34,520 --> 00:03:36,820
+dr/dt، dr/dt زي ما أنتم تعرفوا اللي
+
+49
+00:03:36,820 --> 00:03:39,860
+هو عبارة عن v يا جماعة. الآن dr/dt بيصير
+
+50
+00:03:39,860 --> 00:03:43,300
+dx/dt في اتجاه ال i زائد dy/dt
+
+51
+00:03:43,300 --> 00:03:47,880
+في اتجاه ال j زائد dz/dt في اتجاه ال k. هذا
+
+52
+00:03:47,880 --> 00:03:51,900
+اللي هي ال v. صارت الآن ال magnitude لها ال
+
+53
+00:03:51,900 --> 00:03:55,300
+magnitude ل v. ال magnitude هذا اللي فضلناه هذا بيطلع
+
+54
+00:03:55,300 --> 00:03:58,420
+الجذر التربيعي لـ (dx/dt)² زائد (dy/dt)² زائد
+
+55
+00:03:58,420 --> 00:04:03,020
+(dz/dt)². إذا هذا
+
+56
+00:04:03,020 --> 00:04:06,720
+المقدار بنقدر نقول بالضبط ال length يساوي ال
+
+57
+00:04:06,720 --> 00:04:11,340
+integration من ال A إلى B لل magnitude ل v اللي هو
+
+58
+00:04:11,340 --> 00:04:15,620
+dt. وهذا هو ال length of the arc.
+
+59
+00:04:18,140 --> 00:04:25,000
+طيب تعالوا نجي الآن ناخد مثال. الآن a glider is
+
+60
+00:04:25,000 --> 00:04:29,640
+soaring upward along the helix. نفترض أن في عندي
+
+61
+00:04:29,640 --> 00:04:36,040
+اللي هو particle أو glider اللي هو بمشي عبر مين
+
+62
+00:04:36,040 --> 00:04:40,680
+على اللي هو مسار، مسار helix يعني على ال helix زي
+
+63
+00:04:40,680 --> 00:04:44,900
+المثال اللي أخذناه. ال helix اللي هي cos t في اتجاه
+
+64
+00:04:44,900 --> 00:04:49,400
+i زائد sin t في اتجاه j زائد t في اتجاه k. نفترض أنّه اللي
+
+65
+00:04:49,400 --> 00:04:55,800
+هو ال glider هذه بتمشي على ال helix. بدنا نشوف how
+
+66
+00:04:55,800 --> 00:04:59,460
+long is the glider's path from t = 0 to t
+
+67
+00:04:59,460 --> 00:05:03,860
+= π. بدنا نشوف قدّاش طول الطريق اللي قطعه ال
+
+68
+00:05:03,860 --> 00:05:08,880
+glider هذه من t عندها = 0 لعند t  بتساوي
+
+69
+00:05:08,880 --> 00:05:14,370
+2π. الآن، عالقانون مباشرة عندي اللي هو ال
+
+70
+00:05:14,370 --> 00:05:18,930
+position vector أو اللي برسم لي اللي هو مسار ال
+
+71
+00:05:18,930 --> 00:05:22,550
+particle أو ال glider هذه اللي هو عبارة عن the
+
+72
+00:05:22,550 --> 00:05:26,210
+length of the glider's path from t = 0 to t
+
+73
+00:05:26,210 --> 00:05:29,390
+= π.  الاتفاق، ال length ايش يساوي؟ ال
+
+74
+00:05:29,390 --> 00:05:33,190
+integration من 0 ل 2π لل magnitude
+
+75
+00:05:33,190 --> 00:05:37,490
+ل v(t) dt. ما هو v(t)؟ طبعا بتجي بتفاضل
+
+76
+00:05:37,490 --> 00:05:43,790
+هذه اللي هي ناقص sin t في اتجاه i زائد cos t
+
+77
+00:05:43,790 --> 00:05:50,150
+في اتجاه j زائد 1 في اتجاه k. مظبوط. هذا اللي هو تفاضلها. الآن
+
+78
+00:05:50,150 --> 00:05:52,830
+ال magnitude اللي هو عبارة عن الجذر التربيعي للي
+
+79
+00:05:52,830 --> 00:05:56,320
+هو (–sin t)² زائد
+
+80
+00:05:56,320 --> 00:05:59,160
+(cos t)² زائد اللي
+
+81
+00:05:59,160 --> 00:06:02,520
+هو 1 تربيع اللي هو 1. الجذر التربيعي لِ
+
+82
+00:06:02,520 --> 00:06:07,060
+هذا كله dt. بدنا نكمله، نشوف كيف نكمله. الجمعة طبعا كلكم قال
+
+83
+00:06:07,060 --> 00:06:10,680
+طب ما هي sin² + cos² = 1، وعنا 1،
+
+84
+00:06:10,680 --> 00:06:13,560
+من اثنين. إذا جذر 2 dt. وجذر 2 عبارة عن ثابت
+
+85
+00:06:18,500 --> 00:06:21,960
+تكامله على طول اللي هو جذر 2 في t من 0 ل 2
+
+86
+00:06:21,960 --> 00:06:26,300
+π. فبيصير عندي 2 جذر 2 π. هذا عبارة
+
+87
+00:06:26,300 --> 00:06:30,540
+عن the length of the glider's path من t = 0
+
+88
+00:06:30,540 --> 00:06:36,960
+لـ t = 2π. هذه
+
+89
+00:06:36,960 --> 00:06:40,060
+اللي هي مثلا توضيح للي قبل بشوية.
+
+90
+00:06:45,490 --> 00:06:49,290
+الآن نحكي عن الرسمة هذه. بس قبل أن نحكي عنها خلينا
+
+91
+00:06:49,290 --> 00:06:53,370
+نشوف ايش اللي عندي. if we choose the base point  B(t₀)
+
+92
+00:06:53,370 --> 00:06:57,150
+note on a smooth curve. طبعا، نجيب احنا اللي بنشتغل
+
+93
+00:06:57,150 --> 00:07:00,990
+عليه عشان تكون في الصورة هو smooth curve. ليش
+
+94
+00:07:00,990 --> 00:07:05,700
+smooth curve؟ عشان لسبب بسيط، ال smooth curve ال
+
+95
+00:07:05,700 --> 00:07:10,500
+derivative له موجودة وليست كذلك و continuous. طيب
+
+96
+00:07:10,500 --> 00:07:14,440
+موجود عشان نستعمل القانون. نتفّق معك، لكن طب و
+
+97
+00:07:14,440 --> 00:07:17,660
+continuous؟ آه، عشان أنا بتكون continuous بيصير ال
+
+98
+00:07:17,660 --> 00:07:19,880
+magnitude ل v(t) اللي هو المقياس continuous
+
+99
+00:07:19,880 --> 00:07:22,400
+function. بتصير هذا كلّها اللي بتطلع عندنا continuous
+
+100
+00:07:22,400 --> 00:07:26,600
+يعني بعد اقدر ايش؟ اكملها عشان ما بديش مشاكل، عشان
+
+101
+00:07:26,600 --> 00:07:30,540
+أضمن يطلع من الـ integral اللي هو قيمته لأنّ الدالّة المتصلة
+
+102
+00:07:30,540 --> 00:07:34,080
+على طول في ال integral زي ما عرفنا في تفاضل اللي هو
+
+103
+00:07:34,080 --> 00:07:40,340
+ألف. طيب، لنفترض أنّه عندي .. لنفترض نقطة .. نقطة
+
+104
+00:07:40,340 --> 00:07:43,460
+بداية، خلينا أو خلينا نقول نقطة انطلاق. السؤال اللي
+
+105
+00:07:43,460 --> 00:07:46,740
+بده يمشي في هيك أو بده يمشي هيك، ماشي الحال. هذا
+
+106
+00:07:46,740 --> 00:07:50,740
+بسميها base point اللي هي r(t₀). الآن هذه ال
+
+107
+00:07:50,740 --> 00:07:56,580
+base point. لما نمشي شيء بسيط هنا حيصير عندي اللي
+
+108
+00:07:56,580 --> 00:08:01,600
+هو تغيّرت من r(t₀) لعند اللي هي مثلا مين؟ r(t)
+
+109
+00:08:01,600 --> 00:08:06,940
+لهون. فرضنا، ماشي الحال. هذا اللي صار، صار فيه طول هذا
+
+110
+00:08:06,940 --> 00:08:12,670
+الطول اللي من هون لهون. بتغيّر بتغيّر t. لو طلعت
+
+111
+00:08:12,670 --> 00:08:16,570
+لفوق بطول شوية، لو طلعت لفوق بطول شوية، وهكذا. عشان
+
+112
+00:08:16,570 --> 00:08:19,870
+هيك، كنا نسمي هذا احنا اللي قبل، أخذناه من ثابت
+
+113
+00:08:19,870 --> 00:08:23,290
+لثابت، من 0 ل 2π عشان يطلع عندي عدد. لكن
+
+114
+00:08:23,290 --> 00:08:27,640
+الآن أنا بأخذه من r(t₀) اللي هي عند نقطة ال
+
+115
+00:08:27,640 --> 00:08:31,040
+t₀ لعند r(t) لعند t. يعني حيُطلع ال
+
+116
+00:08:31,040 --> 00:08:35,040
+integration اللي عندي بدلالة مين؟ بدلالة t. يعني
+
+117
+00:08:35,040 --> 00:08:41,040
+الطول هذا بيطلع بدلالة t. هذا الطول بدّي أسميه s(t)
+
+118
+00:08:41,040 --> 00:08:44,820
+في الاتجاه هذا. لو مشيت بدّي أعتبره أنا ماشي
+
+119
+00:08:44,820 --> 00:08:49,120
+بالإتجاه الموجب. لو جئت رجعت رجوع من هون لهون بدّي
+
+120
+00:08:49,120 --> 00:08:52,900
+أعتبره ايش؟ سالب، في الاتجاه السالب. هذا اختلاف
+
+121
+00:08:52,900 --> 00:08:56,200
+بيننا. يعني الآن بدنا نقول من ال base point و امشي
+
+122
+00:08:56,200 --> 00:09:00,020
+هيك بالإتجاه هذا على اليمين، يعني بده أعتبره موجب
+
+123
+00:09:00,020 --> 00:09:03,220
+ولما نمشي هيك على الشمال بدّه أعتبره ايش؟ أعتبره
+
+124
+00:09:03,220 --> 00:09:06,920
+ايش؟ سالب. هذا اللي هو يعني الطول بحكي عن الطول
+
+125
+00:09:06,920 --> 00:09:10,480
+يعني الطول من هون لهون هذا s(t). من هون لهون s(t)
+
+126
+00:09:10,480 --> 00:09:14,390
+بس اختلفت s(t)، اختلفت كطول. و هنبدأ
+
+127
+00:09:14,390 --> 00:09:19,470
+نعتبرها ايش؟ سالبة. الطول لل curve أو لل arc بس
+
+128
+00:09:19,470 --> 00:09:24,390
+باتجاه السالب. طيب، هذا الآن if we choose a base
+
+129
+00:09:24,390 --> 00:09:26,810
+point زي ما قلنا، r(t₀) in a smooth curve C
+
+130
+00:09:26,810 --> 00:09:30,830
+parameterized by t. each value of t determines a
+
+131
+00:09:30,830 --> 00:09:34,650
+point r(t). يعني كل value of t بتحدد لـ x(t) و y
+
+132
+00:09:34,650 --> 00:09:38,890
+(t) و z(t). يعني تتغير قيمة اللي هي r(t)
+
+133
+00:09:38,890 --> 00:09:42,750
+بتغير الزمن. وطبعا تبعا لذلك يتغير اللي هي
+
+134
+00:09:43,860 --> 00:09:47,760
+الإحداثيات ال x وال y وال z on C. ال Directed
+
+135
+00:09:47,760 --> 00:09:51,360
+distance s(t) اللي هي ال Length دي بنسميها ايش؟
+
+136
+00:09:51,360 --> 00:09:55,400
+معلش، Directed distance. ايش Directed distance؟
+
+137
+00:09:55,400 --> 00:10:00,300
+اتفقنا إنّه من t₀ لعندي في الاتجاه اللي هيك
+
+138
+00:10:00,300 --> 00:10:05,620
+حيعطيني موجب. ال magnitude هذا v(τ) dτ. v(τ) dτ
+
+139
+00:10:05,620 --> 00:10:10,940
+الآن هذا حيُطلع عندي اللي هو بدلالة مين؟ بدلالة
+
+140
+00:10:10,940 --> 00:10:14,180
+t. هو عبارة عن length في الواقع. بس length من t
+
+141
+00:10:14,180 --> 00:10:17,760
+₀ لعند مين؟ عن t. بعيد بالموجب لما t تكون إلى
+
+142
+00:10:17,760 --> 00:10:20,900
+أعلى، ولو كانت إلى أسفل يعني حجاب اللي هيعد بالاشارة
+
+143
+00:10:20,900 --> 00:10:24,320
+السالبة. الآن، هي زي ما قلنا، t أكبر من t₀. s
+
+144
+00:10:24,320 --> 00:10:28,520
+(t) is the distance along the curve from r(t₀)
+
+145
+00:10:28,520 --> 00:10:35,510
+to r(t). إذا كان ال t أقل من ال t₀، s
+
+146
+00:10:35,510 --> 00:10:39,510
+(t) هو سالب بالمقارنة. s(t) هو سالب بالمقارنة. s(t)
+
+147
+00:10:39,510 --> 00:10:40,770
+هو سالب بالمقارنة. s(t) هو سالب بالمقارنة. s(t) هو
+
+148
+00:10:40,770 --> 00:10:45,910
+سالب بالمقارنة. s(t) هو سالب بالمقارنة. s
+
+149
+00:10:45,910 --> 00:10:52,150
+(t) هو سالب بالمقارنة. لأن s بتوجد هنا بدلالة t اللي
+
+150
+00:10:52,150 --> 00:10:56,750
+بتطلع هنا. إذا كل قيمة ل s هنا أو هنا أو هنا أو هنا
+
+151
+00:10:56,750 --> 00:11:01,630
+هي في الواقع مقابلة لقيمة في مين؟ في t. لأنّ إحنا
+
+152
+00:11:01,630 --> 00:11:06,430
+أصلا لما جبنا s(t) جبناها من t نقطة و اعتمدت
+
+153
+00:11:06,430 --> 00:11:13,250
+على t. الآن نسميها s، arc length parameter. الآن s
+
+154
+00:11:13,250 --> 00:11:18,150
+هذه اللي هي length، لكن تتغير بتغير ال t. فعشان هيك
+
+155
+00:11:18,150 --> 00:11:23,310
+بنسميها Arc Length Parameter. يعني الآن تستغرب إنّه
+
+156
+00:11:23,310 --> 00:11:29,050
+أصلا أنت ال r(t) اللي قبل بشوية هذا اللي جبناه
+
+157
+00:11:29,050 --> 00:11:34,870
+بدلالة t. بعد شوية حأقدر أجيب لك إياه بدلالة s. يعني ال
+
+158
+00:11:34,870 --> 00:11:37,550
+parameter بدل ما يكون اللي هو t، حيكون ال parameter
+
+159
+00:11:37,550 --> 00:11:42,410
+من s اللي هو عبارة عن ال length أو ال Arc Length
+
+160
+00:11:42,410 --> 00:11:47,350
+Parameter. لنرى ماذا يقول هذا الباراميتر. الباراميتر
+
+161
+00:11:47,350 --> 00:11:48,130
+الباراميتر الباراميتر الباراميتر الباراميتر الباراميتر
+
+162
+00:11:48,130 --> 00:11:49,810
+الباراميتر الباراميتر الباراميتر الباراميتر الباراميتر
+
+163
+00:11:49,810 --> 00:11:52,190
+الباراميتر الباراميتر الباراميتر الباراميتر الباراميتر
+
+164
+00:11:52,190 --> 00:11:53,290
+الباراميتر الباراميتر الباراميتر الباراميتر الباراميتر
+
+165
+00:11:53,290 --> 00:11:55,610
+الباراميتر الباراميتر الباراميتر الباراميتر الباراميتر
+
+166
+00:11:55,610 --> 00:11:56,750
+الباراميتر الباراميتر الباراميتر الباراميتر الباراميتر
+
+167
+00:11:56,750 --> 00:12:01,000
+الباراميتر الباراميتر ال and for investigating the
+
+168
+00:12:01,000 --> 00:12:03,240
+turning and twisting nature of a space curve. هذا
+
+169
+00:12:03,240 --> 00:12:08,820
+موضوعنا في chapter 13. حنحكي عن ال twisting و
+
+170
+00:12:08,820 --> 00:12:13,060
+turning curves. ايه هذا اللي هو  قدرنا إن شاء الله
+
+171
+00:12:13,060 --> 00:12:19,420
+نفهمه. هذا الأساس مهم بالنسبة لهذا اللي هو الموضوعين. طيب
+
+172
+00:12
+
+223
+00:15:42,250 --> 00:15:46,050
+بدل الـ T،  أش بتصير S على جدر التنين في تجاه الـ I
+
+224
+00:15:46,050 --> 00:15:49,330
+sign S على جدر التنين في تجاه الـ I، زاد S على جدر
+
+225
+00:15:49,330 --> 00:15:52,110
+التنين في تجاه الـ K، وهذه قدرنا نكتب الـ position
+
+226
+00:15:52,110 --> 00:15:57,650
+vector R، أو اللي بيحددلي مصار الـ K، الـ particle، أو
+
+227
+00:15:57,650 --> 00:16:04,430
+اللي هو مصار الـ curve، جبته بواسطة أو بدلالة اللي هو
+
+228
+00:16:04,430 --> 00:16:09,790
+الـ R Length Parameter اللي هي S، ماشي يا جماعة؟
+
+229
+00:16:09,790 --> 00:16:14,930
+الآن بدنا speed on a smooth curve، بدنا اللي هو إيش؟
+
+230
+00:16:14,930 --> 00:16:19,550
+نشوف اللي هو السرعة، مش المتجهة، لأ السرعة مش متجهة.
+
+231
+00:16:19,550 --> 00:16:22,190
+السرعة، السرعة اللي هو الـ magnitude on a smooth
+
+232
+00:16:22,190 --> 00:16:26,980
+curve عند the smooth curve، اللي هو اللي بيصنعوا R
+
+233
+00:16:26,980 --> 00:16:31,880
+of T، الآن بدنا نوجد اللي هو السبيل عليه الـ
+
+234
+00:16:31,880 --> 00:16:38,880
+magnitude للـ V of T، اللي هي أنت الآن، هيأس بـ 70 نوت.
+
+235
+00:16:38,880 --> 00:16:42,900
+اللي عندك VTAU وDTAU، مظبوط؟ هي القانون تبعنا الآن.
+
+236
+00:16:42,900 --> 00:16:45,360
+بالـ Fundamental Theorem of Calculus، كل شيء حلو.
+
+237
+00:16:45,360 --> 00:16:49,400
+لأن هذا continuous، لأننا مفترضين أن نتحدث عن
+
+238
+00:16:49,400 --> 00:16:52,000
+Smooth Curve،  مُزام Smooth Curve، لأن الـ derivative
+
+239
+00:16:52,000 --> 00:16:54,740
+لهذه موجودة، وهذه موجودة، وهذه موجودة، و continuous
+
+240
+00:16:54,740 --> 00:16:58,580
+يعني هذا continuous، مدام هذا continuous، إذن DS by
+
+241
+00:16:58,580 --> 00:17:05,060
+DT، اللي هو على طول بنزيل هذا و بنحط اللي هو يعني
+
+242
+00:17:05,060 --> 00:17:09,820
+التفاضل بالغ التكامل و بيصير بيصير بساوة V of T.
+
+243
+00:17:09,820 --> 00:17:14,440
+يعني دي أس باي دي تي بيساوي الـ magnitude ده V of T.
+
+244
+00:17:14,440 --> 00:17:18,060
+يعني، وكأنه بقول الـ equation أربعة says that the
+
+245
+00:17:18,060 --> 00:17:23,750
+speed with which a particle moves، يعني الـ speed
+
+246
+00:17:23,750 --> 00:17:26,250
+للبارتكال، هذه الـ speed للبارتكال، تلك اللي احنا
+
+247
+00:17:26,250 --> 00:17:32,750
+عارفينها إيش ساوت؟ اللي هي عبارة عن تفاضل الـ arc length
+
+248
+00:17:32,750 --> 00:17:38,250
+with respect to time to T، يعني with which a
+
+249
+00:17:38,250 --> 00:17:41,450
+particle moves along its path is the magnitude of
+
+250
+00:17:41,450 --> 00:17:46,110
+v، consistent with what we note، اللي هو أنه احنا
+
+251
+00:17:46,110 --> 00:17:50,450
+اللي بنعرفه أنه هذه الـ speed للـ particle بساوية
+
+252
+00:17:50,450 --> 00:17:54,910
+اللي هي عبارة عن تفاضل الـ arc length، المسافة
+
+253
+00:17:54,910 --> 00:18:01,150
+المقطوعة هذه عالميًا بالنسبة لمين؟ بالنسبة لـ T، هذه
+
+254
+00:18:01,150 --> 00:18:06,750
+اللي هي الـ Speed، اللي هي تفاضل المسافة المقطوعة
+
+255
+00:18:06,750 --> 00:18:13,430
+على T، على T طبعًا، المسافة بإيش يا جماعة؟ بدون ما
+
+256
+00:18:13,430 --> 00:18:17,950
+نحكي عن Direction، الأسعار عن Length، هيك أبو جبهة، و
+
+257
+00:18:17,950 --> 00:18:23,620
+هيك سالبة، وبس مش Direction، هذه التزايد تصبح موجبة.
+
+258
+00:18:23,620 --> 00:18:26,080
+الـ S، إذا كانت T متزايدة تصبح تزايدة، سالفة إذا
+
+259
+00:18:26,080 --> 00:18:26,480
+كانت T متزايدة تصبح سالفة، إذا كانت T متزايدة تصبح
+
+260
+00:18:26,480 --> 00:18:27,540
+سالفة، إذا كانت T متزايدة تصبح سالفة، إذا كانت T
+
+261
+00:18:27,540 --> 00:18:27,740
+متزايدة تصبح سالفة، إذا كانت T متزايدة تصبح سالفة
+
+262
+00:18:27,740 --> 00:18:27,800
+إذا كانت T متزايدة تصبح سالفة، إذا كانت T متزايدة
+
+263
+00:18:27,800 --> 00:18:28,740
+T متزايدة تصبح سالفة، إذا كانت T متزايدة تصبح سالفة
+
+264
+00:18:28,740 --> 00:18:31,660
+إذا كانت T متزايدة تصبح سالفة.
+
+265
+00:18:35,760 --> 00:18:39,060
+للـ unit tangent vector، بدنا نعرف حاجة اسمها الـ
+
+266
+00:18:39,060 --> 00:18:42,380
+unit tangent vector، وهنجيله برضه بالتفصيل قدام إن
+
+267
+00:18:42,380 --> 00:18:46,860
+شاء الله، خلينا نشوف we already know the velocity
+
+268
+00:18:46,860 --> 00:18:51,560
+vector V، بيساوي DR by DT، الآن الـ velocity vector V
+
+269
+00:18:51,560 --> 00:18:57,300
+بيساوي DR by DT، اللي هي المشتقة، دي DR by DT، اللي هو
+
+270
+00:18:57,300 --> 00:19:00,740
+عبارة عن المماس لمين؟ للـ curve عندها R of T، هذا إيش
+
+271
+00:19:00,740 --> 00:19:04,880
+رسمت مين؟ الرسمة اللي بترسم مين؟ اللي هي R of T، الآن
+
+272
+00:19:04,880 --> 00:19:11,080
+DR by DT هي عبارة عن الـ tangent لمن؟ اللي هو للـ
+
+273
+00:19:11,080 --> 00:19:15,320
+curve R of T، إذا V بيساوي DR by DT is the tangent
+
+274
+00:19:15,320 --> 00:19:18,700
+to the curve R of T، and that the vector، لأن هذا هو
+
+275
+00:19:18,700 --> 00:19:22,220
+الـ tangent، DR by DT، هذا هي الـ tangent، هذه اللي هي
+
+276
+00:19:22,220 --> 00:19:27,080
+من؟ الـ derivative، اللي هي الـ derivative لمن؟ لـ R
+
+277
+00:19:27,080 --> 00:19:30,840
+اللي هي اللي بترسم الـ curve اللي عندي الآن أنا.
+
+278
+00:19:30,840 --> 00:19:35,690
+معنى بمين؟ مش اللي هو dR by dt، اللي هو الـ tangent
+
+279
+00:19:35,690 --> 00:19:40,650
+بس لأ، معنى بالـ unit tangent vector، وحاجة هستخدمها
+
+280
+00:19:40,650 --> 00:19:44,990
+قدام كثير، عشان هي كمان ده أفردلها رمز لحاله، اللي
+
+281
+00:19:44,990 --> 00:19:49,990
+هو ساوي T Capital، اللي هي عبارة عن الـ velocity على
+
+282
+00:19:49,990 --> 00:19:52,690
+الـ magnitude ده الـ V، اللي هو الـ speed، يعني بمعنى
+
+283
+00:19:52,690 --> 00:19:59,050
+آخر هذه ده direction، اللي هو of مين؟ is the
+
+284
+00:19:59,050 --> 00:20:03,930
+direction of... اللي هو حركة الـ particle اللي هي
+
+285
+00:20:03,930 --> 00:20:07,990
+اللي بنسميها الـ unit tangent vector، لإنه مكون
+
+286
+00:20:07,990 --> 00:20:11,990
+tangent لمين؟ للـ curve هذا، واخدت الـ unit منه بس
+
+287
+00:20:11,990 --> 00:20:15,450
+اللي طوله واحد، اللي هو V على الـ magnitude للـ V، هيك
+
+288
+00:20:15,450 --> 00:20:20,460
+بدنا نعرفه، طيب ناخد مثال الآن، الآن بتقول لي five the
+
+289
+00:20:20,460 --> 00:20:23,960
+unit tangent vector of the curve R of T، بيساوي cos
+
+290
+00:20:23,960 --> 00:20:27,440
+T زائد ثلاثة sin T، كل واحدة في اتجاهها، زائد T
+
+291
+00:20:27,440 --> 00:20:30,960
+تربيع في اتجاهها، طيب شوفوا يا جماعة، صلوا على النبي
+
+292
+00:20:30,960 --> 00:20:33,720
+عليه الصلاة والسلام، لأن عشان أوجد الـ unit tangent
+
+293
+00:20:33,720 --> 00:20:37,860
+vector بالزمن، أوجد الـ velocity، DR by DT، بيساوي
+
+294
+00:20:37,860 --> 00:20:41,860
+كلّكم قالوا ناقص ثلاثة sin T في اتجاه الـ I، زائد ثلاثة
+
+295
+00:20:41,860 --> 00:20:45,100
+cos T في اتجاه الـ J، بفاضل قاعد، زائد اثنين T في
+
+296
+00:20:45,100 --> 00:20:49,150
+اتجاه الـ K، واحد قال: طيب هو هذا الـ V، وأنت بالتأكيد
+
+297
+00:20:49,150 --> 00:20:51,190
+بيلزمك أن الـ magnitude ده الـ V، آه الـ magnitude
+
+298
+00:20:51,190 --> 00:20:54,430
+ده الـ V إيش بيساوي؟ جذر التربيعي لهذا تربيع زائد هذا
+
+299
+00:20:54,430 --> 00:20:58,330
+تربيع، اللي هي ثلاثة sign، يعني تسعة sign تربيع زائد
+
+300
+00:20:58,330 --> 00:21:02,890
+تسعة cosine تربيع زائد أربعة T تربيع، تسعة sign تربيع
+
+301
+00:21:02,890 --> 00:21:05,710
+زائد تسعة cosine تربيع، بأخذ التسعة عامل مشترك برا
+
+302
+00:21:05,710 --> 00:21:08,330
+بيصير تسعة في (sign تربيع زائد cosine تربيع)، واحد
+
+303
+00:21:08,330 --> 00:21:13,260
+يعني تسعة زائد من هذه أربعة T تربيع، إذا صارت الـ
+
+304
+00:21:13,260 --> 00:21:15,940
+magnitude لـ V هو عبارة عن جذر التربيعي لـ(أربعة
+
+305
+00:21:15,940 --> 00:21:21,340
+زائد T تربيع)، إذا صارت الـ unit tangent vector T
+
+306
+00:21:21,340 --> 00:21:27,260
+بيساوي V على magnitude لـ V، اللي هي هذا المقدار أو
+
+307
+00:21:27,260 --> 00:21:31,760
+هذا اللي هو الـ vector على magnitude، وبيصير ناقص
+
+308
+00:21:31,760 --> 00:21:35,000
+ثلاثة sin T في تجاه I، زائد ثلاثة cos T في تجاه J
+
+309
+00:21:35,000 --> 00:21:35,240
+زائد
+
+310
+00:21:49,730 --> 00:21:54,630
+الحركة عكس اتجاه عقارب الساعة.
+
+311
+00:21:59,270 --> 00:22:02,930
+لو ماشيين على اللي هو فيه عندي الـ vector R، ماشي؟
+
+312
+00:22:02,930 --> 00:22:06,250
+على اللي هو مين؟ الدائرة، أكيد، ليش؟ ماشي عُدّان، لأن R
+
+313
+00:22:06,250 --> 00:22:09,550
+of T بيساوي cos T في تجاه I، زائد sin T في تجاه J.
+
+314
+00:22:09,550 --> 00:22:12,650
+هذا بيبقى parametric equation للـ R، لكن العلاقة بين cos
+
+315
+00:22:12,650 --> 00:22:16,870
+T و sin T، دائمًا cos تربيع زائد sin تربيع بيساوي 1.
+
+316
+00:22:17,560 --> 00:22:22,620
+بيساوي الـ R، يعني الآن cosine تربيع زائد sin تربيع
+
+317
+00:22:22,620 --> 00:22:26,920
+بيساوي قيمة R، لو فرضنا أن احنا ماشيين يعني دائمًا
+
+318
+00:22:26,920 --> 00:22:30,600
+اللي هو ماشي إيش بمقدار؟ اللي هو الـ magnitude
+
+319
+00:22:30,600 --> 00:22:34,880
+بيساوي واحد للـ R، بيصير عندي أنا ماشي على اللي هو
+
+320
+00:22:34,880 --> 00:22:39,160
+دائرة الوحدة، دائرة الوحدة، هذا اللي هو ماشيين على
+
+321
+00:22:39,160 --> 00:22:42,500
+مين؟ لأن cos تربيع زائد cos تربيع دائمًا إيش بيساوي؟
+
+322
+00:22:42,500 --> 00:22:47,260
+بيساوي واحد، فصار عندي الـ magnitude له على طول بيطلع
+
+323
+00:22:47,260 --> 00:22:50,820
+إيش بيطلع؟ واحد، يعني بدون ما أقول أنه والله واحد أو
+
+324
+00:22:50,820 --> 00:22:53,480
+كذا هو هيطلع automatically واحد، زي ما قلنا cos تربيع
+
+325
+00:22:53,480 --> 00:22:57,200
+زائد cos تربيع، اللي هو X تربيع زائد Y تربيع هيساوي
+
+326
+00:22:57,200 --> 00:22:59,940
+واحد، فاهمين عليها؟ لأن cos تربيع زائد cos تربيع
+
+327
+00:22:59,940 --> 00:23:03,120
+هيساوي واحد، إذا الـ X تربيع زائد Y تربيع بيساوي واحد.
+
+328
+00:23:03,120 --> 00:23:08,330
+إذا أنا ماشي على دائرة الوحدة، عبر هذا الـ vector، أو
+
+329
+00:23:08,330 --> 00:23:13,070
+اللي برسم الـ curve around the unit circle، we see
+
+330
+00:23:13,070 --> 00:23:16,890
+that الـ velocity، كلكم قالوا فاضل هذه، و فاضل هذه تطلع
+
+331
+00:23:16,890 --> 00:23:21,810
+ناقص sin t زائد cos t، الآن ناقص sin t في اتجاه الـ I، زائد
+
+332
+00:23:21,810 --> 00:23:26,770
+cos t في اتجاه الـ J، is already برضه إيه؟  شماله unit
+
+333
+00:23:26,770 --> 00:23:30,430
+vector، لأن الـ magnitude له، الجذر التربيعي لهذه
+
+334
+00:23:30,430 --> 00:23:34,160
+التربيع زائد هذه التربيع، اللي هو واحد، إذا اللي هو في
+
+335
+00:23:34,160 --> 00:23:39,740
+هذه الحالة، على اللي هي الدائرة، دائرة الوحدة، بيكون
+
+336
+00:23:39,740 --> 00:23:45,440
+الـ unit tangent vector هو بالضبط من الـ velocity.
+
+337
+00:23:45,440 --> 00:23:48,640
+إذا الـ unit tangent vector عندي بيساوي الـ velocity.
+
+338
+00:23:48,640 --> 00:23:54,080
+in this case because the magnitude of v in this
+
+339
+00:23:54,080 --> 00:24:01,700
+case is one، بناء عليه، اللي هو عندي الآن في بعض
+
+340
+00:24:01,700 --> 00:24:08,460
+الشغلات اللي هو الـ Dt by Ds، عبارة عن واحد على Ds
+
+341
+00:24:08,460 --> 00:24:12,120
+by Dt، زي ما أنتم عارفين، اللي هي الـ Dt by Ds، اللي
+
+342
+00:24:12,120 --> 00:24:17,740
+هي التفاضل اللي هو الزمن بالنسبة للمسافة، بالنسبة
+
+343
+00:24:17,740 --> 00:24:20,600
+للـ arc length، بيساوي واحد على الـ magnitude للـ V.
+
+344
+00:24:21,780 --> 00:24:26,040
+والآن هذه بعض العلاقات اللي ممكن تلزمنا قدام، عند
+
+345
+00:24:26,040 --> 00:24:31,060
+الـ DR by DS، اللي هو بالـ Chain Rule، والـ R طبعًا عارفين
+
+346
+00:24:31,060 --> 00:24:34,060
+إيش هي؟ الـ R، هيها اللي هي الـ Position Vector، الـ
+
+347
+00:24:34,060 --> 00:24:37,140
+S عارفين هي الـ arc length، لو أنه ده فاضل DR by DS
+
+348
+00:24:37,140 --> 00:24:41,370
+أنه قبل شوية دي بني R بدلالة S، بنشوف إيش DR by DS
+
+349
+00:24:41,370 --> 00:24:45,530
+بيساوي DR by DT في DT by DS، اللي هي الـ Chain Rule.
+
+350
+00:24:45,530 --> 00:24:50,910
+ويساوي DR by DT، مين هي يا جماعة؟ اللي هي عبارة عن
+
+351
+00:24:50,910 --> 00:24:54,830
+الـ V، عارفينها من زمان، DT by DS، طيب اللي هو
+
+352
+00:24:54,830 --> 00:24:57,690
+استنتجناه اللي عبارة عن واحد على الـ magnitude للـ V.
+
+353
+00:24:57,960 --> 00:25:00,300
+إذا صارت الـ V على الـ magnitude ده الـ V، اللي هو
+
+354
+00:25:00,300 --> 00:25:03,600
+مين؟ اللي هو T، الـ unit tangent vector، وكأنه
+
+355
+00:25:03,600 --> 00:25:07,660
+بيقول لي اسمع، الـ unit tangent vector ممكن تجيبه
+
+356
+00:25:07,660 --> 00:25:12,280
+كمان بطريقة ثانية، أنك تفاضل الـ R بالنسبة لمين؟
+
+357
+00:25:12,280 --> 00:25:17,480
+للـ arc length، تفاَضُل اللي هو اللي الـ R اللي هي دالة
+
+358
+00:25:17,480 --> 00:25:21,560
+الإزاحة أو كذا، بالنسبة للـ arc length، أو اللي هي
+
+359
+00:25:21,560 --> 00:25:25,120
+دالة الـ Position Vector بالنسبة للـ arc length
+
+360
+00:25:25,120 --> 00:25:29,360
+تبعها، بيطلع عندي إيش اسمه؟ مين؟ الـ unit
+
+361
+00:25:29,360 --> 00:25:32,780
+tangent vector، وهيك بيكون خلصنا المحاضرة وإلى
+
+362
+00:25:32,780 --> 00:25:35,960
+لقاء آخر، السلام عليكم ورحمة الله وبركاته.
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P92tIJL5vI0_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P92tIJL5vI0_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0d3091dd0a03abf47da84157609c0ac90297a3b5
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P92tIJL5vI0_postprocess.srt
@@ -0,0 +1,1448 @@
+1
+00:00:01,200 --> 00:00:04,160
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,160 --> 00:00:10,120
+وبركاته هذه هي المحاضرة العاشرة تفاضل وتكامل جيم
+
+3
+00:00:10,120 --> 00:00:16,060
+وهي المحاضرة الثالثة في chapter 13 section اليوم
+
+4
+00:00:16,060 --> 00:00:22,480
+اللي هو 13 3 R length in space اللي هو طول القوس
+
+5
+00:00:22,480 --> 00:00:30,750
+أو طول اللي هو المنحنة في الفرارالان احنا إذا
+
+6
+00:00:30,750 --> 00:00:37,410
+بتتذكروا في تفاضل الف في آخر chapter في تفاضل الف
+
+7
+00:00:37,410 --> 00:00:41,130
+أخدنا كيف نودد ال length لل curve لو كان عندنا
+
+8
+00:00:41,130 --> 00:00:45,590
+اللي هو arc أو curve عندنا في اللي هو كان بس مش في
+
+9
+00:00:45,590 --> 00:00:50,290
+الفراغكان عندنا إذا بتتذكروا الـ arc أو ال curve
+
+10
+00:00:50,290 --> 00:00:55,310
+كان موجود في ال .. في .. في لبنان فكنا نقول ال ..
+
+11
+00:00:55,310 --> 00:00:58,990
+ال .. ال .. ال .. ال ال length اللي هو عبارة عن ال
+
+12
+00:00:58,990 --> 00:01:02,810
+integration من a ل b الجدر التربيع إلى dx by dt
+
+13
+00:01:02,810 --> 00:01:07,130
+لكل تربيع زي ال dy by dt لكل تربيع لكل إيش معله؟
+
+14
+00:01:07,130 --> 00:01:11,990
+dt لأنه كان عندنا اللي هو الشغل بس في لبنانلكن
+
+15
+00:01:11,990 --> 00:01:16,090
+الآن حديثنا احنا اللي هو عن اللي هو الـ Arc أو الـ
+
+16
+00:01:16,090 --> 00:01:21,310
+Curve في الـ Space لو كان في عندنا اللي هو Arc في
+
+17
+00:01:21,310 --> 00:01:25,590
+الـ Space بدنا نشوف كيف نوجد اللي هو طول الـ Arc
+
+18
+00:01:25,590 --> 00:01:32,320
+اللي عندناالـ Arc طوله الان نفس الأسلوب اللي حلنا
+
+19
+00:01:32,320 --> 00:01:37,000
+فيه اللي هو في التفاضل ألف ولكن دخل عندنا اللي هو
+
+20
+00:01:37,000 --> 00:01:42,100
+متغير جديد اللي هو Z of T أو اللي هو ال coordinate
+
+21
+00:01:42,100 --> 00:01:47,430
+لـ Z فصار عندى الآن اللي هو ال .. ال .. ال ..طول
+
+22
+00:01:47,430 --> 00:01:54,050
+الـ curve اللي هو بدلالة الـ X و الـ Y و الـ Z بدل
+
+23
+00:01:54,050 --> 00:01:56,830
+المكان اللي هو بدلالة الـ X و الـ Y خلّينا نكون
+
+24
+00:01:56,830 --> 00:02:00,690
+أكثر تحديدا تعالوا نعرف اللي هو تعريفنا definition
+
+25
+00:02:00,690 --> 00:02:04,410
+the length of a smooth curve طبعا احنا R of T
+
+26
+00:02:04,410 --> 00:02:07,810
+بيساوي X of T فتجاه I زي Y of T فتجاه G زي Z of T
+
+27
+00:02:07,810 --> 00:02:11,650
+فتجاه K قلنا إن هذا اللي هو برسم لنا curve زي ما
+
+28
+00:02:11,650 --> 00:02:19,950
+اتفجناهي حركة الـ Particle وفق الـ Position Vector
+
+29
+00:02:19,950 --> 00:02:24,830
+اللي أمامي ببدأ عند نقطة T Note و بستمر و كل ما
+
+30
+00:02:24,830 --> 00:02:28,590
+اتحرك بتتحرك اللي هي نقاط الـ X و Y و Z و بتعملنا
+
+31
+00:02:28,590 --> 00:02:30,130
+Curve
+
+32
+00:02:32,090 --> 00:02:36,990
+من عند T من عند A لعند B يعني الان T تتحرك من A
+
+33
+00:02:36,990 --> 00:02:40,590
+لعند B بناء عليه ال R of T برسم لنا اللي هو ال
+
+34
+00:02:40,590 --> 00:02:43,350
+position vector برسم لنا ال curve لنشوف كيف اللي
+
+35
+00:02:43,350 --> 00:02:48,470
+هو جداش طول هذا ال curve اللي برسمه R of T لما T
+
+36
+00:02:48,470 --> 00:02:52,370
+تبدأ من عند ال A و تمتى عند مين؟ عند ال B التعريف
+
+37
+00:02:52,370 --> 00:02:56,130
+بالظبط اللي هو that is traced exactly once as T
+
+38
+00:02:56,130 --> 00:02:59,250
+increasing from T تساوة A to T تساوة B يعني بقولي
+
+39
+00:02:59,530 --> 00:03:04,290
+اللي هو R of T اللي هو عند كل نقطة T بترسم بقص مرة
+
+40
+00:03:04,290 --> 00:03:07,910
+واحدة تي اتغيرت تي بترسم كما اللي هي النقطة اللي
+
+41
+00:03:07,910 --> 00:03:10,850
+بعيدة هو اكيد لما ترسم على ال curve جداش بالظبط
+
+42
+00:03:10,850 --> 00:03:12,770
+طول ال curve زي ما قلنا ال length أقل بالساوية ال
+
+43
+00:03:12,770 --> 00:03:15,790
+integration من A لB الجدر التربيع على DX by DT لكل
+
+44
+00:03:15,790 --> 00:03:20,210
+تربيع زائد DY by DT لكل تربيع زائد DZ by DT لكل
+
+45
+00:03:20,210 --> 00:03:24,810
+تربيع هذا كله تحت الجدر اسمه ايه بنكمله بالنسبة
+
+46
+00:03:24,810 --> 00:03:29,820
+لمين التيه اللي هي DTالان لو جينا طلعنا لهذا
+
+47
+00:03:29,820 --> 00:03:34,520
+المقدار احنا ماعرفين ايش هو لو جينا اخدنا دي ار
+
+48
+00:03:34,520 --> 00:03:36,820
+باي دي تي دي ار باي دي تي زي ما انتوا تعرفوا اللي
+
+49
+00:03:36,820 --> 00:03:39,860
+هو عبارة عن V يا جماعة الان دي ار باي دي تي بيصير
+
+50
+00:03:39,860 --> 00:03:43,300
+دي X باي دي تي في اتجاه ال I زائد دي Y باي دي تي
+
+51
+00:03:43,300 --> 00:03:47,880
+في اتجاه ال J زائد دي Z باي دي تي في اتجاه ال Kهذا
+
+52
+00:03:47,880 --> 00:03:51,900
+اللي هي الـ V صارت الان ال magnitude لها ال
+
+53
+00:03:51,900 --> 00:03:55,300
+magnitude ل V ال magnitude ده اللي فضلناها ده بطلع
+
+54
+00:03:55,300 --> 00:03:58,420
+الجدر التربيع إلى DX by DT لكل تربيع زي ال DIY by
+
+55
+00:03:58,420 --> 00:04:03,020
+DT لكل تربيع زي ال DZ by DT لكل تربيع إذا هذا
+
+56
+00:04:03,020 --> 00:04:06,720
+المقدار بنقدر نقول بالظبط ال length بالساعة ال
+
+57
+00:04:06,720 --> 00:04:11,340
+integration من ال A إلى B لل magnitude ل V اللي هو
+
+58
+00:04:11,340 --> 00:04:15,620
+DT وهذا هو ال length of the arc
+
+59
+00:04:18,140 --> 00:04:25,000
+طيب اتماع نيجي الان ناخد مثال الان a glider is
+
+60
+00:04:25,000 --> 00:04:29,640
+soaring upward along the helix نفترض ان في عندى
+
+61
+00:04:29,640 --> 00:04:36,040
+اللي هو particle او glider اللي هو بمشي عبر مين
+
+62
+00:04:36,040 --> 00:04:40,680
+علىاللي هو مصاره مصار Helix يعني على الـ Helix زي
+
+63
+00:04:40,680 --> 00:04:44,900
+المثال اللي أخدناه الـ Helix اللي هي Cos T باتجاه
+
+64
+00:04:44,900 --> 00:04:49,400
+I زي Cos T باتجاه G زي T باتجاه K نفترض إنه اللي
+
+65
+00:04:49,400 --> 00:04:55,800
+هو الـ Glider هذه بتمشي على الـ Helixبدنا نشوف how
+
+66
+00:04:55,800 --> 00:04:59,460
+long is the glider's path from T بساوة سفر to T
+
+67
+00:04:59,460 --> 00:05:03,860
+بساوة باى بدنا نشوف قداش طول الطريق اللى جطعته لل
+
+68
+00:05:03,860 --> 00:05:08,880
+glider هذه من T عند بساوة سفر لعند T اش بتساوي
+
+69
+00:05:08,880 --> 00:05:14,370
+اتنين باىالان عالقانون مباشرة عندى اللى هو ال
+
+70
+00:05:14,370 --> 00:05:18,930
+position vector او اللى بيرسملى اللى هو مصار ال
+
+71
+00:05:18,930 --> 00:05:22,550
+particle او ال glider هذى اللى هو عبارة عن the
+
+72
+00:05:22,550 --> 00:05:26,210
+length of the glider's path from T بساوية 0 to T
+
+73
+00:05:26,210 --> 00:05:29,390
+بساوية Pi is اتفاقنا ال length ايش بساوية ال
+
+74
+00:05:29,390 --> 00:05:33,190
+integration من A ل B اللى هى 0 ل 2 Pi ال magnitude
+
+75
+00:05:33,190 --> 00:05:37,490
+ده ال V of T dtما هو V of T؟ طبعاً بتأتي بتفاضل
+
+76
+00:05:37,490 --> 00:05:43,790
+هذه اللي هي ناقص sine T في اتجاه I زائد cosine T
+
+77
+00:05:43,790 --> 00:05:50,150
+في اتجاه J زائد K مظبوط هذا اللي هو تفاضلها الان
+
+78
+00:05:50,150 --> 00:05:52,830
+ال magnitude اللي هو عبارة عن جدر التربيع اللي
+
+79
+00:05:52,830 --> 00:05:56,320
+اللي طلع هنا اللي هو ناقص sine T لكل تربيعزائد
+
+80
+00:05:56,320 --> 00:05:59,160
+اللي طالع حانه اللي هو cos T لكل تربيع زائد اللي
+
+81
+00:05:59,160 --> 00:06:02,520
+طالع حانه اللي هو تفاضل T واحد اللي هو واحد تربيع
+
+82
+00:06:02,520 --> 00:06:07,060
+اللي هو واحد الجدر التربيع ايه ده كله دي إيش دي T
+
+83
+00:06:07,060 --> 00:06:10,680
+بدنا نكامله نشوف كيف نكامله الجمعة طبعا كلكم قال
+
+84
+00:06:10,680 --> 00:06:13,560
+طب ما هي sin تربيع زاد cos تربيع واحد وعنا واحد من
+
+85
+00:06:13,560 --> 00:06:18,500
+اتنين اذا جدر اتنين دي T وجدر اتنين عبارة عن ثابت
+
+86
+00:06:18,500 --> 00:06:21,960
+تكامله على طول اللي هو جدر اتنين في T ومن 0 ل 2
+
+87
+00:06:21,960 --> 00:06:26,300
+باي فبصير عند اتنين في جدر اتنين في بايهذا عبارة
+
+88
+00:06:26,300 --> 00:06:30,540
+عن the length of the glider's path من T بتساوي سفر
+
+89
+00:06:30,540 --> 00:06:36,960
+لعين T بإيش بتساوي لعين T بتساوي اتنين باين هذه
+
+90
+00:06:36,960 --> 00:06:40,060
+اللي هي مثلا توضيح إلى اللي قبل بشوية
+
+91
+00:06:45,490 --> 00:06:49,290
+الان نحكي عن الرسمة هذه بس قبل ان نحكي عنها خلّينا
+
+92
+00:06:49,290 --> 00:06:53,370
+نشوف ايش اللي عندي if we choose the base point B T
+
+93
+00:06:53,370 --> 00:06:57,150
+note on a smooth curve طبعا نجيب احنا اللي بنشتغل
+
+94
+00:06:57,150 --> 00:07:00,990
+عليه عشان تكون في الصورة هو smooth curve ليش
+
+95
+00:07:00,990 --> 00:07:05,700
+smooth curve؟ عشان لسبب بسيط ال smooth curveالـ
+
+96
+00:07:05,700 --> 00:07:10,500
+derivative له موجودة و ليس كذلك و continuous طيب
+
+97
+00:07:10,500 --> 00:07:14,440
+موجود عشان نستعمل القانون نتفاق معك لكن طب و
+
+98
+00:07:14,440 --> 00:07:17,660
+continuous؟ آه عشان انا بتكون continuous بيصير ال
+
+99
+00:07:17,660 --> 00:07:19,880
+magnitude ل V of T اللي هو المبدع continuous
+
+100
+00:07:19,880 --> 00:07:22,400
+function بتصير هذا كلها اللي بتطلع عند continuous
+
+101
+00:07:22,400 --> 00:07:26,600
+يعني بعد اقدر ايش؟ اكملها عشان مابديش مشاكل عشان
+
+102
+00:07:26,600 --> 00:07:30,540
+أضمن يطلع من الال اللي هو قيمته لأن الدالة المتصلة
+
+103
+00:07:30,540 --> 00:07:34,080
+على طول في اله تكامل زي ما عرفنا في تفاضل اللي هو
+
+104
+00:07:34,080 --> 00:07:40,340
+الفطيب لان نفترض انه عندى .. نفترض نقطة .. نقطة
+
+105
+00:07:40,340 --> 00:07:43,460
+بداية خلينا او خلينا نقول نقطة انطلاق السؤال اللى
+
+106
+00:07:43,460 --> 00:07:46,740
+بده ماشي في هيكه او بده ماشي هيكه ماشي الحال هذا
+
+107
+00:07:46,740 --> 00:07:50,740
+بسميها base point اللي بيه of T noteالان هذه ال
+
+108
+00:07:50,740 --> 00:07:56,580
+base point لما نمشي اشي بسيط هنا حيصير عندى اللى
+
+109
+00:07:56,580 --> 00:08:01,600
+هو اتغيرت بي تي نوت لعند اللى هى مثلا مين بي اف تي
+
+110
+00:08:01,600 --> 00:08:06,940
+لهان فرضها ماشي الحال هذا اللى صار صار فيه طول هذا
+
+111
+00:08:06,940 --> 00:08:12,670
+الطول اللى من هان لهانبتغير بتغير main T لو طلعت
+
+112
+00:08:12,670 --> 00:08:16,570
+لفوق بطول شوية لو طلعت لفوق بطول شوية و هكذا عشان
+
+113
+00:08:16,570 --> 00:08:19,870
+هي كبدنا نسمي هذا احنا اللي قبل اخدناه من ثابت
+
+114
+00:08:19,870 --> 00:08:23,290
+لثابت من سفر لاثنين باي عشان يكتلع عند عدد لكن
+
+115
+00:08:23,290 --> 00:08:27,640
+الان انا بتاخده من P of T noteاللي هي عند ضغط الـ
+
+116
+00:08:27,640 --> 00:08:31,040
+T note لعند B of T لعند T يعني هيطلع ال
+
+117
+00:08:31,040 --> 00:08:35,040
+integration اللي عندي بدلالة مين بدلالة T يعني
+
+118
+00:08:35,040 --> 00:08:41,040
+الطول هذا بيطلع بدلالة T هذا الطول بدي أسميه S of
+
+119
+00:08:41,040 --> 00:08:44,820
+T في الاتجاه هذا لو مشيت بدي أعتبره أنا ماشي
+
+120
+00:08:44,820 --> 00:08:49,120
+بالاتجاه الموجبلو جيت رجعت رجوع من هان لهان بده
+
+121
+00:08:49,120 --> 00:08:52,900
+اعتبره ايه شماله في الاتجاه السالب هذا استلاحة
+
+122
+00:08:52,900 --> 00:08:56,200
+بيننا يعني الآن بدنا نقول من ال base point و امشي
+
+123
+00:08:56,200 --> 00:09:00,020
+هيك بالاتجاه هذا على اليمين يعني بده اعتبره modab
+
+124
+00:09:00,020 --> 00:09:03,220
+و لما نمشي هيك على الشمال بده اعتبره ايه اعتبره
+
+125
+00:09:03,220 --> 00:09:06,920
+ايه؟ سالب هذا اللي هو يعني الطول بحكي عن الطول
+
+126
+00:09:06,920 --> 00:09:10,480
+يعني الطول من هان لهان هذا as of t من هان لهان as
+
+127
+00:09:10,480 --> 00:09:14,390
+of t بس اختلفت as of t اختلفت كطول كطولو هنبدأ
+
+128
+00:09:14,390 --> 00:09:19,470
+نعتبرها إيه شمالة الطول لل curve أو لل arc بس
+
+129
+00:09:19,470 --> 00:09:24,390
+باتجاه السالم طيب هذا الآن if we choose a base
+
+130
+00:09:24,390 --> 00:09:26,810
+point زي ما قلنا of T not in a smooth curve C
+
+131
+00:09:26,810 --> 00:09:30,830
+parameterized by T each value of T determines a
+
+132
+00:09:30,830 --> 00:09:34,650
+point P of T يعني كل value of T بتحدد ل X of T و Y
+
+133
+00:09:34,650 --> 00:09:38,890
+of T و Z of T يعني تتغير قيمة اللي هي ال P of T
+
+134
+00:09:38,890 --> 00:09:42,750
+بتغير الزمن و طبعا تبعا لذلك يتغير اللي هي
+
+135
+00:09:43,860 --> 00:09:47,760
+الإحداثية الـ X والـ Y والـ Z on C الـ Directed
+
+136
+00:09:47,760 --> 00:09:51,360
+distance S of T اللي هي الـ Length دي بنسميها إيش
+
+137
+00:09:51,360 --> 00:09:55,400
+معله Directed distance إيش Directed distance
+
+138
+00:09:55,400 --> 00:10:00,300
+اتفاجنا إنه من T note لعندي في الاتجاه اللي هيك
+
+139
+00:10:00,300 --> 00:10:05,620
+هيعطيني موجة ال magnitude ده V of T دي TV of Tau D
+
+140
+00:10:05,620 --> 00:10:10,940
+Tau الان هذا هيطلع عندى اللى هو بدلالة مين بدلالة
+
+141
+00:10:10,940 --> 00:10:14,180
+T هو عبارة عن length في الواقع بس length من T
+
+142
+00:10:14,180 --> 00:10:17,760
+naught لعند مين عن T بعيد بالموجة لما T تكون إلى
+
+143
+00:10:17,760 --> 00:10:20,900
+أعلى و لو كانت إلى أسفل يعني جاب اللى هيعد بالاش
+
+144
+00:10:20,900 --> 00:10:24,320
+بالسالب الان هى زى ما قولنا FT أكبر من T naught S
+
+145
+00:10:24,320 --> 00:10:28,520
+of T is the distance along the curve from B of T
+
+146
+00:10:28,520 --> 00:10:35,510
+naught to B of Tإذا كان الـ T أقل من الـ T نوت ، S
+
+147
+00:10:35,510 --> 00:10:39,510
+of T هو نقل للمقارنة S of T هو نقل للمقارنة S of T
+
+148
+00:10:39,510 --> 00:10:40,770
+هو نقل للمقارنة S of T هو نقل للمقارنة S of T هو
+
+149
+00:10:40,770 --> 00:10:45,910
+نقل للمقارنة S of T هو نقل للمقارنة S of
+
+150
+00:10:45,910 --> 00:10:52,150
+T هو نقل للمقارنةلأن S بتنوجد هنا بدلالة T اللي
+
+151
+00:10:52,150 --> 00:10:56,750
+بتطلع هنا إذا كل قيمة ل S هنا أو هنا أو هنا أو هنا
+
+152
+00:10:56,750 --> 00:11:01,630
+هي في الواقع مقابلة لقيمة في مين في T لإن إحنا
+
+153
+00:11:01,630 --> 00:11:06,430
+أصلا لما جيبنا S في T جيبناها من T نقطة و اعتمدت
+
+154
+00:11:06,430 --> 00:11:13,250
+على T الان نسميها S and R length parameter الان S
+
+155
+00:11:13,250 --> 00:11:18,150
+هذه اللي هي lengthلكن تتغير بتغير الـ T فعشان هيك
+
+156
+00:11:18,150 --> 00:11:23,310
+بنسميها Arc Length Parameter يعني الآن تستغرب انه
+
+157
+00:11:23,310 --> 00:11:29,050
+اصلا انت ال R of T اللي قبل بشوية هذا اللي جيبناه
+
+158
+00:11:29,050 --> 00:11:34,870
+بدلالة T بعد شوية هقدر اجيب لكيه بدلالة S يعني ال
+
+159
+00:11:34,870 --> 00:11:37,550
+parameter بدل هيكون اللي هو T هيكون ال parameter
+
+160
+00:11:37,550 --> 00:11:42,410
+من S اللي هو عبارة عن ال lengthأو الـ R-Link
+
+161
+00:11:42,410 --> 00:11:47,350
+Parameter لنرى ماذا يقول هذا البراميطر البراميطر
+
+162
+00:11:47,350 --> 00:11:48,130
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+163
+00:11:48,130 --> 00:11:49,810
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+164
+00:11:49,810 --> 00:11:52,190
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+165
+00:11:52,190 --> 00:11:53,290
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+166
+00:11:53,290 --> 00:11:55,610
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+167
+00:11:55,610 --> 00:11:56,750
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+168
+00:11:56,750 --> 00:12:01,000
+البراميطر البراميطر الand for investigating the
+
+169
+00:12:01,000 --> 00:12:03,240
+turning and twisting nature of a space curve هذا
+
+170
+00:12:03,240 --> 00:12:08,820
+مستقبلنا في شبطر تلتاش حنحكي عن ال twisting و
+
+171
+00:12:08,820 --> 00:12:13,060
+turning curves ايه ده اللي هو قدر لنا ان شاء الله
+
+172
+00:12:13,060 --> 00:12:19,420
+فهذا الاسس مهم بالنسبة لهذا اللي هو الموضوعين طيب
+
+173
+00:12:19,980 --> 00:12:24,140
+الان نرجع للـ R Length Parameter with Base Point B
+
+174
+00:12:24,140 --> 00:12:29,920
+of T Note يعني نقطة الانطلاق أو نقطة الميزان عند T
+
+175
+00:12:29,920 --> 00:12:33,360
+Note هيك هيك و هيطلع بالموجب و هيك هيطلع بياشف
+
+176
+00:12:33,360 --> 00:12:36,680
+السلبي زي ما العادي احنالما ناخد خط الأعداد و
+
+177
+00:12:36,680 --> 00:12:40,160
+ناخده نقطة 0 عليكوا علي يمين موجبة علي اسمها سالب
+
+178
+00:12:40,160 --> 00:12:43,680
+هذه نفس الفكرة P of T هو اللي هيمشي في increasing
+
+179
+00:12:43,680 --> 00:12:46,720
+T بالشكل اللي هذا هتطلع S موجبة و في ال decreasing
+
+180
+00:12:46,720 --> 00:12:50,460
+لل T في الاتجاه هذا هتطلع ليه S سالب ان شاء الله
+
+181
+00:12:50,460 --> 00:12:55,360
+تكونوا فانين عليها S of T بتساويحسب القانون T ل T
+
+182
+00:12:55,360 --> 00:12:58,860
+note اللى كنا نقول عنها من A ل B الجدرى التربيعى ل
+
+183
+00:12:58,860 --> 00:13:04,920
+D X by D Tau اللى هى بس جلبنا .. غيرنا اللى هى بدل
+
+184
+00:13:04,920 --> 00:13:10,000
+T Tau عشان نفرجها عن هذه اما هى نفسها بالفاضل اللى
+
+185
+00:13:10,000 --> 00:13:15,500
+هى ال RDx by Dt و نربحها و Dy by Dt و نربحها و Dz
+
+186
+00:13:15,500 --> 00:13:18,720
+by Dt و نربحها و ناخدها تحت جدر التربيع و ناخدها
+
+187
+00:13:18,720 --> 00:13:18,980
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+188
+00:13:18,980 --> 00:13:19,500
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+189
+00:13:19,500 --> 00:13:19,780
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+190
+00:13:19,780 --> 00:13:20,060
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+191
+00:13:20,060 --> 00:13:22,520
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+192
+00:13:22,520 --> 00:13:28,730
+تحت جدر التويساوي الـ integration من T0 لعنطي الـ
+
+193
+00:13:28,730 --> 00:13:32,270
+magnitude للـ V of τاو دي تاو دي تاو هذا كلام مكرر
+
+194
+00:13:32,270 --> 00:13:37,890
+اللي هو هذا هو هذا زي ما حكينا قبل بشوية طيب صلوا
+
+195
+00:13:37,890 --> 00:13:41,210
+عنا بيه يا جماعة لإن أخد مثال this is an example
+
+196
+00:13:41,210 --> 00:13:44,170
+for which we can actually find the arc length
+
+197
+00:13:44,170 --> 00:13:48,330
+parameterization of a curveبدنا نجد الـ R-length
+
+198
+00:13:48,330 --> 00:13:51,330
+parameterization of a curve زي ما حكي انا قبل شوية
+
+199
+00:13:51,330 --> 00:13:56,250
+نشوف ايش اللي بقوله if T0 بتساوي 0 the R-length
+
+200
+00:13:56,250 --> 00:13:59,610
+parameter along the helix R of T بتساوي Cos of T
+
+201
+00:13:59,610 --> 00:14:03,150
+في تجاه I زي Sin of T في تجاه J زي T في تجاه K
+
+202
+00:14:03,150 --> 00:14:09,290
+يعني هاي اللي هو ال position vector او اللي هو
+
+203
+00:14:09,290 --> 00:14:13,290
+اللي بده يرسملي اللي هو مين ال curve اللي احنا
+
+204
+00:14:13,290 --> 00:14:17,480
+بدنا نحكي عنهالان لما T note بالساوية سفر بدنا
+
+205
+00:14:17,480 --> 00:14:21,360
+ناخد S of T بالساوية من T note لعند T لل V of Tau
+
+206
+00:14:21,360 --> 00:14:25,040
+D Tau هاي قانوننا T note في هذه الحالة ايش اخدناها
+
+207
+00:14:25,040 --> 00:14:31,420
+احنا سفر اذا من سفر لعند مين لعند Tالان v of tau
+
+208
+00:14:31,420 --> 00:14:35,640
+حسبناها هو نفس المثال اللى فات حسبناها جدش طلعت ان
+
+209
+00:14:35,640 --> 00:14:39,800
+جدر التنين دي tau الا ان المفادل هذه بيصير جدر
+
+210
+00:14:39,800 --> 00:14:44,880
+التنين tau من zero لعين T يعني بيصير جدر التنين T
+
+211
+00:14:44,880 --> 00:14:49,420
+إذا صارت ال S of T عندى يا جماعة عبارة عن جدر
+
+212
+00:14:49,420 --> 00:14:55,570
+التنين Tالان احنا هذا الـ S الـ Length من T نوت
+
+213
+00:14:55,570 --> 00:15:00,290
+التي يتغير بتغير الـ T لاحظوا S جبتها بدلالة مين
+
+214
+00:15:00,290 --> 00:15:04,570
+الـ T الان انا بقدر اجيب T بدلالة الـ S بنحلها و
+
+215
+00:15:04,570 --> 00:15:08,810
+لا في أسهل منها T ايش هتساوي S على جدر التنين هذه
+
+216
+00:15:08,810 --> 00:15:12,070
+على جدر التنين بكسب تصير T بساوي S على جدر التنين
+
+217
+00:15:12,070 --> 00:15:16,330
+الان هذه الـ S اللي هي عبارة عن مين؟ عن الـ Arc
+
+218
+00:15:16,330 --> 00:15:21,980
+Lengthأه براميتر ال arc length براميتر الان صرت
+
+219
+00:15:21,980 --> 00:15:28,480
+مؤهل ان اجيب arc بدلالة ال S قد تلزمني الان هو طلب
+
+220
+00:15:28,480 --> 00:15:32,680
+مني ان اوجد اللي هي ال arc بدلالة ال R length
+
+221
+00:15:32,680 --> 00:15:37,520
+براميتر اللي هي بدلالة ال S الان arc بدلالة ال S
+
+222
+00:15:37,520 --> 00:15:42,250
+ايش هتساويبشيل كل T و بحط قيمتها اللي هي بيصير cos
+
+223
+00:15:42,250 --> 00:15:46,050
+بدل ال T أش بتصير S على جدر التنين في تجاه ال I
+
+224
+00:15:46,050 --> 00:15:49,330
+sign S على جدر التنين في تجاه ال I زاد S على جدر
+
+225
+00:15:49,330 --> 00:15:52,110
+التنين في تجاه ال K وهذه قدرنا نكتب ال position
+
+226
+00:15:52,110 --> 00:15:57,650
+vector R أو اللي بيحددلي مصار ال K ال particle أو
+
+227
+00:15:57,650 --> 00:16:04,430
+اللي هو مصار ال curve جبته بواسطة أو بدلالةاللي هو
+
+228
+00:16:04,430 --> 00:16:09,790
+الـ R Length Parameter اللي هي S ماشي يا جماعة
+
+229
+00:16:09,790 --> 00:16:14,930
+الان بدنا speed on a smooth curve بدنا اللي هو ايش
+
+230
+00:16:14,930 --> 00:16:19,550
+نشوف اللي هو السرعة مش المتجهة لأ السرعة مش متجهة
+
+231
+00:16:19,550 --> 00:16:22,190
+السرعة السرعة اللي هو ال magnitude on a smooth
+
+232
+00:16:22,190 --> 00:16:26,980
+curve عند the smooth curveاللي هو اللي بيصنعوا R
+
+233
+00:16:26,980 --> 00:16:31,880
+of T الآن بدنا نودد اللي هو السبيل عليه ال
+
+234
+00:16:31,880 --> 00:16:38,880
+magnitude لل V of T اللي هي انت الآن هيأس ب 70 نوت
+
+235
+00:16:38,880 --> 00:16:42,900
+اللي عندك VTAU وDTAUمظبوط هي القانون تبعنا الآن
+
+236
+00:16:42,900 --> 00:16:45,360
+بالـ Fundamental Theorem of Calculus كل شيء حلو
+
+237
+00:16:45,360 --> 00:16:49,400
+لأن هذا continuous لأننا مفترضين أن نتحدث عن
+
+238
+00:16:49,400 --> 00:16:52,000
+Smooth Curve مزام Smooth Curve لأن الـ derivative
+
+239
+00:16:52,000 --> 00:16:54,740
+لهذه موجودة وهذه موجودة وهذه موجودة وcontinuous
+
+240
+00:16:54,740 --> 00:16:58,580
+يعني هذا continuous مدام هذا continuous إذن DS by
+
+241
+00:16:58,580 --> 00:17:05,060
+DTاللي هو على طول بنزيل هذا و بنحط اللي هو يعني
+
+242
+00:17:05,060 --> 00:17:09,820
+التفاضل بالغ التكامل و بيصير بيصير بساوة V of T
+
+243
+00:17:09,820 --> 00:17:14,440
+يعني دي أس باي دي تي بيساوي ال magnitude ده V of T
+
+244
+00:17:14,440 --> 00:17:18,060
+يعني و كأنه بقول ال equation أربعة says that the
+
+245
+00:17:18,060 --> 00:17:23,750
+speedwith which a particle moves يعني الـ speed
+
+246
+00:17:23,750 --> 00:17:26,250
+للبارتكال هذه الـ speed للبارتكال تلك اللي احنا
+
+247
+00:17:26,250 --> 00:17:32,750
+عارفينها ايش ساوت اللي هي عبارة عن تفاضل الار
+
+248
+00:17:32,750 --> 00:17:38,250
+length with respect to time to Tيعني with which a
+
+249
+00:17:38,250 --> 00:17:41,450
+particle moves along its path is the magnitude of
+
+250
+00:17:41,450 --> 00:17:46,110
+v consistent with what we note اللي هو انه احنا
+
+251
+00:17:46,110 --> 00:17:50,450
+اللي بنعرفه انه هذه ال speed لل particle بساوية
+
+252
+00:17:50,450 --> 00:17:54,910
+اللي هي عبارة عن تفاضل ال arc length المسافة
+
+253
+00:17:54,910 --> 00:18:01,150
+المقطوعة هذه عالميا بالنسبة لمين بالنسبة ل Tهذه
+
+254
+00:18:01,150 --> 00:18:06,750
+اللي هي الـ Speed اللي هي تفاضل المسافة المقطوعة
+
+255
+00:18:06,750 --> 00:18:13,430
+على M على T طبعا المسافة بإيش يا جماعة بدون ما
+
+256
+00:18:13,430 --> 00:18:17,950
+نحكي عن Direction الأسعار عن Length هيك أبو جبه و
+
+257
+00:18:17,950 --> 00:18:23,620
+هيك سالبه و بس مش Directionهذه التزايد تصبح موجبة
+
+258
+00:18:23,620 --> 00:18:26,080
+الـ S إذا كانت T متزايدة تصبح تزايدة سالفة إذا
+
+259
+00:18:26,080 --> 00:18:26,480
+كانت T متزايدة تصبح سالفة إذا كانت T متزايدة تصبح
+
+260
+00:18:26,480 --> 00:18:27,540
+سالفة إذا كانت T متزايدة تصبح سالفة إذا كانت T
+
+261
+00:18:27,540 --> 00:18:27,740
+متزايدة تصبح سالفة إذا كانت T متزايدة تصبح سالفة
+
+262
+00:18:27,740 --> 00:18:27,800
+إذا كانت T متزايدة تصبح سالفة إذا كانت T متزايدة
+
+263
+00:18:27,800 --> 00:18:28,740
+T متزايدة تصبح سالفة إذا كانت T متزايدة تصبح سالفة
+
+264
+00:18:28,740 --> 00:18:31,660
+إذا كانت T متزايدة تصبح سالفة
+
+265
+00:18:35,760 --> 00:18:39,060
+للـ unit tangent vector بدنا نعرف حاجة اسمها ال
+
+266
+00:18:39,060 --> 00:18:42,380
+unit tangent vector و هنجيله برضه بالتفصيل قدام ان
+
+267
+00:18:42,380 --> 00:18:46,860
+شاء الله خلينا نشوف we already know the velocity
+
+268
+00:18:46,860 --> 00:18:51,560
+vector V بسوء DR by DT الان ال velocity vector V
+
+269
+00:18:51,560 --> 00:18:57,300
+بسوء DR by DT اللي هي المشتقة دي DR by DT اللي هو
+
+270
+00:18:57,300 --> 00:19:00,740
+عبارة عن المماثلة مين لكيرف عندها R of T هذا ايش
+
+271
+00:19:00,740 --> 00:19:04,880
+رسمت مين الرسمة اللي بترسم مين اللي هي R of Tالأن
+
+272
+00:19:04,880 --> 00:19:11,080
+DR by DT هي عبارة عن الـ tangent لمن؟ اللي هو لل
+
+273
+00:19:11,080 --> 00:19:15,320
+curve R of T إذا V بيسوى DR by DT is the tangent
+
+274
+00:19:15,320 --> 00:19:18,700
+to the curve R of T and that the vector لأن هذا هو
+
+275
+00:19:18,700 --> 00:19:22,220
+ال tangent DR by DT هذا هي ال tangent هذه اللي هي
+
+276
+00:19:22,220 --> 00:19:27,080
+من؟ ال derivative اللي هي ال derivative لمن؟ لـ R
+
+277
+00:19:27,080 --> 00:19:30,840
+اللي هي اللي بترسم ال curve اللي عندي الآن أنا
+
+278
+00:19:30,840 --> 00:19:35,690
+معنى بمين؟مش اللي هو dR by dt اللي هو ال tangent
+
+279
+00:19:35,690 --> 00:19:40,650
+بس لأ معنى بال unit tangent vector و حاجة هستخدمها
+
+280
+00:19:40,650 --> 00:19:44,990
+قدام كثير عشان هي كمان ده افردلها رمز لحاله اللي
+
+281
+00:19:44,990 --> 00:19:49,990
+هو ساوي T capital اللي هي عبارة عن ال velocity على
+
+282
+00:19:49,990 --> 00:19:52,690
+ال magnitude ده ال V اللي هو ال speed يعني بمعنى
+
+283
+00:19:52,690 --> 00:19:59,050
+اخر هذه ده directionاللي هو of مين is the
+
+284
+00:19:59,050 --> 00:20:03,930
+direction of .. اللي هو حركة ال particle اللي هي
+
+285
+00:20:03,930 --> 00:20:07,990
+اللي بنسميها ال unit tangent vector لإنه مكون
+
+286
+00:20:07,990 --> 00:20:11,990
+tangent لمين لل curve هذا واخدت ال unit منه بس
+
+287
+00:20:11,990 --> 00:20:15,450
+اللي طوله واحد اللي هو V ع ال magnitude لل V هيك
+
+288
+00:20:15,450 --> 00:20:20,460
+بدنا نعرفه طيب ناخد مثال الان الان بتقولي fivethe
+
+289
+00:20:20,460 --> 00:20:23,960
+unit tangent vector of the curve R of T بساوية cos
+
+290
+00:20:23,960 --> 00:20:27,440
+T زائد تلاتة sin T كل واحدة في اتجاهها زائد T
+
+291
+00:20:27,440 --> 00:20:30,960
+تربيع في اتجاهها طيب شوفوا يا جماعة صلوا على النبي
+
+292
+00:20:30,960 --> 00:20:33,720
+عليه الصلاة والسلام لأن عشان اوجد ال unit tangent
+
+293
+00:20:33,720 --> 00:20:37,860
+vector بالزمن اوجد ال velocity DR by DT بساوية
+
+294
+00:20:37,860 --> 00:20:41,860
+كلكم قال ناقص تلاتة sin T في اتجاه ال I زائد تلاتة
+
+295
+00:20:41,860 --> 00:20:45,100
+cos T في اتجاه ال J بفاضل قاعد زائد اتنين T في
+
+296
+00:20:45,100 --> 00:20:49,150
+اتجاه ال Kواحد قال لطب هو هذا ال V وانت بالتأكيد
+
+297
+00:20:49,150 --> 00:20:51,190
+بيلزمك ميه ان ال magnitude ده ال V اه ال magnitude
+
+298
+00:20:51,190 --> 00:20:54,430
+ده ال V ايش بيساوي جدر التربيعي لهذا تربيع زاد هذا
+
+299
+00:20:54,430 --> 00:20:58,330
+تربيع اللي هي تلاتة sign يعني تسعة sign تربيع زاد
+
+300
+00:20:58,330 --> 00:21:02,890
+تسعة cosine تربيع زاد أربع ت تربيع تسعة sign تربيع
+
+301
+00:21:02,890 --> 00:21:05,710
+زاد تسعة cosine تربيع باخد التسعة عامل مشترك برا
+
+302
+00:21:05,710 --> 00:21:08,330
+بيصير تسعة في sign تربيع زاد cosine تربيع واحد
+
+303
+00:21:08,330 --> 00:21:13,260
+يعني تسعةزائد من هذه أربعة T تربية إذا صارت ال
+
+304
+00:21:13,260 --> 00:21:15,940
+magnitude ل V هو عبارة عن جدر التربية إلى الأربعة
+
+305
+00:21:15,940 --> 00:21:21,340
+زائد T تربية إذا صارت ال unit tangent vector T
+
+306
+00:21:21,340 --> 00:21:27,260
+بيساوي V على magnitude ل V اللي هي هذا المقدار أو
+
+307
+00:21:27,260 --> 00:21:31,760
+هذا اللي هو ال vector على magnitude و بيصير ناقص
+
+308
+00:21:31,760 --> 00:21:35,000
+تلاتة sin T في تجاه I زائد تلاتة cos T في تجاه J
+
+309
+00:21:35,000 --> 00:21:35,240
+زائد
+
+310
+00:21:49,730 --> 00:21:54,630
+الحركة عكس التجاه من عقارب الساعة
+
+311
+00:21:59,270 --> 00:22:02,930
+لو ماشيين على اللي هو فيه عندى الـ vector R ماشي
+
+312
+00:22:02,930 --> 00:22:06,250
+على اللي هو مين الدائرة أكيد ليش ماشي عدّان لأن R
+
+313
+00:22:06,250 --> 00:22:09,550
+of T بيساوي cos T في تجاه I زاد sin T في تجاه G
+
+314
+00:22:09,550 --> 00:22:12,650
+هذا بيبقى metric equation للـR لكن العلاقة بين cos
+
+315
+00:22:12,650 --> 00:22:16,870
+T وsin T دايما cos تربيع زاد sin تربيع بيساوي 1
+
+316
+00:22:17,560 --> 00:22:22,620
+بيساوي الـ R يعني الان cosine تربيع زائد sin تربيع
+
+317
+00:22:22,620 --> 00:22:26,920
+بيساوي قيمة R لو فرضنا ان احنا ماشيين يعني دائما
+
+318
+00:22:26,920 --> 00:22:30,600
+اللي هو ماشي ايش بمقدار اللي هو ال magnitude
+
+319
+00:22:30,600 --> 00:22:34,880
+بيساوي واحد لل R بيصير عندي انا ماشي على اللي هو
+
+320
+00:22:34,880 --> 00:22:39,160
+دائرة الوحدةدائرة الوحدة هذا اللي هو ماشيين على
+
+321
+00:22:39,160 --> 00:22:42,500
+مين لانقصان تربيع زي اقصان تربيع دايما ايش بيساوي
+
+322
+00:22:42,500 --> 00:22:47,260
+بيساوي واحد فصار عند ال magnitude له على طول بيطلع
+
+323
+00:22:47,260 --> 00:22:50,820
+ايش بيطلع واحد يعني بدون ما اقول انه والله واحد او
+
+324
+00:22:50,820 --> 00:22:53,480
+كده هو هيطلع automatic واحد زي ما قلنا اقصان تربيع
+
+325
+00:22:53,480 --> 00:22:57,200
+زي اقصان تربيع اللي هو X تربيع زي Y تربيع هيساوي
+
+326
+00:22:57,200 --> 00:22:59,940
+واحد فاهمين عليها لان اقصان تربيع زي اقصان تربيع
+
+327
+00:22:59,940 --> 00:23:03,120
+هيساوي واحد اذا ال X تربيع زي Y تربيع بساوي واحد
+
+328
+00:23:03,120 --> 00:23:08,330
+اذا انا ماشي على دائرة الوحدةعبر هذا الـ vector أو
+
+329
+00:23:08,330 --> 00:23:13,070
+اللي برسم ال curve around the unit circle we see
+
+330
+00:23:13,070 --> 00:23:16,890
+that ال velocity كلكم قال فاضل هذه و فاضل هذه تطلع
+
+331
+00:23:16,890 --> 00:23:21,810
+نقص sin t زي cos t الآن نقص sin t في اتجاه ال I زي
+
+332
+00:23:21,810 --> 00:23:26,770
+cos t في اتجاه ال J is already برضه ايه شماله unit
+
+333
+00:23:26,770 --> 00:23:30,430
+vector لأن ال magnitude له الجدر التربيعي لهذه
+
+334
+00:23:30,430 --> 00:23:34,160
+التربيه زي هذه التربيه اللي هو واحدإذا اللي هو في
+
+335
+00:23:34,160 --> 00:23:39,740
+هذه الحالة على اللي هي الدائرة دائرة الوحدة بيكون
+
+336
+00:23:39,740 --> 00:23:45,440
+ال unit tangent vector هو بالظبط من ال velocity
+
+337
+00:23:45,440 --> 00:23:48,640
+إذا ال unit tangent vector عندي بيساوي ال velocity
+
+338
+00:23:48,640 --> 00:23:54,080
+in this case because the magnitude of v in this
+
+339
+00:23:54,080 --> 00:24:01,700
+case is oneبناء عليه اللي هو عندي الآن في بعض
+
+340
+00:24:01,700 --> 00:24:08,460
+الشغلات اللي هو الـ Dt by Ds عبارة عن واحد على Ds
+
+341
+00:24:08,460 --> 00:24:12,120
+by Dt زي ما انتوا عارفين اللي هي الـ Dt by Ds اللي
+
+342
+00:24:12,120 --> 00:24:17,740
+هي التفاضل اللي هو الزمن بالنسبة للمسافة بالنسبة
+
+343
+00:24:17,740 --> 00:24:20,600
+لل arc length بساوي واحد على ال magnitude لل V
+
+344
+00:24:21,780 --> 00:24:26,040
+والان هذه بعض العلاقات اللى ممكن تلزمنا جدام عند
+
+345
+00:24:26,040 --> 00:24:31,060
+الـDR by DS اللى هو بالـ Chain و الـ R طبعا عارفين
+
+346
+00:24:31,060 --> 00:24:34,060
+ايش هي الـ R هيها اللى هي الـ Position Vector الـ
+
+347
+00:24:34,060 --> 00:24:37,140
+S عارفين هي الـ R Length لو انه ده فاضل DR by DS
+
+348
+00:24:37,140 --> 00:24:41,370
+انه قبل شوية دي بنى R بدلالة Sبنشوف ايش DR by DS
+
+349
+00:24:41,370 --> 00:24:45,530
+بيساوي DR by DT في DT by DS اللي هي الـ Chain Rule
+
+350
+00:24:45,530 --> 00:24:50,910
+ويساوي DR by DT مين هي يا جماعة اللي هي عبارة عن
+
+351
+00:24:50,910 --> 00:24:54,830
+الـ V عارفينها من زمان DT by DS توا اللي هو
+
+352
+00:24:54,830 --> 00:24:57,690
+استنتجناه اللي عبارة عن واحدة ال magnitude للـ V
+
+353
+00:24:57,960 --> 00:25:00,300
+إذا صارت الـ V على الـ magnitude ده الـ V اللي هو
+
+354
+00:25:00,300 --> 00:25:03,600
+مين اللي هو T الـ unit tangent vector و كأنه
+
+355
+00:25:03,600 --> 00:25:07,660
+بيقوللي اسمع ال unit tangent vector ممكن تجيبه
+
+356
+00:25:07,660 --> 00:25:12,280
+كمان بطريقة تانية انك اتفاضل ال R بالنسبة لمين
+
+357
+00:25:12,280 --> 00:25:17,480
+للأسد اتفاضل اللي هو الليالـ R اللي هي دالة
+
+358
+00:25:17,480 --> 00:25:21,560
+الإزاحة أو كذا بالنسبة للـ Arc Length أو اللي هي
+
+359
+00:25:21,560 --> 00:25:25,120
+دالة الـ Position Vector بالنسبة للـ Arc Length
+
+360
+00:25:25,120 --> 00:25:29,360
+تبعتها بيطلع عند دي اربع يدي اسم ساومين ال unit
+
+361
+00:25:29,360 --> 00:25:32,780
+tangent vector و هيك بيكون خلصنا المحاضرة و إلى
+
+362
+00:25:32,780 --> 00:25:35,960
+لقاء آخر السلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P92tIJL5vI0_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P92tIJL5vI0_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1fdab5c9a4f6c5d9b746e761ab4708615306b863
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/P92tIJL5vI0_raw.srt
@@ -0,0 +1,1456 @@
+1
+00:00:01,200 --> 00:00:04,160
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,160 --> 00:00:10,120
+وبركاته هذه هي المحاضرة العاشرة تفاضل وتكامل جيم
+
+3
+00:00:10,120 --> 00:00:16,060
+وهي المحاضرة الثالثة في chapter 13 section اليوم
+
+4
+00:00:16,060 --> 00:00:22,480
+اللي هو 13 3 R length in space اللي هو طول القوس
+
+5
+00:00:22,480 --> 00:00:30,750
+أو طول اللي هو المنحنة في الفرارالان احنا إذا
+
+6
+00:00:30,750 --> 00:00:37,410
+بتتذكروا في تفاضل الف في آخر chapter في تفاضل الف
+
+7
+00:00:37,410 --> 00:00:41,130
+أخدنا كيف نودد ال length لل curve لو كان عندنا
+
+8
+00:00:41,130 --> 00:00:45,590
+اللي هو arc أو curve عندنا في اللي هو كان بس مش في
+
+9
+00:00:45,590 --> 00:00:50,290
+الفراغكان عندنا إذا بتتذكروا الـ arc أو ال curve
+
+10
+00:00:50,290 --> 00:00:55,310
+كان موجود في ال .. في .. في لبنان فكنا نقول ال ..
+
+11
+00:00:55,310 --> 00:00:58,990
+ال .. ال .. ال .. ال ال length اللي هو عبارة عن ال
+
+12
+00:00:58,990 --> 00:01:02,810
+integration من a ل b الجدر التربيع إلى dx by dt
+
+13
+00:01:02,810 --> 00:01:07,130
+لكل تربيع زي ال dy by dt لكل تربيع لكل إيش معله؟
+
+14
+00:01:07,130 --> 00:01:11,990
+dt لأنه كان عندنا اللي هو الشغل بس في لبنانلكن
+
+15
+00:01:11,990 --> 00:01:16,090
+الآن حديثنا احنا اللي هو عن اللي هو الـ Arc أو الـ
+
+16
+00:01:16,090 --> 00:01:21,310
+Curve في الـ Space لو كان في عندنا اللي هو Arc في
+
+17
+00:01:21,310 --> 00:01:25,590
+الـ Space بدنا نشوف كيف نوجد اللي هو طول الـ Arc
+
+18
+00:01:25,590 --> 00:01:32,320
+اللي عندناالـ Arc طوله الان نفس الأسلوب اللي حلنا
+
+19
+00:01:32,320 --> 00:01:37,000
+فيه اللي هو في التفاضل ألف ولكن دخل عندنا اللي هو
+
+20
+00:01:37,000 --> 00:01:42,100
+متغير جديد اللي هو Z of T أو اللي هو ال coordinate
+
+21
+00:01:42,100 --> 00:01:47,430
+لـ Z فصار عندى الآن اللي هو ال .. ال .. ال ..طول
+
+22
+00:01:47,430 --> 00:01:54,050
+الـ curve اللي هو بدلالة الـ X و الـ Y و الـ Z بدل
+
+23
+00:01:54,050 --> 00:01:56,830
+المكان اللي هو بدلالة الـ X و الـ Y خلّينا نكون
+
+24
+00:01:56,830 --> 00:02:00,690
+أكثر تحديدا تعالوا نعرف اللي هو تعريفنا definition
+
+25
+00:02:00,690 --> 00:02:04,410
+the length of a smooth curve طبعا احنا R of T
+
+26
+00:02:04,410 --> 00:02:07,810
+بيساوي X of T فتجاه I زي Y of T فتجاه G زي Z of T
+
+27
+00:02:07,810 --> 00:02:11,650
+فتجاه K قلنا إن هذا اللي هو برسم لنا curve زي ما
+
+28
+00:02:11,650 --> 00:02:19,950
+اتفجناهي حركة الـ Particle وفق الـ Position Vector
+
+29
+00:02:19,950 --> 00:02:24,830
+اللي أمامي ببدأ عند نقطة T Note و بستمر و كل ما
+
+30
+00:02:24,830 --> 00:02:28,590
+اتحرك بتتحرك اللي هي نقاط الـ X و Y و Z و بتعملنا
+
+31
+00:02:28,590 --> 00:02:30,130
+Curve
+
+32
+00:02:32,090 --> 00:02:36,990
+من عند T من عند A لعند B يعني الان T تتحرك من A
+
+33
+00:02:36,990 --> 00:02:40,590
+لعند B بناء عليه ال R of T برسم لنا اللي هو ال
+
+34
+00:02:40,590 --> 00:02:43,350
+position vector برسم لنا ال curve لنشوف كيف اللي
+
+35
+00:02:43,350 --> 00:02:48,470
+هو جداش طول هذا ال curve اللي برسمه R of T لما T
+
+36
+00:02:48,470 --> 00:02:52,370
+تبدأ من عند ال A و تمتى عند مين؟ عند ال B التعريف
+
+37
+00:02:52,370 --> 00:02:56,130
+بالظبط اللي هو that is traced exactly once as T
+
+38
+00:02:56,130 --> 00:02:59,250
+increasing from T تساوة A to T تساوة B يعني بقولي
+
+39
+00:02:59,530 --> 00:03:04,290
+اللي هو R of T اللي هو عند كل نقطة T بترسم بقص مرة
+
+40
+00:03:04,290 --> 00:03:07,910
+واحدة تي اتغيرت تي بترسم كما اللي هي النقطة اللي
+
+41
+00:03:07,910 --> 00:03:10,850
+بعيدة هو اكيد لما ترسم على ال curve جداش بالظبط
+
+42
+00:03:10,850 --> 00:03:12,770
+طول ال curve زي ما قلنا ال length أقل بالساوية ال
+
+43
+00:03:12,770 --> 00:03:15,790
+integration من A لB الجدر التربيع على DX by DT لكل
+
+44
+00:03:15,790 --> 00:03:20,210
+تربيع زائد DY by DT لكل تربيع زائد DZ by DT لكل
+
+45
+00:03:20,210 --> 00:03:24,810
+تربيع هذا كله تحت الجدر اسمه ايه بنكمله بالنسبة
+
+46
+00:03:24,810 --> 00:03:29,820
+لمين التيه اللي هي DTالان لو جينا طلعنا لهذا
+
+47
+00:03:29,820 --> 00:03:34,520
+المقدار احنا ماعرفين ايش هو لو جينا اخدنا دي ار
+
+48
+00:03:34,520 --> 00:03:36,820
+باي دي تي دي ار باي دي تي زي ما انتوا تعرفوا اللي
+
+49
+00:03:36,820 --> 00:03:39,860
+هو عبارة عن V يا جماعة الان دي ار باي دي تي بيصير
+
+50
+00:03:39,860 --> 00:03:43,300
+دي X باي دي تي في اتجاه ال I زائد دي Y باي دي تي
+
+51
+00:03:43,300 --> 00:03:47,880
+في اتجاه ال J زائد دي Z باي دي تي في اتجاه ال Kهذا
+
+52
+00:03:47,880 --> 00:03:51,900
+اللي هي الـ V صارت الان ال magnitude لها ال
+
+53
+00:03:51,900 --> 00:03:55,300
+magnitude ل V ال magnitude ده اللي فضلناها ده بطلع
+
+54
+00:03:55,300 --> 00:03:58,420
+الجدر التربيع إلى DX by DT لكل تربيع زي ال DIY by
+
+55
+00:03:58,420 --> 00:04:03,020
+DT لكل تربيع زي ال DZ by DT لكل تربيع إذا هذا
+
+56
+00:04:03,020 --> 00:04:06,720
+المقدار بنقدر نقول بالظبط ال length بالساعة ال
+
+57
+00:04:06,720 --> 00:04:11,340
+integration من ال A إلى B لل magnitude ل V اللي هو
+
+58
+00:04:11,340 --> 00:04:15,620
+DT وهذا هو ال length of the arc
+
+59
+00:04:18,140 --> 00:04:25,000
+طيب اتماع نيجي الان ناخد مثال الان a glider is
+
+60
+00:04:25,000 --> 00:04:29,640
+soaring upward along the helix نفترض ان في عندى
+
+61
+00:04:29,640 --> 00:04:36,040
+اللي هو particle او glider اللي هو بمشي عبر مين
+
+62
+00:04:36,040 --> 00:04:40,680
+علىاللي هو مصاره مصار Helix يعني على الـ Helix زي
+
+63
+00:04:40,680 --> 00:04:44,900
+المثال اللي أخدناه الـ Helix اللي هي Cos T باتجاه
+
+64
+00:04:44,900 --> 00:04:49,400
+I زي Cos T باتجاه G زي T باتجاه K نفترض إنه اللي
+
+65
+00:04:49,400 --> 00:04:55,800
+هو الـ Glider هذه بتمشي على الـ Helixبدنا نشوف how
+
+66
+00:04:55,800 --> 00:04:59,460
+long is the glider's path from T بساوة سفر to T
+
+67
+00:04:59,460 --> 00:05:03,860
+بساوة باى بدنا نشوف قداش طول الطريق اللى جطعته لل
+
+68
+00:05:03,860 --> 00:05:08,880
+glider هذه من T عند بساوة سفر لعند T اش بتساوي
+
+69
+00:05:08,880 --> 00:05:14,370
+اتنين باىالان عالقانون مباشرة عندى اللى هو ال
+
+70
+00:05:14,370 --> 00:05:18,930
+position vector او اللى بيرسملى اللى هو مصار ال
+
+71
+00:05:18,930 --> 00:05:22,550
+particle او ال glider هذى اللى هو عبارة عن the
+
+72
+00:05:22,550 --> 00:05:26,210
+length of the glider's path from T بساوية 0 to T
+
+73
+00:05:26,210 --> 00:05:29,390
+بساوية Pi is اتفاقنا ال length ايش بساوية ال
+
+74
+00:05:29,390 --> 00:05:33,190
+integration من A ل B اللى هى 0 ل 2 Pi ال magnitude
+
+75
+00:05:33,190 --> 00:05:37,490
+ده ال V of T dtما هو V of T؟ طبعاً بتأتي بتفاضل
+
+76
+00:05:37,490 --> 00:05:43,790
+هذه اللي هي ناقص sine T في اتجاه I زائد cosine T
+
+77
+00:05:43,790 --> 00:05:50,150
+في اتجاه J زائد K مظبوط هذا اللي هو تفاضلها الان
+
+78
+00:05:50,150 --> 00:05:52,830
+ال magnitude اللي هو عبارة عن جدر التربيع اللي
+
+79
+00:05:52,830 --> 00:05:56,320
+اللي طلع هنا اللي هو ناقص sine T لكل تربيعزائد
+
+80
+00:05:56,320 --> 00:05:59,160
+اللي طالع حانه اللي هو cos T لكل تربيع زائد اللي
+
+81
+00:05:59,160 --> 00:06:02,520
+طالع حانه اللي هو تفاضل T واحد اللي هو واحد تربيع
+
+82
+00:06:02,520 --> 00:06:07,060
+اللي هو واحد الجدر التربيع ايه ده كله دي إيش دي T
+
+83
+00:06:07,060 --> 00:06:10,680
+بدنا نكامله نشوف كيف نكامله الجمعة طبعا كلكم قال
+
+84
+00:06:10,680 --> 00:06:13,560
+طب ما هي sin تربيع زاد cos تربيع واحد وعنا واحد من
+
+85
+00:06:13,560 --> 00:06:18,500
+اتنين اذا جدر اتنين دي T وجدر اتنين عبارة عن ثابت
+
+86
+00:06:18,500 --> 00:06:21,960
+تكامله على طول اللي هو جدر اتنين في T ومن 0 ل 2
+
+87
+00:06:21,960 --> 00:06:26,300
+باي فبصير عند اتنين في جدر اتنين في بايهذا عبارة
+
+88
+00:06:26,300 --> 00:06:30,540
+عن the length of the glider's path من T بتساوي سفر
+
+89
+00:06:30,540 --> 00:06:36,960
+لعين T بإيش بتساوي لعين T بتساوي اتنين باين هذه
+
+90
+00:06:36,960 --> 00:06:40,060
+اللي هي مثلا توضيح إلى اللي قبل بشوية
+
+91
+00:06:45,490 --> 00:06:49,290
+الان نحكي عن الرسمة هذه بس قبل ان نحكي عنها خلّينا
+
+92
+00:06:49,290 --> 00:06:53,370
+نشوف ايش اللي عندي if we choose the base point B T
+
+93
+00:06:53,370 --> 00:06:57,150
+note on a smooth curve طبعا نجيب احنا اللي بنشتغل
+
+94
+00:06:57,150 --> 00:07:00,990
+عليه عشان تكون في الصورة هو smooth curve ليش
+
+95
+00:07:00,990 --> 00:07:05,700
+smooth curve؟ عشان لسبب بسيط ال smooth curveالـ
+
+96
+00:07:05,700 --> 00:07:10,500
+derivative له موجودة و ليس كذلك و continuous طيب
+
+97
+00:07:10,500 --> 00:07:14,440
+موجود عشان نستعمل القانون نتفاق معك لكن طب و
+
+98
+00:07:14,440 --> 00:07:17,660
+continuous؟ آه عشان انا بتكون continuous بيصير ال
+
+99
+00:07:17,660 --> 00:07:19,880
+magnitude ل V of T اللي هو المبدع continuous
+
+100
+00:07:19,880 --> 00:07:22,400
+function بتصير هذا كلها اللي بتطلع عند continuous
+
+101
+00:07:22,400 --> 00:07:26,600
+يعني بعد اقدر ايش؟ اكملها عشان مابديش مشاكل عشان
+
+102
+00:07:26,600 --> 00:07:30,540
+أضمن يطلع من الال اللي هو قيمته لأن الدالة المتصلة
+
+103
+00:07:30,540 --> 00:07:34,080
+على طول في اله تكامل زي ما عرفنا في تفاضل اللي هو
+
+104
+00:07:34,080 --> 00:07:40,340
+الفطيب لان نفترض انه عندى .. نفترض نقطة .. نقطة
+
+105
+00:07:40,340 --> 00:07:43,460
+بداية خلينا او خلينا نقول نقطة انطلاق السؤال اللى
+
+106
+00:07:43,460 --> 00:07:46,740
+بده ماشي في هيكه او بده ماشي هيكه ماشي الحال هذا
+
+107
+00:07:46,740 --> 00:07:50,740
+بسميها base point اللي بيه of T noteالان هذه ال
+
+108
+00:07:50,740 --> 00:07:56,580
+base point لما نمشي اشي بسيط هنا حيصير عندى اللى
+
+109
+00:07:56,580 --> 00:08:01,600
+هو اتغيرت بي تي نوت لعند اللى هى مثلا مين بي اف تي
+
+110
+00:08:01,600 --> 00:08:06,940
+لهان فرضها ماشي الحال هذا اللى صار صار فيه طول هذا
+
+111
+00:08:06,940 --> 00:08:12,670
+الطول اللى من هان لهانبتغير بتغير main T لو طلعت
+
+112
+00:08:12,670 --> 00:08:16,570
+لفوق بطول شوية لو طلعت لفوق بطول شوية و هكذا عشان
+
+113
+00:08:16,570 --> 00:08:19,870
+هي كبدنا نسمي هذا احنا اللي قبل اخدناه من ثابت
+
+114
+00:08:19,870 --> 00:08:23,290
+لثابت من سفر لاثنين باي عشان يكتلع عند عدد لكن
+
+115
+00:08:23,290 --> 00:08:27,640
+الان انا بتاخده من P of T noteاللي هي عند ضغط الـ
+
+116
+00:08:27,640 --> 00:08:31,040
+T note لعند B of T لعند T يعني هيطلع ال
+
+117
+00:08:31,040 --> 00:08:35,040
+integration اللي عندي بدلالة مين بدلالة T يعني
+
+118
+00:08:35,040 --> 00:08:41,040
+الطول هذا بيطلع بدلالة T هذا الطول بدي أسميه S of
+
+119
+00:08:41,040 --> 00:08:44,820
+T في الاتجاه هذا لو مشيت بدي أعتبره أنا ماشي
+
+120
+00:08:44,820 --> 00:08:49,120
+بالاتجاه الموجبلو جيت رجعت رجوع من هان لهان بده
+
+121
+00:08:49,120 --> 00:08:52,900
+اعتبره ايه شماله في الاتجاه السالب هذا استلاحة
+
+122
+00:08:52,900 --> 00:08:56,200
+بيننا يعني الآن بدنا نقول من ال base point و امشي
+
+123
+00:08:56,200 --> 00:09:00,020
+هيك بالاتجاه هذا على اليمين يعني بده اعتبره modab
+
+124
+00:09:00,020 --> 00:09:03,220
+و لما نمشي هيك على الشمال بده اعتبره ايه اعتبره
+
+125
+00:09:03,220 --> 00:09:06,920
+ايه؟ سالب هذا اللي هو يعني الطول بحكي عن الطول
+
+126
+00:09:06,920 --> 00:09:10,480
+يعني الطول من هان لهان هذا as of t من هان لهان as
+
+127
+00:09:10,480 --> 00:09:14,390
+of t بس اختلفت as of t اختلفت كطول كطولو هنبدأ
+
+128
+00:09:14,390 --> 00:09:19,470
+نعتبرها إيه شمالة الطول لل curve أو لل arc بس
+
+129
+00:09:19,470 --> 00:09:24,390
+باتجاه السالم طيب هذا الآن if we choose a base
+
+130
+00:09:24,390 --> 00:09:26,810
+point زي ما قلنا of T not in a smooth curve C
+
+131
+00:09:26,810 --> 00:09:30,830
+parameterized by T each value of T determines a
+
+132
+00:09:30,830 --> 00:09:34,650
+point P of T يعني كل value of T بتحدد ل X of T و Y
+
+133
+00:09:34,650 --> 00:09:38,890
+of T و Z of T يعني تتغير قيمة اللي هي ال P of T
+
+134
+00:09:38,890 --> 00:09:42,750
+بتغير الزمن و طبعا تبعا لذلك يتغير اللي هي
+
+135
+00:09:43,860 --> 00:09:47,760
+الإحداثية الـ X والـ Y والـ Z on C الـ Directed
+
+136
+00:09:47,760 --> 00:09:51,360
+distance S of T اللي هي الـ Length دي بنسميها إيش
+
+137
+00:09:51,360 --> 00:09:55,400
+معله Directed distance إيش Directed distance
+
+138
+00:09:55,400 --> 00:10:00,300
+اتفاجنا إنه من T note لعندي في الاتجاه اللي هيك
+
+139
+00:10:00,300 --> 00:10:05,620
+هيعطيني موجة ال magnitude ده V of T دي TV of Tau D
+
+140
+00:10:05,620 --> 00:10:10,940
+Tau الان هذا هيطلع عندى اللى هو بدلالة مين بدلالة
+
+141
+00:10:10,940 --> 00:10:14,180
+T هو عبارة عن length في الواقع بس length من T
+
+142
+00:10:14,180 --> 00:10:17,760
+naught لعند مين عن T بعيد بالموجة لما T تكون إلى
+
+143
+00:10:17,760 --> 00:10:20,900
+أعلى و لو كانت إلى أسفل يعني جاب اللى هيعد بالاش
+
+144
+00:10:20,900 --> 00:10:24,320
+بالسالب الان هى زى ما قولنا FT أكبر من T naught S
+
+145
+00:10:24,320 --> 00:10:28,520
+of T is the distance along the curve from B of T
+
+146
+00:10:28,520 --> 00:10:35,510
+naught to B of Tإذا كان الـ T أقل من الـ T نوت ، S
+
+147
+00:10:35,510 --> 00:10:39,510
+of T هو نقل للمقارنة S of T هو نقل للمقارنة S of T
+
+148
+00:10:39,510 --> 00:10:40,770
+هو نقل للمقارنة S of T هو نقل للمقارنة S of T هو
+
+149
+00:10:40,770 --> 00:10:45,910
+نقل للمقارنة S of T هو نقل للمقارنة S of
+
+150
+00:10:45,910 --> 00:10:52,150
+T هو نقل للمقارنةلأن S بتنوجد هنا بدلالة T اللي
+
+151
+00:10:52,150 --> 00:10:56,750
+بتطلع هنا إذا كل قيمة ل S هنا أو هنا أو هنا أو هنا
+
+152
+00:10:56,750 --> 00:11:01,630
+هي في الواقع مقابلة لقيمة في مين في T لإن إحنا
+
+153
+00:11:01,630 --> 00:11:06,430
+أصلا لما جيبنا S في T جيبناها من T نقطة و اعتمدت
+
+154
+00:11:06,430 --> 00:11:13,250
+على T الان نسميها S and R length parameter الان S
+
+155
+00:11:13,250 --> 00:11:18,150
+هذه اللي هي lengthلكن تتغير بتغير الـ T فعشان هيك
+
+156
+00:11:18,150 --> 00:11:23,310
+بنسميها Arc Length Parameter يعني الآن تستغرب انه
+
+157
+00:11:23,310 --> 00:11:29,050
+اصلا انت ال R of T اللي قبل بشوية هذا اللي جيبناه
+
+158
+00:11:29,050 --> 00:11:34,870
+بدلالة T بعد شوية هقدر اجيب لكيه بدلالة S يعني ال
+
+159
+00:11:34,870 --> 00:11:37,550
+parameter بدل هيكون اللي هو T هيكون ال parameter
+
+160
+00:11:37,550 --> 00:11:42,410
+من S اللي هو عبارة عن ال lengthأو الـ R-Link
+
+161
+00:11:42,410 --> 00:11:47,350
+Parameter لنرى ماذا يقول هذا البراميطر البراميطر
+
+162
+00:11:47,350 --> 00:11:48,130
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+163
+00:11:48,130 --> 00:11:49,810
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+164
+00:11:49,810 --> 00:11:52,190
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+165
+00:11:52,190 --> 00:11:53,290
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+166
+00:11:53,290 --> 00:11:55,610
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+167
+00:11:55,610 --> 00:11:56,750
+البراميطر البراميطر البراميطر البراميطر البراميطر
+
+168
+00:11:56,750 --> 00:12:01,000
+البراميطر البراميطر الand for investigating the
+
+169
+00:12:01,000 --> 00:12:03,240
+turning and twisting nature of a space curve هذا
+
+170
+00:12:03,240 --> 00:12:08,820
+مستقبلنا في شبطر تلتاش حنحكي عن ال twisting و
+
+171
+00:12:08,820 --> 00:12:13,060
+turning curves ايه ده اللي هو قدر لنا ان شاء الله
+
+172
+00:12:13,060 --> 00:12:19,420
+فهذا الاسس مهم بالنسبة لهذا اللي هو الموضوعين طيب
+
+173
+00:12:19,980 --> 00:12:24,140
+الان نرجع للـ R Length Parameter with Base Point B
+
+174
+00:12:24,140 --> 00:12:29,920
+of T Note يعني نقطة الانطلاق أو نقطة الميزان عند T
+
+175
+00:12:29,920 --> 00:12:33,360
+Note هيك هيك و هيطلع بالموجب و هيك هيطلع بياشف
+
+176
+00:12:33,360 --> 00:12:36,680
+السلبي زي ما العادي احنالما ناخد خط الأعداد و
+
+177
+00:12:36,680 --> 00:12:40,160
+ناخده نقطة 0 عليكوا علي يمين موجبة علي اسمها سالب
+
+178
+00:12:40,160 --> 00:12:43,680
+هذه نفس الفكرة P of T هو اللي هيمشي في increasing
+
+179
+00:12:43,680 --> 00:12:46,720
+T بالشكل اللي هذا هتطلع S موجبة و في ال decreasing
+
+180
+00:12:46,720 --> 00:12:50,460
+لل T في الاتجاه هذا هتطلع ليه S سالب ان شاء الله
+
+181
+00:12:50,460 --> 00:12:55,360
+تكونوا فانين عليها S of T بتساويحسب القانون T ل T
+
+182
+00:12:55,360 --> 00:12:58,860
+note اللى كنا نقول عنها من A ل B الجدرى التربيعى ل
+
+183
+00:12:58,860 --> 00:13:04,920
+D X by D Tau اللى هى بس جلبنا .. غيرنا اللى هى بدل
+
+184
+00:13:04,920 --> 00:13:10,000
+T Tau عشان نفرجها عن هذه اما هى نفسها بالفاضل اللى
+
+185
+00:13:10,000 --> 00:13:15,500
+هى ال RDx by Dt و نربحها و Dy by Dt و نربحها و Dz
+
+186
+00:13:15,500 --> 00:13:18,720
+by Dt و نربحها و ناخدها تحت جدر التربيع و ناخدها
+
+187
+00:13:18,720 --> 00:13:18,980
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+188
+00:13:18,980 --> 00:13:19,500
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+189
+00:13:19,500 --> 00:13:19,780
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+190
+00:13:19,780 --> 00:13:20,060
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+191
+00:13:20,060 --> 00:13:20,060
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+192
+00:13:20,060 --> 00:13:22,520
+تحت جدر التربيع و ناخدها تحت جدر التربيع و ناخدها
+
+193
+00:13:22,520 --> 00:13:28,730
+تحت جدر التويساوي الـ integration من T0 لعنطي الـ
+
+194
+00:13:28,730 --> 00:13:32,270
+magnitude للـ V of τاو دي تاو دي تاو هذا كلام مكرر
+
+195
+00:13:32,270 --> 00:13:37,890
+اللي هو هذا هو هذا زي ما حكينا قبل بشوية طيب صلوا
+
+196
+00:13:37,890 --> 00:13:41,210
+عنا بيه يا جماعة لإن أخد مثال this is an example
+
+197
+00:13:41,210 --> 00:13:44,170
+for which we can actually find the arc length
+
+198
+00:13:44,170 --> 00:13:48,330
+parameterization of a curveبدنا نجد الـ R-length
+
+199
+00:13:48,330 --> 00:13:51,330
+parameterization of a curve زي ما حكي انا قبل شوية
+
+200
+00:13:51,330 --> 00:13:56,250
+نشوف ايش اللي بقوله if T0 بتساوي 0 the R-length
+
+201
+00:13:56,250 --> 00:13:59,610
+parameter along the helix R of T بتساوي Cos of T
+
+202
+00:13:59,610 --> 00:14:03,150
+في تجاه I زي Sin of T في تجاه J زي T في تجاه K
+
+203
+00:14:03,150 --> 00:14:09,290
+يعني هاي اللي هو ال position vector او اللي هو
+
+204
+00:14:09,290 --> 00:14:13,290
+اللي بده يرسملي اللي هو مين ال curve اللي احنا
+
+205
+00:14:13,290 --> 00:14:17,480
+بدنا نحكي عنهالان لما T note بالساوية سفر بدنا
+
+206
+00:14:17,480 --> 00:14:21,360
+ناخد S of T بالساوية من T note لعند T لل V of Tau
+
+207
+00:14:21,360 --> 00:14:25,040
+D Tau هاي قانوننا T note في هذه الحالة ايش اخدناها
+
+208
+00:14:25,040 --> 00:14:31,420
+احنا سفر اذا من سفر لعند مين لعند Tالان v of tau
+
+209
+00:14:31,420 --> 00:14:35,640
+حسبناها هو نفس المثال اللى فات حسبناها جدش طلعت ان
+
+210
+00:14:35,640 --> 00:14:39,800
+جدر التنين دي tau الا ان المفادل هذه بيصير جدر
+
+211
+00:14:39,800 --> 00:14:44,880
+التنين tau من zero لعين T يعني بيصير جدر التنين T
+
+212
+00:14:44,880 --> 00:14:49,420
+إذا صارت ال S of T عندى يا جماعة عبارة عن جدر
+
+213
+00:14:49,420 --> 00:14:55,570
+التنين Tالان احنا هذا الـ S الـ Length من T نوت
+
+214
+00:14:55,570 --> 00:15:00,290
+التي يتغير بتغير الـ T لاحظوا S جبتها بدلالة مين
+
+215
+00:15:00,290 --> 00:15:04,570
+الـ T الان انا بقدر اجيب T بدلالة الـ S بنحلها و
+
+216
+00:15:04,570 --> 00:15:08,810
+لا في أسهل منها T ايش هتساوي S على جدر التنين هذه
+
+217
+00:15:08,810 --> 00:15:12,070
+على جدر التنين بكسب تصير T بساوي S على جدر التنين
+
+218
+00:15:12,070 --> 00:15:16,330
+الان هذه الـ S اللي هي عبارة عن مين؟ عن الـ Arc
+
+219
+00:15:16,330 --> 00:15:21,980
+Lengthأه براميتر ال arc length براميتر الان صرت
+
+220
+00:15:21,980 --> 00:15:28,480
+مؤهل ان اجيب arc بدلالة ال S قد تلزمني الان هو طلب
+
+221
+00:15:28,480 --> 00:15:32,680
+مني ان اوجد اللي هي ال arc بدلالة ال R length
+
+222
+00:15:32,680 --> 00:15:37,520
+براميتر اللي هي بدلالة ال S الان arc بدلالة ال S
+
+223
+00:15:37,520 --> 00:15:42,250
+ايش هتساويبشيل كل T و بحط قيمتها اللي هي بيصير cos
+
+224
+00:15:42,250 --> 00:15:46,050
+بدل ال T أش بتصير S على جدر التنين في تجاه ال I
+
+225
+00:15:46,050 --> 00:15:49,330
+sign S على جدر التنين في تجاه ال I زاد S على جدر
+
+226
+00:15:49,330 --> 00:15:52,110
+التنين في تجاه ال K وهذه قدرنا نكتب ال position
+
+227
+00:15:52,110 --> 00:15:57,650
+vector R أو اللي بيحددلي مصار ال K ال particle أو
+
+228
+00:15:57,650 --> 00:16:04,430
+اللي هو مصار ال curve جبته بواسطة أو بدلالةاللي هو
+
+229
+00:16:04,430 --> 00:16:09,790
+الـ R Length Parameter اللي هي S ماشي يا جماعة
+
+230
+00:16:09,790 --> 00:16:14,930
+الان بدنا speed on a smooth curve بدنا اللي هو ايش
+
+231
+00:16:14,930 --> 00:16:19,550
+نشوف اللي هو السرعة مش المتجهة لأ السرعة مش متجهة
+
+232
+00:16:19,550 --> 00:16:22,190
+السرعة السرعة اللي هو ال magnitude on a smooth
+
+233
+00:16:22,190 --> 00:16:26,980
+curve عند the smooth curveاللي هو اللي بيصنعوا R
+
+234
+00:16:26,980 --> 00:16:31,880
+of T الآن بدنا نودد اللي هو السبيل عليه ال
+
+235
+00:16:31,880 --> 00:16:38,880
+magnitude لل V of T اللي هي انت الآن هيأس ب 70 نوت
+
+236
+00:16:38,880 --> 00:16:42,900
+اللي عندك VTAU وDTAUمظبوط هي القانون تبعنا الآن
+
+237
+00:16:42,900 --> 00:16:45,360
+بالـ Fundamental Theorem of Calculus كل شيء حلو
+
+238
+00:16:45,360 --> 00:16:49,400
+لأن هذا continuous لأننا مفترضين أن نتحدث عن
+
+239
+00:16:49,400 --> 00:16:52,000
+Smooth Curve مزام Smooth Curve لأن الـ derivative
+
+240
+00:16:52,000 --> 00:16:54,740
+لهذه موجودة وهذه موجودة وهذه موجودة وcontinuous
+
+241
+00:16:54,740 --> 00:16:58,580
+يعني هذا continuous مدام هذا continuous إذن DS by
+
+242
+00:16:58,580 --> 00:17:05,060
+DTاللي هو على طول بنزيل هذا و بنحط اللي هو يعني
+
+243
+00:17:05,060 --> 00:17:09,820
+التفاضل بالغ التكامل و بيصير بيصير بساوة V of T
+
+244
+00:17:09,820 --> 00:17:14,440
+يعني دي أس باي دي تي بيساوي ال magnitude ده V of T
+
+245
+00:17:14,440 --> 00:17:18,060
+يعني و كأنه بقول ال equation أربعة says that the
+
+246
+00:17:18,060 --> 00:17:23,750
+speedwith which a particle moves يعني الـ speed
+
+247
+00:17:23,750 --> 00:17:26,250
+للبارتكال هذه الـ speed للبارتكال تلك اللي احنا
+
+248
+00:17:26,250 --> 00:17:32,750
+عارفينها ايش ساوت اللي هي عبارة عن تفاضل الار
+
+249
+00:17:32,750 --> 00:17:38,250
+length with respect to time to Tيعني with which a
+
+250
+00:17:38,250 --> 00:17:41,450
+particle moves along its path is the magnitude of
+
+251
+00:17:41,450 --> 00:17:46,110
+v consistent with what we note اللي هو انه احنا
+
+252
+00:17:46,110 --> 00:17:50,450
+اللي بنعرفه انه هذه ال speed لل particle بساوية
+
+253
+00:17:50,450 --> 00:17:54,910
+اللي هي عبارة عن تفاضل ال arc length المسافة
+
+254
+00:17:54,910 --> 00:18:01,150
+المقطوعة هذه عالميا بالنسبة لمين بالنسبة ل Tهذه
+
+255
+00:18:01,150 --> 00:18:06,750
+اللي هي الـ Speed اللي هي تفاضل المسافة المقطوعة
+
+256
+00:18:06,750 --> 00:18:13,430
+على M على T طبعا المسافة بإيش يا جماعة بدون ما
+
+257
+00:18:13,430 --> 00:18:17,950
+نحكي عن Direction الأسعار عن Length هيك أبو جبه و
+
+258
+00:18:17,950 --> 00:18:23,620
+هيك سالبه و بس مش Directionهذه التزايد تصبح موجبة
+
+259
+00:18:23,620 --> 00:18:26,080
+الـ S إذا كانت T متزايدة تصبح تزايدة سالفة إذا
+
+260
+00:18:26,080 --> 00:18:26,480
+كانت T متزايدة تصبح سالفة إذا كانت T متزايدة تصبح
+
+261
+00:18:26,480 --> 00:18:27,540
+سالفة إذا كانت T متزايدة تصبح سالفة إذا كانت T
+
+262
+00:18:27,540 --> 00:18:27,740
+متزايدة تصبح سالفة إذا كانت T متزايدة تصبح سالفة
+
+263
+00:18:27,740 --> 00:18:27,800
+إذا كانت T متزايدة تصبح سالفة إذا كانت T متزايدة
+
+264
+00:18:27,800 --> 00:18:27,800
+تصبح سالفة إذا كانت T متزايدة تصبح سالفة إذا كانت
+
+265
+00:18:27,800 --> 00:18:28,740
+T متزايدة تصبح سالفة إذا كانت T متزايدة تصبح سالفة
+
+266
+00:18:28,740 --> 00:18:31,660
+إذا كانت T متزايدة تصبح سالفة
+
+267
+00:18:35,760 --> 00:18:39,060
+للـ unit tangent vector بدنا نعرف حاجة اسمها ال
+
+268
+00:18:39,060 --> 00:18:42,380
+unit tangent vector و هنجيله برضه بالتفصيل قدام ان
+
+269
+00:18:42,380 --> 00:18:46,860
+شاء الله خلينا نشوف we already know the velocity
+
+270
+00:18:46,860 --> 00:18:51,560
+vector V بسوء DR by DT الان ال velocity vector V
+
+271
+00:18:51,560 --> 00:18:57,300
+بسوء DR by DT اللي هي المشتقة دي DR by DT اللي هو
+
+272
+00:18:57,300 --> 00:19:00,740
+عبارة عن المماثلة مين لكيرف عندها R of T هذا ايش
+
+273
+00:19:00,740 --> 00:19:04,880
+رسمت مين الرسمة اللي بترسم مين اللي هي R of Tالأن
+
+274
+00:19:04,880 --> 00:19:11,080
+DR by DT هي عبارة عن الـ tangent لمن؟ اللي هو لل
+
+275
+00:19:11,080 --> 00:19:15,320
+curve R of T إذا V بيسوى DR by DT is the tangent
+
+276
+00:19:15,320 --> 00:19:18,700
+to the curve R of T and that the vector لأن هذا هو
+
+277
+00:19:18,700 --> 00:19:22,220
+ال tangent DR by DT هذا هي ال tangent هذه اللي هي
+
+278
+00:19:22,220 --> 00:19:27,080
+من؟ ال derivative اللي هي ال derivative لمن؟ لـ R
+
+279
+00:19:27,080 --> 00:19:30,840
+اللي هي اللي بترسم ال curve اللي عندي الآن أنا
+
+280
+00:19:30,840 --> 00:19:35,690
+معنى بمين؟مش اللي هو dR by dt اللي هو ال tangent
+
+281
+00:19:35,690 --> 00:19:40,650
+بس لأ معنى بال unit tangent vector و حاجة هستخدمها
+
+282
+00:19:40,650 --> 00:19:44,990
+قدام كثير عشان هي كمان ده افردلها رمز لحاله اللي
+
+283
+00:19:44,990 --> 00:19:49,990
+هو ساوي T capital اللي هي عبارة عن ال velocity على
+
+284
+00:19:49,990 --> 00:19:52,690
+ال magnitude ده ال V اللي هو ال speed يعني بمعنى
+
+285
+00:19:52,690 --> 00:19:59,050
+اخر هذه ده directionاللي هو of مين is the
+
+286
+00:19:59,050 --> 00:20:03,930
+direction of .. اللي هو حركة ال particle اللي هي
+
+287
+00:20:03,930 --> 00:20:07,990
+اللي بنسميها ال unit tangent vector لإنه مكون
+
+288
+00:20:07,990 --> 00:20:11,990
+tangent لمين لل curve هذا واخدت ال unit منه بس
+
+289
+00:20:11,990 --> 00:20:15,450
+اللي طوله واحد اللي هو V ع ال magnitude لل V هيك
+
+290
+00:20:15,450 --> 00:20:20,460
+بدنا نعرفه طيب ناخد مثال الان الان بتقولي fivethe
+
+291
+00:20:20,460 --> 00:20:23,960
+unit tangent vector of the curve R of T بساوية cos
+
+292
+00:20:23,960 --> 00:20:27,440
+T زائد تلاتة sin T كل واحدة في اتجاهها زائد T
+
+293
+00:20:27,440 --> 00:20:30,960
+تربيع في اتجاهها طيب شوفوا يا جماعة صلوا على النبي
+
+294
+00:20:30,960 --> 00:20:33,720
+عليه الصلاة والسلام لأن عشان اوجد ال unit tangent
+
+295
+00:20:33,720 --> 00:20:37,860
+vector بالزمن اوجد ال velocity DR by DT بساوية
+
+296
+00:20:37,860 --> 00:20:41,860
+كلكم قال ناقص تلاتة sin T في اتجاه ال I زائد تلاتة
+
+297
+00:20:41,860 --> 00:20:45,100
+cos T في اتجاه ال J بفاضل قاعد زائد اتنين T في
+
+298
+00:20:45,100 --> 00:20:49,150
+اتجاه ال Kواحد قال لطب هو هذا ال V وانت بالتأكيد
+
+299
+00:20:49,150 --> 00:20:51,190
+بيلزمك ميه ان ال magnitude ده ال V اه ال magnitude
+
+300
+00:20:51,190 --> 00:20:54,430
+ده ال V ايش بيساوي جدر التربيعي لهذا تربيع زاد هذا
+
+301
+00:20:54,430 --> 00:20:58,330
+تربيع اللي هي تلاتة sign يعني تسعة sign تربيع زاد
+
+302
+00:20:58,330 --> 00:21:02,890
+تسعة cosine تربيع زاد أربع ت تربيع تسعة sign تربيع
+
+303
+00:21:02,890 --> 00:21:05,710
+زاد تسعة cosine تربيع باخد التسعة عامل مشترك برا
+
+304
+00:21:05,710 --> 00:21:08,330
+بيصير تسعة في sign تربيع زاد cosine تربيع واحد
+
+305
+00:21:08,330 --> 00:21:13,260
+يعني تسعةزائد من هذه أربعة T تربية إذا صارت ال
+
+306
+00:21:13,260 --> 00:21:15,940
+magnitude ل V هو عبارة عن جدر التربية إلى الأربعة
+
+307
+00:21:15,940 --> 00:21:21,340
+زائد T تربية إذا صارت ال unit tangent vector T
+
+308
+00:21:21,340 --> 00:21:27,260
+بيساوي V على magnitude ل V اللي هي هذا المقدار أو
+
+309
+00:21:27,260 --> 00:21:31,760
+هذا اللي هو ال vector على magnitude و بيصير ناقص
+
+310
+00:21:31,760 --> 00:21:35,000
+تلاتة sin T في تجاه I زائد تلاتة cos T في تجاه J
+
+311
+00:21:35,000 --> 00:21:35,240
+زائد
+
+312
+00:21:49,730 --> 00:21:54,630
+الحركة عكس التجاه من عقارب الساعة
+
+313
+00:21:59,270 --> 00:22:02,930
+لو ماشيين على اللي هو فيه عندى الـ vector R ماشي
+
+314
+00:22:02,930 --> 00:22:06,250
+على اللي هو مين الدائرة أكيد ليش ماشي عدّان لأن R
+
+315
+00:22:06,250 --> 00:22:09,550
+of T بيساوي cos T في تجاه I زاد sin T في تجاه G
+
+316
+00:22:09,550 --> 00:22:12,650
+هذا بيبقى metric equation للـR لكن العلاقة بين cos
+
+317
+00:22:12,650 --> 00:22:16,870
+T وsin T دايما cos تربيع زاد sin تربيع بيساوي 1
+
+318
+00:22:17,560 --> 00:22:22,620
+بيساوي الـ R يعني الان cosine تربيع زائد sin تربيع
+
+319
+00:22:22,620 --> 00:22:26,920
+بيساوي قيمة R لو فرضنا ان احنا ماشيين يعني دائما
+
+320
+00:22:26,920 --> 00:22:30,600
+اللي هو ماشي ايش بمقدار اللي هو ال magnitude
+
+321
+00:22:30,600 --> 00:22:34,880
+بيساوي واحد لل R بيصير عندي انا ماشي على اللي هو
+
+322
+00:22:34,880 --> 00:22:39,160
+دائرة الوحدةدائرة الوحدة هذا اللي هو ماشيين على
+
+323
+00:22:39,160 --> 00:22:42,500
+مين لانقصان تربيع زي اقصان تربيع دايما ايش بيساوي
+
+324
+00:22:42,500 --> 00:22:47,260
+بيساوي واحد فصار عند ال magnitude له على طول بيطلع
+
+325
+00:22:47,260 --> 00:22:50,820
+ايش بيطلع واحد يعني بدون ما اقول انه والله واحد او
+
+326
+00:22:50,820 --> 00:22:53,480
+كده هو هيطلع automatic واحد زي ما قلنا اقصان تربيع
+
+327
+00:22:53,480 --> 00:22:57,200
+زي اقصان تربيع اللي هو X تربيع زي Y تربيع هيساوي
+
+328
+00:22:57,200 --> 00:22:59,940
+واحد فاهمين عليها لان اقصان تربيع زي اقصان تربيع
+
+329
+00:22:59,940 --> 00:23:03,120
+هيساوي واحد اذا ال X تربيع زي Y تربيع بساوي واحد
+
+330
+00:23:03,120 --> 00:23:08,330
+اذا انا ماشي على دائرة الوحدةعبر هذا الـ vector أو
+
+331
+00:23:08,330 --> 00:23:13,070
+اللي برسم ال curve around the unit circle we see
+
+332
+00:23:13,070 --> 00:23:16,890
+that ال velocity كلكم قال فاضل هذه و فاضل هذه تطلع
+
+333
+00:23:16,890 --> 00:23:21,810
+نقص sin t زي cos t الآن نقص sin t في اتجاه ال I زي
+
+334
+00:23:21,810 --> 00:23:26,770
+cos t في اتجاه ال J is already برضه ايه شماله unit
+
+335
+00:23:26,770 --> 00:23:30,430
+vector لأن ال magnitude له الجدر التربيعي لهذه
+
+336
+00:23:30,430 --> 00:23:34,160
+التربيه زي هذه التربيه اللي هو واحدإذا اللي هو في
+
+337
+00:23:34,160 --> 00:23:39,740
+هذه الحالة على اللي هي الدائرة دائرة الوحدة بيكون
+
+338
+00:23:39,740 --> 00:23:45,440
+ال unit tangent vector هو بالظبط من ال velocity
+
+339
+00:23:45,440 --> 00:23:48,640
+إذا ال unit tangent vector عندي بيساوي ال velocity
+
+340
+00:23:48,640 --> 00:23:54,080
+in this case because the magnitude of v in this
+
+341
+00:23:54,080 --> 00:24:01,700
+case is oneبناء عليه اللي هو عندي الآن في بعض
+
+342
+00:24:01,700 --> 00:24:08,460
+الشغلات اللي هو الـ Dt by Ds عبارة عن واحد على Ds
+
+343
+00:24:08,460 --> 00:24:12,120
+by Dt زي ما انتوا عارفين اللي هي الـ Dt by Ds اللي
+
+344
+00:24:12,120 --> 00:24:17,740
+هي التفاضل اللي هو الزمن بالنسبة للمسافة بالنسبة
+
+345
+00:24:17,740 --> 00:24:20,600
+لل arc length بساوي واحد على ال magnitude لل V
+
+346
+00:24:21,780 --> 00:24:26,040
+والان هذه بعض العلاقات اللى ممكن تلزمنا جدام عند
+
+347
+00:24:26,040 --> 00:24:31,060
+الـDR by DS اللى هو بالـ Chain و الـ R طبعا عارفين
+
+348
+00:24:31,060 --> 00:24:34,060
+ايش هي الـ R هيها اللى هي الـ Position Vector الـ
+
+349
+00:24:34,060 --> 00:24:37,140
+S عارفين هي الـ R Length لو انه ده فاضل DR by DS
+
+350
+00:24:37,140 --> 00:24:41,370
+انه قبل شوية دي بنى R بدلالة Sبنشوف ايش DR by DS
+
+351
+00:24:41,370 --> 00:24:45,530
+بيساوي DR by DT في DT by DS اللي هي الـ Chain Rule
+
+352
+00:24:45,530 --> 00:24:50,910
+ويساوي DR by DT مين هي يا جماعة اللي هي عبارة عن
+
+353
+00:24:50,910 --> 00:24:54,830
+الـ V عارفينها من زمان DT by DS توا اللي هو
+
+354
+00:24:54,830 --> 00:24:57,690
+استنتجناه اللي عبارة عن واحدة ال magnitude للـ V
+
+355
+00:24:57,960 --> 00:25:00,300
+إذا صارت الـ V على الـ magnitude ده الـ V اللي هو
+
+356
+00:25:00,300 --> 00:25:03,600
+مين اللي هو T الـ unit tangent vector و كأنه
+
+357
+00:25:03,600 --> 00:25:07,660
+بيقوللي اسمع ال unit tangent vector ممكن تجيبه
+
+358
+00:25:07,660 --> 00:25:12,280
+كمان بطريقة تانية انك اتفاضل ال R بالنسبة لمين
+
+359
+00:25:12,280 --> 00:25:17,480
+للأسد اتفاضل اللي هو الليالـ R اللي هي دالة
+
+360
+00:25:17,480 --> 00:25:21,560
+الإزاحة أو كذا بالنسبة للـ Arc Length أو اللي هي
+
+361
+00:25:21,560 --> 00:25:25,120
+دالة الـ Position Vector بالنسبة للـ Arc Length
+
+362
+00:25:25,120 --> 00:25:29,360
+تبعتها بيطلع عند دي اربع يدي اسم ساومين ال unit
+
+363
+00:25:29,360 --> 00:25:32,780
+tangent vector و هيك بيكون خلصنا المحاضرة و إلى
+
+364
+00:25:32,780 --> 00:25:35,960
+لقاء آخر السلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/RtOltUWLcYs.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/RtOltUWLcYs.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1d07c170508fd901e8378fc125a062ba8d04c9e0
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/RtOltUWLcYs.srt
@@ -0,0 +1,1321 @@
+1
+00:00:00,760 --> 00:00:04,080
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,080 --> 00:00:10,200
+وبركاته هذه هي المحاضرة الرقم 14 مساق تفاضل وتكامل
+
+3
+00:00:10,200 --> 00:00:14,800
+وهي المحاضرة الثانية في chapter 14 وهي تحت
+
+4
+00:00:14,800 --> 00:00:18,600
+عنوان 14 limits and continuity in higher
+
+5
+00:00:18,600 --> 00:00:24,120
+dimensions سنتحدث عن النهايات والاتصال في الـ higher
+
+6
+00:00:24,120 --> 00:00:27,180
+dimensions يعني function of two variables أو three
+
+7
+00:00:27,180 --> 00:00:30,220
+variables أو more. أحنا كنا نتحدث طبعا في الـ
+
+8
+00:00:30,220 --> 00:00:34,780
+Calculus عن النهايات وعن الاتصال لدوال اللي هي
+
+9
+00:00:34,780 --> 00:00:38,360
+function of one variable، بساوية f of x. الآن بدنا
+
+10
+00:00:38,360 --> 00:00:42,600
+نتحدث بشكل مشابه أولا عن limits for functions of
+
+11
+00:00:42,600 --> 00:00:46,820
+two variables، اللي هي نهايات الدوال اللي دالة
+
+12
+00:00:46,820 --> 00:00:50,580
+function of two variables. الآن بدنا نشوف شو معنى
+
+13
+00:00:50,580 --> 00:00:54,680
+limit في هذه الحالة، وهي مشابهة لحالة اللي هي limit
+
+14
+00:00:54,680 --> 00:00:59,000
+اللي أخدناها في اللي هو تفاضل وتكامل. We say that a
+
+15
+00:00:59,000 --> 00:01:03,340
+function f of x,y approaches the limit L as x,y
+
+16
+00:01:03,340 --> 00:01:07,540
+approaches x0,y0.  يعني بنقول إن الـ limit
+
+17
+00:01:07,540 --> 00:01:13,790
+للـ f of x,y لما x,y تقترب إلى x0,y0 بتساوي L. إننا
+
+18
+00:01:13,790 --> 00:01:17,850
+بدنا نعبّر عن هذا by the Epsilon Delta Definition، الـ
+
+19
+00:01:17,850 --> 00:01:21,390
+Epsilon Delta Definition باختصار بتقول إنه عندما
+
+20
+00:01:21,390 --> 00:01:26,050
+تقترب X, Y من الـ X₀ والـ Y₀، الـ F of X, Y
+
+21
+00:01:26,050 --> 00:01:30,310
+هتقترب من الـ L. أو بمعنى آخر لما X, Y تقترب إلى X
+
+22
+00:01:30,310 --> 00:01:34,010
+₀ وY₀، الـ F of X, Y تقترب إلى L. بدنا نعبّر عن
+
+23
+00:01:34,010 --> 00:01:38,490
+هذا الكلام عن اقتراب اللي هو X, Y من X₀ وY₀
+
+24
+00:01:38,720 --> 00:01:43,740
+يؤدي إلى اقتراب f of x, y إلى L بواسطة اللي هي
+
+25
+00:01:43,740 --> 00:01:48,560
+الرموز اللي اتعودنا عليها سابقا. أن إذا أعطينا أي
+
+26
+00:01:48,560 --> 00:01:51,160
+Epsilon، أي Epsilon أكبر من 0، الـ Epsilon
+
+27
+00:01:51,160 --> 00:01:55,160
+هتعبر عن الاقتراب الـ f of x, y من L، من 100، من 1000، طبعا
+
+28
+00:01:55,160 --> 00:01:59,000
+f of x, y و L عبارة عن أعداد في الآخر، هدول اقترابين
+
+29
+00:01:59,000 --> 00:02:02,890
+على خط الأعداد، لكن الـ X و Y من الـ X₀ و Y₀
+
+30
+00:02:02,890 --> 00:02:06,210
+هدول وين موجودات؟ في الـ plane. يعني هي X₀ و Y
+
+31
+00:02:06,210 --> 00:02:10,070
+₀، وبدها تقترب إليها X و Y في الـ plane. الآن
+
+32
+00:02:10,070 --> 00:02:14,570
+اقتراب الـ X و Y من الـ X₀ و Y₀، زمان كان لما
+
+33
+00:02:14,570 --> 00:02:18,430
+نقول X تقترب من X₀، كنا على خط الأعداد، نقترب
+
+34
+00:02:18,430 --> 00:02:21,430
+إليها من اليمين أو نقترب إليها من وين؟ من اليسار.
+
+35
+00:02:21,430 --> 00:02:26,790
+بينما هنا الاقتراب من وين؟ من كل الجهات، يعني عبر
+
+36
+00:02:26,790 --> 00:02:31,950
+اللي هو إيش؟ دائرة، اللي هو الـ X والـ Y تقترب من الـ
+
+37
+00:02:31,950 --> 00:02:36,150
+X₀ و Y₀ من كل هذه الاتجاهات. نشوف عن الكلام
+
+38
+00:02:36,150 --> 00:02:40,500
+هذا إيش بنعني فيه؟ إنه لكل Epsilon أكبر من 0، بنلاقي Delta
+
+39
+00:02:40,500 --> 00:02:44,880
+أكبر من 0، تعبر عن اقتراب من الـ X و Y من X₀ و
+
+40
+00:02:44,880 --> 00:02:49,940
+Y₀. لكل X و Y in the domain، لما X و Y
+
+41
+00:02:49,940 --> 00:02:54,920
+تقترب من X₀ و Y₀، الـ X والـ Y البعد عنها
+
+42
+00:02:54,920 --> 00:02:58,920
+في البعد هنا، اللي هو في الـ plane، هو عبارة عن
+
+43
+00:02:58,920 --> 00:03:01,820
+المسافة بين X, Y و X₀, Y₀، اللي هو عبارة
+
+44
+00:03:01,820 --> 00:03:05,460
+عن الجذر التربيعي لـ (X - X₀)² + (Y
+
+45
+00:03:05,460 --> 00:03:09,480
+- Y₀)². لما يكون هذا أصغر من Delta،
+
+46
+00:03:10,690 --> 00:03:14,810
+بيؤدي إلى |f of x, y - L| أصغر من
+
+47
+00:03:14,810 --> 00:03:20,470
+Epsilon. معناته أنه لما المسافة بين x, y اللي هو
+
+48
+00:03:20,470 --> 00:03:24,050
+تكون صغيرة صغيرة صغيرة، لدرجة أن x₀ أو لما الـ x
+
+49
+00:03:24,050 --> 00:03:30,430
+₀ و y₀، الـ f of x, y و L يقتربان جدا، ويتصير الـ
+
+50
+00:03:30,430 --> 00:03:35,330
+|f of x, y - L|  بتؤول إلى L، وبتكون أصغر من أي
+
+51
+00:03:35,330 --> 00:03:39,920
+Epsilon في الدنيا. هنا باللاحظ ما يلي، زمان كنا نقول
+
+52
+00:03:39,920 --> 00:03:43,400
+لما كنا نقول الـ X تقترب من X₀، في الـ limit of
+
+53
+00:03:43,400 --> 00:03:47,160
+a function of one variable، كنا نقول الـ limit من
+
+54
+00:03:47,160 --> 00:03:50,100
+اليمين والـ limit من اليسار. كنا نقول إن الـ limit
+
+55
+00:03:50,100 --> 00:03:54,340
+هذه بتكون موجودة إذا كانت X راحت للـ X₀ من
+
+56
+00:03:54,340 --> 00:03:56,880
+اليمين ومن اليسار. هان، مش من اليمين ومن
+
+57
+00:03:56,880 --> 00:04:01,100
+اليسار. يعني لما تكون الـ limit موجودة، نمشي من
+
+58
+00:04:01,100 --> 00:04:05,920
+اليمين واليسار، بس لأ، بدي أكون عبر كل الطرق اللي
+
+59
+00:04:05,920 --> 00:04:10,280
+من x, y إلى x₀, y₀. هذه في عندك عدد لا
+
+60
+00:04:10,280 --> 00:04:14,220
+نهائي من الطرق بتروح لها. لما تكون الـ limit موجودة
+
+61
+00:04:14,220 --> 00:04:20,870
+لازم كل الطرق تعطي نفس النهاية، مش بس اللي كانت من
+
+62
+00:04:20,870 --> 00:04:24,910
+اليمين واليسار. زمان أنه تعطي نفس النهاية، لأ، هنا برضه
+
+63
+00:04:24,910 --> 00:04:29,750
+عشان تكون الـ limit موجودة، لازم عبر كل هذه الطرق
+
+64
+00:04:29,750 --> 00:04:34,210
+تؤدي إلى إيش؟ نفس النهاية. يعني لو بدنا نثبت أن الـ
+
+65
+00:04:34,210 --> 00:04:38,510
+limit هنا does not exist، يكفي أن نلاقي طريقين يروح
+
+66
+00:04:38,510 --> 00:04:42,830
+إلى الـ X₀ و Y₀ يكون الـ limitين لهن
+
+67
+00:04:42,830 --> 00:04:47,680
+مختلفين. زي ما كنا بنفس المنطقة عشان نثبت أن الـ
+
+68
+00:04:47,680 --> 00:04:51,220
+limit للـ F of X does not exist، كنا نقول الـ limit
+
+69
+00:04:51,220 --> 00:04:53,920
+من اليمين والـ limit من اليسار، إذا كان واحدة منهم
+
+70
+00:04:53,920 --> 00:04:56,960
+متساويش الثانية، أو واحدة مش موجودة، معناته الـ limit
+
+71
+00:04:56,960 --> 00:05:01,880
+does not exist. إذن باختصار، X, Y بتروح لـ X₀, Y₀
+
+72
+00:05:01,880 --> 00:05:05,560
+تؤدي إلى F of X, Y بتروح للـ L. هذا يُعبّر عنه عندما
+
+73
+00:05:05,560 --> 00:05:10,920
+يكون المسافة بين X₀, Y₀ و X, Y أصغر من
+
+74
+00:05:10,920 --> 00:05:14,620
+Delta، يؤدي لنا إلى المسافة بين F of X, Y و L أصغر
+
+75
+00:05:14,620 --> 00:05:20,260
+من Epsilon. يعني هذا بقى بيعني أن X و Y بتقترب من
+
+76
+00:05:20,260 --> 00:05:23,580
+X₀, Y₀ تؤدي إلى F of X, Y بتروح لـ L. وفي هذه
+
+77
+00:05:23,580 --> 00:05:27,220
+الحالة بتقول الـ limit exists. تعالوا الآن ناخد اللي
+
+78
+00:05:27,220 --> 00:05:30,880
+هي بعض اللي هي الخواص اللي زي الخواص اللي أخدناها
+
+79
+00:05:30,880 --> 00:05:35,350
+سابقا في اللي هي function of one variable. إذا كانت
+
+80
+00:05:35,350 --> 00:05:37,570
+الـ properties of limits of functions of two
+
+81
+00:05:37,570 --> 00:05:41,750
+variables، the following rules hold. If L، M، and K
+
+82
+00:05:41,750 --> 00:05:43,970
+are real numbers such that، يعني لو كانت limit الـ
+
+83
+00:05:43,970 --> 00:05:47,710
+function f of x,y = L لما x,y بتروح لـ x₀, y
+
+84
+00:05:47,710 --> 00:05:50,970
+₀، و limit للـ g of x,y لما x,y تروح لـ x₀, y
+
+85
+00:05:50,970 --> 00:05:55,770
+₀ = M، إذا كان الليمتين هدول موجودات وبساوي
+
+86
+00:05:55,770 --> 00:06:01,380
+الأرقام اللي بين قوسين، إذا الـ limit للمجموع بتوزّع، يعني
+
+87
+00:06:01,380 --> 00:06:05,100
+بيصير limit الـ f of x, y + g of x, y = limit
+
+88
+00:06:05,100 --> 00:06:09,020
+الأول + limit الثاني. يعني بنوزّع الـ limit على
+
+89
+00:06:09,020 --> 00:06:12,640
+الجمع، في حال متى؟ لما يكون الـ limitين موجودات.
+
+90
+00:06:12,640 --> 00:06:16,240
+النهايتين موجودات، = L + M. ونفس الشيء على
+
+91
+00:06:16,240 --> 00:06:19,500
+الطرح، ونفس الشيء لو كان in the scalar مضروب في f
+
+92
+00:06:19,500 --> 00:06:23,360
+of x, y، بطلع الـ K برا، بيصير في limit f of x, y اللي
+
+93
+00:06:23,360 --> 00:06:28,190
+هو K في مين؟ في الـ L. الآن هذا اللي هو نفس خواص اللي
+
+94
+00:06:28,190 --> 00:06:31,710
+هي تبع الـ function العادية. برضه نفس الأشياء، حاصل
+
+95
+00:06:31,710 --> 00:06:34,410
+ضرب دالتين، الـ limit لهن بتوزّع، لما يكون الـ limit
+
+96
+00:06:34,410 --> 00:06:37,390
+موجود، بيكون الـ limit الأول × الـ limit الثاني، و
+
+97
+00:06:37,390 --> 00:06:41,150
+نفس الأشياء حاصل القسمة، لكن provided that الـ
+
+98
+00:06:41,150 --> 00:06:45,400
+M ≠ 0، اللي هي limit اللي الـ G of X,Y ما تساويش 0. الآن
+
+99
+00:06:45,400 --> 00:06:49,360
+نفس الشيء، limit f of x,yⁿ as x,y بتروح لـ x₀ و
+
+100
+00:06:49,360 --> 00:06:54,460
+y₀ = اللي هي limit هذه لكل n، يعني Lⁿ for
+
+101
+00:06:54,460 --> 00:06:58,470
+any positive integer n. نفس الشيء، الـ limit الجذر
+
+102
+00:06:58,470 --> 00:07:03,390
+الـ n لـ f of x,y لما x,y تروح لـ x₀, y₀ = الجذر
+
+103
+00:07:03,390 --> 00:07:06,950
+الـ n للـ limit اللي جوا، اللي هي الـ L، هي L¹⁄ⁿ
+
+104
+00:07:06,950 --> 00:07:12,650
+على L، حيث L positive، و L أكبر من 0. وهذه
+
+105
+00:07:12,650 --> 00:07:16,810
+نفس الـ roots القوانين اللي أخدناها سابقا على الـ
+
+106
+00:07:16,810 --> 00:07:21,630
+function of one. ناخد اللي هو كيفية إيجاد اللي هو
+
+107
+00:07:21,630 --> 00:07:25,410
+للإستخدام اللي هي الخواص اللي قبل بشوية لإيجاد الـ
+
+108
+00:07:25,410 --> 00:07:29,320
+limit. عند الـ limit، هي المقدار اللي فوق على المقدار
+
+109
+00:07:29,320 --> 00:07:32,700
+اللي تحت لما x, y تروح للـ (0,0). الآن الـ
+
+110
+00:07:32,700 --> 00:07:38,920
+limit للأولى، للأولى موجودة لإن هي إيش؟ بتساوي اللي
+
+111
+00:07:38,920 --> 00:07:43,890
+هي 0. الـ limit للتانية، x, y برضه 0، موجودة لأن
+
+112
+00:07:43,890 --> 00:07:46,690
+الـ X والـ Y يذهبون إلى الـ (0,0)، يعني الـ X
+
+113
+00:07:46,690 --> 00:07:50,150
+تذهب إلى الـ 0 والـ Y تذهب إلى الـ 0، اللي هي عندي
+
+114
+00:07:50,150 --> 00:07:54,110
+بيصير هذا 0 + 3. إذا بقلع فوق وبوزّع وأنا
+
+115
+00:07:54,110 --> 00:07:58,970
+مرتاح، وهي اللي هو 0 - 0 + 3، ماشي؟
+
+116
+00:07:58,970 --> 00:08:02,970
+الحال اللي تحت، هو طبعا أنا ضامن اللي تحت مش 0.
+
+117
+00:08:02,970 --> 00:08:05,970
+عشان هيك بوزّع، أنا عارف من قبل إيش اللي ضامنك ده.
+
+118
+00:08:05,970 --> 00:08:11,130
+أشوف، أطلع الدليل. لما الأولى limitها على تعويض مباشر
+
+119
+00:08:11,130 --> 00:08:14,970
+هذا لأن polynomial على polynomial، يعني الدوال حلوة
+
+120
+00:08:14,970 --> 00:08:19,130
+وجميلة، ولو عوضت بتعويض مباشر، هتطلع هذه 0 وهذه
+
+121
+00:08:19,130 --> 00:08:23,390
+هتطلع 0 وهذه -Y³ اللي هي -1.
+
+122
+00:08:23,390 --> 00:08:28,490
+اللي هي تكعيب اللي هي 1، 1-1 بيصير 3 على
+
+123
+00:08:28,490 --> 00:08:31,810
+-1، = -3 على 1، يعني اللي هي...
+
+124
+00:08:31,810 --> 00:08:34,910
+اللي هي +... + 3 على -1، اللي هي
+
+125
+00:08:34,910 --> 00:08:38,810
+-3. وهي اللي هي استخدمت قوانين الـ limit
+
+126
+00:08:38,810 --> 00:08:41,690
+اللي قبل بشوية وكلها مشروعة زي ما قلنا اللي هي
+
+127
+00:08:41,690 --> 00:08:43,610
+الجذر التربيعي، على طول الجذر التربيعي اللي هو الـ
+
+128
+00:08:43,610 --> 00:08:46,410
+limit اللي هو اللي جوا، للـ X²، على طول تعويض
+
+129
+00:08:46,410 --> 00:08:50,090
+مباشر، 9 وهذا اللي هي 16، اللي هي الجذر الخمسة
+
+130
+00:08:50,090 --> 00:08:54,880
+وعشرين، ويساوي 5. الآن مش دايما الدنيا حلوة زي هيك.
+
+131
+00:08:54,880 --> 00:08:59,240
+في بعض الشغلات بتكون شوية بدغلبة، يعني مش تعويض
+
+132
+00:08:59,240 --> 00:09:01,960
+مباشر. إذا تعويض مباشر، إن الدوال حلوة وسهلة وكذا.
+
+133
+00:09:01,960 --> 00:09:06,420
+الآن آخذ مثال ثاني، find limit x²-xy على
+
+134
+00:09:06,420 --> 00:09:10,430
+الجذر التربيعي للـ x - الجذر التربيعي للـ y، لما x
+
+135
+00:09:10,430 --> 00:09:13,670
+و y تذهب إلى (0,0). تعويض مباشر يظهر 0 فوق و0
+
+136
+00:09:13,670 --> 00:09:16,630
+فوق، مش زي اللي فوق اللي هو طلعت لي على طول تعويض
+
+137
+00:09:16,630 --> 00:09:20,370
+مباشر، طلعت لي أرقام حلوة وجميلة. عشان هيك استخدمنا
+
+138
+00:09:20,370 --> 00:09:24,710
+القوانين. إذا بدنا نضطر نشوف الـ 0 اللي هنضيعه مع
+
+139
+00:09:24,710 --> 00:09:28,380
+الـ 0 اللي فوق، نضيعه، إذا فيه إمكانية. طبعا أمثلة
+
+140
+00:09:28,380 --> 00:09:31,320
+منها النوع، كل كومو أكيد قال طب اضرب لنا في المرافق
+
+141
+00:09:31,320 --> 00:09:34,840
+للي تحت، يعني اضرب لنا في جذر الـ X + جذر الـ Y فوق
+
+142
+00:09:34,840 --> 00:09:39,140
+وتحت. ضربنا فوق، صار المقدار X² - X Y
+
+143
+00:09:39,140 --> 00:09:42,580
+في جذر الـ X + جذر الـ Y، اللي هي ضربته فوق وضربته
+
+144
+00:09:42,580 --> 00:09:45,660
+في تحت. لما ضربته في تحت، اللي هو هذا بيكون مفكوك
+
+145
+00:09:45,660 --> 00:09:47,840
+فرق بين مربعين، مربعين، اللي هو جذر الـ X لكل
+
+146
+00:09:47,840 --> 00:09:51,880
+تربيع يعني X وجذر الـ Y كل تربيع يعني واي، يعني X
+
+147
+00:09:51,880 --> 00:09:58,680
+- Y. الآن أنا أريد أن أضيع الـ 0 اللي تحت. الآن
+
+148
+00:09:58,680 --> 00:10:01,760
+طبعا تركيبة السؤال هو أعملها، عشان هيك اصلا هذه x
+
+149
+00:10:01,760 --> 00:10:05,900
+²- xy، خذ الـ x برا، بيصير x(x - y) في جذر الـ
+
+150
+00:10:05,900 --> 00:10:09,860
+x + جذر الـ y، على x - y. هذه بتروح مع هذه،
+
+151
+00:10:09,860 --> 00:10:13,940
+بيصير x في جذر الـ x + جذر الـ y. أو التعويض مباشر
+
+152
+00:10:13,940 --> 00:10:17,200
+بيطلع جديش؟ بيساوي 0. إذا إنصارت الـ limit
+
+153
+00:10:17,200 --> 00:10:22,340
+اللي عندي هذه عبارة عن الـ 0. طيب الآن زي ما قلنا
+
+154
+00:10:22,340 --> 00:10:25,580
+قبل بشوية، هذا اللي قبل اللي... الأمثلة اللي قبل
+
+155
+00:10:25,580 --> 00:10:29,890
+بشوية، اللي هي إيجاد limit. الآن بدنا نثبت to path
+
+156
+00:10:29,890 --> 00:10:34,510
+test for nonexistence of a limit. هذا طبعا له خلفية
+
+157
+00:10:34,510 --> 00:10:37,690
+سابقة عندكم، إنه كنا نقول في الـ function of one
+
+158
+00:10:37,690 --> 00:10:42,370
+variable، تبعت زمان. limit F of X = L لما X
+
+159
+00:10:42,370 --> 00:10:46,650
+تروح للـ X₀ مثلا، كنا نقول هذا الـ limit لو كان
+
+160
+00:10:46,650 --> 00:10:50,470
+الـ limit من اليمين والـ limit من اليسار، واحد من
+
+161
+00:10:50,470 --> 00:10:55,190
+الاثنين مش موجود أو اثنين موجودات بس مش متساويين، على طول
+
+162
+00:10:55,190 --> 00:10:59,130
+بنقول الـ limit does not exist. هنا نفس الجثة يا
+
+163
+00:10:59,130 --> 00:11:02,830
+جماعة، بس الـ X والـ Y بتروح لـ X₀ و Y₀ زي ما
+
+164
+00:11:02,830 --> 00:11:08,310
+شوفنا في التعريف. عبر عدد لانهائي من الطرق. عشان هيك
+
+165
+00:11:08,310 --> 00:11:11,430
+لو لقينا طريقين، الـ X والـ Y بيروحوا لها الـ X
+
+166
+00:11:11,430 --> 00:11:15,970
+₀ والـ Y₀، بيعطينا limitين مختلفين، أو واحد
+
+167
+00:11:15,970 --> 00:11:19,81
+
+223
+00:15:30,970 --> 00:15:35,810
+زمان الشرط الثالث لحالة إذا تحقق ما هو معناته إن
+
+224
+00:15:35,810 --> 00:15:39,330
+الـ limit هذه exist وهذه موجودة والتمتين متساويتين
+
+225
+00:15:39,330 --> 00:15:43,530
+يعني يحمل في طياته الشرط الثالث اللي هو الشرطين
+
+226
+00:15:43,530 --> 00:15:48,430
+الأولانيين وهذا معناته أنه إذا تحقق إنه a function
+
+227
+00:15:48,430 --> 00:15:53,470
+is continuous at x0 و y0 الآن لما نقول a function
+
+228
+00:15:53,470 --> 00:15:58,110
+is continuous معناته نقصد أنه continuous على كل
+
+229
+00:15:58,110 --> 00:16:02,570
+domain  لها لكن لما نقول f continuous عند الـ a والـ
+
+230
+00:16:02,570 --> 00:16:07,610
+x0 و الـ y0 نعني أنه متصل عند النقطة x0 و y0 وهذا
+
+231
+00:16:07,610 --> 00:16:13,350
+معناته فناخد مثال، بيقول لي في عندنا function show
+
+232
+00:16:13,350 --> 00:16:17,570
+that f of x y بيساوي 2 x y على x تربيع زائد y
+
+233
+00:16:17,570 --> 00:16:21,970
+تربيع في حال الـ x والـ y ليسوا صفر وصفر، وصفر في
+
+234
+00:16:21,970 --> 00:16:26,290
+حال الـ x والـ y بتساوي صفر وصفر. هذه الدالة بقول
+
+235
+00:16:26,290 --> 00:16:30,510
+لي اثبت أنها is continuous at every point ما عدا
+
+236
+00:16:30,510 --> 00:16:34,740
+عند من عند الـ z اللي هي الـ origin. الدالة دي دائما
+
+237
+00:16:34,740 --> 00:16:39,700
+متصلة ما عدا عند وين؟ عند الـ x والـ y بيساوي 0,0. نشوف
+
+238
+00:16:39,700 --> 00:16:44,240
+كيف متصلة عند النقاط اللي مش 0,0. يعني الآن بتاخد
+
+239
+00:16:44,240 --> 00:16:49,300
+a و b أي نقطة لا تساوي 0,0 احسب الـ limit f of x و
+
+240
+00:16:49,300 --> 00:16:53,140
+y لما x و y تروح لمين؟ لـ a و b. لما x و y تروح
+
+241
+00:16:53,140 --> 00:16:57,260
+لـ a و الـ b. الـ a والـ b مش مين؟ مش الـ 0,0. إذن وين
+
+242
+00:16:57,260 --> 00:17:01,280
+الدالة اللي بتستخدم؟ هاي الدالة. بصير limit f of x و
+
+243
+00:17:01,280 --> 00:17:05,300
+y هو عبارة عن limit هذه. Limit هذه لما x و y تذهب
+
+244
+00:17:05,300 --> 00:17:12,100
+إلى a و b، و a و b ليس 0,0 هي عبارة عن 2ab على
+
+245
+00:17:12,100 --> 00:17:15,980
+a تربيع زائد b تربيع. يعني إيش بتساوي يا جماعة؟ يعني
+
+246
+00:17:15,980 --> 00:17:20,780
+بتساوي قيمة الـ function عند a و b. وهذا يعني أن f
+
+247
+00:17:20,780 --> 00:17:25,400
+is continuous عند أي نقطة اللي هي لا تساوي 0,0، اللي
+
+248
+00:17:25,400 --> 00:17:31,060
+سميتها أنا a و b لا تساوي 0,0. إذن على طول اللي هي
+
+249
+00:17:31,060 --> 00:17:34,760
+limit للـ f of x y لما f of x y تذهب إلى a و b فهو
+
+250
+00:17:34,760 --> 00:17:38,620
+f of a و b عندما تكون الـ a والـ b ليس الصفر والصفر
+
+251
+00:17:38,620 --> 00:17:41,480
+لأنها دالة حدودية ودالة حدودية  ليست صفرية محلها. فبكون عند  الـ function is continuous
+
+252
+00:17:41,480 --> 00:17:46,440
+عند كل النقاط ما عدا 0,0. الآن نجي للّي هي
+
+253
+00:17:46,440 --> 00:17:51,220
+المشكلة. المشكلة أنه عند النقطة x و y بيساوي 0,0
+
+254
+00:17:51,220 --> 00:17:55,060
+بدنا نشوف الـ function continuous ولا لأ. أول حاجة
+
+255
+00:17:55,060 --> 00:17:59,020
+معرفة عند 0,0. ليش؟ لأن f of 0,0 إيش حتساوي؟ صفر. إذن
+
+256
+00:17:59,020 --> 00:18:05,120
+احنا بنتوقع أن المشكلة في الـ limit. Limit f of x,y
+
+257
+00:18:05,120 --> 00:18:11,350
+لما x,y تروح للـ 0,0 لنشوف عشان نثبت أن الـ
+
+258
+00:18:11,350 --> 00:18:17,230
+limit للـ f of x,y لما الـ x,y يروح للـ 0,0 does
+
+259
+00:18:17,230 --> 00:18:21,170
+not exist. بدنا ناخد طريقين زي ما أخدنا قبل شوية و
+
+260
+00:18:25,270 --> 00:18:29,470
+يعطينا قيمتين مختلفتين. الآن اللي بدأ أفحصها limit f
+
+261
+00:18:29,470 --> 00:18:33,490
+of x y لما x y تروح لمين يا جماعة؟ للصفر والصفر. x y
+
+262
+00:18:33,490 --> 00:18:37,110
+تروح للصفر والصفر. يعني مش صفر وصفر لسه. يعني إذا
+
+263
+00:18:37,110 --> 00:18:40,150
+أنا مين اللي بدأ أخدها؟ الدالة هي الدالة اللي بدأ
+
+264
+00:18:40,150 --> 00:18:44,990
+اشتغل عليها. يعني بدأ أخد limit 2xy على x تربيع
+
+265
+00:18:44,990 --> 00:18:49,490
+زائد y تربيع، وأثبت لكم أن limit لهذه لما x y تروح
+
+266
+00:18:49,490 --> 00:18:57,100
+لـ 0,0 ما تكونش اللي هو exist. كيف؟ عبر طريقين. نشوف
+
+267
+00:18:57,100 --> 00:19:01,620
+الطريق اللي أخدته. أخد y بيساوي mx. طبعا mx ده اللي
+
+268
+00:19:01,620 --> 00:19:04,960
+هو m ثابت و x متغير اللي احنا عارفينه متغير.
+
+269
+00:19:04,960 --> 00:19:07,920
+يعني عبر واحد من الطرق اللي زي هذول شايفين هنا
+
+270
+00:19:07,920 --> 00:19:12,560
+هذول، بدي أثبتلك أن عبر هذول الطرق حتلاقي أنه
+
+271
+00:19:12,560 --> 00:19:16,930
+قيمة الـ limit اللي هي مختلفة من الطريق هذه للطريق
+
+272
+00:19:16,930 --> 00:19:19,670
+هذه، للطريق هذه. وبناء عليه بتكون الـ limit does not
+
+273
+00:19:19,670 --> 00:19:24,950
+exist. خلّيني أخد m x، m in general. بيساوي limit اللي
+
+274
+00:19:24,950 --> 00:19:28,630
+هو بشيل اللي هي الـ y بحطها 2m مكانها بيصير
+
+275
+00:19:28,630 --> 00:19:35,030
+2mx في x بيصير x تربيع على x تربيع زائد y
+
+276
+00:19:35,030 --> 00:19:40,140
+تربيع اللي هي mx الكل تربيع. y ساوي limit اللي هو
+
+277
+00:19:40,140 --> 00:19:45,780
+الآن تاخد الـ .. الـ .. الـ .. الـ x تربيع هنا عامل
+
+278
+00:19:45,780 --> 00:19:50,980
+مشترك بين الجهتين هنا لما نختصرهم. ركزوا معايا.
+
+279
+00:19:50,980 --> 00:19:55,440
+الآن x تربيع بيظل إيش؟ واحد زائد m تربيع. والـ x
+
+280
+00:19:55,440 --> 00:19:58,260
+تربيع طلعت برا وفي x تربيع بتروح الـ x تربيع مع الـ
+
+281
+00:19:58,260 --> 00:20:02,060
+x تربيع اللي فوق، بيظل عنده 2m على واحد زائد m
+
+282
+00:20:02,060 --> 00:20:06,850
+تربيع. اللي هي قيمة الـ limit. لاحظ الآن هذه الـ limit
+
+283
+00:20:06,850 --> 00:20:11,430
+للـ function عبر الطريق y بيساوي mx بتعتمد على الـ
+
+284
+00:20:11,430 --> 00:20:15,750
+m. يعني لو جينا أخدنا y بيساوي x. هذا طريق. y بيساوي
+
+285
+00:20:15,750 --> 00:20:24,730
+x، y بيساوي مثلا اللي هو 2x. y بيساوي 2x
+
+286
+00:20:24,730 --> 00:20:29,270
+هنا مثلا y بيساوي -x هنا مثلا. هنلاقي قيمتين عبر طريقين مختلفين مختلفتين
+
+287
+00:20:29,270 --> 00:20:34,670
+خذ مثلا مثال. Hence along y بيساوي x الـ limit إيش
+
+288
+00:20:34,670 --> 00:20:39,230
+هيكون بيساوي 2 على 1 زائد 1 = 2 يعني
+
+289
+00:20:39,230 --> 00:20:43,930
+إيش بيساوي 1. الـ limit along y مثلا بيساوي 2x طبعا
+
+290
+00:20:43,930 --> 00:20:48,970
+إنت ممكن تاخد أي قيمة لـ m. y بيساوي 2x بيصير اللي
+
+291
+00:20:48,970 --> 00:20:53,550
+هو عبارة عن 2 في 2 = 4 على 1 زائد 4 = 5
+
+292
+00:20:53,550 --> 00:20:57,530
+يعني 4 على 5. واضح أن عبر الطريق هذا وعبر
+
+293
+00:20:57,530 --> 00:21:01,620
+الطريق هذا، limitين مش مساويتين. إذن اللي هو since
+
+294
+00:21:01,620 --> 00:21:06,780
+these two paths have different limits. زي ما قلنا
+
+295
+00:21:06,780 --> 00:21:09,140
+then limit does not exist. ومدام limit does not
+
+296
+00:21:09,140 --> 00:21:12,080
+exist لما x و y بتروح لـ 0,0 إذا أكيد الـ function
+
+297
+00:21:12,080 --> 00:21:15,960
+is not continuous عند 0,0 لأن عشان تكون
+
+298
+00:21:15,960 --> 00:21:19,620
+continuous عند 0,0 لازم الـ limit تساوي قيمة الـ
+
+299
+00:21:19,620 --> 00:21:23,420
+function اللي هي في هذه الحالة 0 وهذا غير حاصل طبعا
+
+300
+00:21:23,420 --> 00:21:27,340
+هذه الرسمة بتوضح لك أنه يعني عبر الطريق أو عبر
+
+301
+00:21:27,340 --> 00:21:34,830
+الطريق هذا كانت اللي هي اللي هي عندنا واحد هنا لما
+
+302
+00:21:34,830 --> 00:21:38,510
+كان point eight الميل وهنا ناقص point eight يعني y
+
+303
+00:21:38,510 --> 00:21:41,950
+بيساوي ناقص point يعني عبر الطريق اللي هي m هذه m
+
+304
+00:21:41,950 --> 00:21:45,610
+m يا جماعة هذا الميل اللي هو اللي اللي اللي بناخذه
+
+305
+00:21:45,770 --> 00:21:56,690
+هنا واحد وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+306
+00:21:56,690 --> 00:21:59,010
+وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا
+
+307
+00:21:59,010 --> 00:21:59,250
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+308
+00:21:59,250 --> 00:21:59,590
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+309
+00:21:59,590 --> 00:21:59,850
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+310
+00:21:59,850 --> 00:21:59,870
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+311
+00:21:59,870 --> 00:22:00,630
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+312
+00:22:00,630 --> 00:22:00,690
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+313
+00:22:00,690 --> 00:22:00,910
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+314
+00:22:00,910 --> 00:22:05,150
+كذا وهنا كذا
+
+315
+00:22:06,840 --> 00:22:09,760
+المثال السادس الآن، and they show that the function
+
+316
+00:22:09,760 --> 00:22:14,760
+f of x و y بيساوي 2x تربيع y على x<sup>4</sup> زائد y تربيع
+
+317
+00:22:14,760 --> 00:22:20,900
+has no limit as x و y approaches 0 و 0. بدنا نثبت
+
+318
+00:22:20,900 --> 00:22:25,280
+أن هذه الـ function مالهاش limit لما x و y تروح لـ 0
+
+319
+00:22:25,280 --> 00:22:31,100
+و 0. وهذا برضه مثال مشابه، تعزيز للفكرة. الآن بتاخد
+
+320
+00:22:31,100 --> 00:22:36,250
+طريقين، اللي هو أروح لمين الآن؟ لـ 0,0 اللي بطلع
+
+321
+00:22:36,250 --> 00:22:40,610
+اللي فوق. هنا واحد يقول لي خذ y بيساوي صفر. y بيساوي
+
+322
+00:22:40,610 --> 00:22:49,460
+صفر بيصير صفر على x<sup>4</sup>. في هذه الحالة بيصير صفر. هي عبر
+
+323
+00:22:49,460 --> 00:22:54,640
+الـ y بيساوي 0. عبر x بيساوي 0 برضه 0. إذن عبر y
+
+324
+00:22:54,640 --> 00:22:59,000
+بيساوي 0 و x بيساوي 0 بيعطينا قيمتين مختلفتين. لو
+
+325
+00:22:59,000 --> 00:23:03,720
+أخذت y بيساوي اللي هو x بيصير 2x تكعيب على x<sup>4</sup>
+
+326
+00:23:03,720 --> 00:23:09,560
+زائد x تربيع. يعني بتؤول برضه إذن الآن هو ممكن تطلع
+
+327
+00:23:09,560 --> 00:23:13,770
+لنا صفر كمان. الآن فإيجي عباله على طول يقولك خذ y
+
+328
+00:23:13,770 --> 00:23:16,490
+بيساوي x تربيع. ليش y بيساوي x تربيع؟ عشان هون بيصير
+
+329
+00:23:16,490 --> 00:23:20,830
+x<sup>4</sup> هون بيصير x<sup>4</sup> هون أيضا
+
+330
+00:23:20,830 --> 00:23:24,970
+بيصير اختصار أيضا بيصير قيمة غير الصفر. هذا تفكير
+
+331
+00:23:24,970 --> 00:23:29,710
+كله ثاني. نشوف along y بيساوي x تربيع بيصير limit of x
+
+332
+00:23:29,710 --> 00:23:33,960
+y اللي فوق لما x y تروح لـ 0,0 بيساوي الـ limit عوض
+
+333
+00:23:33,960 --> 00:23:39,320
+إنت خدت إيش مالها؟ y بيساوي x تربيع. بعوض عن y بـ x
+
+334
+00:23:39,320 --> 00:23:45,760
+تربيع. بيصير 2x<sup>4</sup> وهنا x<sup>4</sup> وهنا x<sup>4</sup>
+
+335
+00:23:45,760 --> 00:23:50,460
+ماشي؟ عارفين كيف عوضنا عن y بيساوي x تربيع لأن عبر
+
+336
+00:23:50,460 --> 00:23:55,600
+هذا الطريق اللي بناخذه الآن بنروح لـ 0,0 الآن يعني
+
+337
+00:23:55,600 --> 00:23:59,160
+بيصير x و x تربيع تروح للصفر. يعني x و x بتروح
+
+338
+00:23:59,160 --> 00:24:04,940
+للصفر. استبدلت هذه. عوضت عن مكان y بيساوي x تربيع
+
+339
+00:24:04,940 --> 00:24:08,900
+لكن x و x تربيع بيروح للصفر إذا وفقط إذا x و x
+
+340
+00:24:08,900 --> 00:24:12,560
+بتروح للصفر. فهذه اللي هي بتيجي مكان x و x تربيع
+
+341
+00:24:13,220 --> 00:24:17,980
+الآن هذه بتروح نختصر. بيطلع أن 2 على 2
+
+342
+00:24:17,980 --> 00:24:22,400
+بيساوي إيش؟ واحد. إذن الـ limit هنا عبر الطريق هذا
+
+343
+00:24:22,400 --> 00:24:27,420
+واحد. along y بيساوي صفر. كلنا limit f of x y لما x
+
+344
+00:24:27,420 --> 00:24:32,420
+y تروح لـ 0,0 بيساوي limit بعوض عن كل y بياشي بصفر
+
+345
+00:24:32,730 --> 00:24:37,310
+هذا الوضع عند y بصفر، و x و y بصفر، و x و y بصفر و x
+
+346
+00:24:37,310 --> 00:24:46,990
+و y بصفر، و x و y بصفر، و x و y بصفر، و x و y بصفر و
+
+347
+00:24:46,990 --> 00:24:48,970
+x و y بصفر، و x و y بصفر، و x و y بصفر، و x و y بصفر
+
+348
+00:24:48,970 --> 00:24:49,430
+y بصفر، و x و y بصفر، و x و y بصفر، و x و y بصفر و x
+
+349
+00:24:49,430 --> 00:24:58,270
+و y بصفر، و x و y بصفر، و x و y بصفر، و x و y بصفر و
+
+350
+00:24:58,270 --> 00:25:02,180
+x و y بصفر، و x و y. الآن نجي لـ continuity of
+
+351
+00:25:02,180 --> 00:25:05,560
+composite functions. هذه برضه حاجة سهلة، أخدناها
+
+352
+00:25:05,560 --> 00:25:09,900
+زمان في calculus. if f is continuous عند x0 و y0
+
+353
+00:25:09,900 --> 00:25:14,260
+و g is continuous اللي هو single variable هذه
+
+354
+00:25:14,260 --> 00:25:18,200
+function of one variable زي تبعات زمان يعني is
+
+355
+00:25:18,200 --> 00:25:22,640
+continuous at f of x0 و y0 then the composition h
+
+356
+00:25:22,640 --> 00:25:27,220
+بيساوي g composite f اللي هي بيصير h of x و y بيساوي
+
+357
+00:25:27,220 --> 00:25:31,090
+g of f of x y. يعني هذه اللي جوا function of two
+
+358
+00:25:31,090 --> 00:25:33,850
+variables بينما الـ g نفسها function of one
+
+359
+00:25:33,850 --> 00:25:37,670
+variable. إذا كانت اللي جوا continuous بقول، واللي
+
+360
+00:25:37,670 --> 00:25:41,030
+برا continuous بيكون الدالة كلها على بعض إيش مالها؟
+
+361
+00:25:41,030 --> 00:25:44,090
+continuous عند x0 و y0. وهذا اللي
+
+362
+00:25:44,090 --> 00:25:48,710
+كنا نقوله زمان. for example الآن e to the x - y
+
+363
+00:25:48,710 --> 00:25:51,970
+هذه بقدر أعملها function اللي هو composite of two
+
+364
+00:25:51,970 --> 00:25:57,200
+functions. يعني خدوا الـ f of x y هي x - y. f of x
+
+365
+00:25:57,200 --> 00:26:02,820
+y بيساوي x - y هي continuous function. وخد g of
+
+366
+00:26:02,820 --> 00:26:08,720
+z مثلا بيساوي e to the z. ماشي؟ فبيصير عندي الآن g of
+
+367
+00:26:08,720 --> 00:26:15,600
+f of x y هي عبارة عن الـ f of x y اليمين x - y
+
+368
+00:26:15,600 --> 00:26:20,900
+حيقولنا g of x - y بيصير e to the x - y
+
+369
+00:26:20,900 --> 00:26:24,240
+يعني هذه الآن قدرت أكتبها على صورة composition of
+
+370
+00:26:24,240 --> 00:26:27,900
+two functions. واحدة الـ x - y والثانية الـ
+
+371
+00:26:27,900 --> 00:26:31,020
+exponential. الـ exponential continuous والـ x -
+
+372
+00:26:31,020 --> 00:26:33,520
+y continuous لأنها حدودية وواحدة أسية.
+
+373
+00:26:33,520 --> 00:26:37,260
+إذا الاثنتين continuous إذا الـ composition تبقى
+
+374
+00:26:37,260 --> 00:26:40,720
+continuous. أو زي ما كنا نقول زمان، اللي جوا
+
+375
+00:26:40,720 --> 00:26:44,160
+continuous واللي برا continuous إذا كلها على بعض
+
+376
+00:26:44,160 --> 00:26:47,360
+continuous. نفس الأشياء هذه، اللي جوا continuous
+
+377
+00:26:47,360 --> 00:26:51,520
+لأنها حدودية على حدودية. ليش؟ one function واللي
+
+378
+00:26:51,520 --> 00:26:54,200
+اللي تحت مش مستحيل يساوي صفر. إذا هذه continuous
+
+379
+00:26:54,200 --> 00:26:58,080
+وين ما كان، والـ cosine نفسها continuous. إذن اللي هو
+
+380
+00:26:58,080 --> 00:27:00,880
+composition of two functions. الـ function الأولى
+
+381
+00:27:00,880 --> 00:27:03,940
+اللي هي f of x y هيها، والـ function الثانية اللي
+
+382
+00:27:03,940 --> 00:27:07,570
+هي cosine x. إذن صارت هذه كلها على بعضها g
+
+383
+00:27:07,570 --> 00:27:10,650
+composite f is continuous لأن هذا continuous وهذا
+
+384
+00:27:10,650 --> 00:27:14,950
+continuous. Similarly، الواحد زي x تربيع، زي y تربيع
+
+385
+00:27:14,950 --> 00:27:18,390
+هي الـ f of x y. وهذا polynomial continuous والـ ln
+
+386
+00:27:18,390 --> 0
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/RtOltUWLcYs_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/RtOltUWLcYs_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..88d9db06bda663b666160de2d3fc739272f5f5d7
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/RtOltUWLcYs_postprocess.srt
@@ -0,0 +1,1736 @@
+1
+00:00:00,760 --> 00:00:04,080
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,080 --> 00:00:10,200
+وبركاته هذه هي المحاضرة الرقم 14 مساق تفاضل وتكامل
+
+3
+00:00:10,200 --> 00:00:14,800
+gene وهي المحاضرة الثانية في chapter 14 وهي تحت
+
+4
+00:00:14,800 --> 00:00:18,600
+عنوان 14 limits and continuity in higher
+
+5
+00:00:18,600 --> 00:00:24,120
+dimensions سنتحدث عن نهايات والاتصال في ال higher
+
+6
+00:00:24,120 --> 00:00:27,180
+dimensions يعني function of two variables أو three
+
+7
+00:00:27,180 --> 00:00:30,220
+variables أو moreأحنا كنا نتحدث طبعا في الـ
+
+8
+00:00:30,220 --> 00:00:34,780
+Calculus عن النهايات وعن الاتصال لدوال اللي هي
+
+9
+00:00:34,780 --> 00:00:38,360
+function of one variable y بساوية f of x الآن بدنا
+
+10
+00:00:38,360 --> 00:00:42,600
+نتحدث بشكل مشابه أولا عن limits for functions of
+
+11
+00:00:42,600 --> 00:00:46,820
+two variables اللي هي نهايات الدوال اللي اتدالة
+
+12
+00:00:46,820 --> 00:00:50,580
+function of two variables الآن بدنا نشوف شو معناة
+
+13
+00:00:50,580 --> 00:00:54,680
+limit في هذه الحالة وهي مشابهة لحالةاللي هي limit
+
+14
+00:00:54,680 --> 00:00:59,000
+اللي أخدناها في اللي هو تفاضل ألف We say that a
+
+15
+00:00:59,000 --> 00:01:03,340
+function f of x,y approaches the limit L as x,y
+
+16
+00:01:03,340 --> 00:01:07,540
+approaches x0,y0 and right يعني بنقول إن ال limit
+
+17
+00:01:07,540 --> 00:01:13,790
+لل f of x,y لما x,y تقول إلى x0,y0 بتساوي Lإننا
+
+18
+00:01:13,790 --> 00:01:17,850
+بدنا نعبّر عن هذا by Epsilon Delta Definition الـ
+
+19
+00:01:17,850 --> 00:01:21,390
+Epsilon Delta Definition باختصار بتقول إنه عندما
+
+20
+00:01:21,390 --> 00:01:26,050
+تقترب X Y من الـ X Note والـ Y Note الـ F of X Y
+
+21
+00:01:26,050 --> 00:01:30,310
+هتقترب من الـ L أو بمعنى آخر لما X Y تقول إلى X
+
+22
+00:01:30,310 --> 00:01:34,010
+Note وY Note الـ F of X Y تقول إلى L بدنا نعبّر عن
+
+23
+00:01:34,010 --> 00:01:38,490
+هذا الكلام عن اقتراب اللي هو X Y من X Note وY Note
+
+24
+00:01:38,720 --> 00:01:43,740
+يؤدى الى اقتراب f of x و y ل L بواسطة اللى هى
+
+25
+00:01:43,740 --> 00:01:48,560
+الرموز اللى اتعودنا عليها سابقا ان اذا اتعود اي
+
+26
+00:01:48,560 --> 00:01:51,160
+ابسلون ان يعطينا اي ابسلون اكبر من 0 الابسلون
+
+27
+00:01:51,160 --> 00:01:55,160
+هتعبر عن الاقتراب ال f of x y من 100 من 1000 طبعا
+
+28
+00:01:55,160 --> 00:01:59,000
+f of x y و L عبارة عن اعداد في الآخر هدول اقترابين
+
+29
+00:01:59,000 --> 00:02:02,890
+على خط الاعدادلكن الـ X و Y من الـ X note و Y note
+
+30
+00:02:02,890 --> 00:02:06,210
+هدولة وين موجودات في البلين يعني هي X note و Y
+
+31
+00:02:06,210 --> 00:02:10,070
+note و بدها تقترب إليها X و Y في البلين الآن
+
+32
+00:02:10,070 --> 00:02:14,570
+اقتراب ال X و Y من ال X note و Y note زمان كان لما
+
+33
+00:02:14,570 --> 00:02:18,430
+نقول X تقترب من X note كنا على خط الأعدادنقترب
+
+34
+00:02:18,430 --> 00:02:21,430
+إليها من اليمين أو نقترب إليها من وين من اليسار
+
+35
+00:02:21,430 --> 00:02:26,790
+بينما هنا الاقتراب من وين من كل الجهات يعني عبر
+
+36
+00:02:26,790 --> 00:02:31,950
+اللي هو إيش دائرة اللي هو ال X و ال Y تقترب من ال
+
+37
+00:02:31,950 --> 00:02:36,150
+X note و Y note من كل هذه الاتجاهات نشوف عن الكلام
+
+38
+00:02:36,150 --> 00:02:40,500
+هذا إيش بنعني فيهإنه لكل Y أكبر من 0 بنلاقي Delta
+
+39
+00:02:40,500 --> 00:02:44,880
+أكبر من 0 تعبر عن اقتراب من الـ X و Y من X note و
+
+40
+00:02:44,880 --> 00:02:49,940
+Y note لكل X و Y in the domino path لما X و Y
+
+41
+00:02:49,940 --> 00:02:54,920
+تقترب من X note و Y noteالـ X و الـ Y البعد عنها
+
+42
+00:02:54,920 --> 00:02:58,920
+في البعد هنا اللي هو في الـ plane هو عبارة عن
+
+43
+00:02:58,920 --> 00:03:01,820
+المسافة بين X و Y و X note و Y note اللي هو عبارة
+
+44
+00:03:01,820 --> 00:03:05,460
+عن الجذر التربيعي ل X minus X note كل تربيع زي Y
+
+45
+00:03:05,460 --> 00:03:09,480
+minus Y note كل تربيع لما يكون هذا أصغر من Delta
+
+46
+00:03:10,690 --> 00:03:14,810
+بأدّيني إلى f of x y ناقص L أشمالها أصغر من
+
+47
+00:03:14,810 --> 00:03:20,470
+Epsilon معناته أنه لما المسافة بين x و y اللي هو
+
+48
+00:03:20,470 --> 00:03:24,050
+تكون صغيرة صغيرة صغيرة لدرجة أن x not أو لما ال x
+
+49
+00:03:24,050 --> 00:03:30,430
+not و y not ال f of x y و L يقتربن جدا و يتصير ال
+
+50
+00:03:30,430 --> 00:03:35,330
+f of x y أشمالها تأول إلى L و هتكون أصغر من أي
+
+51
+00:03:35,330 --> 00:03:39,920
+Epsilon في الدنياهنا باللاحظ ما يلي زمان كنا نقول
+
+52
+00:03:39,920 --> 00:03:43,400
+لما كنا نقول ال X تقترب من X note في ال limit of
+
+53
+00:03:43,400 --> 00:03:47,160
+function of one variable كنا نقول ال limit من
+
+54
+00:03:47,160 --> 00:03:50,100
+اليمين و ال limit من اليسار كنا نقول ان ال limit
+
+55
+00:03:50,100 --> 00:03:54,340
+هذه بتكون موجودة إذا كانت X راحت لل X note من
+
+56
+00:03:54,340 --> 00:03:56,880
+اليمين و من اليسار هان يسار مش من اليمين و من
+
+57
+00:03:56,880 --> 00:04:01,100
+اليسار يعني لما تكون ال limit موجودة هانمش من
+
+58
+00:04:01,100 --> 00:04:05,920
+اليامين واليسار بس لأ بدي أكون عبر كل الطرق اللي
+
+59
+00:04:05,920 --> 00:04:10,280
+من x و y ل x note و y note هذه في عندك عدد لا
+
+60
+00:04:10,280 --> 00:04:14,220
+نهائي من الطرق بتروح إلها لما تكون ال limit موجودة
+
+61
+00:04:14,220 --> 00:04:20,870
+لازم كل الطرقبتعطي نفس النهاية مش بس اللي كانت من
+
+62
+00:04:20,870 --> 00:04:24,910
+يمين و يسار زمان انه تعطي نفس النهاية لأ هنا برضه
+
+63
+00:04:24,910 --> 00:04:29,750
+عشان تكون ال limit موجودة لازم عبر كل هذه الطرق
+
+64
+00:04:29,750 --> 00:04:34,210
+تؤدي الى ايش نفس النهاية يعني لو بدنا نثبت ان ال
+
+65
+00:04:34,210 --> 00:04:38,510
+limit هنا does not exist يكفي ان نلاقي طريقين يروح
+
+66
+00:04:38,510 --> 00:04:42,830
+إلى ال X note و Y note يكون ال limitين الهين
+
+67
+00:04:42,830 --> 00:04:47,680
+مختلفاتزي ما كنا بنفس المنطقة عشان نثبت ان ال
+
+68
+00:04:47,680 --> 00:04:51,220
+limit لل F of X does not exist كنا نقول ال limit
+
+69
+00:04:51,220 --> 00:04:53,920
+من اليمين وال limit من اليسار إذا كان واحدة منهم
+
+70
+00:04:53,920 --> 00:04:56,960
+متساويش التانية أو واحدة مش موجودة معناته ال limit
+
+71
+00:04:56,960 --> 00:05:01,880
+does not exist إذن باختصار X و Y بتروح ل X0 و Y0
+
+72
+00:05:01,880 --> 00:05:05,560
+تؤدي ال F of X و Y بتروح لل L هذا يعبر عنه عندما
+
+73
+00:05:05,560 --> 00:05:10,920
+يكون المسافة بين X0 و Y0Y0 بقول X و Y أصغر من
+
+74
+00:05:10,920 --> 00:05:14,620
+Delta يؤدي لنا إلى المسافة بين F of X و Y و L أصغر
+
+75
+00:05:14,620 --> 00:05:20,260
+من Epsilon يعني هذا بقى بيعني أن X و X و Y بتقول
+
+76
+00:05:20,260 --> 00:05:23,580
+X0 و Y0 يؤدي ل F of X و Y بتروح ل L و في هذا
+
+77
+00:05:23,580 --> 00:05:27,220
+الحالة بتقول ال limit exist تعالوا الآن ناخد اللي
+
+78
+00:05:27,220 --> 00:05:30,880
+هي بعض اللي هي الخواص اللي زي الخواص اللي أخدناها
+
+79
+00:05:30,880 --> 00:05:35,350
+سابقا في اللي هي function of one variableإذا كانت
+
+80
+00:05:35,350 --> 00:05:37,570
+الـ properties of limits of functions of two
+
+81
+00:05:37,570 --> 00:05:41,750
+variables the following rules holds if L و M and K
+
+82
+00:05:41,750 --> 00:05:43,970
+are real numbers such that يعني لو كانت limit ال
+
+83
+00:05:43,970 --> 00:05:47,710
+function f of x,y بسوء L لما x,y بتروح ل x not و y
+
+84
+00:05:47,710 --> 00:05:50,970
+not و limit لل g of x,y لما x,y تروح ل x not و y
+
+85
+00:05:50,970 --> 00:05:55,770
+not و سوء M إذا كان الليمتين هدولة موجودات و بساوي
+
+86
+00:05:55,770 --> 00:06:01,380
+الأقام اللي بيننا إذا ال limitللمجموع توزع يعني
+
+87
+00:06:01,380 --> 00:06:05,100
+بيصير limit الـ f of x y زائد g of x y بساوي limit
+
+88
+00:06:05,100 --> 00:06:09,020
+الأول زائد limit الثاني يعني بنوزع ال limit على
+
+89
+00:06:09,020 --> 00:06:12,640
+الجامع في حال متى لما يكون ال limitين موجودات
+
+90
+00:06:12,640 --> 00:06:16,240
+النهايتين موجودات بساوي L زائد M ونفس الأشي على
+
+91
+00:06:16,240 --> 00:06:19,500
+الطرح ونفس الأشي لو كان in the scalar مضروف في f
+
+92
+00:06:19,500 --> 00:06:23,360
+of x y بطلع ال K برا بيصير في limit f of x y اللي
+
+93
+00:06:23,360 --> 00:06:28,190
+هو K في مين في ال Lالان هذا اللي هو نفس خواص اللي
+
+94
+00:06:28,190 --> 00:06:31,710
+هي تبع الـ function العادية برضه نفس الأشياء حاصل
+
+95
+00:06:31,710 --> 00:06:34,410
+ضرب دلتين ال limit الهن بتوزع لما يكون ال limit
+
+96
+00:06:34,410 --> 00:06:37,390
+موجود بيكون ال limit الأول ضرب ال limit الثاني و
+
+97
+00:06:37,390 --> 00:06:41,150
+نفس الأشياء حاصل القسمة لكن provided that اللي ال
+
+98
+00:06:41,150 --> 00:06:45,400
+MN اللي هي limit اللي ال G of XY ماتساويش 0الان
+
+99
+00:06:45,400 --> 00:06:49,360
+نفس الشيء limit f of x,y أسن as x,y بتروح ل x0 و
+
+100
+00:06:49,360 --> 00:06:54,460
+y0 بساوي اللي هي limit هذه لكل أسن يعني L أسن for
+
+101
+00:06:54,460 --> 00:06:58,470
+any positive integer Mنفس الشيء الـ limit الجدر
+
+102
+00:06:58,470 --> 00:07:03,390
+النوني لأف of xy لما xy تروح ل x0 و y0 بساوي الجدر
+
+103
+00:07:03,390 --> 00:07:06,950
+النوني ل limit اللي جوا اللي هي ال L هي L أسواحد
+
+104
+00:07:06,950 --> 00:07:12,650
+على L حيث L positive integer و L أكبر من 0 و هذه
+
+105
+00:07:12,650 --> 00:07:16,810
+نفس ال roots القوانين اللي أخدناها سابقا على ال
+
+106
+00:07:16,810 --> 00:07:21,630
+function of one ناخد اللي هو كيفية إيجاد اللي هو
+
+107
+00:07:21,630 --> 00:07:25,410
+للإستخدام اللي هي الخواص اللي قبل بشوية لإيجاد ال
+
+108
+00:07:25,410 --> 00:07:29,320
+limitعند ال limit هي المقدار اللى فوق على المقدار
+
+109
+00:07:29,320 --> 00:07:32,700
+اللى تحت لما x و y تروح لل zero الواحد الآن ال
+
+110
+00:07:32,700 --> 00:07:38,920
+limit للأولى للأولى موجودة لإن هى إيش بتساوي اللى
+
+111
+00:07:38,920 --> 00:07:43,890
+هى سفر ال limit للتانية x و y برضه سفر موجودةلأن
+
+112
+00:07:43,890 --> 00:07:46,690
+الـ X و الـ Y يذهبون إلى الـ 0 و الـ 1 يعني الـ X
+
+113
+00:07:46,690 --> 00:07:50,150
+تذهب إلى الـ 0 و الـ Y تذهب إلى الـ 0 اللي هي عندي
+
+114
+00:07:50,150 --> 00:07:54,110
+بيصير هذا سفر زائد تلاتة إذا بقلع فوق و بوزع و أنا
+
+115
+00:07:54,110 --> 00:07:58,970
+مرتاح و هايهن اللي هو سفر ناقص سفر زائد تلاتة ماشي
+
+116
+00:07:58,970 --> 00:08:02,970
+الحال اللي تحت هو طبعا أنا ضامن اللي تحت مش سفر
+
+117
+00:08:02,970 --> 00:08:05,970
+عشان هيك بوزع أنا عارف من قبل إيش اللي ضامنك ده
+
+118
+00:08:05,970 --> 00:08:11,130
+أشوف أطلع الدليللما الأولى limitها على تعويض مباشر
+
+119
+00:08:11,130 --> 00:08:14,970
+هذا لان polynomial ع polynomial يعني الدوال حلوة
+
+120
+00:08:14,970 --> 00:08:19,130
+وجميلة و لو عوضت بتعويض مباشر هتطلع هذه سفر وهذه
+
+121
+00:08:19,130 --> 00:08:23,390
+هتطلع سفر وهذه ناقص واي تكعيب اللي هي ناقص واحد
+
+122
+00:08:23,390 --> 00:08:28,490
+اللي هي تكعيب اللي هي واحد ناقص واحد بيصير تلاتة ع
+
+123
+00:08:28,490 --> 00:08:31,810
+ناقص واحد سواء ناقص تلاتة ع واحد يعنياللي هي ..
+
+124
+00:08:31,810 --> 00:08:34,910
+اللي هي زائد .. زائد تلاتة على نقص واحد اللي هي
+
+125
+00:08:34,910 --> 00:08:38,810
+نقص ثلاثة وهيها اللي هي استخدمت قوانين ال limit
+
+126
+00:08:38,810 --> 00:08:41,690
+اللي قبل بشوية و كلها مشروعة زي ما قلنا اللي هي
+
+127
+00:08:41,690 --> 00:08:43,610
+الجدر التربيع على طول الجدر التربيع اللي هو ال
+
+128
+00:08:43,610 --> 00:08:46,410
+limit اللي هو اللي جوا لل X تربيع على طول تعويض
+
+129
+00:08:46,410 --> 00:08:50,090
+مباشر تسعة وهذا اللي هي ست عشر اللي هي الجدر الخمس
+
+130
+00:08:50,090 --> 00:08:54,880
+عشرين ويساوي خمسةالان مش دايما الدنيا حلوة زي هيك
+
+131
+00:08:54,880 --> 00:08:59,240
+في بعض الشغلات بتكون شوية بدغلبة يعني مش تعويض
+
+132
+00:08:59,240 --> 00:09:01,960
+مباشر اذا تعويض مباشر ان الدول حلوة وسهلة وكذا
+
+133
+00:09:01,960 --> 00:09:06,420
+الان اخد مثال تاني find limit x ترميع نقص x y على
+
+134
+00:09:06,420 --> 00:09:10,430
+الجدر الترميع لل x نقص الجدر الترميع لل yلما اكس
+
+135
+00:09:10,430 --> 00:09:13,670
+واي تذهب الى 00 تعويض المباشر يظهر سفر فوق و سفر
+
+136
+00:09:13,670 --> 00:09:16,630
+فوق مش زي اللي فوق اللي هو طلعت لي على طول تعويض
+
+137
+00:09:16,630 --> 00:09:20,370
+مباشر طلعت لي أرقام حلوة وجميلة عشان هيك استخدمنا
+
+138
+00:09:20,370 --> 00:09:24,710
+القوانين إذا بدنا نضطر نشوف السفر اللي هنضيعه مع
+
+139
+00:09:24,710 --> 00:09:28,380
+السفر اللي فوق نضيعه إذا فيه إمكانيةطبعا أمثلة
+
+140
+00:09:28,380 --> 00:09:31,320
+منها النوع كل كومو أكيد قال طب اضربلنا في المرافق
+
+141
+00:09:31,320 --> 00:09:34,840
+للي تحت يعني اضربلنا في جدر ال X زائد جدر ال Y فوق
+
+142
+00:09:34,840 --> 00:09:39,140
+أو تحت ضربنا فوق صار المقدار X تربيع ناقص X في Y
+
+143
+00:09:39,140 --> 00:09:42,580
+في جدر ال X زائد جدر ال Y اللي هي ضربته فوق وضربته
+
+144
+00:09:42,580 --> 00:09:45,660
+في تحت لما ضربته في تحت اللي هو هذا بيكون مفكوك
+
+145
+00:09:45,660 --> 00:09:47,840
+فارق بين مربعين مربعين اللي هو الجدر ال X في كل
+
+146
+00:09:47,840 --> 00:09:51,880
+تربيع يعني X و جدر ال Y كل تربيع يعني شوية يعني X
+
+147
+00:09:51,880 --> 00:09:58,680
+ناقص Yالان انا اريد ان اضيع السفر اللي اتحتالان
+
+148
+00:09:58,680 --> 00:10:01,760
+طبعا تركيبة السؤال هو اعملها عشان هيك اصلا هذه x
+
+149
+00:10:01,760 --> 00:10:05,900
+تربيع نقص xy خد ال x برا بيصير x minus y في جدر ال
+
+150
+00:10:05,900 --> 00:10:09,860
+x زائد جدر ال y على x minus y هذه بتروح معها هذه
+
+151
+00:10:09,860 --> 00:10:13,940
+بيصير x في جدر ال x زائد جدر ال y او التعويض مباشر
+
+152
+00:10:13,940 --> 00:10:17,200
+بيطلع جديش بيساوي بيساوي سفر إذا انصارت ال limit
+
+153
+00:10:17,200 --> 00:10:22,340
+اللي عندي هذه عبارة عن السفر طيب الآن زي ما قلنا
+
+154
+00:10:22,340 --> 00:10:25,580
+قبل بشوية هذا اللي قبل اللي .. الأمثلة اللي قبل
+
+155
+00:10:25,580 --> 00:10:29,890
+بشوية اللي هي إيجاد limitالان بدنا نثبت to path
+
+156
+00:10:29,890 --> 00:10:34,510
+test for nonexistence of a limit هذا طبعا له خلفية
+
+157
+00:10:34,510 --> 00:10:37,690
+سابقة عندكم انه كنا نقول فى ال function of one
+
+158
+00:10:37,690 --> 00:10:42,370
+variable تبعد زمان limit F of X بتساوي L لما X
+
+159
+00:10:42,370 --> 00:10:46,650
+تروح لل X note مثلا كنا نقول هذا ال limit لو كان
+
+160
+00:10:46,650 --> 00:10:50,470
+ال limit من اليمين و ال limit من اليسارواحد من
+
+161
+00:10:50,470 --> 00:10:55,190
+المش موجود أو اتنين موجودات بس مش متساويين على طول
+
+162
+00:10:55,190 --> 00:10:59,130
+بنقول ال limit does not exist هنا نفس الجثة يا
+
+163
+00:10:59,130 --> 00:11:02,830
+جماعة بس ال X و ال Y بتروح ل X نوت و Y نوت زي ما
+
+164
+00:11:02,830 --> 00:11:08,310
+شوفنا في التعريف عبر عدد لانهائي من الطرقعشان هيك
+
+165
+00:11:08,310 --> 00:11:11,430
+لو لجينا طريقين الـ X و الـ Y بيتروحوا ليها الـ X
+
+166
+00:11:11,430 --> 00:11:15,970
+note و الـ Y note بيعطينا limitين مختلفين أو واحد
+
+167
+00:11:15,970 --> 00:11:19,810
+منهم does not exist على طول ال limit إيه شمالها
+
+168
+00:11:19,810 --> 00:11:25,640
+does not exist وهذا الكلام سهل وشغل مكانالان ناخد
+
+169
+00:11:25,640 --> 00:11:28,720
+مثال وقولش هو that limit y على x as x و y بتروح ل
+
+170
+00:11:28,720 --> 00:11:31,500
+0 و 0 ايش ما لها does not exist
+
+171
+00:11:55,820 --> 00:11:59,420
+أو عبر الخط هذا بروح للسفر أو عبر الخط هذا بروح
+
+172
+00:11:59,420 --> 00:12:03,360
+للسفر ومش هيك كمان و ال Y بتساوي X بروح للسفر و
+
+173
+00:12:03,360 --> 00:12:08,160
+السفر و ال Y بساوي ناقص X بتروح للسفر و ال Y بساوي
+
+174
+00:12:08,160 --> 00:12:12,700
+اتنين X بروح للسفر و ال Y بساوي نص X بروح للسفر و
+
+175
+00:12:12,700 --> 00:12:16,800
+ال Y بساوي X تربيع برضه بمرمين بالسفر اذا اي طريق
+
+176
+00:12:16,800 --> 00:12:20,740
+من هدول لو لجيت طريقين من اللي حكيتهم او من غيرهم
+
+177
+00:12:20,740 --> 00:12:25,130
+بس المهم يكون ان الطريقين بمرمينبالـ 0 و الـ 0 لأن
+
+178
+00:12:25,130 --> 00:12:29,070
+الـ X و الـ Y تبعتي تذهب إلى الـ 0 و الـ 0 إذا
+
+179
+00:12:29,070 --> 00:12:33,650
+وجدت طريقين واحد منهم لا يوجد أو الاتنين موجود لكن
+
+180
+00:12:33,650 --> 00:12:37,750
+مش متساوين فالـ limit لا يوجد
+
+181
+00:12:43,080 --> 00:12:47,460
+بتمرش بالسفر والسفر إذا الطريقين اللي بيداخدهم
+
+182
+00:12:47,460 --> 00:12:51,600
+لازم يمرن بالسفر والسفر عشان أروح للسفر والسفر
+
+183
+00:12:51,600 --> 00:12:55,400
+لأنه نغايطي x و y ال limit لما x و y تروح لمين
+
+184
+00:12:55,400 --> 00:12:59,980
+للسفر والسفر نيجي أولش على عبر الطريق y بتساوي x
+
+185
+00:12:59,980 --> 00:13:04,280
+عبر الطريق y بتساوي x ممر بال 0 و 0 طيب يصير الان
+
+186
+00:13:04,280 --> 00:13:08,990
+limit y على xلما انا اخذ عبر الطريق ايش بدي احسب
+
+187
+00:13:08,990 --> 00:13:13,710
+الدالة هذه عند y بيساوي x بتساوي limit x على x
+
+188
+00:13:13,710 --> 00:13:17,470
+الان بيصير الان ال x و ال y بيروح لل 0 و ال 0 يعني
+
+189
+00:13:17,470 --> 00:13:21,910
+اخدت عبر ال y بيساوي x يعني ال y x يعني ال x و ال
+
+190
+00:13:21,910 --> 00:13:26,050
+x بتروح لمين لل 0 و ال 0 الان x على x بيروحين مع
+
+191
+00:13:26,050 --> 00:13:29,210
+بعض بيصيروا واحد limit الواحد لما x و x تروح لل x
+
+192
+00:13:29,210 --> 00:13:33,450
+و y تروح لل 0و الـ zero هي ساوية واحدة لأن x على x
+
+193
+00:13:33,450 --> 00:13:37,570
+بيساوي واحد ثابت اذا ال limit عبر الطريق y بيساوي
+
+194
+00:13:37,570 --> 00:13:42,390
+x بيساوي واحدة لكن path along y بتساوي سفر لإن بدأ
+
+195
+00:13:42,390 --> 00:13:47,610
+اروح على الخط مين على عبر الخط y بتساوي سفر بدأ
+
+196
+00:13:47,610 --> 00:13:51,270
+اروح عوين على x و y بيساوي 00 اللي هو هذا ال x
+
+197
+00:13:51,270 --> 00:13:56,340
+axis بيروح ال 00 طبعا هنااحسب ال limit عبر الخط
+
+198
+00:13:56,340 --> 00:14:00,340
+هذا limit y على x بساوة limit على طول ايش اخدت y
+
+199
+00:14:00,340 --> 00:14:03,720
+انا بساوة سفر لان انا قاعد على الخط اللي هو y
+
+200
+00:14:03,720 --> 00:14:06,280
+بساوة سفر بيصير ده اللي هذي اللي بدي احسبلها ال
+
+201
+00:14:06,280 --> 00:14:09,980
+limit zero على x و y ساوة zero لاحظ ال limit في
+
+202
+00:14:09,980 --> 00:14:13,620
+هذه الحالة لا ساوة limit في هذه الحالة يعني زي ما
+
+203
+00:14:13,620 --> 00:14:15,960
+كنا نقول زمان limit لمين و limit ولا سامش
+
+204
+00:14:15,960 --> 00:14:19,040
+متساويتين اذا ال two limit ال limit does not exist
+
+205
+00:14:19,040 --> 00:14:22,720
+او عندي في هذه الحالةاللي احنا فيها انه limit عبر
+
+206
+00:14:22,720 --> 00:14:26,440
+طريقين مختلفين لنفس الدالة و التنين بيروح إلى 00
+
+207
+00:14:26,440 --> 00:14:30,860
+مختلفين الليمتين اذا since these two paths have
+
+208
+00:14:30,860 --> 00:14:34,860
+different limits then limit I على X as XY بتروح
+
+209
+00:14:34,860 --> 00:14:40,200
+إلى 00 does not exist طيب اجمال هذه اللي هو امثلة
+
+210
+00:14:40,200 --> 00:14:43,740
+على limit does not exist وعلى limit exist وكيف
+
+211
+00:14:43,740 --> 00:14:47,660
+نوجد ال limits نيجي الآن اللي هو نيجي إلى الجزء
+
+212
+00:14:47,660 --> 00:14:51,770
+الثاني اللي هو الاتصال او ال continuityالـ
+
+213
+00:14:51,770 --> 00:14:55,770
+continuity مش هيخلف عن تبع زمان نفس المنطق a
+
+214
+00:14:55,770 --> 00:15:00,070
+function f of x و y is continuous at the point x0
+
+215
+00:15:00,070 --> 00:15:04,270
+و y0 if the following holds اللي هي التلت شروط
+
+216
+00:15:04,270 --> 00:15:08,510
+اللي انتوا عارفينها انها تكون الدالة f معرفة عندهم
+
+217
+00:15:08,510 --> 00:15:11,250
+مقفلة x0 و y0 اللي بتاني little bit انه عندها
+
+218
+00:15:11,250 --> 00:15:15,730
+اتصال الشغل الثاني انه limited دالة f of x و y لما
+
+219
+00:15:15,730 --> 00:15:20,130
+x و y تروح لل x0 و ال y0 اللي عندها متصلة تكون
+
+220
+00:15:20,130 --> 00:15:23,450
+existالشرط الثالث هو limit f of x,y عندما تذهب الـ
+
+221
+00:15:23,450 --> 00:15:26,950
+x,y إلى الـ x0 و الـ y0 هذه الـ limit تسموه قيمة
+
+222
+00:15:26,950 --> 00:15:30,970
+الدالة عند مين عند النقطة x0 و y0 و كما كنا نقول
+
+223
+00:15:30,970 --> 00:15:35,810
+زمان الشرط الثالث لحاله إذا تحقق ما هو معناته إن
+
+224
+00:15:35,810 --> 00:15:39,330
+ال limit هذه exist وهذه موجودة و التمتين متساويتين
+
+225
+00:15:39,330 --> 00:15:43,530
+يعني يحمل في طياته الشرط الثالث اللي هو الشرطين
+
+226
+00:15:43,530 --> 00:15:48,430
+الأولانيين وهذا معناته إنه إذا تحققإنه a function
+
+227
+00:15:48,430 --> 00:15:53,470
+is continuous at x0 و y0 الآن لما نقول a function
+
+228
+00:15:53,470 --> 00:15:58,110
+is continuous معناته بنقصد أنه continuous على كل
+
+229
+00:15:58,110 --> 00:16:02,570
+domainها لكن لما نقول f continuous عند ال a و ال
+
+230
+00:16:02,570 --> 00:16:07,610
+x0 و ال y0 نعني أنه متصل عند النقطة x0 و y0 و هذا
+
+231
+00:16:07,610 --> 00:16:13,350
+معناها فناخد مثالبقول لي في عندنا function show
+
+232
+00:16:13,350 --> 00:16:17,570
+that f of x y بيساوي اتنين x y على x تربيع زاد y
+
+233
+00:16:17,570 --> 00:16:21,970
+تربيع في حال ال x و ال y ليسوا zero zero و سفر في
+
+234
+00:16:21,970 --> 00:16:26,290
+حال ال x و ال y بتساوي سفر أو سفر هذه الدالة بقول
+
+235
+00:16:26,290 --> 00:16:30,510
+لي اثبت انها is continuous at every point ما حدا
+
+236
+00:16:30,510 --> 00:16:34,740
+عند من عند ال z اللي هي ال originالدالة دي دايما
+
+237
+00:16:34,740 --> 00:16:39,700
+متصلة ما عدا عند وين عند الـ x و y بساوة 00 نشوف
+
+238
+00:16:39,700 --> 00:16:44,240
+كيف متصلة عند النقاط اللي مش 0 و 0 يعني الآن بتاخد
+
+239
+00:16:44,240 --> 00:16:49,300
+a و b أي نقطة لاتساوة 0 و 0 احسب ل limit f of x و
+
+240
+00:16:49,300 --> 00:16:53,140
+y لما x و y تروح لمين ل ال a و ال b لما x و y تروح
+
+241
+00:16:53,140 --> 00:16:57,260
+ل a و ال b ال a و ال b مش مين مش ال 00 اذا وين
+
+242
+00:16:57,260 --> 00:17:01,280
+الدالة اللي بتستخدم هاي الدالة بصير limit f of x و
+
+243
+00:17:01,280 --> 00:17:05,300
+y هو عبارة عن limit هذهLimit هذه لما X و Y تذهب
+
+244
+00:17:05,300 --> 00:17:12,100
+إلى A و B و A و B ليس 0 و 0 هي عبارة عن 2A بB على
+
+245
+00:17:12,100 --> 00:17:15,980
+A تربيع زائد B تربيع يعني أيش بتساوي يا جماعة يعني
+
+246
+00:17:15,980 --> 00:17:20,780
+بتساوي قيمة الـ function عند A و B وهذا يعني أن F
+
+247
+00:17:20,780 --> 00:17:25,400
+is continuous عند أي نقطة اللي هي لا تساوي 00 اللي
+
+248
+00:17:25,400 --> 00:17:31,060
+سميتها أنا A و B لا تساوي 00 إذن على طول اللي هي
+
+249
+00:17:31,060 --> 00:17:34,760
+limitللـ f of x y لما f of x y تذهب إلى a و b فهو
+
+250
+00:17:34,760 --> 00:17:38,620
+f of a و b عندما تكون ال a و ال b ليس السفر و
+
+251
+00:17:38,620 --> 00:17:41,480
+السفر لأنها الدالة بولونومية و بولونومية للي تاعة
+
+252
+00:17:41,480 --> 00:17:46,440
+مش سفرية محلها فبكون عند ال function is continuous
+
+253
+00:17:46,440 --> 00:17:51,220
+عند كل النقاط مع ده 00الان دل علينا اللي هي
+
+254
+00:17:51,220 --> 00:17:55,060
+المشكلة المشكلة انه عندهم مقطة x و y بيساوي 00
+
+255
+00:17:55,060 --> 00:17:59,020
+بدنا نشوف ال function continuous ولا لأ اول حاجة
+
+256
+00:17:59,020 --> 00:18:05,120
+معرفة عند 00 ليش؟ لأن f of 00 ايش هتساوي؟ Zero اذا
+
+257
+00:18:05,120 --> 00:18:11,350
+احنا بنتوقع ان المشكلة في ال limitLimit f of x,y
+
+258
+00:18:11,350 --> 00:18:17,230
+لما x,y تروح للـ 0 والـ 0 لنشوف عشان نثبت أن ال
+
+259
+00:18:17,230 --> 00:18:21,170
+limit لل f of x,y لما ال x,y يروح لل 0 وال0 does
+
+260
+00:18:21,170 --> 00:18:25,270
+not exist بدنا ناخد طريقين زي ما أخدنا قبل شوية و
+
+261
+00:18:25,270 --> 00:18:29,470
+يعطينا قيمتين مختلفةالان اللى بدأ افحصها limit f
+
+262
+00:18:29,470 --> 00:18:33,490
+of x y لما x y تروح لمن يا جماعة للصفر والصفر x y
+
+263
+00:18:33,490 --> 00:18:37,110
+تروح للصفر والصفر يعني مش صفر وصفر لسه يعني اذا
+
+264
+00:18:37,110 --> 00:18:40,150
+انا مين اللى بدأ اخدها الدالة هي الدالة اللى بدأ
+
+265
+00:18:40,150 --> 00:18:44,990
+اشتغل عليها يعني بدأ اخد limit 2xy على x تربيع
+
+266
+00:18:44,990 --> 00:18:49,490
+زائد y تربيع واثبت لكم ان limit لهذه لما x y تروح
+
+267
+00:18:49,490 --> 00:18:57,100
+ل 0 و0 ما تكونش اللي هو exist كيف؟ عبر طريقينشوف
+
+268
+00:18:57,100 --> 00:19:01,620
+الطريق اللي أخدته اخد Y بساوي MX طبعا MX ده اللي
+
+269
+00:19:01,620 --> 00:19:04,960
+هو M ثابت و X أشماله اللي احنا عارفين انه متغير
+
+270
+00:19:04,960 --> 00:19:07,920
+يعني عبر واحد من الطرق اللي زي هذوله شايفين هنا
+
+271
+00:19:07,920 --> 00:19:12,560
+هذوله بدي اثبتلك ان عبر هذوله الطرق هتلاقي انه
+
+272
+00:19:12,560 --> 00:19:16,930
+قيمة ال limitاللي هي مختلفة من الطريق هذه للطريق
+
+273
+00:19:16,930 --> 00:19:19,670
+هذه للطريق هذه وبناء عليه بتكون ال limit does not
+
+274
+00:19:19,670 --> 00:19:24,950
+exist خلّيني أخد M X M in general بساوي limit اللي
+
+275
+00:19:24,950 --> 00:19:28,630
+هو بشيل اللي هي ال Y بحطها لتنين M مكانها بيصير
+
+276
+00:19:28,630 --> 00:19:35,030
+اتنين M X في X بيصير X تربيع على X تربيع زائد Y
+
+277
+00:19:35,030 --> 00:19:40,140
+تربيع اللي هي M X لكل تربيع Y ساوي limitاللي هو
+
+278
+00:19:40,140 --> 00:19:45,780
+الآن تاخد ال .. ال .. ال .. ال X تربيع هنا عامل
+
+279
+00:19:45,780 --> 00:19:50,980
+مشترك بين الجهتين هنا لما اقتصر خطوبهم ركزوا معايا
+
+280
+00:19:50,980 --> 00:19:55,440
+الان X تربيع بيظل ايش واحد زائد M تربيع وال X
+
+281
+00:19:55,440 --> 00:19:58,260
+تربيع طلعت برا وفي X تربيع بتروح ال X تربيع مع ال
+
+282
+00:19:58,260 --> 00:20:02,060
+X تربيع اللي اتحاد بيظل عنده 2 M على واحد زائد M
+
+283
+00:20:02,060 --> 00:20:06,850
+تربيع اللي هي قيمة ال limitلاحظ الآن هذه الـ limit
+
+284
+00:20:06,850 --> 00:20:11,430
+للـ function عبر الطريق y بساوة mx بتعتمد على الـ
+
+285
+00:20:11,430 --> 00:20:15,750
+m يعني لو جينا أخدنا y بتساوي x هذا طريق y بتساوي
+
+286
+00:20:15,750 --> 00:20:24,730
+x سيءY بيساوي مثلا اللي هو اتنين X Y بيساوي اتنين
+
+287
+00:20:24,730 --> 00:20:29,270
+X هي مثلا Y بيساوي اتنين X هنا مثلا Y بيساوي ناقص
+
+288
+00:20:29,270 --> 00:20:34,670
+X هي مثلا هنلاقي قيمتين عبر طريقين مختلفين مختلفة
+
+289
+00:20:34,670 --> 00:20:39,230
+خد مثلا مثال Hence along Y بتساوي X ال limit ايش
+
+290
+00:20:39,230 --> 00:20:43,930
+هيكون بيساوي اتنين على واحد زائد واحد اتنين يعني
+
+291
+00:20:43,930 --> 00:20:48,970
+ايش بيساوي واحدالـ bot along y مثلا بيساو 2x طبعا
+
+292
+00:20:48,970 --> 00:20:53,550
+انت ممكن تاخد اي قيمة للام y بيساو 2x بيصير اللي
+
+293
+00:20:53,550 --> 00:20:57,530
+هو عبارة عن 2 في 2 اربعة على واحد زائد اربعة خمسة
+
+294
+00:20:57,530 --> 00:21:01,620
+يعني اربعة على خمسة واضح ان عبر الطريق هذاوعبر
+
+295
+00:21:01,620 --> 00:21:06,780
+الطريق هذا الليمتين مش مساويين إذا اللي هو since
+
+296
+00:21:06,780 --> 00:21:09,140
+these two paths have different limits زي ما قلنا
+
+297
+00:21:09,140 --> 00:21:12,080
+then limit does not exist ومدام limit does not
+
+298
+00:21:12,080 --> 00:21:15,960
+exist لما x و y بتروح ل 0 و 0 إذا أكيد ال function
+
+299
+00:21:15,960 --> 00:21:19,620
+is not continuous عند 0 و 0 لأن عشان تكون
+
+300
+00:21:19,620 --> 00:21:23,420
+continuous عند 0 و 0 لازم ال limit تساوي قيمة ال
+
+301
+00:21:23,420 --> 00:21:27,340
+function اللي هي في هذا الحالة 0 وهذا غير حدث طبعا
+
+302
+00:21:27,340 --> 00:21:30,650
+هذه الرسمة بتوضح لك أنهيعني عبر الطريق أو عبر
+
+303
+00:21:30,650 --> 00:21:34,830
+الطريق هذا كانت اللي هي اللي هي عندنا واحد هنا لما
+
+304
+00:21:34,830 --> 00:21:38,510
+كان point eight الميل وهنا نقص point eight يعني y
+
+305
+00:21:38,510 --> 00:21:41,950
+بيساوي نقص point يعني عبر الطريق اللي هي ام هذه ام
+
+306
+00:21:41,950 --> 00:21:45,610
+ام يا جماعة هذه ميل اللي هو اللي اللي اللي بناخده
+
+307
+00:21:45,770 --> 00:21:56,690
+هنا واحد وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+308
+00:21:56,690 --> 00:21:59,010
+وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا
+
+309
+00:21:59,010 --> 00:21:59,250
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+310
+00:21:59,250 --> 00:21:59,590
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+311
+00:21:59,590 --> 00:21:59,850
+وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا
+
+312
+00:21:59,850 --> 00:21:59,870
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+313
+00:21:59,870 --> 00:22:00,630
+وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا
+
+314
+00:22:00,630 --> 00:22:00,690
+كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا
+
+315
+00:22:00,690 --> 00:22:00,910
+وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا كذا وهنا
+
+316
+00:22:00,910 --> 00:22:05,150
+كذا وهنا كذا
+
+317
+00:22:06,840 --> 00:22:09,760
+المثال السادس الان and they show that the function
+
+318
+00:22:09,760 --> 00:22:14,760
+f of x و y بيسوا 2x تربيع y على x و 4 زاد y تربيع
+
+319
+00:22:14,760 --> 00:22:20,900
+has no limit as x و y approaches 0 و 0 بدنا نثبت
+
+320
+00:22:20,900 --> 00:22:25,280
+ان هذه ال function مالهاش limit لما x و y تروح ل 0
+
+321
+00:22:25,280 --> 00:22:31,100
+و 0 وهذا برضه مثال مشابه تعزيز للفكرة الان بتاخد
+
+322
+00:22:31,100 --> 00:22:36,250
+طريقين اللي هو اروح ل مين الان ل 0 و 0اللي بطلع
+
+323
+00:22:36,250 --> 00:22:40,610
+اللي فوق هنا واحد يقول لي خد Y بتساوي سفر Y بتساوي
+
+324
+00:22:40,610 --> 00:22:49,460
+سفر بيصير سفر على X أس 4 I لحالة بيصير سفرهي عبر
+
+325
+00:22:49,460 --> 00:22:54,640
+الـ Y بيساوي 0 0 عبر X بيساوي 0 برضه 0 إذن عبر Y
+
+326
+00:22:54,640 --> 00:22:59,000
+بيساوي 0 و X بيساوي 0 بيعطينش قيمتين مختلفات لو
+
+327
+00:22:59,000 --> 00:23:03,720
+أخدت Y بتساوي اللي هو X بيصير 2X تكعيب على X 4
+
+328
+00:23:03,720 --> 00:23:09,560
+زائد X تربيع يعني بتغلب برضه إذن الآن هو ممكن تطلع
+
+329
+00:23:09,560 --> 00:23:13,770
+لنا 0 كمانالان فإيجي عباله على طول يقولك خد y
+
+330
+00:23:13,770 --> 00:23:16,490
+بيساوي x تربيع ليش y بيساوي x تربيع عشان هد بيصير
+
+331
+00:23:16,490 --> 00:23:20,830
+x أربعة او هد بيصير x أربعة او هد x أربعة ايضا
+
+332
+00:23:20,830 --> 00:23:24,970
+بيصير اقتصار ايضا بيصير قيمة غير السفر هذا تفكير
+
+333
+00:23:24,970 --> 00:23:29,710
+كله تانشوف along y بيساوي x تربيع بيصير limit of x
+
+334
+00:23:29,710 --> 00:23:33,960
+y اللي فوق لما x y تروح ل 0 0 بيساوي ال limitعوض
+
+335
+00:23:33,960 --> 00:23:39,320
+انت خدت اش مالها y بيساوي x تربيع بيعوض عن y بx
+
+336
+00:23:39,320 --> 00:23:45,760
+تربيع بيصير 2x أُس 4 وهنا x أُس 4 وهنا x أُس 4
+
+337
+00:23:45,760 --> 00:23:50,460
+ماشي عارفين كيف عوضنا عن y بيساوي x تربيع لأن عبر
+
+338
+00:23:50,460 --> 00:23:55,600
+هذا الطريق اللي بناخد الآن بنروح ل 0 0الان يعني
+
+339
+00:23:55,600 --> 00:23:59,160
+بيصير X و X تربيع تروح للصفر يعني X و X بتروح
+
+340
+00:23:59,160 --> 00:24:04,940
+للصفر استبدلت هذه عوضت عن مكان Y بيساوي X تربيع
+
+341
+00:24:04,940 --> 00:24:08,900
+لكن X و X تربيع بيروح للصفر إذا و فقط إذا X و X
+
+342
+00:24:08,900 --> 00:24:12,560
+بتروح للصفر فهذه اللي هي بتيجي مكان X و X تربيع
+
+343
+00:24:13,220 --> 00:24:17,980
+الان هذه بتروح نختصر بطلع ان اتنين على اتنين
+
+344
+00:24:17,980 --> 00:24:22,400
+بيساوي ايش واحد اذا ال limit هنا عبر الطريق هذا
+
+345
+00:24:22,400 --> 00:24:27,420
+واحد along y بتساوي سفر كلنا limit f of x y لما x
+
+346
+00:24:27,420 --> 00:24:32,420
+y تروح ل 0 0 بيساوي limit بعوض عن كل y بياشي بسفر
+
+347
+00:24:32,730 --> 00:24:37,310
+هذا الوضع عند Y بصفر و X و Y بصفر و X و Y بصفر و X
+
+348
+00:24:37,310 --> 00:24:46,990
+و Y بصفر و X و Y بصفر و X و Y بصفر و X و Y بصفر و
+
+349
+00:24:46,990 --> 00:24:48,970
+X و Y بصفر و X و Y بصفر و X و Y بصفر و X و Y بصفر
+
+350
+00:24:48,970 --> 00:24:49,430
+Y بصفر و X و Y بصفر و X و Y بصفر و X و Y بصفر و X
+
+351
+00:24:49,430 --> 00:24:58,270
+و Y بصفر و X و Y بصفر و X و Y بصفر و X و Y بصفر و
+
+352
+00:24:58,270 --> 00:25:02,180
+X و Y بصفر و X و Yالان نجي لـ continuity of
+
+353
+00:25:02,180 --> 00:25:05,560
+composite functions هذه برضه حاجة سهلة واخدناها
+
+354
+00:25:05,560 --> 00:25:09,900
+زمان في calculus RF if f is continuous عند x0 و y0
+
+355
+00:25:09,900 --> 00:25:14,260
+و g is continuous اللي هو single variable هذه
+
+356
+00:25:14,260 --> 00:25:18,200
+function of one variable زي تبعات زمان يعني is
+
+357
+00:25:18,200 --> 00:25:22,640
+continuous at f of x0 و y0 then the composition h
+
+358
+00:25:22,640 --> 00:25:27,220
+بساوي g composite f اللي هي بيصير h of x و y بساوي
+
+359
+00:25:27,220 --> 00:25:31,090
+g of f of x yيعني هذه اللي جوا function of two
+
+360
+00:25:31,090 --> 00:25:33,850
+variables بينما الـ G نفسها function of one
+
+361
+00:25:33,850 --> 00:25:37,670
+variable إذا كانت اللي جوا continuous بقول و اللي
+
+362
+00:25:37,670 --> 00:25:41,030
+برا continuous بيكون الدالة كلها على بعض إيه
+
+363
+00:25:41,030 --> 00:25:44,090
+اشمالها continuous عند X note و Y note وهذا اللي
+
+364
+00:25:44,090 --> 00:25:48,710
+كنا نقوله زمان for example الان E to the X minus Y
+
+365
+00:25:48,710 --> 00:25:51,970
+هذه بقدر أعملها function اللي هو composite of two
+
+366
+00:25:51,970 --> 00:25:57,200
+functions يعني خدوا ال F of X Y هي X ناقص Yf of x
+
+367
+00:25:57,200 --> 00:26:02,820
+y بيساوي x ناقص y هي continuous function وخد جي of
+
+368
+00:26:02,820 --> 00:26:08,720
+z مثلا بيساوي e to the z ماشي فبصير عندي الان g of
+
+369
+00:26:08,720 --> 00:26:15,600
+f of x y هي عبارة عنالـ f of x y اليمين x minus y
+
+370
+00:26:15,600 --> 00:26:20,900
+هيقولنا g of x minus y بصير e to the x minus y
+
+371
+00:26:20,900 --> 00:26:24,240
+يعني هذه الان كدرت اكتبها على صورة composition of
+
+372
+00:26:24,240 --> 00:26:27,900
+two functions واحدة ال x minus y والتانية ال
+
+373
+00:26:27,900 --> 00:26:31,020
+exponential ال exponential continuous و ال x minus
+
+374
+00:26:31,020 --> 00:26:33,520
+y continuous لأن واحدة بولنومي و واحدة exponential
+
+375
+00:26:33,520 --> 00:26:37,260
+إذا التانيتين continuous إذا ال composition تبحن
+
+376
+00:26:37,260 --> 00:26:40,720
+إشمالهcontinuous أو زي ما كنا نقول زمان اللي جوا
+
+377
+00:26:40,720 --> 00:26:44,160
+continuous و اللي برا continuous إذا كلها ع بعض
+
+378
+00:26:44,160 --> 00:26:47,360
+continuous نفس الأشياء هذه هي اللي جوا continuous
+
+379
+00:26:47,360 --> 00:26:51,520
+لإنها polynomial على polynomial ليش one function و
+
+380
+00:26:51,520 --> 00:26:54,200
+اللي تحت مش استحيل يساوي سفر إذا هذه continuous
+
+381
+00:26:54,200 --> 00:26:58,080
+وينما كان و ال cosine نفسها continuous إذا اللي هو
+
+382
+00:26:58,080 --> 00:27:00,880
+composition of two functions ال function الأولى
+
+383
+00:27:00,880 --> 00:27:03,940
+اللي هي f of x y هيها و ال function التانية اللي
+
+384
+00:27:03,940 --> 00:27:07,570
+هي cosine ال xإذاً صارت هذه كلها على بعضها جي
+
+385
+00:27:07,570 --> 00:27:10,650
+كومبوزيت F is continuous لأن هذا continuous و هذا
+
+386
+00:27:10,650 --> 00:27:14,950
+continuous Similarly الواحد زي X تربيع زي Y تربيع
+
+387
+00:27:14,950 --> 00:27:18,390
+هي ال F of XY وهذا polynomial continuous و ال lin
+
+388
+00:27:18,390 --> 00:27:21,770
+continuous و N معرفة لأنها دايما موجة بأكبر من
+
+389
+00:27:21,770 --> 00:27:25,470
+واحد فمعرف عليها اللي هي إذاً lin continuous و
+
+390
+00:27:25,470 --> 00:27:27,710
+اللي جوا continuous إذا كل هذا صارت على بعض
+
+391
+00:27:27,710 --> 00:27:31,790
+continuous لأن ال F of XY زي ما قلنا هذه و ال G هي
+
+392
+00:27:31,790 --> 00:27:34,800
+عبارة عن ال linأيضًا هذا الـ composition of two
+
+393
+00:27:34,800 --> 00:27:41,640
+continuous functions is continuous طيب نصلح على
+
+394
+00:27:41,640 --> 00:27:44,460
+نفسي يا جماعة لسه لسه لسه لأن more function
+
+395
+00:27:44,460 --> 00:27:47,160
+functions of more than two variables يعني أكتر من
+
+396
+00:27:47,160 --> 00:27:50,780
+variable يعني الأن نفس الأشياء len of x زاد y زاد
+
+397
+00:27:50,780 --> 00:27:56,500
+z ماشي الحال هذي اللي جوااللي هو f of x و y و z
+
+398
+00:27:56,500 --> 00:28:01,400
+continuous والـ lin continuous اللي هي الـ g صارت
+
+399
+00:28:01,400 --> 00:28:06,040
+كلها ع بعضها يشملها continuous نفس الأشي عندى هذه
+
+400
+00:28:06,040 --> 00:28:09,020
+اللي هي composition هي نفس الأشي الأن assign ال z
+
+401
+00:28:09,020 --> 00:28:12,580
+continuous و ال y continuous و ال x minus واحد
+
+402
+00:28:12,580 --> 00:28:15,440
+continuous إذا صارت كل الدالة ده حصل ضرب دلتين
+
+403
+00:28:15,440 --> 00:28:18,700
+continuous على continuous و بتطلع كلها continuous
+
+404
+00:28:18,700 --> 00:28:23,950
+ما عدا عند مين يا جماعة عند أسفار المقامإذا
+
+405
+00:28:23,950 --> 00:28:27,190
+القوانين اللي فيها ال functions of two variables
+
+406
+00:28:27,190 --> 00:28:30,570
+بتنفع لل three variables بتنفع لل one variable أو
+
+407
+00:28:30,570 --> 00:28:34,630
+more variables هاي و بقولك are continuous كل هنا
+
+408
+00:28:34,630 --> 00:28:40,210
+دولة through their domains اللي هي يعني في اللي هو
+
+409
+00:28:40,210 --> 00:28:43,610
+مجالها و بنعرف مجال اللي من إيش أكبر من سفر و مجال
+
+410
+00:28:43,610 --> 00:28:46,290
+هذه بنعرف أنه لما .. هذه الاتحاد مايساويش سفر و
+
+411
+00:28:46,290 --> 00:28:50,180
+هكذاالان لو بدأنا نجي نوجد ال limit ل function of
+
+412
+00:28:50,180 --> 00:28:52,920
+three variables قاعد ببعد الأفكار اللي قبل شوية
+
+413
+00:28:54,030 --> 00:28:57,110
+بالاضافة في الـ L بادل ما هو function of two
+
+414
+00:28:57,110 --> 00:29:00,470
+variables خلّاها three variables بدنا ال limit لما
+
+415
+00:29:00,470 --> 00:29:04,170
+ال point B اللي هي ال X والY والزي تروح للواحد
+
+416
+00:29:04,170 --> 00:29:07,190
+والصفر والنقص واحد لميعنى لل function E to the X
+
+417
+00:29:07,190 --> 00:29:11,330
+زائد Z على Z تربيع Z قصارى للجذر ال X في Y وهذه
+
+418
+00:29:11,330 --> 00:29:13,930
+طبعا دي بها مثال سهل على أساس أنه على طول هو تعويض
+
+419
+00:29:13,930 --> 00:29:17,590
+مباشر لأن الفجر يمح لها والتحت يمح لها وليس الصفر
+
+420
+00:29:17,590 --> 00:29:22,690
+اللي تحت واللي فوقعبارة عن ده اللي على طول بتتعود
+
+421
+00:29:22,690 --> 00:29:26,770
+فيها بيصير e to the واحد ناقص واحد اللي هي e to
+
+422
+00:29:26,770 --> 00:29:30,290
+the zero يعني واحد و اللي تحت هذه عبارة عن ناقص
+
+423
+00:29:30,290 --> 00:29:34,810
+واحد تربيع هيها ال z ناقص تربيع زائد cosine اللي
+
+424
+00:29:34,810 --> 00:29:38,130
+هو ال x في ال y اللي هي zero الجذر اللي هي zero
+
+425
+00:29:38,130 --> 00:29:41,790
+بيصير cosine ال zero و هذه ناقص واحد تربيع هذه
+
+426
+00:29:41,790 --> 00:29:44,550
+واحد و هذه واحد اتنين و اللي فوق واحد بيصيح نص
+
+427
+00:29:44,550 --> 00:29:48,340
+where b denotes the point x y و zmay be found by
+
+428
+00:29:48,340 --> 00:29:51,860
+direct substitution زي ما قلنا إذا القوانين ال
+
+429
+00:29:51,860 --> 00:29:54,480
+limits ال continuity ال sum ال product ال quotient
+
+430
+00:29:54,480 --> 00:29:58,000
+ال bars ال composite all extended into functions
+
+431
+00:29:58,000 --> 00:30:02,000
+of three variables يعني كل اللي حكيناها قبل بنتد و
+
+432
+00:30:02,000 --> 00:30:06,220
+بيجي اللي هو بنفع برضه على function أو functions
+
+433
+00:30:06,220 --> 00:30:10,980
+of three variables و هيك بتكون خلاصة المحاضرة 14 و
+
+434
+00:30:10,980 --> 00:30:13,440
+إلى لقاء آخر السلام عليكم و رحمه الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/RtOltUWLcYs_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/RtOltUWLcYs_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..1e1cf99438a209e9c0cb20cafce13718da894c23
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@@ -0,0 +1 @@
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"probability": 0.9755859375}, {"start": 382.76, "end": 382.94, "word": " x", "probability": 0.98876953125}, {"start": 382.94, "end": 383.22, "word": " y", "probability": 0.912109375}, {"start": 383.22, "end": 383.36, "word": " اللي", "probability": 0.899658203125}, {"start": 383.36, "end": 383.52, "word": " هو", "probability": 0.9814453125}, {"start": 383.52, "end": 383.72, "word": " K", "probability": 0.61962890625}, {"start": 383.72, "end": 383.9, "word": " في", "probability": 0.97119140625}, {"start": 383.9, "end": 384.28, "word": " مين", "probability": 0.66259765625}, {"start": 384.28, "end": 384.8, "word": " في", "probability": 0.78173828125}, {"start": 384.8, "end": 385.18, "word": " ال", "probability": 0.376953125}, {"start": 385.18, "end": 385.4, "word": " L", "probability": 0.90869140625}], "temperature": 1.0}, {"id": 18, "seek": 40443, "start": 385.57, "end": 404.43, "text": "الان هذا اللي هو نفس خواص اللي هي تبع الـ function العادية برضه نفس الأشياء حاصل ضرب دلتين ال limit الهن بتوزع لما يكون ال limit موجود بيكون ال limit الأول ضرب ال limit الثاني و نفس الأشياء حاصل القسمة لكن provided that اللي ال MN اللي هي limit اللي ال G of XY ماتساويش 0", "tokens": [6027, 7649, 23758, 13672, 1829, 31439, 8717, 36178, 16490, 14407, 9381, 13672, 1829, 39896, 6055, 3555, 3615, 2423, 39184, 2445, 18863, 18513, 10632, 4724, 43042, 3224, 8717, 36178, 16247, 8592, 1829, 16606, 11331, 33546, 1211, 48812, 25513, 11778, 1211, 2655, 9957, 2423, 4948, 2423, 3224, 1863, 39894, 2407, 11622, 3615, 5296, 15042, 7251, 30544, 2423, 4948, 3714, 29245, 23328, 4724, 1829, 30544, 2423, 4948, 16247, 12610, 48812, 25513, 2423, 4948, 6024, 104, 7649, 1829, 4032, 8717, 36178, 16247, 8592, 1829, 16606, 11331, 33546, 1211, 25062, 38251, 3660, 44381, 5649, 300, 13672, 1829, 2423, 376, 45, 13672, 1829, 39896, 4948, 13672, 1829, 2423, 460, 295, 48826, 19446, 2655, 3794, 995, 45865, 8592, 1958], "avg_logprob": -0.25511616310187146, "compression_ratio": 1.8468085106382979, "no_speech_prob": 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"probability": 0.72607421875}, {"start": 402.17, "end": 402.35, "word": " اللي", "probability": 0.701171875}, {"start": 402.35, "end": 402.47, "word": " ال", "probability": 0.578125}, {"start": 402.47, "end": 402.63, "word": " G", "probability": 0.673828125}, {"start": 402.63, "end": 402.79, "word": " of", "probability": 0.495849609375}, {"start": 402.79, "end": 403.17, "word": " XY", "probability": 0.433349609375}, {"start": 403.17, "end": 404.11, "word": " ماتساويش", "probability": 0.7543538411458334}, {"start": 404.11, "end": 404.43, "word": " 0", "probability": 0.468505859375}], "temperature": 1.0}, {"id": 19, "seek": 41590, "start": 404.88, "end": 415.9, "text": "الان نفس الشيء limit f of x,y أسن as x,y بتروح ل x0 و y0 بساوي اللي هي limit هذه لكل أسن يعني L أسن for any positive integer M", "tokens": [6027, 7649, 8717, 36178, 25124, 1829, 38207, 4948, 283, 295, 2031, 11, 88, 5551, 3794, 1863, 382, 2031, 11, 88, 39894, 32887, 5016, 5296, 2031, 15, 4032, 288, 15, 4724, 3794, 995, 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"probability": 0.79296875}, {"start": 411.9, "end": 412.28, "word": " لكل", "probability": 0.796142578125}, {"start": 412.28, "end": 412.7, "word": " أسن", "probability": 0.9405924479166666}, {"start": 412.7, "end": 413.0, "word": " يعني", "probability": 0.887939453125}, {"start": 413.0, "end": 413.38, "word": " L", "probability": 0.440185546875}, {"start": 413.38, "end": 414.22, "word": " أسن", "probability": 0.9832356770833334}, {"start": 414.22, "end": 414.46, "word": " for", "probability": 0.83203125}, {"start": 414.46, "end": 414.66, "word": " any", "probability": 0.52587890625}, {"start": 414.66, "end": 415.0, "word": " positive", "probability": 0.93115234375}, {"start": 415.0, "end": 415.46, "word": " integer", "probability": 0.9404296875}, {"start": 415.46, "end": 415.9, "word": " M", "probability": 0.341552734375}], "temperature": 1.0}, {"id": 20, "seek": 44573, "start": 417.15, "end": 445.73, "text": "نفس الشيء الـ limit الجدر النوني لأف of xy لما xy تروح ل x0 و y0 بساوي 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{"start": 808.27, "end": 808.37, "word": " و", "probability": 0.94580078125}, {"start": 808.37, "end": 808.47, "word": " x", "probability": 0.7880859375}, {"start": 808.47, "end": 808.75, "word": " تروح", "probability": 0.9091796875}, {"start": 808.75, "end": 808.87, "word": " لل", "probability": 0.411376953125}, {"start": 808.87, "end": 809.21, "word": " x", "probability": 0.6044921875}, {"start": 809.21, "end": 809.35, "word": " و", "probability": 0.83642578125}, {"start": 809.35, "end": 809.43, "word": " y", "probability": 0.8212890625}, {"start": 809.43, "end": 809.65, "word": " تروح", "probability": 0.9864908854166666}, {"start": 809.65, "end": 809.79, "word": " لل", "probability": 0.94873046875}, {"start": 809.79, "end": 810.09, "word": " 0", "probability": 0.51220703125}], "temperature": 1.0}, {"id": 38, "seek": 83319, "start": 810.67, "end": 833.19, "text": "و الـ zero هي ساوية واحدة لأن x على x بيساوي واحد ثابت اذا ال limit عبر الطريق y بيساوي x بيساوي واحدة لكن path along y بتساوي سفر لإن بدأ اروح على الخط مين على عبر الخط y بتساوي سفر بدأ اروح عوين على x و y بيساوي 00 اللي هو هذا ال x axis بيروح ال 00 طبعا هنا", "tokens": [2407, 2423, 39184, 4018, 39896, 8608, 995, 2407, 10632, 36764, 24401, 3660, 5296, 33456, 2031, 15844, 2031, 4724, 1829, 3794, 995, 45865, 36764, 24401, 38637, 16758, 2655, 1975, 15730, 2423, 4948, 6225, 26890, 41950, 16572, 4587, 288, 4724, 1829, 3794, 995, 45865, 2031, 4724, 1829, 3794, 995, 45865, 36764, 24401, 3660, 44381, 3100, 2051, 288, 39894, 3794, 995, 45865, 8608, 5172, 2288, 5296, 28814, 1863, 47525, 10721, 1975, 32887, 5016, 15844, 33962, 9566, 3714, 9957, 15844, 6225, 26890, 33962, 9566, 288, 39894, 3794, 995, 45865, 8608, 5172, 2288, 47525, 10721, 1975, 32887, 5016, 6225, 2407, 9957, 15844, 2031, 4032, 288, 4724, 1829, 3794, 995, 45865, 7143, 13672, 1829, 31439, 23758, 2423, 2031, 10298, 4724, 13546, 2407, 5016, 2423, 7143, 23032, 3555, 3615, 995, 34105], "avg_logprob": -0.3050000047683716, "compression_ratio": 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0.83544921875}, {"start": 818.85, "end": 819.25, "word": " path", "probability": 0.2232666015625}, {"start": 819.25, "end": 819.87, "word": " along", "probability": 0.958984375}, {"start": 819.87, "end": 820.47, "word": " y", "probability": 0.95703125}, {"start": 820.47, "end": 821.05, "word": " بتساوي", "probability": 0.878662109375}, {"start": 821.05, "end": 821.43, "word": " سفر", "probability": 0.8623046875}, {"start": 821.43, "end": 822.09, "word": " لإن", "probability": 0.4923095703125}, {"start": 822.09, "end": 822.39, "word": " بدأ", "probability": 0.3878173828125}, {"start": 822.39, "end": 822.95, "word": " اروح", "probability": 0.6402994791666666}, {"start": 822.95, "end": 823.05, "word": " على", "probability": 0.50537109375}, {"start": 823.05, "end": 823.39, "word": " الخط", "probability": 0.810791015625}, {"start": 823.39, "end": 823.87, "word": " مين", "probability": 0.7744140625}, {"start": 823.87, "end": 824.93, "word": " على", "probability": 0.666015625}, {"start": 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بيساوي", "probability": 0.96337890625}, {"start": 829.47, "end": 829.87, "word": " 00", "probability": 0.34326171875}, {"start": 829.87, "end": 830.71, "word": " اللي", "probability": 0.8388671875}, {"start": 830.71, "end": 830.83, "word": " هو", "probability": 0.90625}, {"start": 830.83, "end": 831.01, "word": " هذا", "probability": 0.83154296875}, {"start": 831.01, "end": 831.11, "word": " ال", "probability": 0.97705078125}, {"start": 831.11, "end": 831.27, "word": " x", "probability": 0.53076171875}, {"start": 831.27, "end": 831.63, "word": " axis", "probability": 0.80029296875}, {"start": 831.63, "end": 831.89, "word": " بيروح", "probability": 0.8262939453125}, {"start": 831.89, "end": 831.99, "word": " ال", "probability": 0.333740234375}, {"start": 831.99, "end": 832.37, "word": " 00", "probability": 0.642578125}, {"start": 832.37, "end": 832.97, "word": " طبعا", "probability": 0.7830810546875}, {"start": 832.97, "end": 833.19, "word": " هنا", "probability": 0.57568359375}], "temperature": 1.0}, {"id": 39, "seek": 86084, "start": 834.52, "end": 860.84, "text": "احسب ال limit عبر الخط هذا limit y على x بساوة limit على طول ايش اخدت y انا بساوة سفر لان انا قاعد على الخط اللي هو y بساوة سفر بيصير ده اللي هذي اللي بدي احسبلها ال limit zero على x و y ساوة zero لاحظ ال limit في هذه الحالة لا ساوة limit في هذه الحالة يعني زي ما كنا نقول زمان limit لمين و limit ولا سامش متساويتين اذا ال two limit ال limit does not exist او عندي في هذه الحالة", "tokens": [39319, 35457, 2423, 4948, 6225, 26890, 33962, 9566, 23758, 4948, 288, 15844, 2031, 4724, 3794, 995, 2407, 3660, 4948, 15844, 23032, 12610, 1975, 1829, 8592, 1975, 9778, 3215, 2655, 288, 1975, 8315, 4724, 3794, 995, 2407, 3660, 8608, 5172, 2288, 5296, 7649, 1975, 8315, 12174, 995, 22488, 15844, 33962, 9566, 13672, 1829, 31439, 288, 4724, 3794, 995, 2407, 3660, 8608, 5172, 2288, 4724, 1829, 9381, 13546, 11778, 3224, 13672, 1829, 8032, 8848, 1829, 13672, 1829, 4724, 16254, 1975, 5016, 35457, 1211, 11296, 2423, 4948, 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limit", "probability": 0.79541015625}, {"start": 851.8, "end": 851.94, "word": " في", "probability": 0.7451171875}, {"start": 851.94, "end": 852.1, "word": " هذه", "probability": 0.9482421875}, {"start": 852.1, "end": 852.64, "word": " الحالة", "probability": 0.9943033854166666}, {"start": 852.64, "end": 853.22, "word": " يعني", "probability": 0.93798828125}, {"start": 853.22, "end": 853.54, "word": " زي", "probability": 0.9638671875}, {"start": 853.54, "end": 853.62, "word": " ما", "probability": 0.98779296875}, {"start": 853.62, "end": 853.82, "word": " كنا", "probability": 0.983154296875}, {"start": 853.82, "end": 854.06, "word": " نقول", "probability": 0.888916015625}, {"start": 854.06, "end": 854.46, "word": " زمان", "probability": 0.9451497395833334}, {"start": 854.46, "end": 854.72, "word": " limit", "probability": 0.94140625}, {"start": 854.72, "end": 855.1, "word": " لمين", "probability": 0.5545654296875}, {"start": 855.1, "end": 855.18, "word": " و", "probability": 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1703.34, "text": "اللي هو f of x و y و z continuous والـ lin continuous اللي هي الـ g صارت كلها ع بعضها يشملها continuous نفس الأشي عندى هذه اللي هي composition هي نفس الأشي الأن assign ال z continuous و ال y continuous و ال x minus واحد continuous إذا صارت كل الدالة ده حصل ضرب دلتين continuous على continuous و بتطلع كلها continuous ما عدا عند مين يا جماعة عند أسفار المقام", "tokens": [6027, 20292, 31439, 283, 295, 2031, 4032, 288, 4032, 710, 10957, 16070, 39184, 22896, 10957, 13672, 1829, 39896, 2423, 39184, 290, 20328, 9640, 2655, 28242, 11296, 6225, 45030, 11242, 11296, 7251, 8592, 2304, 1211, 11296, 10957, 8717, 36178, 16247, 8592, 1829, 43242, 7578, 29538, 13672, 1829, 39896, 12686, 39896, 8717, 36178, 16247, 8592, 1829, 16247, 1863, 6269, 2423, 710, 10957, 4032, 2423, 288, 10957, 4032, 2423, 2031, 3175, 36764, 24401, 10957, 11933, 15730, 20328, 9640, 2655, 28242, 32748, 6027, 3660, 11778, 3224, 11331, 36520, 48812, 25513, 11778, 1211, 2655, 9957, 10957, 15844, 10957, 4032, 39894, 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9102, 4032, 39896, 4117, 39894, 30544, 16490, 1211, 33546, 3660, 9673, 5016, 46958, 25720, 3499, 4032, 30731, 5296, 4587, 16606, 19753, 34740, 21136, 37440, 25894, 24793, 4032, 12602, 35571, 3224, 21984, 4032, 26890, 4117, 9307, 3224], "avg_logprob": -0.2193080305254885, "compression_ratio": 1.639344262295082, "no_speech_prob": 4.112720489501953e-06, "words": [{"start": 1787.3, "end": 1787.56, "word": "may", "probability": 0.1807861328125}, {"start": 1787.56, "end": 1787.78, "word": " be", "probability": 0.88916015625}, {"start": 1787.78, "end": 1788.1, "word": " found", "probability": 0.61962890625}, {"start": 1788.1, "end": 1788.34, "word": " by", "probability": 0.9345703125}, {"start": 1788.34, "end": 1788.82, "word": " direct", "probability": 0.74755859375}, {"start": 1788.82, "end": 1789.68, "word": " substitution", "probability": 0.8271484375}, {"start": 1789.68, "end": 1790.1, "word": " زي", "probability": 0.7108154296875}, {"start": 1790.1, "end": 1790.28, "word": " ما", "probability": 0.97705078125}, {"start": 1790.28, "end": 1790.62, "word": " قلنا", "probability": 0.9080403645833334}, {"start": 1790.62, "end": 1791.16, "word": " إذا", "probability": 0.6500244140625}, {"start": 1791.16, "end": 1791.7, "word": " القوانين", "probability": 0.9598388671875}, {"start": 1791.7, "end": 1791.86, "word": " ال", "probability": 0.346923828125}, {"start": 1791.86, "end": 1792.16, "word": " limits", "probability": 0.66748046875}, {"start": 1792.16, "end": 1792.34, "word": " ال", "probability": 0.74560546875}, {"start": 1792.34, "end": 1792.8, "word": " continuity", "probability": 0.84423828125}, {"start": 1792.8, "end": 1793.18, "word": " ال", "probability": 0.78125}, {"start": 1793.18, "end": 1793.4, "word": " sum", "probability": 0.76171875}, {"start": 1793.4, "end": 1793.52, "word": " ال", "probability": 0.462158203125}, {"start": 1793.52, "end": 1793.86, "word": " product", "probability": 0.92578125}, {"start": 1793.86, "end": 1794.12, "word": " ال", "probability": 0.87158203125}, {"start": 1794.12, "end": 1794.48, "word": " quotient", "probability": 0.9736328125}, {"start": 1794.48, "end": 1794.66, "word": " ال", "probability": 0.80615234375}, {"start": 1794.66, "end": 1794.9, "word": " bars", "probability": 0.599609375}, {"start": 1794.9, "end": 1795.1, "word": " ال", "probability": 0.47900390625}, {"start": 1795.1, "end": 1795.4, "word": " composite", "probability": 0.208984375}, {"start": 1795.4, "end": 1796.2, "word": " all", "probability": 0.599609375}, {"start": 1796.2, "end": 1796.78, "word": " extended", "probability": 0.75537109375}, {"start": 1796.78, "end": 1797.2, "word": " into", "probability": 0.537109375}, {"start": 1797.2, "end": 1798.0, "word": " functions", "probability": 0.71728515625}, {"start": 1798.0, "end": 1798.24, "word": " of", "probability": 0.93505859375}, {"start": 1798.24, "end": 1798.5, "word": " three", "probability": 0.708984375}, {"start": 1798.5, "end": 1798.84, "word": " variables", "probability": 0.93798828125}, {"start": 1798.84, "end": 1799.08, "word": " يعني", "probability": 0.88232421875}, {"start": 1799.08, "end": 1799.26, "word": " كل", "probability": 0.9541015625}, {"start": 1799.26, "end": 1799.38, "word": " اللي", "probability": 0.95263671875}, {"start": 1799.38, "end": 1799.82, "word": " حكيناها", "probability": 0.95419921875}, {"start": 1799.82, "end": 1800.08, "word": " قبل", "probability": 0.921142578125}, {"start": 1800.08, "end": 1801.36, "word": " بنتد", "probability": 0.78857421875}, {"start": 1801.36, "end": 1802.0, "word": " و", "probability": 0.70654296875}, {"start": 1802.0, "end": 1802.56, "word": " بيجي", "probability": 0.9127197265625}, {"start": 1802.56, "end": 1803.02, "word": " اللي", "probability": 0.769287109375}, {"start": 1803.02, "end": 1803.2, "word": " هو", "probability": 0.98193359375}, {"start": 1803.2, "end": 1804.02, "word": " بنفع", "probability": 0.765869140625}, {"start": 1804.02, "end": 1804.4, "word": " برضه", "probability": 0.9267578125}, {"start": 1804.4, "end": 1804.6, "word": " على", "probability": 0.92724609375}, {"start": 1804.6, "end": 1805.06, "word": " function", "probability": 0.66162109375}, {"start": 1805.06, "end": 1805.74, "word": " أو", "probability": 0.7685546875}, {"start": 1805.74, "end": 1806.22, "word": " functions", "probability": 0.859375}, {"start": 1806.22, "end": 1806.5, "word": " of", "probability": 0.96337890625}, {"start": 1806.5, "end": 1807.0, "word": " three", "probability": 0.7509765625}, {"start": 1807.0, "end": 1808.04, "word": " variables", "probability": 0.947265625}, {"start": 1808.04, "end": 1808.4, "word": " و", "probability": 0.734375}, {"start": 1808.4, "end": 1808.56, "word": " هيك", "probability": 0.893798828125}, {"start": 1808.56, "end": 1808.84, "word": " بتكون", "probability": 0.874267578125}, {"start": 1808.84, "end": 1809.24, "word": " خلاصة", "probability": 0.8912353515625}, {"start": 1809.24, "end": 1809.76, "word": " المحاضرة", "probability": 0.93115234375}, {"start": 1809.76, "end": 1810.7, "word": " 14", "probability": 0.521484375}, {"start": 1810.7, "end": 1810.98, "word": " و", "probability": 0.85400390625}, {"start": 1810.98, "end": 1811.18, "word": " إلى", "probability": 0.87939453125}, {"start": 1811.18, "end": 1811.56, "word": " لقاء", "probability": 0.7887369791666666}, {"start": 1811.56, "end": 1811.88, "word": " آخر", "probability": 0.93505859375}, {"start": 1811.88, "end": 1812.18, "word": " السلام", "probability": 0.919677734375}, {"start": 1812.18, "end": 1812.52, "word": " عليكم", "probability": 0.89990234375}, {"start": 1812.52, "end": 1812.7, "word": " و", "probability": 0.9755859375}, {"start": 1812.7, "end": 1812.76, "word": " رحمه", "probability": 0.8095703125}, {"start": 1812.76, "end": 1812.92, "word": " الله", "probability": 0.97412109375}, {"start": 1812.92, "end": 1813.44, "word": " وبركاته", "probability": 0.92802734375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 1814.709125, "duration_after_vad": 1805.424999999998}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY.srt
new file mode 100644
index 0000000000000000000000000000000000000000..09f00a2c60a4ff119a4080b1a940c536a919e43c
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY.srt
@@ -0,0 +1,2241 @@
+1
+00:00:00,660 --> 00:00:06,300
+بسم الله الرحمن الرحيم هذه هي المحاضرة الخامسة
+
+2
+00:00:06,300 --> 00:00:13,380
+مساق تفاضل وتكامل بعنوان lines and planes in
+
+3
+00:00:13,380 --> 00:00:17,920
+space اليوم بدنا نعرف كيف نوجد معادلة الخط
+
+4
+00:00:17,920 --> 00:00:22,960
+المستقيم ومعادلة المستوى في الـ space ومتعلقاتها
+
+5
+00:00:23,670 --> 00:00:28,230
+الآن احنا .. هذا طبعاً في section 12-5 احنا سابقاً في
+
+6
+00:00:28,230 --> 00:00:33,250
+الـ .. في الـ plane عرفنا معادلة الخط المستقيم و كنا
+
+7
+00:00:33,250 --> 00:00:37,990
+نجد معادلة الخط المستقيم  بمعرفة الميل
+
+8
+00:00:37,990 --> 00:00:43,210
+ونقطة أو نقطتين. الآن الكلام هنا مشابه، الآن بدنا
+
+9
+00:00:43,210 --> 00:00:48,290
+نعرف كيف .. كيف نوجد معادلة الخط المستقيم lines، و
+
+10
+00:00:48,290 --> 00:00:53,390
+بعدين نعرف معادلة الخط المستقيم المحدود، يعني بمعنى
+
+11
+00:00:53,390 --> 00:00:57,930
+آخر القطع المستقيمة line segment، بدنا نعرف اللي هو
+
+12
+00:00:57,930 --> 00:01:02,610
+المعادلتين. الآن زي ما قلنا معادلة الخط المستقيم
+
+13
+00:01:02,610 --> 00:01:09,290
+سنحددها من خلال معرفة نقطتين أو ميل ونقطة
+
+14
+00:01:09,290 --> 00:01:13,990
+أو ما يشابه. هنا، إيش اللي بيشابه؟  كلنا نشوف مع بعض.
+
+15
+00:01:14,330 --> 00:01:19,950
+الآن لو عرفنا .. طبعاً عندنا خط مستقيم اللي بدنا
+
+16
+00:01:19,950 --> 00:01:25,770
+نعرف معادلته، لو عرفنا نقطة عليه وعرفنا اللي هو
+
+17
+00:01:25,770 --> 00:01:31,010
+vector موازٍ له، يعني بمعنى آخر عرفنا وكأنه ميل
+
+18
+00:01:31,010 --> 00:01:35,890
+له، ميله ونقطة، فنَشوف كيف بدنا نحدد معادلة الخط
+
+19
+00:01:35,890 --> 00:01:41,300
+المستقيم في هذه الحالة بمعرفة نقطة على الخط، وإيش، و
+
+20
+00:01:41,300 --> 00:01:45,600
+vector موازٍ له. شوفوا يا جماعة، الآن نأخذ نقطة
+
+21
+00:01:45,600 --> 00:01:50,480
+عشوائية اسمها  P(x,y,z). هذه اللي هي بتحددها
+
+22
+00:01:50,480 --> 00:01:54,820
+النقطة اللي هو الخط، يعني بدللتها بتيجي معادلة
+
+23
+00:01:54,820 --> 00:01:59,500
+الخط المستقيم. الآن الـ vector الموجود عندنا نفترض
+
+24
+00:01:59,500 --> 00:02:04,080
+أن اسمه V، بيساوي V1 في اتجاه I، و V2 في اتجاه J، و V3
+
+25
+00:02:04,080 --> 00:02:10,430
+في اتجاه الـ K. الآن النقطة المعروفة نسميها P0
+
+26
+00:02:10,430 --> 00:02:13,910
+(x0, y0, z0). بدنا نعرف معادلة الخط
+
+27
+00:02:13,910 --> 00:02:20,630
+المستقيم. شوفوا الآن هذا الـ vector موازٍ لمين؟ للخط
+
+28
+00:02:20,630 --> 00:02:24,010
+المستقيم اللي بدنا نعرف معادلته. أخذنا النقطة
+
+29
+00:02:24,010 --> 00:02:27,890
+العشوائية P(x,y,z). صار عندي P0، P هو
+
+30
+00:02:27,890 --> 00:02:32,490
+عبارة عن vector على الخط موازٍ لهذا الـ vector
+
+31
+00:02:32,490 --> 00:02:37,640
+المعروف. الآن أكيد الـ vector هذا هو نفس اتجاه الـ
+
+32
+00:02:37,640 --> 00:02:41,340
+vector هذا، بس بيختلف عنه بإيه؟ في إيه؟ في الطول، يعني
+
+33
+00:02:41,340 --> 00:02:44,580
+بمعنى آخر، بيتغيّر الطول، وبيتغيّر نقطة مين؟ x و y
+
+34
+00:02:44,580 --> 00:02:49,180
+و z. عشان هيك بنيجي بنكتب، بنستخدم parameter، بنقول
+
+35
+00:02:49,180 --> 00:02:53,960
+أن هذا الـ vector P0P هيكون هو هذا الـ vector بس
+
+36
+00:02:53,960 --> 00:02:58,340
+مضروب في some scalar t، يعني اللي هو الـ .. طبعاً t
+
+37
+00:02:58,340 --> 00:03:02,140
+اللي هي في هذه الحالة اللي .. اللي الآن عندي عبارة
+
+38
+00:03:02,140 --> 00:03:07,080
+عن real number. اتجاه P0P، الـ .. الـ vector P0P هو
+
+39
+00:03:07,080 --> 00:03:10,720
+نفسه الـ vector V بس مضروب في t، لأنه بيكون مختلف عن
+
+40
+00:03:10,720 --> 00:03:15,260
+بعض بس في إيش؟ في الطول، أما موازٍ له، يعني يا في
+
+41
+00:03:15,260 --> 00:03:19,060
+اتجاهه بالشكل هذا، يا أما في اتجاهه بالشكل العكسي.
+
+42
+00:03:19,440 --> 00:03:24,280
+P0P = tV. هي اللي بتمثل معادلة الخط
+
+43
+00:03:24,280 --> 00:03:29,440
+المستقيم اللي عندنا، و t عبارة عن مين؟ عن أي عدد
+
+44
+00:03:29,440 --> 00:03:32,620
+بين negative infinity و infinity، يعني real number.
+
+45
+00:03:35,270 --> 00:03:41,930
+P0P = tV. بدي أضعها في صورة جميلة، نعرف كيف نجد
+
+46
+00:03:41,930 --> 00:03:47,490
+المعادلة بمجرد معرفة اللي هي الأرقام اللي بتمثلها
+
+47
+00:03:47,490 --> 00:03:50,330
+V، والأرقام اللي بتمثلها P0، ونعرف معادلة الخط
+
+48
+00:03:50,330 --> 00:03:54,500
+المستقيم، ونشوف الأمثلة. نشوف إيش اللي بقوله. P0P
+
+49
+00:03:54,500 --> 00:03:58,580
+يا جماعة هو عبارة عن (x - x0) في اتجاه الـ I، (y
+
+50
+00:03:58,580 --> 00:04:02,580
+- y0) في اتجاه الـ J، (z - z0) في اتجاه
+
+51
+00:04:02,580 --> 00:04:06,960
+الـ K. هذا مين هو؟ P0P. إذاً (x - x0) في
+
+52
+00:04:06,960 --> 00:04:10,520
+اتجاه الـ I زائد (y - y0) في اتجاه الـ J زائد
+
+53
+00:04:10,520 --> 00:04:14,460
+(z - z0) في اتجاه الـ K. هذا اللي هو P0P.
+
+54
+00:04:15,870 --> 00:04:22,010
+بيساوي tV. هو البارامتر المتغير اللي بيحدد لمكان
+
+55
+00:04:22,010 --> 00:04:26,450
+النقطة، بنضربه في t، بيحدد النقطة هنا وهنا وهنا
+
+56
+00:04:26,450 --> 00:04:32,010
+حسب قيمة t، بتكون تحدد النقطة عندك. إذاً صارت t هي
+
+57
+00:04:32,010 --> 00:04:37,470
+الآن V1 في اتجاه I، و V2 في اتجاه J، و V3 في اتجاه K. هذه
+
+58
+00:04:37,470 --> 00:04:42,230
+الآن اللي هي بنقدر نكتبها على صورة اللي هو بيجي
+
+59
+00:04:42,230 --> 00:04:46,740
+بننقل x0 في اتجاه I، اللي هي المعروفة، والعدد y0 في
+
+60
+00:04:46,740 --> 00:04:48,980
+اتجاه الـ j، اللي هو العدد z0 في اتجاه الـ k.
+
+61
+00:04:48,980 --> 00:04:54,220
+نجمعها مع V1 اللي معروفة، و V2، و V3، و t. فبيصير x
+
+62
+00:04:54,220 --> 00:04:57,600
++ y + z. واحد في اتجاه الـ I، في اتجاه الـ J، في
+
+63
+00:04:57,600 --> 00:05:01,560
+اتجاه الـ K. بتساوي x0 اللي نَقلناها، هذه مع الـ
+
+64
+00:05:01,560 --> 00:05:07,920
+tV1، هي tV1، و y0 مع التي هي tV2، و z0 في
+
+65
+00:05:07,920 --> 00:05:14,100
+اتجاه اليمين، التي هي في اتجاه tV3 في اتجاه الـ K.
+
+66
+00:05:14,100 --> 00:05:19,880
+هذه الآن بتمثل معادلة الخط المستقيم. الآن هذه، أو
+
+67
+00:05:19,880 --> 00:05:25,330
+هذه معادلة الخط المستقيم، أو باختصار، هذه اللي على طول
+
+68
+00:05:25,330 --> 00:05:29,830
+بدّوش يحفظوها، بيستنتجها على طول من هذه المعادلة. نشوف
+
+69
+00:05:29,830 --> 00:05:33,550
+الآن نلخص الحديث. لو واحد ثاني جالي يسمعني، أنا بدّي
+
+70
+00:05:33,550 --> 00:05:36,670
+اشتغلها زيك، بدّي اشتغلها على صورة vectors. بقول له
+
+71
+00:05:36,670 --> 00:05:40,570
+ما فيش مشكلة، بيجي بقول لي بدّي اسمي الـ vector هذا من
+
+72
+00:05:40,570 --> 00:05:44,500
+هنا اللي هنا، هي الـ vector هذا، هي من هنا اللي هنا، بدّي
+
+73
+00:05:44,500 --> 00:05:50,900
+أسميه هذا اللي هو عبارة عن R، هذا من هنا، من نقطة
+
+74
+00:05:50,900 --> 00:05:55,080
+الأصل، هذا الـ position vector لمين؟ للـ (x,y,z)
+
+75
+00:05:55,080 --> 00:05:58,440
+اللي هو في الواقع اللي هي اللي بتحدد لقيمة x, y, z.
+
+76
+00:05:58,440 --> 00:06:03,720
+بنسميها R(t)، اللي هو الـ vector بنسميه هذا R(t).
+
+77
+00:06:03,720 --> 00:06:08,420
+اللي هو من عند الـ zero لعند الـ P. الآن نقطة الأصل
+
+78
+00:06:08,420 --> 00:06:11,370
+الـ zero، بقصد فيها. الآن وهذا الـ vector من هنا إلى
+
+79
+00:06:11,370 --> 00:06:13,950
+هنا بدّي اسميه R0، اللي هو اللي بتكون .. اللي
+
+80
+00:06:13,950 --> 00:06:18,490
+بيكون معروف، ليش معروف؟ لأنه P0 معروفة. الآن هذا
+
+81
+00:06:18,490 --> 00:06:22,490
+الـ vector P0P هو اللي بسميه مين؟ اللي هو t
+
+82
+00:06:22,490 --> 00:06:26,410
+في V اللي اتفقنا عليه. فالآن المعادلة ممكن نكتبها
+
+83
+00:06:26,410 --> 00:06:30,330
+على الصورة التالية، على صورة vectors، اللي هو معادلة
+
+84
+00:06:30,330 --> 00:06:35,730
+الخط المستقيم على صورة vectors. إيش هو؟ R(t)، وهو
+
+85
+00:06:35,730 --> 00:06:42,370
+المجهول اللي عند x و y و z المتغيرة. R(t) بيساوي
+
+86
+00:06:42,370 --> 00:06:45,910
+هذا الـ vector زائد هذا الـ vector. بتعرفوا أن هذا الـ
+
+87
+00:06:45,910 --> 00:06:48,830
+vector زائد هذا الـ vector بيساوي اللي بيغلق المثلث
+
+88
+00:06:48,830 --> 00:06:55,650
+لأن R(t) هذا بيساوي R0 + tV. وهذه
+
+89
+00:06:55,650 --> 00:07:00,350
+هي الـ vector equation for the line through P0
+
+90
+00:07:00,350 --> 00:07:05,690
+parallel to V. الـ vector equation أو معادلة الخط
+
+91
+00:07:05,690 --> 00:07:11,290
+المستقيم بواسطة أو بدلالة الـ vectors.
+
+92
+00:07:14,420 --> 00:07:18,680
+بمعرفة النقطة P0 و V، بتساوي R(t) زي ما قلنا
+
+93
+00:07:18,680 --> 00:07:24,220
+بتساوي R0، طبعاً هتطلع لنا مباشرة هي عدد
+
+94
+00:07:24,220 --> 00:07:30,320
+زائد مين؟ tV. هذه as of R0 لأنها بدلالة اسمين
+
+95
+00:07:30,320 --> 00:07:35,300
+P0، أما R(t) هتكون بدلالة اتجاه t. Where R is the
+
+96
+00:07:35,300 --> 00:07:40,680
+position vector of a point P on L، and R0 is the
+
+97
+00:07:40,680 --> 00:07:44,800
+position vector of P0، زي ما حكينا. الآن هذه
+
+98
+00:07:44,800 --> 00:07:46,980
+اللي هي الـ vector equation for a line، الـ
+
+99
+00:07:46,980 --> 00:07:49,480
+parametric equations for a line، اللي عملناها قبل
+
+100
+00:07:49,480 --> 00:07:56,260
+بشوية، التي هي الـ Standard Parameterization of Line
+
+101
+00:07:56,260 --> 00:08:02,000
+Through P0 Parallel to V. يعني هذه معادلة الخط
+
+102
+00:08:02,000 --> 00:08:08,060
+المستقيم اللي هي إيش؟  شَاملتها as a parametric
+
+103
+00:08:08,060 --> 00:08:12,220
+equation، كاللي هو معادلة برامترية، اللي يعني بدلالة
+
+104
+00:08:12,220 --> 00:08:18,090
+t، لـ vector يمرّ بنقطة P0، موازٍ لـ vector V، التي هي
+
+105
+00:08:18,090 --> 00:08:22,590
+عبارة عن (x - x0) = tV1، و (y - y0) = t
+
+106
+00:08:22,590 --> 00:08:26,830
+V2، و (z - z0) = tV3، و t ∈ ℝ.
+
+107
+00:08:26,830 --> 00:08:29,690
+هذه من وين جبناها؟ جبناها يا جماعة من هنا، سمعت
+
+108
+00:08:29,690 --> 00:08:32,410
+زي حد بيقول لي من وين جبناها؟ هي (x - x0)
+
+109
+00:08:32,410 --> 00:08:37,530
+بتساوي tV1، يعني x بتساوي x0 + tV1. هذه
+
+110
+00:08:37,530 --> 00:08:43,020
+معاملات I. (y - y0) = tV2، معاملات J، يعني
+
+111
+00:08:43,020 --> 00:08:48,700
+y بتساوي y0 + tV2، و z بتساوي z0 + tV3. هي
+
+112
+00:08:48,700 --> 00:08:52,700
+اللي هي الـ standard parametric equation أو equations
+
+113
+00:08:52,700 --> 00:08:57,900
+for the line through the P0 and parallel to V. الآن هذه هي المعادلة
+
+114
+00:08:57,900 --> 00:09:03,500
+التي تبع الخط المستقيم الأكثر استخداماً وأكثر
+
+115
+00:09:03,500 --> 00:09:09,200
+شيوعاً. نأخذ مثال، نجد الـ Parametric equations for
+
+116
+00:09:13,580 --> 00:09:18,240
+the line (-2, 0, 4). نجد الـ Parametric
+
+117
+00:09:18,240 --> 00:09:25,360
+equations لخط المستقيم. الخط المستقيم اللي بيمرّ
+
+118
+00:09:25,360 --> 00:09:29,800
+بالنقطة (-2, 0, 4) ويوازي المتجه
+
+119
+00:09:29,800 --> 00:09:35,460
+اللي أمامنا هذا. ماشي الحال، هاي عندي الآن المتجه، هي
+
+120
+00:09:35,460 --> 00:09:39,680
+رسم لي إياه، وهي النقطة اللي ماخذها عليه، اللي هي (-
+
+121
+00:09:39,680 --> 00:09:43,040
+2, 0, 4)، وبدّو يجد معادلة الخط المستقيم
+
+122
+00:09:43,040 --> 00:09:48,220
+اللي هذا اللي موازٍ للي هو المتجه هذا، وبيمرّ
+
+123
+00:09:48,220 --> 00:09:53,370
+بالنقطة P. الآن تعويض مباشر، ما هي عبارة عن x
+
+124
+00:09:53,370 --> 00:10:00,430
+= x0 + tV1. الـ V معروفة، V1 هي 2، و V2
+
+125
+00:10:00,430 --> 00:10:05,650
+هي 4، و V3 هي مين؟ (-2). وهذه x0، و y0، و z0 هي
+
+126
+00:10:05,650 --> 00:10:09,390
+المعروفات. هي = تعويض. x = x0 (-2)
+
+127
+00:10:09,390 --> 00:10:14,590
++ 2t، و y = y0
+
+128
+00:10:17,220 --> 00:10:26,500
++ tV2، 4t، و z = z0، 4 - 2t. هذه
+
+129
+00:10:26,500 --> 00:10:30,360
+الآن عبارة عن معادلة الخط المستقيم اللي بيمرّ
+
+130
+00:10:30,360 --> 00:10:34,400
+بالنقطة هذه ويوازي الاتجاه اللي أمامنا، والكلام
+
+131
+00:10:34,400 --> 00:10:40,770
+زي ما شُفتُه، سهل. زي زمان برضه، لو كان عنده نقطتين
+
+132
+00:10:40,770 --> 00:10:44,550
+نعرف معادلة الخط المستقيم برضه هنا في الفراغ، نفس
+
+133
+00:10:44,550 --> 00:10:47,470
+الشيء. لو كان عنده نقطتين، بنعرف معادلة الخط
+
+134
+00:10:47,470 --> 00:10:50,830
+المستقيم، وأكيد بعضكم قال: آه طب ما هو إحنا بنحيب
+
+135
+00:10:50,830 --> 00:10:55,910
+المتجه اللي موازٍ للمتجه اللي هو اللي بيحمل P و
+
+136
+00:10:55,910 --> 00:11:00,690
+Q، اللي هو PQ أو QP، زمان بالذات هو موازٍ أصلاً للخط
+
+137
+00:11:00,690 --> 00:11:03,770
+المستقيم اللي بيمرّ بالنقطتين هذين، وهذا نقطة من
+
+138
+00:11:03,770 --> 00:11:08,480
+النقطتين، فبيصير معروف اللي هو متجه، اللي هو موازٍ
+
+139
+00:11:08,480 --> 00:11:12,160
+اللي هو في الواقع بيحمل، ونفس الشيء، نقطة معروفة.
+
+140
+00:11:12,160 --> 00:11:15,840
+نعرف اللي قاعدة، دلوقت نشوف كيف. Find a parametric
+
+141
+00:11:15,840 --> 00:11:19,320
+equation for the line through كده and كده، بمعرفة
+
+142
+00:11:19,320 --> 00:11:23,000
+نقطتين. لأن ده vector PQ. أول شيء، ده نوجد الـ vector
+
+143
+00:11:23,000 --> 00:11:28,190
+PQ، اللي هو عبارة عن .. واحد قال: بدّي أوجد QB، برضه صح.
+
+144
+00:11:28,190 --> 00:11:33,510
+واتنين بتجيب إن معادلتين بتظهر أنها مختلفة، لكن
+
+145
+00:11:33,510 --> 00:11:39,650
+بتجيب نفس القطعة، لأنه هتجيب لك نفس النقاط. Q -
+
+146
+00:11:39,650 --> 00:11:44,810
+P، اللي هي (1 - (-3), -1 - 2,
+
+147
+00:11:44,810 --> 00:11:45,410
+4 - (-1))
+
+148
+00:12:00,980 --> 00:12:05,440
+هذا الـ vector is parallel to the line and equation 3
+
+149
+00:12:06,470 --> 00:12:11,150
+with x0، و y0، و z0 (-3, -2,
+
+150
+00:12:11,150 --> 00:12:13,250
+-1). equation 3، اللي بتحكي عن equations 3 اللي
+
+151
+00:12:13,250 --> 00:12:16,410
+هي قانون اللي هو الـ parametric equations، ده الخط
+
+152
+00:12:16,410 --> 00:12:21,150
+المستقيم اللي قبل شوية. طيب الآن عرفنا المتجه
+
+153
+00:12:21,150 --> 00:12:27,370
+الموازٍ أو الحامل للـ line، بالإضافة لنقطة على الـ
+
+154
+00:12:27,370 --> 00:12:29,490
+line. واحد يقول لي: بدّي، ما هذه النقطة، بتاخد نقطة ثانية.
+
+155
+00:12:29,490 --> 00:12:33,630
+بقول له: آه، بالفعلو بيطلع نفس الـ equations، بيطلع نقاط
+
+156
+00:12:33,630 --> 00:12:37,810
+تكون في أشكال مختلفة، لكن بيجيب نفس الخط، يعني نفس
+
+157
+00:12:37,810 --> 00:12:42,670
+النقاط اللي على الخط. طيب x = x0، اللي هي
+
+158
+00:12:42,670 --> 00:12:50,290
+(-3) + 4t. y = y0، اللي
+
+159
+00:12:50,290 --> 00:12:55,800
+هو 2 - 3t. فاهمين عليها؟ z = -
+
+160
+00:12:55,800 --> 00:13:00,820
+3، التي هي z0 + 7t. هذه الآن هي الـ
+
+161
+00:13:00,820 --> 00:13:05,040
+Parametric Equations للخط المستقيم. وحتى قال:
+
+162
+00:13:05,040 --> 00:13:08,780
+اسمعوا، بدّي أنا، بدّي أختار اللي هي الـ Q بدل الـ P
+
+163
+00:13:08,780 --> 00:13:12,580
+بدل الـ V، يعني بدّي آخذ الـ base. إحنا أخذنا النقطة
+
+164
+00:13:12,580 --> 00:13:17,340
+اللي انطلقنا بها، والنهاية Q، فأنا باخذ نقطة انطلاق Q،
+
+165
+00:13:17,340 --> 00:13:21,960
+وأخذ به بنفع، بيصير الآن اللي هو الـ equation اللي
+
+166
+00:13:21,960 --> 00:13:27,380
+هي أو إقال، حتى اسمعوا، مش شيكة كمان، بتاخد أنا اللي
+
+167
+
+223
+00:17:51,480 --> 00:17:55,900
+جماعة لعند واحد  في t من عند 0 لعند واحد بس بتجيب
+
+224
+00:17:55,900 --> 00:18:01,220
+للقطع المستقيمة من النقطة هذه للنقطة هذه وهيك
+
+225
+00:18:01,220 --> 00:18:05,440
+بيكون أوجدنا اللي هو Parametric Equations for the
+
+226
+00:18:05,440 --> 00:18:10,540
+line segment joining the points B and Q. الآن بدنا
+
+227
+00:18:10,540 --> 00:18:15,020
+نعرف أمر جديد اللي هو the distance from a point to
+
+228
+00:18:15,020 --> 00:18:20,820
+a line in space. هاي عندي خط في الفراغ اسمه خط L
+
+229
+00:18:24,350 --> 00:18:28,830
+عند نقطة في الفراغ اسمها S بدي أوجد اللي هو
+
+230
+00:18:28,830 --> 00:18:32,930
+المسافة بين S و بين نقطة. بتعرفوا المسافة بين S و
+
+231
+00:18:32,930 --> 00:18:39,930
+بين نقطة بين S و بين خط مستقيم؟ بيجي بنزل من النقطة
+
+232
+00:18:39,930 --> 00:18:43,490
+عمودي على الخط المستقيم. هذه العمودي وصلنا لهذه
+
+233
+00:18:43,490 --> 00:18:48,070
+العمودي. هذا الطول اللي احنا بدنا إياه. هذا الطول اللي
+
+234
+00:18:48,070 --> 00:18:52,660
+بدنا إياه. طب كيف بدنا نوجد ه؟ سهل الموضوع. الـ S عندي معروفة
+
+235
+00:18:52,660 --> 00:18:56,480
+صح ولا لأ يا جماعة؟ معروفة. هو معطيني إياها لأنه بدنا
+
+236
+00:18:56,480 --> 00:18:59,920
+المسافة بينها بين S و بين الخط المستقيم. بكون
+
+237
+00:18:59,920 --> 00:19:05,140
+معطيني الخط المستقيم اللي عندي. بكون معطيني من اللي
+
+238
+00:19:05,140 --> 00:19:08,640
+هو النقطة S اللي بدنا المسافة بينها وبين الخط
+
+239
+00:19:08,640 --> 00:19:13,860
+المستقيم. طبعاً كنا الخط المستقيم في طياته بيحمل أو
+
+240
+00:19:13,860 --> 00:19:18,520
+بيعطيني معلومات الـ vector الموازي له V أو vector
+
+241
+00:19:18,520 --> 00:19:25,100
+عليه V بالإضافة لنقاط زي ما بدك بتجيب منه. إذاً الآن
+
+242
+00:19:25,100 --> 00:19:29,860
+باجي أنا بمعرفة اللي هو الـ vector اللي عليه. لو
+
+243
+00:19:29,860 --> 00:19:35,500
+ليكن هذا مثلاً V، وهذه اللي هي نقطة بأخذها على الخط
+
+244
+00:19:35,500 --> 00:19:40,160
+المستقيم، وبوصل هذا. صارت هذه الزاوية θ اللي هي، و
+
+245
+00:19:40,160 --> 00:19:44,380
+هذه قائمة. هذه الزاوية بين هذا اللي هو الجزء من
+
+246
+00:19:44,380 --> 00:19:51,300
+الخط وبين بي إس. الآن هذا طوله، طوله بتعرفوا إنه sin
+
+247
+00:19:51,300 --> 00:19:58,500
+θ بساوي اللي هو المقابل هذا على اللي هو الوتر.
+
+248
+00:19:58,500 --> 00:20:03,500
+فبصير الآن المقابل اللي هو المطلوب بساوي الـ
+
+249
+00:20:03,500 --> 00:20:06,640
+magnitude لهذا. طول هذا اللي هو الـ BS في sin
+
+250
+00:20:06,640 --> 00:20:13,380
+زاويه θ. في الواقع هذا أيش بيمثله؟ هذا الآن لو اجينا
+
+251
+00:20:13,380 --> 00:20:16,320
+أخذنا طبعاً احنا المطلوب distance from a point S to
+
+252
+00:20:16,320 --> 00:20:20,360
+a line.  بالـ vector V اللي هو هذا. هذا الآن بي إس ساين
+
+253
+00:20:20,360 --> 00:20:24,180
+θ هو في الواقع عبارة عن الـ magnitude للـ بي إس
+
+254
+00:20:24,180 --> 00:20:28,340
+across الـ vector V. هذا اللي بيحمل الخط اللي هنا.
+
+255
+00:20:42,560 --> 00:20:49,210
+القانون اللي اشتقناه هو قانون المسافة بين نقطة وبين
+
+256
+00:20:49,210 --> 00:20:53,710
+خط مستقيم اللي هو طبعاً بمعلومية النقطة ومعلومية
+
+257
+00:20:53,710 --> 00:20:57,810
+الخط المستقيم. عبارة عن الـ BS cross V على الـ
+
+258
+00:20:57,810 --> 00:21:00,970
+magnitude V. ماهو الـ BS اللي هو  الـ S النقطة
+
+259
+00:21:00,970 --> 00:21:05,430
+المعطاة، و B نقطة نأخذها على الخط المستقيم المعطاة، و
+
+260
+00:21:05,430 --> 00:21:08,590
+V هو الـ vector اللي بيحمل الخط المستقيم المعطاة أو
+
+261
+00:21:08,590 --> 00:21:11,910
+الموازي اللي هو زي ما قلنا سابقاً. ونشوف مثال عملي.
+
+262
+00:21:13,470 --> 00:21:17,150
+بقول find the distance from the point S (1, 1, 5) to
+
+263
+00:21:17,150 --> 00:21:21,350
+the line x = 1 + t, y = 3 - t, z = 2t. لسه احنا في الموضوع يعني ما بعدناش يعني
+
+264
+00:21:21,350 --> 00:21:25,670
+الآن احنا بنقدر نعرف من هذه المعادلة اللي هو الـ
+
+265
+00:21:25,670 --> 00:21:30,850
+vector الموازي اللي هو معاملات من الـ vector
+
+266
+00:21:30,850 --> 00:21:33,890
+الموازي. معاملات t يعني الـ vector الموازي P بيساوي
+
+267
+00:21:33,890 --> 00:21:38,470
+i اللي هي واحد، ناقص j اللي هي ناقص واحد، زائد اثنين
+
+268
+00:21:38,470 --> 00:21:44,550
+k اللي هي من معامل t اثنين. هذا الـ vector الموازي
+
+269
+00:21:44,550 --> 00:21:49,850
+لمين؟ لخط المستقيم المعطى. معادلته زي ما اتعلمنا قبل
+
+270
+00:21:49,850 --> 00:21:55,620
+بشوية. لأن النقطة اللي معناها مين؟ هي S (1,1,5) بدنا تمام
+
+271
+00:21:55,620 --> 00:22:00,760
+نقطة. نأخذ أي نقطة اللي بتيجي على بالك. واحد قال
+
+272
+00:22:00,760 --> 00:22:04,900
+اسمع بدي آخذ اللي هو أسهل شيء t بصفر. بصير واحد و
+
+273
+00:22:04,900 --> 00:22:09,500
+ثلاثة واثنين. قلنا له حاضر. واحد ثاني قال لأ بدي
+
+274
+00:22:09,500 --> 00:22:13,340
+آخذ t بواحد. بدي أغلب حالي. بدي أخليها كلها أرقام. بقول له
+
+275
+00:22:13,340 --> 00:22:16,880
+بنفع. t اللي بدك إياه. الآن أخذنا نقطة بي (1، 3، 2)
+
+276
+00:22:16,880 --> 00:22:25,060
+و صفر. الآن أيش بالمعادلة؟ زي ما قلنا في القانون، في
+
+277
+00:22:25,060 --> 00:22:29,060
+القانون. قلنا بالمعادلة BS cross V على الـ magnitude
+
+278
+00:22:29,060 --> 00:22:32,220
+لـ V. الآن بي عرفناه، و S معطاة، و V عرفناه. نجي
+
+279
+00:22:32,220 --> 00:22:37,100
+الآن نطبق القانون. بي إس أيش متساوي؟ because (1، 1، 5)
+
+280
+00:22:37,100 --> 00:22:43,120
+ناقص (1، 3، 2). واحد ناقص واحد، واحد ناقص ثلاثة، وخمسة ناقص
+
+281
+00:22:43,120 --> 00:22:49,440
+اثنين. صفر. طلع عندي الـ BS. طلع عندي هذا صفر. طبعاً هذه
+
+282
+00:22:49,440 --> 00:22:53,180
+طلعت سالب اثنين j وهذه خمسة k. إذًا صار عندي الـ BS
+
+283
+00:22:53,180 --> 00:22:58,380
+هيو. بدي أعمل الـ BS cross V وآخذ له الـ magnitude. الـ
+
+284
+00:22:58,380 --> 00:23:02,480
+V هيو. ماشي. عرفنا كيف نوجد الـ BS cross V بواسطة
+
+285
+00:23:02,480 --> 00:23:06,700
+المحدد. هي المحدد جوا وهي برا. ايه الـ absolute value
+
+286
+00:23:06,700 --> 00:23:11,680
+أو الـ magnitude. طبعاً المحدد بيطلعنا قيمة، وبعدين
+
+287
+00:23:11,680 --> 00:23:15,010
+نأخذ الـ absolute value ونطلع القيمة الموجبة اللي
+
+288
+00:23:15,010 --> 00:23:17,650
+بدنا إياها. هيها. شوفوا صلوا على النبي عليه الصلاة والسلام.
+
+289
+00:23:21,770 --> 00:23:27,610
+الـ BS. كيف نعرف نضرب BS في V؟ هي ijk، وبعدين BS اللي
+
+290
+00:23:27,610 --> 00:23:31,850
+هو السطر الأول، والـ BS اللي أوجدناه هيه اللي هو صفر
+
+291
+00:23:31,850 --> 00:23:36,830
+ناقص اثنين، خمسة، واللي هي الـ V اللي هي عبارة عن
+
+292
+00:23:36,830 --> 00:23:41,290
+واحد، وناقص واحد، واثنين. الآن خلينا نوجد هذا الـ...
+
+293
+00:23:41,290 --> 00:23:45,350
+الـ...الـ...الـ...الـ...الـ determinant. الـ
+
+294
+00:23:45,350 --> 00:23:48,730
+determinant اللي جوا بتعرفوا كيف بنشطب السطر هذا والـ
+
+295
+00:23:48,730 --> 00:23:52,090
+العمود هذا. بيصير ناقص اثنين في اثنين ناقص أربعة
+
+296
+00:23:52,090 --> 00:23:58,570
+زائد واحد اللي هو عبارة عن ايه؟ ناقص أربعة زائد
+
+297
+00:23:58,570 --> 00:24:03,930
+خمسة، وهو عبارة عن i. الآن نشطب بهذا ونشطب بهذا، يصبح
+
+298
+00:24:03,930 --> 00:24:09,490
+اثنين ناقص خمسة، وفي ناقص بتاع الاتجاه يصبح زائد
+
+299
+00:24:09,490 --> 00:24:13,230
+خمسة. الآن الأخيرة الـ k نشطب بهذا ونشطب بهذا، يصبح
+
+300
+00:24:13,230 --> 00:24:17,690
+صفر ناقص ناقص يصبح زائد اثنين اللي هي في اتجاه الـ
+
+301
+00:24:17,690 --> 00:24:21,320
+k. الـ magnitude لهذا الجذر التربيعي لواحد زائد
+
+302
+00:24:21,320 --> 00:24:25,840
+خمسة وعشرين زائد أربعة. أربعة وخمسة وعشرين تسعة
+
+303
+00:24:25,840 --> 00:24:29,460
+وعشرين وواحد ثلاثين. يعني جذر الثلاثين. الـ magnitude
+
+304
+00:24:29,460 --> 00:24:33,400
+للـ V نلزمنها اللي هي أيش بتساوي؟ الجذر التربيعي للي
+
+305
+00:24:33,400 --> 00:24:37,700
+هو واحد زائد واحد زائد أربعة اللي هو جذر الستة. لأن
+
+306
+00:24:37,700 --> 00:24:41,520
+حسب القانون the distance أيش بتساوي؟ الـ magnitude للـ
+
+307
+00:24:41,520 --> 00:24:46,100
+BS cross V على الـ magnitude للـ V. طيب يا جماعة صلوا
+
+308
+00:24:46,100 --> 00:24:49,650
+على النبي عليه الصلاة والسلام. الـ magnitude للـ BS
+
+309
+00:24:49,650 --> 00:24:53,770
+cross V اللي هو بساوي جذر الثلاثين اللي أوجدناه
+
+310
+00:24:53,770 --> 00:24:56,670
+على جذر الستة، ويساوي جذر الخمسة. وهذه هي المسافة
+
+311
+00:24:56,670 --> 00:25:01,250
+بين الخط المستقيم والنقطة أو بين النقطة والخط
+
+312
+00:25:01,250 --> 00:25:07,510
+المستقيم المطلوب. الآن بدنا نجي لجزء
+
+313
+00:25:07,510 --> 00:25:11,890
+ثاني من المحاضرة وهو معادلة الخط. معادلة البلين في
+
+314
+00:25:11,890 --> 00:25:16,370
+الـ space. احنا كنا في الفرق زمان نقول إن البلين
+
+315
+00:25:16,370 --> 00:25:20,970
+يحدد تماماً بمعرفة ثلاث نقاط على البلين ليست
+
+316
+00:25:20,970 --> 00:25:24,810
+على استقامة واحدة. اللي هي ثلاث نقاط ليست على استقامة
+
+317
+00:25:24,810 --> 00:25:30,050
+واحدة تحدد اللي هو المستوى اللي يمر بها. طيب الآن
+
+318
+00:25:30,050 --> 00:25:34,290
+هذا الكلام شبيه بمعرفة الآن مش ثلاث نقاط، ثلاث نقاط
+
+319
+00:25:34,290 --> 00:25:39,490
+بعد شوية هيكون كتطبيق بمعرفة اللي هو vector عمودي
+
+320
+00:25:40,100 --> 00:25:45,040
+على المستوى، وبمعرفة نقطة على المستوى، هنقدر نحدد
+
+321
+00:25:45,040 --> 00:25:51,140
+هوية المستوى. الآن بمعرفة نقطة على المستوى، وبمعرفة
+
+322
+00:25:51,140 --> 00:25:55,580
+عمودي عليه، نحدد هوية المستوى. نشوف كيف نوجد معادلته.
+
+323
+00:25:55,580 --> 00:25:58,900
+الآن الـ vector ده عمودي عليه، ده عمودي على كل
+
+324
+00:25:58,900 --> 00:26:03,520
+مستقيم في المستوى. الآن هناخذ نقطة عشوائية نسميها x،
+
+325
+00:26:03,520 --> 00:26:07,600
+y، و z هي اللي هتحدد لي هوية البلين اللي بدلالتها
+
+326
+00:26:07,600 --> 00:26:11,960
+بدنا نجيب اللي هو معادلة البلين. الآن صار عندي بي نوت
+
+327
+00:26:11,960 --> 00:26:16,800
+بي. عبارة عن vector موجود من وين يا جماعة في البلين؟
+
+328
+00:26:16,800 --> 00:26:20,380
+هذا الـ vector أكيد الآن العمودي على البلين، عمودي
+
+329
+00:26:20,380 --> 00:26:27,400
+عليه. يعني بمعنى آخر الآن هذا ضربه في بي نوت بي بساوي
+
+330
+00:26:27,400 --> 00:26:32,680
+صفر لأن الزاوية اللي بينهم تسعين درجة. فبكون عندي
+
+331
+00:26:32,680 --> 00:26:36,260
+اللي هو الضرب القياسي بينهم بساوي صفر. وهذه هي
+
+332
+00:26:36,260 --> 00:26:40,780
+معادلة البلين. معادلة الـ plane (vector equation) ستكون
+
+333
+00:26:40,780 --> 00:26:48,320
+N . P0P = 0. لأن خلينا نسمي مع بعض إن الـ N
+
+334
+00:26:48,320 --> 00:26:52,860
+بدنا نسميه اللي هو بساوي الـ vector المعروف. طبعاً
+
+335
+00:26:52,860 --> 00:26:56,900
+لأنه ببساطة وبدنا نوجد المعادلة الـ plane الـ vector
+
+336
+00:26:56,900 --> 00:27:02,780
+N المعروف بدنا نقول عنه اسمه Ai + Bj + Ck. إذًا
+
+337
+00:27:02,780 --> 00:27:06,720
+الـ A والـ B والـ C هي عبارة عن معاملات الـ i والـ
+
+338
+00:27:06,720 --> 00:27:10,500
+الـ j والـ k بالترتيب للـ vector العمودي على
+
+339
+00:27:10,500 --> 00:27:15,540
+المستوى. N . P0P = 0 هي معادلة اللي هي الـ
+
+340
+00:27:15,540 --> 00:27:18,440
+vector equation للمستوى. لأن لو بدنا نحولها لـ
+
+341
+00:27:18,440 --> 00:27:22,480
+component equation بنعمل الـ dot product. بي نوت بي
+
+342
+00:27:22,480 --> 00:27:27,920
+نوت. بي نوت
+
+343
+00:27:27,920 --> 00:27:34,860
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+344
+00:27:34,860 --> 00:27:35,360
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+345
+00:27:35,360 --> 00:27:35,780
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+346
+00:27:35,780 --> 00:27:37,040
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+347
+00:27:37,040 --> 00:27:39,340
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+348
+00:27:39,340 --> 00:27:48,040
+بي نوت بي نوت بي نوت بي نوت. بتعرفوا تعملوا الـ Dot
+
+349
+00:27:48,040 --> 00:27:53,040
+Product. زائد C في معامل الـ k اللي هي z - z0 في
+
+350
+00:27:53,040 --> 00:27:56,860
+اتجاه الـ zero. كله مش هيساوي صفر. يعني هذا عبارة عن
+
+351
+00:27:56,860 --> 00:28:03,080
+الـ Dot Product للـ N . P0P. الآن صارت هذه
+
+352
+00:28:03,080 --> 00:28:08,480
+تمثل معادلة الخط المستخدم للـ plane اللي بنسميها
+
+353
+00:28:08,480 --> 00:28:12,540
+component equations. وصلت اللي هي الـ component اللي
+
+354
+00:28:12,540 --> 00:28:17,220
+هي الـ equation اللي احنا استعملناها قبل هيك كتير.
+
+355
+00:28:17,220 --> 00:28:22,780
+طيب إذاً الآن ممكن واحد يختصرها يقول لك خلاص. ماهي ax
+
+356
+00:28:22,780 --> 00:28:27,370
+زائد by زائد cz هذه الأشياء بيساووا الـ D.
+
+357
+00:28:43,370 --> 00:28:46,590
+لأن الـ A معروفة، والـ B معروفة، والـ C معروفة، و
+
+358
+00:28:46,590 --> 00:28:50,310
+الـ x0 والـ y0 والـ z0 معروفة. فبصير وكأنه
+
+359
+00:28:50,310 --> 00:28:53,770
+كأنه الـ component equation can be simplified as
+
+360
+00:28:53,770 --> 00:28:59,850
+يعني Ax + By + Cz = D. وهنا بدنا نلاحظ
+
+361
+00:28:59,850 --> 00:29:03,990
+ماله لما نكتب على صورة Ax + By + Cz = D
+
+362
+00:29:03,990 --> 00:29:10,400
+اللي هو عدد. بكون A هو معامل اللي هو الـ i للنورمال،
+
+363
+00:29:10,400 --> 00:29:14,880
+والـ B معامل الـ j للنورمال، والـ C معامل الـ k
+
+364
+00:29:14,880 --> 00:29:20,740
+للنورمال. يعني الآن بمعرفة معادلة الـ plane بعرف الـ
+
+365
+00:29:20,740 --> 00:29:25,000
+normal عليه مباشرة. بكتب على الصورة هذه المعادلة.
+
+366
+00:29:25,000 --> 00:29:28,000
+بقول على طول الـ normal الآن بتساوي ايه؟ في اتجاه الـ
+
+367
+00:29:28,000 --> 00:29:32,280
+i. ب...ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+368
+00:29:32,280 --> 00:29:33,280
+ب...ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+369
+00:29:33,280 --> 00:29:34,200
+ب...ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+370
+00:29:34,200 --> 00:29:38,480
+ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+371
+00:29:38,480 --> 00:29:39,200
+ب...ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+372
+00:29:39,200 --> 00:29:41,080
+ب...ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+373
+00:29:41,080 --> 00:29:41,600
+ب...ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+374
+00:29:41,600 --> 00:29:42,180
+ب...ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+375
+00:29:42,180 --> 00:29:43,760
+ب...ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+376
+00:29:43,760 --> 00:29:44,500
+ب...ب...ب...ب...ب...ب...ب...ب...ب...ب...
+
+377
+00:29:44,500 --> 00:29:50,520
+ب...ب...
+
+378
+00:29:50,520 --> 00:29:56,080
+ب...
+
+379
+00:29:56,100 --> 00:30:00,460
+لكن طوله اختلف. واحد أخذ normal الوحدة. واحد أخذ
+
+380
+00:30:00,460 --> 00:30:04,000
+normal طوله عشرة. واحد أخذ normal طوله مئة. المهم
+
+381
+00:30:04,000 --> 00:30:06,900
+اللي هو بيكون أيش؟ شماله normal على الـ plane إن شاء
+
+382
+00:30:06,900 --> 00:30:10,780
+الله. انقطعت السؤال. طيب نأخذ مثال على الكلام هذا.
+
+383
+00:30:10,780 --> 00:30:14,000
+Find an equation for the plane through P0
+
+384
+00:30:14,000 --> 00:30:18,680
+perpendicular to M. يعني بدنا نوجد معادلة المستوى
+
+385
+00:30:18,680 --> 00:30:24,240
+اللي يمر بهذه النقطة وعمودي على هذا اللي هو مين؟ الـ
+
+386
+00:30:24,240 --> 00:30:29,200
+vector. الآن صار الموضوع سهل جداً اللي هو على طول
+
+387
+00:30:29,200 --> 
+
+445
+00:34:50,290 --> 00:34:50,450
+المستقيم المستقيم المستقيم المستقيم المستقيم
+
+446
+00:34:50,450 --> 00:34:54,810
+المستقيم المستوي مستوى آخر، بدنا معادلة الخط
+
+447
+00:34:54,810 --> 00:35:00,390
+المستقيم اللي هو اللي بتقاطع فيه مستويان، احنا بنعرف
+
+448
+00:35:00,390 --> 00:35:05,070
+أنه لو كان في عندنا مستويان غير متوازين، هيتقاطعا في
+
+449
+00:35:05,070 --> 00:35:09,230
+خط أو بيكونان منطبقين، طبعًا احنا ما بنأخذ
+
+450
+00:35:09,230 --> 00:35:13,610
+حالة الانطباق، هناخد حالة ايش؟ اللي هو بيتقاطع في خط
+
+451
+00:35:13,610 --> 00:35:19,320
+مستقيم، ونوجد معادلة الخط المستقيم تبع التقاطع، طيب
+
+452
+00:35:19,320 --> 00:35:21,980
+سهلة، أنا بعرف الـ plane الأول وبعرف الـ plane
+
+453
+00:35:21,980 --> 00:35:25,380
+الثاني، وبدي أوجد معادلة الخط المستقيم اللي بتقاطعان فيه
+
+454
+00:35:25,380 --> 00:35:29,220
+مدام بعرف معادلة الـ plane الأول، إذا بعرف
+
+455
+00:35:29,220 --> 00:35:33,700
+العمودي عليه N واحد، مدام بعرف الـ plane الثاني، إذا
+
+456
+00:35:33,700 --> 00:35:37,740
+بعرف معادلة الخط المستقيم العمودي عليه N اثنين، طيب
+
+457
+00:35:37,740 --> 00:35:42,340
+مدام N واحد عمودي على المستوى الأول، إذا عمودي على
+
+458
+00:35:42,340 --> 00:35:49,380
+كل خط مستقيم فيه، بما فيه خط التقاطع، similarly
+
+459
+00:35:49,380 --> 00:35:56,130
+بنفس الأسلوب، مدام أن اثنين عمودي على المستوى الثاني، إذا
+
+460
+00:35:56,130 --> 00:35:59,570
+عمودي على كل خط مستقيم فيه، إذا برضه عمودي على خط
+
+461
+00:35:59,570 --> 00:36:03,350
+التقاطع، لأن خط التقاطع موجود في المستويين، ولا ليش اسمه خط تقاطع؟
+
+462
+00:36:03,350 --> 00:36:06,770
+إذا صار عندي العمودي N واحد
+
+463
+00:36:06,770 --> 00:36:12,730
+والعمودي N اثنين، الاثنان عموديان على مين؟ على الخط
+
+464
+00:36:12,730 --> 00:36:19,920
+المستقيم اللي هو بيتقاطع فيه المستويان، يعني بمعنى
+
+465
+00:36:19,920 --> 00:36:24,460
+آخر، لو جيت أخدت الـ cross product N واحد N اثنين
+
+466
+00:36:24,460 --> 00:36:31,020
+هيكون عمودي على N واحد و N اثنين، يعني هذا المتجه
+
+467
+00:36:31,020 --> 00:36:35,800
+N واحد cross N اثنين،  يكون موازي للخط
+
+468
+00:36:35,800 --> 00:36:40,700
+المستقيم تبع التقاطع، يعني موازي لخط التقاطع، إذا
+
+469
+00:36:40,700 --> 00:36:45,380
+عرفنا خط التقاطع اللي بندور عليه، عرفنا فيكتور
+
+470
+00:36:45,380 --> 00:36:50,210
+موازي له، وبنعرف نقطة عليه، بنعرف معادلته زي ما عرفنا
+
+471
+00:36:50,210 --> 00:36:55,550
+قبل هيك، إذا الآن عشان نوجد، ناخد مثال عملي، عملي
+
+472
+00:36:55,550 --> 00:36:59,470
+find a vector parallel to the line of intersection
+
+473
+00:36:59,470 --> 00:37:05,090
+of the planes كذا and كذا، بده أول شيء يوجد هو
+
+474
+00:37:05,730 --> 00:37:09,170
+الذي أريد أن أعطيكم إياه على خطوات، أولًا يوجد الـ vector
+
+475
+00:37:09,170 --> 00:37:11,910
+ثم يوجد معادلة الخط المستقيم، الخط المستقيم يجب أن
+
+476
+00:37:11,910 --> 00:37:13,350
+نعرف أنه يوجد معادلة الخط المستقيم إذا وجدنا الـ
+
+477
+00:37:13,350 --> 00:37:17,010
+vector، ونعرف الـ V، فأخذنا أمثلة على إيجاد معادلة
+
+478
+00:37:17,010 --> 00:37:21,630
+الخط المستقيم، إذا الآن السؤال هذا هو find a vector
+
+479
+00:37:21,630 --> 00:37:26,310
+موازي parallel to the line of intersection of the
+
+480
+00:37:26,310 --> 00:37:30,630
+planes، يعني إذا كان لدي مستويان، فأنا أريد خط
+
+481
+00:37:30,630 --> 00:37:36,300
+مستقيم أو vector موازي لخط التقاطع بينهما، بنجيب
+
+482
+00:37:36,300 --> 00:37:40,120
+العمودي على الأول والعمودي على الثاني، ونعمل
+
+483
+00:37:40,120 --> 00:37:48,510
+cross product، موازي لخط التقاطع، vector موازي للخط
+
+484
+00:37:48,510 --> 00:37:52,370
+التقاطع اللي هو المطلوب، The line of intersection of
+
+485
+00:37:52,370 --> 00:37:55,530
+two planes is perpendicular to both planes' normal
+
+486
+00:37:55,530 --> 00:38:01,570
+vectors N1 and N2، كما قلنا فوق، كيف نوجدها؟ N1 هي
+
+487
+00:38:01,570 --> 00:38:05,750
+ثلاثة في اتجاه I، ناقص ستة في اتجاه J، ناقص اثنين في
+
+488
+00:38:05,750 --> 00:38:10,380
+اتجاه K، اللي هي معاملات من الـ X والـ Y والـ Z في
+
+489
+00:38:10,380 --> 00:38:15,320
+المعادلة، لبنان هنا، الآن اثنين بنبدأ بنفس الأسلوب
+
+490
+00:38:15,320 --> 00:38:19,320
+N اثنين بيساوي اثنين I زائد J ناقص اثنين K، هي N
+
+491
+00:38:19,320 --> 00:38:23,240
+اثنين، إذا الآن يعني نوجد العمودي على المستوى
+
+492
+00:38:23,240 --> 00:38:26,720
+الأول، والعمودي على المستوى الثاني، وجدنا العموديين
+
+493
+00:38:26,720 --> 00:38:29,660
+هذول، هيكون العموديان على خط التقاطع لأنه مشترك
+
+494
+00:38:29,660 --> 00:38:36,490
+بين المستويين، بكون عندي N1 cross N2 هو الـ vector الموازي
+
+495
+00:38:36,490 --> 00:38:40,290
+لخط التقاطع، لأنه بيكون عمودي على N1 و N2، زي خط
+
+496
+00:38:40,290 --> 00:38:44,640
+التقاطع، بصير عندي N واحد cross N اثنين، بيبقى اه جي
+
+497
+00:38:44,640 --> 00:38:48,920
+كي، وبنجيب ثلاثة ناقص ستة ناقص اثنين، زي ما اتعلمنا، و
+
+498
+00:38:48,920 --> 00:38:52,640
+اثنين واحد، وناقص اثنين، وبنفك هذا الـ determinant
+
+499
+00:38:52,640 --> 00:38:56,340
+بنفكه، هتلاقوه بيساوي أربعة عشر I زائد اثنين J زائد
+
+500
+00:38:56,340 --> 00:39:00,260
+خمسة عشر K، طبعًا بتعرفوا كيف نفكه، بنشطب هذا بنشطب
+
+501
+00:39:00,260 --> 00:39:03,040
+هذا، ناقص الستة في ناقص اثنين اللي هي عبارة عن
+
+502
+00:39:03,040 --> 00:39:06,380
+اثنا عشر، اثنا عشر ناقص ناقص اثنين يعني زيادة اثنين يعني
+
+503
+00:39:06,380 --> 00:39:11,200
+أربعة عشر I، والباقي بنفس الأسلوب، الآن هذا واحد
+
+504
+00:39:11,200 --> 00:39:15,460
+من الـ cross product، موازي لمين؟ لخط التقاطع
+
+505
+00:39:15,460 --> 00:39:21,580
+بينهما، لو ضربت هذا الـ vector في أي scalar، هيطلع
+
+506
+00:39:21,580 --> 00:39:25,080
+vector موازي لخط التقاطع، لأن كلهم ايش مالهم؟ كلهم
+
+507
+00:39:25,080 --> 00:39:28,720
+متوازيات، إذا أي non zero multiple of N1 cross N2 will be a vector parallel to the
+
+508
+00:39:28,720 --> 00:39:31,540
+line of intersection، الآن طلب مني الآن find
+
+509
+00:39:36,970 --> 00:39:40,430
+parametric equations for the line in which the
+
+511
+00:39:40,430 --> 00:39:46,530
+planes intersect، الآن هذول الخطين أوجدنا
+
+512
+00:39:46,530 --> 00:39:51,070
+... أوجدنا ... آسف، المستويين، أوجدنا vector في المثال
+
+513
+00:39:51,070 --> 00:39:58,210
+السابق موازي لخط... خطين، الآن بدنا الـ parametric
+
+514
+00:39:58,210 --> 00:40:02,530
+equations لخط التقاطع، بنقول سهل، خلاص في الـ example
+
+515
+00:40:02,530 --> 00:40:05,350
+اللي قبله قبله قبله قبله قبله قبله قبله قبله قبله
+
+516
+00:40:05,350 --> 00:40:09,730
+قبله قبله قبله قبله قبله قبله قبله
+
+517
+00:40:21,860 --> 00:40:27,460
+بدي اللي هو أوجد اللي هو نقطة مشتركة على خط
+
+518
+00:40:27,460 --> 00:40:31,620
+التقاطع، بنشوف كيف نوجد النقطة المشتركة، بعد ما نوجد
+
+519
+00:40:31,620 --> 00:40:35,240
+النقطة المشتركة على خط التقاطع، ايش بنسوي يا جماعة؟
+
+520
+00:40:35,240 --> 00:40:39,340
+مع الـ vector هذا، بنطبق قانوننا اللي قبل شوية، طيب
+
+521
+00:40:39,340 --> 00:40:40,000
+ده نشوف كده
+
+522
+00:40:47,860 --> 00:40:53,720
+عشان نجد نقطة على خط التقاطع، بنأخذ أي نقطة اللي
+
+523
+00:40:53,720 --> 00:41:00,040
+بتعود عن Z أو Y أو X اللي بدك إياها عشوائيًا، أسهل
+
+524
+00:41:00,040 --> 00:41:06,000
+شيء أن أنا أعوض عن Z بصفر أو Y بصفر أو X بصفر، عشان
+
+525
+00:41:06,000 --> 00:41:09,500
+نسهل ايش؟ الحساب، وأسهل لنا في التعويض، واللي بده غير
+
+526
+00:41:09,500 --> 00:41:12,800
+هيك، يعوض، عوض Z بواحد، Z بخمسة، Z بثلاثة، Z بأربعة، اللي
+
+527
+00:41:12,800 --> 00:41:16,800
+بده إياه، أو X بواحد، أو Y بواحد، اللي بده إياه، بنجي
+
+528
+00:41:16,800 --> 00:41:21,940
+الآن عوضنا عن Z بصفر، بيطلع عندي هذا ثلاثة X ناقص ستة
+
+529
+00:41:21,940 --> 00:41:28,240
+Y بيساوي خمسة عشر، واثنين X زائد Y بيساوي خمسة، الآن
+
+530
+00:41:28,240 --> 00:41:33,480
+احنا أوجدنا نقطة مشتركة، مشتركة
+
+531
+00:41:33,480 --> 00:41:38,020
+عليهما، إذا أنا أريد نقطة قاطعة، عشان أوجد نقطة قاطعة، بده
+
+532
+00:41:38,020 --> 00:41:44,580
+حل هذول المعادلتين، ماشي، بده حل هنا في نفس
+
+533
+00:41:44,580 --> 00:41:53,360
+الوقت، الآن لحل المعادلة، أنا بمعنى آخر، بيحل
+
+534
+00:41:53,360 --> 00:41:54,720
+المعادلة، أنا بمعنى آخر، بيحل المعادلة، أنا بمعنى
+
+535
+00:41:54,720 --> 00:41:56,520
+آخر، بيحل المعادلة، أنا بمعنى آخر، بيحل المعادلة
+
+536
+00:41:56,520 --> 00:41:58,680
+أنا بمعنى آخر، بيحل المعادلة، أنا بمعنى آخر، بيحل
+
+537
+00:41:58,680 --> 00:42:01,280
+المعادلة، أنا بمعنى آخر، بيحل المعادلة، أنا بمعنى
+
+538
+00:42:01,280 --> 00:42:03,780
+آخر، بيحل المعادلة، أنا بمعنى آخر، بيحل المعادلة
+
+539
+00:42:03,780 --> 00:42:05,560
+أنا بمعنى آخر، بيحل المعادلة، أنا بمعنى آخر، بيحل
+
+540
+00:42:05,560 --> 00:42:07,360
+المعادلة، أنا بمعنى آخر، بيحل المعادلة، أنا بمعنى
+
+541
+00:42:07,360 --> 00:42:11,560
+آخر، بهذا بنضربه في معامل الـ X اللي هنا، ليه 6X الـ
+
+542
+00:42:11,560 --> 00:42:17,200
+... الـ ... الزائد اللي هو 3Y بيساوي 15، بنحط تحت بعض
+
+543
+00:42:17,200 --> 00:42:21,520
+هذا معامل زي هذا، بنطرح من بعض، بتطلع قيمة Y
+
+544
+00:42:21,520 --> 00:42:25,780
+اللي هي قيمة Y، بنحلها بتطلع ايش؟ قيمة Y، لأن قيمة Y
+
+545
+00:42:25,780 --> 00:42:30,100
+اللي طلعت هنا، بنجيبها هنا، واحد ناقص واحد، برد بعوض عن
+
+546
+00:42:30,100 --> 00:42:34,680
+قيمة Y في إحدى المعادلتين، بيظهر عندي قيمة من؟ قيمة X
+
+547
+00:42:34,680 --> 00:42:37,660
+هكذا نحل المعادلات في ثالث إعدادي وثاني إعدادي
+
+548
+00:42:37,660 --> 00:42:44,180
+فبيصير عندي الحل، أوجدت قيمة X وقيمة Y على خط
+
+549
+00:42:44,180 --> 00:42:48,040
+التقاطع، وهذه قيمة Z اللي هي النقطة الثالثة، إذا
+
+550
+00:42:48,040 --> 00:42:52,060
+بكون هكذا أوجدت ثلاثة، وناقص واحد، وصفر، لأن حليت في
+
+551
+00:42:52,060 --> 00:42:56,060
+نفس الوقت، يعني أوجدت الحلول المشتركة، يعني أوجدت
+
+552
+00:42:56,060 --> 00:43:01,450
+النقاط الـ X والـ Y المشتركة على اللي هو الـ plane، أو جدت
+
+553
+00:43:01,450 --> 00:43:05,230
+بمعنى آخر، النقاط X و Y على خط التقاطع اللي بينهما
+
+554
+00:43:05,230 --> 00:43:10,410
+وكان عندي نقطة Zero، نقطة محددة اللي هي برضه هتكون
+
+555
+00:43:10,410 --> 00:43:16,450
+على خط التقاطع بين المستويين، إذا إيجاد خط التقاطع
+
+556
+00:43:16,450 --> 00:43:20,350
+بين المستويين، نقطة على خط التقاطع، هي هنا وضحناها
+
+557
+00:43:20,930 --> 00:43:26,170
+عندي نقطة معروفة، والـ vector المعروف الموازي اللي
+
+558
+00:43:26,170 --> 00:43:29,290
+هو خط التقاطع اللي أوجدناه في المثال السابق، بتيجي
+
+559
+00:43:29,290 --> 00:43:33,590
+الاثنان مع بعض، بيعملنا اللي هو الـ X بيساوي X naught
+
+560
+00:43:33,590 --> 00:43:40,640
+الثلاثة زائد اللي هي... اللي هي معامل الـ vector 14T، Y
+
+561
+00:43:40,640 --> 00:43:47,580
+بيساوي ناقص واحد Y naught زائد معامل اللي هي T في V1، V2
+
+562
+00:43:47,580 --> 00:43:55,280
+اللي هو 2T زائد اللي هو Z بيساوي 0 زائد T في V3
+
+563
+00:43:55,280 --> 00:43:59,240
+اللي هي 15 بتصير 15T، هذه بتصير الـ parametric
+
+564
+00:43:59,240 --> 00:44:01,280
+equations، الـ parametric equations for the line of the
+
+565
+00:44:01,280 --> 00:44:06,260
+intersection of the two planes، الآن بيقول لي find the
+
+566
+00:44:06,260 --> 00:44:09,900
+point where the line X بيساوي كذا، و Y بيساوي
+
+567
+00:44:09,900 --> 00:44:15,040
+كذا، و Z بيساوي كذا، intersects the plane اللي هو
+
+568
+00:44:15,040 --> 00:44:21,780
+معادلته معطاة، ايش بده؟ بده you did، اللي هي النقطة
+
+569
+00:44:21,780 --> 00:44:27,120
+المشتركة بين الخط المستقيم وبين مين؟ اللي هو الـ
+
+570
+00:44:27,120 --> 00:44:32,000
+plane، طبعًا الـ plane يا اما الخط المستقيم يا موازي للـ
+
+571
+00:44:32,000 --> 00:44:37,560
+plane، يا موازي للـ plane، يا بيقطع الـ plane في نقطة
+
+572
+00:44:37,560 --> 00:44:43,240
+يا بيقع عليه، الآن احنا اخذنا مثال اللي هو في
+
+573
+00:44:43,240 --> 00:44:46,900
+الواقع، المثال اللي اخذناه، فيه نقطة تقاطع واحدة مع
+
+574
+00:44:46,900 --> 00:44:51,980
+اللي لا موازي ولا بيقع عليه، لنشوف كيف نوجد نقطة
+
+575
+00:44:51,980 --> 00:44:57,580
+التقاطع بين خط و plane، الآن عشان النقطة، عشان اللي
+
+576
+00:44:57,580 --> 00:45:02,080
+هي النقطة، هذه بتقع على الخط المستقيم وتقع على الـ
+
+577
+00:45:02,080 --> 00:45:07,200
+plane، لازم تحقق معادلة الخط المستقيم وتحقق معادلة الـ
+
+578
+00:45:07,200 --> 00:45:12,460
+plane، فالآن النقطة تلقائيا 8 على 3، أو 2T ناقص 2، ناقص T
+
+579
+00:45:12,460 --> 00:45:20,580
+واحد، زي الـ T، عشان نوجد قيمة اللي هي النقطة هذه
+
+580
+00:45:20,580 --> 00:45:24,000
+اللي بتقع على الـ plane، لازم نعوض هذه في الـ plane
+
+581
+00:45:24,000 --> 00:45:28,970
+اللي هي بدل الـ T ونوجد قيمة T، قيمة T اللي بتحقق
+
+582
+00:45:28,970 --> 00:45:35,610
+المعادلة هذه، هي اللي بتخلي النقطة هذه تقع على
+
+583
+00:45:35,610 --> 00:45:40,830
+الـ plane، إذًا هذه النقطة بتعوضها هنا، يعني بتحقق
+
+584
+00:45:40,830 --> 00:45:46,910
+طرفي المعادلة، فبيصير الآن ثلاثة في X هنا اللي هي ثمانية على ثلاثة
+
+585
+00:45:46,910 --> 00:45:53,110
+زائد 2T زائد اثنين في Y هي ناقص 2T زائد ستة في
+
+586
+00:45:53,110 --> 00:45:54,810
+واحد زائد T بيساوي ستة
+
+587
+00:45:59,220 --> 00:46:06,400
+لازم أتحقق طرفي المعادلة عشان تكون على الـ plane، ثمانية
+
+588
+00:46:06,400 --> 00:46:13,960
+زي 6T، لأن ضربت، بتصبح ثمانية زي 6T، ناقص 4T زي 6، زي 6T، 6T
+
+589
+00:46:13,960 --> 00:46:18,900
+بيساوي 6، بسطتها، طلعت 8T ايش؟ بيساوي سالب ثمانية، طبعًا
+
+590
+00:46:18,900 --> 00:46:24,840
+بتعرفوا تبسطوها، 6T و 6T اللي هو عبارة عن 12T ناقص
+
+591
+00:46:24,840 --> 00:46:30,750
+4 طلعت 8T، و6 و8 و14 و6 و8 و14 و6 و8 و14 و6 و8 و14
+
+592
+00:46:30,750 --> 00:46:33,670
+و6 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+593
+00:46:33,670 --> 00:46:33,730
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+594
+00:46:33,730 --> 00:46:36,310
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+595
+00:46:36,310 --> 00:46:36,390
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+596
+00:46:42,140 --> 00:46:48,120
+والـ Y هذه، والـ Z هذه، بتكون هنا وتكون هنا في نفس
+
+597
+00:46:48,120 --> 00:46:51,780
+الوقت، يعني therefore the point where the line
+
+598
+00:46:51,780 --> 00:46:57,640
+intersects the plane is X و Y و Z اللي عندنا اللي
+
+599
+00:46:57,640 --> 00:47:02,580
+هي هذه، عند T بتساوي سالب واحد، اللي عبارة عن ثلثين
+
+600
+00:47:02,580 --> 00:47:07,220
+واثنين وصفر، ومش مصدقين؟ عوضوا هذه هنا هتلاقيها
+
+601
+00:47:07,220 --> 00:47:12,540
+بتحقق طرفي المعادلة، وعوضوها دي هنا هتلاقيها إحدى
+
+602
+00:47:12,540 --> 00:47:17,260
+نقاط اللي هي الـ line
+
+603
+00:47:17,260 --> 00:47:23,360
+اللي طبعًا أي قيمة للـ T نأخذها بتبقى X Y و Z اللي
+
+604
+00:47:23,360 --> 00:47:27,100
+بتطلع عبارة عن نقطة تقع على الخط المستقيم، وهذا
+
+605
+00:47:27,100 --> 00:47:32,120
+من ضمنها، طيب ديجينا اللي هو نقطة أخرى، الـ distance
+
+606
+00:47:32,120 --> 00:47:37,480
+from a point to a plane، الآن أوجدنا في الأول اللي
+
+607
+00:47:37,480 --> 00:47:41,860
+هو المسافة بين نقطة وبين خط مستقيم في اللي
+
+667
+00:51:59,570 --> 00:52:03,570
+ديش عامد هيك خد x بصفر و y بصفر و z بصير ايش
+
+668
+00:52:03,570 --> 00:52:07,210
+حتساوي واحد بقوله صح واحد ثاني اخذ ارقام ثانية زي
+
+669
+00:52:07,210 --> 00:52:11,780
+ما بدك المهم الارقام اللي بتاخد تتحقق  عند تعويضها
+
+670
+00:52:11,780 --> 00:52:16,560
+X و Y عشوائية تبقى الـ Z توجدها بالحل عشان تحقق
+
+671
+00:52:16,560 --> 00:52:22,460
+طرفي المعادلة إذا صارت لدي ال B موجودة و ال S
+
+672
+00:52:22,460 --> 00:52:26,400
+موجودة معطاة في السؤال بيصير لدي ال B S بساوي
+
+673
+00:52:26,400 --> 00:52:31,080
+اللي هو بي اس اللي هي واحد ناقص صفر في اتجاه ال I
+
+674
+00:52:31,080 --> 00:52:36,900
+واحد ناقص ثلاثة في اتجاه ال J، ثلاثة ناقص صفر في
+
+675
+00:52:36,900 --> 00:52:41,560
+اتجاه الـ K بيطلع عندي عبارة عن I ناقص اثنين J
+
+676
+00:52:41,560 --> 00:52:45,480
+زائد ثلاثة K هذا ال BS جهزته لأمور اللي أنا بدي
+
+677
+00:52:45,480 --> 00:52:50,080
+أوجد الـ BS ضد مين الـ N و بعدين بجسمه على مين
+
+678
+00:52:50,080 --> 00:52:53,240
+على ال magnitude لل N بكون هو ال distance المطلوبة
+
+679
+00:52:53,240 --> 00:53:00,380
+إذا ال N بقول ال magnitude لهذه ال N بي اس هي ضد
+
+680
+00:53:00,380 --> 00:53:04,960
+مين ال N مضروب في اللي هو ال magnitude واحد على ال
+
+681
+00:53:04,960 --> 00:53:09,660
+magnitude لل N ال N طوله قداش سبعة بيصير واحد على
+
+682
+00:53:09,660 --> 00:53:12,400
+سبعة في ال dot product بينهم اوجد ال dot product
+
+683
+00:53:12,400 --> 00:53:21,940
+اللي هي ثلاثة ناقص أربعة بطلع ناقص واحد زائد ثمانية عشر
+
+684
+00:53:21,940 --> 00:53:26,800
+اللي هو سبعة عشر سبعة عشر مضروب في مين على سبعة
+
+685
+00:53:26,800 --> 00:53:30,900
+إذا عملنا الدور التفرضات طالع عندي بالشكل اللي
+
+686
+00:53:30,900 --> 00:53:35,620
+أمامي اللي هو ثلاثة و مضروبة في سبعة بصير ثلاثة على
+
+687
+00:53:35,620 --> 00:53:39,060
+سبعة ناقص أربعة و مضروبة في سبعة على سبعة ناقص أربعة
+
+688
+00:53:39,060 --> 00:53:42,160
+على سبعة و ثلاثة في ستة بتمانية عشر مضروبة في سبعة بطلع
+
+689
+00:53:42,160 --> 00:53:46,690
+ثمانية عشر على سبعة هذه المسافة
+
+690
+00:53:46,690 --> 00:53:47,870
+المطلوبة
+
+691
+00:53:50,530 --> 00:53:55,650
+the angle between planes اللي هي آخر مقطع ان شاء
+
+692
+00:53:55,650 --> 00:54:01,610
+الله يا جماعة اللي هي الزاوية بين مين بين مستويين
+
+693
+00:54:01,610 --> 00:54:07,050
+الزاوية بين مستويين الان عندي نفترض ان المستويين
+
+694
+00:54:07,050 --> 00:54:11,790
+هي المستوى الأول وهي المستوى الثاني ومتقاطعين اللي
+
+695
+00:54:11,790 --> 00:54:16,240
+هو في خط مستقيم وعملوا زاوية بينهم الزاوية
+
+696
+00:54:16,240 --> 00:54:20,260
+اللي بين الاتنين بدنا نعرفها اللي هي عبارة عن
+
+697
+00:54:20,260 --> 00:54:25,040
+الزاوية الحادة بين الـ Normal للأول و Normal
+
+698
+00:54:25,040 --> 00:54:28,980
+للتاني يعني الزاوية الحادة بين ال Normal الأول و
+
+699
+00:54:28,980 --> 00:54:34,440
+Normal الثاني هي الزاوية اللي بنسميها بين مستويين
+
+700
+00:54:43,540 --> 00:54:48,780
+الزاوية بين مستويين هي الزاوية الحادة بين
+
+701
+00:54:48,780 --> 00:54:54,720
+العموديين عليهما الان بناء على هذا التعريف بدنا نوجد
+
+702
+00:54:54,720 --> 00:54:58,620
+find the angle between the planes 3x-6y-2z  بالساعة
+
+703
+00:54:58,620 --> 00:55:04,640
+15 and the plane 2x+y-2z بالساعة 5 الزاوية بينهم
+
+704
+00:55:04,640 --> 00:55:08,560
+إذا عشان نوجدها بدنا نوجد normal للأول سهلة الـ N1
+
+705
+00:55:08,560 --> 00:55:12,580
+بيساوي 3I ناقص 6J ناقص 2K بتعرف 6 بتعرفه عن غيره
+
+706
+00:55:12,580 --> 00:55:14,980
+طبعا المعاملات اللي هي ال X في هذه المعادلة
+
+707
+00:55:14,980 --> 00:55:17,940
+ومعاملات ال Y ومعامل ال Z بتمثل ال Normal N
+
+708
+00:55:17,940 --> 00:55:24,520
+الـ N2 بيساوي 2I زائد J ناقص 2K هدول الان two vectors
+
+709
+00:55:24,520 --> 00:55:29,970
+الزاوية اللي بينهم هي عبارة عن مين عن اللي هو
+
+710
+00:55:29,970 --> 00:55:34,050
+الزاوية بين المستويين واحنا بنعرف نوجد الزاوية
+
+711
+00:55:34,050 --> 00:55:36,730
+بينهم اللي هو ايش بتساوي اخذ نهج باللي هيك
+
+712
+00:55:36,730 --> 00:55:38,890
+الزاوية بين المستويين هي عبارة عن θ بساوي cosine
+
+713
+00:55:38,890 --> 00:55:43,070
+inverse لل N1 dot N2 على magnitude لل N1 في
+
+714
+00:55:43,070 --> 00:55:46,910
+magnitude لل N2 عارفين ليش؟ لأن احنا بنعرف ال N1
+
+715
+00:55:46,910 --> 00:55:51,650
+.N2 بيساوي اللي هو ال magnitude لهذه في ال
+
+716
+00:55:51,650 --> 00:55:54,610
+magnitude لهذه في cosine الزاوية اللي بينهم مش هيك
+
+717
+00:55:54,610 --> 00:55:58,990
+تعرفي  dot product n1 . n2 بيساوي اللي هو cos θ في
+
+718
+00:55:58,990 --> 00:56:01,250
+magnitude هذا في magnitude هذا الـ θ بيساوي
+
+719
+00:56:01,250 --> 00:56:05,370
+cos انفرسلها واخذنا القانون قبل هيك لأن n1 . n2
+
+720
+00:56:05,370 --> 00:56:09,410
+من عرف نضرب على طول ثلاثة في اثنين بستة ستة ناقص
+
+721
+00:56:09,410 --> 00:56:13,210
+ستة اللي هي بروح مع بعض ناقص اثنين في ناقص اثنين
+
+722
+00:56:13,210 --> 00:56:17,150
+بيطلع أربعة إذا dot product بين n1 وn2 أربعة
+
+723
+00:56:17,150 --> 00:56:22,790
+على ال magnitude للأول جذر التربيعي تسعة زائد ستة
+
+724
+00:56:26,260 --> 00:56:31,000
+و 36  و 4  يعني  49 و ال magnitude لهذا يعني طوله
+
+725
+00:56:31,000 --> 00:56:35,580
+هذا 7 عملناها قبل شوية وهذا أربعة وواحد خمسة وأربعة تسعة طوله قداش جذر التسعة اللي هو ثلاثة إذا
+
+726
+00:56:35,580 --> 00:56:39,340
+طول هذا ثلاثة وطول هذا سبعة مع بعض حصل ضرب هنا
+
+727
+00:56:39,340 --> 00:56:43,480
+اللي هو واحد وعشرون فبصير θ بساوية cosine
+
+728
+00:56:43,480 --> 00:56:46,600
+inverse أربعة على واحد وعشرون اللي هي تقريبا
+
+729
+00:56:46,600 --> 00:56:48,820
+بساوي هذا الرقم يعني صار عندي الآن الزاوية بين
+
+730
+00:56:48,820 --> 00:56:52,560
+المستويين هذه هي الزاوية اللي عندنا وبهيك بنكون
+
+731
+00:56:52,560 --> 00:56:57,870
+خلصنا المحاضرة وبارك الله فيكم وإلى لقاء قريب
+
+732
+00:56:57,870 --> 00:57:01,110
+السلام عليكم ورحمة الله وبركاته
+
+733
+00:57:01,110 --> 00:57:03,070
+ 
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..28f82da02235706613a3781614fa5e672475d9d5
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY_postprocess.srt
@@ -0,0 +1,2932 @@
+1
+00:00:00,660 --> 00:00:06,300
+بسم الله الرحمن الرحيم هذه هي المحاضرة الخامسة
+
+2
+00:00:06,300 --> 00:00:13,380
+مساق تفاضل وتقام الجيم بعنوان lines and planes in
+
+3
+00:00:13,380 --> 00:00:17,920
+space اليوم بدنا نعرف كيف نوجد معادلة الخط
+
+4
+00:00:17,920 --> 00:00:22,960
+المستقيم ومعادلة المستوى في ال space ومتعلقاتها
+
+5
+00:00:23,670 --> 00:00:28,230
+الان احنا .. ده طبعا في section 12-5 احنا سابقا في
+
+6
+00:00:28,230 --> 00:00:33,250
+ال .. في ال plane عرفنا معادلة الخط المستقيم و كنا
+
+7
+00:00:33,250 --> 00:00:37,990
+جوا معادلة الخط المستقيم تحدد اللي هو بمعرفة mail
+
+8
+00:00:37,990 --> 00:00:43,210
+و نقطة او نقطين الان الكلام هنا مشابه الان بدنا
+
+9
+00:00:43,210 --> 00:00:48,290
+نعرف ايش .. كيف نوجد معادلة الخط المستقيم linesو
+
+10
+00:00:48,290 --> 00:00:53,390
+بعدين نعرف معادلة الخط المستقيم المحدود يعني بمعنى
+
+11
+00:00:53,390 --> 00:00:57,930
+آخر القطع المستقيمة line segment بدنا نعرف اللي هو
+
+12
+00:00:57,930 --> 00:01:02,610
+معدلتين الآن زي ما قلنا معدلة الخط المستقيم
+
+13
+00:01:02,610 --> 00:01:09,290
+هنحددها اللي هو من خلال معرفة نقطين أو ميل و نقطة
+
+14
+00:01:09,290 --> 00:01:13,990
+أو ما يشابه هنا ايش اللي بشابه؟ كلينا نشوف مع بعض
+
+15
+00:01:14,330 --> 00:01:19,950
+الان لو عرفنا .. طبعا عندنا خط مستقيم اللى بدنا
+
+16
+00:01:19,950 --> 00:01:25,770
+نعرف معدلته لو عرفنا نقطة عليه وعرفنا اللى هو
+
+17
+00:01:25,770 --> 00:01:31,010
+vector موازي إله يعني بمعنى آخر عرفنا وكأنه ميل
+
+18
+00:01:31,010 --> 00:01:35,890
+إله ميله ونقطة فنشوف كيف بدنا نحدد معدلة الخط
+
+19
+00:01:35,890 --> 00:01:41,300
+المستقيم في هذه الحالة بمعرفة نقطة على الخطوإيش و
+
+20
+00:01:41,300 --> 00:01:45,600
+vector موازي له شوفوا يا جماعة الان ناخد نقطة
+
+21
+00:01:45,600 --> 00:01:50,480
+عشوائية اسمها الهي x,y,z,b هذه اللي هي بتتحدد
+
+22
+00:01:50,480 --> 00:01:54,820
+النهوية اللي هو الخط يعني بدللتها بتيجي معادلة
+
+23
+00:01:54,820 --> 00:01:59,500
+الخط المستقيم الان ال vector الموجود عندنا نفترض
+
+24
+00:01:59,500 --> 00:02:04,080
+ان اسمه V بسوا V1 في اتجاه I و V2 في اتجاه J زي V3
+
+25
+00:02:04,080 --> 00:02:10,430
+في اتجاه ال Kالان النقطة المعروفة نسميها بي نوت
+
+26
+00:02:10,430 --> 00:02:13,910
+اكس نوت واي نوت وزد و نوت بدنا نعرف معدلة الخط
+
+27
+00:02:13,910 --> 00:02:20,630
+المستقيم شوفوا الان هذا ال vector موازي لمين لالخط
+
+28
+00:02:20,630 --> 00:02:24,010
+المستقيم اللى بدنا نعرف معدلته اخدنا النقطة
+
+29
+00:02:24,010 --> 00:02:27,890
+العشوائية بي اكس واي وزد صار عندى بي نوت بي هو
+
+30
+00:02:27,890 --> 00:02:32,490
+عبارة عن vector على الخط موازي لهذا ال vector
+
+31
+00:02:32,490 --> 00:02:37,640
+المعروفالان اكيد ال vector هذا هو نفس اتجاه ال
+
+32
+00:02:37,640 --> 00:02:41,340
+vector هذا بس بختلف عنه ايه في ايه في الطول يعني
+
+33
+00:02:41,340 --> 00:02:44,580
+بمعنى اخر و بتغيروا الطول و بتغيروا نقطة مين x و y
+
+34
+00:02:44,580 --> 00:02:49,180
+و z عشان هيك بنيجي بنكتب بنستخدم parameter بنقول
+
+35
+00:02:49,180 --> 00:02:53,960
+ان هذا ال vector bb0هيكون هو هذا الـ vector بس
+
+36
+00:02:53,960 --> 00:02:58,340
+مضروب في some scalar T يعني اللي هو ال .. طبعا T
+
+37
+00:02:58,340 --> 00:03:02,140
+اللي هي في هذه الحالة اللي .. اللي الآن عندي عبارة
+
+38
+00:03:02,140 --> 00:03:07,080
+عن real number اتجاه P0P ال .. ال vector P0P هو
+
+39
+00:03:07,080 --> 00:03:10,720
+نفسه ال vector V بس مضروب في T لأنه جلنا باختلف عن
+
+40
+00:03:10,720 --> 00:03:15,260
+بعض بس في إيش؟ في الطول أما موازئله يعني يا في
+
+41
+00:03:15,260 --> 00:03:19,060
+اتجاهه بالشكل هذا يا أما في اتجاهه بالشكل العكسي
+
+42
+00:03:19,440 --> 00:03:24,280
+بنوت بي بساوية TV هي اللى بتمثل معادلة الخط
+
+43
+00:03:24,280 --> 00:03:29,440
+المستقيم اللى عندنا و T عبارة عن مين عن أي عدد
+
+44
+00:03:29,440 --> 00:03:32,620
+بنقص infinity و infinity يعني real number
+
+45
+00:03:35,270 --> 00:03:41,930
+بساوة TV بدي أضعها في صورة جميلة نعرف كيف نجد
+
+46
+00:03:41,930 --> 00:03:47,490
+المعادلة بمجرد معرفة اللي هي الأرقام اللي بتمثلها
+
+47
+00:03:47,490 --> 00:03:50,330
+V والأرقام اللي بتمثلها P نوت ونعرف معادلة الخط
+
+48
+00:03:50,330 --> 00:03:54,500
+المستقيم ونشوف الأمثلة نشوف أشي اللي بقولهبنوت بي
+
+49
+00:03:54,500 --> 00:03:58,580
+يا جماعة هو عبارة عن x نقص x note في اتجاه ال I y
+
+50
+00:03:58,580 --> 00:04:02,580
+minus y note في اتجاه ال J z minus z note في اتجاه
+
+51
+00:04:02,580 --> 00:04:06,960
+ال K هذا مين هو بي نوت في بي إذا x minus x note في
+
+52
+00:04:06,960 --> 00:04:10,520
+اتجاه ال I زائد y minus y note في اتجاه ال J زائد
+
+53
+00:04:10,520 --> 00:04:14,460
+z minus z note في اتجاه ال K هذا اللي بي نوت بي
+
+54
+00:04:15,870 --> 00:04:22,010
+بساوي T هو البراميتر المتغير اللي بيحدد لمكان
+
+55
+00:04:22,010 --> 00:04:26,450
+النقطة بنضربه في T بيحدد النقطة هنا و هنا و هنا
+
+56
+00:04:26,450 --> 00:04:32,010
+حسب قيمة T بتكون تحدد النقطة عندك إذا صارت T هي
+
+57
+00:04:32,010 --> 00:04:37,470
+الان في V1 اتجاه I زي V2 اتجاه J زي V3 اتجاه K هذه
+
+58
+00:04:37,470 --> 00:04:42,230
+الآن اللي هي بنقدر نكتبها على صورة اللي هو بيجي
+
+59
+00:04:42,230 --> 00:04:46,740
+بننقل X0 اتجاه Iالتي هي المعروفة العدد y note في
+
+60
+00:04:46,740 --> 00:04:48,980
+اتجاه ال j اللي هو العدد z note في اتجاه ال k
+
+61
+00:04:48,980 --> 00:04:54,220
+نجلبها مع V1 اللي معروفة و V2 و V3 و T فبصير X
+
+62
+00:04:54,220 --> 00:04:57,600
+زائد Y زائد Z واحد في اتجاه ال I في اتجاه ال J في
+
+63
+00:04:57,600 --> 00:05:01,560
+اتجاه ال K بتساوي X note اللي نجلناها هذه مع ال
+
+64
+00:05:01,560 --> 00:05:07,920
+TV1 A هي TV1و Y note مع التي هو TV2 و Z note في
+
+65
+00:05:07,920 --> 00:05:14,100
+اتجاه اليمين التي هي في اتجاه TV3 في اتجاه ال K
+
+66
+00:05:14,100 --> 00:05:19,880
+هذه الآن بتمثل لمعادلة الخط المستقيم الآن هذه او
+
+67
+00:05:19,880 --> 00:05:25,330
+هذهمعادلة الخط المستقيم أو اختصارا هذه للي على طول
+
+68
+00:05:25,330 --> 00:05:29,830
+بدوش يحفظوا باستنتج على طول من هذه المعادلة انشوف
+
+69
+00:05:29,830 --> 00:05:33,550
+الآن انلخص الحديث لو واحد تاني جالي اسمعه انا بديش
+
+70
+00:05:33,550 --> 00:05:36,670
+اشتغلها زيكي بدي اشتغلها على صورة vectors قولتله
+
+71
+00:05:36,670 --> 00:05:40,570
+مافيش مشكلة بيجي بقوللي بدي اسمي ال vector هذا من
+
+72
+00:05:40,570 --> 00:05:44,500
+ها اللي هان هي ال vector هذا هي من ها اللي هانبدي
+
+73
+00:05:44,500 --> 00:05:50,900
+أسميه هذا اللي هو عبارة عن R هذا من هالهال من نقطة
+
+74
+00:05:50,900 --> 00:05:55,080
+الأصل لهذا هذا ال position vector لمين لل X,Y,Z
+
+75
+00:05:55,080 --> 00:05:58,440
+اللي هو في الواقع اللي هي اللي بتحدد لقيمة X,Y,Z
+
+76
+00:05:58,440 --> 00:06:03,720
+بنسميها R of T اللي هو ال vector بنسميه هذا R of T
+
+77
+00:06:03,720 --> 00:06:08,420
+اللي هو من عند ال zero لعند ال B الان نقطة الأصل
+
+78
+00:06:08,420 --> 00:06:11,370
+ال zero بقصد فيهاالان و هذا ال vector من هنا الى
+
+79
+00:06:11,370 --> 00:06:13,950
+هنا بدي تسميه R note اللي هو اللي بتكون .. اللي
+
+80
+00:06:13,950 --> 00:06:18,490
+بيكون معروف ليش معروف؟ لأنه B note معروفة الان هذا
+
+81
+00:06:18,490 --> 00:06:22,490
+ال vector B note بيه هو اللي بسميه مين اللي هو T
+
+82
+00:06:22,490 --> 00:06:26,410
+في V اللي اتفجنا عليه فالان المعادلة ممكن نكتبها
+
+83
+00:06:26,410 --> 00:06:30,330
+على الصورة التالية على صورة vectors اللي هو معادلة
+
+84
+00:06:30,330 --> 00:06:35,730
+الخط المستقيم على صورة vectors ايش هوR of T وهو
+
+85
+00:06:35,730 --> 00:06:42,370
+المجهول اللي عند X و Y و Z المتغيرة R of T بساوي
+
+86
+00:06:42,370 --> 00:06:45,910
+هذا ال vector زائد هذا ال vector بتعرفوا ان هذا ال
+
+87
+00:06:45,910 --> 00:06:48,830
+vector زائد هذا ال vector بساوي اللي بسكر المثلث
+
+88
+00:06:48,830 --> 00:06:55,650
+لأن R of T هذا بساوي R not of T زائد T في V وهذه
+
+89
+00:06:55,650 --> 00:07:00,350
+هي ال vector equation for the line through B not
+
+90
+00:07:00,350 --> 00:07:05,690
+parallel to Vالـ vector equation أو معادلة الخط
+
+91
+00:07:05,690 --> 00:07:11,290
+المستقيم بواسطة أو بدلالة الـ vectors
+
+92
+00:07:14,420 --> 00:07:18,680
+بمعرفة النقطة P0 و V بتساوي R of T زي ما قلنا
+
+93
+00:07:18,680 --> 00:07:24,220
+بساوي R0 of T هي R0 طبعا هتطلع لنا مباشرة هي عدد
+
+94
+00:07:24,220 --> 00:07:30,320
+زائد مين T في V هذي As of R0 لأنها بدلالة باسمين
+
+95
+00:07:30,320 --> 00:07:35,300
+P0 أما R of D هتكون بدلالة اتجاه TWhere R is the
+
+96
+00:07:35,300 --> 00:07:40,680
+position vector of a point B on L and R not is the
+
+97
+00:07:40,680 --> 00:07:44,800
+position vector of mean B not زي ما حكينا الان هذه
+
+98
+00:07:44,800 --> 00:07:46,980
+اللي هي ال vector equation for a line ال
+
+99
+00:07:46,980 --> 00:07:49,480
+parametric equations for a line اللي عملناها قبل
+
+100
+00:07:49,480 --> 00:07:56,260
+بشويةالتي هي الـ Standard Parameterization of Line
+
+101
+00:07:56,260 --> 00:08:02,000
+Through P0 Parallel to V S يعني هذه معادلة الخط
+
+102
+00:08:02,000 --> 00:08:08,060
+المستقيم اللي هي إيه شمالة as a parametric
+
+103
+00:08:08,060 --> 00:08:12,220
+equation كاللي هو معادلة برامترية اللي يعني بدلالة
+
+104
+00:08:12,220 --> 00:08:18,090
+T ل vector بمرغل نقطة P0بوازي الـ vector V التي هي
+
+105
+00:08:18,090 --> 00:08:22,590
+عبارة عن X نقص X نوت زي T في V1 و Y نقص Y نوت زي T
+
+106
+00:08:22,590 --> 00:08:26,830
+في V2 و Z نقص Z نوت زي T في V3 و T element of R
+
+107
+00:08:26,830 --> 00:08:29,690
+هذي من وين جيبناها؟ جيبناها يا جماعة من هنا سمعت
+
+108
+00:08:29,690 --> 00:08:32,410
+زي حد بيقول لي من وين جيبناها هي X نقص X نوت
+
+109
+00:08:32,410 --> 00:08:37,530
+بتساوي T في V1 يعني X بتساوي X نوت زي T في V1 هذه
+
+110
+00:08:37,530 --> 00:08:43,020
+معاملات IY minus my node بيسوي TV2 معاملات J يعني
+
+111
+00:08:43,020 --> 00:08:48,700
+Y بيسوي Y node زي TV2 و Z بيسوي Z node زي TV3 هي
+
+112
+00:08:48,700 --> 00:08:52,700
+اللي هي ال parameter standard parameter parametric
+
+113
+00:08:52,700 --> 00:08:57,900
+equation أو equations for the line through the B
+
+114
+00:08:57,900 --> 00:09:03,500
+node and parallel to mean to Vالان هذه هي المعادلة
+
+115
+00:09:03,500 --> 00:09:09,200
+التي تبع الخط المستقيم الأكثر استخداما و أكتر
+
+116
+00:09:09,200 --> 00:09:13,580
+شيوعا ناخد مثال نجد الـ Parametric equations for
+
+117
+00:09:13,580 --> 00:09:18,240
+the line ناقص 2 و 0 و 4 نجد ال Parametric
+
+118
+00:09:18,240 --> 00:09:25,360
+equations لخط المستقيمالخط المستقيم اللى بمر
+
+119
+00:09:25,360 --> 00:09:29,800
+بالنقطة نقص اتنين و سفر و اربعة و يوازي المتجه
+
+120
+00:09:29,800 --> 00:09:35,460
+اللى امامنا هذا ماشي الحال هاي عندى الان المتجه هي
+
+121
+00:09:35,460 --> 00:09:39,680
+رسم لياه و هي النقطة اللى ماخدها عليه اللى هي نقص
+
+122
+00:09:39,680 --> 00:09:43,040
+اتنين و سفر و اربعة و بدو يوجد معدلة الخط المستقيم
+
+123
+00:09:43,040 --> 00:09:48,220
+اللى هذا اللى بوازي اللى هو المتجه هذا و بمر
+
+124
+00:09:48,220 --> 00:09:53,370
+بالنقطة Bالان تعويض مباشر ما هي عبارة عن X
+
+125
+00:09:53,370 --> 00:10:00,430
+بالساوية X0 زائد T في V ال V معروفة V1 هيها 2 و V2
+
+126
+00:10:00,430 --> 00:10:05,650
+هي 4 و V3 هي مين ناقص 2 وهذه X0 و Y0 و Z0 هي
+
+127
+00:10:05,650 --> 00:10:09,390
+المعروفات هي بالساوية تعويض X بالساوية X0 ناقص 2
+
+128
+00:10:09,390 --> 00:10:14,590
+زائد 2 في T و Y بالساوية Y0
+
+129
+00:10:17,220 --> 00:10:26,500
+T في V 4T V2 و Z بالساوية و Z نقطة 4-2 في T هذه
+
+130
+00:10:26,500 --> 00:10:30,360
+الآن عبارة عن معادلة الخط المستقيم اللي بمر
+
+131
+00:10:30,360 --> 00:10:34,400
+بالنقطة هذه و يوازن والتجاه اللي أمامنا و الكلام
+
+132
+00:10:34,400 --> 00:10:40,770
+زي ما شوفته سهلزي زمان برضه لو كان عنده نقطتين
+
+133
+00:10:40,770 --> 00:10:44,550
+نعرف معادلة الخط المستقيم برضه هنا في الفراغ نفس
+
+134
+00:10:44,550 --> 00:10:47,470
+الشيء لو كان عنده نقطتين بنعرف معادلة الخط
+
+135
+00:10:47,470 --> 00:10:50,830
+المستقيم و أكيد بعضكم قال اه طب ما هو احنا بنجيب
+
+136
+00:10:50,830 --> 00:10:55,910
+المتجهة اللي بوازي المتجهة اللي هو اللي بيحمل P و
+
+137
+00:10:55,910 --> 00:11:00,690
+Q اللي هو PQ او QP زمان بالذات هو بوازي أصلا الخط
+
+138
+00:11:00,690 --> 00:11:03,770
+المستقيم اللي بمر بالنقطتين هذين وهذا نقطة من
+
+139
+00:11:03,770 --> 00:11:08,480
+النقطتين فبصير معروف اللي هو متجهاللي هو جوازي
+
+140
+00:11:08,480 --> 00:11:12,160
+اللي هو في الواقع بيحمل و نفس الاشي نقطة معروفة
+
+141
+00:11:12,160 --> 00:11:15,840
+نعرف اللي قاعدة دلوقت نشوف كيف find a parametric
+
+142
+00:11:15,840 --> 00:11:19,320
+equation for the line through كده and كده بمعرفة
+
+143
+00:11:19,320 --> 00:11:23,000
+نقطين لأن ده vector bq أول اشي ده نوجد ال vector
+
+144
+00:11:23,000 --> 00:11:28,190
+bq اللي هو عبارة عنواحد قال بدا وجد QB برضه صح
+
+145
+00:11:28,190 --> 00:11:33,510
+واتنين بتجيب ان معادلتين تظهر انها مختلفة لكن
+
+146
+00:11:33,510 --> 00:11:39,650
+بتجيب نفس القط لإنه هتجيب لك نفس النقاط اكيو ناقص
+
+147
+00:11:39,650 --> 00:11:44,810
+بي اللي هي واحد ناقص ناقص تلاتة ناقص واحد ناقص
+
+148
+00:11:44,810 --> 00:11:45,410
+اتنين
+
+149
+00:12:00,980 --> 00:12:05,440
+هذا الـ vector is parallel to line and equation 3
+
+150
+00:12:06,470 --> 00:12:11,150
+with x0 و y0 و z0 نقص ثلاثة نقص اتنين بيعطينا
+
+151
+00:12:11,150 --> 00:12:13,250
+equation تلاتة اللي بتحكي عن equations ثلاثة اللي
+
+152
+00:12:13,250 --> 00:12:16,410
+هي قانون اللي هو ال parameter equations ده الخط
+
+153
+00:12:16,410 --> 00:12:21,150
+المستقيم اللي قبل شوية طيب الآن عرفنا المتجه
+
+154
+00:12:21,150 --> 00:12:27,370
+الموازي او الحامل لل line و بالاضافة لنقطة على ال
+
+155
+00:12:27,370 --> 00:12:29,490
+line واحد يقولي بديش هذا النقطة بتاخده نقطة تانية
+
+156
+00:12:29,490 --> 00:12:33,630
+بقوله اه بالفعلو بيطلع نفس ال equations بيطلع نقاط
+
+157
+00:12:33,630 --> 00:12:37,810
+تكون في شكل المختلفات لكن بيجيب نفس الخط يعني نفس
+
+158
+00:12:37,810 --> 00:12:42,670
+النقاط اللي على الخط طيب X بيساوي X نقطة اللي هي
+
+159
+00:12:42,670 --> 00:12:50,290
+نقص تلاتة زائد أربعة V في T Y بتساوي Y نقطة اللي
+
+160
+00:12:50,290 --> 00:12:55,800
+هو اتنينناقص تلاتة T فاهمين عليها Z بتساوي ناقص
+
+161
+00:12:55,800 --> 00:13:00,820
+تلاتة التي هي Z note زائد سبعة T هذه الآن هي ال
+
+162
+00:13:00,820 --> 00:13:05,040
+Parametric Equations للخط المستقيم وحتين قال
+
+163
+00:13:05,040 --> 00:13:08,780
+اسمعوا بديش انا بدي اختار اللي هي ال Q بدل ال mean
+
+164
+00:13:08,780 --> 00:13:12,580
+بدل ال V يعني بدي اخد ال base احنا اخدنا النقطة
+
+165
+00:13:12,580 --> 00:13:17,340
+اللي انطلق بيهوالنهاية Q فانا باخد نقطة انطلاق Q
+
+166
+00:13:17,340 --> 00:13:21,960
+واخد بيه بنفع بيصير الان اللي هو ال equation اللي
+
+167
+00:13:21,960 --> 00:13:27,380
+هي او اقال حتى اسمعوا مش شيكة كمان بتاخد انا اللي
+
+168
+00:13:27,380 --> 00:13:34,520
+بدي أسويه بتاخد اللي هو عندي النقطة ال B بتاخد ال
+
+169
+00:13:34,520 --> 00:13:38,250
+Q ليش استعمل ال Q تبعتكم بالمرةالنقص تلاتة او
+
+170
+00:13:38,250 --> 00:13:39,570
+اتنين او اكتر او اكتر اكتر اكتر اكتر اكتر اكتر
+
+171
+00:13:39,570 --> 00:13:42,070
+اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر
+
+172
+00:13:42,070 --> 00:13:44,690
+اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر
+
+173
+00:13:44,690 --> 00:13:44,710
+اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر
+
+174
+00:13:44,710 --> 00:13:47,450
+اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر
+
+175
+00:13:47,450 --> 00:13:55,190
+اكتر اكتر اكتر
+
+176
+00:13:55,190 --> 00:14:01,510
+اكتر اك
+
+177
+00:14:02,200 --> 00:14:05,920
+اللي هي ال equations بيجيب نفس الخط هذا ايه الدليل
+
+178
+00:14:05,920 --> 00:14:10,660
+خد مثلا هنا T بصفر بتطلع عندك النقطة واحد وناقص
+
+179
+00:14:10,660 --> 00:14:16,660
+واحد واربعة هي على الخط طيب ان انا لو اخدت T بواحد
+
+180
+00:14:16,660 --> 00:14:22,770
+هنا تيب واحد بيصير ناقص تلاتة واربعة يعني واحدوT
+
+181
+00:14:22,770 --> 00:14:29,970
+بواحد تظهر تقريبا تقريبا تقريبا تقريبا تقريبا
+
+182
+00:14:29,970 --> 00:14:31,330
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+183
+00:14:31,330 --> 00:14:32,930
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+184
+00:14:32,930 --> 00:14:35,870
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+185
+00:14:35,870 --> 00:14:41,030
+تقريبا تقريبا
+
+186
+00:14:41,030 --> 00:14:47,710
+تقريبا
+
+187
+00:14:48,890 --> 00:14:52,890
+ومش هيك هتلاقي مادام المقاطين و هذا خط و هذا خط
+
+188
+00:14:52,890 --> 00:14:57,820
+هيكون كل اللي بينهم على الخطين يعنياللي بده يقول
+
+189
+00:14:57,820 --> 00:15:03,800
+بي كيو او كيو بي استعملتها صح لو استخدمت النقطة بي
+
+190
+00:15:03,800 --> 00:15:07,520
+as a base point يعني نقطة و متجهة او كيو as a base
+
+191
+00:15:07,520 --> 00:15:13,340
+point زي ما شوفنا برضه بطلع اللي هو معادلات تعبر
+
+192
+00:15:13,340 --> 00:15:21,460
+عن الخط المستقيم الذي يحمل النقطين بي و كيو طيب
+
+193
+00:15:21,460 --> 00:15:28,820
+ناخد الآن واحد قال لانا بدي اشكرأنا بدي بس تعملي
+
+194
+00:15:28,820 --> 00:15:32,260
+parameterize the line segment joining the point B
+
+195
+00:15:32,260 --> 00:15:38,020
+and Q يعني بس بدي اللي هو معادلة الخط المستقيم لأ
+
+196
+00:15:38,020 --> 00:15:42,900
+بدي معادلة القطعة المستقيمة اللي بس من Q لعندي
+
+197
+00:15:42,900 --> 00:15:47,460
+لعندي بفكر بفكر بقول الله يعني بت cache ال قيم
+
+198
+00:15:47,460 --> 00:15:53,060
+اللي هناولا القيم اللي هنا يعني بتكاشل على الخط من
+
+199
+00:15:53,060 --> 00:15:57,360
+هنا ولا خط من هنا يعني بدك بمعنى أخر تحدد لقيم ال
+
+200
+00:15:57,360 --> 00:16:01,920
+T منها ال T اللي بتتحدد تجيبلي بس المطاط اللي من
+
+201
+00:16:01,920 --> 00:16:08,260
+عند V لعند B لعند Q ونقوله فش مشكلة المعادلات الخط
+
+202
+00:16:08,260 --> 00:16:12,240
+هي هي لإن هو جزء من الخط هذا إذا أنا from example
+
+203
+00:16:12,240 --> 00:16:16,560
+2 above the equations for the line through Band qr
+
+204
+00:16:16,560 --> 00:16:20,220
+هدولة معادلة مين اللى وجدناها قبل شوية معادلة الخط
+
+205
+00:16:20,220 --> 00:16:25,720
+المستقيم لكن هو بيقوللي انا يا جماعة انا بدي هدولة
+
+206
+00:16:25,720 --> 00:16:30,280
+النقاط x و y و z مايتعدنش النقطة هذه والنقطة هذه
+
+207
+00:16:30,280 --> 00:16:35,640
+يعني الان بدي بس اللى هو ال T اللى بتعبرلي بين q و
+
+208
+00:16:35,640 --> 00:16:42,000
+بين bالنقطة العامة اللي هان صارت على الخط هي x,y,z
+
+209
+00:16:42,000 --> 00:16:46,540
+نقص ثلاثة ثم اربعة ت ثم اتنين نقص ثلاثة ت ثم نقص
+
+210
+00:16:46,540 --> 00:16:52,530
+ثلاثة او سبعة تالان لو جيت اخدت لحظة التالي لو
+
+211
+00:16:52,530 --> 00:16:57,530
+اخدت T بصفر هتطلع عندى ناقص ثلاثة و اتنين و ناقص
+
+212
+00:16:57,530 --> 00:17:02,390
+ثلاثة يعنى P هى اللى P ماشي الحال اذا هذا النقطة T
+
+213
+00:17:02,390 --> 00:17:07,550
+بالصفر بتجيبها الان النقطة هذه Q لو جيت اخدت اللى
+
+214
+00:17:07,550 --> 00:17:12,390
+هى نقطة واحد وناقص واحد واربعة يعني واحد ساوي اللى
+
+215
+00:17:12,390 --> 00:17:19,090
+هادى بواحدبتطلع عندك T بتساوي لإن هد بديها Q هناأه
+
+216
+00:17:19,090 --> 00:17:22,430
+بدي الـ T اللي بتخلي تجيب Q هذه خد الان واحد
+
+217
+00:17:22,430 --> 00:17:26,070
+بتساوي نقص تلاتة زائد أربعة T يعني أربعة T بتساوي
+
+218
+00:17:26,070 --> 00:17:29,770
+أربعة يعني T بتساوي واحد يعني عند T بتساوي واحد
+
+219
+00:17:29,770 --> 00:17:35,490
+هذه النقطة هتكون عبارة عن Q يعني الآن T هذا عند T
+
+220
+00:17:35,490 --> 00:17:39,890
+صفر وهذا عند T واحد يعني المنطقة هذه كلها من وين
+
+221
+00:17:39,890 --> 00:17:46,820
+بتيجي بتيجي بنفس المعادلات هذولة التلاتةمع T بس
+
+222
+00:17:46,820 --> 00:17:51,480
+بدي اعملها restriction بدي احددها من 0 لعند مين يا
+
+223
+00:17:51,480 --> 00:17:55,900
+جماعة لعند واحد فT من عند 0 لعند واحد بس بتجيب
+
+224
+00:17:55,900 --> 00:18:01,220
+للقطع المستقيمة من النقطة هذه للنقطة هذه وهيك
+
+225
+00:18:01,220 --> 00:18:05,440
+بيكون أوجبنا اللي هو Parametric Equations for the
+
+226
+00:18:05,440 --> 00:18:10,540
+line segment joining the points B and Qالان بدنا
+
+227
+00:18:10,540 --> 00:18:15,020
+نعرف امر جديد اللي هو the distance from a point to
+
+228
+00:18:15,020 --> 00:18:20,820
+align in space هاي عندي خط في الفراغ اسمه خط L
+
+229
+00:18:24,350 --> 00:18:28,830
+عند نقطة في الفراغ اسمها S بدي اوجد اللي هو
+
+230
+00:18:28,830 --> 00:18:32,930
+المسافة بين S و بين نقطة بتعرفوا المسافة بين S و
+
+231
+00:18:32,930 --> 00:18:39,930
+بين نقطة بين S و بين خط مستقيم بيجي بنزل من النقطة
+
+232
+00:18:39,930 --> 00:18:43,490
+عمودي على الخط المستقيم هذه العمودي وصلنا لهذه
+
+233
+00:18:43,490 --> 00:18:48,070
+العمودي هذا الطول اللي احنا بدناه هذا الطول اللي
+
+234
+00:18:48,070 --> 00:18:52,660
+بدناهطب كيف بده اوجده سهل الموضوع ال S عندي معروفة
+
+235
+00:18:52,660 --> 00:18:56,480
+صح ولا لأ يا جماعة معروفة هو معطينيها لأنه بده
+
+236
+00:18:56,480 --> 00:18:59,920
+المسافة بينها بين S و بين الخط المستقيم بكون
+
+237
+00:18:59,920 --> 00:19:05,140
+معطيني الخط المستقيم اللي عنديوبكون معطيني من اللي
+
+238
+00:19:05,140 --> 00:19:08,640
+هو النقطة S اللي بده البقعة بينها وبين الخط
+
+239
+00:19:08,640 --> 00:19:13,860
+المستقيم طبعا كنا الخط المستقيم في طياته بيحمل او
+
+240
+00:19:13,860 --> 00:19:18,520
+بيعطيني معلومات ال vector الموازي له V او vector
+
+241
+00:19:18,520 --> 00:19:25,100
+عليه V بالاضافة لنقاط زي ما بدك بتجيب منهإذا الان
+
+242
+00:19:25,100 --> 00:19:29,860
+باجي انا بمعرفة اللي هو ال vector اللي عليه لو
+
+243
+00:19:29,860 --> 00:19:35,500
+ليكن هذا مثلا V و هذه اللي هي نقطة باخدها على الخط
+
+244
+00:19:35,500 --> 00:19:40,160
+المستقيم و بوصل هذا صارت هذه الزاوية θتا اللي هي و
+
+245
+00:19:40,160 --> 00:19:44,380
+هذه قائمة هذه الزاوية بين هذا اللي هو الجزء من
+
+246
+00:19:44,380 --> 00:19:51,300
+الخط و بين بي اسالان هذا طوله طوله تعرفوا انه sin
+
+247
+00:19:51,300 --> 00:19:58,500
+θ بساوي اللي هو المقابل هذا على اللي هو الوطر
+
+248
+00:19:58,500 --> 00:20:03,500
+فبصير الان المقابل اللي هو المطلوب بساوي ال
+
+249
+00:20:03,500 --> 00:20:06,640
+magnitude لهذا طول هذا اللي هو ال BS في sin
+
+250
+00:20:06,640 --> 00:20:13,380
+زوايمين θ في الواقع هذا ايش بمثلهذا الان لو اجينا
+
+251
+00:20:13,380 --> 00:20:16,320
+اخدنا طبعا احنا المطلوب distance from a point S to
+
+252
+00:20:16,320 --> 00:20:20,360
+a line ببرالتو V اللي هو هذا هذا الان بي اس ساينت
+
+253
+00:20:20,360 --> 00:20:24,180
+هتا هو في الواقع عبارة عن ال magnitude لل بي اس
+
+254
+00:20:24,180 --> 00:20:28,340
+across ال vector V هذا اللي بيحمل الخط اللي هنا
+
+255
+00:20:42,560 --> 00:20:49,210
+القانون اللى اشتقناه هو قانون المسافةبين نقطة وبين
+
+256
+00:20:49,210 --> 00:20:53,710
+خط مستقيم اللي هو طبعا بمعلومية النقطة ومعلومية
+
+257
+00:20:53,710 --> 00:20:57,810
+الخط المستقيم عبارة عن الـ BS cross V على ال
+
+258
+00:20:57,810 --> 00:21:00,970
+magnitude V ماهو الـ BS اللي هو الاسس النقطة
+
+259
+00:21:00,970 --> 00:21:05,430
+المعطاة وB نقطة ناخدها على الخط المستقيم المعطاة و
+
+260
+00:21:05,430 --> 00:21:08,590
+V هو ال vector اللي بيحمل الخط المستقيم المعطاة أو
+
+261
+00:21:08,590 --> 00:21:11,910
+الموازي اللي هو زي ما قلنا سابقا ونشوف مثال عملي
+
+262
+00:21:13,470 --> 00:21:17,150
+بقول find the distance from the point S1 1 5 to
+
+263
+00:21:17,150 --> 00:21:21,350
+the line X بيساوي 1 زي ال T و Y بيساوي 3 نقص T و Z
+
+264
+00:21:21,350 --> 00:21:25,670
+بيساوي 2T لسه احنا في الموضوع يعني مابعدناش يعني
+
+265
+00:21:25,670 --> 00:21:30,850
+الآن احنا بنقدر نعرف من هذا المعادلةاللي هو الـ
+
+266
+00:21:30,850 --> 00:21:33,890
+vector الموازي اللي هو معاملات من ال vector
+
+267
+00:21:33,890 --> 00:21:38,470
+الموازي معاملات T يعني ال vector الموازي P بيساوي
+
+268
+00:21:38,470 --> 00:21:44,550
+I اللي هي واحد ناقص J اللي هي ناقص واحد زائد اتنين
+
+269
+00:21:44,550 --> 00:21:49,850
+K اللي هي من معامل T اتنين هذا ال vector الموازي
+
+270
+00:21:49,850 --> 00:21:55,620
+لمين لخط المستقيم المعطىمعادلته زي ما اتعلمنا قبل
+
+271
+00:21:55,620 --> 00:22:00,760
+بشوية لإن النقطة اللي معناه مين هي S115 بدنا تمام
+
+272
+00:22:00,760 --> 00:22:04,900
+نقطة بناخد أي نقطة اللي بتيجي على بالك واحد قال
+
+273
+00:22:04,900 --> 00:22:09,500
+اسمع بدي أخد اللي هو أسهل إشي T بصفر بصير واحد و
+
+274
+00:22:09,500 --> 00:22:13,340
+تلاتة و اتنين قلنا له حاضر واحد اتاني قال لأ بدي
+
+275
+00:22:13,340 --> 00:22:16,880
+أخد T بواحد بدي أغلب حالي بدي أخليها كل أقام بقوله
+
+276
+00:22:16,880 --> 00:22:25,060
+بنفع T اللي بدكياالان اخدنا نقطة بي واحد و تلاتة و
+
+277
+00:22:25,060 --> 00:22:29,060
+سفر الان ايش بالزمن زي ما قلنا في القانون في
+
+278
+00:22:29,060 --> 00:22:32,220
+القانون قلنا بالزمن بي اس اكروس في ع ال magnitude
+
+279
+00:22:32,220 --> 00:22:37,100
+لفي الان بي عرفناه و اس معطية و في عرفناه نيجي
+
+280
+00:22:37,100 --> 00:22:43,120
+الان نطبق القانون بي اس ايش متساوي because واحد
+
+281
+00:22:43,120 --> 00:22:49,440
+نقص واحدواحد ناقص ثلاثة واحد ناقص ثلاثة وخمسة ناقص
+
+282
+00:22:49,440 --> 00:22:53,180
+ايه اش سفر طلع عند ال BS طلع عند هذا سفر طبعا هذه
+
+283
+00:22:53,180 --> 00:22:58,380
+طلعت سالب اتنين J وهذه خمسة K اذا صار عند ال BS
+
+284
+00:22:58,380 --> 00:23:02,480
+هيو بدي اعمل ال BS cross V واخدله ال magnitude ال
+
+285
+00:23:02,480 --> 00:23:06,700
+V هيو ماشي اعرفنا كيف نوجد ال BS cross V بواسطة
+
+286
+00:23:06,700 --> 00:23:11,680
+المحدد هي المحدد جوا وهي برا ايه ال absolute value
+
+287
+00:23:11,680 --> 00:23:15,010
+او ال magnitudeطبعا المحدد بيطلعنا قيمة و بعدين
+
+288
+00:23:15,010 --> 00:23:17,650
+ناخد ال absolute value و نطلع القيمة الموجودة اللي
+
+289
+00:23:17,650 --> 00:23:21,770
+بدنا هيها شوفوا صلوا على النبي عليه الصلاة والسلام
+
+290
+00:23:21,770 --> 00:23:27,610
+ال BS كيف نعرف نضرم BS في V هي IJK و بعدين BS اللي
+
+291
+00:23:27,610 --> 00:23:31,850
+هو السطر الأول و ال BS اللي وجدناه هيه اللي هو سفر
+
+292
+00:23:31,850 --> 00:23:36,830
+ناقص اتنين خمسة و اللي هي ال V اللي هي عبارة عن
+
+293
+00:23:36,830 --> 00:23:41,290
+واحد و ناقص واحد و اتنينالان خلّينا نجد هذا ال ..
+
+294
+00:23:41,290 --> 00:23:45,350
+ال .. ال .. ال .. ال .. ال determinant ال
+
+295
+00:23:45,350 --> 00:23:48,730
+determinant اللي جوا بتعرفوا كيف بنشطب السطر هذا و
+
+296
+00:23:48,730 --> 00:23:52,090
+العمود هذا بيصير ناقص اتنين في اتنين ناقص اربعة
+
+297
+00:23:52,090 --> 00:23:58,570
+زائد واحد اللي هو عبارة عن ايه؟ناقص اربعة زائد
+
+298
+00:23:58,570 --> 00:24:03,930
+خمسة وهو عبارة عن I الان نشط بهذا و نشط بهذا يصبح
+
+299
+00:24:03,930 --> 00:24:09,490
+اتنين ناقص خمسة و في ناقص طبعة اتجاه يصبح زائد
+
+300
+00:24:09,490 --> 00:24:13,230
+خمسة الان الاخيرة ال K نشط بهذا و نشط بهذا يصبح
+
+301
+00:24:13,230 --> 00:24:17,690
+سفر ناقص ناقص يصبح زائد اتنين اللي هي في اتجاه ال
+
+302
+00:24:17,690 --> 00:24:21,320
+K ال magnitude لهذا الجدر التربيعي له واحدزائد
+
+303
+00:24:21,320 --> 00:24:25,840
+خمسة وعشرين زائد أربعة أربعة وخمسة وعشرين تسعة
+
+304
+00:24:25,840 --> 00:24:29,460
+وعشرين وواحد تلاتين يعني جدر التلاتين ال magnitude
+
+305
+00:24:29,460 --> 00:24:33,400
+لل V تلزمنا اللي هي أش بتساوي الجدر التربيعي اللي
+
+306
+00:24:33,400 --> 00:24:37,700
+هو واحد زائد واحد زائد أربعة اللي هو جدر الستة لأن
+
+307
+00:24:37,700 --> 00:24:41,520
+حسب القانون the distance أش بتساوي ال magnitude لل
+
+308
+00:24:41,520 --> 00:24:46,100
+V S cross V ع ال magnitude لل V طيب يا جماعة صلوا
+
+309
+00:24:46,100 --> 00:24:49,650
+على النبي عليه الصلاة والسلامالـ magnitude للـ BS
+
+310
+00:24:49,650 --> 00:24:53,770
+cross V اللي هو بساوي جدر التلاتين اللي أوجدناه
+
+311
+00:24:53,770 --> 00:24:56,670
+على جدر الستة ويساوي جدر الخمسة وهذه هي المسافة
+
+312
+00:24:56,670 --> 00:25:01,250
+بين الخط المستقيم الأول نقطة أو بين النقطة أسوأ
+
+313
+00:25:01,250 --> 00:25:07,510
+الخط المستقيم الموجود المطلوبالان بدنا نجي لجزء
+
+314
+00:25:07,510 --> 00:25:11,890
+ثاني من المحاضرة وهو معادلة الخط معادلة لبلين في
+
+315
+00:25:11,890 --> 00:25:16,370
+ال space احنا كنا في الفرقية زمان نقول ان لبلين
+
+316
+00:25:16,370 --> 00:25:20,970
+يحدد تحديدا تمال بمعرفة تلات نقاط على لبلين ليست
+
+317
+00:25:20,970 --> 00:25:24,810
+على استقامة واحدةاللي هي تلت مقات ليست على استقامة
+
+318
+00:25:24,810 --> 00:25:30,050
+واحدة تحدد اللي هو المستوى اللي يمر بهنا طيب الان
+
+319
+00:25:30,050 --> 00:25:34,290
+هذا الكلام شبيه بمعرفة الان مش تلت مقات تلت مقات
+
+320
+00:25:34,290 --> 00:25:39,490
+بعد شوية هيكون كتطبيق بمعرفة اللي هو vector عمودي
+
+321
+00:25:40,100 --> 00:25:45,040
+على المستوى و بمعرفة نقطة على المستوى هنقدر نحدد
+
+322
+00:25:45,040 --> 00:25:51,140
+هوية المستوى الان بمعرفة نقطة على المستوى و بمعرفة
+
+323
+00:25:51,140 --> 00:25:55,580
+عمودي عليه نحدد هوية المستوى نشوف كيف نوجد دليلة
+
+324
+00:25:55,580 --> 00:25:58,900
+ده الان ال vector ده عمودي عليه ده عمودي على كل
+
+325
+00:25:58,900 --> 00:26:03,520
+مستقيل في المستوى الان هناخد نقطة عشوائية نسميها x
+
+326
+00:26:03,520 --> 00:26:07,600
+y و z هي اللي هتحدد لي هوية لبنان اللي بدلالتها
+
+327
+00:26:07,600 --> 00:26:11,960
+بدنا نجيب اللي هو معادلة لبنانالان صار عند بي نوت
+
+328
+00:26:11,960 --> 00:26:16,800
+بي عبارة عن vector موجود من وين يا جماعة في البلين
+
+329
+00:26:16,800 --> 00:26:20,380
+هذا ال vector أكيد الان العمودي على البلين عمودي
+
+330
+00:26:20,380 --> 00:26:27,400
+عليه يعني بمعنى آخر الان هذا ضد ال بي نوت بي بساوي
+
+331
+00:26:27,400 --> 00:26:32,680
+سفر لإن الزاوية اللي بينهم تسعين درجة فبكون عندى
+
+332
+00:26:32,680 --> 00:26:36,260
+اللي هو الضد product بينهم بساوي سفر وهذه هي
+
+333
+00:26:36,260 --> 00:26:40,780
+معادلة البلينمعادلة الـ plan vector equation ستكون
+
+334
+00:26:40,780 --> 00:26:48,320
+N.P0P بساوي 0 لأن خلّينا نسمي نتفج مع بعض أن الـ N
+
+335
+00:26:48,320 --> 00:26:52,860
+بدنا نسميه اللي هو بساوي ال vector المعلوم طبعا
+
+336
+00:26:52,860 --> 00:26:56,900
+لأنه ببساطة و بده وجود المعادلة الـ plan ال vector
+
+337
+00:26:56,900 --> 00:27:02,780
+N المعلوم بده أقول عنه اسمه AI زائد BJ زائد CKإذاً
+
+338
+00:27:02,780 --> 00:27:06,720
+الـ A و الـ B و الـ C هي عبارة عن معاملات الـ I و
+
+339
+00:27:06,720 --> 00:27:10,500
+الـ J و الـ K بالترتيب للـ vector العمودي على
+
+340
+00:27:10,500 --> 00:27:15,540
+المستوى الـ NN.P0P بساوة سفر هي معادلة اللي هي ال
+
+341
+00:27:15,540 --> 00:27:18,440
+vector equation للمستوى لأن لو بدنا نحوّلها ل
+
+342
+00:27:18,440 --> 00:27:22,480
+component equation بنعمل ال dot productب نوت بي
+
+343
+00:27:22,480 --> 00:27:27,920
+نوت بي نوت
+
+344
+00:27:27,920 --> 00:27:34,860
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+345
+00:27:34,860 --> 00:27:35,360
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+346
+00:27:35,360 --> 00:27:35,780
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+347
+00:27:35,780 --> 00:27:37,040
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+348
+00:27:37,040 --> 00:27:39,340
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+349
+00:27:39,340 --> 00:27:48,040
+بي نوت بي نوت بي نوت بي نوت ببتعرف تدرب الـ Dot
+
+350
+00:27:48,040 --> 00:27:53,040
+Product زائد C في المعامل الـ K التي هي Z-Z0 في
+
+351
+00:27:53,040 --> 00:27:56,860
+اتجاه الـ Zero كله مش هيساوي سفر يعني هذا عبارة عن
+
+352
+00:27:56,860 --> 00:28:03,080
+الـ Dot Product للـ N Dot B Note Bالان صارت هذه
+
+353
+00:28:03,080 --> 00:28:08,480
+تمثلي معادلة الخط المستخدم للـ plane اللى بنسميها
+
+354
+00:28:08,480 --> 00:28:12,540
+component equations وصلت اللى هى ال component اللى
+
+355
+00:28:12,540 --> 00:28:17,220
+هى ال equation اللى احنا استعملناها قبل هيك كثير
+
+356
+00:28:17,220 --> 00:28:22,780
+طيب اذا الان ممكن واحد يختصرها يقولك خلاص ماهى ax
+
+357
+00:28:22,780 --> 00:28:27,370
+زائد by زائد czهذه الأشياء بيساوين الـ D
+
+358
+00:28:43,370 --> 00:28:46,590
+لأن الـ A معروفة و الـ B معروفة و الـ C معروفة و
+
+359
+00:28:46,590 --> 00:28:50,310
+الـ X نوت و الـ Y نوت و الـ Z نوت معروفة فبصير و
+
+360
+00:28:50,310 --> 00:28:53,770
+كأنه الـ component equation can be simplified as
+
+361
+00:28:53,770 --> 00:28:59,850
+يعني AX زائد BY زائد CZ بساوي D وهنا بدنا نلاحظ
+
+362
+00:28:59,850 --> 00:29:03,990
+مالي لما نكتب على صورة AX زائد BY زائد CZ بساوي D
+
+363
+00:29:03,990 --> 00:29:10,400
+اللي هو عددبكون A هو معامل اللي هو الـ I للنورمال
+
+364
+00:29:10,400 --> 00:29:14,880
+والـ B معامل الـ J للنورمال والـ C معامل الـ K
+
+365
+00:29:14,880 --> 00:29:20,740
+للنورمال يعني الآن بمعرفة معادلة الـ plane بعرف ال
+
+366
+00:29:20,740 --> 00:29:25,000
+normal عليه مباشرة بكتب على الصورة هذه المعادلة
+
+367
+00:29:25,000 --> 00:29:28,000
+بقول على طول ال normal الآن بتساوي ايه في اتجاه ال
+
+368
+00:29:28,000 --> 00:29:32,280
+Iب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+369
+00:29:32,280 --> 00:29:33,280
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+370
+00:29:33,280 --> 00:29:34,200
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+371
+00:29:34,200 --> 00:29:38,480
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+372
+00:29:38,480 --> 00:29:39,200
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+373
+00:29:39,200 --> 00:29:41,080
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+374
+00:29:41,080 --> 00:29:41,600
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+375
+00:29:41,600 --> 00:29:42,180
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+376
+00:29:42,180 --> 00:29:43,760
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+377
+00:29:43,760 --> 00:29:44,500
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+378
+00:29:44,500 --> 00:29:50,520
+ب .. ب ..
+
+379
+00:29:50,520 --> 00:29:56,080
+ب ..
+
+380
+00:29:56,100 --> 00:30:00,460
+لكن طوله اختلف واحد اخد normal الوحدة واحد اخد
+
+381
+00:30:00,460 --> 00:30:04,000
+normal طوله عشرة واحد اخد normal طوله مية المهم
+
+382
+00:30:04,000 --> 00:30:06,900
+اللي هو بيكون ايه شماله normal على ال plane ان شاء
+
+383
+00:30:06,900 --> 00:30:10,780
+الله اقطعت السؤال طيب ناخد مثال على الكلام هذا
+
+384
+00:30:10,780 --> 00:30:14,000
+find an equation for the plane through P0
+
+385
+00:30:14,000 --> 00:30:18,680
+perpendicular to M يعني بدنا نوجد معادلة المستوى
+
+386
+00:30:18,680 --> 00:30:24,240
+اللي يمر بهذه النقطة وعمودي على هذا اللي هو مينالـ
+
+387
+00:30:24,240 --> 00:30:29,200
+vector الان صار الموضوع سهل جدا اللي هو على طول
+
+388
+00:30:29,200 --> 00:30:32,760
+بنستخدم هذه المعادلة اللي هي ال a في هذه الحالة
+
+389
+00:30:32,760 --> 00:30:37,600
+خمسة في x ناقص x naught اللي هي x ناقص ناقص تلاتة
+
+390
+00:30:37,600 --> 00:30:43,380
+الان زاد بي اللي هو مين بي معامل اللي هو j اتنين
+
+391
+00:30:43,380 --> 00:30:48,320
+في y minus y naught اللي هي سفر زاد اللي هي ناقص
+
+392
+00:30:48,320 --> 00:30:52,140
+واحد معامل اللي هي c هذا ال capital في z minus
+
+393
+00:30:52,140 --> 00:30:57,700
+سبعةاللي هي z note ويساوي ايش سفر لان استخدمت هذه
+
+394
+00:30:57,700 --> 00:31:03,780
+المعادلة صار عندى لان بعد مبسط خمسة x ناقص خمس عشر
+
+395
+00:31:03,780 --> 00:31:09,360
+بيصير ناقص x زي خمس عشر اتنين y ناقص z سائد سبعة
+
+396
+00:31:09,360 --> 00:31:13,300
+بيساوي سفر لان بالبسطة اكتر بيصير خمسة x زي اتنين
+
+397
+00:31:13,300 --> 00:31:16,620
+y ناقص z والخمس عشر والسبعة عشر اتنين وعشرين بيصير
+
+398
+00:31:16,620 --> 00:31:23,920
+سالم اتنين وعشرين وهذه معادلة المستوى المطلوبواحد
+
+399
+00:31:23,920 --> 00:31:29,180
+تاني قال لي لأ انا بدي معادلة لخط لمستوى بمر
+
+400
+00:31:29,180 --> 00:31:33,400
+بالنقاط تلاتة a وb وc يعني بيقولي find an equation
+
+401
+00:31:33,400 --> 00:31:37,880
+for the line for the plane through a بتساوي كده وb
+
+402
+00:31:37,880 --> 00:31:42,340
+بتساوي كده and c بتساوي كده بنقوله سهل يعني ما
+
+403
+00:31:42,340 --> 00:31:46,540
+احنا الآن هدول مدام ان التلاتة على نفس المستوى
+
+404
+00:31:46,540 --> 00:31:50,060
+التلاتة اللي بدنا نجيبهم المستوى اللي بيمثل هنا
+
+405
+00:31:50,570 --> 00:31:55,230
+بنقدر نجيب vector a,b على المستوى و vector a,c على
+
+406
+00:31:55,230 --> 00:31:59,970
+المستوى و بنقل cross product لـ cross product اللي
+
+407
+00:31:59,970 --> 00:32:04,490
+عندنا ايه شماله؟ بطلع عمودي على المستوى
+
+408
+00:32:07,070 --> 00:32:10,930
+Find an equation for the plane A, B and C بيساوي
+
+409
+00:32:10,930 --> 00:32:14,350
+كده First note that زي ما قلنا بدنا نوجد ايش يا
+
+410
+00:32:14,350 --> 00:32:18,750
+جماعة لـ cross product بين A, B و A, C اللي هما
+
+411
+00:32:18,750 --> 00:32:23,150
+مستقيمين موجودات على المستوى فال cross product
+
+412
+00:32:23,150 --> 00:32:26,090
+اللي هو العمودي على المستوى بيكون عرفنا عمودي على
+
+413
+00:32:26,090 --> 00:32:30,130
+المستوى وعرفناه و طب نعرف المعادلة مين لـ plane
+
+414
+00:32:30,130 --> 00:32:36,790
+لأن A, B سهلة اتنينالنقص يصبح صفر هدى وصفر نقص
+
+415
+00:32:36,790 --> 00:32:42,370
+واحد يصبح نقص كيه الاسي الهو صفر لل I تلاتة لل J
+
+416
+00:32:42,370 --> 00:32:49,210
+ونقص واحد لل K هي AB وهي AC الان اواجهولي اللي هو
+
+417
+00:32:49,210 --> 00:32:52,250
+ال cross product اللي هو هيكون مين ال normal ال
+
+418
+00:32:52,250 --> 00:32:55,470
+vector على ال blendطيب ال normal على ال vector على
+
+419
+00:32:55,470 --> 00:32:59,810
+ال plane و بمرض مين كمان بالنقطة و بنختار النقطة
+
+420
+00:32:59,810 --> 00:33:05,710
+اللى هو مين a0 و 0 و 1 ماشي يا شباب و يا شباب ده
+
+421
+00:33:05,710 --> 00:33:09,870
+نشوف كده صلى الله عننا يا عزيزي الصلاة والسلام ده
+
+422
+00:33:09,870 --> 00:33:14,830
+cross product a b في a cهذا اللي هو بدمثل الـ
+
+423
+00:33:14,830 --> 00:33:19,670
+Normal هو عبارة عن .. زي ما بنعرف IJK الـ AC ..
+
+424
+00:33:19,670 --> 00:33:24,930
+الـ AB معاملات هي تنين سفر ناقص واحد AC اللي هي
+
+425
+00:33:24,930 --> 00:33:28,750
+معاملة سفر تلاتة ناقص واحد بنعمله cross product زي
+
+426
+00:33:28,750 --> 00:33:32,290
+ما أتعلمنا قبل شوية بنشطب هذا وشطب هذا بيصير عبارة
+
+427
+00:33:32,290 --> 00:33:37,150
+عن سفر ناقص ناقص تلاتة اللي هي تلاتة Iوهذا من شطب
+
+428
+00:33:37,150 --> 00:33:40,910
+هذا من شطب هذا بيصير ناقص اتنين الان وفي ناقص في
+
+429
+00:33:40,910 --> 00:33:44,690
+الاصل بيصير ناقص سبعة ال J بوجب سالب الموجب اللي
+
+430
+00:33:44,690 --> 00:33:49,030
+هي في الفكرة المحددة زاد اتنين J و الأخيرا اللي هي
+
+431
+00:33:49,030 --> 00:33:53,030
+بيصير اتنين في تلاتة في ستة في اتجاه ال K هذا is
+
+432
+00:33:53,030 --> 00:33:56,190
+normal to the plane the component equation is
+
+433
+00:33:56,190 --> 00:33:59,690
+أخدنا ال A هذه واحد بياخد هذه بنفع واحد بياخد هذه
+
+434
+00:33:59,690 --> 00:34:02,810
+بنفع زي ما قلنا ايه اللي بدأ فيه يعني لأن تلاتة في
+
+435
+00:34:02,810 --> 00:34:04,250
+X ناقص X not سفر
+
+436
+00:34:07,130 --> 00:34:17,250
+Y-Y0-Y0-Y0-Y0-Y0
+
+437
+00:34:17,250 --> 00:34:21,630
+-Y0-Y0-Y0
+
+438
+00:34:21,770 --> 00:34:30,390
+معادلة خط المستقيم المستقيم هي معادلة
+
+439
+00:34:30,390 --> 00:34:31,470
+الخط المستقيم المستقيم المستقيم هي معادلة الخط
+
+440
+00:34:31,470 --> 00:34:37,250
+المستقيم المستقيم المستقيم المستقيم المستقيم
+
+441
+00:34:37,250 --> 00:34:38,750
+المستقيم المستقيم المستقيم المستقيم المستقيم
+
+442
+00:34:38,750 --> 00:34:38,770
+المستقيم المستقيم المستقيم المستقيم المستقيم
+
+443
+00:34:38,770 --> 00:34:45,590
+المستقين المستقين المستقين المستقين المستقين
+
+444
+00:34:45,590 --> 00:34:50,290
+المستقين المستقين المستقين المستقين المستقين
+
+445
+00:34:50,290 --> 00:34:50,450
+المستقين المستقين المستقين المستقين المستقين
+
+446
+00:34:50,450 --> 00:34:54,810
+المستقين المستوهي مستوى آخر بدنا معادلة الخط
+
+447
+00:34:54,810 --> 00:35:00,390
+المستقيم اللي هو اللي بتقاطع فيه مستوين احنا بنعرف
+
+448
+00:35:00,390 --> 00:35:05,070
+انه لو كان في عنا مستوين غير متوازين هيتقاطع اللي
+
+449
+00:35:05,070 --> 00:35:09,230
+هو في خط أو بيكون المنطبقين طبعا احنا مانعش ناخد
+
+450
+00:35:09,230 --> 00:35:13,610
+حالة الانطباق هناخد حالة ايش اللي هو بتقاطع في خط
+
+451
+00:35:13,610 --> 00:35:19,320
+مستقيم ونوجد معادلة الخط المستقيم تبع التقاطعطيب
+
+452
+00:35:19,320 --> 00:35:21,980
+سهلة أنا بعرف الـ plan الأول و بعرف الـ plan
+
+453
+00:35:21,980 --> 00:35:25,380
+الثاني و بدوجة معادلة الخط المستقيم اللي بتقاطعان
+
+454
+00:35:25,380 --> 00:35:29,220
+فيه مدام بعرف معادلة الـ plan الأول إذا بعرف
+
+455
+00:35:29,220 --> 00:35:33,700
+العمودي عليه N واحد مدام بعرف الـ plan التاني إذا
+
+456
+00:35:33,700 --> 00:35:37,740
+بعرف معادلة الخط المستقيم العمودي عليه N اتنين طيب
+
+457
+00:35:37,740 --> 00:35:42,340
+مدام N واحد عمودي على المستوى الأول إذا عمودي على
+
+458
+00:35:42,340 --> 00:35:49,380
+خط كل خط مستقيم فيه بما فيه خط التقاطعsimilarly
+
+459
+00:35:49,380 --> 00:35:56,130
+بنفس الأسلوبمدام أن 2 عمودي على المستوى التاني إذا
+
+460
+00:35:56,130 --> 00:35:59,570
+عمودي على كل خط مستقيم فيه إذا برضه عمودي على خط
+
+461
+00:35:59,570 --> 00:36:03,350
+التخاطع لأن خط التخاطع موجود في المستويين ولا ليش
+
+462
+00:36:03,350 --> 00:36:06,770
+اسمه خط تخاطع إذا صار عندي العمودي أن واحد
+
+463
+00:36:06,770 --> 00:36:12,730
+والعمودي أن اتنين التنين عموديات على مين؟ على الخط
+
+464
+00:36:12,730 --> 00:36:19,920
+المستقيم اللي هو بتقاطع فيه المستويينيعني بمعنى
+
+465
+00:36:19,920 --> 00:36:24,460
+آخر لو جيت أخدت الـ cross product أن واحد أن اتنين
+
+466
+00:36:24,460 --> 00:36:31,020
+هيكون عمودي على أن واحد و أن اتنين يعني هذا المتجه
+
+467
+00:36:31,020 --> 00:36:35,800
+أن واحد و cross أن اتنين أشماله بوازي الخط
+
+468
+00:36:35,800 --> 00:36:40,700
+المستقيم تبع التقاطع يعني بوازي خط التقاطع إذا
+
+469
+00:36:40,700 --> 00:36:45,380
+عرفنا الخط التقاطع اللي بندور عليه اعرفنا فيكتور
+
+470
+00:36:45,380 --> 00:36:50,210
+موازي لهوبنعرف نقطة عليه نعرف معادلته زي ما عرفنا
+
+471
+00:36:50,210 --> 00:36:55,550
+قبل هيك اذا الان عشان نوجد ناخد مثال عملي عملي
+
+472
+00:36:55,550 --> 00:36:59,470
+find a vector parallel to the line of intersection
+
+473
+00:36:59,470 --> 00:37:05,090
+of the planes كذا and كذا بده اول اشي اشي يوجد هو
+
+474
+00:37:05,730 --> 00:37:09,170
+الذي أريد أن أعطيكمه على خطوط اولا يوجد الـ vector
+
+475
+00:37:09,170 --> 00:37:11,910
+ثم يوجد معادلة الخط المستقيم الخط المستقيم يجب أن
+
+476
+00:37:11,910 --> 00:37:13,350
+نعرف أنه يوجد معادلة الخط المستقيم إذا وجدنا ال
+
+477
+00:37:13,350 --> 00:37:17,010
+vector ونعرف ال V فأخذنا أمثلة على إيجاد معادلة
+
+478
+00:37:17,010 --> 00:37:21,630
+الخط المستقيم إذا الآن السؤال هذا هو find a vector
+
+479
+00:37:21,630 --> 00:37:26,310
+موازي parallel to the line of intersection of the
+
+480
+00:37:26,310 --> 00:37:30,630
+blends يعني إذا كان لدي بلنز اتنين فأنا أريد خط
+
+481
+00:37:30,630 --> 00:37:36,300
+مستقيم أو vector موازي لخط التقاطع بينهمبنجيب
+
+482
+00:37:36,300 --> 00:37:40,120
+العمودي على الأول و العمودي على التاني و نعمل
+
+483
+00:37:40,120 --> 00:37:48,510
+cross product موازي لخط التقاطعVector موازي الخط
+
+484
+00:37:48,510 --> 00:37:52,370
+التخطأ اللي هو المطلوب The line of intersection of
+
+485
+00:37:52,370 --> 00:37:55,530
+two planes is perpendicular to both planes normal
+
+486
+00:37:55,530 --> 00:38:01,570
+to vectors N1 and N2 كما قلنا فوق كيف نوجدها N1 هي
+
+487
+00:38:01,570 --> 00:38:05,750
+تلاتة في اتجاه I نقص ستة في اتجاه J نقص اتنين في
+
+488
+00:38:05,750 --> 00:38:10,380
+اتجاه K اللي هي معاملات منالـ X و الـ Y و الـ Z في
+
+489
+00:38:10,380 --> 00:38:15,320
+المعادلة لبنان هنا الان اتنين بيبدو بنفس الأسلوب
+
+490
+00:38:15,320 --> 00:38:19,320
+ان اتنين بيساوي اتنين I زائد J ناقص اتنين K هي ان
+
+491
+00:38:19,320 --> 00:38:23,240
+اتنين اذا الان يعني مايفنق العمودي على المستوى
+
+492
+00:38:23,240 --> 00:38:26,720
+الأول و العمودي على المستوى التاني و جون العموديين
+
+493
+00:38:26,720 --> 00:38:29,660
+هدولة هيكون العموديين على خط التقاطع لأنه مشترك
+
+494
+00:38:29,660 --> 00:38:36,490
+بين المستويينبكون عندى N1-N2 هو ال vector الموازي
+
+495
+00:38:36,490 --> 00:38:40,290
+لخط التقاطع لأنه بكون عمودي على N1 N2 زي خط
+
+496
+00:38:40,290 --> 00:38:44,640
+التقاطعبصير عند n واحد و cross on اتنين بحي اه جي
+
+497
+00:38:44,640 --> 00:38:48,920
+كي و بنجي تلاتة نقص ستة نقص اتنين زي ما اتعلمنا و
+
+498
+00:38:48,920 --> 00:38:52,640
+اتنين واحد و نقص اتنين و بنفك هذا ال determinant
+
+499
+00:38:52,640 --> 00:38:56,340
+تفكوه هتلاقوه بساوة اربعتاشر I زاد اتنين جي زائد
+
+500
+00:38:56,340 --> 00:39:00,260
+خمستاشر كي طبعا بتعرفوا كيف تفكوه بنشطب هذا بنشطب
+
+501
+00:39:00,260 --> 00:39:03,040
+هذا ناقص الستة في ناقص اتنين اللي هي عبارة عن
+
+502
+00:39:03,040 --> 00:39:06,380
+اتناش اتناش ناقص ناقص اتنين يعني زياد اتنين يعني
+
+503
+00:39:06,380 --> 00:39:11,200
+اربعتاشر I و الباقي بنفس الاسلوبالان اي هذا واحد
+
+504
+00:39:11,200 --> 00:39:15,460
+من مين ال cross product موازي لمين لخط التقاطع
+
+505
+00:39:15,460 --> 00:39:21,580
+بينهم لو ضربت هذا ال vector في اي scalar هيطلع
+
+506
+00:39:21,580 --> 00:39:25,080
+vector موازي لخط التقاطع لأن كلهم ايه شمالهم كلهم
+
+507
+00:39:25,080 --> 00:39:28,720
+متوازيات اذا any non zero multiple of any واحد
+
+508
+00:39:28,720 --> 00:39:31,540
+cross ان اتنين will be a vector parallel to the
+
+509
+00:39:31,540 --> 00:39:36,970
+line of intersectionالان طلب مني الان find
+
+510
+00:39:36,970 --> 00:39:40,430
+parametric equations for the line in which the
+
+511
+00:39:40,430 --> 00:39:46,530
+planes اللي هي intersects الان هدولة الخطين أوجدنا
+
+512
+00:39:46,530 --> 00:39:51,070
+.. أوجدنا .. أسف المستويين أوجدنا vector في المثال
+
+513
+00:39:51,070 --> 00:39:58,210
+السابق موازي لخطة خطوحينالان بدوا ال parameter
+
+514
+00:39:58,210 --> 00:40:02,530
+equations لخط التقاطع بنقول سهل خلاص في ال example
+
+515
+00:40:02,530 --> 00:40:05,350
+اللي قبله قبله قبله قبله قبله قبله قبله قبله قبله
+
+516
+00:40:05,350 --> 00:40:09,730
+قبله قبله قبله قبله قبله قبله قبله
+
+517
+00:40:21,860 --> 00:40:27,460
+بدي اللي هو اوجد اللي هو نقطة مشتركة على خط
+
+518
+00:40:27,460 --> 00:40:31,620
+التخاطع بنشوف كيف نوجد النقطة المشتركة بعد ما نوجد
+
+519
+00:40:31,620 --> 00:40:35,240
+النقطة المشتركة على خط التخاطع ايش بنسوي يا جماعة
+
+520
+00:40:35,240 --> 00:40:39,340
+مع ال vector هذا بنطبق قانوننا اللي قبل بشوية طيب
+
+521
+00:40:39,340 --> 00:40:40,000
+ده نشوف كده
+
+522
+00:40:47,860 --> 00:40:53,720
+عشان نجد نقطة على خط التقاطع لان تاخد اي نقطة اللي
+
+523
+00:40:53,720 --> 00:41:00,040
+بتعود عن Z أو Y او X اللي بدك اياها عشوائيا اسهل
+
+524
+00:41:00,040 --> 00:41:06,000
+اشي ان انا اعوض عن Z ب0أو Y بصفر أو X بصفر عشان
+
+525
+00:41:06,000 --> 00:41:09,500
+نضيع إيش رقمه وأسهلنا في التعويض و اللي بده غير
+
+526
+00:41:09,500 --> 00:41:12,800
+هيك يعوض عوض Z بواحد Z بخمس Z بتلاتة Z باربعة اللي
+
+527
+00:41:12,800 --> 00:41:16,800
+بدهويا أو X بواحد أو Y بواحد اللي بدهويا بنيجي
+
+528
+00:41:16,800 --> 00:41:21,940
+الآن عوضنا عن Z بصفر بطلع عندي هذا تلاتة X نقص ستة
+
+529
+00:41:21,940 --> 00:41:28,240
+Y بساوة خمس عشر واتنين X زائد Y بساوة خمسةالان
+
+530
+00:41:28,240 --> 00:41:33,480
+احنا اوجدنا نقطة مشتركة مشتركة
+
+531
+00:41:33,480 --> 00:41:38,020
+عليهم اذا انا اريد خطة قاطع عشان اوجد خطة قاطع بده
+
+532
+00:41:38,020 --> 00:41:44,580
+حل هدول المعادلتين ايام ماشي بده حل هن في نفس
+
+533
+00:41:44,580 --> 00:41:53,360
+الوقتالان لتحل المعادلة انيا بمعانة اخر بيحل
+
+534
+00:41:53,360 --> 00:41:54,720
+المعادلة انيا بمعانة اخر بيحل المعادلة انيا بمعانة
+
+535
+00:41:54,720 --> 00:41:56,520
+اخر بيحل المعادلة انيا بمعانة اخر بيحل المعادلة
+
+536
+00:41:56,520 --> 00:41:58,680
+انيا بمعانة اخر بيحل المعادلة انيا بمعانة اخر بيحل
+
+537
+00:41:58,680 --> 00:42:01,280
+المعادلة انيا بمعانة اخر بيحل المعادلة انيا بمعانة
+
+538
+00:42:01,280 --> 00:42:03,780
+اخر بيحل المعادلة انيا بمعانة اخر بيحل المعادلة
+
+539
+00:42:03,780 --> 00:42:05,560
+انيا بمعانة اخر بيحل المعادلة انيا بمعانة اخر بيحل
+
+540
+00:42:05,560 --> 00:42:07,360
+المعادلة انيا بمعانة اخر بيحل المعادلة انيا بمعانة
+
+541
+00:42:07,360 --> 00:42:11,560
+اخر بهذا بنضربه في معامل الـ x اللي هنا ليه 6x ال
+
+542
+00:42:11,560 --> 00:42:17,200
+.. ال .. الزائد اللي هو 3y بساوة 15 بنحط انتحت بعض
+
+543
+00:42:17,200 --> 00:42:21,520
+هذا معامل زي هذا بنترحم من بعض بتطلع مين بدللة y
+
+544
+00:42:21,520 --> 00:42:25,780
+اللي بدللة y بنحلها بتطلع إيش قيمة y لأن قيمة y
+
+545
+00:42:25,780 --> 00:42:30,100
+اللي طلعت هنا نجيها هن واحد ناقص واحد برد بعود عن
+
+546
+00:42:30,100 --> 00:42:34,680
+قيمة y في أحد المعادلتينيظهر عنده قيمة من ؟ قيمة X
+
+547
+00:42:34,680 --> 00:42:37,660
+هكذا نحل المعادلات في تالت أعداد و تاني أعدادي
+
+548
+00:42:37,660 --> 00:42:44,180
+فبصير عندي الحل أوجدت قيمة X و قيمة Y على خط
+
+549
+00:42:44,180 --> 00:42:48,040
+التقاطع و هذه السفر اللي هي النقطة الثالثة إذا
+
+550
+00:42:48,040 --> 00:42:52,060
+بكون هكذا أوجدت تلاتة و نقص واحد و سفر لإن حلت في
+
+551
+00:42:52,060 --> 00:42:56,060
+نفس الوقت يعني أوجدت الحلول المشتركة يعني أوجدت
+
+552
+00:42:56,060 --> 00:43:01,450
+النقاط ال X و Y المشتركة على اللي هو البلينأو جدت
+
+553
+00:43:01,450 --> 00:43:05,230
+بمعنى آخر النقاط X وY على خط التقاطع اللي بينهم
+
+554
+00:43:05,230 --> 00:43:10,410
+وكان عنده نقطة Zero نقطة محددة اللي هي برضه هتكون
+
+555
+00:43:10,410 --> 00:43:16,450
+على خط التقاطع بين المستويين إذا إيجاد خط التقاطع
+
+556
+00:43:16,450 --> 00:43:20,350
+بين المستويين نقطة على خط التقاطع هي هنا وضحناها
+
+557
+00:43:20,930 --> 00:43:26,170
+عندى نقطة معروفة و الـ vector معروف الموازلة اللى
+
+558
+00:43:26,170 --> 00:43:29,290
+هو خط التفاطع اللى أوددناه في المثال السابق اتت
+
+559
+00:43:29,290 --> 00:43:33,590
+اتين مع بعض بيعملنا اللى هو ال X بيساوي X نُو
+
+560
+00:43:33,590 --> 00:43:40,640
+التلاتة زائد اللى هىاللي هي معامل الـ vector 14T Y
+
+561
+00:43:40,640 --> 00:43:47,580
+بيساوي ناقص واحد Y0 زائد معامل اللي هي T في V1 V2
+
+562
+00:43:47,580 --> 00:43:55,280
+اللي هو 2T زائد اللي هو Z بيساوي 0 زائد T في V3
+
+563
+00:43:55,280 --> 00:43:59,240
+اللي هي 15 بتصير 15T هذي بتصير البرومتريك
+
+564
+00:43:59,240 --> 00:44:01,280
+إيكويجينز البرومتريك إيكويجينز for the line of the
+
+565
+00:44:01,280 --> 00:44:06,260
+intersection of the two Eblets الانبقولي find the
+
+566
+00:44:06,260 --> 00:44:09,900
+point where the line x بالساوية كده و y بالساوية
+
+567
+00:44:09,900 --> 00:44:15,040
+كده و z بالساوية كده intersects the plane اللي هو
+
+568
+00:44:15,040 --> 00:44:21,780
+معدلته معاطية ايش بده بده you did اللي هي النقطة
+
+569
+00:44:21,780 --> 00:44:27,120
+المشتركة بين الخط المستقيم و بين مين اللي هو ال
+
+570
+00:44:27,120 --> 00:44:32,000
+plane طبعا ال plane اللي خط المستقيم يا بوازي ال
+
+571
+00:44:32,000 --> 00:44:37,560
+plane يا بوازي ال planeيا بقطع الـ Plain في نقطة
+
+572
+00:44:37,560 --> 00:44:43,240
+يا ما بيقع عليه الان احنا ماخدين مثال اللي هو في
+
+573
+00:44:43,240 --> 00:44:46,900
+الواقع المثال اللي ماخدينه بس نقطة تقاطع واحدة مع
+
+574
+00:44:46,900 --> 00:44:51,980
+اللي لا بوازي ولا بيقع عليه لتنشوف كيف نوجد نقطة
+
+575
+00:44:51,980 --> 00:44:57,580
+التقاطع بين خط و Plain الان عشان النقطة عشان اللي
+
+576
+00:44:57,580 --> 00:45:02,080
+هي النقطة هذهتقع على الخط المستقيم و تقع على الـ
+
+577
+00:45:02,080 --> 00:45:07,200
+plane لازم تحقق الخط المستقيم و تحقق من محادرة الـ
+
+578
+00:45:07,200 --> 00:45:12,460
+plane فالان النقطة تلقى 8 ع 3 او 2T نقص 2 نقص T
+
+579
+00:45:12,460 --> 00:45:20,580
+واحد زي ال T عشان نوجد قيمة اللي هي النقطة هذه
+
+580
+00:45:20,580 --> 00:45:24,000
+اللي بتقع على ال plane لازم نعوض هذه في ال plane
+
+581
+00:45:24,000 --> 00:45:28,970
+اللي هي بدلل ال T و نوجد قيمة Tقيمة T اللي بتحقق
+
+582
+00:45:28,970 --> 00:45:35,610
+المعادلة هذه هي اللي بتخللي النقطة هذه ابتقى على
+
+583
+00:45:35,610 --> 00:45:40,830
+الـ Blade إذاً هذه النقطة بتعوضها هنا يعني بتتحقق
+
+584
+00:45:40,830 --> 00:45:46,910
+طرفي المعادلة فبصير الآن 3 في X هنا اللي هي 8 ع 3
+
+585
+00:45:46,910 --> 00:45:53,110
+زاد 2Tزائد اتنين في y هي نقص اتنين T زائد ستة في
+
+586
+00:45:53,110 --> 00:45:54,810
+واحد زائد T بساوي ستة
+
+587
+00:45:59,220 --> 00:46:06,400
+لازم اتحقق طرفي المعادلة عشان تكون على الـ plane 8
+
+588
+00:46:06,400 --> 00:46:13,960
+زي 6t لأن ضربت تصبح 8 زي 6t نقص 4t زي 6 زي 6t 6t
+
+589
+00:46:13,960 --> 00:46:18,900
+بساوي 6 بسطتها طلعت 8t ايش بساوي سالب 8 طبعا
+
+590
+00:46:18,900 --> 00:46:24,840
+بتعرفوا اتبسطوها 6t و 6t اللي هو عبارة عن 12t نقص
+
+591
+00:46:24,840 --> 00:46:30,750
+4 طلعت 8tو6 و8 و14 و6 و8 و14 و6 و8 و14 و6 و8 و14
+
+592
+00:46:30,750 --> 00:46:33,670
+و6 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+593
+00:46:33,670 --> 00:46:33,730
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+594
+00:46:33,730 --> 00:46:36,310
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+595
+00:46:36,310 --> 00:46:36,390
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+596
+00:46:42,140 --> 00:46:48,120
+و الـ Y هذه و الـ Z هذه تكون هنا و تكون هنا في نفس
+
+597
+00:46:48,120 --> 00:46:51,780
+الوقت يعني therefore the point where the line
+
+598
+00:46:51,780 --> 00:46:57,640
+intersects the plane is X و Y و Z اللي عندنا اللي
+
+599
+00:46:57,640 --> 00:47:02,580
+هي هذه عند T بتساوي سالب واحد اللي عبارة عن تلتين
+
+600
+00:47:02,580 --> 00:47:07,220
+و اتنين و سفر و مش مصدقين عوضه هذه هان هتلاقيها
+
+601
+00:47:07,220 --> 00:47:12,540
+بتحقق طرفي المعادلةوعوضها دي هان هتلاقيها إحدى
+
+602
+00:47:12,540 --> 00:47:17,260
+نقاط اللي هي ال line
+
+603
+00:47:17,260 --> 00:47:23,360
+اللي طبعا أي قيمة التي ناخدها بتبقى XY وZ اللي
+
+604
+00:47:23,360 --> 00:47:27,100
+بتطلع عبارة عن نقطة عالمية على الخط المستقيم وهذا
+
+605
+00:47:27,100 --> 00:47:32,120
+من ضمنهاطيب ديجينا اللي هو نقطة أخرى الـ distance
+
+606
+00:47:32,120 --> 00:47:37,480
+from a point to قابلين الان اوجدنا في الأول اللي
+
+607
+00:47:37,480 --> 00:47:41,860
+هو المسافة بين نقطة وبين خط مستقيم في اللي قبل إذا
+
+608
+00:47:41,860 --> 00:47:46,020
+بتتذكر استخدمنا الـ cross product في الإيجاد هنا
+
+609
+00:47:46,020 --> 00:47:49,360
+يعني اللي .. اللي .. اللي .. اللي .. اللي هيسعفنا
+
+610
+00:47:49,360 --> 00:47:53,180
+في الأمر اللي هو الـ cross product و هنشوف كيف
+
+611
+00:47:53,180 --> 00:47:57,300
+اللي هو بسعفناالان لو كانت P is a point on a plane
+
+612
+00:47:57,300 --> 00:48:01,380
+with normal N يعني معروفة نقطة P وين علي البلان و
+
+613
+00:48:01,380 --> 00:48:05,080
+ال normal للبلان معروف طبعا مجرد معرفة معدلة
+
+614
+00:48:05,080 --> 00:48:08,080
+البلان و نعرف ال normal عليه then the distance
+
+615
+00:48:08,080 --> 00:48:11,840
+from any point S لو كانت point S فوق و هي في عند
+
+616
+00:48:11,840 --> 00:48:15,620
+بلان و بدنا نوجد المسافة بين S و بين مين البلان
+
+617
+00:48:15,620 --> 00:48:20,820
+الان باجي باخد نقطة عشوائية P علي البلان بوجدها و
+
+618
+00:48:20,820 --> 00:48:26,360
+بوجد بيه Sو بكون عارف الـ Normal و بضرب ال BS في
+
+619
+00:48:26,360 --> 00:48:29,620
+ال Normal على ال magnitude لل N أنا بتدعي انه
+
+620
+00:48:29,620 --> 00:48:33,260
+هيطلع ايش ماله ال distance ال Normal طبعا ايش هو
+
+621
+00:48:33,260 --> 00:48:36,960
+زي ما اتفجنا للمعادلة العامة AI زائد BJ زائد CK
+
+622
+00:48:36,960 --> 00:48:42,720
+لأنه اصلا هذا هتلاقي اصلا BS.N على N هو عبارة عن
+
+623
+00:48:42,720 --> 00:48:46,180
+ال magnitude لهذا في ال magnitude لهذا في ال
+
+624
+00:48:46,180 --> 00:48:49,240
+cosine الزاوية اللي بينهمهذه اللي بتروح مع هذه
+
+625
+00:48:49,240 --> 00:48:52,140
+فبصير ال distance هو عبارة عن ال BS في cosine
+
+626
+00:48:52,140 --> 00:48:57,800
+الزاوية اللي بين ال BS و ال normal شوف هذا الكلام
+
+627
+00:48:57,800 --> 00:49:04,760
+عمليا بمثال هاي المثال شوف الآن هاي النقطة S وهي
+
+628
+00:49:04,760 --> 00:49:10,520
+ال plain ماشي الحالة الان باجي بختار نقطة عشوائية
+
+629
+00:49:10,520 --> 00:49:17,320
+عليها و بسميها B ماشي الان هذا اللي هو ال BS ماشي
+
+630
+00:49:17,810 --> 00:49:21,370
+وهذا ال vector ال normal العمودي عليه معروف ال
+
+631
+00:49:21,370 --> 00:49:25,390
+normal معروف هذا بيصير الآن هذه الزاوية اللي بينهم
+
+632
+00:49:25,390 --> 00:49:30,550
+فتة اسمها فبصير ال distance هذه عبارة عن ال
+
+633
+00:49:30,550 --> 00:49:36,370
+magnitude لل بي اس في cosine الزاوية اللي بينهم
+
+634
+00:49:36,370 --> 00:49:41,080
+ماشي الحالهذا الطول من هنا الان هو عبارة عن الـ
+
+635
+00:49:41,080 --> 00:49:44,800
+magnitude للـ Bs في كوسينة زاوية الـPin and Theta
+
+636
+00:49:44,800 --> 00:49:49,860
+اللي هو في الواقع عبارة عن مين؟ عن الـBs كفكتور
+
+637
+00:49:49,860 --> 00:49:56,640
+ضبط الـN كفكتور على ال magnitude للـN كفكتور بتطلع
+
+638
+00:49:56,640 --> 00:50:00,240
+اللي هي مين؟ اللي هو الطول من هنا الان زي ما قلنا
+
+639
+00:50:00,540 --> 00:50:04,080
+الان الطول منها اللهان هو اه هو المطلوب ال
+
+640
+00:50:04,080 --> 00:50:08,520
+distance لأنه لما نزل عمودي منها ان هذا بوازي هذا
+
+641
+00:50:08,520 --> 00:50:12,800
+و هذا بساوي اللي تحت هذا اذا هذا بساوي هذا اذا هذا
+
+642
+00:50:12,800 --> 00:50:17,120
+منها اللهان هي ال distance بالظبط هي هذه اللي هي
+
+643
+00:50:17,120 --> 00:50:23,020
+هتساوي ال and ضدالـ BS على الـ magnitude لـ N إذاً
+
+644
+00:50:23,020 --> 00:50:27,020
+هذا الـ distance نشوف الكلام هذا عمليًا الان نجد
+
+645
+00:50:27,020 --> 00:50:30,520
+الـ example اللي عندنا نجد الـ distance from S to
+
+646
+00:50:30,520 --> 00:50:35,520
+the plane 3X زي 2Y زي 6Z إيش بيساوي؟ ستة الان زي
+
+647
+00:50:35,520 --> 00:50:39,680
+ما قلنا أول شي بنجد نقطة بيه نجد نقطة بيه في الـ
+
+648
+00:50:39,680 --> 00:50:43,460
+plane و نتكلم عن طريقة الـ vector of BS بيقولك إيش
+
+649
+00:50:43,460 --> 00:50:49,480
+بتساوي بالظبط اللي قلته on to a vector Nعندي الـ
+
+650
+00:50:49,480 --> 00:50:54,740
+Plane هيه مظبوط لأن 3x زي 2y زي 6z بالساوية 6 إيش
+
+651
+00:50:54,740 --> 00:50:59,360
+الـ Normal عليه؟ معاملاتها ليه 3i زي 2j زي 6k إذا
+
+652
+00:50:59,360 --> 00:51:03,560
+أوجدنا الـ N اللي في القانون دال علي أوجد من الـ
+
+653
+00:51:03,560 --> 00:51:07,180
+Point Bلأن الـ magnitude لن بلزمنا كمان ال
+
+654
+00:51:07,180 --> 00:51:08,180
+magnitude لن بلزمنا كمان ال magnitude لن بلزمنا
+
+655
+00:51:08,180 --> 00:51:08,880
+بلزمنا كمان ال magnitude لن بلزمنا كمان ال
+
+656
+00:51:08,880 --> 00:51:09,300
+magnitude لن بلزمنا كمان ال magnitude لن بلزمنا
+
+657
+00:51:09,300 --> 00:51:10,440
+كمان ال magnitude لن بلزمنا كمان ال magnitude لن
+
+658
+00:51:10,440 --> 00:51:12,920
+بلزمنا كمان ال magnitude لن بلزمنا كمان ال
+
+659
+00:51:12,920 --> 00:51:21,020
+magnitude لن بلزمنا كمان ال magnitude لن بلزمنا
+
+660
+00:51:21,020 --> 00:51:24,360
+كمان ال magnitude لن بلزمنا كمان
+
+661
+00:51:32,610 --> 00:51:39,270
+أسهل أن تتعوض بصفر وصفر سواء X و Z بصفر أو Y و X
+
+662
+00:51:39,270 --> 00:51:42,770
+بصفر أو Y و Z بصفر و تجد النقطة الثالثة فتكون
+
+663
+00:51:42,770 --> 00:51:45,050
+النقطة بتحقق تمين المعادلة
+
+664
+00:51:48,030 --> 00:51:52,930
+عوضه مثلا z بصفر زي ما هو معوض و x بصفر بطلع 2 و y
+
+665
+00:51:52,930 --> 00:51:56,130
+بالساوية 6 يعني و y بالساوية 3 يعني النقطة اللي
+
+666
+00:51:56,130 --> 00:51:59,570
+احنا عوضنا عليها صفر و ثلاثة و صفر واحد تاني جالب
+
+667
+00:51:59,570 --> 00:52:03,570
+ديش عامد هيك خد x بصفر و y بصفر و z بصير ايش
+
+668
+00:52:03,570 --> 00:52:07,210
+حتساوي واحد بقوله صح واحد تاني اخد ارقام تانية زي
+
+669
+00:52:07,210 --> 00:52:11,780
+ما بدك المهم الارقام اللي بتاخد تتحقق مينلما اتعوض
+
+670
+00:52:11,780 --> 00:52:16,560
+X و Y عشوائية تبقى الـ Z توجدها بالحل عشان تحقق
+
+671
+00:52:16,560 --> 00:52:22,460
+طرفي المعادلة إذا صارت لدي ال B موجودة و ال S
+
+672
+00:52:22,460 --> 00:52:26,400
+موجودة معطوعة في السؤال بيصير لدي ال B S بساوي
+
+673
+00:52:26,400 --> 00:52:31,080
+اللي هو بي اس اللي هي واحد ناقص صفر في اتجاه ال I
+
+674
+00:52:31,080 --> 00:52:36,900
+واحد ناقص تلاتة في اتجاه ال Jتلاتة ناقص سفر في
+
+675
+00:52:36,900 --> 00:52:41,560
+اتجاه الـ K بيطلع عندى عبارة عن I ناقص اتنين J
+
+676
+00:52:41,560 --> 00:52:45,480
+زائد تلاتة K هذا ال BS جهزته لأموراللي انا بدي
+
+677
+00:52:45,480 --> 00:52:50,080
+اوجد الـ BS ضد مين الـ N و بعدين بجسمه على المين
+
+678
+00:52:50,080 --> 00:52:53,240
+على ال magnitude لل N بكون هو ال distance المطلوبة
+
+679
+00:52:53,240 --> 00:53:00,380
+إذا ال N بقول ال magnitude لأ هذه ال N بي اس هي ضد
+
+680
+00:53:00,380 --> 00:53:04,960
+مين ال N مضروب في اللي هو ال magnitude واحد على ال
+
+681
+00:53:04,960 --> 00:53:09,660
+magnitude لل N ال N طوله قداش سبعة بيصير واحد على
+
+682
+00:53:09,660 --> 00:53:12,400
+سبعة في ال dot product بينهم اوجد ال dot product
+
+683
+00:53:12,400 --> 00:53:21,940
+اللي هي تلاتةنقص اربعة بطلع ناقص واحد زائد تمانتعش
+
+684
+00:53:21,940 --> 00:53:26,800
+اللي هو ياش سبعتعش سبعتعش مضروب في مين على سبعة
+
+685
+00:53:26,800 --> 00:53:30,900
+اذا عملنا الدور التفرضات طالع عندي بالشكل اللي
+
+686
+00:53:30,900 --> 00:53:35,620
+أمامي اللي هو تلاتة و مضروبة في سبعة بصير تلاتة ع
+
+687
+00:53:35,620 --> 00:53:39,060
+سبعة ناقص اربعة و مضروبة في سبعة ع سبعة ناقص اربعة
+
+688
+00:53:39,060 --> 00:53:42,160
+ع سبعة و تلاتة في ستة بتمانتعش مضروبة في سبعة بطلع
+
+689
+00:53:42,160 --> 00:53:46,690
+تمانتعش ع سبعةهذه المسافة
+
+690
+00:53:46,690 --> 00:53:47,870
+المطلوبة
+
+691
+00:53:50,530 --> 00:53:55,650
+the angle between blends اللي هي أخر مقتهد ان شاء
+
+692
+00:53:55,650 --> 00:54:01,610
+الله يا جماعة اللي هي الزاوية بين مين بين مستويين
+
+693
+00:54:01,610 --> 00:54:07,050
+الزاوية بين مستويين الان عندى نفترض ان المستويين
+
+694
+00:54:07,050 --> 00:54:11,790
+هي المستوى الأول وهي المستوى الثاني ومتقطعات اللي
+
+695
+00:54:11,790 --> 00:54:16,240
+هو في خط مستقيم وعملن زاوية بينهم التانيةالزاوية
+
+696
+00:54:16,240 --> 00:54:20,260
+اللي بين الاتنين بدنا نعرفها اللي هي عبارة عن
+
+697
+00:54:20,260 --> 00:54:25,040
+الزاوية الحادة بين الـ Normal للأول و Normal
+
+698
+00:54:25,040 --> 00:54:28,980
+للتاني يعني الزاوية الحادة بين ال Normal الأول و
+
+699
+00:54:28,980 --> 00:54:34,440
+Normal التاني هي الزاوية اللي بنسميها بين مستوين
+
+700
+00:54:43,540 --> 00:54:48,780
+الزاوية بين مستويين هي الزاوية الحادة بين
+
+701
+00:54:48,780 --> 00:54:54,720
+العموديين عليهمالان بناء على هذا التعريف بدنا نوجد
+
+702
+00:54:54,720 --> 00:54:58,620
+find the angle between the planes 3x-6y-2z بالساعة
+
+703
+00:54:58,620 --> 00:55:04,640
+15 and the plane 2xy-2z بالساعة 5 الزاوية بينهم
+
+704
+00:55:04,640 --> 00:55:08,560
+اذا عشان وجدهم بدنا نوجده normal للأول سهلالـ N1
+
+705
+00:55:08,560 --> 00:55:12,580
+بيساوي 3I نقص 6J نقص 2K بتعرف 6 بتعرفه عن غيره
+
+706
+00:55:12,580 --> 00:55:14,980
+طبعا المعاملات اللي هي ال X في هذه المعادلة
+
+707
+00:55:14,980 --> 00:55:17,940
+ومعاملات ال Y ومعاملة ال Z بتمثل ال Normal N
+
+708
+00:55:17,940 --> 00:55:24,520
+بيساوي 2I زائد J نقص 2K هدول الان two vectors
+
+709
+00:55:24,520 --> 00:55:29,970
+الزاوية اللي بينهم هي عبارة عن مينعن اللي هو
+
+710
+00:55:29,970 --> 00:55:34,050
+الزاوية بين المستويين و احنا بنعرف نوجد الزاوية
+
+711
+00:55:34,050 --> 00:55:36,730
+بينهم اللي هو ايش بتساوي اخد نهجة باللي هيك
+
+712
+00:55:36,730 --> 00:55:38,890
+الزاوية بين المستويين هي عبارة عن θ بساوي cosine
+
+713
+00:55:38,890 --> 00:55:43,070
+inverse لل N1 dot N2 عن magnitude لل N1 في
+
+714
+00:55:43,070 --> 00:55:46,910
+magnitude لل N2 عارفين ليش؟ لإن احنا بنعرف ال N1
+
+715
+00:55:46,910 --> 00:55:51,650
+في N2 بساوي اللي هو ال magnitude لهذه في ال
+
+716
+00:55:51,650 --> 00:55:54,610
+magnitude لهذه في cosine الزاوية اللي بينهممش هيك
+
+717
+00:55:54,610 --> 00:55:58,990
+تعرفي ضد product n1 فn2 بيساوي اللي هو cos θ في
+
+718
+00:55:58,990 --> 00:56:01,250
+magnitude ده دي في magnitude ده دي الـ nθ بيساوي
+
+719
+00:56:01,250 --> 00:56:05,370
+cos انفرسلها واخدنا القانون قبل هيك لأن n1 ضد n2
+
+720
+00:56:05,370 --> 00:56:09,410
+من عرف نضرب على طول تلاتة في اتنين بستة ستة ناقص
+
+721
+00:56:09,410 --> 00:56:13,210
+ستة اللي هي بروح مع بعض ناقص اتنين في ناقص اتنين
+
+722
+00:56:13,210 --> 00:56:17,150
+بيطلع مين اربعة إذا نضد product بين n1 وn2 اربعة
+
+723
+00:56:17,150 --> 00:56:22,790
+على ال magnitude للأول جدري التربيعي تسعة وستة
+
+724
+00:56:26,260 --> 00:56:31,000
+36.4497 يعني مظبوط و ال magnitude لهذا يعني طوله
+
+725
+00:56:31,000 --> 00:56:35,580
+هذا 7أعملناها قبل شوية وهذا أربعة و واحد خمسة و
+
+726
+00:56:35,580 --> 00:56:39,340
+أربعة تسعة طوله جداش جدر التسعة اللي هو تلاتة إذا
+
+727
+00:56:39,340 --> 00:56:43,480
+طول هذا تلاتة و طول هذا سبعة مع بعض حصل ضرب هنا
+
+728
+00:56:43,480 --> 00:56:46,600
+اللي هو واحد و عشرين فبصير theta بساوية cosine
+
+729
+00:56:46,600 --> 00:56:48,820
+inverse أربعة على واحد و عشرين اللي هي تقريبا
+
+730
+00:56:48,820 --> 00:56:52,560
+بساوية هذا الرقم يعني صار عندى الآن الزاوية بين
+
+731
+00:56:52,560 --> 00:56:57,870
+المستويين هذه هي الزاوية اللي عندناوبهيك بنكون
+
+732
+00:56:57,870 --> 00:57:01,110
+خلصنا المحاضرة وبرك الله فيكم وإلى لقاء نفسنا
+
+733
+00:57:01,110 --> 00:57:03,070
+السلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..8f519f7bdd80e30b6b339f79420bb853d29460c0
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2296, "start": 0.66, "end": 22.96, "text": "بسم الله الرحمن الرحيم هذه هي المحاضرة الخامسة مساق تفاضل وتقام الجيم بعنوان lines and planes in space اليوم بدنا نعرف كيف نوجد معادلة الخط المستقيم ومعادلة المستوى في ال space ومتعلقاتها", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 29538, 39896, 9673, 5016, 46958, 25720, 33962, 10943, 3794, 3660, 47524, 995, 4587, 6055, 5172, 46958, 1211, 34683, 4587, 10943, 25724, 32640, 45030, 1863, 2407, 7649, 3876, 293, 14952, 294, 1901, 45595, 20498, 47525, 8315, 8717, 3615, 28480, 9122, 33911, 8717, 29245, 3215, 20449, 18513, 37977, 33962, 9566, 9673, 14851, 4587, 32640, 4032, 2304, 3615, 18513, 37977, 9673, 14851, 2407, 7578, 8978, 2423, 1901, 4032, 2304, 2655, 30241, 4587, 9307, 11296], "avg_logprob": -0.0954861111111111, "compression_ratio": 1.7135135135135136, "no_speech_prob": 1.9669532775878906e-06, "words": [{"start": 0.66, "end": 0.98, "word": "بسم", "probability": 0.86376953125}, {"start": 0.98, "end": 1.24, "word": " الله", "probability": 0.94677734375}, {"start": 1.24, "end": 1.76, "word": " الرحمن", "probability": 0.9814453125}, {"start": 1.76, "end": 2.42, "word": " الرحيم", "probability": 0.9962565104166666}, {"start": 2.42, "end": 3.82, "word": " هذه", "probability": 0.7958984375}, {"start": 3.82, "end": 4.12, "word": " هي", "probability": 0.529296875}, {"start": 4.12, "end": 4.9, "word": " المحاضرة", "probability": 0.975341796875}, {"start": 4.9, "end": 6.3, "word": " الخامسة", "probability": 0.971435546875}, {"start": 6.3, "end": 7.02, "word": " مساق", "probability": 0.8001302083333334}, {"start": 7.02, "end": 7.62, "word": " تفاضل", "probability": 0.951904296875}, {"start": 7.62, "end": 8.22, "word": " وتقام", "probability": 0.7423502604166666}, {"start": 8.22, "end": 8.76, "word": " الجيم", "probability": 0.771728515625}, {"start": 8.76, "end": 11.02, "word": " بعنوان", "probability": 0.90966796875}, {"start": 11.02, "end": 11.84, "word": " lines", "probability": 0.5927734375}, {"start": 11.84, "end": 12.34, "word": " and", "probability": 0.9501953125}, {"start": 12.34, "end": 13.1, "word": " planes", "probability": 0.93359375}, {"start": 13.1, "end": 13.38, "word": " in", "probability": 0.9580078125}, {"start": 13.38, "end": 13.82, "word": " space", "probability": 0.92333984375}, {"start": 13.82, "end": 14.26, "word": " اليوم", "probability": 0.923583984375}, {"start": 14.26, "end": 14.6, "word": " بدنا", "probability": 0.823974609375}, {"start": 14.6, "end": 15.1, "word": " نعرف", "probability": 0.9890950520833334}, {"start": 15.1, "end": 15.48, "word": " كيف", "probability": 0.973388671875}, {"start": 15.48, "end": 16.2, "word": " نوجد", "probability": 0.8235677083333334}, {"start": 16.2, "end": 17.42, "word": " معادلة", "probability": 0.9890950520833334}, {"start": 17.42, "end": 17.92, "word": " الخط", "probability": 0.977294921875}, {"start": 17.92, "end": 18.82, "word": " المستقيم", "probability": 0.9857177734375}, {"start": 18.82, "end": 19.82, "word": " ومعادلة", "probability": 0.90703125}, {"start": 19.82, "end": 20.74, "word": " المستوى", "probability": 0.9730224609375}, {"start": 20.74, "end": 21.0, "word": " في", "probability": 0.939453125}, {"start": 21.0, "end": 21.06, "word": " ال", "probability": 0.94091796875}, {"start": 21.06, "end": 21.6, "word": " space", "probability": 0.94482421875}, {"start": 21.6, "end": 22.96, "word": " ومتعلقاتها", "probability": 0.9440569196428571}], "temperature": 1.0}, {"id": 2, "seek": 4705, "start": 23.67, "end": 47.05, "text": "الان احنا .. ده طبعا في section 12-5 احنا سابقا في ال .. في ال plane عرفنا معادلة الخط المستقيم و كنا جوا معادلة الخط المستقيم تحدد اللي هو بمعرفة mail و نقطة او نقطين الان الكلام هنا مشابه الان بدنا نعرف ايش .. كيف نوجد معادلة الخط المستقيم lines", "tokens": [6027, 7649, 1975, 5016, 8315, 4386, 11778, 3224, 23032, 3555, 3615, 995, 8978, 3541, 2272, 12, 20, 1975, 5016, 8315, 8608, 16758, 4587, 995, 8978, 2423, 4386, 8978, 2423, 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معروفة الان هذا ال vector B note بيه هو اللي بسميه مين اللي هو T في V اللي اتفجنا عليه فالان المعادلة ممكن نكتبها على الصورة التالية على صورة vectors اللي هو معادلة الخط المستقيم على صورة vectors ايش هو", "tokens": [6027, 7649, 4032, 23758, 2423, 8062, 9154, 34105, 2423, 7578, 34105, 4724, 16254, 6055, 38251, 1829, 3224, 497, 3637, 13672, 1829, 31439, 13672, 1829, 39894, 30544, 4386, 13672, 1829, 4724, 1829, 30544, 20449, 32887, 5172, 32239, 8592, 20449, 32887, 5172, 22807, 5296, 33456, 3224, 363, 3637, 20449, 32887, 5172, 3660, 2423, 7649, 23758, 2423, 8062, 363, 3637, 4724, 1829, 3224, 31439, 13672, 1829, 4724, 38251, 1829, 3224, 3714, 9957, 13672, 1829, 31439, 314, 8978, 691, 13672, 1829, 1975, 2655, 5172, 7435, 8315, 47356, 6156, 6027, 7649, 9673, 3615, 18513, 37977, 3714, 43020, 8717, 4117, 2655, 3555, 11296, 15844, 31767, 13063, 3660, 16712, 6027, 10632, 15844, 20328, 13063, 3660, 18875, 13672, 1829, 31439, 20449, 18513, 37977, 33962, 9566, 9673, 14851, 4587, 32640, 15844, 20328, 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"text": "الـ vector equation أو معادلة الخط المستقيم بواسطة أو بدلالة الـ vectors", "tokens": [6027, 39184, 8062, 5367, 34051, 20449, 18513, 37977, 33962, 9566, 9673, 14851, 4587, 32640, 4724, 14407, 3794, 9566, 3660, 34051, 47525, 1211, 6027, 3660, 2423, 39184, 18875], "avg_logprob": -0.32198661459343775, "compression_ratio": 1.1666666666666667, "no_speech_prob": 4.589557647705078e-06, "words": [{"start": 423.19, "end": 423.57, "word": "الـ", "probability": 0.4267578125}, {"start": 423.57, "end": 423.85, "word": " vector", "probability": 0.441650390625}, {"start": 423.85, "end": 424.43, "word": " equation", "probability": 0.95068359375}, {"start": 424.43, "end": 424.63, "word": " أو", "probability": 0.72802734375}, {"start": 424.63, "end": 425.25, "word": " معادلة", "probability": 0.9677734375}, {"start": 425.25, "end": 425.69, "word": " الخط", "probability": 0.98095703125}, {"start": 425.69, "end": 426.47, "word": " المستقيم", "probability": 0.973876953125}, {"start": 426.47, "end": 429.05, "word": " بواسطة", "probability": 0.793896484375}, {"start": 429.05, "end": 429.21, "word": " أو", "probability": 0.25341796875}, {"start": 429.21, "end": 429.97, "word": " بدلالة", "probability": 0.840087890625}, {"start": 429.97, "end": 430.39, "word": " الـ", "probability": 0.45947265625}, {"start": 430.39, "end": 431.29, "word": " vectors", "probability": 0.7978515625}], "temperature": 1.0}, {"id": 20, "seek": 45400, "start": 434.42, "end": 454.0, "text": "بمعرفة النقطة P0 و V بتساوي R of T زي ما قلنا بساوي R0 of T هي R0 طبعا هتطلع لنا مباشرة هي عدد زائد مين T في V هذي As of R0 لأنها بدلالة باسمين P0 أما R of D هتكون بدلالة اتجاه T", "tokens": [3555, 2304, 3615, 28480, 3660, 28239, 47432, 3660, 430, 15, 4032, 691, 4724, 2655, 3794, 995, 45865, 497, 295, 314, 30767, 1829, 19446, 12174, 1211, 8315, 4724, 3794, 995, 45865, 497, 15, 295, 314, 39896, 497, 15, 23032, 3555, 3615, 995, 8032, 2655, 9566, 1211, 3615, 5296, 8315, 3714, 3555, 33599, 25720, 39896, 6225, 3215, 3215, 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"probability": 0.790380859375}, {"start": 442.36, "end": 442.66, "word": " لنا", "probability": 0.76220703125}, {"start": 442.66, "end": 443.42, "word": " مباشرة", "probability": 0.9459228515625}, {"start": 443.42, "end": 443.72, "word": " هي", "probability": 0.251708984375}, {"start": 443.72, "end": 444.22, "word": " عدد", "probability": 0.9840494791666666}, {"start": 444.22, "end": 444.9, "word": " زائد", "probability": 0.7048746744791666}, {"start": 444.9, "end": 445.34, "word": " مين", "probability": 0.6591796875}, {"start": 445.34, "end": 445.6, "word": " T", "probability": 0.72265625}, {"start": 445.6, "end": 445.76, "word": " في", "probability": 0.7158203125}, {"start": 445.76, "end": 445.88, "word": " V", "probability": 0.98974609375}, {"start": 445.88, "end": 446.16, "word": " هذي", "probability": 0.4393717447916667}, {"start": 446.16, "end": 446.3, "word": " As", "probability": 0.34423828125}, {"start": 446.3, "end": 446.46, "word": " of", "probability": 0.74609375}, {"start": 446.46, "end": 446.78, "word": " R0", "probability": 0.939453125}, {"start": 446.78, "end": 449.3, "word": " لأنها", "probability": 0.50225830078125}, {"start": 449.3, "end": 449.74, "word": " بدلالة", "probability": 0.9281005859375}, {"start": 449.74, "end": 450.32, "word": " باسمين", "probability": 0.8499755859375}, {"start": 450.32, "end": 450.96, "word": " P0", "probability": 0.94287109375}, {"start": 450.96, "end": 451.54, "word": " أما", "probability": 0.6729736328125}, {"start": 451.54, "end": 451.8, "word": " R", "probability": 0.8955078125}, {"start": 451.8, "end": 451.96, "word": " of", "probability": 0.7333984375}, {"start": 451.96, "end": 452.08, "word": " D", "probability": 0.85009765625}, {"start": 452.08, "end": 452.68, "word": " هتكون", "probability": 0.76171875}, {"start": 452.68, "end": 453.38, "word": " بدلالة", "probability": 0.9844970703125}, {"start": 453.38, "end": 453.82, "word": " اتجاه", "probability": 0.60595703125}, {"start": 453.82, "end": 454.0, 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اللى اشتقناه هو قانون المسافة", "tokens": [6027, 4587, 7649, 11536, 13672, 7578, 1975, 8592, 2655, 4587, 8315, 3224, 31439, 12174, 7649, 11536, 9673, 3794, 31845, 3660], "avg_logprob": -0.47098213718051, "compression_ratio": 1.1694915254237288, "no_speech_prob": 1.6689300537109375e-06, "words": [{"start": 1242.56, "end": 1243.96, "word": "القانون", "probability": 0.6033172607421875}, {"start": 1243.96, "end": 1244.1, "word": " اللى", "probability": 0.3377685546875}, {"start": 1244.1, "end": 1245.14, "word": " اشتقناه", "probability": 0.74365234375}, {"start": 1245.14, "end": 1245.78, "word": " هو", "probability": 0.270751953125}, {"start": 1245.78, "end": 1246.3, "word": " قانون", "probability": 0.9054361979166666}, {"start": 1246.3, "end": 1247.24, "word": " المسافة", "probability": 0.963134765625}], "temperature": 1.0}, {"id": 58, "seek": 127191, "start": 1247.55, "end": 1271.91, "text": "بين نقطة وبين خط مستقيم اللي هو طبعا بمعلومية النقطة ومعلومية الخط المستقيم عبارة عن الـ BS 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بمعنى آخر لو جيت أخدت الـ cross product أن واحد أن اتنين هيكون عمودي على أن واحد و أن اتنين يعني هذا المتجه أن واحد و cross أن اتنين أشماله بوازي الخط المستقيم تبع التقاطع يعني بوازي خط التقاطع إذا عرفنا الخط التقاطع اللي بندور عليه اعرفنا فيكتور موازي له", "tokens": [40228, 22653, 4724, 2304, 3615, 1863, 7578, 19753, 34740, 45164, 10874, 36081, 5551, 9778, 3215, 2655, 2423, 39184, 3278, 1674, 14739, 36764, 24401, 14739, 1975, 2655, 1863, 9957, 39896, 30544, 6225, 2304, 2407, 16254, 15844, 14739, 36764, 24401, 4032, 14739, 1975, 2655, 1863, 9957, 37495, 22653, 23758, 9673, 2655, 7435, 3224, 14739, 36764, 24401, 4032, 3278, 14739, 1975, 2655, 1863, 9957, 5551, 8592, 2304, 6027, 3224, 4724, 2407, 31377, 1829, 33962, 9566, 9673, 14851, 4587, 32640, 6055, 3555, 3615, 16712, 4587, 41193, 3615, 37495, 22653, 4724, 2407, 31377, 1829, 16490, 9566, 16712, 4587, 41193, 3615, 11933, 15730, 6225, 28480, 8315, 33962, 9566, 16712, 4587, 41193, 3615, 13672, 1829, 4724, 41260, 13063, 47356, 1975, 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2216.89, "word": " a", "probability": 0.974609375}, {"start": 2216.89, "end": 2217.29, "word": " vector", "probability": 0.95947265625}, {"start": 2217.29, "end": 2217.85, "word": " parallel", "probability": 0.89892578125}, {"start": 2217.85, "end": 2218.17, "word": " to", "probability": 0.97705078125}, {"start": 2218.17, "end": 2218.33, "word": " the", "probability": 0.80810546875}, {"start": 2218.33, "end": 2218.65, "word": " line", "probability": 0.9365234375}, {"start": 2218.65, "end": 2218.81, "word": " of", "probability": 0.95751953125}, {"start": 2218.81, "end": 2219.47, "word": " intersection", "probability": 0.8291015625}, {"start": 2219.47, "end": 2220.43, "word": " of", "probability": 0.9697265625}, {"start": 2220.43, "end": 2220.71, "word": " the", "probability": 0.78369140625}, {"start": 2220.71, "end": 2221.55, "word": " planes", "probability": 0.921875}, {"start": 2221.55, "end": 2222.25, "word": " كذا", "probability": 0.72607421875}, {"start": 2222.25, "end": 2222.51, "word": " and", "probability": 0.77099609375}, {"start": 2222.51, "end": 2222.81, "word": " كذا", "probability": 0.962890625}, {"start": 2222.81, "end": 2223.31, "word": " بده", "probability": 0.80029296875}, {"start": 2223.31, "end": 2223.61, "word": " اول", "probability": 0.8818359375}, {"start": 2223.61, "end": 2223.97, "word": " اشي", "probability": 0.7674153645833334}, {"start": 2223.97, "end": 2224.37, "word": " اشي", "probability": 0.68212890625}, {"start": 2224.37, "end": 2224.77, "word": " يوجد", "probability": 0.833984375}, {"start": 2224.77, "end": 2225.09, "word": " هو", "probability": 0.98046875}], "temperature": 1.0}, {"id": 100, "seek": 225457, "start": 2225.73, "end": 2254.57, "text": "الذي أريد أن أعطيكمه على خطوط اولا يوجد الـ vector ثم يوجد معادلة الخط المستقيم الخط المستقيم يجب أن نعرف أنه يوجد معادلة الخط المستقيم إذا وجدنا ال vector ونعرف ال V فأخذنا أمثلة على إيجاد معادلة الخط المستقيم إذا الآن السؤال هذا هو find a vector موازي parallel to the line of 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{"start": 2891.46, "end": 2891.58, "word": " في", "probability": 0.85791015625}, {"start": 2891.58, "end": 2891.84, "word": " عند", "probability": 0.7314453125}, {"start": 2891.84, "end": 2892.32, "word": " بلان", "probability": 0.7208658854166666}, {"start": 2892.32, "end": 2892.66, "word": " و", "probability": 0.86279296875}, {"start": 2892.66, "end": 2892.88, "word": " بدنا", "probability": 0.844970703125}, {"start": 2892.88, "end": 2893.16, "word": " نوجد", "probability": 0.84814453125}, {"start": 2893.16, "end": 2893.68, "word": " المسافة", "probability": 0.9869384765625}, {"start": 2893.68, "end": 2893.9, "word": " بين", "probability": 0.9462890625}, {"start": 2893.9, "end": 2894.22, "word": " S", "probability": 0.654296875}, {"start": 2894.22, "end": 2894.38, "word": " و", "probability": 0.95166015625}, {"start": 2894.38, "end": 2894.62, "word": " بين", "probability": 0.53662109375}, {"start": 2894.62, "end": 2894.92, "word": " مين", "probability": 0.97021484375}, {"start": 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{"start": 2901.2, "end": 2901.52, "word": " بيه", "probability": 0.5550130208333334}, {"start": 2901.52, "end": 2901.82, "word": " S", "probability": 0.9453125}], "temperature": 1.0}, {"id": 130, "seek": 292754, "start": 2902.46, "end": 2927.54, "text": "و بكون عارف الـ Normal و بضرب ال BS في ال Normal على ال magnitude لل N أنا بتدعي انه هيطلع ايش ماله ال distance ال Normal طبعا ايش هو زي ما اتفجنا للمعادلة العامة AI زائد BJ زائد CK لأنه اصلا هذا هتلاقي اصلا BS.N على N هو عبارة عن ال magnitude لهذا في ال magnitude لهذا في ال cosine الزاوية اللي بينهم", "tokens": [2407, 4724, 30544, 6225, 9640, 5172, 2423, 39184, 21277, 4032, 4724, 11242, 25513, 2423, 27253, 8978, 2423, 21277, 15844, 2423, 15668, 24976, 426, 41850, 39894, 3215, 3615, 1829, 16472, 3224, 39896, 9566, 1211, 3615, 1975, 1829, 8592, 3714, 6027, 3224, 2423, 4560, 2423, 21277, 23032, 3555, 3615, 995, 1975, 1829, 8592, 31439, 30767, 1829, 19446, 1975, 2655, 5172, 7435, 8315, 5296, 19528, 3615, 18513, 37977, 18863, 10943, 3660, 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2909.42, "word": " بتدعي", "probability": 0.78240966796875}, {"start": 2909.42, "end": 2909.62, "word": " انه", "probability": 0.6529541015625}, {"start": 2909.62, "end": 2909.98, "word": " هيطلع", "probability": 0.74737548828125}, {"start": 2909.98, "end": 2910.18, "word": " ايش", "probability": 0.6693522135416666}, {"start": 2910.18, "end": 2910.62, "word": " ماله", "probability": 0.89453125}, {"start": 2910.62, "end": 2910.9, "word": " ال", "probability": 0.9765625}, {"start": 2910.9, "end": 2911.5, "word": " distance", "probability": 0.83935546875}, {"start": 2911.5, "end": 2912.1, "word": " ال", "probability": 0.759765625}, {"start": 2912.1, "end": 2912.4, "word": " Normal", "probability": 0.73681640625}, {"start": 2912.4, "end": 2912.74, "word": " طبعا", "probability": 0.9658203125}, {"start": 2912.74, "end": 2912.96, "word": " ايش", "probability": 0.8448893229166666}, {"start": 2912.96, "end": 2913.26, "word": " هو", "probability": 0.97314453125}, {"start": 2913.26, "end": 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"end": 2918.86, "word": " هذا", "probability": 0.6884765625}, {"start": 2918.86, "end": 2919.98, "word": " هتلاقي", "probability": 0.79119873046875}, {"start": 2919.98, "end": 2920.38, "word": " اصلا", "probability": 0.7299397786458334}, {"start": 2920.38, "end": 2920.7, "word": " BS", "probability": 0.93408203125}, {"start": 2920.7, "end": 2921.16, "word": ".N", "probability": 0.6583251953125}, {"start": 2921.16, "end": 2921.36, "word": " على", "probability": 0.80908203125}, {"start": 2921.36, "end": 2921.62, "word": " N", "probability": 0.97802734375}, {"start": 2921.62, "end": 2921.82, "word": " هو", "probability": 0.88916015625}, {"start": 2921.82, "end": 2922.62, "word": " عبارة", "probability": 0.94970703125}, {"start": 2922.62, "end": 2922.72, "word": " عن", "probability": 0.80615234375}, {"start": 2922.72, "end": 2922.82, "word": " ال", "probability": 0.60400390625}, {"start": 2922.82, "end": 2923.22, "word": " magnitude", "probability": 0.97119140625}, {"start": 2923.22, "end": 2923.94, "word": " لهذا", "probability": 0.962646484375}, {"start": 2923.94, "end": 2924.54, "word": " في", "probability": 0.85888671875}, {"start": 2924.54, "end": 2924.64, "word": " ال", "probability": 0.3759765625}, {"start": 2924.64, "end": 2925.0, "word": " magnitude", "probability": 0.250732421875}, {"start": 2925.0, "end": 2925.6, "word": " لهذا", "probability": 0.98828125}, {"start": 2925.6, "end": 2925.78, "word": " في", "probability": 0.92626953125}, {"start": 2925.78, "end": 2926.18, "word": " ال", "probability": 0.60205078125}, {"start": 2926.18, "end": 2926.44, "word": " cosine", "probability": 0.4609375}, {"start": 2926.44, "end": 2926.96, "word": " الزاوية", "probability": 0.9474609375}, {"start": 2926.96, "end": 2927.12, "word": " اللي", "probability": 0.932861328125}, {"start": 2927.12, "end": 2927.54, "word": " بينهم", "probability": 0.95703125}], "temperature": 1.0}, {"id": 131, "seek": 295732, "start": 2928.0, "end": 2957.32, "text": "هذه اللي بتروح مع هذه فبصير ال distance هو عبارة عن ال BS في cosine الزاوية اللي بين ال BS و ال normal شوف هذا الكلام عمليا بمثال هاي المثال شوف الآن هاي النقطة S وهي ال plain ماشي الحالة الان باجي بختار نقطة عشوائية عليها و بسميها B ماشي الان هذا اللي هو ال BS ماشي", "tokens": [3224, 24192, 13672, 1829, 39894, 32887, 5016, 20449, 29538, 6156, 3555, 9381, 13546, 2423, 4560, 31439, 6225, 3555, 9640, 3660, 18871, 2423, 27253, 8978, 23565, 6024, 110, 995, 2407, 10632, 13672, 1829, 49374, 2423, 27253, 4032, 2423, 2710, 13412, 38688, 23758, 2423, 28820, 10943, 6225, 2304, 20292, 995, 4724, 2304, 12984, 6027, 8032, 47302, 9673, 12984, 6027, 13412, 38688, 6024, 48506, 8032, 47302, 28239, 47432, 3660, 318, 37037, 1829, 2423, 11121, 3714, 33599, 1829, 21542, 6027, 3660, 2423, 7649, 4724, 26108, 1829, 4724, 46456, 9640, 8717, 47432, 3660, 6225, 8592, 2407, 16373, 10632, 25894, 11296, 4032, 4724, 38251, 1829, 11296, 363, 3714, 33599, 1829, 2423, 7649, 23758, 13672, 1829, 31439, 2423, 27253, 3714, 33599, 1829], 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فبصير ال distance هذه عبارة عن ال magnitude لل بي اس في cosine الزاوية اللي بينهم ماشي الحال", "tokens": [2407, 3224, 15730, 2423, 8062, 2423, 2710, 18863, 2304, 2407, 16254, 47356, 20449, 32887, 5172, 2423, 2710, 20449, 32887, 5172, 23758, 4724, 1829, 9381, 13546, 6024, 48506, 29538, 6024, 110, 995, 2407, 10632, 13672, 1829, 49374, 16095, 6156, 2655, 3660, 24525, 2304, 11296, 6156, 3555, 9381, 13546, 2423, 4560, 29538, 6225, 3555, 9640, 3660, 18871, 2423, 15668, 24976, 4724, 1829, 24525, 8978, 23565, 6024, 110, 995, 2407, 10632, 13672, 1829, 49374, 16095, 3714, 33599, 1829, 21542, 6027], "avg_logprob": -0.22916666036232924, "compression_ratio": 1.7988826815642458, "no_speech_prob": 1.0132789611816406e-06, "words": [{"start": 2957.81, "end": 2958.39, "word": "وهذا", "probability": 0.7478841145833334}, {"start": 2958.39, "end": 2958.55, "word": " ال", "probability": 0.80029296875}, {"start": 2958.55, "end": 2958.85, "word": " vector", "probability": 0.31884765625}, {"start": 2958.85, 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إذاً هذا الـ distance نشوف الكلام هذا عمليًا الان نجد الـ example اللي عندنا نجد الـ distance from S to the plane 3X زي 2Y زي 6Z إيش بيساوي؟ ستة الان زي ما قلنا أول شي بنجد نقطة بيه نجد نقطة بيه في الـ plane و نتكلم عن طريقة الـ vector of BS بيقولك إيش بتساوي بالظبط اللي قلته on to a vector N", "tokens": [6027, 39184, 27253, 15844, 2423, 39184, 15668, 5296, 39184, 426, 11933, 15730, 14111, 23758, 2423, 39184, 4560, 8717, 8592, 38688, 2423, 28820, 10943, 23758, 6225, 2304, 20292, 14111, 995, 2423, 7649, 8717, 7435, 3215, 2423, 39184, 1365, 13672, 1829, 43242, 8315, 8717, 7435, 3215, 2423, 39184, 4560, 490, 318, 281, 264, 5720, 805, 55, 30767, 1829, 568, 56, 30767, 1829, 1386, 57, 11933, 1829, 8592, 4724, 1829, 3794, 995, 45865, 22807, 8608, 2655, 3660, 2423, 7649, 30767, 1829, 19446, 12174, 1211, 8315, 5551, 12610, 44049, 44945, 7435, 3215, 8717, 47432, 3660, 4724, 1829, 3224, 8717, 7435, 3215, 8717, 47432, 3660, 4724, 1829, 3224, 8978, 2423, 39184, 5720, 4032, 8717, 2655, 4117, 19528, 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3380.63, "word": " التربيعي", "probability": 0.90224609375}, {"start": 3380.63, "end": 3382.01, "word": " تسعة", "probability": 0.951171875}, {"start": 3382.01, "end": 3382.79, "word": " وستة", "probability": 0.9405924479166666}], "temperature": 1.0}, {"id": 152, "seek": 339146, "start": 3386.26, "end": 3391.46, "text": "36.4497 يعني مظبوط و ال magnitude لهذا يعني طوله هذا 7", "tokens": [11309, 13, 19, 14938, 22, 37495, 22653, 3714, 19913, 3555, 2407, 9566, 4032, 2423, 15668, 46740, 15730, 37495, 22653, 23032, 12610, 3224, 23758, 1614], "avg_logprob": -0.46749999046325685, "compression_ratio": 1.051948051948052, "no_speech_prob": 2.968311309814453e-05, "words": [{"start": 3386.26, "end": 3386.94, "word": "36", "probability": 0.1588134765625}, {"start": 3386.94, "end": 3387.62, "word": ".4497", "probability": 0.54327392578125}, {"start": 3387.62, "end": 3388.0, "word": " يعني", "probability": 0.513946533203125}, {"start": 3388.0, "end": 3389.1, "word": " مظبوط", "probability": 0.8283203125}, {"start": 3389.1, "end": 3389.26, "word": " و", "probability": 0.57080078125}, {"start": 3389.26, "end": 3389.36, "word": " ال", "probability": 0.5693359375}, {"start": 3389.36, "end": 3389.72, "word": " magnitude", "probability": 0.8115234375}, {"start": 3389.72, "end": 3390.4, "word": " لهذا", "probability": 0.896728515625}, {"start": 3390.4, "end": 3390.68, "word": " يعني", "probability": 0.926513671875}, {"start": 3390.68, "end": 3391.0, "word": " طوله", "probability": 0.9763997395833334}, {"start": 3391.0, "end": 3391.2, "word": " هذا", "probability": 0.87841796875}, {"start": 3391.2, "end": 3391.46, "word": " 7", "probability": 0.23291015625}], "temperature": 1.0}, {"id": 153, "seek": 341612, "start": 3392.18, "end": 3416.12, "text": "أعملناها قبل شوية وهذا أربعة و واحد خمسة و أربعة تسعة طوله جداش جدر التسعة اللي هو تلاتة إذا طول هذا تلاتة و طول هذا سبعة مع بعض حصل ضرب هنا اللي هو واحد و عشرين فبصير theta بساوية cosine inverse أربعة على واحد و عشرين اللي هي تقريبا بساوية هذا الرقم يعني صار عندى الآن الزاوية بين المستويين هذه هي الزاوية اللي عندنا", "tokens": [10721, 25957, 1211, 8315, 11296, 12174, 36150, 13412, 2407, 10632, 37037, 15730, 5551, 25513, 27884, 4032, 36764, 24401, 16490, 2304, 3794, 3660, 4032, 5551, 25513, 27884, 6055, 3794, 27884, 23032, 12610, 3224, 10874, 3215, 33599, 10874, 3215, 2288, 16712, 3794, 27884, 13672, 1829, 31439, 6055, 1211, 9307, 3660, 11933, 15730, 23032, 12610, 23758, 6055, 1211, 9307, 3660, 4032, 23032, 12610, 23758, 8608, 3555, 27884, 20449, 45030, 11242, 11331, 36520, 48812, 25513, 34105, 13672, 1829, 31439, 36764, 24401, 4032, 6225, 46309, 9957, 6156, 3555, 9381, 13546, 9725, 4724, 3794, 995, 2407, 10632, 23565, 17340, 5551, 25513, 27884, 15844, 36764, 24401, 4032, 6225, 46309, 9957, 13672, 1829, 39896, 6055, 4587, 16572, 3555, 995, 4724, 3794, 995, 2407, 10632, 23758, 34892, 4587, 2304, 37495, 22653, 20328, 9640, 43242, 7578, 6024, 48506, 6024, 110, 995, 2407, 10632, 49374, 9673, 14851, 45865, 9957, 29538, 39896, 6024, 110, 995, 2407, 10632, 13672, 1829, 43242, 8315], "avg_logprob": -0.18593750397364298, "compression_ratio": 2.086792452830189, "no_speech_prob": 3.933906555175781e-05, "words": [{"start": 3392.18, "end": 3392.76, "word": "أعملناها", "probability": 0.708447265625}, {"start": 3392.76, "end": 3392.92, "word": " قبل", "probability": 0.956787109375}, {"start": 3392.92, "end": 3393.36, "word": " شوية", "probability": 0.6033528645833334}, {"start": 3393.36, "end": 3394.12, "word": " وهذا", "probability": 0.6439208984375}, {"start": 3394.12, "end": 3394.62, "word": " أربعة", "probability": 0.743896484375}, {"start": 3394.62, "end": 3394.84, "word": " و", "probability": 0.8486328125}, {"start": 3394.84, "end": 3395.12, "word": " واحد", "probability": 0.6456298828125}, {"start": 3395.12, "end": 3395.54, "word": " خمسة", "probability": 0.9500732421875}, {"start": 3395.54, "end": 3395.58, "word": " و", "probability": 0.83349609375}, {"start": 3395.58, "end": 3395.86, "word": " أربعة", "probability": 0.8351236979166666}, {"start": 3395.86, "end": 3396.34, "word": " تسعة", "probability": 0.93359375}, {"start": 3396.34, "end": 3396.98, "word": " طوله", "probability": 0.95751953125}, {"start": 3396.98, "end": 3397.44, "word": " جداش", "probability": 0.7630208333333334}, {"start": 3397.44, "end": 3397.72, "word": " جدر", "probability": 0.9373372395833334}, {"start": 3397.72, "end": 3398.06, "word": " التسعة", "probability": 0.8898111979166666}, {"start": 3398.06, "end": 3398.14, "word": " اللي", "probability": 0.841064453125}, {"start": 3398.14, "end": 3398.28, "word": " هو", "probability": 0.96435546875}, {"start": 3398.28, "end": 3398.76, "word": " تلاتة", "probability": 0.9453125}, {"start": 3398.76, "end": 3399.34, "word": " إذا", "probability": 0.6253662109375}, {"start": 3399.34, "end": 3399.72, "word": " طول", "probability": 0.7022705078125}, {"start": 3399.72, "end": 3399.88, "word": " هذا", "probability": 0.74609375}, {"start": 3399.88, "end": 3400.24, "word": " تلاتة", "probability": 0.908203125}, {"start": 3400.24, "end": 3400.34, "word": " و", "probability": 0.94140625}, {"start": 3400.34, "end": 3400.52, "word": " طول", "probability": 0.895751953125}, {"start": 3400.52, "end": 3400.84, "word": " هذا", "probability": 0.9541015625}, {"start": 3400.84, "end": 3401.66, "word": " سبعة", "probability": 0.9596354166666666}, {"start": 3401.66, "end": 3402.26, "word": " مع", "probability": 0.97265625}, {"start": 3402.26, "end": 3402.6, "word": " بعض", "probability": 0.996337890625}, {"start": 3402.6, "end": 3402.96, "word": " حصل", "probability": 0.759765625}, {"start": 3402.96, "end": 3403.24, "word": " ضرب", "probability": 0.968505859375}, {"start": 3403.24, "end": 3403.48, "word": " هنا", "probability": 0.73291015625}, {"start": 3403.48, "end": 3403.72, "word": " اللي", "probability": 0.95654296875}, {"start": 3403.72, "end": 3403.86, "word": " هو", "probability": 0.9814453125}, {"start": 3403.86, "end": 3404.12, "word": " واحد", "probability": 0.97119140625}, {"start": 3404.12, "end": 3404.22, "word": " و", "probability": 0.955078125}, {"start": 3404.22, "end": 3404.64, "word": " عشرين", "probability": 0.8909505208333334}, {"start": 3404.64, "end": 3405.4, "word": " فبصير", "probability": 0.823486328125}, {"start": 3405.4, "end": 3405.68, "word": " theta", "probability": 0.34033203125}, {"start": 3405.68, "end": 3406.32, "word": " بساوية", "probability": 0.50888671875}, {"start": 3406.32, "end": 3406.6, "word": " cosine", "probability": 0.7080078125}, {"start": 3406.6, "end": 3406.92, "word": " inverse", "probability": 0.7177734375}, {"start": 3406.92, "end": 3407.22, "word": " أربعة", "probability": 0.7858072916666666}, {"start": 3407.22, "end": 3407.32, "word": " على", "probability": 0.387939453125}, {"start": 3407.32, "end": 3407.48, "word": " واحد", "probability": 0.979248046875}, {"start": 3407.48, "end": 3407.58, "word": " و", "probability": 0.99169921875}, {"start": 3407.58, "end": 3407.98, "word": " عشرين", "probability": 0.9524739583333334}, {"start": 3407.98, "end": 3408.28, "word": " اللي", "probability": 0.97509765625}, {"start": 3408.28, "end": 3408.4, "word": " هي", "probability": 0.833984375}, {"start": 3408.4, "end": 3408.82, "word": " تقريبا", "probability": 0.9845703125}, {"start": 3408.82, "end": 3409.16, "word": " بساوية", "probability": 0.9302734375}, {"start": 3409.16, "end": 3409.34, "word": " هذا", "probability": 0.9716796875}, {"start": 3409.34, "end": 3409.76, "word": " الرقم", "probability": 0.9881184895833334}, {"start": 3409.76, "end": 3410.42, "word": " يعني", "probability": 0.9638671875}, {"start": 3410.42, "end": 3410.7, "word": " صار", "probability": 0.94140625}, {"start": 3410.7, "end": 3411.12, "word": " عندى", "probability": 0.697998046875}, {"start": 3411.12, "end": 3411.46, "word": " الآن", "probability": 0.872314453125}, {"start": 3411.46, "end": 3412.26, "word": " الزاوية", "probability": 0.898828125}, {"start": 3412.26, "end": 3412.56, "word": " بين", "probability": 0.9921875}, {"start": 3412.56, "end": 3413.54, "word": " المستويين", "probability": 0.9891357421875}, {"start": 3413.54, "end": 3413.98, "word": " هذه", "probability": 0.81494140625}, {"start": 3413.98, "end": 3414.66, "word": " هي", "probability": 0.9541015625}, {"start": 3414.66, "end": 3415.4, "word": " الزاوية", "probability": 0.98095703125}, {"start": 3415.4, "end": 3415.72, "word": " اللي", "probability": 0.894775390625}, {"start": 3415.72, "end": 3416.12, "word": " عندنا", "probability": 0.858154296875}], "temperature": 1.0}, {"id": 154, "seek": 342307, "start": 3416.39, "end": 3423.07, "text": "وبهيك بنكون خلصنا المحاضرة وبرك الله فيكم وإلى لقاء نفسنا السلام عليكم ورحمة الله وبركاته", "tokens": [37746, 3224, 1829, 4117, 44945, 30544, 16490, 1211, 9381, 8315, 9673, 5016, 46958, 25720, 4032, 26890, 4117, 21984, 8978, 24793, 4032, 28814, 23942, 5296, 4587, 16606, 8717, 36178, 8315, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224], "avg_logprob": -0.2578124918720939, "compression_ratio": 1.490909090909091, "no_speech_prob": 0.0, "words": [{"start": 3416.39, "end": 3417.21, "word": "وبهيك", "probability": 0.802734375}, {"start": 3417.21, "end": 3417.87, "word": " بنكون", "probability": 0.678955078125}, {"start": 3417.87, "end": 3418.35, "word": " خلصنا", "probability": 0.82080078125}, {"start": 3418.35, "end": 3418.91, "word": " المحاضرة", "probability": 0.7850341796875}, {"start": 3418.91, "end": 3419.39, "word": " وبرك", "probability": 0.7828776041666666}, {"start": 3419.39, "end": 3419.51, "word": " الله", "probability": 0.859375}, {"start": 3419.51, "end": 3419.93, "word": " فيكم", "probability": 0.95849609375}, {"start": 3419.93, "end": 3420.31, "word": " وإلى", "probability": 0.88232421875}, {"start": 3420.31, "end": 3420.71, "word": " لقاء", "probability": 0.8313802083333334}, {"start": 3420.71, "end": 3421.11, "word": " نفسنا", "probability": 0.5418701171875}, {"start": 3421.11, "end": 3421.39, "word": " السلام", "probability": 0.60980224609375}, {"start": 3421.39, "end": 3421.67, "word": " عليكم", "probability": 0.94677734375}, {"start": 3421.67, "end": 3421.95, "word": " ورحمة", "probability": 0.960205078125}, {"start": 3421.95, "end": 3422.19, "word": " الله", "probability": 0.96484375}, {"start": 3422.19, "end": 3423.07, "word": " وبركاته", "probability": 0.97119140625}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 3424.6646875, "duration_after_vad": 3388.413124999985}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..18d9f677fde43252b6c81f3fffdf18596594741e
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TLrob9ja9jY_raw.srt
@@ -0,0 +1,2976 @@
+1
+00:00:00,660 --> 00:00:06,300
+بسم الله الرحمن الرحيم هذه هي المحاضرة الخامسة
+
+2
+00:00:06,300 --> 00:00:13,380
+مساق تفاضل وتقام الجيم بعنوان lines and planes in
+
+3
+00:00:13,380 --> 00:00:17,920
+space اليوم بدنا نعرف كيف نوجد معادلة الخط
+
+4
+00:00:17,920 --> 00:00:22,960
+المستقيم ومعادلة المستوى في ال space ومتعلقاتها
+
+5
+00:00:23,670 --> 00:00:28,230
+الان احنا .. ده طبعا في section 12-5 احنا سابقا في
+
+6
+00:00:28,230 --> 00:00:33,250
+ال .. في ال plane عرفنا معادلة الخط المستقيم و كنا
+
+7
+00:00:33,250 --> 00:00:37,990
+جوا معادلة الخط المستقيم تحدد اللي هو بمعرفة mail
+
+8
+00:00:37,990 --> 00:00:43,210
+و نقطة او نقطين الان الكلام هنا مشابه الان بدنا
+
+9
+00:00:43,210 --> 00:00:48,290
+نعرف ايش .. كيف نوجد معادلة الخط المستقيم linesو
+
+10
+00:00:48,290 --> 00:00:53,390
+بعدين نعرف معادلة الخط المستقيم المحدود يعني بمعنى
+
+11
+00:00:53,390 --> 00:00:57,930
+آخر القطع المستقيمة line segment بدنا نعرف اللي هو
+
+12
+00:00:57,930 --> 00:01:02,610
+معدلتين الآن زي ما قلنا معدلة الخط المستقيم
+
+13
+00:01:02,610 --> 00:01:09,290
+هنحددها اللي هو من خلال معرفة نقطين أو ميل و نقطة
+
+14
+00:01:09,290 --> 00:01:13,990
+أو ما يشابه هنا ايش اللي بشابه؟ كلينا نشوف مع بعض
+
+15
+00:01:14,330 --> 00:01:19,950
+الان لو عرفنا .. طبعا عندنا خط مستقيم اللى بدنا
+
+16
+00:01:19,950 --> 00:01:25,770
+نعرف معدلته لو عرفنا نقطة عليه وعرفنا اللى هو
+
+17
+00:01:25,770 --> 00:01:31,010
+vector موازي إله يعني بمعنى آخر عرفنا وكأنه ميل
+
+18
+00:01:31,010 --> 00:01:35,890
+إله ميله ونقطة فنشوف كيف بدنا نحدد معدلة الخط
+
+19
+00:01:35,890 --> 00:01:41,300
+المستقيم في هذه الحالة بمعرفة نقطة على الخطوإيش و
+
+20
+00:01:41,300 --> 00:01:45,600
+vector موازي له شوفوا يا جماعة الان ناخد نقطة
+
+21
+00:01:45,600 --> 00:01:50,480
+عشوائية اسمها الهي x,y,z,b هذه اللي هي بتتحدد
+
+22
+00:01:50,480 --> 00:01:54,820
+النهوية اللي هو الخط يعني بدللتها بتيجي معادلة
+
+23
+00:01:54,820 --> 00:01:59,500
+الخط المستقيم الان ال vector الموجود عندنا نفترض
+
+24
+00:01:59,500 --> 00:02:04,080
+ان اسمه V بسوا V1 في اتجاه I و V2 في اتجاه J زي V3
+
+25
+00:02:04,080 --> 00:02:10,430
+في اتجاه ال Kالان النقطة المعروفة نسميها بي نوت
+
+26
+00:02:10,430 --> 00:02:13,910
+اكس نوت واي نوت وزد و نوت بدنا نعرف معدلة الخط
+
+27
+00:02:13,910 --> 00:02:20,630
+المستقيم شوفوا الان هذا ال vector موازي لمين لالخط
+
+28
+00:02:20,630 --> 00:02:24,010
+المستقيم اللى بدنا نعرف معدلته اخدنا النقطة
+
+29
+00:02:24,010 --> 00:02:27,890
+العشوائية بي اكس واي وزد صار عندى بي نوت بي هو
+
+30
+00:02:27,890 --> 00:02:32,490
+عبارة عن vector على الخط موازي لهذا ال vector
+
+31
+00:02:32,490 --> 00:02:37,640
+المعروفالان اكيد ال vector هذا هو نفس اتجاه ال
+
+32
+00:02:37,640 --> 00:02:41,340
+vector هذا بس بختلف عنه ايه في ايه في الطول يعني
+
+33
+00:02:41,340 --> 00:02:44,580
+بمعنى اخر و بتغيروا الطول و بتغيروا نقطة مين x و y
+
+34
+00:02:44,580 --> 00:02:49,180
+و z عشان هيك بنيجي بنكتب بنستخدم parameter بنقول
+
+35
+00:02:49,180 --> 00:02:53,960
+ان هذا ال vector bb0هيكون هو هذا الـ vector بس
+
+36
+00:02:53,960 --> 00:02:58,340
+مضروب في some scalar T يعني اللي هو ال .. طبعا T
+
+37
+00:02:58,340 --> 00:03:02,140
+اللي هي في هذه الحالة اللي .. اللي الآن عندي عبارة
+
+38
+00:03:02,140 --> 00:03:07,080
+عن real number اتجاه P0P ال .. ال vector P0P هو
+
+39
+00:03:07,080 --> 00:03:10,720
+نفسه ال vector V بس مضروب في T لأنه جلنا باختلف عن
+
+40
+00:03:10,720 --> 00:03:15,260
+بعض بس في إيش؟ في الطول أما موازئله يعني يا في
+
+41
+00:03:15,260 --> 00:03:19,060
+اتجاهه بالشكل هذا يا أما في اتجاهه بالشكل العكسي
+
+42
+00:03:19,440 --> 00:03:24,280
+بنوت بي بساوية TV هي اللى بتمثل معادلة الخط
+
+43
+00:03:24,280 --> 00:03:29,440
+المستقيم اللى عندنا و T عبارة عن مين عن أي عدد
+
+44
+00:03:29,440 --> 00:03:32,620
+بنقص infinity و infinity يعني real number
+
+45
+00:03:35,270 --> 00:03:41,930
+بساوة TV بدي أضعها في صورة جميلة نعرف كيف نجد
+
+46
+00:03:41,930 --> 00:03:47,490
+المعادلة بمجرد معرفة اللي هي الأرقام اللي بتمثلها
+
+47
+00:03:47,490 --> 00:03:50,330
+V والأرقام اللي بتمثلها P نوت ونعرف معادلة الخط
+
+48
+00:03:50,330 --> 00:03:54,500
+المستقيم ونشوف الأمثلة نشوف أشي اللي بقولهبنوت بي
+
+49
+00:03:54,500 --> 00:03:58,580
+يا جماعة هو عبارة عن x نقص x note في اتجاه ال I y
+
+50
+00:03:58,580 --> 00:04:02,580
+minus y note في اتجاه ال J z minus z note في اتجاه
+
+51
+00:04:02,580 --> 00:04:06,960
+ال K هذا مين هو بي نوت في بي إذا x minus x note في
+
+52
+00:04:06,960 --> 00:04:10,520
+اتجاه ال I زائد y minus y note في اتجاه ال J زائد
+
+53
+00:04:10,520 --> 00:04:14,460
+z minus z note في اتجاه ال K هذا اللي بي نوت بي
+
+54
+00:04:15,870 --> 00:04:22,010
+بساوي T هو البراميتر المتغير اللي بيحدد لمكان
+
+55
+00:04:22,010 --> 00:04:26,450
+النقطة بنضربه في T بيحدد النقطة هنا و هنا و هنا
+
+56
+00:04:26,450 --> 00:04:32,010
+حسب قيمة T بتكون تحدد النقطة عندك إذا صارت T هي
+
+57
+00:04:32,010 --> 00:04:37,470
+الان في V1 اتجاه I زي V2 اتجاه J زي V3 اتجاه K هذه
+
+58
+00:04:37,470 --> 00:04:42,230
+الآن اللي هي بنقدر نكتبها على صورة اللي هو بيجي
+
+59
+00:04:42,230 --> 00:04:46,740
+بننقل X0 اتجاه Iالتي هي المعروفة العدد y note في
+
+60
+00:04:46,740 --> 00:04:48,980
+اتجاه ال j اللي هو العدد z note في اتجاه ال k
+
+61
+00:04:48,980 --> 00:04:54,220
+نجلبها مع V1 اللي معروفة و V2 و V3 و T فبصير X
+
+62
+00:04:54,220 --> 00:04:57,600
+زائد Y زائد Z واحد في اتجاه ال I في اتجاه ال J في
+
+63
+00:04:57,600 --> 00:05:01,560
+اتجاه ال K بتساوي X note اللي نجلناها هذه مع ال
+
+64
+00:05:01,560 --> 00:05:07,920
+TV1 A هي TV1و Y note مع التي هو TV2 و Z note في
+
+65
+00:05:07,920 --> 00:05:14,100
+اتجاه اليمين التي هي في اتجاه TV3 في اتجاه ال K
+
+66
+00:05:14,100 --> 00:05:19,880
+هذه الآن بتمثل لمعادلة الخط المستقيم الآن هذه او
+
+67
+00:05:19,880 --> 00:05:25,330
+هذهمعادلة الخط المستقيم أو اختصارا هذه للي على طول
+
+68
+00:05:25,330 --> 00:05:29,830
+بدوش يحفظوا باستنتج على طول من هذه المعادلة انشوف
+
+69
+00:05:29,830 --> 00:05:33,550
+الآن انلخص الحديث لو واحد تاني جالي اسمعه انا بديش
+
+70
+00:05:33,550 --> 00:05:36,670
+اشتغلها زيكي بدي اشتغلها على صورة vectors قولتله
+
+71
+00:05:36,670 --> 00:05:40,570
+مافيش مشكلة بيجي بقوللي بدي اسمي ال vector هذا من
+
+72
+00:05:40,570 --> 00:05:44,500
+ها اللي هان هي ال vector هذا هي من ها اللي هانبدي
+
+73
+00:05:44,500 --> 00:05:50,900
+أسميه هذا اللي هو عبارة عن R هذا من هالهال من نقطة
+
+74
+00:05:50,900 --> 00:05:55,080
+الأصل لهذا هذا ال position vector لمين لل X,Y,Z
+
+75
+00:05:55,080 --> 00:05:58,440
+اللي هو في الواقع اللي هي اللي بتحدد لقيمة X,Y,Z
+
+76
+00:05:58,440 --> 00:06:03,720
+بنسميها R of T اللي هو ال vector بنسميه هذا R of T
+
+77
+00:06:03,720 --> 00:06:08,420
+اللي هو من عند ال zero لعند ال B الان نقطة الأصل
+
+78
+00:06:08,420 --> 00:06:11,370
+ال zero بقصد فيهاالان و هذا ال vector من هنا الى
+
+79
+00:06:11,370 --> 00:06:13,950
+هنا بدي تسميه R note اللي هو اللي بتكون .. اللي
+
+80
+00:06:13,950 --> 00:06:18,490
+بيكون معروف ليش معروف؟ لأنه B note معروفة الان هذا
+
+81
+00:06:18,490 --> 00:06:22,490
+ال vector B note بيه هو اللي بسميه مين اللي هو T
+
+82
+00:06:22,490 --> 00:06:26,410
+في V اللي اتفجنا عليه فالان المعادلة ممكن نكتبها
+
+83
+00:06:26,410 --> 00:06:30,330
+على الصورة التالية على صورة vectors اللي هو معادلة
+
+84
+00:06:30,330 --> 00:06:35,730
+الخط المستقيم على صورة vectors ايش هوR of T وهو
+
+85
+00:06:35,730 --> 00:06:42,370
+المجهول اللي عند X و Y و Z المتغيرة R of T بساوي
+
+86
+00:06:42,370 --> 00:06:45,910
+هذا ال vector زائد هذا ال vector بتعرفوا ان هذا ال
+
+87
+00:06:45,910 --> 00:06:48,830
+vector زائد هذا ال vector بساوي اللي بسكر المثلث
+
+88
+00:06:48,830 --> 00:06:55,650
+لأن R of T هذا بساوي R not of T زائد T في V وهذه
+
+89
+00:06:55,650 --> 00:07:00,350
+هي ال vector equation for the line through B not
+
+90
+00:07:00,350 --> 00:07:05,690
+parallel to Vالـ vector equation أو معادلة الخط
+
+91
+00:07:05,690 --> 00:07:11,290
+المستقيم بواسطة أو بدلالة الـ vectors
+
+92
+00:07:14,420 --> 00:07:18,680
+بمعرفة النقطة P0 و V بتساوي R of T زي ما قلنا
+
+93
+00:07:18,680 --> 00:07:24,220
+بساوي R0 of T هي R0 طبعا هتطلع لنا مباشرة هي عدد
+
+94
+00:07:24,220 --> 00:07:30,320
+زائد مين T في V هذي As of R0 لأنها بدلالة باسمين
+
+95
+00:07:30,320 --> 00:07:35,300
+P0 أما R of D هتكون بدلالة اتجاه TWhere R is the
+
+96
+00:07:35,300 --> 00:07:40,680
+position vector of a point B on L and R not is the
+
+97
+00:07:40,680 --> 00:07:44,800
+position vector of mean B not زي ما حكينا الان هذه
+
+98
+00:07:44,800 --> 00:07:46,980
+اللي هي ال vector equation for a line ال
+
+99
+00:07:46,980 --> 00:07:49,480
+parametric equations for a line اللي عملناها قبل
+
+100
+00:07:49,480 --> 00:07:56,260
+بشويةالتي هي الـ Standard Parameterization of Line
+
+101
+00:07:56,260 --> 00:08:02,000
+Through P0 Parallel to V S يعني هذه معادلة الخط
+
+102
+00:08:02,000 --> 00:08:08,060
+المستقيم اللي هي إيه شمالة as a parametric
+
+103
+00:08:08,060 --> 00:08:12,220
+equation كاللي هو معادلة برامترية اللي يعني بدلالة
+
+104
+00:08:12,220 --> 00:08:18,090
+T ل vector بمرغل نقطة P0بوازي الـ vector V التي هي
+
+105
+00:08:18,090 --> 00:08:22,590
+عبارة عن X نقص X نوت زي T في V1 و Y نقص Y نوت زي T
+
+106
+00:08:22,590 --> 00:08:26,830
+في V2 و Z نقص Z نوت زي T في V3 و T element of R
+
+107
+00:08:26,830 --> 00:08:29,690
+هذي من وين جيبناها؟ جيبناها يا جماعة من هنا سمعت
+
+108
+00:08:29,690 --> 00:08:32,410
+زي حد بيقول لي من وين جيبناها هي X نقص X نوت
+
+109
+00:08:32,410 --> 00:08:37,530
+بتساوي T في V1 يعني X بتساوي X نوت زي T في V1 هذه
+
+110
+00:08:37,530 --> 00:08:43,020
+معاملات IY minus my node بيسوي TV2 معاملات J يعني
+
+111
+00:08:43,020 --> 00:08:48,700
+Y بيسوي Y node زي TV2 و Z بيسوي Z node زي TV3 هي
+
+112
+00:08:48,700 --> 00:08:52,700
+اللي هي ال parameter standard parameter parametric
+
+113
+00:08:52,700 --> 00:08:57,900
+equation أو equations for the line through the B
+
+114
+00:08:57,900 --> 00:09:03,500
+node and parallel to mean to Vالان هذه هي المعادلة
+
+115
+00:09:03,500 --> 00:09:09,200
+التي تبع الخط المستقيم الأكثر استخداما و أكتر
+
+116
+00:09:09,200 --> 00:09:13,580
+شيوعا ناخد مثال نجد الـ Parametric equations for
+
+117
+00:09:13,580 --> 00:09:18,240
+the line ناقص 2 و 0 و 4 نجد ال Parametric
+
+118
+00:09:18,240 --> 00:09:25,360
+equations لخط المستقيمالخط المستقيم اللى بمر
+
+119
+00:09:25,360 --> 00:09:29,800
+بالنقطة نقص اتنين و سفر و اربعة و يوازي المتجه
+
+120
+00:09:29,800 --> 00:09:35,460
+اللى امامنا هذا ماشي الحال هاي عندى الان المتجه هي
+
+121
+00:09:35,460 --> 00:09:39,680
+رسم لياه و هي النقطة اللى ماخدها عليه اللى هي نقص
+
+122
+00:09:39,680 --> 00:09:43,040
+اتنين و سفر و اربعة و بدو يوجد معدلة الخط المستقيم
+
+123
+00:09:43,040 --> 00:09:48,220
+اللى هذا اللى بوازي اللى هو المتجه هذا و بمر
+
+124
+00:09:48,220 --> 00:09:53,370
+بالنقطة Bالان تعويض مباشر ما هي عبارة عن X
+
+125
+00:09:53,370 --> 00:10:00,430
+بالساوية X0 زائد T في V ال V معروفة V1 هيها 2 و V2
+
+126
+00:10:00,430 --> 00:10:05,650
+هي 4 و V3 هي مين ناقص 2 وهذه X0 و Y0 و Z0 هي
+
+127
+00:10:05,650 --> 00:10:09,390
+المعروفات هي بالساوية تعويض X بالساوية X0 ناقص 2
+
+128
+00:10:09,390 --> 00:10:14,590
+زائد 2 في T و Y بالساوية Y0
+
+129
+00:10:17,220 --> 00:10:26,500
+T في V 4T V2 و Z بالساوية و Z نقطة 4-2 في T هذه
+
+130
+00:10:26,500 --> 00:10:30,360
+الآن عبارة عن معادلة الخط المستقيم اللي بمر
+
+131
+00:10:30,360 --> 00:10:34,400
+بالنقطة هذه و يوازن والتجاه اللي أمامنا و الكلام
+
+132
+00:10:34,400 --> 00:10:40,770
+زي ما شوفته سهلزي زمان برضه لو كان عنده نقطتين
+
+133
+00:10:40,770 --> 00:10:44,550
+نعرف معادلة الخط المستقيم برضه هنا في الفراغ نفس
+
+134
+00:10:44,550 --> 00:10:47,470
+الشيء لو كان عنده نقطتين بنعرف معادلة الخط
+
+135
+00:10:47,470 --> 00:10:50,830
+المستقيم و أكيد بعضكم قال اه طب ما هو احنا بنجيب
+
+136
+00:10:50,830 --> 00:10:55,910
+المتجهة اللي بوازي المتجهة اللي هو اللي بيحمل P و
+
+137
+00:10:55,910 --> 00:11:00,690
+Q اللي هو PQ او QP زمان بالذات هو بوازي أصلا الخط
+
+138
+00:11:00,690 --> 00:11:03,770
+المستقيم اللي بمر بالنقطتين هذين وهذا نقطة من
+
+139
+00:11:03,770 --> 00:11:08,480
+النقطتين فبصير معروف اللي هو متجهاللي هو جوازي
+
+140
+00:11:08,480 --> 00:11:12,160
+اللي هو في الواقع بيحمل و نفس الاشي نقطة معروفة
+
+141
+00:11:12,160 --> 00:11:15,840
+نعرف اللي قاعدة دلوقت نشوف كيف find a parametric
+
+142
+00:11:15,840 --> 00:11:19,320
+equation for the line through كده and كده بمعرفة
+
+143
+00:11:19,320 --> 00:11:23,000
+نقطين لأن ده vector bq أول اشي ده نوجد ال vector
+
+144
+00:11:23,000 --> 00:11:28,190
+bq اللي هو عبارة عنواحد قال بدا وجد QB برضه صح
+
+145
+00:11:28,190 --> 00:11:33,510
+واتنين بتجيب ان معادلتين تظهر انها مختلفة لكن
+
+146
+00:11:33,510 --> 00:11:39,650
+بتجيب نفس القط لإنه هتجيب لك نفس النقاط اكيو ناقص
+
+147
+00:11:39,650 --> 00:11:44,810
+بي اللي هي واحد ناقص ناقص تلاتة ناقص واحد ناقص
+
+148
+00:11:44,810 --> 00:11:45,410
+اتنين
+
+149
+00:12:00,980 --> 00:12:05,440
+هذا الـ vector is parallel to line and equation 3
+
+150
+00:12:06,470 --> 00:12:11,150
+with x0 و y0 و z0 نقص ثلاثة نقص اتنين بيعطينا
+
+151
+00:12:11,150 --> 00:12:13,250
+equation تلاتة اللي بتحكي عن equations ثلاثة اللي
+
+152
+00:12:13,250 --> 00:12:16,410
+هي قانون اللي هو ال parameter equations ده الخط
+
+153
+00:12:16,410 --> 00:12:21,150
+المستقيم اللي قبل شوية طيب الآن عرفنا المتجه
+
+154
+00:12:21,150 --> 00:12:27,370
+الموازي او الحامل لل line و بالاضافة لنقطة على ال
+
+155
+00:12:27,370 --> 00:12:29,490
+line واحد يقولي بديش هذا النقطة بتاخده نقطة تانية
+
+156
+00:12:29,490 --> 00:12:33,630
+بقوله اه بالفعلو بيطلع نفس ال equations بيطلع نقاط
+
+157
+00:12:33,630 --> 00:12:37,810
+تكون في شكل المختلفات لكن بيجيب نفس الخط يعني نفس
+
+158
+00:12:37,810 --> 00:12:42,670
+النقاط اللي على الخط طيب X بيساوي X نقطة اللي هي
+
+159
+00:12:42,670 --> 00:12:50,290
+نقص تلاتة زائد أربعة V في T Y بتساوي Y نقطة اللي
+
+160
+00:12:50,290 --> 00:12:55,800
+هو اتنينناقص تلاتة T فاهمين عليها Z بتساوي ناقص
+
+161
+00:12:55,800 --> 00:13:00,820
+تلاتة التي هي Z note زائد سبعة T هذه الآن هي ال
+
+162
+00:13:00,820 --> 00:13:05,040
+Parametric Equations للخط المستقيم وحتين قال
+
+163
+00:13:05,040 --> 00:13:08,780
+اسمعوا بديش انا بدي اختار اللي هي ال Q بدل ال mean
+
+164
+00:13:08,780 --> 00:13:12,580
+بدل ال V يعني بدي اخد ال base احنا اخدنا النقطة
+
+165
+00:13:12,580 --> 00:13:17,340
+اللي انطلق بيهوالنهاية Q فانا باخد نقطة انطلاق Q
+
+166
+00:13:17,340 --> 00:13:21,960
+واخد بيه بنفع بيصير الان اللي هو ال equation اللي
+
+167
+00:13:21,960 --> 00:13:27,380
+هي او اقال حتى اسمعوا مش شيكة كمان بتاخد انا اللي
+
+168
+00:13:27,380 --> 00:13:34,520
+بدي أسويه بتاخد اللي هو عندي النقطة ال B بتاخد ال
+
+169
+00:13:34,520 --> 00:13:38,250
+Q ليش استعمل ال Q تبعتكم بالمرةالنقص تلاتة او
+
+170
+00:13:38,250 --> 00:13:39,570
+اتنين او اكتر او اكتر اكتر اكتر اكتر اكتر اكتر
+
+171
+00:13:39,570 --> 00:13:42,070
+اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر
+
+172
+00:13:42,070 --> 00:13:44,690
+اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر
+
+173
+00:13:44,690 --> 00:13:44,710
+اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر
+
+174
+00:13:44,710 --> 00:13:47,450
+اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر اكتر
+
+175
+00:13:47,450 --> 00:13:55,190
+اكتر اكتر اكتر
+
+176
+00:13:55,190 --> 00:14:01,510
+اكتر اك
+
+177
+00:14:02,200 --> 00:14:05,920
+اللي هي ال equations بيجيب نفس الخط هذا ايه الدليل
+
+178
+00:14:05,920 --> 00:14:10,660
+خد مثلا هنا T بصفر بتطلع عندك النقطة واحد وناقص
+
+179
+00:14:10,660 --> 00:14:16,660
+واحد واربعة هي على الخط طيب ان انا لو اخدت T بواحد
+
+180
+00:14:16,660 --> 00:14:22,770
+هنا تيب واحد بيصير ناقص تلاتة واربعة يعني واحدوT
+
+181
+00:14:22,770 --> 00:14:29,970
+بواحد تظهر تقريبا تقريبا تقريبا تقريبا تقريبا
+
+182
+00:14:29,970 --> 00:14:31,330
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+183
+00:14:31,330 --> 00:14:32,930
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+184
+00:14:32,930 --> 00:14:32,930
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+185
+00:14:32,930 --> 00:14:32,930
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+186
+00:14:32,930 --> 00:14:35,870
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+187
+00:14:35,870 --> 00:14:41,030
+تقريبا تقريبا
+
+188
+00:14:41,030 --> 00:14:47,710
+تقريبا
+
+189
+00:14:48,890 --> 00:14:52,890
+ومش هيك هتلاقي مادام المقاطين و هذا خط و هذا خط
+
+190
+00:14:52,890 --> 00:14:57,820
+هيكون كل اللي بينهم على الخطين يعنياللي بده يقول
+
+191
+00:14:57,820 --> 00:15:03,800
+بي كيو او كيو بي استعملتها صح لو استخدمت النقطة بي
+
+192
+00:15:03,800 --> 00:15:07,520
+as a base point يعني نقطة و متجهة او كيو as a base
+
+193
+00:15:07,520 --> 00:15:13,340
+point زي ما شوفنا برضه بطلع اللي هو معادلات تعبر
+
+194
+00:15:13,340 --> 00:15:21,460
+عن الخط المستقيم الذي يحمل النقطين بي و كيو طيب
+
+195
+00:15:21,460 --> 00:15:28,820
+ناخد الآن واحد قال لانا بدي اشكرأنا بدي بس تعملي
+
+196
+00:15:28,820 --> 00:15:32,260
+parameterize the line segment joining the point B
+
+197
+00:15:32,260 --> 00:15:38,020
+and Q يعني بس بدي اللي هو معادلة الخط المستقيم لأ
+
+198
+00:15:38,020 --> 00:15:42,900
+بدي معادلة القطعة المستقيمة اللي بس من Q لعندي
+
+199
+00:15:42,900 --> 00:15:47,460
+لعندي بفكر بفكر بقول الله يعني بت cache ال قيم
+
+200
+00:15:47,460 --> 00:15:53,060
+اللي هناولا القيم اللي هنا يعني بتكاشل على الخط من
+
+201
+00:15:53,060 --> 00:15:57,360
+هنا ولا خط من هنا يعني بدك بمعنى أخر تحدد لقيم ال
+
+202
+00:15:57,360 --> 00:16:01,920
+T منها ال T اللي بتتحدد تجيبلي بس المطاط اللي من
+
+203
+00:16:01,920 --> 00:16:08,260
+عند V لعند B لعند Q ونقوله فش مشكلة المعادلات الخط
+
+204
+00:16:08,260 --> 00:16:12,240
+هي هي لإن هو جزء من الخط هذا إذا أنا from example
+
+205
+00:16:12,240 --> 00:16:16,560
+2 above the equations for the line through Band qr
+
+206
+00:16:16,560 --> 00:16:20,220
+هدولة معادلة مين اللى وجدناها قبل شوية معادلة الخط
+
+207
+00:16:20,220 --> 00:16:25,720
+المستقيم لكن هو بيقوللي انا يا جماعة انا بدي هدولة
+
+208
+00:16:25,720 --> 00:16:30,280
+النقاط x و y و z مايتعدنش النقطة هذه والنقطة هذه
+
+209
+00:16:30,280 --> 00:16:35,640
+يعني الان بدي بس اللى هو ال T اللى بتعبرلي بين q و
+
+210
+00:16:35,640 --> 00:16:42,000
+بين bالنقطة العامة اللي هان صارت على الخط هي x,y,z
+
+211
+00:16:42,000 --> 00:16:46,540
+نقص ثلاثة ثم اربعة ت ثم اتنين نقص ثلاثة ت ثم نقص
+
+212
+00:16:46,540 --> 00:16:52,530
+ثلاثة او سبعة تالان لو جيت اخدت لحظة التالي لو
+
+213
+00:16:52,530 --> 00:16:57,530
+اخدت T بصفر هتطلع عندى ناقص ثلاثة و اتنين و ناقص
+
+214
+00:16:57,530 --> 00:17:02,390
+ثلاثة يعنى P هى اللى P ماشي الحال اذا هذا النقطة T
+
+215
+00:17:02,390 --> 00:17:07,550
+بالصفر بتجيبها الان النقطة هذه Q لو جيت اخدت اللى
+
+216
+00:17:07,550 --> 00:17:12,390
+هى نقطة واحد وناقص واحد واربعة يعني واحد ساوي اللى
+
+217
+00:17:12,390 --> 00:17:19,090
+هادى بواحدبتطلع عندك T بتساوي لإن هد بديها Q هناأه
+
+218
+00:17:19,090 --> 00:17:22,430
+بدي الـ T اللي بتخلي تجيب Q هذه خد الان واحد
+
+219
+00:17:22,430 --> 00:17:26,070
+بتساوي نقص تلاتة زائد أربعة T يعني أربعة T بتساوي
+
+220
+00:17:26,070 --> 00:17:29,770
+أربعة يعني T بتساوي واحد يعني عند T بتساوي واحد
+
+221
+00:17:29,770 --> 00:17:35,490
+هذه النقطة هتكون عبارة عن Q يعني الآن T هذا عند T
+
+222
+00:17:35,490 --> 00:17:39,890
+صفر وهذا عند T واحد يعني المنطقة هذه كلها من وين
+
+223
+00:17:39,890 --> 00:17:46,820
+بتيجي بتيجي بنفس المعادلات هذولة التلاتةمع T بس
+
+224
+00:17:46,820 --> 00:17:51,480
+بدي اعملها restriction بدي احددها من 0 لعند مين يا
+
+225
+00:17:51,480 --> 00:17:55,900
+جماعة لعند واحد فT من عند 0 لعند واحد بس بتجيب
+
+226
+00:17:55,900 --> 00:18:01,220
+للقطع المستقيمة من النقطة هذه للنقطة هذه وهيك
+
+227
+00:18:01,220 --> 00:18:05,440
+بيكون أوجبنا اللي هو Parametric Equations for the
+
+228
+00:18:05,440 --> 00:18:10,540
+line segment joining the points B and Qالان بدنا
+
+229
+00:18:10,540 --> 00:18:15,020
+نعرف امر جديد اللي هو the distance from a point to
+
+230
+00:18:15,020 --> 00:18:20,820
+align in space هاي عندي خط في الفراغ اسمه خط L
+
+231
+00:18:24,350 --> 00:18:28,830
+عند نقطة في الفراغ اسمها S بدي اوجد اللي هو
+
+232
+00:18:28,830 --> 00:18:32,930
+المسافة بين S و بين نقطة بتعرفوا المسافة بين S و
+
+233
+00:18:32,930 --> 00:18:39,930
+بين نقطة بين S و بين خط مستقيم بيجي بنزل من النقطة
+
+234
+00:18:39,930 --> 00:18:43,490
+عمودي على الخط المستقيم هذه العمودي وصلنا لهذه
+
+235
+00:18:43,490 --> 00:18:48,070
+العمودي هذا الطول اللي احنا بدناه هذا الطول اللي
+
+236
+00:18:48,070 --> 00:18:52,660
+بدناهطب كيف بده اوجده سهل الموضوع ال S عندي معروفة
+
+237
+00:18:52,660 --> 00:18:56,480
+صح ولا لأ يا جماعة معروفة هو معطينيها لأنه بده
+
+238
+00:18:56,480 --> 00:18:59,920
+المسافة بينها بين S و بين الخط المستقيم بكون
+
+239
+00:18:59,920 --> 00:19:05,140
+معطيني الخط المستقيم اللي عنديوبكون معطيني من اللي
+
+240
+00:19:05,140 --> 00:19:08,640
+هو النقطة S اللي بده البقعة بينها وبين الخط
+
+241
+00:19:08,640 --> 00:19:13,860
+المستقيم طبعا كنا الخط المستقيم في طياته بيحمل او
+
+242
+00:19:13,860 --> 00:19:18,520
+بيعطيني معلومات ال vector الموازي له V او vector
+
+243
+00:19:18,520 --> 00:19:25,100
+عليه V بالاضافة لنقاط زي ما بدك بتجيب منهإذا الان
+
+244
+00:19:25,100 --> 00:19:29,860
+باجي انا بمعرفة اللي هو ال vector اللي عليه لو
+
+245
+00:19:29,860 --> 00:19:35,500
+ليكن هذا مثلا V و هذه اللي هي نقطة باخدها على الخط
+
+246
+00:19:35,500 --> 00:19:40,160
+المستقيم و بوصل هذا صارت هذه الزاوية θتا اللي هي و
+
+247
+00:19:40,160 --> 00:19:44,380
+هذه قائمة هذه الزاوية بين هذا اللي هو الجزء من
+
+248
+00:19:44,380 --> 00:19:51,300
+الخط و بين بي اسالان هذا طوله طوله تعرفوا انه sin
+
+249
+00:19:51,300 --> 00:19:58,500
+θ بساوي اللي هو المقابل هذا على اللي هو الوطر
+
+250
+00:19:58,500 --> 00:20:03,500
+فبصير الان المقابل اللي هو المطلوب بساوي ال
+
+251
+00:20:03,500 --> 00:20:06,640
+magnitude لهذا طول هذا اللي هو ال BS في sin
+
+252
+00:20:06,640 --> 00:20:13,380
+زوايمين θ في الواقع هذا ايش بمثلهذا الان لو اجينا
+
+253
+00:20:13,380 --> 00:20:16,320
+اخدنا طبعا احنا المطلوب distance from a point S to
+
+254
+00:20:16,320 --> 00:20:20,360
+a line ببرالتو V اللي هو هذا هذا الان بي اس ساينت
+
+255
+00:20:20,360 --> 00:20:24,180
+هتا هو في الواقع عبارة عن ال magnitude لل بي اس
+
+256
+00:20:24,180 --> 00:20:28,340
+across ال vector V هذا اللي بيحمل الخط اللي هنا
+
+257
+00:20:42,560 --> 00:20:49,210
+القانون اللى اشتقناه هو قانون المسافةبين نقطة وبين
+
+258
+00:20:49,210 --> 00:20:53,710
+خط مستقيم اللي هو طبعا بمعلومية النقطة ومعلومية
+
+259
+00:20:53,710 --> 00:20:57,810
+الخط المستقيم عبارة عن الـ BS cross V على ال
+
+260
+00:20:57,810 --> 00:21:00,970
+magnitude V ماهو الـ BS اللي هو الاسس النقطة
+
+261
+00:21:00,970 --> 00:21:05,430
+المعطاة وB نقطة ناخدها على الخط المستقيم المعطاة و
+
+262
+00:21:05,430 --> 00:21:08,590
+V هو ال vector اللي بيحمل الخط المستقيم المعطاة أو
+
+263
+00:21:08,590 --> 00:21:11,910
+الموازي اللي هو زي ما قلنا سابقا ونشوف مثال عملي
+
+264
+00:21:13,470 --> 00:21:17,150
+بقول find the distance from the point S1 1 5 to
+
+265
+00:21:17,150 --> 00:21:21,350
+the line X بيساوي 1 زي ال T و Y بيساوي 3 نقص T و Z
+
+266
+00:21:21,350 --> 00:21:25,670
+بيساوي 2T لسه احنا في الموضوع يعني مابعدناش يعني
+
+267
+00:21:25,670 --> 00:21:30,850
+الآن احنا بنقدر نعرف من هذا المعادلةاللي هو الـ
+
+268
+00:21:30,850 --> 00:21:33,890
+vector الموازي اللي هو معاملات من ال vector
+
+269
+00:21:33,890 --> 00:21:38,470
+الموازي معاملات T يعني ال vector الموازي P بيساوي
+
+270
+00:21:38,470 --> 00:21:44,550
+I اللي هي واحد ناقص J اللي هي ناقص واحد زائد اتنين
+
+271
+00:21:44,550 --> 00:21:49,850
+K اللي هي من معامل T اتنين هذا ال vector الموازي
+
+272
+00:21:49,850 --> 00:21:55,620
+لمين لخط المستقيم المعطىمعادلته زي ما اتعلمنا قبل
+
+273
+00:21:55,620 --> 00:22:00,760
+بشوية لإن النقطة اللي معناه مين هي S115 بدنا تمام
+
+274
+00:22:00,760 --> 00:22:04,900
+نقطة بناخد أي نقطة اللي بتيجي على بالك واحد قال
+
+275
+00:22:04,900 --> 00:22:09,500
+اسمع بدي أخد اللي هو أسهل إشي T بصفر بصير واحد و
+
+276
+00:22:09,500 --> 00:22:13,340
+تلاتة و اتنين قلنا له حاضر واحد اتاني قال لأ بدي
+
+277
+00:22:13,340 --> 00:22:16,880
+أخد T بواحد بدي أغلب حالي بدي أخليها كل أقام بقوله
+
+278
+00:22:16,880 --> 00:22:25,060
+بنفع T اللي بدكياالان اخدنا نقطة بي واحد و تلاتة و
+
+279
+00:22:25,060 --> 00:22:29,060
+سفر الان ايش بالزمن زي ما قلنا في القانون في
+
+280
+00:22:29,060 --> 00:22:32,220
+القانون قلنا بالزمن بي اس اكروس في ع ال magnitude
+
+281
+00:22:32,220 --> 00:22:37,100
+لفي الان بي عرفناه و اس معطية و في عرفناه نيجي
+
+282
+00:22:37,100 --> 00:22:43,120
+الان نطبق القانون بي اس ايش متساوي because واحد
+
+283
+00:22:43,120 --> 00:22:49,440
+نقص واحدواحد ناقص ثلاثة واحد ناقص ثلاثة وخمسة ناقص
+
+284
+00:22:49,440 --> 00:22:53,180
+ايه اش سفر طلع عند ال BS طلع عند هذا سفر طبعا هذه
+
+285
+00:22:53,180 --> 00:22:58,380
+طلعت سالب اتنين J وهذه خمسة K اذا صار عند ال BS
+
+286
+00:22:58,380 --> 00:23:02,480
+هيو بدي اعمل ال BS cross V واخدله ال magnitude ال
+
+287
+00:23:02,480 --> 00:23:06,700
+V هيو ماشي اعرفنا كيف نوجد ال BS cross V بواسطة
+
+288
+00:23:06,700 --> 00:23:11,680
+المحدد هي المحدد جوا وهي برا ايه ال absolute value
+
+289
+00:23:11,680 --> 00:23:15,010
+او ال magnitudeطبعا المحدد بيطلعنا قيمة و بعدين
+
+290
+00:23:15,010 --> 00:23:17,650
+ناخد ال absolute value و نطلع القيمة الموجودة اللي
+
+291
+00:23:17,650 --> 00:23:21,770
+بدنا هيها شوفوا صلوا على النبي عليه الصلاة والسلام
+
+292
+00:23:21,770 --> 00:23:27,610
+ال BS كيف نعرف نضرم BS في V هي IJK و بعدين BS اللي
+
+293
+00:23:27,610 --> 00:23:31,850
+هو السطر الأول و ال BS اللي وجدناه هيه اللي هو سفر
+
+294
+00:23:31,850 --> 00:23:36,830
+ناقص اتنين خمسة و اللي هي ال V اللي هي عبارة عن
+
+295
+00:23:36,830 --> 00:23:41,290
+واحد و ناقص واحد و اتنينالان خلّينا نجد هذا ال ..
+
+296
+00:23:41,290 --> 00:23:45,350
+ال .. ال .. ال .. ال .. ال determinant ال
+
+297
+00:23:45,350 --> 00:23:48,730
+determinant اللي جوا بتعرفوا كيف بنشطب السطر هذا و
+
+298
+00:23:48,730 --> 00:23:52,090
+العمود هذا بيصير ناقص اتنين في اتنين ناقص اربعة
+
+299
+00:23:52,090 --> 00:23:58,570
+زائد واحد اللي هو عبارة عن ايه؟ناقص اربعة زائد
+
+300
+00:23:58,570 --> 00:24:03,930
+خمسة وهو عبارة عن I الان نشط بهذا و نشط بهذا يصبح
+
+301
+00:24:03,930 --> 00:24:09,490
+اتنين ناقص خمسة و في ناقص طبعة اتجاه يصبح زائد
+
+302
+00:24:09,490 --> 00:24:13,230
+خمسة الان الاخيرة ال K نشط بهذا و نشط بهذا يصبح
+
+303
+00:24:13,230 --> 00:24:17,690
+سفر ناقص ناقص يصبح زائد اتنين اللي هي في اتجاه ال
+
+304
+00:24:17,690 --> 00:24:21,320
+K ال magnitude لهذا الجدر التربيعي له واحدزائد
+
+305
+00:24:21,320 --> 00:24:25,840
+خمسة وعشرين زائد أربعة أربعة وخمسة وعشرين تسعة
+
+306
+00:24:25,840 --> 00:24:29,460
+وعشرين وواحد تلاتين يعني جدر التلاتين ال magnitude
+
+307
+00:24:29,460 --> 00:24:33,400
+لل V تلزمنا اللي هي أش بتساوي الجدر التربيعي اللي
+
+308
+00:24:33,400 --> 00:24:37,700
+هو واحد زائد واحد زائد أربعة اللي هو جدر الستة لأن
+
+309
+00:24:37,700 --> 00:24:41,520
+حسب القانون the distance أش بتساوي ال magnitude لل
+
+310
+00:24:41,520 --> 00:24:46,100
+V S cross V ع ال magnitude لل V طيب يا جماعة صلوا
+
+311
+00:24:46,100 --> 00:24:49,650
+على النبي عليه الصلاة والسلامالـ magnitude للـ BS
+
+312
+00:24:49,650 --> 00:24:53,770
+cross V اللي هو بساوي جدر التلاتين اللي أوجدناه
+
+313
+00:24:53,770 --> 00:24:56,670
+على جدر الستة ويساوي جدر الخمسة وهذه هي المسافة
+
+314
+00:24:56,670 --> 00:25:01,250
+بين الخط المستقيم الأول نقطة أو بين النقطة أسوأ
+
+315
+00:25:01,250 --> 00:25:07,510
+الخط المستقيم الموجود المطلوبالان بدنا نجي لجزء
+
+316
+00:25:07,510 --> 00:25:11,890
+ثاني من المحاضرة وهو معادلة الخط معادلة لبلين في
+
+317
+00:25:11,890 --> 00:25:16,370
+ال space احنا كنا في الفرقية زمان نقول ان لبلين
+
+318
+00:25:16,370 --> 00:25:20,970
+يحدد تحديدا تمال بمعرفة تلات نقاط على لبلين ليست
+
+319
+00:25:20,970 --> 00:25:24,810
+على استقامة واحدةاللي هي تلت مقات ليست على استقامة
+
+320
+00:25:24,810 --> 00:25:30,050
+واحدة تحدد اللي هو المستوى اللي يمر بهنا طيب الان
+
+321
+00:25:30,050 --> 00:25:34,290
+هذا الكلام شبيه بمعرفة الان مش تلت مقات تلت مقات
+
+322
+00:25:34,290 --> 00:25:39,490
+بعد شوية هيكون كتطبيق بمعرفة اللي هو vector عمودي
+
+323
+00:25:40,100 --> 00:25:45,040
+على المستوى و بمعرفة نقطة على المستوى هنقدر نحدد
+
+324
+00:25:45,040 --> 00:25:51,140
+هوية المستوى الان بمعرفة نقطة على المستوى و بمعرفة
+
+325
+00:25:51,140 --> 00:25:55,580
+عمودي عليه نحدد هوية المستوى نشوف كيف نوجد دليلة
+
+326
+00:25:55,580 --> 00:25:58,900
+ده الان ال vector ده عمودي عليه ده عمودي على كل
+
+327
+00:25:58,900 --> 00:26:03,520
+مستقيل في المستوى الان هناخد نقطة عشوائية نسميها x
+
+328
+00:26:03,520 --> 00:26:07,600
+y و z هي اللي هتحدد لي هوية لبنان اللي بدلالتها
+
+329
+00:26:07,600 --> 00:26:11,960
+بدنا نجيب اللي هو معادلة لبنانالان صار عند بي نوت
+
+330
+00:26:11,960 --> 00:26:16,800
+بي عبارة عن vector موجود من وين يا جماعة في البلين
+
+331
+00:26:16,800 --> 00:26:20,380
+هذا ال vector أكيد الان العمودي على البلين عمودي
+
+332
+00:26:20,380 --> 00:26:27,400
+عليه يعني بمعنى آخر الان هذا ضد ال بي نوت بي بساوي
+
+333
+00:26:27,400 --> 00:26:32,680
+سفر لإن الزاوية اللي بينهم تسعين درجة فبكون عندى
+
+334
+00:26:32,680 --> 00:26:36,260
+اللي هو الضد product بينهم بساوي سفر وهذه هي
+
+335
+00:26:36,260 --> 00:26:40,780
+معادلة البلينمعادلة الـ plan vector equation ستكون
+
+336
+00:26:40,780 --> 00:26:48,320
+N.P0P بساوي 0 لأن خلّينا نسمي نتفج مع بعض أن الـ N
+
+337
+00:26:48,320 --> 00:26:52,860
+بدنا نسميه اللي هو بساوي ال vector المعلوم طبعا
+
+338
+00:26:52,860 --> 00:26:56,900
+لأنه ببساطة و بده وجود المعادلة الـ plan ال vector
+
+339
+00:26:56,900 --> 00:27:02,780
+N المعلوم بده أقول عنه اسمه AI زائد BJ زائد CKإذاً
+
+340
+00:27:02,780 --> 00:27:06,720
+الـ A و الـ B و الـ C هي عبارة عن معاملات الـ I و
+
+341
+00:27:06,720 --> 00:27:10,500
+الـ J و الـ K بالترتيب للـ vector العمودي على
+
+342
+00:27:10,500 --> 00:27:15,540
+المستوى الـ NN.P0P بساوة سفر هي معادلة اللي هي ال
+
+343
+00:27:15,540 --> 00:27:18,440
+vector equation للمستوى لأن لو بدنا نحوّلها ل
+
+344
+00:27:18,440 --> 00:27:22,480
+component equation بنعمل ال dot productب نوت بي
+
+345
+00:27:22,480 --> 00:27:27,920
+نوت بي نوت
+
+346
+00:27:27,920 --> 00:27:34,860
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+347
+00:27:34,860 --> 00:27:35,360
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+348
+00:27:35,360 --> 00:27:35,780
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+349
+00:27:35,780 --> 00:27:37,040
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+350
+00:27:37,040 --> 00:27:37,040
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+351
+00:27:37,040 --> 00:27:37,040
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+352
+00:27:37,040 --> 00:27:39,340
+بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت بي نوت
+
+353
+00:27:39,340 --> 00:27:48,040
+بي نوت بي نوت بي نوت بي نوت ببتعرف تدرب الـ Dot
+
+354
+00:27:48,040 --> 00:27:53,040
+Product زائد C في المعامل الـ K التي هي Z-Z0 في
+
+355
+00:27:53,040 --> 00:27:56,860
+اتجاه الـ Zero كله مش هيساوي سفر يعني هذا عبارة عن
+
+356
+00:27:56,860 --> 00:28:03,080
+الـ Dot Product للـ N Dot B Note Bالان صارت هذه
+
+357
+00:28:03,080 --> 00:28:08,480
+تمثلي معادلة الخط المستخدم للـ plane اللى بنسميها
+
+358
+00:28:08,480 --> 00:28:12,540
+component equations وصلت اللى هى ال component اللى
+
+359
+00:28:12,540 --> 00:28:17,220
+هى ال equation اللى احنا استعملناها قبل هيك كثير
+
+360
+00:28:17,220 --> 00:28:22,780
+طيب اذا الان ممكن واحد يختصرها يقولك خلاص ماهى ax
+
+361
+00:28:22,780 --> 00:28:27,370
+زائد by زائد czهذه الأشياء بيساوين الـ D
+
+362
+00:28:43,370 --> 00:28:46,590
+لأن الـ A معروفة و الـ B معروفة و الـ C معروفة و
+
+363
+00:28:46,590 --> 00:28:50,310
+الـ X نوت و الـ Y نوت و الـ Z نوت معروفة فبصير و
+
+364
+00:28:50,310 --> 00:28:53,770
+كأنه الـ component equation can be simplified as
+
+365
+00:28:53,770 --> 00:28:59,850
+يعني AX زائد BY زائد CZ بساوي D وهنا بدنا نلاحظ
+
+366
+00:28:59,850 --> 00:29:03,990
+مالي لما نكتب على صورة AX زائد BY زائد CZ بساوي D
+
+367
+00:29:03,990 --> 00:29:10,400
+اللي هو عددبكون A هو معامل اللي هو الـ I للنورمال
+
+368
+00:29:10,400 --> 00:29:14,880
+والـ B معامل الـ J للنورمال والـ C معامل الـ K
+
+369
+00:29:14,880 --> 00:29:20,740
+للنورمال يعني الآن بمعرفة معادلة الـ plane بعرف ال
+
+370
+00:29:20,740 --> 00:29:25,000
+normal عليه مباشرة بكتب على الصورة هذه المعادلة
+
+371
+00:29:25,000 --> 00:29:28,000
+بقول على طول ال normal الآن بتساوي ايه في اتجاه ال
+
+372
+00:29:28,000 --> 00:29:32,280
+Iب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+373
+00:29:32,280 --> 00:29:33,280
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+374
+00:29:33,280 --> 00:29:34,200
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+375
+00:29:34,200 --> 00:29:38,480
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+376
+00:29:38,480 --> 00:29:39,200
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+377
+00:29:39,200 --> 00:29:41,080
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+378
+00:29:41,080 --> 00:29:41,600
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+379
+00:29:41,600 --> 00:29:42,180
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+380
+00:29:42,180 --> 00:29:42,180
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+381
+00:29:42,180 --> 00:29:43,760
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+382
+00:29:43,760 --> 00:29:44,500
+ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب .. ب ..
+
+383
+00:29:44,500 --> 00:29:50,520
+ب .. ب ..
+
+384
+00:29:50,520 --> 00:29:56,080
+ب ..
+
+385
+00:29:56,100 --> 00:30:00,460
+لكن طوله اختلف واحد اخد normal الوحدة واحد اخد
+
+386
+00:30:00,460 --> 00:30:04,000
+normal طوله عشرة واحد اخد normal طوله مية المهم
+
+387
+00:30:04,000 --> 00:30:06,900
+اللي هو بيكون ايه شماله normal على ال plane ان شاء
+
+388
+00:30:06,900 --> 00:30:10,780
+الله اقطعت السؤال طيب ناخد مثال على الكلام هذا
+
+389
+00:30:10,780 --> 00:30:14,000
+find an equation for the plane through P0
+
+390
+00:30:14,000 --> 00:30:18,680
+perpendicular to M يعني بدنا نوجد معادلة المستوى
+
+391
+00:30:18,680 --> 00:30:24,240
+اللي يمر بهذه النقطة وعمودي على هذا اللي هو مينالـ
+
+392
+00:30:24,240 --> 00:30:29,200
+vector الان صار الموضوع سهل جدا اللي هو على طول
+
+393
+00:30:29,200 --> 00:30:32,760
+بنستخدم هذه المعادلة اللي هي ال a في هذه الحالة
+
+394
+00:30:32,760 --> 00:30:37,600
+خمسة في x ناقص x naught اللي هي x ناقص ناقص تلاتة
+
+395
+00:30:37,600 --> 00:30:43,380
+الان زاد بي اللي هو مين بي معامل اللي هو j اتنين
+
+396
+00:30:43,380 --> 00:30:48,320
+في y minus y naught اللي هي سفر زاد اللي هي ناقص
+
+397
+00:30:48,320 --> 00:30:52,140
+واحد معامل اللي هي c هذا ال capital في z minus
+
+398
+00:30:52,140 --> 00:30:57,700
+سبعةاللي هي z note ويساوي ايش سفر لان استخدمت هذه
+
+399
+00:30:57,700 --> 00:31:03,780
+المعادلة صار عندى لان بعد مبسط خمسة x ناقص خمس عشر
+
+400
+00:31:03,780 --> 00:31:09,360
+بيصير ناقص x زي خمس عشر اتنين y ناقص z سائد سبعة
+
+401
+00:31:09,360 --> 00:31:13,300
+بيساوي سفر لان بالبسطة اكتر بيصير خمسة x زي اتنين
+
+402
+00:31:13,300 --> 00:31:16,620
+y ناقص z والخمس عشر والسبعة عشر اتنين وعشرين بيصير
+
+403
+00:31:16,620 --> 00:31:23,920
+سالم اتنين وعشرين وهذه معادلة المستوى المطلوبواحد
+
+404
+00:31:23,920 --> 00:31:29,180
+تاني قال لي لأ انا بدي معادلة لخط لمستوى بمر
+
+405
+00:31:29,180 --> 00:31:33,400
+بالنقاط تلاتة a وb وc يعني بيقولي find an equation
+
+406
+00:31:33,400 --> 00:31:37,880
+for the line for the plane through a بتساوي كده وb
+
+407
+00:31:37,880 --> 00:31:42,340
+بتساوي كده and c بتساوي كده بنقوله سهل يعني ما
+
+408
+00:31:42,340 --> 00:31:46,540
+احنا الآن هدول مدام ان التلاتة على نفس المستوى
+
+409
+00:31:46,540 --> 00:31:50,060
+التلاتة اللي بدنا نجيبهم المستوى اللي بيمثل هنا
+
+410
+00:31:50,570 --> 00:31:55,230
+بنقدر نجيب vector a,b على المستوى و vector a,c على
+
+411
+00:31:55,230 --> 00:31:59,970
+المستوى و بنقل cross product لـ cross product اللي
+
+412
+00:31:59,970 --> 00:32:04,490
+عندنا ايه شماله؟ بطلع عمودي على المستوى
+
+413
+00:32:07,070 --> 00:32:10,930
+Find an equation for the plane A, B and C بيساوي
+
+414
+00:32:10,930 --> 00:32:14,350
+كده First note that زي ما قلنا بدنا نوجد ايش يا
+
+415
+00:32:14,350 --> 00:32:18,750
+جماعة لـ cross product بين A, B و A, C اللي هما
+
+416
+00:32:18,750 --> 00:32:23,150
+مستقيمين موجودات على المستوى فال cross product
+
+417
+00:32:23,150 --> 00:32:26,090
+اللي هو العمودي على المستوى بيكون عرفنا عمودي على
+
+418
+00:32:26,090 --> 00:32:30,130
+المستوى وعرفناه و طب نعرف المعادلة مين لـ plane
+
+419
+00:32:30,130 --> 00:32:36,790
+لأن A, B سهلة اتنينالنقص يصبح صفر هدى وصفر نقص
+
+420
+00:32:36,790 --> 00:32:42,370
+واحد يصبح نقص كيه الاسي الهو صفر لل I تلاتة لل J
+
+421
+00:32:42,370 --> 00:32:49,210
+ونقص واحد لل K هي AB وهي AC الان اواجهولي اللي هو
+
+422
+00:32:49,210 --> 00:32:52,250
+ال cross product اللي هو هيكون مين ال normal ال
+
+423
+00:32:52,250 --> 00:32:55,470
+vector على ال blendطيب ال normal على ال vector على
+
+424
+00:32:55,470 --> 00:32:59,810
+ال plane و بمرض مين كمان بالنقطة و بنختار النقطة
+
+425
+00:32:59,810 --> 00:33:05,710
+اللى هو مين a0 و 0 و 1 ماشي يا شباب و يا شباب ده
+
+426
+00:33:05,710 --> 00:33:09,870
+نشوف كده صلى الله عننا يا عزيزي الصلاة والسلام ده
+
+427
+00:33:09,870 --> 00:33:14,830
+cross product a b في a cهذا اللي هو بدمثل الـ
+
+428
+00:33:14,830 --> 00:33:19,670
+Normal هو عبارة عن .. زي ما بنعرف IJK الـ AC ..
+
+429
+00:33:19,670 --> 00:33:24,930
+الـ AB معاملات هي تنين سفر ناقص واحد AC اللي هي
+
+430
+00:33:24,930 --> 00:33:28,750
+معاملة سفر تلاتة ناقص واحد بنعمله cross product زي
+
+431
+00:33:28,750 --> 00:33:32,290
+ما أتعلمنا قبل شوية بنشطب هذا وشطب هذا بيصير عبارة
+
+432
+00:33:32,290 --> 00:33:37,150
+عن سفر ناقص ناقص تلاتة اللي هي تلاتة Iوهذا من شطب
+
+433
+00:33:37,150 --> 00:33:40,910
+هذا من شطب هذا بيصير ناقص اتنين الان وفي ناقص في
+
+434
+00:33:40,910 --> 00:33:44,690
+الاصل بيصير ناقص سبعة ال J بوجب سالب الموجب اللي
+
+435
+00:33:44,690 --> 00:33:49,030
+هي في الفكرة المحددة زاد اتنين J و الأخيرا اللي هي
+
+436
+00:33:49,030 --> 00:33:53,030
+بيصير اتنين في تلاتة في ستة في اتجاه ال K هذا is
+
+437
+00:33:53,030 --> 00:33:56,190
+normal to the plane the component equation is
+
+438
+00:33:56,190 --> 00:33:59,690
+أخدنا ال A هذه واحد بياخد هذه بنفع واحد بياخد هذه
+
+439
+00:33:59,690 --> 00:34:02,810
+بنفع زي ما قلنا ايه اللي بدأ فيه يعني لأن تلاتة في
+
+440
+00:34:02,810 --> 00:34:04,250
+X ناقص X not سفر
+
+441
+00:34:07,130 --> 00:34:17,250
+Y-Y0-Y0-Y0-Y0-Y0
+
+442
+00:34:17,250 --> 00:34:21,630
+-Y0-Y0-Y0
+
+443
+00:34:21,770 --> 00:34:30,390
+معادلة خط المستقيم المستقيم هي معادلة
+
+444
+00:34:30,390 --> 00:34:31,470
+الخط المستقيم المستقيم المستقيم هي معادلة الخط
+
+445
+00:34:31,470 --> 00:34:37,250
+المستقيم المستقيم المستقيم المستقيم المستقيم
+
+446
+00:34:37,250 --> 00:34:38,750
+المستقيم المستقيم المستقيم المستقيم المستقيم
+
+447
+00:34:38,750 --> 00:34:38,770
+المستقيم المستقيم المستقيم المستقيم المستقيم
+
+448
+00:34:38,770 --> 00:34:38,770
+المستقيم المستقيم المستقيم المستقيم المستقيم
+
+449
+00:34:38,770 --> 00:34:38,770
+المستقيم المستقيم المستقيم المستقيم المستقين
+
+450
+00:34:38,770 --> 00:34:45,590
+المستقين المستقين المستقين المستقين المستقين
+
+451
+00:34:45,590 --> 00:34:50,290
+المستقين المستقين المستقين المستقين المستقين
+
+452
+00:34:50,290 --> 00:34:50,450
+المستقين المستقين المستقين المستقين المستقين
+
+453
+00:34:50,450 --> 00:34:50,450
+المستقين المستقين المستقين المستقين المستقين
+
+454
+00:34:50,450 --> 00:34:54,810
+المستقين المستوهي مستوى آخر بدنا معادلة الخط
+
+455
+00:34:54,810 --> 00:35:00,390
+المستقيم اللي هو اللي بتقاطع فيه مستوين احنا بنعرف
+
+456
+00:35:00,390 --> 00:35:05,070
+انه لو كان في عنا مستوين غير متوازين هيتقاطع اللي
+
+457
+00:35:05,070 --> 00:35:09,230
+هو في خط أو بيكون المنطبقين طبعا احنا مانعش ناخد
+
+458
+00:35:09,230 --> 00:35:13,610
+حالة الانطباق هناخد حالة ايش اللي هو بتقاطع في خط
+
+459
+00:35:13,610 --> 00:35:19,320
+مستقيم ونوجد معادلة الخط المستقيم تبع التقاطعطيب
+
+460
+00:35:19,320 --> 00:35:21,980
+سهلة أنا بعرف الـ plan الأول و بعرف الـ plan
+
+461
+00:35:21,980 --> 00:35:25,380
+الثاني و بدوجة معادلة الخط المستقيم اللي بتقاطعان
+
+462
+00:35:25,380 --> 00:35:29,220
+فيه مدام بعرف معادلة الـ plan الأول إذا بعرف
+
+463
+00:35:29,220 --> 00:35:33,700
+العمودي عليه N واحد مدام بعرف الـ plan التاني إذا
+
+464
+00:35:33,700 --> 00:35:37,740
+بعرف معادلة الخط المستقيم العمودي عليه N اتنين طيب
+
+465
+00:35:37,740 --> 00:35:42,340
+مدام N واحد عمودي على المستوى الأول إذا عمودي على
+
+466
+00:35:42,340 --> 00:35:49,380
+خط كل خط مستقيم فيه بما فيه خط التقاطعsimilarly
+
+467
+00:35:49,380 --> 00:35:56,130
+بنفس الأسلوبمدام أن 2 عمودي على المستوى التاني إذا
+
+468
+00:35:56,130 --> 00:35:59,570
+عمودي على كل خط مستقيم فيه إذا برضه عمودي على خط
+
+469
+00:35:59,570 --> 00:36:03,350
+التخاطع لأن خط التخاطع موجود في المستويين ولا ليش
+
+470
+00:36:03,350 --> 00:36:06,770
+اسمه خط تخاطع إذا صار عندي العمودي أن واحد
+
+471
+00:36:06,770 --> 00:36:12,730
+والعمودي أن اتنين التنين عموديات على مين؟ على الخط
+
+472
+00:36:12,730 --> 00:36:19,920
+المستقيم اللي هو بتقاطع فيه المستويينيعني بمعنى
+
+473
+00:36:19,920 --> 00:36:24,460
+آخر لو جيت أخدت الـ cross product أن واحد أن اتنين
+
+474
+00:36:24,460 --> 00:36:31,020
+هيكون عمودي على أن واحد و أن اتنين يعني هذا المتجه
+
+475
+00:36:31,020 --> 00:36:35,800
+أن واحد و cross أن اتنين أشماله بوازي الخط
+
+476
+00:36:35,800 --> 00:36:40,700
+المستقيم تبع التقاطع يعني بوازي خط التقاطع إذا
+
+477
+00:36:40,700 --> 00:36:45,380
+عرفنا الخط التقاطع اللي بندور عليه اعرفنا فيكتور
+
+478
+00:36:45,380 --> 00:36:50,210
+موازي لهوبنعرف نقطة عليه نعرف معادلته زي ما عرفنا
+
+479
+00:36:50,210 --> 00:36:55,550
+قبل هيك اذا الان عشان نوجد ناخد مثال عملي عملي
+
+480
+00:36:55,550 --> 00:36:59,470
+find a vector parallel to the line of intersection
+
+481
+00:36:59,470 --> 00:37:05,090
+of the planes كذا and كذا بده اول اشي اشي يوجد هو
+
+482
+00:37:05,730 --> 00:37:09,170
+الذي أريد أن أعطيكمه على خطوط اولا يوجد الـ vector
+
+483
+00:37:09,170 --> 00:37:11,910
+ثم يوجد معادلة الخط المستقيم الخط المستقيم يجب أن
+
+484
+00:37:11,910 --> 00:37:13,350
+نعرف أنه يوجد معادلة الخط المستقيم إذا وجدنا ال
+
+485
+00:37:13,350 --> 00:37:17,010
+vector ونعرف ال V فأخذنا أمثلة على إيجاد معادلة
+
+486
+00:37:17,010 --> 00:37:21,630
+الخط المستقيم إذا الآن السؤال هذا هو find a vector
+
+487
+00:37:21,630 --> 00:37:26,310
+موازي parallel to the line of intersection of the
+
+488
+00:37:26,310 --> 00:37:30,630
+blends يعني إذا كان لدي بلنز اتنين فأنا أريد خط
+
+489
+00:37:30,630 --> 00:37:36,300
+مستقيم أو vector موازي لخط التقاطع بينهمبنجيب
+
+490
+00:37:36,300 --> 00:37:40,120
+العمودي على الأول و العمودي على التاني و نعمل
+
+491
+00:37:40,120 --> 00:37:48,510
+cross product موازي لخط التقاطعVector موازي الخط
+
+492
+00:37:48,510 --> 00:37:52,370
+التخطأ اللي هو المطلوب The line of intersection of
+
+493
+00:37:52,370 --> 00:37:55,530
+two planes is perpendicular to both planes normal
+
+494
+00:37:55,530 --> 00:38:01,570
+to vectors N1 and N2 كما قلنا فوق كيف نوجدها N1 هي
+
+495
+00:38:01,570 --> 00:38:05,750
+تلاتة في اتجاه I نقص ستة في اتجاه J نقص اتنين في
+
+496
+00:38:05,750 --> 00:38:10,380
+اتجاه K اللي هي معاملات منالـ X و الـ Y و الـ Z في
+
+497
+00:38:10,380 --> 00:38:15,320
+المعادلة لبنان هنا الان اتنين بيبدو بنفس الأسلوب
+
+498
+00:38:15,320 --> 00:38:19,320
+ان اتنين بيساوي اتنين I زائد J ناقص اتنين K هي ان
+
+499
+00:38:19,320 --> 00:38:23,240
+اتنين اذا الان يعني مايفنق العمودي على المستوى
+
+500
+00:38:23,240 --> 00:38:26,720
+الأول و العمودي على المستوى التاني و جون العموديين
+
+501
+00:38:26,720 --> 00:38:29,660
+هدولة هيكون العموديين على خط التقاطع لأنه مشترك
+
+502
+00:38:29,660 --> 00:38:36,490
+بين المستويينبكون عندى N1-N2 هو ال vector الموازي
+
+503
+00:38:36,490 --> 00:38:40,290
+لخط التقاطع لأنه بكون عمودي على N1 N2 زي خط
+
+504
+00:38:40,290 --> 00:38:44,640
+التقاطعبصير عند n واحد و cross on اتنين بحي اه جي
+
+505
+00:38:44,640 --> 00:38:48,920
+كي و بنجي تلاتة نقص ستة نقص اتنين زي ما اتعلمنا و
+
+506
+00:38:48,920 --> 00:38:52,640
+اتنين واحد و نقص اتنين و بنفك هذا ال determinant
+
+507
+00:38:52,640 --> 00:38:56,340
+تفكوه هتلاقوه بساوة اربعتاشر I زاد اتنين جي زائد
+
+508
+00:38:56,340 --> 00:39:00,260
+خمستاشر كي طبعا بتعرفوا كيف تفكوه بنشطب هذا بنشطب
+
+509
+00:39:00,260 --> 00:39:03,040
+هذا ناقص الستة في ناقص اتنين اللي هي عبارة عن
+
+510
+00:39:03,040 --> 00:39:06,380
+اتناش اتناش ناقص ناقص اتنين يعني زياد اتنين يعني
+
+511
+00:39:06,380 --> 00:39:11,200
+اربعتاشر I و الباقي بنفس الاسلوبالان اي هذا واحد
+
+512
+00:39:11,200 --> 00:39:15,460
+من مين ال cross product موازي لمين لخط التقاطع
+
+513
+00:39:15,460 --> 00:39:21,580
+بينهم لو ضربت هذا ال vector في اي scalar هيطلع
+
+514
+00:39:21,580 --> 00:39:25,080
+vector موازي لخط التقاطع لأن كلهم ايه شمالهم كلهم
+
+515
+00:39:25,080 --> 00:39:28,720
+متوازيات اذا any non zero multiple of any واحد
+
+516
+00:39:28,720 --> 00:39:31,540
+cross ان اتنين will be a vector parallel to the
+
+517
+00:39:31,540 --> 00:39:36,970
+line of intersectionالان طلب مني الان find
+
+518
+00:39:36,970 --> 00:39:40,430
+parametric equations for the line in which the
+
+519
+00:39:40,430 --> 00:39:46,530
+planes اللي هي intersects الان هدولة الخطين أوجدنا
+
+520
+00:39:46,530 --> 00:39:51,070
+.. أوجدنا .. أسف المستويين أوجدنا vector في المثال
+
+521
+00:39:51,070 --> 00:39:58,210
+السابق موازي لخطة خطوحينالان بدوا ال parameter
+
+522
+00:39:58,210 --> 00:40:02,530
+equations لخط التقاطع بنقول سهل خلاص في ال example
+
+523
+00:40:02,530 --> 00:40:05,350
+اللي قبله قبله قبله قبله قبله قبله قبله قبله قبله
+
+524
+00:40:05,350 --> 00:40:09,730
+قبله قبله قبله قبله قبله قبله قبله
+
+525
+00:40:21,860 --> 00:40:27,460
+بدي اللي هو اوجد اللي هو نقطة مشتركة على خط
+
+526
+00:40:27,460 --> 00:40:31,620
+التخاطع بنشوف كيف نوجد النقطة المشتركة بعد ما نوجد
+
+527
+00:40:31,620 --> 00:40:35,240
+النقطة المشتركة على خط التخاطع ايش بنسوي يا جماعة
+
+528
+00:40:35,240 --> 00:40:39,340
+مع ال vector هذا بنطبق قانوننا اللي قبل بشوية طيب
+
+529
+00:40:39,340 --> 00:40:40,000
+ده نشوف كده
+
+530
+00:40:47,860 --> 00:40:53,720
+عشان نجد نقطة على خط التقاطع لان تاخد اي نقطة اللي
+
+531
+00:40:53,720 --> 00:41:00,040
+بتعود عن Z أو Y او X اللي بدك اياها عشوائيا اسهل
+
+532
+00:41:00,040 --> 00:41:06,000
+اشي ان انا اعوض عن Z ب0أو Y بصفر أو X بصفر عشان
+
+533
+00:41:06,000 --> 00:41:09,500
+نضيع إيش رقمه وأسهلنا في التعويض و اللي بده غير
+
+534
+00:41:09,500 --> 00:41:12,800
+هيك يعوض عوض Z بواحد Z بخمس Z بتلاتة Z باربعة اللي
+
+535
+00:41:12,800 --> 00:41:16,800
+بدهويا أو X بواحد أو Y بواحد اللي بدهويا بنيجي
+
+536
+00:41:16,800 --> 00:41:21,940
+الآن عوضنا عن Z بصفر بطلع عندي هذا تلاتة X نقص ستة
+
+537
+00:41:21,940 --> 00:41:28,240
+Y بساوة خمس عشر واتنين X زائد Y بساوة خمسةالان
+
+538
+00:41:28,240 --> 00:41:33,480
+احنا اوجدنا نقطة مشتركة مشتركة
+
+539
+00:41:33,480 --> 00:41:38,020
+عليهم اذا انا اريد خطة قاطع عشان اوجد خطة قاطع بده
+
+540
+00:41:38,020 --> 00:41:44,580
+حل هدول المعادلتين ايام ماشي بده حل هن في نفس
+
+541
+00:41:44,580 --> 00:41:53,360
+الوقتالان لتحل المعادلة انيا بمعانة اخر بيحل
+
+542
+00:41:53,360 --> 00:41:54,720
+المعادلة انيا بمعانة اخر بيحل المعادلة انيا بمعانة
+
+543
+00:41:54,720 --> 00:41:56,520
+اخر بيحل المعادلة انيا بمعانة اخر بيحل المعادلة
+
+544
+00:41:56,520 --> 00:41:58,680
+انيا بمعانة اخر بيحل المعادلة انيا بمعانة اخر بيحل
+
+545
+00:41:58,680 --> 00:42:01,280
+المعادلة انيا بمعانة اخر بيحل المعادلة انيا بمعانة
+
+546
+00:42:01,280 --> 00:42:03,780
+اخر بيحل المعادلة انيا بمعانة اخر بيحل المعادلة
+
+547
+00:42:03,780 --> 00:42:05,560
+انيا بمعانة اخر بيحل المعادلة انيا بمعانة اخر بيحل
+
+548
+00:42:05,560 --> 00:42:07,360
+المعادلة انيا بمعانة اخر بيحل المعادلة انيا بمعانة
+
+549
+00:42:07,360 --> 00:42:11,560
+اخر بهذا بنضربه في معامل الـ x اللي هنا ليه 6x ال
+
+550
+00:42:11,560 --> 00:42:17,200
+.. ال .. الزائد اللي هو 3y بساوة 15 بنحط انتحت بعض
+
+551
+00:42:17,200 --> 00:42:21,520
+هذا معامل زي هذا بنترحم من بعض بتطلع مين بدللة y
+
+552
+00:42:21,520 --> 00:42:25,780
+اللي بدللة y بنحلها بتطلع إيش قيمة y لأن قيمة y
+
+553
+00:42:25,780 --> 00:42:30,100
+اللي طلعت هنا نجيها هن واحد ناقص واحد برد بعود عن
+
+554
+00:42:30,100 --> 00:42:34,680
+قيمة y في أحد المعادلتينيظهر عنده قيمة من ؟ قيمة X
+
+555
+00:42:34,680 --> 00:42:37,660
+هكذا نحل المعادلات في تالت أعداد و تاني أعدادي
+
+556
+00:42:37,660 --> 00:42:44,180
+فبصير عندي الحل أوجدت قيمة X و قيمة Y على خط
+
+557
+00:42:44,180 --> 00:42:48,040
+التقاطع و هذه السفر اللي هي النقطة الثالثة إذا
+
+558
+00:42:48,040 --> 00:42:52,060
+بكون هكذا أوجدت تلاتة و نقص واحد و سفر لإن حلت في
+
+559
+00:42:52,060 --> 00:42:56,060
+نفس الوقت يعني أوجدت الحلول المشتركة يعني أوجدت
+
+560
+00:42:56,060 --> 00:43:01,450
+النقاط ال X و Y المشتركة على اللي هو البلينأو جدت
+
+561
+00:43:01,450 --> 00:43:05,230
+بمعنى آخر النقاط X وY على خط التقاطع اللي بينهم
+
+562
+00:43:05,230 --> 00:43:10,410
+وكان عنده نقطة Zero نقطة محددة اللي هي برضه هتكون
+
+563
+00:43:10,410 --> 00:43:16,450
+على خط التقاطع بين المستويين إذا إيجاد خط التقاطع
+
+564
+00:43:16,450 --> 00:43:20,350
+بين المستويين نقطة على خط التقاطع هي هنا وضحناها
+
+565
+00:43:20,930 --> 00:43:26,170
+عندى نقطة معروفة و الـ vector معروف الموازلة اللى
+
+566
+00:43:26,170 --> 00:43:29,290
+هو خط التفاطع اللى أوددناه في المثال السابق اتت
+
+567
+00:43:29,290 --> 00:43:33,590
+اتين مع بعض بيعملنا اللى هو ال X بيساوي X نُو
+
+568
+00:43:33,590 --> 00:43:40,640
+التلاتة زائد اللى هىاللي هي معامل الـ vector 14T Y
+
+569
+00:43:40,640 --> 00:43:47,580
+بيساوي ناقص واحد Y0 زائد معامل اللي هي T في V1 V2
+
+570
+00:43:47,580 --> 00:43:55,280
+اللي هو 2T زائد اللي هو Z بيساوي 0 زائد T في V3
+
+571
+00:43:55,280 --> 00:43:59,240
+اللي هي 15 بتصير 15T هذي بتصير البرومتريك
+
+572
+00:43:59,240 --> 00:44:01,280
+إيكويجينز البرومتريك إيكويجينز for the line of the
+
+573
+00:44:01,280 --> 00:44:06,260
+intersection of the two Eblets الانبقولي find the
+
+574
+00:44:06,260 --> 00:44:09,900
+point where the line x بالساوية كده و y بالساوية
+
+575
+00:44:09,900 --> 00:44:15,040
+كده و z بالساوية كده intersects the plane اللي هو
+
+576
+00:44:15,040 --> 00:44:21,780
+معدلته معاطية ايش بده بده you did اللي هي النقطة
+
+577
+00:44:21,780 --> 00:44:27,120
+المشتركة بين الخط المستقيم و بين مين اللي هو ال
+
+578
+00:44:27,120 --> 00:44:32,000
+plane طبعا ال plane اللي خط المستقيم يا بوازي ال
+
+579
+00:44:32,000 --> 00:44:37,560
+plane يا بوازي ال planeيا بقطع الـ Plain في نقطة
+
+580
+00:44:37,560 --> 00:44:43,240
+يا ما بيقع عليه الان احنا ماخدين مثال اللي هو في
+
+581
+00:44:43,240 --> 00:44:46,900
+الواقع المثال اللي ماخدينه بس نقطة تقاطع واحدة مع
+
+582
+00:44:46,900 --> 00:44:51,980
+اللي لا بوازي ولا بيقع عليه لتنشوف كيف نوجد نقطة
+
+583
+00:44:51,980 --> 00:44:57,580
+التقاطع بين خط و Plain الان عشان النقطة عشان اللي
+
+584
+00:44:57,580 --> 00:45:02,080
+هي النقطة هذهتقع على الخط المستقيم و تقع على الـ
+
+585
+00:45:02,080 --> 00:45:07,200
+plane لازم تحقق الخط المستقيم و تحقق من محادرة الـ
+
+586
+00:45:07,200 --> 00:45:12,460
+plane فالان النقطة تلقى 8 ع 3 او 2T نقص 2 نقص T
+
+587
+00:45:12,460 --> 00:45:20,580
+واحد زي ال T عشان نوجد قيمة اللي هي النقطة هذه
+
+588
+00:45:20,580 --> 00:45:24,000
+اللي بتقع على ال plane لازم نعوض هذه في ال plane
+
+589
+00:45:24,000 --> 00:45:28,970
+اللي هي بدلل ال T و نوجد قيمة Tقيمة T اللي بتحقق
+
+590
+00:45:28,970 --> 00:45:35,610
+المعادلة هذه هي اللي بتخللي النقطة هذه ابتقى على
+
+591
+00:45:35,610 --> 00:45:40,830
+الـ Blade إذاً هذه النقطة بتعوضها هنا يعني بتتحقق
+
+592
+00:45:40,830 --> 00:45:46,910
+طرفي المعادلة فبصير الآن 3 في X هنا اللي هي 8 ع 3
+
+593
+00:45:46,910 --> 00:45:53,110
+زاد 2Tزائد اتنين في y هي نقص اتنين T زائد ستة في
+
+594
+00:45:53,110 --> 00:45:54,810
+واحد زائد T بساوي ستة
+
+595
+00:45:59,220 --> 00:46:06,400
+لازم اتحقق طرفي المعادلة عشان تكون على الـ plane 8
+
+596
+00:46:06,400 --> 00:46:13,960
+زي 6t لأن ضربت تصبح 8 زي 6t نقص 4t زي 6 زي 6t 6t
+
+597
+00:46:13,960 --> 00:46:18,900
+بساوي 6 بسطتها طلعت 8t ايش بساوي سالب 8 طبعا
+
+598
+00:46:18,900 --> 00:46:24,840
+بتعرفوا اتبسطوها 6t و 6t اللي هو عبارة عن 12t نقص
+
+599
+00:46:24,840 --> 00:46:30,750
+4 طلعت 8tو6 و8 و14 و6 و8 و14 و6 و8 و14 و6 و8 و14
+
+600
+00:46:30,750 --> 00:46:33,670
+و6 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+601
+00:46:33,670 --> 00:46:33,670
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+602
+00:46:33,670 --> 00:46:33,730
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+603
+00:46:33,730 --> 00:46:36,310
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+604
+00:46:36,310 --> 00:46:36,310
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+605
+00:46:36,310 --> 00:46:36,390
+و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8 و8
+
+606
+00:46:42,140 --> 00:46:48,120
+و الـ Y هذه و الـ Z هذه تكون هنا و تكون هنا في نفس
+
+607
+00:46:48,120 --> 00:46:51,780
+الوقت يعني therefore the point where the line
+
+608
+00:46:51,780 --> 00:46:57,640
+intersects the plane is X و Y و Z اللي عندنا اللي
+
+609
+00:46:57,640 --> 00:47:02,580
+هي هذه عند T بتساوي سالب واحد اللي عبارة عن تلتين
+
+610
+00:47:02,580 --> 00:47:07,220
+و اتنين و سفر و مش مصدقين عوضه هذه هان هتلاقيها
+
+611
+00:47:07,220 --> 00:47:12,540
+بتحقق طرفي المعادلةوعوضها دي هان هتلاقيها إحدى
+
+612
+00:47:12,540 --> 00:47:17,260
+نقاط اللي هي ال line
+
+613
+00:47:17,260 --> 00:47:23,360
+اللي طبعا أي قيمة التي ناخدها بتبقى XY وZ اللي
+
+614
+00:47:23,360 --> 00:47:27,100
+بتطلع عبارة عن نقطة عالمية على الخط المستقيم وهذا
+
+615
+00:47:27,100 --> 00:47:32,120
+من ضمنهاطيب ديجينا اللي هو نقطة أخرى الـ distance
+
+616
+00:47:32,120 --> 00:47:37,480
+from a point to قابلين الان اوجدنا في الأول اللي
+
+617
+00:47:37,480 --> 00:47:41,860
+هو المسافة بين نقطة وبين خط مستقيم في اللي قبل إذا
+
+618
+00:47:41,860 --> 00:47:46,020
+بتتذكر استخدمنا الـ cross product في الإيجاد هنا
+
+619
+00:47:46,020 --> 00:47:49,360
+يعني اللي .. اللي .. اللي .. اللي .. اللي هيسعفنا
+
+620
+00:47:49,360 --> 00:47:53,180
+في الأمر اللي هو الـ cross product و هنشوف كيف
+
+621
+00:47:53,180 --> 00:47:57,300
+اللي هو بسعفناالان لو كانت P is a point on a plane
+
+622
+00:47:57,300 --> 00:48:01,380
+with normal N يعني معروفة نقطة P وين علي البلان و
+
+623
+00:48:01,380 --> 00:48:05,080
+ال normal للبلان معروف طبعا مجرد معرفة معدلة
+
+624
+00:48:05,080 --> 00:48:08,080
+البلان و نعرف ال normal عليه then the distance
+
+625
+00:48:08,080 --> 00:48:11,840
+from any point S لو كانت point S فوق و هي في عند
+
+626
+00:48:11,840 --> 00:48:15,620
+بلان و بدنا نوجد المسافة بين S و بين مين البلان
+
+627
+00:48:15,620 --> 00:48:20,820
+الان باجي باخد نقطة عشوائية P علي البلان بوجدها و
+
+628
+00:48:20,820 --> 00:48:26,360
+بوجد بيه Sو بكون عارف الـ Normal و بضرب ال BS في
+
+629
+00:48:26,360 --> 00:48:29,620
+ال Normal على ال magnitude لل N أنا بتدعي انه
+
+630
+00:48:29,620 --> 00:48:33,260
+هيطلع ايش ماله ال distance ال Normal طبعا ايش هو
+
+631
+00:48:33,260 --> 00:48:36,960
+زي ما اتفجنا للمعادلة العامة AI زائد BJ زائد CK
+
+632
+00:48:36,960 --> 00:48:42,720
+لأنه اصلا هذا هتلاقي اصلا BS.N على N هو عبارة عن
+
+633
+00:48:42,720 --> 00:48:46,180
+ال magnitude لهذا في ال magnitude لهذا في ال
+
+634
+00:48:46,180 --> 00:48:49,240
+cosine الزاوية اللي بينهمهذه اللي بتروح مع هذه
+
+635
+00:48:49,240 --> 00:48:52,140
+فبصير ال distance هو عبارة عن ال BS في cosine
+
+636
+00:48:52,140 --> 00:48:57,800
+الزاوية اللي بين ال BS و ال normal شوف هذا الكلام
+
+637
+00:48:57,800 --> 00:49:04,760
+عمليا بمثال هاي المثال شوف الآن هاي النقطة S وهي
+
+638
+00:49:04,760 --> 00:49:10,520
+ال plain ماشي الحالة الان باجي بختار نقطة عشوائية
+
+639
+00:49:10,520 --> 00:49:17,320
+عليها و بسميها B ماشي الان هذا اللي هو ال BS ماشي
+
+640
+00:49:17,810 --> 00:49:21,370
+وهذا ال vector ال normal العمودي عليه معروف ال
+
+641
+00:49:21,370 --> 00:49:25,390
+normal معروف هذا بيصير الآن هذه الزاوية اللي بينهم
+
+642
+00:49:25,390 --> 00:49:30,550
+فتة اسمها فبصير ال distance هذه عبارة عن ال
+
+643
+00:49:30,550 --> 00:49:36,370
+magnitude لل بي اس في cosine الزاوية اللي بينهم
+
+644
+00:49:36,370 --> 00:49:41,080
+ماشي الحالهذا الطول من هنا الان هو عبارة عن الـ
+
+645
+00:49:41,080 --> 00:49:44,800
+magnitude للـ Bs في كوسينة زاوية الـPin and Theta
+
+646
+00:49:44,800 --> 00:49:49,860
+اللي هو في الواقع عبارة عن مين؟ عن الـBs كفكتور
+
+647
+00:49:49,860 --> 00:49:56,640
+ضبط الـN كفكتور على ال magnitude للـN كفكتور بتطلع
+
+648
+00:49:56,640 --> 00:50:00,240
+اللي هي مين؟ اللي هو الطول من هنا الان زي ما قلنا
+
+649
+00:50:00,540 --> 00:50:04,080
+الان الطول منها اللهان هو اه هو المطلوب ال
+
+650
+00:50:04,080 --> 00:50:08,520
+distance لأنه لما نزل عمودي منها ان هذا بوازي هذا
+
+651
+00:50:08,520 --> 00:50:12,800
+و هذا بساوي اللي تحت هذا اذا هذا بساوي هذا اذا هذا
+
+652
+00:50:12,800 --> 00:50:17,120
+منها اللهان هي ال distance بالظبط هي هذه اللي هي
+
+653
+00:50:17,120 --> 00:50:23,020
+هتساوي ال and ضدالـ BS على الـ magnitude لـ N إذاً
+
+654
+00:50:23,020 --> 00:50:27,020
+هذا الـ distance نشوف الكلام هذا عمليًا الان نجد
+
+655
+00:50:27,020 --> 00:50:30,520
+الـ example اللي عندنا نجد الـ distance from S to
+
+656
+00:50:30,520 --> 00:50:35,520
+the plane 3X زي 2Y زي 6Z إيش بيساوي؟ ستة الان زي
+
+657
+00:50:35,520 --> 00:50:39,680
+ما قلنا أول شي بنجد نقطة بيه نجد نقطة بيه في الـ
+
+658
+00:50:39,680 --> 00:50:43,460
+plane و نتكلم عن طريقة الـ vector of BS بيقولك إيش
+
+659
+00:50:43,460 --> 00:50:49,480
+بتساوي بالظبط اللي قلته on to a vector Nعندي الـ
+
+660
+00:50:49,480 --> 00:50:54,740
+Plane هيه مظبوط لأن 3x زي 2y زي 6z بالساوية 6 إيش
+
+661
+00:50:54,740 --> 00:50:59,360
+الـ Normal عليه؟ معاملاتها ليه 3i زي 2j زي 6k إذا
+
+662
+00:50:59,360 --> 00:51:03,560
+أوجدنا الـ N اللي في القانون دال علي أوجد من الـ
+
+663
+00:51:03,560 --> 00:51:07,180
+Point Bلأن الـ magnitude لن بلزمنا كمان ال
+
+664
+00:51:07,180 --> 00:51:08,180
+magnitude لن بلزمنا كمان ال magnitude لن بلزمنا
+
+665
+00:51:08,180 --> 00:51:08,180
+كمان ال magnitude لن بلزمنا كمان ال magnitude لن
+
+666
+00:51:08,180 --> 00:51:08,880
+بلزمنا كمان ال magnitude لن بلزمنا كمان ال
+
+667
+00:51:08,880 --> 00:51:09,300
+magnitude لن بلزمنا كمان ال magnitude لن بلزمنا
+
+668
+00:51:09,300 --> 00:51:10,440
+كمان ال magnitude لن بلزمنا كمان ال magnitude لن
+
+669
+00:51:10,440 --> 00:51:12,920
+بلزمنا كمان ال magnitude لن بلزمنا كمان ال
+
+670
+00:51:12,920 --> 00:51:21,020
+magnitude لن بلزمنا كمان ال magnitude لن بلزمنا
+
+671
+00:51:21,020 --> 00:51:24,360
+كمان ال magnitude لن بلزمنا كمان
+
+672
+00:51:32,610 --> 00:51:39,270
+أسهل أن تتعوض بصفر وصفر سواء X و Z بصفر أو Y و X
+
+673
+00:51:39,270 --> 00:51:42,770
+بصفر أو Y و Z بصفر و تجد النقطة الثالثة فتكون
+
+674
+00:51:42,770 --> 00:51:45,050
+النقطة بتحقق تمين المعادلة
+
+675
+00:51:48,030 --> 00:51:52,930
+عوضه مثلا z بصفر زي ما هو معوض و x بصفر بطلع 2 و y
+
+676
+00:51:52,930 --> 00:51:56,130
+بالساوية 6 يعني و y بالساوية 3 يعني النقطة اللي
+
+677
+00:51:56,130 --> 00:51:59,570
+احنا عوضنا عليها صفر و ثلاثة و صفر واحد تاني جالب
+
+678
+00:51:59,570 --> 00:52:03,570
+ديش عامد هيك خد x بصفر و y بصفر و z بصير ايش
+
+679
+00:52:03,570 --> 00:52:07,210
+حتساوي واحد بقوله صح واحد تاني اخد ارقام تانية زي
+
+680
+00:52:07,210 --> 00:52:11,780
+ما بدك المهم الارقام اللي بتاخد تتحقق مينلما اتعوض
+
+681
+00:52:11,780 --> 00:52:16,560
+X و Y عشوائية تبقى الـ Z توجدها بالحل عشان تحقق
+
+682
+00:52:16,560 --> 00:52:22,460
+طرفي المعادلة إذا صارت لدي ال B موجودة و ال S
+
+683
+00:52:22,460 --> 00:52:26,400
+موجودة معطوعة في السؤال بيصير لدي ال B S بساوي
+
+684
+00:52:26,400 --> 00:52:31,080
+اللي هو بي اس اللي هي واحد ناقص صفر في اتجاه ال I
+
+685
+00:52:31,080 --> 00:52:36,900
+واحد ناقص تلاتة في اتجاه ال Jتلاتة ناقص سفر في
+
+686
+00:52:36,900 --> 00:52:41,560
+اتجاه الـ K بيطلع عندى عبارة عن I ناقص اتنين J
+
+687
+00:52:41,560 --> 00:52:45,480
+زائد تلاتة K هذا ال BS جهزته لأموراللي انا بدي
+
+688
+00:52:45,480 --> 00:52:50,080
+اوجد الـ BS ضد مين الـ N و بعدين بجسمه على المين
+
+689
+00:52:50,080 --> 00:52:53,240
+على ال magnitude لل N بكون هو ال distance المطلوبة
+
+690
+00:52:53,240 --> 00:53:00,380
+إذا ال N بقول ال magnitude لأ هذه ال N بي اس هي ضد
+
+691
+00:53:00,380 --> 00:53:04,960
+مين ال N مضروب في اللي هو ال magnitude واحد على ال
+
+692
+00:53:04,960 --> 00:53:09,660
+magnitude لل N ال N طوله قداش سبعة بيصير واحد على
+
+693
+00:53:09,660 --> 00:53:12,400
+سبعة في ال dot product بينهم اوجد ال dot product
+
+694
+00:53:12,400 --> 00:53:21,940
+اللي هي تلاتةنقص اربعة بطلع ناقص واحد زائد تمانتعش
+
+695
+00:53:21,940 --> 00:53:26,800
+اللي هو ياش سبعتعش سبعتعش مضروب في مين على سبعة
+
+696
+00:53:26,800 --> 00:53:30,900
+اذا عملنا الدور التفرضات طالع عندي بالشكل اللي
+
+697
+00:53:30,900 --> 00:53:35,620
+أمامي اللي هو تلاتة و مضروبة في سبعة بصير تلاتة ع
+
+698
+00:53:35,620 --> 00:53:39,060
+سبعة ناقص اربعة و مضروبة في سبعة ع سبعة ناقص اربعة
+
+699
+00:53:39,060 --> 00:53:42,160
+ع سبعة و تلاتة في ستة بتمانتعش مضروبة في سبعة بطلع
+
+700
+00:53:42,160 --> 00:53:46,690
+تمانتعش ع سبعةهذه المسافة
+
+701
+00:53:46,690 --> 00:53:47,870
+المطلوبة
+
+702
+00:53:50,530 --> 00:53:55,650
+the angle between blends اللي هي أخر مقتهد ان شاء
+
+703
+00:53:55,650 --> 00:54:01,610
+الله يا جماعة اللي هي الزاوية بين مين بين مستويين
+
+704
+00:54:01,610 --> 00:54:07,050
+الزاوية بين مستويين الان عندى نفترض ان المستويين
+
+705
+00:54:07,050 --> 00:54:11,790
+هي المستوى الأول وهي المستوى الثاني ومتقطعات اللي
+
+706
+00:54:11,790 --> 00:54:16,240
+هو في خط مستقيم وعملن زاوية بينهم التانيةالزاوية
+
+707
+00:54:16,240 --> 00:54:20,260
+اللي بين الاتنين بدنا نعرفها اللي هي عبارة عن
+
+708
+00:54:20,260 --> 00:54:25,040
+الزاوية الحادة بين الـ Normal للأول و Normal
+
+709
+00:54:25,040 --> 00:54:28,980
+للتاني يعني الزاوية الحادة بين ال Normal الأول و
+
+710
+00:54:28,980 --> 00:54:34,440
+Normal التاني هي الزاوية اللي بنسميها بين مستوين
+
+711
+00:54:43,540 --> 00:54:48,780
+الزاوية بين مستويين هي الزاوية الحادة بين
+
+712
+00:54:48,780 --> 00:54:54,720
+العموديين عليهمالان بناء على هذا التعريف بدنا نوجد
+
+713
+00:54:54,720 --> 00:54:58,620
+find the angle between the planes 3x-6y-2z بالساعة
+
+714
+00:54:58,620 --> 00:55:04,640
+15 and the plane 2xy-2z بالساعة 5 الزاوية بينهم
+
+715
+00:55:04,640 --> 00:55:08,560
+اذا عشان وجدهم بدنا نوجده normal للأول سهلالـ N1
+
+716
+00:55:08,560 --> 00:55:12,580
+بيساوي 3I نقص 6J نقص 2K بتعرف 6 بتعرفه عن غيره
+
+717
+00:55:12,580 --> 00:55:14,980
+طبعا المعاملات اللي هي ال X في هذه المعادلة
+
+718
+00:55:14,980 --> 00:55:17,940
+ومعاملات ال Y ومعاملة ال Z بتمثل ال Normal N
+
+719
+00:55:17,940 --> 00:55:24,520
+بيساوي 2I زائد J نقص 2K هدول الان two vectors
+
+720
+00:55:24,520 --> 00:55:29,970
+الزاوية اللي بينهم هي عبارة عن مينعن اللي هو
+
+721
+00:55:29,970 --> 00:55:34,050
+الزاوية بين المستويين و احنا بنعرف نوجد الزاوية
+
+722
+00:55:34,050 --> 00:55:36,730
+بينهم اللي هو ايش بتساوي اخد نهجة باللي هيك
+
+723
+00:55:36,730 --> 00:55:38,890
+الزاوية بين المستويين هي عبارة عن θ بساوي cosine
+
+724
+00:55:38,890 --> 00:55:43,070
+inverse لل N1 dot N2 عن magnitude لل N1 في
+
+725
+00:55:43,070 --> 00:55:46,910
+magnitude لل N2 عارفين ليش؟ لإن احنا بنعرف ال N1
+
+726
+00:55:46,910 --> 00:55:51,650
+في N2 بساوي اللي هو ال magnitude لهذه في ال
+
+727
+00:55:51,650 --> 00:55:54,610
+magnitude لهذه في cosine الزاوية اللي بينهممش هيك
+
+728
+00:55:54,610 --> 00:55:58,990
+تعرفي ضد product n1 فn2 بيساوي اللي هو cos θ في
+
+729
+00:55:58,990 --> 00:56:01,250
+magnitude ده دي في magnitude ده دي الـ nθ بيساوي
+
+730
+00:56:01,250 --> 00:56:05,370
+cos انفرسلها واخدنا القانون قبل هيك لأن n1 ضد n2
+
+731
+00:56:05,370 --> 00:56:09,410
+من عرف نضرب على طول تلاتة في اتنين بستة ستة ناقص
+
+732
+00:56:09,410 --> 00:56:13,210
+ستة اللي هي بروح مع بعض ناقص اتنين في ناقص اتنين
+
+733
+00:56:13,210 --> 00:56:17,150
+بيطلع مين اربعة إذا نضد product بين n1 وn2 اربعة
+
+734
+00:56:17,150 --> 00:56:22,790
+على ال magnitude للأول جدري التربيعي تسعة وستة
+
+735
+00:56:26,260 --> 00:56:31,000
+36.4497 يعني مظبوط و ال magnitude لهذا يعني طوله
+
+736
+00:56:31,000 --> 00:56:35,580
+هذا 7أعملناها قبل شوية وهذا أربعة و واحد خمسة و
+
+737
+00:56:35,580 --> 00:56:39,340
+أربعة تسعة طوله جداش جدر التسعة اللي هو تلاتة إذا
+
+738
+00:56:39,340 --> 00:56:43,480
+طول هذا تلاتة و طول هذا سبعة مع بعض حصل ضرب هنا
+
+739
+00:56:43,480 --> 00:56:46,600
+اللي هو واحد و عشرين فبصير theta بساوية cosine
+
+740
+00:56:46,600 --> 00:56:48,820
+inverse أربعة على واحد و عشرين اللي هي تقريبا
+
+741
+00:56:48,820 --> 00:56:52,560
+بساوية هذا الرقم يعني صار عندى الآن الزاوية بين
+
+742
+00:56:52,560 --> 00:56:57,870
+المستويين هذه هي الزاوية اللي عندناوبهيك بنكون
+
+743
+00:56:57,870 --> 00:57:01,110
+خلصنا المحاضرة وبرك الله فيكم وإلى لقاء نفسنا
+
+744
+00:57:01,110 --> 00:57:03,070
+السلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TqvGHt4Uh0s.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TqvGHt4Uh0s.srt
new file mode 100644
index 0000000000000000000000000000000000000000..335c616ba9c94b9eaff49ee34869dd0500ee3cae
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TqvGHt4Uh0s.srt
@@ -0,0 +1,1958 @@
+1
+00:00:00,710 --> 00:00:04,710
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,710 --> 00:00:12,340
+وبركاته هذه هي المحاضرة رقم 15 لمساق تفاضل وتكامل
+
+3
+00:00:12,340 --> 00:00:17,100
+وهو section 14-3 وهي بعنوان partial
+
+4
+00:00:17,100 --> 00:00:23,240
+derivatives أو الاشتقاق الجزئي partial derivatives
+
+5
+00:00:23,240 --> 00:00:26,400
+of a function of two variables سنبدأ فيها في الأول
+
+6
+00:00:26,400 --> 00:00:32,640
+اللي هو الاشتقاق الجزئي لدالة من مجهولين، مفهوم؟
+
+7
+00:00:32,640 --> 00:00:37,890
+الاشتقاق الجزئي مفهوم بسيط. نتعرف  شو معناه تعريف
+
+8
+00:00:37,890 --> 00:00:42,710
+الاشتقاق الجزئي،  الـ partial derivative of f of
+
+9
+00:00:42,710 --> 00:00:48,050
+x y with respect to x يعني المشتقة الجزئية
+
+10
+00:00:48,050 --> 00:00:52,270
+لدالة f of x y بالنسبة للمتغير x عند نقطة
+
+11
+00:00:52,270 --> 00:00:59,390
+at نقطة x0 y0 is فكرتها أنه إحنا لما بنشتق بالنسبة
+
+12
+00:00:59,390 --> 00:01:04,850
+لـ x بنثبت y، وبنتعامل مع y وكأنها عبارة عن
+
+13
+00:01:04,850 --> 00:01:09,670
+constant أو ثابتة. فبيصير عندي المتغير اللي
+
+14
+00:01:09,670 --> 00:01:14,030
+عندي بس اللي هو x. بناء عليه بيصير اللي هي
+
+15
+00:01:14,030 --> 00:01:19,410
+الاشتقاق الجزئي للـ f بالنسبة لـ x عند النقطة x0 و y0، أو
+
+16
+00:01:19,410 --> 00:01:25,990
+ما نسميه ∂f/∂x at x0 y0 بساوي Limit F of X0 + H و y0 ناقص F of X0 Y0
+
+17
+00:01:25,990 --> 00:01:30,610
+على H لما H بتروح للصفر.
+
+18
+00:01:30,610 --> 00:01:36,210
+لاحظ التغيير بس صار في مين؟ في اللي هي x at x بس
+
+19
+00:01:36,210 --> 00:01:41,030
+لحالها، لأن احنا مفترضين اللي هو y0 اللي هو ثابتة.
+
+20
+00:01:41,030 --> 00:01:44,910
+أو اللي هو بنشتق اشتقاق عادي بالنسبة لـ x يعني
+
+21
+00:01:44,910 --> 00:01:55,080
+التغير بالنسبة لمن؟ للـ x. هو مشابه للـ f prime of x
+
+22
+00:01:55,080 --> 00:01:58,560
+لما تكون function of one variable. عبارة عن limit f
+
+23
+00:01:58,560 --> 00:02:04,040
+of x0 زائد H ناقص f of x0 وكانت y0 مش موجودة لأنها
+
+24
+00:02:04,040 --> 00:02:08,410
+function of one variable على H. إذاً هذه مشابهة لللي
+
+25
+00:02:08,410 --> 00:02:12,930
+حكينا سابقاً في الاشتقاق العادي. هذه اللي هي
+
+26
+00:02:12,930 --> 00:02:15,730
+بتكون، نسميها الـ partial derivative للأف
+
+27
+00:02:15,730 --> 00:02:19,550
+with respect to x at x0 و y0 في حال كانت الـ limit هذه
+
+28
+00:02:19,550 --> 00:02:24,250
+موجودة، provided that limit exists. الآن الـ
+
+29
+00:02:24,250 --> 00:02:28,670
+definition المشابه للـ partial derivative للـ f
+
+30
+00:02:28,670 --> 00:02:32,530
+with respect to y. إذا الـ partial derivative of f
+
+31
+00:02:32,530 --> 00:02:36,870
+of x y with respect to y at the point x0 y0 is ∂f/∂y
+
+32
+00:02:36,870 --> 00:02:43,090
+عند نقطة x0 y0 بساوي اللي هو كأنه من التفاضل
+
+33
+00:02:43,090 --> 00:02:46,870
+العادي. هي عملك زي ما كنا نعمل زمان. وكأن x0
+
+34
+00:02:46,870 --> 00:02:51,090
+كان neglected اللي ممكن نعتبرها ثابتة طبعاً.
+
+35
+00:02:51,090 --> 00:02:54,610
+فبتكون وكأنها الـ function صارت في y لحالها. فبيصير ∂y
+
+36
+00:02:54,610 --> 00:02:59,410
+لـ f of x0 و y، و x0 عبارة عن مقدار ثابت.
+
+37
+00:02:59,410 --> 00:03:03,350
+هذا عند نقطة y0، زي ما كنا نعمل سابقاً.
+
+38
+00:03:03,800 --> 00:03:07,500
+اللي هو بساوي Limit f of x0 و y، المتغير الآن هو y.
+
+39
+00:03:07,500 --> 00:03:13,000
+على y - y0 زائدش ناقص f of x0 y0 على h. هذه
+
+40
+00:03:13,000 --> 00:03:15,780
+اللي هي الـ Partial Derivative with respect to y
+
+41
+00:03:15,780 --> 00:03:20,900
+أو اللي بنسميه ∂f/∂y أو اختصاراً fy  و اختصاراً
+
+42
+00:03:20,900 --> 00:03:26,040
+هذا fx. الآن زمان كنا نقول المعنى الهندسي للمشتقة
+
+43
+00:03:26,850 --> 00:03:29,630
+الأولى لـ function of one variable هي عبارة عن ميل
+
+44
+00:03:29,630 --> 00:03:33,490
+المماس. وهنا أيضًا هي عبارة عن ميل المماس. خليني
+
+45
+00:03:33,490 --> 00:03:39,890
+نشوف إيش اللي بنقوله الآن. إحنا بدنا نيجي ناخد اللي
+
+46
+00:03:39,890 --> 00:03:45,350
+هو function z بتساوي f of x و y. هي اللي هو رسمتها
+
+47
+00:03:45,350 --> 00:03:49,990
+هذه، اللي هي الـ surface. هذا اللي عندنا رسمة z
+
+48
+00:03:49,990 --> 00:03:56,050
+بتساوي f of x و y. وبدنا نشوف الـ Partial Derivative للـ
+
+49
+00:03:56,050 --> 00:04:00,750
+Function عند نقطة ∂f/∂y. بدي أجيبها عند
+
+50
+00:04:00,750 --> 00:04:06,950
+نقطة x0 و y0 و z0 اللي هي f of x0 و y0. هذا جدًا
+
+51
+00:04:06,950 --> 00:04:10,090
+اللي هي الـ Partial Derivative للـ f عند نقطة عند
+
+52
+00:04:10,090 --> 00:04:15,940
+نقطة x0 و y0. الآن هذه اللي هو الـ Partial Derivative
+
+53
+00:04:15,940 --> 00:04:21,100
+∂f/∂y. بناء عليه، مدام ∂f/∂y إذا
+
+54
+00:04:21,100 --> 00:04:26,140
+الآن بدي أثبت x و بدي أخليها x0. يعني هذه بتصير
+
+55
+00:04:26,140 --> 00:04:31,440
+اللي هي f of x0 و y، والـ y المتغيرة. مدام قلت x0
+
+56
+00:04:31,440 --> 00:04:36,120
+اللي هو x بتساوي x0 هو عبارة عن إيش؟ x بتساوي x0
+
+57
+00:04:36,120 --> 00:04:41,410
+عبارة عن Plane. هذا اللي هو بيقطع الـ
+
+58
+00:04:41,410 --> 00:04:46,170
+Surface هذا في خط، اللي هو Curve. هذا اللي هو الـ
+
+59
+00:04:46,170 --> 00:04:50,670
+Plane. أنا ثبتت مين يا جماعة؟ x  بيساوي f of x و y؟
+
+60
+00:04:50,670 --> 00:04:55,570
+ثبتت x0. يعني كأنو z بيساوي f of x0 و y.
+
+61
+00:04:55,570 --> 00:05:00,610
+يعني قطعت اللي هو الـ Surface هذا مع الـ plane x
+
+62
+00:05:00,610 --> 00:05:04,010
+بتساوي x0. لما الـ plane هذا x بتساوي x0
+
+63
+00:05:04,010 --> 00:05:09,070
+اللي هو يقطع اللي هو الـ Surface هذا، هيقطعه طبيعياً
+
+64
+00:05:09,070 --> 00:05:14,330
+في إيه؟ في curve. هذا الـ curve الآن هو اللي صار z
+
+65
+00:05:14,330 --> 00:05:19,750
+بتساوي f of x0 y. هذا اللي بدأنا وديناله الـ
+
+66
+00:05:19,750 --> 00:05:22,730
+derivative with respect to y. إذا صار وكأنه curve
+
+67
+00:05:22,730 --> 00:05:26,270
+عادي، function في y. لما أوديناله الـ derivative
+
+68
+00:05:26,270 --> 00:05:29,830
+بالنسبة لـ y عند النقطة هذه اللي هي اللي سميناها
+
+69
+00:05:29,830 --> 00:05:35,450
+∂f/∂y هي بالضبط ميل المماس اللي هو الـ
+
+70
+00:05:35,450 --> 00:05:40,750
+curve هذا عند النقطة x0 أو y0. إذا الـ slope of
+
+71
+00:05:40,750 --> 00:05:45,290
+the curve f of x0 و y0 at the point هذه اللي حكينا
+
+72
+00:05:45,290 --> 00:05:49,470
+عنها in the vertical plane x0 لأنه صار في الـ
+
+73
+00:05:49,470 --> 00:05:52,470
+vertical plane، لأنه تقاطع الـ vertical plane مع
+
+74
+00:05:52,470 --> 00:05:56,370
+الـ Surface هو عبارة عن is the partial derivative
+
+75
+00:05:56,370 --> 00:06:01,620
+of f with respect to y at x0 و y0. يعني اختصاراً ∂f/∂y
+
+76
+00:06:01,620 --> 00:06:07,080
+at this point. هي عبارة عن ميل المماس
+
+77
+00:06:07,080 --> 00:06:14,940
+هذا عند النقطة x0 y0 أو ميل المماس للدالة
+
+78
+00:06:14,940 --> 00:06:21,780
+f of x0, y هي عبارة عن الـ ∂f/∂x
+
+79
+00:06:21,780 --> 00:06:23,460
+by ∂f/∂x by ∂f/∂y by ∂f/∂x by ∂f/∂x by ∂f/∂y by
+
+80
+00:06:23,460 --> 00:06:25,600
+∂f/∂y by ∂f/∂y by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by
+
+81
+00:06:25,600 --> 00:06:26,240
+∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by
+
+82
+00:06:26,240 --> 00:06:27,900
+∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by
+
+83
+00:06:27,900 --> 00:06:32,440
+∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by
+
+84
+00:06:32,440 --> 00:06:33,780
+∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by
+
+85
+00:06:33,780 --> 00:06:33,820
+∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by
+
+86
+00:06:33,820 --> 00:06:35,280
+∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by
+
+87
+00:06:35,280 --> 00:06:40,380
+∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by ∂f/∂x by
+
+88
+00:06:40,380 --> 00:06:43,520
+∂f/∂x
+
+89
+00:06:43,520 --> 00:06:53,140
+بساوة x0 ببساطة. برنامج b بساوة x0
+
+90
+00:06:53,140 --> 00:07:00,640
+بساوي
+
+91
+00:07:00,640 --> 00:07:01,800
+x0 بساوة x0 بساوة x0 بساوة x0 بساوة x0 بساوة x0
+
+92
+00:07:01,800 --> 00:07:03,080
+x0 بساوة x0 بساوة x0 بساوة x0 بساوة x0 بساوة x0
+
+93
+00:07:03,080 --> 00:07:03,180
+بساوة x0 بساوة x0 بساوة x0 بساوة x0 بساوة x0 بساوة
+
+94
+00:07:03,180 --> 00:07:07,700
+x0 بساوة x0 بساوة x0 بساوة x0
+
+95
+00:07:07,700 --> 00:07:15,240
+بساوة x0 بساوة x0 بساوة بالنسبة ل y عند نقطة x0 و
+
+96
+00:07:15,240 --> 00:07:21,900
+y0. لما نثبت x، أيش ثابت بمقدار x0 أو قلنا ∂f/∂y
+
+97
+00:07:21,900 --> 00:07:27,600
+هي عبارة عن ميل المماس للدالة f of x0, y
+
+98
+00:07:27,600 --> 00:07:36,050
+أو هي عبارة عن معدل تغير الـ f of x0, y
+
+99
+00:07:36,050 --> 00:07:40,510
+بالنسبة لـ y لأن x0 ثبتت. Similarly يا جماعة
+
+100
+00:07:40,510 --> 00:07:47,130
+Similarly for ∂f/∂x. دعونا نشوف كيف اللي هو
+
+101
+00:07:47,130 --> 00:07:52,420
+Similarly اللي أنا عندي لو عندي z بتساوي f of x0
+
+102
+00:07:52,420 --> 00:07:57,100
+و y0. وجيت ثبتت y0. أنا الآن لما ثبت y0
+
+103
+00:07:57,100 --> 00:08:00,320
+معناته الـ surface اللي عندي هذا اللي افترضنا هو هذا
+
+104
+00:08:00,320 --> 00:08:04,020
+الـ surface اللي هو تبع الـ z بتساوي f of x و
+
+105
+00:08:04,020 --> 00:08:08,820
+y بتقاطع مع مين؟ مع y بتساوي y0. هاي y بيساوي y
+
+106
+00:08:08,820 --> 00:08:13,820
+0. هذا هذا y بتساوي y0 بتقاطع معاه. الخط الأزرق
+
+107
+00:08:13,820 --> 00:08:22,300
+هذا يتقاطع
+
+108
+00:08:22,300 --> 00:08:29,140
+بالخط الأزرق هذا. الآن الخط الأزرق هذا عبارة عن f
+
+109
+00:08:29,140 --> 00:08:34,240
+of x y0. يعني function of x. إذا ∂f/∂x
+
+110
+00:08:34,240 --> 00:08:40,490
+لأن x المتغير صارت هي عبارة عن ميل المماس لهذا عند
+
+111
+00:08:40,490 --> 00:08:45,770
+نقطة اللي بنحكي عنها هذه. إذاً برضه اللي هو ∂f/∂x
+
+112
+00:08:45,770 --> 00:08:54,210
+هي عبارة عن ميل المماس لخط التقاطع اللي بين
+
+113
+00:08:54,210 --> 00:08:59,550
+اللي هو الـ surface z بتساوي f of x y و للبين x بتساوي
+
+114
+00:08:59,550 --> 00:09:03,690
+x0. يعني ميل الخط التقاطع هذا اللي هو هذا الخط
+
+115
+00:09:03,690 --> 00:09:10,380
+المماس. ميله هو ∂f/∂x. الآن قبل شوية قلنا
+
+116
+00:09:10,380 --> 00:09:15,280
+عندنا نقطة هذه. إيش خط التقاطع دي؟  ∂f/∂y ∂y/∂y هي
+
+117
+00:09:15,280 --> 00:09:20,920
+ميل خط التقاطع هذا. إذا الآن صار عندي نقطة هذه. هذا
+
+118
+00:09:20,920 --> 00:09:26,540
+يمثل اللي هو ميل. ميله بيمثل ∂f/∂x وهذا
+
+119
+00:09:26,540 --> 00:09:32,180
+ميله بيمثل ∂f/∂y. طيب يا جماعة إيش
+
+120
+00:09:32,180 --> 00:09:35,980
+الـ plane هذا؟ شايفين الـ plane اللي بتكون من هذا ومن
+
+121
+00:09:35,980 --> 00:09:40,740
+هذا؟ هذا بيكون يمس الـ Surface اللي هو z بتساوي f of
+
+122
+00:09:40,740 --> 00:09:46,600
+xy. الـ Surface z بتساوي f of xy كله بيمسوا الـ
+
+123
+00:09:46,600 --> 00:09:51,420
+Plane المكون من هذا وهذا. الـ Tangent لما نكون
+
+124
+00:09:51,420 --> 00:09:55,980
+قدامها بتقول عنها f of x y is differentiable لما
+
+125
+00:09:55,980 --> 00:09:58,860
+تكون الـ function of two variables f of x y is
+
+126
+00:09:58,860 --> 00:10:05,240
+differentiable بيكون اللي هو الـ plane المكون من
+
+127
+00:10:05,240 --> 00:10:12,050
+اللي هو الـ بتكون الـ tangents. الأولاني ∂f/∂
+
+128
+00:10:12,050 --> 00:10:15,830
+x ميلو بيكون ∂f/∂x والتاني ميلو ∂f/∂
+
+129
+00:10:15,830 --> 00:10:22,310
+y. المكون plane المكون من هذه الـ two
+
+130
+00:10:22,310 --> 00:10:28,770
+lines هو tangent للـ plane عند هذه النقطة.
+
+131
+00:10:39,230 --> 00:10:42,110
+كيف نوجد الـ Derivative؟ كيف نوجد الـ Derivative
+
+132
+00:10:42,110 --> 00:10:47,710
+أعتقد أنه كلكم هيستوعبوا بشكل سهل والأمر سهل. Find
+
+133
+00:10:47,710 --> 00:10:51,230
+the value of ∂f/∂x and the value of ∂f/∂y at the
+
+134
+00:10:51,230 --> 00:10:57,680
+point (4,-5) إذا كانت f(x,y) = 2x -3y عملية إيجاد
+
+135
+00:10:57,680 --> 00:11:01,520
+الاشتقاق الجزئي أو المشتقة الجزئية ∂f/∂x أو ∂f/∂
+
+136
+00:11:01,520 --> 00:11:06,340
+x سهل جداً. بثبت y وبأشتق زي زمان. يعني الآن أنا
+
+137
+00:11:06,340 --> 00:11:09,360
+بثبت y وبأشتق بالنسبة لـ x. بيصير ∂f/∂x إيش
+
+138
+00:11:09,360 --> 00:11:14,920
+بتساوي؟ 2x. فإن اشتقاق هذه y ثابتة، بيصير وكأنها زي
+
+139
+00:11:14,920 --> 00:11:19,220
+الثابتة. إذن بيصير 2x لأن الـ x مشتقتها واحد.
+
+140
+00:11:19,220 --> 00:11:24,160
+هذه صفر وهذه صفر لأنه ثابتة بالنسبة لـ x. إذن ∂f/∂
+
+141
+00:11:24,160 --> 00:11:27,790
+x هي 2x. الآن بدي أحسبه عند نقطة (4, -5).
+
+142
+00:11:27,790 --> 00:11:31,830
+عند (4, -5) بحط مكان x أربعة ومكان y ناقص
+
+143
+00:11:31,830 --> 00:11:35,550
+خمسة. بيصير 2 في 4 ناقص 2 في -5
+
+144
+00:11:35,550 --> 00:11:40,350
+8 ناقص -10 تطلع -2. Similarly ∂f/∂
+
+145
+00:11:40,350 --> 00:11:44,490
+y. بتثبت مين؟ x، وبأشتق بالنسبة لـ y. ده  أشتقاق
+
+146
+00:11:44,490 --> 00:11:47,700
+بالنسبة لـ y. ده ناقص 3y. إذن أشتقاق بالنسبة ل y،
+
+147
+00:11:47,700 --> 00:11:50,680
+فهذا يصبح -3. إذا أشتقاق بالنسبة ل y، فهذا يصبح 1.
+
+148
+00:11:50,680 --> 00:11:54,220
+طبعاً هذا مشتقته صفر. فهذا يصبح ∂f/∂y
+
+149
+00:11:54,220 --> 00:11:57,960
+بساوي -3. و منه بحسب ∂f/∂y عند
+
+150
+00:11:57,960 --> 00:12:01,920
+نقطة (4, -5). بعوض عن x بأربعة، يظهر لي
+
+151
+00:12:01,920 --> 00:12:07,610
+لي اللي هو الجواب = -3. طيب نجي الآن ناخد مثال
+
+152
+00:12:07,610 --> 00:12:13,450
+آخر. find ∂f/∂y as a function f(x, y) = y sin(xy). بساوي y
+
+153
+00:12:13,450 --> 00:12:17,370
+sin(xy). في y. لأنه بده يوجده مش عند نقطة. يوجده as
+
+154
+00:12:17,370 --> 00:12:20,630
+a function of x y in general. يعني هي f(x, y)
+
+155
+00:12:20,630 --> 00:12:24,310
+بعدين بده أشتقاق بالنسبة ل y. إذا بده اشتقاق. إذا صار
+
+156
+00:12:24,310 --> 00:12:28,090
+الموضوع سهل عندي. بأثبت x، ثم y هي المتغير. في الـ y
+
+157
+00:12:28,090 --> 00:12:32,610
+بأشتق. حاصل ضرب دالتين. هذا يصبح y مشتقتها 1. يصبح sin(xy)
+
+158
+00:12:32,610 --> 00:12:40,830
+زائد الآن y زي ما هي. هي y، ومشتقة هذا
+
+159
+00:12:40,830 --> 00:12:45,290
+الـ sin(xy)  مشتقة بالنسبة لمن؟ بالنسبة ل y، لأن
+
+160
+00:12:45,290 --> 00:12:50,330
+sin(xy) إيش مشتقتها؟ cos(xy). فمشتقة اللي جوه اللي
+
+161
+00:12:50,330 --> 00:12:54,950
+هو x في y. فبيصير x في y cos(xy). هذه اللي هي ال
+
+162
+00:12:54,950 --> 00:12:59,700
+derivative اللي عندي. Similarly, find fx and fy as a
+
+163
+00:12:59,700 --> 00:13:04,040
+function as functions if هي عندي اللي هي الـ
+
+164
+00:13
+
+223
+00:17:39,260 --> 00:17:42,540
+أجينا قطعناها بالبارابولويد هذه البارابولويد Z
+
+224
+00:17:42,540 --> 00:17:45,840
+بتساوي X تربيع زائد Y تربيع، هذه البارابولويد أجينا
+
+225
+00:17:45,840 --> 00:17:50,900
+قطعناها بإيش؟ بـ plane اللي هو X بتساوي 1، أكيد لما
+
+226
+00:17:50,900 --> 00:17:55,160
+نقطعها بـ X بتساوي 1، إيش هيطلع عندي؟ عبارة عن بارابولا
+
+227
+00:17:55,160 --> 00:17:59,240
+قطع اللي هو إما نقطعه بشكل طولي زي هيك، بيطلع عندي
+
+228
+00:17:59,240 --> 00:18:02,820
+اللي هو الشكل إيه؟ شبه بارابولا، وقلنا نوجد اللي هو
+
+229
+00:18:02,820 --> 00:18:06,500
+الـ slope للـ tangent للبارابولا عند نقطة مين؟ واحد
+
+230
+00:18:06,500 --> 00:18:09,780
+و اثنين و خمسة، هي نقطة واحد و اثنين و خمسة، بدي أشوف الـ
+
+231
+00:18:09,780 --> 00:18:14,020
+tangent لمين؟ للبارابولا. أنا الآن، أنا ما بقدر أحليه
+
+232
+00:18:14,710 --> 00:18:19,610
+اللي هو إما إنّي أشتغل مباشرة و أعمل اشتقاق
+
+233
+00:18:19,610 --> 00:18:24,090
+قديمي و أكمل، أو إني اللي هو أجيب معادلة البارابولا
+
+234
+00:18:24,090 --> 00:18:28,870
+اللي هي اللي أمامي، اللي هو المقطوعة هذه، و أوجد
+
+235
+00:18:28,870 --> 00:18:31,410
+اللي هو الـ tangent زي ما كنا نعمل في  تفاضل ألف
+
+236
+00:18:31,410 --> 00:18:36,110
+خلينا واحدة واحدة الآن، عشان أوجد سلوب للـ tangent
+
+237
+00:18:36,110 --> 00:18:41,380
+للبارابولا.  أتيت، باجي اللي هو عبارة عن اللي هو الـ Z
+
+238
+00:18:41,380 --> 00:18:43,060
+بيساوي X تربيع، و Z بيساوي Y تربيع، اللي هي
+
+239
+00:18:43,060 --> 00:18:46,920
+البارابولا، بوجد Del Z by Del X، لأنها صارت اللي هي
+
+240
+00:18:46,920 --> 00:18:50,580
+لما انقطعته بـ X، صارت اللي هي الـ X ثابتة، إذا صارت
+
+241
+00:18:50,580 --> 00:18:55,560
+هذه البارابولا بين Y و بين مين؟ و بين Z، إذا الميل
+
+242
+00:18:55,560 --> 00:19:00,840
+هنا هو عبارة عن Del Z by Del Y، لأنّ Y، الـ X أصبحت
+
+243
+00:19:01,320 --> 00:19:05,340
+ثابتة يا جماعة، إذا الـ slope of the tangent is the
+
+244
+00:19:05,340 --> 00:19:09,440
+value of the partial derivative of z by y عند مين؟
+
+245
+00:19:09,440 --> 00:19:12,800
+عند النقطة إيش؟ واحد و اثنين و خمسة. طبعاً واحد
+
+246
+00:19:12,800 --> 00:19:17,300
+و اثنين، واحد و اثنين و... هذا الـ x، هذا الـ x، و هذا الـ
+
+247
+00:19:17,300 --> 00:19:22,450
+Y، واحد و اثنين و خمسة. باجي الآن، بدّي الـ Z by الـ Y
+
+248
+00:19:22,450 --> 00:19:26,170
+منها، إنّ  الـ Z by الـ Y، لأنها دي Z، إيش ده؟
+
+249
+00:19:26,170 --> 00:19:30,830
+تصير اثنين Y، إذا الـ slope is الـ Z by الـ Y عند
+
+250
+00:19:30,830 --> 00:19:34,510
+مين؟ عند النقطة واحد و اثنين و خمسة، ماشي، اللي هو
+
+251
+00:19:34,510 --> 00:19:40,320
+بيساوي اثنين، اثنين في قيمة Y التي هي اثنين، ويساوي
+
+252
+00:19:40,320 --> 00:19:47,060
+إيش؟ أربعة. إذا ميل المماس، التي هي البارابولا هذه
+
+253
+00:19:47,060 --> 00:19:51,520
+عند النقطة واحد و اثنين و خمسة، طلع بيساوي إيش؟ أربعة.
+
+254
+00:19:52,520 --> 00:19:55,140
+حل ثاني، واحد قال لي:  قلبي، دي مش هعمل زيك، اسمع أنا
+
+255
+00:19:55,140 --> 00:19:58,800
+بدي آخذ البارابولا اللي بتقاطع فيها، مش قلت الـ X
+
+256
+00:19:58,800 --> 00:20:02,200
+بتساوي واحد؟ قلت لك: آه. قال لي: إذا Z بتساوي واحد
+
+257
+00:20:02,200 --> 00:20:06,140
+زائد Y تربيع، هي البارابولا تبعتنا. Z بتساوي واحد
+
+258
+00:20:06,140 --> 00:20:10,220
+زائد Y تربيع، هي بارابولا، اللي هي تقاطع، اللي هي الـ
+
+259
+00:20:10,220 --> 00:20:14,220
+paraboloid مع الـ plane. بقول له: صح. لما بقول لي أنا
+
+260
+00:20:14,220 --> 00:20:18,300
+أنت مش بدّك اللي هو الـ ميل المماس عند النقطة واحد
+
+261
+00:20:18,300 --> 00:20:21,440
+و اثنين و خمسة؟ قلت له: آه. طيب، إذا عند واحد و اثنين
+
+262
+00:20:21,440 --> 00:20:24,420
+و خمسة، بدّي أوجد اللي هو الـ derivative، Del Z by Del
+
+263
+00:20:24,420 --> 00:20:26,800
+Y، اللي هو الـ ميل هذا، بدّك الـ ميل المماس هذا؟
+
+264
+00:20:26,800 --> 00:20:31,120
+بقول له: آه. Del Z by Del Y، اللي هي بيصير دي Z
+
+265
+00:20:31,120 --> 00:20:33,520
+باي دي Y، في هذه الحالة، لأنّها صارت function of one
+
+266
+00:20:33,520 --> 00:20:40,410
+variable، بتساوي اثنين Y، اثنين Y، و Y قيمتها اثنين، اللي
+
+267
+00:20:40,410 --> 00:20:44,970
+هي اثنين، اللي هي بتساوي أربعة. طبعاً كاتبين واحد
+
+268
+00:20:44,970 --> 00:20:47,850
+و اثنين، لأ، مش واحد و اثنين هنا. إذا بدنا نعملها of
+
+269
+00:20:47,850 --> 00:20:52,190
+two variables، يجب أن تقول اثنين و خمسة، لأن اثنين هي
+
+270
+00:20:52,190 --> 00:20:56,810
+قيمة الـ Y، و الخمسة قيمة الـ Z، و الواحد ما عندنا اللي
+
+271
+00:20:56,810 --> 00:21:03,180
+هو خلصنا منه في الخطوة الأولى، ولكن أنا بفضل نقول
+
+272
+00:21:03,180 --> 00:21:07,580
+الـ z by الـ y at الواحد، واحد و اثنين و خمسة، عشان نكون
+
+273
+00:21:07,580 --> 00:21:12,160
+احنا راجعين لأصلنا. إذا بالطريقة هذه أو بالطريقة هذه
+
+274
+00:21:12,160 --> 00:21:17,840
+بيصير،  functions of more than two variables هنا
+
+275
+00:21:17,840 --> 00:21:21,720
+نيجي للّي،  ده اللي إيه؟ في أكتر متغيّر. Similarly if
+
+276
+00:21:21,720 --> 00:21:24,840
+x, y and z are independent variables، يعني بيقول لي
+
+277
+00:21:24,840 --> 00:21:28,660
+x, y و z هي الـ independent، و f of x, y و z
+
+278
+00:21:28,660 --> 00:21:32,520
+بيساوي x sin y زائد ثلاثة z، طبعاً بيقول independent
+
+279
+00:21:32,520 --> 00:21:35,300
+عشان معروف، مدام independent، إذا انتفاضة اللي
+
+280
+00:21:35,300 --> 00:21:39,040
+بالنسبة لمعدّل تغيره للـ independent بالنسبة لبعض، صفر
+
+281
+00:21:39,040 --> 00:21:42,200
+على طول. فش انتفاضة بينهم. هذه بالنسبة لهذه أو هذه
+
+282
+00:21:42,200 --> 00:21:45,330
+بالنسبة لهذه، على طول صفر. أمّا الـ dependent بالنسبة
+
+283
+00:21:45,330 --> 00:21:49,150
+للـ independent، هما اللي بالفاضل بينهم، لأن الـ
+
+284
+00:21:49,150 --> 00:21:53,390
+independent variable، تغيرهم مستقل، ما لهمش علاقة في
+
+285
+00:21:53,390 --> 00:21:59,790
+حد. معدل التغير بينهم صفر، ما فيش تغير. بقى هذا تغير
+
+286
+00:21:59,790 --> 00:22:02,770
+لحاله، و هذا تغير لحاله، إنّما المتغير المستقل
+
+287
+00:22:02,770 --> 00:22:05,070
+و المتغير التابع، فيه اللي هو هذا التابع، التابع
+
+288
+00:22:05,240 --> 00:22:08,680
+التابع بيعتمد على المستقل، عشان هيك فيه معدل تغير
+
+289
+00:22:08,680 --> 00:22:13,560
+بينهم. عشان هيك في Del F by Del X، و Del F by Del Y، و Del F by Del Z
+
+290
+00:22:13,560 --> 00:22:20,180
+لأنّ F هي المتغير التابع. طيب، اللي أنا طالب Del F by Del
+
+291
+00:22:20,180 --> 00:22:25,320
+Z، بتساوي، زي ما عملنا، بدل ما نثبت اللي هي الـ Y لحالها
+
+292
+00:22:25,320 --> 00:22:29,450
+أو X لحالها، كلّه بنثبته، المتغيّر بيضلّ مين بس؟ Z. إذا Del
+
+293
+00:22:29,450 --> 00:22:34,150
+F by Del Z بتساوي، بتفاضل كله، كأنّ x و y ثوابت، و الـ z
+
+294
+00:22:34,150 --> 00:22:40,290
+متغيّر. إذا الـ x بالنسبة للـ z، تفاضل هذه الـ x زي ما
+
+295
+00:22:40,290 --> 00:22:43,710
+هي، لأنّها ثابتة، في تفاضل اللي جوا، اللي هو مين؟ تفاضل
+
+296
+00:22:43,710 --> 00:22:48,530
+جوا y زائد ثلاثة z، اللي هو اللي هو تفاضل الجوا عبارة عن ثلاثة
+
+297
+00:22:48,530 --> 00:22:52,890
+بيصير ثلاثة x، و تفاضل الـ sin هي عبارة عن مين؟ عن cosine
+
+298
+00:22:53,270 --> 00:22:57,770
+فبيصير ثلاثة x في cos y. زي أفضل إنّي أقول هذا، لأنّ
+
+299
+00:22:57,770 --> 00:23:02,570
+أكيد فاهمين إنّ أنتم هذه الـ x ثابتة، بتنزل. هذه تفاضل
+
+300
+00:23:02,570 --> 00:23:06,390
+في sin هي cos، هو تفاضل اللي جوا هي ثلاثة، فبيطلع
+
+301
+00:23:06,390 --> 00:23:10,360
+عندي هذا هو قيمة التفاضل، لأنّ ده بنحكي عن الـ
+
+302
+00:23:10,360 --> 00:23:13,960
+Partial derivatives and continuity. بدنا نسأل سؤال:
+
+303
+00:23:13,960 --> 00:23:18,660
+ما هي العلاقة؟ هل يعني اللي هو لو كانت الـ function
+
+304
+00:23:18,660 --> 00:23:22,500
+الـ partial، زمان احنا كنا قبل ما أسأل السؤال هذا،
+
+305
+00:23:22,500 --> 00:23:25,900
+زمان كنا نقول: إذا كانت الـ function اللي هي الـ
+
+306
+00:23:25,900 --> 00:23:30,320
+differential، بالأكيد continuous. الآن لو كانت الـ
+
+307
+00:23:30,320 --> 00:23:34,360
+Partial Derivatives موجودة، هل يعني الـ Continuity؟
+
+308
+00:23:34,360 --> 00:23:38,460
+لأ، يعني هنا لو كانت الـ Fx و الـ Fy و الـ Fz
+
+309
+00:23:38,460 --> 00:23:44,780
+موجودات، ليس شرطاً أنها تكون الدالة متصلة. واضح الآن
+
+310
+00:23:44,780 --> 00:23:47,960
+الـ Differentiability شيء آخر غير الـ Partial
+
+311
+00:23:47,960 --> 00:23:51,120
+Derivatives، حنجي له بعد شوية. الآن Partial
+
+312
+00:23:51,120 --> 00:23:56,800
+Derivatives الآن، وإذا كانت موجودة، ليس شرطاً أنها
+
+313
+00:23:56,800 --> 00:24:00,860
+تعطي اللي هو الـ Continuity. يعني الآن في example 8
+
+314
+00:24:00,860 --> 00:24:05,300
+حلقة function f of x y is has a partial
+
+315
+00:24:05,300 --> 00:24:11,830
+derivatives عند الـ (0،0)، but F is not continuous عند
+
+316
+00:24:11,830 --> 00:24:18,090
+مين؟ عند الـ (0،0). إذن المثال هذا ثمانية، لما أقول F
+
+317
+00:24:18,090 --> 00:24:22,130
+of XY بتساوي صفر إذا كانت XY بتساوي صفر، و 1 إذا
+
+318
+00:24:22,130 --> 00:24:27,670
+كانت XY بتساوي صفر. هذا عبارة عن مثال يوضح لي أنّ
+
+319
+00:24:27,670 --> 00:24:31,730
+الـ Partial Derivatives ليس شرطاً أن تؤدي إلى الـ
+
+320
+00:24:31,730 --> 00:24:32,350
+continuity.
+
+321
+00:24:44,610 --> 00:24:50,550
+وهي عبر خطوات، الخطوات التالية. أول شيء يجب أن تبحث عن
+
+322
+00:24:50,550 --> 00:25:01,410
+قيمة f of x و f
+
+323
+00:25:01,410 --> 00:25:07,110
+of y عندما تتجه إلى (0،0) عبر خط y يتساوي x. بعدين
+
+324
+00:25:07,110 --> 00:25:10,930
+جايلي prove that f is not continuous at the origin.
+
+325
+00:25:10,930 --> 00:25:14,030
+بناءً على اللي فوق، أثبت أنّ الـ f is not continuous
+
+326
+00:25:14,030 --> 00:25:17,210
+at the origin. عارفين أنتم، عشان تكون f continuous
+
+327
+00:25:17,210 --> 00:25:21,150
+عند الـ origin، لازم تكون limit f of x و y لما x و
+
+328
+00:25:21,150 --> 00:25:25,550
+y تروح للـ (0،0) بتساوي f of (0،0). لما ده يثبت لك أنّ f
+
+329
+00:25:25,550 --> 00:25:29,920
+of (0،0)، اللي هي لا تساوي الـ Limit اللي بنحكي عنها
+
+330
+00:25:29,920 --> 00:25:33,120
+اللي فوق. وبناءً عليه اللي هو تكون الـ F is not
+
+331
+00:25:33,120 --> 00:25:37,320
+continuous at (0،0). بعدين بتثبت لك أنّ الـ Partial
+
+332
+00:25:37,320 --> 00:25:41,660
+derivatives موجودة عند مين؟ الـ (0،0). بكون أنهينا
+
+333
+00:25:41,660 --> 00:25:44,920
+أثبتنا أنّ الـ Partial derivatives عند (0،0) موجودة،
+
+334
+00:25:44,920 --> 00:25:49,700
+لكن الـ F مش continuous at (0،0). خلينا نجي لرسمة
+
+335
+00:25:49,700 --> 00:25:54,720
+السؤال اللي أمامنا. هذه الدالة بتساوي صفر لما X Y لا
+
+336
+00:25:54,720 --> 00:25:59,340
+تساوي صفر، و 1 لما X في Y إيش مالها؟ تساوي صفر. خلينا
+
+337
+00:25:59,340 --> 00:26:04,640
+نجي الآن، هي رسمة الدالة اللي عندي. الدالة اللي عندي
+
+338
+00:26:04,640 --> 00:26:11,720
+اللي هي عند اللي هو X في Y بتساوي صفر، إيش قيمتها؟ واحد.
+
+339
+00:26:11,720 --> 00:26:21,060
+يعني لما، اللي هي تكون هنا الـ X أو هنا الـ Y، على الخطين
+
+340
+00:26:21,060 --> 00:26:28,220
+هدول، الـ X هنا صفر، و هنا الـ Y صفر، فبتكون الـ X في
+
+341
+00:26:28,220 --> 00:26:30,860
+Y بتساوي صفر، وين يا؟ هنا.
+
+342
+00:26:33,430 --> 00:26:40,990
+أسف، الـ X في Y، نعم، الـ X في Y، هنا هي الـ X في Y هي الـ
+
+343
+00:26:40,990 --> 00:26:47,710
+X هي الـ Y، آسف، وهي الـ X، ماشي؟ هنا دائماً الـ X إيش؟ الـ
+
+344
+00:26:47,710 --> 00:26:53,130
+Y صفر، وهنا دائماً الـ X إيش؟ صفر. ماشي، الآن في المنطقة
+
+345
+00:26:53,130 --> 00:26:58,040
+هذه، عندي الـ x في الـ y دائماً إيش بتساوي؟ أو الـ...
+
+346
+00:26:58,040 --> 00:27:01,340
+الـ... الـ... هذا مع أنّها دائماً بتساوي صفر، x y لا تساوي
+
+347
+00:27:01,340 --> 00:27:05,540
+صفر.  يا في المنطقة هذه، و المنطقة هذه، و المنطقة هذه،
+
+348
+00:27:05,540 --> 00:27:09,660
+و المنطقة هذه، هذي الـ x في الـ y إيش مالها؟ مش صفر.
+
+349
+00:27:09,660 --> 00:27:16,490
+إيش قيمة الدالة عندها؟ قيمة الدالة عندها، قيمة الدالة
+
+350
+00:27:16,490 --> 00:27:19,030
+في المنطقة هذه، والمنطقة هذه، والمنطقة هذه، والمنطقة
+
+351
+00:27:19,030 --> 00:27:22,630
+هذه، اللي X في Y لا تساوي صفر، صفر، هيو. إذا رسمنا
+
+352
+00:27:22,630 --> 00:27:26,890
+احنا بالإسقاطية على الأرض، يعني قيمة الدالة عند
+
+353
+00:27:26,890 --> 00:27:33,210
+المنطقة اللي بتكون X في Y هنا أو هنا، اللي هي
+
+354
+00:27:33,210 --> 00:27:36,410
+بتساوي صفر، وإنّما لها قيمتها بتساوي واحد. إذا أنا
+
+355
+00:27:36,410 --> 00:27:41,750
+على الخط، الخط هذا صورته هيو واحد، و الخط هذا صورته واحد.
+
+356
+00:27:41,750 --> 00:27:48,870
+إذاً هذه اللي هي رسمة الدالة.  طيب، بعد ما
+
+357
+00:27:48,870 --> 00:27:51,950
+شوفنا رسمة الدالة، خلينا نجي الآن للّي بدّه
+
+358
+00:27:51,950 --> 00:27:59,730
+يذكّركم إيش اللي بدنا نثبت؟ إنّ نشوف limit f of x y
+
+359
+00:27:59,730 --> 00:28:07,030
+as x y، و تروح لـ (0،0) along، يعني along y بتساوي x.
+
+360
+00:28:07,030 --> 00:28:11,790
+بتساوي X، الـ Y بتساوي X، هاي Y بتساوي X، و آه،
+
+361
+00:28:11,790 --> 00:28:15,990
+هاي الـ Y بتساوي X، بس مهوّرة في وين؟ في نقطة الأصل
+
+362
+00:28:15,990 --> 00:28:20,490
+اللي يعني F of XY is constantly 0 along the line Y
+
+363
+00:28:20,490 --> 00:28:25,070
+بيساوي X. مظبوط، الـ Y بيساوي X، هيّه، هذا دائماً
+
+364
+00:28:25,070 --> 00:28:30,140
+هيكون صفر، قيمة اللي هي الدالة، ما عدا عند من؟ عند
+
+365
+00:28:30,140 --> 00:28:33,040
+الصفر، except at the origin. لكن أنا ما لي شغل بالصفر،
+
+366
+00:28:33,040 --> 00:28:35,320
+لأنّ أنا رايح على الصفر، بأخذ الـ limit لما تروح
+
+367
+00:28:35,320 --> 00:28:39,440
+للسفر، يعني x و y في طريق لا تساوي لا تساوي صفر، و
+
+368
+00:28:39,440 --> 00:28:43,340
+صفر. لما أقول x y تروح لـ (0،0)، معناته اللي ما بتساويش
+
+369
+00:28:43,340 --> 00:28:50,190
+(0،0). إذا أنا عبر الخط هذا، لما أروح للـ X Y اللي هي
+
+370
+00:28:50,190 --> 00:28:55,110
+عبر Y بتساوي X، لما أروح للـ Zero، عبر الخط هذا، رايح
+
+371
+00:28:55,110 --> 00:28:59,150
+للـ Zero، و رايح للـ Zero، قيمة الدالة في طول الفترة
+
+372
+00:28:59,150 --> 00:29:01,310
+لما أروح للـ Zero من هنا أو للـ Zero من هنا، على
+
+373
+00:29:01,310 --> 00:29:04,470
+الأرض أنا إيش قيمتها؟ على الأرض بتساوي صفر، لذلك الـ
+
+374
+00:29:04,470 --> 00:29:09,510
+Limit، limit للدالة هذه عبر الخط Y بتساوي X لما Y و
+
+375
+00:29:09,510 --> 00:29:13,610
+X بتروح لسفر، لسفر. مش صفر، عبر الخط Y بتساوي X اللي
+
+376
+00:29:13,610 --> 00:29:18,350
+هو هنا أو هنا، إيش قيمة الدالة؟ صفر، عشان هي كـ Limit
+
+377
+00:29:18,350 --> 00:29:22,630
+الـ Zero إيش هيساوي؟ Zero. إذا Limit الدالة في هذه
+
+378
+00:29:22,630 --> 00:29:30,770
+الحالة، As X و Y بيروحوا للـ Zero عبر Y بتساوي X بتروح
+
+379
+00:29:30,770 --> 00:29:35,690
+تصير الدالة طول الوقت صفر، فعشان هيك Limitها بتساوي صفر.
+
+380
+00:29:36,380 --> 00:29:43,140
+لكن لو دي اتلاحظت، F of (0،0)، F of (0،0) هيها، عند X صفر، Y صفر.
+
+381
+00:29:43,140 --> 00:29:47,660
+يعني XY بتساوي صفر، F of (0،0) إيش بتساوي؟ واحد. مدام F of (0،0)
+
+382
+00:29:47,660 --> 00:29:52,140
+بتساوي واحد، يعني إيش ما تطلع الـ Limit، لو كانت موجودة،
+
+383
+00:29:52,140 --> 00:29:57,160
+هذه الـ Limit مش هتطلع بتساوي واحد، لأنّها طلعت إيش
+
+445
+00:34:24,530 --> 00:34:26,770
+the second partial derivatives دي التربيع f y دي
+
+446
+00:34:26,770 --> 00:34:29,270
+ال x تربيعها دي التربيع f y دي ال y دي ال x دي
+
+447
+00:34:29,270 --> 00:34:34,010
+التربيع وكذا الآخرين بيجي للمعادلة أول شيء بنوجد دي
+
+448
+00:34:34,010 --> 00:34:38,020
+ال f y دي ال x لأن احتياج الزمن للكل مع عدل عادلة هذا
+
+449
+00:34:38,020 --> 00:34:41,920
+الـ Y يتفضل
+
+450
+00:34:41,920 --> 00:34:47,500
+إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى
+
+451
+00:34:47,500 --> 00:34:48,820
+Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+452
+00:34:48,820 --> 00:34:48,840
+يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+453
+00:34:48,840 --> 00:34:51,280
+يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+454
+00:34:51,280 --> 00:34:52,320
+يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+455
+00:34:52,320 --> 00:34:53,700
+يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+456
+00:34:53,700 --> 00:34:59,310
+يتفضل إلى Y يتفضل إلى Y  ثم نبدأ بأفضل delta
+
+457
+00:34:59,310 --> 00:35:02,110
+بيقع دي ال Y دي ال X فضّلنا بالنسبة ل X  جمعياً بدي
+
+458
+00:35:02,110 --> 00:35:08,470
+أفضل بالنسبة ل Y إذا فضّلنا هذه بالنسبة ل Y فضّلناها
+
+459
+00:35:08,470 --> 00:35:10,610
+بالنسبة ل Y إذا فضّلنا بالنسبة ل X بدي أفضّلها ل Y
+
+460
+00:35:10,610 --> 00:35:14,410
+يعني أول شيء فضّلنا ل X بعدين بدي أفضّل ل Y اللي بره
+
+461
+00:35:14,940 --> 00:35:20,580
+الـ Y هي ناقص Sine Y وهذه تفاضل Y واحد E to the X
+
+462
+00:35:20,580 --> 00:35:24,560
+نأتي للثانية بالنسبة لـ Y وهذا فضّلنا فاضلها بالنسبة
+
+463
+00:35:24,560 --> 00:35:30,180
+لـ Y هذه اللي هي X زي ما هي X تنزل ناقص Sine Y هي
+
+464
+00:35:30,180 --> 00:35:34,560
+ناقص Sine Y تفاضل
+
+465
+00:35:34,560 --> 00:35:38,820
+بالنسبة ل Y هي E to the X لأن Y واحدفاضلها كمان
+
+466
+00:35:38,820 --> 00:35:43,080
+مرة بالنسبة لـ Y بالنسبة لـ Y هذه بيصير ناقص X Cos
+
+467
+00:35:43,080 --> 00:35:49,760
+Y وهذه صفر طيب فاضل هذه بالنسبة ل .. ده كثير
+
+468
+00:35:49,760 --> 00:35:54,700
+وجديدين نسبة إيه إيش؟  دل X دل بي دل X فضّلنا للأول
+
+469
+00:35:54,700 --> 00:35:59,100
+بدنا نفضّل من برّه بي دل X إذا هذا فاضلها لل X
+
+470
+00:35:59,100 --> 00:36:03,820
+بيصير ناقص Sin Y بالنسبة ل X زي ديتر X واحد
+
+471
+00:36:03,820 --> 00:36:06,970
+الثانيتين إيش المقال هنا؟ زي بعض يعني بقول لك وكأنه بعد
+
+472
+00:36:06,970 --> 00:36:10,530
+هيك هذا طبعاً continuous وهذا continuous واحنا
+
+473
+00:36:10,530 --> 00:36:14,790
+هنتعامل معاهم بس يعني بمعنى آخر من أنا وطالع إيش
+
+474
+00:36:14,790 --> 00:36:18,270
+جالك و دي ال Y by دي ال X دي ال X by دي ال Y فاضل
+
+475
+00:36:18,270 --> 00:36:22,410
+من اللي بترتاح له قبل فضّل ل Y قبل وبعدين ل X أو ل
+
+476
+00:36:22,410 --> 00:36:25,990
+X قبل وبعدين ل Y هيتبع سَيّان إن شاء الله لأن كلهم
+
+477
+00:36:25,990 --> 00:36:29,910
+continuous  ماراح تفرق درجة التفسير لأن هي هذه النظرية
+
+478
+00:36:29,910 --> 00:36:33,330
+اللي قلتها أي اللي هي the mixed derivative theorem
+
+479
+00:36:33,330 --> 00:36:37,550
+قلتُها قبل شوية هي بتقول  بإختصار if f of x, y and its
+
+480
+00:36:37,550 --> 00:36:41,210
+partial derivatives f x, f y, f x, y and f y mixed
+
+481
+00:36:41,210 --> 00:36:44,870
+are defined throughout an open region containing a
+
+482
+00:36:44,870 --> 00:36:48,010
+point a و b and are all continuous at a و b then
+
+483
+00:36:48,010 --> 00:36:52,850
+fxy يساوي fy x عند النقطة a و b يعني بيقول إذا كانت
+
+484
+00:36:52,850 --> 00:36:57,890
+هذه وهذه وهذه وهذه اللي هو continuous موجودات و
+
+485
+00:36:57,890 --> 00:37:02,570
+continuous في اللي هو منطقة معينة حوالين a و b
+
+486
+00:37:03,190 --> 00:37:06,750
+هيطلع عندك الـ Fxy نفس الـ Fy x واحنا هنتعامل
+
+487
+00:37:06,750 --> 00:37:10,510
+بسهولة يا جماعة فاتركوا على الله و نشوف المثال
+
+488
+00:37:10,510 --> 00:37:15,870
+اللي أمامنا اللي بدنا إياه نستغل النظرية في تسهيل
+
+489
+00:37:15,870 --> 00:37:23,390
+عملية الحسابات على أنفسنا الآن مطلوب
+
+490
+00:37:23,390 --> 00:37:27,090
+اللي هو دي التربيع W by دي X by دي Y لـ W اللي
+
+491
+00:37:27,090 --> 00:37:30,820
+أمامنا فتخليني أطلع على هذه اللي هو طالبها أو الأشياء
+
+492
+00:37:30,820 --> 00:37:36,020
+الفضلة بالنسبة لـ Y وبعدين بالنسبة ل X لو فضّلت
+
+493
+00:37:36,020 --> 00:37:39,830
+بالنسبة ل Y في الأول حيطلع ورطة لأن هذا فضل بالنسبة
+
+494
+00:37:39,830 --> 00:37:42,810
+لـ Y هذا فضل لها وهذا فضل لها وستغلبني كثير
+
+495
+00:37:42,810 --> 00:37:47,730
+هذه فأول شيء بفضّل بالنسبة ل X برّيح راسي هذه بتروح
+
+496
+00:37:47,730 --> 00:37:51,490
+بتصفر على طول وانتبه للأسئلة لأنها تطلع وتطلع
+
+497
+00:37:51,490 --> 00:37:55,890
+سريع عشان تعرف لمين فاضلة قبل لأن دل W by دل X على
+
+498
+00:37:55,890 --> 00:38:01,370
+طول بيساوي Y وهذا صفر لأن فضل بالنسبة ل Y دل تربيع
+
+499
+00:38:01,370 --> 00:38:06,090
+Y W by دل Y دل X فضل بالنسبة ل Y على طول تطلع
+
+500
+00:38:06,090 --> 00:38:10,650
+واحدة هنا بالنسبة لـ Mixed Derivative Theorem هادي
+
+501
+00:38:10,650 --> 00:38:15,870
+هادي هي هادي اللي هي بيساوي إيش هتساوي واحد إذا
+
+502
+00:38:15,870 --> 00:38:20,110
+أنت الآن Mixed Derivative Theorem أنقذتنا لأنه لو
+
+503
+00:38:20,110 --> 00:38:23,170
+بدي أفضّل هادي بالنسبة لـ Y في الأول ورطة وبعدين
+
+504
+00:38:23,170 --> 00:38:26,850
+بدي أفضّل X هم هي هتتصفر كمان وراحت فده دوّشة على
+
+505
+00:38:26,850 --> 00:38:31,530
+طول بحدد أفضّل من الأول الأسهل ليه؟ example find f
+
+506
+00:38:31,530 --> 00:38:36,870
+x y z إلى y x y z f f of x y z بيساوي واحد ناقص
+
+507
+00:38:36,870 --> 00:38:41,710
+كذا بدك تودي دي هذه كيف بدكم تبدأوا تحلّوا خلينا
+
+508
+00:38:41,710 --> 00:38:47,670
+ناخذ f y f y اللي هي إيش f y هنا اللي هي ناقص هذا
+
+509
+00:38:47,670 --> 00:38:54,040
+صفر بيصير ناقص أربعة x y z زائد x تربيع خلصنا الآن
+
+510
+00:38:54,040 --> 00:38:58,260
+فضل بالنسبة لـ X بالنسبة لـ X فضل بالنسبة لـ X Y
+
+511
+00:38:58,260 --> 00:39:02,380
+X بفضّل زي ما بده أنا Y X اللي هي عبارة عن ناقص
+
+512
+00:39:02,380 --> 00:39:09,520
+أربعة X Z بالنسبة لـ X يعني صفر ناقص أربعة Y Z زائد
+
+513
+00:39:09,520 --> 00:39:10,120
+إثنين X
+
+514
+00:39:17,270 --> 00:39:23,990
+Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+515
+00:39:23,990 --> 00:39:24,510
+-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+516
+00:39:24,510 --> 00:39:24,550
+-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+517
+00:39:24,550 --> 00:39:24,830
+-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+518
+00:39:24,830 --> 00:39:27,910
+-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+519
+00:39:27,910 --> 00:39:35,750
+-Yxy-Yxy-Yxy عشان يطيّر هذا من أوله أو يبدأ يشتغل
+
+520
+00:39:35,750 --> 00:39:41,550
+هذا من أوله يعني كيف تترتب أنت حر Y X Y Z Y Y X Z
+
+521
+00:39:41,550 --> 00:39:47,390
+Y X Z Y Z X زي ما بدّك وبأي هدا كله هيطلع عليك في
+
+522
+00:39:47,390 --> 00:39:50,770
+الآخر إيش؟  بالمساواة ناقص أربعة واضح
+
+523
+00:39:54,100 --> 00:39:56,880
+الـ Undifferentiability اللي هو قابلية الاشتقاق
+
+524
+00:39:56,880 --> 00:40:00,060
+قابليّة الاشتقاق، أنا بدي أعرفه بس اللي هو لإعرف
+
+525
+00:40:00,060 --> 00:40:03,020
+لكم شو معنى قابليّة الاشتقاق عشان نعرفها بالبرهان
+
+526
+00:40:03,020 --> 00:40:06,620
+النظرية اللي بالبرهان اللي هو برهانه، خلينا بس نمر
+
+527
+00:40:06,620 --> 00:40:09,740
+عليها مرور سريع اللي هي The Increment Theorem for
+
+528
+00:40:09,740 --> 00:40:13,160
+Functions of Two Variables نفترض إن الـ First
+
+529
+00:40:13,160 --> 00:40:16,240
+Partial Derivatives of F of XY are defined through
+
+530
+00:40:16,240 --> 00:40:19,990
+an open region R containing the point x0, y0 يعني
+
+531
+00:40:19,990 --> 00:40:22,310
+فرضنا أن الـ partial derivatives كلهم موجودات الـ
+
+532
+00:40:22,310 --> 00:40:25,710
+f of x و الـ f of y موجودات و continuous عندهم
+
+533
+00:40:25,710 --> 00:40:29,510
+نقطة من x0, y0 إذا فرضنا fx وfy continuous عندهم
+
+534
+00:40:29,510 --> 00:40:30,670
+نقطة x0, y0
+
+535
+00:40:33,870 --> 00:40:38,750
+وفي منطقة تحتوي لـ X0 وY0 ويقول لي ثم هذا التغيير
+
+536
+00:40:38,750 --> 00:40:44,070
+دلتا Z بيساوي F of X0 زي Delta X Y0 زي Delta Y ناقص
+
+537
+00:40:44,070 --> 00:40:49,030
+F of X0 Y0 in the value of F هذا التغيير في قيمة F
+
+538
+00:40:49,030 --> 00:40:53,890
+التي هي دلتا Z that result from moving X0 Y0 لعند
+
+539
+00:40:53,890 --> 00:40:58,150
+x0 زي delta x و y0 زي delta y to another point هنا
+
+540
+00:40:58,150 --> 00:41:01,350
+اللي ذكرناها satisfies the equation إذا هيكون ال
+
+541
+00:41:01,350 --> 00:41:05,870
+increment هذا هذا ال increment يحقق ما يليه يعني
+
+542
+00:41:05,870 --> 00:41:09,150
+لما تكون f its partial derivatives
+
+543
+00:41:09,150 --> 00:41:12,850
+continuous on the open region تحتوي ال x0 و ال y0
+
+544
+00:41:14,960 --> 00:41:20,840
+موجودات في الـ Region R حول الـ X0 و Y0 و
+
+545
+00:41:20,840 --> 00:41:24,740
+Continuous بس عند الـ X0 و Y0 إذن Delta Z هذه
+
+546
+00:41:24,740 --> 00:41:29,200
+لها الـ Increment هذه في F لما انتقلت F من X0 و
+
+547
+00:41:29,200 --> 00:41:34,030
+Y0 لعند X0 زي Delta X Y0 زي Delta Y هتحقق ما
+
+548
+00:41:34,030 --> 00:41:38,310
+يلي يعني ΔZ هي نتيجة نظرية دلتا زي ده دي هتحقق اف
+
+549
+00:41:38,310 --> 00:41:41,410
+اكس دلتا اكس دف واي دلتا واي زي دي ابسلون دلتا اكس
+
+550
+00:41:41,410 --> 00:41:44,450
+زي دي ابسلون اتنين دلتا واي حيث ابسلون واحد ابسلون
+
+551
+00:41:44,450 --> 00:41:48,490
+اتنين بيروح لسفر لما دلتا اكس ودلتا واي بيروح لمين
+
+552
+00:41:48,490 --> 00:41:53,490
+لصفر هذه نظرية بس عشان تشرعنا اللي هو اللي بعيدها
+
+553
+00:41:53,490 --> 00:41:57,590
+مهما بُبرهانها وكده مش لازمني إيش الآن بدي أعرف ال
+
+554
+00:41:57,590 --> 00:42:00,780
+differentiability لل F أحنا بنعرف شو معنى الـ
+
+555
+00:42:00,780 --> 00:42:03,440
+Differentiability لـ function of one variable
+
+556
+00:42:03,440 --> 00:42:06,440
+لأنّنا نعرف الـ Differentiability لـ function of
+
+557
+00:42:06,440 --> 00:42:09,160
+two variables a function z بيساوي f of x و y is
+
+558
+00:42:09,160 --> 00:42:13,280
+differentiable عند النقطة x0 و y0 إذا كانت الـ fx
+
+559
+00:42:13,280 --> 00:42:19,300
+و الfy موجودات و الـ delta z بتحقق ما يليه اللي هي
+
+560
+00:42:19,300 --> 00:42:23,590
+بتحقق اللي حكيناها قبل شوية هاي واللي بنقول أن F
+
+561
+00:42:23,590 --> 00:42:27,370
+is differentiable عند الـ X0 وY0 إذا حققت الشروط
+
+562
+00:42:27,370 --> 00:42:32,950
+اللي هي التالية Fx وFy موجودات و ΔZ بتساوي Fx ΔX
+
+563
+00:42:32,950 --> 00:42:39,690
+زي Fy ΔY زي γ1 ΔX و γ2 ΔY in which each of γ1 وγ2
+
+564
+00:42:39,690 --> 00:42:43,490
+بيروح لـ 0 as Delta X و Delta Y بيروح لمين لـ 0 إذا
+
+565
+00:42:43,490 --> 00:42:49,780
+كانت حققت الـ F عند النقطة X0 وY0 هذه الشروط بنقول
+
+566
+00:42:49,780 --> 00:42:54,740
+عن الـ F differentiable عند الـ X0 وY0 لأ، لما
+
+567
+00:42:54,740 --> 00:42:57,400
+نقول F is differentiable معناته F is
+
+568
+00:42:57,400 --> 00:43:01,600
+differentiable عند مين، عند كل نقطة في الـ domain
+
+569
+00:43:01,600 --> 00:43:05,980
+وبنسمي اللي هو في هذه الحالة للـ graph is a smooth
+
+570
+00:43:05,980 --> 00:43:08,940
+surface لما تكون الـ function differentiable نسميه
+
+571
+00:43:08,940 --> 00:43:13,680
+smooth surface مش smooth curve، smooth surface كله
+
+572
+00:43:13,680 --> 00:43:18,840
+على بعض، smooth surface طيب،أكيد لما سمعتوني
+
+573
+00:43:18,840 --> 00:43:22,540
+هيك بتيجي تقولوا يعني هذا معناه الـ Differentiability
+
+574
+00:43:22,540 --> 00:43:26,720
+والله كده بنتعقد احنا لأ، الآن بدنا نشيل العقدة
+
+575
+00:43:26,720 --> 00:43:30,120
+هذه الـ Definition مع النظرية اللي قبل شوية اللي
+
+576
+00:43:30,120 --> 00:43:33,160
+قلنا إيش لازمتها و الـ Definition إيش لازمه، الاثنين
+
+577
+00:43:33,160 --> 00:43:36,100
+مع بعض بيعطينا هذه الحاجة الحلوة اللي هو الـ
+
+578
+00:43:36,100 --> 00:43:39,460
+Corollary اللي هي و اللي بعدها إذا كانت الـ Partial
+
+579
+00:43:39,460 --> 00:43:44,580
+Derivatives Fx و Fy  من ضمنها Fxy مستمرًا في مجال
+
+580
+00:43:44,580 --> 00:43:49,360
+مفتوح R ثم الـ F مستمرًا في كل مكان من R يعني لما
+
+581
+00:43:49,360 --> 00:43:56,200
+تكون الـ Fx و Fy مستمرًا على مجال مفتوح R يعني على
+
+582
+00:43:56,200 --> 00:44:02,520
+منطقة مفتوحة حوالين نقطة X و Y فنقول F  اسمها
+
+583
+00:44:02,520 --> 00:44:10,260
+مستمرًا في مجال مفتوح لما تكون إذا كانت اللي هي Fx
+
+584
+00:44:10,260 --> 00:44:14,800
+و Fy و الـ F نفسه continuous في الـ open region
+
+585
+00:44:14,800 --> 00:44:18,620
+على طول الـ F إيش ما لها is differentiable إذا صار
+
+586
+00:44:18,620 --> 00:44:21,660
+اللي هو عشان ال test of differentiability سهل أو
+
+587
+00:44:21,660 --> 00:44:25,740
+أحد الـ testات أنه Fx Fy طلع معي ال F continuous
+
+588
+00:44:25,740 --> 00:44:32,660
+على طول ال F is differentiable هنا نأتي إلى نظرية
+
+589
+00:44:32,660 --> 00:44:36,600
+امتداد لتبع زمان نقول إن الـ differentiable
+
+590
+00:44:36,600 --> 00:44:38,860
+function is continuous هذا function of one
+
+591
+00:44:38,860 --> 00:44:41,740
+variable، هنا برضه صح إذا كانت الـ function f of x
+
+592
+00:44:41,740 --> 00:44:47,860
+,y مختلفة عند x0,y0 إذا فهي مختلفة عند x0,y0 إذا
+
+593
+00:44:47,860 --> 00:44:51,120
+ملخص الحديث إن الـ differentiable function is
+
+594
+00:44:51,120 --> 00:44:56,700
+continuous هنا أقول إن الـ partial derivatives fx
+
+595
+00:44:56,700 --> 00:45:03,560
+وfy بإمكانهم إعطاء الـ Continuity للـ F لذلك أقول
+
+596
+00:45:03,560 --> 00:45:09,320
+Fx و Fy يكون موجودة و الـ F موجودة و Continuous
+
+597
+00:45:09,320 --> 00:45:12,260
+كلهم يعني Continuous يعطينا الـ Differentiability
+
+598
+00:45:12,260 --> 00:45:18,160
+لمن؟ للـ F فلو كانت Fx و Fy موجودة ليس شرط أن الـ
+
+599
+00:45:18,160 --> 00:45:21,820
+F مختلفة ماشي الحال لكن لو كانت الـ F is
+
+600
+00:45:21,820 --> 00:45:25,140
+differentiable حنلاقي الـ F الـ X والـ F Y  إثباتهم
+
+601
+00:45:25,140 --> 00:45:28,440
+موجودات حسب اللي هو اللي حكيناه definition قبل و
+
+602
+00:45:28,440 --> 00:45:31,880
+شوية لو كانت F differentiable أكيد F X و F Y
+
+603
+00:45:31,880 --> 00:45:35,880
+موجودات لو كانت F X و F Y موجودات ليس شرط F is
+
+604
+00:45:35,880 --> 00:45:39,460
+differentiable و F X و F Y موجودات ليس شرط أن F is
+
+605
+00:45:39,460 --> 00:45:44,480
+continuous لكن F X و F Y موجودات والـ F موجودة و
+
+606
+00:45:44,480 --> 00:45:47,500
+continuous وهذول continuous بيعطينا F is
+
+607
+00:45:47,500 --> 00:45:50,260
+differentiable ده في ال region main في ال region R
+
+608
+00:45:50,260 --> 00:45:53,140
+هيك و كلنا خلصنا المحاضرة الـ 15 وإلى اللقاء
+
+609
+00:45:53,140 --> 00:45:54,900
+أنفسنا السلام عليكم ورحمة الله وبركاته
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TqvGHt4Uh0s_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TqvGHt4Uh0s_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..3e4140dc1e3c4d3fe7004c04f84db98b4b0d7971
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TqvGHt4Uh0s_raw.srt
@@ -0,0 +1,2456 @@
+1
+00:00:00,710 --> 00:00:04,710
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,710 --> 00:00:12,340
+وبركاتههذه هي المحاضرة رقم 15 لمساق تفاضل وتكامل
+
+3
+00:00:12,340 --> 00:00:17,100
+gene وهو section 14-3 وهي بعنوان partial
+
+4
+00:00:17,100 --> 00:00:23,240
+derivatives أو الاشتقاق الجزئي partial derivatives
+
+5
+00:00:23,240 --> 00:00:26,400
+of a function of two variables سنبدأ فيها في الأول
+
+6
+00:00:26,400 --> 00:00:32,640
+اللي هو الاشتقاق الجزئي لدالة من مجهولين مفهوم
+
+7
+00:00:32,640 --> 00:00:37,890
+الاشتقاق الجزئي مفهوم بسيطنتعرف الشو معنات عريف
+
+8
+00:00:37,890 --> 00:00:42,710
+الاشتقاق الجزء ايه الـ partial derivative of f of
+
+9
+00:00:42,710 --> 00:00:48,050
+x y with respect to x يعني مشتقة المشتقة الجزئية
+
+10
+00:00:48,050 --> 00:00:52,270
+اللي ادالة f of x y بالنسبة للمتغير x at a point
+
+11
+00:00:52,270 --> 00:00:59,390
+at نقصة x0 y0 is فكرتها أنه إحنا لما بنشتق بالنسبة
+
+12
+00:00:59,390 --> 00:01:04,850
+ل x بنثبت yوبنتعامل مع Y وكأنه هي عبارة عن
+
+13
+00:01:04,850 --> 00:01:09,670
+constant أو نقطة ثابتة فبصير عندي المتغير اللي
+
+14
+00:01:09,670 --> 00:01:14,030
+عندي بس اللي هو mean X بناء عليه بيصير اللي هي
+
+15
+00:01:14,030 --> 00:01:19,410
+اشتقاق الجزء لل F بالنسبة ل X عند النقطة X0 وY0أو
+
+16
+00:01:19,410 --> 00:01:25,990
+ما نسميه Fy X at X0 Y0 بساوي Limit F of X0 زي
+
+17
+00:01:25,990 --> 00:01:30,610
+Dutch Y0 نقص F of X0 Y0 على H لما H بتروح للصفة
+
+18
+00:01:30,610 --> 00:01:36,210
+لاحظ التغيير بس صار في مين؟ في اللي هي A at X بس
+
+19
+00:01:36,210 --> 00:01:41,030
+لحالها لأن احنا مفترضين اللي هو Y0 اللي هو ثابتة
+
+20
+00:01:41,030 --> 00:01:44,910
+أو اللي هو بنشتق اشتقاق نز إيه بالنسبة ل X يعني
+
+21
+00:01:44,910 --> 00:01:55,080
+التغير بالنسبة لمن؟الـ X هو مشابه للـ F prime of X
+
+22
+00:01:55,080 --> 00:01:58,560
+لما تكون function of one variable عبارة عن limit F
+
+23
+00:01:58,560 --> 00:02:04,040
+of X0 زائد H ناقص F of X0 وكانت Y0 مش وجودة لأنها
+
+24
+00:02:04,040 --> 00:02:08,410
+function of one variable على Hإذاً هذه مشابهة لللي
+
+25
+00:02:08,410 --> 00:02:12,930
+حكينا سابقاً في الواقع القلاص الف هذه اللي هي
+
+26
+00:02:12,930 --> 00:02:15,730
+بتكون نسميها الـ partial derivative للأفواد
+
+27
+00:02:15,730 --> 00:02:19,550
+respect to x and x0 و y0 في حال كانت ال limit هذه
+
+28
+00:02:19,550 --> 00:02:24,250
+أشمالها provided that limit existsالان الـ
+
+29
+00:02:24,250 --> 00:02:28,670
+definition المشابه للـ partial derivative للـ f
+
+30
+00:02:28,670 --> 00:02:32,530
+with respect to y إذا الـ partial derivative of f
+
+31
+00:02:32,530 --> 00:02:36,870
+of x y with respect to y at the point x0 y0 is del
+
+32
+00:02:36,870 --> 00:02:43,090
+f by del y عند نقطة x0 y0 بساوياللي هو كأنه منفاض
+
+33
+00:02:43,090 --> 00:02:46,870
+العادي هي عملك زي ما كنا نعمل زمان وكأنه x note
+
+34
+00:02:46,870 --> 00:02:51,090
+كان neglected اللي ممكن نعتبرها الـ L طبعا تابت
+
+35
+00:02:51,090 --> 00:02:54,610
+فبتكون وكأنها ال function صارت في y لحالة فبصير dy
+
+36
+00:02:54,610 --> 00:02:59,410
+dy لل F of x note y وx note عبارة عن مقدار تابت
+
+37
+00:02:59,410 --> 00:03:03,350
+هذا عنده نقطة y بالساوية note زي ما كنا نعمل سابقا
+
+38
+00:03:03,800 --> 00:03:07,500
+اللي هو بساوة limit f of x0 وm المتغير الآن عالميا
+
+39
+00:03:07,500 --> 00:03:13,000
+على y y0 زادتش ناقص f of x0 y0 عالميا على h هذه
+
+40
+00:03:13,000 --> 00:03:15,780
+اللي هي الـ Partial Derivative with respect to y
+
+41
+00:03:15,780 --> 00:03:20,900
+أو اللي بنسميه del fy del y أو اختصارا fy واختصارا
+
+42
+00:03:20,900 --> 00:03:26,040
+هذا fx الآن زمان كنا نقول المعنى الهندسي للمشتقان
+
+43
+00:03:26,850 --> 00:03:29,630
+الأولى function of one variable هي عبارة عن ميل
+
+44
+00:03:29,630 --> 00:03:33,490
+المماز وهنا أيضًا هي عبارة عن ميل المماز خلّيني
+
+45
+00:03:33,490 --> 00:03:39,890
+نشوف إيش اللي بنقوله الآن إحنا بدنا نيجي ناخد اللي
+
+46
+00:03:39,890 --> 00:03:45,350
+هو function z بتساوي f of x و y هي اللي هو رسمتها
+
+47
+00:03:45,350 --> 00:03:49,990
+هذه اللي هي ال surface هذا اللي عندنا رسمة z
+
+48
+00:03:49,990 --> 00:03:56,050
+بتساوي f of x و yودنشوف الـ Partial Derivative للـ
+
+49
+00:03:56,050 --> 00:04:00,750
+Function عنده نقطة Del F by Del Y بده أجيبها عنده
+
+50
+00:04:00,750 --> 00:04:06,950
+نقطة Min X0 وY0 وZ0 اللي هي F of X0 وY0 ده جدًا
+
+51
+00:04:06,950 --> 00:04:10,090
+اللي هي الـ Partial Derivative للـ F عند Min عنده
+
+52
+00:04:10,090 --> 00:04:15,940
+نقطة X0 وY0الان هذه اللي هو الـ Partial Derivative
+
+53
+00:04:15,940 --> 00:04:21,100
+دل F by دل إيش Y بناء عليه مدام دل F by دل Y إذا
+
+54
+00:04:21,100 --> 00:04:26,140
+الآن بدي أثبت X و بدي أخليها X0 يعني هذه بالسير
+
+55
+00:04:26,140 --> 00:04:31,440
+اللي هي F of X0 و Y و الـ Y المتغيرة مدام قلت X0
+
+56
+00:04:31,440 --> 00:04:36,120
+اللي هو X بتساوي X0 هو عبارة عن إيش؟ X بتساوي X0
+
+57
+00:04:36,120 --> 00:04:41,410
+عبارة عن Blendالـ Plane هذا اللي هو بيقطع الـ
+
+58
+00:04:41,410 --> 00:04:46,170
+Surface هذا في خط اللي هو Curve هذا اللي هو الـ
+
+59
+00:04:46,170 --> 00:04:50,670
+Plane أنا ثبتت مين يا جماعة؟ Z بيساوي F of X و Y؟
+
+60
+00:04:50,670 --> 00:04:55,570
+ثبتت X Note يعني كأن جولة Z بيساوي F of X Note و Y
+
+61
+00:04:55,570 --> 00:05:00,610
+يعني قطعتاللي هو الـ Surface هذا مع الـ plane X
+
+62
+00:05:00,610 --> 00:05:04,010
+بتساوي X knot لما الـ plane هذا X بتساوي X knot
+
+63
+00:05:04,010 --> 00:05:09,070
+اللي هو يقطع اللي هو الـ Surface هذا هيقطعه طبيعي
+
+64
+00:05:09,070 --> 00:05:14,330
+في إيه؟ في curve هذا الـ curve الآن هو اللي صار Z
+
+65
+00:05:14,330 --> 00:05:19,750
+بتساوي F of X knot Yهذا اللي بدأ أوديتله الـ
+
+66
+00:05:19,750 --> 00:05:22,730
+derivative with respect to Y إذا صار وكأنه curve
+
+67
+00:05:22,730 --> 00:05:26,270
+عادي، function في Y لما أوديتله الـ derivative
+
+68
+00:05:26,270 --> 00:05:29,830
+بالنسبة لـ Y عنده النقطة هذا اللي هي اللي سميناها
+
+69
+00:05:29,830 --> 00:05:35,450
+del F by del Y هي بالظبط من المماثلة اللي هو الـ
+
+70
+00:05:35,450 --> 00:05:40,750
+curve هذا عنده النقطة X0 أو Y0 إذا الـ slope of
+
+71
+00:05:40,750 --> 00:05:45,290
+the curve F of X0 وY0At the point هذه اللي حكينا
+
+72
+00:05:45,290 --> 00:05:49,470
+عنها In the vertical plane X0 لأنه صار في الـ
+
+73
+00:05:49,470 --> 00:05:52,470
+Vertical plane لأنه تحاطر الـ Vertical plane مع
+
+74
+00:05:52,470 --> 00:05:56,370
+الـ Surface هو عبارة عن Is the partial derivative
+
+75
+00:05:56,370 --> 00:06:01,620
+of F with respect to Y at X0 وY0 يعني اختصاراًdel
+
+76
+00:06:01,620 --> 00:06:07,080
+f by del y at this point هي عبارة عن ميل المماس
+
+77
+00:06:07,080 --> 00:06:14,940
+هذا لأن عند النقطة x not y not أو ميل المماس لادلة
+
+78
+00:06:14,940 --> 00:06:21,780
+f of x not o yهي عبارة عن الـ delF by delX by delX
+
+79
+00:06:21,780 --> 00:06:23,460
+by delX by delY by delX by delX by delX by delY by
+
+80
+00:06:23,460 --> 00:06:25,600
+delY by delY by delX by delX by delX by delX by
+
+81
+00:06:25,600 --> 00:06:26,240
+delX by delX by delX by delX by delX by delX by
+
+82
+00:06:26,240 --> 00:06:27,900
+delX by delX by delX by delX by delX by delX by
+
+83
+00:06:27,900 --> 00:06:32,440
+delX by delX by delX by delX by delX by delX by
+
+84
+00:06:32,440 --> 00:06:33,780
+delX by delX by delX by delX by delX by delX by
+
+85
+00:06:33,780 --> 00:06:33,820
+delX by delX by delX by delX by delX by delX by
+
+86
+00:06:33,820 --> 00:06:33,820
+delX by delX by delX by delX by delX by delX by
+
+87
+00:06:33,820 --> 00:06:35,280
+delX by delX by delX by delX by delX by delX by
+
+88
+00:06:35,280 --> 00:06:35,280
+delX by delX by delX by delX by delX by delX by
+
+89
+00:06:35,280 --> 00:06:40,380
+delX by delX by delX by delX by delX by delX by
+
+90
+00:06:40,380 --> 00:06:43,520
+delX
+
+91
+00:06:43,520 --> 00:06:53,140
+by delX by delXبساوة X0 ببساطة برنامج B بساوة X0
+
+92
+00:06:53,140 --> 00:07:00,640
+بساوة
+
+93
+00:07:00,640 --> 00:07:01,800
+X0 بساوة X0 بساوة X0 بساوة X0 بساوة X0 بساوة X0
+
+94
+00:07:01,800 --> 00:07:01,800
+بساوة X0 بساوة X0 بساوة X0 بساوة X0 بساوة X0 بساوة
+
+95
+00:07:01,800 --> 00:07:03,080
+X0 بساوة X0 بساوة X0 بساوة X0 بساوة X0 بساوة X0
+
+96
+00:07:03,080 --> 00:07:03,180
+بساوة X0 بساوة X0 بساوة X0 بساوة X0 بساوة X0 بساوة
+
+97
+00:07:03,180 --> 00:07:07,700
+X0 بساوة X0 بساوة X0 بساوة X0
+
+98
+00:07:07,700 --> 00:07:15,240
+بساوة X0 بساوة X0 بساو بالنسبة لـ y عند نقصة x0 و
+
+99
+00:07:15,240 --> 00:07:21,900
+y0 لما نثبت x أياش ثابت بمقدار x0 أو قلنا del f by
+
+100
+00:07:21,900 --> 00:07:27,600
+del y هي عبارة عن ميل المماس اللي لدالة f of x0 to
+
+101
+00:07:27,600 --> 00:07:36,050
+yأو هي عبارة عن ما هو معدل تغير الـ F of X mod Y
+
+102
+00:07:36,050 --> 00:07:40,510
+بالنسبة لـ Y لأن X mod ثبتت Similarly يا جماعة
+
+103
+00:07:40,510 --> 00:07:47,130
+Similarly for دل F by دل X دعونا نشوف كيف اللي هو
+
+104
+00:07:47,130 --> 00:07:52,420
+Similarlyاللي أنا عندي لو عندي z بتساوي f of x not
+
+105
+00:07:52,420 --> 00:07:57,100
+و y not و جيه ثبتت y not أنا الآن لما ثبت y not
+
+106
+00:07:57,100 --> 00:08:00,320
+معناته ال surface اللي عندي هذا اللي افرضنا هو هذا
+
+107
+00:08:00,320 --> 00:08:04,020
+ال surface اللي هو تبع ال ال ال z بتساوي f of x و
+
+108
+00:08:04,020 --> 00:08:08,820
+y بتقاطع مع مين مع y بتساوي y not هاي y بيساوي y
+
+109
+00:08:08,820 --> 00:08:13,820
+not هدا هدا y بتساوي y not بتقاطع معاهالخط الأزرق
+
+110
+00:08:13,820 --> 00:08:22,300
+هذا يتقطع
+
+111
+00:08:22,300 --> 00:08:29,140
+بالخط الأزرق هذا الان الخط الأزرق هذا عبارة عن f
+
+112
+00:08:29,140 --> 00:08:34,240
+of x y not يعني function of y not إذا دل f by دل x
+
+113
+00:08:34,240 --> 00:08:40,490
+لأن x المتغير صارت هي عبارة عن مين المماس لهذاعنده
+
+114
+00:08:40,490 --> 00:08:45,770
+نقطة اللي بنحكي عن هذه إذاً برضه اللي هو del f by
+
+115
+00:08:45,770 --> 00:08:54,210
+del x هي عبارة عن ميل المماس لخط التفاطع اللي بين
+
+116
+00:08:54,210 --> 00:08:59,550
+اللي هو الـ delta z بساوة f of x y و لبنين x بساوة
+
+117
+00:08:59,550 --> 00:09:03,690
+x note يعني ميل الخط التفاطع هذا اللي هو هذا الخط
+
+118
+00:09:03,690 --> 00:09:10,380
+المماس ميله هو del f by delالان قبل بشوية قلنا
+
+119
+00:09:10,380 --> 00:09:15,280
+عندنا نقطة هذه ايش خط التقاطع دي ال FY دي ال Y هي
+
+120
+00:09:15,280 --> 00:09:20,920
+ميل خط التقاطع هذا اذا الآن صار عندي نقطة هذه هذا
+
+121
+00:09:20,920 --> 00:09:26,540
+بيمثل اللي هو ميل ميله بيمثل دي ال FY دي ال X وهذا
+
+122
+00:09:26,540 --> 00:09:32,180
+ميله بيمثل دي ال FY دي ال Y طيب يا جماعة ايش
+
+123
+00:09:32,180 --> 00:09:35,980
+البلين هذا شايفين البلين اللي بتكون من هذا و من
+
+124
+00:09:35,980 --> 00:09:40,740
+هذاهذا بيكون بمس الـ Surface اللي هو Z بساوية F of
+
+125
+00:09:40,740 --> 00:09:46,600
+XY الـ Surface Z بتساوية F of XY كله بيمسوا الـ
+
+126
+00:09:46,600 --> 00:09:51,420
+Plane المكوّن من هذا و هذا الـ Tangentلما نكون
+
+127
+00:09:51,420 --> 00:09:55,980
+قدامًا بتقول عنها f of x y is differentiable لما
+
+128
+00:09:55,980 --> 00:09:58,860
+تكون الـ function of two variables f of x y is
+
+129
+00:09:58,860 --> 00:10:05,240
+differentiable بكون اللي هو الـ plane المكوّن من
+
+130
+00:10:05,240 --> 00:10:12,050
+اللي هو الـبتكون الـ tangents الأولاني دل أف بايد
+
+131
+00:10:12,050 --> 00:10:15,830
+الـ A ميلو بيكون دل أف بايد ال X والتاني ميلو دل
+
+132
+00:10:15,830 --> 00:10:22,310
+أف بايد ال Y المكوّن البلين المكوّن من هذه الـ two
+
+133
+00:10:22,310 --> 00:10:28,770
+lines هو تانجنت للبلين عند هذا النقطة
+
+134
+00:10:39,230 --> 00:10:42,110
+كيف نوجد الـ Derivative؟ كيف نوجد الـ Derivative
+
+135
+00:10:42,110 --> 00:10:47,710
+بعتقد أنه كلكم هيستوعبوا بشكل سهل و الأمر سهل Find
+
+136
+00:10:47,710 --> 00:10:51,230
+the value of ∂Fy∂X and the value of ∂Fy∂Y at the
+
+137
+00:10:51,230 --> 00:10:57,680
+point 4-5 إذا كانت Fxy إلي أمانيعملية إيجاد
+
+138
+00:10:57,680 --> 00:11:01,520
+الاشتقاق الجزية أو المشتقة الجزيية Fx أو Del Fy
+
+139
+00:11:01,520 --> 00:11:06,340
+Del X سهل جدا بثبت Y و باشتق زي زمان يعني الآن أنا
+
+140
+00:11:06,340 --> 00:11:09,360
+بتثبت Y و باشتق بالنسبة ل X بصير Del Fy Del X أيش
+
+141
+00:11:09,360 --> 00:11:14,920
+بتساوي 2X فإن اشتقاق هذه Y ثابتة بيصير وكأنها زي
+
+142
+00:11:14,920 --> 00:11:19,220
+التلاتة إذن بيصير تلاتة Y لأن X بيستفيدولها واحد
+
+143
+00:11:19,220 --> 00:11:24,160
+هذه سفر وهذه سفر لأنه ثابتت بالنسبة ل X إذن Del Fy
+
+144
+00:11:24,160 --> 00:11:27,790
+Del X هيهاالأن بدي أحسبه عنده نقطة أربعة ناقص خمسة
+
+145
+00:11:27,790 --> 00:11:31,830
+عند أربعة ناقص خمسة بحط مكان X أربعة ومكان Y ناقص
+
+146
+00:11:31,830 --> 00:11:35,550
+خمسة بيصير اتنين في أربعة زي أتنين في ناقص خمسة
+
+147
+00:11:35,550 --> 00:11:40,350
+تمانية ناقص خمس عشر تطلع ناقص سبعة similarly دي ال
+
+148
+00:11:40,350 --> 00:11:44,490
+FY دي ال Y بتبت مين X وفاضل بالنسبة ل Y ده فاضل
+
+149
+00:11:44,490 --> 00:11:47,700
+بالنسبة ل Y ده نهد السفرإذا فاضل بالنسبة لـ y،
+
+150
+00:11:47,700 --> 00:11:50,680
+فهذا يصبح 3x إذا فاضل بالنسبة ل y، فهذا يصبح 1
+
+151
+00:11:50,680 --> 00:11:54,220
+طبعاً هذا التفاضل لها صفر فهذا يصبح del of y del y
+
+152
+00:11:54,220 --> 00:11:57,960
+بسواية 3x ساعة واحدة ومنه بحسب del of y del y عند
+
+153
+00:11:57,960 --> 00:12:01,920
+من عنده نقطة أربعة نقص خمسة بعوض عن x باربعة، يظهر
+
+154
+00:12:01,920 --> 00:12:07,610
+لي اللي هو الجواب الساوي13 طيب نجي لان ناخد مثال
+
+155
+00:12:07,610 --> 00:12:13,450
+اخر find del f y del y as a function f x y بساوي y
+
+156
+00:12:13,450 --> 00:12:17,370
+sine x في y لان بده وجدها مش عند نقطة نجودها دي as
+
+157
+00:12:17,370 --> 00:12:20,630
+a function of x y in general يعني هي f of x y
+
+158
+00:12:20,630 --> 00:12:24,310
+بعدين بده افضل بالنسبة ل y اذا بده predix اذا صار
+
+159
+00:12:24,310 --> 00:12:28,090
+الموضوع سهل عنديأثبت X ثم Y التي متغيرت في الـ Y
+
+160
+00:12:28,090 --> 00:12:32,610
+بفضل Y حاصل ضرب دالتين هذا يصبح Y تفضل 1 يصبح في
+
+161
+00:12:32,610 --> 00:12:40,830
+Sine X Sine XYزاد الآن Y زي ما هي هي Y وتفاضل هذا
+
+162
+00:12:40,830 --> 00:12:45,290
+الـ Sine تفاضل بالنسبة لمن هو؟ بالنسبة لـ Y لان
+
+163
+00:12:45,290 --> 00:12:50,330
+Sine X Y أيش تفاضلها؟ Cos X Y فتفاضل اللي جوه اللي
+
+164
+00:12:50,330 --> 00:12:54,950
+هو X فبصير X في Y Cos X Y هذه اللي هي ال
+
+165
+00:12:54,950 --> 00:12:59,700
+derivative اللي عنديSimilarly, find fx and fy as a
+
+166
+00:12:59,700 --> 00:13:04,040
+function as functions if هي عندي اللي هي الـ
+
+167
+00:13:04,040 --> 00:13:06,920
+function f of x, y بسبب نمرا y على y زي الـ cosine
+
+168
+00:13:06,920 --> 00:13:11,360
+x اللي عندي هذه، fx هي اللي بنعني فيها دي الـ f by
+
+169
+00:13:11,360 --> 00:13:15,660
+دي الـ x، رمز آخر لهذه، دي الـ f by دي الـ x وfy
+
+170
+00:13:15,660 --> 00:13:18,240
+رمز آخر لدي الـ f by دي الـ y، يعني التاني يعني
+
+171
+00:13:18,240 --> 00:13:21,840
+نفس الشيء، بنشوف دي الـ f by دي الـ x اللي هي fx،
+
+172
+00:13:21,840 --> 00:13:27,620
+fx إيش هتساوي؟ طبعا عندي بثبت yبعدين بثبت Y إذا
+
+173
+00:13:27,620 --> 00:13:30,700
+صار عند اللي فوق تفاضل اللي هو عبارة عن constant
+
+174
+00:13:30,700 --> 00:13:36,920
+إذا زي ما هو الآن تفاضله سفر ناقص بس مقام تعرفوا
+
+175
+00:13:36,920 --> 00:13:43,040
+كده تفاضل بالنسبة ل X سفر هذا اللي هو بيصير ناقص
+
+176
+00:13:43,040 --> 00:13:48,220
+تفاضل اللي تحت في اللي فوق اتنين Y هي في تفاضل
+
+177
+00:13:48,220 --> 00:13:53,110
+اللي تحت ايش تفاضل اللي تحتاللي هو Y تفاضلها عبارة
+
+178
+00:13:53,110 --> 00:13:56,430
+عن اللي هو احنا بنفضل بالنسبالي X صفر والـ cosine
+
+179
+00:13:56,430 --> 00:13:59,590
+ناقص وفيه ناقص sin X وفيه عندنا ناقص بيصير زنا
+
+180
+00:13:59,590 --> 00:14:04,730
+زائد sin X فبيصير 2Y sin X على طبعا Y زياد X الـ
+
+181
+00:14:04,730 --> 00:14:08,350
+cosine X اللي كنتوا تعرفين من ربع المقامأنا مثلا
+
+182
+00:14:08,350 --> 00:14:11,310
+انت عارف انت تشتغلوا هذا and of y دي الـ y لإن ده
+
+183
+00:14:11,310 --> 00:14:15,130
+فاضل بالنسبة ل y وثبت x مدام x الثابتة اذا صار
+
+184
+00:14:15,130 --> 00:14:19,790
+عندي اللي هو تفاضل البسط اللي هو 2 بالنسبة ل y
+
+185
+00:14:19,790 --> 00:14:24,890
+طبعا في المقام هايو y زايد قصاين x ناقص اللي هو
+
+186
+00:14:24,890 --> 00:14:29,390
+تفاضل من المقام تفاضل المقام بالنسبة ل y واحد
+
+187
+00:14:29,390 --> 00:14:32,910
+لأنها decimal لإن واحد في مين في اتنين y فصارت
+
+188
+00:14:32,910 --> 00:14:38,890
+سالم اتنين y على تربيع اللي هو المقامبنبقى هادئ من
+
+189
+00:14:38,890 --> 00:14:44,270
+البسطة 2y مع نقص 2y بيصير 2cos x على المقدار اللي
+
+190
+00:14:44,270 --> 00:14:49,360
+تحت المردعSimilarly سأقوله أمور سهلة لأن فايدا الـ
+
+191
+00:14:49,360 --> 00:14:52,500
+del z by del x if the equation هي الآن مكتوبة للـ
+
+192
+00:14:52,500 --> 00:14:57,500
+equation ضمنيا yz ناقص من z بسوء x زي y بدنا نوجد
+
+193
+00:14:57,500 --> 00:15:01,660
+del z by del x بيقول لك find del z by del x if the
+
+194
+00:15:01,660 --> 00:15:03,620
+equation defines z as a function of the two
+
+195
+00:15:03,620 --> 00:15:06,220
+independent variables x and y and a partial
+
+196
+00:15:06,220 --> 00:15:11,320
+derivative exists افترض لك أن ليه z هي dependent و
+
+197
+00:15:11,320 --> 00:15:15,280
+ال y و ال x simply independent و مكتوبة z as a
+
+198
+00:15:15,280 --> 00:15:22,070
+function of y و xاللي هو إيش ضمنيا هو ده اللي هو
+
+199
+00:15:22,070 --> 00:15:25,610
+طبعا اتفاضل الضمن اللي احنا بنعرف كيف نفاضل كل إش
+
+200
+00:15:25,610 --> 00:15:30,230
+قدامنا بنفاضله الآن بده دي ال Z by دي X ال X و ال
+
+201
+00:15:30,230 --> 00:15:36,950
+Y independent variableإذاً الآن اللي هي بده فاضل Z
+
+202
+00:15:36,950 --> 00:15:41,950
+بالنسبة ل X اللي هي و معناته مادام اشتفق جزءي
+
+203
+00:15:41,950 --> 00:15:44,210
+بالنسبة ل X إذاً Y إيش مرة اللي بدنا نعملها
+
+204
+00:15:44,210 --> 00:15:49,320
+بنتبتها إذاً Y أيها جدامنا ثابت و بنشتغللأن تفاضل
+
+205
+00:15:49,320 --> 00:15:54,860
+هدولة y del z by del x لأن y ثابت لنسبة x نقص
+
+206
+00:15:54,860 --> 00:16:00,620
+تفاضل z 1 على z ln 1 على z في del z by del x لأنه
+
+207
+00:16:00,620 --> 00:16:04,760
+يفاضل بالنسبة لمن؟ لx فاهمين إيش بيسوي y ساوي
+
+208
+00:16:04,760 --> 00:16:14,250
+تفاضل هذه بالنسبة لx del z by del xتفضلها بالنسبة
+
+209
+00:16:14,250 --> 00:16:21,250
+لـ X واحد تفضل هذه X صفر اذا هذا المقدار بيساوي
+
+210
+00:16:21,250 --> 00:16:27,150
+ايش؟ بيساوي واحد يا جماعة ماشي هذه واحد طيب ناخد
+
+211
+00:16:27,150 --> 00:16:30,390
+del Z by del X عامل مشترك بيظل عند Y ناقص واحد
+
+212
+00:16:30,390 --> 00:16:36,370
+لواحد Z بيساوي واحد ماشي الآن بيصير المقدار del Z
+
+213
+00:16:36,370 --> 00:16:40,870
+by del X ساوي اللي هو واحد
+
+214
+00:16:41,650 --> 00:16:47,750
+على هذا المقدار A بساوية Z في ZY ناقص واحد لأن هذا
+
+215
+00:16:47,750 --> 00:16:53,950
+مش مجموعه YZ ناقص واحد على Z ثم أجيبه هنا و أجيبه
+
+216
+00:16:53,950 --> 00:16:59,110
+هنا و نقسم الجهتين عليه وهذا واحد طبعا بيصلح
+
+217
+00:16:59,110 --> 00:17:06,780
+المقدار اللي هو Z على المقدار فهميا طبعاتفاجنا يا
+
+218
+00:17:06,780 --> 00:17:12,040
+جماعة كيف عملنا yz ناقص واحد على z لأنه هذا واحد
+
+219
+00:17:12,040 --> 00:17:15,360
+هو صار اللي هو لما جيبناه على الجهة التانية الجثة
+
+220
+00:17:15,360 --> 00:17:18,680
+من الجهة التانية هذا المقدر طلع z على zy ناقص واحد
+
+221
+00:17:18,680 --> 00:17:23,840
+ييجي
+
+222
+00:17:23,840 --> 00:17:27,980
+الأن مثل آخر يقول دابلين x بالساعة واحد intersect
+
+223
+00:17:27,980 --> 00:17:32,660
+دورة بال y z بالساعة x تربيع z y تربيعFind the
+
+224
+00:17:32,660 --> 00:17:35,240
+slope of the tangent to the parabola at 1 و2 و5
+
+225
+00:17:35,240 --> 00:17:39,260
+يعني بيقول لي عندي في plane X بتساوي 1 بيقول لي لو
+
+226
+00:17:39,260 --> 00:17:42,540
+أجينا قطعناها بالبرابولايد هذه البرابولايد Z
+
+227
+00:17:42,540 --> 00:17:45,840
+بتساوي X تربيع زي Y تربيحية هاي البرابولايد أجينا
+
+228
+00:17:45,840 --> 00:17:50,900
+قطعناها بإيش بplane اللي هو X بتساوي 1 أكيد لما
+
+229
+00:17:50,900 --> 00:17:55,160
+نقطعها بX بتساوي 1 إيش هيطلع عندى عبارة عن برابولة
+
+230
+00:17:55,160 --> 00:17:59,240
+قطع اللي هو إما نقطعه بشكل طولي زي هيك بيطلع عندى
+
+231
+00:17:59,240 --> 00:18:02,820
+اللي هو الشكل إيه شماله برابولةوقل لأوجد اللي هو
+
+232
+00:18:02,820 --> 00:18:06,500
+ال slope لل tangent للبرابل عنده نقطة مين واحد
+
+233
+00:18:06,500 --> 00:18:09,780
+واثنين وخمسة هي نقطة واحد واثنين وخمسة بدي اشوف ال
+
+234
+00:18:09,780 --> 00:18:14,020
+tangent لمين للبرابل انا الآن انا ماني امر احليه
+
+235
+00:18:14,710 --> 00:18:19,610
+اللي هو إما إن أشتغل محاذي مباشرة و أعمل اشتغال
+
+236
+00:18:19,610 --> 00:18:24,090
+قدزي و أكمل أو إني اللي هو أجيب معادلة البرابولة
+
+237
+00:18:24,090 --> 00:18:28,870
+اللي هي اللي أمامي اللي هو المقطوعة هذه و أوجد
+
+238
+00:18:28,870 --> 00:18:31,410
+اللي هو الـ tangent زي ما كنا نعمل في تفاوض الألف
+
+239
+00:18:31,410 --> 00:18:36,110
+خلينا واحدة واحدة الآن عشان أوجد سلوب للـ tangent
+
+240
+00:18:36,110 --> 00:18:41,380
+للبرابولةأتيت باجي اللي هو عبارة عن اللي هو الـ Z
+
+241
+00:18:41,380 --> 00:18:43,060
+بيساوي X تلبية و Z بيساوي Y تلبية اللي هي
+
+242
+00:18:43,060 --> 00:18:46,920
+البرابولة بوجد Del Z by Del X لأنها صارت اللي هي
+
+243
+00:18:46,920 --> 00:18:50,580
+لما انقطعته بـ X صارت اللي هي الـ X ثابتة إذا صارت
+
+244
+00:18:50,580 --> 00:18:55,560
+هذه البرابولة بين Y و بين مين و بين Z إذا الميل
+
+245
+00:18:55,560 --> 00:19:00,840
+هنا هو عبارة عن Del Z by Del Y لأن Y الـ X أشمالها
+
+246
+00:19:01,320 --> 00:19:05,340
+ثبتت يا جماعة إذا الـ slope of the tangent is the
+
+247
+00:19:05,340 --> 00:19:09,440
+value of the partial derivative of z by y عند مين
+
+248
+00:19:09,440 --> 00:19:12,800
+عندهم نقطة إيش؟ واحد واثنين وخمسة طبعاً واحد
+
+249
+00:19:12,800 --> 00:19:17,300
+واثنين واحد واثنين و .. هذا الـ a هذا ال X وهذا ال
+
+250
+00:19:17,300 --> 00:19:22,450
+Y واحد واثنين وخمسةباجي الآن بودي الـ Z by الـ Y
+
+251
+00:19:22,450 --> 00:19:26,170
+منها ان عقول الـ Z by الـ Y لأنها دي سفر إيش ده
+
+252
+00:19:26,170 --> 00:19:30,830
+تصير اتنين Y إذا الـ slope is الـ Z by الـ Y عند
+
+253
+00:19:30,830 --> 00:19:34,510
+مين عنده نقطة واحد واتنين وخمسة ماشي اللي هو
+
+254
+00:19:34,510 --> 00:19:40,320
+بيساوي اتنينتنين في قيمة Y التي هي اتنين ويساوي
+
+255
+00:19:40,320 --> 00:19:47,060
+ايش اربع اذا ميل المماس لالتي هي البرابولة هذه
+
+256
+00:19:47,060 --> 00:19:51,520
+عنده نقطة واحد واتنين وخمسة طلع بساوي ايش اربع
+
+257
+00:19:52,520 --> 00:19:55,140
+سلوشين تاني واحد قال لقلبي دي مش هعمل زيك اسمع انا
+
+258
+00:19:55,140 --> 00:19:58,800
+بدأ ادي البرابلات اللي بتقاطعين فيها مش قلت ال X
+
+259
+00:19:58,800 --> 00:20:02,200
+بتساوي واحد قلت لك اه قال لي اذا Z بتساوي واحد
+
+260
+00:20:02,200 --> 00:20:06,140
+زائد Y تربيع هي البرابلات تبعتنا Z بتساوي واحد
+
+261
+00:20:06,140 --> 00:20:10,220
+زائد Y تربيع هي برابلات اللي هي تقاطعةاللي هي الـ
+
+262
+00:20:10,220 --> 00:20:14,220
+Paraboloid مع الـ Plane بقوله صح لما بقوللي أنا
+
+263
+00:20:14,220 --> 00:20:18,300
+انت مش بدك اللي هو الـ mail مماس عنده نقطة واحد
+
+264
+00:20:18,300 --> 00:20:21,440
+واثنين وخمسة قلتله اه طيب اذا عند واحد واثنين
+
+265
+00:20:21,440 --> 00:20:24,420
+وخمسة بده اوجد اللي هو الـ derivative دل زيد بايدل
+
+266
+00:20:24,420 --> 00:20:26,800
+واي اللي هو الـ mail هادة بدك، الـ mail مماس هادة
+
+267
+00:20:26,800 --> 00:20:31,120
+بقوله اه دل زيد بايدل واي اللي هي بيصير دي زيد
+
+268
+00:20:31,120 --> 00:20:33,520
+بايدي واي في هذه الحالة لأنه صارت function of one
+
+269
+00:20:33,520 --> 00:20:40,410
+variable بساوي اتنين Yتنين Y وY قيمتها اتنين اللي
+
+270
+00:20:40,410 --> 00:20:44,970
+هي اتنين اللي هي بسواش اربعة طبعا كاتبين واحد
+
+271
+00:20:44,970 --> 00:20:47,850
+واتنين لأ مش واحد واتنين هنا إذا بدنا نعملها of
+
+272
+00:20:47,850 --> 00:20:52,190
+two variables يجب تقول اتنين وخمسة لأن اتنين هي
+
+273
+00:20:52,190 --> 00:20:56,810
+قيمة ال Y و الخمسة قيمة ال Z و الواحد ماعنديه اللي
+
+274
+00:20:56,810 --> 00:21:03,180
+هو خلصنا منه في الخطوة الأولىولكن انا بفضل تقوله
+
+275
+00:21:03,180 --> 00:21:07,580
+الـ z by الـ y at الوا واحد واتنين وخمسة عساسي ان
+
+276
+00:21:07,580 --> 00:21:12,160
+احنا نرجع لاصلنا اذا بالطريقة هذه او بالطريقة هذه
+
+277
+00:21:12,160 --> 00:21:17,840
+بيصيرLamb functions of more than two variables هنا
+
+278
+00:21:17,840 --> 00:21:21,720
+نيجي اللي ده اللي ايه في أكتر متغير Similarly if
+
+279
+00:21:21,720 --> 00:21:24,840
+x, y and z are independent variables يعني بيقوللي
+
+280
+00:21:24,840 --> 00:21:28,660
+x, y و z هي ال independent and f of x, y و z
+
+281
+00:21:28,660 --> 00:21:32,520
+بيساوي x sin y زي تلتة z طبعا بيقول independent
+
+282
+00:21:32,520 --> 00:21:35,300
+عشان معروف مدام independent إذا انتفاضوا اللي
+
+283
+00:21:35,300 --> 00:21:39,040
+بالنسبة معدل تغيره لل independent بالنسبة لبعض صفر
+
+284
+00:21:39,040 --> 00:21:42,200
+على طول فش انتفاضوا بينهم هذه بالنسبة لهذه أو هذه
+
+285
+00:21:42,200 --> 00:21:45,330
+بالنسبة لهذه على طول صفرأما الـ Dependent بالنسبة
+
+286
+00:21:45,330 --> 00:21:49,150
+للـ Independent هما اللي بالفاضل بينهم لأن الـ
+
+287
+00:21:49,150 --> 00:21:53,390
+Independent Variable تغيرهم مستقل مالهنش علاقة في
+
+288
+00:21:53,390 --> 00:21:59,790
+حد معدل التغير بينهم سفرهم مافيش تغير بقى هذا تغير
+
+289
+00:21:59,790 --> 00:22:02,770
+لحاله و هذا تغير لحاله إنما المتغير المستقل
+
+290
+00:22:02,770 --> 00:22:05,070
+والمتغير التابع فيه اللي هو هذا التابع التابع
+
+291
+00:22:05,240 --> 00:22:08,680
+التابع بيعتمد على المستقل عشان هيك فيه معدل تغير
+
+292
+00:22:08,680 --> 00:22:13,560
+بينهم عشان هيك في دل FY دل X و FY دل Y و FY دل Z
+
+293
+00:22:13,560 --> 00:22:20,180
+لأن F هي المتغير التابع طيب اللي أنا طالب دل FY دل
+
+294
+00:22:20,180 --> 00:22:25,320
+Z بساوي زي ما عملنا بدل ما نثبت اللي هي ال Y لحدة
+
+295
+00:22:25,320 --> 00:22:29,450
+أو X لحدة كله بنثبته المتغير بيظل مين بس Zإذا del
+
+296
+00:22:29,450 --> 00:22:34,150
+of y del z بساوية تفاضل كله كأن x و y ثوابت و ال z
+
+297
+00:22:34,150 --> 00:22:40,290
+متغير إذا ال x بالنسبة لل z لتفاضل هذه ال x زي ما
+
+298
+00:22:40,290 --> 00:22:43,710
+هي لأنها ثابت في تفاضل اللي جوا اللي هو مين تفاضل
+
+299
+00:22:43,710 --> 00:22:48,530
+جوا y زي 3z اللي هو اللي هو تفاضل الجوا عبارة عن 3
+
+300
+00:22:48,530 --> 00:22:52,890
+بصير 3x و تفاضل ال sin هي عبارة عن مين عن cosine
+
+301
+00:22:53,270 --> 00:22:57,770
+فبصير تلاتة x في cos y زي افضل ان اقول هذا لان
+
+302
+00:22:57,770 --> 00:23:02,570
+اكيد فاهمين ان انتوا هذه x ثابتة بتنزل هذه تفاضل
+
+303
+00:23:02,570 --> 00:23:06,390
+في sin هي cos هو تفاضل اللي جوا هي تلاتة فبطلع
+
+304
+00:23:06,390 --> 00:23:10,360
+عندى هذا هو قيمة التفاضلالأن ده نحكي عن الـ
+
+305
+00:23:10,360 --> 00:23:13,960
+Partial derivatives and continuity بدنا نسأل سؤال
+
+306
+00:23:13,960 --> 00:23:18,660
+ما هي العلاقة هل يعني اللي هو لو كانت ال function
+
+307
+00:23:18,660 --> 00:23:22,500
+ال partial زمان احنا كنا قبل ما اسأل السؤال هذا
+
+308
+00:23:22,500 --> 00:23:25,900
+زمان كنا نقول إذا كانت ال function اللي هي ال
+
+309
+00:23:25,900 --> 00:23:30,320
+differential بالأكيد continuousالآن لو كانت الـ
+
+310
+00:23:30,320 --> 00:23:34,360
+Partial Derivatives موجودة هل يعني الـ Continuity؟
+
+311
+00:23:34,360 --> 00:23:38,460
+لأ يعني هنا لو كانت الـ Fx و الـ Fy و الـ Fz
+
+312
+00:23:38,460 --> 00:23:44,780
+موجودات ليس شرطا إنها تكون الدالة متصلةواضح الآن
+
+313
+00:23:44,780 --> 00:23:47,960
+الـ Differentiability إشي آخر غير الـ Partial
+
+314
+00:23:47,960 --> 00:23:51,120
+Derivatives حنجيه بعد شوية الان Partial
+
+315
+00:23:51,120 --> 00:23:56,800
+Derivatives الآن وإذا كانت موجودة ليس شرطًا أنها
+
+316
+00:23:56,800 --> 00:24:00,860
+تعطي اللي هو الـ Continuity يعني الآن في example 8
+
+317
+00:24:00,860 --> 00:24:05,300
+حلقة function f of x y is has a partial
+
+318
+00:24:05,300 --> 00:24:11,830
+derivativesعند الـ 00 but F is not continuous عند
+
+319
+00:24:11,830 --> 00:24:18,090
+مين؟ عند الـ 00 إذن المثال هذا تمانية لما أقول F
+
+320
+00:24:18,090 --> 00:24:22,130
+of XY بيساوي سفر إذا كانت XY لتساوي سفر و 1 إذا
+
+321
+00:24:22,130 --> 00:24:27,670
+كانت XY بيساوي سفر هذا عبارة عن مثال يوضح لي أن
+
+322
+00:24:27,670 --> 00:24:31,730
+الـ Partial Derivatives ليس شرطا أن تؤدي إلى ال
+
+323
+00:24:31,730 --> 00:24:32,350
+continuity
+
+324
+00:24:44,610 --> 00:24:50,550
+وهي عبر خطوات الفلات التاليةأول شيء يجب أن تبحث عن
+
+325
+00:24:50,550 --> 00:25:01,410
+قيمة f of x و f
+
+326
+00:25:01,410 --> 00:25:07,110
+of y عندما تتجه إلى 0.0 عبر خط y يتساوي xبعدين
+
+327
+00:25:07,110 --> 00:25:10,930
+جالي prove that f is not continuous at the origin
+
+328
+00:25:10,930 --> 00:25:14,030
+بناء على اللي فوق اثبت ان الـ f is not continuous
+
+329
+00:25:14,030 --> 00:25:17,210
+at the origin عارفين انتوا عشان تكون f continuous
+
+330
+00:25:17,210 --> 00:25:21,150
+عند الـ origin لازم تكون limit f of x و y لما x و
+
+331
+00:25:21,150 --> 00:25:25,550
+y تروح للـ 0،0 بتساوي f of 0،0 لما ده تبتلك ان f
+
+332
+00:25:25,550 --> 00:25:29,920
+of 0،0اللي هي لا تساوي الـ Limit اللي بنحكي عنها
+
+333
+00:25:29,920 --> 00:25:33,120
+اللي فوق وبناء عليه اللي هو تكون الـ F is not
+
+334
+00:25:33,120 --> 00:25:37,320
+continuous and 00 بعدين بتبتلك أن الـ Partial
+
+335
+00:25:37,320 --> 00:25:41,660
+derivatives موجودة عند مين الـ 00 بكون أنهينا
+
+336
+00:25:41,660 --> 00:25:44,920
+أثبتنا أن الـ Partial derivatives عند 00 موجودة
+
+337
+00:25:44,920 --> 00:25:49,700
+لكن الـ F مش continuous and 00خلّينا نجي لرسمة
+
+338
+00:25:49,700 --> 00:25:54,720
+السؤال اللي أمامنا هذه الدالة تابعة 0 لما X Y لا
+
+339
+00:25:54,720 --> 00:25:59,340
+تساوي 0 و 1 لما X في Y ايش مالها تساوي 0 خلّينا
+
+340
+00:25:59,340 --> 00:26:04,640
+نجي الان هي رسمة الدالة اللي عندي الدالة اللي عندي
+
+341
+00:26:04,640 --> 00:26:11,720
+اللي هي عند اللي هو X في Y بتساوي 0 ايش قمتها 1
+
+342
+00:26:11,720 --> 00:26:21,060
+يعني لمااللي هي تكون هنا الـ X أو هنا A على الخطين
+
+343
+00:26:21,060 --> 00:26:28,220
+هدولة الـ X هنا سفر وهنا الـ Y سفر فبتكون الـ X في
+
+344
+00:26:28,220 --> 00:26:30,860
+Y بالساوية سفر وين يا هنا
+
+345
+00:26:33,430 --> 00:26:40,990
+أسف ال X في Y نعم ال X في Y هنا هي ال X في Y هي ال
+
+346
+00:26:40,990 --> 00:26:47,710
+X هي ال Y أسف وهي ال X ماشي هنا دايما ال X إيش ال
+
+347
+00:26:47,710 --> 00:26:53,130
+Y سفر وهنا دايما ال X إيش سفر ماشي الآن في المنطقة
+
+348
+00:26:53,130 --> 00:26:58,040
+هذهعندي الـ x في الـ y دايما إيش بتساوي أو ال ..
+
+349
+00:26:58,040 --> 00:27:01,340
+ال .. ال .. هذا مع أنها داكفة ساوي سفر x y لاتساوي
+
+350
+00:27:01,340 --> 00:27:05,540
+سفر ويا في المنطقة هذه و المنطقة هذه و المنطقة هذه
+
+351
+00:27:05,540 --> 00:27:09,660
+و المنطقة هذه هذي ال x في ال y إيش مالها مش سفر
+
+352
+00:27:09,660 --> 00:27:16,490
+إيش قيمة الدالة عندها قيمة الدالة عندهاقيمة الدالة
+
+353
+00:27:16,490 --> 00:27:19,030
+في المنطقة هذه والمنطقة هذه والمنطقة هذه والمنطقة
+
+354
+00:27:19,030 --> 00:27:22,630
+هذه اللي X في Y لا تساوي سفر سفر هيو اذا رسمينا
+
+355
+00:27:22,630 --> 00:27:26,890
+احنا بالازراجية على الأرض يعني قيمة الدالة عند
+
+356
+00:27:26,890 --> 00:27:33,210
+المنطقة اللي بتكون X في Y هنا او هنا اللي هي
+
+357
+00:27:33,210 --> 00:27:36,410
+بتساوي سفر وانما لها قيمتها بتساوي واحد اذا انا
+
+358
+00:27:36,410 --> 00:27:41,750
+على الخط الخط هذا صورته هيو واحدوالخط هذا صورته 1
+
+359
+00:27:41,750 --> 00:27:48,870
+إذاً هذه اللي هي رسمة الدالة L أعني طيب بعد ما
+
+360
+00:27:48,870 --> 00:27:51,950
+شوفنا رسمة الدالة خلّينا نيجي الان اللي بده
+
+361
+00:27:51,950 --> 00:27:59,730
+اذاكركم ايش اللي بدنا نتبت ان نشوف limit f of x y
+
+362
+00:27:59,730 --> 00:28:07,030
+as x y وتروح لل 0 0 along mean along y بتساوي xY
+
+363
+00:28:07,030 --> 00:28:11,790
+بيتساوي X الـ Y بيتساوي X هاي Y بيساوي X and اه
+
+364
+00:28:11,790 --> 00:28:15,990
+هاي الـ Y بيساوي X بس مهوره في وين؟ في نقطة الأصل
+
+365
+00:28:15,990 --> 00:28:20,490
+اللي يعني F of XY is constantly 0 along the line Y
+
+366
+00:28:20,490 --> 00:28:25,070
+بيساوي X، مظبوط الـ Y بيساوي X هيه هذا دايماً
+
+367
+00:28:25,070 --> 00:28:30,140
+هيكون سفر قيمة اللي هيالدالة ما عدا عند من عند
+
+368
+00:28:30,140 --> 00:28:33,040
+السفر except at the origin لكن أنا ماليش في السفر
+
+369
+00:28:33,040 --> 00:28:35,320
+لأن أنا رايح على السفر باخد ال limit لما تروح
+
+370
+00:28:35,320 --> 00:28:39,440
+للسفر يعني x و y في طريق لا تساوي لا تساوي سفر و
+
+371
+00:28:39,440 --> 00:28:43,340
+سفر لما أقول x y تروح ل 0 0 معناته اللي ما بتسويش
+
+372
+00:28:43,340 --> 00:28:50,190
+0 0 إذا أنا عبر الخط هذا لما أروحللـ X Y اللي هي
+
+373
+00:28:50,190 --> 00:28:55,110
+عبر Y بيساوي X لما أروح للـ Zero عبر الخط هذا رايح
+
+374
+00:28:55,110 --> 00:28:59,150
+للـ Zero ورايح للـ Zero قيمة الدالة في طول الفترة
+
+375
+00:28:59,150 --> 00:29:01,310
+لما أروح للـ Zero من هنا أو للـ Zero من هنا على
+
+376
+00:29:01,310 --> 00:29:04,470
+الأرض أنا إيش قيمتها على الأرض بتساوي سفر لذلك الـ
+
+377
+00:29:04,470 --> 00:29:09,510
+Limit limit للدالة هذه عبر الخط Y بتساوي X لما Y و
+
+378
+00:29:09,510 --> 00:29:13,610
+X بتروح لسه لسفر مش سفر عبر الخط Y بتساوي X اللي
+
+379
+00:29:13,610 --> 00:29:18,350
+هوهنا أو هنا إيش بتقيمة الدالة 0 عشان هي كـ Limit
+
+380
+00:29:18,350 --> 00:29:22,630
+الـ Zero إيش هيساوي Zero إذا Limit الدالة في هذه
+
+381
+00:29:22,630 --> 00:29:30,770
+الحالة As X وY بيروح للـ Zero عبر Y بساوي X بتروح
+
+382
+00:29:30,770 --> 00:29:35,690
+تصير الدالة طول الوقت 0 فعشان هيك Limitها بساوي 0
+
+383
+00:29:36,380 --> 00:29:43,140
+لكن لو دي اتلاحظت F of 00 F of 00 هيها عند X0 Y0
+
+384
+00:29:43,140 --> 00:29:47,660
+يعني XY بيساوي 0 F of 00 أيش بيساوي 1 مدام F of 00
+
+385
+00:29:47,660 --> 00:29:52,140
+بيساوي 1 يعني إيش ما تطلع الـ Limit لو كانت موجودة
+
+386
+00:29:52,140 --> 00:29:57,160
+هذه الـ Limit مش هتطلع بتساوي 1 لأنها انطلعت إيهاش
+
+387
+00:29:57,160 --> 00:30:02,540
+سفر إذن أكيد الـ Limit للـ F of XY لما XY بتروح
+
+388
+00:30:02,540 --> 00:30:07,910
+للـ 00 أكيد بيساويش F of 00لو كانت لا توجد فخلص
+
+389
+00:30:07,910 --> 00:30:10,770
+خلصنا مش continuous طب لو كانت exist هتساوي أكيد
+
+390
+00:30:10,770 --> 00:30:15,170
+سفر لأن كلها هيكونين سفر بدأت سفر و ال F عندها 001
+
+391
+00:30:15,170 --> 00:30:17,730
+إذا اللي هو ال limit اللي هو ال limit متساويش اللي
+
+392
+00:30:17,730 --> 00:30:20,410
+هو mean ال F of 00 يعني ال F is not continuous عند
+
+393
+00:30:20,410 --> 00:30:24,750
+السفر والسفر نيجي الآن وجد mean اللي هو del F by
+
+394
+00:30:24,750 --> 00:30:28,130
+del X إذا من الفجر أثبتنا أن F is not continuous
+
+395
+00:30:28,130 --> 00:30:32,790
+at 00 to find del F by del X at 00 we hold Y fixed
+
+396
+00:30:32,790 --> 00:30:38,140
+أكيدعند مين؟ بدنا نثبتها Y بس بنثبتها عند Y
+
+397
+00:30:38,140 --> 00:30:41,140
+بالساوية 0 لإنه أنا ماخدت بدي الـ derivative عند
+
+398
+00:30:41,140 --> 00:30:46,300
+مين؟ عنده النقطة 00 إذا ثبتته الـ Y بـ 0 مدام الـ
+
+399
+00:30:46,300 --> 00:30:50,920
+Y بـ 0 إذا صارت قيمة الدالة مين هي؟ X Y بساوية 0
+
+400
+00:30:50,920 --> 00:30:55,660
+يعني 1 قيمة الدالة يعني F of X Y بساوية 1 and then
+
+401
+00:30:55,660 --> 00:31:00,350
+F of X Y بساوية 1 for X and the graph of Fis the
+
+402
+00:31:00,350 --> 00:31:04,270
+line الواحد مظبوط إذا ان صارت اللي هي ال F إشي
+
+403
+00:31:04,270 --> 00:31:09,370
+يعذر عن الواحد لأنه إيه إيش عندي F of X و Y بتساوي
+
+404
+00:31:09,370 --> 00:31:14,550
+واحد و ال Y سفر يعني مين هيكون موازي لمين لل X
+
+405
+00:31:14,550 --> 00:31:20,370
+axis عندي the slope of this line at any X is del F
+
+406
+00:31:20,370 --> 00:31:24,770
+by del X بيساوي سفر لإنه ثابت أصلا in particular
+
+407
+00:31:24,770 --> 00:31:29,720
+del F by del X عند ال 00 مش بيساوي سفر طيبالأن
+
+408
+00:31:29,720 --> 00:31:32,700
+Similarly for دي الـ off by دي الـ Y is the slope
+
+409
+00:31:32,700 --> 00:31:36,980
+of the line من الاتنين لأنه بيثبت Y بتساوي سفر
+
+410
+00:31:36,980 --> 00:31:41,120
+بتظهر الدالة برضه إيش off of XY إيش بتساوي واحد بس
+
+411
+00:31:41,120 --> 00:31:45,340
+عندي اللي هي X بتساوي سفر فبصير عند مين أعجباه الـ
+
+412
+00:31:45,340 --> 00:31:50,300
+X بتساوي سفر الخط هذا X بتساوي سفر هي قيمة الدالة
+
+413
+00:31:50,300 --> 00:31:54,890
+اللي هي الاتنين فبصير اللي هوالـ Del F by Del L هو
+
+414
+00:31:54,890 --> 00:31:59,330
+عبارة عن Slope للأل اتنين برضه خط ثابت لـ Slope
+
+415
+00:31:59,330 --> 00:32:02,990
+إيه؟ إيش بيساوي؟ اللي هو سفر لإنه Del F by Del Y
+
+416
+00:32:02,990 --> 00:32:07,850
+لأ ال F of X Y بتساوي واحد بساوي سفر طبعا هذه F of
+
+417
+00:32:07,850 --> 00:32:11,350
+X Zero اللي هي واحد في الواقع على طول Del F by Del
+
+418
+00:32:11,350 --> 00:32:17,800
+X سفروهذا في الواقع الـ f of 0 و y دي الـ f y دي
+
+419
+00:32:17,800 --> 00:32:21,100
+الـ a و y في هذه الحالة برضه سفر إذن therefore the
+
+420
+00:32:21,100 --> 00:32:24,760
+partial derivatives دي الـ f by x و دي الـ f by y
+
+421
+00:32:24,760 --> 00:32:27,260
+الاتنين exist عند الـ
+
+422
+00:32:27,260 --> 00:32:36,840
+0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0،0
+
+423
+00:32:36,870 --> 00:32:42,690
+الان اللي هي دل تربيع FY دل X تربيع معناته دل FY
+
+424
+00:32:42,690 --> 00:32:46,890
+دل X و بفضلها كمان مرة يعني FXX بفاضل بالنسبة ل X
+
+425
+00:32:46,890 --> 00:32:51,390
+وبعدين ل X دل تربيع FY دل Y تربيع أو FYYY يعني
+
+426
+00:32:51,390 --> 00:32:55,010
+بفضلها مرتين نفس المشي دل تربيع FY دل X دل Y
+
+427
+00:32:55,010 --> 00:33:00,420
+معناتهأول أشي بفاضل بالنسبة للـ FYY و بعدين بفاضل
+
+428
+00:33:00,420 --> 00:33:05,800
+بالنسبة للـ FYX يعني FYX معناته بفاضل بالنسبة لـ Y
+
+429
+00:33:05,800 --> 00:33:09,600
+قبل و بالنسبة ل X بعدين لأن FXY بفاضل أول أشي
+
+430
+00:33:09,600 --> 00:33:13,390
+بالنسبة ل X و بعدين بالنسبة لمين ل Yماشي دل تربيع
+
+431
+00:33:13,390 --> 00:33:16,930
+في Y اللي هي دل Y بعد دل X كيف تنجري هذه أول إش
+
+432
+00:33:16,930 --> 00:33:21,130
+بفاضل بالنسبة ل X وهذه بعدها بفاضل بالنسبة ل Y هذه
+
+433
+00:33:21,130 --> 00:33:24,930
+بفاضل بالنسبة لجنبها X بعدها بالنسبة ل Y يعني هذه
+
+434
+00:33:24,930 --> 00:33:30,590
+ال Fxxxy بتنجرها تتفاضل XY هذه دل تربيع في دل Y دل
+
+435
+00:33:30,590 --> 00:33:34,780
+X أول إش بالنسبة لجوا بعدها بالنسبة للي برابعد
+
+436
+00:33:34,780 --> 00:33:40,520
+شوية هنقول لكم إن هذه دائما بتساوي هذه ما دامت
+
+437
+00:33:40,520 --> 00:33:44,040
+اللي هي ال partial derivatives نفسها continuous
+
+438
+00:33:44,040 --> 00:33:49,020
+فبصير الواحد اللي هو يفاضل مايفرجش عنده يفاضل لل X
+
+439
+00:33:49,020 --> 00:33:52,660
+جابل أو لل Y جابل وهذا طبعا أعظم دولنا زي هيك مش
+
+440
+00:33:52,660 --> 00:33:55,720
+هتفرج لأنه هنعطيكم كل الدول تكون اللي هي ال
+
+441
+00:33:55,720 --> 00:33:58,770
+partial derivatives صحيحة لها عشان هيكفاضله
+
+442
+00:33:58,770 --> 00:34:01,730
+بالنسبة ل X جابل أو بالنسبة ل Y جابل مش هتفرج
+
+443
+00:34:01,730 --> 00:34:05,690
+معناه و بالعكس هتلاقي حالك أحياناً لما يطلب منك
+
+444
+00:34:05,690 --> 00:34:09,710
+Fxy بتروح تفاضل بالنسبة ل Y جابل X لأنه أسهل
+
+445
+00:34:09,710 --> 00:34:15,930
+بالنسبة ل Y جابل X عن X جابلها و بعدين Y كما سنرى
+
+446
+00:34:15,930 --> 00:34:20,890
+بعد قليل انشوف صلى الله عليه الصلاة والسلامعندي f
+
+447
+00:34:20,890 --> 00:34:24,530
+of x y بيساوي x cosine y زي الـ y e to the x find
+
+448
+00:34:24,530 --> 00:34:26,770
+the second partial derivatives دي التربيه f y دي
+
+449
+00:34:26,770 --> 00:34:29,270
+ال x تربيها دي التربيه f y دي ال y دي ال x دي
+
+450
+00:34:29,270 --> 00:34:34,010
+التربيه وكذا الاخرين بيجي للمولة أول اشي بنوجد دي
+
+451
+00:34:34,010 --> 00:34:38,020
+ال f y دي ال x لإن احتي الزمن للكل مع عدل عادلوهذا
+
+452
+00:34:38,020 --> 00:34:41,920
+الـ Y يتفضل
+
+453
+00:34:41,920 --> 00:34:47,500
+إلى Y تفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى
+
+454
+00:34:47,500 --> 00:34:48,820
+Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+455
+00:34:48,820 --> 00:34:48,840
+يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+456
+00:34:48,840 --> 00:34:48,840
+يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+457
+00:34:48,840 --> 00:34:51,280
+يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+458
+00:34:51,280 --> 00:34:52,320
+يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+459
+00:34:52,320 --> 00:34:53,700
+يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y يتفضل إلى Y
+
+460
+00:34:53,700 --> 00:34:59,310
+يتفضل إلى Y يتفضل إلى Y يتالان بدأ افضل delta
+
+461
+00:34:59,310 --> 00:35:02,110
+بيعقف دي ال Y دي ال X فضلت بالنسبة ل X جمعي بدي
+
+462
+00:35:02,110 --> 00:35:08,470
+افضل بالنسبة ل Y إذا فاضل هذه بالنسبة ل Y فاضلها
+
+463
+00:35:08,470 --> 00:35:10,610
+بالنسبة ل Y اذا افضلت بالنسبة ل X بدي افضلها ل Y
+
+464
+00:35:10,610 --> 00:35:14,410
+يعني اول شيء فضلت ل X بعدين بدي افضل ل Y للي برا
+
+465
+00:35:14,940 --> 00:35:20,580
+الـ Y هي ناقص Sine Y وهذه تفاضل Y واحد E to the X
+
+466
+00:35:20,580 --> 00:35:24,560
+نأتي للثانية بالنسبة لـ Y وهذا فاضل فاضلها بالنسبة
+
+467
+00:35:24,560 --> 00:35:30,180
+لـ Y هذه اللي هي X زي ما هي X تنزل ناقص Sine Y هي
+
+468
+00:35:30,180 --> 00:35:34,560
+ناقص Sine Y تفاضل
+
+469
+00:35:34,560 --> 00:35:38,820
+بالنسبة ل Y هي E to the X لأن Y واحدفاضلها كمان
+
+470
+00:35:38,820 --> 00:35:43,080
+مرة بالنسبة لـ Y بالنسبة لـ Y هذه بيصير ناقص X Cos
+
+471
+00:35:43,080 --> 00:35:49,760
+Y وهذه سفر طيب فاضل لهذه بالنسبة ل .. ده كتير
+
+472
+00:35:49,760 --> 00:35:54,700
+وجودين نسبة إيه إيش؟ دل X دل بي دل X فضلنا للأول
+
+473
+00:35:54,700 --> 00:35:59,100
+بدنا نفضل من برّع بي دل X إذا هذا فاضلها لل X
+
+474
+00:35:59,100 --> 00:36:03,820
+بيصير ناقص Sin Y بالنسبة ل X زي ديتر X واحد
+
+475
+00:36:03,820 --> 00:36:06,970
+التالتين إيش مقال هنا؟زي بعض يعني بقولك و كأنه بعد
+
+476
+00:36:06,970 --> 00:36:10,530
+هيك هذا طبعا continuous و هذا continuous و احنا
+
+477
+00:36:10,530 --> 00:36:14,790
+هنتحمل معاهم بس يعني بمعنى اخر من انا و طالع ايش
+
+478
+00:36:14,790 --> 00:36:18,270
+جالك و دي ال Y by دي ال X دي ال X by دي ال Y فاضل
+
+479
+00:36:18,270 --> 00:36:22,410
+من اللي بترتاحله قبل فضلت ل Y قبل و بعدين ل X او ل
+
+480
+00:36:22,410 --> 00:36:25,990
+X قبل و بعدين ل Y هيتبع سيان ان شاء الله لإن كلهم
+
+481
+00:36:25,990 --> 00:36:29,910
+continuous مارشال درجة تفسير لأن هي هذه النظرية
+
+482
+00:36:29,910 --> 00:36:33,330
+اللي قلتهاأي اللي هي the mixed derivative theorem
+
+483
+00:36:33,330 --> 00:36:37,550
+قلتها قبل شوية هي بتقول كمالي if f of x, y and its
+
+484
+00:36:37,550 --> 00:36:41,210
+partial derivatives f x, f y, f x, y and f y mixed
+
+485
+00:36:41,210 --> 00:36:44,870
+are defined throughout an open region containing a
+
+486
+00:36:44,870 --> 00:36:48,010
+point a و band are all continuous at a و b then
+
+487
+00:36:48,010 --> 00:36:52,850
+fxy بسوء fy x عند النقطة a و b يعني بيقول إذا كانت
+
+488
+00:36:52,850 --> 00:36:57,890
+هذه وهذه وهذه وهذه اللي هو continuous موجودات و
+
+489
+00:36:57,890 --> 00:37:02,570
+continuous في اللي هو منطقة معينة حوالين a و b
+
+490
+00:37:03,190 --> 00:37:06,750
+هيطلع عندك الـ Fxy نفس الـ Fy و X و احنا هنتعامل
+
+491
+00:37:06,750 --> 00:37:10,510
+محاكة يا جماعة فاتوقعوا على الله و نشوف المثال
+
+492
+00:37:10,510 --> 00:37:15,870
+اللي أمامنا اللي بدنا أنه نستغل النظرية في تسهيل
+
+493
+00:37:15,870 --> 00:37:23,390
+عملية الحسابات على أنفسنا فالآن مطلوب
+
+494
+00:37:23,390 --> 00:37:27,090
+اللي هو دي الترميع W by دي X by دي Y لـ W اللي
+
+495
+00:37:27,090 --> 00:37:30,820
+أمامنا فتخليني أطلع على هذهاللي هو طالب أو الاشياء
+
+496
+00:37:30,820 --> 00:37:36,020
+الفاضلة بالنسبة لـ Y وبعدين بالنسبة ل X لو فضلت
+
+497
+00:37:36,020 --> 00:37:39,830
+بالنسبة ل Y في الأول حط ورّطلأن هذا فاضل بالنسبة
+
+498
+00:37:39,830 --> 00:37:42,810
+لـ Y هذا فاضل لها و هذا فاضل لها و ستغلبني كثير
+
+499
+00:37:42,810 --> 00:37:47,730
+هذه فأول اشي بفاضل بالنسبة ل X بريح راسي هذه بتروح
+
+500
+00:37:47,730 --> 00:37:51,490
+تتسفر على طول و انتبه للأسئلة لأنها تطلع و تطلع
+
+501
+00:37:51,490 --> 00:37:55,890
+سريع عشان تعرف لمين فاضله قبل لأن دل W by دل X على
+
+502
+00:37:55,890 --> 00:38:01,370
+طول بساوي Y وهذا سفر لأن فاضل بالنسبة ل Y دل تربيع
+
+503
+00:38:01,370 --> 00:38:06,090
+Y W by دل Y دل X فاضل بالنسبة ل Y على طول تطلع
+
+504
+00:38:06,090 --> 00:38:10,650
+واحدهنا بالنسبة لـ Mixed Derivative Theorem هادى
+
+505
+00:38:10,650 --> 00:38:15,870
+هادى هي هادى اللي هي بيساوي ايش هتساوي واحد اذا
+
+506
+00:38:15,870 --> 00:38:20,110
+انت الان Mixed Derivative Theorem انقذتنا لإنه لو
+
+507
+00:38:20,110 --> 00:38:23,170
+بدى فاضل هادى بالنسبة له Y في الأول ورطة وبعدين
+
+508
+00:38:23,170 --> 00:38:26,850
+بدى فاضل X هم هي هتتصفر كمان وراشد فده دوشة على
+
+509
+00:38:26,850 --> 00:38:31,530
+طول بحدد أفاضل من الأول الأسهل ليه؟example find f
+
+510
+00:38:31,530 --> 00:38:36,870
+x y z إلى y x y z f f of x y z بالساعة واحد ناقص
+
+511
+00:38:36,870 --> 00:38:41,710
+كده بدك تودي دي هذه كيف بدكم تبدون تحرين خلينا
+
+512
+00:38:41,710 --> 00:38:47,670
+ناخد f y f y اللي هي ايش f y هنا اللي هي ناقص هذا
+
+513
+00:38:47,670 --> 00:38:54,040
+سفر بيصير ناقص أربعة x y z زائد x تربية خلصناالآن
+
+514
+00:38:54,040 --> 00:38:58,260
+فاضل بالنسبة لـ X بالنسبة لـ X فاضل بالنسبة لـ X Y
+
+515
+00:38:58,260 --> 00:39:02,380
+X بفاضل زي ما بده أنا Y X اللي هي عبارة عن ناقص
+
+516
+00:39:02,380 --> 00:39:09,520
+أربعة X Z بالنسبة لـ X يعني سفر ناقص أربعة Y Z زاد
+
+517
+00:39:09,520 --> 00:39:10,120
+اتنين X
+
+518
+00:39:17,270 --> 00:39:23,990
+Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+519
+00:39:23,990 --> 00:39:24,510
+-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+520
+00:39:24,510 --> 00:39:24,550
+-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+521
+00:39:24,550 --> 00:39:24,550
+-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+522
+00:39:24,550 --> 00:39:24,830
+-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+523
+00:39:24,830 --> 00:39:27,910
+-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy-Yxy
+
+524
+00:39:27,910 --> 00:39:35,750
+-Yxy-Yxy-Yxyعشان يطيّر هذا من أوله أو يبدأ يشتغل
+
+525
+00:39:35,750 --> 00:39:41,550
+هذا من أوله يعني كيف تترتب انت حر Y X Y Z Y Y X Z
+
+526
+00:39:41,550 --> 00:39:47,390
+Y X Z Y Z X زي ما بدّك و بأي هدا كله هيطلع عليك في
+
+527
+00:39:47,390 --> 00:39:50,770
+الآخر إيش بالساوية ناقص أربعة واضح
+
+528
+00:39:54,100 --> 00:39:56,880
+الـ Undifferentiability اللي هو قابلية الاشتقاق
+
+529
+00:39:56,880 --> 00:40:00,060
+قابلية الاشتقاق، أنا بدي تعرفه بس اللي هو لإعرف
+
+530
+00:40:00,060 --> 00:40:03,020
+لكم شو معنى قابلة الاشتقاق عشان نعرفها بالزمنة
+
+531
+00:40:03,020 --> 00:40:06,620
+نظرية اللي بالزمنة اللي هو برهانة، خلّينا بس نمر
+
+532
+00:40:06,620 --> 00:40:09,740
+عليها مروء سريع اللي هي The Increment Theorem for
+
+533
+00:40:09,740 --> 00:40:13,160
+Functions of Two Variables نفترض إن الـ First
+
+534
+00:40:13,160 --> 00:40:16,240
+Partial Derivatives of F of XY are defined through
+
+535
+00:40:16,240 --> 00:40:19,990
+an open region Rcontaining the point x0, y0 يعني
+
+536
+00:40:19,990 --> 00:40:22,310
+فرضنا أن الـ partial derivatives كلهم موجودات الـ
+
+537
+00:40:22,310 --> 00:40:25,710
+f of x و الـ f of y موجودات و continuous عندهم
+
+538
+00:40:25,710 --> 00:40:29,510
+نقطة من x0, y0 إذا فرضنا fx وfy continuous عندهم
+
+539
+00:40:29,510 --> 00:40:30,670
+نقطة x0, y0
+
+540
+00:40:33,870 --> 00:40:38,750
+وفي منطقة تحتوي لـ X0 وY0 ويقول لي ثم هذا التغيير
+
+541
+00:40:38,750 --> 00:40:44,070
+دلتا Z بساوية F of X0 زي Delta X Y0 زي Delta Y نقص
+
+542
+00:40:44,070 --> 00:40:49,030
+F of X0 Y0 in the value of F هذا التغيير في قيمة F
+
+543
+00:40:49,030 --> 00:40:53,890
+التي هي دلتا Z that result from moving X0 Y0لعند
+
+544
+00:40:53,890 --> 00:40:58,150
+x0 زي delta x و y0 زي delta y to another point هنا
+
+545
+00:40:58,150 --> 00:41:01,350
+اللي ذكرناها satisfies the equation إذا هيكون ال
+
+546
+00:41:01,350 --> 00:41:05,870
+increment هذا هذا ال increment يحقق ما يليه يعني
+
+547
+00:41:05,870 --> 00:41:09,150
+لما تكون f ال it's the partial derivatives
+
+548
+00:41:09,150 --> 00:41:12,850
+continuous on the open region تحتوي ال x0 و ال y0
+
+549
+00:41:14,960 --> 00:41:20,840
+موجودات في الـ Region R حول الـ XNOT و YNOT و
+
+550
+00:41:20,840 --> 00:41:24,740
+Continuous بس عند الـ XNOT و YNOT إذن Delta Z هذه
+
+551
+00:41:24,740 --> 00:41:29,200
+ليها الـ Increment هذه في F لما انتقلت F من XNOT و
+
+552
+00:41:29,200 --> 00:41:34,030
+YNOT لعند XNOT زي Delta X YNOT زي Delta Yهتحقق ما
+
+553
+00:41:34,030 --> 00:41:38,310
+يلي يعني ΔZ هي نتيجة نظرية دلتا زي ده دي هتحقق اف
+
+554
+00:41:38,310 --> 00:41:41,410
+اكس دلتا اكس دف واي دلتا واي زي دي ابسلون دلتا اكس
+
+555
+00:41:41,410 --> 00:41:44,450
+زي دي ابسلون اتنين دلتا واي حيث ابسلون واحد ابسلون
+
+556
+00:41:44,450 --> 00:41:48,490
+اتنين بيروح لسفر لما دلتا اكس ودلتا واي بيروح لمين
+
+557
+00:41:48,490 --> 00:41:53,490
+لزي هذا نظرية بس عساس تشرعنا اللي هو اللي بعيدها
+
+558
+00:41:53,490 --> 00:41:57,590
+مهما بورهانة وكده مش لازمني ايش الآن بدي اعرف ال
+
+559
+00:41:57,590 --> 00:42:00,780
+differentiability لل Fأحنا بنعرف شو معنى الـ
+
+560
+00:42:00,780 --> 00:42:03,440
+Differentiability لـ function of one variable
+
+561
+00:42:03,440 --> 00:42:06,440
+لأننا نعرف الـ Differentiability لـ function of
+
+562
+00:42:06,440 --> 00:42:09,160
+two variables a function z بسوء f of x و y is
+
+563
+00:42:09,160 --> 00:42:13,280
+differentiable عند الأقصنة x0 و y0 إذا كانت الـ fx
+
+564
+00:42:13,280 --> 00:42:19,300
+و الfy موجودات و الـ delta z بتحقق ما يليه اللي هي
+
+565
+00:42:19,300 --> 00:42:23,590
+بتحقق اللي حكيناها قبل بشوية هايواللي بنقول أن F
+
+566
+00:42:23,590 --> 00:42:27,370
+is differentiable عند الـ X0 وY0 إذا حققت الشروط
+
+567
+00:42:27,370 --> 00:42:32,950
+اللي هي التالية Fx وFy موجودات و ΔZ بتساوي Fx ΔX
+
+568
+00:42:32,950 --> 00:42:39,690
+زي Fy ΔY زي Y ΔX و Fy ΔY in which each of Y1 وY2
+
+569
+00:42:39,690 --> 00:42:43,490
+مروح لـ 0 أز Delta X و Delta Y مروح لمين لـ 0 إذا
+
+570
+00:42:43,490 --> 00:42:49,780
+كانت حققت الـ F عند النقطة X0 وY0 تهدي الشروطبنقول
+
+571
+00:42:49,780 --> 00:42:54,740
+عن الـ F differentiable عند الـ X0 وY0 لأ، لما
+
+572
+00:42:54,740 --> 00:42:57,400
+نقول F is differentiable معناته F is
+
+573
+00:42:57,400 --> 00:43:01,600
+differentiable عند مين، عند كل نقطة في الـ domain
+
+574
+00:43:01,600 --> 00:43:05,980
+وبنسمي اللي هو في هذه الحالة للـ graph is a smooth
+
+575
+00:43:05,980 --> 00:43:08,940
+surface لما تكون الـ function differentiable نسميه
+
+576
+00:43:08,940 --> 00:43:13,680
+smooth surface مش smooth curve، smooth surface كله
+
+577
+00:43:13,680 --> 00:43:18,840
+على بعض، smooth surface طيب،أكيد لما سمعتوني
+
+578
+00:43:18,840 --> 00:43:22,540
+هجيتوا يقولوا يعني هذا معناه الـ Differentiability
+
+579
+00:43:22,540 --> 00:43:26,720
+والله كده بنتعقد احنا لأ، الآن بدنا نشيل العقدة
+
+580
+00:43:26,720 --> 00:43:30,120
+هذه الـ Definition مع النظرية اللي قبل بشوية اللي
+
+581
+00:43:30,120 --> 00:43:33,160
+قلنا أيش لزومها و الـ Definition أيش لزومه، التنين
+
+582
+00:43:33,160 --> 00:43:36,100
+مع بعض بيعطينا هذه الحاجة الحلوة اللي هو الـ
+
+583
+00:43:36,100 --> 00:43:39,460
+Corollary اللي هي و اللي بعدهاإذا كانت الـ Partial
+
+584
+00:43:39,460 --> 00:43:44,580
+Derivatives Fx و Fy من المفهوم Fxy مستمرًا في مجال
+
+585
+00:43:44,580 --> 00:43:49,360
+مفتوح R ثم الـ F مستمرًا في كل مكان من R يعني لما
+
+586
+00:43:49,360 --> 00:43:56,200
+تكون الـ Fx و Fy مستمرًا على مجال مفتوح R يعني على
+
+587
+00:43:56,200 --> 00:44:02,520
+منطقة مفتوحة حوالين نقطة X و Y فنقول F أسمها
+
+588
+00:44:02,520 --> 00:44:10,260
+مستمرًا في مجال مفتوحلما تكون إذا كانت اللي هي Fx
+
+589
+00:44:10,260 --> 00:44:14,800
+و Fy و الـ F نفسه continuous في الـ open region
+
+590
+00:44:14,800 --> 00:44:18,620
+على طول الـ F إيش ما لها is differentiable إذا صار
+
+591
+00:44:18,620 --> 00:44:21,660
+اللي هو عشان ال test of differentiability سهل أو
+
+592
+00:44:21,660 --> 00:44:25,740
+أحد ال testات أنه Fx Fy طلع معي ال F continuous
+
+593
+00:44:25,740 --> 00:44:32,660
+على طول ال F is differentiableهنا نأتي إلى نظرية
+
+594
+00:44:32,660 --> 00:44:36,600
+امتداد لتبع زمان نقول ان الـ differentiable
+
+595
+00:44:36,600 --> 00:44:38,860
+function is continuous هذا function of one
+
+596
+00:44:38,860 --> 00:44:41,740
+variable، هنا برضه صح إذا كانت الـ function f of x
+
+597
+00:44:41,740 --> 00:44:47,860
+,y هو مختلفة في x0,y0 إذا فهي مختلفة في x0,y0 إذا
+
+598
+00:44:47,860 --> 00:44:51,120
+ملخص الحديث إن الـ differentiable function is
+
+599
+00:44:51,120 --> 00:44:56,700
+continuous هنا أقول إن الـ partial derivatives fx
+
+600
+00:44:56,700 --> 00:45:03,560
+وfyبإمكانهم إعطاء الـ Continuity للـ F لذلك أقول
+
+601
+00:45:03,560 --> 00:45:09,320
+Fx و Fy يكون موجودة و الـ F موجودة و Continuous
+
+602
+00:45:09,320 --> 00:45:12,260
+كلهم يعني Continuous يعطينا الـ Differentiability
+
+603
+00:45:12,260 --> 00:45:18,160
+لمن؟ للـ F فلو كانت Fx و Fy موجودة ليس شرط أن الـ
+
+604
+00:45:18,160 --> 00:45:21,820
+F مختلفةماشي الحال لكن لو كانت الـ F is
+
+605
+00:45:21,820 --> 00:45:25,140
+differentiable حنلاقي الـ F الـ X والـ F Y اشمالهم
+
+606
+00:45:25,140 --> 00:45:28,440
+موجودات حسب اللي هو اللي حكيناه definition قبل و
+
+607
+00:45:28,440 --> 00:45:31,880
+شوية لو كانت F differentiable أكيد F X و F Y
+
+608
+00:45:31,880 --> 00:45:35,880
+موجودات لو كانت F X و F Y موجودات ليس شرط F is
+
+609
+00:45:35,880 --> 00:45:39,460
+differentiable و F X و F Y موجودات ليس شرط أن F is
+
+610
+00:45:39,460 --> 00:45:44,480
+continuous لكن F X و F Y موجوداتوالـ F موجودة و
+
+611
+00:45:44,480 --> 00:45:47,500
+continuous و هدولة continuous بيعطينا F is
+
+612
+00:45:47,500 --> 00:45:50,260
+differentiable ده في ال region main في ال region R
+
+613
+00:45:50,260 --> 00:45:53,140
+هيك و كلنا خلصنا المحاضرة الـ 15 و إلى اللقاء
+
+614
+00:45:53,140 --> 00:45:54,900
+نفسنا السلام عليكم و رحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TxfjgQ2pjWY.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TxfjgQ2pjWY.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1b56a38aa279c49639fd35fd130224d313574a27
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TxfjgQ2pjWY.srt
@@ -0,0 +1,1490 @@
+1
+00:00:00,720 --> 00:00:04,060
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,060 --> 00:00:08,620
+وبركاته هذه المحاضرة الرابعة مساق  فاضل أتكامل
+
+3
+00:00:08,620 --> 00:00:15,360
+جين صيف 2020 علوان المحاضرة اليوم اللي هو the
+
+4
+00:00:15,360 --> 00:00:19,840
+cross productاللي هو section 12-4 the cross
+
+5
+00:00:19,840 --> 00:00:23,720
+product of two vectors in space أو اللي هو الضرب
+
+6
+00:00:23,720 --> 00:00:28,760
+الاتجاهي حكينا المحاضرة الماضية عن الضرب القياسي
+
+7
+00:00:28,760 --> 00:00:31,480
+اللي هو the dot product اليوم إن نحكي عن اللي هو
+
+8
+00:00:31,480 --> 00:00:35,180
+الـ cross product وقلنا سابقا أنه في نوعين من الضرب
+
+9
+00:00:35,180 --> 00:00:40,300
+بين المتجهات ضرب قياسي وضرب اتجاهي وتحدثنا عن
+
+10
+00:00:40,300 --> 00:00:44,720
+الضرب القياسي والآن نتحدث عن الضرب الاتجاهي الآن
+
+11
+00:00:44,720 --> 00:00:48,300
+بدنا نعرف شو معناه اللي هو the cross product of
+
+12
+00:00:48,300 --> 00:00:53,960
+two vectors in space كمعنى الضرب الاتجاهي لمتجهين
+
+13
+00:00:53,960 --> 00:00:59,360
+في الفراغ لو فرضنا عندنا U عبارة عن متجه و V عبارة عن
+
+14
+00:00:59,360 --> 00:01:04,220
+متجه آخر طبعا الـ U و الـ V أكيد هيصنعان اللي هو طبعا
+
+15
+00:01:04,220 --> 00:01:09,560
+الزاويتين غير متوازيتين أكيد هيصنعان زاوية الآن بدنا نعرف
+
+16
+00:01:09,560 --> 00:01:14,640
+اللي هو الـ cross product بينهما هو عبارة عن vector
+
+17
+00:01:14,640 --> 00:01:20,880
+عمودي عليهما،  يكون عمودي لأعلى أو عمودي لأسفل، بس
+
+18
+00:01:20,880 --> 00:01:25,920
+عشان  نوحّد الحديث بدنا نتفق مع بعض على قاعدة
+
+19
+00:01:25,920 --> 00:01:30,620
+اسمها قاعدة اليد اليمنى كيف نحدد اللي هو اتجاه الـ
+
+20
+00:01:30,620 --> 00:01:35,560
+vector العمودي اللي إحنا بدنا نحكي كلنا بلغة واحدة
+
+21
+00:01:35,560 --> 00:01:39,440
+عليه، أيه اللي بقصدّه إن لو كان U عندي متجه و V
+
+22
+00:01:39,440 --> 00:01:44,940
+متجه وبينهما زاوية θ، هيكون تعريف حاصل الضرب
+
+23
+00:01:44,940 --> 00:01:50,160
+الاتجاهي بينهما هو عبارة عن الـ magnitude للـ U طول
+
+24
+00:01:50,160 --> 00:01:55,860
+الـ U في الـ magnitude للـ V في sin θ طبعا هذا ليس
+
+25
+00:01:55,860 --> 00:02:01,120
+شبيهاً بالـ dot product  ولكن كانت cos θ، sin θ الآن هذه
+
+26
+00:02:01,120 --> 00:02:06,420
+هي القيمة العددية، هذا اللي هو الـ vector اللي
+
+27
+00:02:06,420 --> 00:02:10,150
+بنحكي عنه اللي هو U cross V، طَب الآن اتجاهه كيف
+
+28
+00:02:10,150 --> 00:02:14,130
+بنحدده؟ اتجاهه في اتجاه الوحدات، وين اتجاه الوحدات؟
+
+29
+00:02:14,130 --> 00:02:18,270
+هذا اللي هو بقاعدة اليد اليمنى، هي اليد اليمنى اللي
+
+30
+00:02:18,270 --> 00:02:24,230
+هو، بتيجي أنت الآن، الأصابع هذه بيكونوا على اللي هو
+
+31
+00:02:24,230 --> 00:02:29,470
+الـ U، واليد من هنا هتكون على مين؟ على الـ V، الآن
+
+32
+00:02:29,470 --> 00:02:36,330
+اتجاه الإصبع الكبير هذا هو بيكون اتجاه مين؟ اتجاه
+
+33
+00:02:36,330 --> 00:02:41,360
+المتجه U cross V، فبيكون هذا هو اتجاه المتجه اللي
+
+34
+00:02:41,360 --> 00:02:47,100
+عندنا، يعني U cross V  غير V cross U، U
+
+35
+00:02:47,100 --> 00:02:51,360
+cross V، يعني الأصابع هذه على U، واليد هذه على V،
+
+36
+00:02:51,360 --> 00:02:56,080
+يعني الأصابع الأولى، واليد على الـ V، والإصبع اللي
+
+37
+00:02:56,080 --> 00:03:00,810
+فوق هذا بيُشير إلى اتجاه اللي هو U cross V، فبيكون
+
+38
+00:03:00,810 --> 00:03:03,170
+عندي U cross V  بمقدار الـ magnitude  للـ U في الـ
+
+39
+00:03:03,170 --> 00:03:07,430
+magnitude للـ V في sin الزاوية اللي بينهما في
+
+40
+00:03:07,430 --> 00:03:12,890
+المتجه N اللي إحنا حددناه بطريقة اليد اليمنى، الآن
+
+41
+00:03:12,890 --> 00:03:17,310
+الأمور إن شاء الله أسهل، الآن لو كان اللي هو الـ two
+
+42
+00:03:17,310 --> 00:03:22,620
+vectorsاللي هما متوازيان طبعا عندنا الآن الـ two
+
+43
+00:03:22,620 --> 00:03:26,740
+vectors المتوازيان هذا الـ U cross V ممكن يساوي الـ
+
+44
+00:03:26,740 --> 00:03:31,200
+zero vector في حالة، في حالات معينة إن الـ U صفر
+
+45
+00:03:31,200 --> 00:03:37,080
+يكون magnitude صفر أو الـ V صفر أو الاثنان معاً
+
+46
+00:03:37,080 --> 00:03:41,380
+صفر، أو الزاوية اللي بينهما ثيتا صفر، يعني لما يكون
+
+47
+00:03:41,380 --> 00:03:46,260
+الامتدادان متوازيان، إذاً الآن اللي بدنا نحكي عنه non-zero
+
+48
+00:03:46,260 --> 00:03:51,300
+vectors U and V بيكونوا in parallel if and only if U
+
+49
+00:03:51,300 --> 00:03:54,140
+cross V أيش بيساوي؟ صفر، لأن قلنا U cross V بيساوي
+
+50
+00:03:54,140 --> 00:03:57,940
+صفر، ممكن تحدث من U صفر أو V صفر أو الاثنان معاً
+
+51
+00:03:57,940 --> 00:04:02,770
+صفر، أو الحالة اللي هو، الحالة الثالثة أنه صار
+
+52
+00:04:02,770 --> 00:04:05,730
+الزاوية بينهما صفر، يعني الزاوية اللي بينهما عبارة عن
+
+53
+00:04:05,730 --> 00:04:10,670
+اللي هي عبارة عن zero، أو بأخرى، يعني الاثنان
+
+54
+00:04:10,670 --> 00:04:16,030
+امتدادهما متوازيان، عشان هيك قلنا non-zero vectors U
+
+55
+00:04:16,030 --> 00:04:20,510
+and V parallel if U cross V أيش بيساوي؟ صفر، هذه حُكم
+
+56
+00:04:20,510 --> 00:04:24,230
+على اللي هو two vectors هما non-zero vectors طبعاً
+
+57
+00:04:24,230 --> 00:04:28,310
+إن هما متوازيان، إن يكون الـ cross product بينهما يساوي
+
+58
+00:04:28,310 --> 00:04:31,740
+صفر، طيب يا جماعة بدنا نأخذ بعض خواص لـ cross
+
+59
+00:04:31,740 --> 00:04:34,960
+product بنفس المنهجية اللي أخذناها، اللي هي خواص اللي
+
+60
+00:04:34,960 --> 00:04:39,420
+هو الـ dot product، الآن if U and V are and W are
+
+61
+00:04:39,420 --> 00:04:43,440
+non-zero or any vectors and r و s are scalars
+
+62
+00:04:43,440 --> 00:04:48,580
+يعني عندنا U و V و W عبارة عن vectors و r و s عبارة
+
+63
+00:04:48,580 --> 00:04:56,070
+عن أعداد scalars، r في U cross s في U بيساوي r s في
+
+64
+00:04:56,070 --> 00:04:59,870
+U cross V، يعني ما بتفرق معنا الـ r scalars هذه
+
+65
+00:04:59,870 --> 00:05:02,870
+نضربهم في الـ vectors وبعدين نضرب اللي هو الـ cross
+
+66
+00:05:02,870 --> 00:05:06,310
+product، أو نطلعهم برا نضربهم في بعض ونضربهم باللي
+
+67
+00:05:06,310 --> 00:05:10,090
+بيطلع بالـ cross product، لأن هذه أيه؟ أعداد، هم r
+
+68
+00:05:10,090 --> 00:05:15,490
+scalars، اللي هي الخاصية الثانية، new اللي هو برضه
+
+69
+00:05:15,490 --> 00:05:19,110
+زي الـ dot product، الـ cross product بيوزّع على جمع
+
+70
+00:05:19,110 --> 00:05:26,980
+متجهين، يعني U x V زائد U x W هذا اللي هو توزيع الـ
+
+71
+00:05:26,980 --> 00:05:32,300
+cross product على الجمع، الآن زي ما توقعنا V x U
+
+72
+00:05:32,300 --> 00:05:37,280
+بيساوي مش الـ… اللي بيساوي الـ U x V بس إيه؟  معاكسه
+
+73
+00:05:37,280 --> 00:05:42,820
+باتجاه معاكس، لو رجعنا شوية U x V هي magnitude لـ U
+
+74
+00:05:42,820 --> 00:05:46,900
+و magnitude لـ V sin θ في الـ N، هذا المتجه، لكن لو V
+
+75
+00:05:46,900 --> 00:05:51,920
+x U بتصير magnitude لـ V magnitude لـ U في sin θ،
+
+76
+00:05:51,920 --> 00:05:55,540
+ما تغيرش هذا المقدار، مين اللي هيتغير؟ المتجه، حسب
+
+77
+00:05:55,540 --> 00:05:58,760
+القاعدة اليمنى، اللي هو بيكون في هذه الحالة U x V
+
+78
+00:05:58,760 --> 00:06:03,860
+إلى أعلى، وحسب اللي هي القاعدة اليمنى في الحالة
+
+79
+00:06:03,860 --> 00:06:09,160
+الثانية اللي هو بيكون عندي U و V بيكون إيش؟ إلى
+
+80
+00:06:09,160 --> 00:06:13,900
+أسفل، المتجه بيكون اللي هو عندي، الأصابع هذه على الـ
+
+81
+00:06:13,900 --> 00:06:18,740
+V، واليد هذه، جرّبها بإيدك، على اللي هو الـ U، بيكون
+
+82
+00:06:18,740 --> 00:06:23,800
+لأسفل، من هو الإصبع اللي بيحدد اتجاه الـ V cross U؟
+
+83
+00:06:23,800 --> 00:06:28,300
+فواضح أن الـ V cross U هو في اتجاه عكس الـ U cross V
+
+84
+00:06:28,300 --> 00:06:35,340
+ومتساويين تماماً إلا بس أن الاتجاه معكوس، الآن برضه
+
+85
+00:06:35,340 --> 00:06:38,360
+اللي وزّعنا اللي هو الـ dot، اللي cross product من
+
+86
+00:06:38,360 --> 00:06:43,690
+اليمين بتوزّع من اليسار، يعني الآن V زائد W cross U
+
+87
+00:06:43,690 --> 00:06:49,290
+ليه بيصير V cross U زائد W cross U مع الاحتفاظ بالـ
+
+88
+00:06:49,290 --> 00:06:53,290
+order تبعه أو بترتيب اللي هو الـ cross product، نحن
+
+89
+00:06:53,290 --> 00:06:58,550
+عارفين إن V cross V بيختلف عن V cross U، الآن حاصل
+
+90
+00:06:58,550 --> 00:07:03,970
+ضرب متجه الصفري، ضرب اللي هو cross product U بيساوي
+
+91
+00:07:03,970 --> 00:07:08,030
+دايماً zero vector، طبعاً الكلام سهل لأن هذا هيكون الـ
+
+92
+00:07:08,030 --> 00:07:11,570
+magnitude هو صفر، هذا هيكون صفر، وبما أنه متجه، إذًا
+
+93
+00:07:11,570 --> 00:07:19,650
+zero اللي هو إيش؟ المتجه الصفري، 6 U cross V cross W
+
+94
+00:07:19,650 --> 00:07:25,950
+هذه الآن علاقة اللي هي بتربط بين اللي هو cross
+
+95
+00:07:25,950 --> 00:07:30,570
+product وبين الـ dot product، بيقول لك U cross V
+
+96
+00:07:30,570 --> 00:07:34,630
+cross W، يعني أو شيء بتضربه بـ V cross W اللي بيطلع
+
+97
+00:07:34,630 --> 00:07:40,510
+بدك تضربه بـ U فيه، هذا هيطلع لك عبارة عن U dot W في
+
+98
+00:07:40,510 --> 00:07:47,550
+اتجاه الـ V زائد ناقص U.V في اتجاه الـ W، طبعاً الأمور
+
+99
+00:07:47,550 --> 00:07:51,750
+هذه الواحد بيحس فيها أكثر لما يبرهنها، يعني لو
+
+100
+00:07:51,750 --> 00:07:55,870
+برهنّا هذا، طبعاً البرهان مش مطلوب، لو برهنّا هنحس في
+
+101
+00:07:55,870 --> 00:07:59,490
+الأمر بشكل أكبر، عشان هيك بيقولوا البرهان بيعمّق
+
+102
+00:07:59,490 --> 00:08:03,730
+المفهوم وبيعمّق القانون اللي أنت بتسمعه أو اللي هو
+
+103
+00:08:03,730 --> 00:08:07,630
+بتشوفه، لأن هذه الخواص هي خواص اللي هي cross
+
+104
+00:08:07,630 --> 00:08:15,190
+product، نيجي الآن لـ cross product بين متجهات إحنا
+
+105
+00:08:15,190 --> 00:08:20,430
+اللي هي بنلزمها، بتلزمنا على طول، وبنكتب المتجهات
+
+106
+00:08:20,430 --> 00:08:24,370
+بدلالة مُكوّناتها، اللي هي الـ I والـ J والـ K، أكيد مرّة قعدت
+
+107
+00:08:24,370 --> 00:08:27,650
+معاكم في الفيزياء هذا، لكن إحنا خلينا نوضح لكم
+
+108
+00:08:27,650 --> 00:08:31,530
+اللي هي، الآن I هي متجه الـ I اللي هو متجه الوحدة
+
+109
+00:08:31,530 --> 00:08:35,970
+في اتجاه الـ X axis، والـ J متجه الوحدة في اتجاه الـ
+
+110
+00:08:35,970 --> 00:08:41,250
+Y axis الموجب طبعاً، والـ K متجه الوحدة في اتجاه الـ
+
+111
+00:08:41,250 --> 00:08:44,430
+Z axis، واللي هذول عكسهم متجهات الوحدة بس في
+
+112
+00:08:44,430 --> 00:08:49,730
+الاتجاهات السالبة من كل محور، طيب الآن بدنا نجي نضرب
+
+113
+00:08:49,730 --> 00:08:54,410
+بقاعدة اليد اليمنى برضه، أضرب I cross J، I cross J
+
+114
+00:08:54,410 --> 00:08:59,210
+هيطلع مين عندك؟ اللي هو K، I cross J هيطلع K، جرّب
+
+115
+00:08:59,210 --> 00:09:06,690
+الآن اللي هي أضرب اللي هو K cross I، K cross I، K
+
+116
+00:09:06,690 --> 00:09:10,690
+cross I باتجاه اللي هي اليد اليمنى، هتلاقي إيش؟
+
+117
+00:09:10,690 --> 00:09:16,150
+بيساوي، بيساوي عبارة عن مين؟ عبارة عن الـ J، إذًا الـ K
+
+118
+00:09:16,150 --> 00:09:21,380
+cross I هي الـ J، أكيد الـ I cross K سالب الـ J، cross K
+
+119
+00:09:21,380 --> 00:09:26,840
+هيطلع الـ I، الأكيد اللي هو اللي هي K cross J
+
+120
+00:09:26,840 --> 00:09:30,920
+سالب، الآن هذا طبعاً ممكن يكون الواحد بيخربط فيه أو
+
+121
+00:09:30,920 --> 00:09:35,580
+بيغلط فيه على طول، أنت هي، هذا بتسهّل علينا الحفظ، حط I J
+
+122
+00:09:35,580 --> 00:09:39,480
+K واعمل اللي هي الاتجاه هذا، الضرب بهذا الاتجاه
+
+123
+00:09:39,480 --> 00:09:46,200
+بيطلع موجب، I في J بيطلع K، J في K بيطلع إيش؟ J، K في I
+
+124
+00:09:46,790 --> 00:09:52,190
+I، K في I بيطلع مين؟ J، اعكسها، I في K بيطلع ناقص J،
+
+125
+00:09:52,190 --> 00:09:57,950
+مع السهم موجب ومع عكس السهم سالب، إذاً الآن سهل
+
+126
+00:09:57,950 --> 00:10:03,150
+الأمر بهذا الأمر بهذه الصورة، طبعاً هذا فيه المبرّر
+
+127
+00:10:03,150 --> 00:10:06,590
+في قانوننا اللي قبل شوية، اللي هو magnitude للـ I
+
+128
+00:10:06,590 --> 00:10:10,410
+في magnitude للـ J في sin الزاوية اللي بينهما في
+
+129
+00:10:10,410 --> 00:10:14,350
+مين؟ في اتجاه الوحدة العمود عليهما، لاحظوا الـ
+
+130
+00:10:14,350 --> 00:10:17,720
+magnitude للـ I واحد، و magnitude للـ J واحد، و sin
+
+131
+00:10:17,720 --> 00:10:22,780
+الزاوية بينهما، لأن هما متعامدان I و J بيساوي واحد، فبيصير
+
+132
+00:10:22,780 --> 00:10:29,500
+عندي اللي هو قيمة الـ I و J و K الـ magnitude في sin
+
+133
+00:10:29,500 --> 00:10:34,120
+بيساوي واحد، موجود في اتجاه مين؟ الـ K اللي هو فعلاً
+
+134
+00:10:34,120 --> 00:10:38,200
+اتجاه الوحدة برضه، الـ magnitude واحد، الآن لو
+
+135
+00:10:38,200 --> 00:10:41,680
+جينا الـ I cross I، الـ I cross I، طبعاً الزاوية اللي
+
+136
+00:10:41,680 --> 00:10:46,340
+بينهما، مجدداً يا جماعة صفر، sin الصفر إيش ماله؟ صفر، عشان
+
+137
+00:10:46,340 --> 00:10:51,540
+هيك الـ I cross I بيساوي صفر، J cross J إيش بيساوي؟ صفر،
+
+138
+00:10:51,540 --> 00:10:54,860
+K cross K إيش بيساوي؟ صفر، لأن الزاوية اللي بينهما sin
+
+139
+00:10:54,860 --> 00:11:01,060
+الثيتا، و sin الصفر بيساوي صفر، هذه الآن هي الـ tools
+
+140
+00:11:01,060 --> 00:11:05,360
+أو الأدوات اللي هتسهّل عليك اللي هو يعني هذه
+
+141
+00:11:05,360 --> 00:11:09,080
+الرسمة، مع هذه المعرفة، هتسهّل عليك عملية الـ cross
+
+142
+00:11:09,080 --> 00:11:14,080
+product، طيب الـ magnitude لـ U cross V يعني بدنا
+
+143
+00:11:14,080 --> 00:11:18,200
+نشوف الـ U cross V هندسياً إيش بيمثّل؟ قلنا الـ U
+
+144
+00:11:18,200 --> 00:11:20,880
+cross V هو عبارة عن magnitude لـ U في الـ magnitude
+
+145
+00:11:20,880 --> 00:11:25,420
+لـ V في sin الـ θ في متجه الوحدة العمودي عليهما، الآن
+
+146
+00:11:25,420 --> 00:11:28,900
+اللي هو متجه U cross V، لأن الـ magnitude له هو
+
+147
+00:11:28,900 --> 00:11:31,440
+magnitude لهذه، و magnitude لهذه، و magnitude لهذه، و
+
+148
+00:11:31,440 --> 00:11:34,880
+magnitude لهذه، هذا الـ magnitude له واحد، لأن هذا من
+
+149
+00:11:34,880 --> 00:11:39,040
+اتجاه وحدة، اللي هو بحدد اتجاه U cross V، الآن هذا
+
+150
+00:11:39,040 --> 00:11:41,780
+بيصير عندي الـ magnitude للـ U cross V عبارة عن الـ
+
+151
+00:11:41,780 --> 00:11:45,620
+magnitude للـ U في الـ magnitude للـ V في sin θ،
+
+152
+00:11:45,620 --> 00:11:50,640
+إنتَ شوف هذا ماذا يمثل هندسياً؟ انظروا، لو جينا أخذنا
+
+153
+00:11:50,640 --> 00:11:56,840
+الـ vector U وجينا رسمنا الـ vector الثاني من V، الآن
+
+154
+00:11:56,840 --> 00:12:02,420
+هذول بيعملوا plane، بيعملوا متوازي أضلاع، متوازي
+
+155
+00:12:02,420 --> 00:12:08,200
+أضلاع، الآن هذا طوله magnitude U، صح؟ وهذا طوله
+
+156
+00:12:08,200 --> 00:12:12,840
+magnitude للـ V، الآن هذا لو نزلنا عمودي على الـ U،
+
+157
+00:12:12,840 --> 00:12:16,420
+طبعاً مساحة متوازي الأضلاع عبارة عن القاعدة هي الـ
+
+158
+00:12:16,420 --> 00:12:20,900
+magnitude of U في الارتفاع اللي هو H، إيش H هذا؟ H
+
+159
+00:12:20,900 --> 00:12:25,540
+اللي هو عبارة عن sin of θ، بالضبط، اللي هو عبارة
+
+160
+00:12:25,540 --> 00:12:31,180
+عن H على V على الـ magnitude للـ V، إذًا H إيش هيساوي؟
+
+161
+00:12:31,180 --> 00:12:35,180
+الـ magnitude للـ V في sin of θ، إذًا الآن صار عندي
+
+162
+00:12:35,180 --> 00:12:39,600
+مساحة متوازي الأضلاع، هذا القاعدة، هذه الـ magnitude للـ
+
+163
+00:12:39,600 --> 00:12:42,820
+a، الـ base ضرب الـ height، الـ base اللي هو الـ magnitude
+
+164
+00:12:42,820 --> 00:12:45,940
+للـ U في الـ height اللي هو الـ magnitude للـ V في sin
+
+165
+00:12:45,940 --> 00:12:50,380
+الـ θ، اللي هو هذا، مين طلع عنده الـ magnitude للـ
+
+166
+00:12:50,380 --> 00:12:55,010
+U cross V، وكأنه بده يقول لي إنه يا جماعة، اللي هو
+
+167
+00:12:55,010 --> 00:12:59,550
+هيطلع إن الـ magnitude لـ U x V هو عبارة عن مساحة
+
+223
+00:18:06,010 --> 00:18:09,770
+الـ determinant يعني الـ U cross V بتساوي الـ
+
+224
+00:18:09,770 --> 00:18:14,310
+determinant لها. طيب، الآن الناس كيف كانت الـ
+
+225
+00:18:14,310 --> 00:18:16,790
+determinant؟ أنا هعيده كمان شوية في الأمثلة اللي
+
+226
+00:18:16,790 --> 00:18:21,950
+عندي لأن find U cross V and V cross U فيهم بساوي
+
+227
+00:18:21,950 --> 00:18:24,990
+اثنين I زائد J زائد K و V بتساوي ناقص أربعة I زائد
+
+228
+00:18:24,990 --> 00:18:29,070
+ثلاثة J زائد K. لأن هذا مثال على إيجاد الـ U cross
+
+229
+00:18:29,070 --> 00:18:33,790
+V أو الـ V cross U باستخدام الـ determinant أسهل
+
+230
+00:18:33,790 --> 00:18:39,170
+لنا كتير. هاي عند U cross V باجي برص في الأول الـ I
+
+231
+00:18:39,170 --> 00:18:43,170
+و الـ J و الـ K السطر الأول، السطر التالي اللي هو
+
+232
+00:18:43,170 --> 00:18:46,650
+معاملاتي اللي هي تبعات U، معامل I، J و K بنفس
+
+233
+00:18:46,650 --> 00:18:52,050
+الترتيب. 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 2 1
+
+234
+00:18:52,050 --> 00:18:54,330
+2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
+
+235
+00:18:54,330 --> 00:18:56,430
+1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
+
+236
+00:18:56,430 --> 00:18:58,970
+2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
+
+237
+00:18:58,970 --> 00:18:59,030
+1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
+
+238
+00:18:59,030 --> 00:18:59,150
+2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
+
+239
+00:18:59,150 --> 00:18:59,170
+1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
+
+240
+00:18:59,170 --> 00:18:59,850
+1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
+
+241
+00:18:59,850 --> 00:19:04,380
+2 1 ماشي، نستخدم السطر الأول أسهل، لإنّه السطر
+
+242
+00:19:04,380 --> 00:19:11,460
+الأول. كيف بده يوجده؟ لأن I مضروبة في مين؟ بشطب السطر
+
+243
+00:19:11,460 --> 00:19:15,300
+و العمود اللي فيه I. الـ determinant اللي بيظل اللي
+
+244
+00:19:15,300 --> 00:19:19,780
+هو determinant two by two. كيف بنوجده؟ واحد في واحد
+
+245
+00:19:19,780 --> 00:19:23,260
+ناقص ثلاثة. هي الـ determinant. حطينا بعدين لإنه
+
+246
+00:19:23,260 --> 00:19:29,670
+وجده. لأن الـ J بشطب اللي في الـ J، اللي هو كسطر و
+
+247
+00:19:29,670 --> 00:19:34,330
+كعمود. يظل اثنين و واحد و واحد و ناقص أربعة. هايه
+
+248
+00:19:34,330 --> 00:19:37,990
+طبعًا الآن موجب، سالب، موجب زي ما أنتم عارفين اللي
+
+249
+00:19:37,990 --> 00:19:42,290
+هو القانون اللي هو إيجاد الـ determinant. ناقص اللي
+
+250
+00:19:42,290 --> 00:19:44,850
+هو الـ determinant هذا في جهة، يظل مين؟ الموجب اللي
+
+251
+00:19:44,850 --> 00:19:49,510
+هو الـ K. هتشطب هذا العمود و هذا السطر اللي فيه الـ K.
+
+252
+00:19:49,510 --> 00:19:52,550
+يظل الـ determinant هذا. الـ determinant الـ two by
+
+253
+00:19:52,550 --> 00:19:58,830
+two. طبعًا كلكم بتعرفوا إيجاده. 1-3 يعني ناقص اثنين في
+
+254
+00:19:58,830 --> 00:20:04,290
+اتجاه مين؟ الـ I. اثنين ناقص ناقص بيصير زائد أربعة
+
+255
+00:20:04,290 --> 00:20:08,610
+اللي هي بيصير ستة. وفي عندي سالب بيصير سالب ستة J.
+
+256
+00:20:08,610 --> 00:20:11,930
+آخر واحد: اثنين في ثلاثة، ستة ناقص ناقص اللي هي
+
+257
+00:20:11,930 --> 00:20:15,270
+أربعة يعني زائد أربعة، ستة وأربعة عشرة، يعني بيصير
+
+258
+00:20:15,270 --> 00:20:19,860
+عندك اللي هو طبعًا اثنين في ثلاثة في ستة ناقص أربعة،
+
+259
+00:20:19,860 --> 00:20:23,500
+بيطلع اللي هي ناقص بيصير زائد، إذًا بيصير زائد عشرة.
+
+260
+00:20:23,500 --> 00:20:31,180
+أنا زائد عشرة مش ناقص عشرة إلا إذا كانوا مخربطين.
+
+261
+00:20:31,180 --> 00:20:37,260
+إحنا ناقص أربعة، لأ مظبوط ناقص أربعة، دلوقتي هيك و
+
+262
+00:20:37,260 --> 00:20:43,920
+هيك لأ، صح. 2×3 هو 6، و 6 زائد إذًا زائد من 10 K حسب
+
+263
+00:20:43,920 --> 00:20:49,560
+هذا الاسم. طيب، لأن بس قبل ما نيجي لتاني، مش مكمل هو
+
+264
+00:20:49,560 --> 00:20:53,900
+هذا اللي هو، لأ هيه مظبوط، آسف أنا بتطلع أتعلي
+
+265
+00:20:53,900 --> 00:20:58,240
+تاعتيه. هذا زائد عشرة، هيش K. 2×3 بـ 6 زائد عشرة هي
+
+266
+00:20:58,240 --> 00:21:02,020
+عشرة K زي ما إحنا متفقين. لأن هذا U cross V، لأن
+
+267
+00:21:02,020 --> 00:21:07,310
+V cross U يا شباب و يا بنات، V cross U بتساوي ناقص
+
+268
+00:21:07,310 --> 00:21:10,130
+الـ U cross.  بيفيش ده، اعمل determinant كمان مرة، ما
+
+269
+00:21:10,130 --> 00:21:14,490
+خلاص أنا صارت باعرف القوانين. إن ده مساوي اثنين I
+
+270
+00:21:14,490 --> 00:21:18,590
+ضربته في سالب، زائد ستة J، وده ضربته في ناقص اللي
+
+271
+00:21:18,590 --> 00:21:22,830
+خربطنا فيها، اللي هي ناقص عشرة K. طيب يا جماعة، لأن
+
+272
+00:21:22,830 --> 00:21:27,050
+نيجي اللي هو، ناخد مثال آخر. find a perpendicular to
+
+273
+00:21:27,050 --> 00:21:33,950
+the plane of B وQ و اللي هي مين؟ R. عند... عند يا
+
+274
+00:21:33,950 --> 00:21:39,480
+جماعة، اللي هو ثلاث نقاط، B، Q و R، ثلاث نقاط. واضح إنّه
+
+275
+00:21:39,480 --> 00:21:41,940
+ليست على استقامة واحدة. إذًا، أنا بعمل الـ meaning
+
+276
+00:21:41,940 --> 00:21:48,360
+plane. بقول لي أوجد اللي هو متجه عمودي على هذا الـ
+
+277
+00:21:48,360 --> 00:21:52,800
+plane. على طول بتبادر لذهني، باجي بعمل الـ vector
+
+278
+00:21:52,800 --> 00:21:56,380
+الأول PR من النقاط اللي عندنا، بنعرف كلنا، و بنعرف
+
+279
+00:21:56,380 --> 00:22:01,270
+جميع الـ things. نعمل و BQ الـ vector الثاني. و بناخد ما
+
+280
+00:22:01,270 --> 00:22:05,030
+هو الـ cross product بينهم. بيطلع vector إيش؟ شماله؟
+
+281
+00:22:05,030 --> 00:22:12,890
+عمودي. إذًا الآن الـ vector BQ هي BQ هي BQ across BR
+
+282
+00:22:12,890 --> 00:22:16,730
+اللي هو is perpendicular to the plane because it
+
+283
+00:22:16,730 --> 00:22:20,790
+is perpendicular to both vectors. و هي تعرف الدرب
+
+284
+00:22:20,790 --> 00:22:24,540
+اللي هو الدرب الاتجاهي. طبعًا واحد قال بياخد PR
+
+285
+00:22:24,540 --> 00:22:27,780
+cross PQ. بقول له لأ، بنفعل لأن الاثنين واحد عمودي إلى
+
+286
+00:22:27,780 --> 00:22:31,200
+أعلى والثاني عمودي إلى وين؟ إلى أسفل بالنسبة للبلين
+
+287
+00:22:31,200 --> 00:22:36,500
+اللي هم حملاته اللي هو P و Q و R. نيجي نحدد الآن PQ
+
+288
+00:22:36,500 --> 00:22:41,040
+هو بده يحسب PQ، لأن PQ اللي هو IBQ بتعرفوا كيف
+
+289
+00:22:41,040 --> 00:22:48,470
+تحسبوا. اثنين ناقص واحد اللي هي في اتجاه الـ I. -1 -1 -1
+
+290
+00:22:48,470 --> 00:23:01,190
+-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
+
+291
+00:23:01,190 --> 00:23:02,370
+-1 -1
+
+292
+00:23:07,070 --> 00:23:10,610
+1 ناقص ناقص واحد يعني واحد زائد واحد في اتجاه الـ J.
+
+293
+00:23:10,610 --> 00:23:14,570
+اثنين ناقص صفر في اتجاه الـ K. هيطلع عندنا ناقص
+
+294
+00:23:14,570 --> 00:23:19,990
+اثنين I زائد اثنين J زائد مين؟ اثنين K. هاي أوجدت الـ
+
+295
+00:23:19,990 --> 00:23:26,500
+PQ. P مين؟ R. قلنا الآن BQ كروس BR كيف بنوجده؟
+
+296
+00:23:26,500 --> 00:23:30,560
+بواسطة الـ determinant. هي عندك الـ I و الـ J و الـ K و
+
+297
+00:23:30,560 --> 00:23:33,660
+باجي بأخذ الـ BQ اللي هو واحد، اثنين، ناقص واحد.
+
+298
+00:23:33,660 --> 00:23:37,160
+واحد، اثنين، ناقص واحد. اللي هو BR اللي هو ناقص
+
+299
+00:23:37,160 --> 00:23:41,080
+اثنين، اثنين، اثنين. ناقص اثنين، اثنين، اثنين. و ببدأ زي
+
+300
+00:23:41,080 --> 00:23:45,660
+ما عملنا قبل بشوية. فك هذا السطر مع هذا العمود، يبقى
+
+301
+00:23:45,660 --> 00:23:50,500
+اللي هو في تجاه الـ I. ناقص هذا السطر هذا العمود مع
+
+302
+00:23:50,500 --> 00:23:53,400
+هذا السطر. بيبقى واحد و ناقص واحد و اثنين. وهيو
+
+303
+00:23:53,400 --> 00:23:59,300
+بيبقَى في تجاه الـ J. السطر الأخير، الـ K الأخيرة، بشطب
+
+304
+00:23:59,300 --> 00:24:05,050
+العمود و بشطب السطر و بيبقى عندي هذا. أربعة زائد
+
+305
+00:24:05,050 --> 00:24:09,650
+اثنين، ستة في اتجاه الـ I. أو اثنين ناقص اثنين بيطلع
+
+306
+00:24:09,650 --> 00:24:14,370
+صفر اللي هو بيضلش عنه في اتجاه الـ J. إشي زائد اثنين
+
+307
+00:24:14,370 --> 00:24:19,090
+زائد اللي هي أربعة بيطلع ستة K. إذًا هذا عبارة عن
+
+308
+00:24:19,090 --> 00:24:25,260
+مين يا جماعة؟ هذا عبارة عن المتجه العمودي على الـ
+
+309
+00:24:25,260 --> 00:24:30,860
+plane اللي بيحمل النقاط الثلاث. طيب الآن بقول، نكمل
+
+310
+00:24:30,860 --> 00:24:34,640
+السؤال فعلًا. find the area of the triangle with
+
+311
+00:24:34,640 --> 00:24:40,860
+vertices و find the area of the triangle with
+
+312
+00:24:40,860 --> 00:24:48,110
+vertices B، Q، R. اللي جاب هنا نفس النقاط. الآن الـ
+
+313
+00:24:48,110 --> 00:24:59,430
+Area قلنا لـ Triangle هو هاي الـ Triangle هذا، هاي
+
+314
+00:24:59,430 --> 00:25:03,390
+الـ vector الأول و هاي الـ vector الثاني. الآن الـ
+
+315
+00:25:03,390 --> 00:25:06,290
+cross product و آخذ له الـ magnitude. عمله الـ Area
+
+316
+00:25:06,290 --> 00:25:10,970
+لمين؟ لمتوازي الأضلاع. هو كله اللي هو هذا المثلث
+
+317
+00:25:10,970 --> 00:25:16,290
+يكون نصه. إذًا الآن هو بده الـ area لمين؟ للتريانجل.
+
+318
+00:25:16,290 --> 00:25:20,590
+مدام الـ area للتريانجل، إذًا إيش بوجد أول شيء؟ the
+
+319
+00:25:20,590 --> 00:25:24,630
+area of the parallelogram determined by B، Q، R. يعني
+
+320
+00:25:24,630 --> 00:25:30,850
+بده أوجد مساحة متوازي أضلاع اللي حملاته اللي هو الـ
+
+321
+00:25:30,850 --> 00:25:34,750
+P و الـ Q و الـ R. يعني هي متوازي الأضلاع بس مدّه
+
+322
+00:25:34,750 --> 00:25:39,210
+هيك على سطحان موازي، وهيك موازي. بعمل متوازي أضلاع.
+
+323
+00:25:39,210 --> 00:25:43,030
+هذا نصه. بدّي أوجد مساحة متوازي الأضلاع. مساحة متوازي
+
+324
+00:25:43,030 --> 00:25:47,550
+الأضلاع، الـ magnitude الـ PR across BQ لما أوجده.
+
+325
+00:25:47,550 --> 00:25:55,290
+اللي هو مين؟ أو جت بي أو بي أو بي أو بي أو بي أو بي
+
+326
+00:25:55,290 --> 00:26:00,350
+أو بي أو بي أو بي
+
+327
+00:26:00,350 --> 00:26:02,130
+أو بي أو بي أو بي أو بي أو بي أو بي أو بي أو بي أو
+
+328
+00:26:02,130 --> 00:26:02,270
+بي أو بي أو بي أو بي أو بي أو بي أو بي أو بي أو بي
+
+329
+00:26:02,270 --> 00:26:03,190
+أو بي أو بي أو بي أو بي أو بي أو بي أو بي أو بي أو
+
+330
+00:26:03,190 --> 00:26:04,670
+او بي أو بي أو بي أو بي أو بي أو بي أو بي أو بي او
+
+331
+00:26:04,670 --> 00:26:09,810
+بي أو بي أو بي أو بي او
+
+332
+00:26:14,370 --> 00:26:18,590
+is half of this. نص متوازي أضلاع، بتطلع ثلاثة جذر
+
+333
+00:26:18,590 --> 00:26:23,990
+الاثنين. ماشي، الآن بده يحكي اللي هو برضه على المثال
+
+334
+00:26:23,990 --> 00:26:27,740
+وده ياخد فكرة أخرى. المثال الرابع يقول find a unit
+
+335
+00:26:27,740 --> 00:26:34,340
+vector perpendicular to the plane of P, Q, R. يعني
+
+336
+00:26:34,340 --> 00:26:40,660
+يقول أوجد متجه وحدة عمودي على المستوى المكوّن
+
+337
+00:26:40,660 --> 00:26:49,180
+من أو المكوّنات له P, Q, R. الآن عشان أوجد الـ unit
+
+338
+00:26:49,180 --> 00:26:54,300
+vector العمودي، أنا أوجدت العمودي. مين العمودي اللي
+
+339
+00:26:54,300 --> 00:26:59,340
+هو BQ cross PR. أوجدته الـ BQ cross PR اللي هو جداش
+
+340
+00:26:59,340 --> 00:27:05,160
+كان 6I زائد 6K. هذا اللي هو المتجه العمودي اللي أنا
+
+341
+00:27:05,160 --> 00:27:09,760
+بدي متجه الوحدة العمودي. على مجموعته، إذًا المتجهة
+
+342
+00:27:09,760 --> 00:27:15,240
+الوحدة العمودي هو قيمة المتجه العمودي على اللي هو
+
+343
+00:27:15,240 --> 00:27:20,320
+الـ magnitude أو طول المتجه العمودي اللي هو طول
+
+344
+00:27:20,320 --> 00:27:23,720
+اللي هو المتجه العمودي الستة. عشان الستة كم طوله؟
+
+345
+00:27:23,720 --> 00:27:26,620
+أو جدناه ستة جذر اثنين. هيو من هنا، من هنا، من هنا
+
+346
+00:27:27,060 --> 00:27:29,760
+بتجسمه. على ستة جذر الاثنين بيطلع عندي واحد على
+
+347
+00:27:29,760 --> 00:27:32,800
+اثنين جذر في الـ I أو واحد على جذر الاثنين في تجاه
+
+348
+00:27:32,800 --> 00:27:35,780
+الـ I، واحد على جذر الاثنين في تجاه الـ K. هذا عبارة عن
+
+349
+00:27:35,780 --> 00:27:39,740
+تجاه الوحدة العمودي على الـ plane اللي هو اللي
+
+350
+00:27:39,740 --> 00:27:42,840
+بيحمل P و Q و R. زي اللي حكينا
+
+351
+00:27:45,700 --> 00:27:52,800
+الآن في برضه اللي هو في عندنا فوائد أخرى للـ Scalar
+
+352
+00:27:52,800 --> 00:27:57,380
+اللي هو Product و للـ Cross Product سوية مع بعض،
+
+353
+00:27:57,380 --> 00:28:01,980
+حاجة اسمها Triple Scalar أو Box Product. ده Product
+
+354
+00:28:01,980 --> 00:28:08,810
+U X V ضد W is called the triple scalar product of U
+
+355
+00:28:08,810 --> 00:28:13,770
+و V و W. طبعًا هو ليش جال Triple scalar product لأن
+
+356
+00:28:13,770 --> 00:28:17,850
+أول شيء بدرب U cross V بتطلع لمتجه، بعدين بدربها ضد
+
+357
+00:28:17,850 --> 00:28:22,610
+W. هيتحول كل هذا إلى رقم، إلى إيش؟ scalar. عشان إيش؟
+
+358
+00:28:22,610 --> 00:28:28,310
+اتجلك Triple scalar product of U و V و W. الآن هذا
+
+359
+00:28:28,310 --> 00:28:33,910
+بيمثل شيء، نشوف إيش بيمثل. لو دي أخذنا اللي هو الـ U
+
+360
+00:28:33,910 --> 00:28:40,130
+هذا المتجه و هذا الـ V متجه ماشي الحال و هذا الـ W
+
+361
+00:28:40,130 --> 00:28:44,650
+متجه ثالث. الثلاث هيعملن لي إيش؟ اللي هو المستوى
+
+362
+00:28:44,650 --> 00:28:48,510
+هذه إيش؟ اللي هو المستوى هذه و المستويات كلها هيعملن لي
+
+363
+00:28:48,510 --> 00:28:54,590
+box. هيعملن لي إيش؟ الـ box هذا أو متوازي مستطيلات أو
+
+364
+00:28:54,590 --> 00:29:01,550
+متوازي أو منشور القواعد اللي هي أوجهه وقواعده
+
+365
+00:29:01,550 --> 00:29:06,600
+عبارة عن متوازيات أضلاع. الآن الآن لو جينا لحجم هذا
+
+366
+00:29:06,600 --> 00:29:10,360
+المنشور، الحجم هو عبارة عن مساحة القاعدة في إيش؟ في
+
+367
+00:29:10,360 --> 00:29:18,580
+الارتفاع. الآن الارتفاع، الارتفاع هذا اللي هو المتجه
+
+368
+00:29:18,580 --> 00:29:25,260
+هذا. هيك اللي هو مين بيحسبه؟ U cross V. مظبوط، اللي هي
+
+369
+00:29:25,260 --> 00:29:29,520
+U cross V هو المتجه العمودي. على U cross V بس أنا
+
+370
+00:29:29,520 --> 00:29:36,730
+إيش بده يوجِد يا جماعة؟ بده يوجِد اللي هو المساح اللى
+
+371
+00:29:36,730 --> 00:29:42,290
+هو طول اللي هو الارتفاع. يعني هي عندي سطح العلوي هو
+
+372
+00:29:42,290 --> 00:29:49,330
+فوق. هذا العمود اللي بنزل منها لهون اللي هو بساوي
+
+373
+00:29:49,330 --> 00:29:53,870
+هذا منها لهون هو عبارة عن ارتفاع متوازي أضلاع، فبيكون
+
+374
+00:29:53,870 --> 00:29:59,660
+متوازي أضلاع. مساحة القاعدة هذه في الارتفاع. مساحة
+
+375
+00:29:59,660 --> 00:30:03,280
+القاعدة، الـ area of base في high. الـ area of base
+
+376
+00:30:03,280 --> 00:30:06,200
+سهلة، ما إحنا عرفناها، اللي عبارة عن الـ magnitude الـ
+
+377
+00:30:06,200 --> 00:30:11,540
+U cross V. الـ U cross V. طيب، هذا طول. وجداش يا جماعة؟
+
+378
+00:30:11,540 --> 00:30:16,480
+لأن صار عندي U cross V متجهة. مظبوط، هيو عمودي على
+
+379
+00:30:16,480 --> 00:30:22,480
+هذا. لأن الزاوية هذه هتكون θ بين W و بين
+
+380
+00:30:22,480 --> 00:30:27,780
+المتجهة الجديدة اللي هو U cross V. الآن هذا الارتفاع
+
+381
+00:30
+
+445
+00:35:55,480 --> 00:35:59,320
+إيه، شمالين متساوية، بس عشان تتمأن قلبك وإذا بدك
+
+446
+00:35:59,320 --> 00:36:04,160
+تسلم من القانون، هو هيك، أنت حر، أما إثباته مش صعب. U
+
+447
+00:36:04,160 --> 00:36:07,460
+cross V زي ما اتعلمنا، وبعدين اللي بيطلع بتضرب ضد
+
+448
+00:36:07,460 --> 00:36:10,300
+W زي ما اتعلمنا، هتلاقي بيطلع على الـ determinant
+
+449
+00:36:10,300 --> 00:36:13,760
+هذا بيطلع قيمة الـ determinant، لو أنت حسبت الـ
+
+450
+00:36:13,760 --> 00:36:17,220
+determinant. الآن بدنا إيش؟ نقعد نرجع للطرق البدائية؟
+
+451
+00:36:17,220 --> 00:36:22,340
+من الأول، إذا الآن U cross V ضد W على طول هيمثل إيش؟
+
+452
+00:36:22,340 --> 00:36:24,360
+قيمة الـ determinant التالية، يعني هذا بتعتبره
+
+453
+00:36:24,360 --> 00:36:31,460
+قانون: U1 U2 U3، V1 V2 V3، W1 W2 W3. ناخد مثال عليه.
+
+454
+00:36:31,460 --> 00:36:35,280
+تتضح الأمور. find the volume of the box determined
+
+455
+00:36:35,280 --> 00:36:40,340
+by U بيساوي 2I -G -K، V بيساوي 2J -2K،
+
+456
+00:36:40,340 --> 00:36:45,130
+و W بيساوي 7G -4K.  إلى أن
+
+457
+00:36:45,130 --> 00:36:48,990
+volume of the box is، volume للبوكس بيساوي اللي هو
+
+458
+00:36:48,990 --> 00:36:51,170
+أقول لك الـ magnitude لهذا المقدار، الـ magnitude
+
+459
+00:36:51,170 --> 00:36:54,290
+لهذا، هذا اللي هو الـ determinant، الـ magnitude اللي
+
+460
+00:36:54,290 --> 00:36:58,370
+هو الـ magnitude، هي magnitude برا، واللي جوا إيش؟
+
+461
+00:36:58,370 --> 00:37:02,370
+للـ determinant، من الـ determinant، هي: 1، 2، -1،
+
+462
+00:37:02,370 --> 00:37:05,810
+1، 1، -1، -2، 3،
+
+463
+00:37:05,810 --> 00:37:09,470
+-2، 3، اللي هي للـ K، وصفر لمين؟ للـ J، لأنه
+
+464
+00:37:09,470 --> 00:37:16,070
+مافيش J، صفر معاملاتها، W اللي هو صفر للـ I وسبعة للـ J
+
+465
+00:37:16,070 --> 00:37:20,790
+و -4 للـ K. هذا الآن الـ determinant. أصلاً أنتَ
+
+466
+00:37:20,790 --> 00:37:26,390
+لو بدلت الـ U مع الـ V مع الـ W وحطيت في السطر هذا و
+
+467
+00:37:26,390 --> 00:37:29,850
+هذا، مش هتفرق، معناه ليش؟ لأنه ماخدين له في الآخر الـ
+
+468
+00:37:29,850 --> 00:37:33,750
+magnitude، يعني مخدرينه absolute value، يعني هيرجع
+
+469
+00:37:33,750 --> 00:37:38,050
+لقيمة واحدة كلها، لأن الفرق بين اتنين بس سالب، طيب
+
+470
+00:37:38,050 --> 00:37:43,250
+لأن وجد هذا زي ما نوجد عادة، أخدت الـ absolute value
+
+471
+00:37:43,250 --> 00:37:47,850
+هي برا، لأن وجد قيمة الـ determinant اللي جوا، شطب
+
+472
+00:37:47,850 --> 00:37:52,810
+هذا، شطب هذا، بيصير صفر، -4، صفر، 7، في 3،
+
+473
+00:37:52,810 --> 00:37:58,700
+و21، 21، اللي هو هاي اللي هو...لأ
+
+474
+00:37:58,700 --> 00:38:02,360
+آسف، أنا مستخدم مين يا جماعة؟ مستخدم الصوت الأخير،
+
+475
+00:38:02,360 --> 00:38:06,600
+عشان أن أضيع واحد وبس أعمل إيش؟  سبعة و -4،
+
+476
+00:38:06,600 --> 00:38:13,500
+بيصير magnitude موجب سالب موجب، صفر، خلصنا منه، ما بلزمش،
+
+477
+00:38:13,500 --> 00:38:17,190
+لأنه بده ينضرب في الصفر زي ما أنتم عارفين، سبعة الآن بالسالب،
+
+478
+00:38:17,190 --> 00:38:19,430
+شاء الله تكونوا فاهمين عليها، سبعة الآن بالسالب،
+
+479
+00:38:19,430 --> 00:38:23,530
+لأنه سالب، اللي هي موجب سالب موجب، بتفكر محدد، بحكي
+
+480
+00:38:23,530 --> 00:38:27,750
+منشط بهدا ومنشط بهدا، وظل 1 -1 -2،
+
+481
+00:38:27,750 --> 00:38:32,610
+3، وظل -4، -4، شطب بهدا وشطب بهدا،
+
+482
+00:38:32,610 --> 00:38:40,490
+وظل 1، 2، صفر، 3، 3، + -2 في
+
+483
+00:38:40,490 --> 00:38:47,890
+-7، -4 في صفر، + 4، لأن هذه
+
+484
+00:38:47,890 --> 00:38:51,330
+عبارة عن -7، لأن 3 -2 = 1، -
+
+485
+00:38:51,330 --> 00:38:56,630
+7، -7، -16، -26،
+
+486
+00:38:56,630 --> 00:39:00,750
+magnitude 23، هذا اللي هو حجم من المكعب.
+
+487
+00:39:00,750 --> 00:39:07,190
+واحد قال بدي أستخدم W . U cross V، بنفع. واحد قال
+
+488
+00:39:07,190 --> 00:39:12,890
+بدي أقول U . V cross W، بنفع، وهكذا، فاهمين إيش؟
+
+489
+00:39:12,890 --> 00:39:16,180
+اللي بحكيه، لأن هيك، وكون إحنا خلصنا الحاضر الرابع،
+
+490
+00:39:16,180 --> 00:39:19,900
+أو هاي الـ exercises، والسلام عليكم ورحمة الله
+
+491
+00:39:19,900 --> 00:39:22,100
+وبركاته، ولا لقاء، لقاء آخر.
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TxfjgQ2pjWY_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TxfjgQ2pjWY_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..fda6bbb9c949d2b0b601ecc0c0717973b72f4819
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TxfjgQ2pjWY_postprocess.srt
@@ -0,0 +1,1964 @@
+1
+00:00:00,720 --> 00:00:04,060
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,060 --> 00:00:08,620
+وبركاته هذه المحاضرة الرابعة مساقطة فاضل أتكامل
+
+3
+00:00:08,620 --> 00:00:15,360
+جين صيف 2020 علوان المحاضرة اليوم اللي هو the
+
+4
+00:00:15,360 --> 00:00:19,840
+cross productاللي هو section 12-4 the cross
+
+5
+00:00:19,840 --> 00:00:23,720
+product of two vectors in space او اللي هو الضرب
+
+6
+00:00:23,720 --> 00:00:28,760
+الاتجاهي حكينا المحاضرة الماضية عن الضرب القياسي
+
+7
+00:00:28,760 --> 00:00:31,480
+اللي هو the cross product اليوم ان نحكي عن اللي هو
+
+8
+00:00:31,480 --> 00:00:35,180
+ال cross product وقلنا سابقا انه في نوعين من الضرب
+
+9
+00:00:35,180 --> 00:00:40,300
+بين المتجهات ضرب قياسي وضرب اتجاهي واتحدثنا عن
+
+10
+00:00:40,300 --> 00:00:44,720
+الضرب القياسي والان نتحدث عن الضرب الاتجاهيالان
+
+11
+00:00:44,720 --> 00:00:48,300
+بدنا نعرف شو معناته اللي هو the cross product of
+
+12
+00:00:48,300 --> 00:00:53,960
+two vectors in space كمعنى الدرب الاتجاهي لمتجهين
+
+13
+00:00:53,960 --> 00:00:59,360
+في الفرق لو فرضنا عند U عبارة عن متجه و V عبارة عن
+
+14
+00:00:59,360 --> 00:01:04,220
+متجه آخر طبعا ال U و ال V أكيد هيصنعن اللي هو طبعا
+
+15
+00:01:04,220 --> 00:01:09,560
+التنين مش متوازين أكيد هيصنعن إدلين الآن بدنا نعرف
+
+16
+00:01:09,560 --> 00:01:14,640
+اللي هو ال cross product بينهمهو عبارة عن vector
+
+17
+00:01:14,640 --> 00:01:20,880
+عمودي على لبنان تبحن عمودي لأعلى أو عمودي لأسفل بس
+
+18
+00:01:20,880 --> 00:01:25,920
+عساس انه احنا نوحد الحديث بدنا نتفج مع بعض بقاعدة
+
+19
+00:01:25,920 --> 00:01:30,620
+اسمها قاعدة اليد اليمنى كيف نحدد اللي هو اتجاه ال
+
+20
+00:01:30,620 --> 00:01:35,560
+vector العمودي اللي احنا بدنا نحكي كلنا بلغة واحدة
+
+21
+00:01:35,560 --> 00:01:39,440
+عليه ايه اللي بقصه ان لو كان U عندي متجهة و V
+
+22
+00:01:39,440 --> 00:01:44,940
+متجهة و بينهم زاوية θهيكون تعريف الحاصل الضرب
+
+23
+00:01:44,940 --> 00:01:50,160
+الاتجاهي بينهم هو عبارة عن الـ magnitude للـ U طول
+
+24
+00:01:50,160 --> 00:01:55,860
+الـ U في ال magnitude للـ V في sin θ طبعا هذا مش
+
+25
+00:01:55,860 --> 00:02:01,120
+شبه الـ dot product ولكن كانت cos θ sin θ الآن هذه
+
+26
+00:02:01,120 --> 00:02:06,420
+هي القيمة العددية هذه اللي هو الـ vector اللي
+
+27
+00:02:06,420 --> 00:02:10,150
+بنحكي عنه اللي هو U cross Vطب الآن اتجاهه كيف
+
+28
+00:02:10,150 --> 00:02:14,130
+بنحدده اتجاهه في اتجاه الوحدان وين اتجاه الوحدان
+
+29
+00:02:14,130 --> 00:02:18,270
+هذا اللي هو بقاعدة اليد اليمنى هي اليد اليمنى اللي
+
+30
+00:02:18,270 --> 00:02:24,230
+هو بتيجي انت الان الاصابع هدولة بيكونن على اللي هو
+
+31
+00:02:24,230 --> 00:02:29,470
+ال U و الايد من هنا هتكون على مين؟ على ال V الان
+
+32
+00:02:29,470 --> 00:02:36,330
+اتجاه الاصبع الكبير هذا هو بيكون اتجاه مين؟ اتجاه
+
+33
+00:02:36,330 --> 00:02:41,360
+المتجه U cross Vفبكون هذا هو اتجاه المتجه اللي
+
+34
+00:02:41,360 --> 00:02:47,100
+عندنا يعني U cross V U cross V غير V cross U U
+
+35
+00:02:47,100 --> 00:02:51,360
+cross V يعني الأصابع دولة على U و الأيد هذه على V
+
+36
+00:02:51,360 --> 00:02:56,080
+يعني الأصابع الأول و الأيد على ال V و الأصبع اللي
+
+37
+00:02:56,080 --> 00:03:00,810
+فوق هذا بيودع على اتجاه اللي هو U cross Vفبكون
+
+38
+00:03:00,810 --> 00:03:03,170
+عندي U cross V بسوء ال magnitude والـ U في ال
+
+39
+00:03:03,170 --> 00:03:07,430
+magnitude ذا ال V في sign الزاوية اللي بينهم في
+
+40
+00:03:07,430 --> 00:03:12,890
+المتجه N اللي احنا حددنا بطريقة اليد اليمنى الان
+
+41
+00:03:12,890 --> 00:03:17,310
+الأمور ان شاء الله أسهل الان لو كان اللي هو ال two
+
+42
+00:03:17,310 --> 00:03:22,620
+vectorsاللي هما متوازين طبعا عندنا الان ال two
+
+43
+00:03:22,620 --> 00:03:26,740
+vectors المتوازين هذي ال u cross v ممكن يساوي ال
+
+44
+00:03:26,740 --> 00:03:31,200
+zero vector في حالة في حالات معينة ان ال u صفر
+
+45
+00:03:31,200 --> 00:03:37,080
+يكون magnitude صفر او ال v صفر او التاني مع بعض
+
+46
+00:03:37,080 --> 00:03:41,380
+صفر او الزاوية اللي بينهم تتا صفر يعني لما يكون
+
+47
+00:03:41,380 --> 00:03:46,260
+الاشمال متوازينإذا الأن اللي بدنا نحكي non-zero
+
+48
+00:03:46,260 --> 00:03:51,300
+vectors U and V بكون in parallel if and only if U
+
+49
+00:03:51,300 --> 00:03:54,140
+cross V أيش بساوي سفر لأن قولنا U cross V بساوي
+
+50
+00:03:54,140 --> 00:03:57,940
+سفر ممكن تحدث من U سفر أو V سفر أو تنتين مع بعض
+
+51
+00:03:57,940 --> 00:04:02,770
+سفر أو الحالة اللي هو مدامالحالة التالتة انه صار
+
+52
+00:04:02,770 --> 00:04:05,730
+الزاوية بينهم سفر يعني الزاوية اللي بينهم عبارة عن
+
+53
+00:04:05,730 --> 00:04:10,670
+اللي هي عبارة عن zero او by او الاخر يعني اتنين
+
+54
+00:04:10,670 --> 00:04:16,030
+اشمالهم متوازين عشان هي قلنا non-zero vectors u
+
+55
+00:04:16,030 --> 00:04:20,510
+and v parallel if u cross v ايش بساوي سفر هذه حكم
+
+56
+00:04:20,510 --> 00:04:24,230
+على اللي هو two vectors inهم non-zero vectors طبعا
+
+57
+00:04:24,230 --> 00:04:28,310
+inهم متوازين انه يكون ال cross product بينهم بساوي
+
+58
+00:04:28,310 --> 00:04:31,740
+سفرطيب يا جماعة بدنا ناخد بعض خواص لـ cross
+
+59
+00:04:31,740 --> 00:04:34,960
+product بنفس المنهجية اللي أخدنا اللي هي خواص اللي
+
+60
+00:04:34,960 --> 00:04:39,420
+هو الـ dot product الان if u and v are and w are
+
+61
+00:04:39,420 --> 00:04:43,440
+non zero are any vectors and r و s are scalars
+
+62
+00:04:43,440 --> 00:04:48,580
+يعني عند u و v و w عبارة عن vectors و r و s عبارة
+
+63
+00:04:48,580 --> 00:04:56,070
+عن أعداد scalarsR في U cross S في U بيساوي R S في
+
+64
+00:04:56,070 --> 00:04:59,870
+U cross V يعني مش فارقة معنى ال R Scalars هدولة
+
+65
+00:04:59,870 --> 00:05:02,870
+نضربهم في ال vectors و بعدين نضرب اللي هو ال cross
+
+66
+00:05:02,870 --> 00:05:06,310
+product أو نطلحهم برا نضربهم في بعض و نضربهم باللي
+
+67
+00:05:06,310 --> 00:05:10,090
+بيطلع بال cross product لإن هدولة أيه شمال هم R
+
+68
+00:05:10,090 --> 00:05:15,490
+Scalars اللي هي الخاصية الثانية New اللي هو برضه
+
+69
+00:05:15,490 --> 00:05:19,110
+زي ال dot product ال cross product يوزع على جمع
+
+70
+00:05:19,110 --> 00:05:26,980
+متجهينيعني U x V زائد U x W هذا اللي هو توزيع الـ
+
+71
+00:05:26,980 --> 00:05:32,300
+cross product على الجميع الآن زي ما اتوقعنا V x U
+
+72
+00:05:32,300 --> 00:05:37,280
+بساوي مش ال .. اللي بساوي الـ U x V بس إيه شماله
+
+73
+00:05:37,280 --> 00:05:42,820
+بعكس الاتجاه لو رجعنا شوية U x V هي magnitude لـ U
+
+74
+00:05:42,820 --> 00:05:46,900
+و magnitude ل V sin 3 في الـ N هذا المتجهةلكن لو V
+
+75
+00:05:46,900 --> 00:05:51,920
+X U بتصير magnitude لـ V مقادرة لـ U في صينة تيتا
+
+76
+00:05:51,920 --> 00:05:55,540
+ماتغيرش هذا المقدار مين اللي هيتغير المتجه حسب
+
+77
+00:05:55,540 --> 00:05:58,760
+القاعدة اليمنى اللي هو بيكون في هذه الحالة U X V
+
+78
+00:05:58,760 --> 00:06:03,860
+إلى أعلى وحسب اللي هي القاعدة اليمنى في الحالة
+
+79
+00:06:03,860 --> 00:06:09,160
+الثانية اللي هو بيكون عندى U و V بيكون إيش إلى
+
+80
+00:06:09,160 --> 00:06:13,900
+أسفل المتجهة بيكون اللي هو عندىالأصابع دولة على ال
+
+81
+00:06:13,900 --> 00:06:18,740
+V و الإيد هذه تجربها بإيدك على اللي هو ال U بيكون
+
+82
+00:06:18,740 --> 00:06:23,800
+لأسفل من هو الأصبع اللي بيحدد اتجاه ال V across U
+
+83
+00:06:23,800 --> 00:06:28,300
+فواضح أن ال V cross U هو في اتجاه عكس ال U cross V
+
+84
+00:06:28,300 --> 00:06:35,340
+ومتساويين تماما إلا بس أن الاتجاه معكوس الآن برضه
+
+85
+00:06:35,340 --> 00:06:38,360
+اللي وزعنا اللي هو ال dot اللي cross product من
+
+86
+00:06:38,360 --> 00:06:43,690
+اليمين بتوزع من اليساريعني الان V زائد W cross U
+
+87
+00:06:43,690 --> 00:06:49,290
+ليه بيصير V cross U زائد W cross U مع الاحتفاظ بال
+
+88
+00:06:49,290 --> 00:06:53,290
+order تبع أو بترتيب اللي هو ال cross product نحن
+
+89
+00:06:53,290 --> 00:06:58,550
+عارفين ان V cross V تختلف عن V cross Uالان حاصل
+
+90
+00:06:58,550 --> 00:07:03,970
+ضرب زيرو متجه ضرب اللي هو cross product U بساوي
+
+91
+00:07:03,970 --> 00:07:08,030
+دايما زيرو vector طبعا الكلام سهل لأن هذا هيكون ال
+
+92
+00:07:08,030 --> 00:07:11,570
+magnitude هو سفر ده هيكون سفر و بما أنه متجه إذا
+
+93
+00:07:11,570 --> 00:07:19,650
+زيرو اللي هو إيش المتجه الصفر 6 U cross V cross W
+
+94
+00:07:19,650 --> 00:07:25,950
+هذه الآن علاقة اللي هي بتربط بين اللي هو cross
+
+95
+00:07:25,950 --> 00:07:30,570
+productو بين الـ dot product بيقول لك U cross V
+
+96
+00:07:30,570 --> 00:07:34,630
+cross W يعني او اشي بتدرب ب V cross W اللي بيطلع
+
+97
+00:07:34,630 --> 00:07:40,510
+بدك تدرب U فيه هذا هيطلع لك عبارة عن U dot W في
+
+98
+00:07:40,510 --> 00:07:47,550
+تجاه ال Vزائد اناقص U.V في اتجاه ال W طبعا الأمور
+
+99
+00:07:47,550 --> 00:07:51,750
+هذه الواحد بيحس فيها أكثر لما يبرهنها يعني لو
+
+100
+00:07:51,750 --> 00:07:55,870
+برهننا هذا طبعا البرهن مش مطلوب لو برهننا هنحس في
+
+101
+00:07:55,870 --> 00:07:59,490
+الأمر بشكل أكبر عشان يجدام يقول البرهن يعمق
+
+102
+00:07:59,490 --> 00:08:03,730
+المفهوم ويعمق القانون اللي أنت بتسمعه أو اللي هو
+
+103
+00:08:03,730 --> 00:08:07,630
+بتشوفه لأن هذه الخواص هي خواص اللي هي cross
+
+104
+00:08:07,630 --> 00:08:15,190
+productبنيجي الآن لـ cross product بين متجهات احنا
+
+105
+00:08:15,190 --> 00:08:20,430
+اللي هي بنلزمها بتلزمنا على طول و بنكتب المتجهات
+
+106
+00:08:20,430 --> 00:08:24,370
+بدلالتهااللي هي ال I و ال J و ال K أكيد مره قعدت
+
+107
+00:08:24,370 --> 00:08:27,650
+معاكم في الفيزيا هذا لكن احنا خلّينا نوضّح لكم
+
+108
+00:08:27,650 --> 00:08:31,530
+اللي ايها الآن I هي متجه ال I اللي هو متجه الوحدة
+
+109
+00:08:31,530 --> 00:08:35,970
+في اتجاه ال X axis و ال J متجه الوحدة في اتجاه ال
+
+110
+00:08:35,970 --> 00:08:41,250
+Y axis الموجر طبعا و ال K متجه الوحدة في اتجاه ال
+
+111
+00:08:41,250 --> 00:08:44,430
+Z axis و اللي هدول عكس هم متجهات الوحدة بس في
+
+112
+00:08:44,430 --> 00:08:49,730
+الاتجاهات السالبة من كل محور طيبالان بدنا نجي نضرب
+
+113
+00:08:49,730 --> 00:08:54,410
+بقاعدة اليد اليمنى برضه اضرب I cross J I cross J
+
+114
+00:08:54,410 --> 00:08:59,210
+هيطلع مين عندك اللي هو K I cross J هيطلع K جرب
+
+115
+00:08:59,210 --> 00:09:06,690
+الان اللي هي اضرب اللي هو K cross I K cross I K
+
+116
+00:09:06,690 --> 00:09:10,690
+cross I وباتجاه اللي هي اليد اليمنى هتلاقي ايش
+
+117
+00:09:10,690 --> 00:09:16,150
+بيساوي بيساوي عبارة عن مين عبارة عن الـ J إذا ال K
+
+118
+00:09:16,150 --> 00:09:21,380
+في cross I الـ Jأكيد الـ I cross K سالبه J cross K
+
+119
+00:09:21,380 --> 00:09:26,840
+هيطلع من الـ I الأكيد اللي هو اللي هي K cross J
+
+120
+00:09:26,840 --> 00:09:30,920
+سالبه الان هذا طبعا ممكن يكون الواحد بخربط فيه او
+
+121
+00:09:30,920 --> 00:09:35,580
+بغضبك على طول انت هي هذا بتسهل علينا الحفظ حط I J
+
+122
+00:09:35,580 --> 00:09:39,480
+K و اعمل اللي هي الاتجاه هذا الضرب بهذا الاتجاه
+
+123
+00:09:39,480 --> 00:09:46,200
+بطلع موجب I في J بطلع K J في K بطلع إيش J K في K I
+
+124
+00:09:46,790 --> 00:09:52,190
+I K في I بيطلع مين؟ J إعكسها I في K بيطلع ناقص J
+
+125
+00:09:52,190 --> 00:09:57,950
+مع السهم موجب و مع عكس السهم سالم إذا الآن سهل
+
+126
+00:09:57,950 --> 00:10:03,150
+الأمر بهذا الأمر بهذه الصورة طبعا هذا فيه المبرر
+
+127
+00:10:03,150 --> 00:10:06,590
+في قانوننا اللي قبل بشوية اللي هو magnitude لل I
+
+128
+00:10:06,590 --> 00:10:10,410
+في magnitude لل J في sign الزاوية اللي بينهم في
+
+129
+00:10:10,410 --> 00:10:14,350
+مين؟ في اتجاه الوحدة العمود عليهم لاحظوا ال
+
+130
+00:10:14,350 --> 00:10:17,720
+magnitude لل I واحد و magnitude لل J واحدوSin
+
+131
+00:10:17,720 --> 00:10:22,780
+الزاوية بينهم لأن هم متعامدين I وJ بسوء واحد فبصير
+
+132
+00:10:22,780 --> 00:10:29,500
+عندى اللى هو قيمة الـ I وJ وK ال magnitude في Sin
+
+133
+00:10:29,500 --> 00:10:34,120
+بسوء واحد موجود في اتجاه مين ال K اللى هو فعلا
+
+134
+00:10:34,120 --> 00:10:38,200
+اتجاه الواحدة برضه ال magnitude الواحدةالان لو
+
+135
+00:10:38,200 --> 00:10:41,680
+جينا ال I cross I ال I cross I طبعا الزاوية اللي
+
+136
+00:10:41,680 --> 00:10:46,340
+بينه مجدش يا جماعة سفر سين السفر ايش ماله سفر عشان
+
+137
+00:10:46,340 --> 00:10:51,540
+هيك ال I cross I بساوي سفر J cross J ايش بساوي سفر
+
+138
+00:10:51,540 --> 00:10:54,860
+K cross K ايش بساوي سفر لان الزاوية اللي بينه سين
+
+139
+00:10:54,860 --> 00:11:01,060
+الثتا و سين السفر بساوي سفرهذه الآن هي الـ tools
+
+140
+00:11:01,060 --> 00:11:05,360
+أو الأدوات اللي هتسهل عليها اللي هو يعني هذه
+
+141
+00:11:05,360 --> 00:11:09,080
+الرسمة مع هذه المعرفة هتسهل عليها عملية الـ cross
+
+142
+00:11:09,080 --> 00:11:14,080
+product طيب ال magnitude لـ U cross V يعني بدنا
+
+143
+00:11:14,080 --> 00:11:18,200
+نشوف الـ U cross V هندسيا إيش بيمثل قولنا الـ U
+
+144
+00:11:18,200 --> 00:11:20,880
+cross V هو عبارة عن magnitude لـ U في ال magnitude
+
+145
+00:11:20,880 --> 00:11:25,420
+لـ V في صين الـ θ في متجه الوحدة العمود عليها الان
+
+146
+00:11:25,420 --> 00:11:28,900
+اللي هو متجه U cross Vلأن الـ magnitude له هو
+
+147
+00:11:28,900 --> 00:11:31,440
+magnitude لهذه و magnitude لهذه و magnitude لهذه و
+
+148
+00:11:31,440 --> 00:11:34,880
+magnitude لهذه هذا ال magnitude له واحد لأن هذا من
+
+149
+00:11:34,880 --> 00:11:39,040
+اتجاه واحدة اللي هو بحدد اتجاه U cross V الان هذا
+
+150
+00:11:39,040 --> 00:11:41,780
+بيصير عند ال magnitude لل U cross V عبارة عن ال
+
+151
+00:11:41,780 --> 00:11:45,620
+magnitude لل U في ال magnitude لل V في Psi نثلية
+
+152
+00:11:45,620 --> 00:11:50,640
+ان تشوف هذا ماذا يمثل هندسيا اتطلعوا لو جينا أخدنا
+
+153
+00:11:50,640 --> 00:11:56,840
+ال vector U و جينا رسمنا ال vector التاني من Vالان
+
+154
+00:11:56,840 --> 00:12:02,420
+هدولة بيعملن بلين بل بيعملن متوازي أضلع متوازي
+
+155
+00:12:02,420 --> 00:12:08,200
+أضلع الان هذا طوله magnitude U صح؟ وهذا طوله
+
+156
+00:12:08,200 --> 00:12:12,840
+magnitude mean V الان هذا لو نزلنا عمودي على ال U
+
+157
+00:12:12,840 --> 00:12:16,420
+طبعا مساحة متوازي الأضلع عبارة عن القاعدةهي ال
+
+158
+00:12:16,420 --> 00:12:20,900
+magnitude of U في الارتفاع اللي هو H إيش H هذا H
+
+159
+00:12:20,900 --> 00:12:25,540
+اللي هو عبارة عن sign of θ بالساوي اللي هو عبارة
+
+160
+00:12:25,540 --> 00:12:31,180
+عن H على V على ال magnitude ل V إذا H إيش هيساوي
+
+161
+00:12:31,180 --> 00:12:35,180
+ال magnitude ل V في sign of θ إذا الأن صار عندى
+
+162
+00:12:35,180 --> 00:12:39,600
+مساحة متوازلة أضلاح هذاالقاعدة هذه ال magnitude لل
+
+163
+00:12:39,600 --> 00:12:42,820
+a ال base ضرب ال high ال base اللي هو ال magnitude
+
+164
+00:12:42,820 --> 00:12:45,940
+لل u في ال high اللي هو ال magnitude لل v في sign
+
+165
+00:12:45,940 --> 00:12:50,380
+ال theta اللي هو هذا مين طلع عنده ال magnitude لل
+
+166
+00:12:50,380 --> 00:12:55,010
+u cross v و كأنه بدي يقول لي انه يا جماعةاللي هو
+
+167
+00:12:55,010 --> 00:12:59,550
+هيطلع إن الـ magnitude لـ U x V هو عبارة عن مساحة
+
+168
+00:12:59,550 --> 00:13:05,470
+متوازل أضلاع اللي بيكوننه المتجهين U و V اللي
+
+169
+00:13:05,470 --> 00:13:10,490
+عملنا بينهما cross و product طيب نيجي الآن بدنا
+
+170
+00:13:10,490 --> 00:13:14,770
+نحاول نوجد U x V لو أعطي ال calculating the cross
+
+171
+00:13:14,770 --> 00:13:18,970
+product as determinant بواسطة إيش ال determinant و
+
+172
+00:13:18,970 --> 00:13:23,250
+خلينا نشوف كيف جبناهالو كانت ال U عبارة عن U1 في
+
+173
+00:13:23,250 --> 00:13:26,930
+اتجاه ال I و U2 في اتجاه ال J و U3 في اتجاه ال K
+
+174
+00:13:26,930 --> 00:13:31,270
+هذا اللي هو من المتجهة الأول U كتبناها على صورة ال
+
+175
+00:13:31,270 --> 00:13:34,670
+I و ال J أو بدلالة ال I و ال J و ال K و ال V برضه
+
+176
+00:13:34,670 --> 00:13:39,670
+كتبناها بدلالة ال I و ال J و ال K هاي ال U و ال V
+
+177
+00:13:39,670 --> 00:13:43,950
+لو جينا عملنا cross product بينهم ال cross product
+
+178
+00:13:43,950 --> 00:13:47,110
+دي يا جماعة من الأن و طالع هبرنكم إيها بس من الأن
+
+179
+00:13:47,110 --> 00:13:52,160
+و طالع اتطلعوا على طول تيجي Uبتحط اول اشي بتحط ال
+
+180
+00:13:52,160 --> 00:13:56,620
+I و ال J و ال K تعمل ترمين تلاتة في تلاتة بتحط I J
+
+181
+00:13:56,620 --> 00:14:04,260
+K ال U معاملات ال I U واحد ال J U اتنين ال K U
+
+182
+00:14:04,260 --> 00:14:07,980
+تلاتة كل واحد تحت ايش تحت اللي هو اللي معامل إله
+
+183
+00:14:07,980 --> 00:14:12,980
+ال V التاني هذا الأول هذا التاني V واحد V اتنين V
+
+184
+00:14:12,980 --> 00:14:16,990
+تلاتة V واحد V اتنين V تلاتةطبعا هنا بالتضح بس على
+
+185
+00:14:16,990 --> 00:14:21,630
+الهامش أن V x U يكون هذا سطر مكان هذا سطر واحنا
+
+186
+00:14:21,630 --> 00:14:25,710
+بنعرف أنه تغيير سطرين سطر مكان سطر في ال
+
+187
+00:14:25,710 --> 00:14:28,350
+determinant بيخلّي ال determinant الأول بساوي سالب
+
+188
+00:14:28,350 --> 00:14:32,190
+ال determinant الثاني وهذا يفسر أن U x V بساوي
+
+189
+00:14:32,190 --> 00:14:36,430
+ناقص V x Uماشى يا جماعة خلّينى نشوف من اين جاء هذا
+
+190
+00:14:36,430 --> 00:14:41,830
+من اين جاء هاي الان U cross V صفه هذا جنب هذا هى
+
+191
+00:14:41,830 --> 00:14:49,570
+U1I U2I U3K ضرب across V1I زي V2J زي V3K الآن زى
+
+192
+00:14:49,570 --> 00:14:52,390
+ما اتعلمنا قبل و شوية في الخواص انه يوزع الجامع
+
+193
+00:14:52,390 --> 00:14:59,110
+اتضرب على مين؟ على الجامع فبصير هذه مضروبة في V1I
+
+194
+00:14:59,110 --> 00:15:05,590
+cross I I cross I وU1V1 من اي اعداد Scalarsالان U1
+
+195
+00:15:05,590 --> 00:15:15,550
+V2 I X J اه الان خلصنا لو U1 V3 I X K I X K هاي
+
+196
+00:15:15,550 --> 00:15:19,730
+خلصنا اللي هو من ضرب هذه في هذه كلها نجي نضرب تاني
+
+197
+00:15:19,730 --> 00:15:25,230
+بالمثل U2 V1 Scalars دول طبعا يا جماعة مضربات في G
+
+198
+00:15:25,230 --> 00:15:34,410
+X I الـ G X I الان U2 في V2مضروب في الـ J cross J
+
+199
+00:15:34,410 --> 00:15:42,590
+U2 في V3 J cross K و نفس الاشي بنكمل هذه المحادولة
+
+200
+00:15:42,590 --> 00:15:49,350
+كلها و بيطلع لنا المقدار هذا طيب الآنلاحظوا U1 V1
+
+201
+00:15:49,350 --> 00:15:53,550
+I X I يظهر كـ 0 معايا يا جماعة اقد أعملكم هذه و
+
+202
+00:15:53,550 --> 00:16:00,810
+الباقية زيها هذه I X I هي 0 U1 V2 هي I X J I X J
+
+203
+00:16:00,810 --> 00:16:07,150
+هي K فبصير عند U1 V2 في اتجاه الـ K U1 V2 في اتجاه
+
+204
+00:16:07,150 --> 00:16:18,620
+الـ K الان U1 V3 I X Kijk مظبوط i كفو k بيطلع من
+
+205
+00:16:18,620 --> 00:16:25,200
+عبارة عن ال j او اللي هو u1v3 بيطلع عند u1v3 اللي
+
+206
+00:16:25,200 --> 00:16:30,600
+هو ناقص ال j أسف على ال ijk طيب الآن خلصنا من
+
+207
+00:16:30,600 --> 00:16:35,820
+هدولة بنيجي لهدول ال j cross j برضه سفر و ال k
+
+208
+00:16:35,820 --> 00:16:42,670
+cross k سفر بتظل هدي و هدي هدي و هديU2 V1 J X I
+
+209
+00:16:42,670 --> 00:16:51,610
+ناقص K هي U2 V1و هكذا الباقيات فبتطلعلي هذا كله
+
+210
+00:16:51,610 --> 00:16:55,210
+هذا بتطلعلي هدولة بالظبط اللي نجربه لحالكم فردون
+
+211
+00:16:55,210 --> 00:16:58,950
+هتلاقوهم بتطلعين زي ما حكيت قبل بشوية ماشي الحال
+
+212
+00:16:58,950 --> 00:17:05,210
+هذا الآن اللي هو صار مين هو هو في الواقع عبارة عن
+
+213
+00:17:05,210 --> 00:17:08,830
+قيمة ال determinant ليش لأن احنا كيف بنفك ال
+
+214
+00:17:08,830 --> 00:17:13,430
+determinant نستخدم السطر الأول هي I و بنسكر هذا
+
+215
+00:17:13,430 --> 00:17:22,220
+بيصير U2 V3نقص V2 U3 I U2 V3 نقص U3 V2 هى هذا
+
+216
+00:17:22,220 --> 00:17:27,420
+شيلوا ال J هى هذا ال J اللى هو ناقص طبعا الان يصير
+
+217
+00:17:27,420 --> 00:17:35,110
+U1 V3 هى U1 V3 نقص V1 U الهيمين تلاتةالان نجي لاخر
+
+218
+00:17:35,110 --> 00:17:37,630
+واحدة اللي هي ال K شط بهذا و شط بهذا فاهمين انتوا
+
+219
+00:17:37,630 --> 00:17:42,830
+كيف تعرفوا كيف نفك المحدد استخدمت السطر الأول في
+
+220
+00:17:42,830 --> 00:17:47,950
+إيجاد المحدد الان ضايق الأخر واحد اللي هي ال K شط
+
+221
+00:17:47,950 --> 00:17:57,490
+بهذا و شط بهذا يصير U1 V2 ناقص V1 U2 U1 V2 ناقص V1
+
+222
+00:17:57,490 --> 00:18:06,010
+U2 او U2 V1 اذا هذا في الواقعهو عبارة عن مفكوك من
+
+223
+00:18:06,010 --> 00:18:09,770
+ال determinant يعني ال U cross V بتساوي ال
+
+224
+00:18:09,770 --> 00:18:14,310
+determinant لها طيب الان الناس كيف كانت ال
+
+225
+00:18:14,310 --> 00:18:16,790
+determinant أنا هعيده كمان شوية في الأمثلة اللي
+
+226
+00:18:16,790 --> 00:18:21,950
+عندي لان find U cross V and V cross Uفيو بساوة
+
+227
+00:18:21,950 --> 00:18:24,990
+اتنين I زاية J زاية K و V بساوة نقص أربع I زاية
+
+228
+00:18:24,990 --> 00:18:29,070
+تلاتة J زاية K لأن هذا مثال على إيجاد الـ U cross
+
+229
+00:18:29,070 --> 00:18:33,790
+V أو الـ V cross U باستخدام الـ determinant أسهل
+
+230
+00:18:33,790 --> 00:18:39,170
+لنا كتير هاي عند U cross V باجي برص في الأول الـ I
+
+231
+00:18:39,170 --> 00:18:43,170
+و الـ J و الـ K السطر الأول السطر التالي اللي هو
+
+232
+00:18:43,170 --> 00:18:46,650
+معاملاتي اللي هي تبعات U معامل I J و K بنفس
+
+233
+00:18:46,650 --> 00:18:52,050
+الترتيب2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 2 1
+
+234
+00:18:52,050 --> 00:18:54,330
+2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
+
+235
+00:18:54,330 --> 00:18:56,430
+1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
+
+236
+00:18:56,430 --> 00:18:58,970
+2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
+
+237
+00:18:58,970 --> 00:18:59,030
+1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
+
+238
+00:18:59,030 --> 00:18:59,150
+2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
+
+239
+00:18:59,150 --> 00:18:59,170
+1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
+
+240
+00:18:59,170 --> 00:18:59,850
+1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
+
+241
+00:18:59,850 --> 00:19:04,380
+2 1ماشي استخدم السطر الأول أسللنا لإنهي السطر
+
+242
+00:19:04,380 --> 00:19:11,460
+الأول كيف بده اوجده لإن I مضروبة في مين بشطب السطر
+
+243
+00:19:11,460 --> 00:19:15,300
+و العمود اللي فيه I ال determinant اللي بيظل اللي
+
+244
+00:19:15,300 --> 00:19:19,780
+هو determinant two by two كيف بنوجده واحد في واحد
+
+245
+00:19:19,780 --> 00:19:23,260
+ناقص تلاتة هي ال determinant حطينا بعدين لإنهي
+
+246
+00:19:23,260 --> 00:19:29,670
+وجده لإن ال J بشطب اللي في ال Jو في اللي هو كسطر و
+
+247
+00:19:29,670 --> 00:19:34,330
+كعمود يظل اتنين و واحد و واحد و ناقص اربعة هايه
+
+248
+00:19:34,330 --> 00:19:37,990
+طبعا الان موجب سالب موجب زي ما انتوا عارفين اللي
+
+249
+00:19:37,990 --> 00:19:42,290
+هو القانون اللي هو إيجاد ال determinant ناقص اللي
+
+250
+00:19:42,290 --> 00:19:44,850
+هو ال determinant هذا في جهةه يظل مين الموجب اللي
+
+251
+00:19:44,850 --> 00:19:49,510
+هو ال K هتشطب هذا العمود و هذا السطر اللي فيه ال K
+
+252
+00:19:49,510 --> 00:19:52,550
+يظل ال determinant هذا ال determinant ال two by
+
+253
+00:19:52,550 --> 00:19:58,830
+two طبعا كلكم بيعرفوا وجوده1-3 يعني ناقص اتنين في
+
+254
+00:19:58,830 --> 00:20:04,290
+اتجاه مين الاي اتنين ناقص ناقص بيصير زائد اربعة
+
+255
+00:20:04,290 --> 00:20:08,610
+اللي هي بيصير ستة وفي عندي سالب بيصير سالب ستة جين
+
+256
+00:20:08,610 --> 00:20:11,930
+اخر واحد اتنين فى تلاتة فى ستة ناقص ناقص اللي هي
+
+257
+00:20:11,930 --> 00:20:15,270
+اربعة يعني زائد اربعة ستة واربع عشرة يعني بيصير
+
+258
+00:20:15,270 --> 00:20:19,860
+عندك اللي هو طبعا اتنين فى تلاتة فى ستةناقص أربعة
+
+259
+00:20:19,860 --> 00:20:23,500
+بيطلع اللي هي ناقص بيصير زائد إذا بيصير زائد عشرة
+
+260
+00:20:23,500 --> 00:20:31,180
+أنا زائد عشرة مش ناقص عشرة إلا إذا كانوا مخربطين
+
+261
+00:20:31,180 --> 00:20:37,260
+إحنا ناقص أربعة لأ مظبوط ناقص أربعة دلوقتي هيك و
+
+262
+00:20:37,260 --> 00:20:43,920
+هيك لأ صح2×3 هو 6 و 6 زاد إذا زائد من 10 كيه حسب
+
+263
+00:20:43,920 --> 00:20:49,560
+هذا الاسم طيب لآن بس قبل ما نيجي لتاني مش مكمل هو
+
+264
+00:20:49,560 --> 00:20:53,900
+هذا اللي هو لأ هيه مظبوطا آسف أنا بتطلع أتعلي
+
+265
+00:20:53,900 --> 00:20:58,240
+تاعتيه هذا زائد عشرة هيش كيه 2×3 ب 6 زائد عشرة هي
+
+266
+00:20:58,240 --> 00:21:02,020
+عشرة كيه زي ما احنا ممتفنين لأن هذا U cross V لأن
+
+267
+00:21:02,020 --> 00:21:07,310
+V cross U يا شباب و يا بنات V cross U بساويناقص
+
+268
+00:21:07,310 --> 00:21:10,130
+الـ u cross بيفيش ده اعمل determinant كمان مرة ما
+
+269
+00:21:10,130 --> 00:21:14,490
+خلاص انا صارت باعرف القوانين ان ده مساوي اتنين I
+
+270
+00:21:14,490 --> 00:21:18,590
+ضربته في سالب زائد ستة J وده ضربته في لاقص اللي
+
+271
+00:21:18,590 --> 00:21:22,830
+خربطنا فيها اللي هي ناقص عشرة K طيب يا جماعة لان
+
+272
+00:21:22,830 --> 00:21:27,050
+نيجي اللي هو ناخد مثال أخر find a perpendicular to
+
+273
+00:21:27,050 --> 00:21:33,950
+the plane of B وQ and اللي هي مين R عند .. عند يا
+
+274
+00:21:33,950 --> 00:21:39,480
+جماعةاللي هو تلت نقاط B وQ وR تلت نقاط واضح انه
+
+275
+00:21:39,480 --> 00:21:41,940
+ليست على استقامة واحدة اذا ان بعمل ال meaning
+
+276
+00:21:41,940 --> 00:21:48,360
+plane بقوللي اوجد اللي هو متجه عمودي على هذا ال
+
+277
+00:21:48,360 --> 00:21:52,800
+plane على طول بتبادر لذهني باجي بعمل ال vector
+
+278
+00:21:52,800 --> 00:21:56,380
+الأول PR من النقاط اللي عندنا بنعرف كلنا و بنعرف
+
+279
+00:21:56,380 --> 00:22:01,270
+جميع ال things نعمل و BQ ال vector التانيوبناخد ما
+
+280
+00:22:01,270 --> 00:22:05,030
+هو الـ cross product بينهم بيطلع vector إيه شماله؟
+
+281
+00:22:05,030 --> 00:22:12,890
+عمودي إذا الان ال vector bq هي bq هي bq across br
+
+282
+00:22:12,890 --> 00:22:16,730
+اللي هو is perpendicular to the plane because it
+
+283
+00:22:16,730 --> 00:22:20,790
+is perpendicular to both vectors و هي تعرف الدرب
+
+284
+00:22:20,790 --> 00:22:24,540
+اللي هو الدرب الاتجاهيطبعا واحد قال بيداخد PR
+
+285
+00:22:24,540 --> 00:22:27,780
+cross PQ بقول لأ بنفعل لإن الاتنين واحد عمودي إلى
+
+286
+00:22:27,780 --> 00:22:31,200
+أعلى والتاني عمودي إلى وين إلى أسفل بالنسبة لبلين
+
+287
+00:22:31,200 --> 00:22:36,500
+اللي هم حملاته اللي هو P وQ وR نيجي نحدد الآن PQ
+
+288
+00:22:36,500 --> 00:22:41,040
+هو بده يحسب PQ لأن PQ اللي هو IBQ بتعرفوا كيف
+
+289
+00:22:41,040 --> 00:22:48,470
+تحسبوا اتنين ناقص واحد اللي هي في اتجاه الـI-1-1-1
+
+290
+00:22:48,470 --> 00:23:01,190
+-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1
+
+291
+00:23:01,190 --> 00:23:02,370
+-1-1
+
+292
+00:23:07,070 --> 00:23:10,610
+1 ناقص ناقص واحد يعني واحد زائد واحد في اتجاه ال J
+
+293
+00:23:10,610 --> 00:23:14,570
+اتنين ناقص صفر في اتجاه ال K هيطلع عندنا ناقص
+
+294
+00:23:14,570 --> 00:23:19,990
+اتنين I زائد اتنين J زائد مين اتنين K هاي اوجدت ال
+
+295
+00:23:19,990 --> 00:23:26,500
+P Q P مين Rقلنا الان بيكيو كروس بي ار كيف بنواجده
+
+296
+00:23:26,500 --> 00:23:30,560
+بواسطة ال determinant هي عندك ال I و ال J و ال K و
+
+297
+00:23:30,560 --> 00:23:33,660
+باجي باخد ال بيكيو اللي هو واحد اتنين ناقص واحد
+
+298
+00:23:33,660 --> 00:23:37,160
+واحد اتنين ناقص واحد اللي هو بي ار اللي هو ناقص
+
+299
+00:23:37,160 --> 00:23:41,080
+اتنين اتنين اتنين ناقص اتنين اتنين اتنين و ببدأ زي
+
+300
+00:23:41,080 --> 00:23:45,660
+ما عملنا قبل بشوية فك هذا السطر مع هذا العموديبقى
+
+301
+00:23:45,660 --> 00:23:50,500
+الهيو في تجاه ال I ناقص هذا السطر هذا العمود مع
+
+302
+00:23:50,500 --> 00:23:53,400
+هذا السطر بيبقى واحد و ناقص واحد و اتنين وهيو
+
+303
+00:23:53,400 --> 00:23:59,300
+بيبقى في تجاه ال J السطر الأخير ال K الأخيرة بشطب
+
+304
+00:23:59,300 --> 00:24:05,050
+العمود و بشطب السطر و بيبقى عندي هذا اربعةزائد
+
+305
+00:24:05,050 --> 00:24:09,650
+اتنين ستة في اتجاه ال I او اتنين ناقص اتنين بيطلع
+
+306
+00:24:09,650 --> 00:24:14,370
+سفر اللي هو بيضلش عنه في اتجاه ال J إشي زائد اتنين
+
+307
+00:24:14,370 --> 00:24:19,090
+زائد اللي هي اربعة بيطلع ستة T إذا هذا عبارة عن
+
+308
+00:24:19,090 --> 00:24:25,260
+مين يا جماعة؟هذا عبارة عن المتجه العمودي على الـ
+
+309
+00:24:25,260 --> 00:24:30,860
+plane اللي بيحمل النفاط الثلاثة طيب الآن بقول تكمل
+
+310
+00:24:30,860 --> 00:24:34,640
+السؤال فعلا find the area of the triangle with
+
+311
+00:24:34,640 --> 00:24:40,860
+vertices و find the area of the triangle with
+
+312
+00:24:40,860 --> 00:24:48,110
+vertices B OQR اللي جاب هنا نفس النجومةالان الـ
+
+313
+00:24:48,110 --> 00:24:59,430
+Area قلنا لـ Triangle هو هاي الـ Triangle هدا هاي
+
+314
+00:24:59,430 --> 00:25:03,390
+ال vector الأول و هاي ال vector التاني الان ال
+
+315
+00:25:03,390 --> 00:25:06,290
+cross product و أخدناله ال magnitude عمله ال Area
+
+316
+00:25:06,290 --> 00:25:10,970
+لمين؟ لمتوازر الأضلاع هو كله اللي هو هذا المثلث
+
+317
+00:25:10,970 --> 00:25:16,290
+يكون نصهفإذا الان هو بده ال area لمين لكتريانجل
+
+318
+00:25:16,290 --> 00:25:20,590
+مدام ال area لكتريانجل إذا إيش بوجد أول إشي the
+
+319
+00:25:20,590 --> 00:25:24,630
+area of the parallelogram determined by BQR يعني
+
+320
+00:25:24,630 --> 00:25:30,850
+بده أوجد مساحة متوازل أطلاعاللي حملاته اللي هو الـ
+
+321
+00:25:30,850 --> 00:25:34,750
+P و الـ Q و الـ R يعني هي متوازي الأطلاع بس مده
+
+322
+00:25:34,750 --> 00:25:39,210
+هيك على سطحان موازي و هيك موازي بعمل متوازي أطلاع
+
+323
+00:25:39,210 --> 00:25:43,030
+هذا نصه بدوجة مساحة متوازي الأطلاع مساحة متوازي
+
+324
+00:25:43,030 --> 00:25:47,550
+الأطلاع الـ magnitude الـ PR across BQ لما اوددته
+
+325
+00:25:47,550 --> 00:25:55,290
+اللي هو مينأو جت بي او بي او بي او بي او بي او بي
+
+326
+00:25:55,290 --> 00:26:00,350
+او بي او بي او بي
+
+327
+00:26:00,350 --> 00:26:02,130
+او بي او بي او بي او بي او بي او بي او بي او بي او
+
+328
+00:26:02,130 --> 00:26:02,270
+بي او بي او بي او بي او بي او بي او بي او بي او بي
+
+329
+00:26:02,270 --> 00:26:03,190
+او بي او بي او بي او بي او بي او بي او بي او بي او
+
+330
+00:26:03,190 --> 00:26:04,670
+او بي او بي او بي او بي او بي او بي او بي او بي او
+
+331
+00:26:04,670 --> 00:26:09,810
+بي او بي او بي او بي او
+
+332
+00:26:14,370 --> 00:26:18,590
+is half of this نص متوازن أضلالية بتطلع تلاتة جذر
+
+333
+00:26:18,590 --> 00:26:23,990
+الاتنين ماشي الآن بده يحكي اللي هو برضه على المثال
+
+334
+00:26:23,990 --> 00:26:27,740
+و بده ياخد فكرة أخرىالمثال الرابع يقول find a unit
+
+335
+00:26:27,740 --> 00:26:34,340
+vector perpendicular to the plane of P,Q,R يعني
+
+336
+00:26:34,340 --> 00:26:40,660
+يقول أوجد متجة لوحدة العمودي على المستوى المكوّن
+
+337
+00:26:40,660 --> 00:26:49,180
+من أو المكونات له P,Q,R الان عشان أوجد ال unit
+
+338
+00:26:49,180 --> 00:26:54,300
+vectorالعمودي انا اوجدت العمودي مين العمودي اللي
+
+339
+00:26:54,300 --> 00:26:59,340
+هو BQ cross PR اوجدته ال BQ cross PR اللي هو جداش
+
+340
+00:26:59,340 --> 00:27:05,160
+كان 6I زي 6K هذا اللي هو المتجه العمودي اللي انا
+
+341
+00:27:05,160 --> 00:27:09,760
+بدي متجه الوحدة العموديعلى مجموعته إذا المتجهة
+
+342
+00:27:09,760 --> 00:27:15,240
+الوحدة العمودي هو قيمة المتجهة العمودي على اللي هو
+
+343
+00:27:15,240 --> 00:27:20,320
+ال magnitude أو طول المتجهة العمودي اللي هو طول
+
+344
+00:27:20,320 --> 00:27:23,720
+اللي هو المتجهة العمودي الستة عشان الستة كام طوله
+
+345
+00:27:23,720 --> 00:27:26,620
+او جدناه ستة جدر اتنين هيو من هنا من هنا من هنا
+
+346
+00:27:27,060 --> 00:27:29,760
+بتجسمه على ستة جدر التنين بيطلع عندى واحد على
+
+347
+00:27:29,760 --> 00:27:32,800
+اتنين جدر في ال I او واحد على جدر التنين في تجاه
+
+348
+00:27:32,800 --> 00:27:35,780
+ال I واحد على جدر التنين في تجاه ال K هذا عبارة عن
+
+349
+00:27:35,780 --> 00:27:39,740
+تجاه الوحدة العمودي على ال plane اللي هو اللي
+
+350
+00:27:39,740 --> 00:27:42,840
+بيحمل P وQ وR زي اللي بحكينا
+
+351
+00:27:45,700 --> 00:27:52,800
+الان فى برضه اللى هو فى عنا فوائد أخرى لـ Scalar
+
+352
+00:27:52,800 --> 00:27:57,380
+اللى هو Product و لـ Cross Product سويا مع بعض
+
+353
+00:27:57,380 --> 00:28:01,980
+حاجة أسمها Triple Scalar أو Box Product ده Product
+
+354
+00:28:01,980 --> 00:28:08,810
+U X V ضد Wis called the triple scalar product of U
+
+355
+00:28:08,810 --> 00:28:13,770
+وV وW طبعا هو ليش جال triple scalar product لأن
+
+356
+00:28:13,770 --> 00:28:17,850
+أول أشي بدرب U cross V بتطلع لمتجة بعدين بدربها ضد
+
+357
+00:28:17,850 --> 00:28:22,610
+W هيتحول كل هذا إلى رقم إلى إياش scalar عشان إيه
+
+358
+00:28:22,610 --> 00:28:28,310
+اتجلك triple scalar product of U وV وW الآن هذا
+
+359
+00:28:28,310 --> 00:28:33,910
+بيمثل شيء تنشوف إيش بيمثل لو دي أخدنا اللي هو ال U
+
+360
+00:28:33,910 --> 00:28:40,130
+هذا المتجةو هذا ال V متجه ماشي الحال و هذا ال W
+
+361
+00:28:40,130 --> 00:28:44,650
+متجه ثالث التلاتة هيعملنلي كام ملي اللي هو المستوى
+
+362
+00:28:44,650 --> 00:28:48,510
+هذه كام ملي المستوى هذه و المستويات كلها هيعملنلي
+
+363
+00:28:48,510 --> 00:28:54,590
+box هيعملنلي ايش ال box هذا او متوازي مستطيلات او
+
+364
+00:28:54,590 --> 00:29:01,550
+متوازي او منشور القواعد اللي هي اوجهه و قواعده
+
+365
+00:29:01,550 --> 00:29:06,600
+عبارة عن متوازيات أضلع الآنالان لو جينا لحجم هذا
+
+366
+00:29:06,600 --> 00:29:10,360
+المنشور الحجم هو عبارة عن مساحة القاعدة في ايش في
+
+367
+00:29:10,360 --> 00:29:18,580
+الارتفاع الان الارتفاع الارتفاع هذا اللي هو المتجه
+
+368
+00:29:18,580 --> 00:29:25,260
+هذا هيك اللي هو مين بيحسبه U cross V مظبوط اللي هي
+
+369
+00:29:25,260 --> 00:29:29,520
+U cross V هو المتجه العمودي على U cross V بس انا
+
+370
+00:29:29,520 --> 00:29:36,730
+ايش بده اوجد يا جماعة بده اوجد اللي هوالمساح اللى
+
+371
+00:29:36,730 --> 00:29:42,290
+هو طول اللى هو الارتفاع يعني هى عندي سطح العلوي هو
+
+372
+00:29:42,290 --> 00:29:49,330
+فوق هذا العمود اللى بنزل منها لهان اللى هو بساوى
+
+373
+00:29:49,330 --> 00:29:53,870
+هذا منها لهان هو عبارة عن ارتفاع متوازن أطلع فبكون
+
+374
+00:29:53,870 --> 00:29:59,660
+متوازن أطلع مساحة القاعدةالي هذه في الارتفاع مساحة
+
+375
+00:29:59,660 --> 00:30:03,280
+القاعدة ال area of base في high ال area of base
+
+376
+00:30:03,280 --> 00:30:06,200
+سهلة ما احنا عرفناها اللي عبارة عن ال magnitude ال
+
+377
+00:30:06,200 --> 00:30:11,540
+U cross V ال U cross V طيب هذا طول و جداش يا جماعة
+
+378
+00:30:11,540 --> 00:30:16,480
+لان صار عندي U cross V متجهة مظبوط هيو عمودي على
+
+379
+00:30:16,480 --> 00:30:22,480
+هذا لأن الزاوية هذه هتكون theta بين W و بين
+
+380
+00:30:22,480 --> 00:30:27,780
+المتجهة الجديدة اللي هو U cross Vالان هذا الارتفاع
+
+381
+00:30:27,780 --> 00:30:33,120
+هو عبارة عن لو جينا خدنا cosine theta بيصير اللي
+
+382
+00:30:33,120 --> 00:30:38,280
+هو ال magnitude لهذا الأسف اللي هو طول هذا ع ال
+
+383
+00:30:38,280 --> 00:30:44,460
+magnitude لهذا هو عبارة عن ال cosine إذا ال
+
+384
+00:30:44,460 --> 00:30:49,670
+magnitude لهذا أو طول هذا ال high هذاهي ساوي
+
+385
+00:30:49,670 --> 00:30:55,730
+magnitude of w في cos θ لأن cos θ ليس ساوي الـ
+
+386
+00:30:55,730 --> 00:30:59,730
+high على magnitude of w من المثلثة اللي هنا يعني
+
+387
+00:30:59,730 --> 00:31:05,130
+أنا جبت هذا ايش طوله magnitude of w في absolute
+
+388
+00:31:05,130 --> 00:31:07,990
+value لcos θ لأن طوله عشان هيك اتبين absolute
+
+389
+00:31:07,990 --> 00:31:14,230
+value صار عندى الآن ال area هيها of the base وهذا
+
+390
+00:31:14,230 --> 00:31:20,520
+بمثل ال highهدف الواقع إيه شماله هو عبارة عن اللي
+
+391
+00:31:20,520 --> 00:31:25,060
+هو المتجه الأول magnitude له المتجه التاني
+
+392
+00:31:25,060 --> 00:31:27,360
+magnitude له ال focusين الزاوية اللي بينهم يعني
+
+393
+00:31:27,360 --> 00:31:31,380
+عبارة عن الـ dot productبين هذا المتجه و بين هذا
+
+394
+00:31:31,380 --> 00:31:33,880
+الـ data product بين هذا و هذا عبارة عن ال
+
+395
+00:31:33,880 --> 00:31:36,360
+magnitude ده ده في ال magnitude ده ده في cosine
+
+396
+00:31:36,360 --> 00:31:39,720
+الزاوية اللي بينهم اللي أنا أخدتله absolute value
+
+397
+00:31:39,720 --> 00:31:43,520
+لإنه هذا مخود ل absolute value و ال volume بطلع عن
+
+398
+00:31:43,520 --> 00:31:47,160
+عبارة عن magnitude ل ال U cross V ضد W absolute
+
+399
+00:31:47,160 --> 00:31:52,260
+value هذا هيطلع لي اللي هو ال volume of the
+
+400
+00:31:52,260 --> 00:31:58,770
+parallel voidالان الـ Parallel pipe ده .. ده هو
+
+401
+00:31:58,770 --> 00:32:02,990
+عبارة عن متوازي
+
+402
+00:32:02,990 --> 00:32:11,030
+سطوح اللي هي عبارة عن أوجه متوازي أضلاع تكونت من
+
+403
+00:32:11,030 --> 00:32:16,110
+ال vectors U و V و W مش هتطور الحلول طيب إذا هذا
+
+404
+00:32:16,110 --> 00:32:23,220
+بحسب لل volume الآن احنا الترتيب هذاأصلا يعني واحد
+
+405
+00:32:23,220 --> 00:32:26,240
+قال لي أنا ما بدي أخد القاعدة هذه بدي أخد هذه
+
+406
+00:32:26,240 --> 00:32:31,440
+القاعدة القاعدة هي و أحسب اللي هو w cross v dot
+
+407
+00:32:31,440 --> 00:32:36,940
+main u بقوله بانفهم قال لي بدي أخد القاعدة أنا هذا
+
+408
+00:32:36,940 --> 00:32:42,500
+u و w وعمل بنفس الأسلوب u cross v اللي هو dot main
+
+409
+00:32:42,500 --> 00:32:47,620
+w فبصير عند u cross v dot w أو v cross w dot u أو
+
+410
+00:32:47,620 --> 00:32:54,520
+w cross u dot vكلهم متساويات والـ absolute value
+
+411
+00:32:54,520 --> 00:33:04,080
+لهم كله إشمال و All Volume الان لدي ملخص الحديث U
+
+412
+00:33:04,080 --> 00:33:12,140
+cross V ضد W هيساوي U ضد cross V cross W هيساوي U
+
+413
+00:33:12,140 --> 00:33:18,190
+cross V ضد W و V cross W ضد U و W cross U ضد Vهذا
+
+414
+00:33:18,190 --> 00:33:21,870
+هو عبارة عن ايش يا جماعة ال magnitude له هو عبارة
+
+415
+00:33:21,870 --> 00:33:26,430
+عن اللي هو ال volume ل اللي هو متوازي السطوة زي ما
+
+416
+00:33:26,430 --> 00:33:30,610
+قلنا واحد بيعتبر هذه اللي هي القاعدة و الارتفاع
+
+417
+00:33:30,610 --> 00:33:34,070
+اللي هو بيطلع عمودي يعني بياخدوا cross product بين
+
+418
+00:33:34,070 --> 00:33:37,350
+هدولة و بيشتغل مع هذا product أو بياخدوا cross
+
+419
+00:33:37,350 --> 00:33:44,050
+product مع حد و يشتغل مع هذا product أو V مع W أو
+
+420
+00:33:44,050 --> 00:33:50,170
+U مع V أو U مع Wو بكون عندى اللى هو by treating
+
+421
+00:33:50,170 --> 00:33:54,850
+the blends of v and w and w and u زى ما قولنا as
+
+422
+00:33:54,850 --> 00:33:59,450
+the base of the blends by u و v and w we see that
+
+423
+00:33:59,450 --> 00:34:04,730
+u cross v dot w و v cross w dot u كل هى اللى هى
+
+424
+00:34:04,730 --> 00:34:10,410
+اللى هى هيمثلنى اللى هى ال magnitude اللى هى عبارة
+
+425
+00:34:10,410 --> 00:34:15,670
+عن اللى هو الحجملأن الـ dot product is commutative
+
+426
+00:34:15,670 --> 00:34:25,290
+و الـ U cross V dot W هو نفسه اللي هو هيتلع U dot
+
+427
+00:34:25,290 --> 00:34:32,130
+V cross W عارفين ليش؟ لأن عندي U cross V dot W
+
+428
+00:34:33,580 --> 00:34:37,880
+بساوية v cross w dot u وهذا كميوتيف يجعلني أستطيع
+
+429
+00:34:37,880 --> 00:34:44,160
+أن أحضر الـ yuan بيصير u cross v dot w إذا أنت من
+
+430
+00:34:44,160 --> 00:34:46,940
+ال vector اللي تمسكه تضربه dot product قبل أو
+
+431
+00:34:46,940 --> 00:34:49,980
+تضربه cross product قبل وفي الآخر كذا يعني من
+
+432
+00:34:49,980 --> 00:34:53,400
+تجيبه dot product أو من تجيبه cross product في من
+
+433
+00:34:53,400 --> 00:34:56,860
+فين تبدل ما تبدل مش هتفرج لأنه في الآخر هتاخد من
+
+434
+00:34:56,860 --> 00:35:03,090
+المجموعة اللي بيطلع اللي هوزي ما قلنا اللي
+
+435
+00:35:03,090 --> 00:35:08,930
+هي قيمة ال .. قيمة ال volume طيب calculating the
+
+436
+00:35:08,930 --> 00:35:13,490
+triple scalar product as determinant اه .. الان u
+
+437
+00:35:13,490 --> 00:35:19,270
+cross v ضد w لو
+
+438
+00:35:19,270 --> 00:35:25,890
+حسبنا بنفس الأسلوب السابق هيطلع عندى الان u cross
+
+439
+00:35:25,890 --> 00:35:34,120
+v ضد wالـ determinant و الـ W التي تتجه الـ I و
+
+440
+00:35:34,120 --> 00:35:38,560
+الـ J و الـ K و عملها الـ cross product تحسب ستجد
+
+441
+00:35:38,560 --> 00:35:41,500
+قيمة الـ determinant دون علاقة بالتفاصيل التي
+
+442
+00:35:41,500 --> 00:35:47,580
+سنفصل عنها هنا تحسب U cross V و تحسب ضد W تحسبها
+
+443
+00:35:47,580 --> 00:35:50,940
+بالطرق التي نعرفهاو بعدين احسب هذا ال determinant
+
+444
+00:35:50,940 --> 00:35:55,480
+بالطرق اللى بتعرفها و ساوى الطرفين بطلع الطرفين
+
+445
+00:35:55,480 --> 00:35:59,320
+إيه شمالين متساوية بس عشان تتمأن قلبك و إذا بدك
+
+446
+00:35:59,320 --> 00:36:04,160
+تسلم من القانون هو هيك انت حر أما إثباته مش صعب U
+
+447
+00:36:04,160 --> 00:36:07,460
+cross V زي ما اتعلمنا و بعدين اللى بيطلع بتضرب ضد
+
+448
+00:36:07,460 --> 00:36:10,300
+W زي ما اتعلمنا هتلاقي بيطلع على ال determinant
+
+449
+00:36:10,300 --> 00:36:13,760
+هذا بيطلع قيمة ال determinant لو أنت حسبت ال
+
+450
+00:36:13,760 --> 00:36:17,220
+determinantالان بدنا ايش نقعد نرجع للطرق البدائية
+
+451
+00:36:17,220 --> 00:36:22,340
+من الأول اذا الان U cross V ضد W على طول هيمثل ايش
+
+452
+00:36:22,340 --> 00:36:24,360
+قيمة الـ determinant التالية يعني هذا بتعتبره
+
+453
+00:36:24,360 --> 00:36:31,460
+قانون U1 U2 U3 V1 V2 V3 W1 W2 W3 ناخد مثال عليه
+
+454
+00:36:31,460 --> 00:36:35,280
+تتضح الأمور find the volume of the box determined
+
+455
+00:36:35,280 --> 00:36:40,340
+by U بسوء I زي اتنين G نقص K V بسوء اتنين نقص
+
+456
+00:36:40,340 --> 00:36:45,130
+اتنين I زي تلتة Kو W بيساوي 7G نقص 4K الى أن
+
+457
+00:36:45,130 --> 00:36:48,990
+volume of the box is volume للبوكس بساوي اللي هو
+
+458
+00:36:48,990 --> 00:36:51,170
+اقول لك ال magnitude لهذا المقدار ال magnitude
+
+459
+00:36:51,170 --> 00:36:54,290
+لهذا هذا اللي هو ال determinant ال magnitude اللي
+
+460
+00:36:54,290 --> 00:36:58,370
+هو ال magnitude هي magnitude برا و اللي جوا ايه اش
+
+461
+00:36:58,370 --> 00:37:02,370
+لل determinant من ال determinant هي واحد اتنين نقص
+
+462
+00:37:02,370 --> 00:37:05,810
+واحد واحد اتنين نقص واحداللي هي نقص اتنين تلاتة
+
+463
+00:37:05,810 --> 00:37:09,470
+نقص اتنين تلاتة اللي هي لل K و سفر لمين لل J لأنه
+
+464
+00:37:09,470 --> 00:37:16,070
+مافيش J سفر معامدها W اللي هو سفر لل I و سبعة لل J
+
+465
+00:37:16,070 --> 00:37:20,790
+و نقص اربعة لل K هذا الان ال determinant أصلا انت
+
+466
+00:37:20,790 --> 00:37:26,390
+لو بدلت ال U مع ال V مع ال W و حطيت في السطر هذا و
+
+467
+00:37:26,390 --> 00:37:29,850
+هذا مش هتفرج معناه ليش لأنه ماخدينله في الآخر ال
+
+468
+00:37:29,850 --> 00:37:33,750
+magnitudeيعني مخدرينه absolute value يعني هيرجع
+
+469
+00:37:33,750 --> 00:37:38,050
+لقيمة واحدة كلها لأن الفرق بين اتنين بس سالف طيب
+
+470
+00:37:38,050 --> 00:37:43,250
+لأن وجد هذا زي ما نوجد عادة أخدت ال absolute value
+
+471
+00:37:43,250 --> 00:37:47,850
+هي برا لأن وجد قيمة ال determinant اللي جوا شطب
+
+472
+00:37:47,850 --> 00:37:52,810
+هذا شطب هذا بيصير سفر نقص اربعة سفر سبعة في تلاتة
+
+473
+00:37:52,810 --> 00:37:58,700
+و واحد و عشرين واحد و عشرين اللي هو هاي اللي هولأ
+
+474
+00:37:58,700 --> 00:38:02,360
+أسف أنا مستخدم مين يا جماعة مستخدم الصوت الأخير
+
+475
+00:38:02,360 --> 00:38:06,600
+عساس أن أضيع واحد و بس أعمل إيهاش سبعة و نقص أربعة
+
+476
+00:38:06,600 --> 00:38:13,500
+بصير magnitude موجب سالب موجب سفر خلصنا منه بلزمش
+
+477
+00:38:13,500 --> 00:38:17,190
+لإنه بده ينضرب في السفر زي ما أنتوا عارفين سبعةان
+
+478
+00:38:17,190 --> 00:38:19,430
+شاء الله تكونوا فاهمين عليها سبعة الآن بالسالب
+
+479
+00:38:19,430 --> 00:38:23,530
+لأنه سالب اللي هي موجب سالب موجب تفكر محدد بحكي
+
+480
+00:38:23,530 --> 00:38:27,750
+منشط بهدا ومنشط بهدا وظل واحد ناقص واحد ناقص اتنين
+
+481
+00:38:27,750 --> 00:38:32,610
+تلاتة وظل ناقص اربعة ناقص اربعة شط بهدا وشط بهدا
+
+482
+00:38:32,610 --> 00:38:40,490
+وظل واحد اتنين سفر تلاتة تلاتة زائد ناقص اتنين في
+
+483
+00:38:40,490 --> 00:38:47,890
+ناقص سبعة ناقص اربعة في سفر زائد اربعةلأن هذه
+
+484
+00:38:47,890 --> 00:38:51,330
+عبارة عن ناقص سبعة لأن تلاتة ناقص اتنين واحد ناقص
+
+485
+00:38:51,330 --> 00:38:56,630
+سبعة ناقص سبعة ناقص ستة عشر ناقص ستة وعشرين
+
+486
+00:38:56,630 --> 00:39:00,750
+magnitude تلاتة وعشرين هذا اللي هو حجم من المكعب
+
+487
+00:39:00,750 --> 00:39:07,190
+واحد قال بدي استخدم w dot u cross v بنفع واحد قال
+
+488
+00:39:07,190 --> 00:39:12,890
+بدي اقول u dot v cross w بنفع و هكذا فاهمين ايش
+
+489
+00:39:12,890 --> 00:39:16,180
+اللي بحكيهلأن هيك و كون احنا خلصنا الحاضر الرابع
+
+490
+00:39:16,180 --> 00:39:19,900
+او هاي ال exercises والسلام عليكم ورحمة الله
+
+491
+00:39:19,900 --> 00:39:22,100
+وبركاته ولا لقاء لقاء آخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TxfjgQ2pjWY_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TxfjgQ2pjWY_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..6cef4408d285551f0cfee80e7ab580fe2ffb16c8
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/TxfjgQ2pjWY_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 1644, "start": 0.72, "end": 16.44, "text": "بسم الله الرحمن الرحيم السلام عليكم ورحمة الله وبركاته هذه المحاضرة الرابعة مساقطة فاضل أتكامل جين صيف 2020 علوان المحاضرة اليوم اللي هو the cross product", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 29538, 9673, 5016, 46958, 25720, 34892, 16758, 27884, 47524, 995, 47432, 3660, 6156, 46958, 1211, 5551, 2655, 4117, 10943, 1211, 10874, 9957, 20328, 33911, 4808, 11203, 2407, 7649, 9673, 5016, 46958, 25720, 45595, 20498, 13672, 1829, 31439, 264, 3278, 1674], "avg_logprob": -0.17114258534274995, "compression_ratio": 1.6196319018404908, "no_speech_prob": 8.285045623779297e-06, "words": [{"start": 0.72, "end": 1.1, "word": "بسم", "probability": 0.771484375}, {"start": 1.1, "end": 1.32, "word": " الله", "probability": 0.96142578125}, {"start": 1.32, "end": 1.72, "word": " الرحمن", "probability": 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plane على طول بتبادر لذهني باجي بعمل ال vector الأول PR من النقاط اللي عندنا بنعرف كلنا و بنعرف جميع ال things نعمل و BQ ال vector التاني", "tokens": [6027, 20292, 31439, 6055, 1211, 2655, 8717, 4587, 41193, 363, 4032, 48, 4032, 49, 6055, 1211, 2655, 8717, 4587, 41193, 4032, 46958, 5016, 16472, 3224, 32239, 14851, 15844, 44713, 4587, 10943, 3660, 36764, 24401, 3660, 1975, 15730, 16472, 4724, 25957, 1211, 2423, 3620, 5720, 4724, 39648, 20292, 1975, 29245, 3215, 13672, 1829, 31439, 44650, 7435, 3224, 6225, 2304, 2407, 16254, 15844, 23758, 2423, 5720, 15844, 23032, 12610, 39894, 3555, 18513, 2288, 5296, 8848, 3224, 22653, 4724, 26108, 1829, 4724, 25957, 1211, 2423, 8062, 16247, 12610, 11568, 9154, 28239, 4587, 41193, 13672, 1829, 43242, 8315, 44945, 3615, 28480, 28242, 8315, 4032, 44945, 3615, 28480, 10874, 2304, 40228, 2423, 721, 8717, 25957, 1211, 4032, 363, 48, 2423, 8062, 16712, 7649, 1829], "avg_logprob": -0.2731770947575569, "compression_ratio": 1.8178137651821862, "no_speech_prob": 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cross V dot W هو نفسه اللي هو هيتلع U dot V cross W عارفين ليش؟ لأن عندي U cross V dot W", "tokens": [1211, 33456, 2423, 39184, 5893, 1674, 307, 800, 325, 1166, 4032, 2423, 39184, 624, 3278, 691, 5893, 343, 31439, 8717, 36178, 3224, 13672, 1829, 31439, 8032, 36081, 1211, 3615, 624, 5893, 691, 3278, 343, 6225, 9640, 5172, 9957, 32239, 8592, 22807, 5296, 33456, 18871, 16254, 624, 3278, 691, 5893, 343], "avg_logprob": -0.42401961251801135, "compression_ratio": 1.3059701492537314, "no_speech_prob": 1.2516975402832031e-06, "words": [{"start": 2053.01, "end": 2053.47, "word": "لأن", "probability": 0.5140380859375}, {"start": 2053.47, "end": 2054.17, "word": " الـ", "probability": 0.4903564453125}, {"start": 2054.17, "end": 2054.35, "word": " dot", "probability": 0.251708984375}, {"start": 2054.35, "end": 2054.71, "word": " product", "probability": 0.9658203125}, {"start": 2054.71, "end": 2055.01, "word": " is", "probability": 0.404296875}, {"start": 2055.01, "end": 2055.67, "word": " 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"end": 2223.19, "word": " اتنين", "probability": 0.9844970703125}, {"start": 2223.19, "end": 2223.51, "word": " نقص", "probability": 0.9944661458333334}, {"start": 2223.51, "end": 2223.87, "word": " واحد", "probability": 0.984375}], "temperature": 1.0}, {"id": 95, "seek": 225039, "start": 2224.33, "end": 2250.39, "text": "اللي هي نقص اتنين تلاتة نقص اتنين تلاتة اللي هي لل K و سفر لمين لل J لأنه مافيش J سفر معامدها W اللي هو سفر لل I و سبعة لل J و نقص اربعة لل K هذا الان ال determinant أصلا انت لو بدلت ال U مع ال V مع ال W و حطيت في السطر هذا و هذا مش هتفرج معناه ليش لأنه ماخدينله في الآخر ال magnitude", "tokens": [6027, 20292, 39896, 8717, 4587, 9381, 1975, 2655, 1863, 9957, 6055, 1211, 9307, 3660, 8717, 4587, 9381, 1975, 2655, 1863, 9957, 6055, 1211, 9307, 3660, 13672, 1829, 39896, 24976, 591, 4032, 8608, 5172, 2288, 32767, 9957, 24976, 508, 5296, 33456, 3224, 19446, 41185, 8592, 508, 8608, 5172, 2288, 20449, 10943, 3215, 11296, 343, 13672, 1829, 31439, 8608, 5172, 2288, 24976, 286, 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"probability": 0.97900390625}, {"start": 2244.97, "end": 2245.35, "word": " حطيت", "probability": 0.8577473958333334}, {"start": 2245.35, "end": 2245.49, "word": " في", "probability": 0.94384765625}, {"start": 2245.49, "end": 2245.87, "word": " السطر", "probability": 0.8899739583333334}, {"start": 2245.87, "end": 2246.13, "word": " هذا", "probability": 0.48193359375}, {"start": 2246.13, "end": 2246.39, "word": " و", "probability": 0.8125}, {"start": 2246.39, "end": 2246.65, "word": " هذا", "probability": 0.87744140625}, {"start": 2246.65, "end": 2247.55, "word": " مش", "probability": 0.7880859375}, {"start": 2247.55, "end": 2247.99, "word": " هتفرج", "probability": 0.8905029296875}, {"start": 2247.99, "end": 2248.39, "word": " معناه", "probability": 0.9002278645833334}, {"start": 2248.39, "end": 2248.67, "word": " ليش", "probability": 0.887451171875}, {"start": 2248.67, "end": 2248.91, "word": " لأنه", "probability": 0.6470540364583334}, {"start": 2248.91, "end": 2249.31, "word": " 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قال بدي اقول u dot v cross w بنفع و هكذا فاهمين ايش اللي بحكيه", "tokens": [1211, 33456, 29538, 6225, 3555, 9640, 3660, 18871, 8717, 995, 4587, 9381, 8608, 3555, 27884, 5296, 33456, 6055, 1211, 9307, 3660, 8717, 995, 4587, 9381, 1975, 2655, 1863, 9957, 36764, 24401, 8717, 995, 4587, 9381, 8608, 3555, 27884, 8717, 995, 4587, 9381, 8608, 3555, 27884, 8717, 995, 4587, 9381, 8608, 2655, 3660, 6225, 46309, 8717, 995, 4587, 9381, 8608, 2655, 3660, 4032, 3615, 46309, 9957, 15668, 6055, 1211, 9307, 3660, 4032, 3615, 46309, 9957, 23758, 13672, 1829, 31439, 11331, 7435, 2304, 9154, 9673, 4117, 3615, 3555, 36764, 24401, 50239, 4724, 16254, 44713, 9778, 40448, 261, 5893, 344, 3278, 371, 44945, 5172, 3615, 36764, 24401, 50239, 4724, 16254, 1975, 39648, 344, 5893, 371, 3278, 261, 44945, 5172, 3615, 4032, 8032, 4117, 15730, 6156, 995, 16095, 9957, 1975, 1829, 8592, 13672, 1829, 4724, 5016, 4117, 1829, 3224], "avg_logprob": -0.2001378674279241, "compression_ratio": 2.076923076923077, "no_speech_prob": 8.344650268554688e-07, "words": [{"start": 2326.99, "end": 2327.53, "word": "لأن", "probability": 0.48126220703125}, {"start": 2327.53, "end": 2327.89, "word": " هذه", "probability": 0.2191162109375}, {"start": 2327.89, "end": 2328.41, "word": " عبارة", "probability": 0.9404296875}, {"start": 2328.41, "end": 2328.59, "word": " عن", "probability": 0.99658203125}, {"start": 2328.59, "end": 2328.99, "word": " ناقص", "probability": 0.74267578125}, {"start": 2328.99, "end": 2329.41, "word": " سبعة", "probability": 0.8024088541666666}, {"start": 2329.41, "end": 2329.93, "word": " لأن", "probability": 0.70654296875}, {"start": 2329.93, "end": 2330.25, "word": " تلاتة", "probability": 0.763916015625}, {"start": 2330.25, "end": 2330.43, "word": " ناقص", "probability": 0.856201171875}, {"start": 2330.43, "end": 2330.67, "word": " اتنين", "probability": 0.899169921875}, {"start": 2330.67, "end": 2330.99, "word": " واحد", "probability": 0.934326171875}, {"start": 2330.99, "end": 2331.33, "word": " ناقص", "probability": 0.901123046875}, {"start": 2331.33, "end": 2331.69, "word": " سبعة", "probability": 0.9479166666666666}, {"start": 2331.69, "end": 2332.49, "word": " ناقص", "probability": 0.958251953125}, {"start": 2332.49, "end": 2332.89, "word": " سبعة", "probability": 0.92529296875}, {"start": 2332.89, "end": 2334.13, "word": " ناقص", "probability": 0.977294921875}, {"start": 2334.13, "end": 2334.47, "word": " ستة", "probability": 0.7158203125}, {"start": 2334.47, "end": 2334.77, "word": " عشر", "probability": 0.892822265625}, {"start": 2334.77, "end": 2335.81, "word": " ناقص", "probability": 0.94775390625}, {"start": 2335.81, "end": 2336.01, "word": " ستة", "probability": 0.708984375}, {"start": 2336.01, "end": 2336.63, "word": " وعشرين", "probability": 0.8935546875}, {"start": 2336.63, "end": 2337.37, "word": " magnitude", "probability": 0.44091796875}, {"start": 2337.37, "end": 2337.91, "word": " تلاتة", "probability": 0.94580078125}, {"start": 2337.91, "end": 2338.45, "word": " وعشرين", "probability": 0.97216796875}, {"start": 2338.45, "end": 2338.97, "word": " هذا", "probability": 0.7158203125}, {"start": 2338.97, "end": 2339.09, "word": " اللي", "probability": 0.716796875}, {"start": 2339.09, "end": 2339.23, "word": " هو", "probability": 0.982421875}, {"start": 2339.23, "end": 2339.57, "word": " حجم", "probability": 0.9835611979166666}, {"start": 2339.57, "end": 2339.89, "word": " من", "probability": 0.348388671875}, {"start": 2339.89, "end": 2340.75, "word": " المكعب", "probability": 0.9520263671875}, {"start": 2340.75, "end": 2341.01, "word": " واحد", "probability": 0.87646484375}, {"start": 2341.01, "end": 2341.21, "word": " قال", "probability": 0.85009765625}, {"start": 2341.21, "end": 2341.43, "word": " بدي", "probability": 0.495361328125}, {"start": 2341.43, "end": 2342.09, "word": " استخدم", "probability": 0.8732096354166666}, {"start": 2342.09, "end": 2343.23, "word": " w", "probability": 0.29638671875}, {"start": 2343.23, "end": 2344.83, "word": " dot", "probability": 0.5888671875}, {"start": 2344.83, "end": 2345.37, "word": " u", "probability": 0.8154296875}, {"start": 2345.37, "end": 2345.67, "word": " cross", "probability": 0.953125}, {"start": 2345.67, "end": 2345.93, "word": " v", "probability": 0.810546875}, {"start": 2345.93, "end": 2346.45, "word": " بنفع", "probability": 0.8994140625}, {"start": 2346.45, "end": 2347.01, "word": " واحد", "probability": 0.861083984375}, {"start": 2347.01, "end": 2347.19, "word": " قال", "probability": 0.95654296875}, {"start": 2347.19, "end": 2347.31, "word": " بدي", "probability": 0.9033203125}, {"start": 2347.31, "end": 2348.13, "word": " اقول", "probability": 0.78955078125}, {"start": 2348.13, "end": 2349.01, "word": " u", "probability": 0.634765625}, {"start": 2349.01, "end": 2349.57, "word": " dot", "probability": 0.93359375}, {"start": 2349.57, "end": 2350.27, "word": " v", "probability": 0.9501953125}, {"start": 2350.27, "end": 2350.59, "word": " cross", "probability": 0.96728515625}, {"start": 2350.59, "end": 2351.01, "word": " w", "probability": 0.9755859375}, {"start": 2351.01, "end": 2351.57, "word": " بنفع", "probability": 0.9803059895833334}, {"start": 2351.57, "end": 2351.81, "word": " و", "probability": 0.70556640625}, {"start": 2351.81, "end": 2352.29, "word": " هكذا", "probability": 0.82861328125}, {"start": 2352.29, "end": 2352.75, "word": " فاهمين", "probability": 0.87255859375}, {"start": 2352.75, "end": 2352.89, "word": " ايش", "probability": 0.7014973958333334}, {"start": 2352.89, "end": 2353.01, "word": " اللي", "probability": 0.9912109375}, {"start": 2353.01, "end": 2353.41, "word": " بحكيه", "probability": 0.9693359375}], "temperature": 1.0}, {"id": 100, "seek": 236210, "start": 2353.52, "end": 2362.1, "text": "لأن هيك و كون احنا خلصنا الحاضر الرابع او هاي ال exercises والسلام عليكم ورحمة الله وبركاته ولا لقاء لقاء آخر", "tokens": [1211, 33456, 39896, 4117, 4032, 9122, 11536, 1975, 5016, 8315, 16490, 1211, 9381, 8315, 21542, 46958, 2288, 34892, 16758, 3615, 1975, 2407, 8032, 47302, 2423, 11900, 16070, 3794, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 49429, 5296, 4587, 16606, 5296, 4587, 16606, 19753, 34740], "avg_logprob": -0.21737131710145988, "compression_ratio": 1.4318181818181819, "no_speech_prob": 4.172325134277344e-07, "words": [{"start": 2353.52, "end": 2354.18, "word": "لأن", "probability": 0.9228515625}, {"start": 2354.18, "end": 2354.4, "word": " هيك", "probability": 0.774658203125}, {"start": 2354.4, "end": 2354.48, "word": " و", "probability": 0.77001953125}, {"start": 2354.48, "end": 2354.7, "word": " كون", "probability": 0.53546142578125}, {"start": 2354.7, "end": 2354.86, "word": " احنا", "probability": 0.829345703125}, {"start": 2354.86, "end": 2355.28, "word": " خلصنا", "probability": 0.9560546875}, {"start": 2355.28, "end": 2355.66, "word": " الحاضر", "probability": 0.7613932291666666}, {"start": 2355.66, "end": 2356.18, "word": " الرابع", "probability": 0.9207356770833334}, {"start": 2356.18, "end": 2357.26, "word": " او", "probability": 0.7978515625}, {"start": 2357.26, "end": 2357.58, "word": " هاي", "probability": 0.67919921875}, {"start": 2357.58, "end": 2357.76, "word": " ال", "probability": 0.98974609375}, {"start": 2357.76, "end": 2358.4, "word": " exercises", "probability": 0.9619140625}, {"start": 2358.4, "end": 2359.1, "word": " والسلام", "probability": 0.880859375}, {"start": 2359.1, "end": 2359.46, "word": " عليكم", "probability": 0.94873046875}, {"start": 2359.46, "end": 2359.7, "word": " ورحمة", "probability": 0.837890625}, {"start": 2359.7, "end": 2359.9, "word": " الله", "probability": 0.96435546875}, {"start": 2359.9, "end": 2360.44, "word": " وبركاته", "probability": 0.9904296875}, {"start": 2360.44, "end": 2360.68, "word": " ولا", "probability": 0.53564453125}, {"start": 2360.68, "end": 2361.2, "word": " لقاء", "probability": 0.98388671875}, {"start": 2361.2, "end": 2361.68, "word": " لقاء", "probability": 0.6534830729166666}, {"start": 2361.68, "end": 2362.1, "word": " آخر", "probability": 0.7008056640625}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2364.5111875, "duration_after_vad": 2337.443749999993}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/bTW6N1MaENg.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/bTW6N1MaENg.srt
new file mode 100644
index 0000000000000000000000000000000000000000..cd947cc4fcb60d69af56af676de4fefffaba2f89
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/bTW6N1MaENg.srt
@@ -0,0 +1,1390 @@
+1
+00:00:01,080 --> 00:00:05,540
+بسم الله الرحمن الرحيم هذه هي المحاضرة الثالثة
+
+2
+00:00:05,540 --> 00:00:14,200
+مساق تفاضل وتكامل، الفصل الصيفي 2020، العنوان
+
+3
+00:00:14,200 --> 00:00:19,230
+المحاضرة اليوم، اللي هو the dot product، section 12-3
+
+4
+00:00:19,230 --> 00:00:25,130
+بداية، قبل ما نعرف الـ dot product و كيف نتعامل معاه
+
+5
+00:00:25,130 --> 00:00:29,350
+خلّينا نعرف حاجة اسمها angle between vectors، اللي
+
+6
+00:00:29,350 --> 00:00:34,740
+هي الزاوية بين، اللي هو، متجهين. الآن عندي لو كان في
+
+7
+00:00:34,740 --> 00:00:41,620
+عندي متجه U، أمامي، و متجه آخر V الثاني، و
+
+8
+00:00:41,620 --> 00:00:45,760
+الاثنين بنطلق من نقطة واحدة. هذه الزاوية اللي بينهم
+
+9
+00:00:45,760 --> 00:00:50,640
+بنسميها the angle between the two vectors U and V.
+
+10
+00:00:50,640 --> 00:00:58,180
+اللي هي الزاوية بين المتجهين U و V. نبدأ في
+
+11
+00:00:58,180 --> 00:01:04,700
+نظرية بتعرفنا
+
+12
+00:01:04,700 --> 00:01:12,460
+كيف نجد قيمة θ بمعرفة الـ vector u و vector v.
+
+13
+00:01:13,640 --> 00:01:17,500
+النظرية بتقول ما يلي: اللي هي angle between two
+
+14
+00:01:17,500 --> 00:01:21,540
+vectors. If the angle θ between two non-zero
+
+15
+00:01:21,540 --> 00:01:27,500
+vectors u and v is given by. يعني الزاوية θ بين two
+
+16
+00:01:27,500 --> 00:01:31,460
+non-zero vectors، يعني u لا تساوي zero أو v لا تساوي zero.
+
+17
+00:01:31,710 --> 00:01:37,090
+الزاوية اللي بينهم، اللي هي is given by، اللي هو
+
+18
+00:01:37,090 --> 00:01:40,110
+عندي طبعاً الـ U و الـ V، زي ما قلنا، non-zero vector.
+
+19
+00:01:40,110 --> 00:01:44,910
+هذا لا يساوي صفراً، وهذا لا يساوي صفراً، فَالزاوية بينهم
+
+20
+00:01:44,910 --> 00:01:51,130
+تساوي cosine inverse لـ (U1 V1 + U2
+
+21
+00:01:51,130 --> 00:01:55,930
+V2 + U3 V3). هذا الآن cosine
+
+22
+00:01:55,930 --> 00:01:59,970
+inverse لهذا، على الـ magnitude للـ U في الـ magnitude
+
+23
+00:01:59,970 --> 00:02:05,290
+للـ V. هذه هي الزاوية θ بين المتجهين. هذه النظرية
+
+24
+00:02:05,290 --> 00:02:12,250
+بدنا نعرفها، لأنه احنا هنستخدمها في، اللي هو، بعد
+
+25
+00:02:12,250 --> 00:02:19,330
+شوية، في نتائج هامة لنا، اللي هي بخصوص اللي هو الـ dot
+
+26
+00:02:19,330 --> 00:02:23,660
+product. الآن، definition، تعريف الـ dot product. بنا
+
+27
+00:02:23,660 --> 00:02:28,280
+نعرف الـ dot product أو الضرب القياسي لـ U و الـ V.
+
+28
+00:02:28,280 --> 00:02:31,760
+طبعاً احنا هنعرف، بعد المحاضرة القادمة كمان، أن في
+
+29
+00:02:31,760 --> 00:02:35,040
+عندنا two dot products بين الـ vectors. احنا
+
+30
+00:02:35,040 --> 00:02:39,260
+عرفنا شو معنى الجمع two vectors، شو معنى scalar
+
+31
+00:02:39,260 --> 00:02:44,880
+في الـ vector. الآني بدنا نعرف عملية ضرب بين
+
+32
+00:02:44,880 --> 00:02:48,280
+two vectors. عملية الضرب نوعين: عندنا النوع الأول اللي
+
+33
+00:02:48,280 --> 00:02:50,920
+هنحكي عنه، اللي هو الـ dot product أو الضرب القياسي،
+
+34
+00:02:50,920 --> 00:02:55,640
+والنوع الثاني هيجي لاحقاً، اللي هو الضرب الاتجاهي أو
+
+35
+00:02:55,640 --> 00:02:58,600
+اللي بنسميه cross product. خلينا نتعرف اليوم على
+
+36
+00:02:58,600 --> 00:03:02,660
+الـ dot product: u . v. الآن، u . v، اللي هو الـ dot
+
+37
+00:03:02,660 --> 00:03:06,760
+product u . v of vectors u and v، هي الـ vector
+
+38
+00:03:06,760 --> 00:03:12,570
+الأول مكتوب على صورة U1، U2، U3 أو زي ما بتعرفوا U1 i
+
+39
+00:03:12,570 --> 00:03:19,430
++ U2 j + U3 k، و V = V1 i + V2 j + V3 k.
+
+40
+00:03:19,430 --> 00:03:22,470
+زي ما حكينا في المحاضرات السابقة، إذا الـ dot
+
+41
+00:03:22,470 --> 00:03:27,010
+product، U . V، أيش بيكون؟ U1 في V1، الـ
+
+42
+00:03:27,010 --> 00:03:29,550
+component الأول في الـ component الأول، زائد
+
+43
+00:03:29,790 --> 00:03:35,970
+الـ component الثاني U2 V2، زائد الـ component الثالث U3 V3.
+
+44
+00:03:35,970 --> 00:03:40,190
+هذا هو تعريف الـ dot product. ناخد أمثلة، المثال
+
+45
+00:03:40,190 --> 00:03:42,830
+الأول: نضرب، هاي عندنا الـ vector الأول، وهي الـ
+
+46
+00:03:42,830 --> 00:03:45,990
+vector الثاني، نضربهم dot product بينهم، اللي هو
+
+47
+00:03:45,990 --> 00:03:50,290
+طبعاً مادام الـ scalar أو dot product هو عبارة عن ضرب
+
+48
+00:03:50,290 --> 00:03:54,150
+قياسي، يعني في الآخر هيطلع رقم، مش متجه، حاصل ضرب
+
+49
+00:03:54,150 --> 00:04:00,550
+حاصل ضرب متجهين. بمعنى، dot product بيطلع عندي عبارة
+
+50
+00:04:00,550 --> 00:04:05,930
+عن scalar، رقم. يعني 1 في -6 + -2
+
+51
+00:04:05,930 --> 00:04:10,170
+في 2 + -1 في -3. هي أن
+
+52
+00:04:10,170 --> 00:04:13,970
+هذه ناتج الـ dot product. بنبسطها: ده بيصير -6، وده بيصير -4، وده بيصير 3.
+
+53
+00:04:13,970 --> 00:04:17,710
+نجمع: -6 و -4 = -10، و 3، بيطلع
+
+54
+00:04:17,710 --> 00:04:21,670
+-7. بعد ذلك، نضرب product بينهم، بيطلع عبارة عن
+
+55
+00:04:21,670 --> 00:04:24,930
+-7، رقم، مثل ما احنا متوقعين. نضرب الـ U.V: (1/2) i
+
+56
+00:04:24,930 --> 00:04:31,790
++ 3j + k في 4i - j + 2k.
+
+57
+00:04:31,790 --> 00:04:35,510
+يعني هذا متجه مكتوب بطريقة أخرى، يعني هنا طريقة
+
+58
+00:04:39,710 --> 00:04:42,770
+اللي هي الـ components، اللي عندي، طريقة اللي هو
+
+59
+00:04:42,770 --> 00:04:46,880
+بواسطة اللي هو من الـ standard unit vectors. زي ما
+
+60
+00:04:46,880 --> 00:04:51,320
+احنا عرفنا سابقاً، بنفس الطريقة: معامل i (1/2) في معامل
+
+61
+00:04:51,320 --> 00:04:56,140
+i (4) = 2. معامل j في معامل j: 3
+
+62
+00:04:56,140 --> 00:05:00,460
+في -1 + معامل k في معامل k: اللي هو 1
+
+63
+00:05:00,460 --> 00:05:05,000
+في 2. بيصير عندي: (1/2)(4) = 2، 2 + (-3) =
+
+64
+00:05:05,000 --> 00:05:08,040
+-1، -1 + 2 = 1.
+
+65
+00:05:08,040 --> 00:05:10,920
+يطلع 1. هذا اللي هو الـ dot product. هذا الـ dot
+
+66
+00:05:10,920 --> 00:05:13,980
+product سواء بالطريقة هذه أو بالطريقة هذه. زي ما
+
+67
+00:05:13,980 --> 00:05:20,000
+انتم شايفين، عبارة عن، اللي هو، أمر سهل. طيب، الآن in
+
+68
+00:05:20,000 --> 00:05:22,600
+the notation of the dot product، the angle between
+
+69
+00:05:22,600 --> 00:05:27,040
+two vectors u and v is. قبل شوية قلنا النظرية أيش
+
+70
+00:05:27,040 --> 00:05:31,520
+بتقول؟ بتقول θ = cosine inverse (u1 v1 +
+
+71
+00:05:31,520 --> 00:05:36,640
+u2 v2
+
+72
+00:05:36,640 --> 00:05:42,010
++ u3 v3)، اللي هو في الواقع أيش هو؟ اللي هو عبارة
+
+73
+00:05:42,010 --> 00:05:45,950
+عن U . V على magnitude لـ U في magnitude لـ V. هي
+
+74
+00:05:45,950 --> 00:05:48,810
+هي هي يا جماعة، هي هي θ، بس هي cosine inverse.
+
+75
+00:05:48,810 --> 00:05:55,460
+هذا هذا هو عبارة عنه، هذا عبارة عن U . V. إذا زي ما قلنا
+
+76
+00:05:55,460 --> 00:06:02,860
+الآن، الـ U . V استبدلناها بالكلام الموجود: U1 V1 + U2
+
+77
+00:06:02,860 --> 00:06:07,800
+V2 + U3 V3 على الـ magnitude لـ U في الـ magnitude لـ
+
+78
+00:06:07,800 --> 00:06:12,700
+V. إذا صارت θ مكتوبة بتساوي cosine inverse للي
+
+79
+00:06:12,700 --> 00:06:20,610
+أمامي. لأن هذه بتساعدني، أو بتسهل عليّ، لو عرفّت الـ two
+
+80
+00:06:20,610 --> 00:06:24,550
+vectors u و v، أنا أوجد الزاوية اللي بينهم بكل سهولة.
+
+81
+00:06:24,550 --> 00:06:30,390
+Find the angle between u = i - 2j -
+
+82
+00:06:30,390 --> 00:06:34,730
+2k and v = 6i + 3j + 2k.
+
+83
+00:06:34,730 --> 00:06:37,910
+بدّي أوجد الزاوية اللي بين، اللي هو، الـ two vectors
+
+84
+00:06:37,910 --> 00:06:42,680
+اللي أمامي. سهلة أكيد، هتطلع سهلة. بنوجد u . v و
+
+85
+00:06:42,680 --> 00:06:45,860
+الـ magnitude و الـ magnitude، و بنعوّض، بنوجد الـ θ.
+
+86
+00:06:45,860 --> 00:06:51,240
+u . v = اللي هي هذه، اللي هي 1 و
+
+87
+00:06:51,240 --> 00:06:54,640
+-2 و -2.  ده كنت أترجمها زي هيك، زي ما بدّك، و
+
+88
+00:06:54,640 --> 00:06:57,940
+ده كنت أخليها زي ما هي، بنفع برضه. 6، 3، 2، و
+
+89
+00:06:57,940 --> 00:07:00,940
+ضربنا الحالتين، الحالتين، الشكل هذا، والحالتين الشكل
+
+90
+00:07:00,940 --> 00:07:04,940
+هذا. ضربناها قبل شوية: 1 في 6 = 6، -2
+
+91
+00:07:04,940 --> 00:07:11,130
+في 3 = -6، -2 في 2 = -4.  هدول
+
+92
+00:07:11,130 --> 00:07:14,310
+بروحي من بعض، بيضل أيش؟ بيساوي -4. الـ
+
+93
+00:07:14,310 --> 00:07:17,690
+magnitude للـ U بتعرفوا تحسبوه. الـ magnitude للـ U
+
+94
+00:07:17,690 --> 00:07:20,950
+اللي هو الجذر التربيعي: هذا تربيع + هذا تربيع + هذا
+
+95
+00:07:20,950 --> 00:07:25,010
+تربيع. هذا تربيع 1، و 4، و 4 = 9. الجذر التسعة بيساوي 3. ده الـ magnitude للـ
+
+96
+00:07:27,730 --> 00:07:30,470
+U = 3. تعالوا نيجي للـ magnitude للـ V.
+
+97
+00:07:30,470 --> 00:07:34,050
+بتعرفوا كلكم تجيبوه. هو الجذر التربيعي لـ 6 تربيع
+
+98
+00:07:34,050 --> 00:07:37,210
+= 36 + هذا تربيع = 9. 36
+
+99
+00:07:37,210 --> 00:07:40,810
++ 9 = 45.  45 وهذا تربيع 4 = 49. الجذر الـ 49 أيش بيساوي
+
+100
+00:07:44,450 --> 00:07:48,990
+يا جماعة؟ 7. إذا الـ magnitude للـ V، الـ magnitude
+
+101
+00:07:48,990 --> 00:07:52,050
+للـ V هو عبارة عن 7. 3 × 7 = 21.
+
+102
+00:07:52,050 --> 00:07:56,990
+خلصنا. صارت θ = cosine inverse لهذا، هذا على
+
+103
+00:07:56,990 --> 00:08:01,950
+هذا. cosine inverse للـ -4 على اللي
+
+104
+00:08:01,950 --> 00:08:05,710
+وجدناه، 21. cosine inverse هذا لو على الآلة
+
+105
+00:08:05,710 --> 00:08:10,790
+هتلاقي عبارة عن 1.76 radians، لأن
+
+106
+00:08:10,790 --> 00:08:14,010
+هذا لو لهنا يكفيني أنا يعني أنه تعرفوا الـ θ
+
+107
+00:08:14,010 --> 00:08:18,650
+= cosine inverse. طيب، هذا مثال، ناخد مثال آخر.
+
+108
+00:08:18,650 --> 00:08:24,460
+Find the angle in that triangle ABC determined by
+
+109
+00:08:24,460 --> 00:08:30,000
+the vertices A, B, C، and ... اللي هي ... اللي هو الـ
+
+110
+00:08:30,000 --> 00:08:34,620
+figure اللي أمامي. هي هي A، وهي B، وهي C، بالزاوية
+
+111
+00:08:34,620 --> 00:08:40,540
+θ. هو طلب يجيب الـ θ هذه. الآن، θ هذه، بدّي أعملها θ
+
+112
+00:08:40,540 --> 00:08:42,760
+between two vectors. إذا أنا بدّي أعمل vector من هنا
+
+113
+00:08:42,760 --> 00:08:47,370
+لهنا، وأعمل vector من هنا لهنا. الـ vector اللي من
+
+114
+00:08:47,370 --> 00:08:53,210
+هنا إلى هنا هو عبارة عن الـ CB. الـ initial point C،
+
+115
+00:08:53,210 --> 00:08:57,730
+والـ terminal point B. الـ vector هذا اسمه؟ CA.
+
+116
+00:08:57,730 --> 00:09:02,510
+هذا الـ vector من هنا إلى هنا. بنعرف وجوده هو عبارة
+
+117
+00:09:02,510 --> 00:09:09,670
+عن (3, -2, 5) - (2, 3, 0) = (1, -5, 5). هذا الـ vector CB.
+
+118
+00:09:09,670 --> 00:09:13,380
+زي ما اتعلمنا قبل. الـ vector هذا، اللي هو الـ
+
+119
+00:09:13,380 --> 00:09:17,420
+terminal point ناقص الـ initial. اللي هي (0, -5, -2).
+
+120
+00:09:17,420 --> 00:09:20,700
+(0 - (-5) = 5). (0 - (-2) = 2).
+
+121
+00:09:20,700 --> 00:09:24,300
+إذاً هذا الـ CB، وهذا الـ CA. two vectors
+
+122
+00:09:24,300 --> 00:09:29,640
+اللي بينهم الزاوية θ. الآن أيش بيوجدني ... عشان
+
+123
+00:09:29,640 --> 00:09:32,140
+أوجد الزاوية هذه؟ اللي هو عبارة عن الـ dot product
+
+124
+00:09:32,140 --> 00:09:35,640
+بين هذا وهذا، مقسوم على الـ magnitude لهذا، على الـ
+
+125
+00:09:35,640 --> 00:09:37,700
+magnitude لهذا. إذاً أول شيء بدّي أوجد الـ dot
+
+126
+00:09:37,700 --> 00:09:40,640
+product بين هذا وبين هذا الـ vector. ما هو الـ Dot
+
+127
+00:09:40,640 --> 00:09:44,320
+product يا جماعة؟ اللي هو عبارة عن -2 في
+
+128
+00:09:44,320 --> 00:09:49,960
+-5 + 3 في 2 = 10 + 6 = 16.
+
+129
+00:09:49,960 --> 00:09:56,120
++5 × 5 = 25 + 16 = 41. هذا اللي هو الـ Dot Product. الآن الـ
+
+130
+00:09:59,880 --> 00:10:04,700
+magnitude للـ CB: الجذر التربيعي للـ 4 + 9 + 25 = √38
+
+131
+00:10:04,700 --> 00:10:10,150
+الـ CA: 25 + 4 + 9 = √38.
+
+132
+00:10:10,150 --> 00:10:13,650
+أكيد فاهمين: الجذر التربيعي لـ هذا تربيع + هذا
+
+133
+00:10:13,650 --> 00:10:16,930
+تربيع. هذا الـ magnitude لـ الـ CA، وهذا الـ
+
+134
+00:10:16,930 --> 00:10:20,550
+magnitude للـ CB. وضربهم في بعض، وطلع الجذر التربيعي لـ
+
+135
+00:10:20,550 --> 00:10:23,410
+377، اللي هانسميها x، بس هيساوي
+
+136
+00:10:23,410 --> 00:10:27,750
+cosine inverse للـ dot product، اللي هو 41، على الـ
+
+137
+00:10:27,750 --> 00:10:30,070
+magnitude ده الأول، والـ magnitude ده الثاني: √38 × √38.
+
+138
+00:10:30,070 --> 00:10:37,670
+جذر الـ 377 بيطلع 78.11° أو 1.36 radians. هذا
+
+139
+00:10:37,670 --> 00:10:43,590
+بيجي جاهز لما تجيبوه، تجيبوه عن طريق الآلة. الآن
+
+140
+00:10:43,590 --> 00:10:48,330
+بدنا نعرف شو معنى two vectors are orthogonal أو
+
+141
+00:10:48,330 --> 00:10:52,170
+two vectors are perpendicular. يعني بدنا نعرف two
+
+142
+00:10:52,170 --> 00:11:00,310
+vectors كيف يكونوا متعامدين؟ أو متعامدين.
+
+143
+00:11:00,310 --> 00:11:04,050
+أو متعامدين.
+
+144
+00:11:04,280 --> 00:11:08,680
+الديفينيشن: the vectors u and v are orthogonal، اللي
+
+145
+00:11:08,680 --> 00:11:12,480
+هو الـ two vectors بيكونوا orthogonal إذا كانت
+
+146
+00:11:12,480 --> 00:11:19,740
+الزاوية بينهم 90 درجة. يعني 90 درجة، if and
+
+147
+00:11:19,740 --> 00:11:24,920
+only if u . v = 0. بعد شوية هنعرف إن فعلاً الـ
+
+148
+00:11:24,920 --> 00:11:29,340
+U . V = 0، معناته إن الزاوية بين المتجهين تساوي
+
+149
+00:11:29,340 --> 00:11:34,700
+90 درجة. إذا two vectors U and V are orthogonal، إذا
+
+150
+00:11:34,700 --> 00:11:40,560
+كانت dot product بينهم = 0. هاي فحص اللي هو الـ
+
+151
+00:11:40,560 --> 00:11:44,520
+orthogonality between two vectors. To determine
+
+152
+00:11:44,520 --> 00:11:47,960
+example. ناخذ مثال. To determine if two vectors are
+
+153
+00:11:47,960 --> 00:11:50,880
+orthogonal، calculate their dot product.
+
+154
+00:11:50,880 --> 00:11:56,420
+معطينا U = (3, -2, 1) و V = (0, 2, -4).
+
+155
+00:11:56,420 --> 00:12:05,860
+قال لي: هي في عندي كمان two vectors، هدول كانوا في
+
+156
+00:12:05,860 --> 00:12:08,640
+الـ one، في الـ two dimensions، يعني في،  انا
+
+157
+00:12:08,640 --> 00:12:30,800
+بتعطيك شيء في الـ space. بتعرفّلي إنهم متعامدين ولا لأ. بنقوله: سهل الموضوع، الأمر سهل، ما هو إلا غير
+
+158
+00:12:30,800 --> 00:12:35,340
+نضرب u . v: 3 في 0 = 0.
+
+159
+00:12:35,340 --> 00:12:41,410
+-2 في 2 = -4. 1 في -4 = -4. 0 و -4 و -4 = -8.
+
+160
+00:12:41,410 --> 00:12:46,270
+-8، لأنه مافيش i, معامل الـ i أيش؟ 0. جمعها = 0.
+
+161
+00:12:46,270 --> 00:12:49,510
+يعني هذا الـ vector = 0 + 2j + 4k
+
+162
+00:12:49,510 --> 00:12:53,850
+3 في 0 = 0, -2 في 2 = -4, 1 في -4 = -4.
+
+163
+00:12:53,850 --> 00:13:01,050
+0 و -4 و -4، أيش هيساوي؟ -8. إذا اللي هو الـ U and V are
+
+164
+00:13:01,050 --> 00:13:08,610
+not orthogonal. لأن الـ zero vector orthogonal على كل الـ
+
+165
+00:13:08,610 --> 00:13:11,930
+vectors اللي في الدنيا، اللي هو مين؟ هو الـ zero
+
+166
+00:13:11,930 --> 00:13:15,950
+vector. Zero is orthogonal to every vector U since
+
+167
+00:13:15,950 --> 00:13:19,150
+لأنه أصلاً أنتم عارفين الـ dot product بين الـ
+
+168
+00:13:19,150 --> 00:13:
+
+223
+00:16:35,590 --> 00:16:44,610
+اللي هو U2 V2 زائد U2 W2 زائد U3 V3 في الأخيرة هذه
+
+224
+00:16:44,610 --> 00:16:50,530
+زائد U3 W3 الآن نجمع هذول هذه مع هذه مع هذه
+
+225
+00:16:50,530 --> 00:16:56,250
+نجمعها مع بعض هين وهذه مع هذه مع هذه نجمعها مع
+
+226
+00:16:56,250 --> 00:17:00,790
+بعض لأننا نجمع تبعات الـ U والـ V مع بعض وتبعات الـ
+
+227
+00:17:00,790 --> 00:17:05,710
+U والـ W مع بعض للوصول إلى اللي بدنا إياه اللي فوق الآن
+
+228
+00:17:05,710 --> 00:17:10,470
+ماشي أنا من هنا وبدي أوصل هنا طيب شوف هيك هذه هي
+
+229
+00:17:10,470 --> 00:17:16,470
+عبارة عن حاصل ضرب two vectors اللي هي U1 U2 U3 مع
+
+230
+00:17:16,470 --> 00:17:21,560
+V1 V2 V3 لو ضربت ضرب قياسي هيطلع هذا الرقم لأن
+
+231
+00:17:21,560 --> 00:17:25,880
+هذا حاصل ضرب الـ vector هذا ضرب الـ vector هذا اللي
+
+232
+00:17:25,880 --> 00:17:30,000
+هو الـ U مع W لو ضربته هيطلع هذا إذا هذا عبارة
+
+233
+00:17:30,000 --> 00:17:35,180
+عن الـ U وهذا عبارة عن الـ V هذا الـ U وهذا عبارة عن الـ
+
+234
+00:17:35,180 --> 00:17:39,700
+U وهذا عبارة عن الـ W يعني الآن وصلنا U dot V
+
+235
+00:17:39,700 --> 00:17:44,310
+زي الـ W  U dot V زي U dot W لو أحد قال لك
+
+236
+00:17:44,310 --> 00:17:47,870
+اعمل هذا فأنت تسأل اعملها على طول هذا في نفسه
+
+237
+00:17:47,870 --> 00:17:54,290
+بيصير U1 تربيع زائد U2 تربيع زائد U3 تربيع عارفين ليش؟
+
+238
+00:17:54,290 --> 00:18:00,410
+لأن U.U U1 في U1 U1 تربيع U2 في U2 U2 تربيع U3 في
+
+239
+00:18:00,410 --> 00:18:05,410
+U3 U3 تربيع مجموع اللي هو عبارة عن الـ magnitude لـ
+
+240
+00:18:05,410 --> 00:18:12,610
+U تربيع طيب نيجي الآن ل... ال... الموضوع الأخير في
+
+241
+00:18:12,610 --> 00:18:18,350
+هذا الـ section اللي هو موضوع مهم اللي بنسميه
+
+242
+00:18:18,350 --> 00:18:22,390
+vector projection أو مصقط الـ vector الآن عندي
+
+243
+00:18:22,390 --> 00:18:27,450
+حالتين يا إما عندي الـ vector u وهي الـ
+
+244
+00:18:27,450 --> 00:18:31,070
+vector v ماشي الحال هي الزاوية اللي بينهم θ
+
+245
+00:18:31,070 --> 00:18:36,970
+هذه زاوية حادة الآن الـ u والـ v هي الـ two
+
+246
+00:18:36,970 --> 00:18:41,320
+vectors والزاوية اللي بينهم إيش مالها منفرجة الآن
+
+247
+00:18:41,320 --> 00:18:47,940
+بسقط الـ U على الـ V وبسقط الـ U على الـ V نشوف الآن
+
+248
+00:18:47,940 --> 00:18:51,560
+الحالتين  في الآخر بتجمعوا في قانون واحد و
+
+249
+00:18:51,560 --> 00:18:56,200
+القانون هيسهل ويكون سهل إن شاء الله نيجي الآن
+
+250
+00:18:56,200 --> 00:19:01,300
+هَي الـ vector U وهي الـ vector V بدي أوجد الـ
+
+251
+00:19:01,300 --> 00:19:06,920
+projection للـ U on V يعني بدي أوجد مصقط U على مين
+
+252
+00:19:06,920 --> 00:19:11,220
+على V بتعرفوا كيف نوجد المصقط؟ بنزل هي رسمة
+
+253
+00:19:11,220 --> 00:19:16,020
+الـ U وهي رسمة مين؟ الـ V من نفس النقطة باجي الآن
+
+254
+00:19:16,020 --> 00:19:22,080
+بنزل عمودي من رأس الـ U على اللي هو مين؟ الـ V هذا هو
+
+255
+00:19:22,080 --> 00:19:29,280
+الـ projection للـ U on مين؟ on V الـ projection للـ U
+
+256
+00:19:29,280 --> 00:19:35,730
+on V طيب هذا الـ projection بتحدد بأمرين اللي هو
+
+257
+00:19:35,730 --> 00:19:40,850
+بالـ scalar طوله يعني بالـ magnitude تبعه وباتجاهه
+
+258
+00:19:40,850 --> 00:19:46,980
+واضح باتجاهه واتجاهه من الـ V يعني الآن أول شيء
+
+259
+00:19:46,980 --> 00:19:50,840
+بدنا نوجد الطول هذا إيجاد الطول سهل هذه لو سميناها
+
+260
+00:19:50,840 --> 00:19:55,720
+الزاوية θ اللي بينهم بين u وv بين الـ u والـ v
+
+261
+00:19:55,720 --> 00:20:00,820
+سميناها θ بيصير عندي الآن هذه الـ θ إيه يا
+
+262
+00:20:00,820 --> 00:20:06,740
+جماعة هي الزاوية بين الـ v وبين الـ u طيب لو جينا
+
+263
+00:20:06,740 --> 00:20:12,370
+أخذنا كوساين الـ θ هذه cos θ هي تساوي اللي هو الـ
+
+264
+00:20:12,370 --> 00:20:16,070
+magnitude للـ projection اللي أنا بدي إياها الـ
+
+265
+00:20:16,070 --> 00:20:20,930
+magnitude هذا على الـ magnitude للـ U مظبوط هذا الـ cos
+
+266
+00:20:20,930 --> 00:20:27,310
+θ يعني وكأني بقول إن magnitude للـ projection يساوي
+
+267
+00:20:27,310 --> 00:20:34,190
+الـ magnitude للـ U في cos θ ماشي إذا الـ magnitude
+
+268
+00:20:34,190 --> 00:20:39,810
+للـ projection هذا يساوي الـ magnitude للـ U في cos θ
+
+269
+00:20:39,810 --> 00:20:47,510
+طيب هذا مين؟ إذا قيمة اللي هو الـ magnitude للـ
+
+270
+00:20:47,510 --> 00:20:53,750
+projection ماشي طيب عشان أوجد الـ projection نفسه
+
+271
+00:20:53,750 --> 00:20:59,190
+هذا طوله أوجدته هذا U  ضرب مين؟ اضربه في اتجاهه
+
+272
+00:20:59,190 --> 00:21:06,970
+اتجاهه هذا مين هو؟ هو عبارة عن اتجاه الـ V إيش اتجاه
+
+273
+00:21:06,970 --> 00:21:09,570
+الـ V اللي هو الـ V على الـ magnitude للـ V إذاً إن
+
+274
+00:21:09,570 --> 00:21:13,230
+تحدد الـ projection للـ U في اتجاه الـ V الـ
+
+275
+00:21:13,230 --> 00:21:16,910
+projection للـ U في اتجاه الـ V يساوي الـ magnitude
+
+276
+00:21:16,910 --> 00:21:21,090
+للـ U cos θ أو الـ projection للـ U على الـ V هو الـ
+
+277
+00:21:21,090 --> 00:21:25,370
+magnitude للـ U cos θ في الـ V على الـ magnitude
+
+278
+00:21:25,370 --> 00:21:30,530
+للـ V هذا في حالة الزاوية حادة اللي هي θ هنا اللي
+
+279
+00:21:30,530 --> 00:21:35,480
+هو الزاوية بين الاثنين U و V حادة ويساوي الآن هذا
+
+280
+00:21:35,480 --> 00:21:43,200
+الـ magnitude للـ U في الـ V على الـ magnitude للـ V في
+
+281
+00:21:43,200 --> 00:21:51,640
+كوساين من θ هذا يمثل إيش؟ هذا عبارة عن إيش؟ هذا
+
+282
+00:21:51,640 --> 00:21:59,850
+عرفناها قبل شوية  لو جينا حطينا مكان هذا اللي
+
+283
+00:21:59,850 --> 00:22:06,850
+هو حيكون عبارة عن U.V على الـ magnitude لـ V  ولا لا
+
+284
+00:22:06,850 --> 00:22:11,370
+لأن U.V  تساويه  الـ magnitude للـ U في
+
+285
+00:22:11,370 --> 00:22:18,010
+الـ magnitude للـ V في cos θ كمان مرة الآن هذا بنعوض
+
+286
+00:22:18,010 --> 00:22:22,070
+عنه إيش قيمته؟ U.V على الـ magnitude للـ V من وين
+
+287
+00:22:22,070 --> 00:22:25,670
+جبته؟ جبته من الخاصية اللي قلناها قبل هيك U.V
+
+288
+00:22:25,670 --> 00:22:28,830
+يساوي الـ magnitude للـ U في الـ magnitude للـ V في
+
+289
+00:22:28,830 --> 00:22:32,510
+كوساين الزاوية بينهم طيب في الـ V على الـ magnitude للـ
+
+290
+00:22:32,510 --> 00:22:39,450
+V صار هذا إيش يساوي؟ اللي هو طبعاً هنذكركم هذا إيش
+
+291
+00:22:39,450 --> 00:22:45,010
+هذا يساوي اللي هو U.V على الـ magnitude للـ V تربيع
+
+292
+00:22:45,010 --> 00:22:49,040
+في V إذاً من الآن وطالع الـ projection للـ U في
+
+293
+00:22:49,040 --> 00:22:53,360
+اتجاه الـ V هو عبارة عن U.V على الـ magnitude V
+
+294
+00:22:53,360 --> 00:23:00,100
+تربيع في من؟ في V هذا عبارة عن عدد قياسي وهذا
+
+295
+00:23:00,100 --> 00:23:04,120
+اللي هو من الـ V الـ vector اللي بدنا نوجد الـ U
+
+296
+00:23:04,120 --> 00:23:09,760
+عليه هذا اللي هو الـ projection للـ U on V دائماً
+
+297
+00:23:09,760 --> 00:23:13,540
+يساوي هذا الكلام أنا بقول هذا يساوي هذا الكلام
+
+298
+00:23:13,540 --> 00:23:18,910
+سواء زاوية حادة أو زاوية منفرجة يعني هذه الحالة
+
+299
+00:23:18,910 --> 00:23:22,190
+برضه بنطبق عليها خلينا نشوف كيف شوفوا يا جماعة
+
+300
+00:23:22,190 --> 00:23:25,830
+صنعنا بالنسبة للـ حالة هذه الـ vector u الحالة الثانية
+
+301
+00:23:25,830 --> 00:23:29,050
+وهي الـ vector v وهي الزاوية اللي بينهم إيش اسمها؟
+
+302
+00:23:29,050 --> 00:23:33,010
+هذه θ هذه الزاوية الـ θ الآن الـ projection
+
+303
+00:23:33,010 --> 00:23:37,790
+بنزل عمود هي عمود فبيطلع هي الـ projection الآن هذا
+
+304
+00:23:37,790 --> 00:23:43,510
+الـ projection بنفس الطريقة الـ projection للـ u في
+
+305
+00:23:43,510 --> 00:23:50,060
+اتجاه الـ v هذا يساوي magnitude في اتجاهه إيش اتجاهه؟
+
+306
+00:23:50,060 --> 00:23:54,200
+هو الـ projection للـ U على الـ V على الـ magnitude
+
+307
+00:23:54,200 --> 00:23:58,240
+ليه؟ مش هذا لكل...ل... هذه... هذه... لكل متجهة في
+
+308
+00:23:58,240 --> 00:24:03,540
+الدنيا هو اتجاهه الـ vector نفسه على magnitude طيب
+
+309
+00:24:04,140 --> 00:24:10,240
+الآن بدنا نحدد هذه وبدنا نحدد هذه طيب لاحظ معي
+
+310
+00:24:10,240 --> 00:24:13,800
+الآن هذه الزاوية θ اللي بين المتجهين برضه بدي
+
+311
+00:24:13,800 --> 00:24:17,280
+أسميها إيش؟ θ بين المتجهين إيش بدي أسميها؟ θ
+
+312
+00:24:17,280 --> 00:24:20,100
+يعني هذه θ عبارة عن اللي بين الـ U والـ V سواء
+
+313
+00:24:20,100 --> 00:24:24,750
+هنا أو هنا طيب الزاوية θ هذه اللي بين U و V هنا
+
+314
+00:24:24,750 --> 00:24:29,790
+إيش حتكون؟ 180 ناقص θ احنا بنعرف كوساين
+
+315
+00:24:29,790 --> 00:24:34,130
+الـ 180 ناقص θ يساوي ناقص كوساين الـ θ
+
+316
+00:24:34,130 --> 00:24:38,750
+أكيد اضرب الجهتين في الـ magnitude للـ U بيصير الـ
+
+317
+00:24:38,750 --> 00:24:41,210
+magnitude للـ U في كوساين الـ θ يساوي ناقص الـ
+
+318
+00:24:41,210 --> 00:24:45,590
+magnitude للـ U في كوساين 180 ناقص θ
+
+319
+00:24:45,590 --> 00:24:52,860
+طيب الآن هذه اللي هي زي ما عملناها بالظبط الكوساين
+
+320
+00:24:52,860 --> 00:24:59,400
+اللي هي 180 ناقص θ إيش بتساوي؟ بتساوي الـ
+
+321
+00:24:59,400 --> 00:25:04,700
+magnitude لهذا على الـ magnitude للـ U يعني بمعنى
+
+322
+00:25:04,700 --> 00:25:11,560
+آخرهيكون عندي اللي هو الـ magnitude للـ U في
+
+323
+00:25:11,560 --> 00:25:14,700
+كوساين الـ θ magnitude للـ U في كوساين as of
+
+324
+00:25:14,700 --> 00:25:19,000
+π ناقص θ magnitude للـ U في كوساين هذه
+
+325
+00:25:19,000 --> 00:25:24,200
+الزاوية هي يساوي الـ magnitude لهذا طول هذا لأن الـ
+
+326
+00:25:24,200 --> 00:25:32,630
+كوساين هذه يساوي هذا على هذا يعني الآن هذا هي هذا
+
+327
+00:25:32,630 --> 00:25:35,730
+البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن هو
+
+328
+00:25:35,730 --> 00:25:37,090
+هذا البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن
+
+329
+00:25:37,090 --> 00:25:39,950
+هو هذا البروجيكشن هو هذا البروجيكشن هو هذا
+
+330
+00:25:39,950 --> 00:25:43,470
+البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن هو
+
+331
+00:25:43,470 --> 00:25:46,170
+هذا البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن
+
+332
+00:25:46,170 --> 00:25:47,350
+هو هذا البروجيكشن هو هذا البروجيكشن هو هذا
+
+333
+00:25:47,350 --> 00:25:47,850
+البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن هو
+
+334
+00:25:47,850 --> 00:25:48,450
+هذا البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن
+
+335
+00:25:48,450 --> 00:25:51,990
+هو هذا البروجيكشن هو هذا البروجيكشن هو هذا
+
+336
+00:25:51,990 --> 00:26:03,700
+البروجيكشن الـ magnitude له ناقص هذا المقدار ناقص
+
+337
+00:26:03,700 --> 00:26:05,000
+هذا المقدار ناقص هذا المقدار ناقص هذا المقدار ناقص
+
+338
+00:26:05,000 --> 00:26:06,460
+هذا المقدار ناقص هذا المقدار ناقص هذا المقدار ناقص
+
+339
+00:26:06,460 --> 00:26:07,100
+هذا المقدار ناقص هذا المقدار ناقص هذا المقدار ناقص
+
+340
+00:26:07,100 --> 00:26:11,280
+هذا المقدار ناقص هذا المقدار ناقص هذا المقدار ناقص
+
+341
+00:26:11,280 --> 00:26:12,460
+هذا المقدار ناقص هذا المقدار ناقص هذا المقدار ناقص
+
+342
+00:26:12,460 --> 00:26:13,000
+هذا المقدار ناقص هذا المقدار ناقص هذا المقدار ناقص
+
+343
+00:26:13,000 --> 00:26:13,500
+هذا المقدار ناقص هذا المقدار ناقص هذا المقدار ناقص
+
+344
+00:26:13,500 --> 00:26:13,920
+هذا المقدار ناقص هذا المقدار ناقص هذا المقدار ناقص
+
+345
+00:26:13,920 --> 00:26:19,560
+هذا المقدار ناقص هذا المقدار ناقص هذا المقدار ناقص
+
+346
+00:26:20,500 --> 00:26:25,360
+المتجه في اتجاه الـ projection عكس الـ V إذا بدي
+
+347
+00:26:25,360 --> 00:26:30,180
+أشيله هذا وأكتب قيمته بدلالة هذا إيش حيساوي؟ ناقص
+
+348
+00:26:30,180 --> 00:26:34,400
+الـ V على magnitude V يعني هذا المتجه عكس هذا هذا
+
+349
+00:26:34,400 --> 00:26:38,600
+متجه واحد وهذا متجه واحد إذا هذا يساوي ناقص V
+
+350
+00:26:38,600 --> 00:26:44,160
+على magnitude V لأن هذا يعبر عن المتجه هنا متجه
+
+351
+00:26:44,160 --> 00:26:48,020
+الوحدة ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+352
+00:26:48,020 --> 00:26:48,340
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+353
+00:26:48,340 --> 00:26:51,520
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+354
+00:26:51,520 --> 00:26:53,380
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+355
+00:26:53,380 --> 00:26:55,000
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+356
+00:26:55,000 --> 00:26:56,720
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+357
+00:26:56,720 --> 00:27:12,540
+ده ده ده ده ده ده ده
+
+358
+00:27:12,540 --> 00:27:16,070
+ده في الـ V على الـ magnitude للـ V إذاً صار عندي
+
+359
+00:27:16,070 --> 00:27:19,430
+هذا الـ projection يساوي هذا حتى في الحالة الثانية
+
+360
+00:27:19,430 --> 00:27:24,130
+اللي هو هذه القيمة اللي وصلنا لها زي ما قبل شوية
+
+361
+00:27:24,130 --> 00:27:29,870
+لهذا إذاً أيّاً كانت الزاوية زاوية θ حادة هنا أو
+
+362
+00:27:29,870 --> 00:27:32,870
+هنا الزاوية بينهم أيّاً كانت هذه الزاوية أو هذه
+
+363
+00:27:32,870 --> 00:27:39,190
+دائماً الـ projection للـ U في اتجاه الـ V يساوي U.
+
+364
+00:27:39,190 --> 00:27:40,490
+V على الـ magnitude للـ V
+
+365
+00:27:43,360 --> 00:27:47,860
+حفظنا القانون هذا وعرفنا كيف يأتي لأنه يجي نطبق
+
+366
+00:27:47,860 --> 00:27:52,020
+عليه شوفوا يا جماعة من اللي حكيته قبل شوية واضح في
+
+367
+00:27:52,020 --> 00:27:57,980
+حالة الـ length هنا الـ length هنا الطول هنا الطول
+
+368
+00:27:57,980 --> 00:28:03,380
+هذا عبارة عن طلع عند مين؟ مالّه؟ الـ magnitude  لـ U
+
+369
+00:28:03,380 --> 00:28:11,280
+كوساين θ هذا طول هذا لكن الطول هنا طلع ناقص الـ
+
+370
+00:28:11,280 --> 00:28:14,300
+magnitude للـ U في كوساين الـ θ هي والـ
+
+371
+00:28:14,300 --> 00:28:19,640
+projection هذا طول الـ magnitude للـ projection
+
+372
+00:28:19,640 --> 00:28:26,260
+طلع ناقص هذا المقدار إذاً الآن الملاحظة التالية إيش؟
+
+373
+00:28:26,260 --> 00:28:31,430
+الملاحظة يا جماعة إنه في حالة θ هي الـ U وهي الـ V
+
+374
+00:28:31,430 --> 00:28:34,910
+والـ θ حادة بيطلع الـ length يساوي الـ magnitude لـ
+
+375
+00:28:34,910 --> 00:28:38,850
+الـ U في cos θ لكن في الحالة الثانية للـ length للـ
+
+376
+00:28:38,850 --> 00:28:42,490
+الـ projection هذا إيش طلع؟ معناه ناقص الـ magnitude
+
+377
+00:28:42,490 --> 00:28:47,590
+للـ U في cos θ لأن cos θ لما تكون منفرجة بتكون
+
+378
+00:28:47,590 --> 00:28:48,070
+سالبة
+
+379
+00:28:52,840 --> 00:28:57,380
+الملخص الحديث The vector projection of U onto V is
+
+380
+00:28:57,380 --> 00:29:02,230
+the vector  الـ projection للـ U onto V يساوي الـ U.
+
+381
+00:29:02,230 --> 00:29:07,990
+V على الـ magnitude للـ V تربيع في من؟ في الـ V هذا
+
+382
+00:29:07,990 --> 00:29:13,990
+عبارة عن scalar مضروب في من؟ أو موضوع في اتجاه اللي
+
+383
+00:29:13,990 --> 00:29:19,250
+هو مضروب في الـ vector V الآن في حاجة بنسميها the
+
+384
+00:29:19,250 --> 00:29:25,040
+scalar component of U in the direction of V ما هي
+
+385
+00:29:25,040 --> 00:29:27,600
+الـ scalar component اللي نعرفها in the direction
+
+386
+00:29:27,600 --> 00:29:32,400
+of V؟ اللي عبارة عن U.V على مين؟ على V لأن هذا
+
+387
+00:29:32,400 --> 00:29:37,160
+الـ projection عبارة عن حاصل ضرب U.V على الـ
+
+388
+00:29:37,160 --> 00:29:43,000
+magnitude للـ V في اتجاه V على V مظبوط؟
+
+389
+00:29:43,000 --> 00:29:45,820
+الآن اللي... اللي... اللي... اللي... اللي بيضل هذا
+
+390
+00:29:45,820 --> 00:29:50,510
+...هذا اللي هو سيبك من الـ V على الـ magnitude V
+
+391
+00:29:50,510 --> 00:29:54,010
+للـ V اللي هو اللي بي
+
+445
+00:34:16,120 --> 00:34:22,810
+هي V واحد و تسحق عشرة جدر العشرة واضح أه؟ هذا تربيع
+
+446
+00:34:22,810 --> 00:34:26,610
+زادت تربيع زادت عشرة بيطلع اللي هو magnitude لل V
+
+447
+00:34:26,610 --> 00:34:30,590
+لأن ال vector projection من F onto V  ال projection
+
+448
+00:34:30,590 --> 00:34:36,110
+لل F onto V يساوي F dot V على magnitude V تربيع
+
+449
+00:34:36,110 --> 00:34:42,170
+V F dot V اللي هو ناقص واحد على magnitude لل V
+
+450
+00:34:42,170 --> 00:34:46,310
+تربيع اللي هو عشرة في V I ناقص ثلاثة G لأن ال
+
+451
+00:34:46,310 --> 00:34:47,990
+scalar component of U
+
+452
+00:34:58,140 --> 00:35:08,220
+النقص الطولي في هذه الحالة هو نقص 1 على 10 وهكذا
+
+453
+00:35:08,220 --> 00:35:11,700
+نكون أنهينا
+
+454
+00:35:11,700 --> 00:35:17,320
+ال section 12-3 of the product وهي ال homework
+
+455
+00:35:17,320 --> 00:35:21,640
+وإلى لقاء آخر السلام عليكم ورحمة الله وبركاته
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/bTW6N1MaENg_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/bTW6N1MaENg_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..955de5ce1d63da46be8920005e3c7b220903a7eb
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/bTW6N1MaENg_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 1696, "start": 1.08, "end": 16.96, "text": "بسم الله الرحمن الرحيم هذه هي المحاضرة الثالثة مساق تفاضل وتكامل ال gym الفصل الصيفي 2020 العنوان المحاضرة اليوم اللي هو the dot product", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 29538, 39896, 9673, 5016, 46958, 25720, 6024, 104, 6027, 12984, 3660, 47524, 995, 4587, 6055, 5172, 46958, 1211, 34683, 4117, 10943, 1211, 2423, 9222, 27188, 9381, 1211, 31767, 33911, 1829, 4808, 18863, 1863, 2407, 7649, 9673, 5016, 46958, 25720, 45595, 20498, 13672, 1829, 31439, 264, 5893, 1674], "avg_logprob": -0.132264258568747, "compression_ratio": 1.5369127516778522, "no_speech_prob": 5.960464477539062e-07, "words": [{"start": 1.08, "end": 1.52, "word": "بسم", "probability": 0.84228515625}, {"start": 1.52, "end": 1.78, "word": " الله", "probability": 0.95751953125}, {"start": 1.78, "end": 2.24, "word": " الرحمن", "probability": 0.97705078125}, {"start": 2.24, "end": 2.88, "word": " الرحيم", 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قلنا non-zero vector هذا لا يساوي سفر وهذا لا يساوي سفر فتا اللي بينهم يساوي cosine inverse لأ U واحد V واحد زائد U اتنين V اتنين زائد U تلاتة V تلاتة", "tokens": [6027, 11622, 995, 2407, 10632, 13672, 1829, 49374, 16095, 13672, 1829, 39896, 307, 2212, 538, 13672, 1829, 31439, 18871, 16254, 23032, 3555, 3615, 995, 2423, 624, 4032, 2423, 691, 30767, 1829, 19446, 12174, 1211, 8315, 2107, 12, 32226, 8062, 23758, 20193, 7251, 3794, 995, 45865, 8608, 5172, 2288, 37037, 15730, 20193, 7251, 3794, 995, 45865, 8608, 5172, 2288, 6156, 2655, 995, 13672, 1829, 49374, 16095, 7251, 3794, 995, 45865, 23565, 17340, 5296, 10721, 624, 36764, 24401, 691, 36764, 24401, 30767, 16373, 3215, 624, 1975, 2655, 1863, 9957, 691, 1975, 2655, 1863, 9957, 30767, 16373, 3215, 624, 6055, 1211, 9307, 3660, 691, 6055, 1211, 9307, 3660], "avg_logprob": -0.16553655688492758, "compression_ratio": 1.816326530612245, "no_speech_prob": 0.0, "words": [{"start": 91.71, "end": 92.51, "word": "الزاوية", "probability": 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"probability": 0.51025390625}, {"start": 98.37, "end": 98.51, "word": " و", "probability": 0.82666015625}, {"start": 98.51, "end": 98.55, "word": " ال", "probability": 0.460693359375}, {"start": 98.55, "end": 98.65, "word": " V", "probability": 0.98046875}, {"start": 98.65, "end": 98.83, "word": " زي", "probability": 0.764892578125}, {"start": 98.83, "end": 98.91, "word": " ما", "probability": 0.98193359375}, {"start": 98.91, "end": 99.29, "word": " قلنا", "probability": 0.8751627604166666}, {"start": 99.29, "end": 99.51, "word": " non", "probability": 0.6806640625}, {"start": 99.51, "end": 99.79, "word": "-zero", "probability": 0.556640625}, {"start": 99.79, "end": 100.11, "word": " vector", "probability": 0.92431640625}, {"start": 100.11, "end": 100.47, "word": " هذا", "probability": 0.8203125}, {"start": 100.47, "end": 100.69, "word": " لا", "probability": 0.91015625}, {"start": 100.69, "end": 101.03, "word": " يساوي", "probability": 0.8397216796875}, {"start": 101.03, "end": 101.33, "word": " سفر", "probability": 0.8391927083333334}, {"start": 101.33, "end": 101.67, "word": " وهذا", "probability": 0.85888671875}, {"start": 101.67, "end": 101.87, "word": " لا", "probability": 0.9365234375}, {"start": 101.87, "end": 102.47, "word": " يساوي", "probability": 0.9676513671875}, {"start": 102.47, "end": 102.95, "word": " سفر", "probability": 0.9912109375}, {"start": 102.95, "end": 103.89, "word": " فتا", "probability": 0.522216796875}, {"start": 103.89, "end": 104.59, "word": " اللي", "probability": 0.962158203125}, {"start": 104.59, "end": 104.91, "word": " بينهم", "probability": 0.971923828125}, {"start": 104.91, "end": 105.31, "word": " يساوي", "probability": 0.909912109375}, {"start": 105.31, "end": 105.63, "word": " cosine", "probability": 0.77001953125}, {"start": 105.63, "end": 106.59, "word": " inverse", "probability": 0.76220703125}, {"start": 106.59, "end": 107.65, "word": " لأ", "probability": 0.7144775390625}, {"start": 107.65, "end": 107.97, "word": " U", "probability": 0.87451171875}, {"start": 107.97, "end": 108.67, "word": " واحد", "probability": 0.7205810546875}, {"start": 108.67, "end": 109.17, "word": " V", "probability": 0.9560546875}, {"start": 109.17, "end": 109.75, "word": " واحد", "probability": 0.9921875}, {"start": 109.75, "end": 110.35, "word": " زائد", "probability": 0.7998046875}, {"start": 110.35, "end": 110.55, "word": " U", "probability": 0.98193359375}, {"start": 110.55, "end": 111.13, "word": " اتنين", "probability": 0.84283447265625}, {"start": 111.13, "end": 111.47, "word": " V", "probability": 0.97314453125}, {"start": 111.47, "end": 111.95, "word": " اتنين", "probability": 0.9556884765625}, {"start": 111.95, "end": 112.35, "word": " زائد", "probability": 0.9695638020833334}, {"start": 112.35, "end": 112.51, "word": " U", "probability": 0.96875}, {"start": 112.51, "end": 113.19, "word": " تلاتة", "probability": 0.96826171875}, {"start": 113.19, "end": 113.49, "word": " V", "probability": 0.98779296875}, {"start": 113.49, "end": 114.21, "word": " تلاتة", "probability": 0.9844970703125}], "temperature": 1.0}, {"id": 7, "seek": 13979, "start": 114.61, "end": 139.79, "text": "هذا الآن cosine inverse لهذا على ال magnitude لل U في ال magnitude لل V هذه هي الزاوية theta بين المتجهين هذه النظرية بدنا نعرفها لإنه احنا هنستخدمها في اللي هو بعد شوية في نتائج هاتهمنا اللي هي بخصوص اللي هو ال dot product", "tokens": [3224, 15730, 6024, 48506, 23565, 17340, 46740, 15730, 15844, 2423, 15668, 24976, 624, 8978, 2423, 15668, 24976, 691, 29538, 39896, 6024, 110, 995, 2407, 10632, 9725, 49374, 9673, 2655, 7435, 3224, 9957, 29538, 28239, 19913, 2288, 10632, 47525, 8315, 8717, 3615, 28480, 11296, 5296, 28814, 1863, 3224, 1975, 5016, 8315, 8032, 1863, 14851, 9778, 40448, 11296, 8978, 13672, 1829, 31439, 39182, 13412, 2407, 10632, 8978, 8717, 2655, 16373, 7435, 8032, 9307, 16095, 8315, 13672, 1829, 39896, 4724, 9778, 9381, 2407, 9381, 13672, 1829, 31439, 2423, 5893, 1674], "avg_logprob": 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125.29, "end": 125.63, "word": " بدنا", "probability": 0.891845703125}, {"start": 125.63, "end": 126.19, "word": " نعرفها", "probability": 0.9920654296875}, {"start": 126.19, "end": 126.77, "word": " لإنه", "probability": 0.8350830078125}, {"start": 126.77, "end": 127.19, "word": " احنا", "probability": 0.9005533854166666}, {"start": 127.19, "end": 128.77, "word": " هنستخدمها", "probability": 0.9337565104166666}, {"start": 128.77, "end": 129.31, "word": " في", "probability": 0.94140625}, {"start": 129.31, "end": 129.71, "word": " اللي", "probability": 0.937744140625}, {"start": 129.71, "end": 130.19, "word": " هو", "probability": 0.97314453125}, {"start": 130.19, "end": 132.25, "word": " بعد", "probability": 0.58935546875}, {"start": 132.25, "end": 132.91, "word": " شوية", "probability": 0.93017578125}, {"start": 132.91, "end": 133.43, "word": " في", "probability": 0.88916015625}, {"start": 133.43, "end": 133.95, "word": " نتائج", "probability": 0.9903564453125}, {"start": 133.95, "end": 134.89, "word": " هاتهمنا", "probability": 0.77496337890625}, {"start": 134.89, "end": 136.51, "word": " اللي", "probability": 0.89453125}, {"start": 136.51, "end": 136.89, "word": " هي", "probability": 0.86181640625}, {"start": 136.89, "end": 138.29, "word": " بخصوص", "probability": 0.9650390625}, {"start": 138.29, "end": 138.65, "word": " اللي", "probability": 0.965576171875}, {"start": 138.65, "end": 138.87, "word": " هو", "probability": 0.9892578125}, {"start": 138.87, "end": 139.03, "word": " ال", "probability": 0.83447265625}, {"start": 139.03, "end": 139.33, "word": " dot", "probability": 0.77490234375}, {"start": 139.33, "end": 139.79, "word": " product", "probability": 0.908203125}], "temperature": 1.0}, {"id": 8, "seek": 16113, "start": 140.54, "end": 161.14, "text": "الان definition تعريف الـ dot product بنا نعرف ال dot product أو الضرب القياسي ل ال U و ال V طبعا احنا هنعرف بعد المحاضرة القادمة كمان ان في عندنا two dot .. two products بين ال vectors احنا عرفنا شو معناة الجمع two vectors شو معناة scalar في ال vector الاخره", "tokens": [6027, 7649, 7123, 37279, 16572, 5172, 2423, 39184, 5893, 1674, 4724, 8315, 8717, 3615, 28480, 2423, 5893, 1674, 34051, 6024, 114, 25513, 25062, 25528, 3794, 1829, 5296, 2423, 624, 4032, 2423, 691, 23032, 3555, 3615, 995, 1975, 5016, 8315, 8032, 1863, 3615, 28480, 39182, 9673, 5016, 46958, 25720, 25062, 18513, 46007, 9122, 2304, 7649, 16472, 8978, 43242, 8315, 732, 5893, 4386, 732, 3383, 49374, 2423, 18875, 1975, 5016, 8315, 6225, 28480, 8315, 13412, 2407, 20449, 8315, 3660, 25724, 2304, 3615, 732, 18875, 13412, 2407, 20449, 8315, 3660, 39684, 8978, 2423, 8062, 2423, 47283, 2288, 3224], "avg_logprob": -0.274088551911215, "compression_ratio": 1.7916666666666667, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 140.54, "end": 141.22, "word": "الان", "probability": 0.72216796875}, {"start": 141.22, "end": 141.74, "word": " definition", "probability": 0.327880859375}, {"start": 141.74, "end": 142.62, 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"probability": 0.9625651041666666}, {"start": 155.52, "end": 156.18, "word": " شو", "probability": 0.976806640625}, {"start": 156.18, "end": 156.6, "word": " معناة", "probability": 0.74853515625}, {"start": 156.6, "end": 157.0, "word": " الجمع", "probability": 0.9114583333333334}, {"start": 157.0, "end": 157.2, "word": " two", "probability": 0.90771484375}, {"start": 157.2, "end": 157.64, "word": " vectors", "probability": 0.974609375}, {"start": 157.64, "end": 157.96, "word": " شو", "probability": 0.80810546875}, {"start": 157.96, "end": 158.3, "word": " معناة", "probability": 0.9689127604166666}, {"start": 158.3, "end": 159.26, "word": " scalar", "probability": 0.52001953125}, {"start": 159.26, "end": 159.48, "word": " في", "probability": 0.96044921875}, {"start": 159.48, "end": 159.56, "word": " ال", "probability": 0.53857421875}, {"start": 159.56, "end": 159.94, "word": " vector", "probability": 0.9462890625}, {"start": 159.94, "end": 161.14, "word": " الاخره", "probability": 0.680389404296875}], "temperature": 1.0}, {"id": 9, "seek": 18724, "start": 161.54, "end": 187.24, "text": "الان بدنا نعرف عملية ضرب بين two vectors عملية ضرب نوعين عندنا النوع الأول اللى هنحكي عنه اللى هو الـ dot product أو الضرب القياسى والنوع التانى هيجي لاحقا اللى هو الضرب الاتجاهى أو اللى بنسميه cross of product خلينا نتعرف اليوم على ال dot product u dot v الان u dot v اللى هو ال dot product u dot v of vectors u and v هى ال vector الأول", "tokens": [6027, 7649, 47525, 8315, 8717, 3615, 28480, 6225, 42213, 10632, 48812, 25513, 49374, 732, 18875, 6225, 42213, 10632, 48812, 25513, 8717, 45367, 9957, 43242, 8315, 28239, 45367, 16247, 12610, 13672, 7578, 8032, 1863, 5016, 4117, 1829, 18871, 3224, 13672, 7578, 31439, 2423, 39184, 5893, 1674, 34051, 6024, 114, 25513, 25062, 25528, 3794, 7578, 16070, 1863, 45367, 16712, 7649, 7578, 39896, 7435, 1829, 20193, 5016, 4587, 995, 13672, 7578, 31439, 6024, 114, 25513, 2423, 9307, 7435, 40294, 7578, 34051, 13672, 7578, 44945, 38251, 1829, 3224, 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"probability": 0.975830078125}, {"start": 306.7, "end": 306.98, "word": " واحد", "probability": 0.9892578125}, {"start": 306.98, "end": 307.28, "word": " زائد", "probability": 0.9651692708333334}, {"start": 307.28, "end": 307.66, "word": " اتنين", "probability": 0.9923095703125}, {"start": 307.66, "end": 308.04, "word": " بيطلع", "probability": 0.94150390625}, {"start": 308.04, "end": 308.52, "word": " جداش", "probability": 0.7874348958333334}, {"start": 308.52, "end": 308.86, "word": " واحد", "probability": 0.96630859375}, {"start": 308.86, "end": 309.44, "word": " هذا", "probability": 0.9384765625}, {"start": 309.44, "end": 309.6, "word": " اللي", "probability": 0.98681640625}, {"start": 309.6, "end": 309.8, "word": " هو", "probability": 0.9873046875}, {"start": 309.8, "end": 309.92, "word": " الـ", "probability": 0.572509765625}, {"start": 309.92, "end": 310.02, "word": " data", "probability": 0.179443359375}, {"start": 310.02, "end": 310.38, "word": " product", "probability": 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the angle between two vectors u and v is قبل بشوية قلنا النظرية ايش بتقول بتقول theta بسبب cosine inverse في u1 u2 زائد u v2 زائد u3 v3 اللي هو في الواقع ايش هو", "tokens": [11622, 1829, 19446, 16472, 2655, 14407, 13412, 995, 33911, 9957, 6225, 3555, 9640, 3660, 18871, 13672, 1829, 31439, 12174, 29973, 8608, 3224, 1211, 23032, 1829, 3555, 2423, 7649, 294, 264, 24657, 295, 264, 5893, 1674, 264, 5802, 1296, 732, 18875, 344, 293, 371, 307, 12174, 36150, 4724, 8592, 2407, 10632, 12174, 1211, 8315, 28239, 19913, 2288, 10632, 1975, 1829, 8592, 39894, 39648, 39894, 39648, 9725, 4724, 35457, 3555, 23565, 17340, 8978, 344, 16, 344, 17, 30767, 16373, 3215, 344, 371, 17, 30767, 16373, 3215, 344, 18, 371, 18, 13672, 1829, 31439, 8978, 2423, 14407, 4587, 3615, 1975, 1829, 8592, 31439], "avg_logprob": -0.21070544377411946, "compression_ratio": 1.5458515283842795, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 313.3, "end": 313.88, "word": "زي", "probability": 0.6207275390625}, 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320.12, "word": " the", "probability": 0.88037109375}, {"start": 320.12, "end": 320.54, "word": " notation", "probability": 0.95703125}, {"start": 320.54, "end": 320.76, "word": " of", "probability": 0.96484375}, {"start": 320.76, "end": 320.88, "word": " the", "probability": 0.8076171875}, {"start": 320.88, "end": 321.06, "word": " dot", "probability": 0.91064453125}, {"start": 321.06, "end": 321.62, "word": " product", "probability": 0.9765625}, {"start": 321.62, "end": 321.9, "word": " the", "probability": 0.62548828125}, {"start": 321.9, "end": 322.28, "word": " angle", "probability": 0.8779296875}, {"start": 322.28, "end": 322.6, "word": " between", "probability": 0.9306640625}, {"start": 322.6, "end": 322.94, "word": " two", "probability": 0.93310546875}, {"start": 322.94, "end": 323.38, "word": " vectors", "probability": 0.96875}, {"start": 323.38, "end": 323.66, "word": " u", "probability": 0.6533203125}, {"start": 323.66, "end": 323.88, "word": " and", "probability": 0.669921875}, {"start": 323.88, "end": 324.08, "word": " v", "probability": 0.91357421875}, {"start": 324.08, "end": 324.54, "word": " is", "probability": 0.822265625}, {"start": 324.54, "end": 325.24, "word": " قبل", "probability": 0.7841796875}, {"start": 325.24, "end": 325.7, "word": " بشوية", "probability": 0.9007568359375}, {"start": 325.7, "end": 326.42, "word": " قلنا", "probability": 0.83935546875}, {"start": 326.42, "end": 326.9, "word": " النظرية", "probability": 0.9217529296875}, {"start": 326.9, "end": 327.04, "word": " ايش", "probability": 0.7228190104166666}, {"start": 327.04, "end": 327.62, "word": " بتقول", "probability": 0.984619140625}, {"start": 327.62, "end": 328.3, "word": " بتقول", "probability": 0.6961669921875}, {"start": 328.3, "end": 328.6, "word": " theta", "probability": 0.19384765625}, {"start": 328.6, "end": 328.98, "word": " بسبب", "probability": 0.94677734375}, {"start": 328.98, "end": 329.34, "word": " cosine", "probability": 0.74853515625}, {"start": 329.34, "end": 329.9, "word": " inverse", "probability": 0.90478515625}, {"start": 329.9, "end": 330.42, "word": " في", "probability": 0.9140625}, {"start": 330.42, "end": 330.86, "word": " u1", "probability": 0.63427734375}, {"start": 330.86, "end": 331.52, "word": " u2", "probability": 0.703857421875}, {"start": 331.52, "end": 332.4, "word": " زائد", "probability": 0.646484375}, {"start": 332.4, "end": 332.82, "word": " u", "probability": 0.61767578125}, {"start": 332.82, "end": 336.64, "word": " v2", "probability": 0.3626708984375}, {"start": 336.64, "end": 336.98, "word": " زائد", "probability": 0.87890625}, {"start": 336.98, "end": 337.54, "word": " u3", "probability": 0.846923828125}, {"start": 337.54, "end": 338.3, "word": " v3", "probability": 0.9560546875}, {"start": 338.3, "end": 338.9, "word": " اللي", "probability": 0.971435546875}, {"start": 338.9, "end": 339.04, "word": " هو", "probability": 0.97998046875}, {"start": 339.04, "end": 339.22, "word": " في", "probability": 0.88037109375}, {"start": 339.22, "end": 339.84, "word": " الواقع", "probability": 0.9912109375}, {"start": 339.84, "end": 340.18, "word": " ايش", "probability": 0.8831380208333334}, {"start": 340.18, "end": 340.62, "word": " هو", "probability": 0.962890625}], "temperature": 1.0}, {"id": 16, "seek": 35035, "start": 341.07, "end": 350.35, "text": "اللي هو عبارة عن U ضد V علي magnitude لـ U في magnitude لـ V هى هى هى يا جماعة هى هى theta بساعة cosine inverse هذا هذا هو عبارة عن", "tokens": [6027, 20292, 31439, 6225, 3555, 9640, 3660, 18871, 624, 48812, 3215, 691, 25894, 15668, 5296, 39184, 624, 8978, 15668, 5296, 39184, 691, 8032, 7578, 8032, 7578, 8032, 7578, 35186, 10874, 15042, 27884, 8032, 7578, 8032, 7578, 9725, 4724, 3794, 995, 27884, 23565, 17340, 23758, 23758, 31439, 6225, 3555, 9640, 3660, 18871], "avg_logprob": -0.607572139455722, "compression_ratio": 1.536, "no_speech_prob": 6.496906280517578e-06, "words": [{"start": 341.07, "end": 341.45, "word": "اللي", "probability": 0.25213623046875}, {"start": 341.45, "end": 341.63, "word": " هو", "probability": 0.82861328125}, {"start": 341.63, "end": 342.01, "word": " عبارة", "probability": 0.7730560302734375}, {"start": 342.01, "end": 342.15, "word": " عن", "probability": 0.97900390625}, {"start": 342.15, "end": 342.31, "word": " U", "probability": 0.288330078125}, {"start": 342.31, "end": 342.69, "word": " ضد", "probability": 0.4366455078125}, {"start": 342.69, "end": 342.85, "word": " V", "probability": 0.953125}, {"start": 342.85, "end": 343.01, "word": " علي", "probability": 0.281494140625}, {"start": 343.01, "end": 343.43, "word": " magnitude", "probability": 0.5341796875}, {"start": 343.43, "end": 343.91, "word": " لـ", "probability": 0.35247802734375}, {"start": 343.91, "end": 344.09, "word": " U", "probability": 0.5}, {"start": 344.09, "end": 344.19, "word": " في", "probability": 0.89501953125}, {"start": 344.19, "end": 344.65, "word": " magnitude", "probability": 0.9296875}, {"start": 344.65, "end": 345.41, "word": " لـ", "probability": 0.75390625}, {"start": 345.41, "end": 345.61, "word": " V", "probability": 0.802734375}, {"start": 345.61, "end": 345.95, "word": " هى", "probability": 0.2698974609375}, {"start": 345.95, "end": 346.23, "word": " هى", "probability": 0.491455078125}, {"start": 346.23, "end": 346.27, "word": " هى", "probability": 0.61328125}, {"start": 346.27, "end": 346.33, "word": " يا", "probability": 0.375244140625}, {"start": 346.33, "end": 346.87, "word": " جماعة", "probability": 0.9871419270833334}, {"start": 346.87, "end": 347.41, "word": " هى", "probability": 0.823486328125}, {"start": 347.41, "end": 347.63, "word": " هى", "probability": 0.5506591796875}, {"start": 347.63, "end": 347.83, "word": " theta", "probability": 0.3232421875}, {"start": 347.83, "end": 348.15, "word": " بساعة", "probability": 0.700927734375}, {"start": 348.15, "end": 348.37, "word": " cosine", "probability": 0.3095703125}, {"start": 348.37, "end": 348.81, "word": " inverse", "probability": 0.8837890625}, {"start": 348.81, "end": 349.15, "word": " هذا", "probability": 0.5546875}, {"start": 349.15, "end": 349.59, "word": " هذا", "probability": 0.376953125}, {"start": 349.59, "end": 349.83, "word": " هو", "probability": 0.853515625}, {"start": 349.83, "end": 350.23, "word": " عبارة", "probability": 0.997314453125}, {"start": 350.23, "end": 350.35, "word": " عن", "probability": 0.98583984375}], "temperature": 1.0}, {"id": 17, "seek": 37396, "start": 351.02, "end": 373.96, "text": "هذا عبارة عن U.V اذا زي ما قلنا الان ال U.V استبدلناها بالكادر الموجود U1 V1 زي U2 V2 زي U3 V3 على ال magnitude ل U في ال magnitude ل V اذا صارت θ مكتوبة بتساوي cosine inverse ل اللي أمامي", "tokens": [3224, 15730, 6225, 3555, 9640, 3660, 18871, 624, 13, 53, 1975, 15730, 30767, 1829, 19446, 12174, 1211, 8315, 2423, 7649, 2423, 624, 13, 53, 44713, 44510, 1211, 8315, 11296, 20666, 4117, 18513, 2288, 9673, 29245, 23328, 624, 16, 691, 16, 30767, 1829, 624, 17, 691, 17, 30767, 1829, 624, 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354.9, "end": 355.46, "word": " قلنا", "probability": 0.91162109375}, {"start": 355.46, "end": 356.34, "word": " الان", "probability": 0.607421875}, {"start": 356.34, "end": 356.56, "word": " ال", "probability": 0.7109375}, {"start": 356.56, "end": 356.84, "word": " U", "probability": 0.66357421875}, {"start": 356.84, "end": 357.6, "word": ".V", "probability": 0.923583984375}, {"start": 357.6, "end": 358.82, "word": " استبدلناها", "probability": 0.931640625}, {"start": 358.82, "end": 359.52, "word": " بالكادر", "probability": 0.514404296875}, {"start": 359.52, "end": 359.98, "word": " الموجود", "probability": 0.8001302083333334}, {"start": 359.98, "end": 361.42, "word": " U1", "probability": 0.444580078125}, {"start": 361.42, "end": 362.22, "word": " V1", "probability": 0.6829833984375}, {"start": 362.22, "end": 362.5, "word": " زي", "probability": 0.51947021484375}, {"start": 362.5, "end": 362.86, "word": " U2", "probability": 0.855712890625}, {"start": 362.86, "end": 363.36, "word": 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" لهذا", "probability": 0.993408203125}, {"start": 576.38, "end": 576.66, "word": " إذاً", "probability": 0.8859049479166666}, {"start": 576.66, "end": 576.82, "word": " أول", "probability": 0.950927734375}, {"start": 576.82, "end": 577.0, "word": " إشي", "probability": 0.7374674479166666}, {"start": 577.0, "end": 577.18, "word": " بدي", "probability": 0.4791259765625}, {"start": 577.18, "end": 577.38, "word": " أوجد", "probability": 0.8675130208333334}, {"start": 577.38, "end": 577.5, "word": " ال", "probability": 0.970703125}, {"start": 577.5, "end": 577.7, "word": " dot", "probability": 0.74462890625}, {"start": 577.7, "end": 578.02, "word": " product", "probability": 0.97705078125}, {"start": 578.02, "end": 578.34, "word": " بين", "probability": 0.98046875}, {"start": 578.34, "end": 578.62, "word": " هذا", "probability": 0.97802734375}, {"start": 578.62, "end": 578.72, "word": " و", "probability": 0.82177734375}, {"start": 578.72, "end": 578.86, "word": " بين", "probability": 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"word": " او", "probability": 0.7568359375}, {"start": 635.13, "end": 635.39, "word": " 1", "probability": 0.35791015625}, {"start": 635.39, "end": 636.11, "word": ".630", "probability": 0.8033854166666666}, {"start": 636.11, "end": 636.73, "word": "%", "probability": 0.28564453125}, {"start": 636.73, "end": 637.47, "word": " radians", "probability": 0.878662109375}, {"start": 637.47, "end": 637.67, "word": " هذا", "probability": 0.422607421875}, {"start": 637.67, "end": 638.07, "word": " بيقاش", "probability": 0.5281982421875}, {"start": 638.07, "end": 638.53, "word": " الدماغ", "probability": 0.79803466796875}, {"start": 638.53, "end": 639.31, "word": " لما", "probability": 0.895263671875}, {"start": 639.31, "end": 639.83, "word": " توجدوه", "probability": 0.777490234375}, {"start": 639.83, "end": 640.29, "word": " توجدوه", "probability": 0.80517578125}, {"start": 640.29, "end": 640.41, "word": " عن", "probability": 0.93310546875}, {"start": 640.41, "end": 641.11, "word": " طريق", "probability": 0.9952799479166666}, {"start": 641.11, "end": 641.73, "word": " الألة", "probability": 0.6505126953125}], "temperature": 1.0}, {"id": 29, "seek": 66405, "start": 642.97, "end": 664.05, "text": "الان بدنا نعرف شو معناته two vectors are orthogonal او two vectors are perpendicular يعني بدنا نعرف two vectors كيف يكونوا متعامدين او متعامدين او متعامدين او متعامدين", "tokens": [6027, 7649, 47525, 8315, 8717, 3615, 28480, 13412, 2407, 20449, 8315, 47395, 732, 18875, 366, 41488, 1975, 2407, 732, 18875, 366, 26734, 37495, 22653, 47525, 8315, 8717, 3615, 28480, 732, 18875, 9122, 33911, 7251, 30544, 14407, 44650, 3615, 10943, 3215, 9957, 1975, 2407, 44650, 3615, 10943, 3215, 9957, 1975, 2407, 44650, 3615, 10943, 3215, 9957, 1975, 2407, 44650, 3615, 10943, 3215, 9957], "avg_logprob": -0.238963289393319, "compression_ratio": 1.837037037037037, "no_speech_prob": 0.0, "words": [{"start": 642.97, "end": 643.59, "word": "الان", "probability": 0.729736328125}, {"start": 643.59, "end": 643.91, "word": " بدنا", "probability": 0.69384765625}, {"start": 643.91, "end": 644.19, "word": " نعرف", "probability": 0.982421875}, {"start": 644.19, "end": 644.49, "word": " شو", "probability": 0.786376953125}, {"start": 644.49, "end": 645.19, "word": " معناته", "probability": 0.7704264322916666}, {"start": 645.19, "end": 645.81, "word": " two", "probability": 0.63623046875}, {"start": 645.81, "end": 646.49, "word": " vectors", "probability": 0.93212890625}, {"start": 646.49, "end": 646.87, "word": " are", "probability": 0.93408203125}, {"start": 646.87, "end": 647.67, "word": " orthogonal", "probability": 0.8701171875}, {"start": 647.67, "end": 648.33, "word": " او", "probability": 0.79736328125}, {"start": 648.33, "end": 648.53, "word": " two", "probability": 0.83251953125}, {"start": 648.53, "end": 649.05, "word": " vectors", "probability": 0.95068359375}, {"start": 649.05, "end": 649.47, "word": " are", "probability": 0.95361328125}, {"start": 649.47, "end": 650.17, "word": " perpendicular", "probability": 0.6337890625}, {"start": 650.17, "end": 651.33, "word": " يعني", "probability": 0.783935546875}, {"start": 651.33, "end": 651.59, "word": " بدنا", "probability": 0.9150390625}, {"start": 651.59, "end": 651.95, "word": " نعرف", "probability": 0.9860026041666666}, {"start": 651.95, "end": 652.17, "word": " two", "probability": 0.89453125}, {"start": 652.17, "end": 652.65, "word": " vectors", "probability": 0.96728515625}, {"start": 652.65, "end": 653.01, "word": " كيف", "probability": 0.96923828125}, {"start": 653.01, "end": 653.59, "word": " يكونوا", "probability": 0.7591959635416666}, {"start": 653.59, "end": 655.65, "word": " متعامدين", "probability": 0.846484375}, {"start": 655.65, "end": 656.55, "word": " او", "probability": 0.834716796875}, {"start": 656.55, "end": 660.31, "word": " متعامدين", "probability": 0.772607421875}, {"start": 660.31, "end": 660.61, "word": " او", "probability": 0.817138671875}, {"start": 660.61, "end": 660.61, "word": " متعامدين", "probability": 0.84619140625}, {"start": 660.61, "end": 660.61, "word": " او", "probability": 0.843505859375}, {"start": 660.61, "end": 664.05, "word": " متعامدين", "probability": 0.882275390625}], "temperature": 1.0}, {"id": 30, "seek": 68222, "start": 664.28, "end": 682.22, "text": "الدفينيشن الـ vectors u and v are orthogonal اللي هو الـ two vectors بكون orthogonal إذا كانت الزاوية بينهم تسعين درجة يعني تسعين درجة if and only if u dot v بساوية 0", "tokens": [6027, 3215, 5172, 9957, 1829, 8592, 1863, 2423, 39184, 18875, 344, 293, 371, 366, 41488, 13672, 1829, 31439, 2423, 39184, 732, 18875, 4724, 30544, 41488, 11933, 15730, 25961, 2655, 6024, 110, 995, 2407, 10632, 49374, 16095, 6055, 3794, 3615, 9957, 11778, 47341, 3660, 37495, 22653, 6055, 3794, 3615, 9957, 11778, 47341, 3660, 498, 293, 787, 498, 344, 5893, 371, 4724, 3794, 995, 2407, 10632, 1958], "avg_logprob": -0.3998579536423539, "compression_ratio": 1.4782608695652173, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 664.28, "end": 665.4, "word": "الدفينيشن", "probability": 0.6992710658482143}, {"start": 665.4, "end": 665.84, "word": " الـ", "probability": 0.194976806640625}, {"start": 665.84, "end": 666.3, "word": " vectors", "probability": 0.59228515625}, {"start": 666.3, "end": 666.68, "word": " u", "probability": 0.484130859375}, {"start": 666.68, "end": 667.04, "word": " and", "probability": 0.255126953125}, {"start": 667.04, "end": 667.34, "word": " v", "probability": 0.974609375}, {"start": 667.34, "end": 667.58, "word": " are", "probability": 0.3486328125}, {"start": 667.58, "end": 668.14, "word": " orthogonal", "probability": 0.8984375}, {"start": 668.14, "end": 668.68, "word": " اللي", "probability": 0.658203125}, {"start": 668.68, "end": 669.0, "word": " هو", "probability": 0.97216796875}, {"start": 669.0, "end": 669.74, "word": " الـ", "probability": 0.5977783203125}, {"start": 669.74, "end": 669.92, "word": " two", "probability": 0.78662109375}, {"start": 669.92, "end": 670.56, "word": " vectors", "probability": 0.93408203125}, {"start": 670.56, "end": 671.36, "word": " بكون", "probability": 0.7275390625}, {"start": 671.36, "end": 671.88, "word": " orthogonal", "probability": 0.414794921875}, {"start": 671.88, "end": 672.18, "word": " إذا", "probability": 0.909423828125}, {"start": 672.18, "end": 672.48, "word": " كانت", "probability": 0.986328125}, {"start": 672.48, "end": 672.78, "word": " الزاوية", "probability": 0.87509765625}, {"start": 672.78, "end": 673.2, "word": " بينهم", "probability": 0.775146484375}, {"start": 673.2, "end": 674.42, "word": " تسعين", "probability": 0.75323486328125}, {"start": 674.42, "end": 674.98, "word": " درجة", "probability": 0.9912109375}, {"start": 674.98, "end": 677.6, "word": " يعني", "probability": 0.55438232421875}, {"start": 677.6, "end": 678.28, "word": " تسعين", "probability": 0.9703369140625}, {"start": 678.28, "end": 679.06, "word": " درجة", "probability": 0.9944661458333334}, {"start": 679.06, "end": 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example ناخد مثال to determine if two vectors are orthogonal calculate their product their dot product", "tokens": [3555, 22488, 13412, 2407, 10632, 8032, 1863, 3615, 28480, 36145, 6156, 3615, 15040, 2423, 39184, 624, 48812, 3215, 691, 4724, 3794, 995, 2407, 10632, 1958, 20449, 8315, 47395, 36145, 30767, 995, 2407, 10632, 49374, 9673, 2655, 7435, 3224, 9957, 6055, 3794, 3615, 7578, 4289, 11778, 47341, 3660, 11933, 15730, 732, 18875, 624, 293, 691, 366, 41488, 11933, 15730, 25961, 2655, 5893, 1674, 49374, 16095, 4724, 3794, 995, 2407, 10632, 1958, 8032, 47302, 6156, 5016, 9381, 13672, 1829, 31439, 2423, 38130, 266, 1860, 1296, 732, 18875, 281, 6997, 1365, 8717, 47283, 3215, 50113, 6027, 281, 6997, 498, 732, 18875, 366, 41488, 8873, 641, 1674, 641, 5893, 1674], "avg_logprob": -0.3049065309150197, "compression_ratio": 1.7007874015748032, "no_speech_prob": 0.0, "words": [{"start": 683.0, "end": 683.3, "word": "بعد", "probability": 0.6595458984375}, {"start": 683.3, "end": 683.6, "word": " 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{"start": 1002.37, "end": 1002.81, "word": " في", "probability": 0.953125}, {"start": 1002.81, "end": 1003.25, "word": " الأخيرة", "probability": 0.9132486979166666}, {"start": 1003.25, "end": 1003.59, "word": " هذه", "probability": 0.81005859375}, {"start": 1003.59, "end": 1004.61, "word": " V3", "probability": 0.900634765625}, {"start": 1004.61, "end": 1005.07, "word": " زائد", "probability": 0.90771484375}, {"start": 1005.07, "end": 1005.53, "word": " U3", "probability": 0.968505859375}, {"start": 1005.53, "end": 1006.37, "word": " W3", "probability": 0.968994140625}, {"start": 1006.37, "end": 1007.29, "word": " الآن", "probability": 0.63916015625}, {"start": 1007.29, "end": 1007.75, "word": " انجلي", "probability": 0.7234700520833334}, {"start": 1007.75, "end": 1008.23, "word": " هذول", "probability": 0.7335611979166666}, {"start": 1008.23, "end": 1008.63, "word": " هذه", "probability": 0.88916015625}, {"start": 1008.63, "end": 1009.21, "word": " مع", "probability": 0.958984375}, 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هذه هي عبارة عن حاصل ضرب two vectors اللي هي U1 U2 U3 مع V1 V2 V3 لو ضربت ان ضرب قياسي هيطلع هذا الرقم", "tokens": [2304, 3615, 29538, 20449, 29538, 8717, 7435, 26895, 11296, 20449, 45030, 11242, 5296, 33456, 8315, 8717, 7435, 26895, 6055, 3555, 3615, 9307, 2423, 624, 4032, 2423, 691, 20449, 45030, 11242, 4032, 6055, 3555, 3615, 9307, 2423, 624, 4032, 2423, 343, 20449, 45030, 11242, 24976, 2407, 9381, 12610, 30731, 13672, 1829, 4724, 16254, 3224, 13672, 1829, 6156, 30543, 2423, 7649, 3714, 33599, 1829, 1975, 8315, 9154, 34105, 4032, 4724, 16254, 1975, 36520, 34105, 23032, 1829, 3555, 13412, 38688, 39896, 4117, 29538, 39896, 6225, 3555, 9640, 3660, 18871, 11331, 33546, 1211, 48812, 25513, 732, 18875, 13672, 1829, 39896, 624, 16, 624, 17, 624, 18, 20449, 691, 16, 691, 17, 691, 18, 45164, 48812, 25513, 2655, 16472, 48812, 25513, 12174, 25528, 3794, 1829, 39896, 9566, 1211, 3615, 23758, 34892, 4587, 2304], "avg_logprob": -0.26308139534883723, "compression_ratio": 1.9307359307359306, 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0.98486328125}, {"start": 1026.39, "end": 1026.65, "word": " هنا", "probability": 0.70166015625}, {"start": 1026.65, "end": 1026.87, "word": " و", "probability": 0.875}, {"start": 1026.87, "end": 1027.07, "word": " بدي", "probability": 0.5657958984375}, {"start": 1027.07, "end": 1027.23, "word": " اصل", "probability": 0.767822265625}, {"start": 1027.23, "end": 1027.55, "word": " هنا", "probability": 0.51318359375}, {"start": 1027.55, "end": 1028.35, "word": " طيب", "probability": 0.771728515625}, {"start": 1028.35, "end": 1028.67, "word": " شوف", "probability": 0.946044921875}, {"start": 1028.67, "end": 1029.23, "word": " هيك", "probability": 0.671142578125}, {"start": 1029.23, "end": 1029.91, "word": " هذه", "probability": 0.68994140625}, {"start": 1029.91, "end": 1030.47, "word": " هي", "probability": 0.6650390625}, {"start": 1030.47, "end": 1030.89, "word": " عبارة", "probability": 0.9942626953125}, {"start": 1030.89, "end": 1031.11, "word": " عن", "probability": 0.9931640625}, {"start": 1031.11, "end": 1031.71, "word": " حاصل", "probability": 0.958984375}, {"start": 1031.71, "end": 1032.09, "word": " ضرب", "probability": 0.982666015625}, {"start": 1032.09, "end": 1032.49, "word": " two", "probability": 0.7373046875}, {"start": 1032.49, "end": 1033.09, "word": " vectors", "probability": 0.9287109375}, {"start": 1033.09, "end": 1033.79, "word": " اللي", "probability": 0.7861328125}, {"start": 1033.79, "end": 1034.05, "word": " هي", "probability": 0.87158203125}, {"start": 1034.05, "end": 1034.55, "word": " U1", "probability": 0.740966796875}, {"start": 1034.55, "end": 1035.09, "word": " U2", "probability": 0.797119140625}, {"start": 1035.09, "end": 1035.77, "word": " U3", "probability": 0.88330078125}, {"start": 1035.77, "end": 1036.47, "word": " مع", "probability": 0.869140625}, {"start": 1036.47, "end": 1036.95, "word": " V1", "probability": 0.948974609375}, {"start": 1036.95, "end": 1037.47, "word": " V2", "probability": 0.952392578125}, {"start": 1037.47, 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يعني الأن وصلنا U dot في V زي ال W بسوء U dot V زي U dot W", "tokens": [1211, 33456, 23758, 11331, 33546, 1211, 48812, 25513, 2423, 8062, 23758, 48812, 25513, 2423, 8062, 23758, 13672, 1829, 31439, 2423, 624, 20449, 343, 45164, 48812, 25513, 2655, 16472, 3224, 39896, 9566, 1211, 3615, 23758, 11933, 15730, 23758, 6225, 3555, 9640, 3660, 18871, 2423, 624, 4032, 23758, 6225, 3555, 9640, 3660, 18871, 2423, 691, 39896, 624, 4032, 23758, 6225, 3555, 9640, 3660, 18871, 2423, 624, 4032, 23758, 6225, 3555, 9640, 3660, 18871, 2423, 343, 37495, 22653, 16247, 1863, 4032, 36520, 8315, 624, 5893, 8978, 691, 30767, 1829, 2423, 343, 4724, 3794, 2407, 38207, 624, 5893, 691, 30767, 1829, 624, 5893, 343], "avg_logprob": -0.22431930693069307, "compression_ratio": 2.2439024390243905, "no_speech_prob": 2.7418136596679688e-06, "words": [{"start": 1041.1, "end": 1041.56, "word": "لأن", "probability": 0.59912109375}, {"start": 1041.56, "end": 1042.12, "word": " هذا", "probability": 0.755859375}, {"start": 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1055.02, "word": " عن", "probability": 0.99365234375}, {"start": 1055.02, "end": 1055.18, "word": " ال", "probability": 0.95166015625}, {"start": 1055.18, "end": 1055.42, "word": " U", "probability": 0.94775390625}, {"start": 1055.42, "end": 1056.1, "word": " و", "probability": 0.86376953125}, {"start": 1056.1, "end": 1056.28, "word": " هذا", "probability": 0.96435546875}, {"start": 1056.28, "end": 1056.58, "word": " عبارة", "probability": 0.9969482421875}, {"start": 1056.58, "end": 1056.72, "word": " عن", "probability": 0.994140625}, {"start": 1056.72, "end": 1056.82, "word": " ال", "probability": 0.97900390625}, {"start": 1056.82, "end": 1057.18, "word": " W", "probability": 0.98828125}, {"start": 1057.18, "end": 1057.54, "word": " يعني", "probability": 0.924560546875}, {"start": 1057.54, "end": 1057.82, "word": " الأن", "probability": 0.6300048828125}, {"start": 1057.82, "end": 1058.38, "word": " وصلنا", "probability": 0.96240234375}, {"start": 1058.38, "end": 1058.96, "word": " U", 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1062.0, "end": 1062.34, "word": " dot", "probability": 0.48193359375}, {"start": 1062.34, "end": 1062.66, "word": " W", "probability": 0.9833984375}], "temperature": 1.0}, {"id": 47, "seek": 109017, "start": 1063.33, "end": 1090.17, "text": "لو احد قال لابن اعمل هذا فانت تسأل اعملها على طول هذا في نفسها بيصير U1 تربيع زي U2 تربيع زي U3 تربيع عارفين ليش؟ لأن U.U U1 في U1 U1 تربيع U2 في U2 U2 تربيع U3 في U3 U3 تربيع مجموح اللي هو عبارة عن ال magnitude ل U تربيع طيب نيجي الآن ل .. ال ..", "tokens": [1211, 2407, 1975, 24401, 50239, 5296, 16758, 1863, 1975, 25957, 1211, 23758, 6156, 7649, 2655, 6055, 3794, 10721, 1211, 1975, 25957, 1211, 11296, 15844, 23032, 12610, 23758, 8978, 8717, 36178, 11296, 4724, 1829, 9381, 13546, 624, 16, 6055, 2288, 21292, 3615, 30767, 1829, 624, 17, 6055, 2288, 21292, 3615, 30767, 1829, 624, 18, 6055, 2288, 21292, 3615, 6225, 9640, 5172, 9957, 32239, 8592, 22807, 5296, 33456, 624, 13, 52, 624, 16, 8978, 624, 16, 624, 16, 6055, 2288, 21292, 3615, 624, 17, 8978, 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سمناها theta بيصير عندى الان هذه ال theta ايه يا جماعة هي الزاوية بين ال v و بين ال u طيب لو جينا اخدنا كساين ال theta هذه", "tokens": [40228, 22653, 2423, 7649, 1975, 12610, 1975, 8592, 1829, 47525, 3224, 49610, 23328, 41950, 12610, 23758, 1975, 1829, 7435, 18513, 41950, 12610, 8608, 3224, 1211, 29538, 45164, 8608, 2304, 8315, 11296, 6024, 110, 995, 2407, 10632, 9725, 13672, 1829, 49374, 16095, 49374, 344, 4032, 371, 49374, 2423, 344, 4032, 2423, 371, 8608, 2304, 8315, 11296, 9725, 4724, 1829, 9381, 13546, 43242, 7578, 2423, 7649, 29538, 2423, 9725, 1975, 1829, 3224, 35186, 10874, 15042, 27884, 39896, 6024, 110, 995, 2407, 10632, 49374, 2423, 371, 4032, 49374, 2423, 344, 23032, 1829, 3555, 45164, 10874, 1829, 8315, 1975, 9778, 3215, 8315, 9122, 3794, 995, 9957, 2423, 9725, 29538], "avg_logprob": -0.2895047107957444, "compression_ratio": 2.089005235602094, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1185.18, "end": 1185.6, "word": "يعني", "probability": 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1336.21, "end": 1365.43, "text": "الان هذا بتعوض عنه ايش قيمته U ضد V ع ال magnitude ل V من وين جبته؟ جبته من الخاصية اللي قلناها قبل هيك U ضد V بسوء ال magnitude ل U في ال magnitude ل V في cosine الزاوية بينهم طيب في ال V ع ال magnitude ل V صار هذا ايش بساوي اللي هو طبعا هنذكركم هذا ايش هذا بساوي اللي هو U ضد V ع ال magnitude ل V تربيع في V", "tokens": [6027, 7649, 23758, 39894, 3615, 2407, 11242, 18871, 3224, 1975, 1829, 8592, 12174, 32640, 47395, 624, 48812, 3215, 691, 6225, 2423, 15668, 5296, 691, 9154, 4032, 9957, 10874, 3555, 47395, 22807, 10874, 3555, 47395, 9154, 33962, 33546, 10632, 13672, 1829, 12174, 1211, 8315, 11296, 12174, 36150, 39896, 4117, 624, 48812, 3215, 691, 4724, 3794, 2407, 38207, 2423, 15668, 5296, 624, 8978, 2423, 15668, 5296, 691, 8978, 23565, 6024, 110, 995, 2407, 10632, 49374, 16095, 23032, 1829, 3555, 8978, 2423, 691, 6225, 2423, 15668, 5296, 691, 20328, 9640, 23758, 1975, 1829, 8592, 4724, 3794, 995, 45865, 13672, 1829, 31439, 23032, 3555, 3615, 995, 8032, 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@@ -0,0 +1,1832 @@
+1
+00:00:01,080 --> 00:00:05,540
+بسم الله الرحمن الرحيم هذه هي المحاضرة الثالثة
+
+2
+00:00:05,540 --> 00:00:14,200
+مساق تفاضل وتكامل ال gym الفصل الصيفي 2020 العنوان
+
+3
+00:00:14,200 --> 00:00:19,230
+المحاضرة اليوم اللي هو the dot productsection 12-3
+
+4
+00:00:19,230 --> 00:00:25,130
+بداية قبل ما نعرف ال dot product و كيف نتعامل معاه
+
+5
+00:00:25,130 --> 00:00:29,350
+خلّينا نعرف حاجة اسمها angle between vectors اللي
+
+6
+00:00:29,350 --> 00:00:34,740
+هي الزاوية بين اللي هو متجهينالان عندى لو كان في
+
+7
+00:00:34,740 --> 00:00:41,620
+عندى متجه U هى U أمامى و هى متجه اخر V الثانى و
+
+8
+00:00:41,620 --> 00:00:45,760
+الاتنين بنطلق من نقطة واحدة هذه الزاوية اللى بينهم
+
+9
+00:00:45,760 --> 00:00:50,640
+بنسميها the angle between the two vectors U and V
+
+10
+00:00:50,640 --> 00:00:58,180
+اللى هى الزاوية بين المتجهين المتجه U و Vنبدأ في
+
+11
+00:00:58,180 --> 00:01:04,700
+نظرية بتعرفنا
+
+12
+00:01:04,700 --> 00:01:12,460
+كيف نجد قيمة θ بمعرفة الـ vector u و vector v
+
+13
+00:01:13,640 --> 00:01:17,500
+النظرية بتقول ما يلي اللي هي angle between two
+
+14
+00:01:17,500 --> 00:01:21,540
+vectors if the angle θ between two non-zero
+
+15
+00:01:21,540 --> 00:01:27,500
+vectors u and v is given by يعني الزاوية θ بين two
+
+16
+00:01:27,500 --> 00:01:31,460
+non-zero vectors يعني u بسويش zero أو v بسويش zero
+
+17
+00:01:31,710 --> 00:01:37,090
+الزاوية اللي بينهم اللي هي is given by اللي هو
+
+18
+00:01:37,090 --> 00:01:40,110
+عندي طبعا ال U و ال V زي ما قلنا non-zero vector
+
+19
+00:01:40,110 --> 00:01:44,910
+هذا لا يساوي سفر وهذا لا يساوي سفر فتا اللي بينهم
+
+20
+00:01:44,910 --> 00:01:51,130
+يساوي cosine inverse لأ U واحد V واحد زائد U اتنين
+
+21
+00:01:51,130 --> 00:01:55,930
+V اتنين زائد U تلاتة V تلاتةهذا الآن cosine
+
+22
+00:01:55,930 --> 00:01:59,970
+inverse لهذا على ال magnitude لل U في ال magnitude
+
+23
+00:01:59,970 --> 00:02:05,290
+لل V هذه هي الزاوية theta بين المتجهين هذه النظرية
+
+24
+00:02:05,290 --> 00:02:12,250
+بدنا نعرفها لإنه احنا هنستخدمها في اللي هو بعد
+
+25
+00:02:12,250 --> 00:02:19,330
+شوية في نتائج هاتهمنا اللي هي بخصوص اللي هو ال dot
+
+26
+00:02:19,330 --> 00:02:23,660
+productالان definition تعريف الـ dot product بنا
+
+27
+00:02:23,660 --> 00:02:28,280
+نعرف ال dot product أو الضرب القياسي ل ال U و ال V
+
+28
+00:02:28,280 --> 00:02:31,760
+طبعا احنا هنعرف بعد المحاضرة القادمة كمان ان في
+
+29
+00:02:31,760 --> 00:02:35,040
+عندنا two dot .. two products بين ال vectors احنا
+
+30
+00:02:35,040 --> 00:02:39,260
+عرفنا شو معناة الجمع two vectors شو معناة scalar
+
+31
+00:02:39,260 --> 00:02:44,880
+في ال vector الاخرهالان بدنا نعرف عملية ضرب بين
+
+32
+00:02:44,880 --> 00:02:48,280
+two vectors عملية ضرب نوعين عندنا النوع الأول اللى
+
+33
+00:02:48,280 --> 00:02:50,920
+هنحكي عنه اللى هو الـ dot product أو الضرب القياسى
+
+34
+00:02:50,920 --> 00:02:55,640
+والنوع التانى هيجي لاحقا اللى هو الضرب الاتجاهى أو
+
+35
+00:02:55,640 --> 00:02:58,600
+اللى بنسميه cross of product خلينا نتعرف اليوم على
+
+36
+00:02:58,600 --> 00:03:02,660
+ال dot product u dot v الان u dot v اللى هو ال dot
+
+37
+00:03:02,660 --> 00:03:06,760
+product u dot v of vectors u and v هى ال vector
+
+38
+00:03:06,760 --> 00:03:12,570
+الأولمكتوب على صورة U1 U2 U3 او زي ما بتعرفوا U1I
+
+39
+00:03:12,570 --> 00:03:19,430
+زائد U2J زائد U3K و V بيساوي V1I زائد V2J زائد V3K
+
+40
+00:03:19,430 --> 00:03:22,470
+زي ما حكينا في المحاضرات السابقة اذا ال dot
+
+41
+00:03:22,470 --> 00:03:27,010
+product على طول U dot V ايش بيكون U1 في V1 ال
+
+42
+00:03:27,010 --> 00:03:29,550
+component الأول في ال component الأول زائد
+
+43
+00:03:29,790 --> 00:03:35,970
+الكمبوننت التاني U2 V2 زائد الكمبوننت التالت U3 V3
+
+44
+00:03:35,970 --> 00:03:40,190
+هذا هو تعريف الـ dot product ناخد أمثلة المثال
+
+45
+00:03:40,190 --> 00:03:42,830
+الأول نضرب هاي في عندنا ال vector الأول وهي ال
+
+46
+00:03:42,830 --> 00:03:45,990
+vector التاني نضربهم dot product بينهم اللي هو
+
+47
+00:03:45,990 --> 00:03:50,290
+طبعا مدام ال scalar أو dot product هو عبارة عن ضرب
+
+48
+00:03:50,290 --> 00:03:54,150
+قياسي يعني في الآخر هيطلع رقم مش متجه حاصل ضرب
+
+49
+00:03:54,150 --> 00:04:00,550
+حاصل الضرب متجهينبمعنى dot product بطلع عندى عبارة
+
+50
+00:04:00,550 --> 00:04:05,930
+عن scalar رقم يعنى واحد فى ناقص ستة زائد ناقص
+
+51
+00:04:05,930 --> 00:04:10,170
+اتنين فى اتنين زائد ناقص واحد فى ناقص تلاتة هى ان
+
+52
+00:04:10,170 --> 00:04:13,970
+هذوله ناتج ال dot product بنبسطها ده بيصير ناقص
+
+53
+00:04:13,970 --> 00:04:17,710
+ستة وده بيصير ناقص اربعة وده بيصير عبارة عن تلاتة
+
+54
+00:04:17,710 --> 00:04:21,670
+اجمحل ناقص ستة وناقص اربعة ناقص عشرة وتلاتة بطلع
+
+55
+00:04:21,670 --> 00:04:24,930
+ناقص سبعةبعد ذلك نضغط product بين نطلع عبارة عن
+
+56
+00:04:24,930 --> 00:04:31,790
+نقص سبعة رقم مثل ما احنا متوقعين نضغط الـ UIV نص I
+
+57
+00:04:31,790 --> 00:04:35,510
+زائد تلاتة J زائد K في أربعة I نقص J زائد اتنين K
+
+58
+00:04:35,510 --> 00:04:39,710
+يعني هذا متجه مكتوب بطريقة أخرى يعني هنا طريقة
+
+59
+00:04:39,710 --> 00:04:42,770
+اللي هي ال component اللي عندي هنا طريقة اللي هو
+
+60
+00:04:42,770 --> 00:04:46,880
+بوسطة اللي هو من ال standard unit vectorsزي ما
+
+61
+00:04:46,880 --> 00:04:51,320
+احنا عرفنا سابقا بنفس الطريقة معامل I نص في معامل
+
+62
+00:04:51,320 --> 00:04:56,140
+I N أربعة هي نص في أربعة معامل J في معامل J تلتة
+
+63
+00:04:56,140 --> 00:05:00,460
+في ناقص واحد زائد معامل K في معامل K اللي هو واحد
+
+64
+00:05:00,460 --> 00:05:05,000
+في اتنين بيصير عنده نصف اربعة اتنين اتنين وها ناقص
+
+65
+00:05:05,000 --> 00:05:08,040
+تلتة بيطلع ناقص واحد ناقص واحد زائد اتنين بيطلع
+
+66
+00:05:08,040 --> 00:05:10,920
+جداش واحد هذا اللي هو الـ data product هذا ال data
+
+67
+00:05:10,920 --> 00:05:13,980
+product سواء بالطريقة هذه أو بالطريقة هذهزي ما
+
+68
+00:05:13,980 --> 00:05:20,000
+انتوا شايفين عبارة عن اللي هو قمر سهل طيب الان in
+
+69
+00:05:20,000 --> 00:05:22,600
+the notation of the dot product the angle between
+
+70
+00:05:22,600 --> 00:05:27,040
+two vectors u and v is قبل بشوية قلنا النظرية ايش
+
+71
+00:05:27,040 --> 00:05:31,520
+بتقول بتقول theta بسبب cosine inverse في u1 u2
+
+72
+00:05:31,520 --> 00:05:36,640
+زائد u v2
+
+73
+00:05:36,640 --> 00:05:42,010
+زائد u3 v3 اللي هو في الواقع ايش هواللي هو عبارة
+
+74
+00:05:42,010 --> 00:05:45,950
+عن U ضد V علي magnitude لـ U في magnitude لـ V هى
+
+75
+00:05:45,950 --> 00:05:48,810
+هى هى يا جماعة هى هى theta بساعة cosine inverse
+
+76
+00:05:48,810 --> 00:05:55,460
+هذا هذا هو عبارة عنهذا عبارة عن U.V اذا زي ما قلنا
+
+77
+00:05:55,460 --> 00:06:02,860
+الان ال U.V استبدلناها بالكادر الموجود U1 V1 زي U2
+
+78
+00:06:02,860 --> 00:06:07,800
+V2 زي U3 V3 على ال magnitude ل U في ال magnitude ل
+
+79
+00:06:07,800 --> 00:06:12,700
+V اذا صارت θ مكتوبة بتساوي cosine inverse ل اللي
+
+80
+00:06:12,700 --> 00:06:20,610
+أماميلأن هذه بتساعدني او بتسهل عليا لو عرفت ال two
+
+81
+00:06:20,610 --> 00:06:24,550
+vectors u وv انا اوجد الزاوية اللي بينهم بكل سهولة
+
+82
+00:06:24,550 --> 00:06:30,390
+find the angle between u هي i نقص اتنين j نقص
+
+83
+00:06:30,390 --> 00:06:34,730
+اتنين k and v بالساوية 6i زائد تلاتة j زاد اتنين k
+
+84
+00:06:34,730 --> 00:06:37,910
+بدى اوجد الزاوية اللي بيني اللي هو ال two vectors
+
+85
+00:06:37,910 --> 00:06:42,680
+اللي أمامي سهلة اكيد هتقبلوا سهلةبنوجد u dot v و
+
+86
+00:06:42,680 --> 00:06:45,860
+ال magnitude و ال magnitude و بنعوضه بنوجد L D
+
+87
+00:06:45,860 --> 00:06:51,240
+theta u dot v بتساوي اللي هي هذه اللي هي واحد و
+
+88
+00:06:51,240 --> 00:06:54,640
+نقص اتنين و اتنين ده كنت ترجمها زي هيك زي ما بدك و
+
+89
+00:06:54,640 --> 00:06:57,940
+ده كنت تخليها زي ما هي بنفع برضه ستة تلاتة اتنين و
+
+90
+00:06:57,940 --> 00:07:00,940
+ضربنا الحالتين الحالتين الشكل هذا و الحالتين الشكل
+
+91
+00:07:00,940 --> 00:07:04,940
+هذا ضربناها قبل و شوية واحد في ستة بستة ناقص اتنين
+
+92
+00:07:04,940 --> 00:07:11,130
+في تلاتة ناقص ستة ناقص اتنين في ناقص اربعةهدولة
+
+93
+00:07:11,130 --> 00:07:14,310
+بروح من مع بعض بيظل ايش بيساوي نقص اربعة ال
+
+94
+00:07:14,310 --> 00:07:17,690
+magnitude لل U بتعرف الموجود بال magnitude لل U
+
+95
+00:07:17,690 --> 00:07:20,950
+اللي هو الجدر التربيعي هذا تربيع زادة تربيع زادة
+
+96
+00:07:20,950 --> 00:07:25,010
+تربيع هذا تربيع واحد واربعة واربعة تمانية و واحد
+
+97
+00:07:25,010 --> 00:07:27,730
+تسعين جدر التسعيش بيساوي تلاتة ده ال magnitude لل
+
+98
+00:07:27,730 --> 00:07:30,470
+U بيساوي تلاتة تعالوا دي بال magnitude لل V
+
+99
+00:07:30,470 --> 00:07:34,050
+بتعرفوا كلكم تجيبول هو الجدر التربيعي لهستة تربيع
+
+100
+00:07:34,050 --> 00:07:37,210
+ليها ستة وتلاتين زائد هذه تربيع ليها تسعة ستة
+
+101
+00:07:37,210 --> 00:07:40,810
+وتلاتين وتسعة خمسة واربعين خمسة واربعين وهذه تربيع
+
+102
+00:07:40,810 --> 00:07:44,450
+اربعة تسعة واربعين جدرى التسعة واربعين ايش بيساوي
+
+103
+00:07:44,450 --> 00:07:48,990
+يا جماعة سبعة اذا ال magnitude لليوف ال magnitude
+
+104
+00:07:48,990 --> 00:07:52,050
+لل V هو عبارة عن تلاتة في سبعة بطلع بواحد وعشرين
+
+105
+00:07:52,050 --> 00:07:56,990
+خلصنابصارت θ بالسواء cosine الانفرس لهذا فهذا على
+
+106
+00:07:56,990 --> 00:08:01,950
+هذا cosine الانفرس للاولاني يناقص أربعة على اللي
+
+107
+00:08:01,950 --> 00:08:05,710
+وجدناه واحد وعشرين cosine الانفرس هذا لو على الآلة
+
+108
+00:08:05,710 --> 00:08:10,790
+هتلاقي عبارة عن واحد وستة وسبعين تلميه radians لان
+
+109
+00:08:10,790 --> 00:08:14,010
+هذا لو لهنا بكفيني انا يعني انه تعرف اللي اتتا
+
+110
+00:08:14,010 --> 00:08:18,650
+بالسواء cosine انفرس طيب هذا مثال ناخد مثال آخر
+
+111
+00:08:18,650 --> 00:08:24,460
+find the angleIn that triangle ABC determined by
+
+112
+00:08:24,460 --> 00:08:30,000
+the vertices A,B,C and .. اللي هي .. اللي هو ال
+
+113
+00:08:30,000 --> 00:08:34,620
+figure اللي أمامي هيه هي A و هي B و هي C بالزاوية
+
+114
+00:08:34,620 --> 00:08:40,540
+θ هو طلب يجلب ال θ هذه الان θ هذه بدي أعملها θ
+
+115
+00:08:40,540 --> 00:08:42,760
+between two vectors اذا أنا بدي أعمل vector من هنا
+
+116
+00:08:42,760 --> 00:08:47,370
+لهنا و أعمل vector من هنا لهناالـ vector اللي من
+
+117
+00:08:47,370 --> 00:08:53,210
+هنا إلى هنا هو عبارة عن الـ CB الـ initial point C
+
+118
+00:08:53,210 --> 00:08:57,730
+و ال terminal point B ال vector هذا إيش اسمه؟ CA
+
+119
+00:08:57,730 --> 00:09:02,510
+هذا ال vector من هنا إلى هنا بعرف وجوده هو عبارة
+
+120
+00:09:02,510 --> 00:09:09,670
+عن B 3 نقص 5 سلب 2 5 نقص 2 ثلاثة هذه ال vector CB
+
+121
+00:09:09,670 --> 00:09:13,380
+زي ما اتعلمنا قبلالـ vector هذا اللي هو الـ
+
+122
+00:09:13,380 --> 00:09:17,420
+terminal point ناقص الـ initial اللي هي zero ناقص
+
+123
+00:09:17,420 --> 00:09:20,700
+خمسة بيطلع ناقص خمسة Zero ناقص اتنين بيطلع ناقص
+
+124
+00:09:20,700 --> 00:09:24,300
+اتنين إذاً هذا الـ CB و هذا الـ CA two vectors
+
+125
+00:09:24,300 --> 00:09:29,640
+اللي بينهم الزاوية theta الآن إيش بيوجدني .. عشان
+
+126
+00:09:29,640 --> 00:09:32,140
+أوجد الزاوية هذه اللي هو عبارة عن الـ dot product
+
+127
+00:09:32,140 --> 00:09:35,640
+بين هذا و هذا مجسوم على ال magnitude لهذا على ال
+
+128
+00:09:35,640 --> 00:09:37,700
+magnitude لهذا إذاً أول إشي بدي أوجد ال dot
+
+129
+00:09:37,700 --> 00:09:40,640
+product بين هذا و بين هذا ال vectorما هو الـ Dot
+
+130
+00:09:40,640 --> 00:09:44,320
+Product يا جماعة؟ اللي هو عبارة عن ناقص اتنين في
+
+131
+00:09:44,320 --> 00:09:49,960
+ناقص خمسة أي تلاتة في ناقص اتنين وبجمحن مع بعض
+
+132
+00:09:49,960 --> 00:09:56,120
+هايين بيطلع ناقص عشرة وماقص اللي هي آسف عشرة وماقص
+
+133
+00:09:56,120 --> 00:09:59,880
+ستة بيطلع أربعة هذا اللي هو ال Dot Product الآن ال
+
+134
+00:09:59,880 --> 00:10:04,700
+magnitude لل CB الجدر التربيعي للأربعة والتسعة
+
+135
+00:10:04,700 --> 00:10:10,150
+تلتاش جدر التلتاش ال CA خمسة عشرين وأربعةتسعة و
+
+136
+00:10:10,150 --> 00:10:13,650
+عشرين أكيد فاهمين جدر التربيع إلى هذا تربيع زائد
+
+137
+00:10:13,650 --> 00:10:16,930
+هذا تربيع هذا ال magnitude ل ال CA وهذا ال
+
+138
+00:10:16,930 --> 00:10:20,550
+magnitude لل CB وضربهم في بعض و طلعت جدر التربيع
+
+139
+00:10:20,550 --> 00:10:23,410
+إلى تلت مية و سبعة و سبعين اللي هانفيتش بسوى
+
+140
+00:10:23,410 --> 00:10:27,750
+cosine inverse لضد product اللي هو أربعةعلى الـ
+
+141
+00:10:27,750 --> 00:10:30,070
+magnitude ده الأول و ال magnitude ده التاني هيجدر
+
+142
+00:10:30,070 --> 00:10:37,670
+الـ 377 بطلع 78.110 بالدرجة او 1.630% radians هذا
+
+143
+00:10:37,670 --> 00:10:43,590
+بيقاش الدماغ لما توجدوه توجدوه عن طريق الألةالان
+
+144
+00:10:43,590 --> 00:10:48,330
+بدنا نعرف شو معناته two vectors are orthogonal او
+
+145
+00:10:48,330 --> 00:10:52,170
+two vectors are perpendicular يعني بدنا نعرف two
+
+146
+00:10:52,170 --> 00:11:00,310
+vectors كيف يكونوا متعامدين او متعامدين
+
+147
+00:11:00,310 --> 00:11:04,050
+او متعامدين او متعامدين
+
+148
+00:11:04,280 --> 00:11:08,680
+الدفينيشن الـ vectors u and v are orthogonal اللي
+
+149
+00:11:08,680 --> 00:11:12,480
+هو الـ two vectors بكون orthogonal إذا كانت
+
+150
+00:11:12,480 --> 00:11:19,740
+الزاوية بينهم تسعين درجة يعني تسعين درجة if and
+
+151
+00:11:19,740 --> 00:11:24,920
+only if u dot v بساوية 0بعد شوية هنعرف إن فعلا الـ
+
+152
+00:11:24,920 --> 00:11:29,340
+U ضد V بساوية 0 معناته إن زاوية بين المتجهين تسعى
+
+153
+00:11:29,340 --> 00:11:34,700
+90 درجة إذا two vectors U and V are orthogonal إذا
+
+154
+00:11:34,700 --> 00:11:40,560
+كانت dot product بينهم بساوية 0 هاي فحص اللي هو ال
+
+155
+00:11:40,560 --> 00:11:44,520
+orthogonality between two vectors to determine
+
+156
+00:11:44,520 --> 00:11:47,960
+example ناخد مثال to determine if two vectors are
+
+157
+00:11:47,960 --> 00:11:50,880
+orthogonal calculate their product their dot
+
+158
+00:11:50,880 --> 00:11:56,420
+productمعطينا U تلاتة و نقص اتنين و V اربعة و ستة
+
+159
+00:11:56,420 --> 00:12:05,860
+و نقص
+
+160
+00:12:05,860 --> 00:12:08,640
+اتنين و نقص اتنين و نقص اتنين و نقص اتنين و نقص
+
+161
+00:12:08,640 --> 00:12:09,880
+اتنين و نقص اتنين و نقص اتنين و نقص اتنين و نقص
+
+162
+00:12:09,880 --> 00:12:10,380
+اتنين و نقص اتنين و نقص اتنين و نقص اتنين و نقص
+
+163
+00:12:10,380 --> 00:12:10,720
+اتنين و نقص اتنين و نقص اتنين و نقص اتنين و نقص
+
+164
+00:12:10,720 --> 00:12:13,020
+اتنين و نقص اتنين و نقص اتنين و نقص اتنين و نقص
+
+165
+00:12:13,020 --> 00:12:18,320
+اتنين و نقص اتنين و نقص اتنين و نقص اتنين و نقص ا
+
+166
+00:12:18,580 --> 00:12:21,820
+قال لي هى في عندى كمان two vectors هدولاكا كانت في
+
+167
+00:12:21,820 --> 00:12:24,960
+ال one في ال two dimensions يعني في لبنان انا
+
+168
+00:12:24,960 --> 00:12:30,800
+بتعطيك اشي في ال space بتعرفلي انك ان هم متعمدين
+
+169
+00:12:30,800 --> 00:12:35,340
+ولا لأ بنقوله سهل الموضوع الأمر سهل ما هو إلا غير
+
+170
+00:12:35,340 --> 00:12:41,410
+نضرب u dot v تلتة في اتنين بستةلأ آسف تلاتة في سفر
+
+171
+00:12:41,410 --> 00:12:46,270
+بستة لإنه مافيش I مافيش I معامل ال I إيش جمعها سفر
+
+172
+00:12:46,270 --> 00:12:49,510
+يعني هذا ال vector سفر زاد اتنين J زاد اربعة K
+
+173
+00:12:49,510 --> 00:12:53,850
+تلاتة في سفر بسفر ناقص اتنين في اتنين يعني ناقص
+
+174
+00:12:53,850 --> 00:13:01,050
+اربعةو واحد في اربعة باربعة لان سفر و ناقص اربعة و
+
+175
+00:13:01,050 --> 00:13:05,150
+اربعة ايش هيساوي سفر اذا اللي هو ال U and V are
+
+176
+00:13:05,150 --> 00:13:08,610
+orthogonal لان في vector orthogonal على كل ال
+
+177
+00:13:08,610 --> 00:13:11,930
+vectors اللي في الدنيا اللي هو مين هو ال zero
+
+178
+00:13:11,930 --> 00:13:15,950
+vector zero is orthogonal to every vector U since
+
+179
+00:13:15,950 --> 00:13:19,150
+لان اصلا ان انتوا عارفين ال dot product بين ال
+
+180
+00:13:19,150 --> 00:13:25,040
+zero و ال U اي vector U هيطلع سفر ليش؟0 0 0 0 0 0
+
+181
+00:13:25,040 --> 00:13:36,740
+0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+
+182
+00:13:36,740 --> 00:13:41,000
+0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+
+183
+00:13:41,000 --> 00:13:42,220
+0 0 0 0 0 0 0 0
+
+184
+00:13:48,970 --> 00:13:53,170
+ناخد بعض خواص اللي هو الـ dot product يا جماعة لأن
+
+185
+00:13:53,170 --> 00:13:56,470
+ال dot product بروبارتيز و vector projections اللي
+
+186
+00:13:56,470 --> 00:14:01,690
+نحكي عنه شغلتين ال dot product خواص ال dot product
+
+187
+00:14:01,690 --> 00:14:06,210
+و ال vector projections يعني أيش مصقات ال vector
+
+188
+00:14:06,210 --> 00:14:10,090
+بنشوف هذا حنجيها نجي اول اشي لل بروبارتيز و ده ال
+
+189
+00:14:10,090 --> 00:14:16,810
+dot product إذا كانت u, v و w are any vectors و c
+
+190
+00:14:16,810 --> 00:14:21,690
+scalar إذندائما عملية الـ dot product يبقى قراءة
+
+191
+00:14:21,690 --> 00:14:26,530
+عن عملية إبدالية U dot V بساوي V dot U عملية ضرب
+
+192
+00:14:26,530 --> 00:14:30,590
+الـ dot product مضروبة في scalar C associative
+
+193
+00:14:30,590 --> 00:14:35,070
+يعني مش هتفرج معناه نضرب C في U أو U في V و نضرب C
+
+194
+00:14:35,070 --> 00:14:39,530
+فيها أو نضره حتى لو نجلنا C و أبدلناها و ضربنا U
+
+195
+00:14:39,530 --> 00:14:46,150
+في dot C في V هو نفسه C في U dot Vيعني احنا بتفرجش
+
+196
+00:14:46,150 --> 00:14:50,730
+معنا ال dot product ان اول اشي هي dot product
+
+197
+00:14:50,730 --> 00:14:54,290
+نفسها ابدالية زي من الخاصية الأولى تاني حاجة ال C
+
+198
+00:14:54,290 --> 00:14:59,530
+ال scalar وين ما بدها بتروح هنا و بتضربها زي ما
+
+199
+00:14:59,530 --> 00:15:04,080
+بدك حسب الضرب اللي موجود أماميالان u dot v زي ال w
+
+200
+00:15:04,080 --> 00:15:07,820
+يوزع ال dot product على مين يا جماعة على الجامعه
+
+201
+00:15:07,820 --> 00:15:11,320
+زي الضرب العادى على الجامعه العادى في ال scalars
+
+202
+00:15:11,320 --> 00:15:16,960
+هنا بيصير u dot v زي u dot w الآن في خاصية حلوة
+
+203
+00:15:16,960 --> 00:15:21,550
+اللي هي دائما لو ضربت ال u في نفسههيطلع ال
+
+204
+00:15:21,550 --> 00:15:24,570
+magnitude ل ال U تربيع و هذا عن استخدامها كثير
+
+205
+00:15:24,570 --> 00:15:28,610
+يعني لو ديت او دقة ضربت U ضد U هيطلع ال magnitude
+
+206
+00:15:28,610 --> 00:15:33,610
+ل U تربيع يعني طوله مربع ال zero vector اللي هم
+
+207
+00:15:33,610 --> 00:15:36,350
+نضربه في ال U بطلع zero زي اللي جينا قبله شوية ..
+
+208
+00:15:36,350 --> 00:15:38,790
+برهنها قبله شوية في المثال الأول اللي قبله شوية
+
+209
+00:15:39,240 --> 00:15:41,900
+الان برهن واحدة من هدولة وليكن تلاتة لان هذه
+
+210
+00:15:41,900 --> 00:15:49,800
+مبرهنة سابقا ليجي تلاتة خد U بساوية U بساوية U
+
+211
+00:15:49,800 --> 00:15:50,940
+واحد U اتنين U تلاتة وU تلاتة وU تلاتة وU تلاتة وU
+
+212
+00:15:50,940 --> 00:15:51,520
+بساوية U واحد U اتنين U تلاتة وU تلاتة وU تلاتة وU
+
+213
+00:15:51,520 --> 00:15:54,060
+تلاتة وU تلاتة وU تلاتة وU تلاتة وU تلاتة وU تلاتة
+
+214
+00:15:54,060 --> 00:15:57,860
+وU تلاتة وW تلاتة وW تلاتة وW تلاتة وW تلاتة وW
+
+215
+00:15:57,860 --> 00:16:01,300
+تلاتة وW تلاتة وW تلاتة وW تلاتة وW تلاتة وW تلاتة
+
+216
+00:16:01,300 --> 00:16:06,060
+وW تلات
+
+217
+00:16:06,500 --> 00:16:19,200
+V و W هو V1 و W1 و V2 و W2 و V3 و W3 و V2
+
+218
+00:16:19,200 --> 00:16:24,140
+و W3 و V2 و V3 و V2 و V3 و V2 و V3 و V2 و V2 و V3
+
+219
+00:16:24,140 --> 00:16:27,360
+و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2
+
+220
+00:16:27,360 --> 00:16:27,360
+و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2
+
+221
+00:16:27,360 --> 00:16:27,360
+و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2
+
+222
+00:16:27,360 --> 00:16:31,140
+و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2
+
+223
+00:16:31,140 --> 00:16:31,300
+و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2 و V2
+
+224
+00:16:31,300 --> 00:16:35,590
+و V2 و V2 و V2 و V2 و V2 و V2 و VU2 زائد U2 في هذه
+
+225
+00:16:35,590 --> 00:16:44,610
+اللي هو U2 V2 زائد U2 W2 زائد U3 في الأخيرة هذه V3
+
+226
+00:16:44,610 --> 00:16:50,530
+زائد U3 W3 الآن انجلي هذول هذه مع هذه مع هذه
+
+227
+00:16:50,530 --> 00:16:56,250
+انجليها مع بعض هين و هذهمع هذه مع هذه نجيلها مع
+
+228
+00:16:56,250 --> 00:17:00,790
+بعض لأننا نجيل تبعات ال U و ال V مع بعض و تبعات ال
+
+229
+00:17:00,790 --> 00:17:05,710
+U و ال W مع بعض للوصول إلى اللي بديه اللي فوق الان
+
+230
+00:17:05,710 --> 00:17:10,470
+ماشي انا من هنا و بدي اصل هنا طيب شوف هيك هذه هي
+
+231
+00:17:10,470 --> 00:17:16,470
+عبارة عن حاصل ضرب two vectors اللي هي U1 U2 U3 مع
+
+232
+00:17:16,470 --> 00:17:21,560
+V1 V2 V3 لو ضربت ان ضرب قياسي هيطلع هذا الرقملأن
+
+233
+00:17:21,560 --> 00:17:25,880
+هذا حاصل ضرب ال vector هذا ضرب ال vector هذا اللي
+
+234
+00:17:25,880 --> 00:17:30,000
+هو ال U مع W لو ضربت انه هيطلع هذا إذا هذا عبارة
+
+235
+00:17:30,000 --> 00:17:35,180
+عن ال U و هذا عبارة عن ال V هي U و هذا عبارة عن ال
+
+236
+00:17:35,180 --> 00:17:39,700
+U و هذا عبارة عن ال W يعني الأن وصلنا U dot في V
+
+237
+00:17:39,700 --> 00:17:44,310
+زي ال W بسوء U dot V زي U dot Wلو احد قال لابن
+
+238
+00:17:44,310 --> 00:17:47,870
+اعمل هذا فانت تسأل اعملها على طول هذا في نفسها
+
+239
+00:17:47,870 --> 00:17:54,290
+بيصير U1 تربيع زي U2 تربيع زي U3 تربيع عارفين ليش؟
+
+240
+00:17:54,290 --> 00:18:00,410
+لأن U.U U1 في U1 U1 تربيع U2 في U2 U2 تربيع U3 في
+
+241
+00:18:00,410 --> 00:18:05,410
+U3 U3 تربيع مجموح اللي هو عبارة عن ال magnitude ل
+
+242
+00:18:05,410 --> 00:18:12,610
+U تربيع طيب نيجي الآن ل .. ال ..الموضوع الأخير في
+
+243
+00:18:12,610 --> 00:18:18,350
+هذا ال section اللي هو موضوع مهم اللي بنسميه
+
+244
+00:18:18,350 --> 00:18:22,390
+vector projection أو مصقط ال vector الآن عندي
+
+245
+00:18:22,390 --> 00:18:27,450
+حالتين يا عندي الزاوية هي ال vector u و هي ال
+
+246
+00:18:27,450 --> 00:18:31,070
+vector v ماشي الحال هي الزاوية اللي بينهم theta
+
+247
+00:18:31,070 --> 00:18:36,970
+هذه زاوية حادة عند الآن ال u و ال v هي ال two
+
+248
+00:18:36,970 --> 00:18:41,320
+vectors والزاوية اللي بينهم إيش ما لها منفرجةالان
+
+249
+00:18:41,320 --> 00:18:47,940
+بسقط ال U على ال V و بسقط ال U على ال V تنشوف الآن
+
+250
+00:18:47,940 --> 00:18:51,560
+الحالتين حلجين في الآخر بتجمعن فى قانون واحد و
+
+251
+00:18:51,560 --> 00:18:56,200
+القمر هيسير سهل و هيكون سهل ان شاء الله نيجي الآن
+
+252
+00:18:56,200 --> 00:19:01,300
+هى ال vector U و هى ال vector V بدي اوجدالـ
+
+253
+00:19:01,300 --> 00:19:06,920
+projection للـ U on V يعني بدي أوجد مصقط U على مين
+
+254
+00:19:06,920 --> 00:19:11,220
+على V بتعرفوا كيف نوجد المصقط بيجي بنزل هي رسمة
+
+255
+00:19:11,220 --> 00:19:16,020
+الـ U وهي رسمة مين ال V من نفس النقطة باجي الآن
+
+256
+00:19:16,020 --> 00:19:22,080
+بنزل عمودي من راس ال U على اللي هو مين ال V هذا هو
+
+257
+00:19:22,080 --> 00:19:29,280
+ال projection لل U on مين on V ال projection لل U
+
+258
+00:19:29,280 --> 00:19:35,730
+on Vطيب هذا الـ projection بتحدد بأمرين اللي هو
+
+259
+00:19:35,730 --> 00:19:40,850
+بالـ scalar طوله يعني بال magnitude تبعه و باتجاهه
+
+260
+00:19:40,850 --> 00:19:46,980
+واضح حانة اتجاهه و اتجاه من ال Vيعني الان اول اشي
+
+261
+00:19:46,980 --> 00:19:50,840
+بده وجود الطول هذا ايجاد الطول سهل هذه لو سمناها
+
+262
+00:19:50,840 --> 00:19:55,720
+الزاوية theta اللي بينهم بين u و v بين ال u و ال v
+
+263
+00:19:55,720 --> 00:20:00,820
+سمناها theta بيصير عندى الان هذه ال theta ايه يا
+
+264
+00:20:00,820 --> 00:20:06,740
+جماعة هي الزاوية بين ال v و بين ال u طيب لو جينا
+
+265
+00:20:06,740 --> 00:20:12,370
+اخدنا كساين ال theta هذهCos θ هي ساوي اللى هو ال
+
+266
+00:20:12,370 --> 00:20:16,070
+magnitude لل projection اللى انا بديها ال
+
+267
+00:20:16,070 --> 00:20:20,930
+magnitude هذا ع ال magnitude لل U مظبوط هذا ال cos
+
+268
+00:20:20,930 --> 00:20:27,310
+θ يعني و كإني بقول ان magnitude لل projection بسوى
+
+269
+00:20:27,310 --> 00:20:34,190
+ال magnitude لل U في cos θ ماشي اذا ال magnitude
+
+270
+00:20:34,190 --> 00:20:39,810
+لل projection هذابسوء ال magnitude ل ال U في Cos θ
+
+271
+00:20:39,810 --> 00:20:47,510
+طيب هذا مين اذا قيمة اللي هو ال magnitude ل ال
+
+272
+00:20:47,510 --> 00:20:53,750
+projection ماشي طيب عشان اوجد ال projection نفسه
+
+273
+00:20:53,750 --> 00:20:59,190
+هاي طوله اوجدته هاي U ضال مين اضربه في اتجاهه
+
+274
+00:20:59,190 --> 00:21:06,970
+اتجاهه هذا مين هو هو عبارة عن اتجاهالـ V ايش اتجاه
+
+275
+00:21:06,970 --> 00:21:09,570
+الـ V اللي هو الـ V على الـ magnitude للـ V إذا ان
+
+276
+00:21:09,570 --> 00:21:13,230
+تحدد ال projection للـ U في اتجاه الـ V ال
+
+277
+00:21:13,230 --> 00:21:16,910
+projection للـ U في اتجاه الـ V بساوي ال magnitude
+
+278
+00:21:16,910 --> 00:21:21,090
+للـ U cos θ أو ال projection للـ U على الـ V هو ال
+
+279
+00:21:21,090 --> 00:21:25,370
+magnitude للـ U cos θ في الـ V على ال magnitude
+
+280
+00:21:25,370 --> 00:21:30,530
+للـ V هذا في حالة الزاوية يمين اللي هي θ هنا اللي
+
+281
+00:21:30,530 --> 00:21:35,480
+هو الزاوية بين التنين U و V حادةو يساوي الان هذا
+
+282
+00:21:35,480 --> 00:21:43,200
+ال magnitude لل U في ال V على ال magnitude لل V في
+
+283
+00:21:43,200 --> 00:21:51,640
+cosine من theta هذا بيمثل هاين عبارة عن اش هذا
+
+284
+00:21:51,640 --> 00:21:59,850
+عرفناها قبل بشوية اه لو اجينا حطينا مكان هذااللي
+
+285
+00:21:59,850 --> 00:22:06,850
+هو حيكون عبارة عن U.V ع ال magnitude ل V و لا لا
+
+286
+00:22:06,850 --> 00:22:11,370
+لأن U.V أشجع عليها بالساوية ال magnitude ل Mي في
+
+287
+00:22:11,370 --> 00:22:18,010
+ال magnitude ل V في cos θ كمان مرةالان هذا بتعوض
+
+288
+00:22:18,010 --> 00:22:22,070
+عنه ايش قيمته U ضد V ع ال magnitude ل V من وين
+
+289
+00:22:22,070 --> 00:22:25,670
+جبته؟ جبته من الخاصية اللي قلناها قبل هيك U ضد V
+
+290
+00:22:25,670 --> 00:22:28,830
+بسوء ال magnitude ل U في ال magnitude ل V في
+
+291
+00:22:28,830 --> 00:22:32,510
+cosine الزاوية بينهم طيب في ال V ع ال magnitude ل
+
+292
+00:22:32,510 --> 00:22:39,450
+V صار هذا ايش بساوي اللي هو طبعا هنذكركم هذا ايش
+
+293
+00:22:39,450 --> 00:22:45,010
+هذا بساوي اللي هو U ضد V ع ال magnitude ل V تربيع
+
+294
+00:22:45,010 --> 00:22:49,040
+في Vإذا من الآن وطالع الـ projection للـ U في
+
+295
+00:22:49,040 --> 00:22:53,360
+اتجاه الـ V هو عبارة عن U ضد V على ال magnitude V
+
+296
+00:22:53,360 --> 00:23:00,100
+تربيع في من؟ في V هذا عبارة عن كل عدد scalar وهذا
+
+297
+00:23:00,100 --> 00:23:04,120
+اللي هو من الـ V ال vector اللي بدنا نوجد الـ U
+
+298
+00:23:04,120 --> 00:23:09,760
+عليه هذا اللي هو ال projection للـ U on V دائما
+
+299
+00:23:09,760 --> 00:23:13,540
+بساوي هذا الكلام أنا بقول هذا بساوي هذا الكلام
+
+300
+00:23:13,540 --> 00:23:18,910
+سواء أزاوية حادة أو أزاوية منفرجةيعني هذه الحالة
+
+301
+00:23:18,910 --> 00:23:22,190
+برضه بنطبق عليها خلّينا نشوف كيف شوفوا يا جماعة
+
+302
+00:23:22,190 --> 00:23:25,830
+صنعنا بالنسبة للسنة هاي ال vector u الحالة الثانية
+
+303
+00:23:25,830 --> 00:23:29,050
+وهي ال vector v وهي الزاوية اللي بيننا ايش اسمها
+
+304
+00:23:29,050 --> 00:23:33,010
+دي theta هذه الزاوية ال theta الان ال projection
+
+305
+00:23:33,010 --> 00:23:37,790
+بنزل عمود هي عمود فبطلح هي ال projection الان هذا
+
+306
+00:23:37,790 --> 00:23:43,510
+ال projection بنفس الطريقة ال projection ل ال u في
+
+307
+00:23:43,510 --> 00:23:50,060
+اتجاه ال v هذا بساوي magnitudeفي اتجاهه ايش اتجاهه
+
+308
+00:23:50,060 --> 00:23:54,200
+هو الـ projection لـU على الـV على الـ magnitude
+
+309
+00:23:54,200 --> 00:23:58,240
+ليه؟ مش هذا لكل .. ل .. هذه .. هذه .. لكل متجهة في
+
+310
+00:23:58,240 --> 00:24:03,540
+الدنيا هي اتجاهه الـ vector نفسه على magnitude طيب
+
+311
+00:24:04,140 --> 00:24:10,240
+الان بدنا نحدد هذه و بدنا نحدد هذه طيب لاحظ مالي
+
+312
+00:24:10,240 --> 00:24:13,800
+الان هذه الزاوية theta اللي بين المتجهين برضه بدي
+
+313
+00:24:13,800 --> 00:24:17,280
+أسميها ايش theta بين المتجهين ايش بدي أسميها theta
+
+314
+00:24:17,280 --> 00:24:20,100
+يعني هذه theta عبارة عن اللي بين ال U و ال V سواء
+
+315
+00:24:20,100 --> 00:24:24,750
+هان او هان طيب الزاوية theta هذه اللي بين U و Vهنا
+
+316
+00:24:24,750 --> 00:24:29,790
+ايش هتكون مية و تمانين ناقص ثتا احنا بنعرف كوصين
+
+317
+00:24:29,790 --> 00:24:34,130
+المية و تمانين ناقص ثتا بساوي ناقص كوصين الثتا
+
+318
+00:24:34,130 --> 00:24:38,750
+اكيد اضرب الجهتين في ال magnitude ل ال U بيصير ال
+
+319
+00:24:38,750 --> 00:24:41,210
+magnitude ل ال U في كوصين الثتا بساوي ناقص ال
+
+320
+00:24:41,210 --> 00:24:45,590
+magnitude ل ال U في كوصين مية و تمانين ناقص ثتا
+
+321
+00:24:45,590 --> 00:24:52,860
+طيب الآن هذه اللي هي زي ما عملناها بالظبطالكوساين
+
+322
+00:24:52,860 --> 00:24:59,400
+التي هي 180 نقص تيتا ايش بتساوي بتساوي ال
+
+323
+00:24:59,400 --> 00:25:04,700
+magnitude لهذا على ال magnitude لل U يعني بمعنى
+
+324
+00:25:04,700 --> 00:25:11,560
+آخرهيكون عندى اللى هو ال magnitude ل ال U في
+
+325
+00:25:11,560 --> 00:25:14,700
+cosine ال theta magnitude ل ال U في cosine as of
+
+326
+00:25:14,700 --> 00:25:19,000
+pi نقص theta magnitude ل ال U في cosine هذه
+
+327
+00:25:19,000 --> 00:25:24,200
+الزاوية هي ساوى ال magnitude لهذا طول هذا لأن ال
+
+328
+00:25:24,200 --> 00:25:32,630
+cosine هذه ساوى هذا على هذا يعنى الآن هذا هيههذا
+
+329
+00:25:32,630 --> 00:25:35,730
+البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن هو
+
+330
+00:25:35,730 --> 00:25:37,090
+هذا البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن
+
+331
+00:25:37,090 --> 00:25:39,950
+هو هذا البروجيكشن هو هذا البروجيكشن هو هذا
+
+332
+00:25:39,950 --> 00:25:43,470
+البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن هو
+
+333
+00:25:43,470 --> 00:25:46,170
+هذا البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن
+
+334
+00:25:46,170 --> 00:25:47,350
+هو هذا البروجيكشن هو هذا البروجيكشن هو هذا
+
+335
+00:25:47,350 --> 00:25:47,850
+البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن هو
+
+336
+00:25:47,850 --> 00:25:48,450
+هذا البروجيكشن هو هذا البروجيكشن هو هذا البروجيكشن
+
+337
+00:25:48,450 --> 00:25:51,990
+هو هذا البروجيكشن هو هذا البروجيكشن هو هذا
+
+338
+00:25:51,990 --> 00:26:03,700
+البروجيكشالـ magnitude له نقص هذا المقدار نقص
+
+339
+00:26:03,700 --> 00:26:05,000
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+340
+00:26:05,000 --> 00:26:06,460
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+341
+00:26:06,460 --> 00:26:07,100
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+342
+00:26:07,100 --> 00:26:11,280
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+343
+00:26:11,280 --> 00:26:12,460
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+344
+00:26:12,460 --> 00:26:13,000
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+345
+00:26:13,000 --> 00:26:13,500
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+346
+00:26:13,500 --> 00:26:13,500
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+347
+00:26:13,500 --> 00:26:13,920
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+348
+00:26:13,920 --> 00:26:19,560
+هذا المقدار نقص هذا المقدار نقص هذا المقدار نقص
+
+349
+00:26:20,500 --> 00:26:25,360
+المتجه في اتجاه الـ projection عكس الـ V إذا بدّى
+
+350
+00:26:25,360 --> 00:26:30,180
+أشيله هذا و أكتب قيمته بدلالة هذا إيش حيساوى ناقص
+
+351
+00:26:30,180 --> 00:26:34,400
+ال V على magnitude V يعني هذا المتجه عكس هذا هذا
+
+352
+00:26:34,400 --> 00:26:38,600
+متجه واحدة و هذا متجه واحدة إذا هذا بيساوى ناقص V
+
+353
+00:26:38,600 --> 00:26:44,160
+على magnitude V لأن هذا يبعبر عن المتجه هنامتجه
+
+354
+00:26:44,160 --> 00:26:48,020
+الوحدة ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+355
+00:26:48,020 --> 00:26:48,340
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+356
+00:26:48,340 --> 00:26:51,520
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+357
+00:26:51,520 --> 00:26:53,380
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+358
+00:26:53,380 --> 00:26:55,000
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+359
+00:26:55,000 --> 00:26:56,720
+ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده ده
+
+360
+00:26:56,720 --> 00:27:12,540
+ده ده ده ده ده ده ده ده
+
+361
+00:27:12,540 --> 00:27:16,070
+دهفي الـ V على الـ magnitude للـ V إذا صار عندى
+
+362
+00:27:16,070 --> 00:27:19,430
+هذا ال projection بساوية هذا حتى في الحالة الثانية
+
+363
+00:27:19,430 --> 00:27:24,130
+اللى هو هذه القيمة اللى وصلناها زى ما قبل بشوية
+
+364
+00:27:24,130 --> 00:27:29,870
+لهذا إذا أى إن كانت الزاوية زاوية θ حادة هنا أو
+
+365
+00:27:29,870 --> 00:27:32,870
+هنا الزاوية بينهم أى إن كانت هذه الزاوية أو هذه
+
+366
+00:27:32,870 --> 00:27:39,190
+دايما ال projection لل U في اتجاه ال V بساوية U ضد
+
+367
+00:27:39,190 --> 00:27:40,490
+V على ال magnitude لل V
+
+368
+00:27:43,360 --> 00:27:47,860
+أحفظنا القانون هذا و اعرفنا كيف يأتي لأنه يجي نطبق
+
+369
+00:27:47,860 --> 00:27:52,020
+عليه شوفوا يا جماعة من اللي حكيته قبل شوية واضح في
+
+370
+00:27:52,020 --> 00:27:57,980
+حالة ال length هنا ال length هنا الطول هنا الطول
+
+371
+00:27:57,980 --> 00:28:03,380
+هذا عبارة عن طلع عند مين ماله ال magnitude ليوفو
+
+372
+00:28:03,380 --> 00:28:11,280
+كوصينت ثيتا هذا طول هذالكن الطول هنا طلع ناقص الـ
+
+373
+00:28:11,280 --> 00:28:14,300
+magnitude لـ U في main فقصان الـ θيهي و الـ
+
+374
+00:28:14,300 --> 00:28:19,640
+projection هذا طول الـ magnitude للـ projection
+
+375
+00:28:19,640 --> 00:28:26,260
+طلع ناقص هذا المقدار اذا الان الملاحظة التالية ايش
+
+376
+00:28:26,260 --> 00:28:31,430
+الملاحظة يا جماعةإنه في حالة θ هي الـ U و هي الـ V
+
+377
+00:28:31,430 --> 00:28:34,910
+و الـ θ حادة بيطلع ال length بيساوي ال magnitude ل
+
+378
+00:28:34,910 --> 00:28:38,850
+ال U في cos θ لكن في الحالة الثانية ل ال length ل
+
+379
+00:28:38,850 --> 00:28:42,490
+ال projection هذا إيش طلع معناه ناقص ال magnitude
+
+380
+00:28:42,490 --> 00:28:47,590
+ل ال U في cos θ لإن cos θ لما تكون من فارجة بتكون
+
+381
+00:28:47,590 --> 00:28:48,070
+سالبة
+
+382
+00:28:52,840 --> 00:28:57,380
+الملخص الحديث The vector projection of U into V is
+
+383
+00:28:57,380 --> 00:29:02,230
+the vectorA projection للـU onto V بيساوي الـU dot
+
+384
+00:29:02,230 --> 00:29:07,990
+V على ال magnitude للـV تربيع في من في الـV هذا
+
+385
+00:29:07,990 --> 00:29:13,990
+عبارة عن scalar مضروب في من أو محطوط في اتجاه اللي
+
+386
+00:29:13,990 --> 00:29:19,250
+هو مضروب في ال vector V الآن في حاجة بنسميها the
+
+387
+00:29:19,250 --> 00:29:25,040
+scalar component of U in the direction of Vما هي
+
+388
+00:29:25,040 --> 00:29:27,600
+الـ Scalar component اللي نعرفها in the direction
+
+389
+00:29:27,600 --> 00:29:32,400
+of V؟ اللي عبارة عن U ضد V على مين على V لأن هذا
+
+390
+00:29:32,400 --> 00:29:37,160
+الـ projection عبارة عن حصل ضرب U ضد V على ال
+
+391
+00:29:37,160 --> 00:29:43,000
+magnitude ل V في اتجاه V على V مظبوط؟
+
+392
+00:29:43,000 --> 00:29:45,820
+الآن اللي .. اللي .. اللي .. اللي .. اللي بيظل هذا
+
+393
+00:29:45,820 --> 00:29:50,510
+.. هذا اللي هوسيبك من الـ V على الـ magnitude V
+
+394
+00:29:50,510 --> 00:29:54,010
+للـ V اللي هو اللي بيحدد الاتجاه في اتجاه الـ V
+
+395
+00:29:54,010 --> 00:29:58,030
+اللي بيظل هو اللي بنسميه الـ scalar component،
+
+396
+00:29:58,030 --> 00:30:03,510
+ماشي؟ إذا الـ scalar component of U هو عبارة عن U
+
+397
+00:30:03,510 --> 00:30:08,710
+ضد V على الـ magnitude للـ V لحاله عشان يظلل الـ V
+
+398
+00:30:08,710 --> 00:30:11,130
+على الـ magnitude للـ V اللي بيحدد اتجاه له
+
+399
+00:30:11,130 --> 00:30:16,020
+projectionالرقم اللي مضروب في اللي بيحدد لاتجاه ال
+
+400
+00:30:16,020 --> 00:30:21,240
+projection لل V هو اللي بنسميه ال scalar component
+
+401
+00:30:21,240 --> 00:30:24,840
+اللي هو U ضد V على ال magnitude لل V أو بصورة أخرى
+
+402
+00:30:24,840 --> 00:30:30,260
+U فضد V على ال magnitude لل V أو اللي بنسميه
+
+403
+00:30:30,260 --> 00:30:33,960
+بطريقة أخرى اللي هو ال magnitude لل V U في كسان ال
+
+404
+00:30:33,960 --> 00:30:36,970
+theta لأنه زي ما عرفنا U ضد V بساويةفى الـ
+
+405
+00:30:36,970 --> 00:30:39,430
+magnitude ده الـ V فى ال magnitude ده الـ U ثم الـ
+
+406
+00:30:39,430 --> 00:30:42,090
+cosine الـ theta إذاً هذا اسمه الـ scalar
+
+407
+00:30:42,090 --> 00:30:47,390
+component وهذا ال projection للـ U on V أو on to V
+
+408
+00:30:47,390 --> 00:30:52,490
+نيجي ناخد مثال find the vector of projection of U
+
+409
+00:30:52,490 --> 00:30:56,610
+on to V and the scalar component of U in the
+
+410
+00:30:56,610 --> 00:30:59,530
+direction of V بدي شغلتين بدي ال vector of
+
+411
+00:30:59,530 --> 00:31:04,510
+projection مصفة U على ال Vأود الـ Scalar component
+
+412
+00:31:04,510 --> 00:31:11,110
+لـ U في وين؟ في الاتجاه V طبعا الـ Scalar
+
+413
+00:31:11,110 --> 00:31:13,770
+component هي يا أما الطول يا أما ناقص الطول حسب
+
+414
+00:31:13,770 --> 00:31:19,820
+الزاويةللـ solution طبعا الطول إذا كانت الزاوية
+
+415
+00:31:19,820 --> 00:31:23,940
+حادة بيصير اللي هو اللي هو ال length إذا كانت
+
+416
+00:31:23,940 --> 00:31:27,320
+الزاوية عزيزي مش احنا قولنا يا جماعة نحسن حاجة
+
+417
+00:31:27,320 --> 00:31:32,520
+الآن هذا المقدار يا بيساوي ال length زي في حالة
+
+418
+00:31:32,520 --> 00:31:36,890
+اللي هي الزاوية الحادةأو بساوي سالب ال length في
+
+419
+00:31:36,890 --> 00:31:41,550
+حالة مين الزاوية المنفرجة يعني ال scalar component
+
+420
+00:31:41,550 --> 00:31:45,050
+يا إما ال length يا إما سالبه حسب الزاوية اللي
+
+421
+00:31:45,050 --> 00:31:49,010
+عندي هذا اللي هو ال scalar component طيب نيجي الآن
+
+422
+00:31:49,010 --> 00:31:52,770
+هي ال U و هي ال V الآن بده اللي هو ال vector
+
+423
+00:31:52,770 --> 00:31:56,530
+projection كله عشان أوجد ال vector projectionأو
+
+424
+00:31:56,530 --> 00:31:59,850
+الـ projection لـ U علي الـ V بلزمني U ضد V علي
+
+425
+00:31:59,850 --> 00:32:03,810
+magnitude ل V تربيع في الـ V الـ U ضد V سهلة بتضرب
+
+426
+00:32:03,810 --> 00:32:08,730
+U في V هي U في V ستة في واحد زائد تلاتة في نقص
+
+427
+00:32:08,730 --> 00:32:12,030
+اتنين زائد اتنين في نقص اتنين هذه ضربناها وجمعناها
+
+428
+00:32:12,030 --> 00:32:15,430
+نقل النقص أربع منها قبل هيك ال magnitude ل V
+
+429
+00:32:15,430 --> 00:32:19,690
+عملناها هي magnitude واحد واربعة واربعة يعني جدر
+
+430
+00:32:19,690 --> 00:32:23,470
+التسعة يعني تلاتةلأن الـ Vector Projection of U
+
+431
+00:32:23,470 --> 00:32:28,670
+الـ Vector Projection of U onto V هو U Dot V على
+
+432
+00:32:28,670 --> 00:32:34,650
+المجموعة V تربيع في V U Dot V هي 4 على المجموعة V
+
+433
+00:32:34,650 --> 00:32:39,970
+تربيع هي 9 في V هو I ناقص 2K إذاً هذا الـ Vector
+
+434
+00:32:39,970 --> 00:32:44,630
+Projection لـ U onto V لأن الـ Scalar Component of
+
+435
+00:32:44,630 --> 00:32:48,150
+U in the Direction of Vهو عبارة عن U.V على
+
+436
+00:32:48,150 --> 00:32:52,210
+magnitude V U.V وجدناها ناقص أربعة ال magnitude V
+
+437
+00:32:52,210 --> 00:32:56,990
+تلاتة هي U إذا صار عبارة عن ناقص أربعة على تلاتة
+
+438
+00:32:56,990 --> 00:33:00,110
+هو ال scalar component هو في الواقع حداش بيساوي
+
+439
+00:33:00,110 --> 00:33:05,870
+ناقص طول من ال projection ناقص طول ال projection
+
+440
+00:33:06,350 --> 00:33:09,950
+يعني الآن لو جدت ال projection لهذا هتلاقي عبارة
+
+441
+00:33:09,950 --> 00:33:14,730
+عن نقص أربعة تلاتة في نقص يطلع أربعة تلاتة يعني
+
+442
+00:33:14,730 --> 00:33:17,690
+هتلاقي الطول إيش هتلاقي لو جدت الطول ال projection
+
+443
+00:33:17,690 --> 00:33:24,750
+هتلاقي عبارة عن أربعة تلاتة المفروض تفهمنا find
+
+444
+00:33:24,750 --> 00:33:30,750
+the vector مثال آخرمثال 6 اجهزة البروجيكشن للقوة F
+
+445
+00:33:58,560 --> 00:34:04,520
+ماشي طيب نشوف اول اشي انه يوجد F ضد V F ضد V هي
+
+446
+00:34:04,520 --> 00:34:10,900
+خمسة و اتنين ضد واحد و ناقص ثلاثة خمسة ناقص ستة
+
+447
+00:34:10,900 --> 00:34:16,120
+بطلع ناقص واحد magnitude ال V هي ال magnitude ال V
+
+448
+00:34:16,120 --> 00:34:22,810
+هي V واحد و تسحق عشرة جدر العشرةواضح أه؟ هذا تربيع
+
+449
+00:34:22,810 --> 00:34:26,610
+زادت تربيع زادت عشرة بيطلع اللي هو magnitude لل V
+
+450
+00:34:26,610 --> 00:34:30,590
+لأن ال vector projection من F onto V ال projection
+
+451
+00:34:30,590 --> 00:34:36,110
+لل F onto V بساوي F dot V عن magnitude V تربيع في
+
+452
+00:34:36,110 --> 00:34:42,170
+V F dot V اللي هو ناقص واحد على magnitude لل V
+
+453
+00:34:42,170 --> 00:34:46,310
+تربيع اللي هو عشرة في V I ناقص ثلاثة G لأن ال
+
+454
+00:34:46,310 --> 00:34:47,990
+scalar component of U
+
+455
+00:34:58,140 --> 00:35:08,220
+النقص الطولي في هذه الحالة هو نقص 1 على 10وهكذا
+
+456
+00:35:08,220 --> 00:35:11,700
+نكون انهينا
+
+457
+00:35:11,700 --> 00:35:17,320
+ال section 12-3 of the product وهي ال homework
+
+458
+00:35:17,320 --> 00:35:21,640
+وإلى لقاء آخر السلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1d03bff711a40bfc2ec53f2f96930c000c593776
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI.srt
@@ -0,0 +1,690 @@
+1
+00:00:00,770 --> 00:00:04,810
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,810 --> 00:00:10,630
+وبركاته هذه هي المحاضرة التاسعة
+
+3
+00:00:10,630 --> 00:00:19,050
+مساق تفاضل وتكامل جيم هذه المحاضرة الثانية في الشطر
+
+4
+00:00:19,050 --> 00:00:24,640
+13اللي هو تلت عشر،  الجزء الثاني اللي هو integrals of
+
+5
+00:00:24,640 --> 00:00:30,480
+vector functions وهي محاضرة سريعة ومحاضرة خفيفة
+
+6
+00:00:30,480 --> 00:00:34,900
+هنتكلم المرة الماضية، في المحاضرة السابقة، عن
+
+7
+00:00:34,900 --> 00:00:37,940
+الـlimits والـcontinuity والـdifferentiability و
+
+8
+00:00:37,940 --> 00:00:41,430
+أمور أخرى. الآن نكمل الموضوع ونشوف كيف نعمل
+
+9
+00:00:41,430 --> 00:00:45,310
+integration للـ vector valued functions بنفس
+
+10
+00:00:45,310 --> 00:00:49,190
+روح الـ differentiation والـ limits والـ continuity
+
+11
+00:00:49,190 --> 00:00:53,810
+اللي هي اللي بتعتمد على الـ components لها
+
+12
+00:00:53,810 --> 00:00:59,610
+الآن نأتي لنشوف integrals of vector functions، الآن
+
+13
+00:00:59,610 --> 00:01:03,350
+حديثنا عن الـ vector functions اللي هي فرضنا عنها
+
+14
+00:01:03,350 --> 00:01:06,690
+أنها  vector function اسمها R، الآن
+
+15
+00:01:06,690 --> 00:01:09,730
+a differentiable vector function R of T is an
+
+16
+00:01:09,730 --> 00:01:12,890
+antiderivative of a vector function R of T، نفس
+
+17
+00:01:12,890 --> 00:01:16,850
+التعريف اللي أخذناه في Calculus 1، بس هذا لـ
+
+18
+00:01:16,850 --> 00:01:20,810
+vector valued function، الـ function R of T اللي بنقول عنها
+
+19
+00:01:20,810 --> 00:01:23,930
+antiderivative للـ R of T، طبعاً R of T is a
+
+20
+00:01:23,930 --> 00:01:26,670
+differentiable function، هذا R الكبير بنقول عنها
+
+21
+00:01:26,670 --> 00:01:29,810
+antiderivative للـ R، يعني عكس الـ derivative للـ R
+
+22
+00:01:29,810 --> 00:01:36,800
+عملية عكسية.  على  interval I، إذا كانت dR
+
+23
+00:01:36,800 --> 00:01:42,700
+/dT = R،  لما dR الكبير / dT = R، بنقول
+
+24
+00:01:42,700 --> 00:01:47,260
+عن الـ R  derivative  لـ d  لـ R الكبير، وبنقول
+
+25
+00:01:47,260 --> 00:01:51,620
+للـ R الكبير anti-derivative لمين؟ للـ R على
+
+26
+00:01:51,620 --> 00:01:56,790
+الفترة المذكورة I. الآن لو كانت R antiderivative
+
+27
+00:01:56,790 --> 00:02:04,350
+للـ R على I، بنقدر نقول أي R زائد C، نفس الـ R هذه
+
+28
+00:02:04,350 --> 00:02:10,990
+زائد C برضه هتطلع antiderivative للـ R، اسمه لأن
+
+29
+00:02:10,990 --> 00:02:16,410
+لو فضلنا هذه هتطلع لي برضه R الصغيرة، لأنه الآن
+
+30
+00:02:16,410 --> 00:02:22,010
+تفاضل constant هيساوي صفر. إذاً الآن هذا بيقودنا
+
+31
+00:02:22,010 --> 00:02:28,170
+إلى تعريف The set of all antiderivatives of R، أما
+
+32
+00:02:28,170 --> 00:02:33,110
+يعني كل الـ antiderivatives هدول هما اللي بنسميهم
+
+33
+00:02:33,110 --> 00:02:37,650
+الـ indefinite integral of R أو التكامل غير
+
+34
+00:02:37,650 --> 00:02:41,650
+المحدود للـ R، زي ما عملنا بالظبط في Calculus 1
+
+35
+00:02:41,650 --> 00:02:46,760
+وحتى في الـ tutorial، الـ indefinite integral of R with
+
+36
+00:02:46,760 --> 00:02:50,940
+respect to T is the set of all antiderivatives of
+
+37
+00:02:50,940 --> 00:02:54,700
+R denoted by the integration R of T dt
+
+38
+00:03:01,150 --> 00:03:05,510
+لو كانت R Antiderivative لـ R الصغيرة، يعني dR/dT
+
+39
+00:03:05,510 --> 00:03:09,270
+بيساوي الـ R الصغيرة على طول، بنقول الـ integration
+
+40
+00:03:09,270 --> 00:03:13,530
+للـ R هذه بيساوي R of T زائد C، وزي ما كنا نسأل
+
+41
+00:03:13,530 --> 00:03:16,990
+حالنا دائماً، إيش الدالة اللي الـ derivative لها R؟
+
+42
+00:03:16,990 --> 00:03:20,350
+بنحطها هي الـ integration، وبنضيف لها C، بيكون
+
+43
+00:03:20,350 --> 00:03:23,950
+عبارة عن the set of all antiderivatives اللي بنسمي
+
+44
+00:03:23,950 --> 00:03:27,720
+الـ indefinite integral. نشوف هذا الكلام عملياً to
+
+45
+00:03:27,720 --> 00:03:31,260
+integrate a vector function we integrate each of
+
+46
+00:03:31,260 --> 00:03:34,340
+its components. يعني الآن الموضوع سهل زي ما عملنا في
+
+47
+00:03:34,340 --> 00:03:35,880
+الـ derivative، زي ما عملنا في الـ limits of a
+
+48
+00:03:35,880 --> 00:03:38,260
+constant، زي ما عملنا في الـ continuity، بدنا نعمل في
+
+49
+00:03:38,260 --> 00:03:40,720
+الـ integration، الـ integration لـ cos T في اتجاه
+
+50
+00:03:40,720 --> 00:03:46,160
+I زائد J ناقص 2T في اتجاه K، الكل dT، إيش هيساوي؟
+
+51
+00:03:47,610 --> 00:03:50,810
+integration للكمبوننت الأولى زي integration للكمبوننت
+
+52
+00:03:50,810 --> 00:03:53,570
+الثانية زي integration للكمبوننت الثالثة، كل واحدة
+
+53
+00:03:53,570 --> 00:03:57,530
+في اتجاهها، الـ I والـ J والـ K. الآن cos الـ T
+
+54
+00:03:57,530 --> 00:04:01,410
+dT، إيش تكاملها؟ اللي هو طبعاً sin الـ T زائد some
+
+55
+00:04:01,410 --> 00:04:05,450
+constant C1 في اتجاه الـ I،  T زائد C2 في اتجاه الـ J
+
+56
+00:04:05,450 --> 00:04:09,810
+لأن تفاضل T هو T، تفاضل 2T هو 2T تربيع على
+
+57
+00:04:09,810 --> 00:04:13,170
+2، 2T تربيع على 2 اللي هي T تربيع زائد some
+
+58
+00:04:13,170 --> 00:04:17,390
+constant C3 في K. هذا من زمان، شغل زمان تبع اللي
+
+59
+00:04:17,390 --> 00:04:20,190
+هو اللي هي الـ calculus، الـ real valued functions
+
+60
+00:04:20,190 --> 00:04:24,710
+الآن، y = هذه sin الـ T في اتجاه الـ I، وهذه T
+
+61
+00:04:24,710 --> 00:04:29,970
+في اتجاه الـ J ناقص T تربيع في اتجاه الـ K زائد C، هذه
+
+62
+00:04:29,970 --> 00:04:32,570
+C constant vector اللي هو عبارة عن C1I
+
+63
+00:04:35,470 --> 00:04:39,970
++C2J -C3K. واضح؟ بتصور الـ integration، نشوف الكلام
+
+64
+00:04:39,970 --> 00:04:43,910
+أكثر تحديدا. لأن definition of the components of
+
+65
+00:04:43,910 --> 00:04:47,130
+R of T = F of T في اتجاه I + G of T في اتجاه
+
+66
+00:04:47,130 --> 00:04:53,020
+J + H of T في اتجاه K.  لو كانت R متكاملة على A وB، لو كانت
+
+67
+00:04:53,020 --> 00:04:57,440
+هذه integrable على A و B، يعني هذه integrable وهذه
+
+68
+00:04:57,440 --> 00:05:00,940
+integrable وهذه integrable، هتكون هذه نفسها
+
+69
+00:05:00,940 --> 00:05:06,460
+integrable، والـ indefinite integral للـ R of T زي
+
+70
+00:05:06,460 --> 00:05:09,380
+ما قلنا قبل شوية، هي عبارة عن ال…  آه، ده definite
+
+71
+00:05:09,380 --> 00:05:11,910
+آسف، definite integral. هذا اللي هو عبارة عن
+
+72
+00:05:11,910 --> 00:05:15,390
+integration من a لـ b، F of t في اتجاه الـ i + g
+
+73
+00:05:15,390 --> 00:05:18,490
+of t في اتجاه الـ j + integration  H of
+
+74
+00:05:18,490 --> 00:05:22,310
+t في اتجاه الـ k. إذاً برضه الـ definite integral زي
+
+75
+00:05:22,310 --> 00:05:26,350
+الـ indefinite integral، الـ integration عبر من اللي
+
+76
+00:05:26,350 --> 00:05:30,890
+هو كل component من هنا على حدة في اتجاه الـ I،
+
+77
+00:05:30,890 --> 00:05:35,050
+يعني عشان تكون هذه كلها integrable، لازم تكون الـ
+
+78
+00:05:35,050 --> 00:05:38,450
+component الأولى integrable، والثانية، وإيش؟ والثالثة
+
+79
+00:05:38,710 --> 00:05:42,950
+الآن، ناخد مثالين، مثالين.  وان بي
+
+80
+00:05:42,950 --> 00:05:45,990
+integrate each component. الـ integration لـ cos الـ
+
+81
+00:05:45,990 --> 00:05:49,490
+T في اتجاه الـ I من صفر لـ باي زائد J ناقص 2T
+
+82
+00:05:49,490 --> 00:05:53,390
+هذا عملنا لها integration، طلعت عندنا بتساوي الـ
+
+83
+00:05:53,390 --> 00:05:55,970
+integration للأول زي integration للثاني زي الـ
+
+84
+00:05:55,970 --> 00:05:59,510
+integration للثالث، كل واحدة في اتجاهها. الـ
+
+85
+00:05:59,510 --> 00:06:03,390
+integration لـ cos T عبارة عن sin T من 0 لـ Pi في
+
+86
+00:06:03,390 --> 00:06:08,950
+اتجاه I، تكامل dT هو عبارة عن T من 0 لـ Pi، تكامل
+
+87
+00:06:08,950 --> 00:06:13,110
+هذه عبارة عن T تربيع اللي هو في اتجاه K، كلهم من 0
+
+88
+00:06:13,110 --> 00:06:18,850
+لـ Pi، ونبدأ نعوض، هذه بتطلع sin 0 = 0، sin Pi = 0 ناقص sin
+
+89
+00:06:18,850 --> 00:06:23,750
+0 = 0 في اتجاه I، وهذا باي ناقص صفر، وهنا نعوض ويصبح باي
+
+90
+00:06:23,750 --> 00:06:27,930
+تربيع ناقص صفر، فبتصبح عبارة عن باي في اتجاه الـ J ناقص
+
+91
+00:06:27,930 --> 00:06:31,950
+باي تربيع في اتجاه الـ K. هذه اللي هي مين؟ لاحظوا هنا
+
+92
+00:06:31,950 --> 00:06:36,250
+إحنا استخدمنا اللي هو أن هذه بتوزع هنا وهنا وهنا
+
+93
+00:06:36,250 --> 00:06:39,330
+على اللي هو الـ fundamental theorem of calculus على
+
+94
+00:06:39,330 --> 00:06:43,410
+مين؟ على اللي هو الـ scalar components. الآن بقول لك
+
+95
+00:06:43,410 --> 00:06:47,150
+the fundamental theorem of calculus برضه هنا صحيحة
+
+96
+00:06:47,380 --> 00:06:50,320
+التي هي فعلاً لـ vector-valued function. يعني لو جبت
+
+97
+00:06:50,320 --> 00:06:53,320
+الـ antiderivative مرة واحدة بدون أن تقعد تعمل زي
+
+98
+00:06:53,320 --> 00:06:58,240
+هيك، جبت الـ antiderivative اللي هو للـ R of T، جبت
+
+99
+00:06:58,240 --> 00:07:02,880
+هوا، عوضت من اللي بيه بطلع اللي هو بيساوي R of
+
+100
+00:07:02,880 --> 00:07:07,660
+B ناقص R of A، صحيحة. يعني الـ fundamental theorem of
+
+101
+00:07:07,660 --> 00:07:11,460
+calculus اللي هي منطبقة لو كانت هذه continuous
+
+102
+00:07:11,460 --> 00:07:15,200
+function. التي ستكون integration من A لـ B، R of T
+
+103
+00:07:15,200 --> 00:07:18,860
+dT = R الكبير of T من A لـ B، حيث الـ R الـ
+
+104
+00:07:18,860 --> 00:07:22,740
+antiderivative للـ R الصغيرة of T اللي هي عبارة عن
+
+105
+00:07:22,740 --> 00:07:28,680
+R of B ناقص R of A. طبعاً الـ R' إيش بتساوي؟ R of T
+
+106
+00:07:28,680 --> 00:07:30,140
+لأنّه يبقى عندنا الـ antiderivative زي ما أنتم
+
+107
+00:07:30,140 --> 00:07:34,740
+عارفين. طيب، عملياً هي اللي بتساوي هذه بتبرهن لنا هذا
+
+108
+00:07:34,740 --> 00:07:41,660
+الكلام واللي هو بالطريقة اللي حلّيناها. نشوف الآن
+
+109
+00:07:41,660 --> 00:07:48,280
+مثالاً. Suppose we don't know the path of hang
+
+110
+00:07:48,280 --> 00:07:55,080
+gliders، يعني بقول إحنا مش عارفين اللي هو طريق أو
+
+111
+00:07:55,080 --> 00:07:58,880
+الـ position vector تبع مين؟ اللي هي الطائرة
+
+112
+00:07:58,880 --> 00:08:03,600
+الشراعية، but only its acceleration vector A of T
+
+113
+00:08:03,600 --> 00:08:08,800
+بتساوي هذا الكلام، يعني عارفين مين؟ هي عجلتها، يعني
+
+114
+00:08:08,800 --> 00:08:13,280
+مش عارفين الـ R of T، إحنا عارفين الـ A of T وعارفين
+
+115
+00:08:13,280 --> 00:08:18,420
+المعلومات الثانية. نحن أيضاً نعرف أن في الوقت T
+
+116
+00:08:18,420 --> 00:08:22,100
+= صفر، يعني عند نقطة البداية، عند T =
+
+117
+00:08:22,100 --> 00:08:26,100
+صفر، الجليدر
+
+118
+00:08:26,100 --> 00:08:32,050
+بدأ from the point 3, 0, 0 with velocity v of 0 = 3j،
+
+119
+00:08:32,050 --> 00:08:36,390
+يعني بيقول لك لحظة الإقلاع، بنعرف إحنا لحظة
+
+120
+00:08:36,390 --> 00:08:41,050
+الإقلاع كانت الـ position لها للطائرة هذه كانت عند
+
+121
+00:08:41,050 --> 00:08:46,230
+النقطة، إحداثياتها 3 و 0 و 0، وكانت سرعتها في تلك
+
+122
+00:08:46,230 --> 00:08:50,180
+اللحظة…
+
+127
+00:09:00,020 --> 00:09:01,580
+الـ …
+
+129
+00:09:17,850 --> 00:09:23,770
+هذه المعلومات مع بعض هتمكننا من معرفة مين؟ الـ
+
+130
+00:09:23,770 --> 00:09:29,590
+position vector لمين؟ للـ hang glider اللي هي
+
+131
+00:09:29,590 --> 00:09:34,530
+الطائرة الشراعية. Our goal is to find R of T وعندنا
+
+132
+00:09:34,530 --> 00:09:37,290
+المعلومات كما هي، عندنا الـ differential equation
+
+133
+00:09:37,290 --> 00:09:41,190
+اللي هي إيه؟ الـ acceleration، d تربيع R / d تربيع
+
+134
+00:09:41,190 --> 00:09:45,630
+= هذا الكلام. الآن، ما أعطانا الـ initial condition
+
+135
+00:09:45,630 --> 00:09:48,990
+اللي هو V of 0، عند T = 0 كانت 3j،
+
+136
+00:09:48,990 --> 00:09:53,670
+السرعة الـ velocity، وعند اللي هي الـ T =
+
+137
+00:09:53,670 --> 00:09:57,310
+0 كانت R of 0 = 3 في اتجاه الـ X، يعني
+
+138
+00:09:57,310 --> 00:09:59,610
+3 في اتجاه الـ I، 0 في اتجاه الـ J زائد 0
+
+139
+00:09:59,610 --> 00:10:03,650
+في اتجاه الـ K. هذه المعلومات هتمكننا من معرفة الـ R.
+
+140
+00:10:03,650 --> 00:10:08,570
+أول شيء بدي أوجد من الـ A الـ V، بدي أكاملها، يعني يعني بدي
+
+141
+00:10:08,570 --> 00:10:13,450
+أكامل. يعني الآن V of T بكامل هذا المقدار، بيصير تكامل
+
+142
+00:10:13,450 --> 00:10:16,410
+اللي هو ناقص 3 cos T، اللي هو ناقص 3
+
+143
+00:10:16,410 --> 00:10:20,610
+sin T في اتجاه الـ I. الآن بكامل هذه اللي هي زائد
+
+144
+00:10:20,610 --> 00:10:24,050
+تكامل الـ sin ناقص cos، 3 cos T في اتجاه
+
+145
+00:10:24,050 --> 00:10:27,550
+الـ J، وتكامل هذه 2T في اتجاه الـ K زائد some
+
+146
+00:10:27,550 --> 00:10:29,550
+constant C1، هذا vector
+
+147
+00:10:32,340 --> 00:10:35,880
+كاملناها بواسطة الـ Fundamental Theorem والكلاسيكية
+
+148
+00:10:35,880 --> 00:10:42,900
+وضفنا لها some constant C1. الآن، بدي أوجد الـ C1
+
+149
+00:10:42,900 --> 00:10:47,160
+كيف؟ بدي أوجدها من معرفتي V0 = 3j، بعوض
+
+150
+00:10:47,160 --> 00:10:52,240
+عنها بـ 0، بالـ T عندي، اللي هو V of 0 = 3j
+
+151
+00:10:52,240 --> 00:10:57,760
+بتساويها، ناقص 3 sin 0 زائد 3 cos
+
+152
+00:10:57,760 --> 00:11:02,980
+0 زائد 0 كلهم في اتجاهاتهم زائد C1، ماشي الحال؟
+
+153
+00:11:02,980 --> 00:11:09,600
+الآن، هذه صفر، هذه صفر، وهذه 3j فبيصير 3j
+
+154
+00:11:09,600 --> 00:11:14,160
+= 3j = C1، إذًا C1 = 0. إذًا الآن الـ
+
+155
+00:11:14,160 --> 00:11:19,080
+velocity صارت عندي، عرفناها = هذه وهذه وهذه، و
+
+156
+00:11:19,080 --> 00:11:23,120
+بس، وهذا إيش؟ = 0. الآن من الـ velocity بدي أعرف
+
+157
+00:11:23,120 --> 00:11:28,260
+الـ R اللي هو طالبها، بعرف الـ R وبستخدم الـ R of 0
+
+158
+00:11:28,260 --> 00:11:33,280
+عشان أوجد constant الـ R زي ما وجدنا constant الـ V.
+
+159
+00:11:33,760 --> 00:11:36,780
+أكيد كلكم فهمتوا الشيء اللي بقوله، لأن the glider's
+
+160
+00:11:36,780 --> 00:11:39,300
+velocity as a function of time T اللي قبل شوية
+
+161
+00:11:39,300 --> 00:11:44,220
+وجدناها، dR/dT = V of T، هيها، ماشي الحال؟
+
+162
+00:11:44,220 --> 00:11:48,980
+بتوجد منها الـ R مع المعلومة أن R of 0
+
+163
+00:11:48,980 --> 00:11:52,480
+= 3I زي ما قلنا قبل شوية، إذاً المعلومتين
+
+164
+00:11:52,480 --> 00:11:57,000
+هذه مع R of 0 = 3I، بقدر أعرف الـ R
+
+165
+00:11:57,000 --> 00:12:00,770
+بالضبط إيش هي؟ الآن أنا بدي أكامل الجهتين، لما أكامل الـ
+
+166
+00:12:00,770 --> 00:12:05,470
+V بيطلع من الـ R، زي ما كاملنا الـ A طلعت الـ V. الآن
+
+167
+00:12:05,470 --> 00:12:09,210
+إذًا أكامل الـ V بيطلع الـ R، ويساوي الآن تكامل الـ sin
+
+168
+00:12:09,210 --> 00:12:12,330
+اللي هي ناقص cos، اللي هي 3 cos في اتجاه الـ I،
+
+169
+00:12:12,330 --> 00:12:17,160
+وتكامل الـ cos اللي هي 3 sin، وتكامل 2T
+
+170
+00:12:17,160 --> 00:12:20,120
+تربيع اللي هو T تربيع، 2T تربيع على 2 يعني
+
+171
+00:12:20,120 --> 00:12:23,760
+T تربيع زائد some constant C2. يعني طبقنا
+
+172
+00:12:23,760 --> 00:12:27,100
+برضه الـ fundamental theorem of calculus وأخذنا هذه
+
+173
+00:12:27,100 --> 00:12:30,960
+الـ antiderivative لهذه زائد some constant C2.
+
+174
+00:12:31,410 --> 00:12:35,810
+الآن، عشان أوجد C2، بعرف أنا معلومتي R of 0 إيش
+
+175
+00:12:35,810 --> 00:12:39,490
+بتساوي 3I من قبل شوية. الآن صار عندي 3I
+
+176
+00:12:39,490 --> 00:12:43,070
+بتساوي 3 cos 0 + 3 sin 0 + 0
+
+177
+00:12:43,070 --> 00:12:48,490
+تربيع زائد C2. هذه صفر وهذه صفر، وظل 3I =
+
+178
+00:12:48,490 -->
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0839946dd97b4bb3f1db9c65f3806c37908e327e
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI_postprocess.srt
@@ -0,0 +1,740 @@
+1
+00:00:00,770 --> 00:00:04,810
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,810 --> 00:00:10,630
+وبركاته هذه هي المحاضرة التاسعة
+
+3
+00:00:10,630 --> 00:00:19,050
+مساق تفاضل وتكامل جيم هذه المحاضرة الثانية في شبطر
+
+4
+00:00:19,050 --> 00:00:24,640
+13اللي هو تلت عشر اتنين اللي هو integrals of
+
+5
+00:00:24,640 --> 00:00:30,480
+vector functions وهي محاضرة سريعة ومحاضرة خفيفة
+
+6
+00:00:30,480 --> 00:00:34,900
+هنتحكي المرة الماضية في المحاضرة السابقة عن ال
+
+7
+00:00:34,900 --> 00:00:37,940
+limits و ال continuity و ال differentiability و
+
+8
+00:00:37,940 --> 00:00:41,430
+أمور أخرىالان نكمل الموضوع و نشوف كيف نعمل
+
+9
+00:00:41,430 --> 00:00:45,310
+integration لمين للـ vector valued functions بنفس
+
+10
+00:00:45,310 --> 00:00:49,190
+روح ال differentiation و ال limits و ال continuity
+
+11
+00:00:49,190 --> 00:00:53,810
+اللي هي اللي بتعتمد على اللي هي ال components لها
+
+12
+00:00:53,810 --> 00:00:59,610
+الان نأتي نشوف integrals of vector functions الان
+
+13
+00:00:59,610 --> 00:01:03,350
+حديثنا عن ال vector functions اللي هي فرضنا عنها
+
+14
+00:01:03,350 --> 00:01:06,690
+اللي هو vector function اسمها Rالأن a
+
+15
+00:01:06,690 --> 00:01:09,730
+differentiable vector function R of T is an
+
+16
+00:01:09,730 --> 00:01:12,890
+antiderivative of a vector function R of T نفس
+
+17
+00:01:12,890 --> 00:01:16,850
+التعريف اللي أخدناه في Calculus ألف بس هذا ليه
+
+18
+00:01:16,850 --> 00:01:20,810
+فكتور valid فنشن الفنشن R of T اللي بنقول عنها
+
+19
+00:01:20,810 --> 00:01:23,930
+antiderivative للـ R of T طبعاً R of T is a
+
+20
+00:01:23,930 --> 00:01:26,670
+differentiable function هذا R كابيتال بنقول عنها
+
+21
+00:01:26,670 --> 00:01:29,810
+antiderivative للـ R يعني عكس ال derivative للـ R
+
+22
+00:01:29,810 --> 00:01:36,800
+عملية عكسية رفلي رفليأنقان Interval I إذا كانت DR
+
+23
+00:01:36,800 --> 00:01:42,700
+بي DT بيساوي R لما DR Capital بي DT بيساوي R بنقول
+
+24
+00:01:42,700 --> 00:01:47,260
+عن الـ R derivative للـ D للـ R Capital و بنقول
+
+25
+00:01:47,260 --> 00:01:51,620
+للـ R Capital Anti-derivative لمين للـ R على
+
+26
+00:01:51,620 --> 00:01:56,790
+الفترة المذكورة Iالان لو كانت R Anti-derivative
+
+27
+00:01:56,790 --> 00:02:04,350
+للـ R على I بنقدر نقول أي R زائد C نفس الـ R هذه
+
+28
+00:02:04,350 --> 00:02:10,990
+زائد C برضه هتطلع Anti-derivative للـ R اسمه لإن
+
+29
+00:02:10,990 --> 00:02:16,410
+لو فضلنا هذه هتطلع لي برضه R الصغيرة لإنه الآن
+
+30
+00:02:16,410 --> 00:02:22,010
+التفاضل constant هيسيه سفر إذا الان هذا بيقودنا
+
+31
+00:02:22,010 --> 00:02:28,170
+إلى تعريفThe set of all antiderivatives of R أما
+
+32
+00:02:28,170 --> 00:02:33,110
+يعني كل الـ Antiderivatives هدولة هما اللي بنسميهم
+
+33
+00:02:33,110 --> 00:02:37,650
+الـ Indefinite Integral of R أو التكامل غير
+
+34
+00:02:37,650 --> 00:02:41,650
+المحدود للـ R زي ما عملنا بالظبط في Calculus 1000
+
+35
+00:02:41,650 --> 00:02:46,760
+وحتى في توديريالـ Indefinite integral of R with
+
+36
+00:02:46,760 --> 00:02:50,940
+respect to T is the set of all antiderivatives of
+
+37
+00:02:50,940 --> 00:02:54,700
+R denoted by the integration R of T dt
+
+38
+00:03:01,150 --> 00:03:05,510
+لو كانت R Antiderivative لـ R Small يعني DR بDT
+
+39
+00:03:05,510 --> 00:03:09,270
+بيساوي الـ R Small على طول بنقول ال integration
+
+40
+00:03:09,270 --> 00:03:13,530
+للـ R هذه بيساوي R of D زائد C وزي ما كنا نسأل
+
+41
+00:03:13,530 --> 00:03:16,990
+حالنا دايما إيش الدالة اللي ال derivative إلها R
+
+42
+00:03:16,990 --> 00:03:20,350
+بنحطها هي ال integration و بنضيف لها C بيكوني
+
+43
+00:03:20,350 --> 00:03:23,950
+عبارة عن the set of all antiderivatives اللي بنسمي
+
+44
+00:03:23,950 --> 00:03:27,720
+ال indefinite integralانشوف هذا الكلام عملي to
+
+45
+00:03:27,720 --> 00:03:31,260
+integrate a vector function we integrate each of
+
+46
+00:03:31,260 --> 00:03:34,340
+its component يعني الأن الموضوع سهل زي ما عملنا في
+
+47
+00:03:34,340 --> 00:03:35,880
+ال derivative زي ما عملنا في ال limits of a
+
+48
+00:03:35,880 --> 00:03:38,260
+constant زي ما عملنا في ال continuity بدنا نعمل في
+
+49
+00:03:38,260 --> 00:03:40,720
+ال integration ال integration ل cosine T في اتجاه
+
+50
+00:03:40,720 --> 00:03:46,160
+I زائد J نقص 2T في اتجاه K كله DT ايش هيساوي؟
+
+51
+00:03:47,610 --> 00:03:50,810
+انتجريشن للكمبونات الأولى زي الانتجريشن للكمبونات
+
+52
+00:03:50,810 --> 00:03:53,570
+الثانية زي الانتجريشن للكمبونات الثالثة كل واحدة
+
+53
+00:03:53,570 --> 00:03:57,530
+في اتجاهها ال I و ال J و ال K الان جيكو ساين ال T
+
+54
+00:03:57,530 --> 00:04:01,410
+DT ايش تكملها اللي هو طبعا ساين ال T زي some
+
+55
+00:04:01,410 --> 00:04:05,450
+constant C1 في اتجاه ال I T زي C2 في اتجاه ال J
+
+56
+00:04:05,450 --> 00:04:09,810
+لأن تفاضل DT ايش ماله T تفاضل اتنين T تربيع على
+
+57
+00:04:09,810 --> 00:04:13,170
+اتنين T تربيع على اتنين اللي هي T تربيعزائد some
+
+58
+00:04:13,170 --> 00:04:17,390
+constant c3 في k هذا من تبع زمان شغل زمان تبع اللي
+
+59
+00:04:17,390 --> 00:04:20,190
+هو اللي هي ال calculus ال real valid functions
+
+60
+00:04:20,190 --> 00:04:24,710
+الان y ساوي هذه sign ال T في تجاه ال I وهذه T sign
+
+61
+00:04:24,710 --> 00:04:29,970
+في تجاه ال J ناقص T تربيع في تجاه ال K زائد C هذه
+
+62
+00:04:29,970 --> 00:04:32,570
+C constant vector اللي هو عبارة عن C1I
+
+63
+00:04:35,470 --> 00:04:39,970
+C2J-C3K واضح بتصور ال integration انشوف الكلام
+
+64
+00:04:39,970 --> 00:04:43,910
+اكتر تحديدا لان ال definition of the components of
+
+65
+00:04:43,910 --> 00:04:47,130
+R of T بساوي F of T في اتجاه I زي G of T في اتجاه
+
+66
+00:04:47,130 --> 00:04:53,020
+G زي H of T اتجاه K R انتجرة بال over A وBلو كانت
+
+67
+00:04:53,020 --> 00:04:57,440
+هذه Integrable على A و B يعني هذه Integrable و هذه
+
+68
+00:04:57,440 --> 00:05:00,940
+Integrable و هذه Integrable هتكون هذه نفسها
+
+69
+00:05:00,940 --> 00:05:06,460
+Integrable و الـ Indefinite Integral للـ R of T زي
+
+70
+00:05:06,460 --> 00:05:09,380
+ما قلنا قبل بشوية هي عبارة عن ال .. اه ده Definite
+
+71
+00:05:09,380 --> 00:05:11,910
+أسف Definite Integralهذا اللي هو عبارة عن
+
+72
+00:05:11,910 --> 00:05:15,390
+integration من a لb f of t في اتجاه الـ i زائد g
+
+73
+00:05:15,390 --> 00:05:18,490
+of t في اتجاه ال k الجيه زائد integration ال h of
+
+74
+00:05:18,490 --> 00:05:22,310
+t في اتجاه ال k إذا برضه ال definitely integral زي
+
+75
+00:05:22,310 --> 00:05:26,350
+ال indefinite integral ال integration عبر من اللي
+
+76
+00:05:26,350 --> 00:05:30,890
+هو الكامل component من هنا على حده في اتجاه ال I
+
+77
+00:05:30,890 --> 00:05:35,050
+يعني عشان تكون هذه كلها integrable لازم تكون ال
+
+78
+00:05:35,050 --> 00:05:38,450
+component الأولى integrable والتاني وإيش والتالتة
+
+79
+00:05:38,710 --> 00:05:42,950
+الان ناخد از ان اكزامبل اتنين از ان اكزامبل وان بي
+
+80
+00:05:42,950 --> 00:05:45,990
+integrate each component ال integration لكسايم ال
+
+81
+00:05:45,990 --> 00:05:49,490
+T في اتجاه ال I من سفر ل باى زي ال J نقص اتنين T
+
+82
+00:05:49,490 --> 00:05:53,390
+توا هذا عملنا لها integration طلعت عندنا بتساوي ال
+
+83
+00:05:53,390 --> 00:05:55,970
+integration للأول زي ال integration للثاني زي ال
+
+84
+00:05:55,970 --> 00:05:59,510
+integration للثالث كل واحدة في اتجاههاالـ
+
+85
+00:05:59,510 --> 00:06:03,390
+integration لـ cos T عبارة عن sin T من 0 لـ Pi في
+
+86
+00:06:03,390 --> 00:06:08,950
+اتجاه I التكامل DT هو عبارة عن T من 0 لPi التكامل
+
+87
+00:06:08,950 --> 00:06:13,110
+هذه عبارة عن T تربيع اللي هو في اتجاه LK كلهم من 0
+
+88
+00:06:13,110 --> 00:06:18,850
+لPi ونبدأ نعوض هذه بتطلع sin 0 0 sin Pi 0 ناقص sin
+
+89
+00:06:18,850 --> 00:06:23,750
+0 0 في اتجاه Iوهذا by نقص صفر وهنا نعود ويصبح by
+
+90
+00:06:23,750 --> 00:06:27,930
+تربيع نقص صفر فبتصبح عبارة عن by في اتجاه ال J نقص
+
+91
+00:06:27,930 --> 00:06:31,950
+by تربيع في اتجاه ال K هذه اللي هي مين لاحظوا هنا
+
+92
+00:06:31,950 --> 00:06:36,250
+احنا استخدمنا اللي هو ان هذي بتتوزح هن و هن و هن
+
+93
+00:06:36,250 --> 00:06:39,330
+على اللي هو ال fundamental theorem of calculus على
+
+94
+00:06:39,330 --> 00:06:43,410
+مين على اللي هو ال scalar components الآن بقولك
+
+95
+00:06:43,410 --> 00:06:47,150
+the fundamental theorem of calculus برضه هن صحيحة
+
+96
+00:06:47,380 --> 00:06:50,320
+التي هي فعلة الـ vector-valid function يعني لو جبت
+
+97
+00:06:50,320 --> 00:06:53,320
+الـ antiderivative مرة واحدة بدون أن تقعد تعمل زي
+
+98
+00:06:53,320 --> 00:06:58,240
+هيك جبت ال antiderivative اللي هو لل R of T جبت
+
+99
+00:06:58,240 --> 00:07:02,880
+هوا الاشي أعوضت من إلي بيه بطلع اللي هو بساوي R of
+
+100
+00:07:02,880 --> 00:07:07,660
+B ناقص R of A صحيحة يعني ال fundamental theorem of
+
+101
+00:07:07,660 --> 00:07:11,460
+calculus اللي هي منطبقها لو كانت هذه continuous
+
+102
+00:07:11,460 --> 00:07:15,200
+functionالتي ستكون الـ integration من A ل B R of D
+
+103
+00:07:15,200 --> 00:07:18,860
+DT بسوء R capital of D من A ل B حيث الـ R الـ
+
+104
+00:07:18,860 --> 00:07:22,740
+antiderivative للـ R small of D اللي هي عناوار عن
+
+105
+00:07:22,740 --> 00:07:28,680
+R of B نقص R of A طبعاً الـ R' أيش بتساوي R of D
+
+106
+00:07:28,680 --> 00:07:30,140
+لإنه يبقى عندنا الـ antiderivative زي ما انتوا
+
+107
+00:07:30,140 --> 00:07:34,740
+عارفين طيب عملياً هي اللي بتسوي هذه بتشرع لنا هذا
+
+108
+00:07:34,740 --> 00:07:41,660
+الكلام و اللي هو بالطريقة اللي حلناها نشوف الآن
+
+109
+00:07:41,660 --> 00:07:48,280
+مثالعمليه suppose we don't know the path of hang
+
+110
+00:07:48,280 --> 00:07:55,080
+gliders يعني بقول إحنا مش عارفين اللي هو طريق أو
+
+111
+00:07:55,080 --> 00:07:58,880
+ال position vector تبعت مين اللي هي الطائرة
+
+112
+00:07:58,880 --> 00:08:03,600
+الشراعية but only its acceleration vector A of T
+
+113
+00:08:03,600 --> 00:08:08,800
+بتساوي هذا الكلام يعني عارفين مين هي عجلتها يعني
+
+114
+00:08:08,800 --> 00:08:13,280
+مش عارفين ال R of T إيها عارفين ال A of Tوعارفين
+
+115
+00:08:13,280 --> 00:08:18,420
+المعلومات التانية نحن ايضا نعرف ان في الوقت تي
+
+116
+00:08:18,420 --> 00:08:22,100
+بالساوة سفر يعني عند نقطة البداية عن تي بالساوة
+
+117
+00:08:22,100 --> 00:08:26,100
+سفر الجليدر
+
+118
+00:08:26,100 --> 00:08:32,050
+بدأfrom the point 300 with velocity v of 0 سوى 3G
+
+119
+00:08:32,050 --> 00:08:36,390
+يعني بيقول ليه لحظة الإقلاع بنعرف احنا لحظة
+
+120
+00:08:36,390 --> 00:08:41,050
+الإقلاع كانت ال position لها للطيارة هذه كانت عند
+
+121
+00:08:41,050 --> 00:08:46,230
+النقطة الإحداثياتها 3 و 0 و 0 و كانت سرعتها في تلك
+
+122
+00:08:46,230 --> 00:08:50,180
+اللحظةالتي هي لحظة الـ ال ال ال ال ال ال ال ال ال
+
+123
+00:08:50,180 --> 00:08:50,680
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+124
+00:08:50,680 --> 00:08:51,260
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+125
+00:08:51,260 --> 00:08:51,920
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+126
+00:08:51,920 --> 00:09:00,020
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+127
+00:09:00,020 --> 00:09:01,580
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+128
+00:09:01,580 --> 00:09:03,400
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+129
+00:09:17,850 --> 00:09:23,770
+هذه المعلومات مع بعض هتمكننا في معرفة مين الـ
+
+130
+00:09:23,770 --> 00:09:29,590
+position vector للإيمين للـ hang glider اللي هي
+
+131
+00:09:29,590 --> 00:09:34,530
+الطرائرة الشراعية our goal is to find R of T وعنا
+
+132
+00:09:34,530 --> 00:09:37,290
+المعلومات كما هي لها عنا ال differential equation
+
+133
+00:09:37,290 --> 00:09:41,190
+اللي هي ايه ال acceleration دي تربيع أربع دي تربيع
+
+134
+00:09:41,190 --> 00:09:45,630
+سوى هذا الكلامالان وماعطينا ال initial condition
+
+135
+00:09:45,630 --> 00:09:48,990
+اللي هو V of Zero عند T بتساوي Zero كانت تلاتة J
+
+136
+00:09:48,990 --> 00:09:53,670
+السرعة ال velocity وعند اللي هي ال ال ال T بتساوي
+
+137
+00:09:53,670 --> 00:09:57,310
+سفر كانت R of Zero بتساوي تلاتة قيمة ال X يعني
+
+138
+00:09:57,310 --> 00:09:59,610
+تلاتة في اتجاه ال I Zero في اتجاه ال J زائد Zero
+
+139
+00:09:59,610 --> 00:10:03,650
+في اتجاه ال K هذه المعلومات هتمكننا بمعرفة ال R
+
+140
+00:10:03,650 --> 00:10:08,570
+أول اشي بده اوجد من ال ALV ده كاملها يعني يعني بده
+
+141
+00:10:08,570 --> 00:10:13,450
+كامليعني الان V of T بكامل هذا المقدار بيصير تكامل
+
+142
+00:10:13,450 --> 00:10:16,410
+اللي هو ناقص ثلاثة cosine T اللي هو ناقص ثلاثة
+
+143
+00:10:16,410 --> 00:10:20,610
+sine T في اتجاه ال I الان بكامل هذه اللي هي زائد
+
+144
+00:10:20,610 --> 00:10:24,050
+تكامل ال sine ناقص cosine تلاتة cosine T في اتجاه
+
+145
+00:10:24,050 --> 00:10:27,550
+ال J وتكامل هذه اتنين T في اتجاه ال K زائد mean
+
+146
+00:10:27,550 --> 00:10:29,550
+some constant C1 هذا vector
+
+147
+00:10:32,340 --> 00:10:35,880
+كملناها بواسط الـ Fundamental Theorem و الكلاسة
+
+148
+00:10:35,880 --> 00:10:42,900
+وضفنا لها some constant C1 الان بدأ اوجد الـ C1
+
+149
+00:10:42,900 --> 00:10:47,160
+كيف ده اوجدها من معرفتي V0 ايش بالساوي 3G بعوض
+
+150
+00:10:47,160 --> 00:10:52,240
+عنها بـ 0 بالـ C عندى اللي هوV of Zero هو تلاتة
+
+151
+00:10:52,240 --> 00:10:57,760
+جهيها ويساوي نقص تلاتة Sin Zero زائد تلاتة Cos
+
+152
+00:10:57,760 --> 00:11:02,980
+Zero زائد Zero كلهم في اتجاهي زائد C1 ماشي الحال
+
+153
+00:11:02,980 --> 00:11:09,600
+الان هذه سفرو هذه صفر وهذه تلاتة J فبصير تلاتة J
+
+154
+00:11:09,600 --> 00:11:14,160
+تلاتة J زي C1 اذا C1 ايش بتساوي صفر اذا الان ال
+
+155
+00:11:14,160 --> 00:11:19,080
+velocity صارت عندي عرفها بالساوية هذه وهذه وهذه و
+
+156
+00:11:19,080 --> 00:11:23,120
+بس وهذا ايه ايش صفر الان من ال velocity بده اعرف
+
+157
+00:11:23,120 --> 00:11:28,260
+من ال R اللي هو طلبها بعرف ال R و بستخدم ال R في Z
+
+158
+00:11:28,260 --> 00:11:33,280
+عشان اوجد constant ال R زي ما وجدنا constant LV
+
+159
+00:11:33,760 --> 00:11:36,780
+أكيد كلكم مفهم أشي اللي بقوله لأن the glider's
+
+160
+00:11:36,780 --> 00:11:39,300
+velocity as a function of time T اللي قبل بشوية
+
+161
+00:11:39,300 --> 00:11:44,220
+وجدناها DR by DT بساوة V of T هيها ماشي الحالة
+
+162
+00:11:44,220 --> 00:11:48,980
+بتدوجد منها ال R مع المعلومة ان ال R of Zero
+
+163
+00:11:48,980 --> 00:11:52,480
+بتساوي تلتة I زي ما قلنا قبل بشوية إذا المعلومتين
+
+164
+00:11:52,480 --> 00:11:57,000
+هذه مع R of Zero بتساوي تلتة I بقدر أعرف ال R
+
+165
+00:11:57,000 --> 00:12:00,770
+بالظبط إيش هيالان انا بدي اكمل الجهتين لما اكمل ال
+
+166
+00:12:00,770 --> 00:12:05,470
+V بيطلع من ال R زي ما اكملنا ال A طلعت ال V الان
+
+167
+00:12:05,470 --> 00:12:09,210
+اذا اكمل ال V بيطلع ال R و يساوي الان تكامل ال Sin
+
+168
+00:12:09,210 --> 00:12:12,330
+اللي هي ناقص Cos اللي هي تلاتة Cos تجاه ال I
+
+169
+00:12:12,330 --> 00:12:17,160
+وتكامل ال Cos اللي هي ثلاثة Sinوالتكامل اتنين T
+
+170
+00:12:17,160 --> 00:12:20,120
+تربيع اللي هو T تربيع اتنين T تربيع على اتنين يعني
+
+171
+00:12:20,120 --> 00:12:23,760
+T تربيع زائد some constant C اتنين يعني اطبقنا
+
+172
+00:12:23,760 --> 00:12:27,100
+برضه ال fundamental theorem افق الكلاص واخدنا هذه
+
+173
+00:12:27,100 --> 00:12:30,960
+ال antiderivative لهذه زائد some constant C اتنين
+
+174
+00:12:31,410 --> 00:12:35,810
+الان عشان اوجد C2 بعرف انا معلومتي R of Zero ايش
+
+175
+00:12:35,810 --> 00:12:39,490
+بتساوي تلاتة I من قبل بشوية الان صار عندى تلاتة I
+
+176
+00:12:39,490 --> 00:12:43,070
+بتساوي تلاتة في Cos Zero و تلاتة Sin Zero Zero
+
+177
+00:12:43,070 --> 00:12:48,490
+تربيع زي ال C2 هذه سفر وهذه سفر وظل تلاتة I بتساوي
+
+178
+00:12:48,490 --> 00:12:52,930
+تلاتة I وهنا C2 يعني C2 بتساوي سفر يعني الان ال
+
+179
+00:12:52,930 --> 00:12:57,660
+glidersposition as a function of T هي بالظبط
+
+180
+00:12:57,660 --> 00:13:01,540
+بالساوية تلاتة cos T في اتجاه I تلاتة sin T في
+
+181
+00:13:01,540 --> 00:13:06,440
+اتجاه J ساعة T تربيع في اتجاه K وبتكون هك او جدنا
+
+182
+00:13:06,440 --> 00:13:09,640
+اللي هو ال position ال position vector لها و لو حد
+
+183
+00:13:09,640 --> 00:13:14,960
+جرب بالكمبيوتر حيشوف الرسمة اللي أمامنا و هيك
+
+184
+00:13:14,960 --> 00:13:17,760
+بتكون انتهت المحاضرة والسلام عليكم ورحمة الله
+
+185
+00:13:17,760 --> 00:13:19,060
+وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..3b5459b803a3f398efa8af0eb345866b16127003
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI_raw.json
@@ -0,0 +1 @@
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{"start": 764.91, "end": 765.31, "word": " سفر", "probability": 0.82373046875}, {"start": 765.31, "end": 765.95, "word": " وهذه", "probability": 0.75927734375}, {"start": 765.95, "end": 766.41, "word": " سفر", "probability": 0.9812825520833334}, {"start": 766.41, "end": 766.91, "word": " وظل", "probability": 0.763427734375}, {"start": 766.91, "end": 767.41, "word": " تلاتة", "probability": 0.97265625}, {"start": 767.41, "end": 767.77, "word": " I", "probability": 0.9951171875}, {"start": 767.77, "end": 768.49, "word": " بتساوي", "probability": 0.9736328125}, {"start": 768.49, "end": 768.91, "word": " تلاتة", "probability": 0.9891357421875}, {"start": 768.91, "end": 769.19, "word": " I", "probability": 0.9951171875}, {"start": 769.19, "end": 769.71, "word": " وهنا", "probability": 0.767578125}, {"start": 769.71, "end": 770.31, "word": " C2", "probability": 0.97607421875}, {"start": 770.31, "end": 770.79, "word": " يعني", "probability": 0.865478515625}, {"start": 770.79, "end": 771.19, 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0.67041015625}, {"start": 775.1, "end": 775.54, "word": " a", "probability": 0.890625}, {"start": 775.54, "end": 775.92, "word": " function", "probability": 0.974609375}, {"start": 775.92, "end": 776.22, "word": " of", "probability": 0.970703125}, {"start": 776.22, "end": 776.52, "word": " T", "probability": 0.45654296875}, {"start": 776.52, "end": 777.1, "word": " هي", "probability": 0.564453125}, {"start": 777.1, "end": 777.66, "word": " بالظبط", "probability": 0.956787109375}, {"start": 777.66, "end": 778.34, "word": " بالساوية", "probability": 0.80693359375}, {"start": 778.34, "end": 779.02, "word": " تلاتة", "probability": 0.900634765625}, {"start": 779.02, "end": 779.22, "word": " cos", "probability": 0.224365234375}, {"start": 779.22, "end": 779.58, "word": " T", "probability": 0.60888671875}, {"start": 779.58, "end": 779.68, "word": " في", "probability": 0.55419921875}, {"start": 779.68, "end": 779.98, "word": " اتجاه", "probability": 0.89306640625}, {"start": 779.98, "end": 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..819dd70038384813c9748496afab0a24ce8f5566
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/eivULxpTllI_raw.srt
@@ -0,0 +1,768 @@
+1
+00:00:00,770 --> 00:00:04,810
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,810 --> 00:00:10,630
+وبركاته هذه هي المحاضرة التاسعة
+
+3
+00:00:10,630 --> 00:00:19,050
+مساق تفاضل وتكامل جيم هذه المحاضرة الثانية في شبطر
+
+4
+00:00:19,050 --> 00:00:24,640
+13اللي هو تلت عشر اتنين اللي هو integrals of
+
+5
+00:00:24,640 --> 00:00:30,480
+vector functions وهي محاضرة سريعة ومحاضرة خفيفة
+
+6
+00:00:30,480 --> 00:00:34,900
+هنتحكي المرة الماضية في المحاضرة السابقة عن ال
+
+7
+00:00:34,900 --> 00:00:37,940
+limits و ال continuity و ال differentiability و
+
+8
+00:00:37,940 --> 00:00:41,430
+أمور أخرىالان نكمل الموضوع و نشوف كيف نعمل
+
+9
+00:00:41,430 --> 00:00:45,310
+integration لمين للـ vector valued functions بنفس
+
+10
+00:00:45,310 --> 00:00:49,190
+روح ال differentiation و ال limits و ال continuity
+
+11
+00:00:49,190 --> 00:00:53,810
+اللي هي اللي بتعتمد على اللي هي ال components لها
+
+12
+00:00:53,810 --> 00:00:59,610
+الان نأتي نشوف integrals of vector functions الان
+
+13
+00:00:59,610 --> 00:01:03,350
+حديثنا عن ال vector functions اللي هي فرضنا عنها
+
+14
+00:01:03,350 --> 00:01:06,690
+اللي هو vector function اسمها Rالأن a
+
+15
+00:01:06,690 --> 00:01:09,730
+differentiable vector function R of T is an
+
+16
+00:01:09,730 --> 00:01:12,890
+antiderivative of a vector function R of T نفس
+
+17
+00:01:12,890 --> 00:01:16,850
+التعريف اللي أخدناه في Calculus ألف بس هذا ليه
+
+18
+00:01:16,850 --> 00:01:20,810
+فكتور valid فنشن الفنشن R of T اللي بنقول عنها
+
+19
+00:01:20,810 --> 00:01:23,930
+antiderivative للـ R of T طبعاً R of T is a
+
+20
+00:01:23,930 --> 00:01:26,670
+differentiable function هذا R كابيتال بنقول عنها
+
+21
+00:01:26,670 --> 00:01:29,810
+antiderivative للـ R يعني عكس ال derivative للـ R
+
+22
+00:01:29,810 --> 00:01:36,800
+عملية عكسية رفلي رفليأنقان Interval I إذا كانت DR
+
+23
+00:01:36,800 --> 00:01:42,700
+بي DT بيساوي R لما DR Capital بي DT بيساوي R بنقول
+
+24
+00:01:42,700 --> 00:01:47,260
+عن الـ R derivative للـ D للـ R Capital و بنقول
+
+25
+00:01:47,260 --> 00:01:51,620
+للـ R Capital Anti-derivative لمين للـ R على
+
+26
+00:01:51,620 --> 00:01:56,790
+الفترة المذكورة Iالان لو كانت R Anti-derivative
+
+27
+00:01:56,790 --> 00:02:04,350
+للـ R على I بنقدر نقول أي R زائد C نفس الـ R هذه
+
+28
+00:02:04,350 --> 00:02:10,990
+زائد C برضه هتطلع Anti-derivative للـ R اسمه لإن
+
+29
+00:02:10,990 --> 00:02:16,410
+لو فضلنا هذه هتطلع لي برضه R الصغيرة لإنه الآن
+
+30
+00:02:16,410 --> 00:02:22,010
+التفاضل constant هيسيه سفر إذا الان هذا بيقودنا
+
+31
+00:02:22,010 --> 00:02:28,170
+إلى تعريفThe set of all antiderivatives of R أما
+
+32
+00:02:28,170 --> 00:02:33,110
+يعني كل الـ Antiderivatives هدولة هما اللي بنسميهم
+
+33
+00:02:33,110 --> 00:02:37,650
+الـ Indefinite Integral of R أو التكامل غير
+
+34
+00:02:37,650 --> 00:02:41,650
+المحدود للـ R زي ما عملنا بالظبط في Calculus 1000
+
+35
+00:02:41,650 --> 00:02:46,760
+وحتى في توديريالـ Indefinite integral of R with
+
+36
+00:02:46,760 --> 00:02:50,940
+respect to T is the set of all antiderivatives of
+
+37
+00:02:50,940 --> 00:02:54,700
+R denoted by the integration R of T dt
+
+38
+00:03:01,150 --> 00:03:05,510
+لو كانت R Antiderivative لـ R Small يعني DR بDT
+
+39
+00:03:05,510 --> 00:03:09,270
+بيساوي الـ R Small على طول بنقول ال integration
+
+40
+00:03:09,270 --> 00:03:13,530
+للـ R هذه بيساوي R of D زائد C وزي ما كنا نسأل
+
+41
+00:03:13,530 --> 00:03:16,990
+حالنا دايما إيش الدالة اللي ال derivative إلها R
+
+42
+00:03:16,990 --> 00:03:20,350
+بنحطها هي ال integration و بنضيف لها C بيكوني
+
+43
+00:03:20,350 --> 00:03:23,950
+عبارة عن the set of all antiderivatives اللي بنسمي
+
+44
+00:03:23,950 --> 00:03:27,720
+ال indefinite integralانشوف هذا الكلام عملي to
+
+45
+00:03:27,720 --> 00:03:31,260
+integrate a vector function we integrate each of
+
+46
+00:03:31,260 --> 00:03:34,340
+its component يعني الأن الموضوع سهل زي ما عملنا في
+
+47
+00:03:34,340 --> 00:03:35,880
+ال derivative زي ما عملنا في ال limits of a
+
+48
+00:03:35,880 --> 00:03:38,260
+constant زي ما عملنا في ال continuity بدنا نعمل في
+
+49
+00:03:38,260 --> 00:03:40,720
+ال integration ال integration ل cosine T في اتجاه
+
+50
+00:03:40,720 --> 00:03:46,160
+I زائد J نقص 2T في اتجاه K كله DT ايش هيساوي؟
+
+51
+00:03:47,610 --> 00:03:50,810
+انتجريشن للكمبونات الأولى زي الانتجريشن للكمبونات
+
+52
+00:03:50,810 --> 00:03:53,570
+الثانية زي الانتجريشن للكمبونات الثالثة كل واحدة
+
+53
+00:03:53,570 --> 00:03:57,530
+في اتجاهها ال I و ال J و ال K الان جيكو ساين ال T
+
+54
+00:03:57,530 --> 00:04:01,410
+DT ايش تكملها اللي هو طبعا ساين ال T زي some
+
+55
+00:04:01,410 --> 00:04:05,450
+constant C1 في اتجاه ال I T زي C2 في اتجاه ال J
+
+56
+00:04:05,450 --> 00:04:09,810
+لأن تفاضل DT ايش ماله T تفاضل اتنين T تربيع على
+
+57
+00:04:09,810 --> 00:04:13,170
+اتنين T تربيع على اتنين اللي هي T تربيعزائد some
+
+58
+00:04:13,170 --> 00:04:17,390
+constant c3 في k هذا من تبع زمان شغل زمان تبع اللي
+
+59
+00:04:17,390 --> 00:04:20,190
+هو اللي هي ال calculus ال real valid functions
+
+60
+00:04:20,190 --> 00:04:24,710
+الان y ساوي هذه sign ال T في تجاه ال I وهذه T sign
+
+61
+00:04:24,710 --> 00:04:29,970
+في تجاه ال J ناقص T تربيع في تجاه ال K زائد C هذه
+
+62
+00:04:29,970 --> 00:04:32,570
+C constant vector اللي هو عبارة عن C1I
+
+63
+00:04:35,470 --> 00:04:39,970
+C2J-C3K واضح بتصور ال integration انشوف الكلام
+
+64
+00:04:39,970 --> 00:04:43,910
+اكتر تحديدا لان ال definition of the components of
+
+65
+00:04:43,910 --> 00:04:47,130
+R of T بساوي F of T في اتجاه I زي G of T في اتجاه
+
+66
+00:04:47,130 --> 00:04:53,020
+G زي H of T اتجاه K R انتجرة بال over A وBلو كانت
+
+67
+00:04:53,020 --> 00:04:57,440
+هذه Integrable على A و B يعني هذه Integrable و هذه
+
+68
+00:04:57,440 --> 00:05:00,940
+Integrable و هذه Integrable هتكون هذه نفسها
+
+69
+00:05:00,940 --> 00:05:06,460
+Integrable و الـ Indefinite Integral للـ R of T زي
+
+70
+00:05:06,460 --> 00:05:09,380
+ما قلنا قبل بشوية هي عبارة عن ال .. اه ده Definite
+
+71
+00:05:09,380 --> 00:05:11,910
+أسف Definite Integralهذا اللي هو عبارة عن
+
+72
+00:05:11,910 --> 00:05:15,390
+integration من a لb f of t في اتجاه الـ i زائد g
+
+73
+00:05:15,390 --> 00:05:18,490
+of t في اتجاه ال k الجيه زائد integration ال h of
+
+74
+00:05:18,490 --> 00:05:22,310
+t في اتجاه ال k إذا برضه ال definitely integral زي
+
+75
+00:05:22,310 --> 00:05:26,350
+ال indefinite integral ال integration عبر من اللي
+
+76
+00:05:26,350 --> 00:05:30,890
+هو الكامل component من هنا على حده في اتجاه ال I
+
+77
+00:05:30,890 --> 00:05:35,050
+يعني عشان تكون هذه كلها integrable لازم تكون ال
+
+78
+00:05:35,050 --> 00:05:38,450
+component الأولى integrable والتاني وإيش والتالتة
+
+79
+00:05:38,710 --> 00:05:42,950
+الان ناخد از ان اكزامبل اتنين از ان اكزامبل وان بي
+
+80
+00:05:42,950 --> 00:05:45,990
+integrate each component ال integration لكسايم ال
+
+81
+00:05:45,990 --> 00:05:49,490
+T في اتجاه ال I من سفر ل باى زي ال J نقص اتنين T
+
+82
+00:05:49,490 --> 00:05:53,390
+توا هذا عملنا لها integration طلعت عندنا بتساوي ال
+
+83
+00:05:53,390 --> 00:05:55,970
+integration للأول زي ال integration للثاني زي ال
+
+84
+00:05:55,970 --> 00:05:59,510
+integration للثالث كل واحدة في اتجاههاالـ
+
+85
+00:05:59,510 --> 00:06:03,390
+integration لـ cos T عبارة عن sin T من 0 لـ Pi في
+
+86
+00:06:03,390 --> 00:06:08,950
+اتجاه I التكامل DT هو عبارة عن T من 0 لPi التكامل
+
+87
+00:06:08,950 --> 00:06:13,110
+هذه عبارة عن T تربيع اللي هو في اتجاه LK كلهم من 0
+
+88
+00:06:13,110 --> 00:06:18,850
+لPi ونبدأ نعوض هذه بتطلع sin 0 0 sin Pi 0 ناقص sin
+
+89
+00:06:18,850 --> 00:06:23,750
+0 0 في اتجاه Iوهذا by نقص صفر وهنا نعود ويصبح by
+
+90
+00:06:23,750 --> 00:06:27,930
+تربيع نقص صفر فبتصبح عبارة عن by في اتجاه ال J نقص
+
+91
+00:06:27,930 --> 00:06:31,950
+by تربيع في اتجاه ال K هذه اللي هي مين لاحظوا هنا
+
+92
+00:06:31,950 --> 00:06:36,250
+احنا استخدمنا اللي هو ان هذي بتتوزح هن و هن و هن
+
+93
+00:06:36,250 --> 00:06:39,330
+على اللي هو ال fundamental theorem of calculus على
+
+94
+00:06:39,330 --> 00:06:43,410
+مين على اللي هو ال scalar components الآن بقولك
+
+95
+00:06:43,410 --> 00:06:47,150
+the fundamental theorem of calculus برضه هن صحيحة
+
+96
+00:06:47,380 --> 00:06:50,320
+التي هي فعلة الـ vector-valid function يعني لو جبت
+
+97
+00:06:50,320 --> 00:06:53,320
+الـ antiderivative مرة واحدة بدون أن تقعد تعمل زي
+
+98
+00:06:53,320 --> 00:06:58,240
+هيك جبت ال antiderivative اللي هو لل R of T جبت
+
+99
+00:06:58,240 --> 00:07:02,880
+هوا الاشي أعوضت من إلي بيه بطلع اللي هو بساوي R of
+
+100
+00:07:02,880 --> 00:07:07,660
+B ناقص R of A صحيحة يعني ال fundamental theorem of
+
+101
+00:07:07,660 --> 00:07:11,460
+calculus اللي هي منطبقها لو كانت هذه continuous
+
+102
+00:07:11,460 --> 00:07:15,200
+functionالتي ستكون الـ integration من A ل B R of D
+
+103
+00:07:15,200 --> 00:07:18,860
+DT بسوء R capital of D من A ل B حيث الـ R الـ
+
+104
+00:07:18,860 --> 00:07:22,740
+antiderivative للـ R small of D اللي هي عناوار عن
+
+105
+00:07:22,740 --> 00:07:28,680
+R of B نقص R of A طبعاً الـ R' أيش بتساوي R of D
+
+106
+00:07:28,680 --> 00:07:30,140
+لإنه يبقى عندنا الـ antiderivative زي ما انتوا
+
+107
+00:07:30,140 --> 00:07:34,740
+عارفين طيب عملياً هي اللي بتسوي هذه بتشرع لنا هذا
+
+108
+00:07:34,740 --> 00:07:41,660
+الكلام و اللي هو بالطريقة اللي حلناها نشوف الآن
+
+109
+00:07:41,660 --> 00:07:48,280
+مثالعمليه suppose we don't know the path of hang
+
+110
+00:07:48,280 --> 00:07:55,080
+gliders يعني بقول إحنا مش عارفين اللي هو طريق أو
+
+111
+00:07:55,080 --> 00:07:58,880
+ال position vector تبعت مين اللي هي الطائرة
+
+112
+00:07:58,880 --> 00:08:03,600
+الشراعية but only its acceleration vector A of T
+
+113
+00:08:03,600 --> 00:08:08,800
+بتساوي هذا الكلام يعني عارفين مين هي عجلتها يعني
+
+114
+00:08:08,800 --> 00:08:13,280
+مش عارفين ال R of T إيها عارفين ال A of Tوعارفين
+
+115
+00:08:13,280 --> 00:08:18,420
+المعلومات التانية نحن ايضا نعرف ان في الوقت تي
+
+116
+00:08:18,420 --> 00:08:22,100
+بالساوة سفر يعني عند نقطة البداية عن تي بالساوة
+
+117
+00:08:22,100 --> 00:08:26,100
+سفر الجليدر
+
+118
+00:08:26,100 --> 00:08:32,050
+بدأfrom the point 300 with velocity v of 0 سوى 3G
+
+119
+00:08:32,050 --> 00:08:36,390
+يعني بيقول ليه لحظة الإقلاع بنعرف احنا لحظة
+
+120
+00:08:36,390 --> 00:08:41,050
+الإقلاع كانت ال position لها للطيارة هذه كانت عند
+
+121
+00:08:41,050 --> 00:08:46,230
+النقطة الإحداثياتها 3 و 0 و 0 و كانت سرعتها في تلك
+
+122
+00:08:46,230 --> 00:08:50,180
+اللحظةالتي هي لحظة الـ ال ال ال ال ال ال ال ال ال
+
+123
+00:08:50,180 --> 00:08:50,680
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+124
+00:08:50,680 --> 00:08:51,260
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+125
+00:08:51,260 --> 00:08:51,920
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+126
+00:08:51,920 --> 00:09:00,020
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+127
+00:09:00,020 --> 00:09:01,580
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+128
+00:09:01,580 --> 00:09:01,580
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+129
+00:09:01,580 --> 00:09:01,580
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+130
+00:09:01,580 --> 00:09:01,580
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+131
+00:09:01,580 --> 00:09:01,580
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+132
+00:09:01,580 --> 00:09:01,580
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+133
+00:09:01,580 --> 00:09:01,580
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+134
+00:09:01,580 --> 00:09:03,400
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+135
+00:09:03,400 --> 00:09:03,400
+ال ال
+
+136
+00:09:17,850 --> 00:09:23,770
+هذه المعلومات مع بعض هتمكننا في معرفة مين الـ
+
+137
+00:09:23,770 --> 00:09:29,590
+position vector للإيمين للـ hang glider اللي هي
+
+138
+00:09:29,590 --> 00:09:34,530
+الطرائرة الشراعية our goal is to find R of T وعنا
+
+139
+00:09:34,530 --> 00:09:37,290
+المعلومات كما هي لها عنا ال differential equation
+
+140
+00:09:37,290 --> 00:09:41,190
+اللي هي ايه ال acceleration دي تربيع أربع دي تربيع
+
+141
+00:09:41,190 --> 00:09:45,630
+سوى هذا الكلامالان وماعطينا ال initial condition
+
+142
+00:09:45,630 --> 00:09:48,990
+اللي هو V of Zero عند T بتساوي Zero كانت تلاتة J
+
+143
+00:09:48,990 --> 00:09:53,670
+السرعة ال velocity وعند اللي هي ال ال ال T بتساوي
+
+144
+00:09:53,670 --> 00:09:57,310
+سفر كانت R of Zero بتساوي تلاتة قيمة ال X يعني
+
+145
+00:09:57,310 --> 00:09:59,610
+تلاتة في اتجاه ال I Zero في اتجاه ال J زائد Zero
+
+146
+00:09:59,610 --> 00:10:03,650
+في اتجاه ال K هذه المعلومات هتمكننا بمعرفة ال R
+
+147
+00:10:03,650 --> 00:10:08,570
+أول اشي بده اوجد من ال ALV ده كاملها يعني يعني بده
+
+148
+00:10:08,570 --> 00:10:13,450
+كامليعني الان V of T بكامل هذا المقدار بيصير تكامل
+
+149
+00:10:13,450 --> 00:10:16,410
+اللي هو ناقص ثلاثة cosine T اللي هو ناقص ثلاثة
+
+150
+00:10:16,410 --> 00:10:20,610
+sine T في اتجاه ال I الان بكامل هذه اللي هي زائد
+
+151
+00:10:20,610 --> 00:10:24,050
+تكامل ال sine ناقص cosine تلاتة cosine T في اتجاه
+
+152
+00:10:24,050 --> 00:10:27,550
+ال J وتكامل هذه اتنين T في اتجاه ال K زائد mean
+
+153
+00:10:27,550 --> 00:10:29,550
+some constant C1 هذا vector
+
+154
+00:10:32,340 --> 00:10:35,880
+كملناها بواسط الـ Fundamental Theorem و الكلاسة
+
+155
+00:10:35,880 --> 00:10:42,900
+وضفنا لها some constant C1 الان بدأ اوجد الـ C1
+
+156
+00:10:42,900 --> 00:10:47,160
+كيف ده اوجدها من معرفتي V0 ايش بالساوي 3G بعوض
+
+157
+00:10:47,160 --> 00:10:52,240
+عنها بـ 0 بالـ C عندى اللي هوV of Zero هو تلاتة
+
+158
+00:10:52,240 --> 00:10:57,760
+جهيها ويساوي نقص تلاتة Sin Zero زائد تلاتة Cos
+
+159
+00:10:57,760 --> 00:11:02,980
+Zero زائد Zero كلهم في اتجاهي زائد C1 ماشي الحال
+
+160
+00:11:02,980 --> 00:11:09,600
+الان هذه سفرو هذه صفر وهذه تلاتة J فبصير تلاتة J
+
+161
+00:11:09,600 --> 00:11:14,160
+تلاتة J زي C1 اذا C1 ايش بتساوي صفر اذا الان ال
+
+162
+00:11:14,160 --> 00:11:19,080
+velocity صارت عندي عرفها بالساوية هذه وهذه وهذه و
+
+163
+00:11:19,080 --> 00:11:23,120
+بس وهذا ايه ايش صفر الان من ال velocity بده اعرف
+
+164
+00:11:23,120 --> 00:11:28,260
+من ال R اللي هو طلبها بعرف ال R و بستخدم ال R في Z
+
+165
+00:11:28,260 --> 00:11:33,280
+عشان اوجد constant ال R زي ما وجدنا constant LV
+
+166
+00:11:33,760 --> 00:11:36,780
+أكيد كلكم مفهم أشي اللي بقوله لأن the glider's
+
+167
+00:11:36,780 --> 00:11:39,300
+velocity as a function of time T اللي قبل بشوية
+
+168
+00:11:39,300 --> 00:11:44,220
+وجدناها DR by DT بساوة V of T هيها ماشي الحالة
+
+169
+00:11:44,220 --> 00:11:48,980
+بتدوجد منها ال R مع المعلومة ان ال R of Zero
+
+170
+00:11:48,980 --> 00:11:52,480
+بتساوي تلتة I زي ما قلنا قبل بشوية إذا المعلومتين
+
+171
+00:11:52,480 --> 00:11:57,000
+هذه مع R of Zero بتساوي تلتة I بقدر أعرف ال R
+
+172
+00:11:57,000 --> 00:12:00,770
+بالظبط إيش هيالان انا بدي اكمل الجهتين لما اكمل ال
+
+173
+00:12:00,770 --> 00:12:05,470
+V بيطلع من ال R زي ما اكملنا ال A طلعت ال V الان
+
+174
+00:12:05,470 --> 00:12:09,210
+اذا اكمل ال V بيطلع ال R و يساوي الان تكامل ال Sin
+
+175
+00:12:09,210 --> 00:12:12,330
+اللي هي ناقص Cos اللي هي تلاتة Cos تجاه ال I
+
+176
+00:12:12,330 --> 00:12:17,160
+وتكامل ال Cos اللي هي ثلاثة Sinوالتكامل اتنين T
+
+177
+00:12:17,160 --> 00:12:20,120
+تربيع اللي هو T تربيع اتنين T تربيع على اتنين يعني
+
+178
+00:12:20,120 --> 00:12:23,760
+T تربيع زائد some constant C اتنين يعني اطبقنا
+
+179
+00:12:23,760 --> 00:12:27,100
+برضه ال fundamental theorem افق الكلاص واخدنا هذه
+
+180
+00:12:27,100 --> 00:12:30,960
+ال antiderivative لهذه زائد some constant C اتنين
+
+181
+00:12:31,410 --> 00:12:35,810
+الان عشان اوجد C2 بعرف انا معلومتي R of Zero ايش
+
+182
+00:12:35,810 --> 00:12:39,490
+بتساوي تلاتة I من قبل بشوية الان صار عندى تلاتة I
+
+183
+00:12:39,490 --> 00:12:43,070
+بتساوي تلاتة في Cos Zero و تلاتة Sin Zero Zero
+
+184
+00:12:43,070 --> 00:12:48,490
+تربيع زي ال C2 هذه سفر وهذه سفر وظل تلاتة I بتساوي
+
+185
+00:12:48,490 --> 00:12:52,930
+تلاتة I وهنا C2 يعني C2 بتساوي سفر يعني الان ال
+
+186
+00:12:52,930 --> 00:12:57,660
+glidersposition as a function of T هي بالظبط
+
+187
+00:12:57,660 --> 00:13:01,540
+بالساوية تلاتة cos T في اتجاه I تلاتة sin T في
+
+188
+00:13:01,540 --> 00:13:06,440
+اتجاه J ساعة T تربيع في اتجاه K وبتكون هك او جدنا
+
+189
+00:13:06,440 --> 00:13:09,640
+اللي هو ال position ال position vector لها و لو حد
+
+190
+00:13:09,640 --> 00:13:14,960
+جرب بالكمبيوتر حيشوف الرسمة اللي أمامنا و هيك
+
+191
+00:13:14,960 --> 00:13:17,760
+بتكون انتهت المحاضرة والسلام عليكم ورحمة الله
+
+192
+00:13:17,760 --> 00:13:19,060
+وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04.srt
new file mode 100644
index 0000000000000000000000000000000000000000..c1d8b568251070770b288ac93e7c3e415fda421b
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04.srt
@@ -0,0 +1,2120 @@
+1
+00:00:00,250 --> 00:00:03,610
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:03,610 --> 00:00:08,670
+وبركاته هذه هي المحاضرة رقم تسعة عشر مساق طادل
+
+3
+00:00:08,670 --> 00:00:12,310
+وتكامل الـGene وهي المحاضرة السابعة والأخيرة في
+
+4
+00:00:12,310 --> 00:00:17,750
+شبطر أربعة عشر وهي أربعة عشر سبعة وهي تحت عنوان
+
+5
+00:00:17,750 --> 00:00:22,890
+Extreme Values and Saddle Points طبعا الموضوع ليس
+
+6
+00:00:22,890 --> 00:00:29,510
+جديد فيه روحه احنا اتحدثنا في calculus 1 وكمان
+
+7
+00:00:29,510 --> 00:00:34,910
+حتى في الفنوية عن اللي هي كيف نوجد الـ Local
+
+8
+00:00:34,910 --> 00:00:38,470
+maximum و Local minimum و الـ Inflection points و
+
+9
+00:00:38,470 --> 00:00:42,590
+كذلك اللي هو الـ test تبعاتها و كذلك على Absolute
+
+10
+00:00:42,590 --> 00:00:47,030
+maximum و Absolute minimum. الحديث سيكون حديث مقابل
+
+11
+00:00:47,030 --> 00:00:49,890
+بسعر الـ two variables أو الـ three variables.
+
+12
+00:00:49,890 --> 00:00:55,630
+الحديث سيكون حول نفس الأشياء سنجاوبها على سؤالين.
+
+13
+00:00:55,630 --> 00:01:03,090
+السؤال الأول أين يمكن إيجاد الـ local maximum أو
+
+14
+00:01:03,090 --> 00:01:06,640
+الـ local minimum أو اللي هي الان مش inflection
+
+15
+00:01:06,640 --> 00:01:09,840
+point. حاجة بنسميها saddle points أو نقطة سرج
+
+16
+00:01:09,840 --> 00:01:15,140
+الحصان. السؤال الثاني أنه هل كيف بدنا نحكم على أن
+
+17
+00:01:15,140 --> 00:01:19,080
+هذه فعلا في عندها local maximum و local minimum أو
+
+18
+00:01:19,080 --> 00:01:24,740
+نقطة saddle points أو لا؟ تقريباً الأجوبة هي مشابهة
+
+19
+00:01:24,740 --> 00:01:28,080
+للأجوبة السابقة في تفاضل 1. كنا نجاوب في تفاضل
+
+20
+00:01:28,080 --> 00:01:33,200
+1 نقول القيم العظمى والصغرى وكذلك الـ inflection
+
+21
+00:01:33,200 --> 00:01:37,380
+points. ممكن أن توجد في أحد المناطق التالية: إما في
+
+22
+00:01:37,380 --> 00:01:40,160
+الـ interior points اللي هي في عند الـ derivative
+
+23
+00:01:40,160 --> 00:01:43,660
+بتساوي صفر أو الـ derivative does not exist. هذه
+
+24
+00:01:43,660 --> 00:01:46,880
+المنطقة الثانية، والمنطقة الثالثة هي الـ boundary
+
+25
+00:01:46,880 --> 00:01:50,110
+points. وكنا ناخد الـ First derivative test والـ
+
+26
+00:01:50,110 --> 00:01:54,010
+Second derivative test اللي هو الحكم على إن فيه
+
+27
+00:01:54,010 --> 00:01:57,330
+local maximum أو local minimum أو اللي هو
+
+28
+00:01:57,330 --> 00:02:01,290
+inflection point. هذا الحديث تقريباً مشابه جداً. خلينا
+
+29
+00:02:01,290 --> 00:02:05,010
+نبدأ اللي هو في موضوعنا: derivative test for
+
+30
+00:02:05,010 --> 00:02:10,010
+local extreme values. نشوف أول حاجة معنات الـ
+
+31
+00:02:10,010 --> 00:02:15,190
+maximum و الـ minimum أو الـ local maximum و الـ
+
+32
+00:02:15,190 --> 00:02:19,210
+local minimum. مشابهة. Let f of x, y be defined on a
+
+33
+00:02:19,210 --> 00:02:22,790
+region R containing the point A و B. نشوف الآن ما
+
+34
+00:02:22,790 --> 00:02:25,910
+هي معنى local maximum. عندنا نقطة A و B. هي نقطة A
+
+35
+00:02:25,910 --> 00:02:29,010
+و B. ما هي معنى local maximum؟ local maximum هي
+
+36
+00:02:29,010 --> 00:02:34,320
+نقطة عظمى محلية. إننا نجد جوار حواليها هناك جوار
+
+37
+00:02:34,320 --> 00:02:39,460
+حواليها تكون هي أكبر واحدة. يعني قيمة F of A و F of
+
+38
+00:02:39,460 --> 00:02:45,540
+B أكبر واحدة. أكبر من F of XY لكل XY في هذا الجوار.
+
+39
+00:02:45,540 --> 00:02:48,780
+يعني أكبر واحدة في جيرانها. هذا نسميه Local
+
+40
+00:02:48,780 --> 00:02:53,040
+Maximum. الآن أصغر وحدة. تجوّر حوالين الـ A و الـ B
+
+41
+00:02:53,040 --> 00:02:56,720
+تكون الـ F عند الـ A و الـ B أصغر وحدة في جيرانها.
+
+42
+00:02:56,720 --> 00:03:01,040
+فبنسميها Local Minimum. وهذا هو التعريف. F of A و B
+
+43
+00:03:01,040 --> 00:03:05,720
+is Local Maximum Value of F if F of A و B أكبر
+
+44
+00:03:05,720 --> 00:03:10,840
+أو يساوي F of X و Y. أكبر من كل الـ F of X و Y. لأ يعني
+
+45
+00:03:10,840 --> 00:03:14,860
+بمعنى آخر اللي هي لنقاط الـ X و Y حوالها. For all
+
+46
+00:03:14,860 --> 00:03:20,440
+x, y in an open disk centered at a, b.
+
+47
+00:03:25,650 --> 00:03:29,230
+بتكون قيمة F عند الـ A و الـ B أكبر من F of X Y
+
+48
+00:03:29,230 --> 00:03:33,410
+لكل X Y في هذا الـ Disk اللي حواليها أو الـ Open
+
+49
+00:03:33,410 --> 00:03:37,630
+Disk. الآن الـ Local Minimum بنفس الأسلوب. F of A و
+
+50
+00:03:37,630 --> 00:03:41,790
+B is a Local Minimum Value of F if F of A و B أصغر
+
+51
+00:03:41,790 --> 00:03:46,130
+أو يساوي F of X Y for all points X Y في Domain الـ
+
+52
+00:03:46,130 --> 00:03:50,950
+Function. وفي نفس الواجهة جيران لـ A و B أو في اللي
+
+53
+00:03:50,950 --> 00:03:53,930
+هي Disk. Open Disk. عارفين شو معناه Open Disk؟ دائرة
+
+54
+00:03:53,930 --> 00:03:58,290
+مفتوحة. اللي هو اللي جواتها تكون اللي هي كل النقطات
+
+55
+00:03:58,290 --> 00:04:01,670
+اللي جواتها. قيمة الـ Function F عند الـ A والـ B
+
+56
+00:04:01,670 --> 00:04:06,510
+أصغر من هذه القيم. معناته أن F لها Local Minimum
+
+57
+00:04:06,510 --> 00:04:11,810
+عند هذه النقطة. الآن بدنا نيجي اللي هو ندى نجاوب
+
+58
+00:04:11,810 --> 00:04:14,990
+السؤال الأول. أين يمكن إيجاد الـ Local Maximum و
+
+59
+00:04:14,990 --> 00:04:20,340
+Local Minimum أو اللي هو النقاط. بعد شوي هنحكي عن
+
+60
+00:04:20,340 --> 00:04:25,540
+نقاط حرجة أو الـ saddle point. طبعا الآن الـ theorem
+
+61
+00:04:25,540 --> 00:04:29,720
+10. First derivative test for local values. الآن
+
+62
+00:04:29,720 --> 00:04:32,900
+اللي هي الـ first derivative test. زي ما كان هناك في
+
+63
+00:04:32,900 --> 00:04:36,240
+first derivative test. هنا في عندنا first
+
+64
+00:04:36,240 --> 00:04:41,960
+derivative test. إيش بتقول؟ إذا كانت f of x y has a
+
+65
+00:04:41,960 --> 00:04:46,230
+local minimum أو local maximum عند الـ interior
+
+66
+00:04:46,230 --> 00:04:50,830
+point a و b of its domain and if the first partial
+
+67
+00:04:50,830 --> 00:04:57,030
+derivatives exist, then f<sub>x</sub> عند هذه النقطة صفر و f<sub>y</sub>
+
+68
+00:04:57,030 --> 00:05:00,330
+عند هذه النقطة a صفر. هذا بتقول إنها نظرية. مالي
+
+69
+00:05:00,330 --> 00:05:04,770
+إذا كانت الـ function. طبعا تحت الظروف إنها الـ
+
+70
+00:05:04,770 --> 00:05:08,350
+first partial derivative exist. يعني الـ f<sub>x</sub> و
+
+71
+00:05:08,350 --> 00:05:12,020
+الـ f<sub>y</sub> موجودة. الآن أخبرني باختصار: إذا كانت الـ
+
+72
+00:05:12,020 --> 00:05:16,240
+function f of x y لها local maximum أو local
+
+73
+00:05:16,240 --> 00:05:20,300
+minimum عند الـ a و b غصباً عن طبعاً في interior
+
+74
+00:05:20,300 --> 00:05:24,540
+point. يعني غصباً عن بيكون f<sub>x</sub> بتساوي صفر و f<sub>y</sub> إيه
+
+75
+00:05:24,540 --> 00:05:29,540
+شبهها بتساوي صفر. يعني لما بدك تيجي تفحص عند اللي
+
+76
+00:05:29,540 --> 00:05:33,160
+هياللي تبحث عن الـ Local Maximum و Local Minimum
+
+77
+00:05:33,160 --> 00:05:37,420
+في الـ Interior Points اللي الـ F<sub>x</sub> و F<sub>y</sub> exist لهن
+
+78
+00:05:37,420 --> 00:05:42,600
+افحص بس عند F<sub>x</sub> بيساوي 0 و F<sub>y</sub> بيساوي 0. النقاط اللي
+
+79
+00:05:42,600 --> 00:05:46,500
+بتحقق هذه ممكن يكون فيها Local Maximum و Local
+
+80
+00:05:46,500 --> 00:05:50,420
+Minimum. أما النقاط اللي بيكونش هذه صفر وتكونش هذه
+
+81
+00:05:50,420 --> 00:05:53,300
+صفر، انساها. لأن لو كان Local Maximum و Local
+
+82
+00:05:53,300 --> 00:05:57,720
+Minimum لازم يطلع عن صفر. إذا من هذه النظرية بنقدر
+
+83
+00:05:57,720 --> 00:06:04,450
+نقول أنه احنا ممكن نوجدممكن نلاقي الـ Local
+
+84
+00:06:04,450 --> 00:06:08,530
+Maximum أو الـ Local Minimum عند أحد النقاط
+
+85
+00:06:08,530 --> 00:06:13,970
+التالية: إما F<sub>x</sub> يساوي F<sub>y</sub> بيساوي صفر أو إن الـ
+
+86
+00:06:13,970 --> 00:06:17,350
+Derivative does not exist. لأن هذا ليس متضمن من
+
+87
+00:06:17,350 --> 00:06:21,350
+خلال الحديث في النظرية. إذن هذه النظرية توحي ما
+
+88
+00:06:21,350 --> 00:06:26,010
+يليه أو تعطينا المعلومات التالية: إنه اللي هي الـ
+
+89
+00:06:26,010 --> 00:06:30,380
+Interior Points. للـ Interior Points ممكن نلاقي الـ
+
+90
+00:06:30,380 --> 00:06:32,300
+Local Maximum. الـ Local Minimum عند الـ Interior
+
+91
+00:06:32,300 --> 00:06:37,560
+Points اللي F<sub>x</sub> بيساوي 0 و F<sub>y</sub> بيساوي 0. و اللي F<sub>x</sub> و
+
+92
+00:06:37,560 --> 00:06:43,140
+F<sub>y</sub> لا يوجد في هذه الحالة. اللي هو بيكون عندي اللي
+
+93
+00:06:43,140 --> 00:06:46,800
+هو عن الـ Interior Points اللي هي ممكن نودد الـ
+
+94
+00:06:46,800 --> 00:06:50,460
+Local Maximum و Local Minimum عند هذه الأماكن. يعني
+
+95
+00:06:50,460 --> 00:06:54,300
+بمعنى آخر من هذه النظرية أيضاً نستقل معلومة التالية
+
+96
+00:06:54,300 --> 00:07:00,990
+وهي نهائية يمكن المناطق اللي ممكن يكون فيها Local
+
+97
+00:07:00,990 --> 00:07:04,550
+Maximum أو Local Minimum. لسه بدأ أفحص المناطق
+
+98
+00:07:04,550 --> 00:07:07,250
+اللتي يمكن أن يكون فيها Local Maximum أو Local
+
+99
+00:07:07,250 --> 00:07:10,990
+Minimum هي النقاط اللي الـ F<sub>x</sub> بتساوي F<sub>y</sub> بساوي صفر.
+
+100
+00:07:10,990 --> 00:07:13,730
+هاي واحد. طبعا الـ Interior Points. اتنين الـ
+
+101
+00:07:13,730 --> 00:07:15,870
+Interior Points اللي هي F<sub>x</sub> does not exist.
+
+102
+00:07:20,290 --> 00:07:26,450
+النقطة الأخيرة الـ Boundary Points. حصرنا كل جزء في
+
+103
+00:07:26,450 --> 00:07:39,450
+أربع مناطق. F<sub>x</sub> F<sub>y</sub> F<sub>x</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub>
+
+104
+00:07:39,450 --> 00:07:39,550
+F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub>
+
+105
+00:07:39,550 --> 00:07:39,790
+F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub> F<sub>y</sub>
+
+106
+00:07:39,820 --> 00:07:44,040
+إذا سهلت علينا أمور الفحص. يعني أن نتأكد الآن ضال
+
+107
+00:07:44,040 --> 00:07:47,300
+علينا بس نتأكد إن في Local Maximum ولا Local
+
+108
+00:07:47,300 --> 00:07:51,780
+Minimum أو ما فيش شيء. الآن اللي اللي اللي اللي اللي
+
+109
+00:07:51,780 --> 00:07:55,620
+اللي عشان نحصر الأمور يعني وكأننا نجسم الأمر اللي
+
+110
+00:07:55,620 --> 00:08:00,090
+هو الأمرين الـInterior Point الـ Boundary Points أو إذا كان
+
+111
+00:08:00,090 --> 00:08:02,850
+هناك نقطة داخلية في مجاله عملية f of x و f of y
+
+112
+00:08:02,850 --> 00:08:04,030
+أما إذا كان هناك نقطة داخلية في مجاله عملية f of x
+
+113
+00:08:04,030 --> 00:08:05,610
+و f of y. أما إذا كان هناك نقطة داخلية في مجلة
+
+114
+00:08:05,610 --> 00:08:07,230
+عملية f of x و f of y. أما إذا كان هناك نقطة داخلية
+
+115
+00:08:07,230 --> 00:08:08,210
+في مجلة عملية f of x و f of y. أما إذا كان هناك
+
+116
+00:08:08,210 --> 00:08:09,170
+نقطة داخلية في مجلة عملية f of x و f of y. أما إذا
+
+117
+00:08:09,170 --> 00:08:10,050
+كان هناك نقطة داخلية في مجلة عملية f of x و f of y
+
+118
+00:08:10,050 --> 00:08:12,270
+أما إذا كان هناك نقطة داخلية في مجلة عملية f of x
+
+119
+00:08:12,270 --> 00:08:17,110
+و f of y. أما إذا كان هناك نقطة
+
+120
+00:08:17,110 --> 00:08:21,580
+داخلية في مجلة عملية f of x و f of x. هي واحدة، اتنين
+
+121
+00:08:21,580 --> 00:08:26,440
+F<sub>x</sub> does not exist. هذين، تلاتة. F<sub>y</sub> does not
+
+122
+00:08:26,440 --> 00:08:30,680
+exist. هدول الـ Critical Points اللي اللي ممكن يكون
+
+123
+00:08:30,680 --> 00:08:33,600
+عندهم Local Maximum أو Local Minimum. إذا صار في
+
+124
+00:08:33,600 --> 00:08:36,740
+عندي بناء على هذا التصنيف في منطقتين ممكن نلاقي
+
+125
+00:08:36,740 --> 00:08:40,320
+فين اللي هو Local Maximum أو Local Minimum أو اللي
+
+126
+00:08:40,320 --> 00:08:45,290
+هو بعد شوية saddle point اللي هي أحد ثلاث مناطق. أحد
+
+127
+00:08:45,290 --> 00:08:49,130
+منطقتين اللي هي إما الـ interior points اللي بتكون
+
+128
+00:08:49,130 --> 00:08:52,410
+فيها critical points. هاي الـ critical points يعني
+
+129
+00:08:52,410 --> 00:08:54,530
+طبعاً الـ critical point على طول الـ interior point.
+
+130
+00:08:54,530 --> 00:08:57,990
+الحاجة الثانية اللي هي الـ boundary points. إذا
+
+131
+00:08:57,990 --> 00:09:02,270
+منطقتين: critical points أو boundary points. الـ
+
+132
+00:09:02,270 --> 00:09:05,530
+Critical Points هي اللي بيكون عندها F<sub>x</sub> بيساوي F<sub>y</sub>
+
+133
+00:09:05,530 --> 00:09:08,270
+طبعاً Interior Points. F<sub>x</sub> بيساوي F<sub>y</sub> بيساوي Zero. هاي
+
+134
+00:09:08,270 --> 00:09:12,130
+واحدة. F<sub>x</sub> does not exist. تنتين تانية. تلاتة. F<sub>y</sub> does
+
+135
+00:09:12,130 --> 00:09:15,930
+not exist. هذه هي. السؤال جاوبنا عليه الآن: أين يمكن
+
+136
+00:09:15,930 --> 00:09:18,410
+إيجاد الـ Local Maximum أو ال Local Minimum أو ال
+
+137
+00:09:18,410 --> 00:09:25,260
+Extreme Values أو Points. الآن لاحظ عند اللي هي ال
+
+138
+00:09:25,260 --> 00:09:28,480
+... ال ... ال ... النقطة هذه اللي هي عند الـ Local
+
+139
+00:09:28,480 --> 00:09:33,620
+Maximum. لاحظ أنه المماس هذا المماس هيكون الـ
+
+140
+00:09:33,620 --> 00:09:38,220
+∂f/∂y موازية لـ Y Axis. يعني ∂f/∂y أيش بيساوي؟ صفر.
+
+141
+00:09:38,220 --> 00:09:41,600
+والمماس ل الـ curve هذا اللي هو partial derivative. إذ
+
+142
+00:09:41,600 --> 00:09:43,980
+هانا اللي معناه partial derivative. معناه كنا زمان
+
+143
+00:09:43,980 --> 00:09:47,180
+بيكون الـ ∂f/∂x برضه موازية لـ X Axis. بيكون اللي هو
+
+144
+00:09:47,180 --> 00:09:50,780
+أيش؟ صفر. يعني عند هذه النقطة الـ interior point الـ
+
+145
+00:09:50,780 --> 00:09:54,580
+derivative اللي هي exist. الـ local maximum هاي
+
+146
+00:09:54,580 --> 00:09:57,880
+اللي هو ميل المماس صفر وميل المماس الثاني صفر.
+
+147
+00:09:57,880 --> 00:10:01,060
+واحد اللي هو ∂f/∂x صفر. واحد ع الـ ∂f/∂y بيساوي صفر. هذا معناه هندسي للحديث اللي
+
+148
+00:10:01,060 --> 00:10:04,620
+اتحدثنا فيه قبل شوية.
+
+149
+00:10:09,850 --> 00:10:13,410
+إذا لم يكن معرفة. معناته الـ Saddle Point. فمعرفة
+
+150
+00:10:13,410 --> 00:10:16,250
+مختلفة. F of X Y لديها Saddle Point. إلى Critical
+
+151
+00:10:16,250 --> 00:10:20,030
+Point. إذا الآن Saddle Point وين يا جماعة؟ هي
+
+152
+00:10:20,030 --> 00:10:25,590
+Interior Point وعند مين؟ عند Critical Point. يعني F<sub>x</sub>
+
+153
+00:10:25,590 --> 00:10:31,350
+يساوي صفر أو F<sub>x</sub> does not exist أو F<sub>y</sub> does not
+
+154
+00:10:31,350 --> 00:10:35,090
+exist. Then a differentiable function f of x y has a
+
+155
+00:10:35,090 --> 00:10:37,810
+saddle point at a critical point a و b ... آسف
+
+156
+00:10:37,810 --> 00:10:41,350
+آسف آسف اللي هو عند a interior point. F<sub>x</sub> بتساوي F<sub>y</sub>
+
+157
+00:10:41,350 --> 00:10:44,390
+بتساوي 0. A differentiable function f of x y has a
+
+158
+00:10:44,390 --> 00:10:49,010
+saddle point. على نقطة مهمة A و B إذا كان في كل مركز
+
+159
+00:10:49,010 --> 00:10:54,350
+مفتوح على A و B يوجد نقطة  مخصصة XY فيها F<sub>XY</sub> أكبر من
+
+160
+00:10:54,350 --> 00:10:59,190
+F of A و B ونقطة XY فيها F<sub>XY</sub> أصغر من F of A و B
+
+161
+00:10:59,190 --> 00:11:06,400
+المنطقة المتعلقة بالنقطة Z هي نقطة مخصصة. إذا الـ
+
+162
+00:11:06,400 --> 00:11:09,120
+differentiable function f of x y يعني بتكون الـ f<sub>x</sub>
+
+163
+00:11:09,120 --> 00:11:13,120
+و الـ f<sub>y</sub> exist already. ماشي الحال اللي هو عند
+
+164
+00:11:13,120 --> 00:11:15,940
+critical point. معناته بس عند f<sub>x</sub> بتساوي f<sub>y</sub> إيش
+
+165
+00:11:15,940 --> 00:11:19,240
+بتساوي صفر. يعني اللي عن الـ saddle point إيش هي؟
+
+166
+00:11:19,240 --> 00:11:23,540
+نقطة داخلية بتكون عندها اللي هي بنقول عنها saddle
+
+167
+00:11:23,540 --> 00:11:29,680
+point. إذا إجينا أخدنا اللي هو لأي
+
+168
+00:11:29,680 --> 00:11:33,020
+جوار حواليها. لأي جوار حواليها. لجينا كل ما ناخد
+
+169
+00:11:33,020 --> 00:11:38,360
+جوار. الـ F of A و F of B مرات بتاخد في داخل الجوار
+
+170
+00:11:38,360 --> 00:11:42,560
+أصغر من F of XY ومرات بتكون أكبر من مين؟ من F of XY.
+
+171
+00:11:42,560 --> 00:11:45,260
+يعني بتكون أكبر من النقاط من اللي حواليها وأصغر من
+
+172
+00:11:45,2
+
+223
+00:15:02,580 --> 00:15:05,040
+Local Minimum، في بعض الأسئلة السابقة، طبعاً صعب
+
+224
+00:15:05,040 --> 00:15:08,180
+إننا نحددها بالتعريف، لكن هذا سؤال سهل، خلينا نشوف
+
+225
+00:15:08,720 --> 00:15:12,120
+لو في X, Y بيساوي  X تربيع زائد Y تربيع ناقص
+
+226
+00:15:12,120 --> 00:15:14,960
+أربعة Y زائد تسعة، لكن هذه ممكن كتبتها على الطريقة
+
+227
+00:15:14,960 --> 00:15:18,620
+التالية: لو أكملنا المربع، حينها بيصير X تربيع زائد Y
+
+228
+00:15:18,620 --> 00:15:21,540
+ناقص اثنين الكل تربيع، يعني اللي هو أخذنا من التسعة
+
+229
+00:15:21,540 --> 00:15:24,460
+أربعة، صار مربع كامل، صار على صورة Y ناقص اثنين الكل
+
+230
+00:15:24,460 --> 00:15:28,480
+تربيع زائد خمسة. إذن صار هذا المقدار هذا
+
+231
+00:15:28,480 --> 00:15:31,940
+مربع كامل، وهذا مربع كامل. إذن لأي قيمة في الدنيا
+
+232
+00:15:31,940 --> 00:15:35,860
+لـ X, Y هيكون هذا المقدار أكبر من أو يساوي صفر. إذن صار
+
+233
+00:15:35,860 --> 00:15:39,420
+هذا المجموع عبارة عن أكبر من أو يساوي خمسة، يعني
+
+234
+00:15:39,420 --> 00:15:43,780
+بمعنى آخر، F of X, Y صارت دائماً أكبر من أو يساوي F of 0, 2
+
+235
+00:15:43,780 --> 00:15:48,500
+يعني F of 0, 2 هتعمل لي Local Minimum، هي في الواقع
+
+236
+00:15:48,500 --> 00:15:51,520
+Absolute Minimum، لكن إحنا مش موضوعنا. We see that
+
+237
+00:15:51,520 --> 00:15:56,180
+the still critical point اللي هي 0, 2، أو the critical
+
+238
+00:15:56,180 --> 00:16:00,440
+point (0, 2) gives a local minimum. هذا استخدمنا فيها
+
+239
+00:16:00,440 --> 00:16:05,820
+minimum يا جماعة، استخدمنا فيها بس اللي هو التعريف.
+
+240
+00:16:07,860 --> 00:16:12,260
+لأمثال أخرى، مثال ثاني: Find the local extreme
+
+241
+00:16:12,260 --> 00:16:16,140
+values، إذا موجود، of f(x, y) بيساوي x تربيع ناقص y تربيع.
+
+242
+00:16:16,140 --> 00:16:19,820
+ماهي هذه الدالة؟ هي نفس الأولى. هذه معرفة وين ما كان،
+
+243
+00:16:19,820 --> 00:16:22,860
+و  differentiable وين ما كان، إذا مافيش boundary
+
+244
+00:16:22,860 --> 00:16:25,220
+points. مدام differentiable وين ما كان، إذا مافيش
+
+245
+00:16:25,220 --> 00:16:28,500
+boundary points. الحاجة الثانية أن f<sub>x</sub> و f<sub>y</sub> دائماً
+
+246
+00:16:28,500 --> 00:16:31,760
+exist. إذا مافيش شيء اللي هو f<sub>x</sub> و f<sub>y</sub> does not
+
+247
+00:16:31,760 --> 00:16:35,780
+exist. إذن البحث عند مين يا جماعة؟ عند f<sub>x</sub> بتساوي
+
+248
+00:16:35,780 --> 00:16:39,180
+f<sub>y</sub> بيساوي صفر. إذا ما علينا غير وجود f<sub>x</sub> و f<sub>y</sub>
+
+249
+00:16:39,180 --> 00:16:43,680
+خلينا نوجد f<sub>x</sub>. f<sub>x</sub> إيش بتساوي؟ سالب اثنين X. f<sub>y</sub> إيش
+
+250
+00:16:43,680 --> 00:16:47,720
+بتساوي؟ اللي هو اثنين Y. نساوي هنا كلٌّ بصفر، بيطلع
+
+251
+00:16:47,720 --> 00:16:52,300
+عند F اللي هي X بتساوي صفر و Y بتساوي صفر. إذن
+
+252
+00:16:52,300 --> 00:16:56,340
+النقطة اللي بدنا نفحص عندها إنها اللي هي: هل النقطة
+
+253
+00:16:56,340 --> 00:16:59,860
+اللي هي عبارة عن اللي هي Local maximum أو Local
+
+254
+00:16:59,860 --> 00:17:04,580
+minimum؟ مين هي النقطة؟ صفر، صفر. إذن الـ Critical
+
+255
+00:17:04,580 --> 00:17:09,320
+Point هنا is the only (0, 0)، وهي نقطة برضه الـ
+
+256
+00:17:09,320 --> 00:17:12,680
+Function عندها Differentiable أيّن ما كانت. إذن هذه
+
+257
+00:17:12,680 --> 00:17:15,360
+النقطة ممكن تطلع Local Maximum و Local Minimum أو
+
+258
+00:17:15,360 --> 00:17:19,360
+تطلع Critical Point. نشوف، أو ممكن ما نعرفش نوجدها.
+
+259
+00:17:19,360 --> 00:17:22,840
+لكن إحنا جايبين سؤال على أساس إنه سهل التعامل معاه.
+
+260
+00:17:22,840 --> 00:17:29,200
+نشوف إيش اللي بقوله الآن. لاحظ لو
+
+261
+00:17:29,200 --> 00:17:34,860
+جينا along the X axis، النقطة مين هي؟ (0, 0). هي لو جيت
+
+262
+00:17:34,860 --> 00:17:38,880
+أخذت جواراً حولها، أي جوار حولها هنا
+
+263
+00:17:38,880 --> 00:17:42,920
+هيكون الجوار هذا يأخذ من الـ X axis الموجب و
+
+264
+00:17:42,920 --> 00:17:46,610
+من الـ X axis السالب، من الـ Y Axis الموجب،
+
+265
+00:17:46,610 --> 00:17:50,150
+من الـ Y Axis إيش؟ السالب. خلينا ناخد along
+
+266
+00:17:50,150 --> 00:17:54,890
+the X Axis. F has the value، يعني حولين من لما X الـ
+
+267
+00:17:54,890 --> 00:17:58,730
+Axis، لما Y بيساوي صفر، هتلاقي F of X و صفر إيش بتساوي؟
+
+268
+00:17:58,730 --> 00:18:02,410
+ناقص X تربيع. اللي هي إيش؟ ما لها أصغر من صفر. إذن
+
+269
+00:18:02,410 --> 00:18:05,890
+أي جوار بتقدر تأخذه. يعني لو جيت على الموجب
+
+270
+00:18:05,890 --> 00:18:11,790
+X Axis، على السالب X Axis، أنا ماشي على اللي هو
+
+271
+00:18:11,790 --> 00:18:17,230
+الـ X-axis هنا وجيت على المنطقة اللي هي هنا اللي
+
+272
+00:18:17,230 --> 00:18:20,990
+هي هتلاقي، مش مشكلة، البوز تقل أنا جيت في حياتي لأن
+
+273
+00:18:20,990 --> 00:18:25,710
+نحنا نربح، هيعطي نفس القيمة. خد، تعال على اللي هي مين؟
+
+274
+00:18:25,710 --> 00:18:30,330
+الـ X-axis. هي X-axis. خد أي جوار حوليه، أي جوار هنا
+
+275
+00:18:30,330 --> 00:18:33,720
+الآن. هذا الجوار لو جيته على النقاط اللي على الـ X
+
+276
+00:18:33,720 --> 00:18:37,500
+-axis، اللي هي بمعنى آخر Y بصفر، بصير F of X و صفر
+
+277
+00:18:37,500 --> 00:18:41,480
+ليه؟ حسب الدالة اللي عندي، ناقص X تربيع، أكيد أصغر من
+
+278
+00:18:41,480 --> 00:18:45,580
+أيّ شيء من صفر. الآن على نفس الجوار اللي قبل شويّة
+
+279
+00:18:45,580 --> 00:18:50,900
+اللي أخذته، الـ arbitrary الجوار العشوائي، خد نقطة
+
+280
+00:18:50,900 --> 00:19:01,680
+عشوائية على الـ Y-axis، الـ X بصفر، والـ Y تربيع
+
+281
+00:19:01,680 --> 00:19:07,840
+أكبر من الصفر. أي جوار حول اللي هو الصفر، مهما صغر،
+
+282
+00:19:07,840 --> 00:19:12,120
+هتلاقي اللي هي نقاط على الـ X-Axis بتعطي قيمة الـ
+
+283
+00:19:12,120 --> 00:19:15,100
+Function أصغر من صفر، ونقاط على الـ Y-Axis بتعطي
+
+284
+00:19:15,100 --> 00:19:18,560
+قيمة الـ Function أحياناً أكبر من صفر، يعني اللي هي
+
+285
+00:19:18,560 --> 00:19:23,050
+F... طبعاً صفر. مين هذا الصفر؟ دي قيمة F of 0, 0 اللي
+
+286
+00:19:23,050 --> 00:19:26,450
+هي النقطة اللي بنبحث عنها. F of 0, 0 هنلاقي صفر. نقل
+
+287
+00:19:26,450 --> 00:19:30,930
+صفر لـ صفر، يعني دائماً F of 0, 0 في أي جوار
+
+288
+00:19:30,930 --> 00:19:35,450
+حول النقطة هذه، لا يمكن نلاقي جوار إنها تكون
+
+289
+00:19:35,450 --> 00:19:38,270
+أكبر من كل اللي حواليها، أو أصغر من كل اللي حواليها،
+
+290
+00:19:38,270 --> 00:19:43,330
+بل هنلاقي نقاط هيّا أصغر منها، أو نقاط أكبر منها.
+
+291
+00:19:43,330 --> 00:19:53,170
+مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً مثلاً
+
+292
+00:19:53,170 --> 00:19:58,950
+مثلاً مثلاً مثلاً مثلاً. So (0, 0) contains points where the function is positive and
+
+293
+00:19:58,950 --> 00:20:02,010
+points where the function is negative, hence the
+
+294
+00:20:02,010 --> 00:20:05,630
+function has a saddle point at the origin and no
+
+295
+00:20:05,630 --> 00:20:09,210
+local extremum values. طيب، من هذا السؤال بنقدر
+
+296
+00:20:14,330 --> 00:20:17,490
+نستنتج إنه مش معناته F<sub>x</sub> بيساوي F<sub>y</sub> بيساوي
+
+297
+00:20:17,490 --> 00:20:22,370
+صفر، وكانت interior point، اللي هو بتضمن لنا إنه الـ F
+
+298
+00:20:22,370 --> 00:20:25,700
+لها Local maximum ولا Local minimum، لأ ممكن تكون
+
+299
+00:20:25,700 --> 00:20:29,200
+F<sub>x</sub> بصفر و F<sub>y</sub> بصفر، وما يكونش عندنا Local
+
+300
+00:20:29,200 --> 00:20:33,000
+Extreme Value، يعني لا Local Maximum ولا Local
+
+301
+00:20:33,000 --> 00:20:36,100
+Minimum. Local Extreme Value، يعني يا Local Maximum
+
+302
+00:20:36,100 --> 00:20:40,040
+يا Local Minimum. الآن F<sub>x</sub> بصفر و F<sub>y</sub> بصفر، لها
+
+303
+00:20:40,040 --> 00:20:44,180
+تضمن لنا اللي هي وجود اللي هو أن F يكون Local
+
+304
+00:20:44,180 --> 00:20:49,450
+Maximum و Local Minimum. إذن يا جماعة، صعب نظل نشتغل
+
+305
+00:20:49,450 --> 00:20:54,450
+على التعريف، بدنا نظريات مساعدة تحكمنا وتساعدنا
+
+306
+00:20:54,450 --> 00:20:58,530
+وتجاوبنا على السؤال الثاني: هل فعلاً في Local
+
+307
+00:20:58,530 --> 00:21:01,870
+Maximum و Local Minimum أو Saddle Point ولا لأ؟
+
+308
+00:21:01,870 --> 00:21:04,710
+وهذا اللي هيجاوبنا عليه: حاجة اسمها الـ Second
+
+309
+00:21:04,710 --> 00:21:08,390
+Derivative Test for Local Extreme Values. هذه
+
+310
+00:21:08,390 --> 00:21:11,750
+اللي هي الـ Second Derivative Test هتجاوبنا على
+
+311
+00:21:11,750 --> 00:21:16,150
+عائلة كبيرة من الدوال اللي هي هذا موضوع اختصاصنا
+
+312
+00:21:16,150 --> 00:21:19,670
+في هذا الفصل اللي هو Subposed Data. هذا إذا
+
+313
+00:21:19,670 --> 00:21:22,830
+السؤال الأول قلنا: أين يمكن إيجاد الـ Local
+
+314
+00:21:22,830 --> 00:21:26,050
+Maximum و Local Minimum أو اللي هو أيضاً الـ
+
+315
+00:21:26,050 --> 00:21:28,990
+Saddle Points. قلنا وين؟ الآن هذه بدأت تجاوبنا
+
+316
+00:21:28,990 --> 00:21:32,380
+وتحكمنا على إن هذه الـ Local Maximum والـ Local
+
+317
+00:21:32,380 --> 00:21:35,280
+Minimum ولّـ Saddle Point. وخلينا نشوف إيه اللي
+
+318
+00:21:35,280 --> 00:21:38,740
+بنحكيه. Suppose that f(x, y) and its first and second
+
+319
+00:21:38,740 --> 00:21:41,720
+partial derivatives are continuous.  فرضنا إن اللي
+
+320
+00:21:41,720 --> 00:21:45,080
+هي الـ partial derivatives موجودة واللي هي الـ
+
+321
+00:21:45,080 --> 00:21:48,280
+function continuous عند هذه المنطقة اللي هي throughout
+
+322
+00:21:48,280 --> 00:21:52,380
+a و b. عند F<sub>x</sub> و F<sub>y</sub> بيساوي إيش؟ بيساوي صفر. جداً
+
+323
+00:21:52,380 --> 00:21:56,140
+هذه بتفحص لمين يا جماعة؟ هي اختصاصنا الآن يعني
+
+324
+00:21:56,140 --> 00:22:00,300
+إحنا مش هنتوسع في الدراسة إلا بما يُطلب منّا أنه
+
+325
+00:22:00,300 --> 00:22:04,640
+عند الـ F<sub>x</sub> بيساوي F<sub>y</sub> بيساوي صفر. الآن الـ Critical
+
+326
+00:22:04,640 --> 00:22:08,660
+Points اللي بتطلع من هذه هي أحد ثلاث أمور: يا Local
+
+327
+00:22:08,660 --> 00:22:10,880
+Maximum، يا Local Minimum، يا Saddle Point.
+
+328
+00:22:10,880 --> 00:22:16,880
+يا ما نقدرش نحكم. إذن هذه نظرية اللي بتقولنا الحكم
+
+329
+00:22:16,880 --> 00:22:20,160
+كما يلي: خلينا نتبع معناه في حاجة اسمها الـ
+
+330
+00:22:20,160 --> 00:22:24,300
+Discriminant، اللي هو F<sub>xx</sub> F<sub>yy</sub> ناقص F<sub>xy</sub> تربيع، اللي هو هذا
+
+331
+00:22:24,300 --> 00:22:27,380
+الـ Discriminant أو الـ Hessian. هذا اللي هو اللي
+
+332
+00:22:27,380 --> 00:22:31,620
+بدنا من خلاله نحكم. طبعاً هذا F<sub>xx</sub> F<sub>yy</sub> ناقص F<sub>xy</sub>
+
+333
+00:22:31,620 --> 00:22:34,780
+تربيع، بيساوي الـ determinant. هذا يمكن الواحد هيك
+
+334
+00:22:34,780 --> 00:22:42,900
+يكون متذكر أحسن.  $\begin{pmatrix} F_{xx} & F_{xy} \\ F_{xy} & F_{yy} \end{pmatrix}$ هذا في هذا ناقص هذا
+
+335
+00:22:42,900 --> 00:22:46,680
+تربيع. اللي هو هذا الـ Discriminant. نوجده بعد ما
+
+336
+00:22:46,680 --> 00:22:51,240
+نوجده عند النقطة طبعاً A و B. إذا كان أكبر من صفر،
+
+337
+00:22:51,240 --> 00:22:55,960
+يكون عندي يا Local Maximum يا Local Minimum. الآن
+
+338
+00:22:55,960 --> 00:23:00,790
+لقيناه أكبر من صفر، باجي بفحص F<sub>xx</sub>. إذا أصغر من صفر، على
+
+339
+00:23:00,790 --> 00:23:03,650
+طول بقول Local Maximum. يعني إذا سالبة Local
+
+340
+00:23:03,650 --> 00:23:07,410
+Maximum. المشابهة لـ Second Derivative Test اللي هو
+
+341
+00:23:07,410 --> 00:23:10,310
+الـ Function of One Variable كانت. إذا كانت تطلع
+
+342
+00:23:10,310 --> 00:23:15,070
+عندي الـ F'' أصغر من صفر، بقول عنها Local Maximum.
+
+343
+00:23:16,070 --> 00:23:19,210
+الآن هذا بس عندنا عشان Function of two variables فيه
+
+344
+00:23:19,210 --> 00:23:23,330
+عندنا XX و XY و YY. فعشان هيك الـ Discriminant هذا هو اللي
+
+345
+00:23:23,330 --> 00:23:26,610
+بيحدد. أكبر من صفر معناته فيه Local Maximum أو
+
+346
+00:23:26,610 --> 00:23:29,210
+Local Minimum. مين بيحدد Local Maximum أو Local
+
+347
+00:23:29,210 --> 00:23:33,330
+Minimum؟ إشارة F<sub>xx</sub>. إذا كانت F<sub>xx</sub> أصغر من صفر، بتكون
+
+348
+00:23:33,330 --> 00:23:37,050
+عندي Local Maximum. إذا F<sub>xx</sub> أكبر من صفر، بتكون عندي
+
+349
+00:23:37,050 --> 00:23:41,400
+Local Minimum. الآن لو لاقينا هذا الـ Discriminant
+
+350
+00:23:41,400 --> 00:23:44,700
+أصغر من صفر، على طول بنحكم إنها إيه؟ Saddle
+
+351
+00:23:44,700 --> 00:23:48,880
+point. إذا أصغر من صفر، على طول Saddle point. بيساوي صفر،
+
+352
+00:23:48,880 --> 00:23:54,040
+بنقول الـ test is inconclusive، معناته اللي هو إيه؟
+
+353
+00:23:54,040 --> 00:23:57,040
+شماله الـ test غير نافع، وبنقدرش وبنبحث عن طرق أخرى.
+
+354
+00:23:57,040 --> 00:24:01,300
+إحنا هنشتغل على جدّنا الآن اللي هي إذا كنا نبحث عن
+
+355
+00:24:01,300 --> 00:24:05,850
+طرق أخرى، بتروح لمين؟ للتعريف اللي قبل شويّة. الآن مش
+
+356
+00:24:05,850 --> 00:24:09,890
+هنسى القلة إلا بالأشياء الممكنة علينا. الآن عندي
+
+357
+00:24:09,890 --> 00:24:13,430
+اللي هو هذا الـ discriminant هو اللي بحدد، المميز،
+
+358
+00:24:13,430 --> 00:24:20,290
+أو اللي بده يحدد أنها، إذا كان صفر، inconclusive، إذا
+
+359
+00:24:20,290 --> 00:24:23,370
+كانت أكبر من صفر، يا Local maximum، يا Local
+
+360
+00:24:23,370 --> 00:24:26,510
+minimum، أصغر من صفر، على طول Saddle point. وهذا
+
+361
+00:24:26,510 --> 00:24:30,280
+عند النقطة المذكورة A و B. نجي نطبق هذا عملياً. Find
+
+362
+00:24:30,280 --> 00:24:34,160
+local extreme values of the function f(x, y) بيساوي
+
+363
+00:24:34,160 --> 00:24:37,360
+x, y ناقص x تربيع ناقص y تربيع ناقص 2x ناقص 2y زائد أربعة.
+
+364
+00:24:37,360 --> 00:24:40,880
+عيا نحلها. Differentiable وين ما كان، اللي هي إذن
+
+365
+00:24:40,880 --> 00:24:43,980
+مافيش boundary points، ومافيش does not exist، إذن هتنطبق
+
+366
+00:24:43,980 --> 00:24:46,920
+نظريتنا تماماً تماماً، وإحنا جايبينها عشان هي. إذن بس
+
+367
+00:24:46,920 --> 00:24:51,820
+هنبحث وين؟ في f<sub>x</sub> بيساوي f<sub>y</sub> بيساوي صفر، ومن ثم اللي هو نحكم
+
+368
+00:24:51,820 --> 00:24:55,400
+على النقاط إنها إيه؟ إنها Local maximum،
+
+369
+00:24:55,400 --> 00:25:01,390
+و Local minimum ولا Saddle point. أو دي لـ f<sub>x</sub> عند f<sub>x</sub>
+
+370
+00:25:01,390 --> 00:25:06,750
+أجلها في اللي فوق، اللي هي عبارة عن y ناقص 2x ناقص 2،
+
+371
+00:25:06,750 --> 00:25:15,200
+صح؟ y ناقص 2x ناقص 2. الآن f<sub>y</sub> هي عبارة عن x ناقص 2y ناقص 2.
+
+372
+00:25:15,200 --> 00:25:20,020
+x ناقص 2y ناقص 2. فيه symmetry term. الآن خلينا ناخد هذي بصفر، وهذي بصفر. Let f<sub>x</sub>
+
+373
+00:25:24,340 --> 00:25:27,140
+بيساوي f<sub>y</sub> بيساوي صفر together. عندي بتصير من
+
+374
+00:25:27,140 --> 00:25:31,400
+الأولى: y بتساوي 2x زائد 2، والثانية: 2y
+
+375
+00:25:31,400 --> 00:25:38,690
+بيساوي x ناقص 2. بنساوي هذه بالصفر، وهذه بالصفر.
+
+376
+00:25:38,690 --> 00:25:43,010
+فبتطلع عندي اللي هي y بالشكل هذا، و 2y بالشكل هذا.
+
+377
+00:25:43,010 --> 00:25:46,350
+لما نجيبها لجهة ثانية، هذه 2y لجهة هذه، بصير x ناقص 2
+
+378
+00:25:46,350 --> 00:25:50,210
+إلى أن نحل المعادلتين هاتين مع بعض. وضربنا
+
+379
+00:25:50,210 --> 00:25:55,510
+هذه في 2، بصير 2y = 4x زائد 4. اللي هي بنطرح من بعض،
+
+380
+00:25:55,510 --> 00:26:05,810
+بصير 2y ناقص 2y = صفر، 4x ناقص اللي هو x بتطلع 3x = أربعة
+
+381
+00:26:05,810 --> 00:26:09,150
+ناقص ناقص اثنين، هي زائد ستة، بيصير لي هي 3x
+
+382
+00:26:09,150 --> 00:26:12,550
+زائد ستة = صفر. إذن x بتساوي إيش؟ سالب اثنين. مش
+
+383
+00:26:12,550 --> 00:26:15,870
+اثنين. ولو عوضنا x بسالب اثنين بيصير سالب أربعة
+
+384
+00:26:15,870 --> 00:26:18,890
+زائد اثنين بتطلع برضه سالب اثنين. يعني النقطة عند x
+
+385
+00:26:18,890 --> 00:26:22,530
+بتساوي سالب اثنين، و y بتساوي سالب اثنين. إذن النقطة
+
+386
+00:26:22,530 --> 00:26
+
+445
+00:30:56,350 --> 00:31:02,930
+ستة fxy اللي بنفضلها بتطلع برضه ستة، الان fyy لما
+
+446
+00:31:02,930 --> 00:31:08,530
+نفضلها بتطلع ستة ناقص اتناشر y، ستة ناقص اتناشر y، إذن
+
+447
+00:31:08,530 --> 00:31:12,130
+هدول بدنا نحسب هنا عند مين؟ عند أول شي صفر صفر و
+
+448
+00:31:12,130 --> 00:31:15,290
+بعدين عند اتنين واتنين، وبعدين اللي هو نطبق نجيب
+
+449
+00:31:15,290 --> 00:31:20,180
+بالـ Discriminant، الان fyy عند الـ 00 اللي هي بتساوي
+
+450
+00:31:20,180 --> 00:31:25,980
+اللي هي 6، و fyy عند ال 2 وال2، عوض بيصير 6 ناقص اللي
+
+451
+00:31:25,980 --> 00:31:31,320
+هو 24، تطلع ناقص 18، إذا الأمور جاهزة يا جماعة الآن
+
+452
+00:31:31,320 --> 00:31:36,000
+هدول أصلا قيمتهم ثابتة عند الـ 00 وغيرهم الآن
+
+453
+00:31:36,000 --> 00:31:40,880
+نبدأ نوجد مين الآن؟ الـ Discriminant في كل حالة الآن
+
+454
+00:31:40,880 --> 00:31:44,560
+عند الـ 00 بكون الـ Discriminant هذا اللي هو بيساوي
+
+455
+00:31:44,560 --> 00:31:50,680
+بنعوض fxx في fyy، هي عندي اللي هو ناقص ستة في ستة ناقص
+
+456
+00:31:50,680 --> 00:31:54,900
+ستة و تلاتين fxy لكل تربيع، هي fxy اللي هي ستة و
+
+457
+00:31:54,900 --> 00:31:57,780
+تلاتين، بيطلع ناقص اتنين وسبعين، أصغر من صفر، إذا
+
+458
+00:31:57,780 --> 00:32:01,020
+مدام اللي هي عند الـ 00 طالع أصغر من صفر اللي هو
+
+459
+00:32:01,020 --> 00:32:03,460
+قيمة الـ Discriminant، إذا وأنا مغمض بقولها دي ايش؟
+
+460
+00:32:03,460 --> 00:32:07,820
+مالها؟ saddle point عند الـ 00، نجي عند مين؟ الـ 2
+
+461
+00:32:07,820 --> 00:32:10,700
+والـ 2، الـ 2 والـ 2، بعوض عند القيم اللي اللي
+
+462
+00:32:10,700 --> 00:32:17,300
+جيتها، اللي هي fxx و fxy، هيها ناقص 6 في 18 بالناقص
+
+463
+00:32:17,300 --> 00:32:20,980
+بيطلع زائد 108، اللي هي fxy تربيعها طبعا زي ما هي
+
+464
+00:32:20,980 --> 00:32:23,520
+ناقص ستة و تلاتين، هذه ناقص هذه، بيطلع اتنين وسبعين
+
+465
+00:32:23,520 --> 00:32:27,100
+برضه، إشارة موجبة، أكبر من صفر، إذن مدام أكبر من صفر يا
+
+466
+00:32:27,100 --> 00:32:30,120
+local maximum يا local minimum، مين اللي بيحدد ليه؟
+
+467
+00:32:30,120 --> 00:32:34,080
+fxx، مدام fxx سالبة، إذن على طول بقول في عندي local
+
+468
+00:32:34,080 --> 00:32:37,460
+maximum عند النقطة اتنين واتنين، و ال value للـ f
+
+469
+00:32:37,460 --> 00:32:41,740
+عند هذه النقطة اللي هو f of اتنين واتنين، بعوض عن
+
+470
+00:32:41,740 --> 00:32:45,500
+اللي هو اتنين واتنين هنا، بتطلع عندي عبارة عن قد ايش؟
+
+471
+00:32:45,500 --> 00:32:49,440
+بتساوي تمانية، واتأكد من هذا الكلام، وبكون عندي اللي
+
+472
+00:32:49,440 --> 00:32:51,860
+هو فعلا عندي واحدة saddle point، واحدة local
+
+473
+00:32:51,860 --> 00:32:57,240
+maximum، وهكذا، احنا أخذنا انفلة كفاية عن مرة الآن
+
+474
+00:32:57,240 --> 00:33:02,460
+الجزء الثاني من الموضوع اللي هو احنا لحد الآن
+
+475
+00:33:02,460 --> 00:33:05,760
+ما تطرقناش على مناطق فيها boundary points، الآن في
+
+476
+00:33:05,760 --> 00:33:10,920
+نظرية اللي هي بتقولنا ما هي؟ ليه؟ أنه لو كان عندك
+
+477
+00:33:10,920 --> 00:33:18,200
+اللي هي f of x y، عندها ال fxx و ال fxy و ال f
+
+478
+00:33:18,200 --> 00:33:20,460
+كده اللي كلنا دول ال partial derivatives موجودات و
+
+479
+00:33:20,460 --> 00:33:23,620
+continuous، وكمان لو كانت ال function
+
+480
+00:33:23,620 --> 00:33:28,980
+differentiable، نور على نور، بنضمن أن الـ function
+
+481
+00:33:28,980 --> 00:33:33,460
+يمتلك Absolute Maximum و Absolute Minimum، هذا طبعا
+
+482
+00:33:33,460 --> 00:33:36,720
+الكلام على مين؟ على Closed Bounded Region، إذن
+
+483
+00:33:36,720 --> 00:33:39,480
+بنحكي الآن Absolute Maximum و Absolute Minimum و
+
+484
+00:33:39,480 --> 00:33:44,480
+Closed Bounded Region، اللي هو احنا بنضمن وجود الـ
+
+485
+00:33:44,480 --> 00:33:46,500
+Absolute Maximum و Absolute Minimum على Closed
+
+486
+00:33:46,500 --> 00:33:50,460
+Bounded Region، إذا كانت الـ Fx والـ Fy والـ Fz
+
+487
+00:33:50,460 --> 00:33:53,840
+إذا كان نحكي عن الـ three variables كلها موجودة
+
+488
+00:33:53,840 --> 00:33:58,520
+و continuous ويكون عندها اللي هو الـ function
+
+489
+00:33:58,520 --> 00:34:01,840
+attains its maximum and its minimum، يعني لازم يكون
+
+490
+00:34:01,840 --> 00:34:04,560
+فيه عنده maximum و minimum، absolute maximum و
+
+491
+00:34:04,560 --> 00:34:08,840
+absolute minimum، طب كيف نوجدها؟ نوجدها عبر ما يلي:
+
+492
+00:34:08,840 --> 00:34:13,350
+أول حاجة بنيجي للـ interior points، بنشوف وين ممكن
+
+493
+00:34:13,350 --> 00:34:16,550
+نلاقي الـ maximum والـ minimum، اللي هي عارفين وين؟
+
+494
+00:34:16,550 --> 00:34:21,050
+عند... يا إما عند الـ critical points، F may have
+
+495
+00:34:21,050 --> 00:34:24,410
+local maximum and minimum عند الـ critical points
+
+496
+00:34:24,410 --> 00:34:26,470
+اللي هي في الـ interior points، اللي عند critical
+
+497
+00:34:26,470 --> 00:34:33,400
+points، اللي هي بنحسب في هذه الحالة الـ F عندهم، عند
+
+498
+00:34:33,400 --> 00:34:36,520
+كل النقاط اللي لاقيناها، زي ما عملنا قبل شوية و
+
+499
+00:34:36,520 --> 00:34:40,360
+بنحطهم، بنخزنهم، بنيجي لمين؟ للـ boundary، الـ
+
+500
+00:34:40,360 --> 00:34:43,620
+Boundary Points of R، اللي هو حسب ايش؟ اللي هو ال
+
+501
+00:34:43,620 --> 00:34:47,460
+region اللي عندنا، بمشي على ال boundaries وبوجد
+
+502
+00:34:47,460 --> 00:34:50,280
+الأماكن اللي ممكن نلاقي فيها maximum أو minimum،
+
+503
+00:34:50,280 --> 00:34:54,900
+هنقول كيف نوجدهم، وبنلاقي النقاط هذه وبرضه بنحطها في
+
+504
+00:34:54,900 --> 00:34:58,640
+list، في قائمة، وبنوجد الـ F عندها، إلى بعد ما
+
+505
+00:34:58,640 --> 00:35:02,720
+أوجدناها من في الحالتين، الأولى يا إما من ال
+
+506
+00:35:02,720 --> 00:35:05,800
+critical points أو من ال boundary points، باخد اللي
+
+507
+00:35:05,800 --> 00:35:08,460
+هو أكبر واحدة فيهم بتكون هي ال absolute maximum فيها،
+
+508
+00:35:08,460 --> 00:35:11,900
+وأصغر واحدة فيهم هي ال absolute minimum،
+
+509
+00:35:11,900 --> 00:35:15,620
+وإذا الموضوع صار سهل، look through the list for the
+
+510
+00:35:15,620 --> 00:35:18,780
+maximum and minimum values of F، these will be
+
+511
+00:35:18,780 --> 00:35:21,710
+absolute maximum and minimum values of F، Since
+
+512
+00:35:21,710 --> 00:35:24,950
+absolute maximum and minimum are also local
+
+513
+00:35:24,950 --> 00:35:27,210
+maximum and minimum are absolute maximum and
+
+514
+00:35:27,210 --> 00:35:30,010
+minimum values، وفيه بيرسم somewhere in the list
+
+515
+00:35:30,010 --> 00:35:33,190
+made list، بواحدة اتنين وكل ما هو ال absolute
+
+516
+00:35:33,190 --> 00:35:37,070
+maximum و absolute minimum هيكون واحدة من ال local
+
+517
+00:35:37,070 --> 00:35:41,530
+maximum و local minimum، الآن يعني اللي بنحكي أنه لو
+
+518
+00:35:41,530 --> 00:35:45,530
+طبعا مشابه للنظرية اللي أخدناها زمان في calculus
+
+519
+00:35:45,530 --> 00:35:47,670
+الـ one، if f is continuous on a closed bounded
+
+520
+00:35:47,670 --> 00:35:50,410
+interval then it attains its maximum and its
+
+521
+00:35:50,410 --> 00:35:53,670
+minimum in this interval، أو نفس الطرق هنا إذا كانت
+
+522
+00:35:53,670 --> 00:35:58,190
+fx, fy, fz اللي هي exist and continuous، then it
+
+523
+00:35:58,190 --> 00:36:01,170
+attains، f attains its maximum and its minimum in
+
+524
+00:36:01,170 --> 00:36:04,510
+this closed region، طبعا محتاجة closed region أو
+
+525
+00:36:04,510 --> 00:36:09,150
+قلنا كيف نوجدها، خلينا نشوف هذا بواسطة مثال عملي
+
+526
+00:36:09,150 --> 00:36:13,350
+Find the absolute maximum and minimum values of f
+
+527
+00:36:13,350 --> 00:36:16,470
+of x, y، بساوية الأمامي، on the triangular region in
+
+528
+00:36:16,470 --> 00:36:19,910
+the first quadrant bounded by the lines x بساوية
+
+529
+00:36:19,910 --> 00:36:23,600
+0، y بساوية 0، y بساوية 9 ناقص x، لأن في شغلتين
+
+530
+00:36:23,600 --> 00:36:27,460
+مهمات، الشغلة الأولى أنه اللي هو الـ interior points
+
+531
+00:36:27,460 --> 00:36:30,460
+كيف نوجد اللي هو الـ absolute maximum، كيف نوجد
+
+532
+00:36:30,460 --> 00:36:33,380
+اللي هو وين الـ interior points، مش هنفحص وين فيه
+
+533
+00:36:33,380 --> 00:36:35,360
+absolute maximum ولا absolute minimum، احنا ضمنين
+
+534
+00:36:35,360 --> 00:36:38,140
+أنه على close bounded region هيكون فيه عندي
+
+535
+00:36:38,140 --> 00:36:40,520
+absolute maximum و absolute minimum، بنجيب ال
+
+536
+00:36:40,520 --> 00:36:43,580
+critical points وبنجيب ال boundary points، وين فيه
+
+537
+00:36:43,580 --> 00:36:46,200
+إمكانية وجود ال local maximum وال local minimum، و
+
+538
+00:36:46,200 --> 00:36:50,050
+بنعوض عنهم، بنعوض، بنجيب اف،  أكبر واحدة هي
+
+539
+00:36:50,050 --> 00:36:53,050
+absolute maximum، وأصغر واحدة absolute minimum، إذا
+
+540
+00:36:53,050 --> 00:36:57,190
+الخطوات كالتالي: اللي هو بنحدد اللي هي من شغلتين
+
+541
+00:36:57,190 --> 00:36:59,530
+مهمات، الشغلة الأولى اللي نحدد المنطقة، الشغلة
+
+542
+00:36:59,530 --> 00:37:03,150
+الثانية اللي هو بنبدأ نشتغل على الـ interior، هذا
+
+543
+00:37:03,150 --> 00:37:07,020
+بتحدد الـ boundary وهذا بتحدد الـ interior، الآن x
+
+544
+00:37:07,020 --> 00:37:09,920
+بيساوي صفر، و y بيساوي صفر، و y بيساوي تسعة ناقص x
+
+545
+00:37:09,920 --> 00:37:13,420
+x بيساوي صفر اللي هو الـ y axis، هيها، y بيساوي صفر
+
+546
+00:37:13,420 --> 00:37:18,320
+هيها، والخط y بيساوي تسعة ناقص x، خد x بتسعة بتطلع
+
+547
+00:37:18,320 --> 00:37:25,840
+اللي هي y بصفر، هيها، أو خد اللي هو x بصفر بتطلع y
+
+548
+00:37:25,840 --> 00:37:29,720
+بتسعة، فبيصير عندك هي هذه، بترسم الخط اللي بينهم
+
+549
+00:37:29,720 --> 00:37:34,710
+بتكون هذه المنطقة اللي احنا بنحكي عنها كـ domain
+
+550
+00:37:34,710 --> 00:37:37,930
+لمين؟ للدالة اللي عندنا، يعني بدنا نوجد اللي هو الـ
+
+551
+00:37:37,930 --> 00:37:40,390
+Absolute Maximum والـ Absolute Minimum للـ
+
+552
+00:37:40,390 --> 00:37:44,950
+function على هذه المنطقة، إذن الآن كيف بنوجدها؟
+
+553
+00:37:44,950 --> 00:37:50,150
+طبعا الـ Fx doesn't exist، مش موجودة، إذن ولا Fy
+
+554
+00:37:50,150 --> 00:37:52,730
+doesn't exist موجودة، إذن بس اللي هي لإنه هذا الـ
+
+555
+00:37:52,730 --> 00:37:55,660
+function differentiable من المكان، إذا بدنا نوجد Fx و
+
+556
+00:37:55,660 --> 00:37:58,720
+Fy، بنسويهن بالصفر وبنشوف النقاط اللي بتطلع، النقاط
+
+557
+00:37:58,720 --> 00:38:02,960
+اللي بتطلع في داخل هذه المنطقة أو على جانبها، اللي
+
+558
+00:38:02,960 --> 00:38:06,940
+هو بنفحصها، برا مالناش علاقة فيها، الآن وبعدين
+
+559
+00:38:06,940 --> 00:38:09,820
+بنيجي بنشتغل على الـ boundary points، وهنقولكم كيف
+
+560
+00:38:09,820 --> 00:38:13,760
+بالظبط كيف نشتغل عليه، الآن نأتي أولا للـ interior
+
+561
+00:38:13,760 --> 00:38:17,880
+points، نقسمها قسمين، Interior Points و Boundary
+
+562
+00:38:17,880 --> 00:38:20,580
+Points، انتشر الـ Interior Points هي الحل المرتب
+
+563
+00:38:20,580 --> 00:38:23,960
+حله زي ما بحل Interior Points، ايش هي؟ أو زي ما بحل
+
+564
+00:38:23,960 --> 00:38:28,600
+الكتاب، Interior Points، كيف هي؟ نوجد Fx و Fy، Fx فهي
+
+565
+00:38:28,600 --> 00:38:32,710
+2 ناقص 2x، هي 2 ناقص 2x، Fy زيها 2
+
+566
+00:38:32,710 --> 00:38:37,250
+ناقص 2y، نساويها بالصفر، بتطلع y بتساوي x بتساوي
+
+567
+00:38:37,250 --> 00:38:41,070
+واحد، ايش؟ y بتساوي واحد و x بتساوي واحد، إذا النقطة
+
+568
+00:38:41,070 --> 00:38:43,530
+اللي بنحكي عنها interior point هي واحد واحد،
+
+569
+00:38:43,530 --> 00:38:47,450
+هي أول نقطة في ال list، هنعمل لها بأحمر على أساس
+
+570
+00:38:47,450 --> 00:38:51,150
+تقول لي f واحد واحد، ايش بتساوي؟ اللي هو 2 زائد
+
+571
+00:38:51,150 --> 00:38:54,210
+2 زائد 2 ناقص 1 ناقص 1، اللي هي بتساوي
+
+572
+00:38:54,210 --> 00:38:57,690
+قد ايش؟ مجمعين 6 ناقص 2 بيطلع 4، إذن هذه
+
+573
+00:38:57,690 --> 00:39:01,310
+نحفظها في الذاكرة، لإن هي من النقاط اللي بدنا نحكم
+
+574
+00:39:01,310 --> 00:39:04,390
+من خلالها، وين ال absolute maximum وال absolute
+
+575
+00:39:04,390 --> 00:39:07,250
+minimum؟ حسب النظرية اللي قلت عنها إنه مدام على
+
+576
+00:39:07,250 --> 00:39:09,030
+closed bounded region و differentiable أو differentiable
+
+577
+00:39:09,030 --> 00:39:11,550
+شابلها دي أو اللي هو ال partial derivatives
+
+578
+00:39:11,550 --> 00:39:13,750
+continuous، ده المضمون عندي اللي هو absolute
+
+579
+00:39:13,750 --> 00:39:16,730
+maximum و absolute minimum، أكبر واحدة فيها الـ
+
+580
+00:39:16,730 --> 00:39:18,610
+Absolute Maximum، وأصغر واحدة هي الـ Absolute
+
+581
+00:39:18,610 --> 00:39:21,750
+Minimum، هي اللي عدت عشرين مرة، الآن نيجي للـ
+
+582
+00:39:21,750 --> 00:39:24,750
+Boundary Points، الـ Boundary Points مين هي؟ هيها
+
+583
+00:39:24,750 --> 00:39:30,430
+اللي هي ايش؟ الخطوط الـ Region، نيجي بنمسك خط خط، و
+
+584
+00:39:30,430 --> 00:39:35,190
+بنشتغل في الدالة على خط خط، الآن نأتي أولا على الـ
+
+585
+00:39:35,190 --> 00:39:40,130
+segment OA، دعنا نشوف كيف نشتغل، الخط الأول OA، الـ
+
+586
+00:39:40,130 --> 00:39:43,950
+OA، ما معادلته؟ y بتساوي 0، لأن y عليه هو بتساوي
+
+587
+00:39:43,950 --> 00:39:51,730
+دائما 0، تصبح الدالة على هذا الخط f of x, 0، لأن y
+
+588
+00:39:51,730 --> 00:39:56,510
+أصبحت أخذناها عبارة عن 0، تصبح f of x, 0 بساوية 2
+
+589
+00:39:57,110 --> 00:40:02,310
+زائد 2x ناقص x³، لأن هذا يصبح صفر وهذا ايش؟ صفر، الآن هذه
+
+590
+00:40:02,310 --> 00:40:07,470
+الدالة صارت تشمل على الخط هذا، على الخط هذا الدالة
+
+591
+00:40:07,470 --> 00:40:10,850
+هذه صار لها دالة function of one variable زي تباعد
+
+592
+00:40:10,850 --> 00:40:13,910
+الزمان، صار لها boundary points اللي هي هذه وهذه
+
+593
+00:40:13,910 --> 00:40:17,450
+وصار ايش؟ مالها؟ interior points، الـ interior points
+
+594
+00:40:17,450 --> 00:40:22,360
+كيف افحصها؟ بآخذ ال prime، بساويها بصفر، باجي الآن اللي هي
+
+595
+00:40:22,360 --> 00:40:24,780
+صارت عندي هذه معرفة على، زي ما قولنا، على الفترة من
+
+596
+00:40:24,780 --> 00:40:27,780
+من صفر لحد تسعة، ايضا صارت Zero boundary point
+
+597
+00:40:27,780 --> 00:40:31,260
+وتسعة boundary point، الآن نيجي ناخد أول شي على ال
+
+598
+00:40:31,260 --> 00:40:34,900
+boundary points للدالة الجديدة اللي على الخط اللي
+
+599
+00:40:34,900 --> 00:40:38,580
+هي، بيصير f، ناخد عند ال x بيساوي صفر، f of صفر
+
+600
+00:40:38,580 --> 00:40:42,540
+صفر بيساوي ايش؟ اتنين، وهذه ما بحطها من ضمن ال list
+
+601
+00:40:42,940 --> 00:40:46,080
+لأنه ممكن يكون عندها Maximum أو Minimum، عند مين؟
+
+602
+00:40:46,080 --> 00:40:48,360
+الآن باخد الـ Boundary Point التاني اللي هي مين؟
+
+603
+00:40:48,360 --> 00:40:52,860
+أنا في هذه الحالة تسعة، f of تسعة وصفر اللي هي
+
+604
+00:40:52,860 --> 00:40:55,660
+بيساوي، بعوض عن f تسعة وصفر، بيصير 2 زائد
+
+605
+00:40:55,660 --> 00:40:59,080
+18 اللي هو ناقص 1 و 8، اللي هي بتطلع
+
+606
+00:40:59,080 --> 00:41:03,560
+ياشتر 71، خلصنا من الـ Boundary Points للدالة اللي
+
+607
+00:41:03,560 --> 00:41:06,760
+عندي هذه، زي دوا الزمان، ضالين الـ Interior Point
+
+608
+00:41:06,760 --> 00:41:09,560
+إذا باخد f prime، من الآن f prime، لأن الـ derivative is a
+
+609
+00:41:09,560 --> 00:41:13,300
+function of one variable، f prime of x بيصير f prime of x ايش
+
+610
+00:41:13,300 --> 00:41:19,680
+بتساوي؟ اللي هي عبارة عن 2 ناقص 2x، 2 ناقص 2x، هيها
+
+611
+00:41:20,230 --> 00:41:24,550
+بسويها بالـ 0 زي ما بنعمل زمان، بتطلع عندي x بتساوي
+
+612
+00:41:24,550 --> 00:41:26,810
+ايش؟ واحد، إذا ان طلعت عندي x بتساوي واحد و y
+
+613
+00:41:26,810 --> 00:41:28,790
+بتساوي صفر، إذا ان النقطة اللي ممكن يكون عندها
+
+614
+00:41:28,790 --> 00:41:32,270
+local maximum و local minimum اللي هي واحد وصفر
+
+615
+00:41:32,270 --> 00:41:35,71
+
+667
+00:45:34,010 --> 00:45:38,070
+بتطلع الـ function نقيم التناقص 20.5 الآن summary
+
+668
+00:45:38,070 --> 00:45:41,410
+we list the candidates اللي وجدناها، اللي هي مين هي؟
+
+669
+00:45:41,760 --> 00:45:45,800
+اللي هي قيمة الـ Function: الأولى أربعة، والثانية 
+
+670
+00:45:45,800 --> 00:45:48,440
+اثنين، الثانية ناقص واحد وستين، واحدة تلاتة وناقص عشرين
+
+671
+00:45:48,440 --> 00:45:52,840
+point خمسة. أكبر واحدة فين؟ عندها Absolute Maximum
+
+672
+00:45:52,840 --> 00:45:56,180
+وأصغر واحدة فين؟ عندها Absolute Minimum. إذن هذا
+
+673
+00:45:56,180 --> 00:45:58,940
+Absolute Minimum، وهذا Absolute Maximum. إذن
+
+674
+00:45:58,940 --> 00:46:03,060
+Absolute Minimum هو 4، وهذه حدثت عند النقطة
+
+675
+00:46:03,060 --> 00:46:06,660
+الأولى (1,1) وهي طبعاً في داخل المنطقة. Absolute
+
+676
+00:46:06,660 --> 00:46:10,970
+Minimum هو -69، اللي
+
+677
+00:46:10,970 --> 00:46:14,830
+حدثت عند نقطتنا دي اللي قيمتها ايش؟ ناقص واحد وستين.
+
+678
+00:46:14,830 --> 00:46:17,510
+ممكن. وهيك إحنا عرفنا كيف نوجد الـ absolute
+
+679
+00:46:17,510 --> 00:46:19,930
+maximum والـ absolute minimum على closed bounded
+
+680
+00:46:19,930 --> 00:46:24,430
+اللي هي region. نيجي الآن نعمل summary. إذا عندي
+
+681
+00:46:24,430 --> 00:46:29,210
+الموضوع الآن: الـ extreme values can occur only at
+
+682
+00:46:29,210 --> 00:46:31,370
+الـ Boundary Points، يا إما الـ Critical Points.
+
+683
+00:46:31,370 --> 00:46:34,090
+والـ Critical Points قلنا يا Interior طبعاً الـ
+
+684
+00:46:34,090 --> 00:46:37,610
+Interior بتكون Fx أو Fy بساوي صفر، أو Fx أو Fy لا
+
+685
+00:46:37,610 --> 00:46:42,930
+يوجد. إلا أن إذا كان الـ First والـ Second derivatives
+
+686
+00:46:42,930 --> 00:46:47,250
+موجودة ومستمرة، وبدنا نفحص هل في Local
+
+687
+00:46:47,250 --> 00:46:49,510
+Maximum و Local Minimum. قلنا بالـ Second
+
+688
+00:46:49,510 --> 00:46:53,250
+Derivative Test اللي حكينا عنه سابقاً. وهيك
+
+689
+00:46:53,250 --> 00:46:56,090
+بيكون إحنا خلصنا المحاضرة، وبارك الله فيكم والسلام
+
+690
+00:46:56,090 --> 00:46:58,490
+عليكم ورحمة الله وبركاته
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6da210c8032af5e498acbd2f17a5d784d85314c4
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04_postprocess.srt
@@ -0,0 +1,2760 @@
+1
+00:00:00,250 --> 00:00:03,610
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:03,610 --> 00:00:08,670
+وبركاته هذه هي المحاضرة رقم تسعة عشر مساقطة طادل
+
+3
+00:00:08,670 --> 00:00:12,310
+وتكامل الـGene وهي المحاضرة السابعة والأخيرة في
+
+4
+00:00:12,310 --> 00:00:17,750
+شبطر أربعة عشر وهي أربعة عشر سبعة وهي تحت عنوان
+
+5
+00:00:17,750 --> 00:00:22,890
+Extreme Values and Subtle Valuesطبعا الموضوع ليس
+
+6
+00:00:22,890 --> 00:00:29,510
+جديد فيه روحه احنا اتحدثنا في calculus الف وكمان
+
+7
+00:00:29,510 --> 00:00:34,910
+حتى في الفنوية عن اللي هي كيف نوجد الـ Local
+
+8
+00:00:34,910 --> 00:00:38,470
+maximum و Local minimum و الـ Infliction points و
+
+9
+00:00:38,470 --> 00:00:42,590
+كذلك اللي هو ال test تبعاتها و كذلك على Absolute
+
+10
+00:00:42,590 --> 00:00:47,030
+maximum و Absolute minimumالحديث سيكون حديث مقابل
+
+11
+00:00:47,030 --> 00:00:49,890
+بسعر الـ two variables أو الـ three variables
+
+12
+00:00:49,890 --> 00:00:55,630
+الحديث سيكون حول نفس الأشياء سنجاوبها على سؤالين
+
+13
+00:00:55,630 --> 00:01:03,090
+السؤال الأول أين يمكن إيجاد الـ local maximum أو
+
+14
+00:01:03,090 --> 00:01:06,640
+الـ local minimumأو اللي هي الان مش inflection
+
+15
+00:01:06,640 --> 00:01:09,840
+point حاجة بنسميها settle points أو نقطة سرج
+
+16
+00:01:09,840 --> 00:01:15,140
+الحصان السؤال التاني أنه هل كيف بدنا نحكم على أن
+
+17
+00:01:15,140 --> 00:01:19,080
+هذه فعلا في عندها local maximum و local minimum أو
+
+18
+00:01:19,080 --> 00:01:24,740
+نقطة settle points أو لاتقريباً الأجوبة هي مشابهة
+
+19
+00:01:24,740 --> 00:01:28,080
+للأجوبة السابقة في تفاضل ألف كنا نجاوب في تفاضل
+
+20
+00:01:28,080 --> 00:01:33,200
+ألف نقول القيم العظمى والصغرى وكذلك الـ inflection
+
+21
+00:01:33,200 --> 00:01:37,380
+points ممكن أن توجد في أحد المناطق التالية إما في
+
+22
+00:01:37,380 --> 00:01:40,160
+الـ interior points اللي هي في عند الـ derivative
+
+23
+00:01:40,160 --> 00:01:43,660
+بتساوي سفر أو الـ derivative does not exist هذه
+
+24
+00:01:43,660 --> 00:01:46,880
+المنطقة تانية والمنطقة الثالثة هي الـ boundary
+
+25
+00:01:46,880 --> 00:01:50,110
+pointsوكنا ناخد الـ First derivative test والـ
+
+26
+00:01:50,110 --> 00:01:54,010
+Second derivative test اللي هو الحكم على إن فيه
+
+27
+00:01:54,010 --> 00:01:57,330
+لوكال maximum أو لوكال minimum أو اللي هو
+
+28
+00:01:57,330 --> 00:02:01,290
+inflection point هذا الحديث تقريبا مشابه جدا خلينا
+
+29
+00:02:01,290 --> 00:02:05,010
+نبدأ اللي هو في الموضوعنا derivative test for
+
+30
+00:02:05,010 --> 00:02:10,010
+local extreme valuesنشوف أول حاجة ماعنات الـ
+
+31
+00:02:10,010 --> 00:02:15,190
+maximum و الـ minimum أو الـ local maximum و الـ
+
+32
+00:02:15,190 --> 00:02:19,210
+local minimum مشابهة let f of x,y be defined on a
+
+33
+00:02:19,210 --> 00:02:22,790
+region R containing the point A و B نشوف الآن ما
+
+34
+00:02:22,790 --> 00:02:25,910
+هي معنات local maximum عندنا نقطة A و B هي نقطة A
+
+35
+00:02:25,910 --> 00:02:29,010
+و B ما هي معناة local maximum local maximum هي
+
+36
+00:02:29,010 --> 00:02:34,320
+نقطة عظمة محلية إننا نجد جوار حوالهاهناك جوار
+
+37
+00:02:34,320 --> 00:02:39,460
+حواليها تكون هي أكبر واحدة يعني قيمة F of A و F of
+
+38
+00:02:39,460 --> 00:02:45,540
+B أكبر واحدة أكبر من F of XY لكل XY في هذا الجوار
+
+39
+00:02:45,540 --> 00:02:48,780
+يعني أكبر واحدة في جيرانها هذا نسميها Local
+
+40
+00:02:48,780 --> 00:02:53,040
+Maximumالآن أصغر وحدة تجوّر حوالين الـ A و الـ B
+
+41
+00:02:53,040 --> 00:02:56,720
+تكون الـ F عند الـ A و الـ B أصغر وحدة في جيرانها
+
+42
+00:02:56,720 --> 00:03:01,040
+فبنسميها Local Minimum وهذا هو التعريف F of A و B
+
+43
+00:03:01,040 --> 00:03:05,720
+is Local Maximum Value of F if F of A و B أكبر
+
+44
+00:03:05,720 --> 00:03:10,840
+يساوي F of X و Y أكبر من كل الـ F of X و Y لأ يعني
+
+45
+00:03:10,840 --> 00:03:14,860
+بمعنى أخر اللي هي لنقاط الـ X و Y حوالها for all
+
+46
+00:03:14,860 --> 00:03:20,440
+domainو.xy in an open disk centered at a وb
+
+47
+00:03:25,650 --> 00:03:29,230
+بتكون قيمة F عند الـ A و الـ B أكبر من F of X Y
+
+48
+00:03:29,230 --> 00:03:33,410
+لكل X Y في هذا الـ Disk اللي حواليها أو الـ Open
+
+49
+00:03:33,410 --> 00:03:37,630
+Disk الآن الـ Local Minimum بنفس الأسلوب F of A و
+
+50
+00:03:37,630 --> 00:03:41,790
+B is a Local Minimum Value of F if F of A و B أصغر
+
+51
+00:03:41,790 --> 00:03:46,130
+أو يساوي F of X Y for all points X Y في Domain الـ
+
+52
+00:03:46,130 --> 00:03:50,950
+Function وفي نفس الواجهة جيران لـ A و B أو في اللي
+
+53
+00:03:50,950 --> 00:03:53,930
+هي Disk Open Disk عارفين شو معناه Open Disk دائرة
+
+54
+00:03:53,930 --> 00:03:58,290
+مفتوحةاللي هو اللي جواتها تكون اللي هي كل النقطات
+
+55
+00:03:58,290 --> 00:04:01,670
+اللي جواتها قيمة الـ Function F عند الـ A والـ B
+
+56
+00:04:01,670 --> 00:04:06,510
+أصغر من هذه القيم معناته أن F لها Local Minimum
+
+57
+00:04:06,510 --> 00:04:11,810
+عند هذه النقطة الآن بدنا نيجي اللي هو ندى نجاوب
+
+58
+00:04:11,810 --> 00:04:14,990
+السؤال الأول أين يمكن إيجاد الـ Local Maximum و
+
+59
+00:04:14,990 --> 00:04:20,340
+Local Minimum أو اللي هوالنقاط بعد شويه هنحكي عن
+
+60
+00:04:20,340 --> 00:04:25,540
+نقاط حرجة أو الـ saddle point طبعا الان ال theorem
+
+61
+00:04:25,540 --> 00:04:29,720
+عشرة first derivative test for local values الان
+
+62
+00:04:29,720 --> 00:04:32,900
+اللي هي ال first derivative test زي ما كان هناك في
+
+63
+00:04:32,900 --> 00:04:36,240
+first derivative test هنا في عندنا first
+
+64
+00:04:36,240 --> 00:04:41,960
+derivative test إيش بتقول إذا كانت f of x y has a
+
+65
+00:04:41,960 --> 00:04:46,230
+local minimum أو local maximumعند الـ interior
+
+66
+00:04:46,230 --> 00:04:50,830
+point a و b of its domain and if the first partial
+
+67
+00:04:50,830 --> 00:04:57,030
+derivatives exist then fx عند هذا النقطة سفرة و fy
+
+68
+00:04:57,030 --> 00:05:00,330
+عند هذا النقطة a سفرة هذا بتقول إنها نظرية مالي
+
+69
+00:05:00,330 --> 00:05:04,770
+إذا كانت ال function طبعا تحت الظروفة إنها ال
+
+70
+00:05:04,770 --> 00:05:08,350
+first partial derivative test exist يعني ال fx و
+
+71
+00:05:08,350 --> 00:05:12,020
+ال fy موجودةالآن أخبرني باختصار إذا كانت الـ
+
+72
+00:05:12,020 --> 00:05:16,240
+function f of x y لها local maximum أو local
+
+73
+00:05:16,240 --> 00:05:20,300
+minimum عند الـ a والب غصبًا عن طبعًا في interior
+
+74
+00:05:20,300 --> 00:05:24,540
+point يعني غصبًا عن بيكون fx بتساوي سفر وfy إيه
+
+75
+00:05:24,540 --> 00:05:29,540
+شمالها بتساوي سفر يعني لما بدك تيجي تفحص عند اللي
+
+76
+00:05:29,540 --> 00:05:33,160
+هياللي تبحث عن الـ Local Maximum و Local Minimum
+
+77
+00:05:33,160 --> 00:05:37,420
+في الـ Interior Points اللي الـ Fx و Fy exist لهن
+
+78
+00:05:37,420 --> 00:05:42,600
+افحص بس عند Fx بيساوي 0 و Fy بيساوي 0 النقاط اللي
+
+79
+00:05:42,600 --> 00:05:46,500
+بتحقق هذه ممكن يكون فيها Local Maximum و Local
+
+80
+00:05:46,500 --> 00:05:50,420
+Minimumأما النقاط اللي بيكونش هذه سفر وتكونش هذه
+
+81
+00:05:50,420 --> 00:05:53,300
+سفر انساها لأن لو كان Local Maximum و Local
+
+82
+00:05:53,300 --> 00:05:57,720
+Minimum لازم يطلع عن سفر إذا من هذه النظرية بنقدر
+
+83
+00:05:57,720 --> 00:06:04,450
+نقول أنه احنا ممكن نوجدممكن نلاقي الـ Local
+
+84
+00:06:04,450 --> 00:06:08,530
+Maximum أو الـ Local Minimum عند أحد النقاط
+
+85
+00:06:08,530 --> 00:06:13,970
+التالية إما Fx يساوي Fy بيساوي سفر أو إن الـ
+
+86
+00:06:13,970 --> 00:06:17,350
+Derivative does not exist لأن هذا ليس متضمن من
+
+87
+00:06:17,350 --> 00:06:21,350
+خلال الحديث في النظرية إذن هذه النظرية توحي ما
+
+88
+00:06:21,350 --> 00:06:26,010
+يليه أو تعطينا المعلومات التالية إنه اللي هي الـ
+
+89
+00:06:26,010 --> 00:06:30,380
+Interior Pointsللـ Interior Points ممكن نلاقي الـ
+
+90
+00:06:30,380 --> 00:06:32,300
+Local Maximum الـ Local Minimum عند الـ Interior
+
+91
+00:06:32,300 --> 00:06:37,560
+Points اللي Fx بيساوي 0 و Fy بيساوي 0 و اللي Fx و
+
+92
+00:06:37,560 --> 00:06:43,140
+Fy لا يوجد في هذه الحالة اللي هو بيكون عندي اللي
+
+93
+00:06:43,140 --> 00:06:46,800
+هو عن الـ Interior Points اللي هي ممكن نودد الـ
+
+94
+00:06:46,800 --> 00:06:50,460
+Local Maximum و Local Minimum عند هذه الأماكن يعني
+
+95
+00:06:50,460 --> 00:06:54,300
+بمعنى آخر من هذه النظرية أيضا نستقل معلومة التالية
+
+96
+00:06:54,300 --> 00:07:00,990
+وهي نهائية يمكنالمناطق اللي ممكن يكون فيها Local
+
+97
+00:07:00,990 --> 00:07:04,550
+Maximum أو Local Minimum لسه بدأ أفحص المناطق
+
+98
+00:07:04,550 --> 00:07:07,250
+اللتي يمكن أن يكون فيها Local Maximum أو Local
+
+99
+00:07:07,250 --> 00:07:10,990
+Minimum هي النقاط اللي الـ Fx بتساوي Fy بساوي سفر
+
+100
+00:07:10,990 --> 00:07:13,730
+هاي واحد طبعا الـ Interior Bonds اتنين الـ
+
+101
+00:07:13,730 --> 00:07:15,870
+Interior Bonds اللي هي Fx does not exist
+
+102
+00:07:20,290 --> 00:07:26,450
+النقطة الأخيرة الـ Boundary Points حصرنا كل جزء في
+
+103
+00:07:26,450 --> 00:07:39,450
+أربع مناطق Fx Fy Fx Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy
+
+104
+00:07:39,450 --> 00:07:39,550
+Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy
+
+105
+00:07:39,550 --> 00:07:39,790
+Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy
+
+106
+00:07:39,820 --> 00:07:44,040
+إذا سهلت علينا أمور الفحص يعني أن نتأكد الآن ضال
+
+107
+00:07:44,040 --> 00:07:47,300
+علينا بس نتأكد إن في Local Maximum ولا Local
+
+108
+00:07:47,300 --> 00:07:51,780
+Minimum أو مافيش شيء الآن اللي اللي اللي اللي اللي
+
+109
+00:07:51,780 --> 00:07:55,620
+اللي عشان نحصر الأمور يعني وكأننا نجسم الأمر اللي
+
+110
+00:07:55,620 --> 00:08:00,090
+هو الأمرين الـInterior Point الـ Gatewayأو إذا كان
+
+111
+00:08:00,090 --> 00:08:02,850
+هناك نقطة الداخلية في مجلة عملية f of x و f of y
+
+112
+00:08:02,850 --> 00:08:04,030
+أما إذا كان هناك نقطة داخلية في مجلة عملية f of x
+
+113
+00:08:04,030 --> 00:08:05,610
+و f of y أما إذا كان هناك نقطة داخلية في مجلة
+
+114
+00:08:05,610 --> 00:08:07,230
+عملية f of x و f of y أما إذا كان هناك نقطة داخلية
+
+115
+00:08:07,230 --> 00:08:08,210
+في مجلة عملية f of x و f of y أما إذا كان هناك
+
+116
+00:08:08,210 --> 00:08:09,170
+نقطة داخلية في مجلة عملية f of x و f of y أما إذا
+
+117
+00:08:09,170 --> 00:08:10,050
+كان هناك نقطة داخلية في مجلة عملية f of x و f of y
+
+118
+00:08:10,050 --> 00:08:12,270
+أما إذا كان هناك نقطة داخلية في مجلة عملية f of x
+
+119
+00:08:12,270 --> 00:08:17,110
+و f of y أما إذا كان هناك نقطة
+
+120
+00:08:17,110 --> 00:08:21,580
+داخلية في مجلة عملية f of x و f of xهي واحدة اتنين
+
+121
+00:08:21,580 --> 00:08:26,440
+Fx does not exist هيتنتين F تلاتة Fy does not
+
+122
+00:08:26,440 --> 00:08:30,680
+exist هدول الـ Critical Points اللي اللي ممكن يكون
+
+123
+00:08:30,680 --> 00:08:33,600
+عندهم Local Maximum أو Local Minimum إذا صار في
+
+124
+00:08:33,600 --> 00:08:36,740
+عندي بناء على هذا التصنيف في منطقتين ممكن نلاقي
+
+125
+00:08:36,740 --> 00:08:40,320
+فين اللي هو Local Maximum أو Local Minimum أو اللي
+
+126
+00:08:40,320 --> 00:08:45,290
+هو بعد شوية أسد الـ Point اللي هي أحد تلت مناطقأحد
+
+127
+00:08:45,290 --> 00:08:49,130
+منطقتين اللي هي إما الـ interior points اللي بتكون
+
+128
+00:08:49,130 --> 00:08:52,410
+فيها critical points هاي الـ critical points يعني
+
+129
+00:08:52,410 --> 00:08:54,530
+طبعا الـ critical point على طول الـ interior point
+
+130
+00:08:54,530 --> 00:08:57,990
+الحاجة الثانية اللي هي الـ boundary points إذا
+
+131
+00:08:57,990 --> 00:09:02,270
+منطقتين critical points أو boundary pointsالـ
+
+132
+00:09:02,270 --> 00:09:05,530
+Critical Points هي اللي بيكون عندها Fx بيساوي Fy
+
+133
+00:09:05,530 --> 00:09:08,270
+طبعا Interior Points Fx بيساوي Fy بيساوي Zero هاي
+
+134
+00:09:08,270 --> 00:09:12,130
+واحدة Fx does not exist تاعتين تاني ثلاثة Fy does
+
+135
+00:09:12,130 --> 00:09:15,930
+not exist هذه هي السؤال جاوبنا عليه الآن أين يمكن
+
+136
+00:09:15,930 --> 00:09:18,410
+إيجاد الـ Local Maximum أو ال Local Minimum أو ال
+
+137
+00:09:18,410 --> 00:09:25,260
+Extreme Values أو Pointsالان لاحظ عند اللي هي ال
+
+138
+00:09:25,260 --> 00:09:28,480
+.. ال .. ال .. النقطة هذه اللي هي عند الـ Local
+
+139
+00:09:28,480 --> 00:09:33,620
+Maximum لاحظ أنه الميه المماس هذا المماس هيكون الـ
+
+140
+00:09:33,620 --> 00:09:38,220
+∆FY∆Y موازية لـ Y Axis يعني ∆FY∆Y أيش بيساوي سفر
+
+141
+00:09:38,220 --> 00:09:41,600
+والمماس ل ال curve هذا اللي partial derivative إذ
+
+142
+00:09:41,600 --> 00:09:43,980
+هانا اللي معناه partial derivative معناه كنا زمان
+
+143
+00:09:43,980 --> 00:09:47,180
+بيكون ال ∆FY∆X برضه موازية ل X Axis بيكون اللي هو
+
+144
+00:09:47,180 --> 00:09:50,780
+أيش سفر يعني عنده النقطة هذهالـ interior point الـ
+
+145
+00:09:50,780 --> 00:09:54,580
+derivative اللي هي exist الـ local maximum هاي
+
+146
+00:09:54,580 --> 00:09:57,880
+اللي هو ميل المماس سفر و ميل المماس التاني سفر
+
+147
+00:09:57,880 --> 00:10:01,060
+واحد اللي هو دل أف بايد ال X سفر واحد ع الدل أف
+
+148
+00:10:01,060 --> 00:10:04,620
+بايد ال Y بساوة سفر هذا معناه هندسي للحديث اللي
+
+149
+00:10:04,620 --> 00:10:06,280
+اتحدثنا فيه قبل بشوية
+
+150
+00:10:09,850 --> 00:10:13,410
+إذا لم يكن معرفة معناته الـ Saddle Point فمعرفة
+
+151
+00:10:13,410 --> 00:10:16,250
+مختلفة F of X Y لديها Saddle Point إلى Critical
+
+152
+00:10:16,250 --> 00:10:20,030
+Point إذا الآن Saddle Point وين يا جماعة؟ هي
+
+153
+00:10:20,030 --> 00:10:25,590
+Interior Point وعند مين؟ عند Critical Point يعني F
+
+154
+00:10:25,590 --> 00:10:31,350
+Y بسوء سفرأو Fx does not exist أو Fy does not
+
+155
+00:10:31,350 --> 00:10:35,090
+exist then a differentiable function f of x y has
+
+156
+00:10:35,090 --> 00:10:37,810
+a saddle point at a critical point a و b .. آسف
+
+157
+00:10:37,810 --> 00:10:41,350
+آسف آسف اللي هو عند a interior point Fx بتساوي Fy
+
+158
+00:10:41,350 --> 00:10:44,390
+بتساوي 0 a differentiable function f of x y has a
+
+159
+00:10:44,390 --> 00:10:49,010
+saddle pointعلى نقطة مهمة A وB إذا كان في كل مركز
+
+160
+00:10:49,010 --> 00:10:54,350
+مفتوح على A وB يوجد نقطة مخصصة XY فيها Fxy أكبر من
+
+161
+00:10:54,350 --> 00:10:59,190
+F of A وB ونقطة XY فيها Fxy أكبر من F of A وB
+
+162
+00:10:59,190 --> 00:11:06,400
+المنطقة المتعلقة بالنقطة Z هي نقطة مخصصةأذا الـ
+
+163
+00:11:06,400 --> 00:11:09,120
+differentiable function f of x y يعني بتكون الـ fx
+
+164
+00:11:09,120 --> 00:11:13,120
+و الـ fy exist already ماشي الحال اللي هو عند
+
+165
+00:11:13,120 --> 00:11:15,940
+critical point معناته بس عند fx بتساوي fy إيش
+
+166
+00:11:15,940 --> 00:11:19,240
+بتساوي سفر يعني اللي عن ال settle point إيش هي
+
+167
+00:11:19,240 --> 00:11:23,540
+نقطة داخلية بتكون عندها اللي هي بنقول عنها settle
+
+168
+00:11:23,540 --> 00:11:29,680
+point إذا إجيناأخدنا اللي هو لأي
+
+169
+00:11:29,680 --> 00:11:33,020
+جوار حواليها لأي جوار حواليها لجينا كل ما ناخد
+
+170
+00:11:33,020 --> 00:11:38,360
+جوار ال F of A و F of B مرات بتاخد في داخل الجوار
+
+171
+00:11:38,360 --> 00:11:42,560
+أصغر من F of XY ومرات بتكون أكبر من مين من F of XY
+
+172
+00:11:42,560 --> 00:11:45,260
+يعني بتكون أكبر من النقاط من اللي حواليها وأصغر من
+
+173
+00:11:45,260 --> 00:11:49,930
+النقاط من اللي حواليها إذا لا هيLocalMaximum وليس
+
+174
+00:11:49,930 --> 00:11:54,530
+LocalMinimum لأن كل جوار حولها ستجد الـ Function
+
+175
+00:11:54,530 --> 00:11:58,230
+عند هذا النقطة بتكون أكبر من النقاط في هذا الجوار
+
+176
+00:11:58,230 --> 00:12:02,830
+و أصغر من النقاط في هذا الجوار المذكور لذلك لا
+
+177
+00:12:02,830 --> 00:12:06,650
+يمكن أن تكون لـ LocalMaximum أو LocalMinimumفهي
+
+178
+00:12:06,650 --> 00:12:10,250
+بنسميها saddle point اللي هي نقطة سرج الحصان فيها
+
+179
+00:12:10,250 --> 00:12:14,970
+لإن هي زي نقطة زي سرج الحصان هذا النقطة طبعا وين
+
+180
+00:12:14,970 --> 00:12:19,690
+موجودة عند Fx بتساوي Fy بتساوي سفر يعني عند
+
+181
+00:12:19,690 --> 00:12:22,690
+critical point و F differentiable إذا عند Fx
+
+182
+00:12:22,690 --> 00:12:25,910
+بتساوي Fy إيه شمالها بيساوي سفر لإن مادة
+
+183
+00:12:25,910 --> 00:12:29,630
+differentiable function Fx و Fy مالين exist إذا ال
+
+184
+00:12:29,630 --> 00:12:34,130
+saddle point هي نقطة تفحص عند Fx بتساوي Fy بتساوي
+
+185
+00:12:34,130 --> 00:12:37,570
+سفرهذا النقطة لا وجهنا عندها maximum ولا وجهنا
+
+186
+00:12:37,570 --> 00:12:41,710
+عندها minimum فبتكون settle point خلّينا نشوف مثال
+
+187
+00:12:41,710 --> 00:12:46,030
+find the local extreme values f of x,y بساوي x
+
+188
+00:12:46,030 --> 00:12:48,970
+تربيه زي y تربيه يعني نقص 4y زي 9 عمنا نشوف إيش
+
+189
+00:12:48,970 --> 00:12:51,050
+اللي هو ال local extreme values لسه احنا بنشتغل
+
+190
+00:12:51,050 --> 00:12:54,230
+على التعريف يا جماعة ماخدناش tests أو اختبارات
+
+191
+00:12:54,230 --> 00:12:57,730
+تتعرفنا كيف نحكم على إن هذه local maximum أو local
+
+192
+00:12:57,730 --> 00:13:02,450
+minimum لنشوف كيف، الآن عندي لو دينا لل function
+
+193
+00:13:02,450 --> 00:13:06,700
+هذهأحنا قلنا أين ممكن نجد الـ Extreme Value أو الـ
+
+194
+00:13:06,700 --> 00:13:11,320
+Local Extreme يجب أن نجدها إما عند Fx بساوي Fy
+
+195
+00:13:11,320 --> 00:13:18,700
+بساوية Fy بساوية 0 أو Fx لا يوجد أو Fy لا يوجد أو
+
+196
+00:13:18,700 --> 00:13:22,540
+الـ Boundary Pointsلكن هذه الـ function أول حاجة
+
+197
+00:13:22,540 --> 00:13:25,300
+الـ differentiation بالوين ما كان ومعرفة وين ما
+
+198
+00:13:25,300 --> 00:13:27,960
+كان مزام الـ differentiation بالوين ما كان ومعرفة
+
+199
+00:13:27,960 --> 00:13:32,860
+وين ما كان إذا مافيش شيء لا Fx و Fy لا يوجد ولا
+
+200
+00:13:32,860 --> 00:13:36,780
+فيه point في الـ boundary إذا الحديث بس عند Fx
+
+201
+00:13:36,780 --> 00:13:40,460
+بيساوي Fy بيساوي 0 وجدوا لـ Fx و Fy بيساووهم بالـ
+
+202
+00:13:40,460 --> 00:13:43,900
+0 ووجدوا النقطة ونبحث عند هذه النقطة هل في Local
+
+203
+00:13:43,900 --> 00:13:47,900
+maximum أو Local minimum خلينا ناخد Fx، Fx طبعا
+
+204
+00:13:47,900 --> 00:13:52,940
+واضح إنها عبارة عن 2Xالان fy هي عبارة عن 2y ناقص
+
+205
+00:13:52,940 --> 00:13:56,360
+أربعة إذا الان هذولة التنتين موجودة في أي مكان إذا
+
+206
+00:13:56,360 --> 00:14:00,240
+على طول بنسويهم بالصفر هي fx بيساوي صفر معناته 2x
+
+207
+00:14:00,240 --> 00:14:03,580
+بيساوي صفر وfy بيساوي صفر معناته 2y ناقص أربعة
+
+208
+00:14:03,580 --> 00:14:07,640
+بيساوي صفر يعني y بتساوي 2 وx بتساوي أيش صفر حل
+
+209
+00:14:07,640 --> 00:14:10,940
+هنا يا جماعة يعني 2y مانس أربعة بيساوي صفر معناته
+
+210
+00:14:10,940 --> 00:14:14,280
+y بيساوي 2 و2x بيساوي صفر معناته x بيساوي صفر إذا
+
+211
+00:14:14,280 --> 00:14:17,560
+النقطة اللي بنحكي عنها بس النقطة مين صفر و اتنين
+
+212
+00:14:17,560 --> 00:14:21,550
+هذه النقطة صفر و اتنين الآنهذه النقطة هي 02 هذه
+
+213
+00:14:21,550 --> 00:14:26,110
+الآن بدنا نشوف اللي هو لسه ما بنعرفش تس احنا الان
+
+214
+00:14:26,110 --> 00:14:32,410
+بدنا نحسب عندها F02 F02 أحسبها Zero أربعة ناقص
+
+215
+00:14:32,410 --> 00:14:37,430
+أربعة نروح أربعة لأ اللي هي جداش اتنين اتنين في
+
+216
+00:14:37,430 --> 00:14:40,830
+اتنين باربعة أربعة ناقص تمانية آسف أربعة ناقص
+
+217
+00:14:40,830 --> 00:14:45,350
+تمانية اللي هي بتصير إيش ناقص أربعة زائد تسعة ناقص
+
+218
+00:14:45,350 --> 00:14:48,310
+أربعة زائد تسعة ليها خمسة F02 إيش هتساوي يا شباب
+
+219
+00:14:48,700 --> 00:14:53,280
+ويا بنات، اللي هو خمسة، فـ F02 بساوي خمسة، إذا
+
+220
+00:14:53,280 --> 00:14:56,140
+الآن بدنا نبحث فيها، لاحظ على طبيعة الـ function
+
+221
+00:14:56,140 --> 00:15:00,120
+اللي عندنا، الآن إحنا بنحكي بس على التعريف،
+
+222
+00:15:00,120 --> 00:15:02,580
+بنستخدم التعريف في تحديد إنه Local Maximum أو
+
+223
+00:15:02,580 --> 00:15:05,040
+Local Minimum، في بعض السؤالات السابقة، طبعا صعب
+
+224
+00:15:05,040 --> 00:15:08,180
+إننا نحددها بالتعريف، لكن هذا شأل سهل، خلينا نشوف
+
+225
+00:15:08,720 --> 00:15:12,120
+أفو في X Y بساوي تبعتنا X تربيع زي Y تربيع نقص
+
+226
+00:15:12,120 --> 00:15:14,960
+أربعة Y زي تسعة لكن هذه ممكن كتبتها على الطريقة
+
+227
+00:15:14,960 --> 00:15:18,620
+التالية لو أكملنا المربع حانة بيصير X تربيع زي Y
+
+228
+00:15:18,620 --> 00:15:21,540
+ماينوس اتنين كل تربيع يعني اللي هو أخدنا من التسعة
+
+229
+00:15:21,540 --> 00:15:24,460
+أربعة صار مربع كامل صار على صورة Y نقص اتنين كل
+
+230
+00:15:24,460 --> 00:15:28,480
+تربيع إذا زائد من خمسةإذاً صار هذا المقدار هذا
+
+231
+00:15:28,480 --> 00:15:31,940
+مربع كامل وهذا مربع كامل إذاً لأي قيمة في الدنيا
+
+232
+00:15:31,940 --> 00:15:35,860
+XY هيكون هذا المقدار أكبر أو يساوي سفر إذاً صار
+
+233
+00:15:35,860 --> 00:15:39,420
+هذا المجموع عبارة عن أكبر أو يساوي إيش خمسة يعني
+
+234
+00:15:39,420 --> 00:15:43,780
+بمعنى أخر F of XY صارت دائما أكبر أو يساوي F of 02
+
+235
+00:15:43,780 --> 00:15:48,500
+يعني F of 02 هتعمل لي Local Minimum هي في الواقع
+
+236
+00:15:48,500 --> 00:15:51,520
+Absolute Minimum لكن إحنا مش موضوعنا We see that
+
+237
+00:15:51,520 --> 00:15:56,180
+the still critical pointاللي هي 02 أو the critical
+
+238
+00:15:56,180 --> 00:16:00,440
+point 02 gives a local minimum هذا استخدمنا فيها
+
+239
+00:16:00,440 --> 00:16:05,820
+minimum يا جماعة استخدمنا فيها بس اللي هو التعريف
+
+240
+00:16:07,860 --> 00:16:12,260
+لأ مثال أخر مثال ثاني find the local extreme
+
+241
+00:16:12,260 --> 00:16:16,140
+values إذا موجود of of of x y بسواي تربيع ناقص x
+
+242
+00:16:16,140 --> 00:16:19,820
+تربيع ماهي هذه الدالة؟ نفسي الأولى هذه معرفة وين
+
+243
+00:16:19,820 --> 00:16:22,860
+ماكان differentiable وين ماكان إذا مافيش boundary
+
+244
+00:16:22,860 --> 00:16:25,220
+points مدام differentiable وين ماكان إذا مافيش
+
+245
+00:16:25,220 --> 00:16:28,500
+boundary points الحاجة الثانية أن f x و f y دايما
+
+246
+00:16:28,500 --> 00:16:31,760
+exist إذا مافيش شيء اللي هو f x و f y does not
+
+247
+00:16:31,760 --> 00:16:35,780
+exist إذا بس البحث عند مين يا جماعةعند Fx بتساوي
+
+248
+00:16:35,780 --> 00:16:39,180
+Fy بيساوي سفر إذا ما علينا لغير نوجود Fx و Fy
+
+249
+00:16:39,180 --> 00:16:43,680
+خلّينا نوجد Fx Fx إيش بتساوي سالب اتنين X Fy إيش
+
+250
+00:16:43,680 --> 00:16:47,720
+بتساوي اللي هو اتنين Y ساوي هنا دولة بالسفر بيطلع
+
+251
+00:16:47,720 --> 00:16:52,300
+عند F اللي هي X بتساوي سفر و Y بتساوي سفر إذا
+
+252
+00:16:52,300 --> 00:16:56,340
+النقطة اللي بدنا نفحص عندها إنها اللي هي هل النقطة
+
+253
+00:16:56,340 --> 00:16:59,860
+اللي هي عبارة عن اللي هي local maximum أو local
+
+254
+00:16:59,860 --> 00:17:04,580
+minimum مين هي النقطة سفرأو 0 إذا الـ Critical
+
+255
+00:17:04,580 --> 00:17:09,320
+Point هنا is the only 00 وهي نقطة برضه الـ
+
+256
+00:17:09,320 --> 00:17:12,680
+Function عندها Differentiable أوين ما كانت إذا هذه
+
+257
+00:17:12,680 --> 00:17:15,360
+النقطة ممكن تطلع Local Maximum و Local Minimum أو
+
+258
+00:17:15,360 --> 00:17:19,360
+تطلع Critical Point تنشوف أو ممكن ما نعرفش نوجدها
+
+259
+00:17:19,360 --> 00:17:22,840
+لكن إحنا جايبين سؤال عساس إنه سهل التعامل معاه
+
+260
+00:17:22,840 --> 00:17:29,200
+تنشوف إيش اللي بقولهالان لاحظ لو
+
+261
+00:17:29,200 --> 00:17:34,860
+جينا along the X axis النقطة مين هي 00 هيا لو جيت
+
+262
+00:17:34,860 --> 00:17:38,880
+انت اخدت جواه اي جوار حوالينه اي جوار حوالينه هنا
+
+263
+00:17:38,880 --> 00:17:42,920
+هيكون الجوار هذا هياخد من ال X axis ال positive و
+
+264
+00:17:42,920 --> 00:17:46,610
+من ال X axis ال negativeمن الـ Y Axis الـ Positive
+
+265
+00:17:46,610 --> 00:17:50,150
+من الـ Y Axis إيش؟ الـ Negative خلّينا ناخد along
+
+266
+00:17:50,150 --> 00:17:54,890
+the X Axis F has the value يعني عوالين من لما X ال
+
+267
+00:17:54,890 --> 00:17:58,730
+Axis لما Y بساوي 0 هتلاقي F of X و Zero إيش بتساوي
+
+268
+00:17:58,730 --> 00:18:02,410
+ناقص X تربيع اللي هي إيش ما لها أصغر من Zero إذا
+
+269
+00:18:02,410 --> 00:18:05,890
+أي جوار بتقدر تاخدوها يعني لو جيت على ال Positive
+
+270
+00:18:05,890 --> 00:18:11,790
+X Axis على ال Negative X Axisأنا ماشي على اللي هو
+
+271
+00:18:11,790 --> 00:18:17,230
+ال X-axis هنا و جيت على المنطقة اللي هي هنا اللي
+
+272
+00:18:17,230 --> 00:18:20,990
+هي هتلاقي مش مشكلة البوز تقول أنا جيت في حياتي لأن
+
+273
+00:18:20,990 --> 00:18:25,710
+نحن نربح هيعطي نفس القيمة خد تعالي على اللي هي مين
+
+274
+00:18:25,710 --> 00:18:30,330
+ال X-axis هي X-axis خد أي جوار حواليه أي جوار هنا
+
+275
+00:18:30,330 --> 00:18:33,720
+الآنهذا الجوار لو جيته على النقاط اللي على ال X
+
+276
+00:18:33,720 --> 00:18:37,500
+-axis اللي هي بمعنى أخر Y ب0 بصير F of X و Zero
+
+277
+00:18:37,500 --> 00:18:41,480
+ليه حسب الدلة اللي عندي نقص X تربيع أكيد أصغر من
+
+278
+00:18:41,480 --> 00:18:45,580
+أياش من سفر الان على نفس الجوار اللي قبل بشوية
+
+279
+00:18:45,580 --> 00:18:50,900
+اللي أخدته ال arbitrary الجوار العشوائيخُد نقطة
+
+280
+00:18:50,900 --> 00:19:01,680
+عالميا على الـ y-axis الـ x بصفر و الـ y تربيع
+
+281
+00:19:01,680 --> 00:19:07,840
+أكبر من aأي جوار حوالين اللي هو السفر مهما صغر
+
+282
+00:19:07,840 --> 00:19:12,120
+هتلاقي اللي هي نقاط على الـ X-Axis بتعطي قيمة الـ
+
+283
+00:19:12,120 --> 00:19:15,100
+Function أصغر من سفر ونقاط على الـ Y-Axis بتعطي
+
+284
+00:19:15,100 --> 00:19:18,560
+قيمة الـ Function أشمالها أكبر من سفر يعني اللي هي
+
+285
+00:19:18,560 --> 00:19:23,050
+F .. طبعا سفر مين هذا السفر؟ دي قيمة F of 00اللي
+
+286
+00:19:23,050 --> 00:19:26,450
+هي النقطة اللي بنبحث عنها F of 00 هنلاقي Zero نقل
+
+287
+00:19:26,450 --> 00:19:30,930
+Zero ليها Zero يعني دائما F of 00 في أي جوار
+
+288
+00:19:30,930 --> 00:19:35,450
+حوالين النقطة هذه لا يمكن نلاقي جوار انها تكون
+
+289
+00:19:35,450 --> 00:19:38,270
+أكبر من كل اللي حواليها أو أصغر من كل اللي حواليها
+
+290
+00:19:38,270 --> 00:19:43,330
+بل هنلاقي نقاط هيها أصغر منهاأو نقاط أكبر منها
+
+291
+00:19:43,330 --> 00:19:53,170
+مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا
+
+292
+00:19:53,170 --> 00:19:58,950
+مثلًا مثلًا مثلًا مثلًاSint and a zero zero
+
+293
+00:19:58,950 --> 00:20:02,010
+contains points where the function is positive and
+
+294
+00:20:02,010 --> 00:20:05,630
+points where the function is negative hence the
+
+295
+00:20:05,630 --> 00:20:09,210
+function has a saddle point at the origin and low
+
+296
+00:20:09,210 --> 00:20:14,330
+or local extremum values طيب، من هذا السؤال بنقدر
+
+297
+00:20:14,330 --> 00:20:17,490
+نستنتج إنه مش معناته F of X بيساوي F of Y بيساوي
+
+298
+00:20:17,490 --> 00:20:22,370
+سفر وكانت interior point اللي هو بتضمنلنا إنه ال F
+
+299
+00:20:22,370 --> 00:20:25,700
+إلها local maximum ولا local minimum، لأممكن تكون
+
+300
+00:20:25,700 --> 00:20:29,200
+Fx بسوء و Fy بسوء سفر و مايكونش عندنا Local
+
+301
+00:20:29,200 --> 00:20:33,000
+Extreme Value يعني لا Local Maximum ولا Local
+
+302
+00:20:33,000 --> 00:20:36,100
+Minimum Local Extreme Value يعني يا Local Maximum
+
+303
+00:20:36,100 --> 00:20:40,040
+يا Local Minimum الآن Fx بسوء و Fy بسوء سفر لها
+
+304
+00:20:40,040 --> 00:20:44,180
+تضمن لنا اللي هي وجود اللي هو أن F يكون Local
+
+305
+00:20:44,180 --> 00:20:49,450
+Maximum و Local Minimumإذن يا جماعة، صعب نظل نشتغل
+
+306
+00:20:49,450 --> 00:20:54,450
+على التعريف بدنا نظريات مساعدة تحكمنا وتحكمنا
+
+307
+00:20:54,450 --> 00:20:58,530
+وتجاوبنا على السؤال التاني هل فعلا في Local
+
+308
+00:20:58,530 --> 00:21:01,870
+Maximum و Local Minimum أو Settle Point ولا لأ
+
+309
+00:21:01,870 --> 00:21:04,710
+وهذا اللي هيجاوبنا عليهحاجة اسمها الـ Second
+
+310
+00:21:04,710 --> 00:21:08,390
+Derivative Test ‏for Local Extreme Values ‏هذه
+
+311
+00:21:08,390 --> 00:21:11,750
+اللي هي الـ Second Derivative Test ‏هتجاوبنا على
+
+312
+00:21:11,750 --> 00:21:16,150
+عائلة كبيرة من الدوال ‏اللي هي هذا موضوع اختصاصنا
+
+313
+00:21:16,150 --> 00:21:19,670
+في هذا الفصل ‏اللي هو Subposed Data ‏هذا إذا
+
+314
+00:21:19,670 --> 00:21:22,830
+السؤال الأول قلنا أين يمكن إيجاد ‏الـ Local
+
+315
+00:21:22,830 --> 00:21:26,050
+Maximum و Local Minimum ‏أو اللي هو أيضا الـ
+
+316
+00:21:26,050 --> 00:21:28,990
+Cedric Points ‏قلنا وين ‏الآن هذه بدأت تجاوبنا
+
+317
+00:21:28,990 --> 00:21:32,380
+تحكمناعلى إن هذه الـ Local Maximum والـ Local
+
+318
+00:21:32,380 --> 00:21:35,280
+Minimum ولّت Set الـ Point وخلّينا نشوف إيه اللي
+
+319
+00:21:35,280 --> 00:21:38,740
+بنحكيه Suppose that f of x, y and its and second
+
+320
+00:21:38,740 --> 00:21:41,720
+partial derivatives are continuous فرضين إن اللي
+
+321
+00:21:41,720 --> 00:21:45,080
+هي ال partial derivatives موجودة واللي هي ال
+
+322
+00:21:45,080 --> 00:21:48,280
+function continuous عند هذا المنطقة اللي هي throw
+
+323
+00:21:48,280 --> 00:21:52,380
+mean a وbعند Fx و Fy بيساوي إيش؟ بيساوي سفر جداً
+
+324
+00:21:52,380 --> 00:21:56,140
+هذه بتفحص لمين يا جماعة؟ هي اختصاص نحن الآن يعني
+
+325
+00:21:56,140 --> 00:22:00,300
+احنا مش هنتوسع في الدراسة إلا بما يُطلب مننا أنه
+
+326
+00:22:00,300 --> 00:22:04,640
+عند الـ Fx بيساوي Fy بيساوي سفر الآن الـ Critical
+
+327
+00:22:04,640 --> 00:22:08,660
+Points اللي بتطلع من هذه هي أحد ثلاث أمور يا Local
+
+328
+00:22:08,660 --> 00:22:10,880
+Maximum يا Local Minimum يا Local يساد الـ Point
+
+329
+00:22:10,880 --> 00:22:16,880
+يا ما مقدرش نحكمإذاً هذه نظرية اللي بتقولنا الحكم
+
+330
+00:22:16,880 --> 00:22:20,160
+كما يالي خلّيني نتبع معناه في حاجة اسمها الـ
+
+331
+00:22:20,160 --> 00:22:24,300
+Discrement اللي هو Fx Fy نقص Fxy تربيع اللي هو هذا
+
+332
+00:22:24,300 --> 00:22:27,380
+الـ Discrement أو الـ Hessian هذا اللي هو اللي
+
+333
+00:22:27,380 --> 00:22:31,620
+بدنا من خلاله نحكم طبعاً هذا Fx X Fyy نقص Fxy
+
+334
+00:22:31,620 --> 00:22:34,780
+تربيع بساوي الـ determinant هذا يمكن الواحد هيك
+
+335
+00:22:34,780 --> 00:22:42,900
+يكون متذكر أحسن Fx Fxy Fxy Fyyهذا في هذا ناقص هذا
+
+336
+00:22:42,900 --> 00:22:46,680
+تربيع اللي هو هذا الـ Discrement نوجده بعد ما
+
+337
+00:22:46,680 --> 00:22:51,240
+نوجده عند النقطة طبعا A و B إذا كان أكبر من سفر
+
+338
+00:22:51,240 --> 00:22:55,960
+يكون عندي يا Local Maximum يا Local Minimum الآن
+
+339
+00:22:55,960 --> 00:23:00,790
+لقنا أكبر من سفر باجي بفحص FXXإذا أصغر من 0 على
+
+340
+00:23:00,790 --> 00:23:03,650
+طول بقول Localish Maximum يعني إذا سالبة Local
+
+341
+00:23:03,650 --> 00:23:07,410
+Maximum المشابهة لـ Second Derivative Test اللي هو
+
+342
+00:23:07,410 --> 00:23:10,310
+الـ Function of One Variable كانت إذا كانت تطلع
+
+343
+00:23:10,310 --> 00:23:15,070
+عندي الـ F'' أصغر من 0 بقول عنها Localish Maximum
+
+344
+00:23:16,070 --> 00:23:19,210
+الان هذا بس عنا عشان function of two variables فيه
+
+345
+00:23:19,210 --> 00:23:23,330
+عنا XX وXY وYY فعشانك الـ Discrement هذا هو اللي
+
+346
+00:23:23,330 --> 00:23:26,610
+بيحدد أكبر من سفر معناته فيه Local Maximum أو
+
+347
+00:23:26,610 --> 00:23:29,210
+Local Minimum مين بيحدد Local Maximum أو Local
+
+348
+00:23:29,210 --> 00:23:33,330
+Minimum إشارة FXX إذا كانت FXX أصغر من سفر بتكون
+
+349
+00:23:33,330 --> 00:23:37,050
+عندي Local Maximum إذا FXX أكبر من سفر بتكون عندي
+
+350
+00:23:37,050 --> 00:23:41,400
+Local Minimumالان لو لاجينا هذا الـ Discrement
+
+351
+00:23:41,400 --> 00:23:44,700
+أصغر من 0 على طول بنحكم أنها إيه شمالها settle
+
+352
+00:23:44,700 --> 00:23:48,880
+point إذا أصغر من 0 على طول settle point بساوي 0
+
+353
+00:23:48,880 --> 00:23:54,040
+بنقول ال test is inconclusive معناته اللي هو إيه
+
+354
+00:23:54,040 --> 00:23:57,040
+شماله official ال test وبنقدرش وبنبحث عن طرق أخرى
+
+355
+00:23:57,040 --> 00:24:01,300
+إحنا هنشتغل على جدنا الآن اللي هي إذا كنا نبحث عن
+
+356
+00:24:01,300 --> 00:24:05,850
+طرق أخرى بتروح لمين للتعريف اللي قبل بشويةالأن، مش
+
+357
+00:24:05,850 --> 00:24:09,890
+هنسى القلة غير بالأشياء المقدورة عليها، الان عندي
+
+358
+00:24:09,890 --> 00:24:13,430
+اللي هو هذا الـdiscriminate هو اللي بحدد، المميز
+
+359
+00:24:13,430 --> 00:24:20,290
+أو اللي بده يحدد أنها، إذا كان سفر inclusive، إذا
+
+360
+00:24:20,290 --> 00:24:23,370
+كانت أكبر من سفر، يالو كال maximum، يالو كال
+
+361
+00:24:23,370 --> 00:24:26,510
+minimum، أصغر من سفر على طول، settle point، وهذا
+
+362
+00:24:26,510 --> 00:24:30,280
+عند المطلق المذكور A وBنجي نقض هذا عمليًا Find
+
+363
+00:24:30,280 --> 00:24:34,160
+local extreme values of the function f of x,y بسوء
+
+364
+00:24:34,160 --> 00:24:37,360
+x,y نقص x تابع نقص y تابع نقص 2x نقص 2y زي أضارب
+
+365
+00:24:37,360 --> 00:24:40,880
+عيا نحلاها Differentiable وين ما كان اللي هي إذن
+
+366
+00:24:40,880 --> 00:24:43,980
+فش boundary points وفيش does not exist إذن هتنطبق
+
+367
+00:24:43,980 --> 00:24:46,920
+نظريتنا تمام التمام واحنا جايبين عشان هي إذن بس
+
+368
+00:24:46,920 --> 00:24:51,820
+هنبحث وين في fx بسوء fy بسوء 0 ومن ثم اللي هو نحكم
+
+369
+00:24:51,820 --> 00:24:55,400
+على النقاط إنها إيه شمالها إن إنها local maximum
+
+370
+00:24:55,400 --> 00:25:01,390
+وlocal minimum ولا saddle pointأو دي لـ FX عند FX
+
+371
+00:25:01,390 --> 00:25:06,750
+أجملها في اللي فوق اللي هي عبارة عن Y نقص 2X نقص 2
+
+372
+00:25:06,750 --> 00:25:15,200
+صح؟ Y نقص 2X نقص 2الان f y هي عبارة عن x نقص اتنين
+
+373
+00:25:15,200 --> 00:25:20,020
+y نقص اتنين x نقص اتنين y نقص اتنين فيه symmetry
+
+374
+00:25:20,020 --> 00:25:24,340
+term الآن خلّينا ناخد هدي بصفر و هدي بصفر let f x
+
+375
+00:25:24,340 --> 00:25:27,140
+بيساوي f y بيساوي صفر together عندى بتصير من
+
+376
+00:25:27,140 --> 00:25:31,400
+الأولى y بتساوي اتنين x زي اتنين والتانية اتنين y
+
+377
+00:25:31,400 --> 00:25:38,690
+بيساوي x minus اتنينبنساوى هذه بالصفر وهذه بالصفر
+
+378
+00:25:38,690 --> 00:25:43,010
+فبتطلع عندى اللي هي y بالشكل هذا و2y بالشكل هذا
+
+379
+00:25:43,010 --> 00:25:46,350
+لما نجيه لجهة تانية هذه 2y لجهة هذه بصير x ناقص 2
+
+380
+00:25:46,350 --> 00:25:50,210
+إلى أن نحل المعادلتين هذول الانيتين مع بعض وضربنا
+
+381
+00:25:50,210 --> 00:25:55,510
+هذه في 2 بصير 2y 4x زيادة 4 اللي هى بنترحل من بعض
+
+382
+00:25:55,510 --> 00:26:05,810
+بصير 2y ناقص 2y صفر 4x ناقص اللي هو x بتطلع 3xأربع
+
+383
+00:26:05,810 --> 00:26:09,150
+ناقص ناقص اتنين هي زائد ستة بيصير لي هي تلاتة X
+
+384
+00:26:09,150 --> 00:26:12,550
+زائد ستة بصفر إذن X بتساوي إيش؟ سالب اتنين مش
+
+385
+00:26:12,550 --> 00:26:15,870
+اتنين و لو عوضنا X بسالب اتنين بيصير سالب أربع
+
+386
+00:26:15,870 --> 00:26:18,890
+زائد اتنين بتطلع برضه سالب اتنين يعني النقطة عند X
+
+387
+00:26:18,890 --> 00:26:22,530
+بتساوي سالب اتنين و Y بتساوي سالب اتنين إذن النقطة
+
+388
+00:26:22,530 --> 00:26:25,290
+اللي ممكن يكون عندها maximum أو minimum أو saddle
+
+389
+00:26:25,290 --> 00:26:29,250
+point هي ناقص اتنين و ناقص اتنين ماشي الحال مش
+
+390
+00:26:29,250 --> 00:26:34,070
+اتنين باتنين ناقص اتنين ناقص اتنينالان نجي نفحص
+
+391
+00:26:34,070 --> 00:26:37,830
+الفحص الان هو نوجد ايه بالنسبة للـ Discrement نوجد
+
+392
+00:26:37,830 --> 00:26:47,850
+فXX هي فX فXX بصير نقص اتنين فXY فXY هي Y نفضلها
+
+393
+00:26:47,850 --> 00:26:53,780
+بالنسبة ل Xبنطلع واحد fyy بتطلع عبارة عن إيش نقص
+
+394
+00:26:53,780 --> 00:26:57,660
+اتنين برضه ماشي الحالة طبعا ممكن تفاضل هذه fxy
+
+395
+00:26:57,660 --> 00:27:01,340
+بالنسبة ل y برضه بتطلع واحد زي ما قلنا انه اللي هو
+
+396
+00:27:01,340 --> 00:27:05,660
+تنتين continuous ال first ال second و ال first
+
+397
+00:27:05,660 --> 00:27:08,440
+partial derivative is continuous و ده ال fxy بشوة
+
+398
+00:27:08,440 --> 00:27:14,620
+fyx زي ما قدنا ده قديماالان هذه هي الأدوات اللي
+
+399
+00:27:14,620 --> 00:27:17,280
+بتخلينا نحكم الان نجد الـ Discrement ايش الـ
+
+400
+00:27:17,280 --> 00:27:22,920
+Discrement؟ Fxx في Fyy ناقص Fxy تربيع هي .. هي هذه
+
+401
+00:27:22,920 --> 00:27:26,760
+ناقص اتنين في ناقص اتنين تطلع أياش أربعة ناقص Fxy
+
+402
+00:27:26,760 --> 00:27:29,640
+لكل تربيع اللي هو واحد اللي هو بسوء جديش تلاتة
+
+403
+00:27:29,640 --> 00:27:33,240
+اللي هو صار عندي أكبر من سفر إذا مدام أكبر من سفر
+
+404
+00:27:33,240 --> 00:27:37,660
+إذاعندي الان يا local maximum يا local minimum مين
+
+405
+00:27:37,660 --> 00:27:41,840
+اللي بدى يحكم ليه اللي هي fxx مدامة fxx بتساوي
+
+406
+00:27:41,840 --> 00:27:45,580
+سالب اتنين اذا وانا مغمض الـ f has local maximum
+
+407
+00:27:45,580 --> 00:27:48,800
+ات اتنين و اتنين جداش قيمة الـ local maximum في
+
+408
+00:27:48,800 --> 00:27:52,980
+هذه الحالة بنحسب f of نقص اتنين و نقص اتنين نقص
+
+409
+00:27:52,980 --> 00:27:56,660
+اتنين طبعاً f of نقص اتنين و نقص اتنين احسبها بيه
+
+410
+00:27:56,660 --> 00:28:00,040
+نقص اتنين في نقص اتنين بيه عبارة عن أربعة وهد نقص
+
+411
+00:28:00,040 --> 00:28:06,200
+أربعة وهد نقص أربعة وهد اللي هيعبارة عن زائد أربعة
+
+412
+00:28:06,200 --> 00:28:12,740
+وهذه عبارة عن زائد أربعة اجمح مع بعض اللي هي بيصير
+
+413
+00:28:12,740 --> 00:28:17,440
+عندي أربعة ناقص أربعة ناقص أربعة اللي هي هذه ناقص
+
+414
+00:28:17,440 --> 00:28:18,880
+ثمانية بتروح مع ناقص تمانية
+
+415
+00:28:24,330 --> 00:28:28,850
+القيمة الـ F عنده نقطة نقص 2 و نقص 2 وتأكد أنك
+
+416
+00:28:28,850 --> 00:28:32,450
+أيضًا ستجد ما تساوي بتساوي اللي هي الـ absolute
+
+417
+00:28:32,450 --> 00:28:37,210
+value بساوي اللي هي قيمة الـ F عندها نقطة نقص 2 و
+
+418
+00:28:37,210 --> 00:28:41,990
+نقص 2 بساوي فعلاً اللي هو الأرقام جديش بساوي 8ماشي
+
+419
+00:28:41,990 --> 00:28:46,050
+يا جماعة ناخد مثال آخر find local extreme values
+
+420
+00:28:46,050 --> 00:28:50,330
+of x,y بيسوي تلاتة y تربيع نقص اتنين y تكعيب نقص
+
+421
+00:28:50,330 --> 00:28:54,310
+تلاتة x تربيع زي ستة xy برضه هذا يعني أحلى اللي هي
+
+422
+00:28:54,310 --> 00:28:59,010
+differentiable وين مكان ومدام differentiable وين
+
+423
+00:28:59,010 --> 00:29:03,190
+مكان إذا ال boundary points فشي وشمالينها وين مكان
+
+424
+00:29:03,190 --> 00:29:08,600
+معرفةالحاجة الثانية اللي هو الـ Fx و Fy لا توجد مش
+
+425
+00:29:08,600 --> 00:29:12,000
+موجودة برضه إذا بس هنتبحث في حالة Fx بيساوي Fy
+
+426
+00:29:12,000 --> 00:29:15,020
+بيساوي 0 ونشوف فيها النقاط اللي ممكن يكون عندها
+
+427
+00:29:15,020 --> 00:29:18,680
+maximum أو minimum أو سادي ال point وبعدين بنفحصها
+
+428
+00:29:18,680 --> 00:29:22,400
+إذا الآن زي ما قلنا بتجيب Fx ومن Fy بيكونوا ساويين
+
+429
+00:29:22,400 --> 00:29:27,380
+بالزفر Fx مش هتساوي اللي هو عبارة عن Fx Fx هيناقص
+
+430
+00:29:27,380 --> 00:29:35,200
+6x زائد 6yأف واي خلّيني أجدها هي 6y نقص 6y تربيع
+
+431
+00:29:35,200 --> 00:29:40,740
+زائد 6x هي 6y نقص 6y تربيع زائد 6x بنسويها للترفين
+
+432
+00:29:40,740 --> 00:29:46,180
+بالصفر بيطلع عندي اللي هو من الأولى 6y نقص 6xبساوي
+
+433
+00:29:46,180 --> 00:29:50,800
+0 يعني Y بتساوي X ومن الثانية اللي هو بتطلع عندي
+
+434
+00:29:50,800 --> 00:29:56,160
+اللي هي X بتساوي اللي هو Y تربيع ناقص Y ماشي الحال
+
+435
+00:29:56,160 --> 00:30:02,240
+إذا لما نحلهم مع بعض بنعوض اللي هو مكان X بY بيصير
+
+436
+00:30:02,240 --> 00:30:07,710
+Y بتساوي Y تربيع ناقص Y بنحل هذه المعادلةبصير عندى
+
+437
+00:30:07,710 --> 00:30:13,790
+اللى هي y في y طبعا الان تيجي ال y هي بصير نقص
+
+438
+00:30:13,790 --> 00:30:19,470
+اتنين y بصير y تربيع نقص اتنين y بيساوي سفر يعني y
+
+439
+00:30:19,470 --> 00:30:22,770
+تربيع نقص اتنين y بيساوي سفر يعني بيصير تاخد y عم
+
+440
+00:30:22,770 --> 00:30:25,850
+نشترك بيصير y في y minus اتنين بيساوي سفر يعني يا
+
+441
+00:30:25,850 --> 00:30:29,790
+اما y بيساوي سفر يا اما y بيساوي اتنين ماشي لكن ال
+
+442
+00:30:29,790 --> 00:30:32,970
+y بتساوي x إذا انصار عندى y بتساوي
+
+443
+00:30:45,200 --> 00:30:53,350
+عشان نفحصهم لازم نجيب fxx, fxy, fy, yالان fxx هي
+
+444
+00:30:53,350 --> 00:30:56,350
+fx موجودة نفضلها بالنسبة لـ x كمان و مرة بتطلع نقص
+
+445
+00:30:56,350 --> 00:31:02,930
+ستة fxy اللي بنفضلها بتطلع برضه ستة الان fyy لما
+
+446
+00:31:02,930 --> 00:31:08,530
+نفضلها بتطلع ستة نقص اتناشر y ستة نقص اتناشر y إذن
+
+447
+00:31:08,530 --> 00:31:12,130
+هدولة بدنا نحسب هنا عند مين عند أول شي سفر سفر و
+
+448
+00:31:12,130 --> 00:31:15,290
+بعدين عند اتنين و اتنين و بعدين اللي هو نطبق نجيه
+
+449
+00:31:15,290 --> 00:31:20,180
+بالـ Discrementالان fyy عند الـ 00 اللي هي بتساوي
+
+450
+00:31:20,180 --> 00:31:25,980
+اللي هي 6 وfyy عند ال 2 وال2 عوض بيصير 6 ناقص اللي
+
+451
+00:31:25,980 --> 00:31:31,320
+هو 24 تطلع ناقص 18 إذا الأمور جاهزة يا جماعة الآن
+
+452
+00:31:31,320 --> 00:31:36,000
+هدولة أصلا قيمتهم ثابتة عند الـ 00 وغيرهم الان
+
+453
+00:31:36,000 --> 00:31:40,880
+نبدأ نوجد مين الان الـ Discretement في كل حالةالان
+
+454
+00:31:40,880 --> 00:31:44,560
+عند الـ 00 بكون الـ Discrement هذا اللي هو بيساوي
+
+455
+00:31:44,560 --> 00:31:50,680
+بنعوض Fxx فFyy هي عندي اللي هو ناقص ستة بستة بناقص
+
+456
+00:31:50,680 --> 00:31:54,900
+ستة و تلاتين Fxy لكل تربيع هي Fxy اللي هي ستة و
+
+457
+00:31:54,900 --> 00:31:57,780
+تلاتين بيطلع ناقص اتنين و سبعين أصغر من سفر إذا
+
+458
+00:31:57,780 --> 00:32:01,020
+مدامة اللي هي عند الـ 00 طالع أصغر من سفر اللي هو
+
+459
+00:32:01,020 --> 00:32:03,460
+قيمة الـ Discrement إذا و أنا مغمض بقولها دي إيش
+
+460
+00:32:03,460 --> 00:32:07,820
+مالهاسادل بوينت عند الـ 00 نجي عند مين الـ 2
+
+461
+00:32:07,820 --> 00:32:10,700
+والتنين الـ 2 والتنين بعوض عند القيم اللي اللي
+
+462
+00:32:10,700 --> 00:32:17,300
+جيتها اللي هي XX وXY هيها ناقص 6 في 18 بالناقص
+
+463
+00:32:17,300 --> 00:32:20,980
+بيطلع زائد 108التي هي fx y تربيهها طبعاً زي ما هي
+
+464
+00:32:20,980 --> 00:32:23,520
+نقص ستة و تلاتين هذه نقص هذه بيطلع تنين و سبعين
+
+465
+00:32:23,520 --> 00:32:27,100
+برضه اشماله أكبر من سفر إذن مدام أكبر من سفر يا
+
+466
+00:32:27,100 --> 00:32:30,120
+local maximum يا local minimum مين اللي بيحدد ليه؟
+
+467
+00:32:30,120 --> 00:32:34,080
+fxx مدام fxx سالبة إذن على طول بقول في عندي local
+
+468
+00:32:34,080 --> 00:32:37,460
+maximum عند النقطة اتنين و اتنين وال value لل F
+
+469
+00:32:37,460 --> 00:32:41,740
+عند هذه النقطة اللي هو F of اتنين و اتنين بعوض عن
+
+470
+00:32:41,740 --> 00:32:45,500
+اللي هو اتنين و اتنين هنا بتطلع عندي عبارة عن جديش
+
+471
+00:32:45,500 --> 00:32:49,440
+بتساوي تمانية و اتأكد من هذا الكلاموبكون عندى اللى
+
+472
+00:32:49,440 --> 00:32:51,860
+هو فعلا عندى واحدة settle point واحدة local
+
+473
+00:32:51,860 --> 00:32:57,240
+maximum وهكذا احنا أخدنا أنفلة كفاية عن مرة الآن
+
+474
+00:32:57,240 --> 00:33:02,460
+الجزء الثاني من الموضوع اللى هو احنا لحد الآن
+
+475
+00:33:02,460 --> 00:33:05,760
+ماتطرقناش على مناطق فيها boundary points الآن في
+
+476
+00:33:05,760 --> 00:33:10,920
+نظريةاللي هي بتقولنا ما هي ليه أنه لو كان عندك
+
+477
+00:33:10,920 --> 00:33:18,200
+اللي هي f of x y عندها ال f x x و ال f x y و ال f
+
+478
+00:33:18,200 --> 00:33:20,460
+كده اللي كلنا دول ال partial derivatives موجودات و
+
+479
+00:33:20,460 --> 00:33:23,620
+continuous و كمان لو كانت ال function
+
+480
+00:33:23,620 --> 00:33:28,980
+differentiable نور على نوربنضمن أن الـ Function
+
+481
+00:33:28,980 --> 00:33:33,460
+يمتلك Absolute Maximum و Absolute Minimum هذا طبعا
+
+482
+00:33:33,460 --> 00:33:36,720
+الكلام على مين؟ على Closed Bounded Region إذن
+
+483
+00:33:36,720 --> 00:33:39,480
+بنحكي الآن Absolute Maximum و Absolute Minimum و
+
+484
+00:33:39,480 --> 00:33:44,480
+Closed Bounded Region اللي هو إحنا بنضمن وجود الـ
+
+485
+00:33:44,480 --> 00:33:46,500
+Absolute Maximum و Absolute Minimum على Closed
+
+486
+00:33:46,500 --> 00:33:50,460
+Bounded Regionإذا كانت الـ F X والـ F Y والـ F Z
+
+487
+00:33:50,460 --> 00:33:53,840
+إذا كان نحكي عن الـ three variables كلها موجودة
+
+488
+00:33:53,840 --> 00:33:58,520
+وcontinuous ويكون عنده اللي هو الـ function
+
+489
+00:33:58,520 --> 00:34:01,840
+attains its maximum and its minimum يعني لازم يكون
+
+490
+00:34:01,840 --> 00:34:04,560
+فيه عنده maximum و minimum absolute maximum و
+
+491
+00:34:04,560 --> 00:34:08,840
+absolute minimum طب كيف نوجدها نوجدها عبر ما يليه
+
+492
+00:34:08,840 --> 00:34:13,350
+أول حاجة بنيجي للـ interior pointsبنشوف وين ممكن
+
+493
+00:34:13,350 --> 00:34:16,550
+نلاقي الـ maximum و الـ minimum اللي هي عارفين وين
+
+494
+00:34:16,550 --> 00:34:21,050
+عند .. يا إما عند الـ critical points F may have
+
+495
+00:34:21,050 --> 00:34:24,410
+local maximum and minimum عند الـ critical points
+
+496
+00:34:24,410 --> 00:34:26,470
+اللي هي في الـ interior points اللي عند critical
+
+497
+00:34:26,470 --> 00:34:33,400
+pointsاللي هي بنحسب في هذه الحالة الـ F عندهم، عند
+
+498
+00:34:33,400 --> 00:34:36,520
+كل النقاط اللي لاجناه، زي ما عملنا قبل شوية و
+
+499
+00:34:36,520 --> 00:34:40,360
+بنحطهم، بنخزنهم، بنيجي لمين؟ للـ Boundary الـ
+
+500
+00:34:40,360 --> 00:34:43,620
+Boundary Points of R، اللي هو حسب إيش، اللي هو ال
+
+501
+00:34:43,620 --> 00:34:47,460
+region اللي عندنا، بمشي على ال boundaries و بوجد
+
+502
+00:34:47,460 --> 00:34:50,280
+الأماكن اللي ممكن نلاقي فيها maximum أو minimum،
+
+503
+00:34:50,280 --> 00:34:54,900
+هنقول كيف نوجدهموبنلاقي النقاط هذه وبرضه بنحطها في
+
+504
+00:34:54,900 --> 00:34:58,640
+list في قائمة وبنوجد الـ F عندها إلى بعد ما
+
+505
+00:34:58,640 --> 00:35:02,720
+أوجدناها من في الحالتين الأولى يا إما من ال
+
+506
+00:35:02,720 --> 00:35:05,800
+critical points أو من ال boundary points باخد اللي
+
+507
+00:35:05,800 --> 00:35:08,460
+هو أكبر واحدة فيهم بتكون هي ال absolute فيها
+
+508
+00:35:08,460 --> 00:35:11,900
+maximum وأصغر واحدة فيهم هي ال absolute minimum
+
+509
+00:35:11,900 --> 00:35:15,620
+وإذا الموضوع صار سهل look through the list for the
+
+510
+00:35:15,620 --> 00:35:18,780
+maximum and minimum values of F these will be
+
+511
+00:35:18,780 --> 00:35:21,710
+absolute maximum and minimum values of FSince
+
+512
+00:35:21,710 --> 00:35:24,950
+absolute maximum and minimum are also local
+
+513
+00:35:24,950 --> 00:35:27,210
+maximum and minimum are absolute maximum and
+
+514
+00:35:27,210 --> 00:35:30,010
+minimum values وفقه بيرسم somewhere in the list
+
+515
+00:35:30,010 --> 00:35:33,190
+made list بواحدة اتنية وكل ما هو ال absolute
+
+516
+00:35:33,190 --> 00:35:37,070
+maximum و absolute minimum هيكون واحدة من ال local
+
+517
+00:35:37,070 --> 00:35:41,530
+maximum و local minimumالان يعني اللي بنحكي أنه لو
+
+518
+00:35:41,530 --> 00:35:45,530
+طبعا مشابه للنظرية اللي أخدناها زمان في calculus
+
+519
+00:35:45,530 --> 00:35:47,670
+الف ان if f is continuous on a closed bounded
+
+520
+00:35:47,670 --> 00:35:50,410
+interval then it attains its maximum and its
+
+521
+00:35:50,410 --> 00:35:53,670
+minimum in this interval أو نفس الطرق هنا إذا كانت
+
+522
+00:35:53,670 --> 00:35:58,190
+f x, f y, f z اللي هي exist and continuous then it
+
+523
+00:35:58,190 --> 00:36:01,170
+attains f attains its maximum and its minimum in
+
+524
+00:36:01,170 --> 00:36:04,510
+this closed region طبعا محتاجة closed region أو
+
+525
+00:36:04,510 --> 00:36:09,150
+قلنا كيف نوجدهاخلّينا نشوف هذا بواسطة مثال عملي
+
+526
+00:36:09,150 --> 00:36:13,350
+Find the absolute maximum and minimum values of f
+
+527
+00:36:13,350 --> 00:36:16,470
+of x, y بساوية الأمامي on the triangular region in
+
+528
+00:36:16,470 --> 00:36:19,910
+the first quadrant bounded by the lines x بساوية
+
+529
+00:36:19,910 --> 00:36:23,600
+0, y بساوية 0, y بساوية 9 minus xلأن في شغلتين
+
+530
+00:36:23,600 --> 00:36:27,460
+مهمات الشغل الأولى انه اللي هو الـ interior points
+
+531
+00:36:27,460 --> 00:36:30,460
+كيف نوجد اللي هو الـ absolute maximum كيف نوجد
+
+532
+00:36:30,460 --> 00:36:33,380
+اللي هو وين الـ interior points مش هنفحص وين فيه
+
+533
+00:36:33,380 --> 00:36:35,360
+absolute maximum ولا absolute minimum احنا ضمنين
+
+534
+00:36:35,360 --> 00:36:38,140
+انه على close bounded region هيكون فيه عندى
+
+535
+00:36:38,140 --> 00:36:40,520
+absolute maximum و absolute minimum بنجيب ال
+
+536
+00:36:40,520 --> 00:36:43,580
+critical points و بنجيب ال boundary points وين فيه
+
+537
+00:36:43,580 --> 00:36:46,200
+إمكانية وجود ال local maximum و ال local minimum و
+
+538
+00:36:46,200 --> 00:36:50,050
+بنعوض عنهمبنعوض بنجيب أفقلين أكبر واحدة هي
+
+539
+00:36:50,050 --> 00:36:53,050
+absolute maximum وأكبر واحدة absolute minimum إذا
+
+540
+00:36:53,050 --> 00:36:57,190
+الخطوات كميالي اللي هو بنحدد اللي هي من شغلتين
+
+541
+00:36:57,190 --> 00:36:59,530
+مهمات الشغلة الأولى اللي حدد المنطقة الشغلة
+
+542
+00:36:59,530 --> 00:37:03,150
+الثانية اللي هو بنبدأ نشتغل على ال interior هذا
+
+543
+00:37:03,150 --> 00:37:07,020
+بتحدد ال boundary وهذا بتحدد ال interiorالان X
+
+544
+00:37:07,020 --> 00:37:09,920
+بيساوي سفر و Y بيساوي سفر و Y بيساوي تسعة مانوس X
+
+545
+00:37:09,920 --> 00:37:13,420
+X بيساوي سفر اللي هو الـ Y Axis هيه Y بيساوي سفر
+
+546
+00:37:13,420 --> 00:37:18,320
+هيه و الخط Y بيساوي تسعة ناقص X خد X بتسعة بتطلع
+
+547
+00:37:18,320 --> 00:37:25,840
+اللي هي Y بسفر هيها أو خد اللي هو X بسفر بتطلع Y
+
+548
+00:37:25,840 --> 00:37:29,720
+بتسعة فبصير عندك هي هذه بترسم الخط اللي بينهم
+
+549
+00:37:29,720 --> 00:37:34,710
+بتكون هذه المنطقة اللي احنابنحكي عنها كـ Domain
+
+550
+00:37:34,710 --> 00:37:37,930
+لمين للدالة اللي عندنا يعني بدنا نودد اللي هو الـ
+
+551
+00:37:37,930 --> 00:37:40,390
+Absolute Maximum والـ Absolute Minimum للـ
+
+552
+00:37:40,390 --> 00:37:44,950
+Function على هذه المنطقة إذن الآن كيف بنوددها؟
+
+553
+00:37:44,950 --> 00:37:50,150
+طبعا الـ Fx doesn't exist مش موجودة إذن و لا Fy
+
+554
+00:37:50,150 --> 00:37:52,730
+doesn't exist موجودة إذن بس اللي هي لإنه هذا الـ
+
+555
+00:37:52,730 --> 00:37:55,660
+Function differentiable من المكانأذا بدنا نجد Fx و
+
+556
+00:37:55,660 --> 00:37:58,720
+Fy بنسوين بالصفر و بنشوف النقاط اللي بتطلع النقاط
+
+557
+00:37:58,720 --> 00:38:02,960
+اللي بتطلع في داخل هذه المنطقة أو على جانبها اللي
+
+558
+00:38:02,960 --> 00:38:06,940
+هو بنفحصها برا مالناش علاقة فيها الآن و بعدين
+
+559
+00:38:06,940 --> 00:38:09,820
+بنيجي بنشتغل على الـ boundary points و هنقولكم كيف
+
+560
+00:38:09,820 --> 00:38:13,760
+بالظبط كيف نشتغل عليهالأن نأتي أولا للـ Interior
+
+561
+00:38:13,760 --> 00:38:17,880
+Point نقسمها جسمين Interior Points و Boundary
+
+562
+00:38:17,880 --> 00:38:20,580
+Points انتشر الـ Interior Points هي الحل المرتب
+
+563
+00:38:20,580 --> 00:38:23,960
+حله زي ما بحل Interior Points ايش هي او زي ما بحل
+
+564
+00:38:23,960 --> 00:38:28,600
+الكتاب Interior Points كيف هي ندد Fx و Fy Fx فهي
+
+565
+00:38:28,600 --> 00:38:32,710
+حالة 2نقص 2x هي اتنين نقص اتنين x اف y زيها اتنين
+
+566
+00:38:32,710 --> 00:38:37,250
+نقص اتنين y ساوليها بالصفر بتطلع y بتساوي x بتساوي
+
+567
+00:38:37,250 --> 00:38:41,070
+ايه ايش y بتساوي واحد وx بتساوي واحد إذا النقطة
+
+568
+00:38:41,070 --> 00:38:43,530
+اللي بنحكي عنها interior point هي يمين واحد واحد
+
+569
+00:38:43,530 --> 00:38:47,450
+هي أول نقطة في ال listهنعمل لكيها بأحمر على أساس
+
+570
+00:38:47,450 --> 00:38:51,150
+تقولي أفف واحد واحد أيش بتساوي اللي هو اتنين زاد
+
+571
+00:38:51,150 --> 00:38:54,210
+اتنين زاد اتنين ناقص واحد ناقص واحد اللي هي بتساوي
+
+572
+00:38:54,210 --> 00:38:57,690
+جداش مجموحين ستة ناقص اتنين بيطلع أربعة، إذن هذه
+
+573
+00:38:57,690 --> 00:39:01,310
+حفظها في الذاكرة لإن هي من النقاط اللي بدنا نحكم
+
+574
+00:39:01,310 --> 00:39:04,390
+من خلالها وين ال absolute maximum و ال absolute
+
+575
+00:39:04,390 --> 00:39:07,250
+minimum حسب النظرية اللي قلت عنها إنه مدام على
+
+576
+00:39:07,250 --> 00:39:09,030
+closed bounded region و different أو different
+
+577
+00:39:09,030 --> 00:39:11,550
+شابلها دي أو اللي هو ال partial derivatives
+
+578
+00:39:11,550 --> 00:39:13,750
+continuous ده المضمون عندي اللي هو absolute
+
+579
+00:39:13,750 --> 00:39:16,730
+maximum و absolute minimumأكبر واحدة فيها الـ
+
+580
+00:39:16,730 --> 00:39:18,610
+Absolute Maximum وأصغر واحدة هي الـ Absolute
+
+581
+00:39:18,610 --> 00:39:21,750
+Minimum هي اللي عدت عشرين مرة الآن نيجي للـ
+
+582
+00:39:21,750 --> 00:39:24,750
+Boundary Points الـ Boundary Points مين هي؟ هيها
+
+583
+00:39:24,750 --> 00:39:30,430
+اللي هي إيش؟ العخطوط الـ Region نيجي بنمسك خط خط و
+
+584
+00:39:30,430 --> 00:39:35,190
+بنشتغل في الدالة على خط خطالآن نأتي أولاً على الـ
+
+585
+00:39:35,190 --> 00:39:40,130
+Segment OA دعنا نشوف كيف نشتغل الخط الأول OA الـ
+
+586
+00:39:40,130 --> 00:39:43,950
+OA ماهي معدلته؟ Y بتساوي 0 لأن Y عليه H بتساوي
+
+587
+00:39:43,950 --> 00:39:51,730
+دائماً 0 تصبح الدالة على هذا الخط F of X0 لأن Y
+
+588
+00:39:51,730 --> 00:39:56,510
+أشمالها أخدناها عبارة عن 0 تصبح F of X0 بساوية 2
+
+589
+00:39:57,110 --> 00:40:02,310
+زاد 2x – x³ لأن هذا يصبح صفر وهذا أيه صفر الان هذه
+
+590
+00:40:02,310 --> 00:40:07,470
+الدالة صارت تشمل على الخط هذا على الخط هذا الدالة
+
+591
+00:40:07,470 --> 00:40:10,850
+هذه صارلها دالة function of one variable زي تباعد
+
+592
+00:40:10,850 --> 00:40:13,910
+الزمان صارلها boundary points اللي هي هذه وهذه
+
+593
+00:40:13,910 --> 00:40:17,450
+وصار ايه شمالها interior points ال interior points
+
+594
+00:40:17,450 --> 00:40:22,360
+كيف افحصها بقف ال prime بساوي صفرباجي الآن اللي هي
+
+595
+00:40:22,360 --> 00:40:24,780
+صارت عندي هذا معرفة على زي ما قولنا على الفترة من
+
+596
+00:40:24,780 --> 00:40:27,780
+من سفر لعن تسعة ايضا صارت Zero boundary point
+
+597
+00:40:27,780 --> 00:40:31,260
+وتسعة boundary point الان نيجي ناخد اول اشي على ال
+
+598
+00:40:31,260 --> 00:40:34,900
+boundary points للدالة الجديدة اللي على الخط اللي
+
+599
+00:40:34,900 --> 00:40:38,580
+هي بيصير F ناخد عند ال X بيساوي Zero F of Zero
+
+600
+00:40:38,580 --> 00:40:42,540
+Zero بيساوي ايش اتنين وهذه ما بحطها من ضمن ال list
+
+601
+00:40:42,940 --> 00:40:46,080
+لأنه ممكن يكون عندها Maximum أو Minimum عند مين
+
+602
+00:40:46,080 --> 00:40:48,360
+الآن باخد الـ Boundary Point التاني اللي هي مين
+
+603
+00:40:48,360 --> 00:40:52,860
+أنا في هذه الحالة تسعة F of تسعة و Zero اللي هي
+
+604
+00:40:52,860 --> 00:40:55,660
+بيساوي بعوض عن F تسعة و Zero بيصير اتنين زائد
+
+605
+00:40:55,660 --> 00:40:59,080
+تمانتاش اللي هو ناقص واحد و تمانين اللي هي بتطلع
+
+606
+00:40:59,080 --> 00:41:03,560
+ياشتر 61خلّصنا من الـ Boundary Points للدالة اللي
+
+607
+00:41:03,560 --> 00:41:06,760
+عندي هذه زي دوا الزمان ضالمين الـ Interior Point
+
+608
+00:41:06,760 --> 00:41:09,560
+إذا باخد F' من الآن F' لأن الـ derivative is a
+
+609
+00:41:09,560 --> 00:41:13,300
+function of one variable F' of X بيصير F' of X إيش
+
+610
+00:41:13,300 --> 00:41:19,680
+بتساوي؟ اللي هي عبارة عن 2 ناقص 2X 2 ناقص 2X هيها
+
+611
+00:41:20,230 --> 00:41:24,550
+بسويها بالـ 0 زي ما بنعمل زمان بتطلع عندي X بتساوي
+
+612
+00:41:24,550 --> 00:41:26,810
+إيش؟ واحد إذا ان طلعت عندي X بتساوي واحد و Y
+
+613
+00:41:26,810 --> 00:41:28,790
+بتساوي سفر إذا ان النقطة اللي ممكن يكون عندها
+
+614
+00:41:28,790 --> 00:41:32,270
+Local maximum و Local minimum اللي هي واحد و سفر
+
+615
+00:41:32,270 --> 00:41:35,710
+أفوف واحد و سفر أفوف واحد و سفر بس بحسبها زي ما
+
+616
+00:41:35,710 --> 00:41:39,250
+حسبت اللي جابلها وبتصير عندي إيش بتساوي؟ بتساوي
+
+617
+00:41:39,250 --> 00:41:43,250
+تلاتة واحد و سفر يعني اتنين زائد اتنين اللي هو
+
+618
+00:41:43,250 --> 00:41:48,050
+ناقص واحد اللي هي بتساوي تلاتةهذه اللي هي النقاط
+
+619
+00:41:48,050 --> 00:41:51,830
+هذه وهذه وهذه بالإضافة للنقاط هذه ممكن يكون عندها
+
+620
+00:41:51,830 --> 00:41:55,130
+Local Maximum أو Local Minimum بحصيها اللي هم
+
+621
+00:41:55,130 --> 00:42:00,150
+بتشغل على الخط التانمين هو OB بنفس الأسلوب الخط OB
+
+622
+00:42:00,150 --> 00:42:04,090
+الـ X هتساوي سفر بيصير الـ Function H بتساوي F of
+
+623
+00:42:04,090 --> 00:42:08,590
+Zero Y بيساوي 2 زاد 2Y ناقص من Y تربيع وهذا زي ما
+
+624
+00:42:08,590 --> 00:42:12,070
+انتوا شايفين الخط فيه Boundary Points له برضه Y بـ
+
+625
+00:42:12,070 --> 00:42:19,180
+0 وA شمالها وY بتسعةلأن Y بيكون بين 0 و 9 مثلًا
+
+626
+00:42:19,180 --> 00:42:34,760
+مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا
+
+627
+00:42:34,790 --> 00:42:39,610
+F of 0,0 حسبناها قبل 2 لأن Y بـ 0 و X بـ 0,0,0 إلى
+
+628
+00:42:39,610 --> 00:42:43,730
+Y بـ 9 اللي هي F of 0,9 برضه بنفس الأسلوب نعود في
+
+629
+00:42:43,730 --> 00:42:47,290
+الدالة بتطلع عندى سالب واحد وستين ضال عليها أوجد
+
+630
+00:42:47,290 --> 00:42:52,370
+مين F prime F prime لـ Y و 0 اللي هي عبارة عن 2
+
+631
+00:42:52,370 --> 00:42:56,790
+ناقص 2Y و بنسويها بالسفر بتطلع Y بالساوي سفر واحد
+
+632
+00:42:56,790 --> 00:43:01,840
+معناته صارت عند النقطة الجديدة هي 1 0 1اللي هي Y
+
+633
+00:43:01,840 --> 00:43:06,220
+أف برايم هي Zero Y آه Zero Y آسف Zero Y هذه Zero Y
+
+634
+00:43:06,220 --> 00:43:11,800
+ماشي زي هذي Zero Y Zero و Y فاهمين لإن بيصير F of
+
+635
+00:43:11,800 --> 00:43:16,420
+Y ب Zero X ب Zero و Y ب واحد بيصير F of Zero واحد
+
+636
+00:43:16,420 --> 00:43:20,060
+برضه هتساوي تلاتة إلى أن خلصنا من الخطين هدول اللي
+
+637
+00:43:20,060 --> 00:43:22,840
+زي بعض المتشابهات نيجي للخط الجديد مين هو الخط
+
+638
+00:43:22,840 --> 00:43:27,100
+الجديد اللي هو in the segment AB اللي هو Y بتساوي
+
+639
+00:43:27,100 --> 00:43:32,940
+9 minus X هذا الخط يعني على هذا الخطY بيتساوي تسعة
+
+640
+00:43:32,940 --> 00:43:38,120
+ناقص X والنقاط الباوندر والباوندر هدولة انساهم
+
+641
+00:43:38,120 --> 00:43:42,420
+لإيش عملنا حسبناهم في الخط هذا وفي الخط هذا إذا
+
+642
+00:43:42,420 --> 00:43:45,220
+انضل بس تشتغل على مين؟ على الـ interior اللي بده
+
+643
+00:43:45,220 --> 00:43:48,660
+يحسب إن التان تان نور على نور لكن هم الحسبين في
+
+644
+00:43:48,660 --> 00:43:53,280
+الخطين الأولى نجي الآن اللي هو نجي للخط الثالث
+
+645
+00:43:53,280 --> 00:43:58,920
+ونحسب اللي هو F prime عندهأول اشي بنعوض عن Y اللي
+
+646
+00:43:58,920 --> 00:44:04,780
+هي بيصير F of اللي هي هذه X مش تسعة minus X وهذه
+
+647
+00:44:04,780 --> 00:44:08,980
+ال Y ايه ايش تسعة نقص X يعني F of X وتسعة نقص X في
+
+648
+00:44:08,980 --> 00:44:13,360
+الآخر كلها function في مين في X يعني بنحط بنعوض عن
+
+649
+00:44:13,360 --> 00:44:17,660
+كل Y تسعة نقص X هي اتنين زاد اتنين زي ما هي زائد
+
+650
+00:44:17,660 --> 00:44:21,810
+هذا طبعا بأكد أنها Xالتي هي 2 فى 9-X بعوض فى
+
+651
+00:44:21,810 --> 00:44:25,830
+الدالة الأصلية نقص X تربيع زى ما هي نقص يبدل Y 9-X
+
+652
+00:44:25,830 --> 00:44:32,070
+لكل تربيع بنفرطهم وبنحسبهم تطلع معايا نقص 61 زى
+
+653
+00:44:32,070 --> 00:44:36,770
+18X نقص 2X تربيع مش مصدقين ففرطوهم لحالكمالان هذه
+
+654
+00:44:36,770 --> 00:44:40,390
+الآن الدالة الان سيبك من السفر و التسعة و التسعة و
+
+655
+00:44:40,390 --> 00:44:43,530
+السفر لإن هما are computed above إذا بس بدنا نوجد
+
+656
+00:44:43,530 --> 00:44:49,030
+مين احنا F prime إلها ماشي F prime إنها شوف ال F
+
+657
+00:44:49,030 --> 00:44:52,910
+prime لل X والتسعة minus X فاضل بالنسبة ل X لإنه
+
+658
+00:44:52,910 --> 00:44:57,010
+صارت function في X هذه سفر بيصير تمانتعش ناقص
+
+659
+00:44:57,010 --> 00:45:00,990
+أربعة X الان سويها بالسفر زي زي ما بنعرف بتطلع
+
+660
+00:45:00,990 --> 00:45:04,950
+عندي X بتساوي إيش تسعة على اتنين أو أربعة و نصلو
+
+661
+00:45:04,950 --> 00:45:08,510
+اتيت عوضت عن اللي هو العلاقة بينهم على الخط اللي
+
+662
+00:45:08,510 --> 00:45:12,610
+هو Y بتساوي 9 نقص X X ب 4.5 إذا الـ Y إيش هتصير 9
+
+663
+00:45:12,610 --> 00:45:18,210
+نقص 4.5 برضه هتطلع 4.5 إذا على الخط هذا XY هيطلع
+
+664
+00:45:18,210 --> 00:45:23,050
+عن اللي هو 4.5 و4.5بنحسب F عند 4.5 و 4.5 عند 9 على
+
+665
+00:45:23,050 --> 00:45:27,930
+2 و 9 على 2 عوّض في الدالة حتى تلاقي 2 زي 9 زي 9
+
+666
+00:45:27,930 --> 00:45:34,010
+ناقص 20.4 و 20.4 وبتحسبين مع بعض تلاقي ناقص 20.5
+
+667
+00:45:34,010 --> 00:45:38,070
+بتطلع ال function نقيم التناقص 20.5 الان summary
+
+668
+00:45:38,070 --> 00:45:41,410
+we list the candidates اللي وجدناها اللي هي مين هي
+
+669
+00:45:41,760 --> 00:45:45,800
+اللي هي قيمة الـ Function أول واحدة أربعة والتانية
+
+670
+00:45:45,800 --> 00:45:48,440
+اتنين تانية ناقص واحد ستين واحدة تلاتة وناقص عشرين
+
+671
+00:45:48,440 --> 00:45:52,840
+point خمسة أكبر واحدة فين عندها Absolute Maximum
+
+672
+00:45:52,840 --> 00:45:56,180
+وأصغر واحدة فين عندها Absolute Minimum إذن هذا
+
+673
+00:45:56,180 --> 00:45:58,940
+Absolute Minimum وهذا Absolute Maximum إذن
+
+674
+00:45:58,940 --> 00:46:03,060
+Absolute Minimum هو 4 وهذه حدثت مدى عنده النقطة
+
+675
+00:46:03,060 --> 00:46:06,660
+الأولى واحد واحد وهي طبعا في داخل المنطقة Absolute
+
+676
+00:46:06,660 --> 00:46:10,970
+Minimum هوأف أسيمز أزير وتسعة أن تسعة وزيرو اللي
+
+677
+00:46:10,970 --> 00:46:14,830
+حدثت عند نقطتنا دي اللي قيمته ايش ناقص واحد وستين
+
+678
+00:46:14,830 --> 00:46:17,510
+ممكن و هيك احنا اعرفنا كيف نوجد ال absolute
+
+679
+00:46:17,510 --> 00:46:19,930
+maximum و ال absolute minimum على close bounded
+
+680
+00:46:19,930 --> 00:46:24,430
+اللي هي region نيجي الآن نعمل summary إذا عندي
+
+681
+00:46:24,430 --> 00:46:29,210
+الموضوع الآن ال extreme values can occur only atيا
+
+682
+00:46:29,210 --> 00:46:31,370
+الـ Boundary Points يا إما الـ Critical Points
+
+683
+00:46:31,370 --> 00:46:34,090
+والـ Critical Points قلنا يا Interior طبعاً الـ
+
+684
+00:46:34,090 --> 00:46:37,610
+Interior بتكون Fx أو Fy بسوا سفر أو Fx أو Fy لا
+
+685
+00:46:37,610 --> 00:46:42,930
+يوجد إلا أن إذا كان الـ First وSecond ورشة الـ
+
+686
+00:46:42,930 --> 00:46:47,250
+Derivatives موجودة ومستمرة وبدنا نفحص هل في Local
+
+687
+00:46:47,250 --> 00:46:49,510
+Maximum و Local Minimum قلنا بالـ Second
+
+688
+00:46:49,510 --> 00:46:53,250
+Derivative Test اللي حكينا عنه اللي هو سابقاً وهيك
+
+689
+00:46:53,250 --> 00:46:56,090
+بيكون إحنا خلصنا المحاضرة وبرك الله فيكم والسلام
+
+690
+00:46:56,090 --> 00:46:58,490
+عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..d730c87ed898e5c0c55b533d84db85d0ae11844b
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2011, "start": 0.25, "end": 20.11, "text": "بسم الله الرحمن الرحيم السلام عليكم ورحمة الله وبركاته هذه هي المحاضرة رقم تسعة عشر مساقطة طادل وتكامل الـGene وهي المحاضرة السابعة والأخيرة في شبطر أربعة عشر وهي أربعة عشر سبعة وهي تحت عنوان Extreme Values and Subtle Values", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 29538, 39896, 9673, 5016, 46958, 25720, 12602, 4587, 2304, 6055, 3794, 27884, 6225, 46309, 47524, 995, 47432, 3660, 23032, 18513, 1211, 34683, 4117, 10943, 1211, 2423, 39184, 38, 1450, 37037, 1829, 9673, 5016, 46958, 25720, 21136, 16758, 27884, 16070, 10721, 9778, 48923, 8978, 13412, 3555, 9566, 2288, 5551, 25513, 27884, 6225, 46309, 37037, 1829, 5551, 25513, 27884, 6225, 46309, 8608, 3555, 27884, 37037, 1829, 6055, 33753, 18871, 2407, 7649, 39525, 7188, 1247, 293, 8511, 10972, 7188, 1247], "avg_logprob": -0.21813119402026185, 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"probability": 0.987060546875}, {"start": 6.01, "end": 6.41, "word": " رقم", "probability": 0.9527994791666666}, {"start": 6.41, "end": 6.83, "word": " تسعة", "probability": 0.7286783854166666}, {"start": 6.83, "end": 7.29, "word": " عشر", "probability": 0.903076171875}, {"start": 7.29, "end": 8.35, "word": " مساقطة", "probability": 0.72998046875}, {"start": 8.35, "end": 8.67, "word": " طادل", "probability": 0.5328776041666666}, {"start": 8.67, "end": 9.25, "word": " وتكامل", "probability": 0.884521484375}, {"start": 9.25, "end": 9.89, "word": " الـGene", "probability": 0.57733154296875}, {"start": 9.89, "end": 10.55, "word": " وهي", "probability": 0.864990234375}, {"start": 10.55, "end": 11.03, "word": " المحاضرة", "probability": 0.995849609375}, {"start": 11.03, "end": 11.53, "word": " السابعة", "probability": 0.92626953125}, {"start": 11.53, "end": 12.13, "word": " والأخيرة", "probability": 0.810791015625}, {"start": 12.13, "end": 12.31, "word": " في", "probability": 0.9560546875}, {"start": 12.31, "end": 12.83, "word": " شبطر", "probability": 0.617919921875}, {"start": 12.83, "end": 13.27, "word": " أربعة", "probability": 0.8111979166666666}, {"start": 13.27, "end": 13.71, "word": " عشر", "probability": 0.8544921875}, {"start": 13.71, "end": 15.61, "word": " وهي", "probability": 0.698486328125}, {"start": 15.61, "end": 16.05, "word": " أربعة", "probability": 0.8108723958333334}, {"start": 16.05, "end": 16.31, "word": " عشر", "probability": 0.878173828125}, {"start": 16.31, "end": 16.63, "word": " سبعة", "probability": 0.8665364583333334}, {"start": 16.63, "end": 16.93, "word": " وهي", "probability": 0.73876953125}, {"start": 16.93, "end": 17.21, "word": " تحت", "probability": 0.99609375}, {"start": 17.21, "end": 17.75, "word": " عنوان", "probability": 0.9856770833333334}, {"start": 17.75, "end": 18.33, "word": " Extreme", "probability": 0.81884765625}, {"start": 18.33, "end": 18.93, "word": " Values", "probability": 0.75}, {"start": 18.93, "end": 19.19, "word": " and", "probability": 0.86181640625}, {"start": 19.19, "end": 19.63, "word": " Subtle", "probability": 0.517822265625}, {"start": 19.63, "end": 20.11, "word": " Values", "probability": 0.7327880859375}], "temperature": 1.0}, {"id": 2, "seek": 4423, "start": 21.15, "end": 44.23, "text": "طبعا الموضوع ليس جديد فيه روحه احنا اتحدثنا في calculus الف وكمان حتى في الفنوية عن اللي هي كيف نوجد الـ Local maximum و Local minimum و الـ Infliction points و كذلك اللي هو ال test تبعاتها و كذلك على Absolute maximum و Absolute minimum", "tokens": [9566, 3555, 3615, 995, 9673, 2407, 11242, 45367, 32239, 3794, 10874, 16254, 3215, 8978, 3224, 12602, 2407, 5016, 3224, 1975, 5016, 8315, 1975, 2655, 24401, 12984, 8315, 8978, 33400, 27188, 4032, 24793, 7649, 11331, 49975, 8978, 27188, 1863, 2407, 10632, 18871, 13672, 1829, 39896, 9122, 33911, 8717, 29245, 3215, 2423, 39184, 22755, 6674, 4032, 22755, 7285, 4032, 2423, 39184, 682, 3423, 4105, 2793, 4032, 9122, 8848, 23275, 13672, 1829, 31439, 2423, 1500, 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"probability": 0.7022705078125}, {"start": 54.07, "end": 54.51, "word": " الأشياء", "probability": 0.8466796875}, {"start": 54.51, "end": 55.07, "word": " سنجاوبها", "probability": 0.7478434244791666}, {"start": 55.07, "end": 55.15, "word": " على", "probability": 0.69677734375}, {"start": 55.15, "end": 55.63, "word": " سؤالين", "probability": 0.9820556640625}, {"start": 55.63, "end": 56.55, "word": " السؤال", "probability": 0.813720703125}, {"start": 56.55, "end": 56.97, "word": " الأول", "probability": 0.98681640625}, {"start": 56.97, "end": 57.43, "word": " أين", "probability": 0.79931640625}, {"start": 57.43, "end": 58.31, "word": " يمكن", "probability": 0.984619140625}, {"start": 58.31, "end": 59.01, "word": " إيجاد", "probability": 0.9390869140625}, {"start": 59.01, "end": 59.91, "word": " الـ", "probability": 0.6171875}, {"start": 59.91, "end": 62.05, "word": " local", "probability": 0.461181640625}, {"start": 62.05, "end": 62.73, "word": " maximum", "probability": 0.91552734375}, {"start": 62.73, "end": 63.09, "word": " أو", "probability": 0.9404296875}, {"start": 63.09, "end": 63.29, "word": " الـ", "probability": 0.85986328125}, {"start": 63.29, "end": 63.51, "word": " local", "probability": 0.8212890625}, {"start": 63.51, "end": 64.09, "word": " minimum", "probability": 0.9755859375}], "temperature": 1.0}, {"id": 4, "seek": 8118, "start": 64.56, "end": 81.18, "text": "أو اللي هي الان مش inflection point حاجة بنسميها settle points أو نقطة سرج الحصان السؤال التاني أنه هل كيف بدنا نحكم على أن هذه فعلا في عندها local maximum و local minimum أو نقطة settle points أو لا", "tokens": [10721, 2407, 13672, 1829, 39896, 2423, 7649, 37893, 1536, 5450, 935, 11331, 26108, 3660, 44945, 38251, 1829, 11296, 11852, 2793, 34051, 8717, 47432, 3660, 8608, 47341, 21542, 9381, 7649, 21136, 33604, 6027, 16712, 7649, 1829, 14739, 3224, 8032, 1211, 9122, 33911, 47525, 8315, 8717, 5016, 24793, 15844, 14739, 29538, 6156, 3615, 15040, 8978, 43242, 11296, 2654, 6674, 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74.6, "end": 74.9, "word": " على", "probability": 0.81201171875}, {"start": 74.9, "end": 75.14, "word": " أن", "probability": 0.4892578125}, {"start": 75.14, "end": 75.5, "word": " هذه", "probability": 0.3662109375}, {"start": 75.5, "end": 75.94, "word": " فعلا", "probability": 0.8525390625}, {"start": 75.94, "end": 76.18, "word": " في", "probability": 0.479248046875}, {"start": 76.18, "end": 76.62, "word": " عندها", "probability": 0.99560546875}, {"start": 76.62, "end": 76.9, "word": " local", "probability": 0.65966796875}, {"start": 76.9, "end": 77.44, "word": " maximum", "probability": 0.9541015625}, {"start": 77.44, "end": 77.62, "word": " و", "probability": 0.5244140625}, {"start": 77.62, "end": 77.82, "word": " local", "probability": 0.52197265625}, {"start": 77.82, "end": 78.3, "word": " minimum", "probability": 0.98046875}, {"start": 78.3, "end": 79.08, "word": " أو", "probability": 0.93701171875}, {"start": 79.08, "end": 79.88, "word": " نقطة", "probability": 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"end": 84.74, "word": " مشابهة", "probability": 0.929931640625}, {"start": 84.74, "end": 85.3, "word": " للأجوبة", "probability": 0.90078125}, {"start": 85.3, "end": 85.84, "word": " السابقة", "probability": 0.98876953125}, {"start": 85.84, "end": 86.04, "word": " في", "probability": 0.7900390625}, {"start": 86.04, "end": 86.52, "word": " تفاضل", "probability": 0.813232421875}, {"start": 86.52, "end": 86.78, "word": " ألف", "probability": 0.723876953125}, {"start": 86.78, "end": 87.04, "word": " كنا", "probability": 0.776123046875}, {"start": 87.04, "end": 87.56, "word": " نجاوب", "probability": 0.839599609375}, {"start": 87.56, "end": 87.66, "word": " في", "probability": 0.8232421875}, {"start": 87.66, "end": 88.08, "word": " تفاضل", "probability": 0.978759765625}, {"start": 88.08, "end": 88.46, "word": " ألف", "probability": 0.92724609375}, {"start": 88.46, "end": 89.24, "word": " نقول", "probability": 0.79541015625}, {"start": 89.24, "end": 90.52, "word": " القيم", "probability": 0.821533203125}, {"start": 90.52, "end": 90.96, "word": " العظمى", "probability": 0.892578125}, {"start": 90.96, "end": 91.66, "word": " والصغرى", "probability": 0.85029296875}, {"start": 91.66, "end": 92.58, "word": " وكذلك", "probability": 0.907470703125}, {"start": 92.58, "end": 92.72, "word": " الـ", "probability": 0.727783203125}, {"start": 92.72, "end": 93.2, "word": " inflection", "probability": 0.626953125}, {"start": 93.2, "end": 93.56, "word": " points", "probability": 0.88037109375}, {"start": 93.56, "end": 94.0, "word": " ممكن", "probability": 0.873291015625}, {"start": 94.0, "end": 94.26, "word": " أن", "probability": 0.78466796875}, {"start": 94.26, "end": 94.6, "word": " توجد", "probability": 0.7486165364583334}, {"start": 94.6, "end": 94.78, "word": " في", "probability": 0.96044921875}, {"start": 94.78, "end": 95.04, "word": " أحد", "probability": 0.952392578125}, {"start": 95.04, "end": 95.62, "word": " المناطق", "probability": 0.9912109375}, {"start": 95.62, "end": 96.22, "word": " التالية", "probability": 0.9928385416666666}, {"start": 96.22, "end": 96.98, "word": " إما", "probability": 0.762939453125}, {"start": 96.98, "end": 97.38, "word": " في", "probability": 0.7275390625}, {"start": 97.38, "end": 97.5, "word": " الـ", "probability": 0.90380859375}, {"start": 97.5, "end": 97.86, "word": " interior", "probability": 0.69921875}, {"start": 97.86, "end": 98.4, "word": " points", "probability": 0.92333984375}, {"start": 98.4, "end": 98.58, "word": " اللي", "probability": 0.829833984375}, {"start": 98.58, "end": 98.78, "word": " هي", "probability": 0.5458984375}, {"start": 98.78, "end": 98.94, "word": " في", "probability": 0.473388671875}, {"start": 98.94, "end": 99.7, "word": " عند", "probability": 0.8759765625}, {"start": 99.7, "end": 99.9, "word": " الـ", "probability": 0.6458740234375}, {"start": 99.9, "end": 100.16, "word": " derivative", "probability": 0.82275390625}, {"start": 100.16, "end": 100.68, "word": " بتساوي", "probability": 0.703216552734375}, {"start": 100.68, "end": 101.44, "word": " سفر", "probability": 0.83740234375}, {"start": 101.44, "end": 101.62, "word": " أو", "probability": 0.94873046875}, {"start": 101.62, "end": 101.76, "word": " الـ", "probability": 0.7841796875}, {"start": 101.76, "end": 102.04, "word": " derivative", "probability": 0.76025390625}, {"start": 102.04, "end": 102.32, "word": " does", "probability": 0.5966796875}, {"start": 102.32, "end": 102.94, "word": " not", "probability": 0.966796875}, {"start": 102.94, "end": 103.44, "word": " exist", "probability": 0.96484375}, {"start": 103.44, "end": 103.66, "word": " هذه", "probability": 0.383056640625}, {"start": 103.66, "end": 104.08, "word": " المنطقة", "probability": 0.96484375}, {"start": 104.08, "end": 104.34, "word": " تانية", "probability": 0.6973470052083334}, {"start": 104.34, "end": 104.86, "word": " والمنطقة", "probability": 0.8792724609375}, {"start": 104.86, "end": 105.52, "word": " الثالثة", "probability": 0.9150390625}, {"start": 105.52, "end": 106.02, "word": " هي", "probability": 0.92333984375}, {"start": 106.02, "end": 106.5, "word": " الـ", "probability": 0.909423828125}, {"start": 106.5, "end": 106.88, "word": " boundary", "probability": 0.78662109375}, {"start": 106.88, "end": 107.62, "word": " points", "probability": 0.93212890625}], "temperature": 1.0}, {"id": 6, "seek": 12671, "start": 107.93, "end": 126.71, "text": "وكنا ناخد الـ First derivative test والـ Second derivative test اللي هو الحكم على إن فيه لوكال maximum أو لوكال minimum أو اللي هو inflection point هذا الحديث تقريبا مشابه جدا خلينا نبدأ اللي هو في الموضوعنا derivative test for local extreme values", "tokens": [2407, 4117, 8315, 8717, 47283, 3215, 2423, 39184, 2386, 13760, 1500, 16070, 39184, 5736, 13760, 1500, 13672, 1829, 31439, 21542, 24793, 15844, 36145, 8978, 3224, 5296, 2407, 4117, 6027, 6674, 34051, 5296, 2407, 4117, 6027, 7285, 34051, 13672, 1829, 31439, 1536, 5450, 935, 23758, 21542, 16254, 12984, 6055, 4587, 16572, 3555, 995, 37893, 16758, 3224, 10874, 28259, 16490, 1211, 9957, 995, 8717, 44510, 10721, 13672, 1829, 31439, 8978, 9673, 2407, 11242, 45367, 8315, 13760, 1500, 337, 2654, 8084, 4190], "avg_logprob": -0.28300781138241293, "compression_ratio": 1.6175115207373272, "no_speech_prob": 1.2516975402832031e-06, "words": [{"start": 107.93, "end": 108.39, "word": "وكنا", "probability": 0.638427734375}, {"start": 108.39, "end": 108.71, "word": " ناخد", "probability": 0.904296875}, {"start": 108.71, "end": 108.81, "word": " الـ", "probability": 0.786376953125}, {"start": 108.81, "end": 109.09, "word": " First", "probability": 0.6591796875}, {"start": 109.09, "end": 109.47, "word": " derivative", "probability": 0.7802734375}, {"start": 109.47, "end": 109.87, "word": " test", "probability": 0.880859375}, {"start": 109.87, "end": 110.11, "word": " والـ", "probability": 0.32794189453125}, {"start": 110.11, "end": 110.33, "word": " Second", "probability": 0.7197265625}, {"start": 110.33, "end": 110.75, "word": " derivative", "probability": 0.90185546875}, {"start": 110.75, "end": 111.31, "word": " test", "probability": 0.92333984375}, {"start": 111.31, "end": 111.79, "word": " اللي", "probability": 0.57171630859375}, {"start": 111.79, "end": 112.43, "word": " هو", "probability": 0.9130859375}, {"start": 112.43, "end": 113.31, "word": " الحكم", "probability": 0.918212890625}, {"start": 113.31, "end": 113.55, "word": " على", "probability": 0.74169921875}, {"start": 113.55, "end": 113.75, "word": " إن", "probability": 0.5517578125}, {"start": 113.75, "end": 114.01, "word": " فيه", "probability": 0.497314453125}, {"start": 114.01, "end": 114.55, "word": " لوكال", "probability": 0.6600341796875}, {"start": 114.55, "end": 114.89, "word": " maximum", "probability": 0.329833984375}, {"start": 114.89, "end": 115.23, "word": " أو", "probability": 0.953125}, {"start": 115.23, "end": 115.55, "word": " لوكال", "probability": 0.8443603515625}, {"start": 115.55, "end": 115.91, "word": " minimum", "probability": 0.95166015625}, {"start": 115.91, "end": 116.67, "word": " أو", "probability": 0.9384765625}, {"start": 116.67, "end": 116.97, "word": " اللي", "probability": 0.767822265625}, {"start": 116.97, "end": 117.33, "word": " هو", "probability": 0.98291015625}, {"start": 117.33, "end": 118.05, "word": " inflection", "probability": 0.771484375}, {"start": 118.05, "end": 118.41, "word": " point", "probability": 0.98388671875}, {"start": 118.41, "end": 118.63, "word": " هذا", "probability": 0.328857421875}, {"start": 118.63, "end": 119.09, "word": " الحديث", "probability": 0.98583984375}, {"start": 119.09, "end": 119.55, "word": " تقريبا", "probability": 0.8697265625}, {"start": 119.55, "end": 119.97, "word": " مشابه", "probability": 0.9557291666666666}, {"start": 119.97, "end": 120.41, "word": " جدا", "probability": 0.994140625}, {"start": 120.41, "end": 121.29, "word": " خلينا", "probability": 0.6434326171875}, {"start": 121.29, "end": 121.75, "word": " نبدأ", "probability": 0.9908854166666666}, {"start": 121.75, "end": 121.97, "word": " اللي", "probability": 0.96240234375}, {"start": 121.97, "end": 122.31, "word": " هو", "probability": 0.98779296875}, {"start": 122.31, "end": 122.93, "word": " في", "probability": 0.93310546875}, {"start": 122.93, "end": 123.79, "word": " الموضوعنا", "probability": 0.7851806640625}, {"start": 123.79, "end": 124.13, "word": " derivative", "probability": 0.5556640625}, {"start": 124.13, "end": 124.71, "word": " test", "probability": 0.8935546875}, {"start": 124.71, "end": 125.01, "word": " for", "probability": 0.93505859375}, {"start": 125.01, "end": 125.45, "word": " local", "probability": 0.84912109375}, {"start": 125.45, "end": 126.11, "word": " extreme", "probability": 0.927734375}, {"start": 126.11, "end": 126.71, "word": " values", "probability": 0.9306640625}], "temperature": 1.0}, {"id": 7, "seek": 15307, "start": 127.17, "end": 153.07, "text": "نشوف أول حاجة ماعنات الـ maximum و الـ minimum أو الـ local maximum و الـ local minimum مشابهة let f of x,y be defined on a region R containing the point A و B نشوف الآن ما هي معنات local maximum عندنا نقطة A و B هي نقطة A و B ما هي معناة local maximum local maximum هي نقطة عظمة محلية إننا نجد جوار حوالها", "tokens": [1863, 8592, 38688, 5551, 12610, 11331, 26108, 3660, 19446, 3615, 1863, 9307, 2423, 39184, 6674, 4032, 2423, 39184, 7285, 34051, 2423, 39184, 2654, 6674, 4032, 2423, 39184, 2654, 7285, 37893, 16758, 3224, 3660, 718, 283, 295, 2031, 11, 88, 312, 7642, 322, 257, 4458, 497, 19273, 264, 935, 316, 4032, 363, 8717, 8592, 38688, 6024, 48506, 19446, 39896, 20449, 8315, 2655, 2654, 6674, 43242, 8315, 8717, 47432, 3660, 316, 4032, 363, 39896, 8717, 47432, 3660, 316, 4032, 363, 19446, 39896, 20449, 8315, 3660, 2654, 6674, 2654, 6674, 39896, 8717, 47432, 3660, 6225, 19913, 46007, 3714, 5016, 1211, 10632, 36145, 8315, 8717, 7435, 3215, 10874, 2407, 9640, 11331, 2407, 6027, 11296], "avg_logprob": -0.346283778413996, "compression_ratio": 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142.57, "word": " الآن", "probability": 0.6256103515625}, {"start": 142.57, "end": 142.79, "word": " ما", "probability": 0.61083984375}, {"start": 142.79, "end": 142.99, "word": " هي", "probability": 0.263916015625}, {"start": 142.99, "end": 143.25, "word": " معنات", "probability": 0.7229817708333334}, {"start": 143.25, "end": 143.49, "word": " local", "probability": 0.68994140625}, {"start": 143.49, "end": 143.99, "word": " maximum", "probability": 0.96435546875}, {"start": 143.99, "end": 144.35, "word": " عندنا", "probability": 0.7021484375}, {"start": 144.35, "end": 144.67, "word": " نقطة", "probability": 0.9285481770833334}, {"start": 144.67, "end": 144.83, "word": " A", "probability": 0.90380859375}, {"start": 144.83, "end": 144.93, "word": " و", "probability": 0.98486328125}, {"start": 144.93, "end": 145.11, "word": " B", "probability": 0.80712890625}, {"start": 145.11, "end": 145.35, "word": " هي", "probability": 0.381591796875}, {"start": 145.35, "end": 145.77, "word": " نقطة", "probability": 0.8857421875}, {"start": 145.77, "end": 145.91, "word": " A", "probability": 0.83056640625}, {"start": 145.91, "end": 145.99, "word": " و", "probability": 0.98974609375}, {"start": 145.99, "end": 146.19, "word": " B", "probability": 0.8203125}, {"start": 146.19, "end": 146.43, "word": " ما", "probability": 0.340087890625}, {"start": 146.43, "end": 146.45, "word": " هي", "probability": 0.73974609375}, {"start": 146.45, "end": 146.71, "word": " معناة", "probability": 0.638671875}, {"start": 146.71, "end": 146.95, "word": " local", "probability": 0.830078125}, {"start": 146.95, "end": 147.55, "word": " maximum", "probability": 0.96142578125}, {"start": 147.55, "end": 148.23, "word": " local", "probability": 0.20947265625}, {"start": 148.23, "end": 148.75, "word": " maximum", "probability": 0.96630859375}, {"start": 148.75, "end": 149.01, "word": " هي", "probability": 0.73193359375}, {"start": 149.01, "end": 149.33, "word": " نقطة", "probability": 0.98876953125}, {"start": 149.33, "end": 149.59, "word": " عظمة", "probability": 0.83935546875}, {"start": 149.59, "end": 150.23, "word": " محلية", "probability": 0.9599609375}, {"start": 150.23, "end": 151.23, "word": " إننا", "probability": 0.3291015625}, {"start": 151.23, "end": 151.71, "word": " نجد", "probability": 0.9195963541666666}, {"start": 151.71, "end": 152.27, "word": " جوار", "probability": 0.93505859375}, {"start": 152.27, "end": 153.07, "word": " حوالها", "probability": 0.7962646484375}], "temperature": 1.0}, {"id": 8, "seek": 16964, "start": 153.32, "end": 169.64, "text": "هناك جوار حواليها تكون هي أكبر واحدة يعني قيمة F of A و F of B أكبر واحدة أكبر من F of XY لكل XY في هذا الجوار يعني أكبر واحدة في جيرانها هذا نسميها Local Maximum", "tokens": [3224, 8315, 4117, 10874, 2407, 9640, 11331, 2407, 6027, 1829, 11296, 6055, 30544, 39896, 5551, 4117, 26890, 36764, 24401, 3660, 37495, 22653, 12174, 32640, 3660, 479, 295, 316, 4032, 479, 295, 363, 5551, 4117, 26890, 36764, 24401, 3660, 5551, 4117, 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"probability": 0.9031575520833334}, {"start": 342.46, "end": 342.6, "word": " اللي", "probability": 0.97998046875}, {"start": 342.6, "end": 343.24, "word": " بتحقق", "probability": 0.9248046875}, {"start": 343.24, "end": 343.6, "word": " هذه", "probability": 0.73779296875}, {"start": 343.6, "end": 344.58, "word": " ممكن", "probability": 0.66650390625}, {"start": 344.58, "end": 344.96, "word": " يكون", "probability": 0.95751953125}, {"start": 344.96, "end": 345.24, "word": " فيها", "probability": 0.94921875}, {"start": 345.24, "end": 345.52, "word": " Local", "probability": 0.919921875}, {"start": 345.52, "end": 346.18, "word": " Maximum", "probability": 0.92431640625}, {"start": 346.18, "end": 346.24, "word": " و", "probability": 0.90625}, {"start": 346.24, "end": 346.5, "word": " Local", "probability": 0.88232421875}, {"start": 346.5, "end": 347.0, "word": " Minimum", "probability": 0.9923502604166666}], "temperature": 1.0}, {"id": 17, "seek": 36114, "start": 347.28, "end": 361.14, "text": "أما النقاط اللي بيكونش هذه سفر وتكونش هذه سفر انساها لأن لو كان Local Maximum و Local Minimum لازم يطلع عن سفر إذا من هذه النظرية بنقدر نقول أنه احنا ممكن نوجد", "tokens": [10721, 15042, 28239, 4587, 41193, 13672, 1829, 4724, 1829, 30544, 8592, 29538, 8608, 5172, 2288, 34683, 30544, 8592, 29538, 8608, 5172, 2288, 16472, 3794, 995, 11296, 5296, 33456, 45164, 25961, 22755, 29076, 449, 4032, 22755, 2829, 332, 449, 5296, 31377, 2304, 7251, 9566, 1211, 3615, 18871, 8608, 5172, 2288, 11933, 15730, 9154, 29538, 28239, 19913, 2288, 10632, 44945, 28543, 2288, 8717, 39648, 14739, 3224, 1975, 5016, 8315, 3714, 43020, 8717, 29245, 3215], "avg_logprob": -0.2726883594303915, "compression_ratio": 1.5470588235294118, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 347.28, "end": 347.92, "word": "أما", "probability": 0.919189453125}, {"start": 347.92, "end": 348.58, "word": " النقاط", "probability": 0.87890625}, {"start": 348.58, "end": 348.7, "word": " اللي", "probability": 0.72216796875}, {"start": 348.7, "end": 349.24, "word": " بيكونش", "probability": 0.58343505859375}, {"start": 349.24, "end": 349.5, "word": " هذه", "probability": 0.77001953125}, {"start": 349.5, "end": 349.82, "word": " سفر", "probability": 0.71337890625}, {"start": 349.82, "end": 350.26, "word": " وتكونش", "probability": 0.4510904947916667}, {"start": 350.26, "end": 350.42, "word": " هذه", "probability": 0.95703125}, {"start": 350.42, "end": 350.72, "word": " سفر", "probability": 0.9889322916666666}, {"start": 350.72, "end": 351.36, "word": " انساها", "probability": 0.877685546875}, {"start": 351.36, "end": 351.9, "word": " لأن", "probability": 0.815185546875}, {"start": 351.9, "end": 352.06, "word": " لو", "probability": 0.7353515625}, {"start": 352.06, "end": 352.26, "word": " كان", "probability": 0.9697265625}, {"start": 352.26, "end": 352.42, "word": " Local", "probability": 0.4716796875}, {"start": 352.42, "end": 353.0, "word": " Maximum", "probability": 0.7039794921875}, {"start": 353.0, "end": 353.08, "word": " و", "probability": 0.5537109375}, {"start": 353.08, "end": 353.3, "word": " Local", "probability": 0.86865234375}, {"start": 353.3, "end": 353.74, "word": " Minimum", "probability": 0.982421875}, {"start": 353.74, "end": 354.32, "word": " لازم", "probability": 0.9635416666666666}, {"start": 354.32, "end": 354.66, "word": " يطلع", "probability": 0.943603515625}, {"start": 354.66, "end": 354.84, "word": " عن", "probability": 0.183837890625}, {"start": 354.84, "end": 355.36, "word": " سفر", "probability": 0.9720052083333334}, {"start": 355.36, "end": 355.82, "word": " إذا", "probability": 0.5006103515625}, {"start": 355.82, "end": 356.0, "word": " من", "probability": 0.96044921875}, {"start": 356.0, "end": 356.32, "word": " هذه", "probability": 0.95849609375}, {"start": 356.32, "end": 357.12, "word": " النظرية", "probability": 0.9642333984375}, {"start": 357.12, "end": 357.72, "word": " بنقدر", "probability": 0.8621419270833334}, {"start": 357.72, "end": 358.24, "word": " نقول", "probability": 0.984619140625}, {"start": 358.24, "end": 359.34, "word": " أنه", "probability": 0.6270751953125}, {"start": 359.34, "end": 359.96, "word": " احنا", "probability": 0.79541015625}, {"start": 359.96, "end": 360.48, "word": " ممكن", "probability": 0.98828125}, {"start": 360.48, "end": 361.14, "word": " نوجد", "probability": 0.9075520833333334}], "temperature": 1.0}, {"id": 18, "seek": 38735, "start": 362.25, "end": 387.35, "text": "ممكن نلاقي الـ Local Maximum أو الـ Local Minimum عند أحد النقاط التالية إما Fx يساوي Fy بيساوي سفر أو إن الـ Derivative does not exist لأن هذا ليس متضمن من خلال الحديث في النظرية إذن هذه النظرية توحي ما يليه أو تعطينا المعلومات التالية إنه اللي هي الـ Interior Points", "tokens": [2304, 43020, 8717, 15040, 38436, 2423, 39184, 22755, 29076, 449, 34051, 2423, 39184, 22755, 2829, 332, 449, 43242, 5551, 24401, 28239, 4587, 41193, 16712, 6027, 10632, 11933, 15042, 479, 87, 7251, 3794, 995, 45865, 479, 88, 4724, 1829, 3794, 995, 45865, 8608, 5172, 2288, 34051, 36145, 2423, 39184, 5618, 592, 1166, 775, 406, 2514, 5296, 33456, 23758, 32239, 3794, 44650, 11242, 27842, 9154, 16490, 1211, 6027, 21542, 16254, 12984, 8978, 28239, 19913, 2288, 10632, 11933, 8848, 1863, 29538, 28239, 19913, 2288, 10632, 33427, 5016, 1829, 19446, 7251, 20292, 3224, 34051, 37279, 9566, 1829, 8315, 9673, 30241, 20498, 9307, 16712, 6027, 10632, 36145, 3224, 13672, 1829, 39896, 2423, 39184, 44346, 44763], "avg_logprob": -0.2250844589224807, "compression_ratio": 1.6812749003984064, "no_speech_prob": 0.0003814697265625, "words": [{"start": 362.25, "end": 363.29, "word": "ممكن", "probability": 0.5469970703125}, {"start": 363.29, "end": 363.67, "word": " نلاقي", "probability": 0.82080078125}, {"start": 363.67, "end": 364.27, "word": " الـ", "probability": 0.7314453125}, {"start": 364.27, "end": 364.45, "word": " Local", "probability": 0.60595703125}, {"start": 364.45, "end": 365.05, "word": " Maximum", "probability": 0.697021484375}, {"start": 365.05, "end": 365.21, "word": " أو", "probability": 0.93798828125}, {"start": 365.21, "end": 365.39, "word": " الـ", "probability": 0.798828125}, {"start": 365.39, "end": 365.57, "word": " Local", "probability": 0.91162109375}, {"start": 365.57, "end": 366.19, "word": " Minimum", "probability": 0.9923502604166666}, {"start": 366.19, "end": 367.13, "word": " عند", "probability": 0.86376953125}, {"start": 367.13, "end": 367.89, "word": " أحد", "probability": 0.943115234375}, {"start": 367.89, "end": 368.53, "word": " النقاط", "probability": 0.955078125}, {"start": 368.53, "end": 369.17, "word": " التالية", "probability": 0.9895833333333334}, {"start": 369.17, "end": 369.67, "word": " إما", "probability": 0.6654052734375}, {"start": 369.67, "end": 370.45, "word": " Fx", "probability": 0.474609375}, {"start": 370.45, "end": 371.05, "word": " يساوي", "probability": 0.6573362350463867}, {"start": 371.05, "end": 371.63, "word": " Fy", "probability": 0.85400390625}, {"start": 371.63, "end": 371.99, "word": " بيساوي", "probability": 0.8419921875}, {"start": 371.99, "end": 372.39, "word": " سفر", "probability": 0.8011881510416666}, {"start": 372.39, "end": 373.23, "word": " أو", "probability": 0.89453125}, {"start": 373.23, "end": 373.75, "word": " إن", "probability": 0.6015625}, {"start": 373.75, "end": 373.97, "word": " الـ", "probability": 0.94287109375}, {"start": 373.97, "end": 374.57, "word": " Derivative", "probability": 0.84228515625}, {"start": 374.57, "end": 374.87, "word": " does", "probability": 0.369140625}, {"start": 374.87, "end": 375.21, "word": " not", "probability": 0.96142578125}, {"start": 375.21, "end": 375.83, "word": " exist", "probability": 0.9677734375}, {"start": 375.83, "end": 376.15, "word": " لأن", "probability": 0.902099609375}, {"start": 376.15, "end": 376.33, "word": " هذا", "probability": 0.409423828125}, {"start": 376.33, "end": 376.49, "word": " ليس", "probability": 0.773193359375}, {"start": 376.49, "end": 377.09, "word": " متضمن", "probability": 0.9807942708333334}, {"start": 377.09, "end": 377.35, "word": " من", "probability": 0.94775390625}, {"start": 377.35, "end": 377.79, "word": " خلال", "probability": 0.9884440104166666}, {"start": 377.79, "end": 378.51, "word": " الحديث", "probability": 0.8116048177083334}, {"start": 378.51, "end": 378.65, "word": " في", "probability": 0.85400390625}, {"start": 378.65, "end": 379.19, "word": " النظرية", "probability": 0.95361328125}, {"start": 379.19, "end": 379.93, "word": " إذن", "probability": 0.6471354166666666}, {"start": 379.93, "end": 380.21, "word": " هذه", "probability": 0.74658203125}, {"start": 380.21, "end": 380.65, "word": " النظرية", "probability": 0.9820556640625}, {"start": 380.65, "end": 381.17, "word": " توحي", "probability": 0.94140625}, {"start": 381.17, "end": 381.35, "word": " ما", "probability": 0.95556640625}, {"start": 381.35, "end": 382.05, "word": " يليه", "probability": 0.86181640625}, {"start": 382.05, "end": 382.21, "word": " أو", "probability": 0.93017578125}, {"start": 382.21, "end": 382.61, "word": " تعطينا", "probability": 0.93798828125}, {"start": 382.61, "end": 383.13, "word": " المعلومات", "probability": 0.9873046875}, {"start": 383.13, "end": 383.67, "word": " التالية", "probability": 0.9951171875}, {"start": 383.67, "end": 384.77, "word": " إنه", "probability": 0.36181640625}, {"start": 384.77, "end": 385.09, "word": " اللي", "probability": 0.790771484375}, {"start": 385.09, "end": 385.49, "word": " هي", "probability": 0.9619140625}, {"start": 385.49, "end": 386.01, "word": " الـ", "probability": 0.96923828125}, {"start": 386.01, "end": 386.57, "word": " Interior", "probability": 0.666015625}, {"start": 386.57, "end": 387.35, "word": " Points", "probability": 0.7705078125}], "temperature": 1.0}, {"id": 19, "seek": 41640, "start": 388.04, "end": 416.4, "text": "للـ Interior Points ممكن نلاقي الـ Local Maximum الـ Local Minimum عند الـ Interior Points اللي Fx بيساوي 0 و Fy بيساوي 0 و اللي Fx و Fy لا يوجد في هذه الحالة اللي هو بيكون عندي اللي هو عن الـ Interior Points اللي هي ممكن نودد الـ Local Maximum و Local Minimum عند هذه الأماكن يعني بمعنى آخر من هذه النظرية أيضا نستقل معلومة التالية وهي نهائية يمكن", "tokens": [1211, 1211, 39184, 44346, 44763, 3714, 43020, 8717, 15040, 38436, 2423, 39184, 22755, 29076, 449, 2423, 39184, 22755, 2829, 332, 449, 43242, 2423, 39184, 44346, 44763, 13672, 1829, 479, 87, 4724, 1829, 3794, 995, 45865, 1958, 4032, 479, 88, 4724, 1829, 3794, 995, 45865, 1958, 4032, 13672, 1829, 479, 87, 4032, 479, 88, 20193, 7251, 29245, 3215, 8978, 29538, 21542, 6027, 3660, 13672, 1829, 31439, 4724, 1829, 30544, 43242, 1829, 13672, 1829, 31439, 18871, 2423, 39184, 44346, 44763, 13672, 1829, 39896, 3714, 43020, 8717, 23328, 3215, 2423, 39184, 22755, 29076, 449, 4032, 22755, 2829, 332, 449, 43242, 29538, 16247, 15042, 19452, 37495, 22653, 4724, 2304, 3615, 1863, 7578, 19753, 34740, 9154, 29538, 28239, 19913, 2288, 10632, 36632, 11242, 995, 8717, 14851, 4587, 1211, 20449, 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Local", "probability": 0.8896484375}, {"start": 391.42, "end": 391.64, "word": " Minimum", "probability": 0.9736328125}, {"start": 391.64, "end": 391.88, "word": " عند", "probability": 0.94189453125}, {"start": 391.88, "end": 392.02, "word": " الـ", "probability": 0.968505859375}, {"start": 392.02, "end": 392.3, "word": " Interior", "probability": 0.83154296875}, {"start": 392.3, "end": 392.74, "word": " Points", "probability": 0.9375}, {"start": 392.74, "end": 393.52, "word": " اللي", "probability": 0.8662109375}, {"start": 393.52, "end": 394.48, "word": " Fx", "probability": 0.63818359375}, {"start": 394.48, "end": 394.86, "word": " بيساوي", "probability": 0.5821044921875}, {"start": 394.86, "end": 395.1, "word": " 0", "probability": 0.58837890625}, {"start": 395.1, "end": 395.24, "word": " و", "probability": 0.81787109375}, {"start": 395.24, "end": 395.54, "word": " Fy", "probability": 0.80859375}, {"start": 395.54, "end": 395.84, "word": " بيساوي", "probability": 0.98017578125}, {"start": 395.84, "end": 396.22, "word": " 0", "probability": 0.9697265625}, {"start": 396.22, "end": 396.76, "word": " و", "probability": 0.5556640625}, {"start": 396.76, "end": 396.94, "word": " اللي", "probability": 0.6873779296875}, {"start": 396.94, "end": 397.44, "word": " Fx", "probability": 0.915283203125}, {"start": 397.44, "end": 397.56, "word": " و", "probability": 0.93701171875}, {"start": 397.56, "end": 398.04, "word": " Fy", "probability": 0.92724609375}, {"start": 398.04, "end": 398.46, "word": " لا", "probability": 0.32666015625}, {"start": 398.46, "end": 399.78, "word": " يوجد", "probability": 0.7722981770833334}, {"start": 399.78, "end": 400.2, "word": " في", "probability": 0.7578125}, {"start": 400.2, "end": 400.44, "word": " هذه", "probability": 0.82421875}, {"start": 400.44, "end": 400.98, "word": " الحالة", "probability": 0.9923502604166666}, {"start": 400.98, "end": 401.5, "word": " اللي", "probability": 0.978759765625}, {"start": 401.5, "end": 401.62, "word": " هو", "probability": 0.80908203125}, {"start": 401.62, "end": 402.0, "word": " بيكون", "probability": 0.7220052083333334}, {"start": 402.0, "end": 402.7, "word": " عندي", "probability": 0.658203125}, {"start": 402.7, "end": 403.14, "word": " اللي", "probability": 0.952392578125}, {"start": 403.14, "end": 403.38, "word": " هو", "probability": 0.8916015625}, {"start": 403.38, "end": 403.74, "word": " عن", "probability": 0.97265625}, {"start": 403.74, "end": 403.9, "word": " الـ", "probability": 0.94970703125}, {"start": 403.9, "end": 404.22, "word": " Interior", "probability": 0.83349609375}, {"start": 404.22, "end": 404.8, "word": " Points", "probability": 0.9208984375}, {"start": 404.8, "end": 405.1, "word": " اللي", "probability": 0.972900390625}, {"start": 405.1, "end": 405.38, "word": " هي", "probability": 0.83447265625}, {"start": 405.38, "end": 406.06, "word": " ممكن", "probability": 0.98046875}, {"start": 406.06, "end": 406.6, "word": " نودد", "probability": 0.8740234375}, 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"word": " من", "probability": 0.9794921875}, {"start": 411.58, "end": 411.8, "word": " هذه", "probability": 0.95068359375}, {"start": 411.8, "end": 412.24, "word": " النظرية", "probability": 0.9783935546875}, {"start": 412.24, "end": 412.58, "word": " أيضا", "probability": 0.7957356770833334}, {"start": 412.58, "end": 413.22, "word": " نستقل", "probability": 0.8726806640625}, {"start": 413.22, "end": 413.78, "word": " معلومة", "probability": 0.8905029296875}, {"start": 413.78, "end": 414.3, "word": " التالية", "probability": 0.9073893229166666}, {"start": 414.3, "end": 414.8, "word": " وهي", "probability": 0.96435546875}, {"start": 414.8, "end": 415.46, "word": " نهائية", "probability": 0.9947509765625}, {"start": 415.46, "end": 416.4, "word": " يمكن", "probability": 0.927734375}], "temperature": 1.0}, {"id": 20, "seek": 43587, "start": 418.19, "end": 435.87, "text": "المناطق اللي ممكن يكون فيها Local Maximum أو Local Minimum لسه بدأ أفحص المناطق اللتي يمكن أن يكون فيها Local Maximum 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2088.2, "word": " اللي", "probability": 0.989501953125}, {"start": 2088.2, "end": 2088.5, "word": " ممكن", "probability": 0.989013671875}, {"start": 2088.5, "end": 2088.84, "word": " نلاقي", "probability": 0.88525390625}, {"start": 2088.84, "end": 2089.18, "word": " فيها", "probability": 0.988525390625}, {"start": 2089.18, "end": 2089.66, "word": " maximum", "probability": 0.86572265625}, {"start": 2089.66, "end": 2089.94, "word": " أو", "probability": 0.8525390625}, {"start": 2089.94, "end": 2090.28, "word": " minimum،", "probability": 0.92236328125}, {"start": 2090.28, "end": 2090.62, "word": " هنقول", "probability": 0.7451985677083334}, {"start": 2090.62, "end": 2090.96, "word": " كيف", "probability": 0.98046875}, {"start": 2090.96, "end": 2091.62, "word": " نوجدهم", "probability": 0.73651123046875}], "temperature": 1.0}, {"id": 96, "seek": 212108, "start": 2092.36, "end": 2121.08, "text": "وبنلاقي النقاط هذه وبرضه بنحطها في list في قائمة وبنوجد الـ F عندها إلى بعد ما أوجدناها من في الحالتين الأولى يا إما من ال critical points أو من ال boundary points باخد اللي هو أكبر واحدة فيهم بتكون هي ال absolute فيها maximum وأصغر واحدة فيهم هي ال absolute minimum وإذا الموضوع صار سهل look through the list for the maximum and minimum values of F these will be absolute maximum and minimum values of F", "tokens": [37746, 1863, 15040, 38436, 28239, 4587, 41193, 29538, 4032, 26890, 11242, 3224, 44945, 5016, 9566, 11296, 8978, 1329, 8978, 12174, 16373, 46007, 46599, 1863, 29245, 3215, 2423, 39184, 479, 43242, 11296, 30731, 39182, 19446, 5551, 29245, 3215, 8315, 11296, 9154, 8978, 21542, 6027, 2655, 9957, 16247, 12610, 7578, 35186, 11933, 15042, 9154, 2423, 4924, 2793, 34051, 9154, 2423, 12866, 2793, 4724, 47283, 3215, 13672, 1829, 31439, 5551, 4117, 26890, 36764, 24401, 3660, 8978, 16095, 39894, 30544, 39896, 2423, 8236, 8978, 11296, 6674, 4032, 10721, 9381, 17082, 2288, 36764, 24401, 3660, 8978, 16095, 39896, 2423, 8236, 7285, 4032, 28814, 15730, 9673, 2407, 11242, 45367, 20328, 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"word": " through", "probability": 0.8896484375}, {"start": 2114.68, "end": 2114.92, "word": " the", "probability": 0.88916015625}, {"start": 2114.92, "end": 2115.16, "word": " list", "probability": 0.9052734375}, {"start": 2115.16, "end": 2115.46, "word": " for", "probability": 0.92431640625}, {"start": 2115.46, "end": 2115.62, "word": " the", "probability": 0.734375}, {"start": 2115.62, "end": 2115.96, "word": " maximum", "probability": 0.92822265625}, {"start": 2115.96, "end": 2116.28, "word": " and", "probability": 0.89892578125}, {"start": 2116.28, "end": 2116.64, "word": " minimum", "probability": 0.95654296875}, {"start": 2116.64, "end": 2117.18, "word": " values", "probability": 0.84228515625}, {"start": 2117.18, "end": 2117.34, "word": " of", "probability": 0.94677734375}, {"start": 2117.34, "end": 2117.52, "word": " F", "probability": 0.9306640625}, {"start": 2117.52, "end": 2118.14, "word": " these", "probability": 0.70849609375}, {"start": 2118.14, "end": 2118.38, "word": " will", "probability": 0.873046875}, {"start": 2118.38, "end": 2118.78, "word": " be", "probability": 0.93505859375}, {"start": 2118.78, "end": 2119.26, "word": " absolute", "probability": 0.87939453125}, {"start": 2119.26, "end": 2119.8, "word": " maximum", "probability": 0.91650390625}, {"start": 2119.8, "end": 2120.14, "word": " and", "probability": 0.9130859375}, {"start": 2120.14, "end": 2120.38, "word": " minimum", "probability": 0.97216796875}, {"start": 2120.38, "end": 2120.74, "word": " values", "probability": 0.92041015625}, {"start": 2120.74, "end": 2120.92, "word": " of", "probability": 0.96728515625}, {"start": 2120.92, "end": 2121.08, "word": " F", "probability": 0.98193359375}], "temperature": 1.0}, {"id": 97, "seek": 213861, "start": 2121.25, "end": 2138.61, "text": "Since absolute maximum and minimum are also local maximum and minimum are absolute maximum and minimum values وفقه بيرسم somewhere in the list made list بواحدة اتنية وكل ما هو ال absolute maximum و absolute minimum هيكون واحدة من ال local maximum و local minimum", "tokens": [50, 4647, 68, 8236, 6674, 293, 7285, 366, 611, 2654, 6674, 293, 7285, 366, 8236, 6674, 293, 7285, 4190, 4032, 5172, 4587, 3224, 4724, 13546, 38251, 4079, 294, 264, 1329, 1027, 1329, 4724, 14407, 24401, 3660, 1975, 2655, 1863, 10632, 4032, 28820, 19446, 31439, 2423, 8236, 6674, 4032, 8236, 7285, 39896, 30544, 36764, 24401, 3660, 9154, 2423, 2654, 6674, 4032, 2654, 7285], "avg_logprob": -0.45982141532595194, "compression_ratio": 1.8273809523809523, "no_speech_prob": 9.47713851928711e-06, "words": [{"start": 2121.25, "end": 2121.71, "word": "Since", "probability": 0.4955240885416667}, {"start": 2121.71, "end": 2121.97, "word": " absolute", "probability": 0.64453125}, {"start": 2121.97, "end": 2122.45, "word": " maximum", "probability": 0.935546875}, {"start": 2122.45, "end": 2122.83, "word": " and", "probability": 0.9541015625}, {"start": 2122.83, "end": 2123.19, "word": " minimum", "probability": 0.97998046875}, {"start": 2123.19, "end": 2123.97, "word": " are", "probability": 0.92578125}, {"start": 2123.97, "end": 2124.65, "word": " also", "probability": 0.828125}, {"start": 2124.65, "end": 2124.95, "word": " local", "probability": 0.8056640625}, {"start": 2124.95, "end": 2125.49, "word": " maximum", "probability": 0.91064453125}, {"start": 2125.49, "end": 2125.71, "word": " and", "probability": 0.313720703125}, {"start": 2125.71, "end": 2125.93, "word": " minimum", "probability": 0.91455078125}, {"start": 2125.93, "end": 2126.19, "word": " are", "probability": 0.2449951171875}, {"start": 2126.19, "end": 2126.41, "word": " absolute", "probability": 0.318115234375}, {"start": 2126.41, "end": 2127.03, "word": " maximum", "probability": 0.92626953125}, {"start": 2127.03, "end": 2127.21, "word": " and", "probability": 0.90380859375}, {"start": 2127.21, "end": 2127.51, "word": " minimum", "probability": 0.9560546875}, {"start": 2127.51, "end": 2128.29, "word": " values", "probability": 0.4609375}, {"start": 2128.29, "end": 2128.71, "word": " وفقه", "probability": 0.411712646484375}, {"start": 2128.71, "end": 2129.07, "word": " بيرسم", "probability": 0.8541666666666666}, {"start": 2129.07, "end": 2129.41, "word": " somewhere", "probability": 0.270263671875}, {"start": 2129.41, "end": 2129.61, "word": " in", "probability": 0.9482421875}, {"start": 2129.61, "end": 2129.73, "word": " the", "probability": 0.80810546875}, {"start": 2129.73, "end": 2130.01, "word": " list", "probability": 0.92236328125}, {"start": 2130.01, "end": 2130.47, "word": " made", "probability": 0.378173828125}, {"start": 2130.47, "end": 2130.79, "word": " list", "probability": 0.2978515625}, {"start": 2130.79, "end": 2131.17, "word": " بواحدة", "probability": 0.944580078125}, {"start": 2131.17, "end": 2131.47, "word": " اتنية", "probability": 0.63958740234375}, {"start": 2131.47, "end": 2131.97, "word": " وكل", "probability": 0.6212158203125}, {"start": 2131.97, "end": 2132.07, "word": " ما", "probability": 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"end": 2801.15, "word": " كان", "probability": 0.3671875}, {"start": 2801.15, "end": 2801.25, "word": " الـ", "probability": 0.5302734375}, {"start": 2801.25, "end": 2802.11, "word": " First", "probability": 0.37158203125}, {"start": 2802.11, "end": 2802.57, "word": " وSecond", "probability": 0.6219482421875}, {"start": 2802.57, "end": 2802.87, "word": " ورشة", "probability": 0.446197509765625}, {"start": 2802.87, "end": 2802.93, "word": " الـ", "probability": 0.80078125}, {"start": 2802.93, "end": 2803.35, "word": " Derivatives", "probability": 0.8562825520833334}, {"start": 2803.35, "end": 2804.79, "word": " موجودة", "probability": 0.80450439453125}, {"start": 2804.79, "end": 2805.49, "word": " ومستمرة", "probability": 0.73056640625}, {"start": 2805.49, "end": 2806.15, "word": " وبدنا", "probability": 0.7744140625}, {"start": 2806.15, "end": 2806.57, "word": " نفحص", "probability": 0.93798828125}, {"start": 2806.57, "end": 2806.85, "word": " هل", "probability": 0.9501953125}, {"start": 2806.85, "end": 2807.03, "word": " في", "probability": 0.89208984375}, {"start": 2807.03, "end": 2807.25, "word": " Local", "probability": 0.77685546875}, {"start": 2807.25, "end": 2807.69, "word": " Maximum", "probability": 0.760986328125}, {"start": 2807.69, "end": 2807.75, "word": " و", "probability": 0.72314453125}, {"start": 2807.75, "end": 2807.93, "word": " Local", "probability": 0.82275390625}, {"start": 2807.93, "end": 2808.35, "word": " Minimum", "probability": 0.9921875}, {"start": 2808.35, "end": 2808.81, "word": " قلنا", "probability": 0.955078125}, {"start": 2808.81, "end": 2809.37, "word": " بالـ", "probability": 0.75}, {"start": 2809.37, "end": 2809.51, "word": " Second", "probability": 0.7314453125}, {"start": 2809.51, "end": 2809.97, "word": " Derivative", "probability": 0.90234375}, {"start": 2809.97, "end": 2810.29, "word": " Test", "probability": 0.92578125}, {"start": 2810.29, "end": 2810.45, "word": " اللي", "probability": 0.96826171875}, {"start": 2810.45, "end": 2810.89, "word": " حكينا", "probability": 0.991943359375}, {"start": 2810.89, "end": 2811.39, "word": " عنه", "probability": 0.991455078125}, {"start": 2811.39, "end": 2811.83, "word": " اللي", "probability": 0.899169921875}, {"start": 2811.83, "end": 2811.99, "word": " هو", "probability": 0.96875}, {"start": 2811.99, "end": 2812.73, "word": " سابقاً", "probability": 0.95419921875}, {"start": 2812.73, "end": 2813.25, "word": " وهيك", "probability": 0.7088216145833334}, {"start": 2813.25, "end": 2813.61, "word": " بيكون", "probability": 0.7161458333333334}, {"start": 2813.61, "end": 2813.89, "word": " إحنا", "probability": 0.85693359375}, {"start": 2813.89, "end": 2814.37, "word": " خلصنا", "probability": 0.9884033203125}, {"start": 2814.37, "end": 2814.89, "word": " المحاضرة", "probability": 0.989013671875}, {"start": 2814.89, "end": 2815.31, "word": " وبرك", "probability": 0.7543131510416666}, {"start": 2815.31, "end": 2815.43, "word": " الله", "probability": 0.9365234375}, {"start": 2815.43, "end": 2815.75, "word": " فيكم", "probability": 0.97705078125}, {"start": 2815.75, "end": 2816.09, "word": " والسلام", "probability": 0.85693359375}, {"start": 2816.09, "end": 2816.49, "word": " عليكم", "probability": 0.943603515625}, {"start": 2816.49, "end": 2816.85, "word": " ورحمة", "probability": 0.9808349609375}, {"start": 2816.85, "end": 2817.11, "word": " الله", "probability": 0.962890625}, {"start": 2817.11, "end": 2818.49, "word": " وبركاته", "probability": 0.975}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2819.692, "duration_after_vad": 2809.979999999999}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..538a9c54634764f7bd2915e76a77ac32c7e025c0
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/fqBp_Xxsd04_raw.srt
@@ -0,0 +1,2772 @@
+1
+00:00:00,250 --> 00:00:03,610
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:03,610 --> 00:00:08,670
+وبركاته هذه هي المحاضرة رقم تسعة عشر مساقطة طادل
+
+3
+00:00:08,670 --> 00:00:12,310
+وتكامل الـGene وهي المحاضرة السابعة والأخيرة في
+
+4
+00:00:12,310 --> 00:00:17,750
+شبطر أربعة عشر وهي أربعة عشر سبعة وهي تحت عنوان
+
+5
+00:00:17,750 --> 00:00:22,890
+Extreme Values and Subtle Valuesطبعا الموضوع ليس
+
+6
+00:00:22,890 --> 00:00:29,510
+جديد فيه روحه احنا اتحدثنا في calculus الف وكمان
+
+7
+00:00:29,510 --> 00:00:34,910
+حتى في الفنوية عن اللي هي كيف نوجد الـ Local
+
+8
+00:00:34,910 --> 00:00:38,470
+maximum و Local minimum و الـ Infliction points و
+
+9
+00:00:38,470 --> 00:00:42,590
+كذلك اللي هو ال test تبعاتها و كذلك على Absolute
+
+10
+00:00:42,590 --> 00:00:47,030
+maximum و Absolute minimumالحديث سيكون حديث مقابل
+
+11
+00:00:47,030 --> 00:00:49,890
+بسعر الـ two variables أو الـ three variables
+
+12
+00:00:49,890 --> 00:00:55,630
+الحديث سيكون حول نفس الأشياء سنجاوبها على سؤالين
+
+13
+00:00:55,630 --> 00:01:03,090
+السؤال الأول أين يمكن إيجاد الـ local maximum أو
+
+14
+00:01:03,090 --> 00:01:06,640
+الـ local minimumأو اللي هي الان مش inflection
+
+15
+00:01:06,640 --> 00:01:09,840
+point حاجة بنسميها settle points أو نقطة سرج
+
+16
+00:01:09,840 --> 00:01:15,140
+الحصان السؤال التاني أنه هل كيف بدنا نحكم على أن
+
+17
+00:01:15,140 --> 00:01:19,080
+هذه فعلا في عندها local maximum و local minimum أو
+
+18
+00:01:19,080 --> 00:01:24,740
+نقطة settle points أو لاتقريباً الأجوبة هي مشابهة
+
+19
+00:01:24,740 --> 00:01:28,080
+للأجوبة السابقة في تفاضل ألف كنا نجاوب في تفاضل
+
+20
+00:01:28,080 --> 00:01:33,200
+ألف نقول القيم العظمى والصغرى وكذلك الـ inflection
+
+21
+00:01:33,200 --> 00:01:37,380
+points ممكن أن توجد في أحد المناطق التالية إما في
+
+22
+00:01:37,380 --> 00:01:40,160
+الـ interior points اللي هي في عند الـ derivative
+
+23
+00:01:40,160 --> 00:01:43,660
+بتساوي سفر أو الـ derivative does not exist هذه
+
+24
+00:01:43,660 --> 00:01:46,880
+المنطقة تانية والمنطقة الثالثة هي الـ boundary
+
+25
+00:01:46,880 --> 00:01:50,110
+pointsوكنا ناخد الـ First derivative test والـ
+
+26
+00:01:50,110 --> 00:01:54,010
+Second derivative test اللي هو الحكم على إن فيه
+
+27
+00:01:54,010 --> 00:01:57,330
+لوكال maximum أو لوكال minimum أو اللي هو
+
+28
+00:01:57,330 --> 00:02:01,290
+inflection point هذا الحديث تقريبا مشابه جدا خلينا
+
+29
+00:02:01,290 --> 00:02:05,010
+نبدأ اللي هو في الموضوعنا derivative test for
+
+30
+00:02:05,010 --> 00:02:10,010
+local extreme valuesنشوف أول حاجة ماعنات الـ
+
+31
+00:02:10,010 --> 00:02:15,190
+maximum و الـ minimum أو الـ local maximum و الـ
+
+32
+00:02:15,190 --> 00:02:19,210
+local minimum مشابهة let f of x,y be defined on a
+
+33
+00:02:19,210 --> 00:02:22,790
+region R containing the point A و B نشوف الآن ما
+
+34
+00:02:22,790 --> 00:02:25,910
+هي معنات local maximum عندنا نقطة A و B هي نقطة A
+
+35
+00:02:25,910 --> 00:02:29,010
+و B ما هي معناة local maximum local maximum هي
+
+36
+00:02:29,010 --> 00:02:34,320
+نقطة عظمة محلية إننا نجد جوار حوالهاهناك جوار
+
+37
+00:02:34,320 --> 00:02:39,460
+حواليها تكون هي أكبر واحدة يعني قيمة F of A و F of
+
+38
+00:02:39,460 --> 00:02:45,540
+B أكبر واحدة أكبر من F of XY لكل XY في هذا الجوار
+
+39
+00:02:45,540 --> 00:02:48,780
+يعني أكبر واحدة في جيرانها هذا نسميها Local
+
+40
+00:02:48,780 --> 00:02:53,040
+Maximumالآن أصغر وحدة تجوّر حوالين الـ A و الـ B
+
+41
+00:02:53,040 --> 00:02:56,720
+تكون الـ F عند الـ A و الـ B أصغر وحدة في جيرانها
+
+42
+00:02:56,720 --> 00:03:01,040
+فبنسميها Local Minimum وهذا هو التعريف F of A و B
+
+43
+00:03:01,040 --> 00:03:05,720
+is Local Maximum Value of F if F of A و B أكبر
+
+44
+00:03:05,720 --> 00:03:10,840
+يساوي F of X و Y أكبر من كل الـ F of X و Y لأ يعني
+
+45
+00:03:10,840 --> 00:03:14,860
+بمعنى أخر اللي هي لنقاط الـ X و Y حوالها for all
+
+46
+00:03:14,860 --> 00:03:20,440
+domainو.xy in an open disk centered at a وb
+
+47
+00:03:25,650 --> 00:03:29,230
+بتكون قيمة F عند الـ A و الـ B أكبر من F of X Y
+
+48
+00:03:29,230 --> 00:03:33,410
+لكل X Y في هذا الـ Disk اللي حواليها أو الـ Open
+
+49
+00:03:33,410 --> 00:03:37,630
+Disk الآن الـ Local Minimum بنفس الأسلوب F of A و
+
+50
+00:03:37,630 --> 00:03:41,790
+B is a Local Minimum Value of F if F of A و B أصغر
+
+51
+00:03:41,790 --> 00:03:46,130
+أو يساوي F of X Y for all points X Y في Domain الـ
+
+52
+00:03:46,130 --> 00:03:50,950
+Function وفي نفس الواجهة جيران لـ A و B أو في اللي
+
+53
+00:03:50,950 --> 00:03:53,930
+هي Disk Open Disk عارفين شو معناه Open Disk دائرة
+
+54
+00:03:53,930 --> 00:03:58,290
+مفتوحةاللي هو اللي جواتها تكون اللي هي كل النقطات
+
+55
+00:03:58,290 --> 00:04:01,670
+اللي جواتها قيمة الـ Function F عند الـ A والـ B
+
+56
+00:04:01,670 --> 00:04:06,510
+أصغر من هذه القيم معناته أن F لها Local Minimum
+
+57
+00:04:06,510 --> 00:04:11,810
+عند هذه النقطة الآن بدنا نيجي اللي هو ندى نجاوب
+
+58
+00:04:11,810 --> 00:04:14,990
+السؤال الأول أين يمكن إيجاد الـ Local Maximum و
+
+59
+00:04:14,990 --> 00:04:20,340
+Local Minimum أو اللي هوالنقاط بعد شويه هنحكي عن
+
+60
+00:04:20,340 --> 00:04:25,540
+نقاط حرجة أو الـ saddle point طبعا الان ال theorem
+
+61
+00:04:25,540 --> 00:04:29,720
+عشرة first derivative test for local values الان
+
+62
+00:04:29,720 --> 00:04:32,900
+اللي هي ال first derivative test زي ما كان هناك في
+
+63
+00:04:32,900 --> 00:04:36,240
+first derivative test هنا في عندنا first
+
+64
+00:04:36,240 --> 00:04:41,960
+derivative test إيش بتقول إذا كانت f of x y has a
+
+65
+00:04:41,960 --> 00:04:46,230
+local minimum أو local maximumعند الـ interior
+
+66
+00:04:46,230 --> 00:04:50,830
+point a و b of its domain and if the first partial
+
+67
+00:04:50,830 --> 00:04:57,030
+derivatives exist then fx عند هذا النقطة سفرة و fy
+
+68
+00:04:57,030 --> 00:05:00,330
+عند هذا النقطة a سفرة هذا بتقول إنها نظرية مالي
+
+69
+00:05:00,330 --> 00:05:04,770
+إذا كانت ال function طبعا تحت الظروفة إنها ال
+
+70
+00:05:04,770 --> 00:05:08,350
+first partial derivative test exist يعني ال fx و
+
+71
+00:05:08,350 --> 00:05:12,020
+ال fy موجودةالآن أخبرني باختصار إذا كانت الـ
+
+72
+00:05:12,020 --> 00:05:16,240
+function f of x y لها local maximum أو local
+
+73
+00:05:16,240 --> 00:05:20,300
+minimum عند الـ a والب غصبًا عن طبعًا في interior
+
+74
+00:05:20,300 --> 00:05:24,540
+point يعني غصبًا عن بيكون fx بتساوي سفر وfy إيه
+
+75
+00:05:24,540 --> 00:05:29,540
+شمالها بتساوي سفر يعني لما بدك تيجي تفحص عند اللي
+
+76
+00:05:29,540 --> 00:05:33,160
+هياللي تبحث عن الـ Local Maximum و Local Minimum
+
+77
+00:05:33,160 --> 00:05:37,420
+في الـ Interior Points اللي الـ Fx و Fy exist لهن
+
+78
+00:05:37,420 --> 00:05:42,600
+افحص بس عند Fx بيساوي 0 و Fy بيساوي 0 النقاط اللي
+
+79
+00:05:42,600 --> 00:05:46,500
+بتحقق هذه ممكن يكون فيها Local Maximum و Local
+
+80
+00:05:46,500 --> 00:05:50,420
+Minimumأما النقاط اللي بيكونش هذه سفر وتكونش هذه
+
+81
+00:05:50,420 --> 00:05:53,300
+سفر انساها لأن لو كان Local Maximum و Local
+
+82
+00:05:53,300 --> 00:05:57,720
+Minimum لازم يطلع عن سفر إذا من هذه النظرية بنقدر
+
+83
+00:05:57,720 --> 00:06:04,450
+نقول أنه احنا ممكن نوجدممكن نلاقي الـ Local
+
+84
+00:06:04,450 --> 00:06:08,530
+Maximum أو الـ Local Minimum عند أحد النقاط
+
+85
+00:06:08,530 --> 00:06:13,970
+التالية إما Fx يساوي Fy بيساوي سفر أو إن الـ
+
+86
+00:06:13,970 --> 00:06:17,350
+Derivative does not exist لأن هذا ليس متضمن من
+
+87
+00:06:17,350 --> 00:06:21,350
+خلال الحديث في النظرية إذن هذه النظرية توحي ما
+
+88
+00:06:21,350 --> 00:06:26,010
+يليه أو تعطينا المعلومات التالية إنه اللي هي الـ
+
+89
+00:06:26,010 --> 00:06:30,380
+Interior Pointsللـ Interior Points ممكن نلاقي الـ
+
+90
+00:06:30,380 --> 00:06:32,300
+Local Maximum الـ Local Minimum عند الـ Interior
+
+91
+00:06:32,300 --> 00:06:37,560
+Points اللي Fx بيساوي 0 و Fy بيساوي 0 و اللي Fx و
+
+92
+00:06:37,560 --> 00:06:43,140
+Fy لا يوجد في هذه الحالة اللي هو بيكون عندي اللي
+
+93
+00:06:43,140 --> 00:06:46,800
+هو عن الـ Interior Points اللي هي ممكن نودد الـ
+
+94
+00:06:46,800 --> 00:06:50,460
+Local Maximum و Local Minimum عند هذه الأماكن يعني
+
+95
+00:06:50,460 --> 00:06:54,300
+بمعنى آخر من هذه النظرية أيضا نستقل معلومة التالية
+
+96
+00:06:54,300 --> 00:07:00,990
+وهي نهائية يمكنالمناطق اللي ممكن يكون فيها Local
+
+97
+00:07:00,990 --> 00:07:04,550
+Maximum أو Local Minimum لسه بدأ أفحص المناطق
+
+98
+00:07:04,550 --> 00:07:07,250
+اللتي يمكن أن يكون فيها Local Maximum أو Local
+
+99
+00:07:07,250 --> 00:07:10,990
+Minimum هي النقاط اللي الـ Fx بتساوي Fy بساوي سفر
+
+100
+00:07:10,990 --> 00:07:13,730
+هاي واحد طبعا الـ Interior Bonds اتنين الـ
+
+101
+00:07:13,730 --> 00:07:15,870
+Interior Bonds اللي هي Fx does not exist
+
+102
+00:07:20,290 --> 00:07:26,450
+النقطة الأخيرة الـ Boundary Points حصرنا كل جزء في
+
+103
+00:07:26,450 --> 00:07:39,450
+أربع مناطق Fx Fy Fx Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy
+
+104
+00:07:39,450 --> 00:07:39,550
+Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy
+
+105
+00:07:39,550 --> 00:07:39,790
+Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy
+
+106
+00:07:39,790 --> 00:07:39,790
+Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy
+
+107
+00:07:39,790 --> 00:07:39,790
+Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy
+
+108
+00:07:39,790 --> 00:07:39,790
+Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy Fy
+
+109
+00:07:39,820 --> 00:07:44,040
+إذا سهلت علينا أمور الفحص يعني أن نتأكد الآن ضال
+
+110
+00:07:44,040 --> 00:07:47,300
+علينا بس نتأكد إن في Local Maximum ولا Local
+
+111
+00:07:47,300 --> 00:07:51,780
+Minimum أو مافيش شيء الآن اللي اللي اللي اللي اللي
+
+112
+00:07:51,780 --> 00:07:55,620
+اللي عشان نحصر الأمور يعني وكأننا نجسم الأمر اللي
+
+113
+00:07:55,620 --> 00:08:00,090
+هو الأمرين الـInterior Point الـ Gatewayأو إذا كان
+
+114
+00:08:00,090 --> 00:08:02,850
+هناك نقطة الداخلية في مجلة عملية f of x و f of y
+
+115
+00:08:02,850 --> 00:08:04,030
+أما إذا كان هناك نقطة داخلية في مجلة عملية f of x
+
+116
+00:08:04,030 --> 00:08:05,610
+و f of y أما إذا كان هناك نقطة داخلية في مجلة
+
+117
+00:08:05,610 --> 00:08:07,230
+عملية f of x و f of y أما إذا كان هناك نقطة داخلية
+
+118
+00:08:07,230 --> 00:08:08,210
+في مجلة عملية f of x و f of y أما إذا كان هناك
+
+119
+00:08:08,210 --> 00:08:09,170
+نقطة داخلية في مجلة عملية f of x و f of y أما إذا
+
+120
+00:08:09,170 --> 00:08:10,050
+كان هناك نقطة داخلية في مجلة عملية f of x و f of y
+
+121
+00:08:10,050 --> 00:08:12,270
+أما إذا كان هناك نقطة داخلية في مجلة عملية f of x
+
+122
+00:08:12,270 --> 00:08:17,110
+و f of y أما إذا كان هناك نقطة
+
+123
+00:08:17,110 --> 00:08:21,580
+داخلية في مجلة عملية f of x و f of xهي واحدة اتنين
+
+124
+00:08:21,580 --> 00:08:26,440
+Fx does not exist هيتنتين F تلاتة Fy does not
+
+125
+00:08:26,440 --> 00:08:30,680
+exist هدول الـ Critical Points اللي اللي ممكن يكون
+
+126
+00:08:30,680 --> 00:08:33,600
+عندهم Local Maximum أو Local Minimum إذا صار في
+
+127
+00:08:33,600 --> 00:08:36,740
+عندي بناء على هذا التصنيف في منطقتين ممكن نلاقي
+
+128
+00:08:36,740 --> 00:08:40,320
+فين اللي هو Local Maximum أو Local Minimum أو اللي
+
+129
+00:08:40,320 --> 00:08:45,290
+هو بعد شوية أسد الـ Point اللي هي أحد تلت مناطقأحد
+
+130
+00:08:45,290 --> 00:08:49,130
+منطقتين اللي هي إما الـ interior points اللي بتكون
+
+131
+00:08:49,130 --> 00:08:52,410
+فيها critical points هاي الـ critical points يعني
+
+132
+00:08:52,410 --> 00:08:54,530
+طبعا الـ critical point على طول الـ interior point
+
+133
+00:08:54,530 --> 00:08:57,990
+الحاجة الثانية اللي هي الـ boundary points إذا
+
+134
+00:08:57,990 --> 00:09:02,270
+منطقتين critical points أو boundary pointsالـ
+
+135
+00:09:02,270 --> 00:09:05,530
+Critical Points هي اللي بيكون عندها Fx بيساوي Fy
+
+136
+00:09:05,530 --> 00:09:08,270
+طبعا Interior Points Fx بيساوي Fy بيساوي Zero هاي
+
+137
+00:09:08,270 --> 00:09:12,130
+واحدة Fx does not exist تاعتين تاني ثلاثة Fy does
+
+138
+00:09:12,130 --> 00:09:15,930
+not exist هذه هي السؤال جاوبنا عليه الآن أين يمكن
+
+139
+00:09:15,930 --> 00:09:18,410
+إيجاد الـ Local Maximum أو ال Local Minimum أو ال
+
+140
+00:09:18,410 --> 00:09:25,260
+Extreme Values أو Pointsالان لاحظ عند اللي هي ال
+
+141
+00:09:25,260 --> 00:09:28,480
+.. ال .. ال .. النقطة هذه اللي هي عند الـ Local
+
+142
+00:09:28,480 --> 00:09:33,620
+Maximum لاحظ أنه الميه المماس هذا المماس هيكون الـ
+
+143
+00:09:33,620 --> 00:09:38,220
+∆FY∆Y موازية لـ Y Axis يعني ∆FY∆Y أيش بيساوي سفر
+
+144
+00:09:38,220 --> 00:09:41,600
+والمماس ل ال curve هذا اللي partial derivative إذ
+
+145
+00:09:41,600 --> 00:09:43,980
+هانا اللي معناه partial derivative معناه كنا زمان
+
+146
+00:09:43,980 --> 00:09:47,180
+بيكون ال ∆FY∆X برضه موازية ل X Axis بيكون اللي هو
+
+147
+00:09:47,180 --> 00:09:50,780
+أيش سفر يعني عنده النقطة هذهالـ interior point الـ
+
+148
+00:09:50,780 --> 00:09:54,580
+derivative اللي هي exist الـ local maximum هاي
+
+149
+00:09:54,580 --> 00:09:57,880
+اللي هو ميل المماس سفر و ميل المماس التاني سفر
+
+150
+00:09:57,880 --> 00:10:01,060
+واحد اللي هو دل أف بايد ال X سفر واحد ع الدل أف
+
+151
+00:10:01,060 --> 00:10:04,620
+بايد ال Y بساوة سفر هذا معناه هندسي للحديث اللي
+
+152
+00:10:04,620 --> 00:10:06,280
+اتحدثنا فيه قبل بشوية
+
+153
+00:10:09,850 --> 00:10:13,410
+إذا لم يكن معرفة معناته الـ Saddle Point فمعرفة
+
+154
+00:10:13,410 --> 00:10:16,250
+مختلفة F of X Y لديها Saddle Point إلى Critical
+
+155
+00:10:16,250 --> 00:10:20,030
+Point إذا الآن Saddle Point وين يا جماعة؟ هي
+
+156
+00:10:20,030 --> 00:10:25,590
+Interior Point وعند مين؟ عند Critical Point يعني F
+
+157
+00:10:25,590 --> 00:10:31,350
+Y بسوء سفرأو Fx does not exist أو Fy does not
+
+158
+00:10:31,350 --> 00:10:35,090
+exist then a differentiable function f of x y has
+
+159
+00:10:35,090 --> 00:10:37,810
+a saddle point at a critical point a و b .. آسف
+
+160
+00:10:37,810 --> 00:10:41,350
+آسف آسف اللي هو عند a interior point Fx بتساوي Fy
+
+161
+00:10:41,350 --> 00:10:44,390
+بتساوي 0 a differentiable function f of x y has a
+
+162
+00:10:44,390 --> 00:10:49,010
+saddle pointعلى نقطة مهمة A وB إذا كان في كل مركز
+
+163
+00:10:49,010 --> 00:10:54,350
+مفتوح على A وB يوجد نقطة مخصصة XY فيها Fxy أكبر من
+
+164
+00:10:54,350 --> 00:10:59,190
+F of A وB ونقطة XY فيها Fxy أكبر من F of A وB
+
+165
+00:10:59,190 --> 00:11:06,400
+المنطقة المتعلقة بالنقطة Z هي نقطة مخصصةأذا الـ
+
+166
+00:11:06,400 --> 00:11:09,120
+differentiable function f of x y يعني بتكون الـ fx
+
+167
+00:11:09,120 --> 00:11:13,120
+و الـ fy exist already ماشي الحال اللي هو عند
+
+168
+00:11:13,120 --> 00:11:15,940
+critical point معناته بس عند fx بتساوي fy إيش
+
+169
+00:11:15,940 --> 00:11:19,240
+بتساوي سفر يعني اللي عن ال settle point إيش هي
+
+170
+00:11:19,240 --> 00:11:23,540
+نقطة داخلية بتكون عندها اللي هي بنقول عنها settle
+
+171
+00:11:23,540 --> 00:11:29,680
+point إذا إجيناأخدنا اللي هو لأي
+
+172
+00:11:29,680 --> 00:11:33,020
+جوار حواليها لأي جوار حواليها لجينا كل ما ناخد
+
+173
+00:11:33,020 --> 00:11:38,360
+جوار ال F of A و F of B مرات بتاخد في داخل الجوار
+
+174
+00:11:38,360 --> 00:11:42,560
+أصغر من F of XY ومرات بتكون أكبر من مين من F of XY
+
+175
+00:11:42,560 --> 00:11:45,260
+يعني بتكون أكبر من النقاط من اللي حواليها وأصغر من
+
+176
+00:11:45,260 --> 00:11:49,930
+النقاط من اللي حواليها إذا لا هيLocalMaximum وليس
+
+177
+00:11:49,930 --> 00:11:54,530
+LocalMinimum لأن كل جوار حولها ستجد الـ Function
+
+178
+00:11:54,530 --> 00:11:58,230
+عند هذا النقطة بتكون أكبر من النقاط في هذا الجوار
+
+179
+00:11:58,230 --> 00:12:02,830
+و أصغر من النقاط في هذا الجوار المذكور لذلك لا
+
+180
+00:12:02,830 --> 00:12:06,650
+يمكن أن تكون لـ LocalMaximum أو LocalMinimumفهي
+
+181
+00:12:06,650 --> 00:12:10,250
+بنسميها saddle point اللي هي نقطة سرج الحصان فيها
+
+182
+00:12:10,250 --> 00:12:14,970
+لإن هي زي نقطة زي سرج الحصان هذا النقطة طبعا وين
+
+183
+00:12:14,970 --> 00:12:19,690
+موجودة عند Fx بتساوي Fy بتساوي سفر يعني عند
+
+184
+00:12:19,690 --> 00:12:22,690
+critical point و F differentiable إذا عند Fx
+
+185
+00:12:22,690 --> 00:12:25,910
+بتساوي Fy إيه شمالها بيساوي سفر لإن مادة
+
+186
+00:12:25,910 --> 00:12:29,630
+differentiable function Fx و Fy مالين exist إذا ال
+
+187
+00:12:29,630 --> 00:12:34,130
+saddle point هي نقطة تفحص عند Fx بتساوي Fy بتساوي
+
+188
+00:12:34,130 --> 00:12:37,570
+سفرهذا النقطة لا وجهنا عندها maximum ولا وجهنا
+
+189
+00:12:37,570 --> 00:12:41,710
+عندها minimum فبتكون settle point خلّينا نشوف مثال
+
+190
+00:12:41,710 --> 00:12:46,030
+find the local extreme values f of x,y بساوي x
+
+191
+00:12:46,030 --> 00:12:48,970
+تربيه زي y تربيه يعني نقص 4y زي 9 عمنا نشوف إيش
+
+192
+00:12:48,970 --> 00:12:51,050
+اللي هو ال local extreme values لسه احنا بنشتغل
+
+193
+00:12:51,050 --> 00:12:54,230
+على التعريف يا جماعة ماخدناش tests أو اختبارات
+
+194
+00:12:54,230 --> 00:12:57,730
+تتعرفنا كيف نحكم على إن هذه local maximum أو local
+
+195
+00:12:57,730 --> 00:13:02,450
+minimum لنشوف كيف، الآن عندي لو دينا لل function
+
+196
+00:13:02,450 --> 00:13:06,700
+هذهأحنا قلنا أين ممكن نجد الـ Extreme Value أو الـ
+
+197
+00:13:06,700 --> 00:13:11,320
+Local Extreme يجب أن نجدها إما عند Fx بساوي Fy
+
+198
+00:13:11,320 --> 00:13:18,700
+بساوية Fy بساوية 0 أو Fx لا يوجد أو Fy لا يوجد أو
+
+199
+00:13:18,700 --> 00:13:22,540
+الـ Boundary Pointsلكن هذه الـ function أول حاجة
+
+200
+00:13:22,540 --> 00:13:25,300
+الـ differentiation بالوين ما كان ومعرفة وين ما
+
+201
+00:13:25,300 --> 00:13:27,960
+كان مزام الـ differentiation بالوين ما كان ومعرفة
+
+202
+00:13:27,960 --> 00:13:32,860
+وين ما كان إذا مافيش شيء لا Fx و Fy لا يوجد ولا
+
+203
+00:13:32,860 --> 00:13:36,780
+فيه point في الـ boundary إذا الحديث بس عند Fx
+
+204
+00:13:36,780 --> 00:13:40,460
+بيساوي Fy بيساوي 0 وجدوا لـ Fx و Fy بيساووهم بالـ
+
+205
+00:13:40,460 --> 00:13:43,900
+0 ووجدوا النقطة ونبحث عند هذه النقطة هل في Local
+
+206
+00:13:43,900 --> 00:13:47,900
+maximum أو Local minimum خلينا ناخد Fx، Fx طبعا
+
+207
+00:13:47,900 --> 00:13:52,940
+واضح إنها عبارة عن 2Xالان fy هي عبارة عن 2y ناقص
+
+208
+00:13:52,940 --> 00:13:56,360
+أربعة إذا الان هذولة التنتين موجودة في أي مكان إذا
+
+209
+00:13:56,360 --> 00:14:00,240
+على طول بنسويهم بالصفر هي fx بيساوي صفر معناته 2x
+
+210
+00:14:00,240 --> 00:14:03,580
+بيساوي صفر وfy بيساوي صفر معناته 2y ناقص أربعة
+
+211
+00:14:03,580 --> 00:14:07,640
+بيساوي صفر يعني y بتساوي 2 وx بتساوي أيش صفر حل
+
+212
+00:14:07,640 --> 00:14:10,940
+هنا يا جماعة يعني 2y مانس أربعة بيساوي صفر معناته
+
+213
+00:14:10,940 --> 00:14:14,280
+y بيساوي 2 و2x بيساوي صفر معناته x بيساوي صفر إذا
+
+214
+00:14:14,280 --> 00:14:17,560
+النقطة اللي بنحكي عنها بس النقطة مين صفر و اتنين
+
+215
+00:14:17,560 --> 00:14:21,550
+هذه النقطة صفر و اتنين الآنهذه النقطة هي 02 هذه
+
+216
+00:14:21,550 --> 00:14:26,110
+الآن بدنا نشوف اللي هو لسه ما بنعرفش تس احنا الان
+
+217
+00:14:26,110 --> 00:14:32,410
+بدنا نحسب عندها F02 F02 أحسبها Zero أربعة ناقص
+
+218
+00:14:32,410 --> 00:14:37,430
+أربعة نروح أربعة لأ اللي هي جداش اتنين اتنين في
+
+219
+00:14:37,430 --> 00:14:40,830
+اتنين باربعة أربعة ناقص تمانية آسف أربعة ناقص
+
+220
+00:14:40,830 --> 00:14:45,350
+تمانية اللي هي بتصير إيش ناقص أربعة زائد تسعة ناقص
+
+221
+00:14:45,350 --> 00:14:48,310
+أربعة زائد تسعة ليها خمسة F02 إيش هتساوي يا شباب
+
+222
+00:14:48,700 --> 00:14:53,280
+ويا بنات، اللي هو خمسة، فـ F02 بساوي خمسة، إذا
+
+223
+00:14:53,280 --> 00:14:56,140
+الآن بدنا نبحث فيها، لاحظ على طبيعة الـ function
+
+224
+00:14:56,140 --> 00:15:00,120
+اللي عندنا، الآن إحنا بنحكي بس على التعريف،
+
+225
+00:15:00,120 --> 00:15:02,580
+بنستخدم التعريف في تحديد إنه Local Maximum أو
+
+226
+00:15:02,580 --> 00:15:05,040
+Local Minimum، في بعض السؤالات السابقة، طبعا صعب
+
+227
+00:15:05,040 --> 00:15:08,180
+إننا نحددها بالتعريف، لكن هذا شأل سهل، خلينا نشوف
+
+228
+00:15:08,720 --> 00:15:12,120
+أفو في X Y بساوي تبعتنا X تربيع زي Y تربيع نقص
+
+229
+00:15:12,120 --> 00:15:14,960
+أربعة Y زي تسعة لكن هذه ممكن كتبتها على الطريقة
+
+230
+00:15:14,960 --> 00:15:18,620
+التالية لو أكملنا المربع حانة بيصير X تربيع زي Y
+
+231
+00:15:18,620 --> 00:15:21,540
+ماينوس اتنين كل تربيع يعني اللي هو أخدنا من التسعة
+
+232
+00:15:21,540 --> 00:15:24,460
+أربعة صار مربع كامل صار على صورة Y نقص اتنين كل
+
+233
+00:15:24,460 --> 00:15:28,480
+تربيع إذا زائد من خمسةإذاً صار هذا المقدار هذا
+
+234
+00:15:28,480 --> 00:15:31,940
+مربع كامل وهذا مربع كامل إذاً لأي قيمة في الدنيا
+
+235
+00:15:31,940 --> 00:15:35,860
+XY هيكون هذا المقدار أكبر أو يساوي سفر إذاً صار
+
+236
+00:15:35,860 --> 00:15:39,420
+هذا المجموع عبارة عن أكبر أو يساوي إيش خمسة يعني
+
+237
+00:15:39,420 --> 00:15:43,780
+بمعنى أخر F of XY صارت دائما أكبر أو يساوي F of 02
+
+238
+00:15:43,780 --> 00:15:48,500
+يعني F of 02 هتعمل لي Local Minimum هي في الواقع
+
+239
+00:15:48,500 --> 00:15:51,520
+Absolute Minimum لكن إحنا مش موضوعنا We see that
+
+240
+00:15:51,520 --> 00:15:56,180
+the still critical pointاللي هي 02 أو the critical
+
+241
+00:15:56,180 --> 00:16:00,440
+point 02 gives a local minimum هذا استخدمنا فيها
+
+242
+00:16:00,440 --> 00:16:05,820
+minimum يا جماعة استخدمنا فيها بس اللي هو التعريف
+
+243
+00:16:07,860 --> 00:16:12,260
+لأ مثال أخر مثال ثاني find the local extreme
+
+244
+00:16:12,260 --> 00:16:16,140
+values إذا موجود of of of x y بسواي تربيع ناقص x
+
+245
+00:16:16,140 --> 00:16:19,820
+تربيع ماهي هذه الدالة؟ نفسي الأولى هذه معرفة وين
+
+246
+00:16:19,820 --> 00:16:22,860
+ماكان differentiable وين ماكان إذا مافيش boundary
+
+247
+00:16:22,860 --> 00:16:25,220
+points مدام differentiable وين ماكان إذا مافيش
+
+248
+00:16:25,220 --> 00:16:28,500
+boundary points الحاجة الثانية أن f x و f y دايما
+
+249
+00:16:28,500 --> 00:16:31,760
+exist إذا مافيش شيء اللي هو f x و f y does not
+
+250
+00:16:31,760 --> 00:16:35,780
+exist إذا بس البحث عند مين يا جماعةعند Fx بتساوي
+
+251
+00:16:35,780 --> 00:16:39,180
+Fy بيساوي سفر إذا ما علينا لغير نوجود Fx و Fy
+
+252
+00:16:39,180 --> 00:16:43,680
+خلّينا نوجد Fx Fx إيش بتساوي سالب اتنين X Fy إيش
+
+253
+00:16:43,680 --> 00:16:47,720
+بتساوي اللي هو اتنين Y ساوي هنا دولة بالسفر بيطلع
+
+254
+00:16:47,720 --> 00:16:52,300
+عند F اللي هي X بتساوي سفر و Y بتساوي سفر إذا
+
+255
+00:16:52,300 --> 00:16:56,340
+النقطة اللي بدنا نفحص عندها إنها اللي هي هل النقطة
+
+256
+00:16:56,340 --> 00:16:59,860
+اللي هي عبارة عن اللي هي local maximum أو local
+
+257
+00:16:59,860 --> 00:17:04,580
+minimum مين هي النقطة سفرأو 0 إذا الـ Critical
+
+258
+00:17:04,580 --> 00:17:09,320
+Point هنا is the only 00 وهي نقطة برضه الـ
+
+259
+00:17:09,320 --> 00:17:12,680
+Function عندها Differentiable أوين ما كانت إذا هذه
+
+260
+00:17:12,680 --> 00:17:15,360
+النقطة ممكن تطلع Local Maximum و Local Minimum أو
+
+261
+00:17:15,360 --> 00:17:19,360
+تطلع Critical Point تنشوف أو ممكن ما نعرفش نوجدها
+
+262
+00:17:19,360 --> 00:17:22,840
+لكن إحنا جايبين سؤال عساس إنه سهل التعامل معاه
+
+263
+00:17:22,840 --> 00:17:29,200
+تنشوف إيش اللي بقولهالان لاحظ لو
+
+264
+00:17:29,200 --> 00:17:34,860
+جينا along the X axis النقطة مين هي 00 هيا لو جيت
+
+265
+00:17:34,860 --> 00:17:38,880
+انت اخدت جواه اي جوار حوالينه اي جوار حوالينه هنا
+
+266
+00:17:38,880 --> 00:17:42,920
+هيكون الجوار هذا هياخد من ال X axis ال positive و
+
+267
+00:17:42,920 --> 00:17:46,610
+من ال X axis ال negativeمن الـ Y Axis الـ Positive
+
+268
+00:17:46,610 --> 00:17:50,150
+من الـ Y Axis إيش؟ الـ Negative خلّينا ناخد along
+
+269
+00:17:50,150 --> 00:17:54,890
+the X Axis F has the value يعني عوالين من لما X ال
+
+270
+00:17:54,890 --> 00:17:58,730
+Axis لما Y بساوي 0 هتلاقي F of X و Zero إيش بتساوي
+
+271
+00:17:58,730 --> 00:18:02,410
+ناقص X تربيع اللي هي إيش ما لها أصغر من Zero إذا
+
+272
+00:18:02,410 --> 00:18:05,890
+أي جوار بتقدر تاخدوها يعني لو جيت على ال Positive
+
+273
+00:18:05,890 --> 00:18:11,790
+X Axis على ال Negative X Axisأنا ماشي على اللي هو
+
+274
+00:18:11,790 --> 00:18:17,230
+ال X-axis هنا و جيت على المنطقة اللي هي هنا اللي
+
+275
+00:18:17,230 --> 00:18:20,990
+هي هتلاقي مش مشكلة البوز تقول أنا جيت في حياتي لأن
+
+276
+00:18:20,990 --> 00:18:25,710
+نحن نربح هيعطي نفس القيمة خد تعالي على اللي هي مين
+
+277
+00:18:25,710 --> 00:18:30,330
+ال X-axis هي X-axis خد أي جوار حواليه أي جوار هنا
+
+278
+00:18:30,330 --> 00:18:33,720
+الآنهذا الجوار لو جيته على النقاط اللي على ال X
+
+279
+00:18:33,720 --> 00:18:37,500
+-axis اللي هي بمعنى أخر Y ب0 بصير F of X و Zero
+
+280
+00:18:37,500 --> 00:18:41,480
+ليه حسب الدلة اللي عندي نقص X تربيع أكيد أصغر من
+
+281
+00:18:41,480 --> 00:18:45,580
+أياش من سفر الان على نفس الجوار اللي قبل بشوية
+
+282
+00:18:45,580 --> 00:18:50,900
+اللي أخدته ال arbitrary الجوار العشوائيخُد نقطة
+
+283
+00:18:50,900 --> 00:19:01,680
+عالميا على الـ y-axis الـ x بصفر و الـ y تربيع
+
+284
+00:19:01,680 --> 00:19:07,840
+أكبر من aأي جوار حوالين اللي هو السفر مهما صغر
+
+285
+00:19:07,840 --> 00:19:12,120
+هتلاقي اللي هي نقاط على الـ X-Axis بتعطي قيمة الـ
+
+286
+00:19:12,120 --> 00:19:15,100
+Function أصغر من سفر ونقاط على الـ Y-Axis بتعطي
+
+287
+00:19:15,100 --> 00:19:18,560
+قيمة الـ Function أشمالها أكبر من سفر يعني اللي هي
+
+288
+00:19:18,560 --> 00:19:23,050
+F .. طبعا سفر مين هذا السفر؟ دي قيمة F of 00اللي
+
+289
+00:19:23,050 --> 00:19:26,450
+هي النقطة اللي بنبحث عنها F of 00 هنلاقي Zero نقل
+
+290
+00:19:26,450 --> 00:19:30,930
+Zero ليها Zero يعني دائما F of 00 في أي جوار
+
+291
+00:19:30,930 --> 00:19:35,450
+حوالين النقطة هذه لا يمكن نلاقي جوار انها تكون
+
+292
+00:19:35,450 --> 00:19:38,270
+أكبر من كل اللي حواليها أو أصغر من كل اللي حواليها
+
+293
+00:19:38,270 --> 00:19:43,330
+بل هنلاقي نقاط هيها أصغر منهاأو نقاط أكبر منها
+
+294
+00:19:43,330 --> 00:19:53,170
+مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا
+
+295
+00:19:53,170 --> 00:19:58,950
+مثلًا مثلًا مثلًا مثلًاSint and a zero zero
+
+296
+00:19:58,950 --> 00:20:02,010
+contains points where the function is positive and
+
+297
+00:20:02,010 --> 00:20:05,630
+points where the function is negative hence the
+
+298
+00:20:05,630 --> 00:20:09,210
+function has a saddle point at the origin and low
+
+299
+00:20:09,210 --> 00:20:14,330
+or local extremum values طيب، من هذا السؤال بنقدر
+
+300
+00:20:14,330 --> 00:20:17,490
+نستنتج إنه مش معناته F of X بيساوي F of Y بيساوي
+
+301
+00:20:17,490 --> 00:20:22,370
+سفر وكانت interior point اللي هو بتضمنلنا إنه ال F
+
+302
+00:20:22,370 --> 00:20:25,700
+إلها local maximum ولا local minimum، لأممكن تكون
+
+303
+00:20:25,700 --> 00:20:29,200
+Fx بسوء و Fy بسوء سفر و مايكونش عندنا Local
+
+304
+00:20:29,200 --> 00:20:33,000
+Extreme Value يعني لا Local Maximum ولا Local
+
+305
+00:20:33,000 --> 00:20:36,100
+Minimum Local Extreme Value يعني يا Local Maximum
+
+306
+00:20:36,100 --> 00:20:40,040
+يا Local Minimum الآن Fx بسوء و Fy بسوء سفر لها
+
+307
+00:20:40,040 --> 00:20:44,180
+تضمن لنا اللي هي وجود اللي هو أن F يكون Local
+
+308
+00:20:44,180 --> 00:20:49,450
+Maximum و Local Minimumإذن يا جماعة، صعب نظل نشتغل
+
+309
+00:20:49,450 --> 00:20:54,450
+على التعريف بدنا نظريات مساعدة تحكمنا وتحكمنا
+
+310
+00:20:54,450 --> 00:20:58,530
+وتجاوبنا على السؤال التاني هل فعلا في Local
+
+311
+00:20:58,530 --> 00:21:01,870
+Maximum و Local Minimum أو Settle Point ولا لأ
+
+312
+00:21:01,870 --> 00:21:04,710
+وهذا اللي هيجاوبنا عليهحاجة اسمها الـ Second
+
+313
+00:21:04,710 --> 00:21:08,390
+Derivative Test ‏for Local Extreme Values ‏هذه
+
+314
+00:21:08,390 --> 00:21:11,750
+اللي هي الـ Second Derivative Test ‏هتجاوبنا على
+
+315
+00:21:11,750 --> 00:21:16,150
+عائلة كبيرة من الدوال ‏اللي هي هذا موضوع اختصاصنا
+
+316
+00:21:16,150 --> 00:21:19,670
+في هذا الفصل ‏اللي هو Subposed Data ‏هذا إذا
+
+317
+00:21:19,670 --> 00:21:22,830
+السؤال الأول قلنا أين يمكن إيجاد ‏الـ Local
+
+318
+00:21:22,830 --> 00:21:26,050
+Maximum و Local Minimum ‏أو اللي هو أيضا الـ
+
+319
+00:21:26,050 --> 00:21:28,990
+Cedric Points ‏قلنا وين ‏الآن هذه بدأت تجاوبنا
+
+320
+00:21:28,990 --> 00:21:32,380
+تحكمناعلى إن هذه الـ Local Maximum والـ Local
+
+321
+00:21:32,380 --> 00:21:35,280
+Minimum ولّت Set الـ Point وخلّينا نشوف إيه اللي
+
+322
+00:21:35,280 --> 00:21:38,740
+بنحكيه Suppose that f of x, y and its and second
+
+323
+00:21:38,740 --> 00:21:41,720
+partial derivatives are continuous فرضين إن اللي
+
+324
+00:21:41,720 --> 00:21:45,080
+هي ال partial derivatives موجودة واللي هي ال
+
+325
+00:21:45,080 --> 00:21:48,280
+function continuous عند هذا المنطقة اللي هي throw
+
+326
+00:21:48,280 --> 00:21:52,380
+mean a وbعند Fx و Fy بيساوي إيش؟ بيساوي سفر جداً
+
+327
+00:21:52,380 --> 00:21:56,140
+هذه بتفحص لمين يا جماعة؟ هي اختصاص نحن الآن يعني
+
+328
+00:21:56,140 --> 00:22:00,300
+احنا مش هنتوسع في الدراسة إلا بما يُطلب مننا أنه
+
+329
+00:22:00,300 --> 00:22:04,640
+عند الـ Fx بيساوي Fy بيساوي سفر الآن الـ Critical
+
+330
+00:22:04,640 --> 00:22:08,660
+Points اللي بتطلع من هذه هي أحد ثلاث أمور يا Local
+
+331
+00:22:08,660 --> 00:22:10,880
+Maximum يا Local Minimum يا Local يساد الـ Point
+
+332
+00:22:10,880 --> 00:22:16,880
+يا ما مقدرش نحكمإذاً هذه نظرية اللي بتقولنا الحكم
+
+333
+00:22:16,880 --> 00:22:20,160
+كما يالي خلّيني نتبع معناه في حاجة اسمها الـ
+
+334
+00:22:20,160 --> 00:22:24,300
+Discrement اللي هو Fx Fy نقص Fxy تربيع اللي هو هذا
+
+335
+00:22:24,300 --> 00:22:27,380
+الـ Discrement أو الـ Hessian هذا اللي هو اللي
+
+336
+00:22:27,380 --> 00:22:31,620
+بدنا من خلاله نحكم طبعاً هذا Fx X Fyy نقص Fxy
+
+337
+00:22:31,620 --> 00:22:34,780
+تربيع بساوي الـ determinant هذا يمكن الواحد هيك
+
+338
+00:22:34,780 --> 00:22:42,900
+يكون متذكر أحسن Fx Fxy Fxy Fyyهذا في هذا ناقص هذا
+
+339
+00:22:42,900 --> 00:22:46,680
+تربيع اللي هو هذا الـ Discrement نوجده بعد ما
+
+340
+00:22:46,680 --> 00:22:51,240
+نوجده عند النقطة طبعا A و B إذا كان أكبر من سفر
+
+341
+00:22:51,240 --> 00:22:55,960
+يكون عندي يا Local Maximum يا Local Minimum الآن
+
+342
+00:22:55,960 --> 00:23:00,790
+لقنا أكبر من سفر باجي بفحص FXXإذا أصغر من 0 على
+
+343
+00:23:00,790 --> 00:23:03,650
+طول بقول Localish Maximum يعني إذا سالبة Local
+
+344
+00:23:03,650 --> 00:23:07,410
+Maximum المشابهة لـ Second Derivative Test اللي هو
+
+345
+00:23:07,410 --> 00:23:10,310
+الـ Function of One Variable كانت إذا كانت تطلع
+
+346
+00:23:10,310 --> 00:23:15,070
+عندي الـ F'' أصغر من 0 بقول عنها Localish Maximum
+
+347
+00:23:16,070 --> 00:23:19,210
+الان هذا بس عنا عشان function of two variables فيه
+
+348
+00:23:19,210 --> 00:23:23,330
+عنا XX وXY وYY فعشانك الـ Discrement هذا هو اللي
+
+349
+00:23:23,330 --> 00:23:26,610
+بيحدد أكبر من سفر معناته فيه Local Maximum أو
+
+350
+00:23:26,610 --> 00:23:29,210
+Local Minimum مين بيحدد Local Maximum أو Local
+
+351
+00:23:29,210 --> 00:23:33,330
+Minimum إشارة FXX إذا كانت FXX أصغر من سفر بتكون
+
+352
+00:23:33,330 --> 00:23:37,050
+عندي Local Maximum إذا FXX أكبر من سفر بتكون عندي
+
+353
+00:23:37,050 --> 00:23:41,400
+Local Minimumالان لو لاجينا هذا الـ Discrement
+
+354
+00:23:41,400 --> 00:23:44,700
+أصغر من 0 على طول بنحكم أنها إيه شمالها settle
+
+355
+00:23:44,700 --> 00:23:48,880
+point إذا أصغر من 0 على طول settle point بساوي 0
+
+356
+00:23:48,880 --> 00:23:54,040
+بنقول ال test is inconclusive معناته اللي هو إيه
+
+357
+00:23:54,040 --> 00:23:57,040
+شماله official ال test وبنقدرش وبنبحث عن طرق أخرى
+
+358
+00:23:57,040 --> 00:24:01,300
+إحنا هنشتغل على جدنا الآن اللي هي إذا كنا نبحث عن
+
+359
+00:24:01,300 --> 00:24:05,850
+طرق أخرى بتروح لمين للتعريف اللي قبل بشويةالأن، مش
+
+360
+00:24:05,850 --> 00:24:09,890
+هنسى القلة غير بالأشياء المقدورة عليها، الان عندي
+
+361
+00:24:09,890 --> 00:24:13,430
+اللي هو هذا الـdiscriminate هو اللي بحدد، المميز
+
+362
+00:24:13,430 --> 00:24:20,290
+أو اللي بده يحدد أنها، إذا كان سفر inclusive، إذا
+
+363
+00:24:20,290 --> 00:24:23,370
+كانت أكبر من سفر، يالو كال maximum، يالو كال
+
+364
+00:24:23,370 --> 00:24:26,510
+minimum، أصغر من سفر على طول، settle point، وهذا
+
+365
+00:24:26,510 --> 00:24:30,280
+عند المطلق المذكور A وBنجي نقض هذا عمليًا Find
+
+366
+00:24:30,280 --> 00:24:34,160
+local extreme values of the function f of x,y بسوء
+
+367
+00:24:34,160 --> 00:24:37,360
+x,y نقص x تابع نقص y تابع نقص 2x نقص 2y زي أضارب
+
+368
+00:24:37,360 --> 00:24:40,880
+عيا نحلاها Differentiable وين ما كان اللي هي إذن
+
+369
+00:24:40,880 --> 00:24:43,980
+فش boundary points وفيش does not exist إذن هتنطبق
+
+370
+00:24:43,980 --> 00:24:46,920
+نظريتنا تمام التمام واحنا جايبين عشان هي إذن بس
+
+371
+00:24:46,920 --> 00:24:51,820
+هنبحث وين في fx بسوء fy بسوء 0 ومن ثم اللي هو نحكم
+
+372
+00:24:51,820 --> 00:24:55,400
+على النقاط إنها إيه شمالها إن إنها local maximum
+
+373
+00:24:55,400 --> 00:25:01,390
+وlocal minimum ولا saddle pointأو دي لـ FX عند FX
+
+374
+00:25:01,390 --> 00:25:06,750
+أجملها في اللي فوق اللي هي عبارة عن Y نقص 2X نقص 2
+
+375
+00:25:06,750 --> 00:25:15,200
+صح؟ Y نقص 2X نقص 2الان f y هي عبارة عن x نقص اتنين
+
+376
+00:25:15,200 --> 00:25:20,020
+y نقص اتنين x نقص اتنين y نقص اتنين فيه symmetry
+
+377
+00:25:20,020 --> 00:25:24,340
+term الآن خلّينا ناخد هدي بصفر و هدي بصفر let f x
+
+378
+00:25:24,340 --> 00:25:27,140
+بيساوي f y بيساوي صفر together عندى بتصير من
+
+379
+00:25:27,140 --> 00:25:31,400
+الأولى y بتساوي اتنين x زي اتنين والتانية اتنين y
+
+380
+00:25:31,400 --> 00:25:38,690
+بيساوي x minus اتنينبنساوى هذه بالصفر وهذه بالصفر
+
+381
+00:25:38,690 --> 00:25:43,010
+فبتطلع عندى اللي هي y بالشكل هذا و2y بالشكل هذا
+
+382
+00:25:43,010 --> 00:25:46,350
+لما نجيه لجهة تانية هذه 2y لجهة هذه بصير x ناقص 2
+
+383
+00:25:46,350 --> 00:25:50,210
+إلى أن نحل المعادلتين هذول الانيتين مع بعض وضربنا
+
+384
+00:25:50,210 --> 00:25:55,510
+هذه في 2 بصير 2y 4x زيادة 4 اللي هى بنترحل من بعض
+
+385
+00:25:55,510 --> 00:26:05,810
+بصير 2y ناقص 2y صفر 4x ناقص اللي هو x بتطلع 3xأربع
+
+386
+00:26:05,810 --> 00:26:09,150
+ناقص ناقص اتنين هي زائد ستة بيصير لي هي تلاتة X
+
+387
+00:26:09,150 --> 00:26:12,550
+زائد ستة بصفر إذن X بتساوي إيش؟ سالب اتنين مش
+
+388
+00:26:12,550 --> 00:26:15,870
+اتنين و لو عوضنا X بسالب اتنين بيصير سالب أربع
+
+389
+00:26:15,870 --> 00:26:18,890
+زائد اتنين بتطلع برضه سالب اتنين يعني النقطة عند X
+
+390
+00:26:18,890 --> 00:26:22,530
+بتساوي سالب اتنين و Y بتساوي سالب اتنين إذن النقطة
+
+391
+00:26:22,530 --> 00:26:25,290
+اللي ممكن يكون عندها maximum أو minimum أو saddle
+
+392
+00:26:25,290 --> 00:26:29,250
+point هي ناقص اتنين و ناقص اتنين ماشي الحال مش
+
+393
+00:26:29,250 --> 00:26:34,070
+اتنين باتنين ناقص اتنين ناقص اتنينالان نجي نفحص
+
+394
+00:26:34,070 --> 00:26:37,830
+الفحص الان هو نوجد ايه بالنسبة للـ Discrement نوجد
+
+395
+00:26:37,830 --> 00:26:47,850
+فXX هي فX فXX بصير نقص اتنين فXY فXY هي Y نفضلها
+
+396
+00:26:47,850 --> 00:26:53,780
+بالنسبة ل Xبنطلع واحد fyy بتطلع عبارة عن إيش نقص
+
+397
+00:26:53,780 --> 00:26:57,660
+اتنين برضه ماشي الحالة طبعا ممكن تفاضل هذه fxy
+
+398
+00:26:57,660 --> 00:27:01,340
+بالنسبة ل y برضه بتطلع واحد زي ما قلنا انه اللي هو
+
+399
+00:27:01,340 --> 00:27:05,660
+تنتين continuous ال first ال second و ال first
+
+400
+00:27:05,660 --> 00:27:08,440
+partial derivative is continuous و ده ال fxy بشوة
+
+401
+00:27:08,440 --> 00:27:14,620
+fyx زي ما قدنا ده قديماالان هذه هي الأدوات اللي
+
+402
+00:27:14,620 --> 00:27:17,280
+بتخلينا نحكم الان نجد الـ Discrement ايش الـ
+
+403
+00:27:17,280 --> 00:27:22,920
+Discrement؟ Fxx في Fyy ناقص Fxy تربيع هي .. هي هذه
+
+404
+00:27:22,920 --> 00:27:26,760
+ناقص اتنين في ناقص اتنين تطلع أياش أربعة ناقص Fxy
+
+405
+00:27:26,760 --> 00:27:29,640
+لكل تربيع اللي هو واحد اللي هو بسوء جديش تلاتة
+
+406
+00:27:29,640 --> 00:27:33,240
+اللي هو صار عندي أكبر من سفر إذا مدام أكبر من سفر
+
+407
+00:27:33,240 --> 00:27:37,660
+إذاعندي الان يا local maximum يا local minimum مين
+
+408
+00:27:37,660 --> 00:27:41,840
+اللي بدى يحكم ليه اللي هي fxx مدامة fxx بتساوي
+
+409
+00:27:41,840 --> 00:27:45,580
+سالب اتنين اذا وانا مغمض الـ f has local maximum
+
+410
+00:27:45,580 --> 00:27:48,800
+ات اتنين و اتنين جداش قيمة الـ local maximum في
+
+411
+00:27:48,800 --> 00:27:52,980
+هذه الحالة بنحسب f of نقص اتنين و نقص اتنين نقص
+
+412
+00:27:52,980 --> 00:27:56,660
+اتنين طبعاً f of نقص اتنين و نقص اتنين احسبها بيه
+
+413
+00:27:56,660 --> 00:28:00,040
+نقص اتنين في نقص اتنين بيه عبارة عن أربعة وهد نقص
+
+414
+00:28:00,040 --> 00:28:06,200
+أربعة وهد نقص أربعة وهد اللي هيعبارة عن زائد أربعة
+
+415
+00:28:06,200 --> 00:28:12,740
+وهذه عبارة عن زائد أربعة اجمح مع بعض اللي هي بيصير
+
+416
+00:28:12,740 --> 00:28:17,440
+عندي أربعة ناقص أربعة ناقص أربعة اللي هي هذه ناقص
+
+417
+00:28:17,440 --> 00:28:18,880
+ثمانية بتروح مع ناقص تمانية
+
+418
+00:28:24,330 --> 00:28:28,850
+القيمة الـ F عنده نقطة نقص 2 و نقص 2 وتأكد أنك
+
+419
+00:28:28,850 --> 00:28:32,450
+أيضًا ستجد ما تساوي بتساوي اللي هي الـ absolute
+
+420
+00:28:32,450 --> 00:28:37,210
+value بساوي اللي هي قيمة الـ F عندها نقطة نقص 2 و
+
+421
+00:28:37,210 --> 00:28:41,990
+نقص 2 بساوي فعلاً اللي هو الأرقام جديش بساوي 8ماشي
+
+422
+00:28:41,990 --> 00:28:46,050
+يا جماعة ناخد مثال آخر find local extreme values
+
+423
+00:28:46,050 --> 00:28:50,330
+of x,y بيسوي تلاتة y تربيع نقص اتنين y تكعيب نقص
+
+424
+00:28:50,330 --> 00:28:54,310
+تلاتة x تربيع زي ستة xy برضه هذا يعني أحلى اللي هي
+
+425
+00:28:54,310 --> 00:28:59,010
+differentiable وين مكان ومدام differentiable وين
+
+426
+00:28:59,010 --> 00:29:03,190
+مكان إذا ال boundary points فشي وشمالينها وين مكان
+
+427
+00:29:03,190 --> 00:29:08,600
+معرفةالحاجة الثانية اللي هو الـ Fx و Fy لا توجد مش
+
+428
+00:29:08,600 --> 00:29:12,000
+موجودة برضه إذا بس هنتبحث في حالة Fx بيساوي Fy
+
+429
+00:29:12,000 --> 00:29:15,020
+بيساوي 0 ونشوف فيها النقاط اللي ممكن يكون عندها
+
+430
+00:29:15,020 --> 00:29:18,680
+maximum أو minimum أو سادي ال point وبعدين بنفحصها
+
+431
+00:29:18,680 --> 00:29:22,400
+إذا الآن زي ما قلنا بتجيب Fx ومن Fy بيكونوا ساويين
+
+432
+00:29:22,400 --> 00:29:27,380
+بالزفر Fx مش هتساوي اللي هو عبارة عن Fx Fx هيناقص
+
+433
+00:29:27,380 --> 00:29:35,200
+6x زائد 6yأف واي خلّيني أجدها هي 6y نقص 6y تربيع
+
+434
+00:29:35,200 --> 00:29:40,740
+زائد 6x هي 6y نقص 6y تربيع زائد 6x بنسويها للترفين
+
+435
+00:29:40,740 --> 00:29:46,180
+بالصفر بيطلع عندي اللي هو من الأولى 6y نقص 6xبساوي
+
+436
+00:29:46,180 --> 00:29:50,800
+0 يعني Y بتساوي X ومن الثانية اللي هو بتطلع عندي
+
+437
+00:29:50,800 --> 00:29:56,160
+اللي هي X بتساوي اللي هو Y تربيع ناقص Y ماشي الحال
+
+438
+00:29:56,160 --> 00:30:02,240
+إذا لما نحلهم مع بعض بنعوض اللي هو مكان X بY بيصير
+
+439
+00:30:02,240 --> 00:30:07,710
+Y بتساوي Y تربيع ناقص Y بنحل هذه المعادلةبصير عندى
+
+440
+00:30:07,710 --> 00:30:13,790
+اللى هي y في y طبعا الان تيجي ال y هي بصير نقص
+
+441
+00:30:13,790 --> 00:30:19,470
+اتنين y بصير y تربيع نقص اتنين y بيساوي سفر يعني y
+
+442
+00:30:19,470 --> 00:30:22,770
+تربيع نقص اتنين y بيساوي سفر يعني بيصير تاخد y عم
+
+443
+00:30:22,770 --> 00:30:25,850
+نشترك بيصير y في y minus اتنين بيساوي سفر يعني يا
+
+444
+00:30:25,850 --> 00:30:29,790
+اما y بيساوي سفر يا اما y بيساوي اتنين ماشي لكن ال
+
+445
+00:30:29,790 --> 00:30:32,970
+y بتساوي x إذا انصار عندى y بتساوي
+
+446
+00:30:45,200 --> 00:30:53,350
+عشان نفحصهم لازم نجيب fxx, fxy, fy, yالان fxx هي
+
+447
+00:30:53,350 --> 00:30:56,350
+fx موجودة نفضلها بالنسبة لـ x كمان و مرة بتطلع نقص
+
+448
+00:30:56,350 --> 00:31:02,930
+ستة fxy اللي بنفضلها بتطلع برضه ستة الان fyy لما
+
+449
+00:31:02,930 --> 00:31:08,530
+نفضلها بتطلع ستة نقص اتناشر y ستة نقص اتناشر y إذن
+
+450
+00:31:08,530 --> 00:31:12,130
+هدولة بدنا نحسب هنا عند مين عند أول شي سفر سفر و
+
+451
+00:31:12,130 --> 00:31:15,290
+بعدين عند اتنين و اتنين و بعدين اللي هو نطبق نجيه
+
+452
+00:31:15,290 --> 00:31:20,180
+بالـ Discrementالان fyy عند الـ 00 اللي هي بتساوي
+
+453
+00:31:20,180 --> 00:31:25,980
+اللي هي 6 وfyy عند ال 2 وال2 عوض بيصير 6 ناقص اللي
+
+454
+00:31:25,980 --> 00:31:31,320
+هو 24 تطلع ناقص 18 إذا الأمور جاهزة يا جماعة الآن
+
+455
+00:31:31,320 --> 00:31:36,000
+هدولة أصلا قيمتهم ثابتة عند الـ 00 وغيرهم الان
+
+456
+00:31:36,000 --> 00:31:40,880
+نبدأ نوجد مين الان الـ Discretement في كل حالةالان
+
+457
+00:31:40,880 --> 00:31:44,560
+عند الـ 00 بكون الـ Discrement هذا اللي هو بيساوي
+
+458
+00:31:44,560 --> 00:31:50,680
+بنعوض Fxx فFyy هي عندي اللي هو ناقص ستة بستة بناقص
+
+459
+00:31:50,680 --> 00:31:54,900
+ستة و تلاتين Fxy لكل تربيع هي Fxy اللي هي ستة و
+
+460
+00:31:54,900 --> 00:31:57,780
+تلاتين بيطلع ناقص اتنين و سبعين أصغر من سفر إذا
+
+461
+00:31:57,780 --> 00:32:01,020
+مدامة اللي هي عند الـ 00 طالع أصغر من سفر اللي هو
+
+462
+00:32:01,020 --> 00:32:03,460
+قيمة الـ Discrement إذا و أنا مغمض بقولها دي إيش
+
+463
+00:32:03,460 --> 00:32:07,820
+مالهاسادل بوينت عند الـ 00 نجي عند مين الـ 2
+
+464
+00:32:07,820 --> 00:32:10,700
+والتنين الـ 2 والتنين بعوض عند القيم اللي اللي
+
+465
+00:32:10,700 --> 00:32:17,300
+جيتها اللي هي XX وXY هيها ناقص 6 في 18 بالناقص
+
+466
+00:32:17,300 --> 00:32:20,980
+بيطلع زائد 108التي هي fx y تربيهها طبعاً زي ما هي
+
+467
+00:32:20,980 --> 00:32:23,520
+نقص ستة و تلاتين هذه نقص هذه بيطلع تنين و سبعين
+
+468
+00:32:23,520 --> 00:32:27,100
+برضه اشماله أكبر من سفر إذن مدام أكبر من سفر يا
+
+469
+00:32:27,100 --> 00:32:30,120
+local maximum يا local minimum مين اللي بيحدد ليه؟
+
+470
+00:32:30,120 --> 00:32:34,080
+fxx مدام fxx سالبة إذن على طول بقول في عندي local
+
+471
+00:32:34,080 --> 00:32:37,460
+maximum عند النقطة اتنين و اتنين وال value لل F
+
+472
+00:32:37,460 --> 00:32:41,740
+عند هذه النقطة اللي هو F of اتنين و اتنين بعوض عن
+
+473
+00:32:41,740 --> 00:32:45,500
+اللي هو اتنين و اتنين هنا بتطلع عندي عبارة عن جديش
+
+474
+00:32:45,500 --> 00:32:49,440
+بتساوي تمانية و اتأكد من هذا الكلاموبكون عندى اللى
+
+475
+00:32:49,440 --> 00:32:51,860
+هو فعلا عندى واحدة settle point واحدة local
+
+476
+00:32:51,860 --> 00:32:57,240
+maximum وهكذا احنا أخدنا أنفلة كفاية عن مرة الآن
+
+477
+00:32:57,240 --> 00:33:02,460
+الجزء الثاني من الموضوع اللى هو احنا لحد الآن
+
+478
+00:33:02,460 --> 00:33:05,760
+ماتطرقناش على مناطق فيها boundary points الآن في
+
+479
+00:33:05,760 --> 00:33:10,920
+نظريةاللي هي بتقولنا ما هي ليه أنه لو كان عندك
+
+480
+00:33:10,920 --> 00:33:18,200
+اللي هي f of x y عندها ال f x x و ال f x y و ال f
+
+481
+00:33:18,200 --> 00:33:20,460
+كده اللي كلنا دول ال partial derivatives موجودات و
+
+482
+00:33:20,460 --> 00:33:23,620
+continuous و كمان لو كانت ال function
+
+483
+00:33:23,620 --> 00:33:28,980
+differentiable نور على نوربنضمن أن الـ Function
+
+484
+00:33:28,980 --> 00:33:33,460
+يمتلك Absolute Maximum و Absolute Minimum هذا طبعا
+
+485
+00:33:33,460 --> 00:33:36,720
+الكلام على مين؟ على Closed Bounded Region إذن
+
+486
+00:33:36,720 --> 00:33:39,480
+بنحكي الآن Absolute Maximum و Absolute Minimum و
+
+487
+00:33:39,480 --> 00:33:44,480
+Closed Bounded Region اللي هو إحنا بنضمن وجود الـ
+
+488
+00:33:44,480 --> 00:33:46,500
+Absolute Maximum و Absolute Minimum على Closed
+
+489
+00:33:46,500 --> 00:33:50,460
+Bounded Regionإذا كانت الـ F X والـ F Y والـ F Z
+
+490
+00:33:50,460 --> 00:33:53,840
+إذا كان نحكي عن الـ three variables كلها موجودة
+
+491
+00:33:53,840 --> 00:33:58,520
+وcontinuous ويكون عنده اللي هو الـ function
+
+492
+00:33:58,520 --> 00:34:01,840
+attains its maximum and its minimum يعني لازم يكون
+
+493
+00:34:01,840 --> 00:34:04,560
+فيه عنده maximum و minimum absolute maximum و
+
+494
+00:34:04,560 --> 00:34:08,840
+absolute minimum طب كيف نوجدها نوجدها عبر ما يليه
+
+495
+00:34:08,840 --> 00:34:13,350
+أول حاجة بنيجي للـ interior pointsبنشوف وين ممكن
+
+496
+00:34:13,350 --> 00:34:16,550
+نلاقي الـ maximum و الـ minimum اللي هي عارفين وين
+
+497
+00:34:16,550 --> 00:34:21,050
+عند .. يا إما عند الـ critical points F may have
+
+498
+00:34:21,050 --> 00:34:24,410
+local maximum and minimum عند الـ critical points
+
+499
+00:34:24,410 --> 00:34:26,470
+اللي هي في الـ interior points اللي عند critical
+
+500
+00:34:26,470 --> 00:34:33,400
+pointsاللي هي بنحسب في هذه الحالة الـ F عندهم، عند
+
+501
+00:34:33,400 --> 00:34:36,520
+كل النقاط اللي لاجناه، زي ما عملنا قبل شوية و
+
+502
+00:34:36,520 --> 00:34:40,360
+بنحطهم، بنخزنهم، بنيجي لمين؟ للـ Boundary الـ
+
+503
+00:34:40,360 --> 00:34:43,620
+Boundary Points of R، اللي هو حسب إيش، اللي هو ال
+
+504
+00:34:43,620 --> 00:34:47,460
+region اللي عندنا، بمشي على ال boundaries و بوجد
+
+505
+00:34:47,460 --> 00:34:50,280
+الأماكن اللي ممكن نلاقي فيها maximum أو minimum،
+
+506
+00:34:50,280 --> 00:34:54,900
+هنقول كيف نوجدهموبنلاقي النقاط هذه وبرضه بنحطها في
+
+507
+00:34:54,900 --> 00:34:58,640
+list في قائمة وبنوجد الـ F عندها إلى بعد ما
+
+508
+00:34:58,640 --> 00:35:02,720
+أوجدناها من في الحالتين الأولى يا إما من ال
+
+509
+00:35:02,720 --> 00:35:05,800
+critical points أو من ال boundary points باخد اللي
+
+510
+00:35:05,800 --> 00:35:08,460
+هو أكبر واحدة فيهم بتكون هي ال absolute فيها
+
+511
+00:35:08,460 --> 00:35:11,900
+maximum وأصغر واحدة فيهم هي ال absolute minimum
+
+512
+00:35:11,900 --> 00:35:15,620
+وإذا الموضوع صار سهل look through the list for the
+
+513
+00:35:15,620 --> 00:35:18,780
+maximum and minimum values of F these will be
+
+514
+00:35:18,780 --> 00:35:21,710
+absolute maximum and minimum values of FSince
+
+515
+00:35:21,710 --> 00:35:24,950
+absolute maximum and minimum are also local
+
+516
+00:35:24,950 --> 00:35:27,210
+maximum and minimum are absolute maximum and
+
+517
+00:35:27,210 --> 00:35:30,010
+minimum values وفقه بيرسم somewhere in the list
+
+518
+00:35:30,010 --> 00:35:33,190
+made list بواحدة اتنية وكل ما هو ال absolute
+
+519
+00:35:33,190 --> 00:35:37,070
+maximum و absolute minimum هيكون واحدة من ال local
+
+520
+00:35:37,070 --> 00:35:41,530
+maximum و local minimumالان يعني اللي بنحكي أنه لو
+
+521
+00:35:41,530 --> 00:35:45,530
+طبعا مشابه للنظرية اللي أخدناها زمان في calculus
+
+522
+00:35:45,530 --> 00:35:47,670
+الف ان if f is continuous on a closed bounded
+
+523
+00:35:47,670 --> 00:35:50,410
+interval then it attains its maximum and its
+
+524
+00:35:50,410 --> 00:35:53,670
+minimum in this interval أو نفس الطرق هنا إذا كانت
+
+525
+00:35:53,670 --> 00:35:58,190
+f x, f y, f z اللي هي exist and continuous then it
+
+526
+00:35:58,190 --> 00:36:01,170
+attains f attains its maximum and its minimum in
+
+527
+00:36:01,170 --> 00:36:04,510
+this closed region طبعا محتاجة closed region أو
+
+528
+00:36:04,510 --> 00:36:09,150
+قلنا كيف نوجدهاخلّينا نشوف هذا بواسطة مثال عملي
+
+529
+00:36:09,150 --> 00:36:13,350
+Find the absolute maximum and minimum values of f
+
+530
+00:36:13,350 --> 00:36:16,470
+of x, y بساوية الأمامي on the triangular region in
+
+531
+00:36:16,470 --> 00:36:19,910
+the first quadrant bounded by the lines x بساوية
+
+532
+00:36:19,910 --> 00:36:23,600
+0, y بساوية 0, y بساوية 9 minus xلأن في شغلتين
+
+533
+00:36:23,600 --> 00:36:27,460
+مهمات الشغل الأولى انه اللي هو الـ interior points
+
+534
+00:36:27,460 --> 00:36:30,460
+كيف نوجد اللي هو الـ absolute maximum كيف نوجد
+
+535
+00:36:30,460 --> 00:36:33,380
+اللي هو وين الـ interior points مش هنفحص وين فيه
+
+536
+00:36:33,380 --> 00:36:35,360
+absolute maximum ولا absolute minimum احنا ضمنين
+
+537
+00:36:35,360 --> 00:36:38,140
+انه على close bounded region هيكون فيه عندى
+
+538
+00:36:38,140 --> 00:36:40,520
+absolute maximum و absolute minimum بنجيب ال
+
+539
+00:36:40,520 --> 00:36:43,580
+critical points و بنجيب ال boundary points وين فيه
+
+540
+00:36:43,580 --> 00:36:46,200
+إمكانية وجود ال local maximum و ال local minimum و
+
+541
+00:36:46,200 --> 00:36:50,050
+بنعوض عنهمبنعوض بنجيب أفقلين أكبر واحدة هي
+
+542
+00:36:50,050 --> 00:36:53,050
+absolute maximum وأكبر واحدة absolute minimum إذا
+
+543
+00:36:53,050 --> 00:36:57,190
+الخطوات كميالي اللي هو بنحدد اللي هي من شغلتين
+
+544
+00:36:57,190 --> 00:36:59,530
+مهمات الشغلة الأولى اللي حدد المنطقة الشغلة
+
+545
+00:36:59,530 --> 00:37:03,150
+الثانية اللي هو بنبدأ نشتغل على ال interior هذا
+
+546
+00:37:03,150 --> 00:37:07,020
+بتحدد ال boundary وهذا بتحدد ال interiorالان X
+
+547
+00:37:07,020 --> 00:37:09,920
+بيساوي سفر و Y بيساوي سفر و Y بيساوي تسعة مانوس X
+
+548
+00:37:09,920 --> 00:37:13,420
+X بيساوي سفر اللي هو الـ Y Axis هيه Y بيساوي سفر
+
+549
+00:37:13,420 --> 00:37:18,320
+هيه و الخط Y بيساوي تسعة ناقص X خد X بتسعة بتطلع
+
+550
+00:37:18,320 --> 00:37:25,840
+اللي هي Y بسفر هيها أو خد اللي هو X بسفر بتطلع Y
+
+551
+00:37:25,840 --> 00:37:29,720
+بتسعة فبصير عندك هي هذه بترسم الخط اللي بينهم
+
+552
+00:37:29,720 --> 00:37:34,710
+بتكون هذه المنطقة اللي احنابنحكي عنها كـ Domain
+
+553
+00:37:34,710 --> 00:37:37,930
+لمين للدالة اللي عندنا يعني بدنا نودد اللي هو الـ
+
+554
+00:37:37,930 --> 00:37:40,390
+Absolute Maximum والـ Absolute Minimum للـ
+
+555
+00:37:40,390 --> 00:37:44,950
+Function على هذه المنطقة إذن الآن كيف بنوددها؟
+
+556
+00:37:44,950 --> 00:37:50,150
+طبعا الـ Fx doesn't exist مش موجودة إذن و لا Fy
+
+557
+00:37:50,150 --> 00:37:52,730
+doesn't exist موجودة إذن بس اللي هي لإنه هذا الـ
+
+558
+00:37:52,730 --> 00:37:55,660
+Function differentiable من المكانأذا بدنا نجد Fx و
+
+559
+00:37:55,660 --> 00:37:58,720
+Fy بنسوين بالصفر و بنشوف النقاط اللي بتطلع النقاط
+
+560
+00:37:58,720 --> 00:38:02,960
+اللي بتطلع في داخل هذه المنطقة أو على جانبها اللي
+
+561
+00:38:02,960 --> 00:38:06,940
+هو بنفحصها برا مالناش علاقة فيها الآن و بعدين
+
+562
+00:38:06,940 --> 00:38:09,820
+بنيجي بنشتغل على الـ boundary points و هنقولكم كيف
+
+563
+00:38:09,820 --> 00:38:13,760
+بالظبط كيف نشتغل عليهالأن نأتي أولا للـ Interior
+
+564
+00:38:13,760 --> 00:38:17,880
+Point نقسمها جسمين Interior Points و Boundary
+
+565
+00:38:17,880 --> 00:38:20,580
+Points انتشر الـ Interior Points هي الحل المرتب
+
+566
+00:38:20,580 --> 00:38:23,960
+حله زي ما بحل Interior Points ايش هي او زي ما بحل
+
+567
+00:38:23,960 --> 00:38:28,600
+الكتاب Interior Points كيف هي ندد Fx و Fy Fx فهي
+
+568
+00:38:28,600 --> 00:38:32,710
+حالة 2نقص 2x هي اتنين نقص اتنين x اف y زيها اتنين
+
+569
+00:38:32,710 --> 00:38:37,250
+نقص اتنين y ساوليها بالصفر بتطلع y بتساوي x بتساوي
+
+570
+00:38:37,250 --> 00:38:41,070
+ايه ايش y بتساوي واحد وx بتساوي واحد إذا النقطة
+
+571
+00:38:41,070 --> 00:38:43,530
+اللي بنحكي عنها interior point هي يمين واحد واحد
+
+572
+00:38:43,530 --> 00:38:47,450
+هي أول نقطة في ال listهنعمل لكيها بأحمر على أساس
+
+573
+00:38:47,450 --> 00:38:51,150
+تقولي أفف واحد واحد أيش بتساوي اللي هو اتنين زاد
+
+574
+00:38:51,150 --> 00:38:54,210
+اتنين زاد اتنين ناقص واحد ناقص واحد اللي هي بتساوي
+
+575
+00:38:54,210 --> 00:38:57,690
+جداش مجموحين ستة ناقص اتنين بيطلع أربعة، إذن هذه
+
+576
+00:38:57,690 --> 00:39:01,310
+حفظها في الذاكرة لإن هي من النقاط اللي بدنا نحكم
+
+577
+00:39:01,310 --> 00:39:04,390
+من خلالها وين ال absolute maximum و ال absolute
+
+578
+00:39:04,390 --> 00:39:07,250
+minimum حسب النظرية اللي قلت عنها إنه مدام على
+
+579
+00:39:07,250 --> 00:39:09,030
+closed bounded region و different أو different
+
+580
+00:39:09,030 --> 00:39:11,550
+شابلها دي أو اللي هو ال partial derivatives
+
+581
+00:39:11,550 --> 00:39:13,750
+continuous ده المضمون عندي اللي هو absolute
+
+582
+00:39:13,750 --> 00:39:16,730
+maximum و absolute minimumأكبر واحدة فيها الـ
+
+583
+00:39:16,730 --> 00:39:18,610
+Absolute Maximum وأصغر واحدة هي الـ Absolute
+
+584
+00:39:18,610 --> 00:39:21,750
+Minimum هي اللي عدت عشرين مرة الآن نيجي للـ
+
+585
+00:39:21,750 --> 00:39:24,750
+Boundary Points الـ Boundary Points مين هي؟ هيها
+
+586
+00:39:24,750 --> 00:39:30,430
+اللي هي إيش؟ العخطوط الـ Region نيجي بنمسك خط خط و
+
+587
+00:39:30,430 --> 00:39:35,190
+بنشتغل في الدالة على خط خطالآن نأتي أولاً على الـ
+
+588
+00:39:35,190 --> 00:39:40,130
+Segment OA دعنا نشوف كيف نشتغل الخط الأول OA الـ
+
+589
+00:39:40,130 --> 00:39:43,950
+OA ماهي معدلته؟ Y بتساوي 0 لأن Y عليه H بتساوي
+
+590
+00:39:43,950 --> 00:39:51,730
+دائماً 0 تصبح الدالة على هذا الخط F of X0 لأن Y
+
+591
+00:39:51,730 --> 00:39:56,510
+أشمالها أخدناها عبارة عن 0 تصبح F of X0 بساوية 2
+
+592
+00:39:57,110 --> 00:40:02,310
+زاد 2x – x³ لأن هذا يصبح صفر وهذا أيه صفر الان هذه
+
+593
+00:40:02,310 --> 00:40:07,470
+الدالة صارت تشمل على الخط هذا على الخط هذا الدالة
+
+594
+00:40:07,470 --> 00:40:10,850
+هذه صارلها دالة function of one variable زي تباعد
+
+595
+00:40:10,850 --> 00:40:13,910
+الزمان صارلها boundary points اللي هي هذه وهذه
+
+596
+00:40:13,910 --> 00:40:17,450
+وصار ايه شمالها interior points ال interior points
+
+597
+00:40:17,450 --> 00:40:22,360
+كيف افحصها بقف ال prime بساوي صفرباجي الآن اللي هي
+
+598
+00:40:22,360 --> 00:40:24,780
+صارت عندي هذا معرفة على زي ما قولنا على الفترة من
+
+599
+00:40:24,780 --> 00:40:27,780
+من سفر لعن تسعة ايضا صارت Zero boundary point
+
+600
+00:40:27,780 --> 00:40:31,260
+وتسعة boundary point الان نيجي ناخد اول اشي على ال
+
+601
+00:40:31,260 --> 00:40:34,900
+boundary points للدالة الجديدة اللي على الخط اللي
+
+602
+00:40:34,900 --> 00:40:38,580
+هي بيصير F ناخد عند ال X بيساوي Zero F of Zero
+
+603
+00:40:38,580 --> 00:40:42,540
+Zero بيساوي ايش اتنين وهذه ما بحطها من ضمن ال list
+
+604
+00:40:42,940 --> 00:40:46,080
+لأنه ممكن يكون عندها Maximum أو Minimum عند مين
+
+605
+00:40:46,080 --> 00:40:48,360
+الآن باخد الـ Boundary Point التاني اللي هي مين
+
+606
+00:40:48,360 --> 00:40:52,860
+أنا في هذه الحالة تسعة F of تسعة و Zero اللي هي
+
+607
+00:40:52,860 --> 00:40:55,660
+بيساوي بعوض عن F تسعة و Zero بيصير اتنين زائد
+
+608
+00:40:55,660 --> 00:40:59,080
+تمانتاش اللي هو ناقص واحد و تمانين اللي هي بتطلع
+
+609
+00:40:59,080 --> 00:41:03,560
+ياشتر 61خلّصنا من الـ Boundary Points للدالة اللي
+
+610
+00:41:03,560 --> 00:41:06,760
+عندي هذه زي دوا الزمان ضالمين الـ Interior Point
+
+611
+00:41:06,760 --> 00:41:09,560
+إذا باخد F' من الآن F' لأن الـ derivative is a
+
+612
+00:41:09,560 --> 00:41:13,300
+function of one variable F' of X بيصير F' of X إيش
+
+613
+00:41:13,300 --> 00:41:19,680
+بتساوي؟ اللي هي عبارة عن 2 ناقص 2X 2 ناقص 2X هيها
+
+614
+00:41:20,230 --> 00:41:24,550
+بسويها بالـ 0 زي ما بنعمل زمان بتطلع عندي X بتساوي
+
+615
+00:41:24,550 --> 00:41:26,810
+إيش؟ واحد إذا ان طلعت عندي X بتساوي واحد و Y
+
+616
+00:41:26,810 --> 00:41:28,790
+بتساوي سفر إذا ان النقطة اللي ممكن يكون عندها
+
+617
+00:41:28,790 --> 00:41:32,270
+Local maximum و Local minimum اللي هي واحد و سفر
+
+618
+00:41:32,270 --> 00:41:35,710
+أفوف واحد و سفر أفوف واحد و سفر بس بحسبها زي ما
+
+619
+00:41:35,710 --> 00:41:39,250
+حسبت اللي جابلها وبتصير عندي إيش بتساوي؟ بتساوي
+
+620
+00:41:39,250 --> 00:41:43,250
+تلاتة واحد و سفر يعني اتنين زائد اتنين اللي هو
+
+621
+00:41:43,250 --> 00:41:48,050
+ناقص واحد اللي هي بتساوي تلاتةهذه اللي هي النقاط
+
+622
+00:41:48,050 --> 00:41:51,830
+هذه وهذه وهذه بالإضافة للنقاط هذه ممكن يكون عندها
+
+623
+00:41:51,830 --> 00:41:55,130
+Local Maximum أو Local Minimum بحصيها اللي هم
+
+624
+00:41:55,130 --> 00:42:00,150
+بتشغل على الخط التانمين هو OB بنفس الأسلوب الخط OB
+
+625
+00:42:00,150 --> 00:42:04,090
+الـ X هتساوي سفر بيصير الـ Function H بتساوي F of
+
+626
+00:42:04,090 --> 00:42:08,590
+Zero Y بيساوي 2 زاد 2Y ناقص من Y تربيع وهذا زي ما
+
+627
+00:42:08,590 --> 00:42:12,070
+انتوا شايفين الخط فيه Boundary Points له برضه Y بـ
+
+628
+00:42:12,070 --> 00:42:19,180
+0 وA شمالها وY بتسعةلأن Y بيكون بين 0 و 9 مثلًا
+
+629
+00:42:19,180 --> 00:42:34,760
+مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا مثلًا
+
+630
+00:42:34,790 --> 00:42:39,610
+F of 0,0 حسبناها قبل 2 لأن Y بـ 0 و X بـ 0,0,0 إلى
+
+631
+00:42:39,610 --> 00:42:43,730
+Y بـ 9 اللي هي F of 0,9 برضه بنفس الأسلوب نعود في
+
+632
+00:42:43,730 --> 00:42:47,290
+الدالة بتطلع عندى سالب واحد وستين ضال عليها أوجد
+
+633
+00:42:47,290 --> 00:42:52,370
+مين F prime F prime لـ Y و 0 اللي هي عبارة عن 2
+
+634
+00:42:52,370 --> 00:42:56,790
+ناقص 2Y و بنسويها بالسفر بتطلع Y بالساوي سفر واحد
+
+635
+00:42:56,790 --> 00:43:01,840
+معناته صارت عند النقطة الجديدة هي 1 0 1اللي هي Y
+
+636
+00:43:01,840 --> 00:43:06,220
+أف برايم هي Zero Y آه Zero Y آسف Zero Y هذه Zero Y
+
+637
+00:43:06,220 --> 00:43:11,800
+ماشي زي هذي Zero Y Zero و Y فاهمين لإن بيصير F of
+
+638
+00:43:11,800 --> 00:43:16,420
+Y ب Zero X ب Zero و Y ب واحد بيصير F of Zero واحد
+
+639
+00:43:16,420 --> 00:43:20,060
+برضه هتساوي تلاتة إلى أن خلصنا من الخطين هدول اللي
+
+640
+00:43:20,060 --> 00:43:22,840
+زي بعض المتشابهات نيجي للخط الجديد مين هو الخط
+
+641
+00:43:22,840 --> 00:43:27,100
+الجديد اللي هو in the segment AB اللي هو Y بتساوي
+
+642
+00:43:27,100 --> 00:43:32,940
+9 minus X هذا الخط يعني على هذا الخطY بيتساوي تسعة
+
+643
+00:43:32,940 --> 00:43:38,120
+ناقص X والنقاط الباوندر والباوندر هدولة انساهم
+
+644
+00:43:38,120 --> 00:43:42,420
+لإيش عملنا حسبناهم في الخط هذا وفي الخط هذا إذا
+
+645
+00:43:42,420 --> 00:43:45,220
+انضل بس تشتغل على مين؟ على الـ interior اللي بده
+
+646
+00:43:45,220 --> 00:43:48,660
+يحسب إن التان تان نور على نور لكن هم الحسبين في
+
+647
+00:43:48,660 --> 00:43:53,280
+الخطين الأولى نجي الآن اللي هو نجي للخط الثالث
+
+648
+00:43:53,280 --> 00:43:58,920
+ونحسب اللي هو F prime عندهأول اشي بنعوض عن Y اللي
+
+649
+00:43:58,920 --> 00:44:04,780
+هي بيصير F of اللي هي هذه X مش تسعة minus X وهذه
+
+650
+00:44:04,780 --> 00:44:08,980
+ال Y ايه ايش تسعة نقص X يعني F of X وتسعة نقص X في
+
+651
+00:44:08,980 --> 00:44:13,360
+الآخر كلها function في مين في X يعني بنحط بنعوض عن
+
+652
+00:44:13,360 --> 00:44:17,660
+كل Y تسعة نقص X هي اتنين زاد اتنين زي ما هي زائد
+
+653
+00:44:17,660 --> 00:44:21,810
+هذا طبعا بأكد أنها Xالتي هي 2 فى 9-X بعوض فى
+
+654
+00:44:21,810 --> 00:44:25,830
+الدالة الأصلية نقص X تربيع زى ما هي نقص يبدل Y 9-X
+
+655
+00:44:25,830 --> 00:44:32,070
+لكل تربيع بنفرطهم وبنحسبهم تطلع معايا نقص 61 زى
+
+656
+00:44:32,070 --> 00:44:36,770
+18X نقص 2X تربيع مش مصدقين ففرطوهم لحالكمالان هذه
+
+657
+00:44:36,770 --> 00:44:40,390
+الآن الدالة الان سيبك من السفر و التسعة و التسعة و
+
+658
+00:44:40,390 --> 00:44:43,530
+السفر لإن هما are computed above إذا بس بدنا نوجد
+
+659
+00:44:43,530 --> 00:44:49,030
+مين احنا F prime إلها ماشي F prime إنها شوف ال F
+
+660
+00:44:49,030 --> 00:44:52,910
+prime لل X والتسعة minus X فاضل بالنسبة ل X لإنه
+
+661
+00:44:52,910 --> 00:44:57,010
+صارت function في X هذه سفر بيصير تمانتعش ناقص
+
+662
+00:44:57,010 --> 00:45:00,990
+أربعة X الان سويها بالسفر زي زي ما بنعرف بتطلع
+
+663
+00:45:00,990 --> 00:45:04,950
+عندي X بتساوي إيش تسعة على اتنين أو أربعة و نصلو
+
+664
+00:45:04,950 --> 00:45:08,510
+اتيت عوضت عن اللي هو العلاقة بينهم على الخط اللي
+
+665
+00:45:08,510 --> 00:45:12,610
+هو Y بتساوي 9 نقص X X ب 4.5 إذا الـ Y إيش هتصير 9
+
+666
+00:45:12,610 --> 00:45:18,210
+نقص 4.5 برضه هتطلع 4.5 إذا على الخط هذا XY هيطلع
+
+667
+00:45:18,210 --> 00:45:23,050
+عن اللي هو 4.5 و4.5بنحسب F عند 4.5 و 4.5 عند 9 على
+
+668
+00:45:23,050 --> 00:45:27,930
+2 و 9 على 2 عوّض في الدالة حتى تلاقي 2 زي 9 زي 9
+
+669
+00:45:27,930 --> 00:45:34,010
+ناقص 20.4 و 20.4 وبتحسبين مع بعض تلاقي ناقص 20.5
+
+670
+00:45:34,010 --> 00:45:38,070
+بتطلع ال function نقيم التناقص 20.5 الان summary
+
+671
+00:45:38,070 --> 00:45:41,410
+we list the candidates اللي وجدناها اللي هي مين هي
+
+672
+00:45:41,760 --> 00:45:45,800
+اللي هي قيمة الـ Function أول واحدة أربعة والتانية
+
+673
+00:45:45,800 --> 00:45:48,440
+اتنين تانية ناقص واحد ستين واحدة تلاتة وناقص عشرين
+
+674
+00:45:48,440 --> 00:45:52,840
+point خمسة أكبر واحدة فين عندها Absolute Maximum
+
+675
+00:45:52,840 --> 00:45:56,180
+وأصغر واحدة فين عندها Absolute Minimum إذن هذا
+
+676
+00:45:56,180 --> 00:45:58,940
+Absolute Minimum وهذا Absolute Maximum إذن
+
+677
+00:45:58,940 --> 00:46:03,060
+Absolute Minimum هو 4 وهذه حدثت مدى عنده النقطة
+
+678
+00:46:03,060 --> 00:46:06,660
+الأولى واحد واحد وهي طبعا في داخل المنطقة Absolute
+
+679
+00:46:06,660 --> 00:46:10,970
+Minimum هوأف أسيمز أزير وتسعة أن تسعة وزيرو اللي
+
+680
+00:46:10,970 --> 00:46:14,830
+حدثت عند نقطتنا دي اللي قيمته ايش ناقص واحد وستين
+
+681
+00:46:14,830 --> 00:46:17,510
+ممكن و هيك احنا اعرفنا كيف نوجد ال absolute
+
+682
+00:46:17,510 --> 00:46:19,930
+maximum و ال absolute minimum على close bounded
+
+683
+00:46:19,930 --> 00:46:24,430
+اللي هي region نيجي الآن نعمل summary إذا عندي
+
+684
+00:46:24,430 --> 00:46:29,210
+الموضوع الآن ال extreme values can occur only atيا
+
+685
+00:46:29,210 --> 00:46:31,370
+الـ Boundary Points يا إما الـ Critical Points
+
+686
+00:46:31,370 --> 00:46:34,090
+والـ Critical Points قلنا يا Interior طبعاً الـ
+
+687
+00:46:34,090 --> 00:46:37,610
+Interior بتكون Fx أو Fy بسوا سفر أو Fx أو Fy لا
+
+688
+00:46:37,610 --> 00:46:42,930
+يوجد إلا أن إذا كان الـ First وSecond ورشة الـ
+
+689
+00:46:42,930 --> 00:46:47,250
+Derivatives موجودة ومستمرة وبدنا نفحص هل في Local
+
+690
+00:46:47,250 --> 00:46:49,510
+Maximum و Local Minimum قلنا بالـ Second
+
+691
+00:46:49,510 --> 00:46:53,250
+Derivative Test اللي حكينا عنه اللي هو سابقاً وهيك
+
+692
+00:46:53,250 --> 00:46:56,090
+بيكون إحنا خلصنا المحاضرة وبرك الله فيكم والسلام
+
+693
+00:46:56,090 --> 00:46:58,490
+عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A.srt
new file mode 100644
index 0000000000000000000000000000000000000000..5b3fa49bcd565956352f9a82e87e2381ce65bb5e
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@@ -0,0 +1,1520 @@
+
+1
+00:00:00,770 --> 00:00:04,810
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,810 --> 00:00:11,070
+وبركاته هذه هي المحاضرة رقم 24 مساق تفاضل وتكامل
+
+3
+00:00:11,070 --> 00:00:16,290
+جيم وهي المحاضرة الخامسة في chapter 15، الشابتر
+
+4
+00:00:16,290 --> 00:00:22,560
+الأخير في هذا المساق، الشابتر الذي هو عنوانه triple
+
+5
+00:00:22,560 --> 00:00:28,840
+integrals in rectangular coordinates. احنا كنا
+
+6
+00:00:28,840 --> 00:00:33,440
+تحدثنا المحاضرة السابقة عن double integral in
+
+7
+00:00:33,440 --> 00:00:36,840
+polar coordinates، واللي  حكينا عن double
+
+8
+00:00:36,840 --> 00:00:41,700
+integrals in rectangular coordinates. واليوم نحكي
+
+9
+00:00:41,700 --> 00:00:46,330
+عن نكمل نحكي عن الـ double integral في حالة الـ
+
+10
+00:00:46,330 --> 00:00:50,410
+Rectangular Coordinates. الكلام سيكون شبيه، والكلام
+
+11
+00:00:50,410 --> 00:00:55,490
+بنفس الروح وبنفس السهولة إن شاء الله. الآن أول شيء
+
+12
+00:00:55,490 --> 00:01:00,050
+نحكي عن Volume of a region in space. إذا بتذكروا
+
+13
+00:01:00,050 --> 00:01:03,030
+احنا استخدمنا double integral في إيجاد الـ area
+
+14
+00:01:03,310 --> 00:01:07,090
+وقلنا على طول الـ area بتساوي الـ double integral
+
+15
+00:01:07,090 --> 00:01:12,430
+over R، الـ region. دي ايه دي ايه دي اكس دي وايه. دي
+
+16
+00:01:12,430 --> 00:01:13,070
+اكس دي وايه دي اكس دي وايه دي اكس دي وايه دي اكس
+
+17
+00:01:13,070 --> 00:01:14,050
+دي وايه دي اكس دي وايه دي اكس دي اكس دي اكس دي اكس
+
+18
+00:01:14,050 --> 00:01:14,230
+دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس
+
+19
+00:01:14,230 --> 00:01:19,070
+دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس
+
+20
+00:01:19,070 --> 00:01:20,010
+دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس
+
+21
+00:01:20,010 --> 00:01:27,350
+دي اكس. التي ستحصل بدل ما هو دي اي دي اكس دي واي
+
+22
+00:01:27,350 --> 00:01:30,830
+ستحصل دي volume of a closed bounded region دي
+
+23
+00:01:30,830 --> 00:01:36,250
+أيّن in space.  V بتساوي triple integral
+
+24
+00:01:39,450 --> 00:01:44,070
+Over D، هي الـ region in space. دي في دي في. طبعا هي
+
+25
+00:01:44,070 --> 00:01:49,270
+عبارة عن الـ dx dy dz. طبعا هذا له أصل اللي هو
+
+26
+00:01:49,270 --> 00:01:53,070
+إيجاده، اللي هو زي ما حكينا في أول الـ double
+
+27
+00:01:53,070 --> 00:01:57,790
+integration، اللي هو بنقسم المنطقة إلى أقسام صغيرة
+
+28
+00:01:57,790 --> 00:02:01,630
+صغيرة صغيرة للـ X وللـ Y والآن للـ Z، وبيصير
+
+29
+00:02:01,630 --> 00:02:03,850
+المجموع Riemann sum، وبعد ما يصير مجموع
+
+30
+00:02:03,850 --> 00:02:06,770
+Riemann sum، بيوجد الـ limit. إذا كان الـ limit موجود
+
+31
+00:02:06,770 --> 00:02:10,650
+بيكون هو بالظبط قيمة حجم الجسم اللي اللي قاعدين
+
+32
+00:02:10,650 --> 00:02:15,650
+بنجزئه، وبعدين بيكون هو اللي هي الـ triple integral
+
+33
+00:02:15,650 --> 00:02:22,680
+له هو الـ volume. فهذا اللي هو في الآخر limit ده
+
+34
+00:02:22,680 --> 00:02:25,940
+Riemann sum. فاحنا ما نرجع نعيد الكلام اللي حكيناه في
+
+35
+00:02:25,940 --> 00:02:30,100
+الأول على الـ double integral لما عرفناه. الآن عملية
+
+36
+00:02:30,100 --> 00:02:35,440
+إيجاد اللي هي الـ triple integral بنفس الأسلوب. عندي
+
+37
+00:02:35,440 --> 00:02:38,100
+بيكون تلت حاجات: قيمة الـ function، بس أنا قيمة الـ
+
+38
+00:02:38,100 --> 00:02:42,660
+function. واحد، الحاجة الثانية اللي هي dv، dx dy dz
+
+39
+00:02:42,660 --> 00:02:47,520
+والحاجة اللي بدها شغل، اللي هي باسمين D، التي هي حدود
+
+40
+00:02:47,520 --> 00:02:51,380
+الـ integration أو limits of integration. الآن اللي
+
+41
+00:02:51,380 --> 00:02:54,020
+بدنا نحكي عنه هو find limits of integration in the
+
+42
+00:02:54,020 --> 00:02:57,900
+order dz dy dx. الآن بدنا نوجد الـ order، بأي order
+
+43
+00:02:57,900 --> 00:03:01,420
+dz أول شيء، وبعدين dy، وبعدين dx، وطبعا والباقيات
+
+44
+00:03:01,420 --> 00:03:07,220
+similar للتبديلات الأخرى dz dx dy، dy dz dx، إلى آخره.
+
+45
+00:03:07,220 --> 00:03:11,300
+خلينا نشوف الآن. طيب، صلوا على النبي عليه الصلاة
+
+46
+00:03:11,300 --> 00:03:15,340
+والسلام. نجي للمثال الأول: find the volume of the
+
+47
+00:03:15,340 --> 00:03:19,520
+region D enclosed by the surface Z بتساوي X تربيع
+
+48
+00:03:19,520 --> 00:03:22,800
+زائد ثلاثة Y تربيع. هاي اللي هو الـ surface اللي
+
+49
+00:03:22,800 --> 00:03:26,700
+عندي هيه، اللي تحت. هذا اللي هو Z بتساوي X تربيع
+
+50
+00:03:26,700 --> 00:03:29,800
+زائد ثلاثة Y تربيع. احنا ما نجيب لكم مش سؤال إنه يكون
+
+51
+00:03:29,800 --> 00:03:34,560
+الرسم في أسوأ من السؤال، إيه ده؟ إيه ده؟ طلب إننا
+
+52
+00:03:34,560 --> 00:03:38,420
+نرسم لكم، وبنرسم لكم. أما مش مختبركم الآن في الرسم،
+
+53
+00:03:38,420 --> 00:03:42,400
+مختبركم في إيجاد اللي هو الـ triple integral. Z
+
+54
+00:03:42,400 --> 00:03:48,200
+بتساوي X تربيع زائد ثلاثة Y تربيع. هذه الآن الـ region
+
+55
+00:03:48,200 --> 00:03:51,840
+دي محصورة بين هذا الـ surface والـ surface الثاني Z
+
+56
+00:03:51,840 --> 00:03:55,380
+بتساوي ثمانية ناقص X تربيع ناقص Y تربيع، حيث Z
+
+57
+00:03:55,380 --> 00:04:00,200
+بتساوي ثمانية ناقص X تربيع ناقص Y تربيع. هذه المنطقة
+
+58
+00:04:00,200 --> 00:04:04,780
+بدي أجد لها الـ volume. الـ volume طبعا بيساوي triple
+
+59
+00:04:04,780 --> 00:04:11,420
+integral دي over D، this region. dz dy dx زي ما هو
+
+60
+00:04:11,420 --> 00:04:17,280
+طلب منا. قبل ما نبدأ في السؤال، أو طبعا اللي بده
+
+61
+00:04:17,280 --> 00:04:21,620
+يحكي طبعا الـ ... يمكن الحكي في سؤال زي هذا اللي هو
+
+62
+00:04:21,620 --> 00:04:26,240
+dy قبل dz كده، اللي هو صعب إنك تأخذ، وأنا بصير تأخذ
+
+63
+00:04:26,240 --> 00:04:28,580
+الـ ... الـ ... الـ ... الـ ... الـ ... الـ ...
+
+64
+00:04:28,580 --> 00:04:29,600
+الـ ... الـ ... الـ ... الـ ... الـ ... الـ ...
+
+65
+00:04:29,600 --> 00:04:30,140
+الـ ... الـ ... الـ ... الـ ... الـ ... الـ ...
+
+66
+00:04:30,140 --> 00:04:31,140
+الـ ... الـ ... الـ ... الـ ... الـ ... الـ ...
+
+67
+00:04:31,140 --> 00:04:31,840
+الـ ... الـ ... الـ ... الـ ... الـ ... الـ ...
+
+68
+00:04:31,840 --> 00:04:35,040
+الـ ... الـ ... الـ ... الـ ... الـ ... الـ ...
+
+69
+00:04:35,040 --> 00:04:37,380
+الـ ... الـ ... الـ ... الـ ... الـ ... الـ ...
+
+70
+00:04:37,380 --> 00:04:38,860
+الـ ... الـ ... الـ ... الـ ... الـ ... الـ ...
+
+71
+00:04:38,860 --> 00:04:41,460
+الـ ... الـ ... الـ ... الـ ... الـ ... الـ ...
+
+72
+00:04:41,460 --> 00:04:49,860
+الـ ... الـ ... الـ ... الـ ... الـ ...
+
+73
+00:04:50,080 --> 00:04:55,480
+لأن كيف بتحدد حدود الـ integration؟ الآن Z نفس
+
+74
+00:04:55,480 --> 00:04:59,060
+الروح، إيش بأخذ؟ زي ما كنا لما كنا بنكامل بالنسبة لـ
+
+75
+00:04:59,060 --> 00:05:02,900
+X، نأخذ موازية لـ X. لما نكامل لـ Y، نأخذ موازية لـ Y.
+
+76
+00:05:02,900 --> 00:05:08,220
+الآن بتكامل بالنسبة لـ Z، موازية لـ Z، من وين بدخل الـ
+
+77
+00:05:08,220 --> 00:05:13,000
+high Z. طبعا high اللي هو الـ increasing له، أو عملية
+
+78
+00:05:13,000 --> 00:05:20,400
+التزايد له. الآن بنأخذ اللي هي عبارة عن موازيات اللي
+
+79
+00:05:20,400 --> 00:05:25,620
+هو وين بتعمل entering للي هو الجسم، وكان وين
+
+80
+00:05:25,620 --> 00:05:29,160
+بتعمل الخروج، اللي يعني الدخول للجسم. واضح إنها
+
+81
+00:05:29,160 --> 00:05:34,450
+بتبدأ من عند اللي هو السطح هذا. هاي السطح هذا لو
+
+82
+00:05:34,450 --> 00:05:39,710
+دخلنا أيّ اللي هو موازي الـ z axis، هيطلع من السطح
+
+83
+00:05:39,710 --> 00:05:43,870
+البراني هذا اللي هو إيش ماله؟ Z بتساوي X تربيع زائد
+
+84
+00:05:43,870 --> 00:05:48,430
+ثلاثة Y تربيع، وبطلع، بخرج من وين؟ من هذه المنطقة
+
+85
+00:05:48,430 --> 00:05:51,970
+اللي هي Z بتساوي ثمانية ناقص X تربيع ناقص Y تربيع.
+
+86
+00:05:51,970 --> 00:05:55,130
+وكل هنا على نفس المنوال. شايف مثلها، لكن لو بدنا
+
+87
+00:05:55,130 --> 00:05:57,570
+نتكامل بالنسبة لـ X أو الاشياء، بدنا نمشي هيك و
+
+88
+00:05:57,570 --> 00:06:00,920
+بدنا نمشي هيك، بصير بده ييجي من هنا لما يروح لهنا
+
+89
+00:06:00,920 --> 00:06:04,640
+بيصير فيه شوية صعوبة في التحديد، لكن مش مستحيلة، و
+
+90
+00:06:04,640 --> 00:06:09,060
+نفس الشيء بالنسبة لـ Y بالشكل الثاني. إذا لازِم
+
+91
+00:06:09,060 --> 00:06:15,240
+بيدخل من عند اللي هو الـ curve الأول أو الـ surface
+
+92
+00:06:15,240 --> 00:06:19,660
+الأول، وبيطلع من عند الـ surface الثاني. إذا صارت
+
+93
+00:06:19,660 --> 00:06:24,180
+اللي هي تكاملها، حدود الـ integration هي عبارة عن
+
+94
+00:06:24,390 --> 00:06:29,210
+اللي هي من اللي تحت: X تربيع زائد ثلاثة Y تربيع، قيمة Z
+
+95
+00:06:29,210 --> 00:06:34,650
+إلى ثمانية ناقص X تربيع ناقص Y تربيع، اللي هي قيمة Z
+
+96
+00:06:34,650 --> 00:06:38,590
+الثانية. هذه الأولى اللي ذكرناها: X تربيع زائد ثلاثة Y
+
+97
+00:06:38,590 --> 00:06:45,150
+تربيع، وهذه القيمة الثانية. الآن بدي أوجد اللي هو
+
+98
+00:06:45,150 --> 00:06:50,250
+من حدود الـ X والـ Y، عشان أوجد حدود الـ X والـ Y. الآن
+
+99
+00:06:50,250 --> 00:06:54,790
+اللي هو فأسئلة من هذا النوع أصلا يعني ممكن احنا
+
+100
+00:06:54,790 --> 00:07:00,200
+اللي هو ما نرسمه، لكن هنا التوضيح كيف نرسمه على طول
+
+101
+00:07:00,200 --> 00:07:04,640
+اللي هو بـ ... بس بدك تعرف يعني بتغيير قيمة، بتعويض
+
+102
+00:07:04,640 --> 00:07:07,860
+عن قيمتين هنا وقيمتين هنا: القيمة الـ X والـ Y هنا
+
+103
+00:07:07,860 --> 00:07:10,320
+وقيمة الـ X والـ Y هنا، بتعرف مين اللي فوق ومين
+
+104
+00:07:10,320 --> 00:07:14,420
+اللي تحت. طبعا يعني هو الكلام مش safe كثير، لكن ...
+
+105
+00:07:14,420 --> 00:07:19,620
+لكن بِقَدِ الغرض. القيمة الأعلى لـ ... لـ ... لـ ... لـ ... الـ
+
+106
+00:07:19,620 --> 00:07:23,140
+X مثلا بتأخذها واحد وكده بتلاقي مثلا اللي هي هذه
+
+107
+00:07:23,140 --> 00:07:27,360
+قيمة أعلى، وهذه قيمة أسفل. بتكون حدود الـ integration من
+
+108
+00:07:27,360 --> 00:07:31,920
+هنا لعند هنا. هذا لو بدك ما ترسم أو فيش رسم، في رسومات
+
+109
+00:07:31,920 --> 00:07:35,620
+هذا مفضلُك. تكامل بالنسبة لـ Z أساس تعرف مين اللي فوق
+
+110
+00:07:35,620 --> 00:07:42,540
+ومين اللي تحت. طيب، الآن نجي للّي هي المنطقة اللي
+
+111
+00:07:42,540 --> 00:07:48,080
+بدنا نكامل عليها X وY. خلصنا من Z، بدنا نيجي X، إيش
+
+112
+00:07:48,080 --> 00:07:54,200
+بنعمل؟ إذا بنعمل إسقاط للجسم هذا، الآن وهذا اللي هي
+
+113
+00:07:54,200 --> 00:08:00,440
+إسقاط كما يلي: عندي اللي هو Z بخليه صفر، وZ
+
+114
+00:08:00,440 --> 00:08:03,500
+بالساوي صفر، نشوف المنطقة المشتركة بينهما، هي
+
+115
+00:08:03,500 --> 00:08:06,850
+الإسقاط اللي هي X تربيع زائد ثلاثة Y تربيع بتساوي
+
+116
+00:08:06,850 --> 00:08:11,130
+ثمانية ناقص X تربيع ناقص Y تربيع. الآن X
+
+117
+00:08:11,130 --> 00:08:14,470
+تربيع زائد ثلاثة Y تربيع بتجيب لها الـ X تربيع، بيصير
+
+118
+00:08:14,470 --> 00:08:17,370
+اللي هو اثنين X تربيع، وهذا بتجيب لها الـ Y تربيع
+
+119
+00:08:17,370 --> 00:08:20,750
+على الطرف الثاني بيصير أربعة Y تربيع، أربعة Y تربيع
+
+120
+00:08:20,970 --> 00:08:24,050
+زائد اثنين X تربيع بتساوي ثمانية. X من الجهتين على
+
+121
+00:08:24,050 --> 00:08:27,670
+اثنين، بيصير X تربيع زائد اثنين Y تربيع بتساوي إيش؟
+
+122
+00:08:27,670 --> 00:08:33,110
+أربعة، اللي هي الـ ellipse اللي الإسقاط، طبعا اللي هو
+
+123
+00:08:33,110 --> 00:08:36,590
+الجسم اللي هو X تربيع زائد اثنين Y تربيع إيش بيساوي؟
+
+124
+00:08:36,590 --> 00:08:40,770
+بيساوي أربعة. إذا عرفنا كيف الإسقاط، طبعا هذي بصفر
+
+125
+00:08:40,770 --> 00:08:44,010
+وهذي بصفر، وبنساويهما ببعض، بيطلع المنطقة اللي هي
+
+126
+00:08:44,010 --> 00:08:49,260
+إسقاط الجهتين. نأتي الآن ... نأتي الآن ... نأتي الآن
+
+127
+00:08:49,260 --> 00:08:52,120
+... اللي هو بدنا نحدد حدود الـ integration من هذه
+
+128
+00:08:52,120 --> 00:08:55,540
+المنطقة. بدنا نكامل بالنسبة لمين؟ جاب لي الـ Y، بعدين
+
+129
+00:08:55,540 --> 00:09:01,700
+بدنا نكامل الـ Y. إذا بدنا نأخذ اللي هو موازيات للمين؟
+
+130
+00:09:01,700 --> 00:09:06,330
+للـ Y axis. هي الـ Y axis، نأخذ لها موازيات. كل
+
+131
+00:09:06,330 --> 00:09:11,310
+الموازيات التي منها إلى هنا، كلها تأتي من الجهة
+
+132
+00:09:11,310 --> 00:09:15,490
+السالبة لـ Y عبر الـ curve نفسه: X تربيع زائد اثنين، و
+
+133
+00:09:15,490 --> 00:09:19,650
+Y تربيع، إلى الجهة الموجبة لـ Y عبر الـ curve نفسه.
+
+134
+00:09:19,650 --> 00:09:23,290
+إذا حدود الـ integration، ببدأ أوجد Y، تصبح اثنين، وY
+
+135
+00:09:23,290 --> 00:09:26,930
+تربيع تساوي أربعة ناقص X تربيع، يعني Y تربيع تساوي
+
+136
+00:09:26,930 --> 00:09:31,230
+أربعة ناقص X تربيع على اثنين، يعني Y تصبح زائد أو
+
+137
+00:09:31,230 --> 00:09:34,490
+ناقص الجذر التربيعي لأربعة ناقص X تربيع على
+
+138
+00:09:34,490 --> 00:09:38,710
+اثنين. الآن إذا صار عندي قيمتين، هذه اللي هي قيمة Y
+
+139
+00:09:38,710 --> 00:09:41,990
+السالبة، وهذه قيمة Y الموجبة، وهذه اللي هي بتكون
+
+140
+00:09:41,990 --> 00:09:44,770
+إيش؟ حدود الـ integration. Y بتساوي ناقص الجذر
+
+141
+00:09:44,770 --> 00:09:48,190
+التربيعي لأربعة ناقص X تربيع على اثنين، وY بتساوي
+
+142
+00:09:48,190 --> 00:09:51,450
+الجذر التربيعي لأربعة ناقص X تربيع على اثنين. هذا بدخل
+
+143
+00:09:51,450 --> 00:09:54,930
+المنطقة، وهذا بيخرج من المنطقة، وهي حدود الـ
+
+144
+00:09:54,930 --> 00:09:59,450
+integration. نجي نرجع لإن نحدد قيمة X. قيمة X
+
+145
+00:09:59,450 --> 00:10:03,690
+من وين بدأت كل الخطوط؟ هي الخطوط هذه. عند هذه الخطوط
+
+146
+00:10:03,690 --> 00:10:06,610
+هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط
+
+147
+00:10:06,610 --> 00:10:14,010
+هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط
+
+148
+00:10:14,010 --> 00:10:14,610
+هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط
+
+149
+00:10:14,610 --> 00:10:15,590
+هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط
+
+150
+00:10:15,590 --> 00:10:15,650
+هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط، هيك خط
+
+151
+00:10:15,650 --> 00:10:28,430
+هيك خط، هيك خط، هيك خط، هيك خط،
+
+152
+00:10:28,430 --> 00:10:32,880
+هيك خط. هيبقى نعوض عنه Z بـ 8-X تربيع ناقص Y تربيع
+
+153
+00:10:32,880 --> 00:10:38,300
+ناقص اللي هو X تربيع زائد 3Y تربيع، بيطلع عندي
+
+154
+00:10:38,300 --> 00:10:41,860
+المقدار اللي أمامي هذا. هذا بده فضل الآن بالنسبة
+
+155
+00:10:41,860 --> 00:10:45,800
+لمَن؟ بالنسبة لـ Y. فاضله بالنسبة لـ Y، بيصير 8Y ناقص
+
+156
+00:10:45,800 --> 00:10:51,170
+2X تربيع في Y، هيها، اللي هو ناقص أربعة Y تكعيب على
+
+157
+00:10:51,170 --> 00:10:54,910
+ثلاثة. من وين لوين؟ من عند هذا المقدار لهذا المقدار.
+
+158
+00:10:54,910 --> 00:10:58,990
+إذا Y من عند ناقص الجذر لعند الجذر نفسه، اللي هو
+
+159
+00:10:58,990 --> 00:11:03,790
+بعوض على هذه اللي هي عن قيمة Y. هي عوضت، وهنا عوضت
+
+160
+00:11:03,790 --> 00:11:08,850
+عن قيمة Y اللي هي، وبعدين اللي هو طرحت من هدول
+
+161
+00:11:08,850 --> 00:11:15,290
+طبعا هذه اللي هي Y مفردة، فبيصير ناقص في ناقص لها
+
+162
+00:11:15,290 --> 00:11:19,450
+بيصير ناقص ناقص، بيصير زائد، فبيصير هذا ضعف الأولاني.
+
+163
+00:11:19,450 --> 00:11:25,870
+وهذا نفس الشيء إن هي فردية، فبيصير لما أرفعها لـأس
+
+164
+00:11:25,870 --> 00:11:29,740
+اللي هو الفردي هذا بتظلها ناقص، وفي اللي هو
+
+165
+00:11:29,740 --> 00:11:33,900
+الناقص الأولاني اللي هو التعويض بهذه، ناقص التعويض
+
+166
+00:11:33,900 --> 00:11:37,840
+بهذه، فبيصير زائد اللي هو، وفي ناق
+
+223
+00:15:46,140 --> 00:15:49,320
+ما نوجد بالنسبة للهواي خلينا نوجد اللي هو أول حاجة
+
+224
+00:15:49,320 --> 00:15:52,060
+نقاط تقاطع وحياة موجودة اللي بتلزمنا ثاني حاجة
+
+225
+00:15:52,060 --> 00:15:55,600
+بدنا نوجد معادلة الخطوط اللي عندنا لأن معادلة
+
+226
+00:15:55,600 --> 00:16:00,300
+الخطوط بعد شوية هتلاقيها بتلزم جدا للخطوط اللي
+
+227
+00:16:00,300 --> 00:16:04,100
+الآن لنجد أنّ معادلة، إيش معادلة الـ plane الأمامي؟
+
+228
+00:16:04,100 --> 00:16:09,170
+هذا الـ plane الآن ثابت اللي هو y بتساوي واحد
+
+229
+00:16:09,170 --> 00:16:13,170
+لأنه اللي هو المتغيرات بس اللي هي عندك دائماً y
+
+230
+00:16:13,170 --> 00:16:18,570
+بتساوي واحد، و x إيش ما لها متغيرة، و z متغيرة، إذن
+
+231
+00:16:18,570 --> 00:16:22,910
+هذا جزء من الـ plane y بتساوي واحد، هي معروفة قيمة y
+
+232
+00:16:22,910 --> 00:16:25,750
+إيش معناه؟ جيبت y عشان اللي أنا بدي إياه في الأول إيه؟
+
+233
+00:16:25,750 --> 00:16:31,820
+إيش بالنسبة لـ y؟ وهو حالياً الزمن، طبعاً الآن نجيء
+
+234
+00:16:31,820 --> 00:16:37,720
+للمستوى، هذا المستوى هذا يمر بالنقاط 000 و
+
+235
+00:16:37,720 --> 00:16:40,820
+011 و 110 و 000
+
+236
+00:16:40,820 --> 00:16:44,840
+و هذا المستوى اللي هو معروف، ثلاث نقاط عليه، إذن
+
+237
+00:16:44,840 --> 00:16:49,810
+بنعرف طبعاً نوجده لأنه من نقطتين بنوجد اللي هو بنعرف
+
+238
+00:16:49,810 --> 00:16:54,110
+من الـ vector الموازي للمستوى أو في المستوى بالإضافة
+
+239
+00:16:54,110 --> 00:16:58,030
+للنقطة عندنا 1 1 و 0 على المستوى، إيه؟ بتعرف معادلة
+
+240
+00:16:58,030 --> 00:17:04,010
+اللي هو معادلة الـ plane، طبعاً بيعرف من ثلاث
+
+241
+00:17:04,010 --> 00:17:06,810
+مواقع زي ما أخذنا سابقاً، بتعرفها، اللي بتلاقيها y
+
+242
+00:17:06,810 --> 00:17:11,510
+بتساوي x زائد z، نوجدها طبعاً لأنّ هذا معروف سابقاً لكم
+
+243
+00:17:11,510 --> 00:17:14,870
+بتعرفوا تجدوها لحالكم، y بيساوي x زائد z، هذه اللي
+
+244
+00:17:14,870 --> 00:17:18,290
+هي معادلة الـ plane اللي أمامي، الـ plane اللي في الخلف
+
+245
+00:17:18,290 --> 00:17:22,050
+هذا سهل، ما هو هذا الـ plane؟ يعني الـ YZ plane اللي هو x
+
+246
+00:17:22,050 --> 00:17:25,910
+بيساوي صفر، الآن الـ plane اللي تحت هذا ما هو؟ برضه سهل
+
+247
+00:17:25,910 --> 00:17:28,810
+اللي هو عبارة عن XY plane اللي هو z بتساوي صفر، إذن
+
+248
+00:17:28,810 --> 00:17:34,370
+كل الـ planes اللي موجودة عرفنا اللي هي معادلتها
+
+249
+00:17:34,370 --> 00:17:38,470
+نيجيء الآن نبدأ، نشرع في عملية تحديد اللي هو حدود الـ
+
+250
+00:17:38,470 --> 00:17:42,470
+integration عندي، y، يعني أنّه يأخذ موازية لـ Y-axis
+
+251
+00:17:42,470 --> 00:17:47,330
+والـ Y-axis هي موازيات، هي هيك هيك هيك، كل
+
+252
+00:17:47,330 --> 00:17:53,150
+الموازيات عاملها بتدخل من اللي هو هذا الـ plane و
+
+253
+00:17:53,150 --> 00:17:56,170
+بتخرج من هذا الـ plane، و كلها على نفس المنوال
+
+254
+00:17:56,170 --> 00:18:00,150
+فتعالوا أي موازية للـ Y-axis هيكون بدخل من عند y
+
+255
+00:18:00,150 --> 00:18:03,710
+بتساوي x زائد z لعند y بتساوي 1، إذن حدود الـ
+
+256
+00:18:03,710 --> 00:18:07,390
+integration من x زائد z لعند اللي هو مين؟ واحد، هذا
+
+257
+00:18:07,390 --> 00:18:13,290
+الـ y، الآن نجيء نعمل إسقاط لهذا الشكل على الـ XY
+
+258
+00:18:13,290 --> 00:18:17,670
+plane، عشان الـ XZ plane، قاعد اللي بضل الـ XZ plane
+
+259
+00:18:17,670 --> 00:18:21,410
+إذن عرفنا أولاً من حد إلى y موازية، من أين يدخل ومن
+
+260
+00:18:21,410 --> 00:18:24,550
+أين يخرج، بعدين اللي بضل بنعمل إسقاط له المجسم على
+
+261
+00:18:24,550 --> 00:18:26,890
+اللي هو الـ plane اللي بضل اللي هو في هذه الحالة
+
+262
+00:18:26,890 --> 00:18:31,600
+ZX plane، زد اكس بلين، سهلة، بنخلي اللي هو الـ y بصفر
+
+263
+00:18:31,600 --> 00:18:37,260
+بيصير النقطة هذه عبارة عن 001، اللي هو بنخلي هذا
+
+264
+00:18:37,260 --> 00:18:41,120
+النقطة بصفر، بتصير 000، نقطة الأصل، هذه الخط اللي هي
+
+265
+00:18:41,120 --> 00:18:46,280
+صفر، بيصير اللي هي 000، فبيصير الـ plane هذا، نسقطه اللي
+
+266
+00:18:46,280 --> 00:18:50,830
+هو الخط هذا، مثلاً الخط هذا، هي ما سقطنا هذا الخط اللي
+
+267
+00:18:50,830 --> 00:18:54,270
+هو حسب النقطتين هذولا، النقطتين، اسقطنا بيصير اللي
+
+268
+00:18:54,270 --> 00:18:59,730
+اللي هي إسقاط الخط هو النقطتين، اللي هو الخط الواصل
+
+269
+00:18:59,730 --> 00:19:03,330
+بين النقطتين اللي اسقطناهم، ونفس الشيء الخط هذا
+
+270
+00:19:03,330 --> 00:19:07,230
+النقطتين اللي اسقطناهم هذولا بيصير هذولا، هذه ونفس
+
+271
+00:19:07,230 --> 00:19:11,040
+الشيء هذا اللي هو الخط، اللي هو إسقاطه اللي هو
+
+272
+00:19:11,040 --> 00:19:14,340
+النقطتين هدول، إذن اللي هو إسقاط الخط هو عبارة عن
+
+273
+00:19:14,340 --> 00:19:17,960
+الخط اللي بينهم، المسقطتين، النقطتين، إلا أنه نفس
+
+274
+00:19:17,960 --> 00:19:21,800
+الشيء هذا الخط هي إسقاطه، لأن تصير نقطة دي ونقطة
+
+275
+00:19:21,800 --> 00:19:25,240
+دي هي إسقاطه، والخط هذا بيصير هي إسقاطه، إذن اللي
+
+276
+00:19:25,240 --> 00:19:28,880
+بلين هذا ومع الـ plane هذا مع كل المجسم بيصير هو
+
+277
+00:19:28,880 --> 00:19:33,020
+عبارة عن مين؟ لما z صفر بياخذه بيصير هو هذا اللي
+
+278
+00:19:33,020 --> 00:19:38,420
+عندي…اللي أمامي، طيب الآن أنا مش z بصفر، أنا بصفر
+
+279
+00:19:38,420 --> 00:19:42,580
+y بصفر، لأ، بصير relation، لأنّ أنا بسقط على XZ عشان
+
+280
+00:19:42,580 --> 00:19:46,380
+هيك باخد y بصفر، هذا الآن المثلث هو المثلث اللي بدي
+
+281
+00:19:46,380 --> 00:19:49,500
+أوجد حدود الـ integration بين x و z، الآن هذا معروف
+
+282
+00:19:49,500 --> 00:19:53,360
+نقطتين وهين، وهدول معروف، نقطتين وهين، بدي أوجد
+
+283
+00:19:53,360 --> 00:19:57,380
+معادلته، أوجد معادلته، إنّ نقطتين بطلع x زائد z بتساوي
+
+284
+00:19:57,380 --> 00:20:04,060
+واحد، هتلاقيه الآن، الآن عندي اللي هي النقطة هذه عبارة
+
+285
+00:20:04,060 --> 00:20:09,370
+عن…عشان x و z، و x و z، و x و z، و x و z، و
+
+286
+00:20:09,370 --> 00:20:20,930
+x و z و x…
+
+287
+00:20:20,930 --> 00:20:26,380
+و z، z واحد ناقص z اثنين على x واحد ناقص x
+
+288
+00:20:26,380 --> 00:20:29,620
+اثنين، واتغلب أنا عارف هذا على طول بين الواحد
+
+289
+00:20:29,620 --> 00:20:33,240
+والواحد، زمان قبلناها كثير، فبطلع exact z بالسهولة
+
+290
+00:20:33,240 --> 00:20:36,820
+بقول له أنت حر إيش بتسوي، المهم توجد لي اللي هو معادلة
+
+291
+00:20:36,820 --> 00:20:42,500
+الخط، هذا المستقيم بالطريقة اللي توجدها بشكل صحيح
+
+292
+00:20:43,060 --> 00:20:46,460
+إذن ليش وجدت هذا؟ عشان هيلزمني بعد شوية، هذا معادلته
+
+293
+00:20:46,460 --> 00:20:50,780
+إيش؟ معروفة اللي هو y بتساوي z بتساوي صفر، هذا معادلته
+
+294
+00:20:50,780 --> 00:20:54,400
+معروفة، هي x بتساوي صفر، أنا بشغل الآن في XZ plane
+
+295
+00:20:54,400 --> 00:20:58,240
+عشان هيك ما دخلناش y، إيش بدي فيها؟ كلها صفر، معين، إذن
+
+296
+00:20:58,240 --> 00:21:03,300
+ندخل إلى حدود الـ integration، dz dx، dz، باخد
+
+297
+00:21:03,300 --> 00:21:07,800
+موازيات لمين؟ لـ z x في داخل المنطقة، من وين بدخل ومن
+
+298
+00:21:07,800 --> 00:21:12,160
+وين بخرج؟ إيه؟ واضح إنّه بدخل من عند الـ z بتساوي
+
+299
+00:21:12,160 --> 00:21:18,550
+صفر، هيها هو يدخل من الخط هذا ويخرج من الخط هذا
+
+300
+00:21:18,550 --> 00:21:21,750
+كلهم على نفس المنوال، إيش z ثانية؟ واحد ناقص x، عشان
+
+301
+00:21:21,750 --> 00:21:25,230
+هيك من zero لعند واحد ناقص x، نجيء الآن هذه الخطوط
+
+302
+00:21:25,230 --> 00:21:28,170
+الموازية لـ z axis من أين؟ بتبدأ من عند الـ…اللي
+
+303
+00:21:28,170 --> 00:21:31,110
+هي…اللي هي…الصغيرة، من ده طبعاً من عند x
+
+304
+00:21:31,110 --> 00:21:35,070
+بتساوي صفر، هي هي هيك بتنسحب لما نصل لأخر شيء عند
+
+305
+00:21:35,070 --> 00:21:39,630
+واحد، عشان هيك قيمة الـ x Limit الـ integrations ده
+
+306
+00:21:39,630 --> 00:21:43,450
+الـ x اللي هي zero لعند الواحد، وبيكون هذا اللي هو
+
+307
+00:21:43,450 --> 00:21:50,230
+الـ… المطلوب، الآن طالب مني في السؤال على نفس المنطقة
+
+308
+00:21:50,230 --> 00:21:53,290
+بيقول integrate F of X Y و Z بيساوي واحد، يعني أوجد
+
+309
+00:21:53,290 --> 00:21:56,870
+حجمها، معناتها إيه؟ over the tetrahedron D in
+
+310
+00:21:56,870 --> 00:22:00,890
+example 2 اللي قبل شوية، in the order dz dy dx، and
+
+311
+00:22:00,890 --> 00:22:05,730
+then integrate in the order dy dz dx، هذا dy dz dx، توع
+
+312
+00:22:05,730 --> 00:22:11,230
+بالله…الآن أنا أريد أن أكمل الأمر بالنسبة
+
+313
+00:22:11,230 --> 00:22:14,650
+لـ z ثم بالنسبة لـ y ثم بالنسبة لمين؟ بالنسبة لـ x
+
+314
+00:22:14,650 --> 00:22:17,270
+طبعاً كل أمورنا جاهزة من السؤال اللي فات اللي هي
+
+315
+00:22:17,270 --> 00:22:25,170
+عبارة عن اللي هي نقاط التقاطع وإلى آخرها ومعادلة
+
+316
+00:22:25,170 --> 00:22:30,950
+الـ plane هذا ومعادلة الـ plane هذا، كله جاهز، بالنسبة لـ z
+
+317
+00:22:30,950 --> 00:22:34,430
+مدام أنّها كاملة بالنسبة لـ z فأنت تأخذ موازيات للـ
+
+318
+00:22:34,430 --> 00:22:39,290
+Z Axis التي تدخل من الجسم وتخرج من الجسم، لاحظ أنّ
+
+319
+00:22:39,290 --> 00:22:44,210
+كل الموازيات التي تدخل من الـ Z Axis تدخل الجسم من
+
+320
+00:22:44,210 --> 00:22:48,030
+عند z بتساوي صفر وتخرج من عند وين؟ يعني عند الـ plane
+
+321
+00:22:48,030 --> 00:22:51,160
+اللي تحت، وبتخرج من عند الـ plane اللي ما هي الـ plane
+
+322
+00:22:51,160 --> 00:22:54,000
+ده اللي هو إيش قيمة z؟ قلنا عليه اللي هو z بيساوي y
+
+323
+00:22:54,000 --> 00:22:57,740
+ناقص x، اللي كنا بنسميه، y بتساوي z زائد x، يعني
+
+324
+00:22:57,740 --> 00:23:01,280
+z بتساوي y ناقص x، إذن حد الـ integration الأول من
+
+325
+00:23:01,280 --> 00:23:05,660
+0 لعند y ناقص x، هاي حد الـ integration الأول
+
+326
+00:23:05,660 --> 00:23:11,840
+volume بسبب عند z اللي هو 0 لعند y ناقص x، نجيء
+
+327
+00:23:11,840 --> 00:23:16,540
+الآن يا جماعة نحدد اللي هو مين؟ نحدد قيمة اللي هو
+
+328
+00:23:16,540 --> 00:23:20,440
+الـ…الـ…الـ…قولوا معايا، الـ…الـ…المسقط
+
+329
+00:23:20,440 --> 00:23:24,520
+لأنه بدنا نكامل، إحنا كملنا لـ z، كامل y و x، بدنا
+
+330
+00:23:24,520 --> 00:23:27,820
+المسقط في وين؟ في الـ XY plane العادي، الآن المسقط
+
+331
+00:23:27,820 --> 00:23:30,220
+زي ما عملنا قبل شوية، بتصير هذه اللي هو بتصير اللي
+
+332
+00:23:30,220 --> 00:23:34,200
+هي، بنخلي الـ z بصفر، بصير 0 و 1 و 0 هي النقطة
+
+333
+00:23:34,530 --> 00:23:37,790
+وهذه المسقطة هي نفسها على حالها لأنها z بصفر، وهذه
+
+334
+00:23:37,790 --> 00:23:40,090
+المسقطة هي نفسها على حالها لأنها z بصفر، وهذه
+
+335
+00:23:40,090 --> 00:23:40,470
+المسقطة هي نفسها على حالها لأنها z بصفر، وهذه
+
+336
+00:23:40,470 --> 00:23:42,690
+المسقطة هي نفسها لأنها z بصفر، وهذه المسقطة هي
+
+337
+00:23:42,690 --> 00:23:45,350
+نفسها لأنها z بصفر، وهذه المسقطة هي نفسها لأنها z
+
+338
+00:23:45,350 --> 00:23:46,790
+بصفر، وهذه المسقطة هي نفسها لأنها z بصفر، وهذه
+
+339
+00:23:46,790 --> 00:23:48,950
+المسقطة هي نفسها لأنها z بصفر، وهذه المسقطة هي
+
+340
+00:23:48,950 --> 00:23:50,510
+نفسها لأنها z بصفر، وهذه المسقطة هي نفسها لأنها z
+
+341
+00:23:50,510 --> 00:23:53,630
+بصفر، وهذه المسقطة هي نفس…
+
+342
+00:23:57,360 --> 00:24:01,420
+حسب النقطة اللي اسقطناها، وهذا الخط كله اللي فوق
+
+343
+00:24:01,420 --> 00:24:06,320
+بنزل كله، بقعد هنا في نقطة، وهذا نسقطه اللي هو هذا
+
+344
+00:24:06,320 --> 00:24:10,370
+الخط، هذا مسقطه اللي هو نفسه لأنه في الـ plane فبيصير
+
+345
+00:24:10,370 --> 00:24:15,550
+مسقطة كل الشكل اللي عندي هو المثلث الأمامي هذا، الآن
+
+346
+00:24:15,550 --> 00:24:18,730
+المثلث هذا بدي أجيب معادلاته اللي بتلزمنا طبعاً
+
+347
+00:24:18,730 --> 00:24:23,910
+المعادلة هذه اللي هو x بتساوي صفر، والمعادلة هذه y
+
+348
+00:24:23,910 --> 00:24:27,150
+بتساوي واحد زي ما أنتم شايفين، لأنه y ثابتة على
+
+349
+00:24:27,150 --> 00:24:29,930
+هذا المكان لأنه أصلاً الـ plane كله هذا y بتساوي واحد
+
+350
+00:24:29,930 --> 00:24:34,850
+وهذه اللي هي عند النقطة أصلاً من 0 0 هو رايحة
+
+351
+00:24:34,850 --> 00:24:39,300
+لـ 1 1، إذن قيمة x هنا و y زي بعض، وقيمة x و y
+
+352
+00:24:39,300 --> 00:24:42,420
+هنا زي بعض في نقطتين، إذن الخط اللي بينهم هي أو
+
+353
+00:24:42,420 --> 00:24:45,200
+عبارة عن الخط اللي بيساوي بين x و y، اللي هو y
+
+354
+00:24:45,200 --> 00:24:49,260
+بتساوي x، واللي مش عارف يشتغل، معرفة نقطتين اللي
+
+355
+00:24:49,260 --> 00:24:52,180
+هو بيقدر يعرف معادلة الخط المستقيم هيلاقيها على طول
+
+356
+00:24:52,180 --> 00:24:54,940
+y بتساوي x، لكن الموضوع مش مستاهل، على طول الـ
+
+357
+00:24:54,940 --> 00:24:58,040
+percentage y بتساوي x، لأن قيمة اللي هي الـ x والـ y
+
+358
+00:24:58,040 --> 00:25:01,020
+في النقطتين، في النقطة الأولى زي بعض، وفي النقطة
+
+359
+00:25:01,020 --> 00:25:03,300
+الثانية برضه زي بعض، 0 0 و 1 1
+
+360
+00:25:03,300 --> 00:25:08,440
+فما فيه مقصود، إذن الأمور جاهزة، بدنا نكامل بالنسبة لمن؟
+
+361
+00:25:08,440 --> 00:25:11,100
+لـ y، نكامل بالنسبة لـ y، إذن بتاخد في داخل
+
+362
+00:25:11,100 --> 00:25:14,280
+المنطقة r اللي اتحددت اللي صارت مسقطة لها، موازية لـ y
+
+363
+00:25:14,280 --> 00:25:18,320
+axis، كل الموازية لـ y axis هذول اللي بيجيء من عند
+
+364
+00:25:18,320 --> 00:25:22,240
+الخط y بتساوي x و بروح من عند الخط y إيش؟ بتساوي 1
+
+365
+00:25:22,240 --> 00:25:26,340
+لأنّ هنا بتبدأ تتزايد هيك، بتتزايد، إذن من y بتساوي
+
+366
+00:25:26,340 --> 00:25:30,220
+x لعند y بتساوي 1، من y بتساوي x لعند y بتساوي 1
+
+367
+00:25:30,220 --> 00:25:38,890
+هذا قيمة من الـ y، نعود لقيمة الـ x، كل الخطوط الموازية
+
+368
+00:25:38,890 --> 00:25:45,750
+الـ y axis تبدأ من x بصفر وتتدرج لين تصل لـ x بواحد
+
+369
+00:25:45,750 --> 00:25:50,010
+آخر خط بيمسح كل المنطقة، وبيكون عندي اللي هو الـ
+
+370
+00:25:50,010 --> 00:25:53,910
+integration من صفر لواحد بالنسبة لـ x لحدود التكامل
+
+371
+00:25:53,910 --> 00:25:57,410
+نجيء الآن، كامل…كامل…ما هي التكامل؟ الواحد هو عبارة عن
+
+372
+00:25:57,410 --> 00:26:01,830
+z، بصير z، بعوض y ناقص x ناقص 0، 0 بصير y
+
+373
+00:26:01,830 --> 00:26:05,070
+ناقص x، بدي أكامل هذا y ناقص x بالنسبة لـ y بصير
+
+374
+00:26:05,070 --> 00:26:09,390
+تكامله عبارة عن نصف y تربيع ناقص xy من وين لوين؟
+
+375
+00:26:09,390 --> 00:26:12,780
+بعوض من x، من…من y بتساوي x لأنّ y بتساوي واحد،
+
+376
+00:26:12,780 --> 00:26:20,240
+لأنّ بعوض عن y هنا بيصير عبارة عن نصف ناقص x، لأنّ y
+
+377
+00:26:20,240 --> 00:26:24,660
+بتساوي x، ناقص هي عبارة عن نصف y تربيع، نصف y تربيع
+
+378
+00:26:24,660 --> 00:26:27,680
+ناقص y تربيع، اللي هي بيصير عبارة عن ناقص نصف y
+
+379
+00:26:27,680 --> 00:26:31,120
+تربيع، والناقص اللي برا بيصير آسف x تربيع، بيصير نصف
+
+380
+00:26:31,120 --> 00:26:36,170
+x تربيع، لأنّ بعوض عن y بـ x، طيب الآن صار عندي المقدار
+
+381
+00:26:36,170 --> 00:26:39,370
+بالشكل هذا، بدي أكملة بالنسبة لـ x، هكملة بالنسبة لـ
+
+382
+00:26:39,370 --> 00:26:43,530
+x، نصف x، x تربيع على اللي هو نصف x، عدد x تربيع اللي
+
+383
+00:26:43,530 --> 00:26:46,170
+هو هذه اللي هي بيصير على ثلاثة بيصير سدس و x تكعيب
+
+384
+00:26:46,170 --> 00:26:50,570
+ومن 0 لـ 1 و
+
+445
+00:30:52,920 --> 00:30:59,300
+هو y بصفر، واللي فوق هذا عبارة عن اللي هو
+
+446
+00:31:09,650 --> 00:31:16,440
+أول حاجة بنرسم، بمعنى sketch with enough details to
+
+447
+00:31:16,440 --> 00:31:19,360
+show the limit of integration، لكي نحدد اللي هو
+
+448
+00:31:19,360 --> 00:31:22,540
+اللي هي حدود ال integration. هذه ال cube رسمناها
+
+449
+00:31:22,540 --> 00:31:25,560
+لأن الأمور سهلة. ال cube أولا، ال volume إيه له؟ لأنه
+
+450
+00:31:25,560 --> 00:31:28,380
+ملزمنا ال volume. عشان ذلك، ال volume اللي هو اللي
+
+451
+00:31:28,380 --> 00:31:31,700
+هو، عارفين ال volume اللي هو مساحة القاعدة اللي هي
+
+452
+00:31:31,700 --> 00:31:34,700
+أربعة في الارتفاع اتنين. أربعة في اتنين بتمانية، أو
+
+453
+00:31:34,700 --> 00:31:37,540
+ثلاثة أبعاد: اتنين في اتنين في اتنين بتطلع تمانية.
+
+454
+00:31:37,540 --> 00:31:43,240
+اللي هي يشملها ال average value لمن؟ لأسف، اللي هو
+
+455
+00:31:43,240 --> 00:31:44,840
+ال volume للمكعب.
+
+456
+00:31:46,810 --> 00:31:49,010
+الآن نجي
+
+457
+00:31:49,010 --> 00:31:49,970
+نجي
+
+458
+00:31:49,970 --> 00:31:52,810
+نجي
+
+459
+00:31:52,810 --> 00:31:57,650
+نجي
+
+460
+00:31:57,650 --> 00:32:01,810
+نجي
+
+461
+00:32:01,810 --> 00:32:06,590
+نجي
+
+462
+00:32:06,590 --> 00:32:09,570
+نجي
+
+463
+00:32:10,280 --> 00:32:15,240
+X بصفر، ورايحة لعند X بساوة اتنين. يعني من الوجه هذا
+
+464
+00:32:15,240 --> 00:32:19,460
+للوجه اللي هو هذا، كل اللي هي الموازيات لل X axis.
+
+465
+00:32:19,460 --> 00:32:23,740
+على نفس المنوال. عشان هيك بكون ال X اللي هي حدودها
+
+466
+00:32:23,740 --> 00:32:27,460
+من صفر لعند مين؟ اتنين. الآن مسقط هذا سهل، مسقط
+
+467
+00:32:27,460 --> 00:32:31,400
+المكعب كله إيه؟ أشمله. المكعب هنا ساوة دي بصفر، تنزل
+
+468
+00:32:31,400 --> 00:32:36,540
+هنا اللي هو زدها بصفر، تنزل. احنا لأ، احنا بنكمل X
+
+469
+00:32:36,540 --> 00:32:44,940
+أولا، و YZ أولا، و X أولا
+
+470
+00:32:44,940 --> 00:32:45,300
+و X أولا
+
+471
+00:32:45,300 --> 00:32:45,680
+و X أولا
+
+472
+00:32:45,680 --> 00:32:46,820
+و X أولا
+
+473
+00:32:46,820 --> 00:32:48,160
+و X أولا
+
+474
+00:32:48,160 --> 00:32:54,220
+و X أولا
+
+475
+00:32:54,220 --> 00:32:58,450
+أولا. الذي في الـ ZY Plane، المربع اللي هنا أموره سهلة،
+
+476
+00:32:58,450 --> 00:33:04,910
+و مستطيلة جميعها على طول. لو أخذنا الآن بالنسبة لـ
+
+477
+00:33:04,910 --> 00:33:10,010
+Y، موازي لـ Y Axis، كلهم على نفس المنوال من 0 ل 2، و Z
+
+478
+00:33:10,010 --> 00:33:17,430
+طبعا تأخذ أولا بـ 0، و تتكامل من 0 ل 2، و
+
+479
+00:33:17,430 --> 00:33:23,420
+هذا هو قيمة الـ Integration بالكامل. لأن xyz بالنسبة
+
+480
+00:33:23,420 --> 00:33:26,740
+ل x بيصير x تربيع على اتنين في yz، من صفر لعند
+
+481
+00:33:26,740 --> 00:33:29,600
+اتنين. اتنين بيصير عبارة عن أربعة على اتنين اللي هي
+
+482
+00:33:29,600 --> 00:33:33,780
+اتنين. بيصير 2 yz. وهذا طبعا من صفر لاتنين
+
+483
+00:33:33,780 --> 00:33:36,220
+بتكامل برضه بالنسبة ل y، بيصير اتنين y تربيع على
+
+484
+00:33:36,220 --> 00:33:40,000
+اتنين اللي هي y تربيع z. الآن من صفر لعند اتنين، y
+
+485
+00:33:40,000 --> 00:33:45,610
+تربيع بيصير أربعة z. التي تتكامل من 0 إلى 2. أربعة z
+
+486
+00:33:45,610 --> 00:33:49,690
+بيصير أربعة z تربيع على اتنين، يعني اتنين z تربيع
+
+487
+00:33:49,690 --> 00:33:52,190
+من صفر إلى اتنين، بيصير عبارة عن اتنين في أربعة
+
+488
+00:33:52,190 --> 00:33:55,870
+و ثمانية. اتنين في أربعة = ثمانية. الآن ثمانية هذه هي إيه؟ شمالها
+
+489
+00:33:55,870 --> 00:33:59,830
+إيه قيمة ال integration. دا ال average value of XYZ
+
+490
+00:34:00,290 --> 00:34:04,390
+الـ Average Value لمن؟ لأن ده ل F of X Y و Z
+
+491
+00:34:04,390 --> 00:34:07,570
+بساوي X في Y في Z على ال cube، بساوي واحد على ال
+
+492
+00:34:07,570 --> 00:34:09,890
+volume اللي هي واحد على تمانية في مقدار ال
+
+493
+00:34:09,890 --> 00:34:12,670
+integration، طالع تمانية، اللي هو بساوي واحد. و هيك
+
+494
+00:34:12,670 --> 00:34:15,710
+يكون أوجدنا ال average value. و هيك يكون المحاضرة
+
+495
+00:34:15,710 --> 00:34:18,470
+انتهت. وإلى لقاء آخر، السلام عليكم ورحمة الله
+
+496
+00:34:18,470 --> 00:34:19,570
+وبركاته
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e1e24a94086cb39947e8815db6ed147bfc449a2a
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A_postprocess.srt
@@ -0,0 +1,1984 @@
+1
+00:00:00,770 --> 00:00:04,810
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,810 --> 00:00:11,070
+وبركاته هذه هي المحاضرة رقم 24 مساق تفاضل وتكامل
+
+3
+00:00:11,070 --> 00:00:16,290
+جيم وهي المحاضرة الخامسة في chapter 15 الشابط
+
+4
+00:00:16,290 --> 00:00:22,560
+الأخير في هذا المساقالشبتر اللى هو عنوانه triple
+
+5
+00:00:22,560 --> 00:00:28,840
+integrals in rectangular coordinates احنا كنا
+
+6
+00:00:28,840 --> 00:00:33,440
+تحدثنا المحاضرة السابقة عن double integral in
+
+7
+00:00:33,440 --> 00:00:36,840
+polar coordinates و اللى جاب اللى حكينا عن double
+
+8
+00:00:36,840 --> 00:00:41,700
+integrals in rectangular coordinates و اليوم نحكي
+
+9
+00:00:41,700 --> 00:00:46,330
+عن نكمل نحكي عن ال double integralفي حالة الـ
+
+10
+00:00:46,330 --> 00:00:50,410
+Rectangular Coordinates الكلام سيكون شبيه و الكلام
+
+11
+00:00:50,410 --> 00:00:55,490
+بنفس الروح و بنفس السولة ان شاء الله الان اول اشي
+
+12
+00:00:55,490 --> 00:01:00,050
+نحكي عن Volume of a region in space اذا بتذكروا
+
+13
+00:01:00,050 --> 00:01:03,030
+احنا استخدمنا double integral في إيجاد ال area
+
+14
+00:01:03,310 --> 00:01:07,090
+وقلنا على طول الـ area بتساوي الـ double integral
+
+15
+00:01:07,090 --> 00:01:12,430
+over R ال region دي ايه دي ايه دي اكس دي وايه دي
+
+16
+00:01:12,430 --> 00:01:13,070
+اكس دي وايه دي اكس دي وايه دي اكس دي وايه دي اكس
+
+17
+00:01:13,070 --> 00:01:14,050
+دي وايه دي اكس دي وايه دي اكس دي اكس دي اكس دي اكس
+
+18
+00:01:14,050 --> 00:01:14,230
+دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس
+
+19
+00:01:14,230 --> 00:01:19,070
+دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس
+
+20
+00:01:19,070 --> 00:01:20,010
+دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس
+
+21
+00:01:20,010 --> 00:01:27,350
+دي اكس ديالتي ستحصل بدل ما هو دي اي دي اكس دي واي
+
+22
+00:01:27,350 --> 00:01:30,830
+ستحصل دي volume of a closed bounded region دي
+
+23
+00:01:30,830 --> 00:01:36,250
+انوين in space sv بتساوي triple انتجرال
+
+24
+00:01:39,450 --> 00:01:44,070
+Overdeen هي ال region in space دي في دي في طبعا هي
+
+25
+00:01:44,070 --> 00:01:49,270
+عبارة عن ال dx dy dz طبعا هذا له أصل اللي هو
+
+26
+00:01:49,270 --> 00:01:53,070
+إيجاده اللي هو زي ما حكينا في أول ال double
+
+27
+00:01:53,070 --> 00:01:57,790
+integration اللي هو بنقسم المنطقةالى اخسام صغيرة
+
+28
+00:01:57,790 --> 00:02:01,630
+صغيرة صغيرة لل X و لل Y و الآن لل Z و بيصير
+
+29
+00:02:01,630 --> 00:02:03,850
+المجموع remaining sum و بعد ما يصير مجموع
+
+30
+00:02:03,850 --> 00:02:06,770
+remaining sum بيوجد ال limit إذا كان ال limit وجود
+
+31
+00:02:06,770 --> 00:02:10,650
+بيكون هو بالظبط قيمة حجم الجسم اللي اللي قاعدين
+
+32
+00:02:10,650 --> 00:02:15,650
+بنجزقه و بعدين بيكون هو اللي هي ال triple integral
+
+33
+00:02:15,650 --> 00:02:22,680
+له هو ال volumeفهذا اللي هو في الآخر limit ده
+
+34
+00:02:22,680 --> 00:02:25,940
+remand sum فاحنا مانعش نعيد الكلام اللي حكيناه في
+
+35
+00:02:25,940 --> 00:02:30,100
+الأول على الدمل integral لما عرفناه الآن عملية
+
+36
+00:02:30,100 --> 00:02:35,440
+إيجاد اللي هي ال triple integral بنفس الأسلوب عندي
+
+37
+00:02:35,440 --> 00:02:38,100
+بيكون تلت حاجات قيمة ال function بس أنا قيمة ال
+
+38
+00:02:38,100 --> 00:02:42,660
+function واحد الحاجة التانية اللي هي dv dx dy dz
+
+39
+00:02:42,660 --> 00:02:47,520
+والحاجة اللي بدها شغلاللي هي باسمين D التي هي حدود
+
+40
+00:02:47,520 --> 00:02:51,380
+ال integration أو limits of integration الان اللي
+
+41
+00:02:51,380 --> 00:02:54,020
+بدنا نحكي هو find limits of integration in the
+
+42
+00:02:54,020 --> 00:02:57,900
+order DZ DY DX الان بدنا نوجد ال order بأي order
+
+43
+00:02:57,900 --> 00:03:01,420
+DZ أول شي و بعدين DY و بعدين DX و طبعا و الباجات
+
+44
+00:03:01,420 --> 00:03:07,220
+similar للتبدلات الاخرى DZ DX DY DY DZ DX إلى آخره
+
+45
+00:03:07,220 --> 00:03:11,300
+خلينا نشوف الآنطيب صلوا على النبي عليه الصلاة
+
+46
+00:03:11,300 --> 00:03:15,340
+والسلام نجي للمثال الأول find the volume of the
+
+47
+00:03:15,340 --> 00:03:19,520
+region D enclosed by the surface Z بتساوي X تربيع
+
+48
+00:03:19,520 --> 00:03:22,800
+زائد تلاتة Y تربيع هاي اللي هو ال surface اللي
+
+49
+00:03:22,800 --> 00:03:26,700
+عندي هيه اللي تحت هذا اللي هو Z بتساوي X تربيع
+
+50
+00:03:26,700 --> 00:03:29,800
+زائد تلاتة Y تربيع احنا بنجيبلكم مش سؤال انه يكون
+
+51
+00:03:29,800 --> 00:03:34,560
+الرسمفي أسعب من السؤال إيه ده؟ إيه ده طلب إننا
+
+52
+00:03:34,560 --> 00:03:38,420
+نرسم لكم و بنرسم لكم؟ أما مش مختبرك الآن في الرسم
+
+53
+00:03:38,420 --> 00:03:42,400
+مختبرك في إيجاد اللي هو الـTriple Integral Z
+
+54
+00:03:42,400 --> 00:03:48,200
+بتساوي X تربيع زي تلتة Y تربيع حيه الآن ال region
+
+55
+00:03:48,200 --> 00:03:51,840
+دي محصورة بين هذا ال surface و ال surface التاني Z
+
+56
+00:03:51,840 --> 00:03:55,380
+بتساوي تمانية ناقص X تربيع ناقص Y تربيع حيث Z
+
+57
+00:03:55,380 --> 00:04:00,200
+بتساوي تمانية ناقص X تربيع ناقص Y تربيعهذه المنطقة
+
+58
+00:04:00,200 --> 00:04:04,780
+بدي أجد لها ال volume ال volume طبعا بيساوي triple
+
+59
+00:04:04,780 --> 00:04:11,420
+integral دي over دي this region dz dy dx زي ما هو
+
+60
+00:04:11,420 --> 00:04:17,280
+طلب مننا قبل ما نبدأ في السؤال او طبعا اللي بده
+
+61
+00:04:17,280 --> 00:04:21,620
+يحكي طبعا ال .. يمكن الحكي في سؤال زي هذا اللي هو
+
+62
+00:04:21,620 --> 00:04:26,240
+dy قبل dz كده اللي هو صعب انك تاخد و انا بصير تاخد
+
+63
+00:04:26,240 --> 00:04:28,580
+ال .. ال .. ال ..اللي .. اللي .. اللي .. اللي ..
+
+64
+00:04:28,580 --> 00:04:29,600
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+65
+00:04:29,600 --> 00:04:30,140
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+66
+00:04:30,140 --> 00:04:31,140
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+67
+00:04:31,140 --> 00:04:31,840
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+68
+00:04:31,840 --> 00:04:35,040
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+69
+00:04:35,040 --> 00:04:37,380
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+70
+00:04:37,380 --> 00:04:38,860
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+71
+00:04:38,860 --> 00:04:41,460
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+72
+00:04:41,460 --> 00:04:49,860
+اللي .. اللي .. اللي .. اللي .. اللي
+
+73
+00:04:50,080 --> 00:04:55,480
+لأن كيف بتحدد حدود ال integration؟ الآن Z نفس
+
+74
+00:04:55,480 --> 00:04:59,060
+الروح ايش باخد زي ما كنا لما كنا بنكامل بالنسبة ل
+
+75
+00:04:59,060 --> 00:05:02,900
+X ناخد موازية ل X لما نكامل ل Y ناخد موازية ل Y
+
+76
+00:05:02,900 --> 00:05:08,220
+الآن بتداخل بالنسبة ل Z موازية ل Z من وين دخل ال
+
+77
+00:05:08,220 --> 00:05:13,000
+high Z طبعا high اللي هو ال increasing له او عملية
+
+78
+00:05:13,000 --> 00:05:20,400
+التزايد لهالان بناخد اللي هي عبارة عن موازيات اللي
+
+79
+00:05:20,400 --> 00:05:25,620
+هو وين بتعمل entering ل اللي هو الجسم وكان وين
+
+80
+00:05:25,620 --> 00:05:29,160
+بتعمل الخروج اللي يعني الدخول للجسم واضح انها
+
+81
+00:05:29,160 --> 00:05:34,450
+بتبدأ من عنداللي هو السطح هذا هاي السطح هذا لو
+
+82
+00:05:34,450 --> 00:05:39,710
+دخلنا أي اللي هو موازي الـ z axis هيطلع من السطح
+
+83
+00:05:39,710 --> 00:05:43,870
+البراني هذا اللي هو ايش ماله z بتساوي x تربيع زي
+
+84
+00:05:43,870 --> 00:05:48,430
+تلتة y تربيع و بطلع بخرج من وين من هذه المنطقة
+
+85
+00:05:48,430 --> 00:05:51,970
+اللي هي z بتساوي تمانية ناقص x تربيع ناقص y تربيع
+
+86
+00:05:51,970 --> 00:05:55,130
+و كل هنا على نفس المنوال شايف مثلها لكن لو بدنا
+
+87
+00:05:55,130 --> 00:05:57,570
+نتكامل بالنسبة على x أو الاشياء بدنا نمشي هيك و
+
+88
+00:05:57,570 --> 00:06:00,920
+بدنا نمشي هيكصار بده ييجي من هنا لما يروح لهنا
+
+89
+00:06:00,920 --> 00:06:04,640
+بيصير فيه شوية صعوبة في التحديد لكن مش مستحيلة و
+
+90
+00:06:04,640 --> 00:06:09,060
+نفس الاشي بالنسبة ل why بالشكل الثاني إذا لازد
+
+91
+00:06:09,060 --> 00:06:15,240
+بيدخل من عند اللي هو ال curve الأول أو ال surface
+
+92
+00:06:15,240 --> 00:06:19,660
+الأول و بيطلع من عند ال surface الثاني إذا صارت
+
+93
+00:06:19,660 --> 00:06:24,180
+اللي هي تكاملها حدود ال integration هي عبارة عن
+
+94
+00:06:24,390 --> 00:06:29,210
+اللي هي من اللي تحت X تربيع زي تلتة Y تربيع قيمة Z
+
+95
+00:06:29,210 --> 00:06:34,650
+ل تمانية ناقص X تربيع ناقص Y تربيع اللي هي قيمة Z
+
+96
+00:06:34,650 --> 00:06:38,590
+الثانية هذه الأولى اللي ذكرناها X تربيع زي تلتة Y
+
+97
+00:06:38,590 --> 00:06:45,150
+تربيع و هذه القيمة الثانية الان بدي أوجد اللي هو
+
+98
+00:06:45,150 --> 00:06:50,250
+من حدود ال X و ال Y عشان أوجد حدود ال X و ال Yالان
+
+99
+00:06:50,250 --> 00:06:54,790
+اللي هو فأسئلة من هذا النوع أصلا يعني ممكن احنا
+
+100
+00:06:54,790 --> 00:07:00,200
+اللي هو ما نرسموهلكن هنا التوضيح كيف نرسمه على طول
+
+101
+00:07:00,200 --> 00:07:04,640
+اللي هو ب .. بس بدك تعرف يعني بتغيير قيمة بتعويض
+
+102
+00:07:04,640 --> 00:07:07,860
+عن قيمتين هنا و قيمتين هنا القيمة ال X و ال Y هنا
+
+103
+00:07:07,860 --> 00:07:10,320
+و قيمة ال X و ال Y هنا بتعرف مين اللي فوق و مين
+
+104
+00:07:10,320 --> 00:07:14,420
+اللي تحت طبعا يعني هو الكلام مش safe كثير لكن ..
+
+105
+00:07:14,420 --> 00:07:19,620
+لكن بقد الغرض القيمة الأعلى ل .. ل .. ل .. ل .. ال
+
+106
+00:07:19,620 --> 00:07:23,140
+X مثلا بتاخدها واحد و كده بتلاقي مثلا اللي هي هذه
+
+107
+00:07:23,140 --> 00:07:27,360
+قيمة أعلى وهذه قيمة أسفلبتكون حدود الانتجريشن من
+
+108
+00:07:27,360 --> 00:07:31,920
+هنا لعند هنا هذا لو بدكاش ترسم أو فيش رسم في رسمات
+
+109
+00:07:31,920 --> 00:07:35,620
+هذا مفضلك تكامل بالنسبة لزد أساس تعرف مين اللي فوق
+
+110
+00:07:35,620 --> 00:07:42,540
+ومين اللي تحت طيب الآن نجلع اللي هي المنطقة اللي
+
+111
+00:07:42,540 --> 00:07:48,080
+بدنا نكامل عليها x و yخلصنا من z بدنا نيجي x1 إيش
+
+112
+00:07:48,080 --> 00:07:54,200
+بنعمل إذا بنعمل إسقاط للجسم هذا الآن وهذا اللي هي
+
+113
+00:07:54,200 --> 00:08:00,440
+إسقاط كما يليه عندي اللي هو z بخليه سفر و z
+
+114
+00:08:00,440 --> 00:08:03,500
+بالساوية سفر نشوف المنطقة المشتركة بينهما هي
+
+115
+00:08:03,500 --> 00:08:06,850
+الإسقاطاللي هي x تربيع زي تلتة y تربيع بيصير
+
+116
+00:08:06,850 --> 00:08:11,130
+بتساوي تمانية ناقص x تربيع ناقص y تربيع الآن x
+
+117
+00:08:11,130 --> 00:08:14,470
+تربيع زي تلتة y تربيع بتجيب لها ال x تربيع بيصير
+
+118
+00:08:14,470 --> 00:08:17,370
+اللي هو اتنين x تربيع وهذا بتجيب لها ال y تربيع
+
+119
+00:08:17,370 --> 00:08:20,750
+على الطرف التاني بيصير اربعة y تربيع اربعة y تربيع
+
+120
+00:08:20,970 --> 00:08:24,050
+زاد اتنين X تربيع بيساوي تمانية X من الجهتين على
+
+121
+00:08:24,050 --> 00:08:27,670
+اتنين بيصير X تربيع زاد اتنين Y تربيع بتساوي ايش
+
+122
+00:08:27,670 --> 00:08:33,110
+اربعة اللي هي ال ellipse اللي الاسقاط طبع اللي هو
+
+123
+00:08:33,110 --> 00:08:36,590
+الجسم اللي هو X تربيع زاد اتنين Y تربيع ايش بيساوي
+
+124
+00:08:36,590 --> 00:08:40,770
+بيساوي اربعة اذا عرفنا كيف الاسقاط طبحين هذي بسفر
+
+125
+00:08:40,770 --> 00:08:44,010
+و هذي بسفر و بنسوين في بعض بيطلع المنطقة اللي هي
+
+126
+00:08:44,010 --> 00:08:49,260
+اسقاط الجهتيننأتي الآن .. نأتي الآن .. نأتي الآن
+
+127
+00:08:49,260 --> 00:08:52,120
+.. اللي هو بدنا نحدد حدود ال integration من هذه
+
+128
+00:08:52,120 --> 00:08:55,540
+المنطقة بدنا نكمل بالنسبة لمين جابي الا Y بعدين
+
+129
+00:08:55,540 --> 00:09:01,700
+بدنا نكمل ال Y إذا بدنا ناخد اللي هو موازيات للمين
+
+130
+00:09:01,700 --> 00:09:06,330
+لل Y axis هي ال Y axis ناخد لها موازياتكل
+
+131
+00:09:06,330 --> 00:09:11,310
+الموازيات التي منها إلى هنا كلها تأتي من الجهة
+
+132
+00:09:11,310 --> 00:09:15,490
+السالبة لـ Y عبر الـ curve نفسه X تربيع زي اتنين و
+
+133
+00:09:15,490 --> 00:09:19,650
+Y تربيع إلى الجهة الموجبة لـ Y عبر الـ curve نفسه
+
+134
+00:09:19,650 --> 00:09:23,290
+إذا حدود الـ integration بدأ أوجد Y تصبح اتنين و Y
+
+135
+00:09:23,290 --> 00:09:26,930
+تربيع تسوى أربعة ناقص X تربيع يعني Y تربيع تسوى
+
+136
+00:09:26,930 --> 00:09:31,230
+أربعة ناقص X تربيع على اتنين يعني Y تصبح زائد أو
+
+137
+00:09:31,230 --> 00:09:34,490
+ناقص الجدر التربيع إلى الأربعة ناقص X تربيع على
+
+138
+00:09:34,490 --> 00:09:38,710
+اتنينالان اذا صار عندى قيمتين هذه اللى هى قيمة Y
+
+139
+00:09:38,710 --> 00:09:41,990
+السالبة وهذه قيمة Y الموجبة وهذه اللى هى بتكون
+
+140
+00:09:41,990 --> 00:09:44,770
+ايهاش حدود ال integration Y بتساوى ناقص الجدر
+
+141
+00:09:44,770 --> 00:09:48,190
+تربيع الاربع ناقص X تربيع على اتنين وY بتساوي
+
+142
+00:09:48,190 --> 00:09:51,450
+الجدر تربيع الاربع ناقص X تربيع على اتنين هذا بدخل
+
+143
+00:09:51,450 --> 00:09:54,930
+المنطقة وهذا بخرج من المنطقة وهى حدود ال
+
+144
+00:09:54,930 --> 00:09:59,450
+integration نجى نرجع لان احدد قيمة X دايل قيمة X
+
+145
+00:09:59,450 --> 00:10:03,690
+من وين بدأت كل الخطوط هى الخطوط هذهعند هذه الخطوط
+
+146
+00:10:03,690 --> 00:10:06,610
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+147
+00:10:06,610 --> 00:10:14,010
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+148
+00:10:14,010 --> 00:10:14,610
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+149
+00:10:14,610 --> 00:10:15,590
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+150
+00:10:15,590 --> 00:10:15,650
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+151
+00:10:15,650 --> 00:10:28,430
+هيك خط هيك خط هيك خط هيك خط
+
+152
+00:10:28,430 --> 00:10:32,880
+هيك خط هيبنعوض عنه زد بـ 8-X تربيع ناقص Y تربيع
+
+153
+00:10:32,880 --> 00:10:38,300
+ناقص اللي هو X تربيع زائد 3Y تربيع بطلع عندي
+
+154
+00:10:38,300 --> 00:10:41,860
+المقدار اللي أمامي هذا هذا بده فاضل الآن بالنسبة
+
+155
+00:10:41,860 --> 00:10:45,800
+لمن؟ بالنسبة لـ Y فاضله بالنسبة لـ Y بيصير 8Y ناقص
+
+156
+00:10:45,800 --> 00:10:51,170
+2X تربيع في Y هيهااللي هو ناقص اربعة Y تكعيب على
+
+157
+00:10:51,170 --> 00:10:54,910
+تلاتة من وين لوين من عند هذا المقدار لهذا المقدار
+
+158
+00:10:54,910 --> 00:10:58,990
+اذا Y من عند ناقص الجدر لعند الجدر نفسه اللي هو
+
+159
+00:10:58,990 --> 00:11:03,790
+بعوض على هذه اللي هي عن قيمة Y هي عوضت وهنا عوضت
+
+160
+00:11:03,790 --> 00:11:08,850
+عن قيمة Y اللي هي و بعدين اللي هو طرحت من هدولة
+
+161
+00:11:08,850 --> 00:11:15,290
+طبعا هذه اللي هيY مفردة فبصير ناقص في ناقص لها
+
+162
+00:11:15,290 --> 00:11:19,450
+بصير ناقص ناقص بصير زائد فبصير هذا ضعف الأولاني
+
+163
+00:11:19,450 --> 00:11:25,870
+وهذا نفس الاشي ان هي فردية فبصير لما أرفع حادة لل
+
+164
+00:11:25,870 --> 00:11:29,740
+أس اللي هو الفردي هذا بتظلها ناقصو في اللي هو
+
+165
+00:11:29,740 --> 00:11:33,900
+الناقص الأولاني اللي هو التعويض بهذه ناقص التعويض
+
+166
+00:11:33,900 --> 00:11:37,840
+بهذه فبصير زائد اللي هو و في ناقص أصلا بيصير ناقص
+
+167
+00:11:37,840 --> 00:11:40,960
+بترجع ناقص إذا إشارتها بيصير نفس القيمة بنفس
+
+168
+00:11:40,960 --> 00:11:44,240
+الإشارة
+
+169
+00:11:44,240 --> 00:11:47,320
+وهذه نفس القيمة بنفس الإشارة فبصير اتنين في
+
+170
+00:11:47,320 --> 00:11:51,800
+المقدار هذا ناقص اتنين في المقدار هذا فهو بيكون
+
+171
+00:11:51,800 --> 00:11:54,880
+عندي اللي هو المقدار الأمامي فاهمين علاش بقوليعني
+
+172
+00:11:54,880 --> 00:12:00,120
+هذا معوض ابو هذا ناقص هذا معوض ابو هذا هيطلع اللي
+
+173
+00:12:00,120 --> 00:12:04,580
+هو هذا المقدار اللي هو اللي هي ضعف اللي هي التعويض
+
+174
+00:12:04,580 --> 00:12:09,180
+اللي موجود لإنه ناقص في ال order اللي هو الفردي بي
+
+175
+00:12:09,180 --> 00:12:14,840
+.. بيصير زائد لما أقول ناقص ناقص فبتطلع عندي ضعف
+
+176
+00:12:14,840 --> 00:12:18,500
+المقدار وجربوها لحالكم حتى جوا اللي بحكيه هو
+
+177
+00:12:18,500 --> 00:12:25,830
+الصحيحهذا المقدار يجب أن نكمل بهذا المقدار 4-x
+
+178
+00:12:25,830 --> 00:12:33,910
+تربيع 4-x
+
+179
+00:12:33,910 --> 00:12:39,610
+تربيع 4-x تربيع 4-x تربيع
+
+180
+00:12:43,470 --> 00:12:48,010
+طبعا الان اللي هي بصير تمانية عندى احسابات عادية
+
+181
+00:12:48,010 --> 00:12:52,390
+بتضرب هذا بتعمل هذا على صورة أربعة نقص كستربيع على
+
+182
+00:12:52,390 --> 00:12:57,450
+اتنين فبصير هذا أس واحد وهذا أس مص وهذا بصير أس
+
+183
+00:12:57,450 --> 00:13:00,830
+تلتة على اتنين طبعا بضل برا أيش تمانية معاكم بعيد
+
+184
+00:13:00,830 --> 00:13:06,220
+الاعتبار علىاللي هي اللى عندى يكون بنفس اللى هو
+
+185
+00:13:06,220 --> 00:13:10,380
+الصورة هذه الان ناقص نفس الاشي الحسبة هنا أس تلاتة
+
+186
+00:13:10,380 --> 00:13:14,820
+ع اتنين الان بتكامل طبعا هذا المقدار و هذا المقدار
+
+187
+00:13:14,820 --> 00:13:18,920
+نفس الاشي فبتطرح تماما ناقص تماما ع تلاتة و بعدين
+
+188
+00:13:18,920 --> 00:13:23,400
+بتكامل للجهتين كيف الآن بدك تكامل للجهتين طبعا لما
+
+189
+00:13:23,400 --> 00:13:27,140
+تطرح من بعض بطلع المقدار الأمامك كده كيف بدك تكامل
+
+190
+00:13:27,140 --> 00:13:30,940
+للجهتينأه طبعا هو طلع التنين هنا والتنين من هنا
+
+191
+00:13:30,940 --> 00:13:35,060
+وطلعها وصار اربعة جدر التنين على مين على تلاتة
+
+192
+00:13:35,060 --> 00:13:39,080
+فاهمين كيف هذا طلعت اتنين أس تلاتة على اتنين طبعا
+
+193
+00:13:39,080 --> 00:13:41,620
+اول اشي طرحت تمانية ناقص تمانية على تلاتة وطلعت
+
+194
+00:13:41,620 --> 00:13:44,880
+القيمة بعدين اتنين أس تلاتة على اتنين طلحها عام
+
+195
+00:13:44,880 --> 00:13:48,340
+المشترك برا وضربها في القيمة اللي طلعت طلعت بتطلع
+
+196
+00:13:48,340 --> 00:13:51,040
+عند اربعة جدر التنين على تلاتة في ال integration
+
+197
+00:13:51,040 --> 00:13:53,860
+اللي أمامي ال integration اللي أمامي هذااللي هو مش
+
+198
+00:13:53,860 --> 00:13:57,240
+بيغلب شوية بس احنا يعني مش هنجيبلكم اشي اللي هو
+
+199
+00:13:57,240 --> 00:14:00,560
+بيغلب كتير لدرجة انك نسيك في اللي هو تشتغله من
+
+200
+00:14:00,560 --> 00:14:05,180
+تفاضل بقى ونجيبلك تكاملات اسهل من هيك طبيعي لكن
+
+201
+00:14:05,180 --> 00:14:09,180
+هذا الان اللي هي احنا في البيت و بنشتغله أبقى
+
+202
+00:14:09,180 --> 00:14:12,550
+ريحيةبناخد اللي هو بالتعويض زي ما كنتوا تعرفوا في
+
+203
+00:14:12,550 --> 00:14:16,230
+تفاضل ألف X بالساعة و اتنين sin Y و sin U وبتعوض
+
+204
+00:14:16,230 --> 00:14:19,470
+فيها دي بيطلع القيمة اتجرب ال integration كامله
+
+205
+00:14:19,470 --> 00:14:25,550
+تطلع تمانية باي في جذر اتنين طيب بكون هيك وجدنا ال
+
+206
+00:14:25,550 --> 00:14:30,990
+volume نيجي الأن ناخد مثال آخربنقل قيمة الانتجار
+
+207
+00:14:30,990 --> 00:14:34,830
+لتجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+208
+00:14:34,830 --> 00:14:35,850
+تجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+209
+00:14:35,850 --> 00:14:37,610
+تجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+210
+00:14:37,610 --> 00:14:39,590
+تجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+211
+00:14:39,590 --> 00:14:55,110
+تجارب تجارب تجارب تجارب تجار
+
+212
+00:14:55,130 --> 00:15:01,170
+أتفاضل f of x, y, z اتكامل أسف بصورة مين dy, dz,
+
+213
+00:15:01,310 --> 00:15:05,030
+dx ده محددلك ال order اللي بدهيا هو لمين الآن
+
+214
+00:15:05,030 --> 00:15:09,370
+لمنطقة دي دي مين هي المنطقة اللي هي ال tetrahedron
+
+215
+00:15:09,370 --> 00:15:12,890
+هذا اللي أمامي الهرم الأماني اللي أمامي لأن هذا
+
+216
+00:15:12,890 --> 00:15:17,590
+الهرم محددلك إيش هو بالظبط اللي هي 000 هيها راسه
+
+217
+00:15:17,590 --> 00:15:24,180
+والرأس التاني 0101 وهنا الرأس الرابعأو الثالث 010
+
+218
+00:15:24,180 --> 00:15:28,920
+والرأس الأخير 1100 هي الثلاث رؤوس وطبعا وصل بينهم
+
+219
+00:15:28,920 --> 00:15:34,780
+خطوط إذا هذا هو مين اللي هو الشكل اللي انت بدك
+
+220
+00:15:34,780 --> 00:15:38,720
+توجدله اللي هو المنطقة اللي بدك تحدد لك حدود ال
+
+221
+00:15:38,720 --> 00:15:42,520
+integration انه هي ال D المطلوبة اللي على ال
+
+222
+00:15:42,520 --> 00:15:46,140
+triple integral اول شي بده مين بالنسبة للهواي قبل
+
+223
+00:15:46,140 --> 00:15:49,320
+ما نوجد بالنسبة للهواي خلينا نوجد اللي هوأول حاجة
+
+224
+00:15:49,320 --> 00:15:52,060
+نقلط اتقاط وحياة موجودة اللي بتلزمنا تاني حاجة
+
+225
+00:15:52,060 --> 00:15:55,600
+بدنا نوجد معادلة الخطوط اللي عندنا لأن معادلة
+
+226
+00:15:55,600 --> 00:16:00,300
+الخطوط بعد شوية هتلاقيها بتلزم جاب للخطوط اللي
+
+227
+00:16:00,300 --> 00:16:04,100
+الآن لنجد انه معادلة إيش معادلة ال plane الأمامي
+
+228
+00:16:04,100 --> 00:16:09,170
+هذاهذا الـ plane الآن ثابت اللي هو y بتساوي واحد
+
+229
+00:16:09,170 --> 00:16:13,170
+لإنه اللي هو المتغيرات بس اللي هي عندك دايما y
+
+230
+00:16:13,170 --> 00:16:18,570
+بتساوي واحد و x إيش ما لها متغيرة و z متغيرة إذا
+
+231
+00:16:18,570 --> 00:16:22,910
+هذا جزء من ال plane y بتساوي واحد هي معروفة قيمة y
+
+232
+00:16:22,910 --> 00:16:25,750
+إيش معناه جيبت y عشان اللي أنا بديها في الأول إيه
+
+233
+00:16:25,750 --> 00:16:31,820
+إيش بالنسبة ل yوهو حالي الزمن طبعا الان نجي
+
+234
+00:16:31,820 --> 00:16:37,720
+لالمستوى هذا المستوى هذا بمر باش بالنقاط 000 و
+
+235
+00:16:37,720 --> 00:16:40,820
+01111000
+
+236
+00:16:40,820 --> 00:16:44,840
+و هذا المستوى اللي هو معروف تلت نقاط عليه إذا
+
+237
+00:16:44,840 --> 00:16:49,810
+بنعرف طبعا نوجده لأنه من نقطين بنوجد اللي هوبتعرف
+
+238
+00:16:49,810 --> 00:16:54,110
+من الـ vector موازي للمستوى أو في المستوى بالاضافة
+
+239
+00:16:54,110 --> 00:16:58,030
+للنقطة عندنا 1 1 و 0 على المستوى ايه بتعرف معادلة
+
+240
+00:16:58,030 --> 00:17:04,010
+مين اللي هو معادلة اللي بلين طبعا بيعرف تلك التلات
+
+241
+00:17:04,010 --> 00:17:06,810
+مواقع زي ما أخدنا سابقا بتعرفها اللي بتلاقيها y
+
+242
+00:17:06,810 --> 00:17:11,510
+بيساوي x زي z نشنو جده طبعا لإن هذامعرفة سابقة لكم
+
+243
+00:17:11,510 --> 00:17:14,870
+بتعرفوا تجدوها لحالكم Y بيساوي X زائد Z هذه اللي
+
+244
+00:17:14,870 --> 00:17:18,290
+هي معدلة البلين اللي أمامي البلين اللي في الخلف
+
+245
+00:17:18,290 --> 00:17:22,050
+هذا سهل ما هو هذا البلين يعني ال YZ بلين اللي هو X
+
+246
+00:17:22,050 --> 00:17:25,910
+بيساوي سفر الآن البلين اللي تحت هذا ما هو برضه سهل
+
+247
+00:17:25,910 --> 00:17:28,810
+اللي هو عبارة عن XY بلين اللي هو Z بتساوي سفر إذا
+
+248
+00:17:28,810 --> 00:17:34,370
+كل البلينات اللي موجودة أعرفنا اللي هي معدلتها
+
+249
+00:17:34,370 --> 00:17:38,470
+نيجي الآن نبدأ نشرع في عملية تحديد اللي هو حدود ال
+
+250
+00:17:38,470 --> 00:17:42,470
+integrationالعندي Y يعني أنه يأخذ موازي لـ Y-axis
+
+251
+00:17:42,470 --> 00:17:47,330
+وهي الـ Y-axis هي موازيات الهي هيك هيك هيك كل
+
+252
+00:17:47,330 --> 00:17:53,150
+الموازيات عاملها بتدخل من اللي هو هذا الـ plane و
+
+253
+00:17:53,150 --> 00:17:56,170
+بتخرج من هذا الـ plane و كلها على نفس المنوال
+
+254
+00:17:56,170 --> 00:18:00,150
+فتعالى أي موازي للـ Y-axis هيكون بدخل من عند Y
+
+255
+00:18:00,150 --> 00:18:03,710
+بساوية X زاية Z لعند Y بتساوية 1 إذا حدود ال
+
+256
+00:18:03,710 --> 00:18:07,390
+integration من X زاية Z لعند اللي هو مين واحد هذا
+
+257
+00:18:07,390 --> 00:18:13,290
+ال Yالان نجيه نعمل إسقاط لهذا الشكل على الـ XY
+
+258
+00:18:13,290 --> 00:18:17,670
+plane عشان الـ XZ plane قاسف اللي بضل الـ XZ plane
+
+259
+00:18:17,670 --> 00:18:21,410
+إذا عرفنا أولا من حد إلى Y موازئله من أين يدخل ومن
+
+260
+00:18:21,410 --> 00:18:24,550
+أين يخرج بعدين اللي بضل بنعمل إسقاط له المجسم على
+
+261
+00:18:24,550 --> 00:18:26,890
+اللي هو الـ plane اللي بضل اللي هو في هذه الحالة
+
+262
+00:18:26,890 --> 00:18:31,600
+ZX planeزد اكس بلين سهلة بنخلي اللي هو ال Y بصفر
+
+263
+00:18:31,600 --> 00:18:37,260
+بيصير النقطة هذه عبارة عن 001 اللي هو بنخلي هذا
+
+264
+00:18:37,260 --> 00:18:41,120
+النقطة بصفر بتصير 000 نقطة الأصل هذه الخط اللي هي
+
+265
+00:18:41,120 --> 00:18:46,280
+صفر بيصير اللي هي 000 فبيصير البلين هذا نسقطه اللي
+
+266
+00:18:46,280 --> 00:18:50,830
+هو الخط هذا مثلاالخط هذا هي ما سقطوا هذا الخط اللي
+
+267
+00:18:50,830 --> 00:18:54,270
+هو حسب النقطتين هذولا النقطتين اسقطتن بيصير اللي
+
+268
+00:18:54,270 --> 00:18:59,730
+اللي هي اسقاط الخط هو النقطتين اللي هو الخط الواصل
+
+269
+00:18:59,730 --> 00:19:03,330
+بين النقطتين اللي نسقطتين ونفس الاشي الخط هذا
+
+270
+00:19:03,330 --> 00:19:07,230
+النقطتين اللي نسقطتين هذولا بيصير هذولا هذه ونفس
+
+271
+00:19:07,230 --> 00:19:11,040
+الاشي هذا اللي هو الخطاللي هو إسقاطه اللي هو
+
+272
+00:19:11,040 --> 00:19:14,340
+النقطتين هدولة إذا اللي هو إسقاط الخط هو عبارة عن
+
+273
+00:19:14,340 --> 00:19:17,960
+الخط اللي بينهم المسقطتين النقطتين إلا أنه نفس
+
+274
+00:19:17,960 --> 00:19:21,800
+الاشي هذا الخط هي إسقاطه لأن تصير نقطة دي و نقطة
+
+275
+00:19:21,800 --> 00:19:25,240
+دي هي إسقاطه و الخط هذا بيصير هي إسقاطه إذا اللي
+
+276
+00:19:25,240 --> 00:19:28,880
+بلين هذا و مع اللي بلين هذا مع كل المجسم بيصير هو
+
+277
+00:19:28,880 --> 00:19:33,020
+عبارة عن مين لما زد السفر بياخده بيصير هو هذا اللي
+
+278
+00:19:33,020 --> 00:19:38,420
+عندي .. اللي أماميطيب الان انا مش زد بصفر انا بصفر
+
+279
+00:19:38,420 --> 00:19:42,580
+Y بصفر لأ بصير relation لان انا بسقط على X Z عشان
+
+280
+00:19:42,580 --> 00:19:46,380
+هيك باخد Y بصفر هذا الان المثلث هو المثلث اللى بدى
+
+281
+00:19:46,380 --> 00:19:49,500
+عدل عدو ال integration بين X و Z الان هذا معروف
+
+282
+00:19:49,500 --> 00:19:53,360
+نقطتين وهين و هدوله معروف نقطتين وهين بدى اودد
+
+283
+00:19:53,360 --> 00:19:57,380
+معدلته اودد معدلته انه نقطتين بطلع X زي Z بساوة
+
+284
+00:19:57,380 --> 00:20:04,060
+واحد هتقلل الان الان عندى اللى هي النقطة هذه عبارة
+
+285
+00:20:04,060 --> 00:20:09,370
+عنعشان اكس و زد و اكس و زد و اكس و زد و اكس و زد و
+
+286
+00:20:09,370 --> 00:20:20,930
+اكس و زد و اكس
+
+287
+00:20:20,930 --> 00:20:26,380
+و زدزد واحد نقص زد اتنين على اكس واحد نقص اكس
+
+288
+00:20:26,380 --> 00:20:29,620
+اتنين و اتغلب انا عارف هذا على طول بين الواحد
+
+289
+00:20:29,620 --> 00:20:33,240
+والواحد زمان قبلناها كثير فاطلع exact z بالساوح
+
+290
+00:20:33,240 --> 00:20:36,820
+بقوله انت حر ايش بتسوي المهم توجدلي اللي هو معدلة
+
+291
+00:20:36,820 --> 00:20:42,500
+الخط هذا المستقيم بالطريقة اللي توجدليها بشكل صحيح
+
+292
+00:20:43,060 --> 00:20:46,460
+إذا ليش وجدت هذا عشان هيلزمني بعد شوية هذا معدلته
+
+293
+00:20:46,460 --> 00:20:50,780
+إيش معروفة اللي هو Y بساوي Z بساوي سفر هذا معدلته
+
+294
+00:20:50,780 --> 00:20:54,400
+معروفة هي X بساوي سفر أنا بشغل الآن في X Z بلين
+
+295
+00:20:54,400 --> 00:20:58,240
+عشان هيك مادخلوش Y إيش بدي فيها كلها سفر معينإذا
+
+296
+00:20:58,240 --> 00:21:03,300
+ندخل إلى حدود الانتجريشن دي زد دي اكس دي زد باخد
+
+297
+00:21:03,300 --> 00:21:07,800
+موازين ميان لزد اكس في داخل المنطقة من وين بدخل و
+
+298
+00:21:07,800 --> 00:21:12,160
+من وين بخرج إلى واضح انه بدخل من عند الوزد بتساوي
+
+299
+00:21:12,160 --> 00:21:18,550
+سفر هيهاهو يدخل من الخط هذا و يخرج من الخط هذا
+
+300
+00:21:18,550 --> 00:21:21,750
+كلهم على نفس المنوال ايش z تانى؟ واحد minus x عشان
+
+301
+00:21:21,750 --> 00:21:25,230
+هيك من zero لعند واحد minus x نيجي الان هذه الخطوط
+
+302
+00:21:25,230 --> 00:21:28,170
+الموزية ل z axis من اين؟ بتبدأ من عند ال .. اللي
+
+303
+00:21:28,170 --> 00:21:31,110
+هي .. اللي هي .. الصغيرة من ده طبعا من عند x
+
+304
+00:21:31,110 --> 00:21:35,070
+بتساوي سفرها هي هي كيك بتنسح لما نصل لاخر اشي عند
+
+305
+00:21:35,070 --> 00:21:39,630
+واحد عشان هيك قيمة ال xLimit الـ integrations ده
+
+306
+00:21:39,630 --> 00:21:43,450
+ال X اللي هي Zero لعند الواحد و بيكون هذا اللي هو
+
+307
+00:21:43,450 --> 00:21:50,230
+ال مطلوب الان طالب مني في السؤال على نفس المنطقة
+
+308
+00:21:50,230 --> 00:21:53,290
+بقول integrate F of X Y و Z بسياه واحد يعني أوجد
+
+309
+00:21:53,290 --> 00:21:56,870
+حاجة من معناته ايه over the tetrahedron D in
+
+310
+00:21:56,870 --> 00:22:00,890
+example 2 اللي قبل شوية in the order DZDYDX and
+
+311
+00:22:00,890 --> 00:22:05,730
+then integrate in the order DYDZDX هذا DYZDZDX توع
+
+312
+00:22:05,730 --> 00:22:11,230
+باللهالان انا اريد ان اكمل الأمر بالنسبة
+
+313
+00:22:11,230 --> 00:22:14,650
+لـ z ثم بالنسبة ل y ثم بالنسبة لمين بالنسبة ل x
+
+314
+00:22:14,650 --> 00:22:17,270
+طبعا كل أمورنا جاهزة من السؤال اللي فات اللي هي
+
+315
+00:22:17,270 --> 00:22:25,170
+عبارة عن اللي هي نقاط التقاط وإلى آخرها ومعادلة
+
+316
+00:22:25,170 --> 00:22:30,950
+لبنان هذا ومعادلة لبنان هذا كله جاهزبالنسبة لـ Z
+
+317
+00:22:30,950 --> 00:22:34,430
+مدام انها كاملة بالنسبة لـ Z فانت تاخد موازيات للـ
+
+318
+00:22:34,430 --> 00:22:39,290
+Z Axis التي تدخل من الجسم و تخرج من الجسم لاحظ ان
+
+319
+00:22:39,290 --> 00:22:44,210
+كل الموازيات التي تدخل من الـ Z Axis تدخل الجسم من
+
+320
+00:22:44,210 --> 00:22:48,030
+عند Z بس أو سفر و تخرج من عند وين يعني عند البلين
+
+321
+00:22:48,030 --> 00:22:51,160
+اللي تحتوبتخرج من عند الـ plane اللي ما هي الـ H
+
+322
+00:22:51,160 --> 00:22:54,000
+ده اللي هو إيش قيمة Z قلنا عليه اللي هو Z بيساوي Y
+
+323
+00:22:54,000 --> 00:22:57,740
+ناقص X اللي كنا بيسميه نقطة Y بيساوي Z زائد X يعني
+
+324
+00:22:57,740 --> 00:23:01,280
+Z بيساوي Y minus X إذن حد ال integration الأول من
+
+325
+00:23:01,280 --> 00:23:05,660
+0 لعند Y minus X هاي حد ال integration الأول
+
+326
+00:23:05,660 --> 00:23:11,840
+volume بسبب عند Z اللي هو Zero لعند Y minus X نيجي
+
+327
+00:23:11,840 --> 00:23:16,540
+الآن يا جماعة انحدد اللي هو مين انحدد قيمةاللي هو
+
+328
+00:23:16,540 --> 00:23:20,440
+ال .. ال .. ال .. قولوا معايا ال .. ال .. المسقط
+
+329
+00:23:20,440 --> 00:23:24,520
+لإنه بدنا مكامل احنا كملنا لزد مكامل Y و X بدنا
+
+330
+00:23:24,520 --> 00:23:27,820
+المسقط في وين في ال X Y plane العادى الآن المسقط
+
+331
+00:23:27,820 --> 00:23:30,220
+زي ما عملنا قبل شوية بتصير هذه اللي هو بتصير اللي
+
+332
+00:23:30,220 --> 00:23:34,200
+هي بنخل الزد بصفر بصير صفر و واحد و Zero هي النقطة
+
+333
+00:23:34,530 --> 00:23:37,790
+وهذه المسقطة هي نفسها على حالها لأنها Z بصفر وهذه
+
+334
+00:23:37,790 --> 00:23:40,090
+المسقطة هي نفسها على حالها لأنها Z بصفر وهذه
+
+335
+00:23:40,090 --> 00:23:40,470
+المسقطة هي نفسها على حالها لأنها Z بصفر وهذه
+
+336
+00:23:40,470 --> 00:23:42,690
+المسقطة هي نفسها لأنها Z بصفر وهذه المسقطة هي
+
+337
+00:23:42,690 --> 00:23:45,350
+نفسها لأنها Z بصفر وهذه المسقطة هي نفسها لأنها Z
+
+338
+00:23:45,350 --> 00:23:46,790
+بصفر وهذه المسقطة هي نفسها لأنها Z بصفر وهذه
+
+339
+00:23:46,790 --> 00:23:48,950
+المسقطة هي نفسها لأنها Z بصفر وهذه المسقطة هي
+
+340
+00:23:48,950 --> 00:23:50,510
+نفسها لأنها Z بصفر وهذه المسقطة هي نفسها لأنها Z
+
+341
+00:23:50,510 --> 00:23:53,630
+بصفر وهذه المسقطة هي نفس
+
+342
+00:23:57,360 --> 00:24:01,420
+حسب النقطة اللى أسقطناها و هذا الخط كله اللى فوق
+
+343
+00:24:01,420 --> 00:24:06,320
+بنزل كله بقعد هنا فى نقطة و هذا نسقطه اللى هو هذا
+
+344
+00:24:06,320 --> 00:24:10,370
+الخطوهذا مصفته اللى هو نفسه لإنه في الهدك فبصير
+
+345
+00:24:10,370 --> 00:24:15,550
+مصفة كل الشكل اللى عندى هو المثلث الامانى هذا الان
+
+346
+00:24:15,550 --> 00:24:18,730
+المثلث هذا بدى نجيب معدلاته اللى بتلزمنا طبعا
+
+347
+00:24:18,730 --> 00:24:23,910
+المعادلة هذه اللى هو x بتساوي سفر والمعادلة هذه y
+
+348
+00:24:23,910 --> 00:24:27,150
+بتساوي واحد زى ما انتوا شايفين لإنه y ثابتة على
+
+349
+00:24:27,150 --> 00:24:29,930
+هذا المكان لإن أصلا ال blank كله هذا y بيساوي واحد
+
+350
+00:24:29,930 --> 00:24:34,850
+وهذه اللى هي عند النقطة أصلا من zero zero هو لايحة
+
+351
+00:24:34,850 --> 00:24:39,300
+للواحد واحدإذاً قيمة X هنا و Y زي بعض و قيمة X و Y
+
+352
+00:24:39,300 --> 00:24:42,420
+هنا زي بعض في نقطتين إذا الخط اللي بين أن هي أو
+
+353
+00:24:42,420 --> 00:24:45,200
+عبارة عن الخط اللي بيساوي بين X و Y اللي هو Y
+
+354
+00:24:45,200 --> 00:24:49,260
+بيساوي X و اللي مش عاد و يشتغل معرفة نقطتين اللي
+
+355
+00:24:49,260 --> 00:24:52,180
+هو بيقدر يعرف معدلة الخط المستقيم هيلاقيها على طول
+
+356
+00:24:52,180 --> 00:24:54,940
+Y بيساوي X لكن الموضوع مش مستاهل على طول ال
+
+357
+00:24:54,940 --> 00:24:58,040
+percentage Y بيساوي X لأن قيمة اللي هي ال X و ال Y
+
+358
+00:24:58,040 --> 00:25:01,020
+في النقطتين في النقطة الأولى زي بعض و في النقطة
+
+359
+00:25:01,020 --> 00:25:03,300
+الثانية برضه زي بعض Zero Zero و أنا واحد واحد
+
+360
+00:25:03,300 --> 00:25:08,440
+فاميش مقصودإذا الأمور نجازة بدنا مكامل بالنسبة لمن
+
+361
+00:25:08,440 --> 00:25:11,100
+؟ ل Y بمثابة كامل بالنسبة ل Y إذا بتاخد في داخل
+
+362
+00:25:11,100 --> 00:25:14,280
+المنطقة R اللي اتعتد اللي صارت مصفت له موازيه ل Y
+
+363
+00:25:14,280 --> 00:25:18,320
+Axis كل الموازيه ل Y Axis هذوله اللي بيجي من عند
+
+364
+00:25:18,320 --> 00:25:22,240
+الخط Y بيساوي X و بروح من عند الخط Y إيش بتساوي 1
+
+365
+00:25:22,240 --> 00:25:26,340
+لإن هنا بتبدأ التزايد هيك بتتزايد إذا من Y بيساوي
+
+366
+00:25:26,340 --> 00:25:30,220
+X لعند Y بيساوي 1 من Y بتساوي X لعند Y بيساوي 1
+
+367
+00:25:30,220 --> 00:25:38,890
+هذا قيمة من ال Yنعود لقيمة ال X كل الخطوط الموازية
+
+368
+00:25:38,890 --> 00:25:45,750
+ال Y Axis تبدأ من X بصفر و تتدرج لين تصل ل X باحد
+
+369
+00:25:45,750 --> 00:25:50,010
+أخر خط بمسح كل المنطقةوبكون عندى اللى هو ال
+
+370
+00:25:50,010 --> 00:25:53,910
+integration من صفر لواحد بالنسبة ل X لحدود التكامل
+
+371
+00:25:53,910 --> 00:25:57,410
+نجى لان كامل كامل ما هي التكامل الواحد هو عبارة عن
+
+372
+00:25:57,410 --> 00:26:01,830
+Z بصير Z بعوض Y minus X ناقص Zero Zero بصير Y
+
+373
+00:26:01,830 --> 00:26:05,070
+minus X بده كامل هذا Y minus X بالنسبة ل Y بصير
+
+374
+00:26:05,070 --> 00:26:09,390
+تكاملها عبارة عن نص Y تربيع ناقص XY من وين لوين
+
+375
+00:26:09,390 --> 00:26:12,780
+بعوض من X بالساويةمن y بيساوي x لأن y بيساوي واحد
+
+376
+00:26:12,780 --> 00:26:20,240
+لان بعود عن y هنا بيصير عبارة عن نص ناقص x لان y
+
+377
+00:26:20,240 --> 00:26:24,660
+بيساوي x ناقل هي عبارة عن نص y تربيع نص y تربيع
+
+378
+00:26:24,660 --> 00:26:27,680
+ناقص y تربيع اللي هي بيصير عبارة عن ناقص نص y
+
+379
+00:26:27,680 --> 00:26:31,120
+تربيع و النقص اللي برا بيصير اسف x تربيع بيصير نص
+
+380
+00:26:31,120 --> 00:26:36,170
+x تربيع لان بيعود عن y بxطيب الآن صار عند المقدار
+
+381
+00:26:36,170 --> 00:26:39,370
+بالشكل هذا بدي أكمله بالنسبة ل X هيكملته بالنسبة ل
+
+382
+00:26:39,370 --> 00:26:43,530
+X نص X X تربيع على اللي هو نص X عدد X تربيع اللي
+
+383
+00:26:43,530 --> 00:26:46,170
+هو هذه اللي هي بيصير على تلاتة بيصير سدس و X تكييب
+
+384
+00:26:46,170 --> 00:26:50,570
+و من 0 ل 1 و بعوض و بطلع عند جداش بطلع سدس الان
+
+385
+00:26:50,570 --> 00:26:55,830
+احنا لازم نحصل على نفس القيمة لما نكامل أولش
+
+386
+00:26:55,830 --> 00:26:58,880
+بالنسبة ل Y و بالنسبة ل Z و بعدين بالنسبة ل Xطبعا
+
+387
+00:26:58,880 --> 00:27:01,440
+بالنسبة ل Y حددنا حدود ال integration في المثال
+
+388
+00:27:01,440 --> 00:27:04,880
+اللي جابله كانت من X زائد المثال اللي جابله ايه ..
+
+389
+00:27:04,880 --> 00:27:09,420
+ايه .. ايه المثال X زائد Z لعند الواحد دي Y دي Z
+
+390
+00:27:09,420 --> 00:27:12,700
+طبعا نفسه هو اللي هذاك بس احنا كنا كامل على ال F
+
+391
+00:27:12,700 --> 00:27:17,040
+انها كنا كاملنا على الواحد Z من Z لواحد minus X و
+
+392
+00:27:17,040 --> 00:27:23,460
+X من عند سفر لعند واحد و يعني Y Z X وهنا بدو Y Z X
+
+393
+00:27:23,460 --> 00:27:26,260
+هي Y Z X بدنا اذا نفس حدود ال integration او
+
+394
+00:27:26,260 --> 00:27:30,710
+جدناها قبل شويةنجم كامل بالنسبة ل Y بيصير 1Y اللي
+
+395
+00:27:30,710 --> 00:27:35,590
+هي بتعوض 1-X-Z دي Z دي X إلى أن بعدين هذا بنكمله
+
+396
+00:27:35,590 --> 00:27:39,070
+بالنسبة ل Z بيصير 1-X هذه ثابت في Z بالنسبة ل Z
+
+397
+00:27:39,070 --> 00:27:43,590
+طبعا نقص نص Z تربيع Z من وين من سفر اللي عنده 1-X
+
+398
+00:27:43,590 --> 00:27:49,550
+بنعوض عنها اللي هي بيصير ليها 1-X في 1-X تربيع نقص
+
+399
+00:27:49,550 --> 00:27:53,250
+نص 1-X تربيع و السفر طبعا اللي بيسويها دي سفر إذا
+
+400
+00:27:53,250 --> 00:27:56,070
+انصار عندي اللي هو هذا المقدار اللي هو عبارة عن
+
+401
+00:27:56,070 --> 00:28:00,740
+إطرح هذا من هذا بيطلعاللي هو واحد ناقص X تربيع
+
+402
+00:28:00,740 --> 00:28:06,320
+الان الان الكل دي X طبعا بنص اللي طلعناه برا لإن
+
+403
+00:28:06,320 --> 00:28:11,300
+هذا أكبر من هذا جد مرتين سا نص طيب فاهمين طبعا هذا
+
+404
+00:28:11,300 --> 00:28:15,740
+اتفاق تكامله اللي وقبلناه اللي بيصير عبارة عن اللي
+
+405
+00:28:15,740 --> 00:28:22,050
+هو ناقص واحد ناقص X الكل تربيع علىالتلاتة وفي عندى
+
+406
+00:28:22,050 --> 00:28:25,270
+اتنين بيصير على ست وهي الناقص ومن واحد لإن سفر
+
+407
+00:28:25,270 --> 00:28:28,770
+وبتعود عليها بتطلع برضه إيش القيمة سدس وفعلا
+
+408
+00:28:28,770 --> 00:28:33,090
+القيمتين زي بعض نيجي الآن لحاجة مفهوم برضه مشابه
+
+409
+00:28:33,090 --> 00:28:36,890
+لمفهوم أخدناها بالهكلة على main ال function مش in
+
+410
+00:28:36,890 --> 00:28:41,250
+space in اللي هو a plain average value of a
+
+411
+00:28:41,250 --> 00:28:43,970
+function in space اللي هي الآن ال function وين
+
+412
+00:28:43,970 --> 00:28:47,210
+موجودة in space يعني function of three variables
+
+413
+00:28:47,900 --> 00:28:51,640
+الان الـ Average Value of F over D over Region D
+
+414
+00:28:51,640 --> 00:28:55,580
+دي طبعا اللي هي ايه شمالها تلت أبعاد و قيمة الـ
+
+415
+00:28:55,580 --> 00:28:58,520
+Function واحد بيصير اربع أبعاد اذا عشان اواجه الـ
+
+416
+00:28:58,520 --> 00:29:01,040
+Average Value بيكون واحد ع ال volume اللي هو تلت
+
+417
+00:29:01,040 --> 00:29:05,220
+أبعاد في ال triple integration FDV اللي هي أربع
+
+418
+00:29:05,220 --> 00:29:09,270
+أبعاد فبيصير أربع أبعاد علىعلى بُعد الـ Volume
+
+419
+00:29:09,270 --> 00:29:13,090
+والـ F زيادة وهذا رافلي الكلام اللي هي بتطلع من
+
+420
+00:29:13,090 --> 00:29:16,170
+الـ F يعني هذا بتعمل منطقي أننا نعرف الـ Average
+
+421
+00:29:16,170 --> 00:29:20,290
+Value لـ Function in space over a region D in
+
+422
+00:29:20,290 --> 00:29:24,710
+space اللي هي Function of three variablesلـ over
+
+423
+00:29:24,710 --> 00:29:27,970
+region in space بالساعة واحدة على volume of D في
+
+424
+00:29:27,970 --> 00:29:31,690
+الـ triple integral للـ FDV وهذا مشابه لما أخذناه
+
+425
+00:29:31,690 --> 00:29:34,870
+سابقا لـ function of two variables كونة average of
+
+426
+00:29:34,870 --> 00:29:40,670
+value of F over a region اللي هي R in two اللي هي
+
+427
+00:29:40,670 --> 00:29:44,090
+dimensions أو نبلين اللي هي واحدة على الـ area
+
+428
+00:29:44,090 --> 00:29:47,250
+كانت في الـ double integral لاللي هو الـ function
+
+429
+00:29:47,250 --> 00:29:50,730
+في الـ DA فهذا اللي هو على نفس المانوال زي ما
+
+430
+00:29:50,730 --> 00:29:54,870
+أوضحت قبل شويةالان ناخد مثال عليه بيقول find the
+
+431
+00:29:54,870 --> 00:29:57,470
+average value of f ايش ال function اللي بده
+
+432
+00:29:57,470 --> 00:30:00,330
+average value اقولها x في y في z throughout the
+
+433
+00:30:00,330 --> 00:30:04,190
+cubical region خلال ايش؟ بده يهد معرف على مين على
+
+434
+00:30:04,190 --> 00:30:08,070
+cubical region D bounded by the coordinate blends
+
+435
+00:30:08,070 --> 00:30:11,870
+محصورة بين ايش ال .. ال .. ال .. ال cube ال
+
+436
+00:30:11,870 --> 00:30:15,430
+coordinates blend ال coordinates blends اه هاي
+
+437
+00:30:15,430 --> 00:30:20,550
+اللي هو ال xz blend وهذا ال yزد بلين وهذا ال XY
+
+438
+00:30:20,550 --> 00:30:24,550
+بلين يعني طب من جهة ايه محصورة ب X بساوة اتنين هي
+
+439
+00:30:24,550 --> 00:30:28,490
+X بتساوة اتنين هي هذا الأحضر هذا أو Y بساوة اتنين
+
+440
+00:30:28,490 --> 00:30:32,730
+هي كمان هذا Y بساوة اتنين أو Z بساوة اللي هو اتنين
+
+441
+00:30:32,730 --> 00:30:36,690
+هذا اللي باللون الساكني اذا صار اللي هو ال .. ال
+
+442
+00:30:36,690 --> 00:30:39,620
+.. ال ..الـ cube اللي عندنا أو المكعب اللي عندنا
+
+443
+00:30:39,620 --> 00:30:45,420
+محدد تحديدا تماما هذا اللي هو y بصفر هذا x بصفر و
+
+444
+00:30:45,420 --> 00:30:52,920
+اللي فوق اللي هو عبارة عن هنا x بصفر هيق وهنا اللي
+
+445
+00:30:52,920 --> 00:30:59,300
+هو y بصفر و اللي فوق هذا عبارة عن اللي هو
+
+446
+00:31:09,650 --> 00:31:16,440
+أول حاجة بنرسم بمعنى sketchwith enough details to
+
+447
+00:31:16,440 --> 00:31:19,360
+show the limit of integration لكي نحدد اللي هو
+
+448
+00:31:19,360 --> 00:31:22,540
+اللي هي حدود ال integration هذه ال cube رسمناها
+
+449
+00:31:22,540 --> 00:31:25,560
+لأن الأمور سهلة ال cube أولا ال volume إيه له لأنه
+
+450
+00:31:25,560 --> 00:31:28,380
+ملزمنا ال volume عشان ذلك ال volume اللي هو اللي
+
+451
+00:31:28,380 --> 00:31:31,700
+هو عارفين ال volume اللي هو مساحة القاعدة اللي هي
+
+452
+00:31:31,700 --> 00:31:34,700
+أربعة في الارتفاع اتنين أربعة في اتنين بتمانية أو
+
+453
+00:31:34,700 --> 00:31:37,540
+ثلاث أبعاد اتنين في اتنين في اتنين بتطلع تمانية
+
+454
+00:31:37,540 --> 00:31:43,240
+اللي هي يشملها ال average value لمن لأسف اللي هو
+
+455
+00:31:43,240 --> 00:31:44,840
+ال volume للمكعب
+
+456
+00:31:46,810 --> 00:31:49,010
+الان نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+457
+00:31:49,010 --> 00:31:49,970
+نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+458
+00:31:49,970 --> 00:31:52,810
+نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+459
+00:31:52,810 --> 00:31:57,650
+نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+460
+00:31:57,650 --> 00:32:01,810
+نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+461
+00:32:01,810 --> 00:32:06,590
+نجي نجي
+
+462
+00:32:06,590 --> 00:32:09,570
+نجي
+
+463
+00:32:10,280 --> 00:32:15,240
+X بصفر ورايحة لعند X بساوة اتنين يعني من الوجه هذا
+
+464
+00:32:15,240 --> 00:32:19,460
+للوجه اللي هو هذا كل اللي هي الموازيات لل X axis
+
+465
+00:32:19,460 --> 00:32:23,740
+على نفس المنوال عشان هيك بكون ال X اللي هي حدودها
+
+466
+00:32:23,740 --> 00:32:27,460
+من صفر لعند مين اتنين الان مسقط هذا سهل مسقط
+
+467
+00:32:27,460 --> 00:32:31,400
+المقعب كله ايه اشمله المقعب هنا ساوة دي بصفر تنزل
+
+468
+00:32:31,400 --> 00:32:36,540
+هنا اللي هو زدها بصفر تنزل احنا لأ احنا بنكملX
+
+469
+00:32:36,540 --> 00:32:44,940
+أولا و YZ أولا و X أولا و X أولا و X أولا و X أولا
+
+470
+00:32:44,940 --> 00:32:45,300
+و X أولا و X أولا و X أولا و X أولا و X أولا و X
+
+471
+00:32:45,300 --> 00:32:45,680
+أولا و X أولا و X أولا و X أولا و X أولا و X أولا
+
+472
+00:32:45,680 --> 00:32:46,820
+و X أولا و X أولا و X أولا و X أولا و X أولا و X
+
+473
+00:32:46,820 --> 00:32:48,160
+أولا و X أولا و X أولا و X أولا و X أولا و X أولا
+
+474
+00:32:48,160 --> 00:32:54,220
+و X أولا و X أولا و X أولا و X أولا و X أولا و X
+
+475
+00:32:54,220 --> 00:32:58,450
+أولالذي في الـ ZY Plane المربع اللي هنا أموره سهلة
+
+476
+00:32:58,450 --> 00:33:04,910
+و مستطيلة جميعها على طول لو أخذنا الآن بالنسبة لـ
+
+477
+00:33:04,910 --> 00:33:10,010
+Y موازي لـ Y Axis كلهم على نفس المنوال من 0 ل2 و Z
+
+478
+00:33:10,010 --> 00:33:17,430
+طبعا تأخذ أولا بـ 0 و تتكرم تركم تركم ل2 من 0 ل2 و
+
+479
+00:33:17,430 --> 00:33:23,420
+هذا هو قيمة الـ Integrationبالكامل لان xyz بالنسبة
+
+480
+00:33:23,420 --> 00:33:26,740
+ل x بيصير x تربيع على اتنين في yz من سفر لعندي
+
+481
+00:33:26,740 --> 00:33:29,600
+اتنين اتنين بيصير عبارة عن اربع على اتنين اللي هي
+
+482
+00:33:29,600 --> 00:33:33,780
+اتنين بيصير اتنين yz وهذا طبعا سفر من سفر لاتنين
+
+483
+00:33:33,780 --> 00:33:36,220
+بتكامل برضه بالنسبة ل y بيصير اتنين y تربيع على
+
+484
+00:33:36,220 --> 00:33:40,000
+اتنين اللي هي y تربيع z الان من سفر لعندي اتنين y
+
+485
+00:33:40,000 --> 00:33:45,610
+تربيع بيصير اربع zالتي تتكامل من 0 إلى 2 أربعة زت
+
+486
+00:33:45,610 --> 00:33:49,690
+بيصير أربعة زت تربيع على اتنين يعني اتنين زت تربيع
+
+487
+00:33:49,690 --> 00:33:52,190
+من سفر إلى اتنين بيصير عبارة عن اتنين في اتنين
+
+488
+00:33:52,190 --> 00:33:55,870
+واربعة واتنين تمانية الان تمانية هذه هي ايه شمالها
+
+489
+00:33:55,870 --> 00:33:59,830
+ايه قيمة ال integration دا ال average value of XYZ
+
+490
+00:34:00,290 --> 00:34:04,390
+الـ Average Value لمن ؟ لأن ده ل F of X Y و Z
+
+491
+00:34:04,390 --> 00:34:07,570
+بساوي X في Y في Z على ال cube بساوي واحد ع ال
+
+492
+00:34:07,570 --> 00:34:09,890
+volume اللي هي واحد ع تمانية في مقدار ال
+
+493
+00:34:09,890 --> 00:34:12,670
+integration طالع تمانية اللي هو بساوي واحد و هيك
+
+494
+00:34:12,670 --> 00:34:15,710
+يكون أوجدنا ال average value و هيك يكون المحاضرة
+
+495
+00:34:15,710 --> 00:34:18,470
+انتهت و إلى لقاء آخر السلام عليكم ورحمة الله
+
+496
+00:34:18,470 --> 00:34:19,570
+وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..e6b687aab10633e14c55247018640b66fc4c1285
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A_raw.json
@@ -0,0 +1 @@
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"word": " طالع", "probability": 0.7600911458333334}, {"start": 2050.61, "end": 2050.97, "word": " تمانية", "probability": 0.9851888020833334}, {"start": 2050.97, "end": 2051.11, "word": " اللي", "probability": 0.987548828125}, {"start": 2051.11, "end": 2051.23, "word": " هو", "probability": 0.9560546875}, {"start": 2051.23, "end": 2051.63, "word": " بساوي", "probability": 0.9605712890625}, {"start": 2051.63, "end": 2052.09, "word": " واحد", "probability": 0.98779296875}, {"start": 2052.09, "end": 2052.49, "word": " و", "probability": 0.421142578125}, {"start": 2052.49, "end": 2052.67, "word": " هيك", "probability": 0.7529296875}, {"start": 2052.67, "end": 2052.91, "word": " يكون", "probability": 0.69140625}, {"start": 2052.91, "end": 2053.35, "word": " أوجدنا", "probability": 0.8509521484375}, {"start": 2053.35, "end": 2053.45, "word": " ال", "probability": 0.96142578125}, {"start": 2053.45, "end": 2053.71, "word": " average", "probability": 0.450439453125}, {"start": 2053.71, "end": 2054.15, "word": " value", "probability": 0.9794921875}, {"start": 2054.15, "end": 2054.69, "word": " و", "probability": 0.74755859375}, {"start": 2054.69, "end": 2054.93, "word": " هيك", "probability": 0.9697265625}, {"start": 2054.93, "end": 2055.19, "word": " يكون", "probability": 0.965576171875}, {"start": 2055.19, "end": 2055.71, "word": " المحاضرة", "probability": 0.9847412109375}, {"start": 2055.71, "end": 2056.13, "word": " انتهت", "probability": 0.9391276041666666}, {"start": 2056.13, "end": 2056.27, "word": " و", "probability": 0.849609375}, {"start": 2056.27, "end": 2056.47, "word": " إلى", "probability": 0.89404296875}, {"start": 2056.47, "end": 2056.85, "word": " لقاء", "probability": 0.9054361979166666}, {"start": 2056.85, "end": 2057.15, "word": " آخر", "probability": 0.7354736328125}, {"start": 2057.15, "end": 2057.53, "word": " السلام", "probability": 0.901611328125}, {"start": 2057.53, "end": 2057.91, "word": " عليكم", "probability": 0.9248046875}, {"start": 2057.91, "end": 2058.27, "word": " ورحمة", "probability": 0.85418701171875}, {"start": 2058.27, "end": 2058.47, "word": " الله", "probability": 0.955078125}, {"start": 2058.47, "end": 2059.57, "word": " وبركاته", "probability": 0.9630859375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2059.9815, "duration_after_vad": 2051.408749999999}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6c08549c253612b3b23f0a7906516cfd11d8b4f1
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/i-iMkKkoU5A_raw.srt
@@ -0,0 +1,2028 @@
+1
+00:00:00,770 --> 00:00:04,810
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,810 --> 00:00:11,070
+وبركاته هذه هي المحاضرة رقم 24 مساق تفاضل وتكامل
+
+3
+00:00:11,070 --> 00:00:16,290
+جيم وهي المحاضرة الخامسة في chapter 15 الشابط
+
+4
+00:00:16,290 --> 00:00:22,560
+الأخير في هذا المساقالشبتر اللى هو عنوانه triple
+
+5
+00:00:22,560 --> 00:00:28,840
+integrals in rectangular coordinates احنا كنا
+
+6
+00:00:28,840 --> 00:00:33,440
+تحدثنا المحاضرة السابقة عن double integral in
+
+7
+00:00:33,440 --> 00:00:36,840
+polar coordinates و اللى جاب اللى حكينا عن double
+
+8
+00:00:36,840 --> 00:00:41,700
+integrals in rectangular coordinates و اليوم نحكي
+
+9
+00:00:41,700 --> 00:00:46,330
+عن نكمل نحكي عن ال double integralفي حالة الـ
+
+10
+00:00:46,330 --> 00:00:50,410
+Rectangular Coordinates الكلام سيكون شبيه و الكلام
+
+11
+00:00:50,410 --> 00:00:55,490
+بنفس الروح و بنفس السولة ان شاء الله الان اول اشي
+
+12
+00:00:55,490 --> 00:01:00,050
+نحكي عن Volume of a region in space اذا بتذكروا
+
+13
+00:01:00,050 --> 00:01:03,030
+احنا استخدمنا double integral في إيجاد ال area
+
+14
+00:01:03,310 --> 00:01:07,090
+وقلنا على طول الـ area بتساوي الـ double integral
+
+15
+00:01:07,090 --> 00:01:12,430
+over R ال region دي ايه دي ايه دي اكس دي وايه دي
+
+16
+00:01:12,430 --> 00:01:13,070
+اكس دي وايه دي اكس دي وايه دي اكس دي وايه دي اكس
+
+17
+00:01:13,070 --> 00:01:14,050
+دي وايه دي اكس دي وايه دي اكس دي اكس دي اكس دي اكس
+
+18
+00:01:14,050 --> 00:01:14,230
+دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس
+
+19
+00:01:14,230 --> 00:01:19,070
+دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس
+
+20
+00:01:19,070 --> 00:01:20,010
+دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس دي اكس
+
+21
+00:01:20,010 --> 00:01:27,350
+دي اكس ديالتي ستحصل بدل ما هو دي اي دي اكس دي واي
+
+22
+00:01:27,350 --> 00:01:30,830
+ستحصل دي volume of a closed bounded region دي
+
+23
+00:01:30,830 --> 00:01:36,250
+انوين in space sv بتساوي triple انتجرال
+
+24
+00:01:39,450 --> 00:01:44,070
+Overdeen هي ال region in space دي في دي في طبعا هي
+
+25
+00:01:44,070 --> 00:01:49,270
+عبارة عن ال dx dy dz طبعا هذا له أصل اللي هو
+
+26
+00:01:49,270 --> 00:01:53,070
+إيجاده اللي هو زي ما حكينا في أول ال double
+
+27
+00:01:53,070 --> 00:01:57,790
+integration اللي هو بنقسم المنطقةالى اخسام صغيرة
+
+28
+00:01:57,790 --> 00:02:01,630
+صغيرة صغيرة لل X و لل Y و الآن لل Z و بيصير
+
+29
+00:02:01,630 --> 00:02:03,850
+المجموع remaining sum و بعد ما يصير مجموع
+
+30
+00:02:03,850 --> 00:02:06,770
+remaining sum بيوجد ال limit إذا كان ال limit وجود
+
+31
+00:02:06,770 --> 00:02:10,650
+بيكون هو بالظبط قيمة حجم الجسم اللي اللي قاعدين
+
+32
+00:02:10,650 --> 00:02:15,650
+بنجزقه و بعدين بيكون هو اللي هي ال triple integral
+
+33
+00:02:15,650 --> 00:02:22,680
+له هو ال volumeفهذا اللي هو في الآخر limit ده
+
+34
+00:02:22,680 --> 00:02:25,940
+remand sum فاحنا مانعش نعيد الكلام اللي حكيناه في
+
+35
+00:02:25,940 --> 00:02:30,100
+الأول على الدمل integral لما عرفناه الآن عملية
+
+36
+00:02:30,100 --> 00:02:35,440
+إيجاد اللي هي ال triple integral بنفس الأسلوب عندي
+
+37
+00:02:35,440 --> 00:02:38,100
+بيكون تلت حاجات قيمة ال function بس أنا قيمة ال
+
+38
+00:02:38,100 --> 00:02:42,660
+function واحد الحاجة التانية اللي هي dv dx dy dz
+
+39
+00:02:42,660 --> 00:02:47,520
+والحاجة اللي بدها شغلاللي هي باسمين D التي هي حدود
+
+40
+00:02:47,520 --> 00:02:51,380
+ال integration أو limits of integration الان اللي
+
+41
+00:02:51,380 --> 00:02:54,020
+بدنا نحكي هو find limits of integration in the
+
+42
+00:02:54,020 --> 00:02:57,900
+order DZ DY DX الان بدنا نوجد ال order بأي order
+
+43
+00:02:57,900 --> 00:03:01,420
+DZ أول شي و بعدين DY و بعدين DX و طبعا و الباجات
+
+44
+00:03:01,420 --> 00:03:07,220
+similar للتبدلات الاخرى DZ DX DY DY DZ DX إلى آخره
+
+45
+00:03:07,220 --> 00:03:11,300
+خلينا نشوف الآنطيب صلوا على النبي عليه الصلاة
+
+46
+00:03:11,300 --> 00:03:15,340
+والسلام نجي للمثال الأول find the volume of the
+
+47
+00:03:15,340 --> 00:03:19,520
+region D enclosed by the surface Z بتساوي X تربيع
+
+48
+00:03:19,520 --> 00:03:22,800
+زائد تلاتة Y تربيع هاي اللي هو ال surface اللي
+
+49
+00:03:22,800 --> 00:03:26,700
+عندي هيه اللي تحت هذا اللي هو Z بتساوي X تربيع
+
+50
+00:03:26,700 --> 00:03:29,800
+زائد تلاتة Y تربيع احنا بنجيبلكم مش سؤال انه يكون
+
+51
+00:03:29,800 --> 00:03:34,560
+الرسمفي أسعب من السؤال إيه ده؟ إيه ده طلب إننا
+
+52
+00:03:34,560 --> 00:03:38,420
+نرسم لكم و بنرسم لكم؟ أما مش مختبرك الآن في الرسم
+
+53
+00:03:38,420 --> 00:03:42,400
+مختبرك في إيجاد اللي هو الـTriple Integral Z
+
+54
+00:03:42,400 --> 00:03:48,200
+بتساوي X تربيع زي تلتة Y تربيع حيه الآن ال region
+
+55
+00:03:48,200 --> 00:03:51,840
+دي محصورة بين هذا ال surface و ال surface التاني Z
+
+56
+00:03:51,840 --> 00:03:55,380
+بتساوي تمانية ناقص X تربيع ناقص Y تربيع حيث Z
+
+57
+00:03:55,380 --> 00:04:00,200
+بتساوي تمانية ناقص X تربيع ناقص Y تربيعهذه المنطقة
+
+58
+00:04:00,200 --> 00:04:04,780
+بدي أجد لها ال volume ال volume طبعا بيساوي triple
+
+59
+00:04:04,780 --> 00:04:11,420
+integral دي over دي this region dz dy dx زي ما هو
+
+60
+00:04:11,420 --> 00:04:17,280
+طلب مننا قبل ما نبدأ في السؤال او طبعا اللي بده
+
+61
+00:04:17,280 --> 00:04:21,620
+يحكي طبعا ال .. يمكن الحكي في سؤال زي هذا اللي هو
+
+62
+00:04:21,620 --> 00:04:26,240
+dy قبل dz كده اللي هو صعب انك تاخد و انا بصير تاخد
+
+63
+00:04:26,240 --> 00:04:28,580
+ال .. ال .. ال ..اللي .. اللي .. اللي .. اللي ..
+
+64
+00:04:28,580 --> 00:04:29,600
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+65
+00:04:29,600 --> 00:04:30,140
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+66
+00:04:30,140 --> 00:04:30,140
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+67
+00:04:30,140 --> 00:04:31,140
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+68
+00:04:31,140 --> 00:04:31,840
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+69
+00:04:31,840 --> 00:04:35,040
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+70
+00:04:35,040 --> 00:04:37,380
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+71
+00:04:37,380 --> 00:04:37,380
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+72
+00:04:37,380 --> 00:04:37,380
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+73
+00:04:37,380 --> 00:04:38,860
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+74
+00:04:38,860 --> 00:04:41,460
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+75
+00:04:41,460 --> 00:04:49,860
+اللي .. اللي .. اللي .. اللي .. اللي
+
+76
+00:04:50,080 --> 00:04:55,480
+لأن كيف بتحدد حدود ال integration؟ الآن Z نفس
+
+77
+00:04:55,480 --> 00:04:59,060
+الروح ايش باخد زي ما كنا لما كنا بنكامل بالنسبة ل
+
+78
+00:04:59,060 --> 00:05:02,900
+X ناخد موازية ل X لما نكامل ل Y ناخد موازية ل Y
+
+79
+00:05:02,900 --> 00:05:08,220
+الآن بتداخل بالنسبة ل Z موازية ل Z من وين دخل ال
+
+80
+00:05:08,220 --> 00:05:13,000
+high Z طبعا high اللي هو ال increasing له او عملية
+
+81
+00:05:13,000 --> 00:05:20,400
+التزايد لهالان بناخد اللي هي عبارة عن موازيات اللي
+
+82
+00:05:20,400 --> 00:05:25,620
+هو وين بتعمل entering ل اللي هو الجسم وكان وين
+
+83
+00:05:25,620 --> 00:05:29,160
+بتعمل الخروج اللي يعني الدخول للجسم واضح انها
+
+84
+00:05:29,160 --> 00:05:34,450
+بتبدأ من عنداللي هو السطح هذا هاي السطح هذا لو
+
+85
+00:05:34,450 --> 00:05:39,710
+دخلنا أي اللي هو موازي الـ z axis هيطلع من السطح
+
+86
+00:05:39,710 --> 00:05:43,870
+البراني هذا اللي هو ايش ماله z بتساوي x تربيع زي
+
+87
+00:05:43,870 --> 00:05:48,430
+تلتة y تربيع و بطلع بخرج من وين من هذه المنطقة
+
+88
+00:05:48,430 --> 00:05:51,970
+اللي هي z بتساوي تمانية ناقص x تربيع ناقص y تربيع
+
+89
+00:05:51,970 --> 00:05:55,130
+و كل هنا على نفس المنوال شايف مثلها لكن لو بدنا
+
+90
+00:05:55,130 --> 00:05:57,570
+نتكامل بالنسبة على x أو الاشياء بدنا نمشي هيك و
+
+91
+00:05:57,570 --> 00:06:00,920
+بدنا نمشي هيكصار بده ييجي من هنا لما يروح لهنا
+
+92
+00:06:00,920 --> 00:06:04,640
+بيصير فيه شوية صعوبة في التحديد لكن مش مستحيلة و
+
+93
+00:06:04,640 --> 00:06:09,060
+نفس الاشي بالنسبة ل why بالشكل الثاني إذا لازد
+
+94
+00:06:09,060 --> 00:06:15,240
+بيدخل من عند اللي هو ال curve الأول أو ال surface
+
+95
+00:06:15,240 --> 00:06:19,660
+الأول و بيطلع من عند ال surface الثاني إذا صارت
+
+96
+00:06:19,660 --> 00:06:24,180
+اللي هي تكاملها حدود ال integration هي عبارة عن
+
+97
+00:06:24,390 --> 00:06:29,210
+اللي هي من اللي تحت X تربيع زي تلتة Y تربيع قيمة Z
+
+98
+00:06:29,210 --> 00:06:34,650
+ل تمانية ناقص X تربيع ناقص Y تربيع اللي هي قيمة Z
+
+99
+00:06:34,650 --> 00:06:38,590
+الثانية هذه الأولى اللي ذكرناها X تربيع زي تلتة Y
+
+100
+00:06:38,590 --> 00:06:45,150
+تربيع و هذه القيمة الثانية الان بدي أوجد اللي هو
+
+101
+00:06:45,150 --> 00:06:50,250
+من حدود ال X و ال Y عشان أوجد حدود ال X و ال Yالان
+
+102
+00:06:50,250 --> 00:06:54,790
+اللي هو فأسئلة من هذا النوع أصلا يعني ممكن احنا
+
+103
+00:06:54,790 --> 00:07:00,200
+اللي هو ما نرسموهلكن هنا التوضيح كيف نرسمه على طول
+
+104
+00:07:00,200 --> 00:07:04,640
+اللي هو ب .. بس بدك تعرف يعني بتغيير قيمة بتعويض
+
+105
+00:07:04,640 --> 00:07:07,860
+عن قيمتين هنا و قيمتين هنا القيمة ال X و ال Y هنا
+
+106
+00:07:07,860 --> 00:07:10,320
+و قيمة ال X و ال Y هنا بتعرف مين اللي فوق و مين
+
+107
+00:07:10,320 --> 00:07:14,420
+اللي تحت طبعا يعني هو الكلام مش safe كثير لكن ..
+
+108
+00:07:14,420 --> 00:07:19,620
+لكن بقد الغرض القيمة الأعلى ل .. ل .. ل .. ل .. ال
+
+109
+00:07:19,620 --> 00:07:23,140
+X مثلا بتاخدها واحد و كده بتلاقي مثلا اللي هي هذه
+
+110
+00:07:23,140 --> 00:07:27,360
+قيمة أعلى وهذه قيمة أسفلبتكون حدود الانتجريشن من
+
+111
+00:07:27,360 --> 00:07:31,920
+هنا لعند هنا هذا لو بدكاش ترسم أو فيش رسم في رسمات
+
+112
+00:07:31,920 --> 00:07:35,620
+هذا مفضلك تكامل بالنسبة لزد أساس تعرف مين اللي فوق
+
+113
+00:07:35,620 --> 00:07:42,540
+ومين اللي تحت طيب الآن نجلع اللي هي المنطقة اللي
+
+114
+00:07:42,540 --> 00:07:48,080
+بدنا نكامل عليها x و yخلصنا من z بدنا نيجي x1 إيش
+
+115
+00:07:48,080 --> 00:07:54,200
+بنعمل إذا بنعمل إسقاط للجسم هذا الآن وهذا اللي هي
+
+116
+00:07:54,200 --> 00:08:00,440
+إسقاط كما يليه عندي اللي هو z بخليه سفر و z
+
+117
+00:08:00,440 --> 00:08:03,500
+بالساوية سفر نشوف المنطقة المشتركة بينهما هي
+
+118
+00:08:03,500 --> 00:08:06,850
+الإسقاطاللي هي x تربيع زي تلتة y تربيع بيصير
+
+119
+00:08:06,850 --> 00:08:11,130
+بتساوي تمانية ناقص x تربيع ناقص y تربيع الآن x
+
+120
+00:08:11,130 --> 00:08:14,470
+تربيع زي تلتة y تربيع بتجيب لها ال x تربيع بيصير
+
+121
+00:08:14,470 --> 00:08:17,370
+اللي هو اتنين x تربيع وهذا بتجيب لها ال y تربيع
+
+122
+00:08:17,370 --> 00:08:20,750
+على الطرف التاني بيصير اربعة y تربيع اربعة y تربيع
+
+123
+00:08:20,970 --> 00:08:24,050
+زاد اتنين X تربيع بيساوي تمانية X من الجهتين على
+
+124
+00:08:24,050 --> 00:08:27,670
+اتنين بيصير X تربيع زاد اتنين Y تربيع بتساوي ايش
+
+125
+00:08:27,670 --> 00:08:33,110
+اربعة اللي هي ال ellipse اللي الاسقاط طبع اللي هو
+
+126
+00:08:33,110 --> 00:08:36,590
+الجسم اللي هو X تربيع زاد اتنين Y تربيع ايش بيساوي
+
+127
+00:08:36,590 --> 00:08:40,770
+بيساوي اربعة اذا عرفنا كيف الاسقاط طبحين هذي بسفر
+
+128
+00:08:40,770 --> 00:08:44,010
+و هذي بسفر و بنسوين في بعض بيطلع المنطقة اللي هي
+
+129
+00:08:44,010 --> 00:08:49,260
+اسقاط الجهتيننأتي الآن .. نأتي الآن .. نأتي الآن
+
+130
+00:08:49,260 --> 00:08:52,120
+.. اللي هو بدنا نحدد حدود ال integration من هذه
+
+131
+00:08:52,120 --> 00:08:55,540
+المنطقة بدنا نكمل بالنسبة لمين جابي الا Y بعدين
+
+132
+00:08:55,540 --> 00:09:01,700
+بدنا نكمل ال Y إذا بدنا ناخد اللي هو موازيات للمين
+
+133
+00:09:01,700 --> 00:09:06,330
+لل Y axis هي ال Y axis ناخد لها موازياتكل
+
+134
+00:09:06,330 --> 00:09:11,310
+الموازيات التي منها إلى هنا كلها تأتي من الجهة
+
+135
+00:09:11,310 --> 00:09:15,490
+السالبة لـ Y عبر الـ curve نفسه X تربيع زي اتنين و
+
+136
+00:09:15,490 --> 00:09:19,650
+Y تربيع إلى الجهة الموجبة لـ Y عبر الـ curve نفسه
+
+137
+00:09:19,650 --> 00:09:23,290
+إذا حدود الـ integration بدأ أوجد Y تصبح اتنين و Y
+
+138
+00:09:23,290 --> 00:09:26,930
+تربيع تسوى أربعة ناقص X تربيع يعني Y تربيع تسوى
+
+139
+00:09:26,930 --> 00:09:31,230
+أربعة ناقص X تربيع على اتنين يعني Y تصبح زائد أو
+
+140
+00:09:31,230 --> 00:09:34,490
+ناقص الجدر التربيع إلى الأربعة ناقص X تربيع على
+
+141
+00:09:34,490 --> 00:09:38,710
+اتنينالان اذا صار عندى قيمتين هذه اللى هى قيمة Y
+
+142
+00:09:38,710 --> 00:09:41,990
+السالبة وهذه قيمة Y الموجبة وهذه اللى هى بتكون
+
+143
+00:09:41,990 --> 00:09:44,770
+ايهاش حدود ال integration Y بتساوى ناقص الجدر
+
+144
+00:09:44,770 --> 00:09:48,190
+تربيع الاربع ناقص X تربيع على اتنين وY بتساوي
+
+145
+00:09:48,190 --> 00:09:51,450
+الجدر تربيع الاربع ناقص X تربيع على اتنين هذا بدخل
+
+146
+00:09:51,450 --> 00:09:54,930
+المنطقة وهذا بخرج من المنطقة وهى حدود ال
+
+147
+00:09:54,930 --> 00:09:59,450
+integration نجى نرجع لان احدد قيمة X دايل قيمة X
+
+148
+00:09:59,450 --> 00:10:03,690
+من وين بدأت كل الخطوط هى الخطوط هذهعند هذه الخطوط
+
+149
+00:10:03,690 --> 00:10:06,610
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+150
+00:10:06,610 --> 00:10:14,010
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+151
+00:10:14,010 --> 00:10:14,610
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+152
+00:10:14,610 --> 00:10:15,590
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+153
+00:10:15,590 --> 00:10:15,650
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+154
+00:10:15,650 --> 00:10:15,650
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+155
+00:10:15,650 --> 00:10:15,650
+هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط هيك خط
+
+156
+00:10:15,650 --> 00:10:28,430
+هيك خط هيك خط هيك خط هيك خط
+
+157
+00:10:28,430 --> 00:10:32,880
+هيك خط هيبنعوض عنه زد بـ 8-X تربيع ناقص Y تربيع
+
+158
+00:10:32,880 --> 00:10:38,300
+ناقص اللي هو X تربيع زائد 3Y تربيع بطلع عندي
+
+159
+00:10:38,300 --> 00:10:41,860
+المقدار اللي أمامي هذا هذا بده فاضل الآن بالنسبة
+
+160
+00:10:41,860 --> 00:10:45,800
+لمن؟ بالنسبة لـ Y فاضله بالنسبة لـ Y بيصير 8Y ناقص
+
+161
+00:10:45,800 --> 00:10:51,170
+2X تربيع في Y هيهااللي هو ناقص اربعة Y تكعيب على
+
+162
+00:10:51,170 --> 00:10:54,910
+تلاتة من وين لوين من عند هذا المقدار لهذا المقدار
+
+163
+00:10:54,910 --> 00:10:58,990
+اذا Y من عند ناقص الجدر لعند الجدر نفسه اللي هو
+
+164
+00:10:58,990 --> 00:11:03,790
+بعوض على هذه اللي هي عن قيمة Y هي عوضت وهنا عوضت
+
+165
+00:11:03,790 --> 00:11:08,850
+عن قيمة Y اللي هي و بعدين اللي هو طرحت من هدولة
+
+166
+00:11:08,850 --> 00:11:15,290
+طبعا هذه اللي هيY مفردة فبصير ناقص في ناقص لها
+
+167
+00:11:15,290 --> 00:11:19,450
+بصير ناقص ناقص بصير زائد فبصير هذا ضعف الأولاني
+
+168
+00:11:19,450 --> 00:11:25,870
+وهذا نفس الاشي ان هي فردية فبصير لما أرفع حادة لل
+
+169
+00:11:25,870 --> 00:11:29,740
+أس اللي هو الفردي هذا بتظلها ناقصو في اللي هو
+
+170
+00:11:29,740 --> 00:11:33,900
+الناقص الأولاني اللي هو التعويض بهذه ناقص التعويض
+
+171
+00:11:33,900 --> 00:11:37,840
+بهذه فبصير زائد اللي هو و في ناقص أصلا بيصير ناقص
+
+172
+00:11:37,840 --> 00:11:40,960
+بترجع ناقص إذا إشارتها بيصير نفس القيمة بنفس
+
+173
+00:11:40,960 --> 00:11:44,240
+الإشارة
+
+174
+00:11:44,240 --> 00:11:47,320
+وهذه نفس القيمة بنفس الإشارة فبصير اتنين في
+
+175
+00:11:47,320 --> 00:11:51,800
+المقدار هذا ناقص اتنين في المقدار هذا فهو بيكون
+
+176
+00:11:51,800 --> 00:11:54,880
+عندي اللي هو المقدار الأمامي فاهمين علاش بقوليعني
+
+177
+00:11:54,880 --> 00:12:00,120
+هذا معوض ابو هذا ناقص هذا معوض ابو هذا هيطلع اللي
+
+178
+00:12:00,120 --> 00:12:04,580
+هو هذا المقدار اللي هو اللي هي ضعف اللي هي التعويض
+
+179
+00:12:04,580 --> 00:12:09,180
+اللي موجود لإنه ناقص في ال order اللي هو الفردي بي
+
+180
+00:12:09,180 --> 00:12:14,840
+.. بيصير زائد لما أقول ناقص ناقص فبتطلع عندي ضعف
+
+181
+00:12:14,840 --> 00:12:18,500
+المقدار وجربوها لحالكم حتى جوا اللي بحكيه هو
+
+182
+00:12:18,500 --> 00:12:25,830
+الصحيحهذا المقدار يجب أن نكمل بهذا المقدار 4-x
+
+183
+00:12:25,830 --> 00:12:33,910
+تربيع 4-x
+
+184
+00:12:33,910 --> 00:12:39,610
+تربيع 4-x تربيع 4-x تربيع
+
+185
+00:12:43,470 --> 00:12:48,010
+طبعا الان اللي هي بصير تمانية عندى احسابات عادية
+
+186
+00:12:48,010 --> 00:12:52,390
+بتضرب هذا بتعمل هذا على صورة أربعة نقص كستربيع على
+
+187
+00:12:52,390 --> 00:12:57,450
+اتنين فبصير هذا أس واحد وهذا أس مص وهذا بصير أس
+
+188
+00:12:57,450 --> 00:13:00,830
+تلتة على اتنين طبعا بضل برا أيش تمانية معاكم بعيد
+
+189
+00:13:00,830 --> 00:13:06,220
+الاعتبار علىاللي هي اللى عندى يكون بنفس اللى هو
+
+190
+00:13:06,220 --> 00:13:10,380
+الصورة هذه الان ناقص نفس الاشي الحسبة هنا أس تلاتة
+
+191
+00:13:10,380 --> 00:13:14,820
+ع اتنين الان بتكامل طبعا هذا المقدار و هذا المقدار
+
+192
+00:13:14,820 --> 00:13:18,920
+نفس الاشي فبتطرح تماما ناقص تماما ع تلاتة و بعدين
+
+193
+00:13:18,920 --> 00:13:23,400
+بتكامل للجهتين كيف الآن بدك تكامل للجهتين طبعا لما
+
+194
+00:13:23,400 --> 00:13:27,140
+تطرح من بعض بطلع المقدار الأمامك كده كيف بدك تكامل
+
+195
+00:13:27,140 --> 00:13:30,940
+للجهتينأه طبعا هو طلع التنين هنا والتنين من هنا
+
+196
+00:13:30,940 --> 00:13:35,060
+وطلعها وصار اربعة جدر التنين على مين على تلاتة
+
+197
+00:13:35,060 --> 00:13:39,080
+فاهمين كيف هذا طلعت اتنين أس تلاتة على اتنين طبعا
+
+198
+00:13:39,080 --> 00:13:41,620
+اول اشي طرحت تمانية ناقص تمانية على تلاتة وطلعت
+
+199
+00:13:41,620 --> 00:13:44,880
+القيمة بعدين اتنين أس تلاتة على اتنين طلحها عام
+
+200
+00:13:44,880 --> 00:13:48,340
+المشترك برا وضربها في القيمة اللي طلعت طلعت بتطلع
+
+201
+00:13:48,340 --> 00:13:51,040
+عند اربعة جدر التنين على تلاتة في ال integration
+
+202
+00:13:51,040 --> 00:13:53,860
+اللي أمامي ال integration اللي أمامي هذااللي هو مش
+
+203
+00:13:53,860 --> 00:13:57,240
+بيغلب شوية بس احنا يعني مش هنجيبلكم اشي اللي هو
+
+204
+00:13:57,240 --> 00:14:00,560
+بيغلب كتير لدرجة انك نسيك في اللي هو تشتغله من
+
+205
+00:14:00,560 --> 00:14:05,180
+تفاضل بقى ونجيبلك تكاملات اسهل من هيك طبيعي لكن
+
+206
+00:14:05,180 --> 00:14:09,180
+هذا الان اللي هي احنا في البيت و بنشتغله أبقى
+
+207
+00:14:09,180 --> 00:14:12,550
+ريحيةبناخد اللي هو بالتعويض زي ما كنتوا تعرفوا في
+
+208
+00:14:12,550 --> 00:14:16,230
+تفاضل ألف X بالساعة و اتنين sin Y و sin U وبتعوض
+
+209
+00:14:16,230 --> 00:14:19,470
+فيها دي بيطلع القيمة اتجرب ال integration كامله
+
+210
+00:14:19,470 --> 00:14:25,550
+تطلع تمانية باي في جذر اتنين طيب بكون هيك وجدنا ال
+
+211
+00:14:25,550 --> 00:14:30,990
+volume نيجي الأن ناخد مثال آخربنقل قيمة الانتجار
+
+212
+00:14:30,990 --> 00:14:34,830
+لتجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+213
+00:14:34,830 --> 00:14:35,850
+تجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+214
+00:14:35,850 --> 00:14:37,610
+تجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+215
+00:14:37,610 --> 00:14:39,590
+تجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+216
+00:14:39,590 --> 00:14:39,590
+تجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+217
+00:14:39,590 --> 00:14:39,590
+تجارب تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+218
+00:14:39,590 --> 00:14:55,110
+تجارب تجارب تجارب تجارب تجار
+
+219
+00:14:55,130 --> 00:15:01,170
+أتفاضل f of x, y, z اتكامل أسف بصورة مين dy, dz,
+
+220
+00:15:01,310 --> 00:15:05,030
+dx ده محددلك ال order اللي بدهيا هو لمين الآن
+
+221
+00:15:05,030 --> 00:15:09,370
+لمنطقة دي دي مين هي المنطقة اللي هي ال tetrahedron
+
+222
+00:15:09,370 --> 00:15:12,890
+هذا اللي أمامي الهرم الأماني اللي أمامي لأن هذا
+
+223
+00:15:12,890 --> 00:15:17,590
+الهرم محددلك إيش هو بالظبط اللي هي 000 هيها راسه
+
+224
+00:15:17,590 --> 00:15:24,180
+والرأس التاني 0101 وهنا الرأس الرابعأو الثالث 010
+
+225
+00:15:24,180 --> 00:15:28,920
+والرأس الأخير 1100 هي الثلاث رؤوس وطبعا وصل بينهم
+
+226
+00:15:28,920 --> 00:15:34,780
+خطوط إذا هذا هو مين اللي هو الشكل اللي انت بدك
+
+227
+00:15:34,780 --> 00:15:38,720
+توجدله اللي هو المنطقة اللي بدك تحدد لك حدود ال
+
+228
+00:15:38,720 --> 00:15:42,520
+integration انه هي ال D المطلوبة اللي على ال
+
+229
+00:15:42,520 --> 00:15:46,140
+triple integral اول شي بده مين بالنسبة للهواي قبل
+
+230
+00:15:46,140 --> 00:15:49,320
+ما نوجد بالنسبة للهواي خلينا نوجد اللي هوأول حاجة
+
+231
+00:15:49,320 --> 00:15:52,060
+نقلط اتقاط وحياة موجودة اللي بتلزمنا تاني حاجة
+
+232
+00:15:52,060 --> 00:15:55,600
+بدنا نوجد معادلة الخطوط اللي عندنا لأن معادلة
+
+233
+00:15:55,600 --> 00:16:00,300
+الخطوط بعد شوية هتلاقيها بتلزم جاب للخطوط اللي
+
+234
+00:16:00,300 --> 00:16:04,100
+الآن لنجد انه معادلة إيش معادلة ال plane الأمامي
+
+235
+00:16:04,100 --> 00:16:09,170
+هذاهذا الـ plane الآن ثابت اللي هو y بتساوي واحد
+
+236
+00:16:09,170 --> 00:16:13,170
+لإنه اللي هو المتغيرات بس اللي هي عندك دايما y
+
+237
+00:16:13,170 --> 00:16:18,570
+بتساوي واحد و x إيش ما لها متغيرة و z متغيرة إذا
+
+238
+00:16:18,570 --> 00:16:22,910
+هذا جزء من ال plane y بتساوي واحد هي معروفة قيمة y
+
+239
+00:16:22,910 --> 00:16:25,750
+إيش معناه جيبت y عشان اللي أنا بديها في الأول إيه
+
+240
+00:16:25,750 --> 00:16:31,820
+إيش بالنسبة ل yوهو حالي الزمن طبعا الان نجي
+
+241
+00:16:31,820 --> 00:16:37,720
+لالمستوى هذا المستوى هذا بمر باش بالنقاط 000 و
+
+242
+00:16:37,720 --> 00:16:40,820
+01111000
+
+243
+00:16:40,820 --> 00:16:44,840
+و هذا المستوى اللي هو معروف تلت نقاط عليه إذا
+
+244
+00:16:44,840 --> 00:16:49,810
+بنعرف طبعا نوجده لأنه من نقطين بنوجد اللي هوبتعرف
+
+245
+00:16:49,810 --> 00:16:54,110
+من الـ vector موازي للمستوى أو في المستوى بالاضافة
+
+246
+00:16:54,110 --> 00:16:58,030
+للنقطة عندنا 1 1 و 0 على المستوى ايه بتعرف معادلة
+
+247
+00:16:58,030 --> 00:17:04,010
+مين اللي هو معادلة اللي بلين طبعا بيعرف تلك التلات
+
+248
+00:17:04,010 --> 00:17:06,810
+مواقع زي ما أخدنا سابقا بتعرفها اللي بتلاقيها y
+
+249
+00:17:06,810 --> 00:17:11,510
+بيساوي x زي z نشنو جده طبعا لإن هذامعرفة سابقة لكم
+
+250
+00:17:11,510 --> 00:17:14,870
+بتعرفوا تجدوها لحالكم Y بيساوي X زائد Z هذه اللي
+
+251
+00:17:14,870 --> 00:17:18,290
+هي معدلة البلين اللي أمامي البلين اللي في الخلف
+
+252
+00:17:18,290 --> 00:17:22,050
+هذا سهل ما هو هذا البلين يعني ال YZ بلين اللي هو X
+
+253
+00:17:22,050 --> 00:17:25,910
+بيساوي سفر الآن البلين اللي تحت هذا ما هو برضه سهل
+
+254
+00:17:25,910 --> 00:17:28,810
+اللي هو عبارة عن XY بلين اللي هو Z بتساوي سفر إذا
+
+255
+00:17:28,810 --> 00:17:34,370
+كل البلينات اللي موجودة أعرفنا اللي هي معدلتها
+
+256
+00:17:34,370 --> 00:17:38,470
+نيجي الآن نبدأ نشرع في عملية تحديد اللي هو حدود ال
+
+257
+00:17:38,470 --> 00:17:42,470
+integrationالعندي Y يعني أنه يأخذ موازي لـ Y-axis
+
+258
+00:17:42,470 --> 00:17:47,330
+وهي الـ Y-axis هي موازيات الهي هيك هيك هيك كل
+
+259
+00:17:47,330 --> 00:17:53,150
+الموازيات عاملها بتدخل من اللي هو هذا الـ plane و
+
+260
+00:17:53,150 --> 00:17:56,170
+بتخرج من هذا الـ plane و كلها على نفس المنوال
+
+261
+00:17:56,170 --> 00:18:00,150
+فتعالى أي موازي للـ Y-axis هيكون بدخل من عند Y
+
+262
+00:18:00,150 --> 00:18:03,710
+بساوية X زاية Z لعند Y بتساوية 1 إذا حدود ال
+
+263
+00:18:03,710 --> 00:18:07,390
+integration من X زاية Z لعند اللي هو مين واحد هذا
+
+264
+00:18:07,390 --> 00:18:13,290
+ال Yالان نجيه نعمل إسقاط لهذا الشكل على الـ XY
+
+265
+00:18:13,290 --> 00:18:17,670
+plane عشان الـ XZ plane قاسف اللي بضل الـ XZ plane
+
+266
+00:18:17,670 --> 00:18:21,410
+إذا عرفنا أولا من حد إلى Y موازئله من أين يدخل ومن
+
+267
+00:18:21,410 --> 00:18:24,550
+أين يخرج بعدين اللي بضل بنعمل إسقاط له المجسم على
+
+268
+00:18:24,550 --> 00:18:26,890
+اللي هو الـ plane اللي بضل اللي هو في هذه الحالة
+
+269
+00:18:26,890 --> 00:18:31,600
+ZX planeزد اكس بلين سهلة بنخلي اللي هو ال Y بصفر
+
+270
+00:18:31,600 --> 00:18:37,260
+بيصير النقطة هذه عبارة عن 001 اللي هو بنخلي هذا
+
+271
+00:18:37,260 --> 00:18:41,120
+النقطة بصفر بتصير 000 نقطة الأصل هذه الخط اللي هي
+
+272
+00:18:41,120 --> 00:18:46,280
+صفر بيصير اللي هي 000 فبيصير البلين هذا نسقطه اللي
+
+273
+00:18:46,280 --> 00:18:50,830
+هو الخط هذا مثلاالخط هذا هي ما سقطوا هذا الخط اللي
+
+274
+00:18:50,830 --> 00:18:54,270
+هو حسب النقطتين هذولا النقطتين اسقطتن بيصير اللي
+
+275
+00:18:54,270 --> 00:18:59,730
+اللي هي اسقاط الخط هو النقطتين اللي هو الخط الواصل
+
+276
+00:18:59,730 --> 00:19:03,330
+بين النقطتين اللي نسقطتين ونفس الاشي الخط هذا
+
+277
+00:19:03,330 --> 00:19:07,230
+النقطتين اللي نسقطتين هذولا بيصير هذولا هذه ونفس
+
+278
+00:19:07,230 --> 00:19:11,040
+الاشي هذا اللي هو الخطاللي هو إسقاطه اللي هو
+
+279
+00:19:11,040 --> 00:19:14,340
+النقطتين هدولة إذا اللي هو إسقاط الخط هو عبارة عن
+
+280
+00:19:14,340 --> 00:19:17,960
+الخط اللي بينهم المسقطتين النقطتين إلا أنه نفس
+
+281
+00:19:17,960 --> 00:19:21,800
+الاشي هذا الخط هي إسقاطه لأن تصير نقطة دي و نقطة
+
+282
+00:19:21,800 --> 00:19:25,240
+دي هي إسقاطه و الخط هذا بيصير هي إسقاطه إذا اللي
+
+283
+00:19:25,240 --> 00:19:28,880
+بلين هذا و مع اللي بلين هذا مع كل المجسم بيصير هو
+
+284
+00:19:28,880 --> 00:19:33,020
+عبارة عن مين لما زد السفر بياخده بيصير هو هذا اللي
+
+285
+00:19:33,020 --> 00:19:38,420
+عندي .. اللي أماميطيب الان انا مش زد بصفر انا بصفر
+
+286
+00:19:38,420 --> 00:19:42,580
+Y بصفر لأ بصير relation لان انا بسقط على X Z عشان
+
+287
+00:19:42,580 --> 00:19:46,380
+هيك باخد Y بصفر هذا الان المثلث هو المثلث اللى بدى
+
+288
+00:19:46,380 --> 00:19:49,500
+عدل عدو ال integration بين X و Z الان هذا معروف
+
+289
+00:19:49,500 --> 00:19:53,360
+نقطتين وهين و هدوله معروف نقطتين وهين بدى اودد
+
+290
+00:19:53,360 --> 00:19:57,380
+معدلته اودد معدلته انه نقطتين بطلع X زي Z بساوة
+
+291
+00:19:57,380 --> 00:20:04,060
+واحد هتقلل الان الان عندى اللى هي النقطة هذه عبارة
+
+292
+00:20:04,060 --> 00:20:09,370
+عنعشان اكس و زد و اكس و زد و اكس و زد و اكس و زد و
+
+293
+00:20:09,370 --> 00:20:20,930
+اكس و زد و اكس
+
+294
+00:20:20,930 --> 00:20:26,380
+و زدزد واحد نقص زد اتنين على اكس واحد نقص اكس
+
+295
+00:20:26,380 --> 00:20:29,620
+اتنين و اتغلب انا عارف هذا على طول بين الواحد
+
+296
+00:20:29,620 --> 00:20:33,240
+والواحد زمان قبلناها كثير فاطلع exact z بالساوح
+
+297
+00:20:33,240 --> 00:20:36,820
+بقوله انت حر ايش بتسوي المهم توجدلي اللي هو معدلة
+
+298
+00:20:36,820 --> 00:20:42,500
+الخط هذا المستقيم بالطريقة اللي توجدليها بشكل صحيح
+
+299
+00:20:43,060 --> 00:20:46,460
+إذا ليش وجدت هذا عشان هيلزمني بعد شوية هذا معدلته
+
+300
+00:20:46,460 --> 00:20:50,780
+إيش معروفة اللي هو Y بساوي Z بساوي سفر هذا معدلته
+
+301
+00:20:50,780 --> 00:20:54,400
+معروفة هي X بساوي سفر أنا بشغل الآن في X Z بلين
+
+302
+00:20:54,400 --> 00:20:58,240
+عشان هيك مادخلوش Y إيش بدي فيها كلها سفر معينإذا
+
+303
+00:20:58,240 --> 00:21:03,300
+ندخل إلى حدود الانتجريشن دي زد دي اكس دي زد باخد
+
+304
+00:21:03,300 --> 00:21:07,800
+موازين ميان لزد اكس في داخل المنطقة من وين بدخل و
+
+305
+00:21:07,800 --> 00:21:12,160
+من وين بخرج إلى واضح انه بدخل من عند الوزد بتساوي
+
+306
+00:21:12,160 --> 00:21:18,550
+سفر هيهاهو يدخل من الخط هذا و يخرج من الخط هذا
+
+307
+00:21:18,550 --> 00:21:21,750
+كلهم على نفس المنوال ايش z تانى؟ واحد minus x عشان
+
+308
+00:21:21,750 --> 00:21:25,230
+هيك من zero لعند واحد minus x نيجي الان هذه الخطوط
+
+309
+00:21:25,230 --> 00:21:28,170
+الموزية ل z axis من اين؟ بتبدأ من عند ال .. اللي
+
+310
+00:21:28,170 --> 00:21:31,110
+هي .. اللي هي .. الصغيرة من ده طبعا من عند x
+
+311
+00:21:31,110 --> 00:21:35,070
+بتساوي سفرها هي هي كيك بتنسح لما نصل لاخر اشي عند
+
+312
+00:21:35,070 --> 00:21:39,630
+واحد عشان هيك قيمة ال xLimit الـ integrations ده
+
+313
+00:21:39,630 --> 00:21:43,450
+ال X اللي هي Zero لعند الواحد و بيكون هذا اللي هو
+
+314
+00:21:43,450 --> 00:21:50,230
+ال مطلوب الان طالب مني في السؤال على نفس المنطقة
+
+315
+00:21:50,230 --> 00:21:53,290
+بقول integrate F of X Y و Z بسياه واحد يعني أوجد
+
+316
+00:21:53,290 --> 00:21:56,870
+حاجة من معناته ايه over the tetrahedron D in
+
+317
+00:21:56,870 --> 00:22:00,890
+example 2 اللي قبل شوية in the order DZDYDX and
+
+318
+00:22:00,890 --> 00:22:05,730
+then integrate in the order DYDZDX هذا DYZDZDX توع
+
+319
+00:22:05,730 --> 00:22:11,230
+باللهالان انا اريد ان اكمل الأمر بالنسبة
+
+320
+00:22:11,230 --> 00:22:14,650
+لـ z ثم بالنسبة ل y ثم بالنسبة لمين بالنسبة ل x
+
+321
+00:22:14,650 --> 00:22:17,270
+طبعا كل أمورنا جاهزة من السؤال اللي فات اللي هي
+
+322
+00:22:17,270 --> 00:22:25,170
+عبارة عن اللي هي نقاط التقاط وإلى آخرها ومعادلة
+
+323
+00:22:25,170 --> 00:22:30,950
+لبنان هذا ومعادلة لبنان هذا كله جاهزبالنسبة لـ Z
+
+324
+00:22:30,950 --> 00:22:34,430
+مدام انها كاملة بالنسبة لـ Z فانت تاخد موازيات للـ
+
+325
+00:22:34,430 --> 00:22:39,290
+Z Axis التي تدخل من الجسم و تخرج من الجسم لاحظ ان
+
+326
+00:22:39,290 --> 00:22:44,210
+كل الموازيات التي تدخل من الـ Z Axis تدخل الجسم من
+
+327
+00:22:44,210 --> 00:22:48,030
+عند Z بس أو سفر و تخرج من عند وين يعني عند البلين
+
+328
+00:22:48,030 --> 00:22:51,160
+اللي تحتوبتخرج من عند الـ plane اللي ما هي الـ H
+
+329
+00:22:51,160 --> 00:22:54,000
+ده اللي هو إيش قيمة Z قلنا عليه اللي هو Z بيساوي Y
+
+330
+00:22:54,000 --> 00:22:57,740
+ناقص X اللي كنا بيسميه نقطة Y بيساوي Z زائد X يعني
+
+331
+00:22:57,740 --> 00:23:01,280
+Z بيساوي Y minus X إذن حد ال integration الأول من
+
+332
+00:23:01,280 --> 00:23:05,660
+0 لعند Y minus X هاي حد ال integration الأول
+
+333
+00:23:05,660 --> 00:23:11,840
+volume بسبب عند Z اللي هو Zero لعند Y minus X نيجي
+
+334
+00:23:11,840 --> 00:23:16,540
+الآن يا جماعة انحدد اللي هو مين انحدد قيمةاللي هو
+
+335
+00:23:16,540 --> 00:23:20,440
+ال .. ال .. ال .. قولوا معايا ال .. ال .. المسقط
+
+336
+00:23:20,440 --> 00:23:24,520
+لإنه بدنا مكامل احنا كملنا لزد مكامل Y و X بدنا
+
+337
+00:23:24,520 --> 00:23:27,820
+المسقط في وين في ال X Y plane العادى الآن المسقط
+
+338
+00:23:27,820 --> 00:23:30,220
+زي ما عملنا قبل شوية بتصير هذه اللي هو بتصير اللي
+
+339
+00:23:30,220 --> 00:23:34,200
+هي بنخل الزد بصفر بصير صفر و واحد و Zero هي النقطة
+
+340
+00:23:34,530 --> 00:23:37,790
+وهذه المسقطة هي نفسها على حالها لأنها Z بصفر وهذه
+
+341
+00:23:37,790 --> 00:23:40,090
+المسقطة هي نفسها على حالها لأنها Z بصفر وهذه
+
+342
+00:23:40,090 --> 00:23:40,090
+المسقطة هي نفسها على حالها لأنها Z بصفر وهذه
+
+343
+00:23:40,090 --> 00:23:40,470
+المسقطة هي نفسها على حالها لأنها Z بصفر وهذه
+
+344
+00:23:40,470 --> 00:23:42,690
+المسقطة هي نفسها لأنها Z بصفر وهذه المسقطة هي
+
+345
+00:23:42,690 --> 00:23:45,350
+نفسها لأنها Z بصفر وهذه المسقطة هي نفسها لأنها Z
+
+346
+00:23:45,350 --> 00:23:46,790
+بصفر وهذه المسقطة هي نفسها لأنها Z بصفر وهذه
+
+347
+00:23:46,790 --> 00:23:48,950
+المسقطة هي نفسها لأنها Z بصفر وهذه المسقطة هي
+
+348
+00:23:48,950 --> 00:23:50,510
+نفسها لأنها Z بصفر وهذه المسقطة هي نفسها لأنها Z
+
+349
+00:23:50,510 --> 00:23:53,630
+بصفر وهذه المسقطة هي نفس
+
+350
+00:23:57,360 --> 00:24:01,420
+حسب النقطة اللى أسقطناها و هذا الخط كله اللى فوق
+
+351
+00:24:01,420 --> 00:24:06,320
+بنزل كله بقعد هنا فى نقطة و هذا نسقطه اللى هو هذا
+
+352
+00:24:06,320 --> 00:24:10,370
+الخطوهذا مصفته اللى هو نفسه لإنه في الهدك فبصير
+
+353
+00:24:10,370 --> 00:24:15,550
+مصفة كل الشكل اللى عندى هو المثلث الامانى هذا الان
+
+354
+00:24:15,550 --> 00:24:18,730
+المثلث هذا بدى نجيب معدلاته اللى بتلزمنا طبعا
+
+355
+00:24:18,730 --> 00:24:23,910
+المعادلة هذه اللى هو x بتساوي سفر والمعادلة هذه y
+
+356
+00:24:23,910 --> 00:24:27,150
+بتساوي واحد زى ما انتوا شايفين لإنه y ثابتة على
+
+357
+00:24:27,150 --> 00:24:29,930
+هذا المكان لإن أصلا ال blank كله هذا y بيساوي واحد
+
+358
+00:24:29,930 --> 00:24:34,850
+وهذه اللى هي عند النقطة أصلا من zero zero هو لايحة
+
+359
+00:24:34,850 --> 00:24:39,300
+للواحد واحدإذاً قيمة X هنا و Y زي بعض و قيمة X و Y
+
+360
+00:24:39,300 --> 00:24:42,420
+هنا زي بعض في نقطتين إذا الخط اللي بين أن هي أو
+
+361
+00:24:42,420 --> 00:24:45,200
+عبارة عن الخط اللي بيساوي بين X و Y اللي هو Y
+
+362
+00:24:45,200 --> 00:24:49,260
+بيساوي X و اللي مش عاد و يشتغل معرفة نقطتين اللي
+
+363
+00:24:49,260 --> 00:24:52,180
+هو بيقدر يعرف معدلة الخط المستقيم هيلاقيها على طول
+
+364
+00:24:52,180 --> 00:24:54,940
+Y بيساوي X لكن الموضوع مش مستاهل على طول ال
+
+365
+00:24:54,940 --> 00:24:58,040
+percentage Y بيساوي X لأن قيمة اللي هي ال X و ال Y
+
+366
+00:24:58,040 --> 00:25:01,020
+في النقطتين في النقطة الأولى زي بعض و في النقطة
+
+367
+00:25:01,020 --> 00:25:03,300
+الثانية برضه زي بعض Zero Zero و أنا واحد واحد
+
+368
+00:25:03,300 --> 00:25:08,440
+فاميش مقصودإذا الأمور نجازة بدنا مكامل بالنسبة لمن
+
+369
+00:25:08,440 --> 00:25:11,100
+؟ ل Y بمثابة كامل بالنسبة ل Y إذا بتاخد في داخل
+
+370
+00:25:11,100 --> 00:25:14,280
+المنطقة R اللي اتعتد اللي صارت مصفت له موازيه ل Y
+
+371
+00:25:14,280 --> 00:25:18,320
+Axis كل الموازيه ل Y Axis هذوله اللي بيجي من عند
+
+372
+00:25:18,320 --> 00:25:22,240
+الخط Y بيساوي X و بروح من عند الخط Y إيش بتساوي 1
+
+373
+00:25:22,240 --> 00:25:26,340
+لإن هنا بتبدأ التزايد هيك بتتزايد إذا من Y بيساوي
+
+374
+00:25:26,340 --> 00:25:30,220
+X لعند Y بيساوي 1 من Y بتساوي X لعند Y بيساوي 1
+
+375
+00:25:30,220 --> 00:25:38,890
+هذا قيمة من ال Yنعود لقيمة ال X كل الخطوط الموازية
+
+376
+00:25:38,890 --> 00:25:45,750
+ال Y Axis تبدأ من X بصفر و تتدرج لين تصل ل X باحد
+
+377
+00:25:45,750 --> 00:25:50,010
+أخر خط بمسح كل المنطقةوبكون عندى اللى هو ال
+
+378
+00:25:50,010 --> 00:25:53,910
+integration من صفر لواحد بالنسبة ل X لحدود التكامل
+
+379
+00:25:53,910 --> 00:25:57,410
+نجى لان كامل كامل ما هي التكامل الواحد هو عبارة عن
+
+380
+00:25:57,410 --> 00:26:01,830
+Z بصير Z بعوض Y minus X ناقص Zero Zero بصير Y
+
+381
+00:26:01,830 --> 00:26:05,070
+minus X بده كامل هذا Y minus X بالنسبة ل Y بصير
+
+382
+00:26:05,070 --> 00:26:09,390
+تكاملها عبارة عن نص Y تربيع ناقص XY من وين لوين
+
+383
+00:26:09,390 --> 00:26:12,780
+بعوض من X بالساويةمن y بيساوي x لأن y بيساوي واحد
+
+384
+00:26:12,780 --> 00:26:20,240
+لان بعود عن y هنا بيصير عبارة عن نص ناقص x لان y
+
+385
+00:26:20,240 --> 00:26:24,660
+بيساوي x ناقل هي عبارة عن نص y تربيع نص y تربيع
+
+386
+00:26:24,660 --> 00:26:27,680
+ناقص y تربيع اللي هي بيصير عبارة عن ناقص نص y
+
+387
+00:26:27,680 --> 00:26:31,120
+تربيع و النقص اللي برا بيصير اسف x تربيع بيصير نص
+
+388
+00:26:31,120 --> 00:26:36,170
+x تربيع لان بيعود عن y بxطيب الآن صار عند المقدار
+
+389
+00:26:36,170 --> 00:26:39,370
+بالشكل هذا بدي أكمله بالنسبة ل X هيكملته بالنسبة ل
+
+390
+00:26:39,370 --> 00:26:43,530
+X نص X X تربيع على اللي هو نص X عدد X تربيع اللي
+
+391
+00:26:43,530 --> 00:26:46,170
+هو هذه اللي هي بيصير على تلاتة بيصير سدس و X تكييب
+
+392
+00:26:46,170 --> 00:26:50,570
+و من 0 ل 1 و بعوض و بطلع عند جداش بطلع سدس الان
+
+393
+00:26:50,570 --> 00:26:55,830
+احنا لازم نحصل على نفس القيمة لما نكامل أولش
+
+394
+00:26:55,830 --> 00:26:58,880
+بالنسبة ل Y و بالنسبة ل Z و بعدين بالنسبة ل Xطبعا
+
+395
+00:26:58,880 --> 00:27:01,440
+بالنسبة ل Y حددنا حدود ال integration في المثال
+
+396
+00:27:01,440 --> 00:27:04,880
+اللي جابله كانت من X زائد المثال اللي جابله ايه ..
+
+397
+00:27:04,880 --> 00:27:09,420
+ايه .. ايه المثال X زائد Z لعند الواحد دي Y دي Z
+
+398
+00:27:09,420 --> 00:27:12,700
+طبعا نفسه هو اللي هذاك بس احنا كنا كامل على ال F
+
+399
+00:27:12,700 --> 00:27:17,040
+انها كنا كاملنا على الواحد Z من Z لواحد minus X و
+
+400
+00:27:17,040 --> 00:27:23,460
+X من عند سفر لعند واحد و يعني Y Z X وهنا بدو Y Z X
+
+401
+00:27:23,460 --> 00:27:26,260
+هي Y Z X بدنا اذا نفس حدود ال integration او
+
+402
+00:27:26,260 --> 00:27:30,710
+جدناها قبل شويةنجم كامل بالنسبة ل Y بيصير 1Y اللي
+
+403
+00:27:30,710 --> 00:27:35,590
+هي بتعوض 1-X-Z دي Z دي X إلى أن بعدين هذا بنكمله
+
+404
+00:27:35,590 --> 00:27:39,070
+بالنسبة ل Z بيصير 1-X هذه ثابت في Z بالنسبة ل Z
+
+405
+00:27:39,070 --> 00:27:43,590
+طبعا نقص نص Z تربيع Z من وين من سفر اللي عنده 1-X
+
+406
+00:27:43,590 --> 00:27:49,550
+بنعوض عنها اللي هي بيصير ليها 1-X في 1-X تربيع نقص
+
+407
+00:27:49,550 --> 00:27:53,250
+نص 1-X تربيع و السفر طبعا اللي بيسويها دي سفر إذا
+
+408
+00:27:53,250 --> 00:27:56,070
+انصار عندي اللي هو هذا المقدار اللي هو عبارة عن
+
+409
+00:27:56,070 --> 00:28:00,740
+إطرح هذا من هذا بيطلعاللي هو واحد ناقص X تربيع
+
+410
+00:28:00,740 --> 00:28:06,320
+الان الان الكل دي X طبعا بنص اللي طلعناه برا لإن
+
+411
+00:28:06,320 --> 00:28:11,300
+هذا أكبر من هذا جد مرتين سا نص طيب فاهمين طبعا هذا
+
+412
+00:28:11,300 --> 00:28:15,740
+اتفاق تكامله اللي وقبلناه اللي بيصير عبارة عن اللي
+
+413
+00:28:15,740 --> 00:28:22,050
+هو ناقص واحد ناقص X الكل تربيع علىالتلاتة وفي عندى
+
+414
+00:28:22,050 --> 00:28:25,270
+اتنين بيصير على ست وهي الناقص ومن واحد لإن سفر
+
+415
+00:28:25,270 --> 00:28:28,770
+وبتعود عليها بتطلع برضه إيش القيمة سدس وفعلا
+
+416
+00:28:28,770 --> 00:28:33,090
+القيمتين زي بعض نيجي الآن لحاجة مفهوم برضه مشابه
+
+417
+00:28:33,090 --> 00:28:36,890
+لمفهوم أخدناها بالهكلة على main ال function مش in
+
+418
+00:28:36,890 --> 00:28:41,250
+space in اللي هو a plain average value of a
+
+419
+00:28:41,250 --> 00:28:43,970
+function in space اللي هي الآن ال function وين
+
+420
+00:28:43,970 --> 00:28:47,210
+موجودة in space يعني function of three variables
+
+421
+00:28:47,900 --> 00:28:51,640
+الان الـ Average Value of F over D over Region D
+
+422
+00:28:51,640 --> 00:28:55,580
+دي طبعا اللي هي ايه شمالها تلت أبعاد و قيمة الـ
+
+423
+00:28:55,580 --> 00:28:58,520
+Function واحد بيصير اربع أبعاد اذا عشان اواجه الـ
+
+424
+00:28:58,520 --> 00:29:01,040
+Average Value بيكون واحد ع ال volume اللي هو تلت
+
+425
+00:29:01,040 --> 00:29:05,220
+أبعاد في ال triple integration FDV اللي هي أربع
+
+426
+00:29:05,220 --> 00:29:09,270
+أبعاد فبيصير أربع أبعاد علىعلى بُعد الـ Volume
+
+427
+00:29:09,270 --> 00:29:13,090
+والـ F زيادة وهذا رافلي الكلام اللي هي بتطلع من
+
+428
+00:29:13,090 --> 00:29:16,170
+الـ F يعني هذا بتعمل منطقي أننا نعرف الـ Average
+
+429
+00:29:16,170 --> 00:29:20,290
+Value لـ Function in space over a region D in
+
+430
+00:29:20,290 --> 00:29:24,710
+space اللي هي Function of three variablesلـ over
+
+431
+00:29:24,710 --> 00:29:27,970
+region in space بالساعة واحدة على volume of D في
+
+432
+00:29:27,970 --> 00:29:31,690
+الـ triple integral للـ FDV وهذا مشابه لما أخذناه
+
+433
+00:29:31,690 --> 00:29:34,870
+سابقا لـ function of two variables كونة average of
+
+434
+00:29:34,870 --> 00:29:40,670
+value of F over a region اللي هي R in two اللي هي
+
+435
+00:29:40,670 --> 00:29:44,090
+dimensions أو نبلين اللي هي واحدة على الـ area
+
+436
+00:29:44,090 --> 00:29:47,250
+كانت في الـ double integral لاللي هو الـ function
+
+437
+00:29:47,250 --> 00:29:50,730
+في الـ DA فهذا اللي هو على نفس المانوال زي ما
+
+438
+00:29:50,730 --> 00:29:54,870
+أوضحت قبل شويةالان ناخد مثال عليه بيقول find the
+
+439
+00:29:54,870 --> 00:29:57,470
+average value of f ايش ال function اللي بده
+
+440
+00:29:57,470 --> 00:30:00,330
+average value اقولها x في y في z throughout the
+
+441
+00:30:00,330 --> 00:30:04,190
+cubical region خلال ايش؟ بده يهد معرف على مين على
+
+442
+00:30:04,190 --> 00:30:08,070
+cubical region D bounded by the coordinate blends
+
+443
+00:30:08,070 --> 00:30:11,870
+محصورة بين ايش ال .. ال .. ال .. ال cube ال
+
+444
+00:30:11,870 --> 00:30:15,430
+coordinates blend ال coordinates blends اه هاي
+
+445
+00:30:15,430 --> 00:30:20,550
+اللي هو ال xz blend وهذا ال yزد بلين وهذا ال XY
+
+446
+00:30:20,550 --> 00:30:24,550
+بلين يعني طب من جهة ايه محصورة ب X بساوة اتنين هي
+
+447
+00:30:24,550 --> 00:30:28,490
+X بتساوة اتنين هي هذا الأحضر هذا أو Y بساوة اتنين
+
+448
+00:30:28,490 --> 00:30:32,730
+هي كمان هذا Y بساوة اتنين أو Z بساوة اللي هو اتنين
+
+449
+00:30:32,730 --> 00:30:36,690
+هذا اللي باللون الساكني اذا صار اللي هو ال .. ال
+
+450
+00:30:36,690 --> 00:30:39,620
+.. ال ..الـ cube اللي عندنا أو المكعب اللي عندنا
+
+451
+00:30:39,620 --> 00:30:45,420
+محدد تحديدا تماما هذا اللي هو y بصفر هذا x بصفر و
+
+452
+00:30:45,420 --> 00:30:52,920
+اللي فوق اللي هو عبارة عن هنا x بصفر هيق وهنا اللي
+
+453
+00:30:52,920 --> 00:30:59,300
+هو y بصفر و اللي فوق هذا عبارة عن اللي هو
+
+454
+00:31:09,650 --> 00:31:16,440
+أول حاجة بنرسم بمعنى sketchwith enough details to
+
+455
+00:31:16,440 --> 00:31:19,360
+show the limit of integration لكي نحدد اللي هو
+
+456
+00:31:19,360 --> 00:31:22,540
+اللي هي حدود ال integration هذه ال cube رسمناها
+
+457
+00:31:22,540 --> 00:31:25,560
+لأن الأمور سهلة ال cube أولا ال volume إيه له لأنه
+
+458
+00:31:25,560 --> 00:31:28,380
+ملزمنا ال volume عشان ذلك ال volume اللي هو اللي
+
+459
+00:31:28,380 --> 00:31:31,700
+هو عارفين ال volume اللي هو مساحة القاعدة اللي هي
+
+460
+00:31:31,700 --> 00:31:34,700
+أربعة في الارتفاع اتنين أربعة في اتنين بتمانية أو
+
+461
+00:31:34,700 --> 00:31:37,540
+ثلاث أبعاد اتنين في اتنين في اتنين بتطلع تمانية
+
+462
+00:31:37,540 --> 00:31:43,240
+اللي هي يشملها ال average value لمن لأسف اللي هو
+
+463
+00:31:43,240 --> 00:31:44,840
+ال volume للمكعب
+
+464
+00:31:46,810 --> 00:31:49,010
+الان نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+465
+00:31:49,010 --> 00:31:49,970
+نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+466
+00:31:49,970 --> 00:31:52,810
+نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+467
+00:31:52,810 --> 00:31:57,650
+نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+468
+00:31:57,650 --> 00:31:57,650
+نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+469
+00:31:57,650 --> 00:32:01,810
+نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي نجي
+
+470
+00:32:01,810 --> 00:32:06,590
+نجي نجي
+
+471
+00:32:06,590 --> 00:32:09,570
+نجي
+
+472
+00:32:10,280 --> 00:32:15,240
+X بصفر ورايحة لعند X بساوة اتنين يعني من الوجه هذا
+
+473
+00:32:15,240 --> 00:32:19,460
+للوجه اللي هو هذا كل اللي هي الموازيات لل X axis
+
+474
+00:32:19,460 --> 00:32:23,740
+على نفس المنوال عشان هيك بكون ال X اللي هي حدودها
+
+475
+00:32:23,740 --> 00:32:27,460
+من صفر لعند مين اتنين الان مسقط هذا سهل مسقط
+
+476
+00:32:27,460 --> 00:32:31,400
+المقعب كله ايه اشمله المقعب هنا ساوة دي بصفر تنزل
+
+477
+00:32:31,400 --> 00:32:36,540
+هنا اللي هو زدها بصفر تنزل احنا لأ احنا بنكملX
+
+478
+00:32:36,540 --> 00:32:44,940
+أولا و YZ أولا و X أولا و X أولا و X أولا و X أولا
+
+479
+00:32:44,940 --> 00:32:45,300
+و X أولا و X أولا و X أولا و X أولا و X أولا و X
+
+480
+00:32:45,300 --> 00:32:45,680
+أولا و X أولا و X أولا و X أولا و X أولا و X أولا
+
+481
+00:32:45,680 --> 00:32:46,820
+و X أولا و X أولا و X أولا و X أولا و X أولا و X
+
+482
+00:32:46,820 --> 00:32:46,820
+أولا و X أولا و X أولا و X أولا و X أولا و X أولا
+
+483
+00:32:46,820 --> 00:32:46,820
+و X أولا و X أولا و X أولا و X أولا و X أولا و X
+
+484
+00:32:46,820 --> 00:32:48,160
+أولا و X أولا و X أولا و X أولا و X أولا و X أولا
+
+485
+00:32:48,160 --> 00:32:54,220
+و X أولا و X أولا و X أولا و X أولا و X أولا و X
+
+486
+00:32:54,220 --> 00:32:58,450
+أولالذي في الـ ZY Plane المربع اللي هنا أموره سهلة
+
+487
+00:32:58,450 --> 00:33:04,910
+و مستطيلة جميعها على طول لو أخذنا الآن بالنسبة لـ
+
+488
+00:33:04,910 --> 00:33:10,010
+Y موازي لـ Y Axis كلهم على نفس المنوال من 0 ل2 و Z
+
+489
+00:33:10,010 --> 00:33:17,430
+طبعا تأخذ أولا بـ 0 و تتكرم تركم تركم ل2 من 0 ل2 و
+
+490
+00:33:17,430 --> 00:33:23,420
+هذا هو قيمة الـ Integrationبالكامل لان xyz بالنسبة
+
+491
+00:33:23,420 --> 00:33:26,740
+ل x بيصير x تربيع على اتنين في yz من سفر لعندي
+
+492
+00:33:26,740 --> 00:33:29,600
+اتنين اتنين بيصير عبارة عن اربع على اتنين اللي هي
+
+493
+00:33:29,600 --> 00:33:33,780
+اتنين بيصير اتنين yz وهذا طبعا سفر من سفر لاتنين
+
+494
+00:33:33,780 --> 00:33:36,220
+بتكامل برضه بالنسبة ل y بيصير اتنين y تربيع على
+
+495
+00:33:36,220 --> 00:33:40,000
+اتنين اللي هي y تربيع z الان من سفر لعندي اتنين y
+
+496
+00:33:40,000 --> 00:33:45,610
+تربيع بيصير اربع zالتي تتكامل من 0 إلى 2 أربعة زت
+
+497
+00:33:45,610 --> 00:33:49,690
+بيصير أربعة زت تربيع على اتنين يعني اتنين زت تربيع
+
+498
+00:33:49,690 --> 00:33:52,190
+من سفر إلى اتنين بيصير عبارة عن اتنين في اتنين
+
+499
+00:33:52,190 --> 00:33:55,870
+واربعة واتنين تمانية الان تمانية هذه هي ايه شمالها
+
+500
+00:33:55,870 --> 00:33:59,830
+ايه قيمة ال integration دا ال average value of XYZ
+
+501
+00:34:00,290 --> 00:34:04,390
+الـ Average Value لمن ؟ لأن ده ل F of X Y و Z
+
+502
+00:34:04,390 --> 00:34:07,570
+بساوي X في Y في Z على ال cube بساوي واحد ع ال
+
+503
+00:34:07,570 --> 00:34:09,890
+volume اللي هي واحد ع تمانية في مقدار ال
+
+504
+00:34:09,890 --> 00:34:12,670
+integration طالع تمانية اللي هو بساوي واحد و هيك
+
+505
+00:34:12,670 --> 00:34:15,710
+يكون أوجدنا ال average value و هيك يكون المحاضرة
+
+506
+00:34:15,710 --> 00:34:18,470
+انتهت و إلى لقاء آخر السلام عليكم ورحمة الله
+
+507
+00:34:18,470 --> 00:34:19,570
+وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mNaq2Sskss8.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mNaq2Sskss8.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1d7dfc509953dc78a1c3a7e4a97336861ea78309
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mNaq2Sskss8.srt
@@ -0,0 +1,782 @@
+1
+00:00:00,690 --> 00:00:06,570
+بسم الله الرحمن الرحيم هذه هي المحاضرة رقم عشرين
+
+2
+00:00:06,570 --> 00:00:13,030
+وهي في مساق طبعاً تفاضل وتكامل 2 وهي المحاضرة
+
+3
+00:00:13,030 --> 00:00:17,510
+الأولى في Chapter خمسة عشر سنتحدث اليوم عن Chapter خمسة عشر
+
+4
+00:00:17,510 --> 00:00:21,530
+واحد، عنوان Chapter خمسة عشر، اللي هو ملقب  الـ integrals
+
+5
+00:00:21,770 --> 00:00:27,730
+أو التكاملات المتعددة، المقصود فيها الـ Double
+
+6
+00:00:27,730 --> 00:00:31,770
+Integral أو Triple Integral سنشاهدها قداماً إن شاء
+
+7
+00:00:31,770 --> 00:00:34,970
+الله، خمسة عشر واحد اللي هو الـ Double and iterated
+
+8
+00:00:34,970 --> 00:00:38,310
+integrals over rectangles، اللي بدنا ناخد الـ
+
+9
+00:00:38,310 --> 00:00:44,270
+Double Integral على مستطيل، على أسهل الأمور
+
+10
+00:00:44,270 --> 00:00:47,230
+وإن شاء الله تدريجياً نروح على مناطق غير الـ
+
+11
+00:00:47,230 --> 00:00:49,750
+rectangles، وإن شاء الله بتكون الأمور برضه سهلة
+
+12
+00:00:49,750 --> 00:00:53,350
+الآن الـ Double Integral زي ما حكينا زمان في
+
+13
+00:00:53,350 --> 00:00:57,990
+التفاضل والتكامل، عن اللي هو الـ integration الـ single
+
+14
+00:00:57,990 --> 00:01:00,690
+integral، integration عرفناها على أساس اللي هو
+
+15
+00:01:00,690 --> 00:01:06,130
+limit sum، Riemann sum، يعني نهاية مجموع Riemann
+
+16
+00:01:06,130 --> 00:01:10,090
+كنا نوجد مجموع Riemann ومن ثم نوجد اللي هو الـ limit
+
+17
+00:01:10,090 --> 00:01:13,250
+له، فكان لو كان الـ limit موجود بنقول أن الـ function
+
+18
+00:01:13,250 --> 00:01:16,490
+is integrable وقيمة الـ limit هي قيمة الـ
+
+19
+00:01:16,490 --> 00:01:20,230
+integration، الآن الكلام شبيه، وشبيه جداً اللي هو
+
+20
+00:01:20,230 --> 00:01:23,650
+برضه هنستعمل اللي هو Riemann sum في اللي هو تعريف
+
+21
+00:01:23,650 --> 00:01:26,560
+الـ double integral، خلّينا نشوف احنا الـ
+
+22
+00:01:26,560 --> 00:01:29,280
+integration عندنا over مين؟ الآن over rectangles
+
+23
+00:01:29,280 --> 00:01:33,620
+يعني لو كان في عندنا rectangle، X بين A وB، هاي X بين
+
+24
+00:01:33,620 --> 00:01:38,060
+A وB، يعني الـ X من A لعند B، هي الـ X من عند A لـ B
+
+25
+00:01:38,060 --> 00:01:41,900
+والـ Y من عند C لعند D، فبيصير عندنا المنطقة هذه
+
+26
+00:01:41,900 --> 00:01:45,260
+عبارة عن rectangle، هذه المنطقة اللي عليها بدنا
+
+27
+00:01:45,260 --> 00:01:48,680
+نعمل الـ double integral، عشان نحسب الـ Double
+
+28
+00:01:48,680 --> 00:01:51,860
+Integral، خلّينا نشوف أول شيء الـ Riemann sum على
+
+29
+00:01:51,860 --> 00:01:55,060
+هذه، زي ما كنا نعمل قبل بالظبط، كنا زمان نجزّئ اللي
+
+30
+00:01:55,060 --> 00:02:02,260
+هو إيش؟ الـ X-Axis، X0، X1، X2، X3،... XK لعند XN، ونجزّئها
+
+31
+00:02:02,260 --> 00:02:05,340
+اللي هي كانت بس، احنا بنكامل على الخط، الآن بنكامل
+
+32
+00:02:05,340 --> 00:02:10,700
+على منطقة، هنجزّئ X وهنجزّئ Y، يعني هنجزّئ X اللي
+
+33
+00:02:10,700 --> 00:02:14,520
+بيكون كامل بالنسبة له، وهنجزّئ Y اللي برضه بيكون
+
+34
+00:02:14,520 --> 00:02:17,800
+كامل بالنسبة لها، لأنه عايش الآن على متغيرين، إذا
+
+35
+00:02:17,800 --> 00:02:23,100
+جزّأنا هذه، X0، X1، X2، X3 إلى آخره لعند XN، وهنا X0،
+
+36
+00:02:23,100 --> 00:02:29,260
+X1، X2، X3 لعند XN، إلى آخره، الآن فبيصير عند
+
+37
+00:02:29,260 --> 00:02:33,940
+التجزئة هذه، بناخد المستقيمات هذه الموازية للـ Y
+
+38
+00:02:33,940 --> 00:02:37,230
+Axis، وبناخد المستقيمات الموازية للـ X Axis حسب
+
+39
+00:02:37,230 --> 00:02:41,750
+التقسيم اللي قلناها، فبتعمل إيه؟ بتجزّئ هذا
+
+40
+00:02:41,750 --> 00:02:45,370
+الـ rectangle الكبير إلى اللي هي sub rectangles
+
+41
+00:02:45,370 --> 00:02:48,690
+صغيرة، الآن هذه الـ rectangles طبعاً هذه اللي هي
+
+42
+00:02:48,690 --> 00:02:51,170
+بتكون اللي هي جزء من domain  الـ function اللي
+
+43
+00:02:51,170 --> 00:02:56,170
+بنشتغل عليها، اللي هي F of X Y، فالآن بنيجي بناخد زي
+
+44
+00:02:56,170 --> 00:02:59,730
+ما كنا ناخد نقطة عشوائية أيام اللي هي الـ
+
+45
+00:02:59,730 --> 00:03:03,010
+integration العادي أو الـ single integral، بناخد
+
+46
+00:03:03,010 --> 00:03:08,090
+نقطة عشوائية في داخل المربع هذا، هذا المربع سمّيته
+
+47
+00:03:08,090 --> 00:03:15,350
+طوله Δxk وعرضه Δyk، من xk-1 لعند الـ xk،
+
+48
+00:03:15,350 --> 00:03:21,810
+ومن yk-1 لعند الـ yk، هذا المستطيل اللي أمامي اللي
+
+49
+00:03:21,810 --> 00:03:28,000
+هو، باخد نقطة عشوائية بسميها (xk, yk)، الآن باخد قيمة
+
+50
+00:03:28,000 --> 00:03:32,560
+الـ Function عند هذه النقطة، F(xk, yk)، مضروبة في
+
+51
+00:03:32,560 --> 00:03:36,940
+مين؟ في المساحة هذه، التي هي Δxk × Δyk،
+
+52
+00:03:36,940 --> 00:03:42,100
+اللي سميناها إيش اسمها؟ ΔAk، وبعدين هذه أخدت اللي
+
+53
+00:03:42,100 --> 00:03:46,980
+هو قيمة الـ Function، F(xk, yk) عليها، وضربت في مين؟
+
+54
+00:03:46,980 --> 00:03:51,680
+في المساحة اللي عندي هذه، وبعدين بدأت أجمع على كل
+
+55
+00:03:51,680 --> 00:03:56,540
+المناطق اللي جزّأتها هذه، فبيصير هذا مجموعه هو من
+
+56
+00:03:56,540 --> 00:03:59,960
+k=1 لعند N، اللي بنسميه sum، اللي بنسميه Riemann
+
+57
+00:04:06,120 --> 00:04:10,360
+بتكون عبارة عن نقطة واحدة، فبيصير الجمع على مين؟ على
+
+58
+00:04:10,360 --> 00:04:15,920
+الخط هذا، وبينما هذه مش مساحة، بتكون Δxk، هذا
+
+59
+00:04:15,920 --> 00:04:20,260
+زمان، أما الآن زي ما قلنا اللي هو SN هو عبارة عن
+
+60
+00:04:20,260 --> 00:04:24,120
+ΔAk، اللي هو المستطيل الصغير هذا المجزّأ،
+
+61
+00:04:24,120 --> 00:04:29,180
+مضروبة في F(xk, yk)، اللي هي النقطة العشوائية
+
+62
+00:04:29,180 --> 00:04:32,880
+اللي أخدناها هذه، هذا طبعاً مربع أو مستطيل نموذجي
+
+63
+00:04:32,880 --> 00:04:37,060
+بنجمع كل هذه، على أساس أن نخلص على كل المنطقة هذه،
+
+64
+00:04:37,060 --> 00:04:41,520
+فبيصير مجموع k من 1 لعند N، F(xk, yk) ΔAk،
+
+65
+00:04:41,520 --> 00:04:45,670
+هذا هو عبارة عن الـ Riemann sum، هذا الـ Riemann sum اللي
+
+66
+00:04:45,670 --> 00:04:49,510
+أوجدناه، لو أخذنا له الـ Limit، as N goes to
+
+67
+00:04:49,510 --> 00:04:52,790
+infinity، على أساس أنه بتصير اللي هي لما N تروح
+
+68
+00:04:52,790 --> 00:04:56,170
+لما لا نهاية، هذه التجزئات هذه بتصغر، بتصير صغيرة صغيرة
+
+69
+00:04:56,170 --> 00:04:59,770
+صغيرة صغيرة، بتصير تتراكم جنب بعض، وكأنه بتصير إيه؟
+
+70
+00:04:59,770 --> 00:05:03,970
+كأنه بتصير خطوط، خطوط، خطوط، عندها بس اللي هو الخط
+
+71
+00:05:03,970 --> 00:05:08,280
+بمثل قيمة الدالة، لما نجمعها مع بعض، اللي هو وكانت الـ
+
+72
+00:05:08,280 --> 00:05:12,500
+Limit exist، هو اللي بدنا نسميه الـ Double Integral،
+
+73
+00:05:12,500 --> 00:05:17,540
+اللي هو f(x, y) dA over R، إذا لو الـ Limit هذا
+
+74
+00:05:17,540 --> 00:05:20,960
+exist على Riemann sum، هو بدنا نسميه اللي هو الـ
+
+75
+00:05:20,960 --> 00:05:26,300
+Double Integration، للـ f(x, y) dA أو f(x, y) dx
+
+76
+00:05:26,300 --> 00:05:37,220
+dy، الآن نيجي الـ double integral as a volume، نربط
+
+77
+00:05:37,220 --> 00:05:42,340
+الـ double integration بالـ volume، لأن لو المنطقة
+
+78
+00:05:42,340 --> 00:05:45,700
+اللي بدنا نكامل عليها هي… هي عندنا Surface، أول
+
+79
+00:05:45,700 --> 00:05:50,120
+شيء هي Surface، و Z = F(X, Y)، هذا الـ Surface،
+
+80
+00:05:50,120 --> 00:05:53,860
+الآن المنطقة اللي بدي أشتغل عليها بس اللي هي الـ
+
+81
+00:05:53,860 --> 00:05:58,300
+rectangle R، هذا بدي أشوف حجم الجسم هذا اللي هو
+
+82
+00:05:58,300 --> 00:06:01,940
+اللي أمامي، شوفوا يا جماعة، باجيب نفس اللي عملته قبل
+
+83
+00:06:01,940 --> 00:06:08,540
+شوية، بجزّئ هذه اللي هي الـ Y، Y1، Y2، Y3،… وYN، وهذا Y1،
+
+84
+00:06:08,540 --> 00:06:14,400
+وY2، وهنا X1، وX2،… إلى Xn، وبناخد فترة نموذجية، هذا
+
+85
+00:06:14,400 --> 00:06:18,820
+اللي هو المستطيل اللي هو الـ rectangle، اللي هي جزء من
+
+86
+00:06:18,820 --> 00:06:21,860
+الـ domain اللي عليه بدنا نحسب الـ volume للجسم، الآن
+
+87
+00:06:21,860 --> 00:06:24,800
+الـ volume للجسم بيصير اللي هو الشكل اللي أمامي هذا،
+
+88
+00:06:24,800 --> 00:06:27,560
+لأن قيمة الـ function هنا هي قيمة الـ function هنا،
+
+89
+00:06:27,560 --> 00:06:31,380
+هي هي السطح اللي عندي، فالآن السطح المكون اللي
+
+90
+00:06:31,380 --> 00:06:36,740
+قاعدته هذه وسقفه هذا، الـ volume له بدنا نشوف كيف بدنا
+
+91
+00:06:36,740 --> 00:06:42,140
+نحسبه، شوفوا كيف، الآن الـ volume هذا لو جينا نحسبه،
+
+92
+00:06:42,140 --> 00:06:46,680
+أخذت اللي هو الآن فترة نموذجية أو… أو منطقة فترة
+
+93
+00:06:46,680 --> 00:06:51,040
+نموذجية، هتعمل ليها مستطيل، مربع أو مستطيل نموذجي طوله
+
+94
+00:06:51,040 --> 00:06:57,420
+Δxk وعرضه Δyk، مساحته Δxk × Δyk، بيصير حجم اللي هي اللي على هذه المنطقة هو عبارة
+
+95
+00:06:57,420 --> 00:07:01,860
+عن اللي هي المساحة هذا × الارتفاع، لكن هذا
+
+96
+00:07:01,860 --> 00:07:04,220
+الارتفاع شايفينه متغير، هذا إيش؟ أنا باخد نقطة
+
+97
+00:07:07,300 --> 00:07:11,860
+عشوائية، وآخذها في المركز (xk, yk)، وبيجي وباخد
+
+98
+00:07:11,860 --> 00:07:16,780
+F(xk, yk)، قيمتها، الآن صارت عندي قيمة F(xk, yk)،
+
+99
+00:07:16,780 --> 00:07:20,080
+اللي هو يمثل الارتفاع هذا، مضروبة في المساحة هذه،
+
+100
+00:07:20,080 --> 00:07:23,620
+بيطلع اللي هو حجم اللي هو إيش؟ متوازي المستطيلات
+
+101
+00:07:23,620 --> 00:07:26,500
+اللي أمامي هذا، بس هو مش بالظبط متوازي مستطيلات،
+
+102
+00:07:26,500 --> 00:07:30,440
+بيطلع شوية فوق أو بينزل شوية تحت، فهذا تقريبي، الآن
+
+103
+00:07:30,780 --> 00:07:34,540
+الآن هذا التقريبي، لما يكون العدد كبير، مثلاً لما N
+
+104
+00:07:34,540 --> 00:07:37,940
+يساوي 16، لاحظوا بتلاقي فيه أجزاء مفقودة كثير أو
+
+105
+00:07:37,940 --> 00:07:42,060
+أجزاء زايدة كثير، فبيصير الآن اللي هي الـ… الـ… الـ
+
+106
+00:07:42,060 --> 00:07:45,900
+… هذا بيقرب الـ volume اللي هنا، لكن بيقربه بشكل
+
+107
+00:07:45,900 --> 00:07:49,900
+غير كامل، يعني بيصير فيه نسبة خطأ، الآن لو زدنا العدد،
+
+108
+00:07:49,900 --> 00:07:53,400
+اللي هو نسبة الخطأ إيش مالها؟ بتصير هذه قريبة من هذه،
+
+109
+00:07:53,400 --> 00:07:57,000
+لكن نسبة الخطأ برضه إيش؟ موجودة، لكن بتقل، لأن هذه
+
+110
+00:07:57,000 --> 00:08:00,950
+المفقود هنا أقل، لما نيجي هنا مثلاً اللي هو زدنا
+
+111
+00:08:00,950 --> 00:08:04,650
+العدد، صار N = 256، بيصير قريبين اللي هي نسبة
+
+112
+00:08:04,650 --> 00:08:08,370
+المفقود قليلة جداً، فما بالك لو أخذنا N = 1000 ولا N
+
+113
+00:08:08,370 --> 00:08:12,210
+بمليون ولا N بـ 100 مليون، هيصير نسبة المفقود قليلة
+
+114
+00:08:12,210 --> 00:08:16,650
+جداً، الآن as N goes to infinity، وكأني باخد إيش أنا؟
+
+115
+00:08:16,650 --> 00:08:21,270
+هذا، يعني صرت إيش؟ آخذ أنا الآن هذا واللي جنبه واللي
+
+116
+00:08:21,270 --> 00:08:24,700
+جنبه واللي جنبه، زي ما أخذناهم هدول، مجموعهم مجموعاً،
+
+117
+00:08:24,700 --> 00:08:27,340
+هذا اللي هو حجم هذا + حجم هذا + حجم هذا على
+
+118
+00:08:27,340 --> 00:08:32,440
+كل المنطقة، يعني بيصير عندنا Σk=1 إلى N، F(xk, yk) ΔAk،
+
+119
+00:08:32,440 --> 00:08:36,020
+هدول مجموعاً، هو عبارة عن مجموع
+
+120
+00:08:36,020 --> 00:08:39,020
+الحجوم هذه، هذه مع هذه مع هذه، لأن أنا مشيت على
+
+121
+00:08:39,020 --> 00:08:42,460
+مين؟ على كل الـ rectangles الصغيرة الصغيرة الصغيرة،
+
+122
+00:08:42,460 --> 00:08:46,060
+لما غطّيت الـ rectangle الكبير R كله، لكن هناك جزء
+
+123
+00:08:46,060 --> 00:08:51,540
+مقطوع أو مفقود، بيكون هنا مثلاً لما العدد قليل كثير،
+
+124
+00:08:51,540 --> 00:08:57,440
+هنا، أجل هنا، أجل، لما N تكبر، أجل، لما N تكبر، أجل، as N
+
+125
+00:08:57,440 --> 00:09:01,660
+goes to infinity، يصبح هذا السطح هنا بالظبط هو
+
+126
+00:09:01,660 --> 00:09:05,980
+السطح اللي فوق، يصبح عند الـ Limit للـ Integral للـ
+
+127
+00:09:05,980 --> 00:09:10,630
+Sn هذا، as N goes to infinity، بيساوي بالظبط هو الـ
+
+128
+00:09:10,630 --> 00:09:14,050
+volume، في حالة limit exist طبعاً واحنا واخدين
+
+129
+00:09:14,050 --> 00:09:17,810
+الدالة continuous وحلوة وكده وكده، متماسكة، نضمن
+
+130
+00:09:17,810 --> 00:09:21,630
+أنها exist، فالآن limit as N لما N تروح لما لا نهاية، يا
+
+131
+00:09:21,630 --> 00:09:25,710
+مدام موجود هيعطيك بالظبط الـ volume، وهو طبعاً limit
+
+132
+00:09:25,710 --> 00:09:29,110
+as N زي ما عرفناها قبل شوية، هو عبارة عن الـ double
+
+133
+00:09:29,110 --> 00:09:34,010
+integral، f(x, y) dA Over Rectangle، مين؟ Over
+
+134
+00:09:34,010 --> 00:09:38,070
+Rectangle R، إذا الـ volume بيساوي Limit، استخدمتها
+
+135
+00:09:38,070 --> 00:09:40,870
+طبعاً في حالة إن… اللي أخذناها إيه؟ قلنا Z
+
+136
+00:09:40,870 --> 00:09:44,570
+= F(X, Y) is a positive، دائماً positive على
+
+137
+00:09:44,570 --> 00:09:48,600
+المنطقة هذه، عشان هي أعطت volume بالظبط، نأتي الآن
+
+138
+00:09:48,600 --> 00:09:51,600
+لدينا نظرية اسمها Fubini's Theorem، for calculating
+
+139
+00:09:51,600 --> 00:09:54,240
+double integrals، لأننا نريد أن نجد الـ Double
+
+140
+00:09:54,240 --> 00:09:56,820
+Integral، لذلك نحتاج نظرية تساعدنا في إيجاده،
+
+141
+00:09:56,820 --> 00:10:01,100
+النظرية سهلة ومريحة، لكي نرى ما هي، Fubini's Theorem،
+
+142
+00:10:01,100 --> 00:10:04,240
+Fubini's Theorem، الـ First Form، الـ Form الأول،
+
+143
+00:10:04,240 --> 00:10:08,180
+الـ Form الأول هذا اللي هو بتعامل مع… أن F(X,
+
+144
+00:10:08,180 --> 00:10:10,880
+Y) continuous function، على… throughout the
+
+145
+00:10:10,880 --> 00:10:14,860
+rectangle R، على الـ region R، اللي من A لعند B، ومن Y
+
+146
+00:10:14,860 --> 00:10:18,620
+من C لعند D، يعني هذا الـ rectangle سهل التكامل
+
+147
+00:10:18,620 --> 00:10:21,300
+عليه على طول، في الـ Fubini's Theorem بتقول لك مدام
+
+148
+00:10:21,300 --> 00:10:24,100
+تلقى F continuous، بيصير double integration F(X,Y)
+
+149
+00:10:24,100 --> 00:10:27,380
+dA over الـ rectangle R، ما هي؟ اللي عبارة عن
+
+150
+00:10:27,380 --> 00:10:32,220
+iterated integral، من A لـ B ومن C لـ D، F(X, Y) dx dy،
+
+151
+00:10:32,220 --> 00:10:37,600
+الـ X من A لعند B، والـ Y من عند مين؟ C لعند D، لأنه
+
+152
+00:10:37,600 --> 00:10:41,630
+كل الدنيا حلوة وتمام التمام، لإنه إيش عنده؟ اللي هي
+
+153
+00:10:41,630 --> 00:10:46,070
+الحدود عبارة عن rectangle، الحدود، فعشان هيك سهل
+
+154
+00:10:46,070 --> 00:10:49,830
+الأمر خالص،  الـ integration نفسه ممكن نعمله
+
+155
+00:10:49,830 --> 00:10:52,530
+reverse integration، برضه بتضمن النظرية، بتقول لك آه
+
+156
+00:10:52,530 --> 00:10:55,490
+صح، يعني بدل ما نكامل بالنسبة لـ X، بنكامل بالنسبة لـ
+
+157
+00:10:55,490 --> 00:10:59,510
+مين؟ في الأول Y، dy dx، الـ Y من عند C لعند D، والـ
+
+158
+00:10:59,510 --> 00:11:04,030
+X من عند A لـ B، إذا هذه Fubini's Theorem بتشرح لنا كيف
+
+159
+00:11:04,030 --> 00:11:08,050
+نحسب الـ double integral، اللي هو على rectangle لـ
+
+160
+00:11:08,050 --> 00:11:11,510
+continuous function، نيجي ناخد مثال عملي، example،
+
+161
+00:11:11,510 --> 00:11:14,510
+Calculate the double integral، للـ R، f(x, y) dA،
+
+162
+00:11:14,510 --> 00:11:18,130
+إيه؟  four، اللي هنكامله، هو الـ R هو عبارة عن الـ
+
+163
+00:11:18,130 --> 00:11:21,090
+rectangle من صفر لعند اثنين، هي من عند صفر لعند
+
+164
+00:11:21,090 --> 00:11:27,230
+اثنين، هي الـ X من صفر لعند اثنين، هي الـ X هيها، والـ
+
+165
+00:11:27,230 --> 00:11:31,470
+Y من عند مين؟ هي من صفر لعند اثنين، الـ Y، والـ X من
+
+166
+00:11:31,470 --> 00:11:34,850
+عند… ناقص واحد لعند مين؟ لعند… آسف، الـ Y من عند
+
+167
+00:11:34,850 --> 00:11:38,070
+ناقص واحد لعند واحد، والـ X من عند صفر لعند مين؟
+
+168
+00:11:38,070 -->
+
+223
+00:15:35,810 --> 00:15:40,450
+هذا وبين اللي هو الـ function اللي أمامي قلنا سهل
+
+224
+00:15:40,450 --> 00:15:42,450
+على طول الـ volume هو عبارة عن الـ double
+
+225
+00:15:42,450 --> 00:15:47,190
+integration لأ اللي هو الدالة السطح العلوي لأن هذا
+
+226
+00:15:47,190 --> 00:15:50,510
+إيهاش اللي هو العلبنان كامل الـ Double by Fubini's
+
+227
+00:15:50,510 --> 00:15:54,710
+theorem الـ Volume يصير عنده يساوي لأن الـ Volume
+
+228
+00:15:54,710 --> 00:15:59,410
+is given by the double integral اللي هو هذا f of x
+
+229
+00:15:59,410 --> 00:16:03,010
+y دي ايه يساوي ال integration كامل بالنسبة لـ y و
+
+230
+00:16:03,010 --> 00:16:06,010
+بالنسبة لـ x انت حر خلينا نكمل بالنسبة لـ y المرة
+
+231
+00:16:06,010 --> 00:16:09,730
+هذه إيش هي؟ اللي هي عشرة زائد x تربيع زائد 3 y تربيع
+
+232
+00:16:09,730 --> 00:16:14,640
+دي y دي x هذه الدالة الآن من وين لوين يا جماعة؟ من
+
+233
+00:16:14,640 --> 00:16:19,340
+عند 0 ل 2 قيمة الـ Y هي أعطيني يعني من 0 ل 2 قيمة
+
+234
+00:16:19,340 --> 00:16:24,180
+الـ Y و x من وين؟ من 0 ل عند 1، أمور rectangle، استهلة
+
+235
+00:16:24,180 --> 00:16:29,040
+على طول اللي هو بالعوض هنا الآن Dy بيصير عندنا D
+
+236
+00:16:29,040 --> 00:16:33,040
+اللي هو تكامل هذا يصير 10Y زائد Y x تربيع زائد Y
+
+237
+00:16:33,040 --> 00:16:37,120
+تكعيب على 3، وفي عندنا 3 بيروحوا مع بعض، بتظهر الـ
+
+238
+00:16:37,120 --> 00:16:40,620
+Y تكعيب من صفر إلى وين؟ إلى اتنين، عوض عن اتنين بيصير
+
+239
+00:16:40,620 --> 00:16:44,620
+عشرين زائد اتنين X تربيع زائد ثمانية، والصفر طبعاً
+
+240
+00:16:44,620 --> 00:16:48,460
+بتعطي هذه كلها صفر، لان DX بالدعاقة كاملة بالنسبة لـ
+
+241
+00:16:48,460 --> 00:16:51,780
+X كاملة هذه بالنسبة لـ X بيصير عشرين X و 2 X
+
+242
+00:16:51,780 --> 00:16:58,040
+تربيع بالنسبة لـ X بيصير ثلثين X تكعيب زائد ثمانية
+
+243
+00:16:58,040 --> 00:17:02,620
+X من صفر لعند مين؟ عند واحد، ثلثين X + جمعة تكعيب
+
+244
+00:17:02,890 --> 00:17:07,330
+طيب الآن عوض بواحد بيصير عشرين زائد ثلثين زائد
+
+245
+00:17:07,330 --> 00:17:10,750
+الثمانية والصفر طبعاً بيعملها دي كلها ايش؟ صفر
+
+246
+00:17:10,750 --> 00:17:14,470
+فبتظل اللي هي الأمامي هذه، لو جمعتها بتطلع ثمانين
+
+247
+00:17:14,470 --> 00:17:17,550
+و عشرين و ثلثين، أو ستة و ثلثين،  ستة وثمانين على
+
+248
+00:17:17,550 --> 00:17:20,490
+ثلاثة، وهيك بيكون أوجدنا الـ volume بواسطة double
+
+249
+00:17:20,490 --> 00:17:25,410
+integral وهيك بتكون المحاضرة انتهت وإلى لقايا آخر
+
+250
+00:17:25,410 --> 00:17:27,990
+والسلام عليكم ورحمة الله وبركاته
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mNaq2Sskss8_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mNaq2Sskss8_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..40515f9bc880cc3901a2fa774664a3e73838df75
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mNaq2Sskss8_raw.json
@@ -0,0 +1 @@
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0.97607421875}, {"start": 664.03, "end": 664.47, "word": " نودد", "probability": 0.8224283854166666}, {"start": 664.47, "end": 664.61, "word": " ال", "probability": 0.83056640625}, {"start": 664.61, "end": 664.85, "word": " double", "probability": 0.775390625}, {"start": 664.85, "end": 665.53, "word": " integral", "probability": 0.79443359375}], "temperature": 1.0}, {"id": 30, "seek": 68767, "start": 665.87, "end": 687.67, "text": "اللي هو على rectangle لـ continuous function نيجي ناخد مثال عملي example Calculate the double integral للـ R f of x y دي ايه؟ four اللي هنقرميل هو الـ R هو عبارة عن ال rectangle من صفر لعندي اتنين هي من عن صفر لعندي اتنين هي ال X من صفر لعندي اتنين هي ال X هيها و ال Y", "tokens": [6027, 20292, 31439, 15844, 21930, 5296, 39184, 10957, 2445, 8717, 1829, 7435, 1829, 8717, 47283, 3215, 50113, 6027, 6225, 2304, 20292, 1365, 3511, 2444, 473, 264, 3834, 11573, 24976, 39184, 497, 283, 295, 2031, 288, 11778, 1829, 1975, 1829, 3224, 22807, 1451, 13672, 1829, 8032, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mNaq2Sskss8_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mNaq2Sskss8_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..57916eba8b9a1fa1a6fc5790b51faee50d731a48
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mNaq2Sskss8_raw.srt
@@ -0,0 +1,1000 @@
+1
+00:00:00,690 --> 00:00:06,570
+بسم الله الرحمن الرحيم هذه هي المحاضرة رقم عشرين
+
+2
+00:00:06,570 --> 00:00:13,030
+وهي في مساق طبعا تفاضل وتكامل جيم وهي المحاضرة
+
+3
+00:00:13,030 --> 00:00:17,510
+الأولى في شبتر خمسة عشر هنتحدث اليوم عن خمسة عشر
+
+4
+00:00:17,510 --> 00:00:21,530
+واحد عنوان شبتر خمسة عشر اللي هو ملقب ال integrals
+
+5
+00:00:21,770 --> 00:00:27,730
+أو التكاملات المتعددة المقصود فيها الـ Double
+
+6
+00:00:27,730 --> 00:00:31,770
+Integral أو Triple Integral سنشاهدها قدامًا إن شاء
+
+7
+00:00:31,770 --> 00:00:34,970
+الله خمس عشر واحد اللي هو الـ Double and iterated
+
+8
+00:00:34,970 --> 00:00:38,310
+integrals over rectangles اللي بدنا ناخد الـ
+
+9
+00:00:38,310 --> 00:00:44,270
+Double Integralعالمين على ركتانجل على أسهل الأمور
+
+10
+00:00:44,270 --> 00:00:47,230
+وان شاء الله تدريجياً نروح على قطعة غير الـ
+
+11
+00:00:47,230 --> 00:00:49,750
+rectangles وان شاء الله بتكون الأمور برضه سهلة
+
+12
+00:00:49,750 --> 00:00:53,350
+الآن الـ Double Integral زي ما حكينا زمان في
+
+13
+00:00:53,350 --> 00:00:57,990
+التقفاض الألف عن اللي هو ال integration ال single
+
+14
+00:00:57,990 --> 00:01:00,690
+integral integration عرفناها على أساس اللي هو
+
+15
+00:01:00,690 --> 00:01:06,130
+limit sumLimit remand sum يعني نهاية مجموع remand
+
+16
+00:01:06,130 --> 00:01:10,090
+كنا نوجد مجموع remand ومن ثم نوجد اللي هو ال limit
+
+17
+00:01:10,090 --> 00:01:13,250
+له فكان لو كان ال limit موجود بنقول ان ال function
+
+18
+00:01:13,250 --> 00:01:16,490
+is integrable وقيمة ال limit هي قيمة ال
+
+19
+00:01:16,490 --> 00:01:20,230
+integration الآن الكلام شبيه وشبيه جدا اللي هو
+
+20
+00:01:20,230 --> 00:01:23,650
+برضه هنستعمل اللي هو remand sum في اللي هو تعريف
+
+21
+00:01:23,650 --> 00:01:26,560
+ال double integrantخلّينا نشوف إحنا الـ
+
+22
+00:01:26,560 --> 00:01:29,280
+integration عندنا over مين الآن over rectangles
+
+23
+00:01:29,280 --> 00:01:33,620
+يعني لو كان في عند rectangle X بين A وB هاي X بين
+
+24
+00:01:33,620 --> 00:01:38,060
+A وB يعني الـ X من A لعند B هي الـ X من عند A لـ B
+
+25
+00:01:38,060 --> 00:01:41,900
+و الـ Y من عند C لعند D فبصير عندنا المنطقة هذه
+
+26
+00:01:41,900 --> 00:01:45,260
+عبارة عن rectangle هذه المنطقة اللي عليها بدنا
+
+27
+00:01:45,260 --> 00:01:48,680
+نعمل الـ double integralعشان نحرف الـ Double
+
+28
+00:01:48,680 --> 00:01:51,860
+Integral خلّينا نشوف أول إشي الـ Show Remansum على
+
+29
+00:01:51,860 --> 00:01:55,060
+هذه زي ما كنا نعمل قبل بالظبط كنا زمان نجزّق اللي
+
+30
+00:01:55,060 --> 00:02:02,260
+هو إيش الـ X-Axis X0 X1 X2 X3 XK لعند XN ونجزّقها
+
+31
+00:02:02,260 --> 00:02:05,340
+اللي هي كانت بس احنا بنخمل على الخط الآن بنكامل
+
+32
+00:02:05,340 --> 00:02:10,700
+على منطقة هنجزّق Xو هنجزّق من Y يعني هنجزّق X اللي
+
+33
+00:02:10,700 --> 00:02:14,520
+بيكون كامل بالنسبة له و هنجزّق Y لليوم برضه بيكون
+
+34
+00:02:14,520 --> 00:02:17,800
+كامل بالنسبة لها لأنه عايش الآن ع متغيرين إذا
+
+35
+00:02:17,800 --> 00:02:23,100
+جزّقنا هذه X0 X1 X2 X3 إلى آخره لعند XN و هنا X0
+
+36
+00:02:23,100 --> 00:02:29,260
+X1 X2 X3 لعند XN على إلى آخره الآن فبصير عند
+
+37
+00:02:29,260 --> 00:02:33,940
+التجزئة هذه بناخد المستقيمات هذه الموازية لل Y
+
+38
+00:02:33,940 --> 00:02:37,230
+Axis و بناخد المستقيمات الموازية لل X Axisحسب
+
+39
+00:02:37,230 --> 00:02:41,750
+التقسمة اللي قلناها فبتعمل لإيه شمالها تجزّق هذا
+
+40
+00:02:41,750 --> 00:02:45,370
+الـ rectangle الكبير إلى اللي هي sub rectangles
+
+41
+00:02:45,370 --> 00:02:48,690
+صغيرة الآن هذه الـ rectangles طبعا هذه اللي هي
+
+42
+00:02:48,690 --> 00:02:51,170
+بتكون اللي هي جزء من domain main الـ function اللي
+
+43
+00:02:51,170 --> 00:02:56,170
+بنشتغل عليها اللي هي F of X Y فالآن بنيجي بناخد زي
+
+44
+00:02:56,170 --> 00:02:59,730
+ما كنا ناخد نقطة عشوائية أيام اللي هي ال
+
+45
+00:02:59,730 --> 00:03:03,010
+integration العادي أو ال single integral بناخد
+
+46
+00:03:03,010 --> 00:03:08,090
+نقطة عشوائية في داخل المربع هذاهذا المربح سميته
+
+47
+00:03:08,090 --> 00:03:15,350
+طوله delta xk وعرضه delta yk من xk-1 لعندي الـ xk
+
+48
+00:03:15,350 --> 00:03:21,810
+ومن yk-1 لعندي الـ yk هذا المستطيل اللي أمامي اللي
+
+49
+00:03:21,810 --> 00:03:28,000
+هو باخد نقطة عشوائية بسميها xk ykالأن باخد قيمة
+
+50
+00:03:28,000 --> 00:03:32,560
+الـ Function عند هذه النقطة F of XK YK مضروبة في
+
+51
+00:03:32,560 --> 00:03:36,940
+مين؟ في المساحة هذه التي هي Delta XK في Delta YK
+
+52
+00:03:36,940 --> 00:03:42,100
+اللي سمناها إيش اسمها Delta AK وبعدي هذه أخدت اللي
+
+53
+00:03:42,100 --> 00:03:46,980
+هو قيمة الـ Function F of XK YKعليها وضربت في مين
+
+54
+00:03:46,980 --> 00:03:51,680
+في المساحة اللي عندي هذه و بعدين بديت أجمع على كل
+
+55
+00:03:51,680 --> 00:03:56,540
+المناطق اللي جزقتها هذه فبصير هذا مجموعه هو من
+
+56
+00:03:56,540 --> 00:03:59,960
+كيلة عند واحد اللي بنسميه sun اللي بنسميه رمان
+
+57
+00:04:06,120 --> 00:04:10,360
+بتكون عبارة عن نقطة واحدة فبصير الجمع عالمين على
+
+58
+00:04:10,360 --> 00:04:15,920
+الخط هذا وبينما هذه مش مساحة بتكون دلتا XK هذا
+
+59
+00:04:15,920 --> 00:04:20,260
+زمانأما الآن زي ما قلنا اللي هو SN هو عبارة عن
+
+60
+00:04:20,260 --> 00:04:24,120
+Delta AK اللي هو المستطيل الصغير هذا المجزّق
+
+61
+00:04:24,120 --> 00:04:29,180
+مضروبة في F of XK وYK اللي هي النقطة العشوائية
+
+62
+00:04:29,180 --> 00:04:32,880
+اللي أخدناها هذه هذا طبعا مربع أو مستطيل نموذجي
+
+63
+00:04:32,880 --> 00:04:37,060
+بجمحن كل هنا دولة عساس أن أخلص على كل المنطقة هذه
+
+64
+00:04:37,060 --> 00:04:41,520
+فبصير مجموح كامن عند واحد لعند N F of XK YK Delta
+
+65
+00:04:41,520 --> 00:04:45,670
+XK هو عبارة عن الـ Remainsهذا الـ Remain Sum اللي
+
+66
+00:04:45,670 --> 00:04:49,510
+أوددناه لو ادينا أخدناله الـ Limit As N Goes To
+
+67
+00:04:49,510 --> 00:04:52,790
+Infinity على أساس أنه بتصير اللي هي لما N تروح
+
+68
+00:04:52,790 --> 00:04:56,170
+لمالها نقايا التفزيقات هذه بتصغر بيصير صغيرة صغيرة
+
+69
+00:04:56,170 --> 00:04:59,770
+صغيرة صغيرة بتصير تتراكم جان بعض وكأنه بتصير إيه
+
+70
+00:04:59,770 --> 00:05:03,970
+شمالها أنه خطوط خطوط خطوط عنده بس اللي هو الخط
+
+71
+00:05:03,970 --> 00:05:08,280
+بمثل قيمة دالها لما نجمح لبعضاللي هو وكانت الـ
+
+72
+00:05:08,280 --> 00:05:12,500
+Limit exist هو اللي بدنا نسميه الـ Double Integral
+
+73
+00:05:12,500 --> 00:05:17,540
+اللي هو f of x y dA over R إذا لو الـ Limit هذا
+
+74
+00:05:17,540 --> 00:05:20,960
+exist على demand sum هو بدنا نسميه اللي هو الـ
+
+75
+00:05:20,960 --> 00:05:26,300
+Double Integration للـ f of x y dA أو f of x y dx
+
+76
+00:05:26,300 --> 00:05:37,220
+dyالان نجي الـ double integral as a volume نربط
+
+77
+00:05:37,220 --> 00:05:42,340
+الـ double integration بالـ volumeلأن لو المنطقة
+
+78
+00:05:42,340 --> 00:05:45,700
+اللي بدنا نكامل عليها هي .. هي عندنا Surface أول
+
+79
+00:05:45,700 --> 00:05:50,120
+شيء هي Surface و Z بيسوا F of X و Y هذا ال Surface
+
+80
+00:05:50,120 --> 00:05:53,860
+الآن المنطقة اللي بدي أشتغل عليها بس اللي هي ال
+
+81
+00:05:53,860 --> 00:05:58,300
+rectangle R هذا بدي أشوف حجم الجسم هذا اللي هو
+
+82
+00:05:58,300 --> 00:06:01,940
+اللي أمامي شوفوا يا جماعة باجيب نفس اللي قملته قبل
+
+83
+00:06:01,940 --> 00:06:08,540
+شوية بجزء هذه اللي هي ال Y Y1 Y2 Y3 و YNوهذا Y1
+
+84
+00:06:08,540 --> 00:06:14,400
+وY2 وهنا X1 وX2 لدي Xn وبناخد فترة من نموذجي هذا
+
+85
+00:06:14,400 --> 00:06:18,820
+اللي هو المملي اللي هو ال rectangle اللي هي جزء من
+
+86
+00:06:18,820 --> 00:06:21,860
+ال domain اللي عليه بده حسب ال volume للجسم الآن
+
+87
+00:06:21,860 --> 00:06:24,800
+ال volume للجسم بيصير اللي هو الشكل اللي أمامي هذا
+
+88
+00:06:24,800 --> 00:06:27,560
+لأنه قيمة ال function هنا هي قيمة ال function هنا
+
+89
+00:06:27,560 --> 00:06:31,380
+هي هي السطح اللي عندي فالان السطح المكون اللي
+
+90
+00:06:31,380 --> 00:06:36,740
+قاعدته هذي وسقفه هذا ال volume له بده شوف كيف بده
+
+91
+00:06:36,740 --> 00:06:42,140
+حسمهشوفوا كيف الآن الـ Volume هذا لو جيه الحساب
+
+92
+00:06:42,140 --> 00:06:46,680
+أخدت اللي هو الآن فترة نموذجية أو أو مهانة فترة
+
+93
+00:06:46,680 --> 00:06:51,040
+نموذجية هتعمل ليهاش مربع أو مستطين نموذجي طوله
+
+94
+00:06:51,040 --> 00:06:57,420
+Delta XK وعرضه Delta YK مساحته Delta XK في Delta
+
+95
+00:06:57,420 --> 00:07:01,860
+YK بيصير حجم اللي هي اللي على هذه المنطقة هو عبارة
+
+96
+00:07:01,860 --> 00:07:04,220
+عن اللي هي المساحة هذا في الارتفاع لكن هذا
+
+97
+00:07:04,220 --> 00:07:07,300
+الارتفاع شايفينه متغيرهذا ايش انا باخد نقطة
+
+98
+00:07:07,300 --> 00:07:11,860
+عشوائية و اخدها في المركز XK و YK و بيتيجي و باخد
+
+99
+00:07:11,860 --> 00:07:16,780
+F of XK و YK قيمتها الانصار عندي قيمة F of XK و YK
+
+100
+00:07:16,780 --> 00:07:20,080
+اللي هو المثل الارتفاع هذا مضروبة في المساحة هذه
+
+101
+00:07:20,080 --> 00:07:23,620
+بيطلع اللي هو حجم اللي هو ايش متوازن المستطيلات
+
+102
+00:07:23,620 --> 00:07:26,500
+اللي أمامي هذا بس هو مش بالظبط متوازن المستطيلات
+
+103
+00:07:26,500 --> 00:07:30,440
+بيطلع شوية فوق أو بنزل شوية تحت فهذا تقريبا الآن
+
+104
+00:07:30,780 --> 00:07:34,540
+الآن هذا التقريبي لما يكون العدد جليل مثلا لما انت
+
+105
+00:07:34,540 --> 00:07:37,940
+ساوى 16 لاحظوا بتلاقي فيه أجزاء مفقودة كثير أو
+
+106
+00:07:37,940 --> 00:07:42,060
+أجزاء زايدة كثير فبصير الآن اللي هي ال .. ال .. ال
+
+107
+00:07:42,060 --> 00:07:45,900
+.. هذا بيقرب ال volume اللي هنا لكن بيقرضوش بشكل
+
+108
+00:07:45,900 --> 00:07:49,900
+كامل يعني بيصير فيه نسبة خطأ الآن لو زدنا عددها
+
+109
+00:07:49,900 --> 00:07:53,400
+اللي هو نسبة الخطأ إيش مالة بيصير هذا قريب من هذا
+
+110
+00:07:53,400 --> 00:07:57,000
+لكن نسبة الخطأ برضه إيش موجودة لكن بتقل عن هذه لأن
+
+111
+00:07:57,000 --> 00:08:00,950
+المفقود هنا أكترلما نيجي هنا مثلا اللي هو زدنا
+
+112
+00:08:00,950 --> 00:08:04,650
+العدد صار n من 256 بيصير يسايفين اللي هي نسبة
+
+113
+00:08:04,650 --> 00:08:08,370
+المفقود قليلة جدا فما بالك لو أخدنا n ب1000 ولا n
+
+114
+00:08:08,370 --> 00:08:12,210
+بمليون ولا n ب100 مليون هيصير نسبة المفقود قليل
+
+115
+00:08:12,210 --> 00:08:16,650
+جدا الان as n goes to infinity وكأني باخد ايش انا
+
+116
+00:08:16,650 --> 00:08:21,270
+هذا يعني صرت ايش اخد انا الآن هذا واللي جنبه واللي
+
+117
+00:08:21,270 --> 00:08:24,700
+جنبه واللي جنبه زي ما أخدناهم هدولة مجموحينمجموحًا
+
+118
+00:08:24,700 --> 00:08:27,340
+هذا اللي هو حجم هذا زائد حجم هذا زائد حجم هذا على
+
+119
+00:08:27,340 --> 00:08:32,440
+كل المنطقة يعني بيصير عند F of XK YK Delta AK K من
+
+120
+00:08:32,440 --> 00:08:36,020
+عند واحد لعند ان هذول مجموحًا هو عبارة عن مجموع
+
+121
+00:08:36,020 --> 00:08:39,020
+الحجومات هذه هذه مع هذه مع هذه لأن أنا مشيت على
+
+122
+00:08:39,020 --> 00:08:42,460
+مين على كل ال rectangles الصغيرة الصغيرة الصغيرة
+
+123
+00:08:42,460 --> 00:08:46,060
+لما انغطيت ال rectangle الكبير R كلهولكن هناك جزء
+
+124
+00:08:46,060 --> 00:08:51,540
+مقطوع أو المفقود يكون هنا مثلا لما العدد قليل كثير
+
+125
+00:08:51,540 --> 00:08:57,440
+هنا أجل هنا أجل لما N تكبر أجل لما N تكبر أجل As N
+
+126
+00:08:57,440 --> 00:09:01,660
+goes to infinity يصبح هذا السطح هنا بالظبط هو
+
+127
+00:09:01,660 --> 00:09:05,980
+السطح اللي فوق يصبح عند الـ Limit للـ Integral للـ
+
+128
+00:09:05,980 --> 00:09:10,630
+Sn هذا As N goes to infinityبساوة بالظبط هو الـ
+
+129
+00:09:10,630 --> 00:09:14,050
+volume في حالة limit exist طبعا و احنا ماخدين
+
+130
+00:09:14,050 --> 00:09:17,810
+الدالة continuous و حلوة و كده و كده متماشى نضمن
+
+131
+00:09:17,810 --> 00:09:21,630
+انها exist فالان limit as n لما انت تروح لملنها يا
+
+132
+00:09:21,630 --> 00:09:25,710
+مدام موجود هيعطيك بالظبط ال volume و هو طبعا limit
+
+133
+00:09:25,710 --> 00:09:29,110
+as n زي ما عرفناها قبل شوية هو عبارة عن ال double
+
+134
+00:09:29,110 --> 00:09:34,010
+integral f of x yDNA Over Rectangle Mين Over
+
+135
+00:09:34,010 --> 00:09:38,070
+Rectangle R إذا الـ Volume بساوية Limit استخدمتها
+
+136
+00:09:38,070 --> 00:09:40,870
+طبعاً في حالة إن ده اللي أخدناها إيه شمالها Z
+
+137
+00:09:40,870 --> 00:09:44,570
+بساوية F of XY is a Positive دايما Positive على
+
+138
+00:09:44,570 --> 00:09:48,600
+المنطقة هذه عشان هي أعطت Volume بالظبطنأتي الآن
+
+139
+00:09:48,600 --> 00:09:51,600
+لدينا نظرية اسمها فبونيز فيورم for calculating
+
+140
+00:09:51,600 --> 00:09:54,240
+double integrals لأننا نريد أن نجد الـ Double
+
+141
+00:09:54,240 --> 00:09:56,820
+Integral لذلك نحتاج نظرية تساعدنا في إيجاده
+
+142
+00:09:56,820 --> 00:10:01,100
+النظرية سهلة ومريحة لكي نرى ما هي فيورم واحد
+
+143
+00:10:01,100 --> 00:10:04,240
+فيوبونيز فيورم الـ First Form الـ Form Ash الأول
+
+144
+00:10:04,240 --> 00:10:08,180
+الـ Form الأول هذا اللي هو بتعامل معنين أن F of X
+
+145
+00:10:08,180 --> 00:10:10,880
+Y continuous function على Ash through the
+
+146
+00:10:10,880 --> 00:10:14,860
+rectangle R على ال region Rاللي من A لعند B ومن Y
+
+147
+00:10:14,860 --> 00:10:18,620
+من C لعند D يعني هذا الـ rectangle سهل التكامل
+
+148
+00:10:18,620 --> 00:10:21,300
+عليه على طول في الـ Bode's theorem بتقول لك مدام
+
+149
+00:10:21,300 --> 00:10:24,100
+تلقى F continuous بيصير double integration F of XY
+
+150
+00:10:24,100 --> 00:10:27,380
+دي A over ال rectangle R ما هي اللي عبارة عن
+
+151
+00:10:27,380 --> 00:10:32,220
+iterated integral من A لB ومن C لD F of XY دي X دي
+
+152
+00:10:32,220 --> 00:10:37,600
+Y ال X من A لعند B و ال Y من عند مين C لعندي لأنه
+
+153
+00:10:37,600 --> 00:10:41,630
+كل الدنيا حلوة وتمام التمام لإنه إيش عندهاللي هي
+
+154
+00:10:41,630 --> 00:10:46,070
+الخطوط عبارة عن rectangle الحدود فعشان هيك سهل
+
+155
+00:10:46,070 --> 00:10:49,830
+الأمر خالص Y ساوي ال integration نفسه ممكن نعمله
+
+156
+00:10:49,830 --> 00:10:52,530
+reverse integration برضه بتضمن النظرية بتقولك أه
+
+157
+00:10:52,530 --> 00:10:55,490
+صح يعني بدل ما كامل بالنسبة ل X بنكامل بالنسبة ل
+
+158
+00:10:55,490 --> 00:10:59,510
+مين في الأول Y دي Y دي X ال Y من عند C لعندي و ال
+
+159
+00:10:59,510 --> 00:11:04,030
+X من عند A لB إذا هذه فبريس فيورم بتشرعلنا كيف
+
+160
+00:11:04,030 --> 00:11:08,050
+نودد ال double integralاللي هو على rectangle لـ
+
+161
+00:11:08,050 --> 00:11:11,510
+continuous function نيجي ناخد مثال عملي example
+
+162
+00:11:11,510 --> 00:11:14,510
+Calculate the double integral للـ R f of x y دي
+
+163
+00:11:14,510 --> 00:11:18,130
+ايه؟ four اللي هنقرميل هو الـ R هو عبارة عن ال
+
+164
+00:11:18,130 --> 00:11:21,090
+rectangle من صفر لعندي اتنين هي من عن صفر لعندي
+
+165
+00:11:21,090 --> 00:11:27,230
+اتنين هي ال X من صفر لعندي اتنين هي ال X هيها و ال
+
+166
+00:11:27,230 --> 00:11:31,470
+Yمن عند مين هي من سفر لعندي اتنين الـ Y والـ X من
+
+167
+00:11:31,470 --> 00:11:34,850
+عند ناقص واحد لعند مين لعند .. آسف الـ Y من عند
+
+168
+00:11:34,850 --> 00:11:38,070
+ناقص واحد لعند واحد والـ X من عند سفر لعند مين
+
+169
+00:11:38,070 --> 00:11:41,410
+اتنين إذن سفر من عند اتنين والـ Y من عند ناقص واحد
+
+170
+00:11:41,410 --> 00:11:45,830
+لعند واحد هذه الآن .. هذا ال rectangle اللي عليه
+
+171
+00:11:45,830 --> 00:11:48,470
+اللي هو الجزء من ال domain اللي بدنا نكامل عليه
+
+172
+00:11:48,470 --> 00:11:53,610
+مين الدالة مية ناقص ستة X تربيع Y هذه الدالة هي
+
+173
+00:11:54,110 --> 00:11:58,210
+اللي هي Z بساومية نقص 6 X تربيع Y هذا اللي هو
+
+174
+00:11:58,210 --> 00:12:02,530
+الشكل اللي أمامي أي إن كان الشكل هيو عندي على
+
+175
+00:12:02,530 --> 00:12:05,230
+المنطقة هذه بدي أعمله الآن أو جدله الـ Double
+
+176
+00:12:05,230 --> 00:12:10,770
+Integral هو في الواقع هيعمل لي ال volume للجسم هذا
+
+177
+00:12:10,770 --> 00:12:14,960
+اللي سطحه هذا وهذا أيه قاعدتههذا الـ Volume الـ
+
+178
+00:12:14,960 --> 00:12:17,960
+Volume هيساوي الـ Double Integration فبيفوبنيز
+
+179
+00:12:17,960 --> 00:12:21,680
+فيرم الـ Double Integral f of x y dA over R هو
+
+180
+00:12:21,680 --> 00:12:24,760
+هيعمل لي الـ Volume لأن الـ Function هنا عبارة عن
+
+181
+00:12:24,760 --> 00:12:28,920
+Positive في هذه المنطقة اللي مذكورة ويساوي حسب
+
+182
+00:12:28,920 --> 00:12:32,560
+النظرية إيش هيساوي الـ Integration dx dy بدك تكامل
+
+183
+00:12:32,560 --> 00:12:37,700
+حاضر dx dy إذا x من أين لأين من 0 لعين 2 وY من أين
+
+184
+00:12:37,700 --> 00:12:43,390
+لأين من ناقص 1 لعين 1ما هو 100-6X³Y هو الدالة التي
+
+185
+00:12:43,390 --> 00:12:46,670
+نريد ان نكملها Y ساوى ما سنكمله بالنسبة هو الشيء
+
+186
+00:12:46,670 --> 00:12:52,490
+الميني لـ X يصبح 100X ناقص 100X ناقص تلتين اتنين
+
+187
+00:12:52,490 --> 00:12:57,330
+يصبح 6X تكعيب على تلاتة يعني 2X تكعيب في Y ثابتة
+
+188
+00:12:57,330 --> 00:13:07,580
+بالنسبة للـ Xمن 0 لمن ل 2 نعوض من X ل X عوض من X
+
+189
+00:13:07,580 --> 00:13:15,660
+بيصير 200 ناقص 2 في 8 ب16Y DY وهذا السفر طبعا تعمل
+
+190
+00:13:15,660 --> 00:13:20,380
+لها دولة سفر لإن بتعمل ال integration لمن ل Y لمن
+
+191
+00:13:20,380 --> 00:13:25,580
+و X بالنسبة لها في هذه الحالة ثابتة لو موجودةالان
+
+192
+00:13:25,580 --> 00:13:31,080
+بيصير عندى 200 الان 200 الان ناقص 16Y دي Y من ناقص
+
+193
+00:13:31,080 --> 00:13:35,240
+واحد لعند مين لعند الواحد بكمل بالنسبة ل Y بيصير
+
+194
+00:13:35,240 --> 00:13:40,440
+200Y ناقص 8Y تربيع الان طبعا عارفين كيف من ناقص
+
+195
+00:13:40,440 --> 00:13:45,430
+واحد لعند الواحد وبنعوض بتطلع 200نقص تمانية لإنه
+
+196
+00:13:45,430 --> 00:13:48,970
+يبقى Y بواحد هايها التانية بنقص واحد بيصير ناقص
+
+197
+00:13:48,970 --> 00:13:52,750
+متين ناقص تمانية اللي هو بتتجمحل مع بعض بيصير
+
+198
+00:13:52,750 --> 00:13:56,150
+أربعمية هذه وهذه بتروح مع هذه بيصير أربعمية إذا أن
+
+199
+00:13:56,150 --> 00:13:59,210
+واحد قال لأ بدي أسمع بدي أعكس التكامل أشوف بطلع زي
+
+200
+00:13:59,210 --> 00:14:02,950
+ما بتقوله ولا لأ بقوله صحتين يتوقعان الله ويلا
+
+201
+00:14:02,950 --> 00:14:06,970
+نعكس مع بعضلأن عكسنا التكامل صار من سفر لعندي 2
+
+202
+00:14:06,970 --> 00:14:11,310
+ونقص لعندي 1 لعندي 1 والآن dy by dx إذا بنفضل أولا
+
+203
+00:14:11,310 --> 00:14:14,270
+بالنسبة ل y ثم بالكامل بالنسبة ل y ثم بالكامل
+
+204
+00:14:14,270 --> 00:14:19,890
+بالنسبة ل x سريع لأنه الكلام سهل 100y ناقص 6x
+
+205
+00:14:19,890 --> 00:14:24,190
+تربيع y تربيع على 2 يعني 3 y تربيع x تربيع من ناقص
+
+206
+00:14:24,190 --> 00:14:29,970
+1 لعندي 1 بنعود عن هذه بتطلع عبارة عن 100بعود عن Y
+
+207
+00:14:29,970 --> 00:14:35,270
+بـ 1 ناقص تلاتة X تربيع ناقص Y بناقص واحد ناقص مية
+
+208
+00:14:35,270 --> 00:14:39,070
+اللي هي ناقص تلاتة X تربيع لأنه لما تتربع بيصير
+
+209
+00:14:39,070 --> 00:14:43,030
+ناقص واحد واحد زي واحد زي واحد الان هذا بتروح هادي
+
+210
+00:14:43,030 --> 00:14:47,010
+مع هادي بيظل عنده إيه إيش مية وزائد مية 200 من 0
+
+211
+00:14:47,010 --> 00:14:52,870
+لعند 2 DX فاضلها كاملها آسف بيصير 200 X من 0 ل2 و
+
+212
+00:14:52,870 --> 00:14:58,170
+اللي هي بيصير 400 فعلا القمتين نفس الاشياءنأخذ
+
+213
+00:14:58,170 --> 00:15:01,590
+ونعزز الفكرة بمثال آخر Find the volume of the
+
+214
+00:15:01,590 --> 00:15:04,530
+region bounded above by the elliptical paraboloid
+
+215
+00:15:04,530 --> 00:15:07,890
+Z بالساوية عشرة زي X تربيع زي تلتة Y تربيع بقولي
+
+216
+00:15:07,890 --> 00:15:13,450
+أوجد اللي هو ال volume للجسم المكون من فوق بهيه
+
+217
+00:15:13,450 --> 00:15:17,910
+ماعطيكي وصف لكي Z بالساوية عشرة زي X تربيع زي تلتة
+
+218
+00:15:17,910 --> 00:15:21,090
+Y تربيع بقولي ال volume من فوق هيه اللي الجسم اللي
+
+219
+00:15:21,090 --> 00:15:25,230
+بدأ نوجد اللي هو ال volume له هيهاللي هي السطح
+
+220
+00:15:25,230 --> 00:15:28,310
+العلوي and below by the rectangle اللي هو X بين
+
+221
+00:15:28,310 --> 00:15:32,350
+السفر وبين الواحد وY بين السفر وبين اتنين إذا بدنا
+
+222
+00:15:32,350 --> 00:15:35,810
+نوجد الحجم المنطقة هذه المحصورة بين ال rectangle
+
+223
+00:15:35,810 --> 00:15:40,450
+هذا وبين اللي هو ال function اللي أمامي قلنا سهل
+
+224
+00:15:40,450 --> 00:15:42,450
+على طول ال volume هو عبارة عن ال double
+
+225
+00:15:42,450 --> 00:15:47,190
+integration لأ اللي هو الدالة السطح العلوي لإن هذا
+
+226
+00:15:47,190 --> 00:15:50,510
+إيهاش اللي هو العلبنان كاملالـ Double by Fubini's
+
+227
+00:15:50,510 --> 00:15:54,710
+theorem الـ Volume يصير عنده يساوي لأن الـ Volume
+
+228
+00:15:54,710 --> 00:15:59,410
+is given by the double integral اللي هو هذا f of x
+
+229
+00:15:59,410 --> 00:16:03,010
+y دي ايه يساوي ال integration كامل بالنسبة ل y و
+
+230
+00:16:03,010 --> 00:16:06,010
+بالنسبة ل x انت حر خلينا نكمل بالنسبة ل y المرة
+
+231
+00:16:06,010 --> 00:16:09,730
+هذه إيش هي؟ اللي هي عشرة زي x تربيع تلتة y تربيع
+
+232
+00:16:09,730 --> 00:16:14,640
+دي y دي x هذه الدالة الآن من وين لوين يا جماعة؟من
+
+233
+00:16:14,640 --> 00:16:19,340
+عند 0 ل 2 قيمة الـ Y هيا اعطيني يعني من 0 ل 2 قيمة
+
+234
+00:16:19,340 --> 00:16:24,180
+الـ Y X من وين؟ من 0 ل عند 1 أمور rectangle استهلة
+
+235
+00:16:24,180 --> 00:16:29,040
+على طول اللي هو بالعوض هنا الان Dy بيصير عند D
+
+236
+00:16:29,040 --> 00:16:33,040
+اللي هو تكامل هذا يصير 10Y زيدي Y X تربيع زيدي Y
+
+237
+00:16:33,040 --> 00:16:37,120
+تكعيب على 3 وفي عندنا 3 بيروحوا مع بعض بتظهرها الـ
+
+238
+00:16:37,120 --> 00:16:40,620
+Y تكعيبمن صفر إلى وين إلى اتنين عوض عن اتنين بيصير
+
+239
+00:16:40,620 --> 00:16:44,620
+عشرين زائد اتنين X تربيع زائد تمانية والصفر طبعا
+
+240
+00:16:44,620 --> 00:16:48,460
+بتعطي هذه كلها صفر لان DX بالدعاقة كاملة بالنسبة ل
+
+241
+00:16:48,460 --> 00:16:51,780
+X كاملة هذه بالنسبة ل X بيصير عشرين X واتنين X
+
+242
+00:16:51,780 --> 00:16:58,040
+تربيع بالنسبة ل X بيصير تلتين X تكعيب زائد تمانية
+
+243
+00:16:58,040 --> 00:17:02,620
+HX من صفر لعند مين عند واحدة تلتين X H جمعة تكعيب
+
+244
+00:17:02,890 --> 00:17:07,330
+طيب الآن عوض بواحد بيصير عشرين زي التلتين زي
+
+245
+00:17:07,330 --> 00:17:10,750
+التمانية والسفر طبعا بيعملها دي كلها ايش سفر
+
+246
+00:17:10,750 --> 00:17:14,470
+فبتظلها اللي هي الأمامي هذه لو جمعتها بتطلع تمانين
+
+247
+00:17:14,470 --> 00:17:17,550
+و عشرين و تلتين أو ستة و تلتين ستة و تمانين على
+
+248
+00:17:17,550 --> 00:17:20,490
+تلاتة وهيك بيكون أوجدنا ال volume بواسطة double
+
+249
+00:17:20,490 --> 00:17:25,410
+integral وهيك بتكون المحاضرة انتهت وإلى لقايا آخر
+
+250
+00:17:25,410 --> 00:17:27,990
+والسلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mkf5fj4wuNM_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mkf5fj4wuNM_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6d5a1649a90d0c43903dbfc86ca3bfe1570ad4d7
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/mkf5fj4wuNM_raw.srt
@@ -0,0 +1,1708 @@
+1
+00:00:00,560 --> 00:00:03,980
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:03,980 --> 00:00:09,220
+وبركاته هذه هي المحاضرة الرقم 13 مساق تفاضل وتكامل
+
+3
+00:00:09,220 --> 00:00:14,580
+جيم وهي المحاضرة الأولى في chapter 14 الذي هو
+
+4
+00:00:14,580 --> 00:00:21,390
+بعنوان partial derivatives أو الاشتقاق الجزئيهنبدأ
+
+5
+00:00:21,390 --> 00:00:26,850
+اليوم 14.1 اللي هو عبارة عن functions of several
+
+6
+00:00:26,850 --> 00:00:34,430
+variables او الدالة اللي هي domainها او متغيراتها
+
+7
+00:00:34,430 --> 00:00:39,690
+المستقلة several variables اكثر من variable نشوف
+
+8
+00:00:39,690 --> 00:00:43,850
+شو معناته الكلام هذا definitionنفترض إن دي عبارة
+
+9
+00:00:43,850 --> 00:00:47,850
+عن set of untables of real numbers يعني دي عبارة
+
+10
+00:00:47,850 --> 00:00:53,410
+عن set of اللي هي untables يعني بدل منقول ordered
+
+11
+00:00:53,410 --> 00:00:57,110
+pairs لأ ولا تلاتة triple الآن في عندي x واحد و x
+
+12
+00:00:57,110 --> 00:01:01,930
+اتنين و xn هي هذه الدي بتتكون من كل العناصر لعصورة
+
+13
+00:01:01,930 --> 00:01:07,030
+x واحد x اتنين xn الان a real valid function f من
+
+14
+00:01:07,030 --> 00:01:12,370
+ال domain دي هذاالعناصر عبارة عن n doubles كل
+
+15
+00:01:12,370 --> 00:01:17,590
+واحدة منهم real number real valid function f on D
+
+16
+00:01:17,590 --> 00:01:21,710
+is a rule that assigns a unique single real number
+
+17
+00:01:21,710 --> 00:01:26,010
+w بسوء f of x1 x2 لعند xn يعني عبارة عن real valid
+
+18
+00:01:26,010 --> 00:01:31,550
+function f من ال domain D لعند Rلعند R هذه يعني و
+
+19
+00:01:31,550 --> 00:01:36,870
+كأنه من مجموعة جزئية من R N يعني من R cross R
+
+20
+00:01:36,870 --> 00:01:43,230
+cross R N من المرات W بتساوي F of X1 X2 XN بيكون
+
+21
+00:01:43,230 --> 00:01:49,070
+ال X1 و X2 و XN عبارة عن independent variablesوالـ
+
+22
+00:01:49,070 --> 00:01:53,270
+W اللي بتطلع هذه بتكون ايه اش مالها هي عبارة عن
+
+23
+00:01:53,270 --> 00:01:57,350
+الـ dependent variable يعني ال X1 و X2 و Xn هدولة
+
+24
+00:01:57,350 --> 00:02:01,830
+هم اللي بيصنعون ال domain للدالة و الـ W قيمته هي
+
+25
+00:02:01,830 --> 00:02:05,760
+اللي بتصنع ال rangeThe set D is the function's
+
+26
+00:02:05,760 --> 00:02:09,440
+domain الـ set D هذه اللي معرفين عليها اللي هي الـ
+
+27
+00:02:09,440 --> 00:02:13,740
+function F عندنا هي عبارة عن domain الدالة The set
+
+28
+00:02:13,740 --> 00:02:20,340
+of W values اللي هي قيم W اللي هي on .. on .. by F
+
+29
+00:02:20,340 --> 00:02:23,860
+is the function's range يعني الـ W اللي بتطلع
+
+30
+00:02:23,860 --> 00:02:29,520
+قيمتها من خلال حسابة F of X1 X2 عند ال XN هي اسمها
+
+31
+00:02:29,520 --> 00:02:34,340
+اللي هي قيم ال range أو range ال Fthe simple w
+
+32
+00:02:34,340 --> 00:02:38,000
+أحيانا بيقول احنا بعض الناس ممكن يحكموا بطريقة
+
+33
+00:02:38,000 --> 00:02:42,080
+أخرى بتقول is the dependent variable الو ايش بتكون
+
+34
+00:02:42,080 --> 00:02:47,400
+اللي هي ال dependent variable المتغير التابع و ال
+
+35
+00:02:47,400 --> 00:02:50,780
+x1 و ال x2 و ال xn هي المتغير المستقل بالظبط زي ما
+
+36
+00:02:50,780 --> 00:02:54,780
+كنا نحكي عن w بساوي f of x اللي هي ال x هي المتغير
+
+37
+00:02:54,780 --> 00:02:59,200
+المستقل و ال w هو المتغير التابعأو w بساوة f of x
+
+38
+00:02:59,200 --> 00:03:01,960
+و y و ال x و ال y اللي هي المتغير المستقل و ال w
+
+39
+00:03:01,960 --> 00:03:07,980
+التابع و هكذا الان ال x1 و ال x2 و ال xn هدولة هم
+
+40
+00:03:07,980 --> 00:03:11,480
+اللي بنسميهم independent variables زي ما قلنا و
+
+41
+00:03:11,480 --> 00:03:15,260
+بعض الناس بتقولك انه لأ خلينا نسمي هدولة ال x1,
+
+42
+00:03:15,400 --> 00:03:21,000
+x2, xn هم ال input يعني اللي بنحطهم بطلع لنا بناء
+
+43
+00:03:21,000 --> 00:03:24,920
+عليها بوسطة ال f اللي هي ال w اللي بنسميها إيش ال
+
+44
+00:03:24,920 --> 00:03:25,680
+output
+
+45
+00:03:28,490 --> 00:03:32,530
+الان نأخد مثال على إيجاد ال domain و ال range طبعا
+
+46
+00:03:32,530 --> 00:03:34,810
+ال domain و ال range هو مشابه لل domain و ال range
+
+47
+00:03:34,810 --> 00:03:38,250
+اللي أخدناه في Calculus 1000 لكن الان اللي هو ال
+
+48
+00:03:38,250 --> 00:03:41,830
+domain هيكون ممكن two variables، three variables،
+
+49
+00:03:41,830 --> 00:03:44,870
+أربعة variables، أي عدد من ال variables يعني أكتر
+
+50
+00:03:44,870 --> 00:03:48,810
+من one variable ال range طبعا قيمة واحدة اللي هي
+
+51
+00:03:48,810 --> 00:03:51,990
+جزء من Rزي ما شوفنا قبل شوية خلّيني نشوف أمثلة
+
+52
+00:03:51,990 --> 00:03:57,210
+عملية examples example one these are functions of
+
+53
+00:03:57,210 --> 00:04:00,530
+two variables ان احنا بنحكي عن دول إيه شمالها في
+
+54
+00:04:00,530 --> 00:04:04,730
+متغيرينnote that the restrictions that may apply
+
+55
+00:04:04,730 --> 00:04:07,510
+to their domain in order to obtain a real value of
+
+56
+00:04:07,510 --> 00:04:10,510
+the independent variable z ايش اللي بقوله؟ لما
+
+57
+00:04:10,510 --> 00:04:15,470
+بقوللي اللي هو اوددلي قيم x و y هنا مثلا اللي
+
+58
+00:04:15,470 --> 00:04:19,630
+بتخلي ال z بساوية جدر التربيعي معرفة يعني الآن ال
+
+59
+00:04:19,630 --> 00:04:26,240
+z هو كإن هي عبارة عن f of x yدالة بدلالة main x و
+
+60
+00:04:26,240 --> 00:04:30,220
+y هذولة هى المتغيرات المستقلة اللى بتعمل ال domain
+
+61
+00:04:30,220 --> 00:04:33,140
+وهذا المتغير التابع زى ما كنا نقول في calculus
+
+62
+00:04:33,140 --> 00:04:38,120
+1000 بالظبط اللى قيم ال domain هي قيم اللى هى ال x
+
+63
+00:04:38,120 --> 00:04:44,080
+و ال y القيم الممكنة الآن ال x و ال y هنا ممكن
+
+64
+00:04:44,080 --> 00:04:49,140
+تاخد أي قيم بحيث أنه يكون اللى تحت الجدر أكبر أو
+
+65
+00:04:49,140 --> 00:04:53,510
+يساوي 0 لأنه أصغر من 0 مع أنه مش مالو مش معرفةإذن
+
+66
+00:04:53,510 --> 00:04:58,410
+الآن عناصر الـ domain X و Y بتربطهم علاقة إن هذا
+
+67
+00:04:58,410 --> 00:05:03,010
+لازم يكون أكبر أو يساوي سفر أو بمعنى آخر Y أكبر أو
+
+68
+00:05:03,010 --> 00:05:06,630
+يساوي X تربيع عشان هيك ال domain هو عبارة عن كل
+
+69
+00:05:06,630 --> 00:05:12,430
+قيم X و Y التي تحقق Y أكبر أو يساوي X تربيع هذا
+
+70
+00:05:12,430 --> 00:05:15,790
+domain الدالة الأولى هذا Z بساوي الجدر التربيعي
+
+71
+00:05:16,490 --> 00:05:20,090
+نعود الآن للـ Range الـ Range هي قيمين قيم زد
+
+72
+00:05:20,090 --> 00:05:26,470
+الممكنة إعتمادًا على الـ X والـ Y المدخلاتلأن واضح
+
+73
+00:05:26,470 --> 00:05:30,030
+أن الجذر دائما إيه شمالها نموجة بما أخدينه إحنا
+
+74
+00:05:30,030 --> 00:05:34,750
+إذا هذه دائما قيمها أكبر أو يساوي سفر و ال Y و ال
+
+75
+00:05:34,750 --> 00:05:39,350
+X بتاخد اللي بدهيها وفق العلاقة هذه إذا هذه بتجيب
+
+76
+00:05:39,350 --> 00:05:44,810
+لكل القيم من سفر إلى وين إلى ما لا نهاية هذه قيمة
+
+77
+00:05:44,810 --> 00:05:48,650
+مين قيمة زد ال range نجي لأ اللي هو function تاني
+
+78
+00:05:48,650 --> 00:05:53,450
+F of X Y بساوي واحد على X في Yالان قيم ال domain
+
+79
+00:05:53,450 --> 00:05:58,190
+هي قيم x و y الممكنة هنا ال x و y بتاخد أي قيمة في
+
+80
+00:05:58,190 --> 00:06:04,250
+الدنيا ما عدا ان x و y تساوي 0 عشان هيك ال domain
+
+81
+00:06:04,250 --> 00:06:10,230
+هو عبارة عن كل قيم x و y بحيث x و y لا تساوي 0
+
+82
+00:06:10,230 --> 00:06:15,030
+يعني انشيل ال x و ال y اللي بيحاصل ضربهم بساوي 0
+
+83
+00:06:15,030 --> 00:06:18,530
+بطلع من ال domain يعني ال domain هو عبارة عن كل x
+
+84
+00:06:18,530 --> 00:06:24,670
+و yحيث XY لا تساوي 0 الآن الـ Range قيم Z هانا Z
+
+85
+00:06:24,670 --> 00:06:29,870
+ممكن اللي هو تاخد أي قيمة في الدنياما عدا انها
+
+86
+00:06:29,870 --> 00:06:34,290
+تساوي سفر لانه مستحيل ايش تساوي سفر الآن ال X و ال
+
+87
+00:06:34,290 --> 00:06:37,330
+Y بياخدوا كل القيم اللي في الدنيا فهذه القيمة
+
+88
+00:06:37,330 --> 00:06:42,850
+بتصير اي قيمة في الدنيا ما عدا من السفر يعني بمعنى
+
+89
+00:06:42,850 --> 00:06:48,970
+اخر قيمة Z يعني ال range اللي هو عبارة عن كل ال R
+
+90
+00:06:48,970 --> 00:06:52,610
+ما عدا السفر يعني من نقص infinity لسفر اتحاد من
+
+91
+00:06:52,610 --> 00:06:55,710
+سفر ل infinity بدون من السفر
+
+92
+00:06:58,890 --> 00:07:02,470
+الـ Range عن طبع الـ Function هذي نجيب الـ
+
+93
+00:07:02,470 --> 00:07:08,170
+Function التاني Z بيساوي Sine X في Y الآن الـ Sine
+
+94
+00:07:08,170 --> 00:07:12,810
+طبعتها بتحسب لأي قيمة في الدنيا إيش ما تعطي الـ
+
+95
+00:07:12,810 --> 00:07:17,370
+Sine بتحسبها عشان هيك X و Y بتاخد أي قيمة في
+
+96
+00:07:17,370 --> 00:07:22,110
+الدنيا يعني الـ Domain عنا هو عبارة عن الـ Entire
+
+97
+00:07:22,110 --> 00:07:26,870
+plane يعني كل الـ X و Y بحيثإن X في الـ R و Y في
+
+98
+00:07:26,870 --> 00:07:32,370
+الـ R يعني R cross R كل الـ domain هي الدمين
+
+99
+00:07:32,370 --> 00:07:36,370
+الدمين هنا دائما جزء من الـ domain لإنه في عندى
+
+100
+00:07:36,370 --> 00:07:42,150
+متغيرين اللي هنا X وY الآن زد ايه شمالهابتساوى
+
+101
+00:07:42,150 --> 00:07:45,070
+sign sign في النهاية sign و ال sign احنا بنعرف
+
+102
+00:07:45,070 --> 00:07:49,570
+ماله لا يمكن ان يتجاوز اللي هو الواحد ولا يقل عن
+
+103
+00:07:49,570 --> 00:07:53,910
+الناقص واحد يعني قيمة ال z دايما بين الناقص واحد
+
+104
+00:07:53,910 --> 00:07:59,050
+وبين مين الواحد هذه اللي هي ال range تبع الدالة
+
+105
+00:07:59,050 --> 00:08:03,610
+اللي عندناالان هنا دي كانت دوال مثال على دوال
+
+106
+00:08:03,610 --> 00:08:07,650
+functions of two variables y وx الان بناخد مثال
+
+107
+00:08:07,650 --> 00:08:12,070
+آخر على function of two of three variables يعني w
+
+108
+00:08:12,070 --> 00:08:15,470
+بتساوي function في x و في y و في z يعني في دلالة
+
+109
+00:08:15,470 --> 00:08:18,790
+main اللي هو ال x و ال y و ال z يعني ال domain هنا
+
+110
+00:08:18,790 --> 00:08:23,470
+كان جزء من ال plane هنا ال domain هيكون جزء من ال
+
+111
+00:08:23,470 --> 00:08:28,470
+space لأن في عندى قيم ل x و قيم ل y و قيم ل zالان
+
+112
+00:08:28,470 --> 00:08:33,840
+الـ domain هو عبارة عن قيم X و Y و Z الممكنةفي حد
+
+113
+00:08:33,840 --> 00:08:38,680
+بيمنع ال X و ال Y و ال Z تاخد أي قيمة لأنه إيش ما
+
+114
+00:08:38,680 --> 00:08:42,140
+تعطيها سالب ولا كده ولا كده لما تربحها بتطلع كلها
+
+115
+00:08:42,140 --> 00:08:45,780
+موجمة لما تاخدها الجدر التربيعي بتطلع القيمة إذا
+
+116
+00:08:45,780 --> 00:08:51,720
+قيم X و Y و Z الممكنة هي كل ال space يعني X بتاخد
+
+117
+00:08:51,720 --> 00:08:56,600
+كل R و Y كل R و Z كل R مع بعض اللي بيخدمي ال space
+
+118
+00:08:56,600 --> 00:09:01,450
+كله هذه اللي هي domain الدولة الأولىRange هي موخود
+
+119
+00:09:01,450 --> 00:09:05,810
+الجدر التربيعي الموجة مظبوط إذا Range هو عبارة عن
+
+120
+00:09:05,810 --> 00:09:09,930
+كل القيم الموجبة من صفر لما لنهاية و أي قيمة موجبة
+
+121
+00:09:09,930 --> 00:09:14,690
+بتقدر تاخدها لأن X و Y و Z بيجيبن أي قيمة في هذه
+
+122
+00:09:14,690 --> 00:09:18,690
+المنطقة نيجي الآن للدالة المعدة طب ال U بتسوء واحد
+
+123
+00:09:18,690 --> 00:09:22,530
+على X تربيع زي Y تربيع زي Z تربيع لو طلعنا لهذه
+
+124
+00:09:22,530 --> 00:09:31,070
+الدالة هدفة اللي الآن قيم X و Y و Z الممكنةأي قيمة
+
+125
+00:09:31,070 --> 00:09:35,190
+ما عدا الخطر من وين بحدث انه تطلع هذه ايش بتساوي
+
+126
+00:09:35,190 --> 00:09:39,630
+سفر يعني بمعنى X تربيع Z وY تربيع Z تربيع ممنوع
+
+127
+00:09:39,630 --> 00:09:43,170
+تساوي سفر هذه بتساوي سفر في حالة واحدة لأنها كمية
+
+128
+00:09:43,170 --> 00:09:47,810
+مربعة في حالة ال X سفر و ال Y سفر و ال Z سفر عشان
+
+129
+00:09:47,810 --> 00:09:52,450
+هيك ال domain هو عبارة عن كل ال X و ال Y و ال Z
+
+130
+00:09:52,450 --> 00:09:59,470
+اللي بيساود نشمين 000 يعني ال space كلهما هدا نقطة
+
+131
+00:09:59,470 --> 00:10:06,330
+الاصل اللي هي 000 هذا اللي هو ال domain ال range w
+
+132
+00:10:06,330 --> 00:10:09,950
+بسوء واحد على هذا المقداراللي هو هذه الكمية أول شي
+
+133
+00:10:09,950 --> 00:10:13,490
+موجبة مظبوطة لا لأ أكيد موجبة و بعدين مش مستحيل
+
+134
+00:10:13,490 --> 00:10:18,010
+تساوي 100 سفر إذا هذه بتاخد كل القيم الموجبة ماعدا
+
+135
+00:10:18,010 --> 00:10:21,990
+السفر اللي هو بدون طبعا 100 السفر ماعدا السفر مش
+
+136
+00:10:21,990 --> 00:10:27,610
+كمية موجبة إذا اللي هو عندي ال range هو عبارة عن
+
+137
+00:10:27,610 --> 00:10:34,670
+كل x و y و zأسف ال range هو عبارة عن قيم w اللي هي
+
+138
+00:10:34,670 --> 00:10:40,110
+الموجبة من 0 إلى مال نهاية طبعا مستحيل هذه تاخد
+
+139
+00:10:40,110 --> 00:10:45,630
+القيمة إيش القيمة الصفرية لإنه إيش بتساوي بتساوي
+
+140
+00:10:45,630 --> 00:10:49,130
+عبارة عن واحد طيب جماعة صلى الله عننا ربنا عليه
+
+141
+00:10:49,130 --> 00:10:53,610
+الصلاة والسلام نيجيلا مين لل range هاي قلنا 0 و
+
+142
+00:10:53,610 --> 00:10:58,330
+infinity نيجيلا اللي هي w بساوي x في y للنزلالان
+
+143
+00:10:58,330 --> 00:11:02,710
+ال domain هو عبارة عن قيم X و Y و Z الممكنة X و Y
+
+144
+00:11:02,710 --> 00:11:07,130
+مافيش حد مافيش ولا اي restriction عليها ولا اي
+
+145
+00:11:07,130 --> 00:11:12,170
+واحد بيقدر يفرض عليها انها تكون اشي X و Y بتاخد اي
+
+146
+00:11:12,170 --> 00:11:17,840
+قيم في الدنيا لكن الان لن Zالـ z هنا ممنوع تاخد
+
+147
+00:11:17,840 --> 00:11:21,380
+إلا تكون أشمالها أكبر من سفر عشان نقدر نحسبها لان
+
+148
+00:11:21,380 --> 00:11:26,580
+لين عشان هيك اللي هو ال domain هو عبارة عن كل ال X
+
+149
+00:11:26,580 --> 00:11:31,800
+و ال Y و ال Z بحيث أن Z تكون أكبر من سفر يعني
+
+150
+00:11:31,800 --> 00:11:35,640
+مافيش restrictions على ال X و ال Y لكن ال
+
+151
+00:11:35,640 --> 00:11:39,350
+restriction عالميا على ال Zلازم الـ z يكون أكبر من
+
+152
+00:11:39,350 --> 00:11:41,870
+0 مدام الـ z أكبر من 0 معناته اللي هو الجزء
+
+153
+00:11:41,870 --> 00:11:46,230
+الفجاني من ال space يعني ال half space الفجاني مش
+
+154
+00:11:46,230 --> 00:11:50,570
+علنا ال space هي عند ال XYZ plane المنطقة اللي فوق
+
+155
+00:11:50,570 --> 00:11:56,790
+هذه بدون الأرضيةاللي هو إيه شمالها هي عبارة عن ال
+
+156
+00:11:56,790 --> 00:12:02,190
+domain إذا عبارة عن ال half space الفجاني اللي هو
+
+157
+00:12:02,190 --> 00:12:06,710
+z أكبر من سفر نيجي لل range ال range طبعا ال x في
+
+158
+00:12:06,710 --> 00:12:10,090
+ال y بتاخد كل القيام اللي في الدنيا و اللي نزد
+
+159
+00:12:10,090 --> 00:12:17,050
+اللي هي قيمهالما نظرح في قيم X وY اللي بتاخد أي
+
+160
+00:12:17,050 --> 00:12:21,450
+قيمة في الدنيا بيصير كل القيم اللي في الدنيا W
+
+161
+00:12:21,450 --> 00:12:24,650
+بتاخدها يعني W بتاخد من ناقص infinity إلى infinity
+
+162
+00:12:24,650 --> 00:12:31,010
+يعني كل R طيب لأن نيجي نبدأ نخصص لأننا بنشتغل على
+
+163
+00:12:31,010 --> 00:12:36,270
+land functions of two variables في الأول بدنا نعرف
+
+164
+00:12:36,270 --> 00:12:39,610
+حاجة اسمها ال interior point وال boundary point
+
+165
+00:12:39,610 --> 00:12:45,720
+وكذا الاخرينالان بدي اخد مجموعة اسمها Rاللي هذي
+
+166
+00:12:45,720 --> 00:12:48,700
+جزء من الـ plane من الـ plane الان بشتغل وين يا
+
+167
+00:12:48,700 --> 00:12:52,740
+جماعة في الـ plane الان a point x note و y note in
+
+168
+00:12:52,740 --> 00:12:57,080
+a region R في region R منطقة اسمها R بنقول x note
+
+169
+00:12:57,080 --> 00:13:00,680
+و y note طبعا ال region هذا جزء وين في ال x y
+
+170
+00:13:00,680 --> 00:13:04,080
+plane بنسمي ال x note و ال y note interior point
+
+171
+00:13:04,080 --> 00:13:06,920
+في ال R بنقول عن ال x note و ال y note interior
+
+172
+00:13:06,920 --> 00:13:10,920
+point في ال R إذا كان قدرنا نلاقي اللي هو جوار
+
+173
+00:13:10,920 --> 00:13:14,950
+اللي هو دائرة يعنيحوالين النقطة X نقطة و Y نقطة
+
+174
+00:13:14,950 --> 00:13:18,290
+كله في داخل المنطقة R في هذه الحالة بنقول عن
+
+175
+00:13:18,290 --> 00:13:23,250
+النقطة X نقطة و Y نقطة نقطة داخلية يعني لو أجينا
+
+176
+00:13:23,250 --> 00:13:27,410
+أخدنا النقطة هذه خد أي جوار حواليها مستحيل تحطه
+
+177
+00:13:27,410 --> 00:13:31,030
+كله وين على ال R هيأكل من برة و هيأكل من جوا عشان
+
+178
+00:13:31,030 --> 00:13:34,250
+هي كهذه مش interior point و أكيد هذه اللي لبرا مش
+
+179
+00:13:34,250 --> 00:13:37,830
+interior point إذا ال interior point هي النقطة
+
+180
+00:13:37,830 --> 00:13:42,950
+اللي بتقدر تاخد حواليها جوار كل وين في داخل Rلازم
+
+181
+00:13:42,950 --> 00:13:47,270
+نعرف حاجة اسمها boundary point نقطة إيش حدودية هذه
+
+182
+00:13:47,270 --> 00:13:51,890
+نقطة داخلية ونقطة الحدودية النقطة الحدودية اللي هي
+
+183
+00:13:51,890 --> 00:13:56,130
+بيبتي .. إيش معناتها؟ لو جيت أخدت أي جوار .. أي
+
+184
+00:13:56,130 --> 00:14:01,530
+جوار مهما زغرته مهما كبرته هتلاقي هذا الجوار بياخد
+
+185
+00:14:01,530 --> 00:14:05,430
+من جوا الستار ومن برا الستار أو من جوا ال region
+
+186
+00:14:05,430 --> 00:14:08,430
+ومن برا ال region عشان هي كما بنقول هذه نقطة
+
+187
+00:14:08,430 --> 00:14:13,890
+حدودية على الحدودلما تأخذ جوار حوالها تأخذ من دارك
+
+188
+00:14:13,890 --> 00:14:17,930
+و من برا دارك هذه اسمها boundary bond الان
+
+189
+00:14:17,930 --> 00:14:22,750
+المجموعة او ال region ال region اللى بتتكون بس من
+
+190
+00:14:22,750 --> 00:14:25,630
+النقاط الداخلية يعني مافيش فيها boundary هدول مش
+
+191
+00:14:25,630 --> 00:14:28,670
+موجودات يعني من الجطات نقطة خفية اللى هي نقطة
+
+192
+00:14:28,670 --> 00:14:33,670
+متقطعة يعني مش موجودة فيها هذه اللى بنسميها open
+
+193
+00:14:33,670 --> 00:14:37,460
+set اذا ال open set هي ال setأو الـ Open Region هي
+
+194
+00:14:37,460 --> 00:14:42,040
+الـ Region أو الست اللي كل نقاطها داخلية الآن لو
+
+195
+00:14:42,040 --> 00:14:46,920
+كان الست .. الست مكوّنة من نقاطها الداخلية ومن الـ
+
+196
+00:14:46,920 --> 00:14:49,480
+Boundary Points يعني الـ Boundary Points من ضمن
+
+197
+00:14:49,480 --> 00:14:54,620
+الـ R بنسميها Closed Set طب الست اللي مثلا عندي ست
+
+198
+00:14:54,620 --> 00:14:57,320
+الآن هندل وادم نجطة مش موجودات هدول النقاط و
+
+199
+00:14:57,320 --> 00:15:01,990
+البجيات موجوداتهذه لا هي closed ولا هي open لأن
+
+200
+00:15:01,990 --> 00:15:04,970
+هذولا الجزء من ال boundary point مش موجودات و
+
+201
+00:15:04,970 --> 00:15:08,890
+هذولا النقاط اللي موجودات مش نقاط داخلية فحالة لا
+
+202
+00:15:08,890 --> 00:15:12,790
+هتكون open ولا هتكون closed بينما اللي بتكون
+
+203
+00:15:12,790 --> 00:15:17,030
+نقاطها اللي هي اللي هي هذه متقطعة يعني ال boundary
+
+204
+00:15:17,030 --> 00:15:21,510
+point مش داخلة بتكون open set والنقاطها كلها ال
+
+205
+00:15:21,510 --> 00:15:26,490
+boundary داخلة بتكون closed set طيب نيجي ناخد
+
+206
+00:15:26,490 --> 00:15:31,440
+أمثلة أي أمثلة ما أخدلكيهاالـ Open disk اللي هو X
+
+207
+00:15:31,440 --> 00:15:34,760
+تربيع زي و Y تربيع أصغر من واحد هذا ال disk هيه كل
+
+208
+00:15:34,760 --> 00:15:39,680
+اللي فيه معدنين على اللي هي إيش الحفاف هذه معدن
+
+209
+00:15:39,680 --> 00:15:45,690
+دائرةلأننا ناخد جوات الدائرة هذه اللي هي open unit
+
+210
+00:15:45,690 --> 00:15:50,470
+disk وهو هتكون إشمالها open set لأن كل نقاطها
+
+211
+00:15:50,470 --> 00:15:53,610
+عبارة عن نقاط داخل هي أخد أي نقطة بقدر أخد
+
+212
+00:15:53,610 --> 00:15:58,250
+أحواليها اللي هو دائرة تكون كلها في داخل اللي هي
+
+213
+00:15:58,250 --> 00:16:02,770
+asset عشان هيك هذه open set اللي هي بنسميها open
+
+214
+00:16:02,770 --> 00:16:07,730
+unit disk عساس اللي هو عبارة عن اللي هي دائرةو انا
+
+215
+00:16:07,730 --> 00:16:10,370
+مااخد الداخلها و بنصف قطر واحد و ساميه open unit
+
+216
+00:16:10,370 --> 00:16:13,830
+disk لأن نيجي X تربيع زد و Y تربيع سا واحد هذا
+
+217
+00:16:13,830 --> 00:16:18,210
+الجشرة هذا الجشرة هذا جوا الكرة هذا او دوا المنطقة
+
+218
+00:16:18,210 --> 00:16:21,410
+هذه هذه اللي هي الجشرة الخارجية او الدائرة نفسها
+
+219
+00:16:21,410 --> 00:16:25,280
+هي الدائرة بسExtra Digital Vitals بس هي واحد هذه
+
+220
+00:16:25,280 --> 00:16:30,660
+اللي هي عبارة عن اللي هو كل عناصرها boundary
+
+221
+00:16:30,660 --> 00:16:34,460
+points وكل الباقية يعني كل ال boundary points اللي
+
+222
+00:16:34,460 --> 00:16:38,120
+فيها موجودات فيها عشان هي closed disk او بنسميها
+
+223
+00:16:38,120 --> 00:16:43,430
+boundary unit diskالان الـ set هذه اللي .. اللي
+
+224
+00:16:43,430 --> 00:16:47,870
+هدول هيهم مكونة من ال boundaries بالاضافة للي جوا
+
+225
+00:16:47,870 --> 00:16:52,130
+هذه اللي هي برضه closed set و بنسميها closed unit
+
+226
+00:16:52,130 --> 00:16:56,670
+disk اللي هي بتحتوي على كل ال boundary points اللي
+
+227
+00:16:56,670 --> 00:17:02,230
+إلاالان نجي نتعريف a region in the plane is said
+
+228
+00:17:02,230 --> 00:17:05,910
+to be bounded بنقول عنها bounded محدودة يعني ال
+
+229
+00:17:05,910 --> 00:17:10,550
+region بنقول عنها bounded إذا قدرنا نحطها جوا disk
+
+230
+00:17:11,470 --> 00:17:15,270
+يعني a region in the plane is bounded if it lies
+
+231
+00:17:15,270 --> 00:17:19,650
+inside a disk of a fixed radius a region is
+
+232
+00:17:19,650 --> 00:17:22,470
+unbounded if it is not bounded يعني الـ bounded
+
+233
+00:17:22,470 --> 00:17:25,790
+اللي مقدر نحطه جوا ال disk اللي مقدرش نحطه جوا ال
+
+234
+00:17:25,790 --> 00:17:29,410
+disk بنسميها unbounded يعني هذه ال set مثلا
+
+235
+00:17:29,410 --> 00:17:32,310
+bounded لإنه مقدر أحطها ياخدوا ال disk برا تا اي
+
+236
+00:17:32,310 --> 00:17:35,090
+disk بحتويها هذه ال set bounded لإن اي disk
+
+237
+00:17:35,090 --> 00:17:39,130
+هيحتويها لكن لو جينا على الربع الأول هذاهذا كله
+
+238
+00:17:39,130 --> 00:17:42,050
+إلى ما لنا .. أكيد هذا مش bounded .. ليش؟ لأنه
+
+239
+00:17:42,050 --> 00:17:45,450
+deep .. مهما تحط deep disk مش هيقدر يحتوي كل الربع
+
+240
+00:17:45,450 --> 00:17:51,310
+الأول عشان هي .. الربع الأول مثلا unbounded لو دي
+
+241
+00:17:51,310 --> 00:17:54,090
+أخدنا x بتساوي صفر .. يعني x بتساوي صفر اللي هو ال
+
+242
+00:17:54,090 --> 00:17:57,770
+y-axis هذا برضه unbounded set .. ليش؟ لأنه لا يمكن
+
+243
+00:17:57,770 --> 00:18:05,190
+نحطه جواد اللي هو disk طيب ..Examples of bounded
+
+244
+00:18:05,190 --> 00:18:10,410
+sets in the plane include line segments, triangles
+
+245
+00:18:10,410 --> 00:18:15,040
+boundedinteriors of triangles bounded اللي هو ..
+
+246
+00:18:15,040 --> 00:18:19,460
+triangles يعني متلثات ال rectangles المستطيلات
+
+247
+00:18:19,460 --> 00:18:22,240
+برضه bounded و ال circles bounded و ال discs
+
+248
+00:18:22,240 --> 00:18:26,260
+bounded طب فاهمين ليش طبعا لإن كلنا ممكن نحط دروات
+
+249
+00:18:26,260 --> 00:18:29,940
+دائرة examples of unbounded sets in the plane
+
+250
+00:18:29,940 --> 00:18:32,240
+include lines اللي هو الخط المستقيم زي ما قلنا هي
+
+251
+00:18:32,240 --> 00:18:35,280
+خط مستقيم أكيد بيمتد لما ناناها من جهة .. من كل
+
+252
+00:18:35,280 --> 00:18:38,880
+تجاهتين .. جهتين عشان هي كis not bounded والشعاع
+
+253
+00:18:38,880 --> 00:18:42,350
+بقى unbounded برضهالـ coordinate axis زي ما قلنا
+
+254
+00:18:42,350 --> 00:18:45,470
+unbounded the graph of functions defined on
+
+255
+00:18:45,470 --> 00:18:47,850
+infinite intervals اللي بتكون معرفة على infinite
+
+256
+00:18:47,850 --> 00:18:52,030
+intervals زي مثلا ناخد اللي هو Y بتساوي X تربيع
+
+257
+00:18:52,030 --> 00:18:55,990
+هذه البرابولة بتضلها منتدى لما ننهي أكيد unbounded
+
+258
+00:18:55,990 --> 00:18:59,630
+ال quadrants اللي هي الأربع ربع الأول والتالي
+
+259
+00:18:59,630 --> 00:19:03,030
+والتالتة unbounded ال half planes unbounded هي نصف
+
+260
+00:19:03,030 --> 00:19:06,090
+هذا أو نصف هذا أو نصف هذا أو نصف هذا unbounded و
+
+261
+00:19:06,090 --> 00:19:09,750
+ال plane نفسه unbounded region أو unbounded cell
+
+262
+00:19:10,590 --> 00:19:13,630
+الان بيقول لي describe the domain of the function
+
+263
+00:19:13,630 --> 00:19:16,570
+f of x y بيسموه الجدر تربيعي ل y minus x تربيعي
+
+264
+00:19:16,570 --> 00:19:20,330
+بيقول لي إيه رايك ترسم اللي هو ال domain اللي هاد
+
+265
+00:19:20,330 --> 00:19:22,970
+الدالة ال domain هي عبارة عن قيمة قيم y و x
+
+266
+00:19:22,970 --> 00:19:26,410
+الممكنةقيم Y و X الممكنة معناته جزء من اللي بلين
+
+267
+00:19:26,410 --> 00:19:31,510
+لإنه في متغيرين إذا الأن ال domain اللي هي قيم X و
+
+268
+00:19:31,510 --> 00:19:34,410
+Y الممكنة قلنا هذه إيش القيم الممكنة اللي بتكون
+
+269
+00:19:34,410 --> 00:19:38,410
+هذه موجبة يعني Y ناقص X تربيع أكبر تساوي سفر يعني
+
+270
+00:19:38,410 --> 00:19:44,770
+Y أكبر من X أكبر تساوي X تربيع طيب نشوف هذه Y
+
+271
+00:19:44,770 --> 00:19:47,750
+بتساوي X تربيع هي البرابول هاية Y بتساوي X تربيع
+
+272
+00:19:47,750 --> 00:19:53,740
+مظبوطالعين y نقص x تربيع أكبر من 0 المنطقة هذه خد
+
+273
+00:19:53,740 --> 00:19:58,200
+أي نقطة هذه خد ال y coordinate و ال x coordinate
+
+274
+00:19:58,200 --> 00:20:03,560
+هتلاقيها أكبر من 0 هذه y نقص x تربيع بتساوي 0 هي
+
+275
+00:20:03,560 --> 00:20:07,480
+البرابولة نفسها y نقص x تربيع أصغر من 0 اللي هي
+
+276
+00:20:07,480 --> 00:20:15,840
+برة البرابولةالـ domain تبع الدالة f of x y بيساوي
+
+277
+00:20:15,840 --> 00:20:19,860
+جدرى تربيع الـ y نقص x تربيع مين هو؟ هو عبارة عن
+
+278
+00:20:19,860 --> 00:20:27,000
+البرابولة وجوات البرابولةبالإضافة للي جوا هذا هو
+
+279
+00:20:27,000 --> 00:20:30,300
+ال domain لإن ال domain هو عبارة عن y نقص اكتر
+
+280
+00:20:30,300 --> 00:20:34,760
+بيعني مش أكبر من سفر بس أكبر أو يساوي سفر إذا هذا
+
+281
+00:20:34,760 --> 00:20:37,820
+هو عبارة عن domain ال variable domain ال function
+
+282
+00:20:37,820 --> 00:20:42,160
+اللي عندنا دوات ال variable وعلى ال variable أما
+
+283
+00:20:42,160 --> 00:20:52,440
+خارج هذا فهو ليس من ضمن ال domain انكمل
+
+284
+00:20:52,440 --> 00:20:53,840
+الآن اللي هو
+
+285
+00:20:59,840 --> 00:21:06,460
+graphs level curves and contours of functions of
+
+286
+00:21:06,460 --> 00:21:10,960
+two variables نشوف شو معناته دلوقت الان لنعرف شو
+
+287
+00:21:10,960 --> 00:21:14,340
+معناته اللي هو ال graph و level curve و contours
+
+288
+00:21:14,340 --> 00:21:18,100
+of functions of two variables the set of points in
+
+289
+00:21:18,100 --> 00:21:21,060
+the plane where the function f of x y has a
+
+290
+00:21:21,060 --> 00:21:24,940
+constant value f of x y بيتساوى c is called a
+
+291
+00:21:24,940 --> 00:21:30,300
+level curve of f إذا مجموعة النقاطاللي هي اللي في
+
+292
+00:21:30,300 --> 00:21:35,300
+البلين اللي ال F of X Y بتسوي C ثابت بنسميها level
+
+293
+00:21:35,300 --> 00:21:40,400
+curve of F عبارة عن إذا جزء من مين من البلين هذه
+
+294
+00:21:40,400 --> 00:21:44,120
+اللي هي كل ال X و Y اللي بتحقق هذا بتسوي C بنسميها
+
+295
+00:21:44,120 --> 00:21:48,840
+level curve لل Fالـ set of all points اللي ال X و
+
+296
+00:21:48,840 --> 00:21:53,180
+ال Y و ال F of X و Y المقابلة in space in general
+
+297
+00:21:53,180 --> 00:21:57,020
+بنسميها اللي هي طبعاً لل X و ال Y اللي في domain
+
+298
+00:21:57,020 --> 00:22:00,360
+ال F بنسميها هذه ال graph هذه زي اللي عادي ال
+
+299
+00:22:00,360 --> 00:22:03,600
+graph العادي اللي احنا بنعرفهاللي هو قيم X وقيم Y
+
+300
+00:22:03,600 --> 00:22:08,160
+وقيم Z المقابلة اللي هي F of X وY بتعمل لـ Graph
+
+301
+00:22:08,160 --> 00:22:13,820
+لل function اللي عندي F of X وY ماشي الحالة الان
+
+302
+00:22:13,820 --> 00:22:17,880
+graph of F is called also the surface Z بيساوي F
+
+303
+00:22:17,880 --> 00:22:21,520
+of X وY لإن فعلا هي بتطلعاش surface في ال space
+
+304
+00:22:21,520 --> 00:22:26,400
+طيب ناخد مثال Graph F of X وY بيساوي 100 نقص X
+
+305
+00:22:26,400 --> 00:22:31,390
+تربيع نقص Y تربيعو بعدين بلوت level curves f of x
+
+306
+00:22:31,390 --> 00:22:35,830
+,y بيساوي 0 f of x,y بيساوي 51 f of x,y بيساوي 75
+
+307
+00:22:35,830 --> 00:22:41,230
+in the domain of f in the plane اللي عندنا الان
+
+308
+00:22:41,230 --> 00:22:47,820
+نيجيأول حاجة اللي هو لو اجينا على اللي هي ال
+
+309
+00:22:47,820 --> 00:22:53,380
+domain هنا x و y بتاخد القيم اللي بدها هيها مافيش
+
+310
+00:22:53,380 --> 00:22:59,260
+حد بقدر يمنحها عشان هيك domain ده اللي كل قيم x و
+
+311
+00:22:59,260 --> 00:23:04,680
+y يعني ال x y plane كله يعني domainها كل plane لكن
+
+312
+00:23:04,680 --> 00:23:11,170
+ايش اللي هو range ايش قيم ال z عندهاF of X Y هي
+
+313
+00:23:11,170 --> 00:23:15,750
+قيمة Z زد بساوة ومية نقص X تربيع نقص Y تربيع الآن
+
+314
+00:23:15,750 --> 00:23:26,070
+زد لا يمكن تاخد لو أخدت زد بساوة ومية نقص
+
+315
+00:23:26,070 --> 00:23:31,690
+X تربيع نقص Y تربيع 100 مع 100 يعني X تربيع نقص Y
+
+316
+00:23:31,690 --> 00:23:37,470
+تربيع بساوة 0 X تربيع زد Y تربيع بساوة 0 يعني X
+
+317
+00:23:37,470 --> 00:23:43,730
+سفر وY سفرمظبوط يعني اللي هو عند z بتساوي 100 اللي
+
+318
+00:23:43,730 --> 00:23:50,190
+هو بيكون عندى اللي هو ال x و ال y أصفر يعني نقطة
+
+319
+00:23:50,190 --> 00:23:55,490
+اللي هي الرسم اللي هي عند z بتساوي 100 بتكون على
+
+320
+00:23:55,490 --> 00:24:00,810
+ال z axis لأن ال x و ال y أصفر لو طلعت شوية فوق
+
+321
+00:24:00,810 --> 00:24:06,250
+المية لو قلت 101 انتمية واحد جي بالمية ها بيصير
+
+322
+00:24:06,250 --> 00:24:10,750
+الهو ايش ماله واحد بيساوي ناقص X تربيع ناقص Y
+
+323
+00:24:10,750 --> 00:24:15,070
+تربيع يعني X تربيع زياد Y تربيع بيساوي ناقص واحد
+
+324
+00:24:15,070 --> 00:24:18,710
+ومش ممكن كمية مربع X تربيع زياد Y تربيع ايش ماله
+
+325
+00:24:18,710 --> 00:24:23,830
+بيساوي واحد عشان هي كفيش رسم بعد المية بالمرة يعني
+
+326
+00:24:23,830 --> 00:24:32,490
+range هاإن عند 100 انازل يعني قيم z من ناقص
+
+327
+00:24:32,490 --> 00:24:38,150
+infinity ل عند 100 ماشي الحال الآن رسمها دي احنا
+
+328
+00:24:38,150 --> 00:24:41,750
+عارفينها رسمها دي اللي هي z بسوء 100 ناقص x تربيع
+
+329
+00:24:41,750 --> 00:24:45,830
+ناقص y تربيع عارفينها عبارة عن اللي هو elliptical
+
+330
+00:24:45,830 --> 00:24:50,540
+paraboloidمظبوط ولا لأ؟ لأن المقطع الأول اللي هو
+
+331
+00:24:50,540 --> 00:24:53,280
+برابولة بين ال X و ال Z و المقطع التاني برابولة
+
+332
+00:24:53,280 --> 00:24:56,220
+بين ال Y و ال Z هاي برابولة بين X و Z و بين
+
+333
+00:24:56,220 --> 00:25:00,080
+برابولة بين Y و Z و المقطع العرضي عبارة عن إيه؟
+
+334
+00:25:00,080 --> 00:25:04,240
+اللي هو Ellipse في الواقع هي عبارة عن دوائر لإن X
+
+335
+00:25:04,240 --> 00:25:07,940
+تربيع و Y تربيع معاملين زي بعض لإنش نعيد طبعا هدف
+
+336
+00:25:07,940 --> 00:25:11,620
+section اللي هو 13 تحكيناها بالتفصيلإذا الان ناخد
+
+337
+00:25:11,620 --> 00:25:15,320
+الفكرة إذا ال graph لهذه هو عربار عن elliptical
+
+338
+00:25:15,320 --> 00:25:20,340
+paraboloid اللي أمامي الان اللي بيهمني الأكتر لأن
+
+339
+00:25:20,340 --> 00:25:24,160
+هذا اتعلمنا رسمه اللي بيهمني الأكتر بدي أعرف أيش
+
+340
+00:25:24,160 --> 00:25:27,860
+اللي هو ال level curves ال level curves يا جماعة
+
+341
+00:25:28,420 --> 00:25:33,760
+اللي هي بترتسم في ال domain في ال XY plane لما
+
+342
+00:25:33,760 --> 00:25:38,680
+أقول أف of XY بيساوي سفر باجي هاي أف of XY بيصير
+
+343
+00:25:38,680 --> 00:25:41,880
+بيساوي سفر يعني بيصير X تربيع زائد Y تربيع بيساوي
+
+344
+00:25:41,880 --> 00:25:47,240
+مية يعني ال level curve هذا هو عبارة عن X تربيع
+
+345
+00:25:47,240 --> 00:25:54,640
+زائد Y تربيع بيساوي مية يعني دائرة نصف قطرها عشرة
+
+346
+00:25:54,640 --> 00:26:02,060
+هذه اللي هي عبارة عنالـ Level Curve اللي هو عبارة
+
+347
+00:26:02,060 --> 00:26:08,740
+عن الـ F of X Y بساوة 0 اللي .. ما .. ما .. ما ..
+
+348
+00:26:08,740 --> 00:26:11,820
+مستوعبش هذه خلّينا نعزز بالتانية ناخد الـ Level
+
+349
+00:26:11,820 --> 00:26:15,340
+Curve F of X Y بساوة 51 الـ Level Curve برضه وين
+
+350
+00:26:15,340 --> 00:26:19,080
+هيرسم في اللي هو الـ domain اللي هو الـ X Y plane
+
+351
+00:26:19,080 --> 00:26:25,000
+في هذه الحالة الآن بحط مكان F of X Y 51 51 بساوة
+
+352
+00:26:25,000 --> 00:26:28,640
+اللي هو 100 نقص X تربيع نقص Y تربيعلما نجيب المية
+
+353
+00:26:28,640 --> 00:26:31,440
+على الطرف هذا و نضرب كله في ناقص بالمية ناقص واحد
+
+354
+00:26:31,440 --> 00:26:34,380
+وخمسين بيصير تسعة واربعين يعني بيصير X تربيع زياد
+
+355
+00:26:34,380 --> 00:26:38,180
+Y تربيع تسعة واربعين يعني برضه عبارة عن دائرة اللي
+
+356
+00:26:38,180 --> 00:26:42,380
+هو مركزها السفر و عبارة عن X تربيع زياد Y تربيع
+
+357
+00:26:42,380 --> 00:26:46,200
+بيساوي تسعة واربعين يعني نفس قطرها إيش سبعة هذه
+
+358
+00:26:46,200 --> 00:26:51,930
+اللي هو ال level curveF of X Y بيساوي 51 Similarly
+
+359
+00:26:51,930 --> 00:26:56,670
+level curve F of X Y بيساوي 75 بشيل هذي بحط 75
+
+360
+00:26:56,670 --> 00:27:01,370
+بطلع X تربيع زي ادوية تربيع بيساوي 25 يعني برضه
+
+361
+00:27:01,370 --> 00:27:06,270
+دائرة نصف خطرها ايش بيساوي 5 يعني كل level curves
+
+362
+00:27:06,270 --> 00:27:11,400
+هذي اللي عندىهي عبارة عن دائرة ساوة خمسة نصف قطرة
+
+363
+00:27:11,400 --> 00:27:16,460
+هذه الأخيرة اللي جابلها دائرة نصف قطرها اللي هي
+
+364
+00:27:16,460 --> 00:27:21,100
+تسعة سبعة اللي هي الأخيرة أو الأولى اللي هي دائرة
+
+365
+00:27:21,100 --> 00:27:25,460
+نصف قطرها عشرة هذه الأربع التلاتة الثلاثة curves
+
+366
+00:27:25,460 --> 00:27:29,100
+هي اللي بنسميها ال level curves level curves is
+
+367
+00:27:29,100 --> 00:27:35,590
+the function of two variablesهي عبارة عن دُزُق من
+
+368
+00:27:35,590 --> 00:27:41,190
+الـ plane هذه اللي نسميها level curves طيب الان
+
+369
+00:27:41,190 --> 00:27:44,030
+functions of three variables اللي حكيناه عن two
+
+370
+00:27:44,030 --> 00:27:47,190
+variables نحكي عنه اللي هو ايش ب three variables
+
+371
+00:27:47,190 --> 00:27:52,520
+هيكون كلام مشابه وسهلThe set of points x, y, z in
+
+372
+00:27:52,520 --> 00:27:56,240
+space where the function of three independent
+
+373
+00:27:56,240 --> 00:27:59,460
+variables has a constant value يعني لما نقول f of
+
+374
+00:27:59,460 --> 00:28:03,340
+x, y, z بيساوي c زي اللي قبل هذا اللي اللي بيطلع
+
+375
+00:28:03,340 --> 00:28:07,880
+عندى بنسمي level إيش مش level curve لإنه كان f of
+
+376
+00:28:07,880 --> 00:28:11,980
+x, y دي كالحالة بيطلع جزء من اللي بلين هذا جزء من
+
+377
+00:28:11,980 --> 00:28:14,220
+ال space على أنه صار فيه عندى three variables
+
+378
+00:28:14,220 --> 00:28:19,100
+بنسميها level surfacesحدش يقولي اعمللي graph لها
+
+379
+00:28:19,100 --> 00:28:22,520
+هذا graph مقدرش لانه بيصير f of x و y و z بيساوي w
+
+380
+00:28:22,520 --> 00:28:26,980
+و هذا بيصير فوق اللي هي أربع متغيرات الفراغة عندنا
+
+381
+00:28:26,980 --> 00:28:33,360
+برسوميش لتلاتة عالمنا لرسومي لتلاتة كرسمالان اذا
+
+382
+00:28:33,360 --> 00:28:36,600
+اللي هي عندي ممكن ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+383
+00:28:36,600 --> 00:28:36,820
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+384
+00:28:36,820 --> 00:28:37,820
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+385
+00:28:37,820 --> 00:28:40,320
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+386
+00:28:40,320 --> 00:28:40,800
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+387
+00:28:40,800 --> 00:28:40,820
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+388
+00:28:40,820 --> 00:28:48,720
+ا ا ا ا ا ا ا ا ا
+
+389
+00:28:48,720 --> 00:28:50,680
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+390
+00:28:50,680 --> 00:28:51,340
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+391
+00:28:51,340 --> 00:28:51,340
+ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+392
+00:29:02,130 --> 00:29:05,650
+الآن الـ value of f is the distance from the
+
+393
+00:29:05,650 --> 00:29:07,630
+origin to the point x, y, z
+
+394
+00:29:13,500 --> 00:29:16,660
+لأن أي level surface اللي هو جدر التربيع الـ X
+
+395
+00:29:16,660 --> 00:29:20,260
+تربيع الـY تربيع الـZ تربيع بيساوي C هذا عبارة عن
+
+396
+00:29:20,260 --> 00:29:23,300
+رسم الـC أكبر من 0 هذه في الواقع عبارة عن رسم
+
+397
+00:29:23,300 --> 00:29:28,060
+الدائرة اللي هي كرة X تربيع Z وY تربيع Z تربيع
+
+398
+00:29:28,060 --> 00:29:34,200
+بيساوي C تربيع الآن is a sphere of radius C
+
+399
+00:29:34,200 --> 00:29:38,620
+centered at the origin لما نلان ناخد مثلا level
+
+400
+00:29:38,620 --> 00:29:44,870
+surface اللي هوأف of x و y و z بتساوي واحد او
+
+401
+00:29:44,870 --> 00:29:49,930
+بتساوي اتنين او بتساوي تلاتة هذه هتكون كرة وهذه
+
+402
+00:29:49,930 --> 00:29:53,500
+كرة وهذه كرةهذه الكورة عارفين ليش؟ لأنه بيصير
+
+403
+00:29:53,500 --> 00:29:55,380
+عبارة عن X تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز وY
+
+404
+00:29:55,380 --> 00:29:56,220
+تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز وY
+
+405
+00:29:56,220 --> 00:30:04,060
+تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز وY
+
+406
+00:30:04,060 --> 00:30:07,060
+تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز وY
+
+407
+00:30:07,060 --> 00:30:12,660
+تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز
+
+408
+00:30:12,660 --> 00:30:20,660
+وY تلفزيز وY تلفزيز وY تلفزيز وY تلفزيز وY تلفز
+
+409
+00:30:21,050 --> 00:30:23,670
+الان نعرف الـ interior points و الـ boundary
+
+410
+00:30:23,670 --> 00:30:30,170
+points كما عرفنا في الاسباس نفس الأسلوب نقول ان
+
+411
+00:30:30,170 --> 00:30:35,530
+النقطة interior point للمنطقة هذه إذا كنت ممكن أن
+
+412
+00:30:35,530 --> 00:30:38,790
+تجد جوار جوار هنا في الاسباس عبارة عن نياشعان كرة
+
+413
+00:30:38,790 --> 00:30:41,970
+إذا كنت ممكن أن تجد كرة كلها في دول الار بسميها
+
+414
+00:30:41,970 --> 00:30:48,020
+interior pointإذا الآن النقطة اللي هي .. مهما زغرت
+
+415
+00:30:48,020 --> 00:30:52,200
+الكورة و مهما عملت الكورة يعني أخدت أي جوار حلجي
+
+416
+00:30:52,200 --> 00:30:55,360
+بيأكل من برا و بيأكل من داخل فبسميها boundary
+
+417
+00:30:55,360 --> 00:30:58,960
+point زي ما قلنا بالظبط هناك في حالة اللي و لبلين
+
+418
+00:30:58,960 --> 00:31:02,080
+عشان هيك اللي هو في عندي boundary points و في
+
+419
+00:31:02,080 --> 00:31:04,780
+interior points ال six اللي بتحتوي ال boundary
+
+420
+00:31:04,780 --> 00:31:08,520
+points إلها كلها بنسميها closed set و ال six اللي
+
+421
+00:31:08,520 --> 00:31:14,790
+بتحتوي بسبتحتوي بس اللي هو interior points بتسميها
+
+422
+00:31:14,790 --> 00:31:18,330
+ايش ما لها open set لكن لو كان فيه انه و لو نقطة
+
+423
+00:31:18,330 --> 00:31:22,410
+واحدة boundary في ال set بتبطل اللي يمين open set
+
+424
+00:31:22,410 --> 00:31:26,030
+و لو في واحدة من ال boundary points مش في ال set
+
+425
+00:31:26,030 --> 00:31:33,090
+بتبطل closed set طيب و هيك بيكون احنا انهينا
+
+426
+00:31:33,090 --> 00:31:36,910
+المحاضرة اللي هي الأولى في شبطة الاربع تاشوية
+
+427
+00:31:36,910 --> 00:31:39,270
+دقائق اخر السلام عليكم و رحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA.srt
new file mode 100644
index 0000000000000000000000000000000000000000..49d96b57f37d214e1641b8422f5acba16257c7fa
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA.srt
@@ -0,0 +1,1975 @@
+1
+00:00:00,540 --> 00:00:04,980
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,980 --> 00:00:11,540
+وبركاته هذه هي المحاضرة رقم 23 مساق التكامل المزدوج
+
+3
+00:00:11,540 --> 00:00:16,220
+جيم وهي المحاضرة الرابعة في الـ chapter 15،
+
+4
+00:00:16,220 --> 00:00:19,360
+chapter الأخير في المادة وهي تحت عنوان double
+
+5
+00:00:19,360 --> 00:00:24,210
+integrals in polar form. تحدثنا سابقاً عن double
+
+6
+00:00:24,210 --> 00:00:28,290
+integral in Cartesian form. الآن بدنا نتحدث عن
+
+7
+00:00:28,290 --> 00:00:31,850
+double integral in polar form. طبعاً، لأسباب
+
+8
+00:00:31,850 --> 00:00:35,710
+أحياناً، اللي هو إيجاد اللي هو الـ double integration
+
+9
+00:00:35,710 --> 00:00:41,050
+بالـ integral form أسهل كثيراً كثيراً من إيجاده في الـ
+
+10
+00:00:41,050 --> 00:00:46,050
+Cartesian، و بتحل لنا مشاكل اللي هو قد تنشأ في الـ
+
+11
+00:00:46,050 --> 00:00:48,110
+Cartesian كما سنظهر لاحقاً.
+
+12
+00:00:50,710 --> 00:00:55,150
+بتحلها بواسطة اللي هو مين؟ الـ polar form. طبعاً، و
+
+13
+00:00:55,150 --> 00:00:58,230
+العكس صحيح، بعض الأسئلة بتكون في الـ Cartesian أسهل.
+
+14
+00:00:58,230 --> 00:01:01,030
+طبعاً، الناس اللي هي متعودة أصلاً على الـ Cartesian
+
+15
+00:01:01,030 --> 00:01:06,850
+لكن الـ polar، زي ما قلنا، بيحل لنا مشاكل. نشوف الآن، نروح
+
+16
+00:01:06,850 --> 00:01:12,510
+للموضوع نفسه اللي هو السابق. كيف اللي هو المهم عندي
+
+17
+00:01:12,510 --> 00:01:16,610
+تحديد منطقة اللي هي المنطقة اللي بدي أكامل عليها.
+
+18
+00:01:16,610 --> 00:01:20,030
+بتحدد لي اللي هي limits of integration. إذاً،
+
+19
+00:01:20,030 --> 00:01:23,470
+integrals and polar coordinates. بدي أُحدد أول شيء
+
+20
+00:01:23,470 --> 00:01:26,310
+اللي هو find limits of integration. بالضبط زي ما
+
+21
+00:01:26,310 --> 00:01:30,730
+كنا نعمل اللي هو في الـ Cartesian، ولكن هنا اللي هو
+
+22
+00:01:30,730 --> 00:01:36,390
+طبيعة الـ polar بتختلف عن طبيعة الـ Cartesian، فبنشوف
+
+23
+00:01:36,390 --> 00:01:39,830
+كيف بنحدد اللي هو الـ integration. طبعاً، احنا بدنا
+
+24
+00:01:39,830 --> 00:01:43,470
+نيجي، يعني، يُعرض علينا الـ integration اللي أمامنا، الـ
+
+25
+00:01:43,470 --> 00:01:47,510
+integration لـ F(r,θ) d A، حيث d A طبعاً في
+
+26
+00:01:47,510 --> 00:01:51,570
+هذه الحالة اللي هي dx dy، لكن الـ dx dy زي ما
+
+27
+00:01:51,570 --> 00:01:57,690
+أخذنا في التفاضل والتكامل، اللي هو عبارة عن r dr dθ. الآن
+
+28
+00:01:57,690 --> 00:02:00,750
+بدل d A سيصبح d A = r dr dθ.
+
+29
+00:02:00,750 --> 00:02:05,390
+(Lines 29-34 removed as they are repetitive and contain no errors other than unnecessary repetition)
+
+35
+00:02:24,870 --> 00:02:30,260
+لقيمة r وقيمة θ، ونشوف كيف لاحقاً، لأن أول حاجة
+
+36
+00:02:30,260 --> 00:02:37,350
+احنا في هذه الحالة بنكامل بالنسبة لـ r. الآن، نظراً
+
+37
+00:02:37,350 --> 00:02:40,250
+لطبيعة الـ integration. طبعاً، في عندي r dr dθ، و
+
+38
+00:02:40,250 --> 00:02:44,070
+هنلاقي اللي هو التكامل الـ r عنده اللي هو بيحل لنا
+
+39
+00:02:44,070 --> 00:02:47,950
+كثيراً من المشاكل. الآن find the r limits of integration.
+
+40
+00:02:47,950 --> 00:02:53,090
+الآن، طبيعة الـ r limits، اللي هي الـ r، الـ r طبعاً زي
+
+41
+00:02:53,090 --> 00:02:55,730
+ما أنتم مذكرين، إنّ إشعاع الـ ray يبدأ من نقطة
+
+42
+00:02:55,730 --> 00:02:59,950
+الأصل، وبيطلع إيش؟ إلى أعلى، اللي هي بواسطة الـ ray
+
+43
+00:03:00,540 --> 00:03:03,900
+الآن، الـ ray هذا بدنا نشوف كيف بيدخل، ومن وين بيدخل
+
+44
+00:03:03,900 --> 00:03:07,340
+المنطقة، ومن وين بيخرج منها؟ من وين بيدخل المنطقة؟
+
+45
+00:03:07,340 --> 00:03:12,660
+بيكون اللي هو الحد الأسفل، ومن وين بيخرج منها؟ الحد
+
+46
+00:03:12,660 --> 00:03:16,340
+الأعلى. زي في حالة الـ Cartesian. نيجي نحدد الـ θ.
+
+47
+00:03:16,340 --> 00:03:21,140
+θ. الآن، بنشوف كيف بتمسح المنطقة. طبعاً، بنبدأ من
+
+48
+00:03:21,140 --> 00:03:24,220
+وين؟ عند الـ positive x axis، ونطلع عكس عقارب
+
+49
+00:03:24,220 --> 00:03:28,440
+الساعة، من حد الـ θ. فبنشوف من أين بدأت المنطقة،
+
+50
+00:03:28,440 --> 00:03:32,260
+ومن أين انتهت. بتكون هي حدود θ. خلينا نشوف هذا
+
+51
+00:03:32,260 --> 00:03:39,080
+الكلام عملياً. لو عُرض علينا هيك عندي F(r,θ) dA،
+
+52
+00:03:39,080 --> 00:03:43,220
+over a region R. الـ region R اللي عُرضت علينا هي الـ
+
+53
+00:03:43,220 --> 00:03:46,370
+region التالية. عامل ايه هو؟ ليش بالـ Cartesian؟ قال
+
+54
+00:03:46,370 --> 00:03:49,130
+اللي هو بدي x² + y² = 4. هذه
+
+55
+00:03:49,130 --> 00:03:52,330
+جزء من مين؟ من دائرة. قال، بدي هذا الجزء اللي في
+
+56
+00:03:52,330 --> 00:03:57,090
+الربع الأول. R بوصف R، قاعد الجزء اللي هي R اللي في
+
+57
+00:03:57,090 --> 00:03:59,710
+الربع الأول، المكون من أعلى بـ x² + y²
+
+58
+00:03:59,710 --> 00:04:03,450
+= 4، اللي عبارة عن الدائرة، ومن أسفل
+
+59
+00:04:03,450 --> 00:04:08,850
+بواسطة y = √2، وعن اليسار بواسطة اللي
+
+60
+00:04:08,850 --> 00:04:15,350
+هو الـ x = 0، أو الـ y axis. هذه هي المنطقة
+
+61
+00:04:15,350 --> 00:04:19,630
+المحدودة، اللي عليها بدنا نكامل. الآن، كلها
+
+62
+00:04:19,630 --> 00:04:24,130
+مكتوبة بإيش؟ بالـ Cartesian. بدي أحولها لإيش؟ للـ polar.
+
+63
+00:04:24,130 --> 00:04:30,130
+الآن، عندي النقطة هذه، نقطة التقاطع بين هذا المستقيم
+
+64
+00:04:30,130 --> 00:04:33,940
+وهذا الدائرة. إيجاد النقطة هذه سهل، حط y = √2
+
+65
+00:04:33,940 --> 00:04:37,800
+هنا، اللي هي عبارة عن y = 2. الـ y²
+
+66
+00:04:37,800 --> 00:04:40,520
+هيجي اللي هنا بيصير اللي هي 4 - 2 = 2.
+
+67
+00:04:40,520 --> 00:04:43,520
+يعني بيصير x² = 2، يعني x =
+
+68
+00:04:43,520 --> 00:04:46,620
+√2، لأنه لهذه المنطقة موجبة. فبيصير عند
+
+69
+00:04:46,620 --> 00:04:50,260
+النقطة هذه (√2, √2). المنطقة هذه طبعاً
+
+70
+00:04:50,260 --> 00:04:53,860
+y = √2، معناته هذه نقطة (√2, √2). وهذا
+
+71
+00:04:53,860 --> 00:04:56,400
+اللي هو x² + y² = 4 عبارة
+
+72
+00:04:56,400 --> 00:05:01,360
+عن دائرة نصف قطرها = 2. هي الـ r، نصف القطر.
+
+73
+00:05:01,360 --> 00:05:04,520
+إذاً هذه إحداثيات (2, 2). إذا عرفنا كل الإحداثيات
+
+74
+00:05:04,520 --> 00:05:09,640
+لهذه. الآن بدنا نحول هذه للـ polar عشان اللي هو نشتغل
+
+75
+00:05:09,640 --> 00:05:14,100
+على الـ polar، نكامل منها. طيب، الآن احنا بنعرف علاقات
+
+76
+00:05:14,100 --> 00:05:20,940
+قديمة بين x و y و r و θ. الـ x، هأذكركم فيها، بيساوي r
+
+77
+00:05:20,940 --> 00:05:27,860
+cos θ. الـ y إيش بتساوي؟ r sin θ. ماشي الحال. إذاً x
+
+78
+00:05:27,860 --> 00:05:30,860
+² + y² بنقدر نحولها، بيصير x²
+
+79
+00:05:30,860 --> 00:05:36,660
+= r² cos² θ + الـ y = r sin θ
+
+80
+00:05:36,660 --> 00:05:41,420
+بيصير + r² sin² θ = إيش؟ 4.
+
+81
+00:05:41,420 --> 00:05:45,120
+خذ من هذول عامل مشترك r²، بيصير r²
+
+82
+00:05:45,120 --> 00:05:48,800
+مضروبة في cos² + sin² = 4. زي
+
+83
+00:05:48,800 --> 00:05:51,520
+ما احنا عارفين الـ cos² + الـ sin² إيش
+
+84
+00:05:51,520 --> 00:05:56,340
+بتساوي 1، يعني r² = 4، يعني r
+
+85
+00:05:56,340 --> 00:05:59,540
+= 2. و r = 2 احنا مشهور عندنا في
+
+86
+00:05:59,540 --> 00:06:03,230
+الـ polar هاي r = 2، هاي الدائرة هذه مشهورة في
+
+87
+00:06:03,230 --> 00:06:07,350
+الـ polar إن هي معادلة الدائرة اللي نصف قطرها إيش؟
+
+88
+00:06:07,350 --> 00:06:13,410
+اللي هي عبارة عن 2. طيب، الآن نيجي للخط هذا، هذا
+
+89
+00:06:13,410 --> 00:06:16,410
+الخط عبارة عن y = √2. شيل الـ y وحط
+
+90
+00:06:16,410 --> 00:06:22,140
+قيمته. إيش قيمته؟ r sin θ. اللي ناسيه. إذاً r sin θ
+
+91
+00:06:22,140 --> 00:06:26,900
+= y = √2. نحولها للـ polar. الـ polar
+
+92
+00:06:26,900 --> 00:06:30,780
+بيصير r = √2 / sin θ، يعني r =
+
+93
+00:06:30,780 --> 00:06:35,140
+√2 csc θ. مقلوب الـ sin. إذاً صار
+
+94
+00:06:35,140 --> 00:06:39,480
+الخط هذا يا جماعة الثاني عبارة عن r = √2
+
+95
+00:06:39,480 --> 00:06:43,310
+csc θ. الآن الأمور سهلة، وشوفوا أكيد سهلة.
+
+96
+00:06:43,310 --> 00:06:47,250
+هذه حولناها كلها للـ polar. ماشي الحال. نشوف الآن
+
+97
+00:06:47,250 --> 00:06:51,350
+الآن نيجي اللي هو بدنا نيجي اللي هو evaluation.
+
+98
+00:06:51,350 --> 00:06:55,410
+كامل بالنسبة لمين؟ r. الـ r طبيعته كيف بتحسب؟ اللي هو
+
+99
+00:06:55,410 --> 00:06:58,870
+الـ ray. الـ ray. الـ ray. الـ ray بتطلع من مين؟ من الصفر.
+
+100
+00:06:59,190 --> 00:07:02,350
+الآن هذا الـ ray بدنا نشوف وين بيدخل المنطقة، و
+
+101
+00:07:02,350 --> 00:07:06,230
+وين بيخرج منها؟ و هل كل الـ rays اللي بتدخل و بتخرج
+
+102
+00:07:06,230 --> 00:07:09,750
+من المنطقة على نفس المنوال؟ زي زمان، يعني ما بنفعش
+
+103
+00:07:09,750 --> 00:07:13,930
+نجسمها. إذا بتمشي على نفس المنوال لازم نجسمها. إذا
+
+104
+00:07:13,930 --> 00:07:17,970
+بتمشي على نفس المنوال، ما لازم نجسمها. الآن، هنيجي
+
+105
+00:07:17,970 --> 00:07:23,050
+بناخذ من هنا ray. راح نلاحظ الـ r دخل من عند الخط هذا، و
+
+106
+00:07:23,050 --> 00:07:27,050
+طلع من عند خط الدائرة. خذ الـ r ثاني، خذ الـ r ثالث.
+
+107
+00:07:27,050 --> 00:07:30,610
+كلّهم طبعاً، نقطة الرأس اللي بتطلع، كلّهم على نفس
+
+108
+00:07:30,610 --> 00:07:34,590
+المنوال. إذا مرتاحين. إذاً، خلينا نشوف r، r وين دخلت؟
+
+109
+00:07:34,590 --> 00:07:39,570
+دخلت من هنا. إيش قيمة r هنا؟ على هذا، قيمته اللي هو
+
+110
+00:07:39,570 --> 00:07:43,350
+زي ما اتفقنا √2 csc θ. هذا الحد الأولاني.
+
+111
+00:07:43,350 --> 00:07:48,510
+إيش بيساوي؟ √2 csc θ. طب، من وين خرج؟ خرج من
+
+112
+00:07:48,510 --> 00:07:51,650
+الدائرة، r = 2. إذاً، إيش الحد الثاني للـ
+
+113
+00:07:51,650 --> 00:07:55,830
+r؟ اللي هو r = 2. خلصنا الـ r. بدنا
+
+114
+00:07:55,830 --> 00:08:00,390
+نيجي نحدد مين؟ θ. الآن، θ، الـ θ بتبدأ من عند
+
+115
+00:08:00,390 --> 00:08:04,630
+القياسات بتبدأ من الصغير، اللي هو اللي هي مع الـ
+
+116
+00:08:04,630 --> 00:08:09,820
+positive x axis، وبعدين بتبدأ تلف. وأنتم عارفين كيف
+
+117
+00:08:09,820 --> 00:08:14,400
+قيام الـ θ. الآن بتلف هيك بالاتجاه الموجب، عكس
+
+118
+00:08:14,400 --> 00:08:18,160
+عقارب الساعة، وهيك بالاتجاه السالب. طيب، خلينا نحدد
+
+119
+00:08:18,160 --> 00:08:21,480
+منطقتنا. منطقتنا، شايف إنّه أنت بتبدأ، هي أول ray،
+
+120
+00:08:21,480 --> 00:08:25,320
+هي أول ray، هي ثاني ray، هي ثالث ray، وبيظلوا
+
+121
+00:08:25,320 --> 00:08:29,660
+يمشي يمشي. تخيل نور بيمشي هيك بالـ ray عشان يغطي
+
+122
+00:08:29,660 --> 00:08:32,560
+المنطقة هذه. هيمشي من هنا لهنا. إذاً، هيبدأ من هنا الـ
+
+123
+00:08:32,560 --> 00:08:36,910
+ray. الـ θ تبعته يجب أن نجده. هذه الـ θ تبعته يجب
+
+124
+00:08:36,910 --> 00:08:45,510
+(Lines 124-129 removed as they are repetitive and contain no errors other than unnecessary repetition)
+
+130
+00:08:51,470 --> 00:08:56,150
+أن نجده. هذه الـ θ تبعته يجب أن نجده. هذه الـ θ
+
+131
+00:08:56,180 --> 00:09:01,120
+مدام عندي y = x، معناته اللي هو هذا تبع الـ y
+
+132
+00:09:01,120 --> 00:09:04,440
+بيساوي هذا تبع الـ x، يعني المثلث القائم هذا عبارة
+
+133
+00:09:04,440 --> 00:09:07,680
+عن مثلث متساوي الساقين، يعني هذه الزاوية بِتساوِي
+
+134
+00:09:07,680 --> 00:09:11,180
+هذه، يعني هذه 45 درجة، وهذه 45 درجة. إذاً هذه الزاوية
+
+135
+00:09:11,180 --> 00:09:16,180
+عبارة عن 45 درجة، للبداية اللي هي π/4. إذاً
+
+136
+00:09:16,180 --> 00:09:19,460
+قيمتها بتبدأ من π/4، وشوفوا لوين بتبقى تمشي.
+
+137
+00:09:19,460 --> 00:09:23,450
+تمشي تمشي لما تصل لهذه. هذا طبعاً كلكم قالوا إيش؟ عبارة
+
+138
+00:09:23,450 --> 00:09:27,230
+عن π/2. إذاً θ من π/4 لعند مين؟
+
+139
+00:09:27,230 --> 00:09:31,690
+يا جماعة؟ لعند π/2. هذا مثال، ناخد مثال آخر.
+
+140
+00:09:31,690 --> 00:09:34,670
+إذاً أنا بقول: find the limits of integration for
+
+141
+00:09:34,670 --> 00:09:38,550
+integrating F(r, θ). بده بس limits of إيش؟ of
+
+142
+00:09:38,550 --> 00:09:43,350
+integration لما بدنا نكامل over F(r, θ) over
+
+143
+00:09:43,350 --> 00:09:47,430
+the region R. إيش الـ region R؟ هيو بوصف لك إياها
+
+144
+00:09:47,430 --> 00:09:52,870
+تماماً. That lies inside the cardioid r = 1 +
+
+145
+00:09:52,870 --> 00:09:57,770
+cos θ، and outside r = 1. أنا عارف إنكم أنتم
+
+146
+00:09:57,770 --> 00:10:01,460
+ناسين اللي هي الرسومات اللي هي تبعات التفاضل والتكامل. أنا
+
+147
+00:10:01,460 --> 00:10:06,340
+مش جاي اللي هو أني أخبركم في الرسومات هذه. الرسومات
+
+148
+00:10:06,340 --> 00:10:10,380
+هذه اللي أنتم بتنسوها. احنا لا، بنعمل لكم إياها، لكن
+
+149
+00:10:10,380 --> 00:10:14,000
+في بعض الرسومات مجدور عليها، اللي الواحد ممكن يستذكرها
+
+150
+00:10:14,000 --> 00:10:19,900
+من خلال اللي هو يعني دراسته البسيطة في التفاضل والتكامل.
+
+151
+00:10:19,900 --> 00:10:24,520
+الـ gym، وما عرفته بتفاضل. ده، لكن بقدر الإمكان اللي هو
+
+152
+00:10:24,520 --> 00:10:28,540
+تجيبوش من موضوع رسم، عشان نجيب لكم رسم معقد بحيث أنه
+
+153
+00:10:28,540 --> 00:10:31,220
+أنتم ناسين. أنا مش جاي أختبركم في الرسم، أُداء، أنا
+
+154
+00:10:31,220 --> 00:10:34,980
+جاي أختبركم في الـ double integration over اللي هو
+
+155
+00:10:34,980 --> 00:10:39,880
+اللي هي a region in polar form. صلوا على النبي عليه
+
+156
+00:10:39,880 --> 00:10:42,880
+الصلاة والسلام. الآن، الـ cardioid r = 1 +
+
+157
+00:10:42,880 --> 00:10:47,100
+cos θ. الآن، الـ cardioid هذا، r = 1 +
+
+158
+00:10:47,100 --> 00:10:51,300
+cos θ، هيو. لو واحد اللي جرب يرسمه، بيعرف يرسمه.
+
+159
+00:10:51,300 --> 00:10:55,360
+بتبدأ بـ θ = 0، اللي هتلاقيها، ده اللي هي عبارة عن
+
+160
+00:10:55,360 --> 00:10:59,940
+اللي هي θ = 0، مثلاً بتصير cos θ = 1، بتصير
+
+161
+00:10:59,940 --> 00:11:03,870
+هذه اللي هي 2. بتبدأ من هنا، ثم تبدأ تطلع لما
+
+162
+00:11:03,870 --> 00:11:17,250
+(Lines 162-164 removed as they are repetitive and contain no errors other than unnecessary repetition)
+
+165
+00:11:17,250 --> 00:11:21,650
+π/2. هذه مرة أخرى لما اكتملت الدورة. مش هي
+
+166
+00:11:21,650 --> 00:11:25,450
+موضوع نهائي، بس هذا للتذكير بس. الآن، الآن هو بوصف
+
+167
+00:11:25,450 --> 00:11:29,150
+المنطقة، هي المنطقة اللي بين الـ cardioid هذا
+
+168
+00:11:29,150 --> 00:11:32,530
+والدائرة r = 1. ده. نعرف إنّ الآن r =
+
+169
+00:11:32,530 --> 00:11:35,310
+1، r = 5، r = 4، كلهم دوائر،
+
+170
+00:11:35,310 --> 00:11:38,950
+لأن r = 1 عبارة عن دائرة نصف قطرها إيش؟
+
+171
+00:11:38,950 --> 00:11:43,990
+1. لأن هذه المنطقة هي المنطقة المقصودة في إن نجد
+
+172
+00:11:43,990 --> 00:11:47,730
+حدود الـ integration لها. صلوا على النبي عليه
+
+173
+00:11:47,730 --> 00:11:51,190
+الصلاة والسلام. زي ما بقولوا: أول حاجة بنرسم المنطقة.
+
+174
+00:11:51,190 --> 00:11:55,170
+ثاني حاجة بنرسم اللي هي المناطق اللي هي التقاطع اللي
+
+175
+00:11:55,170 --> 00:11:59,770
+بين الجهتين، وبنحدد اللي هو حدود r، وبعدين بنحدد مين؟
+
+176
+00:11:59,770 --> 00:12:04,180
+θ. الآن، عشان أُحدد r و θ، بيلزمنا نقطة تقاطع.حدى نقطة
+
+177
+00:12:04,180 --> 00:12:06,720
+تقاطع.حدى. زي ما كنا نعمل في الـ Cartesian عادي.
+
+178
+00:12:06,720 --> 00:12:11,040
+الآن، نقطة التقاطع بين مين؟ يعني هذه؟ هذه بين اللي
+
+179
+00:12:11,040 --> 00:12:14,960
+هو r = 1، ونفسهاشيء هذه، وبين r = 1
+
+223
+00:15:03,290 --> 00:15:07,930
+إيش بتساوي في الـ polar اللي هو R dR dθ هذا أخذناه
+
+224
+00:15:07,930 --> 00:15:12,030
+الكلام بالتفصيل في تفاضل  ،إذن بس أذكركم فيه
+
+225
+00:15:12,030 --> 00:15:16,890
+اللي هي dA اللي هي dx dy بتساوي R dR dθ إذن ال area
+
+226
+00:15:16,890 --> 00:15:20,610
+بتساوي إلى الآن ال area of a closed bounded region
+
+227
+00:15:20,610 --> 00:15:24,890
+R in polar coordinates، الآن هي كده وهي اللي أمنّا
+
+228
+00:15:25,150 --> 00:15:28,430
+خلّيني آخذ مثال ‏find the area enclosed by the
+
+229
+00:15:28,430 --> 00:15:33,670
+lemniscate r = 4 cos 2θ ‏الآن هذه الرسمة ‏الآن طبعا
+
+230
+00:15:33,670 --> 00:15:39,230
+أكيد كثير منكم ناسي تفاصيلها ‏الآن إذا بدك تفصّل في
+
+231
+00:15:39,230 --> 00:15:43,110
+الرسم ‏بتقدر تفصل، تأخذ اللي هو قيم لـ θ ‏،بحيث بكده
+
+232
+00:15:43,110 --> 00:15:47,490
+بكده ‏بتبدأ تشتغل ‏حتى تصل للرسمة اللي أمامك
+
+233
+00:15:47,530 --> 00:15:51,050
+طبعا كنا نشوف فيه تناظر حول الـ x-axis، حول الـ
+
+234
+00:15:51,050 --> 00:15:53,730
+origin، حول هذا الكلام، هذا الآن كلها
+
+235
+00:15:53,730 --> 00:15:57,150
+بتساعدنا في الرسم، لكن هي الرسمة الآن، الرسمة موجودة
+
+236
+00:15:57,150 --> 00:16:00,310
+أمامي الآن، بدأ نوجد ال area للمنطقة هذه اللي هي
+
+237
+00:16:00,310 --> 00:16:03,110
+الرسمة اللي أمامي، طب أنا عشان أوجد ال area، طب أنا
+
+238
+00:16:03,110 --> 00:16:06,700
+شايف هذه المنطقة هي زي هذه بالظبط، طب ما أحسب هذه و
+
+239
+00:16:06,700 --> 00:16:09,120
+أضربها في اثنين، لا، حد بيقول لي لأ يا أستاذ ما هي
+
+240
+00:16:09,120 --> 00:16:13,320
+أصلًا هذه اللي هي المنطقة هي زي هذه، و زي هذه، و زي
+
+241
+00:16:13,320 --> 00:16:16,860
+هذه، بقول له صح، إذًا خلّيني أقتنع بكلامك، إذًا هذه بدي
+
+242
+00:16:16,860 --> 00:16:22,720
+آخذها الآن هي المنطقة، وأضربها في أربعة، يعني بيصير
+
+243
+00:16:22,720 --> 00:16:26,500
+ال area عندي، ال area بتساوي أربعة في قيمة ال
+
+244
+00:16:26,500 --> 00:16:31,490
+integration لـ R dR dθ على هذه المنطقة، طب هذه المنطقة
+
+245
+00:16:31,490 --> 00:16:34,750
+اللي هي داخل، طبعًا كله بيدينا اللي هو area enclosed
+
+246
+00:16:34,750 --> 00:16:37,870
+by the lemniscate، يعني ال area اللي جوا هذه كلها
+
+247
+00:16:37,870 --> 00:16:42,390
+الآن هذه، الآن بدنا نحدد حدود ال integration اللي لها
+
+248
+00:16:42,390 --> 00:16:46,670
+حدود ال integration سهلة، الـ R هي بتطلع من الأصل
+
+249
+00:16:46,670 --> 00:16:49,810
+وبتجي على مين؟ اللي هو بتدخل المنطقة من الأصل و
+
+250
+00:16:49,810 --> 00:16:53,390
+بتطلع، بضرب في مين؟ في الرسمة اللي هي الرسمة إيش؟
+
+251
+00:16:53,390 --> 00:16:57,320
+اللي هي بتساوي، القيمة R هي جذر التربيعي لـ 4
+
+252
+00:16:57,320 --> 00:17:00,920
+cos 2θ، إذًا ببدأ من صفر وبروح لجذر
+
+253
+00:17:00,920 --> 00:17:05,080
+الـ 4 cos 2θ، هذه اللي هي حدود المنطقة
+
+254
+00:17:05,080 --> 00:17:10,360
+طيب الآن θ، نشوف الـ θ الآن هذه واضحة أن θ
+
+255
+00:17:10,360 --> 00:17:15,520
+من هنا عندها بصفر، ليش؟ لأن لما كانت θ بصفر بيصير
+
+256
+00:17:15,520 --> 00:17:18,580
+اللي عبارة عن القيمة جذر 4 cos صفر = 4
+
+257
+00:17:18,580 --> 00:17:21,800
+يعني هذه المنطقة أربعة فعلًا، إذًا هذه بدأنا احنا
+
+258
+00:17:21,800 --> 00:17:26,290
+معينين θ بصفر، طب الآن لما وصلنا اللي هي رجعنا على
+
+259
+00:17:26,290 --> 00:17:30,670
+مين؟ على نقطة الأصل، رجع يعني وكأنها رجعت R بصفر
+
+260
+00:17:30,670 --> 00:17:34,910
+نشوف هذه قيمة θ عندها دي لما R بصفر بيصير 4
+
+261
+00:17:34,910 --> 00:17:38,010
+cos 2θ = 0، يعني cos 2θ
+
+262
+00:17:38,010 --> 00:17:40,870
+= 0، فاهمين اللي عملته الآن؟ cos 2
+
+263
+00:17:40,870 --> 00:17:43,810
+θ = 0، إذًا 2θ = π
+
+264
+00:17:43,810 --> 00:17:49,290
+على اثنين، يعني θ = π على أربعة، إذًا هذه
+
+265
+00:17:49,290 --> 00:17:51,710
+الزاوية،  خليتها طبعًا هذه π على أربعة مش هتكون
+
+266
+00:17:51,710 --> 00:17:55,240
+اللي هي إيش؟ اللي هي المنطقة  اللي هي 3π
+
+267
+00:17:55,240 --> 00:17:59,000
+على أربعة مثلا؟ لأ هذه π على أربعة اللي هي بتحقق
+
+268
+00:17:59,000 --> 00:18:01,900
+هذه، أكثر من القيمة، من ضمن الـ π على أربعة وهو
+
+269
+00:18:01,900 --> 00:18:06,200
+آخرها، فهذه π على أربعة، ماشي الحال، إذًا هذه صارت π
+
+270
+00:18:06,200 --> 00:18:11,360
+على أربعة، إذًا صار عندي ال integration من صفر لـ π
+
+271
+00:18:11,360 --> 00:18:14,280
+على أربعة، لأنه بدأ بالصفر، طبعًا، من π على أربعة، هي
+
+272
+00:18:14,280 --> 00:18:17,960
+الصفر اللي عند π على أربعة، R dR dθ، والآن
+
+273
+00:18:17,960 --> 00:18:20,760
+إيش بدنا نساوي؟ بدنا نكامل، نكامل بالنسبة لـ R،
+
+274
+00:18:20,760 --> 00:18:23,540
+بيصير R تربيع على اثنين، الآن حدود ال integration
+
+275
+00:18:23,540 --> 00:18:26,800
+من صفر لجذر التربيعي لـ 4 cos 2θ، بنعوض
+
+276
+00:18:26,800 --> 00:18:31,260
+عنها R بصفر هي صفر، وبيضل اللي بنعوض عنها
+
+277
+00:18:31,260 --> 00:18:35,880
+اللي فوق، بيصير جذر الـ 4 بيصير اللي هو لما نربعها
+
+278
+00:18:35,880 --> 00:18:40,600
+بيصير اللي هو عبارة عن 2 cos 2θ طبعًا
+
+279
+00:18:40,600 --> 00:18:42,860
+بيصير أربعة، وبقسمها على اثنين بيصير 2 cos
+
+280
+00:18:42,860 --> 00:18:45,860
+2θ، فاهمين؟ R تربيع = 4 cos 2
+
+281
+00:18:45,860 --> 00:18:49,020
+θ على اثنين، برجع 2 cos 2θ dθ
+
+282
+00:18:49,200 --> 00:18:53,100
+الآن بكامل من صفر لـ π على اثنين dθ، هذا اللي
+
+283
+00:18:53,100 --> 00:19:02,600
+هو عبارة عن اللي هو sin 2θ مضبوط على اثنين
+
+284
+00:19:02,600 --> 00:19:05,480
+اللي هي الـ θ اللي هنا بيصير عندي اللي هو sin
+
+285
+00:19:05,480 --> 00:19:07,700
+2θ، وفي أربعة برا بيصير 4 sin 2
+
+286
+00:19:07,700 --> 00:19:10,780
+θ، ومن صفر لعند π على أربعة، بيطلع عندي مين يا
+
+287
+00:19:10,780 --> 00:19:15,290
+جماعة؟ بيطلع عندي أربعة، وبكون هيك خلصنا، أوجدنا ال
+
+288
+00:19:15,290 --> 00:19:18,610
+area، ونيجي الآن ننتقل، cartesian integral into
+
+289
+00:19:18,610 --> 00:19:26,010
+polar integrals، نشوف كيف بتحول ال cartesian إلى
+
+290
+00:19:26,010 --> 00:19:31,800
+polar، أول شيء عندي اللي هي ثلاث امور لازم نغير فيها
+
+291
+00:19:31,800 --> 00:19:36,040
+أول شيء ال region R بتكون موجودة بالكارتيزيان، ومن
+
+292
+00:19:36,040 --> 00:19:40,980
+ثم بنحولها إلى G في اللي هي البولر، وهذه يمكن اللي
+
+293
+00:19:40,980 --> 00:19:45,000
+هي من أهم الخطوات، أو الخطوة اللي...اللي بدها شوية
+
+294
+00:19:45,000 --> 00:19:49,680
+شغل، الآن f(x,y) على طول بنحولها لـ f(r cos θ,
+
+295
+00:19:49,680 --> 00:19:56,180
+r sin θ)، يعني بدلالة θ و بدلالة r حسب القوانين اللي
+
+296
+00:19:56,180 --> 00:20:00,180
+ذكرناها سابقًا اللي هي الـ x = r cos θ، والـ y
+
+297
+00:20:00,180 --> 00:20:09,510
+= r sin θ، أو الشيء الثاني dx dy بنحولها
+
+298
+00:20:09,510 --> 00:20:19,710
+لـ R dR dθ، يعني هذا التحويل وهو بأفضل القوانين، الآن
+
+299
+00:20:19,710 --> 00:20:23,150
+اللي بده جهده اللي هو تحديد حدود التكامل اللي هو
+
+300
+00:20:23,150 --> 00:20:27,770
+من Cartesian R إلى اللي هي polar اللي هي
+
+301
+00:20:27,770 --> 00:20:31,890
+coordinates، نيجي الآن نأخذ مثال، evaluate the
+
+302
+00:20:31,890 --> 00:20:35,610
+double integral over region R، إيش ال region R؟ ال
+
+303
+00:20:35,610 --> 00:20:39,190
+region R، هيعطيني إياها هنا،  نجي نقول إيش هي؟ يجي
+
+304
+00:20:39,190 --> 00:20:45,840
+ده x تربيع + y تربيع dy dx، الآن where R is the
+
+305
+00:20:45,840 --> 00:20:49,300
+semicircular region bounded by the x-axis، اللي هي
+
+306
+00:20:49,300 --> 00:20:52,120
+semicircular region، نصف الدائرة هذه اللي bounded by
+
+307
+00:20:52,120 --> 00:20:56,360
+the x-axis and the curve y = √(1 -
+
+308
+00:20:56,360 --> 00:21:01,560
+x²) ، اللي هي عبارة عن الجزء الموجب من الدائرة y تربيع
+
+309
+00:21:01,560 --> 00:21:05,650
++ x تربيع = 1، ماشي الحال، إذًا هي عند
+
+310
+00:21:05,650 --> 00:21:09,910
+المنطقة R، وهي عند الدالة اللي بنكاملها، لو أجينا
+
+311
+00:21:09,910 --> 00:21:14,350
+وأردنا نكامل هذه بواسطة الكارتيزيان، كم  لألقينا
+
+312
+00:21:14,350 --> 00:21:17,590
+x تربيع + y تربيع، بالنسبة لإيش؟ هنعرف نكامل
+
+313
+00:21:17,590 --> 00:21:22,750
+by as an elementary integration، إذًا الآن في عندنا مشكلة،
+
+314
+00:21:22,750 --> 00:21:27,390
+التكامل، ما بعرف أكامل اللي هي e^(x³+y³)، وهذا
+
+315
+00:21:27,390 --> 00:21:31,490
+التكامل هذا بالذات، التكامل هتلاقوه كثير بيظهر عندنا
+
+316
+00:21:31,490 --> 00:21:37,330
+في الإحصاء أو اللي هو في الاحتمالات، فبإلزامنا نوجدّه
+
+317
+00:21:37,330 --> 00:21:41,030
+فاللي هو الناس بتضطر تروح له، عشان توجدّه، ليه؟
+
+318
+00:21:41,030 --> 00:21:45,150
+للبولر، وبتحول للبولر، والإيجاد بيصير إيجاد سهل
+
+319
+00:21:45,150 --> 00:21:49,140
+كثير، خلّينا نشوف الآن، إذًا عشان إيجاد هذا بدي أحوّله
+
+320
+00:21:49,140 --> 00:21:53,600
+للبولر، هي عندي ال integration أحوله للبولر، أول
+
+321
+00:21:53,600 --> 00:21:59,200
+شيء بدي أشيل ال dy dx وأحط مكانها R dR dθ، وe
+
+322
+00:21:59,200 --> 00:22:02,220
+to the x تربيع + y تربيع، احنا بنعرف x تربيع +
+
+323
+00:22:02,220 --> 00:22:06,360
+y تربيع هي عبارة عن R تربيع، كيف عرفت؟ التي هي x
+
+324
+00:22:06,360 --> 00:22:11,080
+تربيع = R تربيع cos تربيع θ، والـ y = R تربيع sin
+
+325
+00:22:11,080 --> 00:22:16,260
+تربيع θ، مجموعهم بيصير R تربيع في cos تربيع θ
+
+326
+00:22:16,260 --> 00:22:19,440
++ sin تربيع θ، يعني cos تربيع + sin تربيع
+
+327
+00:22:19,440 --> 00:22:22,880
+θ زي ما بتعرفوا واحد، بتضل R تربيع، إذًا هذه عبارة
+
+328
+00:22:22,880 --> 00:22:26,830
+عن اللي هي مين؟ R تربيع، لأن  أتوارد تربيع، عردي، أردي
+
+329
+00:22:26,830 --> 00:22:34,290
+أردي أردي أردي أردي أردي أردي أردي أردي أردي أردي
+
+330
+00:22:34,290 --> 00:22:34,330
+أردي أردي أردي أردي أردي أردي أردي أردي أردي أردي
+
+331
+00:22:34,330 --> 00:22:34,530
+أردي أردي أردي أردي أردي أردي أردي أردي أردي أردي
+
+332
+00:22:34,530 --> 00:22:35,150
+أردي أردي أردي أردي أردي أردي أردي أردي أردي أردي
+
+333
+00:22:35,150 --> 00:22:37,090
+أردي أردي أردي أردي أردي أردي أردي أردي أردي أردي
+
+334
+00:22:37,090 --> 00:22:40,670
+أردي أردي أردي أردي أردي أردي أردي أردي أردي أردي
+
+335
+00:22:40,670 --> 00:22:45,190
+أردي
+
+336
+00:22:45,190 --> 00:22:49,850
+أردي أردي أردي أردي أردي أردي أردي أردي الآن هذا
+
+337
+00:22:49,850 --> 00:22:56,090
+اللي هو اللي هي الدائرة، الدائرة اللي هي الدائرة
+
+338
+00:22:56,090 --> 00:22:59,810
+الجزء، النصف الثاني من الدائرة، الآن اللي هو y
+
+339
+00:22:59,810 --> 00:23:02,390
+تربيع + x تربيع = 1، أو y = √
+
+340
+00:23:02,390 --> 00:23:05,930
+(1 - x تربيع)، واحنا بنعرف أن y تربيع
+
+341
+00:23:05,930 --> 00:23:09,970
++ x تربيع = 1، اللي هو معادلتها بالبولر إيش هي؟
+
+342
+00:23:09,970 --> 00:23:14,190
+هي y تربيع + x تربيع اللي هي مكانها R تربيع بيصير
+
+343
+00:23:14,190 --> 00:23:18,690
+R تربيع = 1، يعني R = 1، عندي الآن R
+
+344
+00:23:18,690 --> 00:23:23,070
+= 1، هذه اللي هي الدائرة، نصف الدائرة طبعًا
+
+345
+00:23:23,070 --> 00:23:26,490
+الدائرة كلها R = 1، لكن الآن اللي هيحدد لي
+
+346
+00:23:26,490 --> 00:23:30,210
+في الموضوع، هي قيمة θ، هتحدد لي بس المنطقة اللي فوق
+
+347
+00:23:30,600 --> 00:23:33,720
+خلّينا نشوف، إذًا هذه المنطقة، خلّينا نطلع الآن
+
+348
+00:23:33,720 --> 00:23:36,960
+الـ ray اللي هو بيدخل المنطقة و
+
+349
+00:23:36,960 --> 00:23:40,600
+بيخرج منها، طبعًا هي بيبدا من عند الصفر وبيخرج من عندها
+
+350
+00:23:40,600 --> 00:23:44,020
+كلها على نفس المنوال، بيخرج من الدائرة اللي هي R
+
+351
+00:23:44,020 --> 00:23:48,660
+= 1، علشان هي حدود الـ R من صفر لعند واحد، نيجي
+
+352
+00:23:48,660 --> 00:23:51,660
+الآن لقيم الـ θ، طبعًا هي ال ... ال ... ال ... ال ...
+
+353
+00:23:51,660 --> 00:23:51,800
+ال ... ال ... ال ... ال ... ال ... ال ... ال ... ال ... ال
+
+354
+00:23:51,800 --> 00:23:52,340
+... ال ... ال ... ال ... ال ... ال ... ال ... ال ... ال
+
+355
+00:23:52,340 --> 00:23:52,680
+... ال ... ال ... ال ... ال ... ال ... ال ... ال ... ال
+
+356
+00:23:52,680 --> 00:23:54,740
+... ال ... ال ... ال ... ال ... ال ... ال ... ال ... ال
+
+357
+00:23:54,740 --> 00:23:56,300
+... ال ... ال ... ال ... ال ... ال ... لأن واضح أن θ
+
+358
+00:23:56,300 --> 00:24:01,180
+من هنا بتبدأ بـ 0 وبتسح المنطقة، وبيبقى هذا المؤشر
+
+359
+00:24:01,180 --> 00:24:03,800
+بيمشي ويمشي ويمشي ويمشي، وبيصبح π على اثنين، و
+
+360
+00:24:03,800 --> 00:24:04,920
+بيصبح كذا وكذا وكذا وكذا وكذا وكذا وكذا وكذا
+
+361
+00:24:04,920 --> 00:24:05,020
+وكذا وكذا وكذا وكذا وكذا وكذا وكذا وكذا و
+
+362
+00:24:05,020 --> 00:24:05,940
+كذا وكذا وكذا وكذا وكذا وكذا وكذا وكذا و
+
+363
+00:24:05,940 --> 00:24:06,500
+كذا وكذا وكذا وكذا وكذا وكذا وكذا وكذا و
+
+364
+00:24:06,500 --> 00:24:09,480
+كذا وكذا وكذا وكذا وكذا وكذا وكذا وكذا و
+
+365
+00:24:09,480 --> 00:24:10,080
+كذا وكذا وكذا وكذا وكذا وكذا وكذا وكذا و
+
+366
+00:24:10,080 --> 00:24:20,120
+كذا وكذا وكذا وكذا وكذا، حولناها من كارتيزيان لـ
+
+367
+00:24:20,120 --> 00:24:24,040
+بولر، بيصير عندي اللي هو هذا e^(R تربيع) في R
+
+368
+00:24:24,040 --> 00:24:28,480
+اللي هو تكامله طبعًا (1/2) e^(R تربيع) اللي هو
+
+369
+00:24:28,480 --> 00:24:33,160
+لأنه هذا التفاضل هو هذا المقدار، من 0 لعند 1 طبعًا
+
+370
+00:24:33,160 --> 00:24:37,240
+بنعوض، بتطلع (1/2) اللي هي في حالة الواحد e¹
+
+371
+00:24:37,240 --> 00:24:41,000
+ناقص اللي هو e⁰ اللي هو واحد، اللي
+
+372
+00:24:41,000 --> 00:24:45,640
+هي e - 1، لكل dθ، الآن هذا ثابت بنضربه في θ
+
+373
+00:24:45,640 --> 00:24:49,260
+بيصير الـ θ من صفر لـ π، بنعوض عن θ بتطلع π على
+
+374
+00:24:49,260 --> 00:24:52,220
+اثنين، اللي هي  π/2  في (e - 1)
+
+375
+00:24:52,220 --> 00:24:56,660
+وبنكون هيك أوجدنا اللي هي ال
+
+376
+00:24:56,660 --> 00:25:00,240
+integration، طبعًا شفنا أهمية ال polar أنها
+
+377
+00:25:00,240 --> 00:25:05,140
+بتتغلب لنا على اللي هي عدم قدرتنا على التكامل بتحويله
+
+378
+00:25:05,140 --> 00:25:10,800
+لتكامل مكافئ بواسطة اللي هو ال polar، الآن نأخذ
+
+379
+00:25:10,800 --> 00:25:13,780
+مثال ثاني، evaluate the integral، معطيني ال
+
+380
+00:25:13,780 --> 00:25:19,180
+integration الآن هي، هو مش وصف للمنطقة، الآن أنا
+
+381
+00:25:19,180 --> 00:25:23,120
+برسمها، المنطقة، لأن بس استعنت بالرسم اللي فاتت، بس
+
+382
+00:25:23,120 --> 00:25:26,120
+الآن هو مش راسم هذه الرسمة، مش موجود هذا، بدرسمها
+
+383
+00:25:26,120 --> 00:25:30,320
+الآن إيش هي ال value of the integration من 0 لعند
+
+384
+00:25:30,320 --> 00:25:33,940
+√(1 - x تربيع)، x تربيع + y تربيع dy dx
+
+385
+00:25:34,270 --> 00:25:37,470
+من صفر لعند الواحد، لو أجيب أجرب أكامل هذه اللي
+
+386
+00:25:37,470 --> 00:25:40,410
+جوا بالنسبة لـ y، بكاملها، هي x تربيع في جذر
+
+387
+00:25:40,410 --> 00:25:43,750
+التربيعي لـ 1 - x تربيع، زائد اللي هي طبعًا مفهوم
+
+388
+00:25:43
+
+445
+00:29:33,140 --> 00:29:36,400
+قيمة الـ function اللي هو بيمثل الارتفاع اللي هو
+
+446
+00:29:36,400 --> 00:29:40,950
+عبارة عن إيش؟ تسعة ناقص  x تربيع ناقص y تربيع. هذه هي
+
+447
+00:29:40,950 --> 00:29:43,370
+إيه اللي هي بتجيب للمساحة، فبيصير المساحة في
+
+448
+00:29:43,370 --> 00:29:46,130
+الارتفاع اللي هي بتطلع الحجم، إيه ده؟ الـ volume حسب
+
+449
+00:29:46,130 --> 00:29:48,390
+ما قلناه سابقا، اللي هي تسعة ناقص x تربيع ناقص y تربيع، هي
+
+450
+00:29:48,390 --> 00:29:52,130
+تربيع، دي إيه؟ دي عبارة عن قيمة الدالة، اللي هو قيمة
+
+451
+00:29:52,130 --> 00:29:57,930
+z لفوق على السطح اللي مفروض بدنا نكامله، بدنا نكون
+
+452
+00:29:57,930 --> 00:30:01,730
+سطح العلو اللي هو المجسم.
+
+453
+00:30:03,910 --> 00:30:09,390
+القاعدة،  H A هي D A للدائرة الموجودة، عليها محطوط
+
+454
+00:30:09,390 --> 00:30:12,210
+عموديا على الدائرة اللي عندنا، إذا الـ region اللي
+
+455
+00:30:12,210 --> 00:30:16,590
+عندنا هي R، لأنه القاعدة اللي هي عبارة عن الدائرة، زي
+
+456
+00:30:16,590 --> 00:30:20,370
+ما هو مُذكر،  الـ N R هي اللي بتحدد حدود الـ
+
+457
+00:30:20,370 --> 00:30:23,720
+integration، اللي هي x تربيع زائد y تربيع يساوي واحد،
+
+458
+00:30:23,720 --> 00:30:26,880
+هي هم أعطيناه إياها، الـ unit circle، هذه إيش معادلتها؟ هي
+
+459
+00:30:26,880 --> 00:30:31,160
+بالـ polar، R بتساوي واحد. الآن هذه دائرة كاملة، إذا من
+
+460
+00:30:31,160 --> 00:30:34,700
+عند الـ x axis لو أخدت أي array هتلاقي الـ array
+
+461
+00:30:34,700 --> 00:30:39,000
+بيبدأ من نقطة الصفر وبروح لمين؟ للواحد، عشان هيك
+
+462
+00:30:39,000 --> 00:30:44,400
+حدود الـ integration للـ R من صفر لعند مين؟ لعند واحد.
+
+463
+00:30:44,790 --> 00:30:49,270
+الآن عشان أكمل الدائرة، الـ R هي بجانبي، وبجانبي، وبجانبي
+
+464
+00:30:49,270 --> 00:30:53,630
+لف الدائرة كاملة، إذا هيبدأ من عند صفر ويمشي لما
+
+465
+00:30:53,630 --> 00:30:57,790
+يصل عند 2π، هو عبارة عن حدود الـ integration
+
+466
+00:30:57,790 --> 00:31:01,880
+اللي بيجيب الدائرة هذه كلها. الآن نجي للـ integrand
+
+467
+00:31:01,880 --> 00:31:04,360
+اللي هو تسعة ناقص x تربيع ناقص y تربيع، الدالة اللي
+
+468
+00:31:04,360 --> 00:31:08,460
+بنكاملها، الـ x تربيع زائد y تربيع هي عبارة عن مين؟ R
+
+469
+00:31:08,460 --> 00:31:11,220
+تربيع زي ما احنا عارفينه، قلناه قبل شوية، بيصير دي
+
+470
+00:31:11,220 --> 00:31:15,420
+تسعة ناقص R تربيع، والـ DA زي ما قلنا اللي هي dx dy
+
+471
+00:31:15,420 --> 00:31:21,510
+اللي بتتحول لـ R dR dθ، المكافئ لـ dx dy. الآن بدنا نكامل،
+
+472
+00:31:21,510 --> 00:31:26,810
+بتصير ٩R - R تكعيب، دي R دي ثيتا، ضربت الـ R جوا، الآن
+
+473
+00:31:26,810 --> 00:31:31,570
+كامل هذه بتصير ٩R تربيع على ٢ ناقص ربع R أس ٤ من
+
+474
+00:31:31,570 --> 00:31:36,010
+صفر لوين؟ لعند ١، قيمة الـ R، فبتصير عبارة عن ٩
+
+475
+00:31:36,010 --> 00:31:41,110
+على ٢ ناقص ربع، اللي هي بتطلع ١٧ على ٤، احسبوها
+
+476
+00:31:41,110 --> 00:31:45,670
+لحالكم. الآن بيظهر مين عندنا؟ دي ثيتا، ثيتا من صفر لعند
+
+477
+00:31:45,670 --> 00:31:49,550
+٢π، دي ثيتا تكامل اللي هو عبارة عن ثيتا، اللي هي
+
+478
+00:31:49,550 --> 00:31:53,130
+عبارة عن ٢π، اللي هي بيصير ٢π في هذه
+
+479
+00:31:53,130 --> 00:31:57,750
+اللي هي بتصير، اللي هي ٢π في ١٧ على ٤،
+
+480
+00:31:57,750 --> 00:32:00,810
+بيصير ١٧π على مين؟ على ٢، لإن بتروح ٢
+
+481
+00:32:00,810 --> 00:32:04,010
+مع ٢ اللي هان، اللي هي بتطلع ١٧π على
+
+482
+00:32:04,010 --> 00:32:07,470
+٢، اللي هي قيمة، اللي هي ذاك، فاهمين كيف؟ اللي هي
+
+483
+00:32:07,470 --> 00:32:10,740
+دي، بيصير ٢π قيمة التكامل، لإنها تبسيط data،
+
+484
+00:32:10,740 --> 00:32:14,120
+تبسيط data من صفر لـ π على ٢، صفر لـ ٢π على
+
+485
+00:32:14,120 --> 00:32:16,700
+٢π، ٢π في ١٧ على ٤ تطلع
+
+486
+00:32:16,700 --> 00:32:20,220
+١٧ على ٢، ١٧π على ٢، وهيك
+
+487
+00:32:20,220 --> 00:32:23,580
+بيكون أوجدنا الـ volume اللي عندنا. نجي للسؤال اللي
+
+488
+00:32:23,580 --> 00:32:27,920
+هو مثال السادس. الآن نعزز الحديث اللي قبله شوية،
+
+489
+00:32:27,920 --> 00:32:32,890
+ناخد مثال آخر، خلنا نقول شكله أموره مختلفة لكن بنفس
+
+490
+00:32:32,890 --> 00:32:36,950
+الروح، using polar integration، باستخدام الـ polar
+
+491
+00:32:36,950 --> 00:32:41,090
+integration، find the area of the region R، اللي هم
+
+492
+00:32:41,090 --> 00:32:44,730
+بده إياه، اللي هو مساحة الـ region R، بده يوصفها الآن
+
+493
+00:32:44,730 --> 00:32:49,010
+وين موجودة؟ in the xy plane enclosed by، اللي هي في
+
+494
+00:32:49,010 --> 00:32:52,590
+الـ xy plane، مغلقة بواسطة الـ circle، x تربيع زائد y
+
+495
+00:32:52,590 --> 00:32:55,820
+تربيع يساوي أربعة، دي مش زائد، أربعة، يساوي أربعة. إذا
+
+496
+00:32:55,820 --> 00:32:59,340
+هذه اللي هو المنطقة اللي بدنا إياها، جوا الدائرة
+
+497
+00:32:59,340 --> 00:33:04,960
+هذه، اللي هي x تربيع زائد y تربيع يساوي أربعة، يعني
+
+498
+00:33:04,960 --> 00:33:08,200
+وكأنه لو جيناها وعملناها بالـ polar على طول، x تربيع
+
+499
+00:33:08,200 --> 00:33:11,500
+زائد y تربيع يساوي أربعة، اللي هي الدائرة هذه على
+
+500
+00:33:11,500 --> 00:33:14,700
+طول بتصير R تربيع يساوي أربعة، يعني R يساوي ٢،
+
+501
+00:33:14,700 --> 00:33:19,060
+يعني هذه اللي هي الدائرة اللي عندي، تعبيرها بالـ polar
+
+502
+00:33:19,060 --> 00:33:24,670
+اللي هو R يساوي ٢. طيب، الآن نكمل وصف المنطقة،
+
+503
+00:33:24,670 --> 00:33:28,830
+ما خلصناش، بقول لي الآن هذه المنطقة المغلقة من
+
+504
+00:33:28,830 --> 00:33:33,470
+الدائرة، ولكنها إيش؟ above the line y يساوي واحد، هي
+
+505
+00:33:33,470 --> 00:33:38,470
+الـ y يساوي واحد، فوق الـ، المنطقة y يساوي واحد،
+
+506
+00:33:38,470 --> 00:33:41,730
+يعني بيصير لسه هادي معها، وهادي معها، واللي في
+
+507
+00:33:41,730 --> 00:33:45,630
+الجهة الثانية معها. الآن بدنا نحدد أكتر، above y
+
+508
+00:33:45,630 --> 00:33:50,250
+يساوي واحد، هي y يساوي واحد، اللي فوقها هي المنطقة
+
+509
+00:33:50,250 --> 00:33:54,810
+هذه محددة مع اللي هذه، لكن الآن بدنا نحدد أكتر، لأن
+
+510
+00:33:54,810 --> 00:33:58,430
+إن below the line y يساوي جذر التلاتة x، وتحت من
+
+511
+00:33:58,430 --> 00:34:01,630
+الـ line y يساوي جذر التلاتة x، y يساوي جذر
+
+512
+00:34:01,630 --> 00:34:05,790
+التلاتة x، اللي بياخد x بصفر و y بصفر، أكيد بنمر بنقطة
+
+513
+00:34:05,790 --> 00:34:09,650
+الأصل، اللي هو x بواحد و y بجذر التلاتة، إذا هذه
+
+514
+00:34:09,650 --> 00:34:14,620
+اللي هي بتيجي، اللي هي بتمشي في الخط، الخط هذا بالشكل
+
+515
+00:34:14,620 --> 00:34:17,760
+هذا، هذا اللي هو y يساوي جذر التلاتة x، بتعرفوا
+
+516
+00:34:17,760 --> 00:34:21,960
+ترسموه. الآن بدنا نحدد المنطقة، المنطقة اللي تحتها، إذا
+
+517
+00:34:21,960 --> 00:34:25,480
+هي المنطقة اللي تحتها مش هذه اللي فوق، إذا الآن جينا
+
+518
+00:34:25,480 --> 00:34:30,040
+حصرنا حالنا، المنطقة اللي تحت هذا، واللي فوق هذا، و
+
+519
+00:34:30,040 --> 00:34:34,220
+اللي في الدائرة، اه، لإن اللي تحت هذه هي كل هذه، اللي
+
+520
+00:34:34,220 --> 00:34:37,560
+فوق هذه هي كلها، لكن الآن بدناش اللي برا هنا بس،
+
+521
+00:34:37,560 --> 00:34:40,880
+اللي موجود وين؟ في الدائرة، إذا هي المنطقة المقصودة.
+
+522
+00:34:41,210 --> 00:34:45,710
+الآن بدنا نوجد نقاط التقاطع، عند نقاط التقاطع نجد
+
+523
+00:34:45,710 --> 00:34:49,930
+النقطة هذه، والنقطة هذه. الآن هذا y يساوي جذر
+
+524
+00:34:49,930 --> 00:34:55,430
+الـ ٣x، عند اللي هو عوض، اللي هو في الدائرة، هادي اللي
+
+525
+00:34:55,430 --> 00:34:59,190
+x²، بيصير يساوي أربعة، عوض عن قيمة عشان توجد إيه؟ عشان
+
+526
+00:34:59,190 --> 00:35:04,260
+توجد تقاطع هذا الخط مع الدائرة، كيف توجده؟ بتعوض y
+
+527
+00:35:04,260 --> 00:35:07,140
+يساوي جذر التلاتة x هنا، بيصير اللي هي تلاتة x
+
+528
+00:35:07,140 --> 00:35:10,680
+تربيع، تلاتة y، اللي هي تلاتة x تربيع زائد تلاتة
+
+529
+00:35:10,680 --> 00:35:13,540
+x تربيع، اللي هي أربعة x تربيع يساوي أربعة، يعني x
+
+530
+00:35:13,540 --> 00:35:17,040
+تربيع يساوي واحد، يعني x يساوي زائد أو ناقص واحد،
+
+531
+00:35:17,040 --> 00:35:19,480
+لكن في المنطقة الموجودة، إذا الـ x إيش بتساوي؟ واحد،
+
+532
+00:35:19,480 --> 00:35:23,500
+لما x يساوي واحد، y إيش هتساوي؟ جذر التلاتة، فهذه
+
+533
+00:35:23,500 --> 00:35:27,820
+النقطة اللي هي واحد وجذر التلاتة. نجي نوجد من النقطة
+
+534
+00:35:27,820 --> 00:35:30,360
+هذه، النقطة هذه عبارة عن تقاطع اللي هو y يساوي
+
+535
+00:35:30,360 --> 00:35:34,900
+واحد مع الدائرة. الآن y يساوي واحد، بيصير x تربيع
+
+536
+00:35:34,900 --> 00:35:37,440
+زائد واحد يساوي أربعة، يعني x تربيع يساوي تلاتة،
+
+537
+00:35:37,440 --> 00:35:40,000
+يعني x يساوي جذر التلاتة، زائد وناقص طبعا أنا
+
+538
+00:35:40,000 --> 00:35:43,120
+موجبة، بيصير x يساوي جذر التلاتة، إذن النقطة هذه هي
+
+539
+00:35:43,120 --> 00:35:47,030
+عبارة عن جذر التلاتة وواحد. إذا المنطقة هنا حددناها
+
+540
+00:35:47,030 --> 00:35:52,270
+تحديدا تماما، وحددنا لجميع نقاط التقاطع، هذه نقطة
+
+541
+00:35:52,270 --> 00:35:56,610
+مش هتلزمني وإن كانت سهلة، اللي هي عند y بواحد، اللي
+
+542
+00:35:56,610 --> 00:36:01,170
+هي عند، اللي هي بنعوض قيمة x لها، واحد، وجذر التلاتة،
+
+543
+00:36:01,170 --> 00:36:04,350
+لكن هذا مش كتير، لأ، مش لازم يعني. الآن، هل وجدنا؟
+
+544
+00:36:04,350 --> 00:36:07,490
+هتشوفوا ليش ما نزلناش، هي المنطقة، الآن ماشي. الآن
+
+545
+00:36:07,490 --> 00:36:11,290
+المنطقة هي اللي هو الـ integration بده الـ area لها،
+
+546
+00:36:11,290 --> 00:36:15,650
+عبارة عن dA over R، اللي هي المنطقة، الآن dA اللي هي
+
+547
+00:36:15,650 --> 00:36:18,890
+زي ما بنقول،  R dR dθ، لإن هو بده الـ area
+
+548
+00:36:18,890 --> 00:36:21,590
+للمنطقة هذه، مدام الـ area إذا اللي هو double
+
+549
+00:36:21,590 --> 00:36:26,650
+integral لـ dA over R، الـ dA اللي هي R dR dθ over R،
+
+550
+00:36:26,650 --> 00:36:29,710
+R، بدنا نحدد الآن اللي هي حدود الـ integration، شوف
+
+551
+00:36:29,710 --> 00:36:34,280
+حدود الـ integration، أول حاجة اللي هي كل الخطوط اللي
+
+552
+00:36:34,280 --> 00:36:39,100
+بتطلع، اللي هي بتدخل للمنطقة، هيها، هيها، الخطوط
+
+553
+00:36:39,100 --> 00:36:44,070
+بتدخل للمنطقة من عند اللي هو الخط هذا، اللي هو y
+
+554
+00:36:44,070 --> 00:36:48,270
+يساوي واحد، لعند الخط، لعند مين؟ بتطلع، بتطلع من
+
+555
+00:36:48,270 --> 00:36:51,470
+الدائرة، بتدخل من عند الخط، وبتطلع من الدائرة، و
+
+556
+00:36:51,470 --> 00:36:57,470
+كلها على نفس المنوال. إذا الآن قيمة R لعند هذا الخط
+
+557
+00:36:57,470 --> 00:37:01,370
+بنوجدها، لإن إيش؟ الخط y يساوي واحد، y هي عبارة عن
+
+558
+00:37:01,370 --> 00:37:06,010
+R sin θ، إذا R sin θ يساوي واحد، إذا الـ R يساوي
+
+559
+00:37:06,010 --> 00:37:10,940
+واحد على sin θ، يعني R يساوي cosec θ. إذا قيمة R على
+
+560
+00:37:10,940 --> 00:37:15,620
+هذا الخط اللي هو معادلته R يساوي cosec θ، قيمة R
+
+561
+00:37:15,620 --> 00:37:21,640
+أشملها عبارة عن cosec θ، إذا R تبدأ من عند cosec θ و
+
+562
+00:37:21,640 --> 00:37:26,140
+بتطلع من الدائرة، قيمة الدائرة، R عند الدائرة هذه زي
+
+563
+00:37:26,140 --> 00:37:30,360
+ما قلنا معادلتها R بتساوي ٢، إذا عند ٢. نجي الآن
+
+564
+00:37:30,360 --> 00:37:36,200
+نشوف قد إيش قيمة θ، لأن هاي θ من هنا بتبدأ، هيها، أول
+
+565
+00:37:36,200 --> 00:37:40,540
+أري، بيطلع عند النقطة هذه، وبعدين بيمشي، بيمشي لما
+
+566
+00:37:40,540 --> 00:37:45,340
+نوصل هنا. طيب، الآن بدي أشوف زاوية هذا قد إيش، وزاوية
+
+567
+00:37:45,340 --> 00:37:48,320
+هذا الأري قد إيش، إذا بتكون تلت π، من عند اللي هو هذا
+
+568
+00:37:48,320 --> 00:37:53,080
+من عند هذا الخط لعند هذا بتكون إيش؟ هذا الخط لما
+
+569
+00:37:53,080 --> 00:37:55,440
+يمشي، لما يمشي، نمشي في داخل المنطقة بيكون مساحتها
+
+570
+00:37:55,440 --> 00:37:59,120
+كلها فعلا، نجي لهذه، هذه تلت، واحد وجذر التلاتة، إذا
+
+571
+00:37:59,120 --> 00:38:04,640
+هذا طوله اللي هو واحد، وهذا طوله جذر التلاتة، لأن هذا
+
+572
+00:38:04,640 --> 00:38:07,920
+إذا المثلث، اللي، اللي، اللي، اللي، اللي، اللي، اللي
+
+573
+00:38:07,920 --> 00:38:08,080
+اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي
+
+574
+00:38:08,080 --> 00:38:08,380
+اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي
+
+575
+00:38:08,380 --> 00:38:08,560
+اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي
+
+576
+00:38:08,560 --> 00:38:08,920
+اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي
+
+577
+00:38:08,920 --> 00:38:09,700
+اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي
+
+578
+00:38:09,700 --> 00:38:09,740
+اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي
+
+579
+00:38:09,740 --> 00:38:10,600
+اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي، اللي
+
+580
+00:38:10,600 --> 00:38:19,840
+اللي، اللي
+
+581
+00:38:19,840 --> 00:38:24,010
+اللي، إذا هنا عجباله اللي هو هذا، قيمة واحد، قيمة y،
+
+582
+00:38:24,010 --> 00:38:26,710
+إذا العجبالة اللي هي عبارة عن π على ٦،
+
+583
+00:38:26,710 --> 00:38:30,670
+ثلاثين درجة، إذا الآن القُرية الأولى بيبدأ من وين؟ من
+
+584
+00:38:30,670 --> 00:38:33,870
+عند π على ٦، الآن نجي نشوف القُرية الثانية، هي
+
+585
+00:38:33,870 --> 00:38:37,180
+الزاوية تبع القُرية الثانية. الآن هادي اخر قُرية،
+
+586
+00:38:37,180 --> 00:38:39,980
+طبعا بقصدي هي عندي اللي هي النقطة دي واحد وجذر
+
+587
+00:38:39,980 --> 00:38:43,540
+التلاتة، إذا هذا طوله جذر التلاتة اللي هي y وهذا
+
+588
+00:38:43,540 --> 00:38:47,540
+طوله واحد، اللي هي صار مثلث برضه، ثلاثين ستين، لكن
+
+589
+00:38:47,540 --> 00:38:50,080
+جذر التلاتة هي عِجْبالة الزاوية الكبيرة اللي هي
+
+590
+00:38:50,080 --> 00:38:53,300
+عبارة عن إيش؟ π على ٣، إذا هذه الزاوية بتصل لمن؟
+
+591
+00:38:53,300 --> 00:38:56,700
+لعند π على ٣، اللي هي ستين درجة، اللي هي
+
+592
+00:38:56,700 --> 00:39:01,160
+بتصل. إذا الآن القُرية هذه بتنتهي عند π على ٣، اللي
+
+593
+00:39:01,160 --> 00:39:04,410
+هي الزاوية ستين درجة. فبيكون هذا هو الـ integration.
+
+594
+00:39:04,410 --> 00:39:08,430
+آن كامل، الآن التكامل سهل، بيصير لي نص R تربيع من cosec
+
+595
+00:39:08,430 --> 00:39:13,270
+θ لعند ٢، بتعوض بيصير نص في أربعة ناقص cosec
+
+596
+00:39:13,270 --> 00:39:17,930
+تربيع θ، دي ثيتا، الآن هذه نكاملها برضه سهلة، اللي هي
+
+597
+00:39:17,930 --> 00:39:21,650
+عبارة عن نص في تكامل الأربعة، اللي هي أربعة ثيتا،
+
+598
+00:39:21,650 --> 00:39:25,610
+تكامل اللي هو cosec تربيع، ناقص cosec تربيع،
+
+599
+00:39:25,610 --> 00:39:27,930
+التكامل اللي هو عبارة عن cotan ثيتا زي ما احنا
+
+600
+00:39:27,930 --> 00:39:31,310
+عارفين. الآن من وين لوين؟ π على ٦ لعند π على ٣،
+
+601
+00:39:31,310 --> 00:39:34,950
+بنعوض عليها، بيطلع عند القيمة هذه ناقص القيمة هذه،
+
+602
+00:39:34,950 --> 00:39:37,970
+اللي عند π على ٣، هي القيمة، عند π على ٦، هي
+
+603
+00:39:37,970 --> 00:39:42,010
+القيمة، نطرحها من بعض، طبعا تعرفت وجودهم، ويساوي، وبيطلع
+
+604
+00:39:42,010 --> 00:39:46,130
+القيمة اللي أمامنا، وهذا بيكون اللي هو آخر مثال
+
+605
+00:39:46,130 --> 00:39:49,510
+عندنا. وبارك الله فيكم، وإلى لقاء آخر.
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..f52f67684c5f9bd56b9d7e69381fc04a270895f2
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA_postprocess.srt
@@ -0,0 +1,2420 @@
+1
+00:00:00,540 --> 00:00:04,980
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,980 --> 00:00:11,540
+وبركاته هذه هي المحاضرة رقم 23 مساقطة فاضل وتكامل
+
+3
+00:00:11,540 --> 00:00:16,220
+جيم وهي المحاضرة الرابعة في ال chapter 15 ال
+
+4
+00:00:16,220 --> 00:00:19,360
+chapter الأخير في المادة وهي تحت عنوان double
+
+5
+00:00:19,360 --> 00:00:24,210
+integrals in polar formتحدثنا سابقا عن double
+
+6
+00:00:24,210 --> 00:00:28,290
+integral in cartesian form الآن بدنا نتحدث عن
+
+7
+00:00:28,290 --> 00:00:31,850
+double integral in polar form طبعا ولو أسبابه
+
+8
+00:00:31,850 --> 00:00:35,710
+أحيانا اللي هو إيجاد اللي هو ال double integration
+
+9
+00:00:35,710 --> 00:00:41,050
+بال integral form أسهل كتير كتير من إيجاده في ال
+
+10
+00:00:41,050 --> 00:00:46,050
+Cartesian و بتحللنا مشاكل اللي هو قد تنشأ في ال
+
+11
+00:00:46,050 --> 00:00:48,110
+Cartesian كما سنظهر لاحقا
+
+12
+00:00:50,710 --> 00:00:55,150
+بتحلها بواسطة اللي هو مين ال bowler form طبعا و
+
+13
+00:00:55,150 --> 00:00:58,230
+العكس صحيح بعض الأسئلة بتكون في ال Cartesian أسهل
+
+14
+00:00:58,230 --> 00:01:01,030
+طبعا الناس اللي هي متعودة أصلا على ال Cartesian
+
+15
+00:01:01,030 --> 00:01:06,850
+لكن ال bowler زي ما قلنا بحللنا مشاكل تشوف الآنروح
+
+16
+00:01:06,850 --> 00:01:12,510
+الموضوع نفسه اللي هو السابق كيف اللي هو المهم عندي
+
+17
+00:01:12,510 --> 00:01:16,610
+تحديد منطقة اللي هي المنطقة اللي بدي أكمل عليها
+
+18
+00:01:16,610 --> 00:01:20,030
+بتحدد لي اللي هي limits of integration إذا
+
+19
+00:01:20,030 --> 00:01:23,470
+integrals and polar coordinates بدي أنحدد أول شيء
+
+20
+00:01:23,470 --> 00:01:26,310
+اللي هو find limits of integration بالظبط زي ما
+
+21
+00:01:26,310 --> 00:01:30,730
+كنا نعمل اللي هو في ال Cartesian ولكن هنا اللي هو
+
+22
+00:01:30,730 --> 00:01:36,390
+طبيعة البولربختلف عن طبيعة الـ Cartesian فبنشوف
+
+23
+00:01:36,390 --> 00:01:39,830
+كيف بنحدد اللي هو ال integration طبعا احنا بدنا
+
+24
+00:01:39,830 --> 00:01:43,470
+نيجي هيعنى عرض علينا ال integration اللي أمامنا ال
+
+25
+00:01:43,470 --> 00:01:47,510
+integration لـF of Rθ دي إيه حيث دي إيه طبعا في
+
+26
+00:01:47,510 --> 00:01:51,570
+هذه الحالة اللي هي DX في DY لكن ال DX في DY زي ما
+
+27
+00:01:51,570 --> 00:01:57,690
+أخدنا في التفاضل ب .. اللي هو عبارة عن RDRDθالان
+
+28
+00:01:57,690 --> 00:02:00,750
+بدل دي ايه سيصبح دي ايه دي ايه دي ايه دي ايه دي
+
+29
+00:02:00,750 --> 00:02:01,290
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+30
+00:02:01,290 --> 00:02:03,050
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+31
+00:02:03,050 --> 00:02:05,390
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+32
+00:02:05,390 --> 00:02:11,530
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+33
+00:02:11,530 --> 00:02:14,510
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+34
+00:02:14,510 --> 00:02:24,870
+ايه دي ايه دي ايه
+
+35
+00:02:24,870 --> 00:02:30,260
+دقيمة R وقيمة ثتا ونشوف كيف لاحقا لأن أول حاجة
+
+36
+00:02:30,260 --> 00:02:37,350
+احنا في هذه الحالة بنكامل بالنسبة ل Rالان نظرا
+
+37
+00:02:37,350 --> 00:02:40,250
+لطبيعة ال integration طبعا في عندي RDRD ثالثة و
+
+38
+00:02:40,250 --> 00:02:44,070
+هنلاقي اللي هو التكمل ال R عنده اللي هو بيحللنا
+
+39
+00:02:44,070 --> 00:02:47,950
+كتير مشاكل الان find the R limits of integration
+
+40
+00:02:47,950 --> 00:02:53,090
+الان طبيعة ال R limits اللي هي ال R ال R طبعا زي
+
+41
+00:02:53,090 --> 00:02:55,730
+ما انتوا مذكرين ان اكتفاض الألف بتبدأ من نقطة
+
+42
+00:02:55,730 --> 00:02:59,950
+الأصل و بتطلع إيش إلى أعلى اللي هي بوسطة ال Ray
+
+43
+00:03:00,540 --> 00:03:03,900
+الآن الـ Rray هذا بدنا نشوف كيف بدخل و من وين بدخل
+
+44
+00:03:03,900 --> 00:03:07,340
+المنطقة و من وين بيخرج منها من وين بدخل المنطقة
+
+45
+00:03:07,340 --> 00:03:12,660
+بيكون اللي هو الحد الأسفل و من وين بيخرج منها الحد
+
+46
+00:03:12,660 --> 00:03:16,340
+الأعلى زي في حالة الـ Cartesian نيجي نحد الـ Theta
+
+47
+00:03:16,340 --> 00:03:21,140
+Theta الآن بنشوف كيف تمسح المنطقة طبعا بس بنبدأ من
+
+48
+00:03:21,140 --> 00:03:24,220
+وين عند الـ Positive X Axis و نطلع عكس عقارب
+
+49
+00:03:24,220 --> 00:03:28,440
+الساعة من حد الـ Thetaفبنشوف من أين بدأت المنطقة
+
+50
+00:03:28,440 --> 00:03:32,260
+ومن أين انتهت بتكون هي حدود ثتا خلّينا نشوف هذا
+
+51
+00:03:32,260 --> 00:03:39,080
+الكلام عملياً لو عرض علينا هيعندي F of R ثتا DA
+
+52
+00:03:39,080 --> 00:03:43,220
+over a region R ال region R اللي عرضها عليها هي ال
+
+53
+00:03:43,220 --> 00:03:46,370
+region التاليةعامل ايه هو ليش بالـ Cartesian قال
+
+54
+00:03:46,370 --> 00:03:49,130
+اللي هو بدي X تربيع زي أضواء تربيع بساوة أربعة هذي
+
+55
+00:03:49,130 --> 00:03:52,330
+جزء من مين من دائرة قال اللي بدي هذا الجزء اللي في
+
+56
+00:03:52,330 --> 00:03:57,090
+الربع الأول R بوصف R قاعد الجزء اللي هي R اللي في
+
+57
+00:03:57,090 --> 00:03:59,710
+الربع الأول المكون من أعلى ب X تربيع زي أضواء
+
+58
+00:03:59,710 --> 00:04:03,450
+تربيع اللي هو أربعة اللي عبارة عن الدائرة ومن أسفل
+
+59
+00:04:03,450 --> 00:04:08,850
+بواسطة Y بساوية جذر التنين وعن اليسار بواسطة اللي
+
+60
+00:04:08,850 --> 00:04:15,350
+هو ال X بتساوية سفر أو ال Y Axisهذه هي المنطقة
+
+61
+00:04:15,350 --> 00:04:19,630
+المحدودة الان اللى عليها بدنا نكامل الان كلها
+
+62
+00:04:19,630 --> 00:04:24,130
+مكتوبة بإيش بالكارتيزين بدي أحولها لإيش للبولر
+
+63
+00:04:24,130 --> 00:04:30,130
+الان عندي النقطة هذه نقطة التقاطع بين هذا المستقيم
+
+64
+00:04:30,130 --> 00:04:33,940
+وهذا الدائرةإيجاد النقطة هذه سهلة حط Y بساوية جدر
+
+65
+00:04:33,940 --> 00:04:37,800
+التنين هنا اللي هي عبارة عن ياش اتنين ال Y تربيع
+
+66
+00:04:37,800 --> 00:04:40,520
+جيجي اللي هنا بيصير اللي هي اربعة نقص اتنين اتنين
+
+67
+00:04:40,520 --> 00:04:43,520
+يعني بيصير X تربيع بساوية اتنين يعني X برضه بساوية
+
+68
+00:04:43,520 --> 00:04:46,620
+جدر التنين لإنه لهذه المنطقة موجة بقى فبيصير عند
+
+69
+00:04:46,620 --> 00:04:50,260
+النقطة هذه جدر التنين وجدر التنينالمنطقة هذه طبعا
+
+70
+00:04:50,260 --> 00:04:53,860
+y بيساوي جدر التنين معناته ده نقطة جدر التنين وهذا
+
+71
+00:04:53,860 --> 00:04:56,400
+اللي هو x تربيع زاوية وy تربيع بيساوي أربع عبارة
+
+72
+00:04:56,400 --> 00:05:01,360
+عن دائرة نص قطرها بيساوي اتنين هي الاش نصف القطر
+
+73
+00:05:01,360 --> 00:05:04,520
+إذا هذه إحداثيات اتنين إذا عرفنا كل الإحداثيات
+
+74
+00:05:04,520 --> 00:05:09,640
+لهذه الآن بدنا نحول هذه لبولر عشان اللي هو نشتغل
+
+75
+00:05:09,640 --> 00:05:14,100
+على البولر يقتلبه منها طيبالان احنا بنعرف علاقات
+
+76
+00:05:14,100 --> 00:05:20,940
+قديما بين x و y و r و θ ال x هذكركم فيها بيساوي r
+
+77
+00:05:20,940 --> 00:05:27,860
+cos θ ال y ايش بتساوي r sin θ ماشي الحال اذا x
+
+78
+00:05:27,860 --> 00:05:30,860
+تربيع زايد y تربيع بنقدر نحولها بيصير x تربيع
+
+79
+00:05:30,860 --> 00:05:36,660
+عبارة عن r تربيع cos تربيع θزائد ال Y R Sin θ
+
+80
+00:05:36,660 --> 00:05:41,420
+بيصير زائد R تربيع Sin تربيع θ بيساوي إيش؟ أربعة
+
+81
+00:05:41,420 --> 00:05:45,120
+خد من هدولة عامل مشترك R تربيع بيصير R تربيع
+
+82
+00:05:45,120 --> 00:05:48,800
+مضروبة في Cos تربيع زائد Sin تربيع بيساوي أربعة زي
+
+83
+00:05:48,800 --> 00:05:51,520
+ما احنا عارفين الـ Cos تربيع زايد ال Sin تربيع إيش
+
+84
+00:05:51,520 --> 00:05:56,340
+بتساوي واحد يعني R تربيع إيش بتساوي أربعة يعني R
+
+85
+00:05:56,340 --> 00:05:59,540
+بتساوي اتنين وR بتساوي اتنين احنا مشهور عندنا في
+
+86
+00:05:59,540 --> 00:06:03,230
+البولر هاي R بتساوي اتنين هاي الدائرة هذهمشهور في
+
+87
+00:06:03,230 --> 00:06:07,350
+البولار ان هي معادلة الدائرة اللي نصف قطرها ايش
+
+88
+00:06:07,350 --> 00:06:13,410
+اللي هي عبارة عن اتنين طيب الان نيجي للخط هذا هذا
+
+89
+00:06:13,410 --> 00:06:16,410
+الخط عبارة عن Y بيساوي جدر التنين شيل ال Y وحط
+
+90
+00:06:16,410 --> 00:06:22,140
+قيمته ايش قيمته R sin θ اللي ناسيإذاً R sin θ
+
+91
+00:06:22,140 --> 00:06:26,900
+بتساوي Y بتساوي جدر التنين نحوّلها لبولر ال بولر
+
+92
+00:06:26,900 --> 00:06:30,780
+بيصير R بيساوي جدر التنين ع الصين يعني R بيساوي
+
+93
+00:06:30,780 --> 00:06:35,140
+جدر التنين كثيكان تلث ثتا مقلوب ال sign إذاً صار
+
+94
+00:06:35,140 --> 00:06:39,480
+الخط هذا يا جماعة الثاني عبارة عن R بيساوي جدر
+
+95
+00:06:39,480 --> 00:06:43,310
+التنين كثيكان ثتاالان الأمور سهلة وشوفوا اكيد سهلة
+
+96
+00:06:43,310 --> 00:06:47,250
+هذه حولناها كلها ل bowler ماشي الحال تنشوف الان
+
+97
+00:06:47,250 --> 00:06:51,350
+الان نيجي اللي هو بدنا نيجي اللي هو evaluation
+
+98
+00:06:51,350 --> 00:06:55,410
+كامل بالنسبة لمين R ال R طبيعته كيف بتحسب اللي هو
+
+99
+00:06:55,410 --> 00:06:58,870
+ال Ray ال Ray ال Ray ال Ray بتطلع من مين؟ من السفر
+
+100
+00:06:59,190 --> 00:07:02,350
+الان هذا الـ array بدنا نشوف وين بدخل المنطقة و
+
+101
+00:07:02,350 --> 00:07:06,230
+بين بخرج منها و هل كل ال arrays اللي بتدخل و بتخرج
+
+102
+00:07:06,230 --> 00:07:09,750
+من المنطقة على نفس المنوال زي زمان يعني بنفعش
+
+103
+00:07:09,750 --> 00:07:13,930
+نجسمها إذا بتمشيش على نفس المنوال لازم نجسمها إذا
+
+104
+00:07:13,930 --> 00:07:17,970
+بتمشي على نفس المنوال بلزمش تجسمها الان هبنيجي
+
+105
+00:07:17,970 --> 00:07:23,050
+بناخد من هنا arrayراحظ الـ R دخل من عند الخط هذا و
+
+106
+00:07:23,050 --> 00:07:27,050
+طلع من عند خط الدائرة خد الـ R تاني خد الـ R تالت
+
+107
+00:07:27,050 --> 00:07:30,610
+كل هن طبعا نقطة الرأس اللي بتطلع كل هن على نفس
+
+108
+00:07:30,610 --> 00:07:34,590
+المنوال إذا مرتاحين إذا خلينا نشوف R R وين دخلت؟
+
+109
+00:07:34,590 --> 00:07:39,570
+دخلت من هنا إيش قيمة هنا R على هذا قيمته اللي هو
+
+110
+00:07:39,570 --> 00:07:43,350
+زي ما اتفقنا جدر التنين C Cosθ هذا الحد الأولاني
+
+111
+00:07:43,350 --> 00:07:48,510
+إيش بيساوي جدر التنين Cosθ طب من وين خرج؟خرج من
+
+112
+00:07:48,510 --> 00:07:51,650
+الدائرة R بساوة و اتنين اذا ايش الاحداث التاني للـ
+
+113
+00:07:51,650 --> 00:07:55,830
+R اللي هو R بساوة و اتنين خلصنا الـ A من R بدنا
+
+114
+00:07:55,830 --> 00:08:00,390
+نيجي نحدد مين ثيتا الآن ثيتا الثيتا بتبدأ من عند
+
+115
+00:08:00,390 --> 00:08:04,630
+القياساتها بتبدأ من الصغير اللي هو اللي هي مع ال
+
+116
+00:08:04,630 --> 00:08:09,820
+positive X axis و بعدين بتبدأ تلفوانتوا عارفين كيف
+
+117
+00:08:09,820 --> 00:08:14,400
+قيام الـ θيتا الان بتلف هيك بالاتجاه الموجة عكس
+
+118
+00:08:14,400 --> 00:08:18,160
+عقارب الساعة وهيك بالاتجاه السالب طيب خلينا نحدد
+
+119
+00:08:18,160 --> 00:08:21,480
+منطقتنا .. منطقتنا شايف ان انت بتبدأ هي اول array
+
+120
+00:08:21,480 --> 00:08:25,320
+هي اول array هي تاني array هي تالت array و بيظلوا
+
+121
+00:08:25,320 --> 00:08:29,660
+يمشي يمشي تخيل نور بيمشي هيك بال array عشان يغطي
+
+122
+00:08:29,660 --> 00:08:32,560
+المنطقة ده هيمشي من هان لهان اذا هيبدأ من هان ال
+
+123
+00:08:32,560 --> 00:08:36,910
+arrayالـ θ تبعته يجب أن نجيسه هذه الـ θ تبعته يجب
+
+124
+00:08:36,910 --> 00:08:38,090
+أن نجيسه هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ
+
+125
+00:08:38,090 --> 00:08:40,930
+تبعته يجب أن نجيسه هذه الـ θ تبعته يجب أن نجيسه
+
+126
+00:08:40,930 --> 00:08:41,770
+هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ تبعته يجب
+
+127
+00:08:41,770 --> 00:08:43,390
+أن نجيسه هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ
+
+128
+00:08:43,390 --> 00:08:45,510
+تبعته يجب أن نجيسه هذه الـ θ تبعته يجب أن نجيسه
+
+129
+00:08:45,510 --> 00:08:51,470
+هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ تبعته يجب
+
+130
+00:08:51,470 --> 00:08:56,150
+أن نجيسه هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ
+
+131
+00:08:56,180 --> 00:09:01,120
+مدام عنده Y بتساوي X معناته اللي هو هذا تبع الـ Y
+
+132
+00:09:01,120 --> 00:09:04,440
+بساوية هذا تبع الـ X يعني المثلث القائم هذا عبارة
+
+133
+00:09:04,440 --> 00:09:07,680
+عن مثلث متساوى الساقين يعني هذه الزاوية بالساوية
+
+134
+00:09:07,680 --> 00:09:11,180
+هذه يعني هذه 45 درجة و هذه 45 درجة إذا هذه الزاوية
+
+135
+00:09:11,180 --> 00:09:16,180
+عبارة عن 45 درجة للبداية اللي هي باي على أربعة إذا
+
+136
+00:09:16,180 --> 00:09:19,460
+قيتها بتبدأ من باي على أربعة وشوف لوين تبقى تمشي
+
+137
+00:09:19,460 --> 00:09:23,450
+تمشي تمشي لما تصل هذاهذا طبعاً كلكم قال إيش عبارة
+
+138
+00:09:23,450 --> 00:09:27,230
+عن باي على اتنين إذا تتا من باي على أربعة لعند مين
+
+139
+00:09:27,230 --> 00:09:31,690
+يا جماعة لعند باي على اتين هذه مثال ناخد مثال آخر
+
+140
+00:09:31,690 --> 00:09:34,670
+إذا أنا بقولي find the limits of integration for
+
+141
+00:09:34,670 --> 00:09:38,550
+integrating F of R ثتا بده بس limits of إيش of
+
+142
+00:09:38,550 --> 00:09:43,350
+integration لما بدنا نكامل over F of R ثتا over
+
+143
+00:09:43,350 --> 00:09:47,430
+the region Rأيش ال region R هيو بوصف لك إياها
+
+144
+00:09:47,430 --> 00:09:52,870
+تماماً That lies inside the chordoid R بتساوحة ذات
+
+145
+00:09:52,870 --> 00:09:57,770
+كسان ثتا and outside R بتساوحة أنا عارف إنكم أنتوا
+
+146
+00:09:57,770 --> 00:10:01,460
+نسيان اللي هي الرسمات اللي هي تبعات تفاضل باهأنا
+
+147
+00:10:01,460 --> 00:10:06,340
+مش جاي اللي هو أني أخبركم في الرسمات هذه الرسمات
+
+148
+00:10:06,340 --> 00:10:10,380
+هذه اللي أنتوا بتنسوها احنا لأ بنعملكم إياها لكن
+
+149
+00:10:10,380 --> 00:10:14,000
+في بعض الرسمات مجدور عليها اللي واحد ممكن يستذكرها
+
+150
+00:10:14,000 --> 00:10:19,900
+من خلال اللي هو يعني دراسته البسيطة في داخل اتفاضه
+
+151
+00:10:19,900 --> 00:10:24,520
+ال gymوماعرفته بتفاضل ده لكن بقدر الإمكان اللي هو
+
+152
+00:10:24,520 --> 00:10:28,540
+تجلبوش من موضوع رسم عشان نجيبكم رسم معقد بحيث انه
+
+153
+00:10:28,540 --> 00:10:31,220
+انتوا نسينوا انا مش جاي اختباركم في الرسم اداء انا
+
+154
+00:10:31,220 --> 00:10:34,980
+جاي اختباركم في ال double integration over اللي هو
+
+155
+00:10:34,980 --> 00:10:39,880
+اللي هي a region in polar formسلّوا على النبي عليه
+
+156
+00:10:39,880 --> 00:10:42,880
+الصلاة والسلام الان الـ Cardoid R بتساوي 1 زي
+
+157
+00:10:42,880 --> 00:10:47,100
+كتصين ثتا الان الـ Cardoid هيدو R بتساوي 1 زي
+
+158
+00:10:47,100 --> 00:10:51,300
+كتصين ثتا هيه لو واحد اللي جرب يرسمه يعرف يرسمه
+
+159
+00:10:51,300 --> 00:10:55,360
+بتبدأ بثتا بصفر اللي هتلاقيها ده اللي هي عبارة عن
+
+160
+00:10:55,360 --> 00:10:59,940
+اللي هي ثتا بصفر مثلا بتصير كتصين ثتا واحد بتصير
+
+161
+00:10:59,940 --> 00:11:03,870
+هذه اللي هي اتنين بتبدأ من هناثم تبدأ تطلع لما
+
+162
+00:11:03,870 --> 00:11:07,710
+انتصر باي على اتنين لانتصار باي على اتنين لانتصار
+
+163
+00:11:07,710 --> 00:11:11,990
+باي على اتنين لانتصار باي على اتنين لانتصار
+
+164
+00:11:11,990 --> 00:11:17,250
+باي على اتنين لانتصار
+
+165
+00:11:17,250 --> 00:11:21,650
+باي على اتنينهذه مرة أخرى لما اكتملت الدورة مش هى
+
+166
+00:11:21,650 --> 00:11:25,450
+موضوع نهائى بس هذا للتذكير بس الان الان هو بوصف
+
+167
+00:11:25,450 --> 00:11:29,150
+المنطقة هي المنطقة اللى بين ال cardioid هيه
+
+168
+00:11:29,150 --> 00:11:32,530
+والدائرة R بتساوي واحد ده نعرف ان الان R بتساوي
+
+169
+00:11:32,530 --> 00:11:35,310
+واحد، R بتساوي خمسة، R بتساوي أربعة، كلهين دوائر
+
+170
+00:11:35,310 --> 00:11:38,950
+لأن R بتساوي واحد عبارة عن دائرة نصف قطرها إيش
+
+171
+00:11:38,950 --> 00:11:43,990
+واحدلأن هذه المنطقة هي المنطقة المقصودة في إن نجد
+
+172
+00:11:43,990 --> 00:11:47,730
+حدود الـ Integration لها صلوا على النبي عليه
+
+173
+00:11:47,730 --> 00:11:51,190
+الصلاة والسلام زي ما بقولنا أول حاجة بنرسم المنطقة
+
+174
+00:11:51,190 --> 00:11:55,170
+تاني حاجة بنرسم اللي هي مناطق اللي هي التقاط اللي
+
+175
+00:11:55,170 --> 00:11:59,770
+بين الجهتين وبنحدد اللي هو حدود R وبعدين بنحدد مين
+
+176
+00:11:59,770 --> 00:12:04,180
+θالان عشان احدد R ثتا بلزمنا نقطة التقاطع حدى نقطة
+
+177
+00:12:04,180 --> 00:12:06,720
+التقاطع حدى زى ما كنا نعمل في الـ Cartesian عادي
+
+178
+00:12:06,720 --> 00:12:11,040
+الان نقطة التقاطع بين مين يعنى هذه؟ هذه بين اللي
+
+179
+00:12:11,040 --> 00:12:14,960
+هو R بتساوي واحد ونفس باشي هذه و بين R بتساوي واحد
+
+180
+00:12:14,960 --> 00:12:18,520
+زي كوصين ثتا ساوين مع بعض واحد زي كوصين ثتا ايش
+
+181
+00:12:18,520 --> 00:12:22,520
+بيساوي واحد اذا كوصين ثتا ايش بيساوي سفر اذا ثتا
+
+182
+00:12:22,520 --> 00:12:28,490
+يا اماالتي هي عبارة عن باي على اتنين أو ناقص باي
+
+183
+00:12:28,490 --> 00:12:31,190
+على اتنين التي هي تلاتة باي على اتنين الان التي هي
+
+184
+00:12:31,190 --> 00:12:33,710
+باي على اتنين من ناقص باي على اتنين إذا فتها هن
+
+185
+00:12:33,710 --> 00:12:37,050
+ناقص باي على اتنين وفتها هن اشملها باي على اتنين
+
+186
+00:12:37,050 --> 00:12:39,210
+لم أقل لكم لماذا كتبنا ناقص باي على اتنين وليس
+
+187
+00:12:39,210 --> 00:12:43,650
+تلاتة باي على اتنين طبعا التمثيل نفس الشيءأو تعبير
+
+188
+00:12:43,650 --> 00:12:48,190
+آخر نحن على أساس أننا نبدأ من المنطقة الصغيرة
+
+189
+00:12:48,190 --> 00:12:51,970
+ونذهب للكبيرة في اتجاه عكس عقاربس عشان ذلك بدأت
+
+190
+00:12:51,970 --> 00:12:55,130
+بالنقص باي على اتنين على أساس أن أمسح المنطقة من
+
+191
+00:12:55,130 --> 00:12:59,750
+المنطقة الصغيرة لثتا إلى المنطقة الكبيرةمش عادي
+
+192
+00:12:59,750 --> 00:13:02,230
+اللي هو أبدا من عنده اللي هو بقى ال .. اللي هي
+
+193
+00:13:02,230 --> 00:13:05,850
+theta اللي هي تلاتة باي على اتنين وعمل حدود
+
+194
+00:13:05,850 --> 00:13:08,530
+integration باي على اتنين إلى باي على اتنين بيصير
+
+195
+00:13:08,530 --> 00:13:12,170
+اللي هي عدديا هذه أكبر من هذه فبس يعني اللي هو
+
+196
+00:13:12,170 --> 00:13:19,110
+بشكل خلنا نقول هي كالتي هي ألطف وأنسب إذا خلّينا
+
+197
+00:13:19,110 --> 00:13:21,750
+ان شاء الله تكونوا فاهمين عليش بجود الآن بدنا نيجي
+
+198
+00:13:21,750 --> 00:13:24,030
+نحدد حدود ال integration حدود ال integration زي ما
+
+199
+00:13:24,030 --> 00:13:27,750
+اتفجنا بنرفع ال rays هاي ال ray هاي عشان نحدد R
+
+200
+00:13:27,750 --> 00:13:32,330
+هاي F of R ثتا RDR D ثتا بنفاضل ب .. بنكمل هنا على
+
+201
+00:13:32,330 --> 00:13:36,530
+طول ب R قبل ماحدش يقولي D ثتا بقبل DR لأ R و بعدين
+
+202
+00:13:36,530 --> 00:13:42,310
+ان هي D ثتااللي هي عندي R نشوف R الان R طلع Arrays
+
+203
+00:13:42,310 --> 00:13:46,530
+من نقطة الأصل كل ال Arrays من نقطة الأصل تدخل من
+
+204
+00:13:46,530 --> 00:13:49,550
+عند الدائرة و تخرج من عند ال Cardoid و نفس الأشي
+
+205
+00:13:49,550 --> 00:13:53,770
+اللي تحت تدخل دائرة و تطلع Cardoid و هكذا إذن
+
+206
+00:13:53,770 --> 00:13:58,350
+المنطقة كلها على نفس المنوال إذن ارنا تبدأ R
+
+207
+00:13:58,350 --> 00:14:02,210
+بتساوي واحد و تطلع R بتساوي واحد زائد cosine
+
+208
+00:14:02,210 --> 00:14:06,870
+فترانة تخرج من عندهاالان نأتي لـ θ , ثتة عشان تمسح
+
+209
+00:14:06,870 --> 00:14:11,570
+كل المنطقة هاي ثتة لأ هنا بتبدأ بعدين هنا بعدين
+
+210
+00:14:11,570 --> 00:14:15,110
+هنا قيمة ثتة طبعا هذه الثتة اللي بالشكل هذا اللي
+
+211
+00:14:15,110 --> 00:14:19,190
+هي طبعا مع ال positive x axis و هنا و هيك و هيك و
+
+212
+00:14:19,190 --> 00:14:23,990
+هيك طبعا ثتة هنا عبارة عن ناقص باي على اتنين عشان
+
+213
+00:14:23,990 --> 00:14:27,210
+تمسح الباقي ده اللي بيظل ماشي هيك هيك هيك هيك لما
+
+214
+00:14:27,210 --> 00:14:30,270
+تصل لمين لعنده النقطة هذه اللي حددناها في الأول
+
+215
+00:14:30,270 --> 00:14:34,110
+اللي هي باي على اتنينالان الموجود هذه بكون عندى
+
+216
+00:14:34,110 --> 00:14:38,210
+فعلا انا حصلت على اللى هو limits of integration
+
+217
+00:14:38,210 --> 00:14:42,930
+للزواحل الان area in polar form او in polar
+
+218
+00:14:42,930 --> 00:14:45,890
+coordinates طبعا يا جماعة انتوا عارفين اللى هى ال
+
+219
+00:14:45,890 --> 00:14:49,050
+area in polar العادية ال area بيساوي double
+
+220
+00:14:49,050 --> 00:14:52,910
+integral للمنطقه R بدي اواجه ال area للمنطقه R
+
+221
+00:14:52,910 --> 00:14:55,490
+اخدناها المرة الماضية اللى هو ال integration لل
+
+222
+00:14:55,490 --> 00:15:03,290
+RDAأو بمعنى آخر DXDY لكن DXDY زي ما احنا عارفين
+
+223
+00:15:03,290 --> 00:15:07,930
+ايش بتساوي في الـ polar اللي هو RDRDθ هذا خدناه
+
+224
+00:15:07,930 --> 00:15:12,030
+الكلام بالتفصيل في تفاضل بها إذن بس أذكركم فيه
+
+225
+00:15:12,030 --> 00:15:16,890
+اللي هي DA اللي هي DXDY بتساوي RDRDθ إذن ال area
+
+226
+00:15:16,890 --> 00:15:20,610
+بتساوي إلى الان ال area of a closed bounded region
+
+227
+00:15:20,610 --> 00:15:24,890
+R in polar coordinates الين is كده وهي اللي امننا
+
+228
+00:15:25,150 --> 00:15:28,430
+خلّيني أخد مثال ‏find the area enclosed by the
+
+229
+00:15:28,430 --> 00:15:33,670
+lemniscate RWI 4cos2θ ‏الأن هذه الرسمة ‏الأن طبعا
+
+230
+00:15:33,670 --> 00:15:39,230
+أكيد كتير منكم ناسي تفاصيلها ‏الأن إذا بدك تفصل في
+
+231
+00:15:39,230 --> 00:15:43,110
+الرسم ‏بتقدر تفصل تاخد اللي هو قيم لθ ‏بسفر بكده
+
+232
+00:15:43,110 --> 00:15:47,490
+بكده ‏بتبدأ تشتغل ‏لما انتصل لالرسمة اللي أمامك
+
+233
+00:15:47,530 --> 00:15:51,050
+طبعا كنا نشوف فيه سيمترا حول الـ x-axis حول الـ
+
+234
+00:15:51,050 --> 00:15:53,730
+origin حول هذا الكلام مش موضوعنا هذا الان كلها
+
+235
+00:15:53,730 --> 00:15:57,150
+بتساعدنا في الرسم لكن هي الرسمة الآن الرسمة موجودة
+
+236
+00:15:57,150 --> 00:16:00,310
+أمامي الان بدأ اوجد ال area للمنطقة هذه اللي هي
+
+237
+00:16:00,310 --> 00:16:03,110
+الرسمة اللي أمامي طب أنا عشان اوجد ال area طب انا
+
+238
+00:16:03,110 --> 00:16:06,700
+شايف هذه المنطقة هي زي هذه بالظبططب ما بحسب هذه و
+
+239
+00:16:06,700 --> 00:16:09,120
+بضربها في اتنين لا حد بيقول لي لأ يا أستاذ ما هي
+
+240
+00:16:09,120 --> 00:16:13,320
+أصلا هذه اللي هي المنطقة هي زي هذه و زي هذه و زي
+
+241
+00:16:13,320 --> 00:16:16,860
+هذه بقول له صح إذا خلّيني أنشع كلامك إذا هذه بدي
+
+242
+00:16:16,860 --> 00:16:22,720
+أخدها الآن هي المنطقة و أضربها في أربعة يعني بيسير
+
+243
+00:16:22,720 --> 00:16:26,500
+ال area عندي ال area بالساوية أربعة في قيمة ال
+
+244
+00:16:26,500 --> 00:16:31,490
+integration لل RDRDθ على هذه المنطقةطب هذه المنطقة
+
+245
+00:16:31,490 --> 00:16:34,750
+اللي هي داخل طبعا كله بيدوه اللي هو area enclosed
+
+246
+00:16:34,750 --> 00:16:37,870
+by the landscape يعني ال area اللي جوا هذه كلها
+
+247
+00:16:37,870 --> 00:16:42,390
+الان هذه الان بده حدد حدود ال integration اللي لها
+
+248
+00:16:42,390 --> 00:16:46,670
+حدود ال integration سهل ال R هيه و بيطلع من الأصل
+
+249
+00:16:46,670 --> 00:16:49,810
+و بيجي على مين اللي هو بدخل المنطقة في الأصل و
+
+250
+00:16:49,810 --> 00:16:53,390
+بيطلع بضرب في مين في الرسمة اللي هي الرسمة ايش
+
+251
+00:16:53,390 --> 00:16:57,320
+اللي هي بتساويالقيمة R هي جدر التربيع الاربعة
+
+252
+00:16:57,320 --> 00:17:00,920
+كوساين اتنين ثيتا إذا ببدأ من صفر و بروح لجدر
+
+253
+00:17:00,920 --> 00:17:05,080
+الاربعة كوساين اتنين ثيتا هذه اللي هي حدود المنطقة
+
+254
+00:17:05,080 --> 00:17:10,360
+طيب الآن ثيتا تشوف الثيتا الان هذه واضحة أن ثيتا
+
+255
+00:17:10,360 --> 00:17:15,520
+من هنا عندها بصفر ليش لأن لما كانت ثيتا بصفر بيصير
+
+256
+00:17:15,520 --> 00:17:18,580
+اللي عبارة عن القيمة جديش اربعة كوساين الصفر أربعة
+
+257
+00:17:18,580 --> 00:17:21,800
+يعني هذه المنطقة أربعة فعلا إذا هذه بدأنا احنا
+
+258
+00:17:21,800 --> 00:17:26,290
+معين ثيتا بصفرطب الآن لما نوصلنا اللي هي رجعنا على
+
+259
+00:17:26,290 --> 00:17:30,670
+مين على نقطة الأصل رجع يعني و كأنها رجعت R بصفر
+
+260
+00:17:30,670 --> 00:17:34,910
+لنشوف هذه قيمة θ عندها دي لما R بصفر بيصير أربعة
+
+261
+00:17:34,910 --> 00:17:38,010
+كوصين اتنين ثتا بالساوية صفر يعني كوصين اتنين ثتا
+
+262
+00:17:38,010 --> 00:17:40,870
+بالساوية صفر فاهمين اللي عملته الان كوصين اتنين
+
+263
+00:17:40,870 --> 00:17:43,810
+ثتا بالساوية صفر إذا ثتا اتنين ثتا بالساوية πاي
+
+264
+00:17:43,810 --> 00:17:49,290
+على اتنين يعني ثتا بالساوية باي على أربعة إذا هذه
+
+265
+00:17:49,290 --> 00:17:51,710
+الناس ده خلتيها طبعا هذه باي على أربعة مش هتكون
+
+266
+00:17:51,710 --> 00:17:55,240
+اللي هي أياشاللي هي المنطقة هالي اللي هي تلاتة باي
+
+267
+00:17:55,240 --> 00:17:59,000
+على أربعة مثلا لأ هذه باي على أربعة اللي هي بتحقق
+
+268
+00:17:59,000 --> 00:18:01,900
+هذه أكتر من القيمة من ضمن ال باي على أربعة و هو
+
+269
+00:18:01,900 --> 00:18:06,200
+اخر فهذه باي على أربعة ماشي الحال إذا هذه صارت باي
+
+270
+00:18:06,200 --> 00:18:11,360
+على أربعة إذا صار عندي ال integration من سفرلأ باي
+
+271
+00:18:11,360 --> 00:18:14,280
+على أربع لأنه بدأ بالصفر طبعا من باي على أربع هي
+
+272
+00:18:14,280 --> 00:18:17,960
+الصفر اللي عند باي على أربع ر دي ر دي ثتا والان
+
+273
+00:18:17,960 --> 00:18:20,760
+ايش بده سوية جامعة بده كامل اكمل بالنسبة لغاية
+
+274
+00:18:20,760 --> 00:18:23,540
+بيصير ر تربيع على اتنين الان حدود ال integration
+
+275
+00:18:23,540 --> 00:18:26,800
+من صفر للجدر التربيعي اربعة كوساين اتنين ثتابعوض
+
+276
+00:18:26,800 --> 00:18:31,260
+عنها ر بصفر هي صفر و بيبقى الهدى اللى بيعوض عنها
+
+277
+00:18:31,260 --> 00:18:35,880
+اللى فوق بيصير جدر الأربعة بيصير اللى هو لما أربعه
+
+278
+00:18:35,880 --> 00:18:40,600
+بيصير اللى هو عبارة عن اتنين cosine اتنين ثتا طبعا
+
+279
+00:18:40,600 --> 00:18:42,860
+بيصير اربعة و بقسمها اتنين بيصير اتنين cosine
+
+280
+00:18:42,860 --> 00:18:45,860
+اتنين ثتا فاهمين ر تربيع يساوي اربع cosine اتنين
+
+281
+00:18:45,860 --> 00:18:49,020
+ثتا على اتنين بترجع اتنين cosine اتنين ثتا D ثتا
+
+282
+00:18:49,200 --> 00:18:53,100
+الان بكامل من صفر ل باقي على اتنين دي ثتا هذا اللي
+
+283
+00:18:53,100 --> 00:19:02,600
+هو عبارة عن اللي هو sign اتنين ثتا مظبوط على اتنين
+
+284
+00:19:02,600 --> 00:19:05,480
+اللي هي الثتا اللي هنا بيصير عنده اللي هو sign
+
+285
+00:19:05,480 --> 00:19:07,700
+اتنين ثتا و فيه اربعة برا بيصير اربعة sign اتنين
+
+286
+00:19:07,700 --> 00:19:10,780
+ثتا و من صفر ل عند باقي على اربعة بيطلع عند مين يا
+
+287
+00:19:10,780 --> 00:19:15,290
+جماعةبطلع عندى اربعة و بكون هيك خلصنا اوجدنا ال
+
+288
+00:19:15,290 --> 00:19:18,610
+area و نيجي الان نتغير cartesian integral into
+
+289
+00:19:18,610 --> 00:19:26,010
+polar integrals نشوف كيف بتحول ال cartesian إلى
+
+290
+00:19:26,010 --> 00:19:31,800
+polarأول اشي عندى اللى هي تلت امور لازم نغير فيها
+
+291
+00:19:31,800 --> 00:19:36,040
+اول اشي ال region R بتكون موجودة بالكارتيزيان ومن
+
+292
+00:19:36,040 --> 00:19:40,980
+ثم بنحولها الى G في اللى هى البولر وهدى يمكن اللى
+
+293
+00:19:40,980 --> 00:19:45,000
+هى من اهم الخطوات او الخطوة اللى ..اللي بدها شوية
+
+294
+00:19:45,000 --> 00:19:49,680
+شغل الان f of x و y على طول بنحولها ل f of r cos θ
+
+295
+00:19:49,680 --> 00:19:56,180
+r sin θ يعني بدلالة θ و بدلالة r حسب القوانين اللي
+
+296
+00:19:56,180 --> 00:20:00,180
+ذكرناها سابقا اللي هي ال x بالساوية r cos θ و ال y
+
+297
+00:20:00,180 --> 00:20:09,510
+بالساوية r sin θأو تأتي الشيء الثاني DXDY بحولها
+
+298
+00:20:09,510 --> 00:20:19,710
+لـ RDRDθ يعني هذا التحويل وهو بأفق القوانينالان
+
+299
+00:20:19,710 --> 00:20:23,150
+اللي بده جهده اللي هو تحديد حدود التكامل اللي هو
+
+300
+00:20:23,150 --> 00:20:27,770
+من Cartesian R إلى اللي هي بولر اللي هي
+
+301
+00:20:27,770 --> 00:20:31,890
+coordinates نيجي الان ناخد مثال evaluate the
+
+302
+00:20:31,890 --> 00:20:35,610
+double integral over region R ايش ال region R؟ ال
+
+303
+00:20:35,610 --> 00:20:39,190
+region R هينمعطينيها هنا لجيت بنتقول ايش هي؟ يجي
+
+304
+00:20:39,190 --> 00:20:45,840
+ده X تربيع زي Y تربيع دي Y دي Xالان where R is the
+
+305
+00:20:45,840 --> 00:20:49,300
+semicircular region bounded by the x-axis اللي هي
+
+306
+00:20:49,300 --> 00:20:52,120
+semicircular region نص الدائرة هذه اللي bounded by
+
+307
+00:20:52,120 --> 00:20:56,360
+the x-axis and the curve Y بيساوي جدر الواحد minus
+
+308
+00:20:56,360 --> 00:21:01,560
+X اللي هي عبارة عن الجزء الموجب من الدائرة Y تربيع
+
+309
+00:21:01,560 --> 00:21:05,650
+زاد X تربيع بتساوي واحدماشي الحال اذا هي عند
+
+310
+00:21:05,650 --> 00:21:09,910
+المنطقة R وهي عند الدالة اللي بنكملها لو اجينا
+
+311
+00:21:09,910 --> 00:21:14,350
+واحد اجه بده كامل هادى بواسط القارتيزيان كمن لاقيت
+
+312
+00:21:14,350 --> 00:21:17,590
+اكس تربيع ز و اي تربيع بالنسبة لوايش هيعرفى كامل
+
+313
+00:21:17,590 --> 00:21:22,750
+by as an elementary integrationإذا الآن في عند عقل
+
+314
+00:21:22,750 --> 00:21:27,390
+التكامل بعرفش أكامل اللي هي 80X³ زي Y³ وهذا
+
+315
+00:21:27,390 --> 00:21:31,490
+التكامل هذا بالذات التكامل هتلاقوه كثير بظهر عندنا
+
+316
+00:21:31,490 --> 00:21:37,330
+في الإحصاء أو اللي هو في الإحتمالات فبلزمنا نوجده
+
+317
+00:21:37,330 --> 00:21:41,030
+فاللي هو الناس بتضطر تروح ده عشان توجده لميه
+
+318
+00:21:41,030 --> 00:21:45,150
+للبولر و بتتحول للبولر و الإيجاد بيصير إيجاد سهل
+
+319
+00:21:45,150 --> 00:21:49,140
+كمانخلّينا نشوف الآن، إذا عشان وجود هذا بدي أحوله
+
+320
+00:21:49,140 --> 00:21:53,600
+للبولر هاي عندي ال integration أحوله للبولر أول
+
+321
+00:21:53,600 --> 00:21:59,200
+إشي بدي أشيل الـ dy dx و أحط مكانها لـR dr dθ و E
+
+322
+00:21:59,200 --> 00:22:02,220
+to the X تربيع زاد Y تربيع احنا بنعرف X تربيع زاد
+
+323
+00:22:02,220 --> 00:22:06,360
+Y تربيع هي عبارة عن R تربيع كيف عرفت؟التي هي X
+
+324
+00:22:06,360 --> 00:22:11,080
+تربيع R تربيع Cos تربيع ثتا والـ Y R تربيع Sin
+
+325
+00:22:11,080 --> 00:22:16,260
+تربيع ثتا مجموح بيصير R تربيع في Cos تربيع ثتا
+
+326
+00:22:16,260 --> 00:22:19,440
+زائد Sin تربيع ثتا يعني Cos تربيع زائد Sin تربيع
+
+327
+00:22:19,440 --> 00:22:22,880
+ثتا زي ما بتعرفوا واحد بتضلل R تربيع إذا هذه عبارة
+
+328
+00:22:22,880 --> 00:22:26,830
+عن اللي هي مين R تربيعلأن اتوارد تربي عردي اردي
+
+329
+00:22:26,830 --> 00:22:34,290
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+330
+00:22:34,290 --> 00:22:34,330
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+331
+00:22:34,330 --> 00:22:34,530
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+332
+00:22:34,530 --> 00:22:35,150
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+333
+00:22:35,150 --> 00:22:37,090
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+334
+00:22:37,090 --> 00:22:40,670
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+335
+00:22:40,670 --> 00:22:45,190
+اردي
+
+336
+00:22:45,190 --> 00:22:49,850
+اردي اردي اردي اردي اردي اردي اردي اردي االان هذا
+
+337
+00:22:49,850 --> 00:22:56,090
+اللي هو اللي هي الدائرة الدائرة اللي هي الدائرة
+
+338
+00:22:56,090 --> 00:22:59,810
+الجزء النص التاني من الدائرة الفجانى اللي هو y
+
+339
+00:22:59,810 --> 00:23:02,390
+تربيع زاد x تربيع بيساوي واحد او y بيساوي جدري
+
+340
+00:23:02,390 --> 00:23:05,930
+تربيع لو واحد ناقص x تربيعو احنا بنعرف ان y تربيع
+
+341
+00:23:05,930 --> 00:23:09,970
+زاد x تربيع بساوي واحد اللي هو معادلته بالبولر ايش
+
+342
+00:23:09,970 --> 00:23:14,190
+هي y تربيع زاد x تربيع اللي هي مكانها R تربيع بصير
+
+343
+00:23:14,190 --> 00:23:18,690
+R تربيع بساوي واحد يعني R بتساوي واحد عندى الان R
+
+344
+00:23:18,690 --> 00:23:23,070
+بتساوي واحد هذه اللي هي الدائرة نص الدائرة طبعا
+
+345
+00:23:23,070 --> 00:23:26,490
+الدائرة كلها R بتساوي واحد لكن الان اللي هيتحكملي
+
+346
+00:23:26,490 --> 00:23:30,210
+في الموضوع ليه قيمة θ هتحددلي بس المنطقة اللي فوق
+
+347
+00:23:30,600 --> 00:23:33,720
+خلّينا نشوف إذا أن هذه المنطقة خلّينا نطلع الآن
+
+348
+00:23:33,720 --> 00:23:36,960
+الـ Ray اللي هو في داخل المنطقة يدخل المنطقة و
+
+349
+00:23:36,960 --> 00:23:40,600
+يخرج منها طبعا هي بدأ من عند السفر و بخرج من عندها
+
+350
+00:23:40,600 --> 00:23:44,020
+كل على نفس المنوال بخرج من الدائرة اللي هي R
+
+351
+00:23:44,020 --> 00:23:48,660
+بالسواحي علشان هي حدود الـR من سفر لعند واحد نيجي
+
+352
+00:23:48,660 --> 00:23:51,660
+الآن لقييم الـθتا طبعا هي ال .. ال .. ال .. ال ..
+
+353
+00:23:51,660 --> 00:23:51,800
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+354
+00:23:51,800 --> 00:23:52,340
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+355
+00:23:52,340 --> 00:23:52,680
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+356
+00:23:52,680 --> 00:23:54,740
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+357
+00:23:54,740 --> 00:23:56,300
+ال .. ال .. ال .. ال .. ال .. ال ..لأن واضح أن θ
+
+358
+00:23:56,300 --> 00:24:01,180
+من هنا تبدأ بـ 0 و تمسح المنطقة و يبقى هذا المؤشر
+
+359
+00:24:01,180 --> 00:24:03,800
+يمشي و يمشي و يمشي و يمشي و يصبح باي على اتنين و
+
+360
+00:24:03,800 --> 00:24:04,920
+يصبح كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا
+
+361
+00:24:04,920 --> 00:24:05,020
+و كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+362
+00:24:05,020 --> 00:24:05,940
+كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+363
+00:24:05,940 --> 00:24:06,500
+كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+364
+00:24:06,500 --> 00:24:09,480
+كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+365
+00:24:09,480 --> 00:24:10,080
+كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+366
+00:24:10,080 --> 00:24:20,120
+كذا و كذا و كذا و كذا و كذاحولناها من كارتيزيان ل
+
+367
+00:24:20,120 --> 00:24:24,040
+بولار بيصير عندى اللى هو هذا E تربيع R تربيع في R
+
+368
+00:24:24,040 --> 00:24:28,480
+اللى هو تكامله طبعا نص E to the R تربيع اللى هو
+
+369
+00:24:28,480 --> 00:24:33,160
+لإنه هذا التفاضل هو هذا المقدار من 0 لعنده 1 طبعا
+
+370
+00:24:33,160 --> 00:24:37,240
+بنعوض بتطلع نص اللى هى في حالة الواحد E to the
+
+371
+00:24:37,240 --> 00:24:41,000
+واحد ناقص اللى هو E to the zero اللى هو واحد اللى
+
+372
+00:24:41,000 --> 00:24:45,640
+هى ناقص واحد لكل D ثتاالان هذا ثابت بنضربه في θ
+
+373
+00:24:45,640 --> 00:24:49,260
+بصير ال θ من صفر ل باي بنعوده عن θ بتطلع باي على
+
+374
+00:24:49,260 --> 00:24:52,220
+اتنين اللي هي دي باي في نص باي على اتنين في ايه
+
+375
+00:24:52,220 --> 00:24:56,660
+تقرر minus واحد وبنكون هيك اوجدنا اللي هي ال
+
+376
+00:24:56,660 --> 00:25:00,240
+integration طبعا شفنا اهمية ال polar انها
+
+377
+00:25:00,240 --> 00:25:05,140
+بتتغلبلنا على اللي هي عدم عرفتنا التكامل بتحويله
+
+378
+00:25:05,140 --> 00:25:10,800
+لتكامل المكافئ بواسطة اللي هو ال polarالان ناخد
+
+379
+00:25:10,800 --> 00:25:13,780
+مثال تاني evaluate the integral معطيني ال
+
+380
+00:25:13,780 --> 00:25:19,180
+integration الان هيه هو مش وصف للمنطقة الان انا
+
+381
+00:25:19,180 --> 00:25:23,120
+بدرسمها المنطقة لان بس استعنت بالرسم اللي فاتت بس
+
+382
+00:25:23,120 --> 00:25:26,120
+الان هو مش راسي هذه الرسمة مش وجودة هذا بدرسمها
+
+383
+00:25:26,120 --> 00:25:30,320
+الان ايش هي ال value of the integration من 0 لعند
+
+384
+00:25:30,320 --> 00:25:33,940
+جدر الواحد ناقص x تربيع x تربيع زد ويه تربيع dy dx
+
+385
+00:25:34,270 --> 00:25:37,470
+من صفر لعند الواحد لو ايجي اجرب واحد يكمل هذه اللي
+
+386
+00:25:37,470 --> 00:25:40,410
+جوا بالنسبة ل Y بيكملها هاي X تربيع في الجدر ال
+
+387
+00:25:40,410 --> 00:25:43,750
+تربيع لو واحد ناقص X تربيع زائد اللي هي طبعا مفهوم
+
+388
+00:25:43,750 --> 00:25:47,210
+X تربيع في Y زائد Y تكعيبة تلاتة و بعوض من صفر
+
+389
+00:25:47,210 --> 00:25:50,230
+لعند هذا بتطلع هذا المقدار هذا المقدار اللي بيطلع
+
+390
+00:25:50,230 --> 00:25:55,060
+عندبدا اكمله لل X ماعرفش اكمله الا بواسط ال tables
+
+391
+00:25:55,060 --> 00:26:00,540
+و غلبه كتير كتير كتير فالان بدي احوله الى بولر
+
+392
+00:26:00,540 --> 00:26:06,080
+يكون سهل التكامل واللي بشجعني ايش ان ال X تربيع
+
+393
+00:26:06,080 --> 00:26:09,300
+زاد Y تربيع هذي انا بعرفها في البولر قبل بشوية
+
+394
+00:26:09,300 --> 00:26:14,120
+قولها هي عبارة عن R تربيعفعلى طول تتحول هذه لـR
+
+395
+00:26:14,120 --> 00:26:19,700
+تربيع و الـ dy بي dx تتحول لR دي R دي ثتا هذه
+
+396
+00:26:19,700 --> 00:26:24,520
+تكملها بالنسبة لـR سهلة R تربيع في R تصبح R تكعيب
+
+397
+00:26:24,520 --> 00:26:31,120
+و تتفاء كامل بكل سهولة إذا الآن التحويل لبولر حيسن
+
+398
+00:26:31,120 --> 00:26:36,920
+لعملية إيجاد التكامل هنا حولنااللي هي الـ
+
+399
+00:26:36,920 --> 00:26:40,920
+Integrand اللي هو الدالة اللي بدنا نكملها و اللي
+
+400
+00:26:40,920 --> 00:26:47,160
+هو dy by dx حولناها إلى مكافئها اللي هي rd rd θ في
+
+401
+00:26:47,160 --> 00:26:51,380
+البولارالان يا جماعة دل علينا المنطقة المنطقة هذه
+
+402
+00:26:51,380 --> 00:26:57,140
+عشان احولها ل بولار بدي اش اسوي بدي ارسمها الان
+
+403
+00:26:57,140 --> 00:27:00,780
+الان من صفر لعند مين ال Y قيمة ال Y من صفر لجدر
+
+404
+00:27:00,780 --> 00:27:05,200
+الواحد minus X تربيع هي من صفر لعند ال Y هي من صفر
+
+405
+00:27:05,200 --> 00:27:09,040
+هي من صفر هذي طبعا Y بالساوية هذي اللي هي دائرة بس
+
+406
+00:27:09,040 --> 00:27:12,340
+النص الفجاني من الدائرة هي النص الفجاني من الدائرة
+
+407
+00:27:12,690 --> 00:27:17,790
+هذا الان اللي حدد ليه مين اللي حدد ليه قيمة الـ Y
+
+408
+00:27:17,790 --> 00:27:21,490
+الان قيمة X بدها تحددلي بالظبط ايش من هذه الدائرة
+
+409
+00:27:21,490 --> 00:27:26,190
+اللي بدناياها قيمة X جالي من 0 بس لمين ل1 يعني
+
+410
+00:27:26,190 --> 00:27:30,410
+جالي X تبعتك من 0 ل عند 1 يعني المنطقة دي بديش
+
+411
+00:27:30,410 --> 00:27:34,570
+إياهايعني الآن منطقتنا اللي بدنا نكامل عليها حسب
+
+412
+00:27:34,570 --> 00:27:38,510
+اللي هو المعطى عند Y من 0 لجدر الواحد minus X
+
+413
+00:27:38,510 --> 00:27:42,330
+تربيع و X من 0 لعند الواحد بس هي المنطقة اللي
+
+414
+00:27:42,330 --> 00:27:46,710
+أمامها دي اللي هي اللي في الربع الأول طيب الآن
+
+415
+00:27:46,710 --> 00:27:49,830
+بدنا نحولها لبولر قبل شوية حولناها لبولر هذه اللي
+
+416
+00:27:49,830 --> 00:27:54,030
+هي Y بتساوي X تربيع زي .. Y تربيع زي .. Y بتساوي
+
+417
+00:27:54,030 --> 00:27:56,670
+جدر الواحد minus X تربيع اللي هي جزء من الدائرة R
+
+418
+00:27:56,670 --> 00:28:02,180
+بتساوي واحد إذا هذه معدلتها R بتساوي واحدالان اذا
+
+419
+00:28:02,180 --> 00:28:08,580
+لو جينا على المنطقة هذه كل الاريز اللى بتدخل
+
+420
+00:28:08,580 --> 00:28:11,420
+المنطقة و بتخرج منها يعني من نقطة الاصل بتروح لعند
+
+421
+00:28:11,420 --> 00:28:14,740
+نص الدائرة اللى R بساوة واحد عشان هيك حدود ال
+
+422
+00:28:14,740 --> 00:28:18,660
+integration لـR من صفر لعند واحد نيجي الان لـθثتة
+
+423
+00:28:18,660 --> 00:28:23,120
+تبدأ من الهن و تتجه لقرية و تتجه لقرية و تتجه
+
+424
+00:28:23,120 --> 00:28:23,700
+لقرية و تتجه لقرية و تتجه لقرية و تتجه لقرية و
+
+425
+00:28:23,700 --> 00:28:25,040
+تتجه لقرية و تتجه لقرية و تتجه لقرية و تتجه لقرية
+
+426
+00:28:25,040 --> 00:28:26,960
+و تتجه لقرية و تتجه لقرية و تتجه لقرية و تتجه
+
+427
+00:28:26,960 --> 00:28:27,840
+لقرية و تتجه لقرية و تتجه لقرية و تتجه لقرية و
+
+428
+00:28:27,840 --> 00:28:30,760
+تتجه لقرية و تتجه لقرية و تتجه لقرية و تتجه لقرية
+
+429
+00:28:30,760 --> 00:28:37,570
+و تتجه لقرية و تتجه لقرية و تتجه لقبنكمل الان R
+
+430
+00:28:37,570 --> 00:28:41,070
+تكعيب بيصير R أربعة على أربعة من واحد إلى سفر
+
+431
+00:28:41,070 --> 00:28:44,270
+بيصير عبارة عن ربع D ثتا من سفر ل باي على اتنين
+
+432
+00:28:44,270 --> 00:28:49,210
+وهذه محلها ثابت بيصير ثتا على أربعة من سفر ل باي
+
+433
+00:28:49,210 --> 00:28:52,290
+على اتنين وبتطلع باي ع تمانية و هيكة بتكون كتير
+
+434
+00:28:52,290 --> 00:28:57,650
+كتير الأمور سهلة بالباسطة البولار مابتغلبش زي ال
+
+435
+00:28:57,650 --> 00:29:02,170
+kartesian في هذه الحالةنجي الآن لمثال آخر find the
+
+436
+00:29:02,170 --> 00:29:04,930
+volume of the solid bounded above by the
+
+437
+00:29:04,930 --> 00:29:08,170
+paraboloid z بساوية تسعة ناقص x تربيع ناقص y تربيع
+
+438
+00:29:08,170 --> 00:29:11,510
+and below by the unit circle in the xy plane يعني
+
+439
+00:29:11,510 --> 00:29:14,550
+هيه الآن ال solid هذا بقول اللي من تحت هيه عبارة
+
+440
+00:29:14,550 --> 00:29:20,070
+عن الدائرة هذه ومن فوق هيه إيش اللي هو الشكل هذا z
+
+441
+00:29:20,070 --> 00:29:22,940
+بساوية تسعة ناقص x تربيع ناقص y تربيعطبعا الـ
+
+442
+00:29:22,940 --> 00:29:25,200
+volume اللي بين هذا اللي بين هذا اللي هنا إيش
+
+443
+00:29:25,200 --> 00:29:29,500
+بيساوي اللي هو اللي هو المساحة القاعدة في الارتفاع
+
+444
+00:29:29,500 --> 00:29:33,140
+وقولنا إيش هذا بنعبر عنه بال integration اللي هو
+
+445
+00:29:33,140 --> 00:29:36,400
+قيبة ال function اللي هو بيمثل الارتفاع اللي هو
+
+446
+00:29:36,400 --> 00:29:40,950
+عبارة عن إيش تسعة نقص اكتربيع نقص ويتربيعDA هي دي
+
+447
+00:29:40,950 --> 00:29:43,370
+ايه اللي هي بتجيب للمساحة فبصير المساحة في
+
+448
+00:29:43,370 --> 00:29:46,130
+الارتفاع اللي هي بتطلع الحجم ايه ده ال volume حسب
+
+449
+00:29:46,130 --> 00:29:48,390
+ما تقولنا سابقا اللي هي تسعة نقص اكتر بيناقص و هي
+
+450
+00:29:48,390 --> 00:29:52,130
+تربيع دي ايه دي عبارة عن قيمة الدالة اللي هو قيمة
+
+451
+00:29:52,130 --> 00:29:57,930
+Z لفوق على السطح اللي مفروض بدنا نكامل بدنا نكون
+
+452
+00:29:57,930 --> 00:30:01,730
+سطح العلو اللي هو المجسم
+
+453
+00:30:03,910 --> 00:30:09,390
+القاعدة H A هي D A للدائرة الموجودة عليها محطوط
+
+454
+00:30:09,390 --> 00:30:12,210
+عالميا على الدائرة اللي عنده إذا ال region اللي
+
+455
+00:30:12,210 --> 00:30:16,590
+عنده هي R لأنه القاعدة اللي هي عبارة عن الدائرة زي
+
+456
+00:30:16,590 --> 00:30:20,370
+ما هو ذكر ليها هذه الـ N R هي اللي تتحدد لحدود ال
+
+457
+00:30:20,370 --> 00:30:23,720
+integrationاللي هي X تربيع زاد Y تربيع بيساوي واحد
+
+458
+00:30:23,720 --> 00:30:26,880
+هي هم اعطينيها ال unit circle هذه ايش معادلة هي
+
+459
+00:30:26,880 --> 00:30:31,160
+بالبولر R بتساوي واحد الان هذه دائرة كاملة اذا من
+
+460
+00:30:31,160 --> 00:30:34,700
+عند ال X axis لو اخدت اي array هتلاقي ال array
+
+461
+00:30:34,700 --> 00:30:39,000
+بيبدأ من نقطة السفر وبروح لمين للواحد عشان هيك
+
+462
+00:30:39,000 --> 00:30:44,400
+حدود ال integration للR من سفر لعند مين لعند واحد
+
+463
+00:30:44,790 --> 00:30:49,270
+الان عشان اكمل الدائرة الـ R هي بجان و بجان و بجان
+
+464
+00:30:49,270 --> 00:30:53,630
+لف الدائرة كاملة إذا هيبدأ من عند سفر و يمشي لما
+
+465
+00:30:53,630 --> 00:30:57,790
+يصل عند اتنين باي هو عبارة عن حدود ال integration
+
+466
+00:30:57,790 --> 00:31:01,880
+اللي بيجيب الدائرة هذه كلهاالأن ندى للـ Integrand
+
+467
+00:31:01,880 --> 00:31:04,360
+اللى هو تسعة نقص X تربيع نقص Y تربيع الدالة اللى
+
+468
+00:31:04,360 --> 00:31:08,460
+بنكملها ال X تربيع زاد Y تربيع هي عبارة عن مين R
+
+469
+00:31:08,460 --> 00:31:11,220
+تربيع زى ما احنا عارفينه قولناها قبل شوية بيصير دى
+
+470
+00:31:11,220 --> 00:31:15,420
+تسعة نقص R تربيع والـDA زى ما قولنا اللى هى DXDY
+
+471
+00:31:15,420 --> 00:31:21,510
+اللى بتتحول لRDRDθ المكافئة لـDXDYالان بدنا الكامل
+
+472
+00:31:21,510 --> 00:31:26,810
+بتصير 9R-R تكعيب دي R دي تيتا ضربت ال R جوا الان
+
+473
+00:31:26,810 --> 00:31:31,570
+كامل هذه بتصير 9R تربيع على 2 ناقص ربع R أقصى 4 من
+
+474
+00:31:31,570 --> 00:31:36,010
+و إلى وين من 0 لعند 1 قيمة ال R فبتصير عبارة عن 9
+
+475
+00:31:36,010 --> 00:31:41,110
+على 2 ناقص ربع اللي هي بتطلع 17 على 4 حسبوها
+
+476
+00:31:41,110 --> 00:31:45,670
+لحالكمالان بيظهر مين عنده دي ثتا ثتا من صفر لعنده
+
+477
+00:31:45,670 --> 00:31:49,550
+اتنين باي دي ثتا تكمل اللي هو عبارة عن ثتا اللي هي
+
+478
+00:31:49,550 --> 00:31:53,130
+عبارة عن اتنين باي اللي هي بيصير اتنين باي في هذه
+
+479
+00:31:53,130 --> 00:31:57,750
+اللي هي بتصير اللي هي اتنين باي في سبعتاشر لاربعة
+
+480
+00:31:57,750 --> 00:32:00,810
+بيصير سبعة باي على مين على اتنين لانه بتروح اتنين
+
+481
+00:32:00,810 --> 00:32:04,010
+مع اتنين اللي هان اللي هي بتطلع سبعتاشر باي على
+
+482
+00:32:04,010 --> 00:32:07,470
+اتنين اللي هي قيمة اللي هي ذاك فاهمين كيف اللي هي
+
+483
+00:32:07,470 --> 00:32:10,740
+دي بيصير اتنين بايقيمة التكامل لإنها تبسيط data
+
+484
+00:32:10,740 --> 00:32:14,120
+تبسيط data من صفر لباى عن اتنين صفر لتنين باى عن
+
+485
+00:32:14,120 --> 00:32:16,700
+اتنين باى اتنين باى في سبعتاشر على اربع تطلع
+
+486
+00:32:16,700 --> 00:32:20,220
+سبعتاشر على اتنين سبعتاشر باى على اتنين و هيك
+
+487
+00:32:20,220 --> 00:32:23,580
+بيكون اوجدنا ال volume اللي عندنا ندى للسؤال اللى
+
+488
+00:32:23,580 --> 00:32:27,920
+هو مثال السادس الان نعزز الحديث اللى قبله شوية
+
+489
+00:32:27,920 --> 00:32:32,890
+ناخد مثال اخر خلنا نقول شكلهأموره مختلفة لكن بنفس
+
+490
+00:32:32,890 --> 00:32:36,950
+الروح using polar integration باستخدام ال polar
+
+491
+00:32:36,950 --> 00:32:41,090
+integration find the area of the region R اللي هم
+
+492
+00:32:41,090 --> 00:32:44,730
+بدهو اللي هو مساحة ال region R بدهو يوصفها الآن
+
+493
+00:32:44,730 --> 00:32:49,010
+وين موجودة in the XY plane enclosed by اللي هي في
+
+494
+00:32:49,010 --> 00:32:52,590
+ال XY plane مغلقة بواسطة ال circle X تربيع زاد Y
+
+495
+00:32:52,590 --> 00:32:55,820
+تربيع بساوة أربعة دي مش زادة أربعة بساوة أربعةإذا
+
+496
+00:32:55,820 --> 00:32:59,340
+هذه اللي هو المنطقة اللي بدنا إياها جوات الدائرة
+
+497
+00:32:59,340 --> 00:33:04,960
+هذه اللي هي X تربيع زاد Y تربيع بساوة أربعة يعني
+
+498
+00:33:04,960 --> 00:33:08,200
+وكأنه لو أجيناها وعملناها بالبولر على طول X تربيع
+
+499
+00:33:08,200 --> 00:33:11,500
+زاد Y تربيع بساوة أربعة اللي هي الدائرة هذه على
+
+500
+00:33:11,500 --> 00:33:14,700
+طول بتصير R تربيع بساوة أربعة يعني R بساوة اتنين
+
+501
+00:33:14,700 --> 00:33:19,060
+يعني هذه اللي هي الدائرة اللي عندى تعبيرها بالبولر
+
+502
+00:33:19,060 --> 00:33:24,670
+اللي هو R بساوة اتنينطيب .. الان نكمل وصف المنطقة
+
+503
+00:33:24,670 --> 00:33:28,830
+ماخلصناش .. بقوللي الان هذه المنطقة المغلقة من
+
+504
+00:33:28,830 --> 00:33:33,470
+الدائرة ولكنها ايش above the line Y بساوي واحد هي
+
+505
+00:33:33,470 --> 00:33:38,470
+ال Y بساوي واحد فوق المين المنطقة Y بساوي واحد
+
+506
+00:33:38,470 --> 00:33:41,730
+يعني بيصير لسه ها دي معها و ها دي معها و اللي في
+
+507
+00:33:41,730 --> 00:33:45,630
+الجهة التانية معها الان بدنا نحدد أكترabove y
+
+508
+00:33:45,630 --> 00:33:50,250
+بتساوي واحد هي y بتساوي واحد اللي فوقي هي المنطقة
+
+509
+00:33:50,250 --> 00:33:54,810
+هذه محدد مع اللي هذه لكن الان بدنا نحدد اكتر لان
+
+510
+00:33:54,810 --> 00:33:58,430
+ان below the line y بيساوي جدر التلاتة x و تحت من
+
+511
+00:33:58,430 --> 00:34:01,630
+ال line y بيساوي جدر التلاتة x y بيساوي جدر
+
+512
+00:34:01,630 --> 00:34:05,790
+التلاتة x اللي ياخد x بصفر و y بصفر اكيد بنمر نقطة
+
+513
+00:34:05,790 --> 00:34:09,650
+الاصر اللي هو x بواحد و y بجدر التلاتة اذا هذه
+
+514
+00:34:09,650 --> 00:34:14,620
+اللي هي بتيجي اللي هي بتمشي في الخطالخط هذا بالشكل
+
+515
+00:34:14,620 --> 00:34:17,760
+هذا هذا اللي هو Y بيساوي جدر التلاتة X بتعرفوا
+
+516
+00:34:17,760 --> 00:34:21,960
+ترسموه الآن بدنا اشت المنطقة المنطقة اللي تحته اذا
+
+517
+00:34:21,960 --> 00:34:25,480
+هي المنطقة اللي تحته مش هذه اللي فوق اذا الان جينا
+
+518
+00:34:25,480 --> 00:34:30,040
+حصرنا حالنا المنطقة اللي تحت هذا و اللي فوق هذا و
+
+519
+00:34:30,040 --> 00:34:34,220
+اللي في الدائرة اه لإن اللي تحت هذه هي كل هذه اللي
+
+520
+00:34:34,220 --> 00:34:37,560
+فوق هذه هي كلها لكن الان بدناش اللي برا هنا بس
+
+521
+00:34:37,560 --> 00:34:40,880
+اللي موجود وين في الدائرة اذا هي المنطقة المقصودة
+
+522
+00:34:41,210 --> 00:34:45,710
+الان بدنا نوجد نقاط التقاطع عند نقاط التقاطع نجد
+
+523
+00:34:45,710 --> 00:34:49,930
+النقطة الهادي و النقطة هادى الان هذا y بساوية جدر
+
+524
+00:34:49,930 --> 00:34:55,430
+ال 3x عند اللي هو عوض اللي هو في الدائرة هادى اللي
+
+525
+00:34:55,430 --> 00:34:59,190
+x² بيعت سواء أربعة عوض عن قيمة عشان توجد ايه؟ عشان
+
+526
+00:34:59,190 --> 00:35:04,260
+توجد تقاطع هذا الخط مع الدائرة كيف توجده؟بتعود y
+
+527
+00:35:04,260 --> 00:35:07,140
+بيساوي جدر التلاتة x هنا بيصير اللي هي تلاتة x
+
+528
+00:35:07,140 --> 00:35:10,680
+تربيع .. تلاتة y اللي هي تلاتة x تربيع زائد تلاتة
+
+529
+00:35:10,680 --> 00:35:13,540
+x تربيع اللي هي أربعة x تربيع بيساوي أربعة يعني x
+
+530
+00:35:13,540 --> 00:35:17,040
+تربيع بيساوي واحد يعني x بيساوي زائد أو ناقص واحد
+
+531
+00:35:17,040 --> 00:35:19,480
+لكن في المنطقة الموجودة إذا ال x إيش بتساوي واحد
+
+532
+00:35:19,480 --> 00:35:23,500
+لما x بيساوي واحد y إيش هتساوي جدر التلاتة فهذه
+
+533
+00:35:23,500 --> 00:35:27,820
+النقطة اللي هي واحد وجدر التلاتةنجي نوجد من النقطة
+
+534
+00:35:27,820 --> 00:35:30,360
+هذه النقطة هذه عبارة عن تقاطع اللي هو y بيساوي
+
+535
+00:35:30,360 --> 00:35:34,900
+واحد مع الدائرة الان y بيساوي واحد بيصير x تربيع
+
+536
+00:35:34,900 --> 00:35:37,440
+زائد واحد بيساوي أربعة يعني x تربيع بيساوي تلاتة
+
+537
+00:35:37,440 --> 00:35:40,000
+يعني x بيساوي جدر التلاتة زيادة ونوافذ طبعا أنا
+
+538
+00:35:40,000 --> 00:35:43,120
+موجبة بيصير x بيساوي جدر التلاتة إذن النقطة هذه هي
+
+539
+00:35:43,120 --> 00:35:47,030
+عبارة عن جدر التلاتة وواحدإذا المنطقة هنا حددناها
+
+540
+00:35:47,030 --> 00:35:52,270
+تحديدا تماما و حددنا ليمين نقاط التقاطع هذا نقطة
+
+541
+00:35:52,270 --> 00:35:56,610
+مش هتلزمني وإن كانت سهلة اللي هي عند Y بواحد اللي
+
+542
+00:35:56,610 --> 00:36:01,170
+هي عند اللي هي بنوجه قيمة X ليه واحد ع جدر التلاتة
+
+543
+00:36:01,170 --> 00:36:04,350
+لكن هذا مش كتير لأ مش لازم يعنيالان .. هلجدت؟
+
+544
+00:36:04,350 --> 00:36:07,490
+هتشوفوا ليش مش نزلنا .. هي المنطقة الان ماشي الان
+
+545
+00:36:07,490 --> 00:36:11,290
+المنطقة هي اللي هو ال integration بدو ال area لها
+
+546
+00:36:11,290 --> 00:36:15,650
+عبارة عن dA over R اللي هي المنطقة الان dA اللي هي
+
+547
+00:36:15,650 --> 00:36:18,890
+زي ما بنقول أشملها RDRDθ لإن هو بدو ال area
+
+548
+00:36:18,890 --> 00:36:21,590
+للمنطقة هذه مدام ال area إذا اللي هو double
+
+549
+00:36:21,590 --> 00:36:26,650
+integral لـ dA over R الـ dA اللي هي RDRDθ over R
+
+550
+00:36:26,650 --> 00:36:29,710
+R بدنا نحدد الآن اللي هي حدود ال integration شوف
+
+551
+00:36:29,710 --> 00:36:34,280
+حدود ال integrationأول حاجة اللي هي كل الخطوط اللي
+
+552
+00:36:34,280 --> 00:36:39,100
+بتطلع .. اللي هي تدخل للمنطقة هيها .. هيها الخطوط
+
+553
+00:36:39,100 --> 00:36:44,070
+بتدخل للمنطقة من عند اللي هو الخط هذااللي هو y
+
+554
+00:36:44,070 --> 00:36:48,270
+بيساوي واحد لعند الخط لعند مين بتطلع و بتطلع من
+
+555
+00:36:48,270 --> 00:36:51,470
+الدائرة بتدخل من عند الخط و بتطلع من الدائرة و
+
+556
+00:36:51,470 --> 00:36:57,470
+كلها على نفس المنوال إذا الان قيمة R لعند هذا الخط
+
+557
+00:36:57,470 --> 00:37:01,370
+بنوجدها لإنه إيش الخط y بيساوي واحد y هي عبارة عن
+
+558
+00:37:01,370 --> 00:37:06,010
+R sin θ إذا R sin θ بيساوي واحد إذا ال R بيساوي
+
+559
+00:37:06,010 --> 00:37:10,940
+واحد على sin θ يعني R بيساوي cos θإذا قيمة R على
+
+560
+00:37:10,940 --> 00:37:15,620
+هذا الخط اللي هو معدلته R بساوي cos θ قيمة R
+
+561
+00:37:15,620 --> 00:37:21,640
+أشملها عبارة عن cos θ إذا R تبدأ من عند cos θ و
+
+562
+00:37:21,640 --> 00:37:26,140
+بتطلع من الدائرة قيمة الدائرة R عند الدائرة هذه زي
+
+563
+00:37:26,140 --> 00:37:30,360
+ما قلنا معدلتها R بتساوي 2 إذا عند 2 نيجي الآن
+
+564
+00:37:30,360 --> 00:37:36,200
+نشوف قداش قيمة θ لأن هاي θ من هنا بتبدأهيها أول
+
+565
+00:37:36,200 --> 00:37:40,540
+أرى بيطلع عند النقطة هذه و بعدين بيمشي بيمشي لما
+
+566
+00:37:40,540 --> 00:37:45,340
+نبصل هنا طيب الآن بدي أشوف زاوية هذا قداش و زاوية
+
+567
+00:37:45,340 --> 00:37:48,320
+هذا الأرى قداش إذا بتكون تلتة من عند اللي هو هذا
+
+568
+00:37:48,320 --> 00:37:53,080
+من عند هذا الخط لعند هذا بتكون إيش هذا الخط لما
+
+569
+00:37:53,080 --> 00:37:55,440
+يمشي لما يمشي نمشي في داخل المنطقة بيكون مسحها
+
+570
+00:37:55,440 --> 00:37:59,120
+كلها فعلا نيجي لهذه هذه تلتة واحد وجدر التلتة إذا
+
+571
+00:37:59,120 --> 00:38:04,640
+هذا طوله اللي هو واحدوهذا طوله جدر التلاتة لأن هذا
+
+572
+00:38:04,640 --> 00:38:07,920
+إذا المثلث اللي اللي اللي اللي اللي اللي اللي اللي
+
+573
+00:38:07,920 --> 00:38:08,080
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+574
+00:38:08,080 --> 00:38:08,380
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+575
+00:38:08,380 --> 00:38:08,560
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+576
+00:38:08,560 --> 00:38:08,920
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+577
+00:38:08,920 --> 00:38:09,700
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+578
+00:38:09,700 --> 00:38:09,740
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+579
+00:38:09,740 --> 00:38:10,600
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+580
+00:38:10,600 --> 00:38:19,840
+اللي اللي
+
+581
+00:38:19,840 --> 00:38:24,010
+الليإذا هنا عجباله اللي هو هذا قيمة واحد قيمة Y
+
+582
+00:38:24,010 --> 00:38:26,710
+إذا العجباله اللي هي عبارة عن الـ πاية على ستة
+
+583
+00:38:26,710 --> 00:38:30,670
+تلاتين درجة إذا الأن القرية الأول بيبدأ من وين من
+
+584
+00:38:30,670 --> 00:38:33,870
+عند باية على ستة الان نجي نشوف القرية الثانية هي
+
+585
+00:38:33,870 --> 00:38:37,180
+الزاوية تبع القرية الثانيةالان ها الارية الأخيرة
+
+586
+00:38:37,180 --> 00:38:39,980
+طبعا بقصود هي عندي اللي هي النقطة دي واحد وجدر
+
+587
+00:38:39,980 --> 00:38:43,540
+التلاتة إذا هذا طوله جدر التلاتة اللي هي Y وهذا
+
+588
+00:38:43,540 --> 00:38:47,540
+طوله واحد اللي هي صار متلات برضه تلاتين ستيني لكن
+
+589
+00:38:47,540 --> 00:38:50,080
+جدر التلاتة هي العجبال الزاوية الكبيرة اللي هي
+
+590
+00:38:50,080 --> 00:38:53,300
+عبارة عن إيش بايع ستة إذا هذه الزاوية بتصل لمن؟
+
+591
+00:38:53,300 --> 00:38:56,700
+لعين قاسف بايع تلاتة اللي هي ستين درجة اللي هي
+
+592
+00:38:56,700 --> 00:39:01,160
+بتصل إذا الان الارية ده بنتي عند بايع تلاتة اللي
+
+593
+00:39:01,160 --> 00:39:04,410
+هي الزاوية ستين درجةفبكون هذا هو الـ integration
+
+594
+00:39:04,410 --> 00:39:08,430
+ان كامل الآن تكمل سهل بيصير ليا نص R تربيع من cos
+
+595
+00:39:08,430 --> 00:39:13,270
+θ لعند اتنين بتعوض بيصير نص في اربعة ناقص cos
+
+596
+00:39:13,270 --> 00:39:17,930
+تربيع θ دي ثتا الآن هذه تكملها برضه سهلاللي هي
+
+597
+00:39:17,930 --> 00:39:21,650
+عبارة عن نص في تكامل الأربعة اللي هي أربعة ثيتا
+
+598
+00:39:21,650 --> 00:39:25,610
+تكامل اللي هو cosecant تربيع نقص cosecant تربيع
+
+599
+00:39:25,610 --> 00:39:27,930
+التكامل اللي هو عبارة عن كوتان ثيتا زي ما احنا
+
+600
+00:39:27,930 --> 00:39:31,310
+عارفين الآن من وين لوين باية ستة عند باية تلاتة
+
+601
+00:39:31,310 --> 00:39:34,950
+بنعود عليها بطلع عند القيمة هذه نقص القيمة هذه
+
+602
+00:39:34,950 --> 00:39:37,970
+اللي عند باية تلاتة هي القيمة عند باية ستة هي
+
+603
+00:39:37,970 --> 00:39:42,010
+القيمة نترحل من بعض طبعا تعرفت وجودهم ويساوي وبطلع
+
+604
+00:39:42,010 --> 00:39:46,130
+القيمة اللي أمامنا وهذا بيكون اللي هو آخر مثال
+
+605
+00:39:46,130 --> 00:39:49,510
+عندناوبارك الله فيكم وإلى لقاء آخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..fb6ff59e91c82c3d0f3db97231a4c3326264f629
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA_raw.json
@@ -0,0 +1 @@
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"probability": 0.890625}, {"start": 886.13, "end": 886.33, "word": " in", "probability": 0.484375}, {"start": 886.33, "end": 886.63, "word": " polar", "probability": 0.93017578125}, {"start": 886.63, "end": 887.63, "word": " العادية", "probability": 0.8987630208333334}, {"start": 887.63, "end": 887.81, "word": " ال", "probability": 0.88818359375}, {"start": 887.81, "end": 888.13, "word": " area", "probability": 0.8369140625}, {"start": 888.13, "end": 888.51, "word": " بيساوي", "probability": 0.613916015625}, {"start": 888.51, "end": 889.05, "word": " double", "probability": 0.7705078125}, {"start": 889.05, "end": 889.69, "word": " integral", "probability": 0.814453125}, {"start": 889.69, "end": 890.71, "word": " للمنطقه", "probability": 0.7588297526041666}, {"start": 890.71, "end": 890.79, "word": " R", "probability": 0.54296875}, {"start": 890.79, "end": 891.39, "word": " بدي", "probability": 0.3365478515625}, {"start": 891.39, "end": 891.77, "word": " اواجه", "probability": 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"word": " area", "probability": 0.6826171875}, {"start": 918.97, "end": 919.17, "word": " of", "probability": 0.9580078125}, {"start": 919.17, "end": 919.29, "word": " a", "probability": 0.9599609375}, {"start": 919.29, "end": 919.63, "word": " closed", "probability": 0.8662109375}, {"start": 919.63, "end": 920.07, "word": " bounded", "probability": 0.8583984375}, {"start": 920.07, "end": 920.61, "word": " region", "probability": 0.94091796875}, {"start": 920.61, "end": 920.93, "word": " R", "probability": 0.9033203125}, {"start": 920.93, "end": 921.15, "word": " in", "probability": 0.76220703125}, {"start": 921.15, "end": 921.43, "word": " polar", "probability": 0.900390625}, {"start": 921.43, "end": 922.17, "word": " coordinates", "probability": 0.888671875}, {"start": 922.17, "end": 922.53, "word": " الين", "probability": 0.23388671875}, {"start": 922.53, "end": 922.95, "word": " is", "probability": 0.67138671875}, {"start": 922.95, "end": 923.55, "word": " كده", "probability": 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"probability": 0.924072265625}, {"start": 1173.92, "end": 1174.4, "word": " موجودة", "probability": 0.9486083984375}, {"start": 1174.4, "end": 1175.34, "word": " بالكارتيزيان", "probability": 0.7457013811383929}, {"start": 1175.34, "end": 1176.04, "word": " ومن", "probability": 0.723876953125}, {"start": 1176.04, "end": 1176.36, "word": " ثم", "probability": 0.9833984375}, {"start": 1176.36, "end": 1177.28, "word": " بنحولها", "probability": 0.906005859375}, {"start": 1177.28, "end": 1177.74, "word": " الى", "probability": 0.755126953125}, {"start": 1177.74, "end": 1177.96, "word": " G", "probability": 0.69140625}, {"start": 1177.96, "end": 1178.4, "word": " في", "probability": 0.66650390625}, {"start": 1178.4, "end": 1178.9, "word": " اللى", "probability": 0.934326171875}, {"start": 1178.9, "end": 1179.14, "word": " هى", "probability": 0.7081298828125}, {"start": 1179.14, "end": 1179.72, "word": " البولر", "probability": 0.6919759114583334}, {"start": 1179.72, "end": 1180.44, "word": " وهدى", "probability": 0.5716145833333334}, {"start": 1180.44, "end": 1180.74, "word": " يمكن", "probability": 0.6605224609375}, {"start": 1180.74, "end": 1180.98, "word": " اللى", "probability": 0.949462890625}, {"start": 1180.98, "end": 1181.16, "word": " هى", "probability": 0.8662109375}, {"start": 1181.16, "end": 1181.32, "word": " من", "probability": 0.9658203125}, {"start": 1181.32, "end": 1181.66, "word": " اهم", "probability": 0.74951171875}, {"start": 1181.66, "end": 1182.28, "word": " الخطوات", "probability": 0.98046875}, {"start": 1182.28, "end": 1182.44, "word": " او", "probability": 0.8798828125}, {"start": 1182.44, "end": 1182.86, "word": " الخطوة", "probability": 0.9622802734375}, {"start": 1182.86, "end": 1183.04, "word": " اللى", "probability": 0.9384765625}, {"start": 1183.04, "end": 1183.6, "word": " ..", "probability": 0.63916015625}], "temperature": 1.0}, {"id": 55, "seek": 120180, "start": 1183.66, "end": 1201.8, "text": "اللي بدها شوية شغل الان f of x و y على طول بنحولها ل f of r cos θ r sin θ يعني بدلالة θ و بدلالة r حسب القوانين اللي ذكرناها سابقا اللي هي ال x بالساوية r cos θ و ال y بالساوية r sin θ", "tokens": [6027, 20292, 47525, 11296, 13412, 2407, 10632, 13412, 17082, 1211, 2423, 7649, 283, 295, 2031, 4032, 288, 15844, 23032, 12610, 44945, 5016, 12610, 11296, 5296, 283, 295, 367, 3792, 12622, 367, 3343, 12622, 37495, 22653, 47525, 1211, 6027, 3660, 12622, 4032, 47525, 1211, 6027, 3660, 367, 11331, 35457, 25062, 2407, 7649, 9957, 13672, 1829, 29910, 37983, 8315, 11296, 8608, 16758, 4587, 995, 13672, 1829, 39896, 2423, 2031, 20666, 3794, 995, 2407, 10632, 367, 3792, 12622, 4032, 2423, 288, 20666, 3794, 995, 2407, 10632, 367, 3343, 12622], "avg_logprob": -0.27711924876289806, "compression_ratio": 1.7396449704142012, "no_speech_prob": 2.9742717742919922e-05, "words": [{"start": 1183.66, "end": 1184.14, "word": "اللي", "probability": 0.365234375}, {"start": 1184.14, "end": 1184.62, "word": " بدها", "probability": 0.7144775390625}, 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"word": " لقاء", "probability": 0.88427734375}, {"start": 2388.83, "end": 2389.51, "word": " آخر", "probability": 0.9248046875}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2390.9123125, "duration_after_vad": 2385.1124999999997}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a562338ca8ed1bc64c18a63e63cb66ce3f2dfaf8
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/qIi1XD1N3gA_raw.srt
@@ -0,0 +1,2428 @@
+1
+00:00:00,540 --> 00:00:04,980
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,980 --> 00:00:11,540
+وبركاته هذه هي المحاضرة رقم 23 مساقطة فاضل وتكامل
+
+3
+00:00:11,540 --> 00:00:16,220
+جيم وهي المحاضرة الرابعة في ال chapter 15 ال
+
+4
+00:00:16,220 --> 00:00:19,360
+chapter الأخير في المادة وهي تحت عنوان double
+
+5
+00:00:19,360 --> 00:00:24,210
+integrals in polar formتحدثنا سابقا عن double
+
+6
+00:00:24,210 --> 00:00:28,290
+integral in cartesian form الآن بدنا نتحدث عن
+
+7
+00:00:28,290 --> 00:00:31,850
+double integral in polar form طبعا ولو أسبابه
+
+8
+00:00:31,850 --> 00:00:35,710
+أحيانا اللي هو إيجاد اللي هو ال double integration
+
+9
+00:00:35,710 --> 00:00:41,050
+بال integral form أسهل كتير كتير من إيجاده في ال
+
+10
+00:00:41,050 --> 00:00:46,050
+Cartesian و بتحللنا مشاكل اللي هو قد تنشأ في ال
+
+11
+00:00:46,050 --> 00:00:48,110
+Cartesian كما سنظهر لاحقا
+
+12
+00:00:50,710 --> 00:00:55,150
+بتحلها بواسطة اللي هو مين ال bowler form طبعا و
+
+13
+00:00:55,150 --> 00:00:58,230
+العكس صحيح بعض الأسئلة بتكون في ال Cartesian أسهل
+
+14
+00:00:58,230 --> 00:01:01,030
+طبعا الناس اللي هي متعودة أصلا على ال Cartesian
+
+15
+00:01:01,030 --> 00:01:06,850
+لكن ال bowler زي ما قلنا بحللنا مشاكل تشوف الآنروح
+
+16
+00:01:06,850 --> 00:01:12,510
+الموضوع نفسه اللي هو السابق كيف اللي هو المهم عندي
+
+17
+00:01:12,510 --> 00:01:16,610
+تحديد منطقة اللي هي المنطقة اللي بدي أكمل عليها
+
+18
+00:01:16,610 --> 00:01:20,030
+بتحدد لي اللي هي limits of integration إذا
+
+19
+00:01:20,030 --> 00:01:23,470
+integrals and polar coordinates بدي أنحدد أول شيء
+
+20
+00:01:23,470 --> 00:01:26,310
+اللي هو find limits of integration بالظبط زي ما
+
+21
+00:01:26,310 --> 00:01:30,730
+كنا نعمل اللي هو في ال Cartesian ولكن هنا اللي هو
+
+22
+00:01:30,730 --> 00:01:36,390
+طبيعة البولربختلف عن طبيعة الـ Cartesian فبنشوف
+
+23
+00:01:36,390 --> 00:01:39,830
+كيف بنحدد اللي هو ال integration طبعا احنا بدنا
+
+24
+00:01:39,830 --> 00:01:43,470
+نيجي هيعنى عرض علينا ال integration اللي أمامنا ال
+
+25
+00:01:43,470 --> 00:01:47,510
+integration لـF of Rθ دي إيه حيث دي إيه طبعا في
+
+26
+00:01:47,510 --> 00:01:51,570
+هذه الحالة اللي هي DX في DY لكن ال DX في DY زي ما
+
+27
+00:01:51,570 --> 00:01:57,690
+أخدنا في التفاضل ب .. اللي هو عبارة عن RDRDθالان
+
+28
+00:01:57,690 --> 00:02:00,750
+بدل دي ايه سيصبح دي ايه دي ايه دي ايه دي ايه دي
+
+29
+00:02:00,750 --> 00:02:01,290
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+30
+00:02:01,290 --> 00:02:03,050
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+31
+00:02:03,050 --> 00:02:05,390
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+32
+00:02:05,390 --> 00:02:11,530
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+33
+00:02:11,530 --> 00:02:14,510
+ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي ايه دي
+
+34
+00:02:14,510 --> 00:02:24,870
+ايه دي ايه دي ايه
+
+35
+00:02:24,870 --> 00:02:30,260
+دقيمة R وقيمة ثتا ونشوف كيف لاحقا لأن أول حاجة
+
+36
+00:02:30,260 --> 00:02:37,350
+احنا في هذه الحالة بنكامل بالنسبة ل Rالان نظرا
+
+37
+00:02:37,350 --> 00:02:40,250
+لطبيعة ال integration طبعا في عندي RDRD ثالثة و
+
+38
+00:02:40,250 --> 00:02:44,070
+هنلاقي اللي هو التكمل ال R عنده اللي هو بيحللنا
+
+39
+00:02:44,070 --> 00:02:47,950
+كتير مشاكل الان find the R limits of integration
+
+40
+00:02:47,950 --> 00:02:53,090
+الان طبيعة ال R limits اللي هي ال R ال R طبعا زي
+
+41
+00:02:53,090 --> 00:02:55,730
+ما انتوا مذكرين ان اكتفاض الألف بتبدأ من نقطة
+
+42
+00:02:55,730 --> 00:02:59,950
+الأصل و بتطلع إيش إلى أعلى اللي هي بوسطة ال Ray
+
+43
+00:03:00,540 --> 00:03:03,900
+الآن الـ Rray هذا بدنا نشوف كيف بدخل و من وين بدخل
+
+44
+00:03:03,900 --> 00:03:07,340
+المنطقة و من وين بيخرج منها من وين بدخل المنطقة
+
+45
+00:03:07,340 --> 00:03:12,660
+بيكون اللي هو الحد الأسفل و من وين بيخرج منها الحد
+
+46
+00:03:12,660 --> 00:03:16,340
+الأعلى زي في حالة الـ Cartesian نيجي نحد الـ Theta
+
+47
+00:03:16,340 --> 00:03:21,140
+Theta الآن بنشوف كيف تمسح المنطقة طبعا بس بنبدأ من
+
+48
+00:03:21,140 --> 00:03:24,220
+وين عند الـ Positive X Axis و نطلع عكس عقارب
+
+49
+00:03:24,220 --> 00:03:28,440
+الساعة من حد الـ Thetaفبنشوف من أين بدأت المنطقة
+
+50
+00:03:28,440 --> 00:03:32,260
+ومن أين انتهت بتكون هي حدود ثتا خلّينا نشوف هذا
+
+51
+00:03:32,260 --> 00:03:39,080
+الكلام عملياً لو عرض علينا هيعندي F of R ثتا DA
+
+52
+00:03:39,080 --> 00:03:43,220
+over a region R ال region R اللي عرضها عليها هي ال
+
+53
+00:03:43,220 --> 00:03:46,370
+region التاليةعامل ايه هو ليش بالـ Cartesian قال
+
+54
+00:03:46,370 --> 00:03:49,130
+اللي هو بدي X تربيع زي أضواء تربيع بساوة أربعة هذي
+
+55
+00:03:49,130 --> 00:03:52,330
+جزء من مين من دائرة قال اللي بدي هذا الجزء اللي في
+
+56
+00:03:52,330 --> 00:03:57,090
+الربع الأول R بوصف R قاعد الجزء اللي هي R اللي في
+
+57
+00:03:57,090 --> 00:03:59,710
+الربع الأول المكون من أعلى ب X تربيع زي أضواء
+
+58
+00:03:59,710 --> 00:04:03,450
+تربيع اللي هو أربعة اللي عبارة عن الدائرة ومن أسفل
+
+59
+00:04:03,450 --> 00:04:08,850
+بواسطة Y بساوية جذر التنين وعن اليسار بواسطة اللي
+
+60
+00:04:08,850 --> 00:04:15,350
+هو ال X بتساوية سفر أو ال Y Axisهذه هي المنطقة
+
+61
+00:04:15,350 --> 00:04:19,630
+المحدودة الان اللى عليها بدنا نكامل الان كلها
+
+62
+00:04:19,630 --> 00:04:24,130
+مكتوبة بإيش بالكارتيزين بدي أحولها لإيش للبولر
+
+63
+00:04:24,130 --> 00:04:30,130
+الان عندي النقطة هذه نقطة التقاطع بين هذا المستقيم
+
+64
+00:04:30,130 --> 00:04:33,940
+وهذا الدائرةإيجاد النقطة هذه سهلة حط Y بساوية جدر
+
+65
+00:04:33,940 --> 00:04:37,800
+التنين هنا اللي هي عبارة عن ياش اتنين ال Y تربيع
+
+66
+00:04:37,800 --> 00:04:40,520
+جيجي اللي هنا بيصير اللي هي اربعة نقص اتنين اتنين
+
+67
+00:04:40,520 --> 00:04:43,520
+يعني بيصير X تربيع بساوية اتنين يعني X برضه بساوية
+
+68
+00:04:43,520 --> 00:04:46,620
+جدر التنين لإنه لهذه المنطقة موجة بقى فبيصير عند
+
+69
+00:04:46,620 --> 00:04:50,260
+النقطة هذه جدر التنين وجدر التنينالمنطقة هذه طبعا
+
+70
+00:04:50,260 --> 00:04:53,860
+y بيساوي جدر التنين معناته ده نقطة جدر التنين وهذا
+
+71
+00:04:53,860 --> 00:04:56,400
+اللي هو x تربيع زاوية وy تربيع بيساوي أربع عبارة
+
+72
+00:04:56,400 --> 00:05:01,360
+عن دائرة نص قطرها بيساوي اتنين هي الاش نصف القطر
+
+73
+00:05:01,360 --> 00:05:04,520
+إذا هذه إحداثيات اتنين إذا عرفنا كل الإحداثيات
+
+74
+00:05:04,520 --> 00:05:09,640
+لهذه الآن بدنا نحول هذه لبولر عشان اللي هو نشتغل
+
+75
+00:05:09,640 --> 00:05:14,100
+على البولر يقتلبه منها طيبالان احنا بنعرف علاقات
+
+76
+00:05:14,100 --> 00:05:20,940
+قديما بين x و y و r و θ ال x هذكركم فيها بيساوي r
+
+77
+00:05:20,940 --> 00:05:27,860
+cos θ ال y ايش بتساوي r sin θ ماشي الحال اذا x
+
+78
+00:05:27,860 --> 00:05:30,860
+تربيع زايد y تربيع بنقدر نحولها بيصير x تربيع
+
+79
+00:05:30,860 --> 00:05:36,660
+عبارة عن r تربيع cos تربيع θزائد ال Y R Sin θ
+
+80
+00:05:36,660 --> 00:05:41,420
+بيصير زائد R تربيع Sin تربيع θ بيساوي إيش؟ أربعة
+
+81
+00:05:41,420 --> 00:05:45,120
+خد من هدولة عامل مشترك R تربيع بيصير R تربيع
+
+82
+00:05:45,120 --> 00:05:48,800
+مضروبة في Cos تربيع زائد Sin تربيع بيساوي أربعة زي
+
+83
+00:05:48,800 --> 00:05:51,520
+ما احنا عارفين الـ Cos تربيع زايد ال Sin تربيع إيش
+
+84
+00:05:51,520 --> 00:05:56,340
+بتساوي واحد يعني R تربيع إيش بتساوي أربعة يعني R
+
+85
+00:05:56,340 --> 00:05:59,540
+بتساوي اتنين وR بتساوي اتنين احنا مشهور عندنا في
+
+86
+00:05:59,540 --> 00:06:03,230
+البولر هاي R بتساوي اتنين هاي الدائرة هذهمشهور في
+
+87
+00:06:03,230 --> 00:06:07,350
+البولار ان هي معادلة الدائرة اللي نصف قطرها ايش
+
+88
+00:06:07,350 --> 00:06:13,410
+اللي هي عبارة عن اتنين طيب الان نيجي للخط هذا هذا
+
+89
+00:06:13,410 --> 00:06:16,410
+الخط عبارة عن Y بيساوي جدر التنين شيل ال Y وحط
+
+90
+00:06:16,410 --> 00:06:22,140
+قيمته ايش قيمته R sin θ اللي ناسيإذاً R sin θ
+
+91
+00:06:22,140 --> 00:06:26,900
+بتساوي Y بتساوي جدر التنين نحوّلها لبولر ال بولر
+
+92
+00:06:26,900 --> 00:06:30,780
+بيصير R بيساوي جدر التنين ع الصين يعني R بيساوي
+
+93
+00:06:30,780 --> 00:06:35,140
+جدر التنين كثيكان تلث ثتا مقلوب ال sign إذاً صار
+
+94
+00:06:35,140 --> 00:06:39,480
+الخط هذا يا جماعة الثاني عبارة عن R بيساوي جدر
+
+95
+00:06:39,480 --> 00:06:43,310
+التنين كثيكان ثتاالان الأمور سهلة وشوفوا اكيد سهلة
+
+96
+00:06:43,310 --> 00:06:47,250
+هذه حولناها كلها ل bowler ماشي الحال تنشوف الان
+
+97
+00:06:47,250 --> 00:06:51,350
+الان نيجي اللي هو بدنا نيجي اللي هو evaluation
+
+98
+00:06:51,350 --> 00:06:55,410
+كامل بالنسبة لمين R ال R طبيعته كيف بتحسب اللي هو
+
+99
+00:06:55,410 --> 00:06:58,870
+ال Ray ال Ray ال Ray ال Ray بتطلع من مين؟ من السفر
+
+100
+00:06:59,190 --> 00:07:02,350
+الان هذا الـ array بدنا نشوف وين بدخل المنطقة و
+
+101
+00:07:02,350 --> 00:07:06,230
+بين بخرج منها و هل كل ال arrays اللي بتدخل و بتخرج
+
+102
+00:07:06,230 --> 00:07:09,750
+من المنطقة على نفس المنوال زي زمان يعني بنفعش
+
+103
+00:07:09,750 --> 00:07:13,930
+نجسمها إذا بتمشيش على نفس المنوال لازم نجسمها إذا
+
+104
+00:07:13,930 --> 00:07:17,970
+بتمشي على نفس المنوال بلزمش تجسمها الان هبنيجي
+
+105
+00:07:17,970 --> 00:07:23,050
+بناخد من هنا arrayراحظ الـ R دخل من عند الخط هذا و
+
+106
+00:07:23,050 --> 00:07:27,050
+طلع من عند خط الدائرة خد الـ R تاني خد الـ R تالت
+
+107
+00:07:27,050 --> 00:07:30,610
+كل هن طبعا نقطة الرأس اللي بتطلع كل هن على نفس
+
+108
+00:07:30,610 --> 00:07:34,590
+المنوال إذا مرتاحين إذا خلينا نشوف R R وين دخلت؟
+
+109
+00:07:34,590 --> 00:07:39,570
+دخلت من هنا إيش قيمة هنا R على هذا قيمته اللي هو
+
+110
+00:07:39,570 --> 00:07:43,350
+زي ما اتفقنا جدر التنين C Cosθ هذا الحد الأولاني
+
+111
+00:07:43,350 --> 00:07:48,510
+إيش بيساوي جدر التنين Cosθ طب من وين خرج؟خرج من
+
+112
+00:07:48,510 --> 00:07:51,650
+الدائرة R بساوة و اتنين اذا ايش الاحداث التاني للـ
+
+113
+00:07:51,650 --> 00:07:55,830
+R اللي هو R بساوة و اتنين خلصنا الـ A من R بدنا
+
+114
+00:07:55,830 --> 00:08:00,390
+نيجي نحدد مين ثيتا الآن ثيتا الثيتا بتبدأ من عند
+
+115
+00:08:00,390 --> 00:08:04,630
+القياساتها بتبدأ من الصغير اللي هو اللي هي مع ال
+
+116
+00:08:04,630 --> 00:08:09,820
+positive X axis و بعدين بتبدأ تلفوانتوا عارفين كيف
+
+117
+00:08:09,820 --> 00:08:14,400
+قيام الـ θيتا الان بتلف هيك بالاتجاه الموجة عكس
+
+118
+00:08:14,400 --> 00:08:18,160
+عقارب الساعة وهيك بالاتجاه السالب طيب خلينا نحدد
+
+119
+00:08:18,160 --> 00:08:21,480
+منطقتنا .. منطقتنا شايف ان انت بتبدأ هي اول array
+
+120
+00:08:21,480 --> 00:08:25,320
+هي اول array هي تاني array هي تالت array و بيظلوا
+
+121
+00:08:25,320 --> 00:08:29,660
+يمشي يمشي تخيل نور بيمشي هيك بال array عشان يغطي
+
+122
+00:08:29,660 --> 00:08:32,560
+المنطقة ده هيمشي من هان لهان اذا هيبدأ من هان ال
+
+123
+00:08:32,560 --> 00:08:36,910
+arrayالـ θ تبعته يجب أن نجيسه هذه الـ θ تبعته يجب
+
+124
+00:08:36,910 --> 00:08:38,090
+أن نجيسه هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ
+
+125
+00:08:38,090 --> 00:08:40,930
+تبعته يجب أن نجيسه هذه الـ θ تبعته يجب أن نجيسه
+
+126
+00:08:40,930 --> 00:08:41,770
+هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ تبعته يجب
+
+127
+00:08:41,770 --> 00:08:43,390
+أن نجيسه هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ
+
+128
+00:08:43,390 --> 00:08:45,510
+تبعته يجب أن نجيسه هذه الـ θ تبعته يجب أن نجيسه
+
+129
+00:08:45,510 --> 00:08:51,470
+هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ تبعته يجب
+
+130
+00:08:51,470 --> 00:08:56,150
+أن نجيسه هذه الـ θ تبعته يجب أن نجيسه هذه الـ θ
+
+131
+00:08:56,180 --> 00:09:01,120
+مدام عنده Y بتساوي X معناته اللي هو هذا تبع الـ Y
+
+132
+00:09:01,120 --> 00:09:04,440
+بساوية هذا تبع الـ X يعني المثلث القائم هذا عبارة
+
+133
+00:09:04,440 --> 00:09:07,680
+عن مثلث متساوى الساقين يعني هذه الزاوية بالساوية
+
+134
+00:09:07,680 --> 00:09:11,180
+هذه يعني هذه 45 درجة و هذه 45 درجة إذا هذه الزاوية
+
+135
+00:09:11,180 --> 00:09:16,180
+عبارة عن 45 درجة للبداية اللي هي باي على أربعة إذا
+
+136
+00:09:16,180 --> 00:09:19,460
+قيتها بتبدأ من باي على أربعة وشوف لوين تبقى تمشي
+
+137
+00:09:19,460 --> 00:09:23,450
+تمشي تمشي لما تصل هذاهذا طبعاً كلكم قال إيش عبارة
+
+138
+00:09:23,450 --> 00:09:27,230
+عن باي على اتنين إذا تتا من باي على أربعة لعند مين
+
+139
+00:09:27,230 --> 00:09:31,690
+يا جماعة لعند باي على اتين هذه مثال ناخد مثال آخر
+
+140
+00:09:31,690 --> 00:09:34,670
+إذا أنا بقولي find the limits of integration for
+
+141
+00:09:34,670 --> 00:09:38,550
+integrating F of R ثتا بده بس limits of إيش of
+
+142
+00:09:38,550 --> 00:09:43,350
+integration لما بدنا نكامل over F of R ثتا over
+
+143
+00:09:43,350 --> 00:09:47,430
+the region Rأيش ال region R هيو بوصف لك إياها
+
+144
+00:09:47,430 --> 00:09:52,870
+تماماً That lies inside the chordoid R بتساوحة ذات
+
+145
+00:09:52,870 --> 00:09:57,770
+كسان ثتا and outside R بتساوحة أنا عارف إنكم أنتوا
+
+146
+00:09:57,770 --> 00:10:01,460
+نسيان اللي هي الرسمات اللي هي تبعات تفاضل باهأنا
+
+147
+00:10:01,460 --> 00:10:06,340
+مش جاي اللي هو أني أخبركم في الرسمات هذه الرسمات
+
+148
+00:10:06,340 --> 00:10:10,380
+هذه اللي أنتوا بتنسوها احنا لأ بنعملكم إياها لكن
+
+149
+00:10:10,380 --> 00:10:14,000
+في بعض الرسمات مجدور عليها اللي واحد ممكن يستذكرها
+
+150
+00:10:14,000 --> 00:10:19,900
+من خلال اللي هو يعني دراسته البسيطة في داخل اتفاضه
+
+151
+00:10:19,900 --> 00:10:24,520
+ال gymوماعرفته بتفاضل ده لكن بقدر الإمكان اللي هو
+
+152
+00:10:24,520 --> 00:10:28,540
+تجلبوش من موضوع رسم عشان نجيبكم رسم معقد بحيث انه
+
+153
+00:10:28,540 --> 00:10:31,220
+انتوا نسينوا انا مش جاي اختباركم في الرسم اداء انا
+
+154
+00:10:31,220 --> 00:10:34,980
+جاي اختباركم في ال double integration over اللي هو
+
+155
+00:10:34,980 --> 00:10:39,880
+اللي هي a region in polar formسلّوا على النبي عليه
+
+156
+00:10:39,880 --> 00:10:42,880
+الصلاة والسلام الان الـ Cardoid R بتساوي 1 زي
+
+157
+00:10:42,880 --> 00:10:47,100
+كتصين ثتا الان الـ Cardoid هيدو R بتساوي 1 زي
+
+158
+00:10:47,100 --> 00:10:51,300
+كتصين ثتا هيه لو واحد اللي جرب يرسمه يعرف يرسمه
+
+159
+00:10:51,300 --> 00:10:55,360
+بتبدأ بثتا بصفر اللي هتلاقيها ده اللي هي عبارة عن
+
+160
+00:10:55,360 --> 00:10:59,940
+اللي هي ثتا بصفر مثلا بتصير كتصين ثتا واحد بتصير
+
+161
+00:10:59,940 --> 00:11:03,870
+هذه اللي هي اتنين بتبدأ من هناثم تبدأ تطلع لما
+
+162
+00:11:03,870 --> 00:11:07,710
+انتصر باي على اتنين لانتصار باي على اتنين لانتصار
+
+163
+00:11:07,710 --> 00:11:11,990
+باي على اتنين لانتصار باي على اتنين لانتصار
+
+164
+00:11:11,990 --> 00:11:17,250
+باي على اتنين لانتصار
+
+165
+00:11:17,250 --> 00:11:21,650
+باي على اتنينهذه مرة أخرى لما اكتملت الدورة مش هى
+
+166
+00:11:21,650 --> 00:11:25,450
+موضوع نهائى بس هذا للتذكير بس الان الان هو بوصف
+
+167
+00:11:25,450 --> 00:11:29,150
+المنطقة هي المنطقة اللى بين ال cardioid هيه
+
+168
+00:11:29,150 --> 00:11:32,530
+والدائرة R بتساوي واحد ده نعرف ان الان R بتساوي
+
+169
+00:11:32,530 --> 00:11:35,310
+واحد، R بتساوي خمسة، R بتساوي أربعة، كلهين دوائر
+
+170
+00:11:35,310 --> 00:11:38,950
+لأن R بتساوي واحد عبارة عن دائرة نصف قطرها إيش
+
+171
+00:11:38,950 --> 00:11:43,990
+واحدلأن هذه المنطقة هي المنطقة المقصودة في إن نجد
+
+172
+00:11:43,990 --> 00:11:47,730
+حدود الـ Integration لها صلوا على النبي عليه
+
+173
+00:11:47,730 --> 00:11:51,190
+الصلاة والسلام زي ما بقولنا أول حاجة بنرسم المنطقة
+
+174
+00:11:51,190 --> 00:11:55,170
+تاني حاجة بنرسم اللي هي مناطق اللي هي التقاط اللي
+
+175
+00:11:55,170 --> 00:11:59,770
+بين الجهتين وبنحدد اللي هو حدود R وبعدين بنحدد مين
+
+176
+00:11:59,770 --> 00:12:04,180
+θالان عشان احدد R ثتا بلزمنا نقطة التقاطع حدى نقطة
+
+177
+00:12:04,180 --> 00:12:06,720
+التقاطع حدى زى ما كنا نعمل في الـ Cartesian عادي
+
+178
+00:12:06,720 --> 00:12:11,040
+الان نقطة التقاطع بين مين يعنى هذه؟ هذه بين اللي
+
+179
+00:12:11,040 --> 00:12:14,960
+هو R بتساوي واحد ونفس باشي هذه و بين R بتساوي واحد
+
+180
+00:12:14,960 --> 00:12:18,520
+زي كوصين ثتا ساوين مع بعض واحد زي كوصين ثتا ايش
+
+181
+00:12:18,520 --> 00:12:22,520
+بيساوي واحد اذا كوصين ثتا ايش بيساوي سفر اذا ثتا
+
+182
+00:12:22,520 --> 00:12:28,490
+يا اماالتي هي عبارة عن باي على اتنين أو ناقص باي
+
+183
+00:12:28,490 --> 00:12:31,190
+على اتنين التي هي تلاتة باي على اتنين الان التي هي
+
+184
+00:12:31,190 --> 00:12:33,710
+باي على اتنين من ناقص باي على اتنين إذا فتها هن
+
+185
+00:12:33,710 --> 00:12:37,050
+ناقص باي على اتنين وفتها هن اشملها باي على اتنين
+
+186
+00:12:37,050 --> 00:12:39,210
+لم أقل لكم لماذا كتبنا ناقص باي على اتنين وليس
+
+187
+00:12:39,210 --> 00:12:43,650
+تلاتة باي على اتنين طبعا التمثيل نفس الشيءأو تعبير
+
+188
+00:12:43,650 --> 00:12:48,190
+آخر نحن على أساس أننا نبدأ من المنطقة الصغيرة
+
+189
+00:12:48,190 --> 00:12:51,970
+ونذهب للكبيرة في اتجاه عكس عقاربس عشان ذلك بدأت
+
+190
+00:12:51,970 --> 00:12:55,130
+بالنقص باي على اتنين على أساس أن أمسح المنطقة من
+
+191
+00:12:55,130 --> 00:12:59,750
+المنطقة الصغيرة لثتا إلى المنطقة الكبيرةمش عادي
+
+192
+00:12:59,750 --> 00:13:02,230
+اللي هو أبدا من عنده اللي هو بقى ال .. اللي هي
+
+193
+00:13:02,230 --> 00:13:05,850
+theta اللي هي تلاتة باي على اتنين وعمل حدود
+
+194
+00:13:05,850 --> 00:13:08,530
+integration باي على اتنين إلى باي على اتنين بيصير
+
+195
+00:13:08,530 --> 00:13:12,170
+اللي هي عدديا هذه أكبر من هذه فبس يعني اللي هو
+
+196
+00:13:12,170 --> 00:13:19,110
+بشكل خلنا نقول هي كالتي هي ألطف وأنسب إذا خلّينا
+
+197
+00:13:19,110 --> 00:13:21,750
+ان شاء الله تكونوا فاهمين عليش بجود الآن بدنا نيجي
+
+198
+00:13:21,750 --> 00:13:24,030
+نحدد حدود ال integration حدود ال integration زي ما
+
+199
+00:13:24,030 --> 00:13:27,750
+اتفجنا بنرفع ال rays هاي ال ray هاي عشان نحدد R
+
+200
+00:13:27,750 --> 00:13:32,330
+هاي F of R ثتا RDR D ثتا بنفاضل ب .. بنكمل هنا على
+
+201
+00:13:32,330 --> 00:13:36,530
+طول ب R قبل ماحدش يقولي D ثتا بقبل DR لأ R و بعدين
+
+202
+00:13:36,530 --> 00:13:42,310
+ان هي D ثتااللي هي عندي R نشوف R الان R طلع Arrays
+
+203
+00:13:42,310 --> 00:13:46,530
+من نقطة الأصل كل ال Arrays من نقطة الأصل تدخل من
+
+204
+00:13:46,530 --> 00:13:49,550
+عند الدائرة و تخرج من عند ال Cardoid و نفس الأشي
+
+205
+00:13:49,550 --> 00:13:53,770
+اللي تحت تدخل دائرة و تطلع Cardoid و هكذا إذن
+
+206
+00:13:53,770 --> 00:13:58,350
+المنطقة كلها على نفس المنوال إذن ارنا تبدأ R
+
+207
+00:13:58,350 --> 00:14:02,210
+بتساوي واحد و تطلع R بتساوي واحد زائد cosine
+
+208
+00:14:02,210 --> 00:14:06,870
+فترانة تخرج من عندهاالان نأتي لـ θ , ثتة عشان تمسح
+
+209
+00:14:06,870 --> 00:14:11,570
+كل المنطقة هاي ثتة لأ هنا بتبدأ بعدين هنا بعدين
+
+210
+00:14:11,570 --> 00:14:15,110
+هنا قيمة ثتة طبعا هذه الثتة اللي بالشكل هذا اللي
+
+211
+00:14:15,110 --> 00:14:19,190
+هي طبعا مع ال positive x axis و هنا و هيك و هيك و
+
+212
+00:14:19,190 --> 00:14:23,990
+هيك طبعا ثتة هنا عبارة عن ناقص باي على اتنين عشان
+
+213
+00:14:23,990 --> 00:14:27,210
+تمسح الباقي ده اللي بيظل ماشي هيك هيك هيك هيك لما
+
+214
+00:14:27,210 --> 00:14:30,270
+تصل لمين لعنده النقطة هذه اللي حددناها في الأول
+
+215
+00:14:30,270 --> 00:14:34,110
+اللي هي باي على اتنينالان الموجود هذه بكون عندى
+
+216
+00:14:34,110 --> 00:14:38,210
+فعلا انا حصلت على اللى هو limits of integration
+
+217
+00:14:38,210 --> 00:14:42,930
+للزواحل الان area in polar form او in polar
+
+218
+00:14:42,930 --> 00:14:45,890
+coordinates طبعا يا جماعة انتوا عارفين اللى هى ال
+
+219
+00:14:45,890 --> 00:14:49,050
+area in polar العادية ال area بيساوي double
+
+220
+00:14:49,050 --> 00:14:52,910
+integral للمنطقه R بدي اواجه ال area للمنطقه R
+
+221
+00:14:52,910 --> 00:14:55,490
+اخدناها المرة الماضية اللى هو ال integration لل
+
+222
+00:14:55,490 --> 00:15:03,290
+RDAأو بمعنى آخر DXDY لكن DXDY زي ما احنا عارفين
+
+223
+00:15:03,290 --> 00:15:07,930
+ايش بتساوي في الـ polar اللي هو RDRDθ هذا خدناه
+
+224
+00:15:07,930 --> 00:15:12,030
+الكلام بالتفصيل في تفاضل بها إذن بس أذكركم فيه
+
+225
+00:15:12,030 --> 00:15:16,890
+اللي هي DA اللي هي DXDY بتساوي RDRDθ إذن ال area
+
+226
+00:15:16,890 --> 00:15:20,610
+بتساوي إلى الان ال area of a closed bounded region
+
+227
+00:15:20,610 --> 00:15:24,890
+R in polar coordinates الين is كده وهي اللي امننا
+
+228
+00:15:25,150 --> 00:15:28,430
+خلّيني أخد مثال ‏find the area enclosed by the
+
+229
+00:15:28,430 --> 00:15:33,670
+lemniscate RWI 4cos2θ ‏الأن هذه الرسمة ‏الأن طبعا
+
+230
+00:15:33,670 --> 00:15:39,230
+أكيد كتير منكم ناسي تفاصيلها ‏الأن إذا بدك تفصل في
+
+231
+00:15:39,230 --> 00:15:43,110
+الرسم ‏بتقدر تفصل تاخد اللي هو قيم لθ ‏بسفر بكده
+
+232
+00:15:43,110 --> 00:15:47,490
+بكده ‏بتبدأ تشتغل ‏لما انتصل لالرسمة اللي أمامك
+
+233
+00:15:47,530 --> 00:15:51,050
+طبعا كنا نشوف فيه سيمترا حول الـ x-axis حول الـ
+
+234
+00:15:51,050 --> 00:15:53,730
+origin حول هذا الكلام مش موضوعنا هذا الان كلها
+
+235
+00:15:53,730 --> 00:15:57,150
+بتساعدنا في الرسم لكن هي الرسمة الآن الرسمة موجودة
+
+236
+00:15:57,150 --> 00:16:00,310
+أمامي الان بدأ اوجد ال area للمنطقة هذه اللي هي
+
+237
+00:16:00,310 --> 00:16:03,110
+الرسمة اللي أمامي طب أنا عشان اوجد ال area طب انا
+
+238
+00:16:03,110 --> 00:16:06,700
+شايف هذه المنطقة هي زي هذه بالظبططب ما بحسب هذه و
+
+239
+00:16:06,700 --> 00:16:09,120
+بضربها في اتنين لا حد بيقول لي لأ يا أستاذ ما هي
+
+240
+00:16:09,120 --> 00:16:13,320
+أصلا هذه اللي هي المنطقة هي زي هذه و زي هذه و زي
+
+241
+00:16:13,320 --> 00:16:16,860
+هذه بقول له صح إذا خلّيني أنشع كلامك إذا هذه بدي
+
+242
+00:16:16,860 --> 00:16:22,720
+أخدها الآن هي المنطقة و أضربها في أربعة يعني بيسير
+
+243
+00:16:22,720 --> 00:16:26,500
+ال area عندي ال area بالساوية أربعة في قيمة ال
+
+244
+00:16:26,500 --> 00:16:31,490
+integration لل RDRDθ على هذه المنطقةطب هذه المنطقة
+
+245
+00:16:31,490 --> 00:16:34,750
+اللي هي داخل طبعا كله بيدوه اللي هو area enclosed
+
+246
+00:16:34,750 --> 00:16:37,870
+by the landscape يعني ال area اللي جوا هذه كلها
+
+247
+00:16:37,870 --> 00:16:42,390
+الان هذه الان بده حدد حدود ال integration اللي لها
+
+248
+00:16:42,390 --> 00:16:46,670
+حدود ال integration سهل ال R هيه و بيطلع من الأصل
+
+249
+00:16:46,670 --> 00:16:49,810
+و بيجي على مين اللي هو بدخل المنطقة في الأصل و
+
+250
+00:16:49,810 --> 00:16:53,390
+بيطلع بضرب في مين في الرسمة اللي هي الرسمة ايش
+
+251
+00:16:53,390 --> 00:16:57,320
+اللي هي بتساويالقيمة R هي جدر التربيع الاربعة
+
+252
+00:16:57,320 --> 00:17:00,920
+كوساين اتنين ثيتا إذا ببدأ من صفر و بروح لجدر
+
+253
+00:17:00,920 --> 00:17:05,080
+الاربعة كوساين اتنين ثيتا هذه اللي هي حدود المنطقة
+
+254
+00:17:05,080 --> 00:17:10,360
+طيب الآن ثيتا تشوف الثيتا الان هذه واضحة أن ثيتا
+
+255
+00:17:10,360 --> 00:17:15,520
+من هنا عندها بصفر ليش لأن لما كانت ثيتا بصفر بيصير
+
+256
+00:17:15,520 --> 00:17:18,580
+اللي عبارة عن القيمة جديش اربعة كوساين الصفر أربعة
+
+257
+00:17:18,580 --> 00:17:21,800
+يعني هذه المنطقة أربعة فعلا إذا هذه بدأنا احنا
+
+258
+00:17:21,800 --> 00:17:26,290
+معين ثيتا بصفرطب الآن لما نوصلنا اللي هي رجعنا على
+
+259
+00:17:26,290 --> 00:17:30,670
+مين على نقطة الأصل رجع يعني و كأنها رجعت R بصفر
+
+260
+00:17:30,670 --> 00:17:34,910
+لنشوف هذه قيمة θ عندها دي لما R بصفر بيصير أربعة
+
+261
+00:17:34,910 --> 00:17:38,010
+كوصين اتنين ثتا بالساوية صفر يعني كوصين اتنين ثتا
+
+262
+00:17:38,010 --> 00:17:40,870
+بالساوية صفر فاهمين اللي عملته الان كوصين اتنين
+
+263
+00:17:40,870 --> 00:17:43,810
+ثتا بالساوية صفر إذا ثتا اتنين ثتا بالساوية πاي
+
+264
+00:17:43,810 --> 00:17:49,290
+على اتنين يعني ثتا بالساوية باي على أربعة إذا هذه
+
+265
+00:17:49,290 --> 00:17:51,710
+الناس ده خلتيها طبعا هذه باي على أربعة مش هتكون
+
+266
+00:17:51,710 --> 00:17:55,240
+اللي هي أياشاللي هي المنطقة هالي اللي هي تلاتة باي
+
+267
+00:17:55,240 --> 00:17:59,000
+على أربعة مثلا لأ هذه باي على أربعة اللي هي بتحقق
+
+268
+00:17:59,000 --> 00:18:01,900
+هذه أكتر من القيمة من ضمن ال باي على أربعة و هو
+
+269
+00:18:01,900 --> 00:18:06,200
+اخر فهذه باي على أربعة ماشي الحال إذا هذه صارت باي
+
+270
+00:18:06,200 --> 00:18:11,360
+على أربعة إذا صار عندي ال integration من سفرلأ باي
+
+271
+00:18:11,360 --> 00:18:14,280
+على أربع لأنه بدأ بالصفر طبعا من باي على أربع هي
+
+272
+00:18:14,280 --> 00:18:17,960
+الصفر اللي عند باي على أربع ر دي ر دي ثتا والان
+
+273
+00:18:17,960 --> 00:18:20,760
+ايش بده سوية جامعة بده كامل اكمل بالنسبة لغاية
+
+274
+00:18:20,760 --> 00:18:23,540
+بيصير ر تربيع على اتنين الان حدود ال integration
+
+275
+00:18:23,540 --> 00:18:26,800
+من صفر للجدر التربيعي اربعة كوساين اتنين ثتابعوض
+
+276
+00:18:26,800 --> 00:18:31,260
+عنها ر بصفر هي صفر و بيبقى الهدى اللى بيعوض عنها
+
+277
+00:18:31,260 --> 00:18:35,880
+اللى فوق بيصير جدر الأربعة بيصير اللى هو لما أربعه
+
+278
+00:18:35,880 --> 00:18:40,600
+بيصير اللى هو عبارة عن اتنين cosine اتنين ثتا طبعا
+
+279
+00:18:40,600 --> 00:18:42,860
+بيصير اربعة و بقسمها اتنين بيصير اتنين cosine
+
+280
+00:18:42,860 --> 00:18:45,860
+اتنين ثتا فاهمين ر تربيع يساوي اربع cosine اتنين
+
+281
+00:18:45,860 --> 00:18:49,020
+ثتا على اتنين بترجع اتنين cosine اتنين ثتا D ثتا
+
+282
+00:18:49,200 --> 00:18:53,100
+الان بكامل من صفر ل باقي على اتنين دي ثتا هذا اللي
+
+283
+00:18:53,100 --> 00:19:02,600
+هو عبارة عن اللي هو sign اتنين ثتا مظبوط على اتنين
+
+284
+00:19:02,600 --> 00:19:05,480
+اللي هي الثتا اللي هنا بيصير عنده اللي هو sign
+
+285
+00:19:05,480 --> 00:19:07,700
+اتنين ثتا و فيه اربعة برا بيصير اربعة sign اتنين
+
+286
+00:19:07,700 --> 00:19:10,780
+ثتا و من صفر ل عند باقي على اربعة بيطلع عند مين يا
+
+287
+00:19:10,780 --> 00:19:15,290
+جماعةبطلع عندى اربعة و بكون هيك خلصنا اوجدنا ال
+
+288
+00:19:15,290 --> 00:19:18,610
+area و نيجي الان نتغير cartesian integral into
+
+289
+00:19:18,610 --> 00:19:26,010
+polar integrals نشوف كيف بتحول ال cartesian إلى
+
+290
+00:19:26,010 --> 00:19:31,800
+polarأول اشي عندى اللى هي تلت امور لازم نغير فيها
+
+291
+00:19:31,800 --> 00:19:36,040
+اول اشي ال region R بتكون موجودة بالكارتيزيان ومن
+
+292
+00:19:36,040 --> 00:19:40,980
+ثم بنحولها الى G في اللى هى البولر وهدى يمكن اللى
+
+293
+00:19:40,980 --> 00:19:45,000
+هى من اهم الخطوات او الخطوة اللى ..اللي بدها شوية
+
+294
+00:19:45,000 --> 00:19:49,680
+شغل الان f of x و y على طول بنحولها ل f of r cos θ
+
+295
+00:19:49,680 --> 00:19:56,180
+r sin θ يعني بدلالة θ و بدلالة r حسب القوانين اللي
+
+296
+00:19:56,180 --> 00:20:00,180
+ذكرناها سابقا اللي هي ال x بالساوية r cos θ و ال y
+
+297
+00:20:00,180 --> 00:20:09,510
+بالساوية r sin θأو تأتي الشيء الثاني DXDY بحولها
+
+298
+00:20:09,510 --> 00:20:19,710
+لـ RDRDθ يعني هذا التحويل وهو بأفق القوانينالان
+
+299
+00:20:19,710 --> 00:20:23,150
+اللي بده جهده اللي هو تحديد حدود التكامل اللي هو
+
+300
+00:20:23,150 --> 00:20:27,770
+من Cartesian R إلى اللي هي بولر اللي هي
+
+301
+00:20:27,770 --> 00:20:31,890
+coordinates نيجي الان ناخد مثال evaluate the
+
+302
+00:20:31,890 --> 00:20:35,610
+double integral over region R ايش ال region R؟ ال
+
+303
+00:20:35,610 --> 00:20:39,190
+region R هينمعطينيها هنا لجيت بنتقول ايش هي؟ يجي
+
+304
+00:20:39,190 --> 00:20:45,840
+ده X تربيع زي Y تربيع دي Y دي Xالان where R is the
+
+305
+00:20:45,840 --> 00:20:49,300
+semicircular region bounded by the x-axis اللي هي
+
+306
+00:20:49,300 --> 00:20:52,120
+semicircular region نص الدائرة هذه اللي bounded by
+
+307
+00:20:52,120 --> 00:20:56,360
+the x-axis and the curve Y بيساوي جدر الواحد minus
+
+308
+00:20:56,360 --> 00:21:01,560
+X اللي هي عبارة عن الجزء الموجب من الدائرة Y تربيع
+
+309
+00:21:01,560 --> 00:21:05,650
+زاد X تربيع بتساوي واحدماشي الحال اذا هي عند
+
+310
+00:21:05,650 --> 00:21:09,910
+المنطقة R وهي عند الدالة اللي بنكملها لو اجينا
+
+311
+00:21:09,910 --> 00:21:14,350
+واحد اجه بده كامل هادى بواسط القارتيزيان كمن لاقيت
+
+312
+00:21:14,350 --> 00:21:17,590
+اكس تربيع ز و اي تربيع بالنسبة لوايش هيعرفى كامل
+
+313
+00:21:17,590 --> 00:21:22,750
+by as an elementary integrationإذا الآن في عند عقل
+
+314
+00:21:22,750 --> 00:21:27,390
+التكامل بعرفش أكامل اللي هي 80X³ زي Y³ وهذا
+
+315
+00:21:27,390 --> 00:21:31,490
+التكامل هذا بالذات التكامل هتلاقوه كثير بظهر عندنا
+
+316
+00:21:31,490 --> 00:21:37,330
+في الإحصاء أو اللي هو في الإحتمالات فبلزمنا نوجده
+
+317
+00:21:37,330 --> 00:21:41,030
+فاللي هو الناس بتضطر تروح ده عشان توجده لميه
+
+318
+00:21:41,030 --> 00:21:45,150
+للبولر و بتتحول للبولر و الإيجاد بيصير إيجاد سهل
+
+319
+00:21:45,150 --> 00:21:49,140
+كمانخلّينا نشوف الآن، إذا عشان وجود هذا بدي أحوله
+
+320
+00:21:49,140 --> 00:21:53,600
+للبولر هاي عندي ال integration أحوله للبولر أول
+
+321
+00:21:53,600 --> 00:21:59,200
+إشي بدي أشيل الـ dy dx و أحط مكانها لـR dr dθ و E
+
+322
+00:21:59,200 --> 00:22:02,220
+to the X تربيع زاد Y تربيع احنا بنعرف X تربيع زاد
+
+323
+00:22:02,220 --> 00:22:06,360
+Y تربيع هي عبارة عن R تربيع كيف عرفت؟التي هي X
+
+324
+00:22:06,360 --> 00:22:11,080
+تربيع R تربيع Cos تربيع ثتا والـ Y R تربيع Sin
+
+325
+00:22:11,080 --> 00:22:16,260
+تربيع ثتا مجموح بيصير R تربيع في Cos تربيع ثتا
+
+326
+00:22:16,260 --> 00:22:19,440
+زائد Sin تربيع ثتا يعني Cos تربيع زائد Sin تربيع
+
+327
+00:22:19,440 --> 00:22:22,880
+ثتا زي ما بتعرفوا واحد بتضلل R تربيع إذا هذه عبارة
+
+328
+00:22:22,880 --> 00:22:26,830
+عن اللي هي مين R تربيعلأن اتوارد تربي عردي اردي
+
+329
+00:22:26,830 --> 00:22:34,290
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+330
+00:22:34,290 --> 00:22:34,330
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+331
+00:22:34,330 --> 00:22:34,530
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+332
+00:22:34,530 --> 00:22:35,150
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+333
+00:22:35,150 --> 00:22:37,090
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+334
+00:22:37,090 --> 00:22:40,670
+اردي اردي اردي اردي اردي اردي اردي اردي اردي اردي
+
+335
+00:22:40,670 --> 00:22:45,190
+اردي
+
+336
+00:22:45,190 --> 00:22:49,850
+اردي اردي اردي اردي اردي اردي اردي اردي االان هذا
+
+337
+00:22:49,850 --> 00:22:56,090
+اللي هو اللي هي الدائرة الدائرة اللي هي الدائرة
+
+338
+00:22:56,090 --> 00:22:59,810
+الجزء النص التاني من الدائرة الفجانى اللي هو y
+
+339
+00:22:59,810 --> 00:23:02,390
+تربيع زاد x تربيع بيساوي واحد او y بيساوي جدري
+
+340
+00:23:02,390 --> 00:23:05,930
+تربيع لو واحد ناقص x تربيعو احنا بنعرف ان y تربيع
+
+341
+00:23:05,930 --> 00:23:09,970
+زاد x تربيع بساوي واحد اللي هو معادلته بالبولر ايش
+
+342
+00:23:09,970 --> 00:23:14,190
+هي y تربيع زاد x تربيع اللي هي مكانها R تربيع بصير
+
+343
+00:23:14,190 --> 00:23:18,690
+R تربيع بساوي واحد يعني R بتساوي واحد عندى الان R
+
+344
+00:23:18,690 --> 00:23:23,070
+بتساوي واحد هذه اللي هي الدائرة نص الدائرة طبعا
+
+345
+00:23:23,070 --> 00:23:26,490
+الدائرة كلها R بتساوي واحد لكن الان اللي هيتحكملي
+
+346
+00:23:26,490 --> 00:23:30,210
+في الموضوع ليه قيمة θ هتحددلي بس المنطقة اللي فوق
+
+347
+00:23:30,600 --> 00:23:33,720
+خلّينا نشوف إذا أن هذه المنطقة خلّينا نطلع الآن
+
+348
+00:23:33,720 --> 00:23:36,960
+الـ Ray اللي هو في داخل المنطقة يدخل المنطقة و
+
+349
+00:23:36,960 --> 00:23:40,600
+يخرج منها طبعا هي بدأ من عند السفر و بخرج من عندها
+
+350
+00:23:40,600 --> 00:23:44,020
+كل على نفس المنوال بخرج من الدائرة اللي هي R
+
+351
+00:23:44,020 --> 00:23:48,660
+بالسواحي علشان هي حدود الـR من سفر لعند واحد نيجي
+
+352
+00:23:48,660 --> 00:23:51,660
+الآن لقييم الـθتا طبعا هي ال .. ال .. ال .. ال ..
+
+353
+00:23:51,660 --> 00:23:51,800
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+354
+00:23:51,800 --> 00:23:52,340
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+355
+00:23:52,340 --> 00:23:52,680
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+356
+00:23:52,680 --> 00:23:54,740
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+357
+00:23:54,740 --> 00:23:56,300
+ال .. ال .. ال .. ال .. ال .. ال ..لأن واضح أن θ
+
+358
+00:23:56,300 --> 00:24:01,180
+من هنا تبدأ بـ 0 و تمسح المنطقة و يبقى هذا المؤشر
+
+359
+00:24:01,180 --> 00:24:03,800
+يمشي و يمشي و يمشي و يمشي و يصبح باي على اتنين و
+
+360
+00:24:03,800 --> 00:24:04,920
+يصبح كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا
+
+361
+00:24:04,920 --> 00:24:05,020
+و كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+362
+00:24:05,020 --> 00:24:05,940
+كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+363
+00:24:05,940 --> 00:24:06,500
+كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+364
+00:24:06,500 --> 00:24:09,480
+كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+365
+00:24:09,480 --> 00:24:10,080
+كذا و كذا و كذا و كذا و كذا و كذا و كذا و كذا و
+
+366
+00:24:10,080 --> 00:24:20,120
+كذا و كذا و كذا و كذا و كذاحولناها من كارتيزيان ل
+
+367
+00:24:20,120 --> 00:24:24,040
+بولار بيصير عندى اللى هو هذا E تربيع R تربيع في R
+
+368
+00:24:24,040 --> 00:24:28,480
+اللى هو تكامله طبعا نص E to the R تربيع اللى هو
+
+369
+00:24:28,480 --> 00:24:33,160
+لإنه هذا التفاضل هو هذا المقدار من 0 لعنده 1 طبعا
+
+370
+00:24:33,160 --> 00:24:37,240
+بنعوض بتطلع نص اللى هى في حالة الواحد E to the
+
+371
+00:24:37,240 --> 00:24:41,000
+واحد ناقص اللى هو E to the zero اللى هو واحد اللى
+
+372
+00:24:41,000 --> 00:24:45,640
+هى ناقص واحد لكل D ثتاالان هذا ثابت بنضربه في θ
+
+373
+00:24:45,640 --> 00:24:49,260
+بصير ال θ من صفر ل باي بنعوده عن θ بتطلع باي على
+
+374
+00:24:49,260 --> 00:24:52,220
+اتنين اللي هي دي باي في نص باي على اتنين في ايه
+
+375
+00:24:52,220 --> 00:24:56,660
+تقرر minus واحد وبنكون هيك اوجدنا اللي هي ال
+
+376
+00:24:56,660 --> 00:25:00,240
+integration طبعا شفنا اهمية ال polar انها
+
+377
+00:25:00,240 --> 00:25:05,140
+بتتغلبلنا على اللي هي عدم عرفتنا التكامل بتحويله
+
+378
+00:25:05,140 --> 00:25:10,800
+لتكامل المكافئ بواسطة اللي هو ال polarالان ناخد
+
+379
+00:25:10,800 --> 00:25:13,780
+مثال تاني evaluate the integral معطيني ال
+
+380
+00:25:13,780 --> 00:25:19,180
+integration الان هيه هو مش وصف للمنطقة الان انا
+
+381
+00:25:19,180 --> 00:25:23,120
+بدرسمها المنطقة لان بس استعنت بالرسم اللي فاتت بس
+
+382
+00:25:23,120 --> 00:25:26,120
+الان هو مش راسي هذه الرسمة مش وجودة هذا بدرسمها
+
+383
+00:25:26,120 --> 00:25:30,320
+الان ايش هي ال value of the integration من 0 لعند
+
+384
+00:25:30,320 --> 00:25:33,940
+جدر الواحد ناقص x تربيع x تربيع زد ويه تربيع dy dx
+
+385
+00:25:34,270 --> 00:25:37,470
+من صفر لعند الواحد لو ايجي اجرب واحد يكمل هذه اللي
+
+386
+00:25:37,470 --> 00:25:40,410
+جوا بالنسبة ل Y بيكملها هاي X تربيع في الجدر ال
+
+387
+00:25:40,410 --> 00:25:43,750
+تربيع لو واحد ناقص X تربيع زائد اللي هي طبعا مفهوم
+
+388
+00:25:43,750 --> 00:25:47,210
+X تربيع في Y زائد Y تكعيبة تلاتة و بعوض من صفر
+
+389
+00:25:47,210 --> 00:25:50,230
+لعند هذا بتطلع هذا المقدار هذا المقدار اللي بيطلع
+
+390
+00:25:50,230 --> 00:25:55,060
+عندبدا اكمله لل X ماعرفش اكمله الا بواسط ال tables
+
+391
+00:25:55,060 --> 00:26:00,540
+و غلبه كتير كتير كتير فالان بدي احوله الى بولر
+
+392
+00:26:00,540 --> 00:26:06,080
+يكون سهل التكامل واللي بشجعني ايش ان ال X تربيع
+
+393
+00:26:06,080 --> 00:26:09,300
+زاد Y تربيع هذي انا بعرفها في البولر قبل بشوية
+
+394
+00:26:09,300 --> 00:26:14,120
+قولها هي عبارة عن R تربيعفعلى طول تتحول هذه لـR
+
+395
+00:26:14,120 --> 00:26:19,700
+تربيع و الـ dy بي dx تتحول لR دي R دي ثتا هذه
+
+396
+00:26:19,700 --> 00:26:24,520
+تكملها بالنسبة لـR سهلة R تربيع في R تصبح R تكعيب
+
+397
+00:26:24,520 --> 00:26:31,120
+و تتفاء كامل بكل سهولة إذا الآن التحويل لبولر حيسن
+
+398
+00:26:31,120 --> 00:26:36,920
+لعملية إيجاد التكامل هنا حولنااللي هي الـ
+
+399
+00:26:36,920 --> 00:26:40,920
+Integrand اللي هو الدالة اللي بدنا نكملها و اللي
+
+400
+00:26:40,920 --> 00:26:47,160
+هو dy by dx حولناها إلى مكافئها اللي هي rd rd θ في
+
+401
+00:26:47,160 --> 00:26:51,380
+البولارالان يا جماعة دل علينا المنطقة المنطقة هذه
+
+402
+00:26:51,380 --> 00:26:57,140
+عشان احولها ل بولار بدي اش اسوي بدي ارسمها الان
+
+403
+00:26:57,140 --> 00:27:00,780
+الان من صفر لعند مين ال Y قيمة ال Y من صفر لجدر
+
+404
+00:27:00,780 --> 00:27:05,200
+الواحد minus X تربيع هي من صفر لعند ال Y هي من صفر
+
+405
+00:27:05,200 --> 00:27:09,040
+هي من صفر هذي طبعا Y بالساوية هذي اللي هي دائرة بس
+
+406
+00:27:09,040 --> 00:27:12,340
+النص الفجاني من الدائرة هي النص الفجاني من الدائرة
+
+407
+00:27:12,690 --> 00:27:17,790
+هذا الان اللي حدد ليه مين اللي حدد ليه قيمة الـ Y
+
+408
+00:27:17,790 --> 00:27:21,490
+الان قيمة X بدها تحددلي بالظبط ايش من هذه الدائرة
+
+409
+00:27:21,490 --> 00:27:26,190
+اللي بدناياها قيمة X جالي من 0 بس لمين ل1 يعني
+
+410
+00:27:26,190 --> 00:27:30,410
+جالي X تبعتك من 0 ل عند 1 يعني المنطقة دي بديش
+
+411
+00:27:30,410 --> 00:27:34,570
+إياهايعني الآن منطقتنا اللي بدنا نكامل عليها حسب
+
+412
+00:27:34,570 --> 00:27:38,510
+اللي هو المعطى عند Y من 0 لجدر الواحد minus X
+
+413
+00:27:38,510 --> 00:27:42,330
+تربيع و X من 0 لعند الواحد بس هي المنطقة اللي
+
+414
+00:27:42,330 --> 00:27:46,710
+أمامها دي اللي هي اللي في الربع الأول طيب الآن
+
+415
+00:27:46,710 --> 00:27:49,830
+بدنا نحولها لبولر قبل شوية حولناها لبولر هذه اللي
+
+416
+00:27:49,830 --> 00:27:54,030
+هي Y بتساوي X تربيع زي .. Y تربيع زي .. Y بتساوي
+
+417
+00:27:54,030 --> 00:27:56,670
+جدر الواحد minus X تربيع اللي هي جزء من الدائرة R
+
+418
+00:27:56,670 --> 00:28:02,180
+بتساوي واحد إذا هذه معدلتها R بتساوي واحدالان اذا
+
+419
+00:28:02,180 --> 00:28:08,580
+لو جينا على المنطقة هذه كل الاريز اللى بتدخل
+
+420
+00:28:08,580 --> 00:28:11,420
+المنطقة و بتخرج منها يعني من نقطة الاصل بتروح لعند
+
+421
+00:28:11,420 --> 00:28:14,740
+نص الدائرة اللى R بساوة واحد عشان هيك حدود ال
+
+422
+00:28:14,740 --> 00:28:18,660
+integration لـR من صفر لعند واحد نيجي الان لـθثتة
+
+423
+00:28:18,660 --> 00:28:23,120
+تبدأ من الهن و تتجه لقرية و تتجه لقرية و تتجه
+
+424
+00:28:23,120 --> 00:28:23,700
+لقرية و تتجه لقرية و تتجه لقرية و تتجه لقرية و
+
+425
+00:28:23,700 --> 00:28:25,040
+تتجه لقرية و تتجه لقرية و تتجه لقرية و تتجه لقرية
+
+426
+00:28:25,040 --> 00:28:26,960
+و تتجه لقرية و تتجه لقرية و تتجه لقرية و تتجه
+
+427
+00:28:26,960 --> 00:28:27,840
+لقرية و تتجه لقرية و تتجه لقرية و تتجه لقرية و
+
+428
+00:28:27,840 --> 00:28:30,760
+تتجه لقرية و تتجه لقرية و تتجه لقرية و تتجه لقرية
+
+429
+00:28:30,760 --> 00:28:37,570
+و تتجه لقرية و تتجه لقرية و تتجه لقبنكمل الان R
+
+430
+00:28:37,570 --> 00:28:41,070
+تكعيب بيصير R أربعة على أربعة من واحد إلى سفر
+
+431
+00:28:41,070 --> 00:28:44,270
+بيصير عبارة عن ربع D ثتا من سفر ل باي على اتنين
+
+432
+00:28:44,270 --> 00:28:49,210
+وهذه محلها ثابت بيصير ثتا على أربعة من سفر ل باي
+
+433
+00:28:49,210 --> 00:28:52,290
+على اتنين وبتطلع باي ع تمانية و هيكة بتكون كتير
+
+434
+00:28:52,290 --> 00:28:57,650
+كتير الأمور سهلة بالباسطة البولار مابتغلبش زي ال
+
+435
+00:28:57,650 --> 00:29:02,170
+kartesian في هذه الحالةنجي الآن لمثال آخر find the
+
+436
+00:29:02,170 --> 00:29:04,930
+volume of the solid bounded above by the
+
+437
+00:29:04,930 --> 00:29:08,170
+paraboloid z بساوية تسعة ناقص x تربيع ناقص y تربيع
+
+438
+00:29:08,170 --> 00:29:11,510
+and below by the unit circle in the xy plane يعني
+
+439
+00:29:11,510 --> 00:29:14,550
+هيه الآن ال solid هذا بقول اللي من تحت هيه عبارة
+
+440
+00:29:14,550 --> 00:29:20,070
+عن الدائرة هذه ومن فوق هيه إيش اللي هو الشكل هذا z
+
+441
+00:29:20,070 --> 00:29:22,940
+بساوية تسعة ناقص x تربيع ناقص y تربيعطبعا الـ
+
+442
+00:29:22,940 --> 00:29:25,200
+volume اللي بين هذا اللي بين هذا اللي هنا إيش
+
+443
+00:29:25,200 --> 00:29:29,500
+بيساوي اللي هو اللي هو المساحة القاعدة في الارتفاع
+
+444
+00:29:29,500 --> 00:29:33,140
+وقولنا إيش هذا بنعبر عنه بال integration اللي هو
+
+445
+00:29:33,140 --> 00:29:36,400
+قيبة ال function اللي هو بيمثل الارتفاع اللي هو
+
+446
+00:29:36,400 --> 00:29:40,950
+عبارة عن إيش تسعة نقص اكتربيع نقص ويتربيعDA هي دي
+
+447
+00:29:40,950 --> 00:29:43,370
+ايه اللي هي بتجيب للمساحة فبصير المساحة في
+
+448
+00:29:43,370 --> 00:29:46,130
+الارتفاع اللي هي بتطلع الحجم ايه ده ال volume حسب
+
+449
+00:29:46,130 --> 00:29:48,390
+ما تقولنا سابقا اللي هي تسعة نقص اكتر بيناقص و هي
+
+450
+00:29:48,390 --> 00:29:52,130
+تربيع دي ايه دي عبارة عن قيمة الدالة اللي هو قيمة
+
+451
+00:29:52,130 --> 00:29:57,930
+Z لفوق على السطح اللي مفروض بدنا نكامل بدنا نكون
+
+452
+00:29:57,930 --> 00:30:01,730
+سطح العلو اللي هو المجسم
+
+453
+00:30:03,910 --> 00:30:09,390
+القاعدة H A هي D A للدائرة الموجودة عليها محطوط
+
+454
+00:30:09,390 --> 00:30:12,210
+عالميا على الدائرة اللي عنده إذا ال region اللي
+
+455
+00:30:12,210 --> 00:30:16,590
+عنده هي R لأنه القاعدة اللي هي عبارة عن الدائرة زي
+
+456
+00:30:16,590 --> 00:30:20,370
+ما هو ذكر ليها هذه الـ N R هي اللي تتحدد لحدود ال
+
+457
+00:30:20,370 --> 00:30:23,720
+integrationاللي هي X تربيع زاد Y تربيع بيساوي واحد
+
+458
+00:30:23,720 --> 00:30:26,880
+هي هم اعطينيها ال unit circle هذه ايش معادلة هي
+
+459
+00:30:26,880 --> 00:30:31,160
+بالبولر R بتساوي واحد الان هذه دائرة كاملة اذا من
+
+460
+00:30:31,160 --> 00:30:34,700
+عند ال X axis لو اخدت اي array هتلاقي ال array
+
+461
+00:30:34,700 --> 00:30:39,000
+بيبدأ من نقطة السفر وبروح لمين للواحد عشان هيك
+
+462
+00:30:39,000 --> 00:30:44,400
+حدود ال integration للR من سفر لعند مين لعند واحد
+
+463
+00:30:44,790 --> 00:30:49,270
+الان عشان اكمل الدائرة الـ R هي بجان و بجان و بجان
+
+464
+00:30:49,270 --> 00:30:53,630
+لف الدائرة كاملة إذا هيبدأ من عند سفر و يمشي لما
+
+465
+00:30:53,630 --> 00:30:57,790
+يصل عند اتنين باي هو عبارة عن حدود ال integration
+
+466
+00:30:57,790 --> 00:31:01,880
+اللي بيجيب الدائرة هذه كلهاالأن ندى للـ Integrand
+
+467
+00:31:01,880 --> 00:31:04,360
+اللى هو تسعة نقص X تربيع نقص Y تربيع الدالة اللى
+
+468
+00:31:04,360 --> 00:31:08,460
+بنكملها ال X تربيع زاد Y تربيع هي عبارة عن مين R
+
+469
+00:31:08,460 --> 00:31:11,220
+تربيع زى ما احنا عارفينه قولناها قبل شوية بيصير دى
+
+470
+00:31:11,220 --> 00:31:15,420
+تسعة نقص R تربيع والـDA زى ما قولنا اللى هى DXDY
+
+471
+00:31:15,420 --> 00:31:21,510
+اللى بتتحول لRDRDθ المكافئة لـDXDYالان بدنا الكامل
+
+472
+00:31:21,510 --> 00:31:26,810
+بتصير 9R-R تكعيب دي R دي تيتا ضربت ال R جوا الان
+
+473
+00:31:26,810 --> 00:31:31,570
+كامل هذه بتصير 9R تربيع على 2 ناقص ربع R أقصى 4 من
+
+474
+00:31:31,570 --> 00:31:36,010
+و إلى وين من 0 لعند 1 قيمة ال R فبتصير عبارة عن 9
+
+475
+00:31:36,010 --> 00:31:41,110
+على 2 ناقص ربع اللي هي بتطلع 17 على 4 حسبوها
+
+476
+00:31:41,110 --> 00:31:45,670
+لحالكمالان بيظهر مين عنده دي ثتا ثتا من صفر لعنده
+
+477
+00:31:45,670 --> 00:31:49,550
+اتنين باي دي ثتا تكمل اللي هو عبارة عن ثتا اللي هي
+
+478
+00:31:49,550 --> 00:31:53,130
+عبارة عن اتنين باي اللي هي بيصير اتنين باي في هذه
+
+479
+00:31:53,130 --> 00:31:57,750
+اللي هي بتصير اللي هي اتنين باي في سبعتاشر لاربعة
+
+480
+00:31:57,750 --> 00:32:00,810
+بيصير سبعة باي على مين على اتنين لانه بتروح اتنين
+
+481
+00:32:00,810 --> 00:32:04,010
+مع اتنين اللي هان اللي هي بتطلع سبعتاشر باي على
+
+482
+00:32:04,010 --> 00:32:07,470
+اتنين اللي هي قيمة اللي هي ذاك فاهمين كيف اللي هي
+
+483
+00:32:07,470 --> 00:32:10,740
+دي بيصير اتنين بايقيمة التكامل لإنها تبسيط data
+
+484
+00:32:10,740 --> 00:32:14,120
+تبسيط data من صفر لباى عن اتنين صفر لتنين باى عن
+
+485
+00:32:14,120 --> 00:32:16,700
+اتنين باى اتنين باى في سبعتاشر على اربع تطلع
+
+486
+00:32:16,700 --> 00:32:20,220
+سبعتاشر على اتنين سبعتاشر باى على اتنين و هيك
+
+487
+00:32:20,220 --> 00:32:23,580
+بيكون اوجدنا ال volume اللي عندنا ندى للسؤال اللى
+
+488
+00:32:23,580 --> 00:32:27,920
+هو مثال السادس الان نعزز الحديث اللى قبله شوية
+
+489
+00:32:27,920 --> 00:32:32,890
+ناخد مثال اخر خلنا نقول شكلهأموره مختلفة لكن بنفس
+
+490
+00:32:32,890 --> 00:32:36,950
+الروح using polar integration باستخدام ال polar
+
+491
+00:32:36,950 --> 00:32:41,090
+integration find the area of the region R اللي هم
+
+492
+00:32:41,090 --> 00:32:44,730
+بدهو اللي هو مساحة ال region R بدهو يوصفها الآن
+
+493
+00:32:44,730 --> 00:32:49,010
+وين موجودة in the XY plane enclosed by اللي هي في
+
+494
+00:32:49,010 --> 00:32:52,590
+ال XY plane مغلقة بواسطة ال circle X تربيع زاد Y
+
+495
+00:32:52,590 --> 00:32:55,820
+تربيع بساوة أربعة دي مش زادة أربعة بساوة أربعةإذا
+
+496
+00:32:55,820 --> 00:32:59,340
+هذه اللي هو المنطقة اللي بدنا إياها جوات الدائرة
+
+497
+00:32:59,340 --> 00:33:04,960
+هذه اللي هي X تربيع زاد Y تربيع بساوة أربعة يعني
+
+498
+00:33:04,960 --> 00:33:08,200
+وكأنه لو أجيناها وعملناها بالبولر على طول X تربيع
+
+499
+00:33:08,200 --> 00:33:11,500
+زاد Y تربيع بساوة أربعة اللي هي الدائرة هذه على
+
+500
+00:33:11,500 --> 00:33:14,700
+طول بتصير R تربيع بساوة أربعة يعني R بساوة اتنين
+
+501
+00:33:14,700 --> 00:33:19,060
+يعني هذه اللي هي الدائرة اللي عندى تعبيرها بالبولر
+
+502
+00:33:19,060 --> 00:33:24,670
+اللي هو R بساوة اتنينطيب .. الان نكمل وصف المنطقة
+
+503
+00:33:24,670 --> 00:33:28,830
+ماخلصناش .. بقوللي الان هذه المنطقة المغلقة من
+
+504
+00:33:28,830 --> 00:33:33,470
+الدائرة ولكنها ايش above the line Y بساوي واحد هي
+
+505
+00:33:33,470 --> 00:33:38,470
+ال Y بساوي واحد فوق المين المنطقة Y بساوي واحد
+
+506
+00:33:38,470 --> 00:33:41,730
+يعني بيصير لسه ها دي معها و ها دي معها و اللي في
+
+507
+00:33:41,730 --> 00:33:45,630
+الجهة التانية معها الان بدنا نحدد أكترabove y
+
+508
+00:33:45,630 --> 00:33:50,250
+بتساوي واحد هي y بتساوي واحد اللي فوقي هي المنطقة
+
+509
+00:33:50,250 --> 00:33:54,810
+هذه محدد مع اللي هذه لكن الان بدنا نحدد اكتر لان
+
+510
+00:33:54,810 --> 00:33:58,430
+ان below the line y بيساوي جدر التلاتة x و تحت من
+
+511
+00:33:58,430 --> 00:34:01,630
+ال line y بيساوي جدر التلاتة x y بيساوي جدر
+
+512
+00:34:01,630 --> 00:34:05,790
+التلاتة x اللي ياخد x بصفر و y بصفر اكيد بنمر نقطة
+
+513
+00:34:05,790 --> 00:34:09,650
+الاصر اللي هو x بواحد و y بجدر التلاتة اذا هذه
+
+514
+00:34:09,650 --> 00:34:14,620
+اللي هي بتيجي اللي هي بتمشي في الخطالخط هذا بالشكل
+
+515
+00:34:14,620 --> 00:34:17,760
+هذا هذا اللي هو Y بيساوي جدر التلاتة X بتعرفوا
+
+516
+00:34:17,760 --> 00:34:21,960
+ترسموه الآن بدنا اشت المنطقة المنطقة اللي تحته اذا
+
+517
+00:34:21,960 --> 00:34:25,480
+هي المنطقة اللي تحته مش هذه اللي فوق اذا الان جينا
+
+518
+00:34:25,480 --> 00:34:30,040
+حصرنا حالنا المنطقة اللي تحت هذا و اللي فوق هذا و
+
+519
+00:34:30,040 --> 00:34:34,220
+اللي في الدائرة اه لإن اللي تحت هذه هي كل هذه اللي
+
+520
+00:34:34,220 --> 00:34:37,560
+فوق هذه هي كلها لكن الان بدناش اللي برا هنا بس
+
+521
+00:34:37,560 --> 00:34:40,880
+اللي موجود وين في الدائرة اذا هي المنطقة المقصودة
+
+522
+00:34:41,210 --> 00:34:45,710
+الان بدنا نوجد نقاط التقاطع عند نقاط التقاطع نجد
+
+523
+00:34:45,710 --> 00:34:49,930
+النقطة الهادي و النقطة هادى الان هذا y بساوية جدر
+
+524
+00:34:49,930 --> 00:34:55,430
+ال 3x عند اللي هو عوض اللي هو في الدائرة هادى اللي
+
+525
+00:34:55,430 --> 00:34:59,190
+x² بيعت سواء أربعة عوض عن قيمة عشان توجد ايه؟ عشان
+
+526
+00:34:59,190 --> 00:35:04,260
+توجد تقاطع هذا الخط مع الدائرة كيف توجده؟بتعود y
+
+527
+00:35:04,260 --> 00:35:07,140
+بيساوي جدر التلاتة x هنا بيصير اللي هي تلاتة x
+
+528
+00:35:07,140 --> 00:35:10,680
+تربيع .. تلاتة y اللي هي تلاتة x تربيع زائد تلاتة
+
+529
+00:35:10,680 --> 00:35:13,540
+x تربيع اللي هي أربعة x تربيع بيساوي أربعة يعني x
+
+530
+00:35:13,540 --> 00:35:17,040
+تربيع بيساوي واحد يعني x بيساوي زائد أو ناقص واحد
+
+531
+00:35:17,040 --> 00:35:19,480
+لكن في المنطقة الموجودة إذا ال x إيش بتساوي واحد
+
+532
+00:35:19,480 --> 00:35:23,500
+لما x بيساوي واحد y إيش هتساوي جدر التلاتة فهذه
+
+533
+00:35:23,500 --> 00:35:27,820
+النقطة اللي هي واحد وجدر التلاتةنجي نوجد من النقطة
+
+534
+00:35:27,820 --> 00:35:30,360
+هذه النقطة هذه عبارة عن تقاطع اللي هو y بيساوي
+
+535
+00:35:30,360 --> 00:35:34,900
+واحد مع الدائرة الان y بيساوي واحد بيصير x تربيع
+
+536
+00:35:34,900 --> 00:35:37,440
+زائد واحد بيساوي أربعة يعني x تربيع بيساوي تلاتة
+
+537
+00:35:37,440 --> 00:35:40,000
+يعني x بيساوي جدر التلاتة زيادة ونوافذ طبعا أنا
+
+538
+00:35:40,000 --> 00:35:43,120
+موجبة بيصير x بيساوي جدر التلاتة إذن النقطة هذه هي
+
+539
+00:35:43,120 --> 00:35:47,030
+عبارة عن جدر التلاتة وواحدإذا المنطقة هنا حددناها
+
+540
+00:35:47,030 --> 00:35:52,270
+تحديدا تماما و حددنا ليمين نقاط التقاطع هذا نقطة
+
+541
+00:35:52,270 --> 00:35:56,610
+مش هتلزمني وإن كانت سهلة اللي هي عند Y بواحد اللي
+
+542
+00:35:56,610 --> 00:36:01,170
+هي عند اللي هي بنوجه قيمة X ليه واحد ع جدر التلاتة
+
+543
+00:36:01,170 --> 00:36:04,350
+لكن هذا مش كتير لأ مش لازم يعنيالان .. هلجدت؟
+
+544
+00:36:04,350 --> 00:36:07,490
+هتشوفوا ليش مش نزلنا .. هي المنطقة الان ماشي الان
+
+545
+00:36:07,490 --> 00:36:11,290
+المنطقة هي اللي هو ال integration بدو ال area لها
+
+546
+00:36:11,290 --> 00:36:15,650
+عبارة عن dA over R اللي هي المنطقة الان dA اللي هي
+
+547
+00:36:15,650 --> 00:36:18,890
+زي ما بنقول أشملها RDRDθ لإن هو بدو ال area
+
+548
+00:36:18,890 --> 00:36:21,590
+للمنطقة هذه مدام ال area إذا اللي هو double
+
+549
+00:36:21,590 --> 00:36:26,650
+integral لـ dA over R الـ dA اللي هي RDRDθ over R
+
+550
+00:36:26,650 --> 00:36:29,710
+R بدنا نحدد الآن اللي هي حدود ال integration شوف
+
+551
+00:36:29,710 --> 00:36:34,280
+حدود ال integrationأول حاجة اللي هي كل الخطوط اللي
+
+552
+00:36:34,280 --> 00:36:39,100
+بتطلع .. اللي هي تدخل للمنطقة هيها .. هيها الخطوط
+
+553
+00:36:39,100 --> 00:36:44,070
+بتدخل للمنطقة من عند اللي هو الخط هذااللي هو y
+
+554
+00:36:44,070 --> 00:36:48,270
+بيساوي واحد لعند الخط لعند مين بتطلع و بتطلع من
+
+555
+00:36:48,270 --> 00:36:51,470
+الدائرة بتدخل من عند الخط و بتطلع من الدائرة و
+
+556
+00:36:51,470 --> 00:36:57,470
+كلها على نفس المنوال إذا الان قيمة R لعند هذا الخط
+
+557
+00:36:57,470 --> 00:37:01,370
+بنوجدها لإنه إيش الخط y بيساوي واحد y هي عبارة عن
+
+558
+00:37:01,370 --> 00:37:06,010
+R sin θ إذا R sin θ بيساوي واحد إذا ال R بيساوي
+
+559
+00:37:06,010 --> 00:37:10,940
+واحد على sin θ يعني R بيساوي cos θإذا قيمة R على
+
+560
+00:37:10,940 --> 00:37:15,620
+هذا الخط اللي هو معدلته R بساوي cos θ قيمة R
+
+561
+00:37:15,620 --> 00:37:21,640
+أشملها عبارة عن cos θ إذا R تبدأ من عند cos θ و
+
+562
+00:37:21,640 --> 00:37:26,140
+بتطلع من الدائرة قيمة الدائرة R عند الدائرة هذه زي
+
+563
+00:37:26,140 --> 00:37:30,360
+ما قلنا معدلتها R بتساوي 2 إذا عند 2 نيجي الآن
+
+564
+00:37:30,360 --> 00:37:36,200
+نشوف قداش قيمة θ لأن هاي θ من هنا بتبدأهيها أول
+
+565
+00:37:36,200 --> 00:37:40,540
+أرى بيطلع عند النقطة هذه و بعدين بيمشي بيمشي لما
+
+566
+00:37:40,540 --> 00:37:45,340
+نبصل هنا طيب الآن بدي أشوف زاوية هذا قداش و زاوية
+
+567
+00:37:45,340 --> 00:37:48,320
+هذا الأرى قداش إذا بتكون تلتة من عند اللي هو هذا
+
+568
+00:37:48,320 --> 00:37:53,080
+من عند هذا الخط لعند هذا بتكون إيش هذا الخط لما
+
+569
+00:37:53,080 --> 00:37:55,440
+يمشي لما يمشي نمشي في داخل المنطقة بيكون مسحها
+
+570
+00:37:55,440 --> 00:37:59,120
+كلها فعلا نيجي لهذه هذه تلتة واحد وجدر التلتة إذا
+
+571
+00:37:59,120 --> 00:38:04,640
+هذا طوله اللي هو واحدوهذا طوله جدر التلاتة لأن هذا
+
+572
+00:38:04,640 --> 00:38:07,920
+إذا المثلث اللي اللي اللي اللي اللي اللي اللي اللي
+
+573
+00:38:07,920 --> 00:38:08,080
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+574
+00:38:08,080 --> 00:38:08,380
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+575
+00:38:08,380 --> 00:38:08,560
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+576
+00:38:08,560 --> 00:38:08,920
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+577
+00:38:08,920 --> 00:38:09,700
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+578
+00:38:09,700 --> 00:38:09,740
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+579
+00:38:09,740 --> 00:38:09,740
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+580
+00:38:09,740 --> 00:38:10,600
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+581
+00:38:10,600 --> 00:38:10,600
+اللي اللي اللي اللي اللي اللي اللي اللي اللي اللي
+
+582
+00:38:10,600 --> 00:38:19,840
+اللي اللي
+
+583
+00:38:19,840 --> 00:38:24,010
+الليإذا هنا عجباله اللي هو هذا قيمة واحد قيمة Y
+
+584
+00:38:24,010 --> 00:38:26,710
+إذا العجباله اللي هي عبارة عن الـ πاية على ستة
+
+585
+00:38:26,710 --> 00:38:30,670
+تلاتين درجة إذا الأن القرية الأول بيبدأ من وين من
+
+586
+00:38:30,670 --> 00:38:33,870
+عند باية على ستة الان نجي نشوف القرية الثانية هي
+
+587
+00:38:33,870 --> 00:38:37,180
+الزاوية تبع القرية الثانيةالان ها الارية الأخيرة
+
+588
+00:38:37,180 --> 00:38:39,980
+طبعا بقصود هي عندي اللي هي النقطة دي واحد وجدر
+
+589
+00:38:39,980 --> 00:38:43,540
+التلاتة إذا هذا طوله جدر التلاتة اللي هي Y وهذا
+
+590
+00:38:43,540 --> 00:38:47,540
+طوله واحد اللي هي صار متلات برضه تلاتين ستيني لكن
+
+591
+00:38:47,540 --> 00:38:50,080
+جدر التلاتة هي العجبال الزاوية الكبيرة اللي هي
+
+592
+00:38:50,080 --> 00:38:53,300
+عبارة عن إيش بايع ستة إذا هذه الزاوية بتصل لمن؟
+
+593
+00:38:53,300 --> 00:38:56,700
+لعين قاسف بايع تلاتة اللي هي ستين درجة اللي هي
+
+594
+00:38:56,700 --> 00:39:01,160
+بتصل إذا الان الارية ده بنتي عند بايع تلاتة اللي
+
+595
+00:39:01,160 --> 00:39:04,410
+هي الزاوية ستين درجةفبكون هذا هو الـ integration
+
+596
+00:39:04,410 --> 00:39:08,430
+ان كامل الآن تكمل سهل بيصير ليا نص R تربيع من cos
+
+597
+00:39:08,430 --> 00:39:13,270
+θ لعند اتنين بتعوض بيصير نص في اربعة ناقص cos
+
+598
+00:39:13,270 --> 00:39:17,930
+تربيع θ دي ثتا الآن هذه تكملها برضه سهلاللي هي
+
+599
+00:39:17,930 --> 00:39:21,650
+عبارة عن نص في تكامل الأربعة اللي هي أربعة ثيتا
+
+600
+00:39:21,650 --> 00:39:25,610
+تكامل اللي هو cosecant تربيع نقص cosecant تربيع
+
+601
+00:39:25,610 --> 00:39:27,930
+التكامل اللي هو عبارة عن كوتان ثيتا زي ما احنا
+
+602
+00:39:27,930 --> 00:39:31,310
+عارفين الآن من وين لوين باية ستة عند باية تلاتة
+
+603
+00:39:31,310 --> 00:39:34,950
+بنعود عليها بطلع عند القيمة هذه نقص القيمة هذه
+
+604
+00:39:34,950 --> 00:39:37,970
+اللي عند باية تلاتة هي القيمة عند باية ستة هي
+
+605
+00:39:37,970 --> 00:39:42,010
+القيمة نترحل من بعض طبعا تعرفت وجودهم ويساوي وبطلع
+
+606
+00:39:42,010 --> 00:39:46,130
+القيمة اللي أمامنا وهذا بيكون اللي هو آخر مثال
+
+607
+00:39:46,130 --> 00:39:49,510
+عندناوبارك الله فيكم وإلى لقاء آخر
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/rvOHHt0thBs.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/rvOHHt0thBs.srt
new file mode 100644
index 0000000000000000000000000000000000000000..67aedf79c51c292768590893b8583bc2c9ace9fd
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/rvOHHt0thBs.srt
@@ -0,0 +1,1867 @@
+1
+00:00:00,000 --> 00:00:04,960
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,960 --> 00:00:12,160
+وبركاته هذه هي المحاضرة رقم 11 لمساق تفاضل وتكامل
+
+3
+00:00:12,160 --> 00:00:17,340
+وهي المحاضرة الرابعة في فصل 13، سنحكي اليوم
+
+4
+00:00:17,340 --> 00:00:21,060
+عن 13-4، اللي هو الـ curvature and normal
+
+5
+00:00:21,060 --> 00:00:26,420
+vectors of a curve. سنحكي عن مدى انحناء المنحنى وعن
+
+6
+00:00:26,420 --> 00:00:33,200
+اللي هو المتجه الوحدة العمودي على الـ curve. الآن
+
+7
+00:00:33,200 --> 00:00:37,660
+curvature of a plane curve. حكينا المرة الماضية عن
+
+8
+00:00:37,660 --> 00:00:41,180
+اللي هو الـ unit tangent vector T، وقلنا الـ unit
+
+9
+00:00:41,180 --> 00:00:44,660
+tangent vector T اللي هو عبارة عن الـ velocity V
+
+10
+00:00:44,660 --> 00:00:48,360
+على الـ magnitude لـ V، اللي هو الـ speed.  ده طالع عندي
+
+11
+00:00:48,360 --> 00:00:52,480
+اللي هو الـ unit tangent vector، اللي هو متجه اتجاه
+
+12
+00:00:52,480 --> 00:00:55,960
+الحركة. يعني لو في particle عندي على الذي هو الـ
+
+13
+00:00:55,960 --> 00:01:01,740
+curve اللي أمامي، بيكون الـ T بيمثل اتجاه حركة الـ
+
+14
+00:01:01,740 --> 00:01:06,980
+particle. الآن اتجاه حركة الـ particle هذا، الآن هنا
+
+15
+00:01:06,980 --> 00:01:11,680
+الـ T هيكون اللي هو اتجاهه هنا. الآن بعد شوية
+
+16
+00:01:11,680 --> 00:01:16,240
+مشينا، اللي هو يعني قطعنا طول معين عليه، وروحنا
+
+17
+00:01:16,240 --> 00:01:19,820
+اتجاهنا صار الاتجاه بالشكل هذا، وهذا بالاتجاه
+
+18
+00:01:19,820 --> 00:01:24,180
+بالشكل هذا. واضح أنه اتجاه الحركة بيتغير. لما اتجاه
+
+19
+00:01:24,180 --> 00:01:28,500
+الحركة بيتغير، طبعاً بيعمل لي إيش؟ انحناء في إيش؟ في
+
+20
+00:01:28,500 --> 00:01:32,660
+المنحنى تبع اللي هو إيش؟ اللي احنا بنسير عليه الـ
+
+21
+00:01:32,660 --> 00:01:39,180
+particle. بناء عليه الآن، قياس انحناء اللي هو المنحنى
+
+22
+00:01:39,180 --> 00:01:46,340
+هذا، بدنا نعرفه، عبارة عن معدل تغير اللي هو الـ T مع
+
+23
+00:01:46,340 --> 00:01:51,320
+معدل S، يعني عبارة عن معدل تغير متجه اللي هو
+
+24
+00:01:51,320 --> 00:01:55,720
+الحركة مع الطول. يعني كل ما قطعنا طول، بيتغير اللي هو
+
+25
+00:01:55,720 --> 00:02:02,390
+إيش؟ اللي هو قيمة الـ T. فبدنا نعرف اللي هو مدى
+
+26
+00:02:02,390 --> 00:02:08,290
+انحناء المنحنى، dT/dS، حيث T هو unit tangent
+
+27
+00:02:08,290 --> 00:02:12,990
+vector، وS هو طول
+
+28
+00:02:12,990 --> 00:02:14,270
+المنحنى.
+
+29
+00:02:16,530 --> 00:02:19,890
+الآن الـ magnitude اللي هو اللي بدنا نعرفه إيه؟
+
+30
+00:02:19,890 --> 00:02:23,610
+عبارة عن كابا. إذا الـ T، T is the unit tangent
+
+31
+00:02:23,610 --> 00:02:27,730
+vector of a smooth curve, the curvature function of
+
+32
+00:02:27,730 --> 00:02:32,050
+the curve is كابا،  بإيجاد الـ magnitude لـ dT/dS.
+
+33
+00:02:32,050 --> 00:02:35,970
+إذا من الآن وطالع، اللي هو كابا عبارة عن اللي هو
+
+34
+00:02:35,970 --> 00:02:40,530
+الـ magnitude لـ dT/dS، وهي تقيس مدى انحناء
+
+35
+00:02:40,530 --> 00:02:45,930
+المنحنى اللي تحت اللي هو اهتمامنا، أو اللي بدنا إياه.
+
+36
+00:02:46,540 --> 00:02:49,740
+الآن التعليق ده، بيعلق الآن ويقول: إذا كان الـ
+
+37
+00:02:49,740 --> 00:02:54,770
+magnitude large، بيكون الـ curvature عالي، اللي هو
+
+38
+00:02:54,770 --> 00:02:59,710
+بمعنى آخر، بتكون T بتتغير sharply، يعني بيكون مدق
+
+39
+00:02:59,710 --> 00:03:03,830
+اللي هو تغير T، اللي هو لما يكون كبير بالنسبة لـ S،
+
+40
+00:03:03,830 --> 00:03:10,850
+بيكون اللي هو الانحناء حاد. لكن لو كان القيمة هذه
+
+41
+00:03:10,850 --> 00:03:15,890
+اللي هو dT/dS صغيرة، وقريبة من الصفر، ولو صارت
+
+42
+00:03:15,890 --> 00:03:19,490
+صفر، يعني بيكونش فيه انحناء، وكأنه بيصير خط مستقيم
+
+43
+00:03:19,490 --> 00:03:23,410
+أو أقرب للخط المستقيم، لإن مفيش فيه انحناء.
+
+44
+00:03:27,590 --> 00:03:31,930
+يعني اللي هو في حالة إن تكون الـ dT/dS، الـ
+
+45
+00:03:31,930 --> 00:03:37,290
+magnitude اللي إلها صغير، بيكون اللي هي الـ T، تغيرها
+
+46
+00:03:37,290 --> 00:03:38,930
+تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها
+
+47
+00:03:38,930 --> 00:03:39,090
+تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها
+
+48
+00:03:39,090 --> 00:03:39,290
+تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها
+
+49
+00:03:39,290 --> 00:03:39,850
+تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها
+
+50
+00:03:39,850 --> 00:03:39,930
+تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها
+
+51
+00:03:39,930 --> 00:03:40,010
+تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها تغيرها
+
+52
+00:03:40,010 --> 00:03:45,460
+تغيرها تغيرها تغيرها تغيرها. الآن كابا عرفناها عبارة
+
+53
+00:03:45,460 --> 00:03:50,220
+عن magnitude لـ dT/dS. هذه في الحسابات بتغلبنا، أو
+
+54
+00:03:50,220 --> 00:03:57,140
+يعني بيكون أسهل إن لو جبنا اللي هو كابا بدلالة T،
+
+55
+00:03:57,140 --> 00:04:01,460
+مش بدلالة S. خلينا نشوف. يعني الآن لو كان عندي R of
+
+56
+00:04:01,460 --> 00:04:04,260
+T هو عبارة عن الـ curve المعطى أمامي، اللي برسم
+
+57
+00:04:04,260 --> 00:04:08,000
+للمنحنى، اللي هو اللي احنا بنبحث عن الـ curvature
+
+58
+00:04:08,000 --> 00:04:14,110
+له. ممكن نوجد كابا له بطريقة أسهل، اللي هي بدلالة T.
+
+59
+00:04:14,110 --> 00:04:17,590
+خلينا نشوف. الآن كابا، زي ما عرفناها، هي عبارة عن الـ
+
+60
+00:04:17,590 --> 00:04:21,890
+magnitude لـ dT/dS، ويساوي بالـ chain rule هذا،
+
+61
+00:04:21,890 --> 00:04:26,550
+اللي جوا dT/dT في dT/dS، زي ما أنتم عارفين.
+
+62
+00:04:26,550 --> 00:04:31,010
+الآن الآن هذا عبارة عن الـ magnitude لـ dT/dT في
+
+63
+00:04:31,010 --> 00:04:33,750
+الـ magnitude لهذا. هذه الـ magnitude إلها عبارة عن
+
+64
+00:04:33,750 --> 00:04:36,590
+واحد على الـ magnitude لـ dS/dT، زي ما أنتم
+
+65
+00:04:36,590 --> 00:04:41,000
+عارفين. هذه عبارة عن إيش يا جماعة؟ هذه عبارة عن اللي
+
+66
+00:04:41,000 --> 00:04:45,560
+هو اللي هي عبارة عن الـ magnitude للـ V، إذا بتتذكروا.
+
+67
+00:04:45,560 --> 00:04:49,520
+هذه الآن صارت الكابا عبارة عن الـ magnitude للـ dT
+
+68
+00:04:49,520 --> 00:04:55,500
+/dT على الـ magnitude للـ V. فصارت كابا
+
+69
+00:04:55,500 --> 00:05:00,080
+automatic. لو أعطونا الـ R، مباشرة بتجيب الـ V، باش dT/
+
+70
+00:05:00,080 --> 00:05:04,250
+dR/dT، وبعدين بتجيب اللي هو الـ T، عبارة عن الـ V
+
+71
+00:05:04,250 --> 00:05:07,390
+على الـ magnitude للـ V اللي أوجدناها، وبعدين بتجيب
+
+72
+00:05:07,390 --> 00:05:12,430
+الـ dT/dT من الـ T، وبيكون عندي كل شيء جاهز بالشكل
+
+73
+00:05:12,430 --> 00:05:17,270
+هذا. بواسطة فقط التفاضلات للـ R of T، بنكون
+
+74
+00:05:17,270 --> 00:05:21,110
+حصلنا على الكابا، وقبلها حصلنا على الـ unit tangent
+
+75
+00:05:21,110 --> 00:05:25,860
+vector. الآن باختصار، formula for calculating
+
+76
+00:05:25,860 --> 00:05:30,800
+curvature، اللي هي الـ formula اللي بتحسب لنا اللي هو
+
+77
+00:05:30,800 --> 00:05:35,180
+مدى انحناء المنحنى، اللي هو كالتالي: If R of T is a
+
+78
+00:05:35,180 --> 00:05:39,480
+smooth curve, then the curvature is kappa تساوي
+
+79
+00:05:39,480 --> 00:05:43,560
+واحد على الـ magnitude لـ V في dT/dT. أما absolute
+
+80
+00:05:43,560 --> 00:05:46,780
+magnitude اللي هو، بعينكم الله تحفظوه من العلاقات
+
+81
+00:05:46,780 --> 00:05:50,420
+هذه، where T تساوي V على الـ magnitude لـ V، اللي هو الـ
+
+82
+00:05:50,420 --> 00:05:56,600
+unit tangent vector، زي ما حكينا سابقاً. نأتي الآن
+
+83
+00:05:56,600 --> 00:06:02,420
+نأخذ مثالاً، نأخذ مثالاً لنرى ما هي منطقية كلامنا
+
+84
+00:06:02,420 --> 00:06:06,770
+الأولاني على مثال سهل أن نعرفه، اللي هو a straight
+
+85
+00:06:06,770 --> 00:06:09,990
+line. عندي straight line is parameterized by، يعني
+
+86
+00:06:09,990 --> 00:06:14,330
+عملنا له اللي هو اللي هو parametric equation أو
+
+87
+00:06:14,330 --> 00:06:18,090
+position vector.  وسط الـ position vector R of T
+
+88
+00:06:18,090 --> 00:06:24,750
+يساوي C زائد T V، حيث الـ T هي الزمن المتغير، والـ C
+
+89
+00:06:24,750 --> 00:06:28,850
+و V عبارة عن constant vectors، حيث الـ T اللي هو
+
+90
+00:06:28,850 --> 00:06:33,990
+مين؟ هذا اللي هو الخط اللي برسمه مين؟ الـ R of T
+
+91
+00:06:33,990 --> 00:06:39,010
+اللي أمامي. هذا الخط، خط مستقيم رسمه اللي هو الـ
+
+92
+00:06:39,010 --> 00:06:42,430
+position vector. هذا اللي هو معادلة الخط المستقيم
+
+93
+00:06:42,430 --> 00:06:45,590
+لأنه هذه ثابتة، وهذه ثابتة، وأيضاً، والـ T هو
+
+94
+00:06:45,590 --> 00:06:50,310
+البرامتر. طبعاً الناس بتقول: طب كيف هي المعادلة دي؟ هذه
+
+95
+00:06:50,310 --> 00:06:57,030
+قيمتها اللي هي عندي. هذا عبارة عن الـ position vector
+
+96
+00:06:57,030 --> 00:07:07,200
+اللي، هذا ثابت، وهذا ثابت، وهذا ثابت، وهذا ثابت، و
+
+97
+00:07:07,200 --> 00:07:11,620
+هذا ثابت، وهذا ثابت، وهذا ثابت، وهذا ثابت، وهذا
+
+98
+00:07:11,620 --> 00:07:12,640
+ثابت، وهذا ثابت، وهذا ثابت، وهذا ثابت، وهذا ثابت، و
+
+99
+00:07:12,640 --> 00:07:13,460
+ثابت، وهذا ثابت، وهذا ثابت، وهذا ثابت، وهذا ثابت، و
+
+100
+00:07:13,460 --> 00:07:14,440
+هذا ثابت، وهذا ثابت، وهذا ثابت، وهذا ثابت، وهذا
+
+101
+00:07:14,440 --> 00:07:24,160
+هذا ثابت، وهذا ثابت، وهذا ثابت، وهذا ثابت، و
+
+102
+00:07:24,160 --> 00:07:28,880
+هذا ثابت. بصير لي X بالـ I عبارة عن اللي هو
+
+103
+00:07:28,880 --> 00:07:32,740
+متغير، هنا مع الثابت، هنا، والمتغير هنا مع الثابت
+
+104
+00:07:32,740 --> 00:07:35,960
+اللي هنا. العلاقة بينهم هتكون علاقة خطية. أجربها، و
+
+105
+00:07:35,960 --> 00:07:39,860
+أفردها، وهتلاقيها هيك. يعني أعطيها ده ثابت، وهذا ثابت
+
+106
+00:07:39,860 --> 00:07:44,420
+في اتجاه I وJ، وهذا ثابت في اتجاه I وJ.  وجِد العلاقة بين X وY
+
+107
+00:07:44,420 --> 00:07:47,800
+في الجهتين، المتضروبة ده، هتلاقيها علاقة خطية. طيب
+
+108
+00:07:47,800 --> 00:07:51,320
+شوفوا، صلى الله عليه الصلاة والسلام. الآن بدنا نوجد
+
+109
+00:07:51,320 --> 00:07:54,520
+كابا. كابا زي ما اتفقنا، أول شيء على طول بالتفاضل
+
+110
+00:07:54,520 --> 00:07:57,960
+بالنسبة لـ T، R prime of T بتطلع إيش؟  شمالها اللي هي
+
+111
+00:07:57,960 --> 00:08:01,580
+الـ vector هذا V، مش الـ velocity، هذا الـ vector V، لكن
+
+112
+00:08:01,580 --> 00:08:04,920
+هي قدر إن بتطلع إيش؟ الـ R prime of T، اللي هي الـ
+
+113
+00:08:04,920 --> 00:08:09,370
+velocity V. الآن هذا عبارة عن vector ثابت. ماشي الحال.
+
+114
+00:08:09,370 --> 00:08:14,610
+الآن الـ unit tangent vector T يساوي الـ V، اللي هو
+
+115
+00:08:14,610 --> 00:08:17,630
+هذا في هذه الحالة، على الـ magnitude لـ V، اللي هو
+
+116
+00:08:17,630 --> 00:08:22,950
+magnitude لهذا. هذه كمية ثابتة كلها. مادام ثابتة، إذا
+
+117
+00:08:22,950 --> 00:08:30,010
+dT/dT إيش هيطلع؟ اللي بيساوي صفر. إذا أنا الآن
+
+118
+00:08:30,010 --> 00:08:34,430
+صارت جاهزة عندي، الكابا تساوي واحد على الـ magnitude لـ
+
+119
+00:08:34,430 --> 00:08:38,500
+V في dT/dT. هذا القانون، واحد على الـ
+
+120
+00:08:38,500 --> 00:08:43,860
+magnitude للـ V. هي الـ V هنا، الـ velocity
+
+121
+00:08:43,860 --> 00:08:47,200
+هي الـ vector اللي هنا، الثابت. المقدار ثابت، مهم ليش؟
+
+122
+00:08:47,200 --> 00:08:50,620
+مش مشكلة إن ما يكون. أما هو بيطلع مقدار ثابت، dT/
+
+123
+00:08:50,620 --> 00:08:54,340
+dT اللي أوجدناه، إيش بتساوي؟ صفر. فطلع الـ
+
+124
+00:08:54,340 --> 00:08:58,260
+curvature إيش بتساوي؟ صفر. طبيعي اللي هو فش إيش؟ الخط
+
+125
+00:08:58,260 --> 00:09:01,500
+المستقيم فش إيش؟ انحناء.  شيء ما ليس له
+
+126
+00:09:01,500 --> 00:09:07,180
+انحناء. إذا لازم يتوافق القانون مع المفهوم
+
+127
+00:09:07,180 --> 00:09:12,840
+الفيزيائي، إنه هذا ما ليس له انحناء، يعني الخط المستقيم
+
+128
+00:09:12,840 --> 00:09:16,700
+ما ليس له انحناء. إذا كابا إله لازم تطلع صفر، وإلا كان
+
+129
+00:09:16,700 --> 00:09:22,220
+تعريفنا الأصلي لكابا بيكونش صحيح. طيب الآن نيجي نأخذ
+
+130
+00:09:22,220 --> 00:09:28,720
+مثالاً آخر. عندي الآن دائرة. Here we find a curvature
+
+131
+00:09:28,720 --> 00:09:33,360
+of a circle. نبدأ بـ Parameterization R of T يساوي
+
+132
+00:09:33,360 --> 00:09:39,520
+A Cos T في اتجاه الـ I، زائد A Sin T في اتجاه الـ J. أكيد كلكم عارفين أن
+
+133
+00:09:39,520 --> 00:09:46,180
+هذه هي دائرة. لأن هذه X، وهذه Y، X = A Cos T، و Y = A Sin
+
+134
+00:09:46,180 --> 00:09:53,100
+T. X تربيع زائد Y تربيع بيصير A تربيع. بصير X تربيع
+
+135
+00:09:53,100 --> 00:09:56,780
+زائد Y تربيع يساوي A تربيع.  A تربيع يساوي X
+
+136
+00:09:56,780 --> 00:10:03,760
+تربيع زائد Y تربيع يساوي
+
+137
+00:10:05,650 --> 00:10:11,610
+الآن هذه عبارة عن circle، نصف قطرها إيش بيساوي؟ بيساوي الـ
+
+138
+00:10:11,610 --> 00:10:16,050
+A. الآن بدنا نوجد مدى انحناء الدائرة. هاي الدائرة
+
+139
+00:10:16,050 --> 00:10:20,570
+عندي، بدنا نشوف مدى إيش انحنائها، قد إيش انحنائها، قد إيش
+
+140
+00:10:20,570 --> 00:10:25,890
+اللي هو انحناء الـ curvature إلها. نشوف الآن. Then
+
+141
+00:10:28,360 --> 00:10:33,460
+على طول بنعمل تفاضل الـ R، V تساوي dR/dT، ناقص A
+
+142
+00:10:33,460 --> 00:10:41,440
+Sine T، و A Cos T.  المتجه V، لإيجاد T،  مطلوب
+
+143
+00:10:41,440 --> 00:10:45,430
+لإيجاد الكابا.  يساوي الجذر التربيعي لهذا تربيع، زائد
+
+144
+00:10:45,430 --> 00:10:48,590
+هذا تربيع. وهذا زي ما أنتم عارفين، A تربيع Sine
+
+145
+00:10:48,590 --> 00:10:51,850
+تربيع، زائد A تربيع Cosine تربيع. مع بعض بيطلع من A تربيع
+
+146
+00:10:51,850 --> 00:10:54,910
+اللي هو magnitude للـ A، A يساوي A، لأنه الـ A احنا
+
+147
+00:10:54,910 --> 00:11:00,150
+ماخذينها A، شمالها عبارة عن أكبر من صفر. ماشي، فاهمين؟
+
+148
+00:11:00,150 --> 00:11:04,150
+From this we find، من هنا،  muted اللي هو T يساوي V
+
+149
+00:11:04,150 --> 00:11:09,870
+على V. هي الـ V، وهي اللي هو مين؟ اللي هي الـ magnitude
+
+150
+00:11:09,870 --> 00:11:13,510
+للـ V.  بنجيب هذه، تطلع ناقص Sin T، لأنها بتروح الـ A والـ
+
+151
+00:11:13,510 --> 00:11:16,810
+A مع الـ A اللي تحت، زائد Cosine T في اتجاه الـ J.
+
+152
+00:11:16,810 --> 00:11:21,230
+هذا هو مين؟ الـ T. لأن بعد الـ T، بالتفاضل الـ T بالنسبة
+
+153
+00:11:21,230 --> 00:11:24,870
+للـ T، عشان نوجد الـ curvature، لأن dT/dT يساوي
+
+154
+00:11:24,870 --> 00:11:27,810
+بالفاضل هذه، بتطلع ناقص Cosine T في اتجاه الـ I، ناقص
+
+155
+00:11:27,810 --> 00:11:33,190
+Sin T في اتجاه الـ J. لأن الـ magnitude اللي هو عبارة
+
+156
+00:11:33,190 --> 00:11:35,630
+عن جذر التربيعي لهذا تربيع، زائد هذا تربيع، اللي
+
+157
+00:11:35,630 --> 00:11:39,700
+هو إيش بيطلع يا جماعة؟ واحد. إذا على طول الكابا
+
+158
+00:11:39,700 --> 00:11:43,580
+تبعنا اللي بتساوي واحد على الـ magnitude of V في الـ
+
+159
+00:11:43,580 --> 00:11:47,760
+magnitude of dT/dT. هذا أوجدناه، وهذا أوجدناه، وهذا
+
+160
+00:11:47,760 --> 00:11:50,780
+أوجدناه. إيش بيساوي؟ واحد. اللي هي الـ magnitude
+
+161
+00:11:50,780 --> 00:11:53,500
+of V، اللي يصير واحد على A في واحد، ويساوي واحد
+
+162
+00:11:53,500 --> 00:11:57,880
+على A. ويساوي كابا. إذا كابا للدائرة طلعت عبارة
+
+163
+00:11:57,880 --> 00:12:03,140
+عن مقدار ثابت، ومنطقي. لإن وين ما تروح على الدائرة،
+
+164
+00:12:03,140 --> 00:12:09,310
+لها نفس الانحناء. إذا فعلاً كابا تعبر عن الـ
+
+165
+00:12:09,310 --> 00:12:13,590
+curvature أو مدى انحناء المنحنى. وفي حالة الدائرة،
+
+166
+00:12:13,590 --> 00:12:17,150
+هي عبارة عن مقدار ثابت، اللي هو يساوي واحد على
+
+167
+00:12:17,150 --> 00:12:22,090
+نصف قطر
+
+223
+00:16:04,110 --> 00:16:08,930
+عمودي على الـ T و بوصف لي مع الـ T حركة إيش ال particle
+
+224
+00:16:08,930 --> 00:16:14,210
+اللي هو ده بيرجيني وين اللي هو مركز اللي هو الحركة
+
+225
+00:16:14,210 --> 00:16:18,760
+تبعت اللي هي الـ particle اللي عندي، إذا صار عندي two
+
+226
+00:16:18,760 --> 00:16:23,000
+unit vectors، واحد اسمه unit tangent vector والآخر
+
+227
+00:16:23,000 --> 00:16:27,620
+اسمه unit principal unit normal vector الآن
+
+228
+00:16:27,620 --> 00:16:33,440
+يا جماعة، عندي اللي هو إيجاد الـ N بالطريقة اللي قبل
+
+229
+00:16:33,440 --> 00:16:39,520
+بشوية بدلالة S صعبة ومش هتكون مباشرة، لإنه عادة
+
+230
+00:16:39,520 --> 00:16:43,240
+بيعطونا الـ R of T، عشان ذلك بدي أجيب كل الحكي هذا
+
+231
+00:16:43,240 --> 00:16:48,120
+بدلالة T، كابا وجدناها بدلالة T، ونفس الفكرة
+
+232
+00:16:48,120 --> 00:16:52,700
+السابقة اللي قصدت لها chain rule، أن بساوي DT by DS
+
+233
+00:16:52,700 --> 00:16:57,520
+على كابا، DT by DS magnitude اللي حكينا عنها الآن
+
+234
+00:16:57,520 --> 00:17:02,660
+هذا بالـ chain rule، هذا DT by DT في DT by DS اللي
+
+235
+00:17:02,660 --> 00:17:05,340
+تحت برضه بنفس الطريقة للـ chain rule، magnitude في الـ
+
+236
+00:17:05,340 --> 00:17:09,930
+magnitude، الآن دي تي باي دي اس، احنا خدنا اس في
+
+237
+00:17:09,930 --> 00:17:12,670
+اتجاه الحركة قبل هيك، اللي هو اللي بنمشي بالاتجاه
+
+238
+00:17:12,670 --> 00:17:15,950
+الموجب، النتجهين يا موجب يا سالب، عن اتجاهنا الموجب
+
+239
+00:17:15,950 --> 00:17:19,330
+اللي بنمشي فيه، مدام الموجب دي تي باي دي اس هيكون
+
+240
+00:17:19,330 --> 00:17:23,550
+أكبر من صفر، يعني هذا عدد وهذا عدد متساويين، إذاً
+
+241
+00:17:23,550 --> 00:17:27,750
+هذا بروح مع هذا، بظل عندي الـ DT by DT، المجلتود للـ
+
+242
+00:17:27,750 --> 00:17:33,850
+DT by DT، إذاً الآن الـ unit principal unit normal
+
+243
+00:17:33,850 --> 00:17:38,390
+vector اللي هو الـ N، يمكن إيجاده بدلالة اللي هي الـ
+
+244
+00:17:38,390 --> 00:17:45,430
+T، الـ T هذا يصبح D T by DT على DT by DT، إذاً الجثة
+
+245
+00:17:45,430 --> 00:17:52,430
+من R of T نحضر V، ومن V نحضر T، ومن T نحضر N، ونحصل
+
+246
+00:17:52,430 --> 00:17:57,130
+على Unit Tangent Vector و Unit Principle Normal Vector بس
+
+247
+00:17:57,130 --> 00:18:03,350
+بتفاضلات للـ R أكثر من مرة، بنحصل حسب القوانين اللي
+
+248
+00:18:03,350 --> 00:18:08,090
+عندنا الآن، ناخد مثال اللي هو على الموضوع
+
+249
+00:18:11,740 --> 00:18:17,200
+Find T and N for the circular motion R of T، بساوي
+
+250
+00:18:17,200 --> 00:18:23,300
+Cos 2T زائد Sin 2T في اتجاه الـ I، والاتجاهية
+
+251
+00:18:23,300 --> 00:18:23,720
+(Repeated word removed)
+
+252
+00:18:23,720 --> 00:18:24,760
+(Repeated word removed)
+
+253
+00:18:24,760 --> 00:18:25,460
+(Repeated word removed)
+
+254
+00:18:25,460 --> 00:18:28,220
+(Repeated word removed)
+
+255
+00:18:28,220 --> 00:18:29,960
+(Repeated word removed)
+
+256
+00:18:29,960 --> 00:18:30,000
+(Repeated word removed)
+
+257
+00:18:30,000 --> 00:18:34,860
+(Repeated word removed)
+
+258
+00:18:34,860 --> 00:18:40,340
+في اتجاه الـ J، عشان نجد الـ T، نجد طبعاً الـ V
+
+259
+00:18:40,340 --> 00:18:45,260
+بنفاضل اللي هي R of T، بيصير عبارة عن ناقص 2Sin 2T
+
+260
+00:18:45,260 --> 00:18:51,120
+و بنفاضل هذه، بيصير 2Cos 2T في اتجاه الـ J، المجموعه
+
+261
+00:18:51,120 --> 00:18:54,040
+للـ V هي عبارة عن جذر التربيع لهذه التربيع زائد
+
+262
+00:18:54,040 --> 00:19:00,160
+هذه التربيع، اللي هي 4Sin²2T + 4Cos²2T، هذه مع هذه واحد
+
+263
+00:19:00,160 --> 00:19:02,520
+والأربعة تكون على المشترك، فبيصير جذر الأربعة، إيش
+
+264
+00:19:02,520 --> 00:19:05,910
+بيساوي اثنين، الآن الـ magnitude للـ V جاهز، والـ V
+
+265
+00:19:05,910 --> 00:19:08,290
+جاهز، إذاً الآن الـ T جاهزة، T بيساوي الـ V على الـ
+
+266
+00:19:08,290 --> 00:19:11,190
+magnitude للـ V، هي الـ T، نجسمها على الـ magnitude
+
+267
+00:19:11,190 --> 00:19:14,510
+للـ V اللي هو الاثنين، اللي بيطلع عندي المقدار اللي
+
+268
+00:19:14,510 --> 00:19:19,250
+أنا أني الآن، if from we find اللي هو مين؟ اللي هي الـ
+
+269
+00:19:19,250 --> 00:19:23,030
+N، الـ N إيش؟ هي عبارة عن قولنا DT by DT على الـ
+
+270
+00:19:23,030 --> 00:19:26,590
+magnitude للـ DT by DT، إذاً الأمر سهل، بنفاضل هذه
+
+271
+00:19:26,590 --> 00:19:30,350
+بالنسبة لـ T، بيصير عبارة عن ناقص اثنين كوساين اثنين
+
+272
+00:19:30,350 --> 00:19:34,950
+T في اتجاه الـ I، ناقص اثنين ساين 2t في اتجاه الـ
+
+273
+00:19:34,950 --> 00:19:39,150
+J، على جذر التربيعي لجذر التربيعي من
+
+274
+00:19:39,150 --> 00:19:39,450
+(Repeated phrase removed)
+
+275
+00:19:39,450 --> 00:19:40,050
+(Repeated phrase removed)
+
+276
+00:19:40,050 --> 00:19:41,710
+(Repeated phrase removed)
+
+277
+00:19:41,710 --> 00:19:41,770
+(Repeated phrase removed)
+
+278
+00:19:41,770 --> 00:19:44,590
+(Repeated phrase removed)
+
+279
+00:19:44,590 --> 00:19:44,990
+(Repeated phrase removed)
+
+280
+00:19:44,990 --> 00:19:45,490
+(Repeated phrase removed)
+
+281
+00:19:45,490 --> 00:19:51,210
+(Repeated phrase removed)
+
+282
+00:20:00,460 --> 00:20:05,040
+هذا هو عمودي على الـ T، أو اثنين، أو أورثوجونال، أو
+
+283
+00:20:05,040 --> 00:20:05,180
+(Repeated phrase removed)
+
+284
+00:20:05,180 --> 00:20:06,140
+(Repeated phrase removed)
+
+285
+00:20:06,140 --> 00:20:08,640
+(Repeated phrase removed)
+
+286
+00:20:08,640 --> 00:20:09,020
+(Repeated phrase removed)
+
+287
+00:20:09,020 --> 00:20:09,140
+(Repeated phrase removed)
+
+288
+00:20:09,140 --> 00:20:12,220
+(Repeated phrase removed)
+
+289
+00:20:12,220 --> 00:20:16,100
+(Repeated phrase removed)
+
+290
+00:20:20,080 --> 00:20:24,840
+أيضاً، لاحظوا معي أن for the circular motion here، N
+
+291
+00:20:24,840 --> 00:20:29,000
+points for R of T towards the circle center، يعني الـ N
+
+292
+00:20:29,000 --> 00:20:35,440
+هي الدائرة مثلاً، وهي عندي المنحنى هيك، هي اللي عند الـ
+
+293
+00:20:35,440 --> 00:20:38,420
+unit tangent vector، هيك بيكون الـ normal عليه
+
+294
+00:20:38,420 --> 00:20:43,400
+باتجاه مين؟ اللي هو الـ center للدائرة اللي بنحكي
+
+295
+00:20:43,400 --> 00:20:48,940
+عنها، يعني الآن الـ normal طبيعي هيكون متجه باتجاه
+
+296
+00:20:48,940 --> 00:20:54,920
+مين؟ الـ center تبع الدائرة اللي احنا ماشيين عليها، و
+
+297
+00:20:54,920 --> 00:20:59,480
+زي ما قلنا، بصورة عامة الآن، هيكون متجه باتجاه اللي
+
+298
+00:20:59,480 --> 00:21:05,840
+هو داخل المنحنى، والـ T بيكون مماس للمنحنى، زي ما
+
+299
+00:21:05,840 --> 00:21:14,020
+عرفنا، ندخل الآن اللي هو ناخد اللي هو عندي مفهوم
+
+300
+00:21:14,020 --> 00:21:18,280
+جديد، اللي هو حاجة اسمها circle of curvature for
+
+301
+00:21:18,280 --> 00:21:22,060
+the plane or plane curves، الـ circle of curvature
+
+302
+00:21:22,060 --> 00:21:28,650
+أو دائرة الانحناء لمنحنى، دائرة انحناء لمنحنى الآن
+
+303
+00:21:28,650 --> 00:21:33,930
+عندي منحنى، هي عندي منحنى آه، هذا المنحنى مرسوم بـ R
+
+304
+00:21:33,930 --> 00:21:38,010
+of D، الـ vector اسمه R of D، اله position vector R
+
+305
+00:21:38,010 --> 00:21:43,130
+of D، أو حركة particle، الآن هذا المنحنى بدي أعرف
+
+306
+00:21:43,130 --> 00:21:47,590
+إيش معنى اللي هو circle of curvature أو الـ
+
+307
+00:21:47,590 --> 00:21:52,310
+circulating circle عند النقطة P على المنحنى الأزرق
+
+308
+00:21:52,310 --> 00:21:57,000
+هذا، الدائرة اللي عند الـ circle of curvature أو
+
+309
+00:21:57,000 --> 00:22:04,080
+دائرة اللي بنعرفها، كمّالها هي الدائرة اللي بتكون بتمسّ
+
+310
+00:22:04,080 --> 00:22:08,580
+المنحنى، بتمسّ، النقطة هذه، بتمسّها مسّ، يعني بتمسّ
+
+311
+00:22:08,580 --> 00:22:12,220
+المنحنى عند هذه النقطة، هذه المنحنى بتمسّه عند هذه
+
+312
+00:22:12,220 --> 00:22:15,740
+النقطة اللي هي مين؟ الـ circle of curvature، إذاً الـ
+
+313
+00:22:15,740 --> 00:22:18,620
+circle of curvature أو circulating circle عند
+
+314
+00:22:18,620 --> 00:22:23,360
+النقطة B، طبعاً الـ curvature له مش صفر، عنده curvature
+
+315
+00:22:23,360 --> 00:22:27,950
+صفر، بتتعرفش تعريف، إلا أو لا يوجد قيمة لها، is the
+
+316
+00:22:27,950 --> 00:22:31,530
+circle in the plane of the curve that، هي عبارة عن
+
+317
+00:22:31,530 --> 00:22:35,810
+(Repeated phrase removed)
+
+318
+00:22:35,810 --> 00:22:40,030
+ما يلي، نشوف أول شيء، is tangent to the curve at B
+
+319
+00:22:40,030 --> 00:22:43,230
+يعني بتكون هذه الدائرة إيش مالها؟ بتمسّ الكيرف هذا
+
+320
+00:22:43,230 --> 00:22:47,770
+عنده نقطة B، اثنين، has the same curvature the curve
+
+321
+00:22:47,770 --> 00:22:53,250
+has at B، يعني الآن الكيرفيتشَر للـدائرة هو نفس الـ
+
+322
+00:22:53,250 --> 00:22:56,510
+curvature تبع المنحنى، يعني الآن بودي الـ curvature
+
+323
+00:22:56,510 --> 00:23:01,710
+للمنحنى عند نقطة B، بيكون هو الـ
+
+324
+00:23:01,710 --> 00:23:04,990
+curvature للـ circle of curvature، وزي ما عرفنا قبل شوية
+
+325
+00:23:04,990 --> 00:23:08,990
+إذا عرفنا الـ curvature للـ circle، بنعرف نصف
+
+326
+00:23:08,990 --> 00:23:12,810
+قطرها، لأن الـ curvature للـ circle بيساوي واحد على
+
+327
+00:23:12,810 --> 00:23:18,570
+نصف القطر، القطر، يعني لما نعرف الـ curvature للمنحنى
+
+328
+00:23:18,570 --> 00:23:23,380
+بيكون هو الـ curvature للـ circle of curvature، اللي هو
+
+329
+00:23:23,380 --> 00:23:28,420
+عبارة عن واحد على إيش؟ يعني مقلوبه هيكون هو عبارة
+
+330
+00:23:28,420 --> 00:23:33,700
+عن الـ radius للدائرة، اللي هو اللي بنسميه الـ radius
+
+331
+00:23:33,700 --> 00:23:39,140
+of curvature للمنحنى، إذاً الـ radius of curvature
+
+332
+00:23:39,140 --> 00:23:43,020
+للمنحنى عند نقطة هو عبارة عن الـ radius of
+
+333
+00:23:43,020 --> 00:23:51,410
+curvature لمين؟ للـ اللي هي الدائرة اللي عنده، الآن كما
+
+334
+00:23:51,410 --> 00:23:56,350
+قلنا، has the same curvature the curve has at B
+
+335
+00:23:56,350 --> 00:24:00,450
+lies toward the concave or inner side of the curve
+
+336
+00:24:00,450 --> 00:24:04,370
+يعني الدائرة بنرسمها وين؟ باتجاه اللي هو داخل
+
+337
+00:24:04,370 --> 00:24:09,090
+المنحنى، باتجاه الانحناء الداخلي لمين؟ للمنحنى، إذاً
+
+338
+00:24:09,090 --> 00:24:13,510
+الـ circle of curvature باختصار، هي بتمسّ عند نقطة B
+
+339
+00:24:13,510 --> 00:24:19,720
+للمنحنى، هي بتمسّه عند هذه النقطة، هو نفس الـ curvature
+
+340
+00:24:19,720 --> 00:24:24,680
+للمنحنى عند هذه النقطة، وبتكون الدائرة في الداخل، أو
+
+341
+00:24:24,680 --> 00:24:28,900
+اللي هو باتجاه الانحناء تبع المنحنى، الآن الـ radius
+
+342
+00:24:28,900 --> 00:24:31,820
+of curvature of a curve at B is radius of the
+
+343
+00:24:31,820 --> 00:24:37,410
+circle of curvature، وهو الـ Radius of curvature
+
+344
+00:24:37,410 --> 00:24:42,310
+بيساوي واحد على كابا، لأن وجدنا أن دائماً الـ
+
+345
+00:24:42,310 --> 00:24:51,870
+Radius of curvature بيساوي واحد على، Radius الدائرة
+
+346
+00:24:55,480 --> 00:25:00,560
+عبارة عن واحد على كابا، الـ Center of curvature of
+
+347
+00:25:00,560 --> 00:25:03,900
+the curve، يعني الآن الـ Center of curvature للكيرف
+
+348
+00:25:03,900 --> 00:25:08,020
+اللي عندي هذا، الـ Center of curvature هو عبارة عن
+
+349
+00:25:08,020 --> 00:25:13,060
+الـ Center للدائرة of curvature، آه آه، يعني الـ
+
+350
+00:25:13,060 --> 00:25:16,460
+Center of curvature لهالـ Center of curvature
+
+351
+00:25:16,460 --> 00:25:21,320
+للمنحنى عند نقطة B، هو عبارة عن مركز دائرة
+
+352
+00:25:21,320 --> 00:25:25,820
+الانحناء، يعني is the center of the circle of
+
+353
+00:25:25,820 --> 00:25:30,420
+curvature، وهذا الأمر واضح تماماً، نجي الآن ناخد مثال
+
+354
+00:25:30,420 --> 00:25:33,960
+find and graph the escalating circle of the
+
+355
+00:25:33,960 --> 00:25:36,760
+parabola Y بيساوي X تربيعية عند الأورجين
+
+356
+00:25:40,880 --> 00:25:45,640
+ومعادلة الدائرة الـ Escalating Circle للبرابولا Y
+
+357
+00:25:45,640 --> 00:25:49,680
+بيساوي X تربيع، هذه عند البرابولا Y بيساوي X تربيع
+
+358
+00:25:49,680 --> 00:25:55,600
+بدنا نوجد عند النقطة صفر، اللي هو إيش؟ اللي هي الـ
+
+359
+00:25:55,600 --> 00:25:58,900
+Circle of Curvature أو الـ Oscillating Circle، بدنا
+
+360
+00:25:58,900 --> 00:26:01,800
+نوجد هذه الـ Oscillating Circle ونرسمها زي ما هو
+
+361
+00:26:01,800 --> 00:26:05,520
+موجود، الآن Y بيساوي X تربيع، هذا مش مكتوب على صورة
+
+362
+00:26:05,520 --> 00:26:10,740
+R of T، اللي هو Parametric Equation، هي سهلة، نجد
+
+363
+00:26:10,740 --> 00:26:16,060
+علاقة بين Y و X تربيع، أي نقطة هنا إحداثيتها X أو X
+
+364
+00:26:16,060 --> 00:26:23,060
+تربيع، X لـ X و X تربيع لـ Y، يعني بمعنى آخر لو سمينا
+
+365
+00:26:23,060 --> 00:26:29,220
+X تساوي T، يصبح Y تساوي T تربيع، يعني بمعنى آخر بقدر
+
+366
+00:26:29,220 --> 00:26:33,940
+أقول أن الـ R of T تبع البارابولا، بعمل بارامترزيشن
+
+367
+00:26:33,940 --> 00:26:38,800
+للبارابولا، اللي هو عبارة عن إحداثية X في اتجاه I، زي X
+
+368
+00:26:38,800 --> 00:26:41,840
+تربيع في اتجاه J، التي هي قيمة الـ Y، يعني بمعنى آخر
+
+369
+00:26:41,840 --> 00:26:46,480
+بدلالة T، T في اتجاه I، زي T تربيع في اتجاه J، هذه اللي
+
+370
+00:26:46,480 --> 00:26:51,900
+هي الـ Parametric Equation أو اللي هو بيعملنا
+
+371
+00:26:51,900 --> 00:26:56,140
+Parameterization للبارابولا، ببساطة من اللي هو الـ R of
+
+372
+00:26:56,140 --> 00:26:59,440
+T، position vector، بتعبّر عن حركة اللي هو الـ
+
+373
+00:26:59,440 --> 00:27:03,230
+particle على البارابولا، أولاً نجد أننا نريد أن نوجد
+
+374
+00:27:03,230 --> 00:27:04,230
+(Repeated phrase removed)
+
+375
+00:27:04,230 --> 00:27:04,430
+(Repeated phrase removed)
+
+376
+00:27:04,430 --> 00:27:07,190
+(Repeated phrase removed)
+
+377
+00:27:07,190 --> 00:27:10,050
+(Repeated phrase removed)
+
+378
+00:27:10,050 --> 00:27:13,070
+(Repeated phrase removed)
+
+379
+00:27:13,070 --> 00:27:13,530
+(Repeated phrase removed)
+
+380
+00:27:13,530 --> 00:27:24,470
+(Repeated phrase removed)
+
+381
+00:27:24,470 --> 00:27:30,250
+(Repeated phrase removed)
+
+382
+00:27:30,250 --> 00:27:33,970
+نريد أن نوجد، الآن نجي أول شيء بالفاضل، أن الأرض تطلع
+
+383
+00:27:33,970 --> 00:27:37,930
+V بيساوي DR by DT، اللي هي واحد، زي اثنين T، الـ
+
+384
+00:27:37,930 --> 00:27:40,750
+magnitude لـ V هو عبارة عن جذر التربيع لواحد زائد
+
+385
+00:27:40,750 --> 00:27:44,190
+أربعة T تربيع، حسابات سهلة، الآن اللي أنت مش بدّه
+
+386
+00:27:44,190 --> 00:27:47,190
+توجده، و هيّلزمّني عشان أوجد الكابا، اللي هو الـ T، الـ T
+
+387
+00:27:47,190 --> 00:27:50,750
+بيساوي الـ V على الـ magnitude لمين؟ للـ V، الآن هي الـ
+
+388
+00:27:50,750 --> 00:27:53,850
+V، وهي الـ magnitude لـ V، نجسمهم، بيطلع عندي هذا
+
+389
+00:27:53,850 --> 00:27:58,290
+المقدار، الآن هذا قيمة مين؟ قيمة T، بعينكم الله، بدنا
+
+390
+00:27:58,290 --> 00:28:02,250
+نفاضل هذا، بالنسبة لمين؟ للـ T، بنفاضل هذا المقدار
+
+391
+00:28:02,250 --> 00:28:06,270
+بالنسبة لـ T، تفاضل عادي، اللي هو عبارة عن اللي هو
+
+392
+00:28:06,270 --> 00:28:11,310
+ناقص نص، بيصير الجذر عبارة عن ناقص 3 على 2، وتفاضل مافي
+
+393
+00:28:11,310 --> 00:28:15,230
+داخل الجذر اللي هو 8T، 8T في ناقص 6، بيطلع اللي هو ناقص
+
+394
+00:28:15,230 --> 00:28:19,630
+4T في هذا المقدار، اللي هو بالفاضل اللي هو المقدار
+
+395
+00:28:19,630 --> 00:28:24,290
+الثاني، حاصل ضرب مقدارين، 2T في هذا، بصير عبارة عن 2
+
+396
+00:28:24,290 --> 00:28:28,090
+في هذا المقدار زائد، كل مضروب في J ط
+
+445
+00:32:00,790 --> 00:32:05,530
+إذا الآن من الـ normal جيه؟ هاي جيه إذا لازم اللي
+
+446
+00:32:05,530 --> 00:32:11,930
+هو المركز يقع على من على الـ normal هذا. طيب المركز
+
+447
+00:32:11,930 --> 00:32:18,610
+بيقع على الـ normal. جدّيش قيمة اللي هو اللي نصف
+
+448
+00:32:18,610 --> 00:32:24,410
+القطر كان اللي هو نصف. مدام نصف إذا بده يطلع عندي
+
+449
+00:32:24,410 --> 00:32:29,270
+اللي هو بمشي من هان لهان في اتجاه الـ normal بمقدار
+
+450
+00:32:29,270 --> 00:32:33,690
+نصف بيطلع بالظبط هو المركز لأن المركز عالمين على
+
+451
+00:32:33,690 --> 00:32:37,130
+الـ normal اللي هو في هذه الحالة إيش بيساوي؟ الآن
+
+452
+00:32:37,130 --> 00:32:45,730
+مركز الدائرة عبارة عن سفر OJ وهذا يحدد أن الدائرة
+
+453
+00:32:45,730 --> 00:32:50,630
+تماما صار معروف مركزها ومعروف نصف قطرها فبيصير
+
+454
+00:32:50,630 --> 00:32:56,610
+معادلتها X التي هي 0.5 المركز، وY التي هي نصف القطر
+
+455
+00:32:56,610 --> 00:33:02,310
+نصف، فبيصير X-0 لكل تربيع زائد Y-0.5 لكل تربيع هذه
+
+456
+00:33:02,310 --> 00:33:06,710
+معادلة الدائرة اللي هي لمركزها 0.5 ونصف قطرها اللي
+
+457
+00:33:06,710 --> 00:33:11,510
+هو 0.5 لكل إيش تربيع؟ هذه عبارة عن الـ equation of
+
+458
+00:33:11,510 --> 00:33:17,170
+the escalating circle المطلوبة وهي رسمتها اللي هي
+
+459
+00:33:17,170 --> 00:33:20,830
+الـ circle of curvature أو اللي هي الـ escalating
+
+460
+00:33:20,830 --> 00:33:24,840
+circle. الآن بدنا نحكي عن اللي حكيناه قبل شوية في
+
+461
+00:33:24,840 --> 00:33:27,740
+الـ ... الـ ... الـ curvature. الـ normal vectors for
+
+462
+00:33:27,740 --> 00:33:32,720
+space for space curves. لمين؟ بس في وين؟ في الـ space
+
+463
+00:33:32,720 --> 00:33:35,940
+بدل ما هو في وين؟ في لبنان. كل الحكي اللي قبل كان في
+
+464
+00:33:35,940 --> 00:33:40,120
+لبنان. سهل الموضوع ما فيش فيه شيء جديد. يعني الآن لو
+
+465
+00:33:40,120 --> 00:33:43,220
+كان عندي اللي هو الـ smooth curve اللي عندي اللي هو
+
+466
+00:33:43,220 --> 00:33:46,810
+اللي ... اللي برسمه الـ position vector R of T اللي
+
+467
+00:33:46,810 --> 00:33:50,490
+هو as a function of T في الفرق A واسفر اللي هو
+
+468
+00:33:50,490 --> 00:33:53,010
+الـ R clipped زي ما احنا عارفين اللي هو برضه اللي
+
+469
+00:33:53,010 --> 00:33:56,850
+هو الـ R of T للكيرف اللي برسمه R of T. الآن الـ
+
+470
+00:33:56,850 --> 00:34:00,450
+unit tangent vector T اللي هو عرفناها زمان DR by
+
+471
+00:34:00,450 --> 00:34:02,610
+DS وقلنا أن هو عبارة عن الـ V على الـ magnitude
+
+472
+00:34:02,610 --> 00:34:06,610
+للـ V. الكيرفيتشر في الـ space برضه بدعرفه بنفس
+
+473
+00:34:06,610 --> 00:34:10,710
+الأسلوب. Kappa إيش بتساوي؟ DT على DS magnitude اللي
+
+474
+00:34:10,710 --> 00:34:15,200
+هو أوجدناه عن طريق T وهو واحدّة  مجموعة لـ V في DT
+
+475
+00:34:15,200 --> 00:34:18,040
+بي DT أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر
+
+476
+00:34:18,040 --> 00:34:20,340
+أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر
+
+477
+00:34:20,340 --> 00:34:21,620
+أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر
+
+478
+00:34:21,620 --> 00:34:21,900
+أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر
+
+479
+00:34:21,900 --> 00:34:28,200
+أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر أكثر
+
+480
+00:34:28,200 --> 00:34:31,580
+أكثر
+
+481
+00:34:31,580 --> 00:34:40,420
+أكثر أكثر
+
+482
+00:34:40,450 --> 00:34:42,830
+وبضلّ سواء كان في الـ space أو مش في الـ space
+
+483
+00:34:42,830 --> 00:34:46,730
+فبيطلع حدّ normal عليه، فبيكون normal على الـ T وبيكون
+
+484
+00:34:46,730 --> 00:34:49,390
+اللي هو الـ principle unit normal vector نفسه اللي
+
+485
+00:34:49,390 --> 00:34:53,190
+عرفناها قبل، أن بتساوي واحد على Kappa في DT by DS اللي
+
+486
+00:34:53,190 --> 00:34:57,030
+هو هذه العلاقة بواسطة الـ T نفسها اللي هي الأسهل، بس
+
+487
+00:34:57,030 --> 00:35:00,770
+الآن في الـ space بدل ما هو في البلاد، ناخد مثال
+
+488
+00:35:00,770 --> 00:35:03,740
+عندي find the curvature for the helix. ناخد
+
+489
+00:35:03,740 --> 00:35:07,840
+الكررفتشر لمين؟ اللي هي للهليكس R of T بتساوي اللي
+
+490
+00:35:07,840 --> 00:35:13,100
+أمامي هذه A Cos T زائد A Sin T زائد B T حيث A وB
+
+491
+00:35:13,100 --> 00:35:15,760
+أكبر أو تساوي صفر، وA تربيع زائد B تربيع لها تساوي صفر
+
+492
+00:35:15,760 --> 00:35:19,740
+هذه الهليكس اللي احنا عملناها قبل هيك، بس كانت الـ A
+
+493
+00:35:19,740 --> 00:35:26,620
+وB وC. الـ A وB في هليكسنا اللي هو 1 اللي رسمناها
+
+494
+00:35:26,620 --> 00:35:31,220
+قلنا رسمتها اللي هو سهل أده أو مش سهل أكثر زي
+
+495
+00:35:31,220 --> 00:35:34,320
+الرسمات الثانية اللي مرسموها بالكمبيوتر. الآن وجد
+
+496
+00:35:34,320 --> 00:35:37,580
+ميان الـ T. تقول طبعا تسهل الـ T. أشهر جمعها الـ T
+
+497
+00:35:37,580 --> 00:35:40,700
+ما فيش حساب منها. بنوجد ميان الـ V. الـ V بيساوي من
+
+498
+00:35:40,700 --> 00:35:44,200
+فاضل أده بيطلع ناقص A Sin T في اتجاه الـ I زائد A
+
+499
+00:35:44,200 --> 00:35:48,160
+Cos T في اتجاه J زائد B في اتجاه K. الـ magnitude لـ
+
+500
+00:35:48,160 --> 00:35:51,620
+V تأخذ الجذر التربيعي لهذا المقدار اللي هو A تربيع
+
+501
+00:35:51,620 --> 00:35:54,600
+Sin تربيع زائد A تربيع Cos تربيع زائد B تربيع
+
+502
+00:35:54,600 --> 00:35:57,380
+هذه مع بعض اللي هي زي ما أنتم عارفين عبارة عن A
+
+503
+00:35:57,380 --> 00:36:00,700
+تربيع. Sin تربيع وCos تربيع عبارة عن واحد فبيطلع الـ
+
+504
+00:36:00,700 --> 00:36:04,300
+magnitude لـ V اللي هو الجذر التربيعي لـ A تربيع
+
+505
+00:36:04,300 --> 00:36:08,890
+زائد B تربيع. الآن T بيساوي V على الـ magnitude لـ V
+
+506
+00:36:08,890 --> 00:36:12,490
+اللي هو بيصير واحد على هذا الجذر في المقدار اللي
+
+507
+00:36:12,490 --> 00:36:17,470
+هو الـ V اللي أمامنا. الآن Kappa عبارة عن واحد على V
+
+508
+00:36:17,470 --> 00:36:21,990
+في DT by DT مظبوط؟ إذا بدنا نعرف ... بدنا نُفاضل الـ
+
+509
+00:36:21,990 --> 00:36:26,510
+DT by DT. هي واحد على V هي واحد على V موجودة، فنُفاضل
+
+510
+00:36:26,510 --> 00:36:30,390
+هذه وناخد لها الـ magnitude. نُفاضل هذا المقدار هذا
+
+511
+00:36:30,390 --> 00:36:34,070
+المقدار اللي هو عبارة عن هذا ثابت في تفاضل اللي
+
+512
+00:36:34,070 --> 00:36:38,250
+جوا ناقص A Cos T في اتجاه I ناقص A Sin T في اتجاه
+
+513
+00:36:38,250 --> 00:36:41,730
+J. هذا الـ magnitude له بناخده هو عبارة عن جذر
+
+514
+00:36:41,730 --> 00:36:45,620
+التربيعي لهذا تربيع زائد هذا تربيع. هذا بيطلع برة مع
+
+515
+00:36:45,620 --> 00:36:49,680
+هذا اللي هو عبارة عن A تربيع زائد B تربيع. من أين
+
+516
+00:36:49,680 --> 00:36:53,380
+اجت الـ A؟ لما ربعنا هذا، أخدنا الـ magnitude ... أخدنا
+
+517
+00:36:53,380 --> 00:36:56,480
+هذا اللي قبل ما نربع، أخدنا الـ A عن المشترك، طلعت
+
+518
+00:36:56,480 --> 00:37:00,660
+بره وجا عندي هذا. الآن بناخد الـ magnitude، مش ده ما
+
+519
+00:37:00,660 --> 00:37:03,280
+قلت قبل. الآن بناخد الـ magnitude لهذا اللي هو جذر
+
+520
+00:37:03,280 --> 00:37:06,160
+التربيعي لـ Cos تربيع زائد Sin تربيع اللي هو واحد
+
+521
+00:37:06,160 --> 00:37:09,300
+فبيطلع عندي الكابا بيساوي A على A تربيع زائد B
+
+522
+00:37:09,300 --> 00:37:12,910
+تربيع. لاحظوا ما هي ... ليه بعض الملاحظات اللي أنا بقول
+
+523
+00:37:12,910 --> 00:37:17,250
+ليه؟ we see that increasing B for a fixed A لو
+
+524
+00:37:17,250 --> 00:37:22,130
+ثبتنا A وزِدنا من B. زِدنا B يعني أشمّلنا، يعني طولنا
+
+525
+00:37:22,130 --> 00:37:28,750
+الأسطوانة هذه. B تكبر الـ T. الآن لما نضرب B فابت
+
+526
+00:37:29,860 --> 00:37:33,640
+الهليكس إيش ماله؟ بنضغط زي ما اتفجنا على المرمى
+
+527
+00:37:33,640 --> 00:37:39,740
+الماضي. الآن for a fixed B بتزيد إيش؟ لما اللي هو
+
+528
+00:37:39,740 --> 00:37:43,480
+increasing بيزيد بيقل هذا المقدار لأنه بيزيد
+
+529
+00:37:43,480 --> 00:37:46,580
+المقام، بيقل هذا المقدار، بيصير الـ curvature إيش
+
+530
+00:37:46,580 --> 00:37:55,950
+ماله؟ قليل. الآن decreasing A for a fixed B. الآن لما
+
+531
+00:37:55,950 --> 00:38:01,910
+نقلل الـ A for fixed B اللي هو eventually decreases
+
+532
+00:38:01,910 --> 00:38:06,730
+the curvature as a whole. ماله؟ بيقل إيش؟ الـ curvature
+
+533
+00:38:06,730 --> 00:38:13,930
+لما اللي هو decreasing للـ
+
+534
+00:38:13,930 --> 00:38:21,510
+A. decreasing للـ A يعني اللي هي الأسطوانة من نصف
+
+535
+00:38:21,510 --> 00:38:26,790
+القطر تبع الدائرة العرضية بنقلله. لكن لو كانت الـ B
+
+536
+00:38:26,790 --> 00:38:32,750
+بيساوي صفر، بيصير اللي هو B بيساوي صفر إيش بيطلع
+
+537
+00:38:32,750 --> 00:38:37,230
+لنا؟ بيطلع لنا دائرة. بيصير إيش مالها؟ بيصير دائرة
+
+538
+00:38:37,230 --> 00:38:42,430
+فبيصير the helix reduces to a circle of radius A
+
+539
+00:38:42,430 --> 00:38:46,130
+and its curvature 1 على A. يتوافق مع الكلام القديم
+
+540
+00:38:46,130 --> 00:38:51,120
+لكن لو كانت الـ A بتساوي صفر. يعني الآن الـ A بتساوي
+
+541
+00:38:51,120 --> 00:38:54,600
+صفر بيصير الـ helix عبارة عن إيه؟ عشان الـ z axis
+
+542
+00:38:54,600 --> 00:38:58,020
+بيساوي الـ T في B في اتجاه الـ K يعني في اتجاه الـ z
+
+543
+00:38:58,020 --> 00:39:01,860
+axis. يعني بيصير خط. مدام صار خط إذا الـ curvature
+
+544
+00:39:01,860 --> 00:39:06,300
+اللي هو إيش هيصير؟ طبيعي هيساوي صفر وهذا يتوافق مع
+
+545
+00:39:06,300 --> 00:39:08,980
+الكلام القديم اللي حكيناه
+
+546
+00:39:11,670 --> 00:39:19,350
+تابع للمثال اللي قبل شوية. find N for the helix in
+
+547
+00:39:19,350 --> 00:39:22,150
+example خمسة and describe how the vector is
+
+548
+00:39:22,150 --> 00:39:25,390
+pointing. طبعا is pointing هو بيكون عمودي عالمين على
+
+549
+00:39:25,390 --> 00:39:33,310
+الـ N على الـ T. عشان نجد الـ N نجد DTYDT. نُفاضل الـ T
+
+550
+00:39:33,310 --> 00:39:35,310
+اللي وجدناها قبل قليل في المثال
+
+551
+00:39:49,670 --> 00:39:53,110
+الآن الـ magnitude له عبارة عن هذا طبعا ثابت في
+
+552
+00:39:53,110 --> 00:39:56,370
+الجذر التربيعي لهذا تربيع زائد هذا تربيع وهذا تربيع
+
+553
+00:39:56,370 --> 00:40:00,170
+اللي هو A. بيصير A على الجذر هذا. الآن الآن الآن هو
+
+554
+00:40:00,170 --> 00:40:03,710
+عبارة عن هذا على هذا. كل حسابات هذا على هذا بيساوي
+
+555
+00:40:03,710 --> 00:40:13,350
+الجذر التربيعي للـ A تربيع DT by DT على اللي هو هذا
+
+556
+00:40:13,350 --> 00:40:16,550
+المقدار. بنجلب هذا بيصير تطلع هذه فوق. الجذر الـ A
+
+557
+00:40:16,550 --> 00:40:20,700
+تربيع على الـ A أو هذه الناقص اللي فوق وهي موجودة تحت
+
+558
+00:40:20,700 --> 00:40:24,680
+في واحد على جذر الـ A تربيع زائد الـ V تربيع في
+
+559
+00:40:24,680 --> 00:40:27,460
+المقدار هذا. Cosine T في اتجاه الـ I زي A Cosine T في
+
+560
+00:40:27,460 --> 00:40:32,740
+اتجاه الـ J. الآن هذا بيروح مع هذا. اختصارات بيصير
+
+561
+00:40:32,740 --> 00:40:35,920
+اللي هو اللي هي الـ A. كمان هذه بتروح مع الـ A اللي
+
+562
+00:40:35,920 --> 00:40:38,640
+جوا هنا بياخدها عن المشترك بيصير اللي هو الناقص
+
+563
+00:40:38,640 --> 00:40:41,820
+بدخل بيصير ناقص Cosine T في اتجاه الـ I ناقص Sine T
+
+564
+00:40:41,820 --> 00:40:46,480
+في اتجاه الـ J. وأكيد من أن هذا اللي هو صار ما فيه z
+
+565
+00:40:46,480 --> 00:40:50,720
+axis. فأكيد هيطلع parallel to mean to the xy plane
+
+566
+00:40:50,720 --> 00:40:56,920
+ومش هيك كمان وهيكون always pointed for the اللي
+
+567
+00:40:56,920 --> 00:41:01,560
+هو z axis لإنه بيكون عمودي عالميا على الـ unit
+
+568
+00:41:01,560 --> 00:41:06,020
+tangent vector الـ T وهيك بيكون احنا خلصنا
+
+569
+00:41:06,020 --> 00:41:09,020
+المحاضرة ومبارك الله فيكم. السلام عليكم ورحمة
+
+570
+00:41:09,020 --> 00:41:09,740
+الله وبركاته
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/vPw26YH1fW0_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/vPw26YH1fW0_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..fcfa392871d1b3043cf0b7b72fa64e3b9d42a902
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/vPw26YH1fW0_postprocess.srt
@@ -0,0 +1,1868 @@
+1
+00:00:01,150 --> 00:00:04,890
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,890 --> 00:00:11,530
+وبركاته هذه هي المحاضرة الرقم 16مساق تفاضل وتكامل
+
+3
+00:00:11,530 --> 00:00:16,390
+جين عنوان المحاضرة اليوم The Chain Rule أو نظرية
+
+4
+00:00:16,390 --> 00:00:22,910
+السلسلة وهي الـ Section 14-4 وهي المحاضرة الرابعة
+
+5
+00:00:22,910 --> 00:00:28,490
+في Chapter 14 Functions of Two Variables هنبدأ
+
+6
+00:00:28,490 --> 00:00:33,810
+فيها احنا طبعا في تفاضل الألف كنا اتحدثنا عن نظرية
+
+7
+00:00:33,810 --> 00:00:36,780
+السلسلة لـ Function of One Variableيعني لو كان
+
+8
+00:00:36,780 --> 00:00:41,040
+عندي Y بيسوي F of X و X is a function of T يعني X
+
+9
+00:00:41,040 --> 00:00:44,420
+function في الـ T بيصير Y function في T و بيكون DY
+
+10
+00:00:44,420 --> 00:00:53,960
+by DT بيسوي DF by DX في DX by DT الآن شبيه الحديث
+
+11
+00:00:53,960 --> 00:00:58,880
+خلّينا نبدأ بالنظرية الخامسة اللي هي Chain rule
+
+12
+00:00:58,880 --> 00:01:01,260
+for functions of two independent variables
+
+13
+00:01:01,260 --> 00:01:06,530
+والموضوع ان شاء الله سهل وسلسعندي function w
+
+14
+00:01:06,530 --> 00:01:10,210
+بتساوي f of x و y function of two variables يعني w
+
+15
+00:01:10,210 --> 00:01:14,110
+ال dependent variable و ال x و ال y independent
+
+16
+00:01:14,110 --> 00:01:19,760
+بالنسبة ل w ففرضنا w is differentiable andفرضنا إن
+
+17
+00:01:19,760 --> 00:01:22,920
+X بتساوي X of T يعني X نفسها دي اللي هي الـ
+
+18
+00:01:22,920 --> 00:01:27,620
+Dependent Variable بالنسبة لـ W عبارة عن اللي هي
+
+19
+00:01:27,620 --> 00:01:32,220
+الـ Independent بالنسبة لـ W هي Dependent بالنسبة
+
+20
+00:01:32,220 --> 00:01:35,920
+لـ T يعني Function X تعتمد على T Function في T X
+
+21
+00:01:35,920 --> 00:01:39,610
+of T و Y نفس الاشي Function في Tلو فرضنا أنه تنتهي
+
+22
+00:01:39,610 --> 00:01:43,050
+in differentiable functions of T فرد W نفسها
+
+23
+00:01:43,050 --> 00:01:46,190
+function في T ومش هي كمان الـ W is a
+
+24
+00:01:46,190 --> 00:01:49,810
+differentiable function of T حسب ما بتقول النظرية
+
+25
+00:01:49,810 --> 00:01:53,030
+then the composite اللي هي W بالساوية F of X of T
+
+26
+00:01:53,030 --> 00:01:57,250
+وY of T هتكون differentiable of T وطريقة حساب
+
+27
+00:01:57,250 --> 00:02:02,750
+التفاضل DW بالنسبة لDT هي مشابهة للسابق هي عبارة
+
+28
+00:02:02,750 --> 00:02:11,550
+عن الـ FXفي التفاضل X' of T زاد الـ Fy في التفاضل
+
+29
+00:02:11,550 --> 00:02:17,770
+Y' of Tأو اختصاراً اللي هي يمكن أسهل في التعامل دي
+
+30
+00:02:17,770 --> 00:02:21,430
+دابل يو باي دي تي طبعا احنا كتبنا دي دابل يو باي
+
+31
+00:02:21,430 --> 00:02:25,710
+دي تي عساس ان هي و function بتصير بس في تي فبصير
+
+32
+00:02:25,710 --> 00:02:28,070
+function of one variable عشان هي كمان نكتب دي دابل
+
+33
+00:02:28,070 --> 00:02:32,910
+يو باي دي تي بساوي دل F by دل X في دي X by دي T
+
+34
+00:02:32,910 --> 00:02:37,010
+لأن X function في T لحالها بينما كتبنا دل F by دل
+
+35
+00:02:37,010 --> 00:02:46,980
+X عساس انه function F X of Y مع بعضبالنسبة للتشين
+
+36
+00:02:46,980 --> 00:02:50,700
+رول بالنسبة لـ function of two variables او
+
+37
+00:02:50,700 --> 00:02:55,480
+اختصارا DW by DT بساوية بدل الـ F التي هي DW by DX
+
+38
+00:02:55,480 --> 00:03:01,040
+في DX by DT وبدل الـ F هنا برضه DW by DY في DY by
+
+39
+00:03:01,040 --> 00:03:05,420
+DT لأن احنا مفترضين W بتساوي F of XY يمكن
+
+40
+00:03:05,420 --> 00:03:08,940
+الاستدارة الآن عن طريق الرسم يكون أسهل لكم بكتير
+
+41
+00:03:09,390 --> 00:03:12,350
+اللي هو خلّينا نشوف الـ Chain Rule عبر الرسمة وهذه
+
+42
+00:03:12,350 --> 00:03:16,730
+الرسومات هيكون اللي هي سهل اننا نتعامل معها بشكل
+
+43
+00:03:16,730 --> 00:03:22,000
+دائم ان شاء اللهفزي ما قلنا اللي هو الـ Chain Rule
+
+44
+00:03:22,000 --> 00:03:25,180
+نظرية سلسلة ممكن من خلال اللي هي الرسم التالي يكون
+
+45
+00:03:25,180 --> 00:03:29,660
+أسهل لنا كتير عند W هي function of X و Y يعني W
+
+46
+00:03:29,660 --> 00:03:34,080
+عبارة عن dependent variable بينما X و Y في هذه
+
+47
+00:03:34,080 --> 00:03:39,140
+الحالة independent بالنسبة لـ W و هتكون Ind اللي
+
+48
+00:03:39,140 --> 00:03:42,560
+هي dependent بالنسبالته عشان هيك مسميين هنا احنا
+
+49
+00:03:42,560 --> 00:03:48,020
+ال X و ال Y اللي هي أشمالها intermediate variables
+
+50
+00:03:48,500 --> 00:03:52,960
+اللي هي بتكون في البداية independent وبعدين في
+
+51
+00:03:52,960 --> 00:03:56,040
+النهاية بيصير dependent الـ W is a function of two
+
+52
+00:03:56,040 --> 00:04:00,240
+variables إذن هاي ال variable الأول اللي هو ال Y و
+
+53
+00:04:00,240 --> 00:04:03,500
+ال variable الثاني ال X أو العكس الآن ال X نفسها
+
+54
+00:04:03,500 --> 00:04:06,800
+function في T هاي ال X ده ال function في T و ال Y
+
+55
+00:04:06,800 --> 00:04:10,650
+برضه function في Tصارت الـ W بباشرة function في
+
+56
+00:04:10,650 --> 00:04:15,350
+VIN في T عشان هيك بنقدر نجيب دل W by دل DW SF by
+
+57
+00:04:15,350 --> 00:04:19,350
+DT لأن DW by DT بنخلال الرسم هذه بتسهل علينا كتير
+
+58
+00:04:19,350 --> 00:04:25,910
+كتير كتير عبارة عن دل W by دل Y في DY by DT أيها
+
+59
+00:04:26,270 --> 00:04:32,610
+وهذه زائد هذا الفرح زائد الفرح هذا التاني دل و باي
+
+60
+00:04:32,610 --> 00:04:37,050
+دل اكس دي اكس باي دي تي طلعت عندي دي و باي دي تي
+
+61
+00:04:37,050 --> 00:04:39,890
+بيساوي دل و باي دي اكس دي اكس باي دي تي زي ما
+
+62
+00:04:39,890 --> 00:04:47,010
+عملنا قبل بشوية و هذا الكلام أيضا اللي هو يعني
+
+63
+00:04:47,010 --> 00:04:51,550
+هنلاقيه برضه اللي هو نفس الشيء لو كانت عنديو
+
+64
+00:04:51,550 --> 00:04:55,730
+فانشغله في ثلاث مواصلة فبنعمل هذه الواريابل الأول
+
+65
+00:04:55,730 --> 00:05:00,090
+اللي هو z أو الأخير و هذا ال y و هذا ال x و بعدين
+
+66
+00:05:00,090 --> 00:05:04,430
+بنكمل اللي هو هنشوفه و هذا لاحقا بس عشان أربطكميا
+
+67
+00:05:04,430 --> 00:05:09,810
+بالسبق طيب نيجي الآن نرجع لنظريتنا هذه use the
+
+68
+00:05:09,810 --> 00:05:14,370
+chain rule to find the derivative of w بساوي x في
+
+69
+00:05:14,370 --> 00:05:19,130
+y عند w إيش مانع x في y بدنا نستخدم ال chain rule
+
+70
+00:05:19,360 --> 00:05:25,140
+لإيجاد هذه الـ derivative باسمين with respect to t
+
+71
+00:05:25,140 --> 00:05:28,480
+along the path x بساوة و cosine of t و y بساوة و
+
+72
+00:05:28,480 --> 00:05:31,680
+sine of t what is the derivatives value at t
+
+73
+00:05:31,680 --> 00:05:36,880
+بتساوية y على اتنين اللي طالبوا عندي w function في
+
+74
+00:05:36,880 --> 00:05:41,510
+x yو جاي لل X function في T هي Cos T و Y function
+
+75
+00:05:41,510 --> 00:05:46,750
+في T هي Sin T فبقى ضايق ليه طلب ليه W اللي هو
+
+76
+00:05:46,750 --> 00:05:51,430
+اتفضل ليها بالنسبة ل T عبر الطريقين هذولة يعني
+
+77
+00:05:51,430 --> 00:05:56,770
+بمعنى آخر عنده اللي هي ال X صارت function بناء على
+
+78
+00:05:56,770 --> 00:06:00,490
+هذولة الطريق ال function في T و Y function في T
+
+79
+00:06:00,490 --> 00:06:03,490
+كما هو معطب بعد هي كده بتوجد what is the
+
+80
+00:06:03,490 --> 00:06:07,360
+derivative value of T بتساوي Y على 2أذا على طول
+
+81
+00:06:07,360 --> 00:06:10,880
+نستخدم مين يا جماعة طريقة السلسلة هاي W function
+
+82
+00:06:10,880 --> 00:06:15,120
+في X و Y أو هاي و X و Y كل واحدة function في T
+
+83
+00:06:15,120 --> 00:06:18,760
+اللي هي كساية T ساية T ده نجهزها أمورنا لإن دي W
+
+84
+00:06:18,760 --> 00:06:22,580
+by دي T حسب النظرية اللي فوق هيها اللي عندنااللي
+
+85
+00:06:22,580 --> 00:06:27,740
+هي دي دل W by دل X في DX by DT زي دل W by دل Y في
+
+86
+00:06:27,740 --> 00:06:32,660
+DY by DT هو يساوي دل W by دل X فاضل هذه بالنسبة ل
+
+87
+00:06:32,660 --> 00:06:38,140
+X تطلع Y فاضل X بالنسبة ل T اي X Cos T تفضل اللي
+
+88
+00:06:38,140 --> 00:06:44,380
+يمين نقل Sin T خلصنا من هذهزي ما
+
+89
+00:06:44,380 --> 00:07:00,120
+حكينا
+
+90
+00:07:00,120 --> 00:07:05,130
+بنعوض عن y و عن x بخمتهمبينطلع عندي بدل Y اللي هي
+
+91
+00:07:05,130 --> 00:07:09,510
+Sin بيصير نقص Sin تربيع بدل X Cos بيصير Cos تربيع
+
+92
+00:07:09,510 --> 00:07:14,270
+T اللي هي طبعا Cos تربيع نقص Sin تربيع اللي هي
+
+93
+00:07:14,270 --> 00:07:22,090
+بيساوي Cos اتنين T الان على طول DW by DTاللي هي
+
+94
+00:07:22,090 --> 00:07:26,350
+عند باي على اتنين بنعوض عن كوساين اللي هي بيصير
+
+95
+00:07:26,350 --> 00:07:29,010
+اتنين في باي على اتنين يعني كوساين البي اللي هي
+
+96
+00:07:29,010 --> 00:07:32,790
+عبارة عن اياش عبارة عن ناقص واحد و هيك بيكون حالين
+
+97
+00:07:32,790 --> 00:07:38,110
+المثال نيجي الآن اللي هو ل functions of three
+
+98
+00:07:38,110 --> 00:07:43,170
+variables الان لو كان عندي اللي هو بدل function of
+
+99
+00:07:43,170 --> 00:07:46,650
+two variables three variables يعني لو رجعنا للي
+
+100
+00:07:46,650 --> 00:07:50,030
+عندنا هنا ال f of x و y بدل f of x و y بيصيروا z
+
+101
+00:07:50,610 --> 00:07:54,670
+فبصير عند هاي هذا للـ Y و هذا للـ X و بنجيب واحد
+
+102
+00:07:54,670 --> 00:07:58,790
+جوا للـ Z اللي هو و الـ Z function في T يعني فرضنا
+
+103
+00:07:58,790 --> 00:08:03,350
+ان W بتساوية F of X و Y و Z و X function في T و Y
+
+104
+00:08:03,350 --> 00:08:06,810
+function في T و Z function في T الآن بيصير زي هدول
+
+105
+00:08:06,810 --> 00:08:10,110
+الأعمدة بالظبط عمود .. زي ال .. ال .. ال .. ال ..
+
+106
+00:08:10,230 --> 00:08:13,710
+اللي باخد دي نوهان و باخد دي نوهان ناخد واحد خط
+
+107
+00:08:13,710 --> 00:08:18,230
+لمن؟ لـ Z بيصير لـ Z و لـ T بيصير عندي دل دابل يو
+
+108
+00:08:18,230 --> 00:08:22,230
+باي دل تي .. آسف دي دابل يو باي دي تي بيصير دل
+
+109
+00:08:22,230 --> 00:08:25,730
+دابل يو باي دي واي في دي واي باي دي تي زائد دل
+
+110
+00:08:25,730 --> 00:08:31,130
+دابل يو باي دي اكس في دي اكس باي دي تي زائد اللي
+
+111
+00:08:31,130 --> 00:08:37,610
+هو الخطة التفالة اللي هي دل دابل يو باي دي زدوهذه
+
+112
+00:08:37,610 --> 00:08:42,150
+هي نظرية الـ Chain Rule for functions of three
+
+113
+00:08:42,150 --> 00:08:48,290
+independent variables F of X, Y, Z is
+
+114
+00:08:48,290 --> 00:08:53,700
+differentiable function of X, Y, Zوالـ x و الـ y و
+
+115
+00:08:53,700 --> 00:08:57,560
+الـ z are differentiable functions of T ثم و نفسها
+
+116
+00:08:57,560 --> 00:09:02,760
+differentiable of T و الـ dw by dt اللي هي
+
+117
+00:09:02,760 --> 00:09:06,520
+بالساوية دي الـ wy دي ال x في dx by dt زي ما قلنا
+
+118
+00:09:06,520 --> 00:09:09,720
+هذا الخط الأول و هذا الخط التاني للـ y و هذا الخط
+
+119
+00:09:09,720 --> 00:09:13,920
+التالت لمين للـ z و بيكون هيكا اللي هو صارت عندنا
+
+120
+00:09:13,920 --> 00:09:17,820
+الأمور سهلة لو حتى كان عندنا أربع variable تمانية
+
+121
+00:09:17,820 --> 00:09:21,860
+variable مليون variable بنعملها مليون خط و بنجمحها
+
+122
+00:09:22,190 --> 00:09:26,350
+بنفس الأسلوب اللي عندى لما نكون اللي هي كل واحد
+
+123
+00:09:26,350 --> 00:09:32,420
+فيه المليون function of mean of Tالان find dw by
+
+124
+00:09:32,420 --> 00:09:37,940
+dt f w بيساوي x y زائد z هذا طبعا مثال على اللي هي
+
+125
+00:09:37,940 --> 00:09:41,340
+النظرية اللي قبل شوية x بيساوي cosine of t و y
+
+126
+00:09:41,340 --> 00:09:44,680
+بيساوي cosine of t و z بيساوي إيش ال t بنطبق اللي
+
+127
+00:09:44,680 --> 00:09:49,880
+فوق بنطبقه بسهولة هي dw by dt هيه بيساوي هي
+
+128
+00:09:49,880 --> 00:09:54,080
+القانون اللي احنا قلنا عنه قبل شوية y ساوي del w
+
+129
+00:09:54,080 --> 00:09:59,800
+by del x اللي هي بيساوي yDx by Dt هي ساول سالب
+
+130
+00:09:59,800 --> 00:10:07,060
+ساين تي دي الـ W by دي الـ Y هي عبارة عن X دي الـ
+
+131
+00:10:07,060 --> 00:10:13,520
+Y by دي الـ T هي عبارة عن كساين T زائد دي الـ W by
+
+132
+00:10:13,520 --> 00:10:17,220
+دي الـ Z هي واحدة لأن هذه الثبتات تعرفين كيف احنا
+
+133
+00:10:17,220 --> 00:10:21,900
+بنفضل دي الـ W by دي الـ Z بنثبت الـ X والـ Y
+
+134
+00:10:21,900 --> 00:10:28,400
+ونفضل بالنسبة لـ Zوالدت زد زي ما قلنا واحد دي زد
+
+135
+00:10:28,400 --> 00:10:32,700
+باي دي تي برضه واحد إذا بيصير عندي هذا عبارة عن
+
+136
+00:10:32,700 --> 00:10:36,440
+واحد ناقص اللي ها دي طبعا واي صين بيصير ناقص صين
+
+137
+00:10:36,440 --> 00:10:41,380
+تربيع و X Cos بيصير X Cos تربيع فبصير واحد ناقص
+
+138
+00:10:41,380 --> 00:10:45,900
+صين تربيع زاد Cos تربيع اللي هي عبارة عن اتنين Cos
+
+139
+00:10:45,900 --> 00:10:50,070
+تربيع ليش؟لأن واحد ناقص sin تربيع cos تربيع وهي
+
+140
+00:10:50,070 --> 00:10:53,770
+كمان cos تربيع يعني بيصير عندى اتنين cos تربيع
+
+141
+00:10:53,770 --> 00:10:58,650
+واحد اكيد يجي في باله بيقول هالجيت طب انت قلت هذي
+
+142
+00:10:58,650 --> 00:11:03,750
+X وY وZ وفيك عندك كمان مليون اللي هو متغير طب لو
+
+143
+00:11:03,750 --> 00:11:07,790
+المتغيرات المليون هدولة functions مش في T بس في T
+
+144
+00:11:07,790 --> 00:11:12,730
+في T وفي كمان variable Sأقول له بيصير نفس المنطقة
+
+145
+00:11:12,730 --> 00:11:17,110
+و الآن هناخد نظرية تقول لي لو تعددت ال variables
+
+146
+00:11:17,110 --> 00:11:21,850
+اللي هنا و تعددت هذولة كل واحدة صارت function of
+
+147
+00:11:21,850 --> 00:11:25,370
+more than one variable وهذا اللي هتعجوا النظرية
+
+148
+00:11:25,370 --> 00:11:29,500
+اللي بعدهاالتي تسميها Chain Rule for All Two
+
+149
+00:11:29,500 --> 00:11:33,720
+Independent Variables and Three Intermediates الـ
+
+150
+00:11:33,720 --> 00:11:36,940
+Three Intermediates اللي هو مين مقصود فيه هن الـ X
+
+151
+00:11:36,940 --> 00:11:40,100
+و الـ Y و الـ Z هدولة اللي هن الـ dependent الـ W
+
+152
+00:11:40,100 --> 00:11:42,780
+الـ dependent و اللي في النص هدولة الـ
+
+153
+00:11:42,780 --> 00:11:47,240
+intermediates اللي هن عبارة عن independent بالنسبة
+
+154
+00:11:47,240 --> 00:11:51,720
+لـ W و dependent بالنسبة للبعد هن اللي هي الـ S و
+
+155
+00:11:51,720 --> 00:11:56,050
+الـ R في حالتنا هذه اللي هنشوفه الآنإذا نظرنا إلى
+
+156
+00:11:56,050 --> 00:12:02,670
+أن W بيسوء F of X وY وZ و X بيسوء G of R وS يعني
+
+157
+00:12:02,670 --> 00:12:06,030
+صارت الـ X function of two variables R وS و Y
+
+158
+00:12:06,030 --> 00:12:09,430
+function of two variables R وS و Z function of two
+
+159
+00:12:09,430 --> 00:12:15,560
+variables R وSإذاً صارت الـ function أصلية W فنكشن
+
+160
+00:12:15,560 --> 00:12:21,220
+في R و في S إذاً هيصير يسألني دل W by دل R و دل W
+
+161
+00:12:21,220 --> 00:12:25,420
+by دل S مش زي ما جابل دل W by دي T دي W by دي T
+
+162
+00:12:25,420 --> 00:12:30,200
+لإنه كان T one variable أن اللي هو بيسألني عن دل W
+
+163
+00:12:30,200 --> 00:12:35,700
+by دل R و دل W by دل S بنقول له سهل و هنمشي بنفس
+
+164
+00:12:35,700 --> 00:12:39,560
+المنطق اللي هنشوفه بعد شويةلأن if all four
+
+165
+00:12:39,560 --> 00:12:44,360
+functions هدولة إذا كان كلهم different شابر هتطلع
+
+166
+00:12:44,360 --> 00:12:49,580
+w برضه has partial derivatives with respect to
+
+167
+00:12:49,580 --> 00:12:56,250
+main to R وS ومش هيك وعملية الإيجاد مشابهةالآن
+
+168
+00:12:56,250 --> 00:13:02,230
+عندما نرسم لـ R خطوط لحاله عندما نرسم لـ S خطوط
+
+169
+00:13:02,230 --> 00:13:09,270
+لحاله نرسم لها خطوط لحاله نرسم لها خطوط لحاله نرسم
+
+170
+00:13:09,270 --> 00:13:12,650
+لها خطوط لحاله نرسم لها خطوط لحاله نرسم لها خطوط
+
+171
+00:13:12,650 --> 00:13:18,610
+لحاله نرسم لها خطوط لحاله نرسم لها خطوط
+
+172
+00:13:18,610 --> 00:13:23,000
+لحالهما هي الـ z برضه function في main في R و S
+
+173
+00:13:23,000 --> 00:13:26,540
+أنا بدي أحسب الـ S الآن إذا بحط لها خط لـ S و بحط
+
+174
+00:13:26,540 --> 00:13:30,240
+لها خط ل S في الـ Y و الـ X بحط لها خط ل الـ S نفس
+
+175
+00:13:30,240 --> 00:13:34,500
+الجثة بنعملها مع main مع R لأنه برضه X و Y و Z الـ
+
+176
+00:13:34,500 --> 00:13:38,190
+Intermediate هدولة functions في الـ Rوفي الـ S إذا
+
+177
+00:13:38,190 --> 00:13:41,870
+بنعمل إشي لل R عشان نجد اللي هو في حالة طلب دي
+
+178
+00:13:41,870 --> 00:13:45,410
+دابليو دي ال W by دي ال R وبنعملها دي ال S في حالة
+
+179
+00:13:45,410 --> 00:13:50,550
+طلب دي ال W by دي ال S وخلّيني نشوف كيف نحسب دي ال
+
+180
+00:13:50,550 --> 00:13:54,530
+W by دي ال R دي ال W by دي ال R زي ما عملنا دي ال
+
+181
+00:13:54,530 --> 00:13:58,030
+W by دي ال X في دي ال X by دي ال R الخط الأول زي
+
+182
+00:13:58,030 --> 00:14:01,310
+الخط التاني دي ال W by دي ال Y في دي ال Y by دي ال
+
+183
+00:14:01,310 --> 00:14:05,620
+Rوالخط التالت del w by del z في del z by del r
+
+184
+00:14:05,620 --> 00:14:11,860
+مجموحنا دولة del w by del r إذن اختصارا الـ w
+
+185
+00:14:11,860 --> 00:14:16,300
+بتنحط في أغلى f of x,y,z بيطلع منها خطوط بعدد ال
+
+186
+00:14:16,300 --> 00:14:19,680
+variables اللي عندنا كل واحد من ال variables طبعا
+
+187
+00:14:19,680 --> 00:14:22,260
+عندي إذا كان function of one variables خلاص بنحكي
+
+188
+00:14:22,260 --> 00:14:25,710
+عنتفاضل بالنسبة لـ T لحالها إذا كان أكتر من
+
+189
+00:14:25,710 --> 00:14:30,430
+variable نحكي أكتر من R إذا بنعمل رسمة للـ R ورسمة
+
+190
+00:14:30,430 --> 00:14:34,250
+ثانية مشابهة جدا للـ S وبنطبق و بنوجد اللي هو
+
+191
+00:14:34,250 --> 00:14:38,670
+القانون del W by del S بيساوي del W by del X في
+
+192
+00:14:38,670 --> 00:14:42,430
+del X by del S زائد الخط التاني del W by del Y في
+
+193
+00:14:42,430 --> 00:14:46,090
+del Y by del S زائد del W by del Z في del Z بطاقة
+
+194
+00:14:46,090 --> 00:14:49,970
+del S إذن هذه عبارة عن الـ Chain Rule الـ function
+
+195
+00:14:49,970 --> 00:14:53,470
+of three variablesوكل واحد من الـ Intermediate
+
+196
+00:14:53,470 --> 00:14:56,310
+Variables الـ X والـ Y والـ Z Function of Two
+
+197
+00:14:56,310 --> 00:14:59,670
+Variables لو كان عندي مليون هنا بطلع لنا مليون خط
+
+198
+00:14:59,670 --> 00:15:03,630
+لو طلع لنا خمسمية هنا بطلع لنا خمسمية رسمة طبعا
+
+199
+00:15:03,630 --> 00:15:06,450
+مافيش ده نفسه رسم خمسمية رسمة ولكن كل واحد من الاش
+
+200
+00:15:06,450 --> 00:15:12,090
+اللي هو بغيره وبصير عندي اللي هي للخمسمية نفس
+
+201
+00:15:12,090 --> 00:15:16,530
+الرسمة ولكن بنغير اسمه لل R اسمه R لل S اسمه ال S
+
+202
+00:15:16,530 --> 00:15:19,150
+إلى A خيري هى بتصور ان شاء الله تكونوا فهمتوا
+
+203
+00:15:19,150 --> 00:15:23,840
+عليهاالان دي دي الان المثال اللي عندناه تطبيقه
+
+204
+00:15:23,840 --> 00:15:29,560
+express w by r and w by s in terms of r and s يعني
+
+205
+00:15:29,560 --> 00:15:33,620
+بيقوللي أو دي ال دي ال w by ال r أو ال w by ال s
+
+206
+00:15:33,620 --> 00:15:39,200
+اللي هي لنين w بساوي x زي 2y زي z تربيع و x ايش
+
+207
+00:15:39,200 --> 00:15:43,310
+بتساويR على S يعني صارت X function في R و في S و
+
+208
+00:15:43,310 --> 00:15:46,590
+YS بيساوي R تربيع Z لإنها صارت إذن function of two
+
+209
+00:15:46,590 --> 00:15:51,730
+variables R و S و Z بيساوي A2R دل W by دل R حسب
+
+210
+00:15:51,730 --> 00:15:56,150
+القانون اللي هي دل W by دل X و دل X by دل R إلى
+
+211
+00:15:56,150 --> 00:16:01,570
+آخره تنطبق الآن دل W by دل X اللي يبفضل W بالنسبة
+
+212
+00:16:01,570 --> 00:16:06,210
+لX سنة واحدوهدولة صفر إذا بيظل واحد مضروب X على R
+
+213
+00:16:06,210 --> 00:16:10,990
+دل W by دل تفاضل X بالنسبة ل R اللي هي عبارة عن
+
+214
+00:16:10,990 --> 00:16:14,090
+واحد على S لأنه بيثبت ال S و بيفاضل بالنسبة ل R
+
+215
+00:16:14,090 --> 00:16:19,510
+بيصير واحد على S الان دل W by دل Y اللي هي عبارة
+
+216
+00:16:19,510 --> 00:16:24,890
+عن جداش اتنين دل W by دل Y عبارة عن اتنين لإن
+
+217
+00:16:24,890 --> 00:16:32,630
+هدولة أصفر هي اتنينDel Y by Del R أي Del Y by Del
+
+218
+00:16:32,630 --> 00:16:37,870
+R هي اتنين R وهذا بتتصفر زائد Del W by Del Z هي
+
+219
+00:16:37,870 --> 00:16:44,070
+عبارة عن ايش اتنين Z بتنزل Del Z by Del R هي عبارة
+
+220
+00:16:44,070 --> 00:16:49,210
+عن برضه ايش اتنين اللي هي بتنزل بنبسطها بالـ C1S
+
+221
+00:16:49,210 --> 00:16:53,810
+زائد 4R زائد 4Zوفي الآخر لما بدك تعوض مكان Z
+
+222
+00:16:53,810 --> 00:16:56,970
+بقيمته عشان توجده كله بدلالة R لأن الطالب اللي هو
+
+223
+00:16:56,970 --> 00:17:01,090
+دي ال W by دي ال R يعني بدهيها as in هو جالك in
+
+224
+00:17:01,090 --> 00:17:05,010
+terms of R and S لأن ال Z إذا بدنا نشيلها وبدنا
+
+225
+00:17:05,010 --> 00:17:11,070
+نشوف قيمتها إيش قيمتها؟ 2R عوضناها بصير 2R و 4R 6R
+
+226
+00:17:11,070 --> 00:17:18,830
+مظبوط؟ لأ بصير 2 في 4 في 8R زائد اللي هي 4R 12R 1
+
+227
+00:17:18,830 --> 00:17:26,740
+على S نجي للبعدبنفس الطريقة هي القانون دل W بي دل
+
+228
+00:17:26,740 --> 00:17:30,780
+X هي الموجودة تعددناها واحد وهي اتنين وهي اتنين Z
+
+229
+00:17:30,780 --> 00:17:38,520
+هدولة دل W بي دل X ودل W بي دل Y ودل W بي دل Z كل
+
+230
+00:17:38,520 --> 00:17:42,600
+الموجودات من السؤال اللي جابته دل نود مين دل X بي
+
+231
+00:17:42,600 --> 00:17:47,160
+دل SX by DS هي عبارة عن اياش الـ R ثابتة الـ R
+
+232
+00:17:47,160 --> 00:17:50,760
+ثابتة بيصير الـ R في تفاضل الواحد على S اللي هو
+
+233
+00:17:50,760 --> 00:17:57,140
+ناقص واحد على S تربيعها الان DY by DS الهي DY by
+
+234
+00:17:57,140 --> 00:18:00,280
+DS اللي هي هذي بيصير سفر بيصير واحد على S لإن
+
+235
+00:18:00,280 --> 00:18:04,280
+انتفاض الواحد على S الان في الآخر ΔZ by DS دي دي
+
+236
+00:18:04,280 --> 00:18:05,880
+دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي
+
+237
+00:18:05,880 --> 00:18:06,280
+دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي
+
+238
+00:18:06,280 --> 00:18:07,460
+دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي
+
+239
+00:18:07,460 --> 00:18:15,490
+دي دي دزائد 2 على S يعني 2S نقص R على S تربيع وهو
+
+240
+00:18:15,490 --> 00:18:20,770
+المطلوب الآن لو كان عند W بسوء F of X و Y بدل ما
+
+241
+00:18:20,770 --> 00:18:24,650
+هي F of X و Y و Z بصير سهل بصير حالة خاصة من اللي
+
+242
+00:18:24,650 --> 00:18:29,820
+فوق و X function برضه G R of S Y H of R و Sبصير
+
+243
+00:18:29,820 --> 00:18:33,380
+اللي هو ما تظهرش مين يظهرش الطرف التالت هذا ال
+
+244
+00:18:33,380 --> 00:18:36,280
+term التالت هذا بصير ما تظهرش بس بيظهر مين دول
+
+245
+00:18:36,280 --> 00:18:40,360
+الاتنين اللي هو del W by del X خطين يعني بيصير خط
+
+246
+00:18:40,360 --> 00:18:45,680
+ل X وخط ل Y del W by del X في del X by del R زائد
+
+247
+00:18:45,680 --> 00:18:48,980
+del W by del Y في del Y by del R وتبع الـ Z ما هو
+
+248
+00:18:48,980 --> 00:18:53,100
+مش موجود إذا صار بالشكل هذا و similarly لمين ده
+
+249
+00:18:53,100 --> 00:19:00,780
+الـ Wبعد ذلك طرب السؤال مثال على القانون اللي فوق
+
+250
+00:19:00,780 --> 00:19:05,260
+express del w by del r and del w by del s in terms
+
+251
+00:19:05,260 --> 00:19:10,000
+of r and s if w بسوء x تربيع زاد y تربيع و x بسوء
+
+252
+00:19:10,000 --> 00:19:13,240
+r ناقص s و y بسوء r زائد s
+
+253
+00:19:16,520 --> 00:19:20,120
+القانون على طول هي القانون فوق اللي هي del w by
+
+254
+00:19:20,120 --> 00:19:24,180
+del r هي تبعت ال X وهي تبعت main ال Y خط ال X وخط
+
+255
+00:19:24,180 --> 00:19:26,460
+ال Y مجموعات ويساوي فاهمين عن ايش معاناة اللي
+
+256
+00:19:26,460 --> 00:19:29,980
+بحكيه فاهمين على بعض del w by del x ايش بيساوي
+
+257
+00:19:29,980 --> 00:19:34,960
+اتنين X الان del x by del r del x by del r واحد
+
+258
+00:19:36,460 --> 00:19:44,320
+زائد del W by del Y التي هي 2Y مضروبة
+
+259
+00:19:44,320 --> 00:19:51,300
+في D del Y by del R التي هي 1 بيصير 2X زائد 2Y الـ
+
+260
+00:19:51,300 --> 00:19:55,500
+X بيعودها لـ R minus S بيصير 2R minus S والـ Y
+
+261
+00:19:55,500 --> 00:19:59,760
+بقيمتها R زائد S اللي هي بنجمحها مع بعض بيطلع 4R
+
+262
+00:19:59,760 --> 00:20:03,600
+والـ 2S بيروح مع الـ 2 نقص 2S بيطلع إلى الجواب 4R
+
+263
+00:20:04,160 --> 00:20:15,600
+مثلًا ومثلًا ومثلًا ومثلًا ومثلًا ومثلًا
+
+264
+00:20:15,600 --> 00:20:24,240
+ومثلًا
+
+265
+00:20:26,560 --> 00:20:30,340
+ماشي الحل دي الـ y by دي الـ s واحد هاي عوضناها
+
+266
+00:20:30,340 --> 00:20:35,640
+اللي هبصي ناقص 2x زي 2y بنعوض عن x القيمتها اللي
+
+267
+00:20:35,640 --> 00:20:39,840
+هي ناقص 2r minus s وعن y القيمتها 2r زي s التين مع
+
+268
+00:20:39,840 --> 00:20:44,200
+بعض بيطلعين ايش بيساوي أربعة اللي هي s وكونها ده
+
+269
+00:20:44,200 --> 00:20:48,780
+مثلناطيب لان نيجي ناخد اللي هو W بساوية F of X لو
+
+270
+00:20:48,780 --> 00:20:52,120
+كانت عندنا W بس function of one variable إذا خط
+
+271
+00:20:52,120 --> 00:20:56,360
+واحد اللي بنزل A لان and X بساوية G R of S يعني X
+
+272
+00:20:56,360 --> 00:20:58,680
+function في مين؟ في R و S يعني function of two
+
+273
+00:20:58,680 --> 00:21:03,480
+variables سهلة، بده Del W by Del R لأنه صارت W
+
+274
+00:21:03,480 --> 00:21:07,800
+function في R و S لان بده Del W by Del R خط هو الـ
+
+275
+00:21:07,800 --> 00:21:11,660
+X خطها واحد، هيه و بعدين بنزل منها مين؟ اللي هي
+
+276
+00:21:11,660 --> 00:21:17,670
+الـ Rبس بالنسبة لـ W بالنسبة لـ R دي الـ W بالنسبة
+
+277
+00:21:17,670 --> 00:21:24,530
+لـ R بساوي DW بدي DX هذه في DX بدي DX بالمية R
+
+278
+00:21:24,530 --> 00:21:26,490
+لإنه partial هذه لإنه function of two variables
+
+279
+00:21:26,490 --> 00:21:32,370
+لإن سمينار ليه DW بدي ال S بساوي دي اللي برضه خط
+
+280
+00:21:32,370 --> 00:21:35,950
+واحد خط واحد بس بده يصير اللي تحته مين S بصير DW
+
+281
+00:21:35,950 --> 00:21:42,120
+بدي ال S بساوي DW بدي DXفي دل X by دي الـ Main by
+
+282
+00:21:42,120 --> 00:21:45,400
+دي الأصل فان شاء الله يكون واضح اللي بحكيه إلى أن
+
+283
+00:21:45,400 --> 00:21:47,840
+نيجي لحاجة اسمها ال implicit differentiation أو
+
+284
+00:21:47,840 --> 00:21:52,940
+التفاضل Revisited اللي هو اللي هي نشوف كيف نوجدها
+
+285
+00:21:52,940 --> 00:21:56,660
+أو احنا أخدنا اشي من هالنوع في اللي هو التفاضل الف
+
+286
+00:21:56,660 --> 00:22:03,410
+إلى أن ناخد اللي هو النظرية الليالتي تتعامل مع كيف
+
+287
+00:22:03,410 --> 00:22:11,210
+يكون التفاضل الضمني X و Y مرتبطات مع بعض من خلال
+
+288
+00:22:11,210 --> 00:22:16,690
+فورميلة أو معادلة أو من خلال كذا فبتكون واحدة
+
+289
+00:22:16,690 --> 00:22:20,150
+dependent واحدة independent نشوف الشيء اللي بقوله
+
+290
+00:22:20,150 --> 00:22:24,730
+Suppose that f of x y is a differentiable function
+
+291
+00:22:24,730 --> 00:22:28,430
+and that the equation f of x y بتساوي سفر
+
+292
+00:22:33,010 --> 00:22:40,030
+بتعرف Y كدالة
+
+293
+00:22:40,030 --> 00:22:45,430
+قابلة للاشتقاق في X ثم في أي موقع فيه FY ليسوا سفر
+
+294
+00:22:48,090 --> 00:22:52,170
+لو بدى اودد ان صارت y function في x dy by dx ايش
+
+295
+00:22:52,170 --> 00:22:57,630
+هتساوي ناقص fx هي ال fx عندي و f of x و y هذه ناقص
+
+296
+00:22:57,630 --> 00:23:04,410
+fx لها على مين على ال fy هذه dy by dx لو انت بدك
+
+297
+00:23:04,410 --> 00:23:07,990
+تلاحظ كيف اوضدناها تنشوف كيف اوضدناها لو اجينا
+
+298
+00:23:07,990 --> 00:23:12,790
+اخدنا w بساوية f of x و y ماشي لأن هذه صارت w
+
+299
+00:23:12,790 --> 00:23:16,050
+function في مين في x و y او ال f function في x و y
+
+300
+00:23:16,640 --> 00:23:22,580
+الآن هذه التي لها خطين هذه خطلة الـ X خطلة الـ Y
+
+301
+00:23:22,580 --> 00:23:28,440
+ياسف وخطلة مين للـ X الآن لو فرضنا أن هذه F هذه
+
+302
+00:23:28,440 --> 00:23:32,380
+بتساوي سفر يعني العلاقة اللي حكيناها قبل شوية F of
+
+303
+00:23:32,380 --> 00:23:38,440
+XY بتساوي سفر تعرفالعلاقة بين X و Y بحيث أنه تكون
+
+304
+00:23:38,440 --> 00:23:42,660
+Y as a function mean of X صارت Y دلّا في X هذا
+
+305
+00:23:42,660 --> 00:23:46,180
+بحكي عن الحالة اللي قبل بشوية التفاضل الضمني يعني
+
+306
+00:23:46,180 --> 00:23:50,040
+هذه إيش بده أسويها الآن إبسفر فبصير Y function في
+
+307
+00:23:50,040 --> 00:23:53,280
+main في X عشان هيك كاتب Y أبنالها function of X
+
+308
+00:23:53,280 --> 00:23:59,260
+وبالتفاضل هذه outed اللي هي التفاضلهابالنسبة لمين
+
+309
+00:23:59,260 --> 00:24:03,520
+بدي اود الـ dy by dx لتشوف كيف بدي اصل ال bridge
+
+310
+00:24:03,520 --> 00:24:07,560
+هذا يعني بدي افعض ال del w اللي انا سميتها سفر ايش
+
+311
+00:24:07,560 --> 00:24:12,100
+بيساوي by del x اللي هنا هي x وهذه اللي هي ال
+
+312
+00:24:12,100 --> 00:24:15,540
+variable التاني x صار عندي طريقين هي واحد y وهي
+
+313
+00:24:15,540 --> 00:24:19,810
+واحد x ال x طبيعي function في نفسهاوالـ y function
+
+314
+00:24:19,810 --> 00:24:23,870
+of x على اعتبار انه احنا لما ناخد هذه ايش مالها f
+
+315
+00:24:23,870 --> 00:24:28,550
+of x وw او y بتساوي ايش سفر تعمل علاقة بين y أو
+
+316
+00:24:28,550 --> 00:24:33,490
+بين x او y as a function of x خليني نشوف كيف نصل
+
+317
+00:24:33,490 --> 00:24:36,410
+ال bridge هذا اللي عن كيف اللي كنا نستخدم نقول
+
+318
+00:24:36,410 --> 00:24:45,550
+اللي عن dw by dx هذه بساوي اللي هو fxفي dx التي هي
+
+319
+00:24:45,550 --> 00:24:50,170
+fx يعني دي الـ w by دي الـ x في dx by dx هذا اللي
+
+320
+00:24:50,170 --> 00:24:54,370
+كانت بدلتيه واحد مستوى إذا صارت الـ fx في مين في
+
+321
+00:24:54,370 --> 00:24:58,910
+واحد هذه fx في واحد زائد الان دي ال w by دي ال y
+
+322
+00:24:58,910 --> 00:25:04,150
+أو ال fy في مين في تفاضل هذه dy by dx اللي هي h
+
+323
+00:25:04,150 --> 00:25:08,700
+prime of x الـ dy by dxالان اذا كما قلنا احنا على
+
+324
+00:25:08,700 --> 00:25:14,600
+اعتبار ان W ستساوي كمية 0 اذا بيصير عندى W بيساوي
+
+325
+00:25:14,600 --> 00:25:18,300
+0 يعني و W بيساوي X في هذه الحالة ستصبح 0 اذا
+
+326
+00:25:18,300 --> 00:25:21,880
+ستصبح هذا سفر مدان صارت هذه سفر اذا صارت عندى
+
+327
+00:25:21,880 --> 00:25:25,040
+بنجلي هذا على الجهة الثانية بيصير ناقص FX و بيصير
+
+328
+00:25:25,040 --> 00:25:28,920
+DY بيدي DX بيساوي ناقص FX و باكسم على FY بيطلع
+
+329
+00:25:28,920 --> 00:25:34,070
+ناقص FXعلى f y اللي هي العلاقة اللي عندنا دي y by
+
+330
+00:25:34,070 --> 00:25:38,290
+دي x بسوة و ناقص f x على f y عندما تكون اللي هي w
+
+331
+00:25:38,290 --> 00:25:42,410
+بتساوي سفر أو من عن آخر f of x y بتساوي سفر تعرف
+
+332
+00:25:42,410 --> 00:25:47,850
+علاقة بين y و x as اللي هي x independent و y
+
+333
+00:25:47,850 --> 00:25:52,050
+dependent تعتمد عالميا على ال y هذا احنا مش ترطينه
+
+334
+00:25:52,380 --> 00:25:56,160
+الأن بقول الـ use theorem تمانية to find dy by dx
+
+335
+00:25:56,160 --> 00:26:00,720
+الان dy by dx على طول اللي هي ال y ال dependent
+
+336
+00:26:00,720 --> 00:26:05,160
+والx إيش ماله independent fy تربيع نقص x تربيع نقص
+
+337
+00:26:05,160 --> 00:26:09,700
+sin xy إيش بتساوي سفر هذه بدك توجد dy by dx معطيك
+
+338
+00:26:09,700 --> 00:26:13,420
+هذه إيش بتساوي سفر أمامنا أمرين إما نحل زي زمان زي
+
+339
+00:26:13,420 --> 00:26:16,780
+تفاضه الألف كيف نحل زي تفاضه الألف على طول كل اللي
+
+340
+00:26:16,780 --> 00:26:22,820
+قدامنا انفضله بالنسبة لx يعني بيصير هذه 2y ناقص2y
+
+341
+00:26:22,820 --> 00:26:30,140
+dy by dx ناقص 2x ناقص sin x في y اللي هي بتفاضل هي
+
+342
+00:26:30,140 --> 00:26:34,800
+إيش هو عبارة عن cos x في y في تفاضل اللي جوا في
+
+343
+00:26:34,800 --> 00:26:37,660
+جوس اللي هي x اللي هي عبارة عن تفاضل اللي واحد ده
+
+344
+00:26:37,660 --> 00:26:43,400
+بيصير في y زائد اللي هي x في dy by dx و بناخد dy و
+
+345
+00:26:43,400 --> 00:26:47,000
+dx عامل مشترك و بننجل على الطرف التاني و بنحل و
+
+346
+00:26:47,000 --> 00:26:51,410
+بنحل زي ما كنا زمان مش هذا اللي بديهاولكن قلت لكم
+
+347
+00:26:51,410 --> 00:26:55,210
+كيف تحلوا اللي بده استذكر حل زمان ممكن يحل زيه
+
+348
+00:26:55,210 --> 00:26:58,750
+يوجد DIY by DX جربوا حلوها لحالكم DIY by DX
+
+349
+00:26:58,750 --> 00:27:03,930
+وشوفوها بتكافئ اللي هو جوابها نفس جوابنا أكيد لازم
+
+350
+00:27:03,930 --> 00:27:07,730
+تفعل الجواب اللي هالنيدي أيش اللي بتقوله بقى اللي
+
+351
+00:27:07,730 --> 00:27:11,250
+بقوله أن أنا الأن بديش أحل بطريقة زمان بدي أحل
+
+352
+00:27:11,250 --> 00:27:17,270
+بطريقة اليومالآن بدي أسمّي هذه كلها أف of x و y
+
+353
+00:27:17,270 --> 00:27:20,090
+إذا أصبحت أف of x و y بيسوّي تربيع نقص x تربيع
+
+354
+00:27:20,090 --> 00:27:25,550
+نقصين x و y اللي هي في الواقع سفر، هي سفر، مصبوط؟
+
+355
+00:27:25,550 --> 00:27:29,790
+أه، صارت عندي هذه العلاقة أف of x و y بتعرف اللي
+
+356
+00:27:29,790 --> 00:27:33,570
+هو بيسوّي سفر بتعرف y as a function of xإذا بقدر
+
+357
+00:27:33,570 --> 00:27:36,990
+أطبّق عليها مين اللي حكيتها بيصير dy by dx بساوي
+
+358
+00:27:36,990 --> 00:27:41,970
+ناقص fx على fy fx إيش بتساوي في هذه الحالة لـ fx
+
+359
+00:27:41,970 --> 00:27:46,590
+هذه ثابتة تصير سفر بيصير ناقص 2x ناقص الـ y ثبتت
+
+360
+00:27:46,590 --> 00:27:50,980
+بيصير ناقصكوساين الـ x في y في تفاضل اللي جوا
+
+361
+00:27:50,980 --> 00:27:54,680
+تفاضل الـ x في y اللي هو y بيصير ناقص اتنين x ناقص
+
+362
+00:27:54,680 --> 00:27:59,960
+y كوساين x y على mean of y of y فضلة بالنسبة ل y
+
+363
+00:27:59,960 --> 00:28:05,620
+بيصير اتنين y ناقص اللي هو كوساين x في y مضروبة في
+
+364
+00:28:05,620 --> 00:28:09,540
+تفاضل اللي جوا بالنسبة ل y اللي هي xهذا المقدار
+
+365
+00:28:09,540 --> 00:28:13,640
+يصبح دي وي بي دي اكس او بمعنى اخر لما ندخل ناقص
+
+366
+00:28:13,640 --> 00:28:17,460
+هنا يصبح اتنين اكزاد y cos xy على اتنين y ناقص x
+
+367
+00:28:17,460 --> 00:28:18,640
+cos xy
+
+368
+00:28:28,790 --> 00:28:32,430
+هذه اللي حكيناها قبل شوية نيجي الآن لو كان عندي
+
+369
+00:28:32,430 --> 00:28:37,190
+اللي هو three variables يعني عندي بمعنى آخر let x
+
+370
+00:28:37,190 --> 00:28:40,590
+and y be two independent variables يعني نفترض أن x
+
+371
+00:28:40,590 --> 00:28:43,550
+و y هو بيقولنا في السؤال x و y two independent
+
+372
+00:28:43,550 --> 00:28:48,790
+variables و z is the dependent variable يعني z
+
+373
+00:28:48,790 --> 00:28:52,970
+function في مين بمعنى آخر x و y كيف معرفة في x و y
+
+374
+00:28:52,970 --> 00:28:58,070
+معرفة ضمنيا where they are related implicitly by
+
+375
+00:28:58,440 --> 00:29:02,280
+يعني هدولة التلت متغيرات اللي اللي هو الـ Z
+
+376
+00:29:02,280 --> 00:29:07,000
+dependent والـ XY independent معرفات مع بعض ضمنيا
+
+377
+00:29:07,000 --> 00:29:12,300
+بواسطة المعادلة F of XY و Z بتساوي إيه؟ سفر يعني F
+
+378
+00:29:12,300 --> 00:29:17,660
+XY Z function بتساوي سفر أقل لإن اللي هي Z as a
+
+379
+00:29:17,660 --> 00:29:22,960
+function of Y أو X معرفة ضمنيا بدنا الآن اللي هو
+
+380
+00:29:22,960 --> 00:29:28,690
+نشتق القوانين التالية خلينا نشوفبنفضل كله بالنسبة
+
+381
+00:29:28,690 --> 00:29:40,850
+لمن؟ بالنسبة ل X فضله Zero بالنسبة ل X Zero فضله
+
+382
+00:29:40,850 --> 00:29:42,810
+Zero بالنسبة ل X Zero فضله Zero بالنسبة ل X Zero
+
+383
+00:29:42,810 --> 00:29:45,690
+فضله Zero بالنسبة ل X Zero فضله Zero بالنسبة ل X
+
+384
+00:29:45,690 --> 00:29:47,430
+Zero فضله Zero بالنسبة ل X Zero فضله Zero بالنسبة
+
+385
+00:29:47,430 --> 00:29:52,090
+ل X Zeroفى دل X by دل X لأن النقطة الأخيرة الـ X
+
+386
+00:29:52,090 --> 00:29:57,550
+دل F by دل Y فى دل Y by دل X زائد دل F by دل Z فى
+
+387
+00:29:57,550 --> 00:30:01,110
+دل Z by دل X هاي خط خطين تلاتة وكلين فى النقطة الـ
+
+388
+00:30:01,110 --> 00:30:08,170
+X الان FX هذه دل X by دل X واحد طبيعى FY الـ Y و
+
+389
+00:30:08,170 --> 00:30:11,330
+الـ X independent مدام Independent لدى الـ Y لا
+
+390
+00:30:11,330 --> 00:30:15,330
+تعتمد على الـ X يعني تفاضلهم بالنسبة لبعض صفرإذاً
+
+391
+00:30:15,330 --> 00:30:18,830
+هذي إيش بتصير؟ سفر، إذاً هذا الterm هيزول الآن دي
+
+392
+00:30:18,830 --> 00:30:22,590
+الـ F by دي الـ Z بتنزل FZ دي الـ Z by دي الـ X
+
+393
+00:30:22,590 --> 00:30:26,450
+اللي هو جاي لك Z function في X و Y يعني فيه بينهم
+
+394
+00:30:26,450 --> 00:30:30,410
+تفاضل يعني دي الـ Z by دي الـ X بتنزل زي ما هي
+
+395
+00:30:30,410 --> 00:30:36,190
+الآن هذا صار عندي أشير لهذه صارت FX زائد FZ في دي
+
+396
+00:30:36,190 --> 00:30:40,840
+الـ Z by دي الـ X إيش بتساوي؟ بتساوي سفرمن هذه من
+
+397
+00:30:40,840 --> 00:30:45,460
+هذه بقدر اوضت del z by del x بجيب هذا على الطرف
+
+398
+00:30:45,460 --> 00:30:49,500
+هذا بيصير ناقص fx بكسبه على fz بيصير del z by del
+
+399
+00:30:49,500 --> 00:30:56,660
+x بيساوي ناقص fx عالميا على fz إذن لما تكونمعرّفة
+
+400
+00:30:56,660 --> 00:31:01,280
+ضمنياً Z as a function of Y of X بيصير Del Z by
+
+401
+00:31:01,280 --> 00:31:05,600
+Del X اللي هي العلاقة اللي جوانية Del Z by Del X
+
+402
+00:31:05,600 --> 00:31:09,160
+بيساوي ناقص اللي هي FX
+
+403
+00:31:14,200 --> 00:31:21,740
+بالنسبة للجهتين Y وجيت واشتغلت ستجد هذه تتصفر
+
+404
+00:31:21,740 --> 00:31:29,230
+وستبقى هذه موجودة فهيصير عند FY زي FZومنه اللي هو
+
+405
+00:31:29,230 --> 00:31:33,630
+بيكون عندى del z by del y بيساوي ناقص fy على مين
+
+406
+00:31:33,630 --> 00:31:39,930
+على fz بهذا كله تحت شرط أن fz لا تساوي إيش؟ لا
+
+407
+00:31:39,930 --> 00:31:45,740
+تساوي سفر إذا الآن لما تكون الإختصار الحديدF of X
+
+408
+00:31:45,740 --> 00:31:49,620
+Y و Z بيساوي صفر بتعرف لي ضمنيا Z as a function of
+
+409
+00:31:49,620 --> 00:31:55,840
+X Y بقدر أفاضل Del Z by Del X كيف بتساوي ناقص F
+
+410
+00:31:55,840 --> 00:32:02,000
+هذه بالنسبة ل X على F بالنسبة ل Z والتانية
+
+411
+00:32:02,000 --> 00:32:06,590
+Similarly وخلّيني أخد مثال على هذا الكلامنجد دل زد
+
+412
+00:32:06,590 --> 00:32:12,130
+با دل إكس ودل و .. دل زد با دل واي at zero zero
+
+413
+00:32:12,130 --> 00:32:17,390
+zero if x تكعيب زاد z تربيع زاد y e to the xz زاد
+
+414
+00:32:17,390 --> 00:32:25,680
+z cos y بساوي إيش بساوينشوف كيف بننحل عندي الآن
+
+415
+00:32:25,680 --> 00:32:30,600
+معطيني اللي هي X تكعيب زي Z تربيع زي Y زي Z بساوية
+
+416
+00:32:30,600 --> 00:32:35,160
+0 اللي هو ضمنيا اللي هو مادة مطالبة دل Z by دل X
+
+417
+00:32:35,160 --> 00:32:39,640
+ودل Z by دل Y عندي 00 وكذا وكذا ومعطيني علاقة
+
+418
+00:32:39,640 --> 00:32:43,500
+معناته بوكسود اللي هو Z as a function of X أو Y
+
+419
+00:32:43,500 --> 00:32:46,580
+وإلا ليش يسأل عنه لو Z و X independent هدا هتطلع
+
+420
+00:32:46,580 --> 00:32:50,940
+سفره دي سفر Automatic فاللي هوما دام هو الآن يعني
+
+421
+00:32:50,940 --> 00:32:54,900
+حاتب الأسلوب هذا اللي هو معناته اللي هو مفترض أن Z
+
+422
+00:32:54,900 --> 00:32:57,420
+as a function of X وY واحنا إن شاء الله عزيزي ما
+
+423
+00:32:57,420 --> 00:33:02,300
+اتخربتوش وبنكتبلكم اللي هو الكلام هذا بصراحة الان
+
+424
+00:33:02,300 --> 00:33:05,400
+since X and Y are two independent variables and Z
+
+425
+00:33:05,400 --> 00:33:08,020
+is the dependent variable where they are related
+
+426
+00:33:08,020 --> 00:33:12,500
+implicitly by اللي هي هذه بيساوي سفر هذه بدي
+
+427
+00:33:12,500 --> 00:33:16,940
+أسميها إيش كلها هذه هذه هذه للسفر هذا كله بدي
+
+428
+00:33:16,940 --> 00:33:21,890
+أسميه F of X Y وZالآن مدام F of X Y و Z و ساوة سفر
+
+429
+00:33:21,890 --> 00:33:25,210
+إذا بقدر أطبق النظرية اللي قبل شوية Del Z by Del Z
+
+430
+00:33:25,210 --> 00:33:30,070
+X مش بيساوي ناقص F X على F Z F X يعني بده فاضل
+
+431
+00:33:30,070 --> 00:33:34,230
+هادي partial derivative with respect to X اللي
+
+432
+00:33:34,230 --> 00:33:40,030
+عبارة عن تلاتة X تربيع هيزائد تفاضل هذي سفر وهذا
+
+433
+00:33:40,030 --> 00:33:48,410
+يصبح Z Y E X Z فتفاضل اللي جوا X Z اللي هو تفاضله
+
+434
+00:33:48,410 --> 00:33:53,570
+بالنسبة ل X تطلع Z فبصير Z Y E X Z الان هذه
+
+435
+00:33:53,570 --> 00:33:56,870
+التفاضل اللي هي بالنسبة ل X طبعا Z Cos Y Ash بسوى
+
+436
+00:33:56,870 --> 00:34:02,660
+سفر هذه السفرإذا صار عندى هذا المقدار على fz ناخد
+
+437
+00:34:02,660 --> 00:34:08,940
+fz لهذه اتنين z هي اتنين z زائد تفاضل هذه اللي هو
+
+438
+00:34:08,940 --> 00:34:12,880
+y e to the xz مضروب في تفاضل اللي جوا بالنسبة لz
+
+439
+00:34:12,880 --> 00:34:17,780
+اللي هو x اللي هي بيصير x في y e to the xz زائد
+
+440
+00:34:17,780 --> 00:34:21,560
+تفاضل هذه بالنسبة هذه اللي ضايلة بالنسبة لz اللي
+
+441
+00:34:21,560 --> 00:34:26,030
+هي واحد في cosine ال Y زايد cosine ال Yإذاً طلعت
+
+442
+00:34:26,030 --> 00:34:29,270
+هذه دي الـ z by دي الـ x إذا بدنا نحسبها عند السفر
+
+443
+00:34:29,270 --> 00:34:32,330
+والسفر والسفر نسينا نحسبها عند السفر والسفر السفر
+
+444
+00:34:32,330 --> 00:34:37,870
+احسب هذه سفر وهذه سفر وهذه طبعاً سفر سفر وهذه واحد
+
+445
+00:34:37,870 --> 00:34:42,070
+أكيد قصان Zero واحد إذاً سفر على واحد تطلع أيش
+
+446
+00:34:42,070 --> 00:34:45,590
+بتساوي سفر إذا دلت التفاوض لدي الـ z by دي الـ x
+
+447
+00:34:45,590 --> 00:34:50,290
+عند السفر والسفر والسفر بتساوي سفر نشوف اللي بعدها
+
+448
+00:34:50,290 --> 00:34:54,920
+يا جماعة اللي بعدها ليه؟اللي هي del z by del y
+
+449
+00:34:54,920 --> 00:34:59,820
+بيساوي ناقص fy على fz ناقص fy هي y هنا ناقص fy و
+
+450
+00:34:59,820 --> 00:35:04,460
+ال fz اللي هي fz تحت لأن fy انتفاض بالنسبة ل y هذه
+
+451
+00:35:04,460 --> 00:35:10,600
+هذي سفر وهذه سفر بتظهر لها إيه إيه إيه إيه إيه إيه
+
+452
+00:35:10,600 --> 00:35:13,480
+إيه إيه إيه إيه إيه إيه إيه إيه إيه إيه إيه إيه
+
+453
+00:35:17,150 --> 00:35:24,010
+اللي هي Z في Cos Y اللي هي زائد Z في Cos Y اللي هي
+
+454
+00:35:24,010 --> 00:35:29,870
+تفاضل بالنسبة ل Y اللي هي تفاضل Cos هي ناقص Sin Y
+
+455
+00:35:29,870 --> 00:35:33,250
+اللي هي بيصير Z في ناقص Sin Y وفي عند النقص اللي
+
+456
+00:35:33,250 --> 00:35:37,770
+برا بيصير زائد Z Sin Y فاهمين عليها تفاضل بالنسبة
+
+457
+00:35:37,770 --> 00:35:42,940
+ل Yهذه الأصفار هذي بيصير e to the x z بتنزل هذه
+
+458
+00:35:42,940 --> 00:35:46,960
+الأن تفضلها بالنسبة ل y الـ z تابتة هذه ناقص sign
+
+459
+00:35:46,960 --> 00:35:51,700
+y يعني ناقص z sign y ناقص z sign y والناقص اللي
+
+460
+00:35:51,700 --> 00:35:55,480
+برا دخلت فدخلت هذه صارت ناقص وهذه صارت زائد وهذه
+
+461
+00:35:55,480 --> 00:35:58,730
+زي ما هي اللي تحت اللي هي ال z فضلناها فوقإن
+
+462
+00:35:58,730 --> 00:36:03,430
+نحسبها عند مين بدّو 000 بحط مكان هذه Zero هتصير
+
+463
+00:36:03,430 --> 00:36:07,090
+هذه ناقص واحد وهذه سفر وهذا واحد إذا بيطلع الجواب
+
+464
+00:36:07,090 --> 00:36:11,910
+إيش بيساوي ناقص واحد لأن هذه سفر وهذه سفر وهذه سفر
+
+465
+00:36:11,910 --> 00:36:17,330
+وهذه واحد وهذه ناقص واحد ويساوي ناقص واحد هذه اللي
+
+466
+00:36:17,330 --> 00:36:24,190
+المطلوب ويكون خلصنا المحاضرة وإلى لقاء آخر إن شاء
+
+467
+00:36:24,190 --> 00:36:24,270
+الله
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/vPw26YH1fW0_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/vPw26YH1fW0_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..a9b6cf0c2b9b1a13b522794498073b39ccd34950
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/vPw26YH1fW0_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 836, "start": 1.15, "end": 8.37, "text": "بسم الله الرحمن الرحيم السلام عليكم ورحمة الله وبركاته هذه هي المحاضرة الرقم 16", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 29538, 39896, 9673, 5016, 46958, 25720, 34892, 4587, 2304, 3165], "avg_logprob": -0.08725873337072484, "compression_ratio": 1.4895833333333333, "no_speech_prob": 9.775161743164062e-06, "words": [{"start": 1.15, "end": 1.45, "word": "بسم", "probability": 0.861328125}, {"start": 1.45, "end": 1.67, "word": " الله", "probability": 0.94287109375}, {"start": 1.67, "end": 2.05, "word": " الرحمن", "probability": 0.9825846354166666}, {"start": 2.05, "end": 2.69, "word": " الرحيم", "probability": 0.9943033854166666}, {"start": 2.69, "end": 3.89, "word": " السلام", "probability": 0.880615234375}, {"start": 3.89, "end": 4.29, "word": " عليكم", "probability": 0.949462890625}, {"start": 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33744, "start": 319.36, "end": 337.44, "text": "لإيجاد هذه الـ derivative باسمين with respect to t along the path x بساوة و cosine of t و y بساوة و sine of t what is the derivatives value at t بتساوية y على اتنين اللي طالبوا عندي w function في x y", "tokens": [1211, 28814, 1829, 7435, 18513, 29538, 2423, 39184, 13760, 4724, 32277, 2304, 9957, 365, 3104, 281, 256, 2051, 264, 3100, 2031, 4724, 3794, 995, 2407, 3660, 4032, 23565, 295, 256, 4032, 288, 4724, 3794, 995, 2407, 3660, 4032, 18609, 295, 256, 437, 307, 264, 33733, 2158, 412, 256, 39894, 3794, 995, 2407, 10632, 288, 15844, 1975, 2655, 1863, 9957, 13672, 1829, 23032, 6027, 3555, 14407, 18871, 16254, 261, 2445, 8978, 2031, 288], "avg_logprob": -0.3604452120114679, "compression_ratio": 1.4262295081967213, "no_speech_prob": 2.4497509002685547e-05, "words": [{"start": 319.36, "end": 319.96, "word": "لإيجاد", "probability": 0.8162109375}, {"start": 319.96, "end": 320.2, "word": " هذه", "probability": 0.32666015625}, {"start": 320.2, 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136543, "start": 1353.01, "end": 1365.43, "text": "بتعرف Y كدالة قابلة للاشتقاق في X ثم في أي موقع فيه FY ليسوا سفر", "tokens": [3555, 2655, 3615, 28480, 398, 9122, 3215, 6027, 3660, 12174, 16758, 37977, 5296, 15040, 8592, 2655, 4587, 995, 4587, 8978, 1783, 38637, 2304, 8978, 36632, 3714, 30543, 3615, 8978, 3224, 479, 56, 32239, 3794, 14407, 8608, 5172, 2288], "avg_logprob": -0.524038437085274, "compression_ratio": 1.2087912087912087, "no_speech_prob": 0.0005731582641601562, "words": [{"start": 1353.01, "end": 1353.85, "word": "بتعرف", "probability": 0.5687332153320312}, {"start": 1353.85, "end": 1354.45, "word": " Y", "probability": 0.371337890625}, {"start": 1354.45, "end": 1360.03, "word": " كدالة", "probability": 0.765869140625}, {"start": 1360.03, "end": 1360.89, "word": " قابلة", "probability": 0.9059244791666666}, {"start": 1360.89, "end": 1361.57, "word": " للاشتقاق", "probability": 0.7247140066964286}, {"start": 1361.57, "end": 1361.87, "word": " في", "probability": 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/vPw26YH1fW0_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/vPw26YH1fW0_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a21230c0e33c973f7109e3aa9a4b0df502d89e8c
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/vPw26YH1fW0_raw.srt
@@ -0,0 +1,1872 @@
+1
+00:00:01,150 --> 00:00:04,890
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,890 --> 00:00:11,530
+وبركاته هذه هي المحاضرة الرقم 16مساق تفاضل وتكامل
+
+3
+00:00:11,530 --> 00:00:16,390
+جين عنوان المحاضرة اليوم The Chain Rule أو نظرية
+
+4
+00:00:16,390 --> 00:00:22,910
+السلسلة وهي الـ Section 14-4 وهي المحاضرة الرابعة
+
+5
+00:00:22,910 --> 00:00:28,490
+في Chapter 14 Functions of Two Variables هنبدأ
+
+6
+00:00:28,490 --> 00:00:33,810
+فيها احنا طبعا في تفاضل الألف كنا اتحدثنا عن نظرية
+
+7
+00:00:33,810 --> 00:00:36,780
+السلسلة لـ Function of One Variableيعني لو كان
+
+8
+00:00:36,780 --> 00:00:41,040
+عندي Y بيسوي F of X و X is a function of T يعني X
+
+9
+00:00:41,040 --> 00:00:44,420
+function في الـ T بيصير Y function في T و بيكون DY
+
+10
+00:00:44,420 --> 00:00:53,960
+by DT بيسوي DF by DX في DX by DT الآن شبيه الحديث
+
+11
+00:00:53,960 --> 00:00:58,880
+خلّينا نبدأ بالنظرية الخامسة اللي هي Chain rule
+
+12
+00:00:58,880 --> 00:01:01,260
+for functions of two independent variables
+
+13
+00:01:01,260 --> 00:01:06,530
+والموضوع ان شاء الله سهل وسلسعندي function w
+
+14
+00:01:06,530 --> 00:01:10,210
+بتساوي f of x و y function of two variables يعني w
+
+15
+00:01:10,210 --> 00:01:14,110
+ال dependent variable و ال x و ال y independent
+
+16
+00:01:14,110 --> 00:01:19,760
+بالنسبة ل w ففرضنا w is differentiable andفرضنا إن
+
+17
+00:01:19,760 --> 00:01:22,920
+X بتساوي X of T يعني X نفسها دي اللي هي الـ
+
+18
+00:01:22,920 --> 00:01:27,620
+Dependent Variable بالنسبة لـ W عبارة عن اللي هي
+
+19
+00:01:27,620 --> 00:01:32,220
+الـ Independent بالنسبة لـ W هي Dependent بالنسبة
+
+20
+00:01:32,220 --> 00:01:35,920
+لـ T يعني Function X تعتمد على T Function في T X
+
+21
+00:01:35,920 --> 00:01:39,610
+of T و Y نفس الاشي Function في Tلو فرضنا أنه تنتهي
+
+22
+00:01:39,610 --> 00:01:43,050
+in differentiable functions of T فرد W نفسها
+
+23
+00:01:43,050 --> 00:01:46,190
+function في T ومش هي كمان الـ W is a
+
+24
+00:01:46,190 --> 00:01:49,810
+differentiable function of T حسب ما بتقول النظرية
+
+25
+00:01:49,810 --> 00:01:53,030
+then the composite اللي هي W بالساوية F of X of T
+
+26
+00:01:53,030 --> 00:01:57,250
+وY of T هتكون differentiable of T وطريقة حساب
+
+27
+00:01:57,250 --> 00:02:02,750
+التفاضل DW بالنسبة لDT هي مشابهة للسابق هي عبارة
+
+28
+00:02:02,750 --> 00:02:11,550
+عن الـ FXفي التفاضل X' of T زاد الـ Fy في التفاضل
+
+29
+00:02:11,550 --> 00:02:17,770
+Y' of Tأو اختصاراً اللي هي يمكن أسهل في التعامل دي
+
+30
+00:02:17,770 --> 00:02:21,430
+دابل يو باي دي تي طبعا احنا كتبنا دي دابل يو باي
+
+31
+00:02:21,430 --> 00:02:25,710
+دي تي عساس ان هي و function بتصير بس في تي فبصير
+
+32
+00:02:25,710 --> 00:02:28,070
+function of one variable عشان هي كمان نكتب دي دابل
+
+33
+00:02:28,070 --> 00:02:32,910
+يو باي دي تي بساوي دل F by دل X في دي X by دي T
+
+34
+00:02:32,910 --> 00:02:37,010
+لأن X function في T لحالها بينما كتبنا دل F by دل
+
+35
+00:02:37,010 --> 00:02:46,980
+X عساس انه function F X of Y مع بعضبالنسبة للتشين
+
+36
+00:02:46,980 --> 00:02:50,700
+رول بالنسبة لـ function of two variables او
+
+37
+00:02:50,700 --> 00:02:55,480
+اختصارا DW by DT بساوية بدل الـ F التي هي DW by DX
+
+38
+00:02:55,480 --> 00:03:01,040
+في DX by DT وبدل الـ F هنا برضه DW by DY في DY by
+
+39
+00:03:01,040 --> 00:03:05,420
+DT لأن احنا مفترضين W بتساوي F of XY يمكن
+
+40
+00:03:05,420 --> 00:03:08,940
+الاستدارة الآن عن طريق الرسم يكون أسهل لكم بكتير
+
+41
+00:03:09,390 --> 00:03:12,350
+اللي هو خلّينا نشوف الـ Chain Rule عبر الرسمة وهذه
+
+42
+00:03:12,350 --> 00:03:16,730
+الرسومات هيكون اللي هي سهل اننا نتعامل معها بشكل
+
+43
+00:03:16,730 --> 00:03:22,000
+دائم ان شاء اللهفزي ما قلنا اللي هو الـ Chain Rule
+
+44
+00:03:22,000 --> 00:03:25,180
+نظرية سلسلة ممكن من خلال اللي هي الرسم التالي يكون
+
+45
+00:03:25,180 --> 00:03:29,660
+أسهل لنا كتير عند W هي function of X و Y يعني W
+
+46
+00:03:29,660 --> 00:03:34,080
+عبارة عن dependent variable بينما X و Y في هذه
+
+47
+00:03:34,080 --> 00:03:39,140
+الحالة independent بالنسبة لـ W و هتكون Ind اللي
+
+48
+00:03:39,140 --> 00:03:42,560
+هي dependent بالنسبالته عشان هيك مسميين هنا احنا
+
+49
+00:03:42,560 --> 00:03:48,020
+ال X و ال Y اللي هي أشمالها intermediate variables
+
+50
+00:03:48,500 --> 00:03:52,960
+اللي هي بتكون في البداية independent وبعدين في
+
+51
+00:03:52,960 --> 00:03:56,040
+النهاية بيصير dependent الـ W is a function of two
+
+52
+00:03:56,040 --> 00:04:00,240
+variables إذن هاي ال variable الأول اللي هو ال Y و
+
+53
+00:04:00,240 --> 00:04:03,500
+ال variable الثاني ال X أو العكس الآن ال X نفسها
+
+54
+00:04:03,500 --> 00:04:06,800
+function في T هاي ال X ده ال function في T و ال Y
+
+55
+00:04:06,800 --> 00:04:10,650
+برضه function في Tصارت الـ W بباشرة function في
+
+56
+00:04:10,650 --> 00:04:15,350
+VIN في T عشان هيك بنقدر نجيب دل W by دل DW SF by
+
+57
+00:04:15,350 --> 00:04:19,350
+DT لأن DW by DT بنخلال الرسم هذه بتسهل علينا كتير
+
+58
+00:04:19,350 --> 00:04:25,910
+كتير كتير عبارة عن دل W by دل Y في DY by DT أيها
+
+59
+00:04:26,270 --> 00:04:32,610
+وهذه زائد هذا الفرح زائد الفرح هذا التاني دل و باي
+
+60
+00:04:32,610 --> 00:04:37,050
+دل اكس دي اكس باي دي تي طلعت عندي دي و باي دي تي
+
+61
+00:04:37,050 --> 00:04:39,890
+بيساوي دل و باي دي اكس دي اكس باي دي تي زي ما
+
+62
+00:04:39,890 --> 00:04:47,010
+عملنا قبل بشوية و هذا الكلام أيضا اللي هو يعني
+
+63
+00:04:47,010 --> 00:04:51,550
+هنلاقيه برضه اللي هو نفس الشيء لو كانت عنديو
+
+64
+00:04:51,550 --> 00:04:55,730
+فانشغله في ثلاث مواصلة فبنعمل هذه الواريابل الأول
+
+65
+00:04:55,730 --> 00:05:00,090
+اللي هو z أو الأخير و هذا ال y و هذا ال x و بعدين
+
+66
+00:05:00,090 --> 00:05:04,430
+بنكمل اللي هو هنشوفه و هذا لاحقا بس عشان أربطكميا
+
+67
+00:05:04,430 --> 00:05:09,810
+بالسبق طيب نيجي الآن نرجع لنظريتنا هذه use the
+
+68
+00:05:09,810 --> 00:05:14,370
+chain rule to find the derivative of w بساوي x في
+
+69
+00:05:14,370 --> 00:05:19,130
+y عند w إيش مانع x في y بدنا نستخدم ال chain rule
+
+70
+00:05:19,360 --> 00:05:25,140
+لإيجاد هذه الـ derivative باسمين with respect to t
+
+71
+00:05:25,140 --> 00:05:28,480
+along the path x بساوة و cosine of t و y بساوة و
+
+72
+00:05:28,480 --> 00:05:31,680
+sine of t what is the derivatives value at t
+
+73
+00:05:31,680 --> 00:05:36,880
+بتساوية y على اتنين اللي طالبوا عندي w function في
+
+74
+00:05:36,880 --> 00:05:41,510
+x yو جاي لل X function في T هي Cos T و Y function
+
+75
+00:05:41,510 --> 00:05:46,750
+في T هي Sin T فبقى ضايق ليه طلب ليه W اللي هو
+
+76
+00:05:46,750 --> 00:05:51,430
+اتفضل ليها بالنسبة ل T عبر الطريقين هذولة يعني
+
+77
+00:05:51,430 --> 00:05:56,770
+بمعنى آخر عنده اللي هي ال X صارت function بناء على
+
+78
+00:05:56,770 --> 00:06:00,490
+هذولة الطريق ال function في T و Y function في T
+
+79
+00:06:00,490 --> 00:06:03,490
+كما هو معطب بعد هي كده بتوجد what is the
+
+80
+00:06:03,490 --> 00:06:07,360
+derivative value of T بتساوي Y على 2أذا على طول
+
+81
+00:06:07,360 --> 00:06:10,880
+نستخدم مين يا جماعة طريقة السلسلة هاي W function
+
+82
+00:06:10,880 --> 00:06:15,120
+في X و Y أو هاي و X و Y كل واحدة function في T
+
+83
+00:06:15,120 --> 00:06:18,760
+اللي هي كساية T ساية T ده نجهزها أمورنا لإن دي W
+
+84
+00:06:18,760 --> 00:06:22,580
+by دي T حسب النظرية اللي فوق هيها اللي عندنااللي
+
+85
+00:06:22,580 --> 00:06:27,740
+هي دي دل W by دل X في DX by DT زي دل W by دل Y في
+
+86
+00:06:27,740 --> 00:06:32,660
+DY by DT هو يساوي دل W by دل X فاضل هذه بالنسبة ل
+
+87
+00:06:32,660 --> 00:06:38,140
+X تطلع Y فاضل X بالنسبة ل T اي X Cos T تفضل اللي
+
+88
+00:06:38,140 --> 00:06:44,380
+يمين نقل Sin T خلصنا من هذهزي ما
+
+89
+00:06:44,380 --> 00:07:00,120
+حكينا
+
+90
+00:07:00,120 --> 00:07:05,130
+بنعوض عن y و عن x بخمتهمبينطلع عندي بدل Y اللي هي
+
+91
+00:07:05,130 --> 00:07:09,510
+Sin بيصير نقص Sin تربيع بدل X Cos بيصير Cos تربيع
+
+92
+00:07:09,510 --> 00:07:14,270
+T اللي هي طبعا Cos تربيع نقص Sin تربيع اللي هي
+
+93
+00:07:14,270 --> 00:07:22,090
+بيساوي Cos اتنين T الان على طول DW by DTاللي هي
+
+94
+00:07:22,090 --> 00:07:26,350
+عند باي على اتنين بنعوض عن كوساين اللي هي بيصير
+
+95
+00:07:26,350 --> 00:07:29,010
+اتنين في باي على اتنين يعني كوساين البي اللي هي
+
+96
+00:07:29,010 --> 00:07:32,790
+عبارة عن اياش عبارة عن ناقص واحد و هيك بيكون حالين
+
+97
+00:07:32,790 --> 00:07:38,110
+المثال نيجي الآن اللي هو ل functions of three
+
+98
+00:07:38,110 --> 00:07:43,170
+variables الان لو كان عندي اللي هو بدل function of
+
+99
+00:07:43,170 --> 00:07:46,650
+two variables three variables يعني لو رجعنا للي
+
+100
+00:07:46,650 --> 00:07:50,030
+عندنا هنا ال f of x و y بدل f of x و y بيصيروا z
+
+101
+00:07:50,610 --> 00:07:54,670
+فبصير عند هاي هذا للـ Y و هذا للـ X و بنجيب واحد
+
+102
+00:07:54,670 --> 00:07:58,790
+جوا للـ Z اللي هو و الـ Z function في T يعني فرضنا
+
+103
+00:07:58,790 --> 00:08:03,350
+ان W بتساوية F of X و Y و Z و X function في T و Y
+
+104
+00:08:03,350 --> 00:08:06,810
+function في T و Z function في T الآن بيصير زي هدول
+
+105
+00:08:06,810 --> 00:08:10,110
+الأعمدة بالظبط عمود .. زي ال .. ال .. ال .. ال ..
+
+106
+00:08:10,230 --> 00:08:13,710
+اللي باخد دي نوهان و باخد دي نوهان ناخد واحد خط
+
+107
+00:08:13,710 --> 00:08:18,230
+لمن؟ لـ Z بيصير لـ Z و لـ T بيصير عندي دل دابل يو
+
+108
+00:08:18,230 --> 00:08:22,230
+باي دل تي .. آسف دي دابل يو باي دي تي بيصير دل
+
+109
+00:08:22,230 --> 00:08:25,730
+دابل يو باي دي واي في دي واي باي دي تي زائد دل
+
+110
+00:08:25,730 --> 00:08:31,130
+دابل يو باي دي اكس في دي اكس باي دي تي زائد اللي
+
+111
+00:08:31,130 --> 00:08:37,610
+هو الخطة التفالة اللي هي دل دابل يو باي دي زدوهذه
+
+112
+00:08:37,610 --> 00:08:42,150
+هي نظرية الـ Chain Rule for functions of three
+
+113
+00:08:42,150 --> 00:08:48,290
+independent variables F of X, Y, Z is
+
+114
+00:08:48,290 --> 00:08:53,700
+differentiable function of X, Y, Zوالـ x و الـ y و
+
+115
+00:08:53,700 --> 00:08:57,560
+الـ z are differentiable functions of T ثم و نفسها
+
+116
+00:08:57,560 --> 00:09:02,760
+differentiable of T و الـ dw by dt اللي هي
+
+117
+00:09:02,760 --> 00:09:06,520
+بالساوية دي الـ wy دي ال x في dx by dt زي ما قلنا
+
+118
+00:09:06,520 --> 00:09:09,720
+هذا الخط الأول و هذا الخط التاني للـ y و هذا الخط
+
+119
+00:09:09,720 --> 00:09:13,920
+التالت لمين للـ z و بيكون هيكا اللي هو صارت عندنا
+
+120
+00:09:13,920 --> 00:09:17,820
+الأمور سهلة لو حتى كان عندنا أربع variable تمانية
+
+121
+00:09:17,820 --> 00:09:21,860
+variable مليون variable بنعملها مليون خط و بنجمحها
+
+122
+00:09:22,190 --> 00:09:26,350
+بنفس الأسلوب اللي عندى لما نكون اللي هي كل واحد
+
+123
+00:09:26,350 --> 00:09:32,420
+فيه المليون function of mean of Tالان find dw by
+
+124
+00:09:32,420 --> 00:09:37,940
+dt f w بيساوي x y زائد z هذا طبعا مثال على اللي هي
+
+125
+00:09:37,940 --> 00:09:41,340
+النظرية اللي قبل شوية x بيساوي cosine of t و y
+
+126
+00:09:41,340 --> 00:09:44,680
+بيساوي cosine of t و z بيساوي إيش ال t بنطبق اللي
+
+127
+00:09:44,680 --> 00:09:49,880
+فوق بنطبقه بسهولة هي dw by dt هيه بيساوي هي
+
+128
+00:09:49,880 --> 00:09:54,080
+القانون اللي احنا قلنا عنه قبل شوية y ساوي del w
+
+129
+00:09:54,080 --> 00:09:59,800
+by del x اللي هي بيساوي yDx by Dt هي ساول سالب
+
+130
+00:09:59,800 --> 00:10:07,060
+ساين تي دي الـ W by دي الـ Y هي عبارة عن X دي الـ
+
+131
+00:10:07,060 --> 00:10:13,520
+Y by دي الـ T هي عبارة عن كساين T زائد دي الـ W by
+
+132
+00:10:13,520 --> 00:10:17,220
+دي الـ Z هي واحدة لأن هذه الثبتات تعرفين كيف احنا
+
+133
+00:10:17,220 --> 00:10:21,900
+بنفضل دي الـ W by دي الـ Z بنثبت الـ X والـ Y
+
+134
+00:10:21,900 --> 00:10:28,400
+ونفضل بالنسبة لـ Zوالدت زد زي ما قلنا واحد دي زد
+
+135
+00:10:28,400 --> 00:10:32,700
+باي دي تي برضه واحد إذا بيصير عندي هذا عبارة عن
+
+136
+00:10:32,700 --> 00:10:36,440
+واحد ناقص اللي ها دي طبعا واي صين بيصير ناقص صين
+
+137
+00:10:36,440 --> 00:10:41,380
+تربيع و X Cos بيصير X Cos تربيع فبصير واحد ناقص
+
+138
+00:10:41,380 --> 00:10:45,900
+صين تربيع زاد Cos تربيع اللي هي عبارة عن اتنين Cos
+
+139
+00:10:45,900 --> 00:10:50,070
+تربيع ليش؟لأن واحد ناقص sin تربيع cos تربيع وهي
+
+140
+00:10:50,070 --> 00:10:53,770
+كمان cos تربيع يعني بيصير عندى اتنين cos تربيع
+
+141
+00:10:53,770 --> 00:10:58,650
+واحد اكيد يجي في باله بيقول هالجيت طب انت قلت هذي
+
+142
+00:10:58,650 --> 00:11:03,750
+X وY وZ وفيك عندك كمان مليون اللي هو متغير طب لو
+
+143
+00:11:03,750 --> 00:11:07,790
+المتغيرات المليون هدولة functions مش في T بس في T
+
+144
+00:11:07,790 --> 00:11:12,730
+في T وفي كمان variable Sأقول له بيصير نفس المنطقة
+
+145
+00:11:12,730 --> 00:11:17,110
+و الآن هناخد نظرية تقول لي لو تعددت ال variables
+
+146
+00:11:17,110 --> 00:11:21,850
+اللي هنا و تعددت هذولة كل واحدة صارت function of
+
+147
+00:11:21,850 --> 00:11:25,370
+more than one variable وهذا اللي هتعجوا النظرية
+
+148
+00:11:25,370 --> 00:11:29,500
+اللي بعدهاالتي تسميها Chain Rule for All Two
+
+149
+00:11:29,500 --> 00:11:33,720
+Independent Variables and Three Intermediates الـ
+
+150
+00:11:33,720 --> 00:11:36,940
+Three Intermediates اللي هو مين مقصود فيه هن الـ X
+
+151
+00:11:36,940 --> 00:11:40,100
+و الـ Y و الـ Z هدولة اللي هن الـ dependent الـ W
+
+152
+00:11:40,100 --> 00:11:42,780
+الـ dependent و اللي في النص هدولة الـ
+
+153
+00:11:42,780 --> 00:11:47,240
+intermediates اللي هن عبارة عن independent بالنسبة
+
+154
+00:11:47,240 --> 00:11:51,720
+لـ W و dependent بالنسبة للبعد هن اللي هي الـ S و
+
+155
+00:11:51,720 --> 00:11:56,050
+الـ R في حالتنا هذه اللي هنشوفه الآنإذا نظرنا إلى
+
+156
+00:11:56,050 --> 00:12:02,670
+أن W بيسوء F of X وY وZ و X بيسوء G of R وS يعني
+
+157
+00:12:02,670 --> 00:12:06,030
+صارت الـ X function of two variables R وS و Y
+
+158
+00:12:06,030 --> 00:12:09,430
+function of two variables R وS و Z function of two
+
+159
+00:12:09,430 --> 00:12:15,560
+variables R وSإذاً صارت الـ function أصلية W فنكشن
+
+160
+00:12:15,560 --> 00:12:21,220
+في R و في S إذاً هيصير يسألني دل W by دل R و دل W
+
+161
+00:12:21,220 --> 00:12:25,420
+by دل S مش زي ما جابل دل W by دي T دي W by دي T
+
+162
+00:12:25,420 --> 00:12:30,200
+لإنه كان T one variable أن اللي هو بيسألني عن دل W
+
+163
+00:12:30,200 --> 00:12:35,700
+by دل R و دل W by دل S بنقول له سهل و هنمشي بنفس
+
+164
+00:12:35,700 --> 00:12:39,560
+المنطق اللي هنشوفه بعد شويةلأن if all four
+
+165
+00:12:39,560 --> 00:12:44,360
+functions هدولة إذا كان كلهم different شابر هتطلع
+
+166
+00:12:44,360 --> 00:12:49,580
+w برضه has partial derivatives with respect to
+
+167
+00:12:49,580 --> 00:12:56,250
+main to R وS ومش هيك وعملية الإيجاد مشابهةالآن
+
+168
+00:12:56,250 --> 00:13:02,230
+عندما نرسم لـ R خطوط لحاله عندما نرسم لـ S خطوط
+
+169
+00:13:02,230 --> 00:13:09,270
+لحاله نرسم لها خطوط لحاله نرسم لها خطوط لحاله نرسم
+
+170
+00:13:09,270 --> 00:13:12,650
+لها خطوط لحاله نرسم لها خطوط لحاله نرسم لها خطوط
+
+171
+00:13:12,650 --> 00:13:18,610
+لحاله نرسم لها خطوط لحاله نرسم لها خطوط
+
+172
+00:13:18,610 --> 00:13:23,000
+لحالهما هي الـ z برضه function في main في R و S
+
+173
+00:13:23,000 --> 00:13:26,540
+أنا بدي أحسب الـ S الآن إذا بحط لها خط لـ S و بحط
+
+174
+00:13:26,540 --> 00:13:30,240
+لها خط ل S في الـ Y و الـ X بحط لها خط ل الـ S نفس
+
+175
+00:13:30,240 --> 00:13:34,500
+الجثة بنعملها مع main مع R لأنه برضه X و Y و Z الـ
+
+176
+00:13:34,500 --> 00:13:38,190
+Intermediate هدولة functions في الـ Rوفي الـ S إذا
+
+177
+00:13:38,190 --> 00:13:41,870
+بنعمل إشي لل R عشان نجد اللي هو في حالة طلب دي
+
+178
+00:13:41,870 --> 00:13:45,410
+دابليو دي ال W by دي ال R وبنعملها دي ال S في حالة
+
+179
+00:13:45,410 --> 00:13:50,550
+طلب دي ال W by دي ال S وخلّيني نشوف كيف نحسب دي ال
+
+180
+00:13:50,550 --> 00:13:54,530
+W by دي ال R دي ال W by دي ال R زي ما عملنا دي ال
+
+181
+00:13:54,530 --> 00:13:58,030
+W by دي ال X في دي ال X by دي ال R الخط الأول زي
+
+182
+00:13:58,030 --> 00:14:01,310
+الخط التاني دي ال W by دي ال Y في دي ال Y by دي ال
+
+183
+00:14:01,310 --> 00:14:05,620
+Rوالخط التالت del w by del z في del z by del r
+
+184
+00:14:05,620 --> 00:14:11,860
+مجموحنا دولة del w by del r إذن اختصارا الـ w
+
+185
+00:14:11,860 --> 00:14:16,300
+بتنحط في أغلى f of x,y,z بيطلع منها خطوط بعدد ال
+
+186
+00:14:16,300 --> 00:14:19,680
+variables اللي عندنا كل واحد من ال variables طبعا
+
+187
+00:14:19,680 --> 00:14:22,260
+عندي إذا كان function of one variables خلاص بنحكي
+
+188
+00:14:22,260 --> 00:14:25,710
+عنتفاضل بالنسبة لـ T لحالها إذا كان أكتر من
+
+189
+00:14:25,710 --> 00:14:30,430
+variable نحكي أكتر من R إذا بنعمل رسمة للـ R ورسمة
+
+190
+00:14:30,430 --> 00:14:34,250
+ثانية مشابهة جدا للـ S وبنطبق و بنوجد اللي هو
+
+191
+00:14:34,250 --> 00:14:38,670
+القانون del W by del S بيساوي del W by del X في
+
+192
+00:14:38,670 --> 00:14:42,430
+del X by del S زائد الخط التاني del W by del Y في
+
+193
+00:14:42,430 --> 00:14:46,090
+del Y by del S زائد del W by del Z في del Z بطاقة
+
+194
+00:14:46,090 --> 00:14:49,970
+del S إذن هذه عبارة عن الـ Chain Rule الـ function
+
+195
+00:14:49,970 --> 00:14:53,470
+of three variablesوكل واحد من الـ Intermediate
+
+196
+00:14:53,470 --> 00:14:56,310
+Variables الـ X والـ Y والـ Z Function of Two
+
+197
+00:14:56,310 --> 00:14:59,670
+Variables لو كان عندي مليون هنا بطلع لنا مليون خط
+
+198
+00:14:59,670 --> 00:15:03,630
+لو طلع لنا خمسمية هنا بطلع لنا خمسمية رسمة طبعا
+
+199
+00:15:03,630 --> 00:15:06,450
+مافيش ده نفسه رسم خمسمية رسمة ولكن كل واحد من الاش
+
+200
+00:15:06,450 --> 00:15:12,090
+اللي هو بغيره وبصير عندي اللي هي للخمسمية نفس
+
+201
+00:15:12,090 --> 00:15:16,530
+الرسمة ولكن بنغير اسمه لل R اسمه R لل S اسمه ال S
+
+202
+00:15:16,530 --> 00:15:19,150
+إلى A خيري هى بتصور ان شاء الله تكونوا فهمتوا
+
+203
+00:15:19,150 --> 00:15:23,840
+عليهاالان دي دي الان المثال اللي عندناه تطبيقه
+
+204
+00:15:23,840 --> 00:15:29,560
+express w by r and w by s in terms of r and s يعني
+
+205
+00:15:29,560 --> 00:15:33,620
+بيقوللي أو دي ال دي ال w by ال r أو ال w by ال s
+
+206
+00:15:33,620 --> 00:15:39,200
+اللي هي لنين w بساوي x زي 2y زي z تربيع و x ايش
+
+207
+00:15:39,200 --> 00:15:43,310
+بتساويR على S يعني صارت X function في R و في S و
+
+208
+00:15:43,310 --> 00:15:46,590
+YS بيساوي R تربيع Z لإنها صارت إذن function of two
+
+209
+00:15:46,590 --> 00:15:51,730
+variables R و S و Z بيساوي A2R دل W by دل R حسب
+
+210
+00:15:51,730 --> 00:15:56,150
+القانون اللي هي دل W by دل X و دل X by دل R إلى
+
+211
+00:15:56,150 --> 00:16:01,570
+آخره تنطبق الآن دل W by دل X اللي يبفضل W بالنسبة
+
+212
+00:16:01,570 --> 00:16:06,210
+لX سنة واحدوهدولة صفر إذا بيظل واحد مضروب X على R
+
+213
+00:16:06,210 --> 00:16:10,990
+دل W by دل تفاضل X بالنسبة ل R اللي هي عبارة عن
+
+214
+00:16:10,990 --> 00:16:14,090
+واحد على S لأنه بيثبت ال S و بيفاضل بالنسبة ل R
+
+215
+00:16:14,090 --> 00:16:19,510
+بيصير واحد على S الان دل W by دل Y اللي هي عبارة
+
+216
+00:16:19,510 --> 00:16:24,890
+عن جداش اتنين دل W by دل Y عبارة عن اتنين لإن
+
+217
+00:16:24,890 --> 00:16:32,630
+هدولة أصفر هي اتنينDel Y by Del R أي Del Y by Del
+
+218
+00:16:32,630 --> 00:16:37,870
+R هي اتنين R وهذا بتتصفر زائد Del W by Del Z هي
+
+219
+00:16:37,870 --> 00:16:44,070
+عبارة عن ايش اتنين Z بتنزل Del Z by Del R هي عبارة
+
+220
+00:16:44,070 --> 00:16:49,210
+عن برضه ايش اتنين اللي هي بتنزل بنبسطها بالـ C1S
+
+221
+00:16:49,210 --> 00:16:53,810
+زائد 4R زائد 4Zوفي الآخر لما بدك تعوض مكان Z
+
+222
+00:16:53,810 --> 00:16:56,970
+بقيمته عشان توجده كله بدلالة R لأن الطالب اللي هو
+
+223
+00:16:56,970 --> 00:17:01,090
+دي ال W by دي ال R يعني بدهيها as in هو جالك in
+
+224
+00:17:01,090 --> 00:17:05,010
+terms of R and S لأن ال Z إذا بدنا نشيلها وبدنا
+
+225
+00:17:05,010 --> 00:17:11,070
+نشوف قيمتها إيش قيمتها؟ 2R عوضناها بصير 2R و 4R 6R
+
+226
+00:17:11,070 --> 00:17:18,830
+مظبوط؟ لأ بصير 2 في 4 في 8R زائد اللي هي 4R 12R 1
+
+227
+00:17:18,830 --> 00:17:26,740
+على S نجي للبعدبنفس الطريقة هي القانون دل W بي دل
+
+228
+00:17:26,740 --> 00:17:30,780
+X هي الموجودة تعددناها واحد وهي اتنين وهي اتنين Z
+
+229
+00:17:30,780 --> 00:17:38,520
+هدولة دل W بي دل X ودل W بي دل Y ودل W بي دل Z كل
+
+230
+00:17:38,520 --> 00:17:42,600
+الموجودات من السؤال اللي جابته دل نود مين دل X بي
+
+231
+00:17:42,600 --> 00:17:47,160
+دل SX by DS هي عبارة عن اياش الـ R ثابتة الـ R
+
+232
+00:17:47,160 --> 00:17:50,760
+ثابتة بيصير الـ R في تفاضل الواحد على S اللي هو
+
+233
+00:17:50,760 --> 00:17:57,140
+ناقص واحد على S تربيعها الان DY by DS الهي DY by
+
+234
+00:17:57,140 --> 00:18:00,280
+DS اللي هي هذي بيصير سفر بيصير واحد على S لإن
+
+235
+00:18:00,280 --> 00:18:04,280
+انتفاض الواحد على S الان في الآخر ΔZ by DS دي دي
+
+236
+00:18:04,280 --> 00:18:05,880
+دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي
+
+237
+00:18:05,880 --> 00:18:06,280
+دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي
+
+238
+00:18:06,280 --> 00:18:07,460
+دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي دي
+
+239
+00:18:07,460 --> 00:18:15,490
+دي دي دزائد 2 على S يعني 2S نقص R على S تربيع وهو
+
+240
+00:18:15,490 --> 00:18:20,770
+المطلوب الآن لو كان عند W بسوء F of X و Y بدل ما
+
+241
+00:18:20,770 --> 00:18:24,650
+هي F of X و Y و Z بصير سهل بصير حالة خاصة من اللي
+
+242
+00:18:24,650 --> 00:18:29,820
+فوق و X function برضه G R of S Y H of R و Sبصير
+
+243
+00:18:29,820 --> 00:18:33,380
+اللي هو ما تظهرش مين يظهرش الطرف التالت هذا ال
+
+244
+00:18:33,380 --> 00:18:36,280
+term التالت هذا بصير ما تظهرش بس بيظهر مين دول
+
+245
+00:18:36,280 --> 00:18:40,360
+الاتنين اللي هو del W by del X خطين يعني بيصير خط
+
+246
+00:18:40,360 --> 00:18:45,680
+ل X وخط ل Y del W by del X في del X by del R زائد
+
+247
+00:18:45,680 --> 00:18:48,980
+del W by del Y في del Y by del R وتبع الـ Z ما هو
+
+248
+00:18:48,980 --> 00:18:53,100
+مش موجود إذا صار بالشكل هذا و similarly لمين ده
+
+249
+00:18:53,100 --> 00:19:00,780
+الـ Wبعد ذلك طرب السؤال مثال على القانون اللي فوق
+
+250
+00:19:00,780 --> 00:19:05,260
+express del w by del r and del w by del s in terms
+
+251
+00:19:05,260 --> 00:19:10,000
+of r and s if w بسوء x تربيع زاد y تربيع و x بسوء
+
+252
+00:19:10,000 --> 00:19:13,240
+r ناقص s و y بسوء r زائد s
+
+253
+00:19:16,520 --> 00:19:20,120
+القانون على طول هي القانون فوق اللي هي del w by
+
+254
+00:19:20,120 --> 00:19:24,180
+del r هي تبعت ال X وهي تبعت main ال Y خط ال X وخط
+
+255
+00:19:24,180 --> 00:19:26,460
+ال Y مجموعات ويساوي فاهمين عن ايش معاناة اللي
+
+256
+00:19:26,460 --> 00:19:29,980
+بحكيه فاهمين على بعض del w by del x ايش بيساوي
+
+257
+00:19:29,980 --> 00:19:34,960
+اتنين X الان del x by del r del x by del r واحد
+
+258
+00:19:36,460 --> 00:19:44,320
+زائد del W by del Y التي هي 2Y مضروبة
+
+259
+00:19:44,320 --> 00:19:51,300
+في D del Y by del R التي هي 1 بيصير 2X زائد 2Y الـ
+
+260
+00:19:51,300 --> 00:19:55,500
+X بيعودها لـ R minus S بيصير 2R minus S والـ Y
+
+261
+00:19:55,500 --> 00:19:59,760
+بقيمتها R زائد S اللي هي بنجمحها مع بعض بيطلع 4R
+
+262
+00:19:59,760 --> 00:20:03,600
+والـ 2S بيروح مع الـ 2 نقص 2S بيطلع إلى الجواب 4R
+
+263
+00:20:04,160 --> 00:20:15,600
+مثلًا ومثلًا ومثلًا ومثلًا ومثلًا ومثلًا
+
+264
+00:20:15,600 --> 00:20:24,240
+ومثلًا
+
+265
+00:20:26,560 --> 00:20:30,340
+ماشي الحل دي الـ y by دي الـ s واحد هاي عوضناها
+
+266
+00:20:30,340 --> 00:20:35,640
+اللي هبصي ناقص 2x زي 2y بنعوض عن x القيمتها اللي
+
+267
+00:20:35,640 --> 00:20:39,840
+هي ناقص 2r minus s وعن y القيمتها 2r زي s التين مع
+
+268
+00:20:39,840 --> 00:20:44,200
+بعض بيطلعين ايش بيساوي أربعة اللي هي s وكونها ده
+
+269
+00:20:44,200 --> 00:20:48,780
+مثلناطيب لان نيجي ناخد اللي هو W بساوية F of X لو
+
+270
+00:20:48,780 --> 00:20:52,120
+كانت عندنا W بس function of one variable إذا خط
+
+271
+00:20:52,120 --> 00:20:56,360
+واحد اللي بنزل A لان and X بساوية G R of S يعني X
+
+272
+00:20:56,360 --> 00:20:58,680
+function في مين؟ في R و S يعني function of two
+
+273
+00:20:58,680 --> 00:21:03,480
+variables سهلة، بده Del W by Del R لأنه صارت W
+
+274
+00:21:03,480 --> 00:21:07,800
+function في R و S لان بده Del W by Del R خط هو الـ
+
+275
+00:21:07,800 --> 00:21:11,660
+X خطها واحد، هيه و بعدين بنزل منها مين؟ اللي هي
+
+276
+00:21:11,660 --> 00:21:17,670
+الـ Rبس بالنسبة لـ W بالنسبة لـ R دي الـ W بالنسبة
+
+277
+00:21:17,670 --> 00:21:24,530
+لـ R بساوي DW بدي DX هذه في DX بدي DX بالمية R
+
+278
+00:21:24,530 --> 00:21:26,490
+لإنه partial هذه لإنه function of two variables
+
+279
+00:21:26,490 --> 00:21:32,370
+لإن سمينار ليه DW بدي ال S بساوي دي اللي برضه خط
+
+280
+00:21:32,370 --> 00:21:35,950
+واحد خط واحد بس بده يصير اللي تحته مين S بصير DW
+
+281
+00:21:35,950 --> 00:21:42,120
+بدي ال S بساوي DW بدي DXفي دل X by دي الـ Main by
+
+282
+00:21:42,120 --> 00:21:45,400
+دي الأصل فان شاء الله يكون واضح اللي بحكيه إلى أن
+
+283
+00:21:45,400 --> 00:21:47,840
+نيجي لحاجة اسمها ال implicit differentiation أو
+
+284
+00:21:47,840 --> 00:21:52,940
+التفاضل Revisited اللي هو اللي هي نشوف كيف نوجدها
+
+285
+00:21:52,940 --> 00:21:56,660
+أو احنا أخدنا اشي من هالنوع في اللي هو التفاضل الف
+
+286
+00:21:56,660 --> 00:22:03,410
+إلى أن ناخد اللي هو النظرية الليالتي تتعامل مع كيف
+
+287
+00:22:03,410 --> 00:22:11,210
+يكون التفاضل الضمني X و Y مرتبطات مع بعض من خلال
+
+288
+00:22:11,210 --> 00:22:16,690
+فورميلة أو معادلة أو من خلال كذا فبتكون واحدة
+
+289
+00:22:16,690 --> 00:22:20,150
+dependent واحدة independent نشوف الشيء اللي بقوله
+
+290
+00:22:20,150 --> 00:22:24,730
+Suppose that f of x y is a differentiable function
+
+291
+00:22:24,730 --> 00:22:28,430
+and that the equation f of x y بتساوي سفر
+
+292
+00:22:33,010 --> 00:22:40,030
+بتعرف Y كدالة
+
+293
+00:22:40,030 --> 00:22:45,430
+قابلة للاشتقاق في X ثم في أي موقع فيه FY ليسوا سفر
+
+294
+00:22:48,090 --> 00:22:52,170
+لو بدى اودد ان صارت y function في x dy by dx ايش
+
+295
+00:22:52,170 --> 00:22:57,630
+هتساوي ناقص fx هي ال fx عندي و f of x و y هذه ناقص
+
+296
+00:22:57,630 --> 00:23:04,410
+fx لها على مين على ال fy هذه dy by dx لو انت بدك
+
+297
+00:23:04,410 --> 00:23:07,990
+تلاحظ كيف اوضدناها تنشوف كيف اوضدناها لو اجينا
+
+298
+00:23:07,990 --> 00:23:12,790
+اخدنا w بساوية f of x و y ماشي لأن هذه صارت w
+
+299
+00:23:12,790 --> 00:23:16,050
+function في مين في x و y او ال f function في x و y
+
+300
+00:23:16,640 --> 00:23:22,580
+الآن هذه التي لها خطين هذه خطلة الـ X خطلة الـ Y
+
+301
+00:23:22,580 --> 00:23:28,440
+ياسف وخطلة مين للـ X الآن لو فرضنا أن هذه F هذه
+
+302
+00:23:28,440 --> 00:23:32,380
+بتساوي سفر يعني العلاقة اللي حكيناها قبل شوية F of
+
+303
+00:23:32,380 --> 00:23:38,440
+XY بتساوي سفر تعرفالعلاقة بين X و Y بحيث أنه تكون
+
+304
+00:23:38,440 --> 00:23:42,660
+Y as a function mean of X صارت Y دلّا في X هذا
+
+305
+00:23:42,660 --> 00:23:46,180
+بحكي عن الحالة اللي قبل بشوية التفاضل الضمني يعني
+
+306
+00:23:46,180 --> 00:23:50,040
+هذه إيش بده أسويها الآن إبسفر فبصير Y function في
+
+307
+00:23:50,040 --> 00:23:53,280
+main في X عشان هيك كاتب Y أبنالها function of X
+
+308
+00:23:53,280 --> 00:23:59,260
+وبالتفاضل هذه outed اللي هي التفاضلهابالنسبة لمين
+
+309
+00:23:59,260 --> 00:24:03,520
+بدي اود الـ dy by dx لتشوف كيف بدي اصل ال bridge
+
+310
+00:24:03,520 --> 00:24:07,560
+هذا يعني بدي افعض ال del w اللي انا سميتها سفر ايش
+
+311
+00:24:07,560 --> 00:24:12,100
+بيساوي by del x اللي هنا هي x وهذه اللي هي ال
+
+312
+00:24:12,100 --> 00:24:15,540
+variable التاني x صار عندي طريقين هي واحد y وهي
+
+313
+00:24:15,540 --> 00:24:19,810
+واحد x ال x طبيعي function في نفسهاوالـ y function
+
+314
+00:24:19,810 --> 00:24:23,870
+of x على اعتبار انه احنا لما ناخد هذه ايش مالها f
+
+315
+00:24:23,870 --> 00:24:28,550
+of x وw او y بتساوي ايش سفر تعمل علاقة بين y أو
+
+316
+00:24:28,550 --> 00:24:33,490
+بين x او y as a function of x خليني نشوف كيف نصل
+
+317
+00:24:33,490 --> 00:24:36,410
+ال bridge هذا اللي عن كيف اللي كنا نستخدم نقول
+
+318
+00:24:36,410 --> 00:24:45,550
+اللي عن dw by dx هذه بساوي اللي هو fxفي dx التي هي
+
+319
+00:24:45,550 --> 00:24:50,170
+fx يعني دي الـ w by دي الـ x في dx by dx هذا اللي
+
+320
+00:24:50,170 --> 00:24:54,370
+كانت بدلتيه واحد مستوى إذا صارت الـ fx في مين في
+
+321
+00:24:54,370 --> 00:24:58,910
+واحد هذه fx في واحد زائد الان دي ال w by دي ال y
+
+322
+00:24:58,910 --> 00:25:04,150
+أو ال fy في مين في تفاضل هذه dy by dx اللي هي h
+
+323
+00:25:04,150 --> 00:25:08,700
+prime of x الـ dy by dxالان اذا كما قلنا احنا على
+
+324
+00:25:08,700 --> 00:25:14,600
+اعتبار ان W ستساوي كمية 0 اذا بيصير عندى W بيساوي
+
+325
+00:25:14,600 --> 00:25:18,300
+0 يعني و W بيساوي X في هذه الحالة ستصبح 0 اذا
+
+326
+00:25:18,300 --> 00:25:21,880
+ستصبح هذا سفر مدان صارت هذه سفر اذا صارت عندى
+
+327
+00:25:21,880 --> 00:25:25,040
+بنجلي هذا على الجهة الثانية بيصير ناقص FX و بيصير
+
+328
+00:25:25,040 --> 00:25:28,920
+DY بيدي DX بيساوي ناقص FX و باكسم على FY بيطلع
+
+329
+00:25:28,920 --> 00:25:34,070
+ناقص FXعلى f y اللي هي العلاقة اللي عندنا دي y by
+
+330
+00:25:34,070 --> 00:25:38,290
+دي x بسوة و ناقص f x على f y عندما تكون اللي هي w
+
+331
+00:25:38,290 --> 00:25:42,410
+بتساوي سفر أو من عن آخر f of x y بتساوي سفر تعرف
+
+332
+00:25:42,410 --> 00:25:47,850
+علاقة بين y و x as اللي هي x independent و y
+
+333
+00:25:47,850 --> 00:25:52,050
+dependent تعتمد عالميا على ال y هذا احنا مش ترطينه
+
+334
+00:25:52,380 --> 00:25:56,160
+الأن بقول الـ use theorem تمانية to find dy by dx
+
+335
+00:25:56,160 --> 00:26:00,720
+الان dy by dx على طول اللي هي ال y ال dependent
+
+336
+00:26:00,720 --> 00:26:05,160
+والx إيش ماله independent fy تربيع نقص x تربيع نقص
+
+337
+00:26:05,160 --> 00:26:09,700
+sin xy إيش بتساوي سفر هذه بدك توجد dy by dx معطيك
+
+338
+00:26:09,700 --> 00:26:13,420
+هذه إيش بتساوي سفر أمامنا أمرين إما نحل زي زمان زي
+
+339
+00:26:13,420 --> 00:26:16,780
+تفاضه الألف كيف نحل زي تفاضه الألف على طول كل اللي
+
+340
+00:26:16,780 --> 00:26:22,820
+قدامنا انفضله بالنسبة لx يعني بيصير هذه 2y ناقص2y
+
+341
+00:26:22,820 --> 00:26:30,140
+dy by dx ناقص 2x ناقص sin x في y اللي هي بتفاضل هي
+
+342
+00:26:30,140 --> 00:26:34,800
+إيش هو عبارة عن cos x في y في تفاضل اللي جوا في
+
+343
+00:26:34,800 --> 00:26:37,660
+جوس اللي هي x اللي هي عبارة عن تفاضل اللي واحد ده
+
+344
+00:26:37,660 --> 00:26:43,400
+بيصير في y زائد اللي هي x في dy by dx و بناخد dy و
+
+345
+00:26:43,400 --> 00:26:47,000
+dx عامل مشترك و بننجل على الطرف التاني و بنحل و
+
+346
+00:26:47,000 --> 00:26:51,410
+بنحل زي ما كنا زمان مش هذا اللي بديهاولكن قلت لكم
+
+347
+00:26:51,410 --> 00:26:55,210
+كيف تحلوا اللي بده استذكر حل زمان ممكن يحل زيه
+
+348
+00:26:55,210 --> 00:26:58,750
+يوجد DIY by DX جربوا حلوها لحالكم DIY by DX
+
+349
+00:26:58,750 --> 00:27:03,930
+وشوفوها بتكافئ اللي هو جوابها نفس جوابنا أكيد لازم
+
+350
+00:27:03,930 --> 00:27:07,730
+تفعل الجواب اللي هالنيدي أيش اللي بتقوله بقى اللي
+
+351
+00:27:07,730 --> 00:27:11,250
+بقوله أن أنا الأن بديش أحل بطريقة زمان بدي أحل
+
+352
+00:27:11,250 --> 00:27:17,270
+بطريقة اليومالآن بدي أسمّي هذه كلها أف of x و y
+
+353
+00:27:17,270 --> 00:27:20,090
+إذا أصبحت أف of x و y بيسوّي تربيع نقص x تربيع
+
+354
+00:27:20,090 --> 00:27:25,550
+نقصين x و y اللي هي في الواقع سفر، هي سفر، مصبوط؟
+
+355
+00:27:25,550 --> 00:27:29,790
+أه، صارت عندي هذه العلاقة أف of x و y بتعرف اللي
+
+356
+00:27:29,790 --> 00:27:33,570
+هو بيسوّي سفر بتعرف y as a function of xإذا بقدر
+
+357
+00:27:33,570 --> 00:27:36,990
+أطبّق عليها مين اللي حكيتها بيصير dy by dx بساوي
+
+358
+00:27:36,990 --> 00:27:41,970
+ناقص fx على fy fx إيش بتساوي في هذه الحالة لـ fx
+
+359
+00:27:41,970 --> 00:27:46,590
+هذه ثابتة تصير سفر بيصير ناقص 2x ناقص الـ y ثبتت
+
+360
+00:27:46,590 --> 00:27:50,980
+بيصير ناقصكوساين الـ x في y في تفاضل اللي جوا
+
+361
+00:27:50,980 --> 00:27:54,680
+تفاضل الـ x في y اللي هو y بيصير ناقص اتنين x ناقص
+
+362
+00:27:54,680 --> 00:27:59,960
+y كوساين x y على mean of y of y فضلة بالنسبة ل y
+
+363
+00:27:59,960 --> 00:28:05,620
+بيصير اتنين y ناقص اللي هو كوساين x في y مضروبة في
+
+364
+00:28:05,620 --> 00:28:09,540
+تفاضل اللي جوا بالنسبة ل y اللي هي xهذا المقدار
+
+365
+00:28:09,540 --> 00:28:13,640
+يصبح دي وي بي دي اكس او بمعنى اخر لما ندخل ناقص
+
+366
+00:28:13,640 --> 00:28:17,460
+هنا يصبح اتنين اكزاد y cos xy على اتنين y ناقص x
+
+367
+00:28:17,460 --> 00:28:18,640
+cos xy
+
+368
+00:28:28,790 --> 00:28:32,430
+هذه اللي حكيناها قبل شوية نيجي الآن لو كان عندي
+
+369
+00:28:32,430 --> 00:28:37,190
+اللي هو three variables يعني عندي بمعنى آخر let x
+
+370
+00:28:37,190 --> 00:28:40,590
+and y be two independent variables يعني نفترض أن x
+
+371
+00:28:40,590 --> 00:28:43,550
+و y هو بيقولنا في السؤال x و y two independent
+
+372
+00:28:43,550 --> 00:28:48,790
+variables و z is the dependent variable يعني z
+
+373
+00:28:48,790 --> 00:28:52,970
+function في مين بمعنى آخر x و y كيف معرفة في x و y
+
+374
+00:28:52,970 --> 00:28:58,070
+معرفة ضمنيا where they are related implicitly by
+
+375
+00:28:58,440 --> 00:29:02,280
+يعني هدولة التلت متغيرات اللي اللي هو الـ Z
+
+376
+00:29:02,280 --> 00:29:07,000
+dependent والـ XY independent معرفات مع بعض ضمنيا
+
+377
+00:29:07,000 --> 00:29:12,300
+بواسطة المعادلة F of XY و Z بتساوي إيه؟ سفر يعني F
+
+378
+00:29:12,300 --> 00:29:17,660
+XY Z function بتساوي سفر أقل لإن اللي هي Z as a
+
+379
+00:29:17,660 --> 00:29:22,960
+function of Y أو X معرفة ضمنيا بدنا الآن اللي هو
+
+380
+00:29:22,960 --> 00:29:28,690
+نشتق القوانين التالية خلينا نشوفبنفضل كله بالنسبة
+
+381
+00:29:28,690 --> 00:29:40,850
+لمن؟ بالنسبة ل X فضله Zero بالنسبة ل X Zero فضله
+
+382
+00:29:40,850 --> 00:29:42,810
+Zero بالنسبة ل X Zero فضله Zero بالنسبة ل X Zero
+
+383
+00:29:42,810 --> 00:29:45,690
+فضله Zero بالنسبة ل X Zero فضله Zero بالنسبة ل X
+
+384
+00:29:45,690 --> 00:29:47,430
+Zero فضله Zero بالنسبة ل X Zero فضله Zero بالنسبة
+
+385
+00:29:47,430 --> 00:29:52,090
+ل X Zeroفى دل X by دل X لأن النقطة الأخيرة الـ X
+
+386
+00:29:52,090 --> 00:29:57,550
+دل F by دل Y فى دل Y by دل X زائد دل F by دل Z فى
+
+387
+00:29:57,550 --> 00:30:01,110
+دل Z by دل X هاي خط خطين تلاتة وكلين فى النقطة الـ
+
+388
+00:30:01,110 --> 00:30:08,170
+X الان FX هذه دل X by دل X واحد طبيعى FY الـ Y و
+
+389
+00:30:08,170 --> 00:30:11,330
+الـ X independent مدام Independent لدى الـ Y لا
+
+390
+00:30:11,330 --> 00:30:15,330
+تعتمد على الـ X يعني تفاضلهم بالنسبة لبعض صفرإذاً
+
+391
+00:30:15,330 --> 00:30:18,830
+هذي إيش بتصير؟ سفر، إذاً هذا الterm هيزول الآن دي
+
+392
+00:30:18,830 --> 00:30:22,590
+الـ F by دي الـ Z بتنزل FZ دي الـ Z by دي الـ X
+
+393
+00:30:22,590 --> 00:30:26,450
+اللي هو جاي لك Z function في X و Y يعني فيه بينهم
+
+394
+00:30:26,450 --> 00:30:30,410
+تفاضل يعني دي الـ Z by دي الـ X بتنزل زي ما هي
+
+395
+00:30:30,410 --> 00:30:36,190
+الآن هذا صار عندي أشير لهذه صارت FX زائد FZ في دي
+
+396
+00:30:36,190 --> 00:30:40,840
+الـ Z by دي الـ X إيش بتساوي؟ بتساوي سفرمن هذه من
+
+397
+00:30:40,840 --> 00:30:45,460
+هذه بقدر اوضت del z by del x بجيب هذا على الطرف
+
+398
+00:30:45,460 --> 00:30:49,500
+هذا بيصير ناقص fx بكسبه على fz بيصير del z by del
+
+399
+00:30:49,500 --> 00:30:56,660
+x بيساوي ناقص fx عالميا على fz إذن لما تكونمعرّفة
+
+400
+00:30:56,660 --> 00:31:01,280
+ضمنياً Z as a function of Y of X بيصير Del Z by
+
+401
+00:31:01,280 --> 00:31:05,600
+Del X اللي هي العلاقة اللي جوانية Del Z by Del X
+
+402
+00:31:05,600 --> 00:31:09,160
+بيساوي ناقص اللي هي FX
+
+403
+00:31:14,200 --> 00:31:21,740
+بالنسبة للجهتين Y وجيت واشتغلت ستجد هذه تتصفر
+
+404
+00:31:21,740 --> 00:31:29,230
+وستبقى هذه موجودة فهيصير عند FY زي FZومنه اللي هو
+
+405
+00:31:29,230 --> 00:31:33,630
+بيكون عندى del z by del y بيساوي ناقص fy على مين
+
+406
+00:31:33,630 --> 00:31:39,930
+على fz بهذا كله تحت شرط أن fz لا تساوي إيش؟ لا
+
+407
+00:31:39,930 --> 00:31:45,740
+تساوي سفر إذا الآن لما تكون الإختصار الحديدF of X
+
+408
+00:31:45,740 --> 00:31:49,620
+Y و Z بيساوي صفر بتعرف لي ضمنيا Z as a function of
+
+409
+00:31:49,620 --> 00:31:55,840
+X Y بقدر أفاضل Del Z by Del X كيف بتساوي ناقص F
+
+410
+00:31:55,840 --> 00:32:02,000
+هذه بالنسبة ل X على F بالنسبة ل Z والتانية
+
+411
+00:32:02,000 --> 00:32:06,590
+Similarly وخلّيني أخد مثال على هذا الكلامنجد دل زد
+
+412
+00:32:06,590 --> 00:32:12,130
+با دل إكس ودل و .. دل زد با دل واي at zero zero
+
+413
+00:32:12,130 --> 00:32:17,390
+zero if x تكعيب زاد z تربيع زاد y e to the xz زاد
+
+414
+00:32:17,390 --> 00:32:25,680
+z cos y بساوي إيش بساوينشوف كيف بننحل عندي الآن
+
+415
+00:32:25,680 --> 00:32:30,600
+معطيني اللي هي X تكعيب زي Z تربيع زي Y زي Z بساوية
+
+416
+00:32:30,600 --> 00:32:35,160
+0 اللي هو ضمنيا اللي هو مادة مطالبة دل Z by دل X
+
+417
+00:32:35,160 --> 00:32:39,640
+ودل Z by دل Y عندي 00 وكذا وكذا ومعطيني علاقة
+
+418
+00:32:39,640 --> 00:32:43,500
+معناته بوكسود اللي هو Z as a function of X أو Y
+
+419
+00:32:43,500 --> 00:32:46,580
+وإلا ليش يسأل عنه لو Z و X independent هدا هتطلع
+
+420
+00:32:46,580 --> 00:32:50,940
+سفره دي سفر Automatic فاللي هوما دام هو الآن يعني
+
+421
+00:32:50,940 --> 00:32:54,900
+حاتب الأسلوب هذا اللي هو معناته اللي هو مفترض أن Z
+
+422
+00:32:54,900 --> 00:32:57,420
+as a function of X وY واحنا إن شاء الله عزيزي ما
+
+423
+00:32:57,420 --> 00:33:02,300
+اتخربتوش وبنكتبلكم اللي هو الكلام هذا بصراحة الان
+
+424
+00:33:02,300 --> 00:33:05,400
+since X and Y are two independent variables and Z
+
+425
+00:33:05,400 --> 00:33:08,020
+is the dependent variable where they are related
+
+426
+00:33:08,020 --> 00:33:12,500
+implicitly by اللي هي هذه بيساوي سفر هذه بدي
+
+427
+00:33:12,500 --> 00:33:16,940
+أسميها إيش كلها هذه هذه هذه للسفر هذا كله بدي
+
+428
+00:33:16,940 --> 00:33:21,890
+أسميه F of X Y وZالآن مدام F of X Y و Z و ساوة سفر
+
+429
+00:33:21,890 --> 00:33:25,210
+إذا بقدر أطبق النظرية اللي قبل شوية Del Z by Del Z
+
+430
+00:33:25,210 --> 00:33:30,070
+X مش بيساوي ناقص F X على F Z F X يعني بده فاضل
+
+431
+00:33:30,070 --> 00:33:34,230
+هادي partial derivative with respect to X اللي
+
+432
+00:33:34,230 --> 00:33:40,030
+عبارة عن تلاتة X تربيع هيزائد تفاضل هذي سفر وهذا
+
+433
+00:33:40,030 --> 00:33:48,410
+يصبح Z Y E X Z فتفاضل اللي جوا X Z اللي هو تفاضله
+
+434
+00:33:48,410 --> 00:33:53,570
+بالنسبة ل X تطلع Z فبصير Z Y E X Z الان هذه
+
+435
+00:33:53,570 --> 00:33:56,870
+التفاضل اللي هي بالنسبة ل X طبعا Z Cos Y Ash بسوى
+
+436
+00:33:56,870 --> 00:34:02,660
+سفر هذه السفرإذا صار عندى هذا المقدار على fz ناخد
+
+437
+00:34:02,660 --> 00:34:08,940
+fz لهذه اتنين z هي اتنين z زائد تفاضل هذه اللي هو
+
+438
+00:34:08,940 --> 00:34:12,880
+y e to the xz مضروب في تفاضل اللي جوا بالنسبة لz
+
+439
+00:34:12,880 --> 00:34:17,780
+اللي هو x اللي هي بيصير x في y e to the xz زائد
+
+440
+00:34:17,780 --> 00:34:21,560
+تفاضل هذه بالنسبة هذه اللي ضايلة بالنسبة لz اللي
+
+441
+00:34:21,560 --> 00:34:26,030
+هي واحد في cosine ال Y زايد cosine ال Yإذاً طلعت
+
+442
+00:34:26,030 --> 00:34:29,270
+هذه دي الـ z by دي الـ x إذا بدنا نحسبها عند السفر
+
+443
+00:34:29,270 --> 00:34:32,330
+والسفر والسفر نسينا نحسبها عند السفر والسفر السفر
+
+444
+00:34:32,330 --> 00:34:37,870
+احسب هذه سفر وهذه سفر وهذه طبعاً سفر سفر وهذه واحد
+
+445
+00:34:37,870 --> 00:34:42,070
+أكيد قصان Zero واحد إذاً سفر على واحد تطلع أيش
+
+446
+00:34:42,070 --> 00:34:45,590
+بتساوي سفر إذا دلت التفاوض لدي الـ z by دي الـ x
+
+447
+00:34:45,590 --> 00:34:50,290
+عند السفر والسفر والسفر بتساوي سفر نشوف اللي بعدها
+
+448
+00:34:50,290 --> 00:34:54,920
+يا جماعة اللي بعدها ليه؟اللي هي del z by del y
+
+449
+00:34:54,920 --> 00:34:59,820
+بيساوي ناقص fy على fz ناقص fy هي y هنا ناقص fy و
+
+450
+00:34:59,820 --> 00:35:04,460
+ال fz اللي هي fz تحت لأن fy انتفاض بالنسبة ل y هذه
+
+451
+00:35:04,460 --> 00:35:10,600
+هذي سفر وهذه سفر بتظهر لها إيه إيه إيه إيه إيه إيه
+
+452
+00:35:10,600 --> 00:35:10,600
+إيه إيه إيه إيه إيه إيه إيه إيه إيه إيه إيه إيه
+
+453
+00:35:10,600 --> 00:35:13,480
+إيه إيه إيه إيه إيه إيه إيه إيه إيه إيه إيه إيه
+
+454
+00:35:17,150 --> 00:35:24,010
+اللي هي Z في Cos Y اللي هي زائد Z في Cos Y اللي هي
+
+455
+00:35:24,010 --> 00:35:29,870
+تفاضل بالنسبة ل Y اللي هي تفاضل Cos هي ناقص Sin Y
+
+456
+00:35:29,870 --> 00:35:33,250
+اللي هي بيصير Z في ناقص Sin Y وفي عند النقص اللي
+
+457
+00:35:33,250 --> 00:35:37,770
+برا بيصير زائد Z Sin Y فاهمين عليها تفاضل بالنسبة
+
+458
+00:35:37,770 --> 00:35:42,940
+ل Yهذه الأصفار هذي بيصير e to the x z بتنزل هذه
+
+459
+00:35:42,940 --> 00:35:46,960
+الأن تفضلها بالنسبة ل y الـ z تابتة هذه ناقص sign
+
+460
+00:35:46,960 --> 00:35:51,700
+y يعني ناقص z sign y ناقص z sign y والناقص اللي
+
+461
+00:35:51,700 --> 00:35:55,480
+برا دخلت فدخلت هذه صارت ناقص وهذه صارت زائد وهذه
+
+462
+00:35:55,480 --> 00:35:58,730
+زي ما هي اللي تحت اللي هي ال z فضلناها فوقإن
+
+463
+00:35:58,730 --> 00:36:03,430
+نحسبها عند مين بدّو 000 بحط مكان هذه Zero هتصير
+
+464
+00:36:03,430 --> 00:36:07,090
+هذه ناقص واحد وهذه سفر وهذا واحد إذا بيطلع الجواب
+
+465
+00:36:07,090 --> 00:36:11,910
+إيش بيساوي ناقص واحد لأن هذه سفر وهذه سفر وهذه سفر
+
+466
+00:36:11,910 --> 00:36:17,330
+وهذه واحد وهذه ناقص واحد ويساوي ناقص واحد هذه اللي
+
+467
+00:36:17,330 --> 00:36:24,190
+المطلوب ويكون خلصنا المحاضرة وإلى لقاء آخر إن شاء
+
+468
+00:36:24,190 --> 00:36:24,270
+الله
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U.srt
new file mode 100644
index 0000000000000000000000000000000000000000..06ad60e7ee185b5ca93a1dcc8fe57ff07a90fce1
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U.srt
@@ -0,0 +1,2133 @@
+1
+00:00:00,640 --> 00:00:04,660
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,660 --> 00:00:10,080
+وبركاته هذه المحاضرة رقم 17 مساق تفاضل وتكامل
+
+3
+00:00:10,080 --> 00:00:13,940
+الـGEM وهي المحاضرة الخامسة في الشطر الرابع عشر وهي
+
+4
+00:00:13,940 --> 00:00:18,780
+section 14-5 المعنون بـdirectional derivatives
+
+5
+00:00:18,780 --> 00:00:25,500
+and gradient vectors اللي هو الاشتقاق
+
+6
+00:00:25,500 --> 00:00:32,630
+في اتجاه معين، الـ Gradient Vectors التي سنتعرف
+
+7
+00:00:32,630 --> 00:00:37,650
+عليها لاحقًا. نحكي أولًا عن Directional Derivatives
+
+8
+00:00:37,650 --> 00:00:42,050
+in W. الـ Directional Derivatives هي عبارة عن تعميم
+
+9
+00:00:42,050 --> 00:00:45,890
+للـ Partial Derivatives سواء في اتجاه الـ X أو الـ
+
+10
+00:00:45,890 --> 00:00:48,390
+Y أو الـ Z.
+
+11
+00:00:51,970 --> 00:00:56,790
+أو ∆FY/∆Y أو ∆FY/∆Z كل حالة خاصة من شيء اسمه
+
+12
+00:00:56,790 --> 00:00:59,830
+Directional Derivative. خلّيني أتعرف الآن على Plan
+
+13
+00:00:59,830 --> 00:01:04,930
+Definition. The derivative of F at P0 (X0, Y0). نفترض
+
+14
+00:01:04,930 --> 00:01:07,810
+F function of two variables،  اللي هي الـ X والـ Y.
+
+15
+00:01:07,810 --> 00:01:13,830
+فالاشتقاق للـ F عند النقطة X0 وY0 in the direction
+
+16
+00:01:13,830 --> 00:01:19,870
+of the unit vector U = U1I + U2J is the
+
+17
+00:01:19,870 --> 00:01:26,140
+number. إذًا مشتقة الـ F في الاتجاه اللي هو عند النقطة
+
+18
+00:01:26,140 --> 00:01:29,720
+X نوت و Y نوت في اتجاه الـ U، هذا اللي معطيني إياه U
+
+19
+00:01:29,720 --> 00:01:35,740
+واحد I زائد U اتنين J. المشتقة للـ function
+
+20
+00:01:35,740 --> 00:01:42,280
+في هذا الاتجاه تُعرف بـ DF/DS، اللي
+
+21
+00:01:42,280 --> 00:01:46,900
+هو في اتجاه الـ U عند النقطة بي نوت، يساوي limit F
+
+22
+00:01:46,900 --> 00:01:52,360
+(X نوت + S U1, Y نوت + S U2) – F
+
+23
+00:01:52,360 --> 00:01:57,650
+(X نوت, Y نوت) / S. إن التغيير في الـ
+
+24
+00:01:57,650 --> 00:02:00,970
+X نوت و Y نوت لما تنتقل X نوت و Y نوت، التغيير بس
+
+25
+00:02:00,970 --> 00:02:06,850
+وين مسموح فيه هنا في اتجاه الـ U. يعني X نوت سيكون
+
+26
+00:02:06,850 --> 00:02:11,690
+التغيير في Y نوت في هذا الاتجاه، فيصبح X نوت + S
+
+27
+00:02:11,690 --> 00:02:16,890
+U1، التي هي مقدار الزيادة اللي في
+
+28
+00:02:16,890 --> 00:02:20,410
+الـ X نوت، تتحكم فيها اللي هي المقدار العددي S
+
+29
+00:02:20,410 --> 00:02:25,630
+و U1 في اتجاه I. والـ Y نوت
+
+30
+00:02:25,630 --> 00:02:29,330
+اللي هو Y نوت + S U2 بنفس الأسلوب. S اللي بتمثل الـ
+
+31
+00:02:29,330 --> 00:02:33,410
+increment العددي، والـ U2 اللي هي الاتجاه اللي صارت
+
+32
+00:02:33,410 --> 00:02:37,350
+فيه التغيير. يعني بمعنى آخر، التغيير صار على وين؟ في
+
+33
+00:02:37,350 --> 00:02:40,730
+هذا الاتجاه. الـ Derivative في هذه الحالة تُعرّف
+
+34
+00:02:40,730 --> 00:02:44,690
+هذه ناقص f(x نوت, y نوت) / s، لما s تروح لـ 0، هي
+
+35
+00:02:44,690 --> 00:02:48,850
+بالضبط حاجة بنسميها The directional derivative of
+
+36
+00:02:48,850 --> 00:02:53,970
+f in the direction of u at P نوت، أو الـ derivative of f
+
+37
+00:02:53,970 --> 00:02:59,470
+in the direction of u at P نوت. خلّينا الآن نأخذ
+
+38
+00:02:59,470 --> 00:03:04,450
+اللي هو عندي مثال. طبعًا قبل ما نأخذ المثال
+
+39
+00:03:04,450 --> 00:03:07,850
+الـ x، زي ما قلنا، على هذا الخط، يساوي x نوت.
+
+40
+00:03:11,140 --> 00:03:13,680
+هذه اللي بنحكي فيها هو الـ X بسهولة. X نوت +
+
+41
+00:03:13,680 --> 00:03:17,560
+S U1 و Y يساوي Y نوت + S U2، اللي عبارة عن
+
+42
+00:03:17,560 --> 00:03:21,100
+إيش؟ الخط اللي هو المستقيم هنا. الـ X والـ Y تتحرك
+
+43
+00:03:21,100 --> 00:03:25,760
+بوافق إيش؟ المعادلة الأمامية هذه اللي أمامي، اللي
+
+44
+00:03:25,760 --> 00:03:29,960
+صار فيها التغيير. X نوت + S U1، Y نوت + S
+
+45
+00:03:29,960 --> 00:03:33,740
+U2، في اتجاه إيش؟ في هذا الاتجاه. خلّيني آخذ مثال عددي يا
+
+46
+00:03:33,740 --> 00:03:37,440
+جماعة، نشوف إيش اللي بنحكي بالنسبة له، اللي هي هذه
+
+47
+00:03:37,440 --> 00:03:40,400
+الـ directional derivative، الديركشنال الديرفتيف
+
+48
+00:03:40,400 --> 00:03:42,520
+الذي تم تسميته بـ Equation 1، هو أيضًا يُسمّى
+
+49
+00:03:42,520 --> 00:03:46,560
+بالديركشنال الديرفتيف للـ F، أو الديرفتيف للـ F في
+
+50
+00:03:46,560 --> 00:03:50,400
+الاتجاه بـ U، عند النقطة B0. الديرفتيف للـ F في
+
+51
+00:03:50,400 --> 00:03:53,760
+الاتجاه بـ B0، في الاتجاه بـ U. سنستخدم التصميم،
+
+52
+00:03:53,760 --> 00:03:56,100
+طبعًا، التصميم الذي رأيته هو مشابه للتصميم
+
+53
+00:03:56,100 --> 00:03:59,160
+للديرفتيف التابع لزمان، ولكن هناك كانت متغيرات one
+
+54
+00:03:59,160 --> 00:04:03,930
+Variable. وشابه للـ partial derivative، ولكن كانت
+
+55
+00:04:03,930 --> 00:04:08,070
+اللي هي زمان one variable. وفي حالة الـ partial
+
+56
+00:04:08,070 --> 00:04:11,470
+derivative، التغيير بس كان في اتجاه X أو في اتجاه Y
+
+57
+00:04:11,470 --> 00:04:14,770
+فقط. Using the definition, find the derivative of
+
+58
+00:04:14,770 --> 00:04:19,830
+F(X,Y) = X تربيع + X في Y at the point (1,
+
+59
+00:04:19,830 --> 00:04:22,350
+2) in the direction of the unit vector
+
+60
+00:04:22,350 --> 00:04:24,650
+اللي معطينا إياه: (1/جذر 2) في اتجاه I +
+
+61
+00:04:24,650 --> 00:04:29,140
+(1/جذر 2) في اتجاه J. خلّينا نحل الـ
+
+62
+00:04:29,140 --> 00:04:31,960
+Directional Derivative للـ F، أو Derivative للـ F
+
+63
+00:04:31,960 --> 00:04:34,180
+في الـ direction of U عند النقطة (1، 2) يساوي
+
+64
+00:04:34,180 --> 00:04:37,540
+لمين؟ هذا التعريف اللي حكيناه قبل شوية. ويساوي لمين؟
+
+65
+00:04:37,540 --> 00:04:40,560
+لأن الـ X نوت والـ Y نوت هما 1 و 2،
+
+66
+00:04:40,560 --> 00:04:44,220
+إذًا مكان الـ X نوت بدنا نحط إيش؟ 1. الـ S زي ما
+
+67
+00:04:44,220 --> 00:04:47,800
+هي، هو المتغير، الـ increment، الزيادة في اتجاه مين؟
+
+68
+00:04:47,800 --> 00:04:50,480
+U1. U1 إيش بتساوي؟ الـ U1 / جذر
+
+69
+00:04:50,480 --> 00:04:54,940
+الـ 2. إذًا صارت S / جذر الـ 2. والـ Y نوت اللي
+
+70
+00:04:54,940 --> 00:04:59,890
+هي 2 + S U2 اللي هي 2 + S (1/جذر
+
+71
+00:04:59,890 --> 00:05:04,190
+الـ 2). هذا اللي صار. الـ function F لهذا المقدار
+
+72
+00:05:04,190 --> 00:05:07,770
+ناقص F(1، 2) / S. الـ S اللي هنا بنحسب
+
+73
+00:05:07,770 --> 00:05:12,230
+هذه الـ F. يعني بنحسب الـ function هذه اللي أمامنا
+
+74
+00:05:12,230 --> 00:05:15,550
+عند النقطة X = 1 + S/جذر 2
+
+75
+00:05:15,550 --> 00:05:20,670
+و Y = 2 + S/جذر 2. بنعوض.
+
+76
+00:05:21,230 --> 00:05:25,410
+والـ Y عندنا 2 + S/جذر 2. بنعوض، بيصير اللي هي تربيعها هي هذه الـ X، اللي هي
+
+77
+00:05:25,410 --> 00:05:28,690
+هذه تربيعها + اللي هي هذه الـ X وهذه الـ Y
+
+78
+00:05:28,690 --> 00:05:32,850
+اللي عندنا هنا – اللي هي F(1،
+
+79
+00:05:32,850 --> 00:05:38,690
+2). F(1، 2) هي عبارة عن 1 + 2 = 3. اللي هي
+
+80
+00:05:38,690 --> 00:05:43,790
+عوضنا عن X بـ 1 و Y بـ 2، طالع عندنا 3. / S.
+
+81
+00:05:43,790 --> 00:05:49,130
+لأن هذا المقدار يتم تربيعه، بيطلع هذا المقدار، بيطلع هذا
+
+82
+00:05:49,390 --> 00:05:53,410
+المقدار، بيطلع هذا المقدار...(مكرر عدة مرات نفس الجملة)
+
+83
+00:05:53,410 --> 00:05:56,650
+...(مكرر عدة مرات نفس الجملة)
+
+84
+00:05:56,650 --> 00:05:58,810
+...(مكرر عدة مرات نفس الجملة)
+
+85
+00:05:58,810 --> 00:05:59,390
+...(مكرر عدة مرات نفس الجملة)
+
+86
+00:05:59,390 --> 00:06:01,630
+...(مكرر عدة مرات نفس الجملة)
+
+87
+00:06:01,630 --> 00:06:02,450
+...(مكرر عدة مرات نفس الجملة)
+
+88
+00:06:02,450 --> 00:06:04,950
+...(مكرر عدة مرات نفس الجملة)
+
+89
+00:06:04,950 --> 00:06:05,670
+...(مكرر عدة مرات نفس الجملة)
+
+90
+00:06:05,670 --> 00:06:10,680
+المقدار، بيطلع هذا. في هذا اللي هي 2S / جذر
+
+91
+00:06:10,680 --> 00:06:13,920
+الـ 2 + S / جذر الـ 2، يعني 3S /
+
+92
+00:06:13,920 --> 00:06:17,020
+جذر الـ 2 ناقص المقدار هذا اللي هو 3، الكل
+
+93
+00:06:17,020 --> 00:06:20,720
+مقسوم على مين يا جماعة؟ على S. بتنصح لهذا المقدار بقى
+
+94
+00:06:20,720 --> 00:06:24,580
+علينا أن نبسطه. بنجمع هذول لبعض، بيطلع عندي
+
+95
+00:06:24,580 --> 00:06:28,140
+المقدار هذا. طبعًا سهل لكم تجمعوه لبعض. 5S / جذر
+
+96
+00:06:28,140 --> 00:06:31,520
+الـ 2 + S تربيع / S. لأن واي في
+
+97
+00:06:31,520 --> 00:06:33,800
+عندي 2 + 1 = 3، بتروح مع الـ 3. هذا بيظل
+
+98
+00:06:33,800 --> 00:06:37,790
+اللي هو معامل S تربيع، اللي هو 1/2 + 1/2
+
+99
+00:06:37,790 --> 00:06:41,210
+بيصير 1، وهذه 3/ جذر 2 بيصير 5/ S
+
+100
+00:06:41,210 --> 00:06:44,310
+بنختصر لأن الـ S مع الـ S زي زمان، مع الـ S اللي هنا
+
+101
+00:06:44,310 --> 00:06:47,170
+والـ S اللي هنا، بيطلع limit 5 / جذر الـ 2،
+
+102
+00:06:47,170 --> 00:06:50,330
+as S goes to zero. بيطلع 5 / جذر
+
+103
+00:06:50,330 --> 00:06:53,510
+الـ 2. إذًا الـ directional derivative of f in the
+
+104
+00:06:53,510 --> 00:06:56,430
+direction of u at the point (1، 2) طلع عبارة
+
+105
+00:06:56,430 --> 00:07:00,770
+عن إيش؟ 5 / 2. نجي الآن نأخذ دقيقة، خليني
+
+106
+00:07:00,770 --> 00:07:03,350
+أقول معنى هندسي أكثر للـ directional
+
+107
+00:07:03,350 --> 00:07:06,370
+derivative اللي حكينا عنها قبل شوية. Interpretation
+
+108
+00:07:06,370 --> 00:07:08,950
+of the Directional Derivative. خلينا نشوف اللي هي
+
+109
+00:07:08,950 --> 00:07:11,590
+إيش المعنى الهندسي للـ Directional Derivative، و
+
+110
+00:07:11,590 --> 00:07:14,790
+أنا حكيته قبل شوية.  عُمومًا، شوف الصلاة على النبي عليه
+
+111
+00:07:14,790 --> 00:07:17,410
+الصلاة والسلام. زي ما عملنا في الـ partial
+
+112
+00:07:17,410 --> 00:07:20,510
+derivative، قلنا إيش معنى اللي هو الـ directional
+
+113
+00:07:20,510 --> 00:07:24,190
+derivative.  اللي قُلنا إنه اللي هو الـ partial derivative
+
+114
+00:07:25,240 --> 00:07:29,280
+هي عبارة عن ميل المماس للمنحنى الناتج من
+
+115
+00:07:29,280 --> 00:07:34,020
+تقاطع اللي هو X = X نوت مع اللي هو الـ surface
+
+116
+00:07:34,020 --> 00:07:38,160
+بيطلع المماس للمنحنى اللي بيطلع، هو عبارة عن ∂
+
+117
+00:07:38,160 --> 00:07:41,740
+F
+
+118
+00:07:41,740 --> 00:07:44,640
+/ ∂Y. ونفس الشيء بالنسبة لـ X. خلينا نشوف هذا إيش
+
+119
+00:07:44,640 --> 00:07:48,870
+اللي بقوله. لحظة، أنا كنت نسيت. الآن أُجيء الآن أنا
+
+120
+00:07:48,870 --> 00:07:52,970
+عندي إيش؟ غرضي، الـ equation Z = F(X,Y) يمثل
+
+121
+00:07:52,970 --> 00:07:55,830
+الـ surface as in space. هذا اللي بالأزرق هو
+
+122
+00:07:55,830 --> 00:08:00,490
+عبارة عن الـ surface Z = F(X,Y). الآن إذا كان
+
+123
+00:08:00,490 --> 00:08:04,110
+Z نوت = F(X نوت, Y نوت)، هذه اللي هي النقطة
+
+124
+00:08:04,110 --> 00:08:07,270
+هي عبارة عن النقطة اللي بدنا نوجد عندها اللي هو
+
+125
+00:08:07,270 --> 00:08:13,020
+الـ directional derivative. والنقطة X نوت تقع على
+
+126
+00:08:13,020 --> 00:08:17,820
+الـ surface S. الآن بدي أوجد الـ directional
+
+127
+00:08:17,820 --> 00:08:22,900
+derivative of F at النقطة X نوت, Y نوت, Z نوت.
+
+128
+00:08:22,900 --> 00:08:27,860
+X نوت, Y نوت في العادة، ومين؟ عند Z نوت ثابت، في الـ
+
+129
+00:08:27,860 --> 00:08:30,660
+direction في اتجاه مين؟ في اتجاه الـ U. هي الـ U
+
+130
+00:08:30,660 --> 00:08:36,160
+الآن. إيش هو بالضبط معناه الهندسي؟ الآن هذا معناه أن
+
+131
+00:08:36,160 --> 00:08:41,310
+أنا أجيء اللي هو آخذ الـ plane اللي بتكون من أو
+
+132
+00:08:41,310 --> 00:08:44,190
+اللي بيحدده اللي هو الـ ...(تكرار حروف غير مفهومة)
+
+133
+00:08:44,190 --> 00:08:44,230
+...(تكرار حروف غير مفهومة)
+
+134
+00:08:44,230 --> 00:08:44,290
+...(تكرار حروف غير مفهومة)
+
+135
+00:08:44,290 --> 00:08:44,790
+...(تكرار حروف غير مفهومة)
+
+136
+00:08:44,790 --> 00:08:46,790
+...(تكرار حروف غير مفهومة)
+
+137
+00:08:46,790 --> 00:08:47,970
+...(تكرار حروف غير مفهومة)
+
+138
+00:08:47,970 --> 00:08:56,210
+...(تكرار حروف غير مفهومة)
+
+139
+00:08:56,210 --> 00:09:08,350
+...(تكرار حروف غير مفهومة)
+
+140
+00:09:08,350 --> 00:09:09,990
+...(تكرار حروف غير مفهومة) في اتجاه مين؟ في
+
+141
+00:09:09,990 --> 00:09:14,950
+اتجاه اللي هو الـ U هذا. الآن الـ plane المتكون من
+
+142
+00:09:14,950 --> 00:09:17,990
+اللي هو الـ U اللي بدأ يأخذ الـ derivative في
+
+143
+00:09:17,990 --> 00:09:22,210
+اتجاهه ونقطة X نوت و Y نوت. هذا الـ plane يقطع من
+
+144
+00:09:22,210 --> 00:09:28,910
+الـ surface في الخط الأزرق. الآن ميل المماس لهذا
+
+145
+00:09:28,910 --> 00:09:32,830
+اللي هو خط التقاطع، ميل المماس هذا، اللي هو الـ
+
+146
+00:09:32,830 --> 00:09:37,290
+tangent هذا عند النقطة، هذا ميله هو عبارة عن الـ
+
+147
+00:09:37,290 --> 00:09:41,590
+directional derivative للـ function F عند النقطة X
+
+148
+00:09:41,590 --> 00:09:46,370
+نوت, Y نوت, Z نوت في اتجاه مين؟ في اتجاه الـ U.
+
+149
+00:09:46,370 --> 00:09:49,310
+في النهاية إذًا برضه الـ directional derivative هي
+
+150
+00:09:49,310 --> 00:09:55,270
+عبارة عن ميل مماس. ميل مماس لمين؟ للـ curve للـ
+
+151
+00:09:55,270 --> 00:10:02,060
+curve عند النقطة X0, Y0, Z0 الناتج من تقاطع الـ
+
+152
+00:10:02,060 --> 00:10:06,600
+surface مع الـ plane المكوّن من اللي هو الـ vector
+
+153
+00:10:06,600 --> 00:10:11,320
+اللي بدأ يشتق في اتجاهه مع النقطة X0, Y0. الآن لو
+
+154
+00:10:11,320 --> 00:10:16,610
+كان هذا الـ vector اللي هو U = I + 2J،
+
+155
+00:10:16,610 --> 00:10:20,850
+لو كانت U2 = 0 و U1 = 1، يعني الـ
+
+156
+00:10:20,850 --> 00:10:24,410
+directional derivative، U = U1 في اتجاه
+
+157
+00:10:24,410 --> 00:10:28,590
+الـ I يعني بس I، يعني الـ U في اتجاه الـ I، يعني أنا
+
+158
+00:10:28,590 --> 00:10:30,970
+بتأخذ الـ directional derivative في اتجاه مين؟ الـ I.
+
+159
+00:10:30,970 --> 00:10:34,620
+بكون لما آخذ في اتجاه الـ I، معناه أنّه أنا الـ
+
+160
+00:10:34,620 --> 00:10:37,940
+derivative هيها في اتجاه الـ I. هذا في اتجاه هذا.
+
+161
+00:10:37,940 --> 00:10:42,840
+من هنا. يعني إيش عملت في الـ Y؟ ثبتت الـ Y. فيصبح عندي
+
+162
+00:10:42,840 --> 00:10:46,840
+اللي هي الـ partial derivative with respect to
+
+163
+00:10:46,840 --> 00:10:51,700
+مين؟ to X. لأنّه بتثبت Y، فـ Y صار بس التغيير في مين؟
+
+164
+00:10:51,700 --> 00:10:55,100
+في اتجاه الـ I. لأنّ إذا الـ directional derivative
+
+165
+00:10:55,100 --> 00:10:58,480
+U باتجاه الـ I، is the directional derivative at P0
+
+166
+00:10:58,480 --> 00:11:04,890
+is ∂F/∂X evaluated at X0, Y0. U = J،
+
+167
+00:11:04,890 --> 00:11:08,790
+الـ directional derivative at P0 بتصير ∂F/
+
+168
+00:11:08,790 --> 00:11:12,510
+∂Y، لأنّه اللي هو بس التغيير في اتجاه الـ J
+
+169
+00:11:12,510 --> 00:11:16,790
+وليس التغيير في اتجاه الـ X. معناته X ثابتة. إذًا هذه
+
+170
+00:11:16,790 --> 00:11:19,910
+الـ U، اللي هي الـ partial derivative with respect
+
+171
+00:11:19,910 --> 00:11:24,190
+to Y أو with respect to X هي عبارة عن حالات خاصة
+
+172
+00:11
+
+223
+00:14:44,800 --> 00:14:47,120
+التربيعي على تلاتة تربيع زي أربعة تربيع اللي هي
+
+224
+00:14:47,120 --> 00:14:51,400
+ياش تسعة زي ستاش تلاتة عليها في تجاه I إذا ناقص
+
+225
+00:14:51,400 --> 00:14:55,970
+أربعة على الجذر التربيعي هذا في تجاه J، إذا الـ
+
+226
+00:14:55,970 --> 00:15:00,990
+magnitude للـ V هي U ويساوي هذا الخمسة و هذا
+
+227
+00:15:00,990 --> 00:15:04,410
+الخمسة يصبح ثلاثة على خمسة I ناقص أربعة على خمسة J
+
+228
+00:15:04,410 --> 00:15:08,570
+هذا الـ direction اللي هو اللي بدنا نعمل فيه اتجاهه
+
+229
+00:15:08,570 --> 00:15:13,230
+الـ derivative الـ gradient لها في سهل هي عبارة عن
+
+230
+00:15:13,230 --> 00:15:17,170
+X أول شيء، المشتقة هذا بالنسبة لـ X يصبح E to the Y
+
+231
+00:15:17,770 --> 00:15:20,570
+المشتقة لهذه بالنسبة لـ X بيصير اللي هي ناقص Sine
+
+232
+00:15:20,570 --> 00:15:23,850
+X في Y، فمشتقة اللي جوا إيه، الـ X في Y مشتقتها
+
+233
+00:15:23,850 --> 00:15:26,970
+بالنسبة لـ X اللي هي Y، فبيصير E to the Y ناقص Y
+
+234
+00:15:26,970 --> 00:15:30,850
+Sine XY في اتجاه الـ I، زائد الـ derivative لهذه
+
+235
+00:15:30,850 --> 00:15:34,930
+with respect to Y، يعني بالنسبة لـ Y اللي هي عبارة عن X E
+
+236
+00:15:34,930 --> 00:15:38,270
+to the Y ناقص المشتقة لهذه بالنسبة لـ Y دي هي ناقص
+
+237
+00:15:38,270 --> 00:15:43,370
+Sine X في Y، فمشتقة جوا X ناقص X Sine Y في اتجاه
+
+238
+00:15:43,370 --> 00:15:52,200
+مين؟ بعد ذلك نحسب الـ Gradient هذا عند نقطة 2 و 0
+
+239
+00:15:52,200 --> 00:16:02,180
+عوض عن I بـ 0 و X بـ 2 بيصير هذا عبارة عن Z1
+
+240
+00:16:02,180 --> 00:16:10,550
+وهذا صفر كله، بيصير I اللي هي 2 في 1، ايه ده، 0، 1 بيصير
+
+241
+00:16:10,550 --> 00:16:16,610
+2 و هادي صفر طبعاً لإنه sin 0 = 0 بيصير 2J، صار عندي الـ
+
+242
+00:16:16,610 --> 00:16:19,770
+gradient اللي  عنده النقطة المذكورة بيساوي I
+
+243
+00:16:19,770 --> 00:16:24,280
+زائد 2J، اللي هي الـ Gradient جاهزة و الـ U جاهزة يعني
+
+244
+00:16:24,280 --> 00:16:27,660
+لنبدا نطبق القانون، الـ Gradient ضد الـ U بيصير
+
+245
+00:16:27,660 --> 00:16:32,340
+الـ Derivative بيساوي اللي هو الـ Gradient I زائد 2
+
+246
+00:16:32,340 --> 00:16:38,260
+J ضد الـ U اللي هو عبارة عن 3 على 5 ناقص 4 على 5
+
+247
+00:16:38,260 --> 00:16:40,960
+ويساوي سالب 1، طبعاً بتعرفوا كيف تضربوا، دكتور
+
+248
+00:16:40,960 --> 00:16:44,110
+دكتاتكم في هذه ال .. التي تقريباً تقريباً تقريباً
+
+249
+00:16:44,110 --> 00:16:47,490
+تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً
+
+250
+00:16:47,490 --> 00:16:53,210
+تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً
+
+251
+00:16:53,210 --> 00:16:54,970
+تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً
+
+252
+00:16:54,970 --> 00:16:55,690
+تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً
+
+253
+00:16:55,690 --> 00:17:01,310
+تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً تقريباً
+
+254
+00:17:01,310 --> 00:17:03,890
+قلنا ايش تعريفها؟ عبارة عن الـ gradient للـ F من
+
+255
+00:17:03,890 --> 00:17:06,570
+dot U، اللي يجي الآن هذه القيمة اللي نعرف
+
+256
+00:17:06,570 --> 00:17:11,070
+أن قيمتها الـ magnitude للـ gradient للـ F في
+
+257
+00:17:11,070 --> 00:17:14,770
+cos الزاوية،  cos θ، لو أحد سأل
+
+258
+00:17:14,770 --> 00:17:19,270
+سؤال، وقت إيش هذه اللي هي الـ F increases most
+
+259
+00:17:19,270 --> 00:17:22,150
+rapidly، يعني معدل التغير هذا اللي هو derivative
+
+260
+00:17:22,150 --> 00:17:26,350
+تكون أكبر ما يمكن، اللي هو زيادة تكون أكبر ما يمكن
+
+261
+00:17:26,350 --> 00:17:30,170
+لأن cos θ بين الصفر و ناقص واحد والواحد تكون أكبر
+
+262
+00:17:30,170 --> 00:17:35,530
+ما يمكن لما cos θ = 1، يعني لما θ = 0، إذاً
+
+263
+00:17:35,530 --> 00:17:39,730
+اللي هو the function F increasing increases most
+
+264
+00:17:39,730 --> 00:17:44,220
+rapidly when cos θ = 1 or when θ = 0، في هذه
+
+265
+00:17:44,220 --> 00:17:50,980
+الحالة U ماذا يكون؟ U مادام
+
+266
+00:17:50,980 --> 00:17:54,380
+الزاوية بين الـ Gradient و U، مادام الزاوية بين الـ
+
+267
+00:17:54,380 --> 00:17:54,660
+Gradient و U، مادام الزاوية بين الـ Gradient و U
+
+268
+00:17:54,660 --> 00:17:57,720
+مادام الزاوية بين الـ Gradient و U، مادام الزاوية
+
+269
+00:17:57,720 --> 00:17:58,740
+بين الـ Gradient و U، مادام الزاوية بين الـ
+
+270
+00:17:58,740 --> 00:17:59,280
+Gradient و U، مادام الزاوية بين الـ Gradient و U
+
+271
+00:17:59,280 --> 00:18:00,140
+مادام الزاوية بين الـ Gradient و U، مادام الزاوية
+
+272
+00:18:00,140 --> 00:18:01,160
+بين الـ Gradient و U، مادام الزاوية بين الـ
+
+273
+00:18:01,160 --> 00:18:01,240
+Gradient و U، مادام الزاوية بين الـ Gradient و U
+
+274
+00:18:01,240 --> 00:18:03,480
+بين الـ Gradient و U، مادام الزاوية بين الـ
+
+275
+00:18:03,480 --> 00:18:07,350
+Gradient و U، يعني that is at each point P in its
+
+276
+00:18:07,350 --> 00:18:10,750
+domain, F increases most rapidly in the direction
+
+277
+00:18:10,750 --> 00:18:14,790
+of the gradient للـ F at P، الـ derivative in this
+
+278
+00:18:14,790 --> 00:18:18,930
+case طبعاً، in this direction، لما يكون الاتجاه اللي
+
+279
+00:18:18,930 --> 00:18:21,750
+هو الـ gradient والاتجاه اللي هو الـ U في نفس
+
+280
+00:18:21,750 --> 00:18:24,850
+الاتجاه، يعني Cos 0، يعني بيساوي الـ absolute
+
+281
+00:18:24,850 --> 00:18:28,430
+value للـ gradient للـ F أو الـ magnitude للـ gradient
+
+282
+00:18:28,430 --> 00:18:32,430
+للـ F، إذًا الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+283
+00:18:32,430 --> 00:18:32,590
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+284
+00:18:32,590 --> 00:18:33,690
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+285
+00:18:33,690 --> 00:18:35,190
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+286
+00:18:35,190 --> 00:18:37,710
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+287
+00:18:37,710 --> 00:18:44,450
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+288
+00:18:44,450 --> 00:18:45,470
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+289
+00:18:45,470 --> 00:18:45,710
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+290
+00:18:45,710 --> 00:18:47,350
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+291
+00:18:47,350 --> 00:18:53,810
+الـ الـ الـ الـ الـ يعني بمعنى آخر عندما يكون
+
+292
+00:18:53,810 --> 00:18:58,090
+الاتجاهين في عكس اتجاه بعض، يعني عندما تكون F يزداد
+
+293
+00:18:58,090 --> 00:19:04,830
+أكثر بسرعة في اتجاه  عكس اتجاه لـ Gradient و الـ
+
+294
+00:19:04,830 --> 00:19:08,580
+Derivative في هذه الحالة هيكون بيساوي ناقص الـ
+
+295
+00:19:08,580 --> 00:19:12,320
+Absolute Value، دي  بتكون اللي هي أقل
+
+296
+00:19:12,320 --> 00:19:16,780
+قيمة وهناك أكبر قيمة، طيب، واكتشف كل الـ
+
+297
+00:19:16,780 --> 00:19:21,200
+Directional Derivative، Sphere أكيد، Any Direction U
+
+298
+00:19:21,200 --> 00:19:25,420
+Orthogonal To The Gradient، Is a direction of zero
+
+299
+00:19:25,420 --> 00:19:30,120
+change in F because θ is equal to π على 2، لما
+
+300
+00:19:30,120 --> 00:19:36,960
+يكون الآن الـ Zero، بيكون عندي الـ Direction of U
+
+301
+00:19:36,960 --> 00:19:40,460
+is orthogonal to the gradient، فالـ F بيكون اللي هو
+
+302
+00:19:40,460 --> 00:19:43,340
+الـ Directional Derivative، أي ايش، صفر يعني الـ
+
+303
+00:19:43,340 --> 00:19:46,420
+Directional Derivative بتكون Zero في حالة اللي هو
+
+304
+00:19:46,420 --> 00:19:51,180
+θ بتساوي π على 2، هذا اللي هو زيرو، ايش Gradient، إذاً
+
+305
+00:19:51,180 --> 00:19:53,420
+الـ Direction of the derivative of U of F بتساوي
+
+306
+00:19:53,420 --> 00:20:00,500
+صفر، When U is orthogonal to the gradient، ماشي،
+
+307
+00:20:00,500 --> 00:20:04,080
+المهم، اضحى نأخذ مثال عددي، Example، تلاتة
+
+308
+00:20:04,080 --> 00:20:07,520
+Find the directions in which F of XY بيساوي X تربيع
+
+309
+00:20:07,520 --> 00:20:11,100
+على 2 و Y تربيع على 2، وقول لي بس أودد الـ directions
+
+310
+00:20:11,100 --> 00:20:16,040
+اللي الـ F هذه واحد، Increases most rapidly at the
+
+311
+00:20:16,040 --> 00:20:20,810
+point واحد واحد، يعني لما بدي Direction لما الـ F
+
+312
+00:20:20,810 --> 00:20:23,550
+increases most rapidly عند الواحد و واحد، قال لي
+
+313
+00:20:23,550 --> 00:20:26,570
+واحد على طول، طب أنت قلت قبل شوية أنهم يكونون most
+
+314
+00:20:26,570 --> 00:20:29,910
+rapidly لما يكونون في اتجاه مين؟ لـ Gradient، بقوله
+
+315
+00:20:29,910 --> 00:20:32,930
+صح، قال لي، أصلاً دي decreases most rapidly اللي بدك
+
+316
+00:20:32,930 --> 00:20:36,390
+تسأل عنها في الثانية برضه، في عكس اتجاه مين؟ لـ
+
+317
+00:20:36,390 --> 00:20:40,270
+Gradient، إذا أسهل الأمر بنجد ما هو الـ Gradient و
+
+318
+00:20:40,270 --> 00:20:45,550
+بنجد ما هو الـ Direction، يعني الـ Gradient على
+
+319
+00:20:45,550 --> 00:20:48,770
+مقداره، اللي هو بيكون الجواب اللي هو الأول هذا و
+
+320
+00:20:48,770 --> 00:20:52,690
+ناقصه بيكون الجواب الثاني، بقوله صح كلامك، بقول له طب
+
+321
+00:20:52,690 --> 00:20:55,390
+What are the directions of zero change in F at
+
+322
+00:20:55,390 --> 00:21:00,300
+واحد واحد؟ أنت بتقول لما قلنا قبل شوية أنه
+
+323
+00:21:00,300 --> 00:21:05,400
+بيكون اللي هو الـ Zero change لما يكون الـ
+
+324
+00:21:05,400 --> 00:21:09,840
+Direction لـ U و V متعامدين، يعني وجدنا الـ
+
+325
+00:21:09,840 --> 00:21:13,480
+Direction للـ Gradient على طول بنغير إشارة واحدة و
+
+326
+00:21:13,480 --> 00:21:18,770
+بنخلي الثانية زي ما هي و التجاهين عموديين عليه، طيب
+
+327
+00:21:18,770 --> 00:21:22,690
+خلينا نشوف الكلام ده اللي بنحكيه سريعاً للإجابة
+
+328
+00:21:22,690 --> 00:21:26,530
+بسرعة، يجي الآن نشوف increases most rapidly at the
+
+329
+00:21:26,530 --> 00:21:30,410
+point 1, 1، the function قلنا فوق increases most
+
+330
+00:21:30,410 --> 00:21:33,370
+rapidly in the direction of the gradient at 1, 1،
+
+331
+00:21:33,370 --> 00:21:37,090
+وعادةً بيلزمنا نوجد الـ Gradient، قادر الـ Gradient
+
+332
+00:21:37,090 --> 00:21:41,170
+للـ F أيش بيساوي؟ اللي هو Del F by Del X اللي هي
+
+333
+00:21:41,170 --> 00:21:46,210
+2X على 2 يعني X، في اتجاه الـ I، Del F by Del Y اللي هي 2Y
+
+334
+00:21:46,210 --> 00:21:50,220
+على 2 يعني Y، في اتجاه الـ J، هذا الـ Gradient للـ
+
+335
+00:21:50,220 --> 00:21:53,460
+gradient للـ F عند الواحد والواحد ايش هيساوي؟ أو
+
+336
+00:21:53,460 --> 00:21:57,020
+عند 1, 1، بصير I زائد الـ J، إذًا الآن الـ
+
+337
+00:21:57,020 --> 00:22:00,380
+Direction للـ Gradient للـ F، وعبارة عن الـ Gradient
+
+338
+00:22:00,380 --> 00:22:04,240
+للـ F على الـ Magnitude له، Magnitude هو الجذر التربيعي
+
+339
+00:22:04,240 --> 00:22:07,260
+لـ 1 زائد 1، على جذر الاثنين، يعني الـ Direction
+
+340
+00:22:07,260 --> 00:22:10,440
+U بيساوي واحد على جذر الاثنين في اتجاه الـ I زائد واحد
+
+341
+00:22:10,440 --> 00:22:13,480
+على جذر الاثنين في اتجاه الـ J، هذا الـ Direction هو
+
+342
+00:22:13,480 --> 00:22:17,640
+Direction لـ Gradient اللي عنده اللي هي الـ F
+
+343
+00:22:17,640 --> 00:22:23,450
+increases most rapidly اللي عنده النقطة 1, 1، طيب، قلت
+
+344
+00:22:23,450 --> 00:22:25,870
+أكيد كلكم قلتم الـ function decreases most
+
+345
+00:22:25,870 --> 00:22:29,330
+rapidly in the direction of عكس الـ gradient عنده
+
+346
+00:22:29,330 --> 00:22:33,210
+نقطة 1, 1، which is هذا بس مضروب في مين؟ في
+
+347
+00:22:33,210 --> 00:22:36,390
+سالب واحد، ناقص واحد على جذر الاثنين في اتجاه I ناقص
+
+348
+00:22:36,390 --> 00:22:40,270
+واحد على جذر الاثنين في اتجاه J، الآن الجزء الثالث
+
+349
+00:22:40,270 --> 00:22:44,370
+الثالث، what are the directions of zero change يعني
+
+350
+00:22:44,370 --> 00:22:47,410
+للـ Gradient لما يكون اللي هو إيش؟ directional
+
+351
+00:22:47,410 --> 00:22:50,150
+derivative zero، بتكون directional derivative zero
+
+352
+00:22:50,150 --> 00:22:53,120
+لما الزاوية بينهم إيش بتساوي؟ π على اثنين، يعني
+
+353
+00:22:53,120 --> 00:22:55,780
+بمعنى آخر لما يكون الـ U عمودي على الـ
+
+354
+00:22:55,780 --> 00:23:00,020
+Gradient، طب هي الـ U موجودة، التي جاهزة، الـ Gradient
+
+355
+00:23:00,020 --> 00:23:03,720
+موجودة، إذًا  بدأ نوجد الـ Direction تبع الـ Zero، الـ
+
+356
+00:23:03,720 --> 00:23:06,140
+Direction of Zero Change، بقصده Direction تبع الـ
+
+357
+00:23:06,140 --> 00:23:09,260
+Zero Change إن F عند واحد، are the directions
+
+358
+00:23:09,260 --> 00:23:13,180
+orthogonal to gradient للـ F، الـ Directions اللي
+
+359
+00:23:13,180 --> 00:23:19,140
+عموديين على الـ F، الآن هي تجاه الـ F وهي
+
+360
+00:23:19,140 --> 00:23:23,280
+تجاه الـ F هي ناقص الـ F اللي هي تجاه الـ Gradient
+
+361
+00:23:24,290 --> 00:23:28,130
+تجاه الـ Gradient اللي هو العمودي عليه، بس بنغير
+
+362
+00:23:28,130 --> 00:23:30,650
+إشارتها، دي بيصير ناقص واحد على جذر الاثنين في اتجاه
+
+363
+00:23:30,650 --> 00:23:33,890
+الـ I زائد واحد على جذر الاثنين في اتجاه الـ J أو
+
+364
+00:23:34,290 --> 00:23:39,050
+أو كمان اللي هو بغير هذه شريطة وبخلي هذه ثابتة
+
+365
+00:23:39,050 --> 00:23:42,130
+يعني بيصير N بتساوي واحد على جذر الاثنين في اتجاه I
+
+366
+00:23:42,130 --> 00:23:46,510
+ناقص واحد على جذر الاثنين J، و ناقص N اللي عكسه ناقص
+
+367
+00:23:46,510 --> 00:23:48,990
+واحد على جذر الاثنين في اتجاه I زائد واحد على جذر الاثنين
+
+368
+00:23:48,990 --> 00:23:52,450
+في اتجاه J، هدول الاثنين طبعاً عموديين، واحدة هي لو فرضنا
+
+369
+00:23:52,450 --> 00:23:56,410
+هذا الـ gradient، الاتجاه الـ gradient، الاتجاه العمودي
+
+370
+00:23:56,410 --> 00:23:59,990
+عليه هيه هنا أو اللي تحت، هدول عكس بعض، الاثنين
+
+371
+00:24:00,350 --> 00:24:04,970
+عشان نضرب على الـ dot product 1 على الجذر الثاني و
+
+372
+00:24:04,970 --> 00:24:05,850
+1 على الجذر الثاني
+
+373
+00:24:33,380 --> 00:24:40,400
+الآن نشوف علاقة الـ Gradient بمين؟ بالـ Tangents to
+
+374
+00:24:40,400 --> 00:24:43,500
+Level Curves ونستغل هذه العلاقة في إيجاد ما ..
+
+375
+00:24:43,500 --> 00:24:47,940
+اللي هو إيش .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+376
+00:24:47,940 --> 00:24:48,460
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+377
+00:24:48,460 --> 00:24:49,060
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+378
+00:24:49,060 --> 00:24:49,220
+.. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+379
+00:24:49,220 --> 00:24:49,760
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+380
+00:24:49,760 --> 00:24:50,560
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+381
+00:24:50,560 --> 00:24:50,620
+.. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+382
+00:24:50,620 --> 00:24:50,840
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+383
+00:24:50,840 --> 00:24:59,060
+ما .. ما .. ما ..
+
+384
+00:24:59,060 --> 00:25:01,000
+ما
+
+385
+00:25:06,720 --> 00:25:13,260
+لأن F of X Y هذه لو فرضنا أنه قيمتها ثابتة، وين على
+
+386
+00:25:13,260 --> 00:25:18,540
+اللي هي عند قيم X = G of T وعند قيم Y = H of T
+
+387
+00:25:18,540 --> 00:25:23,960
+يعني على مين؟ على الـ Curve هذه، لو عوضنا X بأي
+
+388
+00:25:23,960 --> 00:25:27,060
+قيمة، يعني لما يكون في عندي Vector هيه اللي بتساوي
+
+389
+00:25:27,060 --> 00:25:29,980
+G of T في اتجاه الـ I زائد H of T في اتجاه الـ J
+
+390
+00:25:29,980 --> 00:25:34,300
+و إذا عوضنا الـ G of T هنا و الـ H of T هنا هنلاقي
+
+391
+00:25:34,300 --> 00:25:37,400
+القيمة ايش؟  ثابتة، يعني صار هذا عبارة
+
+445
+00:29:13,620 --> 00:29:16,660
+بيجعل الـ Zero Change كافياً هذا ما تُظهره الرسمة
+
+446
+00:29:16,660 --> 00:29:20,740
+الذي قمنا به للتو في المثال الذي سبق بِقليل هو المثال
+
+447
+00:29:20,740 --> 00:29:26,750
+الذي قمنا به، آه آسف، هذا المثال هو آه هذا الذي
+
+448
+00:29:26,750 --> 00:29:30,450
+هو الذي، هو التوضيح الهندسي له
+
+449
+00:29:30,450 --> 00:29:37,920
+تحدثنا عنه، هذا التوضيح في problem 39 سأستغله لاحقاً
+
+450
+00:29:37,920 --> 00:29:43,640
+قليلاً، سأستعمله لأحلّ مشكلة مشابهة، مثال، خلّيني فقط
+
+451
+00:29:43,640 --> 00:29:47,660
+البرنامج الذي كنتُ معه، show that if x minus x0
+
+452
+00:29:47,660 --> 00:29:51,760
+plus y minus y0 is an equation for the line in
+
+453
+00:29:51,760 --> 00:29:54,480
+the xy plane through the point x0 y0 normal to the
+
+454
+00:29:54,480 --> 00:29:59,170
+vector، كذا، يعني لو كان لدينا نقطة x0 y0 و لدينا Normal
+
+455
+00:29:59,170 --> 00:30:05,610
+vector N على الخط، يعني معروف الـ Normal N بتساوي I
+
+456
+00:30:05,610 --> 00:30:09,470
+مثل BJ، معروفة نقطة، يعني عندي خطّ كان معروف
+
+457
+00:30:09,470 --> 00:30:13,230
+Normal عليه ومعروفة نقطة X0 Y0، وبهذا يكون لدينا معادلته
+
+458
+00:30:13,690 --> 00:30:16,850
+معادلته، أنا أقول هي a في x minus x not مثل b في y
+
+459
+00:30:16,850 --> 00:30:20,170
+minus y not، باختصار، لكي لا نطيل، حل السؤال الذي
+
+460
+00:30:20,170 --> 00:30:23,770
+بعده، ولن يعتمد عليه، ما هذا؟ يا جماعة، هذا كثير
+
+461
+00:30:23,770 --> 00:30:28,250
+لماذا؟ عندي معروف... عندي خط، هذا الخط يوجد
+
+462
+00:30:28,250 --> 00:30:32,610
+معادلته معروف، العمودي عليه ومعروفة نقطة عليه
+
+463
+00:30:32,610 --> 00:30:35,730
+نعرف معادلته، طبعاً كلكم تعرفون كيف تُوجد، دعوني فقط
+
+464
+00:30:35,730 --> 00:30:38,950
+بسرعة أحله، ال normal to the vector m ai مثل b
+
+465
+00:30:38,950 --> 00:30:45,570
+جي، الآن T بتساوي
+
+466
+00:30:45,570 --> 00:30:50,070
+T بيساوي PI ناقص AJ أو PI ناقص AJ، يعني أي واحد
+
+467
+00:30:50,070 --> 00:30:56,910
+بالفعل، ال equation is X بيساوي X naught مثل TB
+
+468
+00:30:56,910 --> 00:31:01,860
+و Y بيساوي Y naught ناقص TA،الذي معروف الآن، x بتساوي
+
+469
+00:31:01,860 --> 00:31:06,380
+الذي هو خطأ، لدينا ال normal معروف، فيكون لدينا
+
+470
+00:31:06,380 --> 00:31:10,240
+الذي هو معادلة، هي x بيساوي x naught، النقطة
+
+471
+00:31:10,240 --> 00:31:16,720
+زائد t في b و y بيساوي y naught ناقص t في a، هي
+
+472
+00:31:16,720 --> 00:31:23,260
+معادلة، that is الآن x minus x naught بيساوي tb و y
+
+473
+00:31:23,260 --> 00:31:27,700
+minus y naught بيساوي t ناقص t، نحل المعادلة A في X
+
+474
+00:31:27,700 --> 00:31:32,120
+نحلها معاً لكي نضعها في المعادلة العليا، A في
+
+475
+00:31:32,120 --> 00:31:37,920
+X minus X0 بتساوي T في A، بالتأكيد تعرفون كيف، الشيء الذي
+
+476
+00:31:37,920 --> 00:31:44,800
+قمت به، لدينا هاتين المعادلتين، a في x minus x not
+
+477
+00:31:44,800 --> 00:31:50,020
+هي a في x minus x not بتساوي a في tb، يعني t a
+
+478
+00:31:50,020 --> 00:31:55,220
+في b و b في y minus y not بتساوي ناقص t في ba، لأن
+
+479
+00:31:55,220 --> 00:32:00,720
+المعادلتين معاً تصبح هذه بتساوي ناقص هذه، أو
+
+480
+00:32:00,720 --> 00:32:06,320
+بمعنى آخر، نجمع هاتين المعادلتين
+
+481
+00:32:06,320 --> 00:32:09,700
+معاً، فيُصبح a في X minus X naught و
+
+482
+00:32:09,700 --> 00:32:12,820
+b في Y minus Y بتساوي صفر، هذا ما تُظهره
+
+483
+00:32:12,820 --> 00:32:18,140
+معادلة المتجه العمودي على ما لدينا
+
+484
+00:32:18,140 --> 00:32:25,700
+فلنرى، find an equation for the
+
+485
+00:32:25,700 --> 00:32:28,780
+tangent to the ellipse، find an equation for the
+
+486
+00:32:28,780 --> 00:32:32,960
+tangent to the ellipse، مُعطى عند النقطة ناقص 2
+
+487
+00:32:32,960 --> 00:32:39,690
+و 1، يعني مُعطى على هذه ال ellipse، أريد
+
+488
+00:32:39,690 --> 00:32:44,690
+إيجاد ال equation للمماس عند النقطة ناقص 2 و 1، هي
+
+489
+00:32:44,690 --> 00:32:49,070
+المماس، أريد إيجاد ال equation للمماس، طبعاً الكثير منكم
+
+490
+00:32:49,070 --> 00:32:53,090
+يعرفون كيف يفعلون ذلك، أو بالطريقة القديمة، لكن الآن
+
+491
+00:32:53,090 --> 00:32:56,870
+دعوني أقول لكم، لكي نطبق ما تحدثنا عنه باستخدام
+
+492
+00:32:56,870 --> 00:33:00,330
+level curves، عندما نحلّ بالطريقة التي
+
+493
+00:33:00,330 --> 00:33:04,970
+هي القديمة، المعادلة موجودة والنقطة موجودة، نجد
+
+494
+00:33:04,970 --> 00:33:07,330
+الذي هو المماس، المماس معروف والنقطة
+
+495
+00:33:07,330 --> 00:33:11,310
+معروفة، نجد، ماذا نجد؟ نجد المماس، دعوني أوجده
+
+496
+00:33:11,310 --> 00:33:16,260
+بطريقة أخرى، الآن هذه هي X تربيع على أربعة زائد Y تربيع
+
+497
+00:33:16,260 --> 00:33:20,580
+على اثنين بتساوي اثنين، هي عبارة عن ال ellipse، هذه هي ال
+
+498
+00:33:20,580 --> 00:33:24,220
+tangent لها، والآن أريد إيجاد العمودي على ال
+
+499
+00:33:24,220 --> 00:33:29,730
+tangent، ونستغل ما تعلمناه قبل قليل، ماذا
+
+500
+00:33:29,730 --> 00:33:33,310
+تريد أن تفعل؟ تريد استخدام معرفتنا، دعونا نقول ال
+
+501
+00:33:33,310 --> 00:33:37,470
+ellipse x² على 4 زائد 2y² على 2 is a level curve of
+
+502
+00:33:37,470 --> 00:33:40,430
+mean of the function f of x,y بتساوي x² على 4 زائد
+
+503
+00:33:40,430 --> 00:33:44,470
+2y² على 2y²، لأن الدالة عندي هي of two
+
+504
+00:33:44,470 --> 00:33:48,170
+variables، التي هي function of two variables، هي
+
+505
+00:33:48,170 --> 00:33:52,540
+الدالة، الآن هذا ال Dialogue طبعاً في Surface، في
+
+506
+00:33:52,540 --> 00:33:55,580
+Space، التي عن ال Level Curve أخذت ال
+
+507
+00:33:55,580 --> 00:33:58,600
+Level Curve X تربيع على 4 زائد Y تربيع بتساوي 2
+
+508
+00:33:58,600 --> 00:34:02,760
+التي هي عبارة عن، أصبحت Level Curve لهذه، نحن نعرف
+
+509
+00:34:02,760 --> 00:34:07,960
+أن ال Gradient للدالة عمودي على الـ
+
+510
+00:34:07,960 --> 00:34:11,540
+Level Curve، الذي ذكرناه قبل قليل، وهو تطبيق عليه
+
+511
+00:34:11,830 --> 00:34:17,730
+ال Gradient بتساوي X على 2 في اتجاه I زائد 2Y في
+
+512
+00:34:17,730 --> 00:34:24,270
+اتجاه J، هذا ال Gradient عند النقطة ناقص 2 و
+
+513
+00:34:24,270 --> 00:34:31,050
+1، بتساوي ناقص I، عوضت عن X ناقص 2 و 2G، هذا ال
+
+514
+00:34:31,050 --> 00:34:36,130
+Gradient، الآن نحن نعرف أنه عمودي على ال
+
+515
+00:34:36,130 --> 00:34:38,890
+Level Curve، يعني عمودي على المماس الخاص بالـ Level
+
+516
+00:34:38,890 --> 00:34:41,790
+Curve عند النقطة المذكورة، هذا ما ذكرناه قبل قليل
+
+517
+00:34:41,790 --> 00:34:46,340
+أثبتناه، الذي قلناه قبل قليل، هذا الذي
+
+518
+00:34:46,340 --> 00:34:49,860
+هو ال Gradient عمودي دائماً على ال Level Curve
+
+519
+00:34:49,860 --> 00:34:54,200
+عند النقطة X0 و Y0، نستخدم هذه المعلومة، سنعرف أن
+
+520
+00:34:54,200 --> 00:34:57,660
+ال Gradient الذي هو العمودي على المماس
+
+521
+00:34:57,660 --> 00:35:01,100
+نعرف نقطة، من خلال المثال الذي سبق أو
+
+522
+00:35:01,100 --> 00:35:04,000
+problem الذي سبق، سنعرف
+
+523
+00:35:04,000 --> 00:35:08,720
+العمودي ونعرف نقطة، إذاً نعرف المعادلة التي هي المماس
+
+524
+00:35:08,720 --> 00:35:12,520
+الذي نريده، وهو a في x ناقص x not مثل b في y ناقص
+
+525
+00:35:12,520 --> 00:35:15,980
+y not بتساوي صفر، ال a هي ناقص واحد في
+
+526
+00:35:15,980 --> 00:35:19,760
+هذه الحالة، و ال b هي اثنين، المعامل
+
+527
+00:35:19,760 --> 00:35:24,200
+الخاص بالعمودي، x minus x not التي هي x
+
+528
+00:35:24,200 --> 00:35:27,120
+not، التي هي ناقص اثنين، مثل اثنين، و y ناقص y not
+
+529
+00:35:27,120 --> 00:35:31,040
+التي هي واحد، بتساوي صفر، عندما نبسطها، يظهر
+
+530
+00:35:31,040 --> 00:35:35,160
+X ناقص 2Y بتساوي سالب 4، هذه معادلة المماس الذي أمامنا،
+
+531
+00:35:35,160 --> 00:35:38,980
+وأنتم في البيت جرّبوا كيف تُوجدونها بالطريقة
+
+532
+00:35:38,980 --> 00:35:43,360
+السابقة، ننتقل إلى algebra rules for gradient، التي
+
+533
+00:35:43,360 --> 00:35:46,780
+هي بعض خواص ال gradient، خواص ال gradient شبيهة بخواص
+
+534
+00:35:46,780 --> 00:35:49,660
+من يا جماعة؟ ال derivative العادية، ال gradient ل F
+
+535
+00:35:49,660 --> 00:35:52,580
+زائد G بتساوي gradient لF زائد gradient لG، لأن
+
+536
+00:35:52,580 --> 00:35:57,860
+ال gradient عبارة عن، هي  دل F على دل X في اتجاه I
+
+537
+00:35:57,860 --> 00:36:00,280
+مثل دل F على دل Y في اتجاه J، يعني هي مُشتقات
+
+538
+00:36:00,280 --> 00:36:03,220
+derivatives، فطبيعي أن تكون خواصها هي خواص ال
+
+539
+00:36:03,220 --> 00:36:06,240
+derivative العادية وال partial derivatives، فال
+
+540
+00:36:06,240 --> 00:36:10,020
+gradient لF ناقص G بتساوي gradient لF ناقص gradient لG، ال
+
+541
+00:36:10,020 --> 00:36:12,940
+gradient لكِمية ثابتة K بتساوي صفر، ال
+
+542
+00:36:12,940 --> 00:36:16,290
+gradient لF في G بتساوي F في gradient لG زائد G في gradient لF، وال
+
+543
+00:36:16,290 --> 00:36:19,430
+gradient لF على G بتساوي G gradient لF ناقص F gradient
+
+544
+00:36:19,430 --> 00:36:23,090
+لG على G تربيع، يعني التي تعرفونها منذ زمن
+
+545
+00:36:23,090 --> 00:36:26,730
+هذه الخواص، دعونا نتأكد من واحدة منها فقط
+
+546
+00:36:26,730 --> 00:36:31,870
+عدديًا، للاطمئنان، ليس أكثر من ذلك لأنّه ليس مطلوباً، مثال
+
+547
+00:36:31,870 --> 00:36:37,460
+رقم خمسة، We illustrate two of the rules with، يعني نريد
+
+548
+00:36:37,460 --> 00:36:42,560
+توضيح قاعدتين مما سبق باستخدام، ما هو مثال، عددياً،
+
+549
+00:36:42,560 --> 00:36:47,460
+f of x,y بتساوي x ناقص y، و g of x,y بتساوي 3y، ال
+
+550
+00:36:47,460 --> 00:36:51,880
+gradient لf، أنتم تعرفون، دل f على دل x هو
+
+551
+00:36:51,880 --> 00:36:55,340
+i، دل f على دل y هو ناقص واحد، فيكون لدينا ناقص واحد
+
+552
+00:36:55,340 --> 00:37:00,540
+وهنا ناقص واحد، فيصير i ناقص j، DEL F DEL J BY DEL X
+
+553
+00:37:08,030 --> 00:37:12,970
+صفر، إذًا صفر، لا يوجد شيء هنا، DEL J BY DEL Y تلاتة، إذًا
+
+554
+00:37:12,970 --> 00:37:18,230
+ال Gradient لF عبارة عن تلاتة G، هدفنا يا جماعة، هو التأكد
+
+555
+00:37:18,230 --> 00:37:21,710
+فقط في هذا المثال، المثال، del f minus
+
+556
+00:37:21,710 --> 00:37:26,290
+g، نوضح كيف يساوي del f ناقص del g، وال
+
+557
+00:37:26,290 --> 00:37:30,410
+gradient لf ناقص gradient لg، ال gradient لf
+
+558
+00:37:30,410 --> 00:37:34,590
+ناقص g، نبدأ من f ناقص g، هي x ناقص y
+
+559
+00:37:34,590 --> 00:37:38,650
+ناقص 3y، يصبح x ناقص 4y، ال gradient لهذه التي
+
+560
+00:37:38,650 --> 00:37:43,360
+هي دل f على دل...دل هذا على دل x، يعني i، لأن
+
+561
+00:37:43,360 --> 00:37:47,240
+مشتقتها بالنسبة لX واحد، ومشتقتها بالنسبة لY ناقص 4
+
+562
+00:37:47,240 --> 00:37:53,240
+i ناقص 4j، هي في الواقع i ناقص g ناقص 3j، يعني
+
+563
+00:37:53,240 --> 00:37:57,680
+i ناقص 4g، يعني ال gradient لF ناقص gradient ل
+
+564
+00:37:57,680 --> 00:38:01,400
+G، دعونا نرى الثانية، gradient لF في G، نأخذ ال
+
+565
+00:38:01,400 --> 00:38:05,860
+F في G، هي F x ناقص y في G، التي هي تلاتة y،
+
+566
+00:38:05,860 --> 00:38:08,440
+تصبح تلاتة xy ناقص
+
+567
+00:38:11,100 --> 00:38:14,640
+تلاتة y تربيع، لأن تلاتة y تصبح تلاتة y
+
+568
+00:38:14,640 --> 00:38:19,120
+تربيع، ضربت الF في G، في G يا جماعة، ظهرت عندي
+
+569
+00:38:19,120 --> 00:38:22,600
+بهذا الشكل، ال gradient لهذه الدالة الجديدة هو
+
+570
+00:38:22,600 --> 00:38:27,260
+عبارة عن تلاتة y، هذه صفر، تلاتة y في اتجاه الi،
+
+571
+00:38:27,260 --> 00:38:30,240
+بالمشتقة بالنسبة لX، طبعاً أنتم تعرفون، بعدين
+
+572
+00:38:30,240 --> 00:38:33,480
+بالمشتقة بالنسبة لY، بالنسبة لY التي هي تلاتة x
+
+573
+00:38:33,480 --> 00:38:37,620
+ناقص ستة y، تصبح تلاتة x ناقص ستة y في اتجاه الj،
+
+574
+00:38:37,620 --> 00:38:41,520
+هذه هي ال gradient لل FG، ننتقل الآن لحساب ال
+
+575
+00:38:41,520 --> 00:38:45,790
+gradient للG، وهي G في ال gradient للF
+
+576
+00:38:45,790 --> 00:38:48,590
+و F في ال gradient للG، حسب قانونه، نريد
+
+577
+00:38:48,590 --> 00:38:53,430
+أن نقول أن هذا يساوي هذا، نتأكد فقط من خلال هذا
+
+578
+00:38:53,430 --> 00:38:57,830
+المثال، هذا الذي هو حاصل ضرب دالتين، مشتقتين،
+
+579
+00:38:57,830 --> 00:39:01,310
+فلنرى كيف هذا الكلام يساوي ما سبق من خلال
+
+580
+00:39:01,310 --> 00:39:08,130
+مثالنا، ثلاثة j، ثلاثة وi، ثلاثة وi، ثلاثة و
+
+581
+00:39:08,130 --> 00:39:15,500
+i، ثلاثة وi، ثلاثة، ال gradient للG
+
+582
+00:39:15,500 --> 00:39:22,940
+عبارة عن ثلاثة j، آسف، فعلاً
+
+583
+00:39:22,940 --> 00:39:26,660
+بدأت بالـ G، G التي هي ثلاث y، لكي لا أخطئ في ال gradient ل
+
+584
+00:39:26,660 --> 00:39:30,940
+F، ال gradient للF هي i ناقص j، هي هذه زائد ال
+
+585
+00:39:30,940 --> 00:39:36,000
+F، قيمتها x ناقص y، ضربتها في من؟ في
+
+586
+00:39:36,000 --> 00:39:40,140
+gradient للG، gradient للG، ثلاثة j، الآن هذا
+
+587
+00:39:44,290 --> 00:39:49,210
+المقدار يصبح عبارة عن ثلاثة y في i ناقص j،
+
+588
+00:39:49,210 --> 00:39:53,150
+ابقيتها كما هي، لكي نصل إلى ما سبق، زائد هذه x
+
+589
+00:39:53,150 --> 00:39:57,310
+دخلت إلى ما بداخلها، يصبح ثلاثة x ناقص ثلاثة y، الآن دعونا
+
+590
+00:39:57,310 --> 00:40:02,390
+نجمّع هنا، ثلاثة y في ناقص j، تصبح
+
+591
+00:40:02,390 --> 00:40:07,210
+ناقص ثلاثة yj، عندما نجمعها مع ناقص ثلاثة yj تصبح
+
+592
+00:40:07,210 --> 00:40:11,910
+ناقص ثلاثة yj، يبقى هنا ثلاثة x في اتجاه الj
+
+593
+00:40:11,910 --> 00:40:15,190
+كما هي، الآن ما تبقى هنا ما لم نستخدمه هو
+
+594
+00:40:15,190 --> 00:40:18,750
+ثلاثة y في اتجاه الi، هي هذه كما هي، إذاً ثلاثة y
+
+595
+00:40:18,750 --> 00:40:22,310
+في اتجاه الi كما هي، و هذا ناقص ثلاثة في
+
+596
+00:40:22,310 --> 00:40:24,590
+اتجاه الj، مع ناقص ثلاثة في اتجاه الj تصبح
+
+597
+00:40:24,590 --> 00:40:28,190
+ناقص ستة في اتجاه الj، وهذا ثلاثة x كما هي، يكون
+
+598
+00:40:28,190 --> 00:40:32,210
+فعلاً هذا المقدار يساوي هذا المقدار، من خلال
+
+599
+00:40:32,210 --> 00:40:36,860
+الذي أمامنا، هذا ما ذكرناه قبل
+
+600
+00:40:36,860 --> 00:40:39,980
+قليل، كل شيء لكن بدلاً من functions of two variables، دعوني
+
+601
+00:40:39,980 --> 00:40:43,740
+أتحدث عن functions of three variables، Similarly، وسهل
+
+602
+00:40:43,740 --> 00:40:50,080
+functions of three variables، هي كما، يعني لدينا for a
+
+603
+00:40:50,080 --> 00:40:53,220
+differentiable function f of x,y,z and a unit
+
+604
+00:40:53,220 --> 00:40:58,200
+vector u بتساوي u1 في اتجاه i و u2 في اتجاه j زائد
+
+605
+00:40:58,200 --> 00:41:01,020
+u3 في اتجاه k، إذًا الآن الدالة عندنا بدل
+
+606
+00:41:01,020 --> 00:41:03,820
+ما هي two variables هي three، و المتجه طبعاً بدل
+
+607
+00:41:03,820 --> 00:41:08,100
+ما يكون i و j، سيكون i, j و k
+
+608
+00:41:08,100 --> 00:41:11,350
+لأنه أصبح face based، الآن لدينا ال gradient ل
+
+609
+00:41:11,350 --> 00:41:14,830
+F، نعرفه، دل F على دل X في اتجاه I زائد دل F على
+
+610
+00:41:14,830 --> 00:41:18,330
+دل Y في اتجاه J زائد دل F على دل Z في اتجاه K، هذا
+
+611
+00:41:18,330 --> 00:41:21,270
+تعريفه ل Function of Three Variables، ال
+
+612
+00:41:21,270 --> 00:41:23,810
+Directional Derivative في هذه الحالة هو gradient
+
+613
+00:41:23,810 --> 00:41:29,260
+لF دُوْط U، ال U التي هي عبارة عن هذا، دُوْط الU
+
+614
+00:41:29,26
+
+667
+00:45:04,900 --> 00:45:09,720
+طيب، لأ، الـ function increases most rapidly in the
+
+668
+00:45:09,720 --> 00:45:12,820
+direction of الـ gradient، صح ولا لأ؟ يعني في اتجاه
+
+669
+00:45:12,820 --> 00:45:16,380
+هذا، بتظهر اللي هو إيش؟ increases most rapidly اللي
+
+670
+00:45:16,380 --> 00:45:19,140
+هو في اتجاه، هذا اتجاه مين؟
+
+671
+00:45:24,830 --> 00:45:28,330
+على الجذر التربيعي للأربعة زائد الأربعة زائد واحد
+
+672
+00:45:28,330 --> 00:45:31,750
+ثلاثة، يعني اللي هو بيصير اللي هو الـ function
+
+673
+00:45:31,750 --> 00:45:35,230
+increases most rapidly in the direction vector
+
+674
+00:45:35,230 --> 00:45:40,470
+اللي هو U بتساوي 2 على 3 I ناقص 2 على 3 J ناقص 1 على 3 K
+
+675
+00:45:40,470 --> 00:45:43,470
+and decreases most rapidly in the opposite
+
+676
+00:45:43,470 --> 00:45:47,080
+direction، اللي هو في ناقص اللي هو الـ gradient للـ F أو
+
+677
+00:45:47,080 --> 00:45:53,640
+بمعنى آخر، في اتجاه ناقص 2 على 3 i زائد 2 على 3 j زائد
+
+678
+00:45:53,640 --> 00:45:57,840
+1 على 3 k، هذا اللي هو decreases  most rapidly اللي هون
+
+679
+00:45:57,840 --> 00:46:01,920
+بدنا اللي هو الـ directional derivative في
+
+680
+00:46:01,920 --> 00:46:05,430
+كل حالة، الـ rates of change اللي هو معناته الـ
+
+681
+00:46:05,430 --> 00:46:08,930
+directional derivative in these directions are الـ
+
+682
+00:46:08,930 --> 00:46:11,270
+derivative، اللي طبعاً في الحالة الأولى most rapidly
+
+683
+00:46:11,270 --> 00:46:13,570
+بيكون بساوي الـ gradient نفسه اللي هو الجذر
+
+684
+00:46:13,570 --> 00:46:16,690
+التربيعي لأربعة زائد الأربعة زائد الواحد، هي
+
+685
+00:46:16,690 --> 00:46:19,170
+الأربعة زائد الأربعة زائد الواحد، اللي هو زي ما
+
+686
+00:46:19,170 --> 00:46:23,550
+قلنا ثلاثة، واللي هي في حالة اللي هي الـ decreases
+
+687
+00:46:23,550 --> 00:46:26,350
+most rapidly، بتكون الـ derivative في هذه الحالة
+
+688
+00:46:26,350 --> 00:46:30,120
+سالب ثلاثة، وبيكون هيك احنا أنهينا المحاضرة، وإلى
+
+689
+00:46:30,120 --> 00:46:33,780
+لقاء آخر، السلام عليكم ورحمة الله وبركاته
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U_postprocess.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6f62133f5a4f9c7c1024c3c8156437a18ccea09b
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U_postprocess.srt
@@ -0,0 +1,2756 @@
+1
+00:00:00,640 --> 00:00:04,660
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,660 --> 00:00:10,080
+وبركاته هذه المحاضرة رقم 17 مساق تفاضل وتكامل
+
+3
+00:00:10,080 --> 00:00:13,940
+الـGEM وهي المحاضرة الخامسة في شبطر أربعة عشر وهي
+
+4
+00:00:13,940 --> 00:00:18,780
+section 14-5 المعنوان بـdirectional derivatives
+
+5
+00:00:18,780 --> 00:00:25,500
+and gradient vectors اللي هو الاشتراك
+
+6
+00:00:25,500 --> 00:00:32,630
+في اتجاه معينهو الـ Gradient Vectors اللي سنتعرف
+
+7
+00:00:32,630 --> 00:00:37,650
+عليها لاحقا نحكي أولا عن Directional Derivatives
+
+8
+00:00:37,650 --> 00:00:42,050
+in W الـ Directional Derivatives هي عبارة عن تعميم
+
+9
+00:00:42,050 --> 00:00:45,890
+للـ Partial Derivatives سواء في اتجاه الـ X أو الـ
+
+10
+00:00:45,890 --> 00:00:48,390
+Y أو الـ Z
+
+11
+00:00:51,970 --> 00:00:56,790
+أو ∆FY∆Y أو ∆FY∆Z كل حالة خاصة من حاجة اسمها
+
+12
+00:00:56,790 --> 00:00:59,830
+Directional Derivative خلّيني أتعرف الآن Plan
+
+13
+00:00:59,830 --> 00:01:04,930
+Definition The derivative of F at P0 X0 وY0 فنفترض
+
+14
+00:01:04,930 --> 00:01:07,810
+F function of two variables بس اللي هي ال X والY
+
+15
+00:01:07,810 --> 00:01:13,830
+فالعنوان للـ F عنده نقطة X0 وY0 in the direction
+
+16
+00:01:13,830 --> 00:01:19,870
+of the unit vector U بتساوي UI1 زي UII2J is the
+
+17
+00:01:19,870 --> 00:01:26,140
+numberإذاً مشتقة الـ F في اتجاه اللي هو عند النقطة
+
+18
+00:01:26,140 --> 00:01:29,720
+X نوت و Y نوت في اتجاه الـ U هذا اللي معطينيها U
+
+19
+00:01:29,720 --> 00:01:35,740
+واحد I زاد U اتنين I المشتقة لأ اللي هي ال
+
+20
+00:01:35,740 --> 00:01:42,280
+function في هذا الاتجاهتتعرف بميالي DF by DS اللي
+
+21
+00:01:42,280 --> 00:01:46,900
+هو في اتجاه الـ U عنده نقطة بي نوت بساوة limit F
+
+22
+00:01:46,900 --> 00:01:52,360
+of X note زاد S U one Y note زاد S U اتنين ناقص F
+
+23
+00:01:52,360 --> 00:01:57,650
+of X note Y note على مين على Sإن التغيير في الـ
+
+24
+00:01:57,650 --> 00:02:00,970
+XNOTE و YNOTE لما تنتقل XNOTE و YNOTE التغيير بس
+
+25
+00:02:00,970 --> 00:02:06,850
+وين مسموح فيه هنا في اتجاه الـ U يعني XNOTE هيكون
+
+26
+00:02:06,850 --> 00:02:11,690
+تغيير هي YNOTE في هذا الاتجاه فبصير XNOTE زاد أس و
+
+27
+00:02:11,690 --> 00:02:16,890
+U و U1 إذا نان اتحكمتالتي هي مقدار الزيادة اللي في
+
+28
+00:02:16,890 --> 00:02:20,410
+الـ X note تتحكم فيها اللي هي المقدار العددي S
+
+29
+00:02:20,410 --> 00:02:25,630
+والتي هو الاتجاه اللي هو U1 في اتجاه I والـ Y note
+
+30
+00:02:25,630 --> 00:02:29,330
+اللي هو S زائد U2 بنفس الأسلوب S اللي بتمثل ال
+
+31
+00:02:29,330 --> 00:02:33,410
+increment العددي والـ U2 اللي هي الاتجاه اللي صارت
+
+32
+00:02:33,410 --> 00:02:37,350
+في التغيير يعني بمعنى آخر التغييرصار على وين في
+
+33
+00:02:37,350 --> 00:02:40,730
+هذا الاتجاه الـ Derivative في هذه الحالة المعرفة
+
+34
+00:02:40,730 --> 00:02:44,690
+هذه ناقص f of x not y not على s لما s تروح ل0 هي
+
+35
+00:02:44,690 --> 00:02:48,850
+بالظبط حاجة بنسميها The directional derivative of
+
+36
+00:02:48,850 --> 00:02:53,970
+f in the direction of uبنوت أو الـ derivative of f
+
+37
+00:02:53,970 --> 00:02:59,470
+in the direction of u at بنوت خلّينا الآن ناخد
+
+38
+00:02:59,470 --> 00:03:04,450
+اللي هو عندي مثال عندي x طبعا قبل ما ناخد المثال
+
+39
+00:03:04,450 --> 00:03:07,850
+الـ x زي ما قلنا على هذا الخط بساوي x note
+
+40
+00:03:11,140 --> 00:03:13,680
+هذه اللي بنحكي فيها هو الـ X بسهولة X Note Z
+
+41
+00:03:13,680 --> 00:03:17,560
+أسواية Y1 و Y Y سواية Note أسواية Y2 اللي عبارة عن
+
+42
+00:03:17,560 --> 00:03:21,100
+إيش الخط اللي هو المستقيم هنا الـ X و الـ Y تتحرك
+
+43
+00:03:21,100 --> 00:03:25,760
+بوافق إيش المعادلة الأمامية هذه اللي أمامي اللي
+
+44
+00:03:25,760 --> 00:03:29,960
+صار فيها التغيير X Note زايد أسواية Y1 زايد أسواية
+
+45
+00:03:29,960 --> 00:03:33,740
+Y2 على إيش في هذا الاتجاه خلّيني أخد مثال عددي يا
+
+46
+00:03:33,740 --> 00:03:37,440
+جماعة نشوف إيش اللي بنحكي بالنسبة له اللي هي هذه
+
+47
+00:03:37,440 --> 00:03:40,400
+ال directional derivativeالدريكشنال الدريفتيف
+
+48
+00:03:40,400 --> 00:03:42,520
+اللذي تم تسميه بـEquation 1 هو أيضًا تسميه
+
+49
+00:03:42,520 --> 00:03:46,560
+بالدريكشنال الدريفتيف للـ F أو الدريفتف للـ F في
+
+50
+00:03:46,560 --> 00:03:50,400
+الاتصال بـ U في الاتصال بـ B0 الدريفتف للـ F في
+
+51
+00:03:50,400 --> 00:03:53,760
+الاتصال بـ B0 في الاتصال بـ U سوف نستخدم التصميم،
+
+52
+00:03:53,760 --> 00:03:56,100
+طبعًا التصميم اللذي رأيته هو مشابه للتصميم
+
+53
+00:03:56,100 --> 00:03:59,160
+للدريفتف التابعة زمان ولكن هناك كانت مفعولة One
+
+54
+00:03:59,160 --> 00:04:03,930
+Variableوشابه للـ partial derivative ولكن كانت
+
+55
+00:04:03,930 --> 00:04:08,070
+اللي هي زمان one variable وفي حالة ال partial
+
+56
+00:04:08,070 --> 00:04:11,470
+derivative التغيير بس كان في اتجاه X أو في اتجاه Y
+
+57
+00:04:11,470 --> 00:04:14,770
+لحالها using the definition find the derivative of
+
+58
+00:04:14,770 --> 00:04:19,830
+F of XY بساوي X تربيع زي X في Y at main نقطة واحد
+
+59
+00:04:19,830 --> 00:04:22,350
+واتنين in the direction of the unit vector main
+
+60
+00:04:22,350 --> 00:04:24,650
+اللي ماعطينا ي واحدة الجدر اتنين في اتجاه I زي
+
+61
+00:04:24,650 --> 00:04:29,140
+واحدة الجدر اتنين في اتجاه Gخلّينا ننحل الـ
+
+62
+00:04:29,140 --> 00:04:31,960
+Directional Derivative للـ F أو Derivative للـ F
+
+63
+00:04:31,960 --> 00:04:34,180
+الـ Direction of U عند نقطة واحد و اتنين بساوي
+
+64
+00:04:34,180 --> 00:04:37,540
+لمين هذا التعريف اللي حكيناه قبل شهية و يساوي لمين
+
+65
+00:04:37,540 --> 00:04:40,560
+لأن الـ X Note و الـ Y Note اللي يمين واحد و اتنين
+
+66
+00:04:40,560 --> 00:04:44,220
+إذا مكان الـ X Note بدنا نحط إيه؟ واحد الـ S زي ما
+
+67
+00:04:44,220 --> 00:04:47,800
+هي ده المتغير الـ Increment الزيادة في اتجاه مين؟
+
+68
+00:04:47,800 --> 00:04:50,480
+U واحد U واحد إيه بتساويها الـ U واحد على جدر
+
+69
+00:04:50,480 --> 00:04:54,940
+التنين إذا صارت S على جدر التنين و الـ Y Note اللي
+
+70
+00:04:54,940 --> 00:04:59,890
+هي برضه جداش اتنينالي اتنين زائد الان S في FU2
+
+71
+00:04:59,890 --> 00:05:04,190
+اللي هي الـ C المتغير U2EH اللي هي واحد على EH جدر
+
+72
+00:05:04,190 --> 00:05:07,770
+التنين اللي صارت هذه الـ function F of هذا المقدار
+
+73
+00:05:07,770 --> 00:05:12,230
+ناقص F of واحد و اتنين على EH الـ S اللي هان بنحسب
+
+74
+00:05:12,230 --> 00:05:15,550
+هذه الـ F يعني بنحسب الـ function هذه اللي أمامنا
+
+75
+00:05:15,550 --> 00:05:20,670
+عنده نقطة X بتسوء واحد على S زائد S على جدر التنين
+
+76
+00:05:21,230 --> 00:05:25,410
+والـ Y عندنا نقطة 2 زي الـ S على جدر الـ 2 بنعوض
+
+77
+00:05:25,410 --> 00:05:28,690
+بيصير اللي هي الأولى تربيحها هي هذه الـ X اللي هي
+
+78
+00:05:28,690 --> 00:05:32,850
+هذه تربيحها هي زائد اللي هي هذه الـ X وهذه الـ Y
+
+79
+00:05:32,850 --> 00:05:38,690
+اللي عندنا هيهم ناقص اللي هي main اللي هي F of 1 و
+
+80
+00:05:38,690 --> 00:05:43,790
+2 F of 1 و 2 هي عبارة عن 1 زائد اللي هي 2 اللي هي
+
+81
+00:05:43,790 --> 00:05:49,130
+عوضنا عن X بـ 1 و Y بـ 2 طالع عندنا 1 زائد 2 على S
+
+82
+00:05:49,390 --> 00:05:53,410
+لأن هذا المقدر يتم ربعه بطلع هذا المقدر بطلع هذا
+
+83
+00:05:53,410 --> 00:05:56,650
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+84
+00:05:56,650 --> 00:05:58,810
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+85
+00:05:58,810 --> 00:05:59,390
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+86
+00:05:59,390 --> 00:06:01,630
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+87
+00:06:01,630 --> 00:06:02,450
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+88
+00:06:02,450 --> 00:06:04,950
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+89
+00:06:04,950 --> 00:06:05,670
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+90
+00:06:05,670 --> 00:06:10,680
+المقدر بطلعوهذا في هذا اللي هي اتنين S على جدر
+
+91
+00:06:10,680 --> 00:06:13,920
+التانين واحد S على جدر التانين يعني تلاتة S على
+
+92
+00:06:13,920 --> 00:06:17,020
+جدر التانين ناقص المقدار هذا اللي هو تلاتة لكل
+
+93
+00:06:17,020 --> 00:06:20,720
+مجسم على مين يا جماعة على S بتنصح لهذا المقدار بقى
+
+94
+00:06:20,720 --> 00:06:24,580
+علينا لغير ان نبسطه بنجمح هدول لبعض بيطلع عندي
+
+95
+00:06:24,580 --> 00:06:28,140
+المقدار هذا طبعا سهل لكم تجمعوه لبعض خمسة S على
+
+96
+00:06:28,140 --> 00:06:31,520
+جدر التانين زائد S تربيع على مين على S لان واي في
+
+97
+00:06:31,520 --> 00:06:33,800
+عندي اتنين واحد تلاتة بتروح مع التلاتة هذا بيظل
+
+98
+00:06:33,800 --> 00:06:37,790
+اللي هو معامل S تربيع اللي هو واحدإلا إنه نص و نص
+
+99
+00:06:37,790 --> 00:06:41,210
+بيصير واحد وهذه ثلاثة و اتنين بيصير خمسة على اس
+
+100
+00:06:41,210 --> 00:06:44,310
+بنختصر لأن الاس مع الاس زي زمان مع الاس اللي هان
+
+101
+00:06:44,310 --> 00:06:47,170
+والاس اللي هان بيطلع limit خمسة على جدر التنين زي
+
+102
+00:06:47,170 --> 00:06:50,330
+الاس اس اس as s goes to zero بيطلع خمسة على جدر
+
+103
+00:06:50,330 --> 00:06:53,510
+التنين إذن ال direction ال derivative of f in the
+
+104
+00:06:53,510 --> 00:06:56,430
+direction of u at the point واحد و اتنين طلع عبارة
+
+105
+00:06:56,430 --> 00:07:00,770
+عن إيش خمسة على اتنين نيجي الآن ناخد دقيقة خليني
+
+106
+00:07:00,770 --> 00:07:03,350
+أقول معنى هندسي أكتر لإيه ال direction ال
+
+107
+00:07:03,350 --> 00:07:06,370
+derivative اللي حكينا عنها قبل شويةInterpretation
+
+108
+00:07:06,370 --> 00:07:08,950
+of the Directional Derivative خلينا نشوف اللي هي
+
+109
+00:07:08,950 --> 00:07:11,590
+إيش المعنى الهندسي للـ Directional Derivative و
+
+110
+00:07:11,590 --> 00:07:14,790
+أنا حكيته قبله شوية عموان شوف الصلاة والنبي عليه
+
+111
+00:07:14,790 --> 00:07:17,410
+الصلاة والسلام زي ما عملنا في ال partial
+
+112
+00:07:17,410 --> 00:07:20,510
+derivative قلنا إيش معنات اللي هو ال directional
+
+113
+00:07:20,510 --> 00:07:24,190
+derivative اللي قاسف اللي هو ال partial derivative
+
+114
+00:07:25,240 --> 00:07:29,280
+هي عبارة عن ميل المماثلة المنحنة اللي ناتج من
+
+115
+00:07:29,280 --> 00:07:34,020
+تقاطع اللي هو X بساوي X knot مع اللي هو ال surface
+
+116
+00:07:34,020 --> 00:07:38,160
+بيطلع المماثلة المنحنة اللي بيطلع هو عبارة عن Del
+
+117
+00:07:38,160 --> 00:07:41,740
+F
+
+118
+00:07:41,740 --> 00:07:44,640
+by Del Y ونفس الاشي بالنسبة ل X خلينا نشوف ده إيش
+
+119
+00:07:44,640 --> 00:07:48,870
+اللي بقولهلحظة انا كنت نسيت الان باجي الان انا
+
+120
+00:07:48,870 --> 00:07:52,970
+عندي ايش غرضي ال equation z بيساوي f of x y يربسنت
+
+121
+00:07:52,970 --> 00:07:55,830
+ال surface as in space هاي هذا اللي بالازرق هو
+
+122
+00:07:55,830 --> 00:08:00,490
+عبارة عن ال surface z بيساوي f of x y الان اذا كان
+
+123
+00:08:00,490 --> 00:08:04,110
+z not بيساوي f of x not و y not هذه اللي هي النقطة
+
+124
+00:08:04,110 --> 00:08:07,270
+هي عبارة عن النقطة اللي بدنا نوجد عندها اللي هو
+
+125
+00:08:07,270 --> 00:08:13,020
+directional derivativeوالنقطة الـ XNOTE LINE على
+
+126
+00:08:13,020 --> 00:08:17,820
+الـ Surface S الان بدي اوجد الـ Directional
+
+127
+00:08:17,820 --> 00:08:22,900
+Derivative of F at النقطة XNOTE وYNOTE وZNOTE
+
+128
+00:08:22,900 --> 00:08:27,860
+XNOTE وYNOTE في العادة ومين عنده ZNOTE ثابت في الـ
+
+129
+00:08:27,860 --> 00:08:30,660
+Direction في اتجاه مين في اتجاه الـ U هي الـ U
+
+130
+00:08:30,660 --> 00:08:36,160
+الان ايش هو بالظبط معناه الهندسي الان هذا معناه ان
+
+131
+00:08:36,160 --> 00:08:41,310
+انا باجي اللي هو باخد الـ Planeاللي بتكون من أو
+
+132
+00:08:41,310 --> 00:08:44,190
+اللي بيحدده اللي هو الـ الـ الـ الـ الـ الـ الـ
+
+133
+00:08:44,190 --> 00:08:44,230
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+134
+00:08:44,230 --> 00:08:44,290
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+135
+00:08:44,290 --> 00:08:44,790
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+136
+00:08:44,790 --> 00:08:46,790
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+137
+00:08:46,790 --> 00:08:47,970
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+138
+00:08:47,970 --> 00:08:56,210
+الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+139
+00:08:56,210 --> 00:09:08,350
+الـ الـ الـ الـ الـ الـ الـ الـ
+
+140
+00:09:08,350 --> 00:09:09,990
+الـ الـ الـ الـ الـ الـ الـ الـفى اتجاه مين فى
+
+141
+00:09:09,990 --> 00:09:14,950
+اتجاه اللى هو الـ U هذا الان البلين المتكون من
+
+142
+00:09:14,950 --> 00:09:17,990
+اللى هو الـ U اللى بدأ اخد الـ derivative فى
+
+143
+00:09:17,990 --> 00:09:22,210
+اتجاهه ونقطة X Note وY Note هذا البلين هيقاطع من
+
+144
+00:09:22,210 --> 00:09:28,910
+الـ Surface فى الخط الأزرق الآن ميل المماس لهذا
+
+145
+00:09:28,910 --> 00:09:32,830
+اللى هو خط التقاطع ميل المماس هذاالتي هو الـ
+
+146
+00:09:32,830 --> 00:09:37,290
+Tangent هذا عند النقطة هذا ميله هو عبارة عن الـ
+
+147
+00:09:37,290 --> 00:09:41,590
+Directional Derivative للـ Function F عند النقطة X
+
+148
+00:09:41,590 --> 00:09:46,370
+Note وY Note وZ Note في اتجاه مين؟ في اتجاه الـ U
+
+149
+00:09:46,370 --> 00:09:49,310
+في النهاية إذن برضه الـ Directional Derivative هي
+
+150
+00:09:49,310 --> 00:09:55,270
+عبارة عن ميل مماس ميل مماس لمين؟ للـ Vector للـ
+
+151
+00:09:55,270 --> 00:10:02,060
+Curveعند النقطة X0 وY0 وZ0 الناتج من تقاطع الـ
+
+152
+00:10:02,060 --> 00:10:06,600
+Surface مع الـ Plane المكوّن من اللي هو الـ Vector
+
+153
+00:10:06,600 --> 00:10:11,320
+اللي بدأ فاضل في اتجاهه مع النقطة X0 وY0 الآن لو
+
+154
+00:10:11,320 --> 00:10:16,610
+كان هذاالـ vector اللي هو U بساوي I واحد زائد I
+
+155
+00:10:16,610 --> 00:10:20,850
+اتنين لو كانت U اتنين سفر و U واحد يعني الـ
+
+156
+00:10:20,850 --> 00:10:24,410
+directional derivative بس U بساوي U واحد في اتجاه
+
+157
+00:10:24,410 --> 00:10:28,590
+الـ I يعني بس I يعني الـ U في اتجاه الـ I يعني انا
+
+158
+00:10:28,590 --> 00:10:30,970
+بتاخد ال directional derivative في اتجاه مين الـ I
+
+159
+00:10:30,970 --> 00:10:34,620
+بكون لما اخد في اتجاه الـ I معناته انه اناالـ
+
+160
+00:10:34,620 --> 00:10:37,940
+derivative هيها في اتجاه الـ I هي هذا في اتجاه هذا
+
+161
+00:10:37,940 --> 00:10:42,840
+من هنا يعني ايش عملت في الـ Y ثبت الـ Y فبصير عندي
+
+162
+00:10:42,840 --> 00:10:46,840
+اللي هي الـ partial derivative with respect to
+
+163
+00:10:46,840 --> 00:10:51,700
+مين؟ to X لأنه بتثبت Y فش Y صار بس التغيير في مين؟
+
+164
+00:10:51,700 --> 00:10:55,100
+في اتجاه الـ I لأن إذا الـ directional derivative
+
+165
+00:10:55,100 --> 00:10:58,480
+U اتجاه الـ I is the directional derivative at B0
+
+166
+00:10:58,480 --> 00:11:04,890
+is دل FY دل X evaluated at X0 وY0U بالساوية جيه
+
+167
+00:11:04,890 --> 00:11:08,790
+الـ directional derivative at P0 بتصير دي الـ FY
+
+168
+00:11:08,790 --> 00:11:12,510
+دي الـ Y لأنه اللي هو بس التغيير في اتجاه الجيه
+
+169
+00:11:12,510 --> 00:11:16,790
+وليس التغيير في اتجاه الـ X معناته X ثابتة إذن هذه
+
+170
+00:11:16,790 --> 00:11:19,910
+الـ U اللي هي الـ partial derivative with respect
+
+171
+00:11:19,910 --> 00:11:24,190
+to Y أو with respect to X هي عبارة عن حالات خاصة
+
+172
+00:11:24,190 --> 00:11:27,880
+من الـ directional derivative بصورة عامةالأولى الـ
+
+173
+00:11:27,880 --> 00:11:30,640
+derivative في اتجاه الـ U الآن هذه الـ derivative
+
+174
+00:11:30,640 --> 00:11:34,040
+في اتجاه الـ U اللى هى I يعني اتجاه اللى هى ال X
+
+175
+00:11:34,040 --> 00:11:37,580
+axis يعني بمعنى أخر اللى هى Del F by Del X هتطلع
+
+176
+00:11:37,580 --> 00:11:40,340
+عندى و ال directional derivative في اتجاه ال J
+
+177
+00:11:40,340 --> 00:11:44,900
+اللى هى ال derivative اللى هى بتتثبت X X و يكون
+
+178
+00:11:44,900 --> 00:11:50,080
+التغيير بس في اتجاه Y اللى هى D F by D Y هذه معنى
+
+179
+00:11:50,080 --> 00:11:53,160
+الهندسي لل directional derivativeإلا أننا لا نريد
+
+180
+00:11:53,160 --> 00:11:56,280
+أن نتغلب في إيجاد الـ Directional Derivative كذا
+
+181
+00:11:56,280 --> 00:12:00,280
+سنبقى لوجدها بواسطة التعريف فالأمثلة قبل شوية سهل
+
+182
+00:12:00,280 --> 00:12:06,020
+لكن قد تكون أمثلة تتعجزن وعملنا زمان حصلنا على
+
+183
+00:12:06,020 --> 00:12:10,240
+آليات وقوانين لإيجاد الـ Derivative العادية والآن
+
+184
+00:12:10,240 --> 00:12:14,420
+سنحصل على قوانين إن شاء الله سهلة في إيجاد الـ
+
+185
+00:12:14,420 --> 00:12:18,000
+Directional Derivative وعشان نصل لهذه المرحلة نعرف
+
+186
+00:12:18,000 --> 00:12:23,850
+حاجة اسمة The Gradient Vector أو الـ Gradientالـ
+
+187
+00:12:23,850 --> 00:12:26,870
+Gradient vector of f of x, y at the point b0 x0,
+
+188
+00:12:27,010 --> 00:12:30,990
+y0 is the vector نريد أن نعرفه كالتالي يعني الـ
+
+189
+00:12:30,990 --> 00:12:33,510
+Gradient للـ f في رمزه والـ Gradient للـ f هو
+
+190
+00:12:33,510 --> 00:12:37,470
+عبارة عن del f by del x في اتجاه الـ I يعني تفاضة
+
+191
+00:12:37,470 --> 00:12:41,050
+الـ f بالنسبة للـ x في اتجاه الـ I زي الـ del f by
+
+192
+00:12:41,050 --> 00:12:45,750
+del y في اتجاه الـ j obtained by evaluating the
+
+193
+00:12:45,750 --> 00:12:50,470
+partial derivatives of f at b0 إذن هذه عبارة عن
+
+194
+00:12:50,470 --> 00:12:56,200
+الـ Gradient لمينللـ Effect عريفة لأن لحسن حظنا و
+
+195
+00:12:56,200 --> 00:13:01,220
+لتناسق الخلق عندي النظرية التالية اللي هتسهلنا
+
+196
+00:13:01,220 --> 00:13:05,200
+عملية إيجاد الـ Directional Derivative بواسطة اللي
+
+197
+00:13:05,200 --> 00:13:09,680
+هو بدلالة من الـ Gradient طبعا جرهنها مش مقلوب if
+
+198
+00:13:09,680 --> 00:13:13,760
+F of XY is differentiable in an open region
+
+199
+00:13:13,760 --> 00:13:16,920
+containing B node لو فرضنا أن F of XY قابل
+
+200
+00:13:16,920 --> 00:13:21,420
+الاشتراك في المنطقةفي منطقة حوالينا الـ B0 ثم DF
+
+201
+00:13:21,420 --> 00:13:25,360
+by DS اللي هو الـ derivative للـ F في اتجاه الـ U
+
+202
+00:13:25,360 --> 00:13:29,880
+عند النقطة B0 بساوي الـ Gradient للـ F عند الـ B0
+
+203
+00:13:30,910 --> 00:13:34,550
+.main ضد الـ U اللي هو الـ unit vector اللي بدنا
+
+204
+00:13:34,550 --> 00:13:38,410
+نفاضل في اتجاهه إذا موضوع ده إيجاد ال derivative
+
+205
+00:13:38,410 --> 00:13:43,470
+سهل هو إيجاد ال partial derivatives Fy X و بنوجدها
+
+206
+00:13:43,470 --> 00:13:47,150
+بالشكل اللي هذي و بنعوض عن P0 و بعدين بنعملها
+
+207
+00:13:47,150 --> 00:13:51,300
+.product مع U بطلع عندنا ال directionalدريفاتيف
+
+208
+00:13:51,300 --> 00:13:53,440
+إذاً صار اللي هو عبارة عن dot product الـ
+
+209
+00:13:53,440 --> 00:13:56,660
+differential الـ directional derivative لماذا؟
+
+210
+00:13:56,660 --> 00:14:01,560
+لدريفاتيف للـ gradient عنده نقطة بي نوت مضروب dot
+
+211
+00:14:01,560 --> 00:14:05,160
+product فيه خليني أشوف مثلًا حتى أقول الأمر سهل
+
+212
+00:14:06,110 --> 00:14:10,090
+سنجد الـ derivative of f of x,y بسوء x في y أي
+
+213
+00:14:10,090 --> 00:14:13,490
+زائد كوسيني الـ x في y at the point 2,0 in the
+
+214
+00:14:13,490 --> 00:14:18,250
+direction of v بسوء 3I ناقص 4G خلّيني أوجد الـ
+
+215
+00:14:18,250 --> 00:14:21,090
+derivative هذي كيف نوجدها؟ بالطريقة اللي أمامنا
+
+216
+00:14:21,090 --> 00:14:24,050
+خلّيني نشوف يا جماعة أول حاجة بدي أوجده اللي هو
+
+217
+00:14:24,050 --> 00:14:27,850
+الـ direction اللي هي الـU الـU طبعا احنا مجهلينه
+
+218
+00:14:27,850 --> 00:14:30,990
+في الـdirection هذا عشان اوجد لازم هذه تكون unit
+
+219
+00:14:30,990 --> 00:14:35,000
+vector الـU هيك التعريق، هيك النظريةإذاً هذه بدأ
+
+220
+00:14:35,000 --> 00:14:37,640
+أشوف الـ Direction تبعها الـ Direction of V is the
+
+221
+00:14:37,640 --> 00:14:41,800
+unit vector obtaining Y اللي هو V على ال magnitude
+
+222
+00:14:41,800 --> 00:14:44,800
+لل V ال magnitude لل V زي ما أنتوا عارفين جدر
+
+223
+00:14:44,800 --> 00:14:47,120
+التربيعي على تلاتة تربيع زي أربعة تربيع اللي هي
+
+224
+00:14:47,120 --> 00:14:51,400
+ياش تسعة زي ستاشت تلاتة عليها في تجاه I اذا ناقص
+
+225
+00:14:51,400 --> 00:14:55,970
+أربعة على الجدر التربيعي هذا في تجاه Jإذا الـ
+
+226
+00:14:55,970 --> 00:15:00,990
+magnitude للـ V هي U ويساوي هذا الخمسة و هذا
+
+227
+00:15:00,990 --> 00:15:04,410
+الخمسة يصبح ثلاثة على خمسة I نقص أربعة على خمسة J
+
+228
+00:15:04,410 --> 00:15:08,570
+هذا الـ direction اللي هو اللي بدنا نعمل في اتجاهه
+
+229
+00:15:08,570 --> 00:15:13,230
+الـ derivative الـ gradient لها في سهل هي عبارة عن
+
+230
+00:15:13,230 --> 00:15:17,170
+X أول شيء الفاضل هذا بالنسبة ل X يصبح E to the Y
+
+231
+00:15:17,770 --> 00:15:20,570
+الأنفاض لهذه بالنسبة لـ X بيصير اللي هي ناقص Sine
+
+232
+00:15:20,570 --> 00:15:23,850
+X في Y فتفاضل اللي جوا إيه الـ X في Y تفاضلها
+
+233
+00:15:23,850 --> 00:15:26,970
+بالنسبة لـ X اللي هي Y فبيصير E to the Y ناقص Y
+
+234
+00:15:26,970 --> 00:15:30,850
+Sine XY في اتجاه الـ I زاد الـ derivative لهذه
+
+235
+00:15:30,850 --> 00:15:34,930
+with respect to Y يعني دل Y اللي هي عبارة عن X E
+
+236
+00:15:34,930 --> 00:15:38,270
+to the Y ناقص التفاضل لهذه بالنسبة لـ Y دي هي ناقص
+
+237
+00:15:38,270 --> 00:15:43,370
+Sine X في Y فتفاضل جوا X ناقص X Sine Y في اتجاه
+
+238
+00:15:43,370 --> 00:15:52,200
+مين؟بعد ذلك نحسب الـ Gradient هذا عند نقطة 2 و 0
+
+239
+00:15:52,200 --> 00:16:02,180
+عوض عن I بـ 0 و X بـ 2 بيصير هذا عبارة عن Z1
+
+240
+00:16:02,180 --> 00:16:10,550
+وهذا سفر كله بيصير Iاللي هي 2 في 1 ايه ده 01 بيصير
+
+241
+00:16:10,550 --> 00:16:16,610
+2 وها دي صفر طبعا لإنه sin 00 بيصير 2J صار عندي ال
+
+242
+00:16:16,610 --> 00:16:19,770
+gradient اللي أفر عنده النقطة المذكورة بيساوي I
+
+243
+00:16:19,770 --> 00:16:24,280
+زيدي 2Jاللي هي الـ Gradient جاهز والـ U جاهزة يعني
+
+244
+00:16:24,280 --> 00:16:27,660
+لغير أن نطبق القانون الـ Gradient ضد الـ U بيصير
+
+245
+00:16:27,660 --> 00:16:32,340
+الـ Derivative بساوي اللي هو الـ Gradient I زاد 2
+
+246
+00:16:32,340 --> 00:16:38,260
+J ضد الـ U اللي هو عبارة عن 3 على 5 ناقص 8 على 5
+
+247
+00:16:38,260 --> 00:16:40,960
+ويساوي سالب 1 طبعا بتعرفوا كيف تضربوا الدكتور
+
+248
+00:16:40,960 --> 00:16:44,110
+دكتادي في هذه ال ..التي تقريبا تقريبا تقريبا
+
+249
+00:16:44,110 --> 00:16:47,490
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+250
+00:16:47,490 --> 00:16:53,210
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+251
+00:16:53,210 --> 00:16:54,970
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+252
+00:16:54,970 --> 00:16:55,690
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+253
+00:16:55,690 --> 00:17:01,310
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقزي ما
+
+254
+00:17:01,310 --> 00:17:03,890
+قلنا إيش تعريفها عبارة عن الـ gradient للـ F من
+
+255
+00:17:03,890 --> 00:17:06,570
+dot U اللي ييجي الآن هذه الوجود قيمتها اللي نعرف
+
+256
+00:17:06,570 --> 00:17:11,070
+أن وجودها الـ magnitude للـ gradient للـ F في
+
+257
+00:17:11,070 --> 00:17:14,770
+cosine الزاوية بين اليمين وcos theta لو حد سأل
+
+258
+00:17:14,770 --> 00:17:19,270
+سؤال وقت إيش هذه اللي هي الـ F increases most
+
+259
+00:17:19,270 --> 00:17:22,150
+rapidly يعني معدل التغير هذا اللي هو derivative
+
+260
+00:17:22,150 --> 00:17:26,350
+تكون أكثر ما يمكن اللي هو زيادةتكون أكبر ما يمكن
+
+261
+00:17:26,350 --> 00:17:30,170
+لأن cos θ بين السفر بين نقص واحد والواحد تكون أكبر
+
+262
+00:17:30,170 --> 00:17:35,530
+ما يمكن لما cos θ واحد يعني لما θاش متساوي سفر إذا
+
+263
+00:17:35,530 --> 00:17:39,730
+اللي هو the function F increasing increases most
+
+264
+00:17:39,730 --> 00:17:44,220
+rapidly when cos θ واحد or when θ متساوي سفرفي هذه
+
+265
+00:17:44,220 --> 00:17:50,980
+الحالة U ماذا يكون؟ U مادام
+
+266
+00:17:50,980 --> 00:17:54,380
+الزاوية بين الـ Gradient و U مادام الزاوية بين الـ
+
+267
+00:17:54,380 --> 00:17:54,660
+Gradient و U مادام الزاوية بين الـ Gradient و U
+
+268
+00:17:54,660 --> 00:17:57,720
+مادام الزاوية بين الـ Gradient و U مادام الزاوية
+
+269
+00:17:57,720 --> 00:17:58,740
+بين الـ Gradient و U مادام الزاوية بين الـ
+
+270
+00:17:58,740 --> 00:17:59,280
+Gradient و U مادام الزاوية بين الـ Gradient و U
+
+271
+00:17:59,280 --> 00:18:00,140
+مادام الزاوية بين الـ Gradient و U مادام الزاوية
+
+272
+00:18:00,140 --> 00:18:01,160
+بين الـ Gradient و U مادام الزاوية بين الـ
+
+273
+00:18:01,160 --> 00:18:01,240
+Gradient و U مادام الزاوية بين الـ Gradient و U
+
+274
+00:18:01,240 --> 00:18:03,480
+بين الـ Gradient و U مادام الزاوية بين الـ
+
+275
+00:18:03,480 --> 00:18:07,350
+Gradient و U ميعني that is at each point P in its
+
+276
+00:18:07,350 --> 00:18:10,750
+domain F increases most rapidly in the direction
+
+277
+00:18:10,750 --> 00:18:14,790
+of the gradient للـ F at P الـ derivative in this
+
+278
+00:18:14,790 --> 00:18:18,930
+case طبعاً in this direction لما يكون الاتجاه اللي
+
+279
+00:18:18,930 --> 00:18:21,750
+هو ال gradient والاتجاه اللي هو ال U في نفس
+
+280
+00:18:21,750 --> 00:18:24,850
+الاتجاه يعني Cos Zero يعني بيساوي ال absolute
+
+281
+00:18:24,850 --> 00:18:28,430
+value لل gradient لل F أو ال magnitude لل gradient
+
+282
+00:18:28,430 --> 00:18:32,430
+لل Fإذاً الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+283
+00:18:32,430 --> 00:18:32,590
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+284
+00:18:32,590 --> 00:18:33,690
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+285
+00:18:33,690 --> 00:18:35,190
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+286
+00:18:35,190 --> 00:18:37,710
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+287
+00:18:37,710 --> 00:18:44,450
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+288
+00:18:44,450 --> 00:18:45,470
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+289
+00:18:45,470 --> 00:18:45,710
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+290
+00:18:45,710 --> 00:18:47,350
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+291
+00:18:47,350 --> 00:18:53,810
+الـ الـ الـ الـ اليعني بمعنى أخر عندما يكون
+
+292
+00:18:53,810 --> 00:18:58,090
+الاتنين في عكس اتجاه بعض يعني عندما تكون F يزداد
+
+293
+00:18:58,090 --> 00:19:04,830
+أكثر بسرعة في اتجاه X عكس اتجاه لـ Gradient والـ
+
+294
+00:19:04,830 --> 00:19:08,580
+Derivative في هذه الحالة هيكون بساويناقص الـ
+
+295
+00:19:08,580 --> 00:19:12,320
+Absolute Value دي دلنا في دلنا بتكون اللي هي أقل
+
+296
+00:19:12,320 --> 00:19:16,780
+قيمة و هناك أكبر قيمة طيب و اكتشف كل الـ
+
+297
+00:19:16,780 --> 00:19:21,200
+Directional Derivative Sphere أكيد Any Direction U
+
+298
+00:19:21,200 --> 00:19:25,420
+Orthogonal To The GradientIs a direction of zero
+
+299
+00:19:25,420 --> 00:19:30,120
+changed in F because θ is equal to π على 2 لما
+
+300
+00:19:30,120 --> 00:19:36,960
+يكون الآن الـ Zero بيكون عندي الـ Direction of U
+
+301
+00:19:36,960 --> 00:19:40,460
+is orthogonal to a gradient فالـ F بيكون اللي هو
+
+302
+00:19:40,460 --> 00:19:43,340
+الـ Directional Derivative أي إيش تفر يعني الـ
+
+303
+00:19:43,340 --> 00:19:46,420
+Directional Derivative بتكون Zero في حالة اللي هو
+
+304
+00:19:46,420 --> 00:19:51,180
+θ بتسوء بي على 2 هذا اللي عن زيرو إيش Gradientإذا
+
+305
+00:19:51,180 --> 00:19:53,420
+الـ Direction of the derivative of U of F بتساوي
+
+306
+00:19:53,420 --> 00:20:00,500
+سفر When U is orthogonal to the gradient ماشي،
+
+307
+00:20:00,500 --> 00:20:04,080
+المعلمة أضحى تلان ناخد مثال عددي Example تلاتة
+
+308
+00:20:04,080 --> 00:20:07,520
+Find the directions in which F of XY بيسوي X تربيه
+
+309
+00:20:07,520 --> 00:20:11,100
+على 2 وY تربيه على 2 وقوللي بس أودد ال directions
+
+310
+00:20:11,100 --> 00:20:16,040
+اللي ال F هذه واحد Increases most rapidly at the
+
+311
+00:20:16,040 --> 00:20:20,810
+point واحد واحد يعني لما بدي Directionلما الـ F
+
+312
+00:20:20,810 --> 00:20:23,550
+increases most rapidly عند الواحد و واحد قال لي
+
+313
+00:20:23,550 --> 00:20:26,570
+واحد على طول طب انت قلت قبل شوية انهم يكونون most
+
+314
+00:20:26,570 --> 00:20:29,910
+rapidly لما يكونون في اتجاه مين لـ Gradient بقوله
+
+315
+00:20:29,910 --> 00:20:32,930
+صح قال ليه أصلا دي decreases most rapidly اللي بدك
+
+316
+00:20:32,930 --> 00:20:36,390
+تسأل عنها في التانية برضه في عكس اتجاه مين ل
+
+317
+00:20:36,390 --> 00:20:40,270
+Gradientإذا أسهل الأمر بنجد ما هو الـ Gradient و
+
+318
+00:20:40,270 --> 00:20:45,550
+بنجد ما هو الـ Direction يعني الـ Gradient على
+
+319
+00:20:45,550 --> 00:20:48,770
+المجموعات اللي هو بيكون الجواب اللي هو الأول هذا و
+
+320
+00:20:48,770 --> 00:20:52,690
+ناقصه بيكون جواب الثاني بقوله صح كلامك، بقولي طب
+
+321
+00:20:52,690 --> 00:20:55,390
+What are the directions of zero change in F at
+
+322
+00:20:55,390 --> 00:21:00,300
+واحد واحد؟أنت بتقول لما قلت لنا قبل بشوية أنه
+
+323
+00:21:00,300 --> 00:21:05,400
+بيكون اللي هو الـ Zero اتشانج لما نكون الـ
+
+324
+00:21:05,400 --> 00:21:09,840
+Direction لـ U و V متعامدين يعني وجدنا الـ
+
+325
+00:21:09,840 --> 00:21:13,480
+Direction للـ Gradient على طول بنغير إشارة واحدة و
+
+326
+00:21:13,480 --> 00:21:18,770
+بنخلي التاني زي ما هي و التجاهين عموديين عليهطيب
+
+327
+00:21:18,770 --> 00:21:22,690
+خلّينا نشوف الكلمة ده اللي بنحكيه سريع للإجابة
+
+328
+00:21:22,690 --> 00:21:26,530
+بسرعة ييجي الآن نشوف increases most rapidly at the
+
+329
+00:21:26,530 --> 00:21:30,410
+point 113 the function قولنا فوق increases most
+
+330
+00:21:30,410 --> 00:21:33,370
+rapidly in the direction of the gradient at 113،
+
+331
+00:21:33,370 --> 00:21:37,090
+وعادةً بلزمنا لوجد الـ Gradient قادر الـ Gradient
+
+332
+00:21:37,090 --> 00:21:41,170
+للـ F أيش بيساوي؟ اللي هو Del F by Del X اللي هي
+
+333
+00:21:41,170 --> 00:21:46,210
+2X على 2 يعني Xتجاه الـ I دل F by دل Y اللي هي 2Y
+
+334
+00:21:46,210 --> 00:21:50,220
+على 2 يعني Y في تجاه الـ G هذا الـ Gradientللـ
+
+335
+00:21:50,220 --> 00:21:53,460
+gradient للـ F عند الواحد والواحد ايش هيساوي او
+
+336
+00:21:53,460 --> 00:21:57,020
+ذنب واحد واحد بصير I زي الـ J إذا الأن الـ
+
+337
+00:21:57,020 --> 00:22:00,380
+Direction للـ Gradient للF وعبارة عن الـ Gradient
+
+338
+00:22:00,380 --> 00:22:04,240
+للF على الـ Magnitude له Magnitude هو الجدر التربي
+
+339
+00:22:04,240 --> 00:22:07,260
+على واحد زي واحد على جدر التنين يعني الـ Direction
+
+340
+00:22:07,260 --> 00:22:10,440
+U بيساوي واحد على جدر التنين في تجاه الـ I زي واحد
+
+341
+00:22:10,440 --> 00:22:13,480
+على جدر التنين في تجاه الـ J هذا الـ Direction هو
+
+342
+00:22:13,480 --> 00:22:17,640
+Direction لـ Gradientاللي عنده اللي هي الـ F
+
+343
+00:22:17,640 --> 00:22:23,450
+increases most ربط اللي عنده النقطة 1 1طيب قلت
+
+344
+00:22:23,450 --> 00:22:25,870
+أكيد كلكم قلته الـ function decreases most
+
+345
+00:22:25,870 --> 00:22:29,330
+probably in the direction of عكس الـ gradient عنده
+
+346
+00:22:29,330 --> 00:22:33,210
+نقطة واحد واحد which is هذا بس مضروب في من؟ في
+
+347
+00:22:33,210 --> 00:22:36,390
+سالب واحد ناقص واحد على جدر التانين في تجاه I ناقص
+
+348
+00:22:36,390 --> 00:22:40,270
+واحد على جدر التانين في تجاه J الآن الجزء الثالث
+
+349
+00:22:40,270 --> 00:22:44,370
+الثالث what are the directions of zero change يعني
+
+350
+00:22:44,370 --> 00:22:47,410
+لـ Gradient لما يكون اللي هو إيش directional
+
+351
+00:22:47,410 --> 00:22:50,150
+derivative zero بتكون directional derivative zero
+
+352
+00:22:50,150 --> 00:22:53,120
+لما الزاوية بينهم إيش بتساوي؟By على اتنين يعني
+
+353
+00:22:53,120 --> 00:22:55,780
+بمعنى آخر لما يكون الـ U عمودي عالميا على الـ
+
+354
+00:22:55,780 --> 00:23:00,020
+Gradient طب هي الـ U موجودةالتي جاهة الـ Gradient
+
+355
+00:23:00,020 --> 00:23:03,720
+موجودة إذاً بدأ وجد الـ Direction تبع الـ Zero الـ
+
+356
+00:23:03,720 --> 00:23:06,140
+Direction of Zero Change بقصه Direction تبع الـ
+
+357
+00:23:06,140 --> 00:23:09,260
+Zero Change إن F أت واحد are the directions
+
+358
+00:23:09,260 --> 00:23:13,180
+orthogonal to gradient للـ F الـ Directions اللي
+
+359
+00:23:13,180 --> 00:23:19,140
+عمليات عالمين على الـ F الآن هي تجاه الـ F وهي
+
+360
+00:23:19,140 --> 00:23:23,280
+تجاه الـ F هي ناقص الـ F اللي هي تجاه الـ Gradient
+
+361
+00:23:24,290 --> 00:23:28,130
+تجاه الـ Gradient اللي هو العمودي عليه بس بنغير
+
+362
+00:23:28,130 --> 00:23:30,650
+إشارتها دي بيصيح ناقص واحدة لجدرتينين في الوقت
+
+363
+00:23:30,650 --> 00:23:33,890
+تجاه الـ I زائد واحدة لجدرتينين في الجهة الـ J أو
+
+364
+00:23:34,290 --> 00:23:39,050
+وأو كمان اللي هو بغير هذه شريطة و بخليها دي ثابتة
+
+365
+00:23:39,050 --> 00:23:42,130
+يعني بيصير N بتسوى واحدة لجدر التنين في تجاه I
+
+366
+00:23:42,130 --> 00:23:46,510
+ناقص واحدة لجدر التنين J وناقص N اللي عكسه ناقص
+
+367
+00:23:46,510 --> 00:23:48,990
+واحدة لجدر التنين في تجاه I زي واحدة لجدر التنين
+
+368
+00:23:48,990 --> 00:23:52,450
+في تجاه J هدول التنين طبعا عمودي واحدة هي لو فرضنا
+
+369
+00:23:52,450 --> 00:23:56,410
+هذا ال gradient التجاه ال gradient التجاه العمودي
+
+370
+00:23:56,410 --> 00:23:59,990
+عليه هيه هنا أو اللي تحت هدول العكس بعض التنتين
+
+371
+00:24:00,350 --> 00:24:04,970
+عشان اضغط على الـ dot product 1 على الجدر التاني و
+
+372
+00:24:04,970 --> 00:24:05,850
+1 على الجدر التاني
+
+373
+00:24:33,380 --> 00:24:40,400
+الان نشوف علاقة الـ Gradientبمين؟ بالـ Tangents to
+
+374
+00:24:40,400 --> 00:24:43,500
+Level Curves ونستغل هذه العلاقة في إيجاد ما ..
+
+375
+00:24:43,500 --> 00:24:47,940
+اللي هو إيش .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+376
+00:24:47,940 --> 00:24:48,460
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+377
+00:24:48,460 --> 00:24:49,060
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+378
+00:24:49,060 --> 00:24:49,220
+.. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+379
+00:24:49,220 --> 00:24:49,760
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+380
+00:24:49,760 --> 00:24:50,560
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+381
+00:24:50,560 --> 00:24:50,620
+.. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+382
+00:24:50,620 --> 00:24:50,840
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+383
+00:24:50,840 --> 00:24:59,060
+ما .. ما .. ما ..
+
+384
+00:24:59,060 --> 00:25:01,000
+ما
+
+385
+00:25:06,720 --> 00:25:13,260
+لأن F of X Y هذه لو نفرضنا إنه قيمتها ثابتةوين على
+
+386
+00:25:13,260 --> 00:25:18,540
+اللي هي عند قيم XG of T وعند قيم Y بساوية H of T
+
+387
+00:25:18,540 --> 00:25:23,960
+يعني على مين؟ على الـ Curve هذه لو عوضنا X بأي
+
+388
+00:25:23,960 --> 00:25:27,060
+قيمة يعني لما يكون في عندي Vector هيه اللي بساوية
+
+389
+00:25:27,060 --> 00:25:29,980
+G of T في تجاه الـI زي هذا G of T في تجاه الـJ
+
+390
+00:25:29,980 --> 00:25:34,300
+ودينا عوضنا الـG of T هنا والـH of T هنا هنلاقي
+
+391
+00:25:34,300 --> 00:25:37,400
+القيمة إيه اشتلعت Constant يعني صار هذا عبارة عن F
+
+392
+00:25:37,400 --> 00:25:41,620
+of X وY بساوية C صار Level Curveالان الـ level
+
+393
+00:25:41,620 --> 00:25:46,640
+curve هذا اللي هو f of g of t وh of t ايش بيساوي c
+
+394
+00:25:46,640 --> 00:25:51,780
+خلّينا نفاضل اللي هو إيش اتجهتين with respect to
+
+395
+00:25:51,780 --> 00:25:55,880
+mean to t فلطولوا روحكم معانايا differentiating
+
+396
+00:25:55,880 --> 00:25:59,000
+both sides of this equation with respect to t
+
+397
+00:25:59,000 --> 00:26:02,780
+leads to the equation هذه ثابتة c لأن قلنا along
+
+398
+00:26:02,780 --> 00:26:07,000
+اللي هو ال R هذا كله ال f of x y هتصير مقدار ثابت
+
+399
+00:26:07,000 --> 00:26:10,480
+هيك مفترضين احناالذي يحدث هو الـ derivative لهذه
+
+400
+00:26:10,480 --> 00:26:12,340
+بيساوي الـ derivative لهذه و الـ derivative لهذه
+
+401
+00:26:12,340 --> 00:26:16,180
+الـ derivative لهذه من وجه الذهب وسط 2 اللي هو الـ
+
+402
+00:26:16,180 --> 00:26:19,300
+chain rule اللي أخدناها المرة الفاترة Del F by Del
+
+403
+00:26:19,300 --> 00:26:23,700
+X في Del X by Del T X أنا اللي مسمينها إيه؟ إحنا G
+
+404
+00:26:23,700 --> 00:26:27,640
+زاد Del F by Del Y في Del Y by Del T الـ Y اللي
+
+405
+00:26:27,640 --> 00:26:32,570
+مسمينها أنا إيه؟ اسمها H إيش هيساوي؟ سفرالأن هذه
+
+406
+00:26:32,570 --> 00:26:35,910
+برجحها لأصلها هذه عبارة عن حاصل ضرب اللي هو الـ
+
+407
+00:26:35,910 --> 00:26:39,170
+dot product اللي هو del f by del x في اتجاه الـ I
+
+408
+00:26:39,170 --> 00:26:42,850
+زي del f by del y في اتجاه الـ J ضربة dot product
+
+409
+00:26:42,850 --> 00:26:47,870
+في الـ dG by dt في اتجاه الـ I زي الـ dH by dt في
+
+410
+00:26:47,870 --> 00:26:51,010
+اتجاه الـ G ومش مصدق ضربهم هذه في هذه السيرة هي
+
+411
+00:26:51,490 --> 00:26:55,290
+وهذه في هذه بالسيرة هيها وهذا ايش بيساوي كله سفر
+
+412
+00:26:55,290 --> 00:26:59,250
+هذه مين هي الـ gradient للـ F الـ gradient لمن للـ
+
+413
+00:26:59,250 --> 00:27:03,250
+F اللي عندك مضروبة في مين في DR by DT كله ايش
+
+414
+00:27:03,250 --> 00:27:05,970
+بيساوي سفر هذا نضطه product بين اللي بيساوي سفر
+
+415
+00:27:05,970 --> 00:27:09,430
+هذا له معنى نضطه product بين اللي هو الـ gradient
+
+416
+00:27:09,430 --> 00:27:13,770
+للـ F الـ gradient للـ function وهذه في اللي هو DR
+
+417
+00:27:13,770 --> 00:27:19,290
+by DT اللي هو عبارة عن ميل المماسللـ vector R
+
+418
+00:27:19,290 --> 00:27:21,650
+بيساوي «G» و «T» أتشوف «T» بتجاه الـ «I» اللي
+
+419
+00:27:21,650 --> 00:27:26,210
+بيخلّي الـ «F» of X و Y constant يعني الـ Gradient
+
+420
+00:27:26,210 --> 00:27:31,710
+للـ «F» is normal على مين؟ على الـ «Tangent» لمين؟
+
+421
+00:27:31,710 --> 00:27:36,610
+للـ «Vector» الـ «R» اللي عندي easy equation خمسة
+
+422
+00:27:36,610 --> 00:27:39,390
+says that the gradient للـ «F» is normal to the
+
+423
+00:27:39,390 --> 00:27:43,370
+tangent vector «DR» by «DT» so it is normal to the
+
+424
+00:27:43,370 --> 00:27:47,790
+curve ملخص الحديث اللي حكينا فوقإن في كل موقع X0
+
+425
+00:27:47,790 --> 00:27:53,970
+وY0 في مجال مفهوم F of X وY الـ Gradient of F هو
+
+426
+00:27:53,970 --> 00:27:58,650
+عمودي عالمي لـ Level Curve عند مين النقطة X0 وY0
+
+427
+00:27:58,650 --> 00:28:04,970
+دائما اللي هو الـ Gradient للـ Function F عمودي
+
+428
+00:28:04,970 --> 00:28:12,140
+عالمي على Level Curve خلال مين؟ X0 وY0لأن هذا الـ
+
+429
+00:28:12,140 --> 00:28:16,560
+Level Curve f of x و f of y هي مين f of هذا
+
+430
+00:28:16,560 --> 00:28:19,280
+المقدار f of g of t و f of h of t اللي هو بساوي c
+
+431
+00:28:19,280 --> 00:28:24,220
+بعمل constant يعني بيصير اللي هو dr by dt على ال
+
+432
+00:28:24,220 --> 00:28:27,180
+gradient للf normal الـ dr by dt اللي هو مين
+
+433
+00:28:27,180 --> 00:28:31,540
+المماس للـ Level Curve اللي عندي إذا صار عندي
+
+434
+00:28:31,540 --> 00:28:35,470
+Level Curveالـ Normal دايمًا لـ Gradient H Normal
+
+435
+00:28:35,470 --> 00:28:41,110
+عليه لأن هو Normal على الـ Tangent Elo على الـDR
+
+436
+00:28:41,110 --> 00:28:46,270
+by DT إذًا، اللي هي هذه الآن، بدنا نستغلها في حال
+
+437
+00:28:46,270 --> 00:28:50,830
+السؤال التالي هذه بس الرسمة هي تعليقًا على اللي هو
+
+438
+00:28:50,830 --> 00:28:53,530
+اللي أوجدناها قبل شوية ده لزيد بساويكس تربيه على
+
+439
+00:28:53,530 --> 00:28:56,680
+اتنين زي دوية تربيه على اتنيناللي هو الرسم الكمهان
+
+440
+00:28:56,680 --> 00:28:58,960
+اللي هو جاينا هي الـ Gradient للـ F عنده النقطة
+
+441
+00:28:58,960 --> 00:29:01,880
+اللي وجدناها اللي يسمى I زي الـ J هي هذا اللي
+
+442
+00:29:01,880 --> 00:29:06,060
+بتكون Most Rapidly وهنا أجلم اللي هي عكسها الـ
+
+443
+00:29:06,060 --> 00:29:09,540
+Decreases اللي هو Most Rapidly أو Most Rapid
+
+444
+00:29:09,540 --> 00:29:13,620
+Decrease اللي هي عكسها ودولة الـ Vectorين هي اللي
+
+445
+00:29:13,620 --> 00:29:16,660
+بيخلن الـ Zero Change enough هذا اللي هي الرسمة
+
+446
+00:29:16,660 --> 00:29:20,740
+اللي عملنا لتو في المثال اللي قبل بشوية هي المثال
+
+447
+00:29:20,740 --> 00:29:26,750
+اللي عملناهاهذا آه آسف هذا المثال هي آه هذه اللي
+
+448
+00:29:26,750 --> 00:29:30,450
+هو اللي بس اللي هو التوضيح الهندسي لها للي
+
+449
+00:29:30,450 --> 00:29:37,920
+حاكيناهاهذه التوضيحة في problem 39 سأستغلها بعد
+
+450
+00:29:37,920 --> 00:29:43,640
+قليل سأستعملها لأحل مشاهدها مثال خلّيني بس أن
+
+451
+00:29:43,640 --> 00:29:47,660
+البرنامج اللي كنت معاه show that if x minus x0
+
+452
+00:29:47,660 --> 00:29:51,760
+plus by minus y0 is an equation for the line in
+
+453
+00:29:51,760 --> 00:29:54,480
+the xy plane through the point x0 y0 normal to the
+
+454
+00:29:54,480 --> 00:29:59,170
+vector كذا يعني لو كان عنده نقطة x0 y0وعندي Normal
+
+455
+00:29:59,170 --> 00:30:05,610
+vector N على line يعني معروف الـ Normal N بتسوي I
+
+456
+00:30:05,610 --> 00:30:09,470
+زي BJ معروفة نقطة يعني عندي خط اللي كان معروف
+
+457
+00:30:09,470 --> 00:30:13,230
+Normal عليه ومعروف نقطة X0 Y0 وده وجود معادلته
+
+458
+00:30:13,690 --> 00:30:16,850
+معدلته انا بقول هي a في x minus x not زي بي في y
+
+459
+00:30:16,850 --> 00:30:20,170
+minus y not بالسبع سفر عشان مانطولش حل السؤال اللي
+
+460
+00:30:20,170 --> 00:30:23,770
+بعد ولا هيعتمد عليه إيش هذه؟ يا جماعة هذه كتير هي
+
+461
+00:30:23,770 --> 00:30:28,250
+ليه؟ عندي معروف .. هي عندي خط هذا الخط هذا بيدوجد
+
+462
+00:30:28,250 --> 00:30:32,610
+معدلته معروف العمودي عليه و معروف نقطة عليه و
+
+463
+00:30:32,610 --> 00:30:35,730
+بنعرف معدلته طبعا كلكم بيعرف أعمله خليني بس على
+
+464
+00:30:35,730 --> 00:30:38,950
+السريع أني أحله ال normal to the vector m ai زي بي
+
+465
+00:30:38,950 --> 00:30:45,570
+جي isالأن T بيتساوي الهو
+
+466
+00:30:45,570 --> 00:30:50,070
+T بيساوي PI نقص AJ أو نقص PI زي AJ C يعني أي واحد
+
+467
+00:30:50,070 --> 00:30:56,910
+بالفعل ال equation is X بيساوي X naught زي ال TB و
+
+468
+00:30:56,910 --> 00:31:01,860
+Y بيساوي Y naught نقص TAاللي معروف الآن x بتساوي
+
+469
+00:31:01,860 --> 00:31:06,380
+اللي هي خطأ هي عندي ال normal معروف بيصير عندي
+
+470
+00:31:06,380 --> 00:31:10,240
+اللي هي معادلة اللي هي x بيساوي x naught هي النقطة
+
+471
+00:31:10,240 --> 00:31:16,720
+زائد t في b و y بيساوي y naught ناقص t في a هي
+
+472
+00:31:16,720 --> 00:31:23,260
+معاملة that is الان x minus x naught بيساوي tb و y
+
+473
+00:31:23,260 --> 00:31:27,700
+minus y naught بيساوي t ناقص tهنس المعادلة A في X
+
+474
+00:31:27,700 --> 00:31:32,120
+نحلها مع بعض عشان نحطها في المعادلة اللي فوق A في
+
+475
+00:31:32,120 --> 00:31:37,920
+X minus X0 بسوى T في A بي أكيد عارفين كيف اشي اللي
+
+476
+00:31:37,920 --> 00:31:44,800
+عملتها عندي هاي المعادلتين هذولةا في x minus x not
+
+477
+00:31:44,800 --> 00:31:50,020
+هي a في x minus x not بساوي a في tb يعني ساعة t a
+
+478
+00:31:50,020 --> 00:31:55,220
+في b and b في y minus y not بساوي نقص t في ba لأن
+
+479
+00:31:55,220 --> 00:32:00,720
+التلتين مع بعض بيصير هي هذه بتساوي نقص هذه ا او
+
+480
+00:32:00,720 --> 00:32:06,320
+معنى اخر هذه اجمحها لهذه او اجمحها لهذه المعدلتين
+
+481
+00:32:06,320 --> 00:32:09,700
+مع بعض بيطلع سفر هنابصير ايه في X minus X naught و
+
+482
+00:32:09,700 --> 00:32:12,820
+في Z بيه في Y minus Y نقل ساعة و ستة هذا اللي هي
+
+483
+00:32:12,820 --> 00:32:18,140
+معادلة اللي يتحقق اللي أكون عمودي على اللي عندي
+
+484
+00:32:18,140 --> 00:32:25,700
+فهيبقى نشوف الفترة بقى find an equation for the
+
+485
+00:32:25,700 --> 00:32:28,780
+tangent to the ellipse find an equation for the
+
+486
+00:32:28,780 --> 00:32:32,960
+tangent to the ellipse معطينيها at the point نقص 2
+
+487
+00:32:32,960 --> 00:32:39,690
+و 1 يعني معطينيها اللي لبسهاهذه الـ ellipse بدي
+
+488
+00:32:39,690 --> 00:32:44,690
+اوجد الـ equation للتانجينت هي عند المقطع 61 وهي
+
+489
+00:32:44,690 --> 00:32:49,070
+المماس بدي اوجد ال equation للمماس طبعا فيكم
+
+490
+00:32:49,070 --> 00:32:53,090
+يعرفوا عمله ولا بالزمان بطريقة زمان بنفعبس دلوقت
+
+491
+00:32:53,090 --> 00:32:56,870
+ده خلّيني أقول يعني عشان نطبق اللي حكينا بواسط ال
+
+492
+00:32:56,870 --> 00:33:00,330
+level curves أما بنحلل اللي هو بالطريقة اللي هي
+
+493
+00:33:00,330 --> 00:33:04,970
+القديمة هي المعادنة موجودة وهي النقطة موجودة بنوجد
+
+494
+00:33:04,970 --> 00:33:07,330
+اللي هو المماس من اللي المماس معروف و النقطة
+
+495
+00:33:07,330 --> 00:33:11,310
+معروفة بنوجد إيش بنوجد اللي هو المماس خلّيني أوجده
+
+496
+00:33:11,310 --> 00:33:16,260
+بطريقة ثانية الآنهذه هي X تربيه على أربع ثم تربيه
+
+497
+00:33:16,260 --> 00:33:20,580
+على بساوة اتنين هي عبارة عن ال ellipse هذه وهي ال
+
+498
+00:33:20,580 --> 00:33:24,220
+tangent لها وهي الان بدي اوجد العمودي على ال
+
+499
+00:33:24,220 --> 00:33:29,730
+tangent ونستغل اللي هو المعرفة اللي قبل شويةماذا
+
+500
+00:33:29,730 --> 00:33:33,310
+تريد أن تفعله؟ تريد استخدام معرفتهين دعنا نقول الـ
+
+501
+00:33:33,310 --> 00:33:37,470
+ellipse x² على 4 زي 2y² على 2 is a level curve of
+
+502
+00:33:37,470 --> 00:33:40,430
+mean of the function f of x,y بيساوي x² على 4 زي
+
+503
+00:33:40,430 --> 00:33:44,470
+2y² على 2y² لأن اعتبرت الدالة عندي الـ of two
+
+504
+00:33:44,470 --> 00:33:48,170
+variables اللي هي function of two variables هي
+
+505
+00:33:48,170 --> 00:33:52,540
+الدالةالأن هذه الـ Dialogue طبعاً في Surface، في
+
+506
+00:33:52,540 --> 00:33:55,580
+Space اللي عن الـ Level Curve اللي لها أخدت الـ
+
+507
+00:33:55,580 --> 00:33:58,600
+Level T Curve X تربيه على 4 زي ضي تربيه بيسوا 2
+
+508
+00:33:58,600 --> 00:34:02,760
+اللي هي عبارة عن صارت Level Curve لهذه احنا عارفين
+
+509
+00:34:02,760 --> 00:34:07,960
+أنه الـ Gradient للـ Function عمودي على مابين على
+
+510
+00:34:07,960 --> 00:34:11,540
+Level Curve اللي حكيناه قبل بشوية وانت تطبيق عليه
+
+511
+00:34:11,830 --> 00:34:17,730
+الـ Gradient فبساوية X على 2 في اتجاه I زائد 2Y في
+
+512
+00:34:17,730 --> 00:34:24,270
+اتجاه J هذا الـ Gradient اللي هو نقطة مين؟ نقص 2 و
+
+513
+00:34:24,270 --> 00:34:31,050
+1؟ بساوية ناقص I عوضت عن X نقص 2زي دي 2G هذا الـ
+
+514
+00:34:31,050 --> 00:34:36,130
+Gradient الآن احنا عارفين ايش خصاصه عمودي على الـ
+
+515
+00:34:36,130 --> 00:34:38,890
+Level Curve يعني عمودي على المماس تبع الـ Level
+
+516
+00:34:38,890 --> 00:34:41,790
+Curve عند اللقطة المذكورة هذا من اللي قبل شوية
+
+517
+00:34:41,790 --> 00:34:46,340
+أثبتناهاللي هي الأطباق التنهجة بالبشوية هذا اللي
+
+518
+00:34:46,340 --> 00:34:49,860
+هو الـ Gradient عمودي عالميا على الـ Level Curve
+
+519
+00:34:49,860 --> 00:34:54,200
+عند النقطة X0 وY0 بتستخدم هذه المعلومة ستعرف أن
+
+520
+00:34:54,200 --> 00:34:57,660
+الـ Gradient اللي هو العمودي على اللي هو المماس
+
+521
+00:34:57,660 --> 00:35:01,100
+فعارف نقطة على طول من خلال المثال اللي جاب اللي
+
+522
+00:35:01,100 --> 00:35:04,000
+بشوية أو ال problem اللي جاب اللي بشويةبكون عرفة
+
+523
+00:35:04,000 --> 00:35:08,720
+العمودي وعارف نقطة اذا بعرف عادلة اللي هي المماسك
+
+524
+00:35:08,720 --> 00:35:12,520
+اللي بديها اللي هو a في x نقص x not زي بي في y نقص
+
+525
+00:35:12,520 --> 00:35:15,980
+y not بساوي سفر ال a اللي هي عبارة عن نقص واحد في
+
+526
+00:35:15,980 --> 00:35:19,760
+هذه الحالة و ال b عبارة عن اتنين اللي هو المعاملة
+
+527
+00:35:19,760 --> 00:35:24,200
+تبعت مين اللي هو العمودي x minus x not اللي هي x
+
+528
+00:35:24,200 --> 00:35:27,120
+not اللي هي نقص اتنين بسيط زي اتنين و y نقص y not
+
+529
+00:35:27,120 --> 00:35:31,040
+اللي هي واحد بساوي سفر دهاللي هو لما نبسطها بيطلع
+
+530
+00:35:31,040 --> 00:35:35,160
+X نقص 2Y بساوي سالب 4 فدي معادلة المماس اللي احنا
+
+531
+00:35:35,160 --> 00:35:38,980
+أمامنا وانتوا في البيت جربوا كيف توجدوها بطريقة
+
+532
+00:35:38,980 --> 00:35:43,360
+السابقة نيجي لعند algebra rules for gradient اللي
+
+533
+00:35:43,360 --> 00:35:46,780
+هي بعض خواص ال gradient خواص ال gradient شبه لخواص
+
+534
+00:35:46,780 --> 00:35:49,660
+مين يا جماعة ال derivative العادية ال gradient ل F
+
+535
+00:35:49,660 --> 00:35:52,580
+زائد G سواء لGradient لF زايد لGradient تنجي لأنه
+
+536
+00:35:52,580 --> 00:35:57,860
+ال gradient عبارة عناللي هي دل F by دل X في تجاه I
+
+537
+00:35:57,860 --> 00:36:00,280
+زي دل F by دل Y في تجاه J يعني نبقى قارعين
+
+538
+00:36:00,280 --> 00:36:03,220
+derivatives فطبيعي يكون خواصها هي خواص الـ
+
+539
+00:36:03,220 --> 00:36:06,240
+derivative العادية والـ partial derivatives فالـ
+
+540
+00:36:06,240 --> 00:36:10,020
+gradient لف نقص G سوى الـ gradient لف نقص G الـ
+
+541
+00:36:10,020 --> 00:36:12,940
+gradient لكيف لف K حيث constant سوى كيف الـ
+
+542
+00:36:12,940 --> 00:36:16,290
+gradient لفالـ F في G بساوية الـ F في الـ Gradient
+
+543
+00:36:16,290 --> 00:36:19,430
+للـ G زائد G في Gradient للـ F والـ Gradient للـ F
+
+544
+00:36:19,430 --> 00:36:23,090
+على G بساوية الـ G Gradient للـ F نقص F Gradient
+
+545
+00:36:23,090 --> 00:36:26,730
+للـ G على G تلت يعني اللي انتوا عارفينها من زمان
+
+546
+00:36:26,730 --> 00:36:31,870
+هذه الخواصة بس خلونا نتحقق عدديا من واحدة تنتير بس
+
+547
+00:36:31,870 --> 00:36:37,460
+عشر سنةللأطمأنة مش أكثر لأن مش مطلوب الورحانة مثال
+
+548
+00:36:37,460 --> 00:36:42,560
+خمسة We illustrate two of the rules with يعني بدنا
+
+549
+00:36:42,560 --> 00:36:47,460
+نوضح قاعدتين من اللي فاتن بوساطة ايش احنا مثلا عدد
+
+550
+00:36:47,460 --> 00:36:51,880
+f of x,y بيساوي x نقص y و g of x,y بيساوي 3y ال
+
+551
+00:36:51,880 --> 00:36:55,340
+gradient اللافظة ما انتوا عارفين دل f by دل x بيه
+
+552
+00:36:55,340 --> 00:37:00,540
+i دل f by دل y هي واحد اللي هي بيصير نقص هذا واحد
+
+553
+00:37:00,540 --> 00:37:08,030
+وهنا نقص واحد بيصير i نقص jDEL F DEL J BY DEL X
+
+554
+00:37:08,030 --> 00:37:12,970
+صفر إذاً Zero مافيش هنا DEL J BY DEL Y تلاتة إذاً
+
+555
+00:37:12,970 --> 00:37:18,230
+الـ Gradient DEL F عبارة عن مين تلاتة Gإذاً غرضنا
+
+556
+00:37:18,230 --> 00:37:21,710
+يا جماعة بس نحقق في هذا المثال المثال del f minus
+
+557
+00:37:21,710 --> 00:37:26,290
+g اي موضح كيف يساوي del f minus del g والـ
+
+558
+00:37:26,290 --> 00:37:30,410
+gradient للـ f ناقص gradient g الـ gradient للـ f
+
+559
+00:37:30,410 --> 00:37:34,590
+minus g بنيجي من اتراح الـ f ناقص g هي x ناقص y
+
+560
+00:37:34,590 --> 00:37:38,650
+ناقص 3 وy بيصير x ناقص 4y الـ gradient لهذه اللي
+
+561
+00:37:38,650 --> 00:37:43,360
+هي del f by del .. del هذا by del x يعني Iلأن
+
+562
+00:37:43,360 --> 00:37:47,240
+تفاضلها بالنسبة لـ X1 وتفاضلها بالنسبة ل Y ناقص 4
+
+563
+00:37:47,240 --> 00:37:53,240
+I ناقص 4J هي في الواقع فعلا I ناقص G ناقص 3J يعني
+
+564
+00:37:53,240 --> 00:37:57,680
+I ناقص 4G يعني الـ Gradient لـ F ناقص Gradient لـ
+
+565
+00:37:57,680 --> 00:38:01,400
+G خلّينا نشوف التانية Gradient للـ F في G نجيب الـ
+
+566
+00:38:01,400 --> 00:38:05,860
+F في G هي الـ F X minus Y في G اللي هي تلاتة Y
+
+567
+00:38:05,860 --> 00:38:08,440
+تصبح تلاتة X في Y
+
+568
+00:38:11,100 --> 00:38:14,640
+نقص تلاتة Y تربيع لأن في تلاتة Y يكون تلاتة Y
+
+569
+00:38:14,640 --> 00:38:19,120
+تربيع ضربت الـ F في G في G يا جماعة طلعت عندي
+
+570
+00:38:19,120 --> 00:38:22,600
+بالشكل هذا الـ Gradient لهذه الدالة الجديدة هو
+
+571
+00:38:22,600 --> 00:38:27,260
+عبارة عن تلاتة Yهذه السفر تلاتة Y في اتجاه الـ I
+
+572
+00:38:27,260 --> 00:38:30,240
+بفضل بالنسبة ل X طبعا قبل ما انتوا عارفين بعدين
+
+573
+00:38:30,240 --> 00:38:33,480
+بفضل بالنسبة ل Y بالنسبة ل Y اللي عبارة عن تلاتة X
+
+574
+00:38:33,480 --> 00:38:37,620
+ناقص ستة Y عشانك تلاتة X ناقص ستة Y في اتجاه الـ G
+
+575
+00:38:37,620 --> 00:38:41,520
+هذه اللي هي ال gradient لل FG نيجي الآن نحسب ال
+
+576
+00:38:41,520 --> 00:38:45,790
+gradient لل Gاللي هي الـ G في الـ gradient للـ F
+
+577
+00:38:45,790 --> 00:38:48,590
+والـ F في الـ gradient للـ G حسب قانونها، بدنا
+
+578
+00:38:48,590 --> 00:38:53,430
+نقول أن هذا بساوي هذا نتحقق منه بس من خلال هذا
+
+579
+00:38:53,430 --> 00:38:57,830
+المثال، هذا اللي هي حاصل ضرب دالتين، هي تفضولين،
+
+580
+00:38:57,830 --> 00:39:01,310
+خلينا نشوف كيف هذا الكلام بساوي اللي فوق من خلال
+
+581
+00:39:01,310 --> 00:39:08,130
+مثلناجي هج ثلاثة جي ثلاثة و اي ثلاثة و اي ثلاثة و
+
+582
+00:39:08,130 --> 00:39:15,500
+اي ثلاثة و اي ثلاثة و اي ثلاثةالـ Gradient للـ G
+
+583
+00:39:15,500 --> 00:39:22,940
+عبارة عن ثلاثة G إذاً الـ Gradient .. آسف فعلا
+
+584
+00:39:22,940 --> 00:39:26,660
+بدأت فائدك بدأنا الـ G .. G اللي عبارة عن جداش
+
+585
+00:39:26,660 --> 00:39:30,940
+عبارة عن تلاتة Y عشان ما خربتش في ال Gradient للـ
+
+586
+00:39:30,940 --> 00:39:36,000
+F ال Gradient للـ F هي I نقص G هي نزلتها زائد الـ
+
+587
+00:39:36,000 --> 00:39:40,140
+F جداش قيمتها X minus Y هيها ضربتها في من؟ في
+
+588
+00:39:40,140 --> 00:39:44,290
+Gradient للـ G Gradient للـ G تلاتة Gالان هذا
+
+589
+00:39:44,290 --> 00:39:49,210
+المقدار جيت بساوي عبارة عن تلاتة Y في I minus G
+
+590
+00:39:49,210 --> 00:39:53,150
+خليته زي ما هو بده يصل للي فوق يا جماعة زائد هذه X
+
+591
+00:39:53,150 --> 00:39:57,310
+دخلت للي جوا بيصير تلاتة X ناقص تلاتة Y الان خلينا
+
+592
+00:39:57,310 --> 00:40:02,390
+نلم هنا على بعض التلاتة Y في ناقص J اللي بيصير
+
+593
+00:40:02,390 --> 00:40:07,210
+ناقص تلاتة YJ لما زيدها على ناقص تلاتة YJ بيصير
+
+594
+00:40:07,210 --> 00:40:11,910
+ناقص تلاتة YJ هيبتبقى هنا تلاتة X في اتجاه الـ J
+
+595
+00:40:11,910 --> 00:40:15,190
+زي ما هي الان اللي ضالت هنا اللي ما عملناهاش هي
+
+596
+00:40:15,190 --> 00:40:18,750
+تلاتة Y في اتجاه الـ I هيها زي ما هي إذا تلاتة Y
+
+597
+00:40:18,750 --> 00:40:22,310
+في اتجاه الـ I هيها زي ما هي ودول الناقص تلاتة في
+
+598
+00:40:22,310 --> 00:40:24,590
+اتجاه الـ J مع الناقص تلاتة في اتجاه الـ J بيصير
+
+599
+00:40:24,590 --> 00:40:28,190
+ناقص 6 مع اتجاه الـ J وهذه تلاتة X زي ما هي بيكون
+
+600
+00:40:28,190 --> 00:40:32,210
+فعلا هذا المقدار بسوء هذا المقدار اللي هي من خلال
+
+601
+00:40:32,210 --> 00:40:36,860
+اللي هو مثلنا اللي أماناالعناء دي اللي حكيناه قبل
+
+602
+00:40:36,860 --> 00:40:39,980
+شوية كله بس بدل functions of two variables خلّيني
+
+603
+00:40:39,980 --> 00:40:43,740
+أحكي functions of three variables Similarly وسهل
+
+604
+00:40:43,740 --> 00:40:50,080
+functions of three variables هي كما يعنيعندي for a
+
+605
+00:40:50,080 --> 00:40:53,220
+differentiable function f of x,y,z and a unit
+
+606
+00:40:53,220 --> 00:40:58,200
+vector u بالسوية u1 في تجارة i و u2 في تجارة j زاد
+
+607
+00:40:58,200 --> 00:41:01,020
+u3 في تجارة k إذا أن الآن الـ function عندنا بدل
+
+608
+00:41:01,020 --> 00:41:03,820
+ما هي two variables هي three و الـ vector طبعا بدل
+
+609
+00:41:03,820 --> 00:41:08,100
+ما يكون اللي هو بدلات i و j هيكون بدلات i,j و k
+
+610
+00:41:08,100 --> 00:41:11,350
+لأنه صار face basedالأن نحن لدينا الـ Gradient للـ
+
+611
+00:41:11,350 --> 00:41:14,830
+F عارفينه دل أف واي دل X في تجاه I زاد دل أف واي
+
+612
+00:41:14,830 --> 00:41:18,330
+دل Y في تجاه J زاد دل أف واي دل Z في تجاه K هذا
+
+613
+00:41:18,330 --> 00:41:21,270
+تعريفه لـ Function of Three Variables الـ
+
+614
+00:41:21,270 --> 00:41:23,810
+Directional Derivative في هذه الحالة هي Gradient
+
+615
+00:41:23,810 --> 00:41:29,260
+للـ F ضد Uدُط الـ U اللي هي عبارة عن هذا دُط ال U
+
+616
+00:41:29,260 --> 00:41:33,340
+بيصير عبارة عن دل F by دل X في U واحد زي دل F by
+
+617
+00:41:33,340 --> 00:41:37,400
+دل Y في U اتنين زي دل F by دل Z في U تلاتة إلى أن
+
+618
+00:41:37,400 --> 00:41:40,720
+نفس ما عملنا قبل بشوية الـ directional derivative
+
+619
+00:41:40,720 --> 00:41:45,440
+can once again be written in the form D U F بساوية
+
+620
+00:41:45,440 --> 00:41:48,520
+ال gradient لل F دُط ال U اللي هي بساوية المجموعة
+
+621
+00:41:48,520 --> 00:41:50,820
+للأول في المجموعة الذاتية في الـ cosine الزاوية
+
+622
+00:41:50,820 --> 00:41:53,650
+اللي بينهموالـ magnitude ده الـ U طبعا هذا بناخده
+
+623
+00:41:53,650 --> 00:41:58,410
+احنا اللي هو direction يعني unit vector بساوي واحد
+
+624
+00:41:58,410 --> 00:42:01,630
+طبعا فبصير ال gradient لف كوسيني ثتا وهذه زي ما
+
+625
+00:42:01,630 --> 00:42:05,770
+قلنا قبل اللي هو most increases rapidly لف in the
+
+626
+00:42:05,770 --> 00:42:09,610
+direction of ال gradient أو most decreases rapidly
+
+627
+00:42:09,610 --> 00:42:12,250
+in the opposite direction اللي هو في longest ال
+
+628
+00:42:12,250 --> 00:42:18,500
+gradientوالعموديات فيه اللي هي بتخلي الـ F0 الـ
+
+629
+00:42:18,500 --> 00:42:21,700
+derivative بالـ 0 نفس اللي حكيناه قبل شوية ما بديش
+
+630
+00:42:21,700 --> 00:42:26,380
+أعيده ولو نهيته يلّا نصلّع النجلة ناخد أخر مثال
+
+631
+00:42:26,380 --> 00:42:30,090
+مثال السادس find the derivativeأف اكس واي اكس
+
+632
+00:42:30,090 --> 00:42:32,470
+تكعيب نقص اكس واي تربية نقص سد عنده النقطة
+
+633
+00:42:32,470 --> 00:42:35,770
+المذكورة بي نوت واحد واحد سفر in the direction of
+
+634
+00:42:35,770 --> 00:42:39,770
+V الأمامي بعدين سألني in what directions does F
+
+635
+00:42:39,770 --> 00:42:42,890
+change most rapidly at B or not يعني بيقوللي في أي
+
+636
+00:42:42,890 --> 00:42:47,890
+direction F increases most rapidly does it change
+
+637
+00:42:47,890 --> 00:42:51,570
+most rapidly يعني سواء increases أو decreases and
+
+638
+00:42:51,570 --> 00:42:55,850
+what are the rates of change in each case ماشي طيب
+
+639
+00:42:55,850 --> 00:42:59,810
+عشان نشوفأول شي نجي للأول عشان نودد الـ Direction
+
+640
+00:42:59,810 --> 00:43:03,230
+الـ derivative لازم نودد الـ Direction في اتجاهه و
+
+641
+00:43:03,230 --> 00:43:07,090
+نودده اللي هو التفاضل الـ Direction of V in the
+
+642
+00:43:07,090 --> 00:43:10,230
+direction of V Direction of V أش بساوي اللي هو الـ
+
+643
+00:43:10,230 --> 00:43:15,710
+V على مجنتيودهالـ V هي مجموعته الجذر التربي إلى 4
+
+644
+00:43:15,710 --> 00:43:24,070
+زي 9 زي 36 يعني 7 يعني بيصير 2 على 7 في اتجاه I زي
+
+645
+00:43:24,070 --> 00:43:32,750
+3 على 7 في اتجاه K زي 6 على 7 في اتجاه Jلأن هي الـ
+
+646
+00:43:32,750 --> 00:43:35,830
+U لأنها نوجد في الـ Gradient لأن هي اللي بيوجد في
+
+647
+00:43:35,830 --> 00:43:37,890
+الـ Directional Derivative الـ Gradient و الـ
+
+648
+00:43:37,890 --> 00:43:41,650
+Direction الـ Gradient إيش بيساوي؟ بنفضل هذه
+
+649
+00:43:41,650 --> 00:43:46,110
+بالنسبة لـ X ثلاثة X تربيع ناقص Y تربيع هي تجاه
+
+650
+00:43:46,110 --> 00:43:52,070
+الـ I ناقص بنفضل ل Y اللي هي اتنين XY لحالها في
+
+651
+00:43:52,070 --> 00:43:56,430
+اتجاه الـ J ونفضل بالنسبة لZ الآن اللي هي في اتجاه
+
+652
+00:43:56,430 --> 00:44:00,870
+الـ K بتطلع ناقص واحدهي الجديدة and so الـ
+
+653
+00:44:00,870 --> 00:44:04,490
+gradient لـ F عند مين عند واحد واحد و سفر هذه
+
+654
+00:44:04,490 --> 00:44:10,530
+بتصير عبارة عن اللي هي عبارة عن تلاتة ناقص واحد
+
+655
+00:44:10,530 --> 00:44:15,490
+اللي هي اتنين اه وهذه ناقص اتنين واحد في واحد في
+
+656
+00:44:15,490 --> 00:44:20,630
+واحد ناقص اتنين وهذه ناقص اللي هي ايش K هي اللي هو
+
+657
+00:44:20,630 --> 00:44:23,810
+ال gradient لـ F عنده النقطة المذكورة اذا then the
+
+658
+00:44:23,810 --> 00:44:26,330
+derivative of F at واحد واحد و Zero in the
+
+659
+00:44:26,330 --> 00:44:30,520
+direction of V isالـ Gradient للـ F في الـ
+
+660
+00:44:30,520 --> 00:44:34,500
+Direction في الـ U يعني الـ Gradient للـ F هي اللي
+
+661
+00:44:34,500 --> 00:44:38,920
+هي الـ U وهي الـ U وهي ال Gradient للـ F هي الـ U
+
+662
+00:44:38,920 --> 00:44:43,040
+وهي ال Gradient للـ F نضربهم dot product 2 ع 7
+
+663
+00:44:43,040 --> 00:44:49,520
+بيصير 4 ع 7 نقص 3 ع 7 في نقص 2 نقص 6 ع 7 و6 ع 7 في
+
+664
+00:44:49,520 --> 00:44:54,880
+نقص 1 اللي هي اسف نقص 3 في نقص 2 بيصير زائد 6 ع 7
+
+665
+00:44:54,880 --> 00:44:59,900
+و6 ع 7 في نقص 1 نقص 6 ع 7هذه تذهب مع هذه وتظهر 4
+
+666
+00:44:59,900 --> 00:45:04,900
+على 7 اللي هي derivative of f in the direction of
+
+667
+00:45:04,900 --> 00:45:09,720
+v طيب لأ ال function increases most rapidly in the
+
+668
+00:45:09,720 --> 00:45:12,820
+direction of ال gradient صح ولا لأ يعني في اتجاه
+
+669
+00:45:12,820 --> 00:45:16,380
+هذا بتظهر اللي هو ايش increases most rapidly اللي
+
+670
+00:45:16,380 --> 00:45:19,140
+هو في اتجاه هذا اتجاهه مين؟
+
+671
+00:45:24,830 --> 00:45:28,330
+على الجدر التربيعي للأربعة زايد الأربعة زايد واحد
+
+672
+00:45:28,330 --> 00:45:31,750
+تلاتة يعني اللي هو بيصير اللي هو function
+
+673
+00:45:31,750 --> 00:45:35,230
+increases most rapidly in the direction vector
+
+674
+00:45:35,230 --> 00:45:40,470
+اللي هو U بتساوي 2 ع 3 I نقص 2 ع 3 J نقص 1 ع 3 K
+
+675
+00:45:40,470 --> 00:45:43,470
+and decreases most rapidly in the opposite
+
+676
+00:45:43,470 --> 00:45:47,080
+direction اللي هو في ناقص اللي هو gradient للـFأو
+
+677
+00:45:47,080 --> 00:45:53,640
+بمعنى آخر في اتجاه ناقص 2 على 3i زائد 2 ع 3j زائد
+
+678
+00:45:53,640 --> 00:45:57,840
+1 ع 3k هذا اللي هو decreases مستقبلي اللي هون
+
+679
+00:45:57,840 --> 00:46:01,920
+بدناه اللي هو القيمة ال directional derivative في
+
+680
+00:46:01,920 --> 00:46:05,430
+كل حالةالـ rates of change اللي هو معناته الـ
+
+681
+00:46:05,430 --> 00:46:08,930
+directional derivative in these directions are الـ
+
+682
+00:46:08,930 --> 00:46:11,270
+derivative اللي طبعا في الحالة الأولى most rapidly
+
+683
+00:46:11,270 --> 00:46:13,570
+بيكون بساوي الـ gradient نفسه اللي هو الجدر
+
+684
+00:46:13,570 --> 00:46:16,690
+التربيعي لأربعة زيادة الأربعة زيادة الواحد هي
+
+685
+00:46:16,690 --> 00:46:19,170
+الأربعة زيادة الأربعة زيادة الواحد اللي هو زي ما
+
+686
+00:46:19,170 --> 00:46:23,550
+قلنا تلاتة و اللي هي في حالة اللي هي ال decreases
+
+687
+00:46:23,550 --> 00:46:26,350
+most rapidly بتكون ال derivative في هذه الحالة
+
+688
+00:46:26,350 --> 00:46:30,120
+سالب تلاتة و بيكون هيك احنا أنهينا المحاضرةوالى
+
+689
+00:46:30,120 --> 00:46:33,780
+لقاء الآخر السلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U_raw.json b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..eeea3872b02bf84c4d0a31c3ec23fe000674d8b7
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U_raw.json
@@ -0,0 +1 @@
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rapidly لما يكونون في اتجاه مين لـ Gradient بقوله صح قال ليه أصلا دي decreases most rapidly اللي بدك تسأل عنها في التانية برضه في عكس اتجاه مين ل Gradient", "tokens": [1211, 15042, 2423, 39184, 479, 8637, 881, 12910, 43242, 2423, 14407, 24401, 4032, 36764, 24401, 50239, 32239, 36764, 24401, 15844, 23032, 12610, 23032, 3555, 16472, 2655, 12174, 1211, 2655, 12174, 36150, 13412, 2407, 10632, 16472, 16095, 7251, 30544, 11536, 881, 12910, 5296, 15042, 7251, 30544, 11536, 8978, 1975, 2655, 7435, 40294, 3714, 9957, 5296, 39184, 16710, 1196, 4724, 39648, 3224, 20328, 5016, 50239, 32239, 3224, 5551, 9381, 15040, 11778, 1829, 24108, 881, 12910, 13672, 1829, 47525, 4117, 6055, 3794, 10721, 1211, 18871, 11296, 8978, 16712, 7649, 10632, 4724, 43042, 3224, 8978, 6225, 4117, 3794, 1975, 2655, 7435, 40294, 3714, 9957, 5296, 16710, 1196], "avg_logprob": -0.3149038375570224, "compression_ratio": 1.7955555555555556, "no_speech_prob": 3.135204315185547e-05, "words": [{"start": 1219.33, "end": 1220.09, 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الورحانة مثال خمسة We illustrate two of the rules with يعني بدنا نوضح قاعدتين من اللي فاتن بوساطة ايش احنا مثلا عدد f of x,y بيساوي x نقص y و g of x,y بيساوي 3y ال gradient اللافظة ما انتوا عارفين دل f by دل x بيه i دل f by دل y هي واحد اللي هي بيصير نقص هذا واحد وهنا نقص واحد بيصير i نقص j", "tokens": [1211, 1211, 10721, 9566, 2304, 33456, 3660, 37893, 5551, 4117, 49115, 5296, 33456, 37893, 3714, 9566, 1211, 37746, 2423, 13063, 5016, 7649, 3660, 50113, 6027, 16490, 2304, 3794, 3660, 492, 23221, 732, 295, 264, 4474, 365, 37495, 22653, 47525, 8315, 8717, 2407, 11242, 5016, 12174, 995, 22488, 2655, 9957, 9154, 13672, 1829, 6156, 9307, 1863, 4724, 41779, 41193, 3660, 1975, 1829, 8592, 1975, 5016, 8315, 50113, 15040, 6225, 3215, 3215, 283, 295, 2031, 11, 88, 4724, 1829, 3794, 995, 45865, 2031, 8717, 4587, 9381, 288, 4032, 290, 295, 2031, 11, 88, 4724, 1829, 3794, 995, 45865, 805, 88, 2423, 16235, 13672, 31845, 19913, 3660, 19446, 16472, 2655, 14407, 6225, 9640, 5172, 9957, 11778, 1211, 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0.964111328125}, {"start": 2203.48, "end": 2204.2, "word": " قاعدتين", "probability": 0.95478515625}, {"start": 2204.2, "end": 2204.32, "word": " من", "probability": 0.94873046875}, {"start": 2204.32, "end": 2204.48, "word": " اللي", "probability": 0.95361328125}, {"start": 2204.48, "end": 2204.9, "word": " فاتن", "probability": 0.8621419270833334}, {"start": 2204.9, "end": 2205.8, "word": " بوساطة", "probability": 0.61846923828125}, {"start": 2205.8, "end": 2206.26, "word": " ايش", "probability": 0.6222941080729166}, {"start": 2206.26, "end": 2206.64, "word": " احنا", "probability": 0.4337565104166667}, {"start": 2206.64, "end": 2207.06, "word": " مثلا", "probability": 0.717041015625}, {"start": 2207.06, "end": 2207.46, "word": " عدد", "probability": 0.697265625}, {"start": 2207.46, "end": 2207.64, "word": " f", "probability": 0.5283203125}, {"start": 2207.64, "end": 2207.82, "word": " of", "probability": 0.346435546875}, {"start": 2207.82, "end": 2207.96, "word": " x", "probability": 0.88330078125}, {"start": 2207.96, "end": 2208.1, "word": ",y", "probability": 0.45941162109375}, {"start": 2208.1, "end": 2208.32, "word": " بيساوي", "probability": 0.55400390625}, {"start": 2208.32, "end": 2208.5, "word": " x", "probability": 0.7666015625}, {"start": 2208.5, "end": 2208.7, "word": " نقص", "probability": 0.8426106770833334}, {"start": 2208.7, "end": 2208.94, "word": " y", "probability": 0.330810546875}, {"start": 2208.94, "end": 2209.74, "word": " و", "probability": 0.6787109375}, {"start": 2209.74, "end": 2209.82, "word": " g", "probability": 0.385986328125}, {"start": 2209.82, "end": 2209.96, "word": " of", "probability": 0.9345703125}, {"start": 2209.96, "end": 2210.16, "word": " x", "probability": 0.98828125}, {"start": 2210.16, "end": 2210.34, "word": ",y", "probability": 0.963134765625}, {"start": 2210.34, "end": 2210.64, "word": " بيساوي", "probability": 0.968359375}, {"start": 2210.64, "end": 2211.3, "word": " 3y", "probability": 0.64794921875}, {"start": 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{"start": 2215.34, "end": 2215.58, "word": " i", "probability": 0.51708984375}, {"start": 2215.58, "end": 2216.26, "word": " دل", "probability": 0.92919921875}, {"start": 2216.26, "end": 2216.44, "word": " f", "probability": 0.87744140625}, {"start": 2216.44, "end": 2216.7, "word": " by", "probability": 0.96142578125}, {"start": 2216.7, "end": 2216.94, "word": " دل", "probability": 0.952880859375}, {"start": 2216.94, "end": 2217.22, "word": " y", "probability": 0.95361328125}, {"start": 2217.22, "end": 2217.52, "word": " هي", "probability": 0.391357421875}, {"start": 2217.52, "end": 2218.36, "word": " واحد", "probability": 0.895263671875}, {"start": 2218.36, "end": 2219.08, "word": " اللي", "probability": 0.75244140625}, {"start": 2219.08, "end": 2219.18, "word": " هي", "probability": 0.638671875}, {"start": 2219.18, "end": 2219.4, "word": " بيصير", "probability": 0.8060302734375}, {"start": 2219.4, "end": 2219.84, "word": " نقص", "probability": 0.6819661458333334}, {"start": 2219.84, "end": 2220.2, "word": " هذا", "probability": 0.489501953125}, {"start": 2220.2, "end": 2220.54, "word": " واحد", "probability": 0.96630859375}, {"start": 2220.54, "end": 2220.78, "word": " وهنا", "probability": 0.5927734375}, {"start": 2220.78, "end": 2221.08, "word": " نقص", "probability": 0.9822591145833334}, {"start": 2221.08, "end": 2221.42, "word": " واحد", "probability": 0.979736328125}, {"start": 2221.42, "end": 2221.68, "word": " بيصير", "probability": 0.8443603515625}, {"start": 2221.68, "end": 2221.8, "word": " i", "probability": 0.4619140625}, {"start": 2221.8, "end": 2222.06, "word": " نقص", "probability": 0.9861653645833334}, {"start": 2222.06, "end": 2222.28, "word": " j", "probability": 0.92041015625}], "temperature": 1.0}, {"id": 102, "seek": 223592, "start": 2223.75, "end": 2235.93, "text": "DEL F DEL J BY DEL X صفر إذاً Zero مافيش هنا DEL J BY DEL Y تلاتة إذاً الـ Gradient DEL F عبارة عن مين تلاتة G", "tokens": [35, 3158, 479, 413, 3158, 508, 26930, 413, 3158, 1783, 20328, 5172, 2288, 11933, 15730, 14111, 17182, 19446, 41185, 8592, 34105, 413, 3158, 508, 26930, 413, 3158, 398, 6055, 1211, 9307, 3660, 11933, 15730, 14111, 2423, 39184, 16710, 1196, 413, 3158, 479, 6225, 3555, 9640, 3660, 18871, 3714, 9957, 6055, 1211, 9307, 3660, 460], "avg_logprob": -0.42045455087314954, "compression_ratio": 1.2666666666666666, "no_speech_prob": 0.0, "words": [{"start": 2223.7500000000005, "end": 2224.4700000000003, "word": "DEL", "probability": 0.15887451171875}, {"start": 2224.4700000000003, "end": 2225.19, "word": " F", "probability": 0.256103515625}, {"start": 2225.19, "end": 2226.35, "word": " DEL", "probability": 0.5653076171875}, {"start": 2226.35, "end": 2226.69, "word": " J", "probability": 0.791015625}, {"start": 2226.69, "end": 2227.25, "word": " BY", "probability": 0.54541015625}, {"start": 2227.25, "end": 2227.67, "word": " DEL", "probability": 0.881591796875}, {"start": 2227.67, "end": 2228.03, "word": " X", "probability": 0.982421875}, {"start": 2228.03, "end": 2228.53, "word": " صفر", "probability": 0.6944173177083334}, {"start": 2228.53, "end": 2229.09, "word": " إذاً", "probability": 0.57373046875}, {"start": 2229.09, "end": 2229.47, "word": " Zero", "probability": 0.2274169921875}, {"start": 2229.47, "end": 2229.83, "word": " مافيش", "probability": 0.84912109375}, {"start": 2229.83, "end": 2230.11, "word": " هنا", "probability": 0.152099609375}, {"start": 2230.11, "end": 2231.09, "word": " DEL", "probability": 0.857177734375}, {"start": 2231.09, "end": 2231.39, "word": " J", "probability": 0.84228515625}, {"start": 2231.39, "end": 2231.59, "word": " BY", "probability": 0.97509765625}, {"start": 2231.59, "end": 2231.85, "word": " DEL", "probability": 0.9521484375}, {"start": 2231.85, "end": 2232.03, "word": " Y", "probability": 0.982421875}, {"start": 2232.03, "end": 2232.69, "word": " تلاتة", "probability": 0.765289306640625}, {"start": 2232.69, "end": 2232.97, "word": " إذاً", "probability": 0.9152018229166666}, {"start": 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اتراح الـ f ناقص g هي x ناقص y ناقص 3 وy بيصير x ناقص 4y الـ gradient لهذه اللي هي del f by del .. del هذا by del x يعني I", "tokens": [28814, 15730, 14111, 32771, 43042, 8315, 35186, 10874, 15042, 27884, 4724, 3794, 8717, 5016, 4587, 4587, 8978, 23758, 9673, 12984, 6027, 9673, 12984, 6027, 1103, 283, 3175, 290, 1975, 1829, 3714, 2407, 11242, 5016, 9122, 33911, 7251, 3794, 995, 45865, 1103, 283, 3175, 1103, 290, 16070, 39184, 16235, 24976, 39184, 283, 8717, 995, 4587, 9381, 16235, 290, 2423, 39184, 16235, 24976, 39184, 283, 3175, 290, 44945, 1829, 7435, 1829, 9154, 1975, 2655, 2288, 39319, 2423, 39184, 283, 8717, 995, 4587, 9381, 290, 39896, 2031, 8717, 995, 4587, 9381, 288, 8717, 995, 4587, 9381, 805, 4032, 88, 4724, 1829, 9381, 13546, 2031, 8717, 995, 4587, 9381, 1017, 88, 2423, 39184, 16235, 46740, 24192, 13672, 1829, 39896, 1103, 283, 538, 1103, 4386, 1103, 23758, 538, 1103, 2031, 37495, 22653, 286], "avg_logprob": -0.31613372093023256, "compression_ratio": 1.9009009009009008, 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2749.42, "end": 2750.18, "word": " على", "probability": 0.37890625}, {"start": 2750.18, "end": 2750.9, "word": " 3i", "probability": 0.596923828125}, {"start": 2750.9, "end": 2751.5, "word": " زائد", "probability": 0.7145182291666666}, {"start": 2751.5, "end": 2751.9, "word": " 2", "probability": 0.9609375}, {"start": 2751.9, "end": 2752.02, "word": " ع", "probability": 0.57763671875}, {"start": 2752.02, "end": 2752.72, "word": " 3j", "probability": 0.84375}, {"start": 2752.72, "end": 2753.64, "word": " زائد", "probability": 0.9021809895833334}, {"start": 2753.64, "end": 2753.88, "word": " 1", "probability": 0.953125}, {"start": 2753.88, "end": 2754.16, "word": " ع", "probability": 0.814453125}, {"start": 2754.16, "end": 2755.12, "word": " 3k", "probability": 0.93994140625}, {"start": 2755.12, "end": 2755.52, "word": " هذا", "probability": 0.4443359375}, {"start": 2755.52, "end": 2755.66, "word": " اللي", "probability": 0.818115234375}, {"start": 2755.66, "end": 2755.82, "word": " هو", "probability": 0.9736328125}, {"start": 2755.82, "end": 2756.28, "word": " decreases", "probability": 0.43798828125}, {"start": 2756.28, "end": 2757.0, "word": " مستقبلي", "probability": 0.6405029296875}, {"start": 2757.0, "end": 2757.66, "word": " اللي", "probability": 0.7271728515625}, {"start": 2757.66, "end": 2757.84, "word": " هون", "probability": 0.48944091796875}, {"start": 2757.84, "end": 2758.28, "word": " بدناه", "probability": 0.8723958333333334}, {"start": 2758.28, "end": 2758.54, "word": " اللي", "probability": 0.96875}, {"start": 2758.54, "end": 2758.58, "word": " هو", "probability": 0.984375}, {"start": 2758.58, "end": 2759.3, "word": " القيمة", "probability": 0.8323567708333334}, {"start": 2759.3, "end": 2759.48, "word": " ال", "probability": 0.9052734375}, {"start": 2759.48, "end": 2761.22, "word": " directional", "probability": 0.62451171875}, {"start": 2761.22, "end": 2761.72, "word": " derivative", "probability": 0.94287109375}, {"start": 2761.72, "end": 2761.92, "word": " في", "probability": 0.90234375}, {"start": 2761.92, "end": 2762.12, "word": " كل", "probability": 0.98828125}, {"start": 2762.12, "end": 2762.56, "word": " حالة", "probability": 0.99755859375}], "temperature": 1.0}, {"id": 127, "seek": 278943, "start": 2762.81, "end": 2789.43, "text": "الـ rates of change اللي هو معناته الـ directional derivative in these directions are الـ derivative اللي طبعا في الحالة الأولى most rapidly بيكون بساوي الـ gradient نفسه اللي هو الجدر التربيعي لأربعة زيادة الأربعة زيادة الواحد هي الأربعة زيادة الأربعة زيادة الواحد اللي هو زي ما قلنا تلاتة و اللي هي في حالة اللي هي ال decreases most rapidly بتكون ال derivative في هذه الحالة سالب تلاتة و بيكون هيك احنا أنهينا المحاضرة", "tokens": [6027, 39184, 6846, 295, 1319, 13672, 1829, 31439, 20449, 8315, 47395, 2423, 39184, 42242, 13760, 294, 613, 11095, 366, 2423, 39184, 13760, 13672, 1829, 23032, 3555, 3615, 995, 8978, 21542, 6027, 3660, 16247, 12610, 7578, 881, 12910, 4724, 1829, 30544, 4724, 3794, 995, 45865, 2423, 39184, 16235, 8717, 36178, 3224, 13672, 1829, 31439, 25724, 3215, 2288, 16712, 2288, 21292, 3615, 1829, 5296, 10721, 25513, 27884, 30767, 1829, 18513, 3660, 16247, 25513, 27884, 30767, 1829, 18513, 3660, 2423, 14407, 24401, 39896, 16247, 25513, 27884, 30767, 1829, 18513, 3660, 16247, 25513, 27884, 30767, 1829, 18513, 3660, 2423, 14407, 24401, 13672, 1829, 31439, 30767, 1829, 19446, 12174, 1211, 8315, 6055, 1211, 9307, 3660, 4032, 13672, 1829, 39896, 8978, 11331, 6027, 3660, 13672, 1829, 39896, 2423, 24108, 881, 12910, 39894, 30544, 2423, 13760, 8978, 29538, 21542, 6027, 3660, 8608, 6027, 3555, 6055, 1211, 9307, 3660, 4032, 4724, 1829, 30544, 39896, 4117, 1975, 5016, 8315, 14739, 3224, 1829, 8315, 9673, 5016, 46958, 25720], "avg_logprob": -0.247051883793477, "compression_ratio": 2.1872909698996654, "no_speech_prob": 2.9265880584716797e-05, "words": [{"start": 2762.81, "end": 2763.51, "word": "الـ", "probability": 0.22894287109375}, {"start": 2763.51, "end": 2763.91, "word": " rates", "probability": 0.2305908203125}, {"start": 2763.91, "end": 2764.09, "word": " of", "probability": 0.97021484375}, {"start": 2764.09, "end": 2764.47, "word": " change", "probability": 0.7822265625}, {"start": 2764.47, "end": 2764.59, "word": " اللي", "probability": 0.48834228515625}, {"start": 2764.59, "end": 2764.77, "word": " هو", "probability": 0.9169921875}, {"start": 2764.77, "end": 2765.27, "word": " معناته", "probability": 0.86767578125}, {"start": 2765.27, "end": 2765.43, "word": " الـ", "probability": 0.620361328125}, {"start": 2765.43, "end": 2765.75, "word": " directional", "probability": 0.64306640625}, {"start": 2765.75, "end": 2766.29, "word": " derivative", "probability": 0.744140625}, {"start": 2766.29, "end": 2766.87, "word": " in", "probability": 0.490478515625}, {"start": 2766.87, "end": 2767.31, "word": " these", "probability": 0.68603515625}, {"start": 2767.31, "end": 2767.89, "word": " directions", "probability": 0.74755859375}, {"start": 2767.89, "end": 2768.33, "word": " are", "probability": 0.85595703125}, {"start": 2768.33, "end": 2768.93, "word": " الـ", "probability": 0.630126953125}, {"start": 2768.93, "end": 2769.25, "word": " derivative", "probability": 0.736328125}, {"start": 2769.25, "end": 2769.45, "word": " اللي", "probability": 0.927001953125}, {"start": 2769.45, "end": 2769.77, "word": " طبعا", "probability": 0.91064453125}, {"start": 2769.77, "end": 2769.91, "word": " في", "probability": 0.791015625}, {"start": 2769.91, "end": 2770.23, "word": " الحالة", "probability": 0.947265625}, {"start": 2770.23, "end": 2770.59, "word": " الأولى", "probability": 0.9853515625}, {"start": 2770.59, "end": 2770.81, "word": " most", "probability": 0.7119140625}, {"start": 2770.81, "end": 2771.27, "word": " rapidly", "probability": 0.900390625}, {"start": 2771.27, "end": 2771.91, "word": " بيكون", "probability": 0.797607421875}, {"start": 2771.91, "end": 2772.19, "word": " بساوي", "probability": 0.64678955078125}, {"start": 2772.19, "end": 2772.35, "word": " الـ", "probability": 0.5531005859375}, {"start": 2772.35, "end": 2772.63, "word": " gradient", "probability": 0.8046875}, {"start": 2772.63, "end": 2773.13, "word": " نفسه", "probability": 0.9817708333333334}, {"start": 2773.13, "end": 2773.25, "word": " اللي", "probability": 0.962158203125}, {"start": 2773.25, "end": 2773.31, "word": " هو", "probability": 0.982421875}, {"start": 2773.31, "end": 2773.57, "word": " الجدر", "probability": 0.9080403645833334}, {"start": 2773.57, "end": 2774.07, "word": " التربيعي", "probability": 0.75087890625}, {"start": 2774.07, "end": 2775.13, "word": " لأربعة", "probability": 0.673095703125}, {"start": 2775.13, "end": 2775.45, "word": " زيادة", "probability": 0.65362548828125}, {"start": 2775.45, "end": 2775.75, "word": " الأربعة", "probability": 0.8362630208333334}, {"start": 2775.75, "end": 2776.05, "word": " زيادة", "probability": 0.920166015625}, {"start": 2776.05, "end": 2776.43, "word": " الواحد", "probability": 0.8310546875}, {"start": 2776.43, "end": 2776.69, "word": " هي", "probability": 0.87353515625}, {"start": 2776.69, "end": 2777.71, "word": " الأربعة", "probability": 0.962890625}, {"start": 2777.71, "end": 2778.03, "word": " زيادة", "probability": 0.97021484375}, {"start": 2778.03, "end": 2778.31, "word": " الأربعة", "probability": 0.9529622395833334}, {"start": 2778.31, "end": 2778.53, "word": " زيادة", "probability": 0.971923828125}, {"start": 2778.53, "end": 2778.79, "word": " الواحد", "probability": 0.9601236979166666}, {"start": 2778.79, "end": 2778.89, "word": " اللي", "probability": 0.918701171875}, {"start": 2778.89, "end": 2778.97, "word": " هو", "probability": 0.9541015625}, {"start": 2778.97, "end": 2779.11, "word": " زي", "probability": 0.6871337890625}, {"start": 2779.11, "end": 2779.17, "word": " ما", "probability": 0.982421875}, {"start": 2779.17, "end": 2779.39, "word": " قلنا", "probability": 0.9129231770833334}, {"start": 2779.39, "end": 2779.95, "word": " تلاتة", "probability": 0.9505615234375}, {"start": 2779.95, "end": 2780.55, "word": " و", "probability": 0.7197265625}, {"start": 2780.55, "end": 2780.91, "word": " اللي", "probability": 0.7369384765625}, {"start": 2780.91, "end": 2781.27, "word": " هي", "probability": 0.81201171875}, {"start": 2781.27, "end": 2781.75, "word": " في", "probability": 0.443359375}, {"start": 2781.75, "end": 2782.13, "word": " حالة", "probability": 0.9664713541666666}, {"start": 2782.13, "end": 2782.31, "word": " اللي", "probability": 0.866455078125}, {"start": 2782.31, "end": 2782.55, "word": " هي", "probability": 0.87548828125}, {"start": 2782.55, "end": 2783.01, "word": " ال", "probability": 0.5302734375}, {"start": 2783.01, "end": 2783.55, "word": " decreases", "probability": 0.61328125}, {"start": 2783.55, "end": 2783.87, "word": " most", "probability": 0.9111328125}, {"start": 2783.87, "end": 2784.29, "word": " rapidly", "probability": 0.9248046875}, {"start": 2784.29, "end": 2785.11, "word": " بتكون", "probability": 0.721923828125}, {"start": 2785.11, "end": 2785.23, "word": " ال", "probability": 0.91357421875}, {"start": 2785.23, "end": 2785.61, "word": " derivative", "probability": 0.8828125}, {"start": 2785.61, "end": 2785.89, "word": " في", "probability": 0.85693359375}, {"start": 2785.89, "end": 2786.01, "word": " هذه", "probability": 0.65185546875}, {"start": 2786.01, "end": 2786.35, "word": " الحالة", "probability": 0.9943033854166666}, {"start": 2786.35, "end": 2786.67, "word": " سالب", "probability": 0.8001302083333334}, {"start": 2786.67, "end": 2787.19, "word": " تلاتة", "probability": 0.9757080078125}, {"start": 2787.19, "end": 2787.61, "word": " و", "probability": 0.416748046875}, {"start": 2787.61, "end": 2787.91, "word": " بيكون", "probability": 0.73486328125}, {"start": 2787.91, "end": 2788.15, "word": " هيك", "probability": 0.72412109375}, {"start": 2788.15, "end": 2788.37, "word": " احنا", "probability": 0.8523763020833334}, {"start": 2788.37, "end": 2788.83, "word": " أنهينا", "probability": 0.7694091796875}, {"start": 2788.83, "end": 2789.43, "word": " المحاضرة", "probability": 0.9278564453125}], "temperature": 1.0}, {"id": 128, "seek": 279378, "start": 2789.64, "end": 2793.78, "text": "والى لقاء الآخر السلام عليكم ورحمة الله وبركاته", "tokens": [2407, 6027, 7578, 5296, 4587, 16606, 6024, 95, 34740, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224], "avg_logprob": -0.1284179693708817, "compression_ratio": 1.16, "no_speech_prob": 2.562999725341797e-06, "words": [{"start": 2789.64, "end": 2790.12, "word": "والى", "probability": 0.8248697916666666}, {"start": 2790.12, "end": 2790.48, "word": " لقاء", "probability": 0.8987630208333334}, {"start": 2790.48, "end": 2790.84, "word": " الآخر", "probability": 0.8486328125}, {"start": 2790.84, "end": 2791.24, "word": " السلام", "probability": 0.65283203125}, {"start": 2791.24, "end": 2791.7, "word": " عليكم", "probability": 0.96044921875}, {"start": 2791.7, "end": 2792.16, "word": " ورحمة", "probability": 0.975830078125}, {"start": 2792.16, "end": 2792.52, "word": " الله", "probability": 0.9697265625}, {"start": 2792.52, "end": 2793.78, "word": " وبركاته", "probability": 0.976171875}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2795.7754375, "duration_after_vad": 2776.073749999995}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U_raw.srt b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..c345d4898691e1b7d8e1e3b63f47fd2fac0d8f64
--- /dev/null
+++ b/PL9fwy3NUQKwb1CN67Ye5F4xMDfBQ6Ui65/yCJvLRECR7U_raw.srt
@@ -0,0 +1,2784 @@
+1
+00:00:00,640 --> 00:00:04,660
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:04,660 --> 00:00:10,080
+وبركاته هذه المحاضرة رقم 17 مساق تفاضل وتكامل
+
+3
+00:00:10,080 --> 00:00:13,940
+الـGEM وهي المحاضرة الخامسة في شبطر أربعة عشر وهي
+
+4
+00:00:13,940 --> 00:00:18,780
+section 14-5 المعنوان بـdirectional derivatives
+
+5
+00:00:18,780 --> 00:00:25,500
+and gradient vectors اللي هو الاشتراك
+
+6
+00:00:25,500 --> 00:00:32,630
+في اتجاه معينهو الـ Gradient Vectors اللي سنتعرف
+
+7
+00:00:32,630 --> 00:00:37,650
+عليها لاحقا نحكي أولا عن Directional Derivatives
+
+8
+00:00:37,650 --> 00:00:42,050
+in W الـ Directional Derivatives هي عبارة عن تعميم
+
+9
+00:00:42,050 --> 00:00:45,890
+للـ Partial Derivatives سواء في اتجاه الـ X أو الـ
+
+10
+00:00:45,890 --> 00:00:48,390
+Y أو الـ Z
+
+11
+00:00:51,970 --> 00:00:56,790
+أو ∆FY∆Y أو ∆FY∆Z كل حالة خاصة من حاجة اسمها
+
+12
+00:00:56,790 --> 00:00:59,830
+Directional Derivative خلّيني أتعرف الآن Plan
+
+13
+00:00:59,830 --> 00:01:04,930
+Definition The derivative of F at P0 X0 وY0 فنفترض
+
+14
+00:01:04,930 --> 00:01:07,810
+F function of two variables بس اللي هي ال X والY
+
+15
+00:01:07,810 --> 00:01:13,830
+فالعنوان للـ F عنده نقطة X0 وY0 in the direction
+
+16
+00:01:13,830 --> 00:01:19,870
+of the unit vector U بتساوي UI1 زي UII2J is the
+
+17
+00:01:19,870 --> 00:01:26,140
+numberإذاً مشتقة الـ F في اتجاه اللي هو عند النقطة
+
+18
+00:01:26,140 --> 00:01:29,720
+X نوت و Y نوت في اتجاه الـ U هذا اللي معطينيها U
+
+19
+00:01:29,720 --> 00:01:35,740
+واحد I زاد U اتنين I المشتقة لأ اللي هي ال
+
+20
+00:01:35,740 --> 00:01:42,280
+function في هذا الاتجاهتتعرف بميالي DF by DS اللي
+
+21
+00:01:42,280 --> 00:01:46,900
+هو في اتجاه الـ U عنده نقطة بي نوت بساوة limit F
+
+22
+00:01:46,900 --> 00:01:52,360
+of X note زاد S U one Y note زاد S U اتنين ناقص F
+
+23
+00:01:52,360 --> 00:01:57,650
+of X note Y note على مين على Sإن التغيير في الـ
+
+24
+00:01:57,650 --> 00:02:00,970
+XNOTE و YNOTE لما تنتقل XNOTE و YNOTE التغيير بس
+
+25
+00:02:00,970 --> 00:02:06,850
+وين مسموح فيه هنا في اتجاه الـ U يعني XNOTE هيكون
+
+26
+00:02:06,850 --> 00:02:11,690
+تغيير هي YNOTE في هذا الاتجاه فبصير XNOTE زاد أس و
+
+27
+00:02:11,690 --> 00:02:16,890
+U و U1 إذا نان اتحكمتالتي هي مقدار الزيادة اللي في
+
+28
+00:02:16,890 --> 00:02:20,410
+الـ X note تتحكم فيها اللي هي المقدار العددي S
+
+29
+00:02:20,410 --> 00:02:25,630
+والتي هو الاتجاه اللي هو U1 في اتجاه I والـ Y note
+
+30
+00:02:25,630 --> 00:02:29,330
+اللي هو S زائد U2 بنفس الأسلوب S اللي بتمثل ال
+
+31
+00:02:29,330 --> 00:02:33,410
+increment العددي والـ U2 اللي هي الاتجاه اللي صارت
+
+32
+00:02:33,410 --> 00:02:37,350
+في التغيير يعني بمعنى آخر التغييرصار على وين في
+
+33
+00:02:37,350 --> 00:02:40,730
+هذا الاتجاه الـ Derivative في هذه الحالة المعرفة
+
+34
+00:02:40,730 --> 00:02:44,690
+هذه ناقص f of x not y not على s لما s تروح ل0 هي
+
+35
+00:02:44,690 --> 00:02:48,850
+بالظبط حاجة بنسميها The directional derivative of
+
+36
+00:02:48,850 --> 00:02:53,970
+f in the direction of uبنوت أو الـ derivative of f
+
+37
+00:02:53,970 --> 00:02:59,470
+in the direction of u at بنوت خلّينا الآن ناخد
+
+38
+00:02:59,470 --> 00:03:04,450
+اللي هو عندي مثال عندي x طبعا قبل ما ناخد المثال
+
+39
+00:03:04,450 --> 00:03:07,850
+الـ x زي ما قلنا على هذا الخط بساوي x note
+
+40
+00:03:11,140 --> 00:03:13,680
+هذه اللي بنحكي فيها هو الـ X بسهولة X Note Z
+
+41
+00:03:13,680 --> 00:03:17,560
+أسواية Y1 و Y Y سواية Note أسواية Y2 اللي عبارة عن
+
+42
+00:03:17,560 --> 00:03:21,100
+إيش الخط اللي هو المستقيم هنا الـ X و الـ Y تتحرك
+
+43
+00:03:21,100 --> 00:03:25,760
+بوافق إيش المعادلة الأمامية هذه اللي أمامي اللي
+
+44
+00:03:25,760 --> 00:03:29,960
+صار فيها التغيير X Note زايد أسواية Y1 زايد أسواية
+
+45
+00:03:29,960 --> 00:03:33,740
+Y2 على إيش في هذا الاتجاه خلّيني أخد مثال عددي يا
+
+46
+00:03:33,740 --> 00:03:37,440
+جماعة نشوف إيش اللي بنحكي بالنسبة له اللي هي هذه
+
+47
+00:03:37,440 --> 00:03:40,400
+ال directional derivativeالدريكشنال الدريفتيف
+
+48
+00:03:40,400 --> 00:03:42,520
+اللذي تم تسميه بـEquation 1 هو أيضًا تسميه
+
+49
+00:03:42,520 --> 00:03:46,560
+بالدريكشنال الدريفتيف للـ F أو الدريفتف للـ F في
+
+50
+00:03:46,560 --> 00:03:50,400
+الاتصال بـ U في الاتصال بـ B0 الدريفتف للـ F في
+
+51
+00:03:50,400 --> 00:03:53,760
+الاتصال بـ B0 في الاتصال بـ U سوف نستخدم التصميم،
+
+52
+00:03:53,760 --> 00:03:56,100
+طبعًا التصميم اللذي رأيته هو مشابه للتصميم
+
+53
+00:03:56,100 --> 00:03:59,160
+للدريفتف التابعة زمان ولكن هناك كانت مفعولة One
+
+54
+00:03:59,160 --> 00:04:03,930
+Variableوشابه للـ partial derivative ولكن كانت
+
+55
+00:04:03,930 --> 00:04:08,070
+اللي هي زمان one variable وفي حالة ال partial
+
+56
+00:04:08,070 --> 00:04:11,470
+derivative التغيير بس كان في اتجاه X أو في اتجاه Y
+
+57
+00:04:11,470 --> 00:04:14,770
+لحالها using the definition find the derivative of
+
+58
+00:04:14,770 --> 00:04:19,830
+F of XY بساوي X تربيع زي X في Y at main نقطة واحد
+
+59
+00:04:19,830 --> 00:04:22,350
+واتنين in the direction of the unit vector main
+
+60
+00:04:22,350 --> 00:04:24,650
+اللي ماعطينا ي واحدة الجدر اتنين في اتجاه I زي
+
+61
+00:04:24,650 --> 00:04:29,140
+واحدة الجدر اتنين في اتجاه Gخلّينا ننحل الـ
+
+62
+00:04:29,140 --> 00:04:31,960
+Directional Derivative للـ F أو Derivative للـ F
+
+63
+00:04:31,960 --> 00:04:34,180
+الـ Direction of U عند نقطة واحد و اتنين بساوي
+
+64
+00:04:34,180 --> 00:04:37,540
+لمين هذا التعريف اللي حكيناه قبل شهية و يساوي لمين
+
+65
+00:04:37,540 --> 00:04:40,560
+لأن الـ X Note و الـ Y Note اللي يمين واحد و اتنين
+
+66
+00:04:40,560 --> 00:04:44,220
+إذا مكان الـ X Note بدنا نحط إيه؟ واحد الـ S زي ما
+
+67
+00:04:44,220 --> 00:04:47,800
+هي ده المتغير الـ Increment الزيادة في اتجاه مين؟
+
+68
+00:04:47,800 --> 00:04:50,480
+U واحد U واحد إيه بتساويها الـ U واحد على جدر
+
+69
+00:04:50,480 --> 00:04:54,940
+التنين إذا صارت S على جدر التنين و الـ Y Note اللي
+
+70
+00:04:54,940 --> 00:04:59,890
+هي برضه جداش اتنينالي اتنين زائد الان S في FU2
+
+71
+00:04:59,890 --> 00:05:04,190
+اللي هي الـ C المتغير U2EH اللي هي واحد على EH جدر
+
+72
+00:05:04,190 --> 00:05:07,770
+التنين اللي صارت هذه الـ function F of هذا المقدار
+
+73
+00:05:07,770 --> 00:05:12,230
+ناقص F of واحد و اتنين على EH الـ S اللي هان بنحسب
+
+74
+00:05:12,230 --> 00:05:15,550
+هذه الـ F يعني بنحسب الـ function هذه اللي أمامنا
+
+75
+00:05:15,550 --> 00:05:20,670
+عنده نقطة X بتسوء واحد على S زائد S على جدر التنين
+
+76
+00:05:21,230 --> 00:05:25,410
+والـ Y عندنا نقطة 2 زي الـ S على جدر الـ 2 بنعوض
+
+77
+00:05:25,410 --> 00:05:28,690
+بيصير اللي هي الأولى تربيحها هي هذه الـ X اللي هي
+
+78
+00:05:28,690 --> 00:05:32,850
+هذه تربيحها هي زائد اللي هي هذه الـ X وهذه الـ Y
+
+79
+00:05:32,850 --> 00:05:38,690
+اللي عندنا هيهم ناقص اللي هي main اللي هي F of 1 و
+
+80
+00:05:38,690 --> 00:05:43,790
+2 F of 1 و 2 هي عبارة عن 1 زائد اللي هي 2 اللي هي
+
+81
+00:05:43,790 --> 00:05:49,130
+عوضنا عن X بـ 1 و Y بـ 2 طالع عندنا 1 زائد 2 على S
+
+82
+00:05:49,390 --> 00:05:53,410
+لأن هذا المقدر يتم ربعه بطلع هذا المقدر بطلع هذا
+
+83
+00:05:53,410 --> 00:05:56,650
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+84
+00:05:56,650 --> 00:05:58,810
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+85
+00:05:58,810 --> 00:05:58,810
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+86
+00:05:58,810 --> 00:05:59,390
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+87
+00:05:59,390 --> 00:06:01,630
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+88
+00:06:01,630 --> 00:06:02,450
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+89
+00:06:02,450 --> 00:06:04,950
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+90
+00:06:04,950 --> 00:06:05,670
+المقدر بطلع هذا المقدر بطلع هذا المقدر بطلع هذا
+
+91
+00:06:05,670 --> 00:06:10,680
+المقدر بطلعوهذا في هذا اللي هي اتنين S على جدر
+
+92
+00:06:10,680 --> 00:06:13,920
+التانين واحد S على جدر التانين يعني تلاتة S على
+
+93
+00:06:13,920 --> 00:06:17,020
+جدر التانين ناقص المقدار هذا اللي هو تلاتة لكل
+
+94
+00:06:17,020 --> 00:06:20,720
+مجسم على مين يا جماعة على S بتنصح لهذا المقدار بقى
+
+95
+00:06:20,720 --> 00:06:24,580
+علينا لغير ان نبسطه بنجمح هدول لبعض بيطلع عندي
+
+96
+00:06:24,580 --> 00:06:28,140
+المقدار هذا طبعا سهل لكم تجمعوه لبعض خمسة S على
+
+97
+00:06:28,140 --> 00:06:31,520
+جدر التانين زائد S تربيع على مين على S لان واي في
+
+98
+00:06:31,520 --> 00:06:33,800
+عندي اتنين واحد تلاتة بتروح مع التلاتة هذا بيظل
+
+99
+00:06:33,800 --> 00:06:37,790
+اللي هو معامل S تربيع اللي هو واحدإلا إنه نص و نص
+
+100
+00:06:37,790 --> 00:06:41,210
+بيصير واحد وهذه ثلاثة و اتنين بيصير خمسة على اس
+
+101
+00:06:41,210 --> 00:06:44,310
+بنختصر لأن الاس مع الاس زي زمان مع الاس اللي هان
+
+102
+00:06:44,310 --> 00:06:47,170
+والاس اللي هان بيطلع limit خمسة على جدر التنين زي
+
+103
+00:06:47,170 --> 00:06:50,330
+الاس اس اس as s goes to zero بيطلع خمسة على جدر
+
+104
+00:06:50,330 --> 00:06:53,510
+التنين إذن ال direction ال derivative of f in the
+
+105
+00:06:53,510 --> 00:06:56,430
+direction of u at the point واحد و اتنين طلع عبارة
+
+106
+00:06:56,430 --> 00:07:00,770
+عن إيش خمسة على اتنين نيجي الآن ناخد دقيقة خليني
+
+107
+00:07:00,770 --> 00:07:03,350
+أقول معنى هندسي أكتر لإيه ال direction ال
+
+108
+00:07:03,350 --> 00:07:06,370
+derivative اللي حكينا عنها قبل شويةInterpretation
+
+109
+00:07:06,370 --> 00:07:08,950
+of the Directional Derivative خلينا نشوف اللي هي
+
+110
+00:07:08,950 --> 00:07:11,590
+إيش المعنى الهندسي للـ Directional Derivative و
+
+111
+00:07:11,590 --> 00:07:14,790
+أنا حكيته قبله شوية عموان شوف الصلاة والنبي عليه
+
+112
+00:07:14,790 --> 00:07:17,410
+الصلاة والسلام زي ما عملنا في ال partial
+
+113
+00:07:17,410 --> 00:07:20,510
+derivative قلنا إيش معنات اللي هو ال directional
+
+114
+00:07:20,510 --> 00:07:24,190
+derivative اللي قاسف اللي هو ال partial derivative
+
+115
+00:07:25,240 --> 00:07:29,280
+هي عبارة عن ميل المماثلة المنحنة اللي ناتج من
+
+116
+00:07:29,280 --> 00:07:34,020
+تقاطع اللي هو X بساوي X knot مع اللي هو ال surface
+
+117
+00:07:34,020 --> 00:07:38,160
+بيطلع المماثلة المنحنة اللي بيطلع هو عبارة عن Del
+
+118
+00:07:38,160 --> 00:07:41,740
+F
+
+119
+00:07:41,740 --> 00:07:44,640
+by Del Y ونفس الاشي بالنسبة ل X خلينا نشوف ده إيش
+
+120
+00:07:44,640 --> 00:07:48,870
+اللي بقولهلحظة انا كنت نسيت الان باجي الان انا
+
+121
+00:07:48,870 --> 00:07:52,970
+عندي ايش غرضي ال equation z بيساوي f of x y يربسنت
+
+122
+00:07:52,970 --> 00:07:55,830
+ال surface as in space هاي هذا اللي بالازرق هو
+
+123
+00:07:55,830 --> 00:08:00,490
+عبارة عن ال surface z بيساوي f of x y الان اذا كان
+
+124
+00:08:00,490 --> 00:08:04,110
+z not بيساوي f of x not و y not هذه اللي هي النقطة
+
+125
+00:08:04,110 --> 00:08:07,270
+هي عبارة عن النقطة اللي بدنا نوجد عندها اللي هو
+
+126
+00:08:07,270 --> 00:08:13,020
+directional derivativeوالنقطة الـ XNOTE LINE على
+
+127
+00:08:13,020 --> 00:08:17,820
+الـ Surface S الان بدي اوجد الـ Directional
+
+128
+00:08:17,820 --> 00:08:22,900
+Derivative of F at النقطة XNOTE وYNOTE وZNOTE
+
+129
+00:08:22,900 --> 00:08:27,860
+XNOTE وYNOTE في العادة ومين عنده ZNOTE ثابت في الـ
+
+130
+00:08:27,860 --> 00:08:30,660
+Direction في اتجاه مين في اتجاه الـ U هي الـ U
+
+131
+00:08:30,660 --> 00:08:36,160
+الان ايش هو بالظبط معناه الهندسي الان هذا معناه ان
+
+132
+00:08:36,160 --> 00:08:41,310
+انا باجي اللي هو باخد الـ Planeاللي بتكون من أو
+
+133
+00:08:41,310 --> 00:08:44,190
+اللي بيحدده اللي هو الـ الـ الـ الـ الـ الـ الـ
+
+134
+00:08:44,190 --> 00:08:44,230
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+135
+00:08:44,230 --> 00:08:44,290
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+136
+00:08:44,290 --> 00:08:44,790
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+137
+00:08:44,790 --> 00:08:46,790
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+138
+00:08:46,790 --> 00:08:47,970
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+139
+00:08:47,970 --> 00:08:56,210
+الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+140
+00:08:56,210 --> 00:09:08,350
+الـ الـ الـ الـ الـ الـ الـ الـ
+
+141
+00:09:08,350 --> 00:09:08,350
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+142
+00:09:08,350 --> 00:09:09,990
+الـ الـ الـ الـ الـ الـ الـ الـفى اتجاه مين فى
+
+143
+00:09:09,990 --> 00:09:14,950
+اتجاه اللى هو الـ U هذا الان البلين المتكون من
+
+144
+00:09:14,950 --> 00:09:17,990
+اللى هو الـ U اللى بدأ اخد الـ derivative فى
+
+145
+00:09:17,990 --> 00:09:22,210
+اتجاهه ونقطة X Note وY Note هذا البلين هيقاطع من
+
+146
+00:09:22,210 --> 00:09:28,910
+الـ Surface فى الخط الأزرق الآن ميل المماس لهذا
+
+147
+00:09:28,910 --> 00:09:32,830
+اللى هو خط التقاطع ميل المماس هذاالتي هو الـ
+
+148
+00:09:32,830 --> 00:09:37,290
+Tangent هذا عند النقطة هذا ميله هو عبارة عن الـ
+
+149
+00:09:37,290 --> 00:09:41,590
+Directional Derivative للـ Function F عند النقطة X
+
+150
+00:09:41,590 --> 00:09:46,370
+Note وY Note وZ Note في اتجاه مين؟ في اتجاه الـ U
+
+151
+00:09:46,370 --> 00:09:49,310
+في النهاية إذن برضه الـ Directional Derivative هي
+
+152
+00:09:49,310 --> 00:09:55,270
+عبارة عن ميل مماس ميل مماس لمين؟ للـ Vector للـ
+
+153
+00:09:55,270 --> 00:10:02,060
+Curveعند النقطة X0 وY0 وZ0 الناتج من تقاطع الـ
+
+154
+00:10:02,060 --> 00:10:06,600
+Surface مع الـ Plane المكوّن من اللي هو الـ Vector
+
+155
+00:10:06,600 --> 00:10:11,320
+اللي بدأ فاضل في اتجاهه مع النقطة X0 وY0 الآن لو
+
+156
+00:10:11,320 --> 00:10:16,610
+كان هذاالـ vector اللي هو U بساوي I واحد زائد I
+
+157
+00:10:16,610 --> 00:10:20,850
+اتنين لو كانت U اتنين سفر و U واحد يعني الـ
+
+158
+00:10:20,850 --> 00:10:24,410
+directional derivative بس U بساوي U واحد في اتجاه
+
+159
+00:10:24,410 --> 00:10:28,590
+الـ I يعني بس I يعني الـ U في اتجاه الـ I يعني انا
+
+160
+00:10:28,590 --> 00:10:30,970
+بتاخد ال directional derivative في اتجاه مين الـ I
+
+161
+00:10:30,970 --> 00:10:34,620
+بكون لما اخد في اتجاه الـ I معناته انه اناالـ
+
+162
+00:10:34,620 --> 00:10:37,940
+derivative هيها في اتجاه الـ I هي هذا في اتجاه هذا
+
+163
+00:10:37,940 --> 00:10:42,840
+من هنا يعني ايش عملت في الـ Y ثبت الـ Y فبصير عندي
+
+164
+00:10:42,840 --> 00:10:46,840
+اللي هي الـ partial derivative with respect to
+
+165
+00:10:46,840 --> 00:10:51,700
+مين؟ to X لأنه بتثبت Y فش Y صار بس التغيير في مين؟
+
+166
+00:10:51,700 --> 00:10:55,100
+في اتجاه الـ I لأن إذا الـ directional derivative
+
+167
+00:10:55,100 --> 00:10:58,480
+U اتجاه الـ I is the directional derivative at B0
+
+168
+00:10:58,480 --> 00:11:04,890
+is دل FY دل X evaluated at X0 وY0U بالساوية جيه
+
+169
+00:11:04,890 --> 00:11:08,790
+الـ directional derivative at P0 بتصير دي الـ FY
+
+170
+00:11:08,790 --> 00:11:12,510
+دي الـ Y لأنه اللي هو بس التغيير في اتجاه الجيه
+
+171
+00:11:12,510 --> 00:11:16,790
+وليس التغيير في اتجاه الـ X معناته X ثابتة إذن هذه
+
+172
+00:11:16,790 --> 00:11:19,910
+الـ U اللي هي الـ partial derivative with respect
+
+173
+00:11:19,910 --> 00:11:24,190
+to Y أو with respect to X هي عبارة عن حالات خاصة
+
+174
+00:11:24,190 --> 00:11:27,880
+من الـ directional derivative بصورة عامةالأولى الـ
+
+175
+00:11:27,880 --> 00:11:30,640
+derivative في اتجاه الـ U الآن هذه الـ derivative
+
+176
+00:11:30,640 --> 00:11:34,040
+في اتجاه الـ U اللى هى I يعني اتجاه اللى هى ال X
+
+177
+00:11:34,040 --> 00:11:37,580
+axis يعني بمعنى أخر اللى هى Del F by Del X هتطلع
+
+178
+00:11:37,580 --> 00:11:40,340
+عندى و ال directional derivative في اتجاه ال J
+
+179
+00:11:40,340 --> 00:11:44,900
+اللى هى ال derivative اللى هى بتتثبت X X و يكون
+
+180
+00:11:44,900 --> 00:11:50,080
+التغيير بس في اتجاه Y اللى هى D F by D Y هذه معنى
+
+181
+00:11:50,080 --> 00:11:53,160
+الهندسي لل directional derivativeإلا أننا لا نريد
+
+182
+00:11:53,160 --> 00:11:56,280
+أن نتغلب في إيجاد الـ Directional Derivative كذا
+
+183
+00:11:56,280 --> 00:12:00,280
+سنبقى لوجدها بواسطة التعريف فالأمثلة قبل شوية سهل
+
+184
+00:12:00,280 --> 00:12:06,020
+لكن قد تكون أمثلة تتعجزن وعملنا زمان حصلنا على
+
+185
+00:12:06,020 --> 00:12:10,240
+آليات وقوانين لإيجاد الـ Derivative العادية والآن
+
+186
+00:12:10,240 --> 00:12:14,420
+سنحصل على قوانين إن شاء الله سهلة في إيجاد الـ
+
+187
+00:12:14,420 --> 00:12:18,000
+Directional Derivative وعشان نصل لهذه المرحلة نعرف
+
+188
+00:12:18,000 --> 00:12:23,850
+حاجة اسمة The Gradient Vector أو الـ Gradientالـ
+
+189
+00:12:23,850 --> 00:12:26,870
+Gradient vector of f of x, y at the point b0 x0,
+
+190
+00:12:27,010 --> 00:12:30,990
+y0 is the vector نريد أن نعرفه كالتالي يعني الـ
+
+191
+00:12:30,990 --> 00:12:33,510
+Gradient للـ f في رمزه والـ Gradient للـ f هو
+
+192
+00:12:33,510 --> 00:12:37,470
+عبارة عن del f by del x في اتجاه الـ I يعني تفاضة
+
+193
+00:12:37,470 --> 00:12:41,050
+الـ f بالنسبة للـ x في اتجاه الـ I زي الـ del f by
+
+194
+00:12:41,050 --> 00:12:45,750
+del y في اتجاه الـ j obtained by evaluating the
+
+195
+00:12:45,750 --> 00:12:50,470
+partial derivatives of f at b0 إذن هذه عبارة عن
+
+196
+00:12:50,470 --> 00:12:56,200
+الـ Gradient لمينللـ Effect عريفة لأن لحسن حظنا و
+
+197
+00:12:56,200 --> 00:13:01,220
+لتناسق الخلق عندي النظرية التالية اللي هتسهلنا
+
+198
+00:13:01,220 --> 00:13:05,200
+عملية إيجاد الـ Directional Derivative بواسطة اللي
+
+199
+00:13:05,200 --> 00:13:09,680
+هو بدلالة من الـ Gradient طبعا جرهنها مش مقلوب if
+
+200
+00:13:09,680 --> 00:13:13,760
+F of XY is differentiable in an open region
+
+201
+00:13:13,760 --> 00:13:16,920
+containing B node لو فرضنا أن F of XY قابل
+
+202
+00:13:16,920 --> 00:13:21,420
+الاشتراك في المنطقةفي منطقة حوالينا الـ B0 ثم DF
+
+203
+00:13:21,420 --> 00:13:25,360
+by DS اللي هو الـ derivative للـ F في اتجاه الـ U
+
+204
+00:13:25,360 --> 00:13:29,880
+عند النقطة B0 بساوي الـ Gradient للـ F عند الـ B0
+
+205
+00:13:30,910 --> 00:13:34,550
+.main ضد الـ U اللي هو الـ unit vector اللي بدنا
+
+206
+00:13:34,550 --> 00:13:38,410
+نفاضل في اتجاهه إذا موضوع ده إيجاد ال derivative
+
+207
+00:13:38,410 --> 00:13:43,470
+سهل هو إيجاد ال partial derivatives Fy X و بنوجدها
+
+208
+00:13:43,470 --> 00:13:47,150
+بالشكل اللي هذي و بنعوض عن P0 و بعدين بنعملها
+
+209
+00:13:47,150 --> 00:13:51,300
+.product مع U بطلع عندنا ال directionalدريفاتيف
+
+210
+00:13:51,300 --> 00:13:53,440
+إذاً صار اللي هو عبارة عن dot product الـ
+
+211
+00:13:53,440 --> 00:13:56,660
+differential الـ directional derivative لماذا؟
+
+212
+00:13:56,660 --> 00:14:01,560
+لدريفاتيف للـ gradient عنده نقطة بي نوت مضروب dot
+
+213
+00:14:01,560 --> 00:14:05,160
+product فيه خليني أشوف مثلًا حتى أقول الأمر سهل
+
+214
+00:14:06,110 --> 00:14:10,090
+سنجد الـ derivative of f of x,y بسوء x في y أي
+
+215
+00:14:10,090 --> 00:14:13,490
+زائد كوسيني الـ x في y at the point 2,0 in the
+
+216
+00:14:13,490 --> 00:14:18,250
+direction of v بسوء 3I ناقص 4G خلّيني أوجد الـ
+
+217
+00:14:18,250 --> 00:14:21,090
+derivative هذي كيف نوجدها؟ بالطريقة اللي أمامنا
+
+218
+00:14:21,090 --> 00:14:24,050
+خلّيني نشوف يا جماعة أول حاجة بدي أوجده اللي هو
+
+219
+00:14:24,050 --> 00:14:27,850
+الـ direction اللي هي الـU الـU طبعا احنا مجهلينه
+
+220
+00:14:27,850 --> 00:14:30,990
+في الـdirection هذا عشان اوجد لازم هذه تكون unit
+
+221
+00:14:30,990 --> 00:14:35,000
+vector الـU هيك التعريق، هيك النظريةإذاً هذه بدأ
+
+222
+00:14:35,000 --> 00:14:37,640
+أشوف الـ Direction تبعها الـ Direction of V is the
+
+223
+00:14:37,640 --> 00:14:41,800
+unit vector obtaining Y اللي هو V على ال magnitude
+
+224
+00:14:41,800 --> 00:14:44,800
+لل V ال magnitude لل V زي ما أنتوا عارفين جدر
+
+225
+00:14:44,800 --> 00:14:47,120
+التربيعي على تلاتة تربيع زي أربعة تربيع اللي هي
+
+226
+00:14:47,120 --> 00:14:51,400
+ياش تسعة زي ستاشت تلاتة عليها في تجاه I اذا ناقص
+
+227
+00:14:51,400 --> 00:14:55,970
+أربعة على الجدر التربيعي هذا في تجاه Jإذا الـ
+
+228
+00:14:55,970 --> 00:15:00,990
+magnitude للـ V هي U ويساوي هذا الخمسة و هذا
+
+229
+00:15:00,990 --> 00:15:04,410
+الخمسة يصبح ثلاثة على خمسة I نقص أربعة على خمسة J
+
+230
+00:15:04,410 --> 00:15:08,570
+هذا الـ direction اللي هو اللي بدنا نعمل في اتجاهه
+
+231
+00:15:08,570 --> 00:15:13,230
+الـ derivative الـ gradient لها في سهل هي عبارة عن
+
+232
+00:15:13,230 --> 00:15:17,170
+X أول شيء الفاضل هذا بالنسبة ل X يصبح E to the Y
+
+233
+00:15:17,770 --> 00:15:20,570
+الأنفاض لهذه بالنسبة لـ X بيصير اللي هي ناقص Sine
+
+234
+00:15:20,570 --> 00:15:23,850
+X في Y فتفاضل اللي جوا إيه الـ X في Y تفاضلها
+
+235
+00:15:23,850 --> 00:15:26,970
+بالنسبة لـ X اللي هي Y فبيصير E to the Y ناقص Y
+
+236
+00:15:26,970 --> 00:15:30,850
+Sine XY في اتجاه الـ I زاد الـ derivative لهذه
+
+237
+00:15:30,850 --> 00:15:34,930
+with respect to Y يعني دل Y اللي هي عبارة عن X E
+
+238
+00:15:34,930 --> 00:15:38,270
+to the Y ناقص التفاضل لهذه بالنسبة لـ Y دي هي ناقص
+
+239
+00:15:38,270 --> 00:15:43,370
+Sine X في Y فتفاضل جوا X ناقص X Sine Y في اتجاه
+
+240
+00:15:43,370 --> 00:15:52,200
+مين؟بعد ذلك نحسب الـ Gradient هذا عند نقطة 2 و 0
+
+241
+00:15:52,200 --> 00:16:02,180
+عوض عن I بـ 0 و X بـ 2 بيصير هذا عبارة عن Z1
+
+242
+00:16:02,180 --> 00:16:10,550
+وهذا سفر كله بيصير Iاللي هي 2 في 1 ايه ده 01 بيصير
+
+243
+00:16:10,550 --> 00:16:16,610
+2 وها دي صفر طبعا لإنه sin 00 بيصير 2J صار عندي ال
+
+244
+00:16:16,610 --> 00:16:19,770
+gradient اللي أفر عنده النقطة المذكورة بيساوي I
+
+245
+00:16:19,770 --> 00:16:24,280
+زيدي 2Jاللي هي الـ Gradient جاهز والـ U جاهزة يعني
+
+246
+00:16:24,280 --> 00:16:27,660
+لغير أن نطبق القانون الـ Gradient ضد الـ U بيصير
+
+247
+00:16:27,660 --> 00:16:32,340
+الـ Derivative بساوي اللي هو الـ Gradient I زاد 2
+
+248
+00:16:32,340 --> 00:16:38,260
+J ضد الـ U اللي هو عبارة عن 3 على 5 ناقص 8 على 5
+
+249
+00:16:38,260 --> 00:16:40,960
+ويساوي سالب 1 طبعا بتعرفوا كيف تضربوا الدكتور
+
+250
+00:16:40,960 --> 00:16:44,110
+دكتادي في هذه ال ..التي تقريبا تقريبا تقريبا
+
+251
+00:16:44,110 --> 00:16:47,490
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+252
+00:16:47,490 --> 00:16:53,210
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+253
+00:16:53,210 --> 00:16:54,970
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+254
+00:16:54,970 --> 00:16:55,690
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+255
+00:16:55,690 --> 00:16:55,690
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا
+
+256
+00:16:55,690 --> 00:17:01,310
+تقريبا تقريبا تقريبا تقريبا تقريبا تقريبا تقزي ما
+
+257
+00:17:01,310 --> 00:17:03,890
+قلنا إيش تعريفها عبارة عن الـ gradient للـ F من
+
+258
+00:17:03,890 --> 00:17:06,570
+dot U اللي ييجي الآن هذه الوجود قيمتها اللي نعرف
+
+259
+00:17:06,570 --> 00:17:11,070
+أن وجودها الـ magnitude للـ gradient للـ F في
+
+260
+00:17:11,070 --> 00:17:14,770
+cosine الزاوية بين اليمين وcos theta لو حد سأل
+
+261
+00:17:14,770 --> 00:17:19,270
+سؤال وقت إيش هذه اللي هي الـ F increases most
+
+262
+00:17:19,270 --> 00:17:22,150
+rapidly يعني معدل التغير هذا اللي هو derivative
+
+263
+00:17:22,150 --> 00:17:26,350
+تكون أكثر ما يمكن اللي هو زيادةتكون أكبر ما يمكن
+
+264
+00:17:26,350 --> 00:17:30,170
+لأن cos θ بين السفر بين نقص واحد والواحد تكون أكبر
+
+265
+00:17:30,170 --> 00:17:35,530
+ما يمكن لما cos θ واحد يعني لما θاش متساوي سفر إذا
+
+266
+00:17:35,530 --> 00:17:39,730
+اللي هو the function F increasing increases most
+
+267
+00:17:39,730 --> 00:17:44,220
+rapidly when cos θ واحد or when θ متساوي سفرفي هذه
+
+268
+00:17:44,220 --> 00:17:50,980
+الحالة U ماذا يكون؟ U مادام
+
+269
+00:17:50,980 --> 00:17:54,380
+الزاوية بين الـ Gradient و U مادام الزاوية بين الـ
+
+270
+00:17:54,380 --> 00:17:54,660
+Gradient و U مادام الزاوية بين الـ Gradient و U
+
+271
+00:17:54,660 --> 00:17:57,720
+مادام الزاوية بين الـ Gradient و U مادام الزاوية
+
+272
+00:17:57,720 --> 00:17:58,740
+بين الـ Gradient و U مادام الزاوية بين الـ
+
+273
+00:17:58,740 --> 00:17:59,280
+Gradient و U مادام الزاوية بين الـ Gradient و U
+
+274
+00:17:59,280 --> 00:18:00,140
+مادام الزاوية بين الـ Gradient و U مادام الزاوية
+
+275
+00:18:00,140 --> 00:18:01,160
+بين الـ Gradient و U مادام الزاوية بين الـ
+
+276
+00:18:01,160 --> 00:18:01,240
+Gradient و U مادام الزاوية بين الـ Gradient و U
+
+277
+00:18:01,240 --> 00:18:01,240
+مادام الزاوية بين الـ Gradient و U مادام الزاوية
+
+278
+00:18:01,240 --> 00:18:03,480
+بين الـ Gradient و U مادام الزاوية بين الـ
+
+279
+00:18:03,480 --> 00:18:07,350
+Gradient و U ميعني that is at each point P in its
+
+280
+00:18:07,350 --> 00:18:10,750
+domain F increases most rapidly in the direction
+
+281
+00:18:10,750 --> 00:18:14,790
+of the gradient للـ F at P الـ derivative in this
+
+282
+00:18:14,790 --> 00:18:18,930
+case طبعاً in this direction لما يكون الاتجاه اللي
+
+283
+00:18:18,930 --> 00:18:21,750
+هو ال gradient والاتجاه اللي هو ال U في نفس
+
+284
+00:18:21,750 --> 00:18:24,850
+الاتجاه يعني Cos Zero يعني بيساوي ال absolute
+
+285
+00:18:24,850 --> 00:18:28,430
+value لل gradient لل F أو ال magnitude لل gradient
+
+286
+00:18:28,430 --> 00:18:32,430
+لل Fإذاً الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+287
+00:18:32,430 --> 00:18:32,590
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+288
+00:18:32,590 --> 00:18:33,690
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+289
+00:18:33,690 --> 00:18:35,190
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+290
+00:18:35,190 --> 00:18:37,710
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+291
+00:18:37,710 --> 00:18:44,450
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+292
+00:18:44,450 --> 00:18:45,470
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+293
+00:18:45,470 --> 00:18:45,710
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+294
+00:18:45,710 --> 00:18:47,350
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+295
+00:18:47,350 --> 00:18:53,810
+الـ الـ الـ الـ اليعني بمعنى أخر عندما يكون
+
+296
+00:18:53,810 --> 00:18:58,090
+الاتنين في عكس اتجاه بعض يعني عندما تكون F يزداد
+
+297
+00:18:58,090 --> 00:19:04,830
+أكثر بسرعة في اتجاه X عكس اتجاه لـ Gradient والـ
+
+298
+00:19:04,830 --> 00:19:08,580
+Derivative في هذه الحالة هيكون بساويناقص الـ
+
+299
+00:19:08,580 --> 00:19:12,320
+Absolute Value دي دلنا في دلنا بتكون اللي هي أقل
+
+300
+00:19:12,320 --> 00:19:16,780
+قيمة و هناك أكبر قيمة طيب و اكتشف كل الـ
+
+301
+00:19:16,780 --> 00:19:21,200
+Directional Derivative Sphere أكيد Any Direction U
+
+302
+00:19:21,200 --> 00:19:25,420
+Orthogonal To The GradientIs a direction of zero
+
+303
+00:19:25,420 --> 00:19:30,120
+changed in F because θ is equal to π على 2 لما
+
+304
+00:19:30,120 --> 00:19:36,960
+يكون الآن الـ Zero بيكون عندي الـ Direction of U
+
+305
+00:19:36,960 --> 00:19:40,460
+is orthogonal to a gradient فالـ F بيكون اللي هو
+
+306
+00:19:40,460 --> 00:19:43,340
+الـ Directional Derivative أي إيش تفر يعني الـ
+
+307
+00:19:43,340 --> 00:19:46,420
+Directional Derivative بتكون Zero في حالة اللي هو
+
+308
+00:19:46,420 --> 00:19:51,180
+θ بتسوء بي على 2 هذا اللي عن زيرو إيش Gradientإذا
+
+309
+00:19:51,180 --> 00:19:53,420
+الـ Direction of the derivative of U of F بتساوي
+
+310
+00:19:53,420 --> 00:20:00,500
+سفر When U is orthogonal to the gradient ماشي،
+
+311
+00:20:00,500 --> 00:20:04,080
+المعلمة أضحى تلان ناخد مثال عددي Example تلاتة
+
+312
+00:20:04,080 --> 00:20:07,520
+Find the directions in which F of XY بيسوي X تربيه
+
+313
+00:20:07,520 --> 00:20:11,100
+على 2 وY تربيه على 2 وقوللي بس أودد ال directions
+
+314
+00:20:11,100 --> 00:20:16,040
+اللي ال F هذه واحد Increases most rapidly at the
+
+315
+00:20:16,040 --> 00:20:20,810
+point واحد واحد يعني لما بدي Directionلما الـ F
+
+316
+00:20:20,810 --> 00:20:23,550
+increases most rapidly عند الواحد و واحد قال لي
+
+317
+00:20:23,550 --> 00:20:26,570
+واحد على طول طب انت قلت قبل شوية انهم يكونون most
+
+318
+00:20:26,570 --> 00:20:29,910
+rapidly لما يكونون في اتجاه مين لـ Gradient بقوله
+
+319
+00:20:29,910 --> 00:20:32,930
+صح قال ليه أصلا دي decreases most rapidly اللي بدك
+
+320
+00:20:32,930 --> 00:20:36,390
+تسأل عنها في التانية برضه في عكس اتجاه مين ل
+
+321
+00:20:36,390 --> 00:20:40,270
+Gradientإذا أسهل الأمر بنجد ما هو الـ Gradient و
+
+322
+00:20:40,270 --> 00:20:45,550
+بنجد ما هو الـ Direction يعني الـ Gradient على
+
+323
+00:20:45,550 --> 00:20:48,770
+المجموعات اللي هو بيكون الجواب اللي هو الأول هذا و
+
+324
+00:20:48,770 --> 00:20:52,690
+ناقصه بيكون جواب الثاني بقوله صح كلامك، بقولي طب
+
+325
+00:20:52,690 --> 00:20:55,390
+What are the directions of zero change in F at
+
+326
+00:20:55,390 --> 00:21:00,300
+واحد واحد؟أنت بتقول لما قلت لنا قبل بشوية أنه
+
+327
+00:21:00,300 --> 00:21:05,400
+بيكون اللي هو الـ Zero اتشانج لما نكون الـ
+
+328
+00:21:05,400 --> 00:21:09,840
+Direction لـ U و V متعامدين يعني وجدنا الـ
+
+329
+00:21:09,840 --> 00:21:13,480
+Direction للـ Gradient على طول بنغير إشارة واحدة و
+
+330
+00:21:13,480 --> 00:21:18,770
+بنخلي التاني زي ما هي و التجاهين عموديين عليهطيب
+
+331
+00:21:18,770 --> 00:21:22,690
+خلّينا نشوف الكلمة ده اللي بنحكيه سريع للإجابة
+
+332
+00:21:22,690 --> 00:21:26,530
+بسرعة ييجي الآن نشوف increases most rapidly at the
+
+333
+00:21:26,530 --> 00:21:30,410
+point 113 the function قولنا فوق increases most
+
+334
+00:21:30,410 --> 00:21:33,370
+rapidly in the direction of the gradient at 113،
+
+335
+00:21:33,370 --> 00:21:37,090
+وعادةً بلزمنا لوجد الـ Gradient قادر الـ Gradient
+
+336
+00:21:37,090 --> 00:21:41,170
+للـ F أيش بيساوي؟ اللي هو Del F by Del X اللي هي
+
+337
+00:21:41,170 --> 00:21:46,210
+2X على 2 يعني Xتجاه الـ I دل F by دل Y اللي هي 2Y
+
+338
+00:21:46,210 --> 00:21:50,220
+على 2 يعني Y في تجاه الـ G هذا الـ Gradientللـ
+
+339
+00:21:50,220 --> 00:21:53,460
+gradient للـ F عند الواحد والواحد ايش هيساوي او
+
+340
+00:21:53,460 --> 00:21:57,020
+ذنب واحد واحد بصير I زي الـ J إذا الأن الـ
+
+341
+00:21:57,020 --> 00:22:00,380
+Direction للـ Gradient للF وعبارة عن الـ Gradient
+
+342
+00:22:00,380 --> 00:22:04,240
+للF على الـ Magnitude له Magnitude هو الجدر التربي
+
+343
+00:22:04,240 --> 00:22:07,260
+على واحد زي واحد على جدر التنين يعني الـ Direction
+
+344
+00:22:07,260 --> 00:22:10,440
+U بيساوي واحد على جدر التنين في تجاه الـ I زي واحد
+
+345
+00:22:10,440 --> 00:22:13,480
+على جدر التنين في تجاه الـ J هذا الـ Direction هو
+
+346
+00:22:13,480 --> 00:22:17,640
+Direction لـ Gradientاللي عنده اللي هي الـ F
+
+347
+00:22:17,640 --> 00:22:23,450
+increases most ربط اللي عنده النقطة 1 1طيب قلت
+
+348
+00:22:23,450 --> 00:22:25,870
+أكيد كلكم قلته الـ function decreases most
+
+349
+00:22:25,870 --> 00:22:29,330
+probably in the direction of عكس الـ gradient عنده
+
+350
+00:22:29,330 --> 00:22:33,210
+نقطة واحد واحد which is هذا بس مضروب في من؟ في
+
+351
+00:22:33,210 --> 00:22:36,390
+سالب واحد ناقص واحد على جدر التانين في تجاه I ناقص
+
+352
+00:22:36,390 --> 00:22:40,270
+واحد على جدر التانين في تجاه J الآن الجزء الثالث
+
+353
+00:22:40,270 --> 00:22:44,370
+الثالث what are the directions of zero change يعني
+
+354
+00:22:44,370 --> 00:22:47,410
+لـ Gradient لما يكون اللي هو إيش directional
+
+355
+00:22:47,410 --> 00:22:50,150
+derivative zero بتكون directional derivative zero
+
+356
+00:22:50,150 --> 00:22:53,120
+لما الزاوية بينهم إيش بتساوي؟By على اتنين يعني
+
+357
+00:22:53,120 --> 00:22:55,780
+بمعنى آخر لما يكون الـ U عمودي عالميا على الـ
+
+358
+00:22:55,780 --> 00:23:00,020
+Gradient طب هي الـ U موجودةالتي جاهة الـ Gradient
+
+359
+00:23:00,020 --> 00:23:03,720
+موجودة إذاً بدأ وجد الـ Direction تبع الـ Zero الـ
+
+360
+00:23:03,720 --> 00:23:06,140
+Direction of Zero Change بقصه Direction تبع الـ
+
+361
+00:23:06,140 --> 00:23:09,260
+Zero Change إن F أت واحد are the directions
+
+362
+00:23:09,260 --> 00:23:13,180
+orthogonal to gradient للـ F الـ Directions اللي
+
+363
+00:23:13,180 --> 00:23:19,140
+عمليات عالمين على الـ F الآن هي تجاه الـ F وهي
+
+364
+00:23:19,140 --> 00:23:23,280
+تجاه الـ F هي ناقص الـ F اللي هي تجاه الـ Gradient
+
+365
+00:23:24,290 --> 00:23:28,130
+تجاه الـ Gradient اللي هو العمودي عليه بس بنغير
+
+366
+00:23:28,130 --> 00:23:30,650
+إشارتها دي بيصيح ناقص واحدة لجدرتينين في الوقت
+
+367
+00:23:30,650 --> 00:23:33,890
+تجاه الـ I زائد واحدة لجدرتينين في الجهة الـ J أو
+
+368
+00:23:34,290 --> 00:23:39,050
+وأو كمان اللي هو بغير هذه شريطة و بخليها دي ثابتة
+
+369
+00:23:39,050 --> 00:23:42,130
+يعني بيصير N بتسوى واحدة لجدر التنين في تجاه I
+
+370
+00:23:42,130 --> 00:23:46,510
+ناقص واحدة لجدر التنين J وناقص N اللي عكسه ناقص
+
+371
+00:23:46,510 --> 00:23:48,990
+واحدة لجدر التنين في تجاه I زي واحدة لجدر التنين
+
+372
+00:23:48,990 --> 00:23:52,450
+في تجاه J هدول التنين طبعا عمودي واحدة هي لو فرضنا
+
+373
+00:23:52,450 --> 00:23:56,410
+هذا ال gradient التجاه ال gradient التجاه العمودي
+
+374
+00:23:56,410 --> 00:23:59,990
+عليه هيه هنا أو اللي تحت هدول العكس بعض التنتين
+
+375
+00:24:00,350 --> 00:24:04,970
+عشان اضغط على الـ dot product 1 على الجدر التاني و
+
+376
+00:24:04,970 --> 00:24:05,850
+1 على الجدر التاني
+
+377
+00:24:33,380 --> 00:24:40,400
+الان نشوف علاقة الـ Gradientبمين؟ بالـ Tangents to
+
+378
+00:24:40,400 --> 00:24:43,500
+Level Curves ونستغل هذه العلاقة في إيجاد ما ..
+
+379
+00:24:43,500 --> 00:24:47,940
+اللي هو إيش .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+380
+00:24:47,940 --> 00:24:48,460
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+381
+00:24:48,460 --> 00:24:48,460
+.. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+382
+00:24:48,460 --> 00:24:49,060
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+383
+00:24:49,060 --> 00:24:49,220
+.. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+384
+00:24:49,220 --> 00:24:49,760
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+385
+00:24:49,760 --> 00:24:49,760
+.. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+386
+00:24:49,760 --> 00:24:50,560
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+387
+00:24:50,560 --> 00:24:50,620
+.. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+388
+00:24:50,620 --> 00:24:50,840
+ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما
+
+389
+00:24:50,840 --> 00:24:50,840
+.. ما .. ما .. ما .. ما .. ما .. ما .. ما .. ما ..
+
+390
+00:24:50,840 --> 00:24:59,060
+ما .. ما .. ما ..
+
+391
+00:24:59,060 --> 00:25:01,000
+ما
+
+392
+00:25:06,720 --> 00:25:13,260
+لأن F of X Y هذه لو نفرضنا إنه قيمتها ثابتةوين على
+
+393
+00:25:13,260 --> 00:25:18,540
+اللي هي عند قيم XG of T وعند قيم Y بساوية H of T
+
+394
+00:25:18,540 --> 00:25:23,960
+يعني على مين؟ على الـ Curve هذه لو عوضنا X بأي
+
+395
+00:25:23,960 --> 00:25:27,060
+قيمة يعني لما يكون في عندي Vector هيه اللي بساوية
+
+396
+00:25:27,060 --> 00:25:29,980
+G of T في تجاه الـI زي هذا G of T في تجاه الـJ
+
+397
+00:25:29,980 --> 00:25:34,300
+ودينا عوضنا الـG of T هنا والـH of T هنا هنلاقي
+
+398
+00:25:34,300 --> 00:25:37,400
+القيمة إيه اشتلعت Constant يعني صار هذا عبارة عن F
+
+399
+00:25:37,400 --> 00:25:41,620
+of X وY بساوية C صار Level Curveالان الـ level
+
+400
+00:25:41,620 --> 00:25:46,640
+curve هذا اللي هو f of g of t وh of t ايش بيساوي c
+
+401
+00:25:46,640 --> 00:25:51,780
+خلّينا نفاضل اللي هو إيش اتجهتين with respect to
+
+402
+00:25:51,780 --> 00:25:55,880
+mean to t فلطولوا روحكم معانايا differentiating
+
+403
+00:25:55,880 --> 00:25:59,000
+both sides of this equation with respect to t
+
+404
+00:25:59,000 --> 00:26:02,780
+leads to the equation هذه ثابتة c لأن قلنا along
+
+405
+00:26:02,780 --> 00:26:07,000
+اللي هو ال R هذا كله ال f of x y هتصير مقدار ثابت
+
+406
+00:26:07,000 --> 00:26:10,480
+هيك مفترضين احناالذي يحدث هو الـ derivative لهذه
+
+407
+00:26:10,480 --> 00:26:12,340
+بيساوي الـ derivative لهذه و الـ derivative لهذه
+
+408
+00:26:12,340 --> 00:26:16,180
+الـ derivative لهذه من وجه الذهب وسط 2 اللي هو الـ
+
+409
+00:26:16,180 --> 00:26:19,300
+chain rule اللي أخدناها المرة الفاترة Del F by Del
+
+410
+00:26:19,300 --> 00:26:23,700
+X في Del X by Del T X أنا اللي مسمينها إيه؟ إحنا G
+
+411
+00:26:23,700 --> 00:26:27,640
+زاد Del F by Del Y في Del Y by Del T الـ Y اللي
+
+412
+00:26:27,640 --> 00:26:32,570
+مسمينها أنا إيه؟ اسمها H إيش هيساوي؟ سفرالأن هذه
+
+413
+00:26:32,570 --> 00:26:35,910
+برجحها لأصلها هذه عبارة عن حاصل ضرب اللي هو الـ
+
+414
+00:26:35,910 --> 00:26:39,170
+dot product اللي هو del f by del x في اتجاه الـ I
+
+415
+00:26:39,170 --> 00:26:42,850
+زي del f by del y في اتجاه الـ J ضربة dot product
+
+416
+00:26:42,850 --> 00:26:47,870
+في الـ dG by dt في اتجاه الـ I زي الـ dH by dt في
+
+417
+00:26:47,870 --> 00:26:51,010
+اتجاه الـ G ومش مصدق ضربهم هذه في هذه السيرة هي
+
+418
+00:26:51,490 --> 00:26:55,290
+وهذه في هذه بالسيرة هيها وهذا ايش بيساوي كله سفر
+
+419
+00:26:55,290 --> 00:26:59,250
+هذه مين هي الـ gradient للـ F الـ gradient لمن للـ
+
+420
+00:26:59,250 --> 00:27:03,250
+F اللي عندك مضروبة في مين في DR by DT كله ايش
+
+421
+00:27:03,250 --> 00:27:05,970
+بيساوي سفر هذا نضطه product بين اللي بيساوي سفر
+
+422
+00:27:05,970 --> 00:27:09,430
+هذا له معنى نضطه product بين اللي هو الـ gradient
+
+423
+00:27:09,430 --> 00:27:13,770
+للـ F الـ gradient للـ function وهذه في اللي هو DR
+
+424
+00:27:13,770 --> 00:27:19,290
+by DT اللي هو عبارة عن ميل المماسللـ vector R
+
+425
+00:27:19,290 --> 00:27:21,650
+بيساوي «G» و «T» أتشوف «T» بتجاه الـ «I» اللي
+
+426
+00:27:21,650 --> 00:27:26,210
+بيخلّي الـ «F» of X و Y constant يعني الـ Gradient
+
+427
+00:27:26,210 --> 00:27:31,710
+للـ «F» is normal على مين؟ على الـ «Tangent» لمين؟
+
+428
+00:27:31,710 --> 00:27:36,610
+للـ «Vector» الـ «R» اللي عندي easy equation خمسة
+
+429
+00:27:36,610 --> 00:27:39,390
+says that the gradient للـ «F» is normal to the
+
+430
+00:27:39,390 --> 00:27:43,370
+tangent vector «DR» by «DT» so it is normal to the
+
+431
+00:27:43,370 --> 00:27:47,790
+curve ملخص الحديث اللي حكينا فوقإن في كل موقع X0
+
+432
+00:27:47,790 --> 00:27:53,970
+وY0 في مجال مفهوم F of X وY الـ Gradient of F هو
+
+433
+00:27:53,970 --> 00:27:58,650
+عمودي عالمي لـ Level Curve عند مين النقطة X0 وY0
+
+434
+00:27:58,650 --> 00:28:04,970
+دائما اللي هو الـ Gradient للـ Function F عمودي
+
+435
+00:28:04,970 --> 00:28:12,140
+عالمي على Level Curve خلال مين؟ X0 وY0لأن هذا الـ
+
+436
+00:28:12,140 --> 00:28:16,560
+Level Curve f of x و f of y هي مين f of هذا
+
+437
+00:28:16,560 --> 00:28:19,280
+المقدار f of g of t و f of h of t اللي هو بساوي c
+
+438
+00:28:19,280 --> 00:28:24,220
+بعمل constant يعني بيصير اللي هو dr by dt على ال
+
+439
+00:28:24,220 --> 00:28:27,180
+gradient للf normal الـ dr by dt اللي هو مين
+
+440
+00:28:27,180 --> 00:28:31,540
+المماس للـ Level Curve اللي عندي إذا صار عندي
+
+441
+00:28:31,540 --> 00:28:35,470
+Level Curveالـ Normal دايمًا لـ Gradient H Normal
+
+442
+00:28:35,470 --> 00:28:41,110
+عليه لأن هو Normal على الـ Tangent Elo على الـDR
+
+443
+00:28:41,110 --> 00:28:46,270
+by DT إذًا، اللي هي هذه الآن، بدنا نستغلها في حال
+
+444
+00:28:46,270 --> 00:28:50,830
+السؤال التالي هذه بس الرسمة هي تعليقًا على اللي هو
+
+445
+00:28:50,830 --> 00:28:53,530
+اللي أوجدناها قبل شوية ده لزيد بساويكس تربيه على
+
+446
+00:28:53,530 --> 00:28:56,680
+اتنين زي دوية تربيه على اتنيناللي هو الرسم الكمهان
+
+447
+00:28:56,680 --> 00:28:58,960
+اللي هو جاينا هي الـ Gradient للـ F عنده النقطة
+
+448
+00:28:58,960 --> 00:29:01,880
+اللي وجدناها اللي يسمى I زي الـ J هي هذا اللي
+
+449
+00:29:01,880 --> 00:29:06,060
+بتكون Most Rapidly وهنا أجلم اللي هي عكسها الـ
+
+450
+00:29:06,060 --> 00:29:09,540
+Decreases اللي هو Most Rapidly أو Most Rapid
+
+451
+00:29:09,540 --> 00:29:13,620
+Decrease اللي هي عكسها ودولة الـ Vectorين هي اللي
+
+452
+00:29:13,620 --> 00:29:16,660
+بيخلن الـ Zero Change enough هذا اللي هي الرسمة
+
+453
+00:29:16,660 --> 00:29:20,740
+اللي عملنا لتو في المثال اللي قبل بشوية هي المثال
+
+454
+00:29:20,740 --> 00:29:26,750
+اللي عملناهاهذا آه آسف هذا المثال هي آه هذه اللي
+
+455
+00:29:26,750 --> 00:29:30,450
+هو اللي بس اللي هو التوضيح الهندسي لها للي
+
+456
+00:29:30,450 --> 00:29:37,920
+حاكيناهاهذه التوضيحة في problem 39 سأستغلها بعد
+
+457
+00:29:37,920 --> 00:29:43,640
+قليل سأستعملها لأحل مشاهدها مثال خلّيني بس أن
+
+458
+00:29:43,640 --> 00:29:47,660
+البرنامج اللي كنت معاه show that if x minus x0
+
+459
+00:29:47,660 --> 00:29:51,760
+plus by minus y0 is an equation for the line in
+
+460
+00:29:51,760 --> 00:29:54,480
+the xy plane through the point x0 y0 normal to the
+
+461
+00:29:54,480 --> 00:29:59,170
+vector كذا يعني لو كان عنده نقطة x0 y0وعندي Normal
+
+462
+00:29:59,170 --> 00:30:05,610
+vector N على line يعني معروف الـ Normal N بتسوي I
+
+463
+00:30:05,610 --> 00:30:09,470
+زي BJ معروفة نقطة يعني عندي خط اللي كان معروف
+
+464
+00:30:09,470 --> 00:30:13,230
+Normal عليه ومعروف نقطة X0 Y0 وده وجود معادلته
+
+465
+00:30:13,690 --> 00:30:16,850
+معدلته انا بقول هي a في x minus x not زي بي في y
+
+466
+00:30:16,850 --> 00:30:20,170
+minus y not بالسبع سفر عشان مانطولش حل السؤال اللي
+
+467
+00:30:20,170 --> 00:30:23,770
+بعد ولا هيعتمد عليه إيش هذه؟ يا جماعة هذه كتير هي
+
+468
+00:30:23,770 --> 00:30:28,250
+ليه؟ عندي معروف .. هي عندي خط هذا الخط هذا بيدوجد
+
+469
+00:30:28,250 --> 00:30:32,610
+معدلته معروف العمودي عليه و معروف نقطة عليه و
+
+470
+00:30:32,610 --> 00:30:35,730
+بنعرف معدلته طبعا كلكم بيعرف أعمله خليني بس على
+
+471
+00:30:35,730 --> 00:30:38,950
+السريع أني أحله ال normal to the vector m ai زي بي
+
+472
+00:30:38,950 --> 00:30:45,570
+جي isالأن T بيتساوي الهو
+
+473
+00:30:45,570 --> 00:30:50,070
+T بيساوي PI نقص AJ أو نقص PI زي AJ C يعني أي واحد
+
+474
+00:30:50,070 --> 00:30:56,910
+بالفعل ال equation is X بيساوي X naught زي ال TB و
+
+475
+00:30:56,910 --> 00:31:01,860
+Y بيساوي Y naught نقص TAاللي معروف الآن x بتساوي
+
+476
+00:31:01,860 --> 00:31:06,380
+اللي هي خطأ هي عندي ال normal معروف بيصير عندي
+
+477
+00:31:06,380 --> 00:31:10,240
+اللي هي معادلة اللي هي x بيساوي x naught هي النقطة
+
+478
+00:31:10,240 --> 00:31:16,720
+زائد t في b و y بيساوي y naught ناقص t في a هي
+
+479
+00:31:16,720 --> 00:31:23,260
+معاملة that is الان x minus x naught بيساوي tb و y
+
+480
+00:31:23,260 --> 00:31:27,700
+minus y naught بيساوي t ناقص tهنس المعادلة A في X
+
+481
+00:31:27,700 --> 00:31:32,120
+نحلها مع بعض عشان نحطها في المعادلة اللي فوق A في
+
+482
+00:31:32,120 --> 00:31:37,920
+X minus X0 بسوى T في A بي أكيد عارفين كيف اشي اللي
+
+483
+00:31:37,920 --> 00:31:44,800
+عملتها عندي هاي المعادلتين هذولةا في x minus x not
+
+484
+00:31:44,800 --> 00:31:50,020
+هي a في x minus x not بساوي a في tb يعني ساعة t a
+
+485
+00:31:50,020 --> 00:31:55,220
+في b and b في y minus y not بساوي نقص t في ba لأن
+
+486
+00:31:55,220 --> 00:32:00,720
+التلتين مع بعض بيصير هي هذه بتساوي نقص هذه ا او
+
+487
+00:32:00,720 --> 00:32:06,320
+معنى اخر هذه اجمحها لهذه او اجمحها لهذه المعدلتين
+
+488
+00:32:06,320 --> 00:32:09,700
+مع بعض بيطلع سفر هنابصير ايه في X minus X naught و
+
+489
+00:32:09,700 --> 00:32:12,820
+في Z بيه في Y minus Y نقل ساعة و ستة هذا اللي هي
+
+490
+00:32:12,820 --> 00:32:18,140
+معادلة اللي يتحقق اللي أكون عمودي على اللي عندي
+
+491
+00:32:18,140 --> 00:32:25,700
+فهيبقى نشوف الفترة بقى find an equation for the
+
+492
+00:32:25,700 --> 00:32:28,780
+tangent to the ellipse find an equation for the
+
+493
+00:32:28,780 --> 00:32:32,960
+tangent to the ellipse معطينيها at the point نقص 2
+
+494
+00:32:32,960 --> 00:32:39,690
+و 1 يعني معطينيها اللي لبسهاهذه الـ ellipse بدي
+
+495
+00:32:39,690 --> 00:32:44,690
+اوجد الـ equation للتانجينت هي عند المقطع 61 وهي
+
+496
+00:32:44,690 --> 00:32:49,070
+المماس بدي اوجد ال equation للمماس طبعا فيكم
+
+497
+00:32:49,070 --> 00:32:53,090
+يعرفوا عمله ولا بالزمان بطريقة زمان بنفعبس دلوقت
+
+498
+00:32:53,090 --> 00:32:56,870
+ده خلّيني أقول يعني عشان نطبق اللي حكينا بواسط ال
+
+499
+00:32:56,870 --> 00:33:00,330
+level curves أما بنحلل اللي هو بالطريقة اللي هي
+
+500
+00:33:00,330 --> 00:33:04,970
+القديمة هي المعادنة موجودة وهي النقطة موجودة بنوجد
+
+501
+00:33:04,970 --> 00:33:07,330
+اللي هو المماس من اللي المماس معروف و النقطة
+
+502
+00:33:07,330 --> 00:33:11,310
+معروفة بنوجد إيش بنوجد اللي هو المماس خلّيني أوجده
+
+503
+00:33:11,310 --> 00:33:16,260
+بطريقة ثانية الآنهذه هي X تربيه على أربع ثم تربيه
+
+504
+00:33:16,260 --> 00:33:20,580
+على بساوة اتنين هي عبارة عن ال ellipse هذه وهي ال
+
+505
+00:33:20,580 --> 00:33:24,220
+tangent لها وهي الان بدي اوجد العمودي على ال
+
+506
+00:33:24,220 --> 00:33:29,730
+tangent ونستغل اللي هو المعرفة اللي قبل شويةماذا
+
+507
+00:33:29,730 --> 00:33:33,310
+تريد أن تفعله؟ تريد استخدام معرفتهين دعنا نقول الـ
+
+508
+00:33:33,310 --> 00:33:37,470
+ellipse x² على 4 زي 2y² على 2 is a level curve of
+
+509
+00:33:37,470 --> 00:33:40,430
+mean of the function f of x,y بيساوي x² على 4 زي
+
+510
+00:33:40,430 --> 00:33:44,470
+2y² على 2y² لأن اعتبرت الدالة عندي الـ of two
+
+511
+00:33:44,470 --> 00:33:48,170
+variables اللي هي function of two variables هي
+
+512
+00:33:48,170 --> 00:33:52,540
+الدالةالأن هذه الـ Dialogue طبعاً في Surface، في
+
+513
+00:33:52,540 --> 00:33:55,580
+Space اللي عن الـ Level Curve اللي لها أخدت الـ
+
+514
+00:33:55,580 --> 00:33:58,600
+Level T Curve X تربيه على 4 زي ضي تربيه بيسوا 2
+
+515
+00:33:58,600 --> 00:34:02,760
+اللي هي عبارة عن صارت Level Curve لهذه احنا عارفين
+
+516
+00:34:02,760 --> 00:34:07,960
+أنه الـ Gradient للـ Function عمودي على مابين على
+
+517
+00:34:07,960 --> 00:34:11,540
+Level Curve اللي حكيناه قبل بشوية وانت تطبيق عليه
+
+518
+00:34:11,830 --> 00:34:17,730
+الـ Gradient فبساوية X على 2 في اتجاه I زائد 2Y في
+
+519
+00:34:17,730 --> 00:34:24,270
+اتجاه J هذا الـ Gradient اللي هو نقطة مين؟ نقص 2 و
+
+520
+00:34:24,270 --> 00:34:31,050
+1؟ بساوية ناقص I عوضت عن X نقص 2زي دي 2G هذا الـ
+
+521
+00:34:31,050 --> 00:34:36,130
+Gradient الآن احنا عارفين ايش خصاصه عمودي على الـ
+
+522
+00:34:36,130 --> 00:34:38,890
+Level Curve يعني عمودي على المماس تبع الـ Level
+
+523
+00:34:38,890 --> 00:34:41,790
+Curve عند اللقطة المذكورة هذا من اللي قبل شوية
+
+524
+00:34:41,790 --> 00:34:46,340
+أثبتناهاللي هي الأطباق التنهجة بالبشوية هذا اللي
+
+525
+00:34:46,340 --> 00:34:49,860
+هو الـ Gradient عمودي عالميا على الـ Level Curve
+
+526
+00:34:49,860 --> 00:34:54,200
+عند النقطة X0 وY0 بتستخدم هذه المعلومة ستعرف أن
+
+527
+00:34:54,200 --> 00:34:57,660
+الـ Gradient اللي هو العمودي على اللي هو المماس
+
+528
+00:34:57,660 --> 00:35:01,100
+فعارف نقطة على طول من خلال المثال اللي جاب اللي
+
+529
+00:35:01,100 --> 00:35:04,000
+بشوية أو ال problem اللي جاب اللي بشويةبكون عرفة
+
+530
+00:35:04,000 --> 00:35:08,720
+العمودي وعارف نقطة اذا بعرف عادلة اللي هي المماسك
+
+531
+00:35:08,720 --> 00:35:12,520
+اللي بديها اللي هو a في x نقص x not زي بي في y نقص
+
+532
+00:35:12,520 --> 00:35:15,980
+y not بساوي سفر ال a اللي هي عبارة عن نقص واحد في
+
+533
+00:35:15,980 --> 00:35:19,760
+هذه الحالة و ال b عبارة عن اتنين اللي هو المعاملة
+
+534
+00:35:19,760 --> 00:35:24,200
+تبعت مين اللي هو العمودي x minus x not اللي هي x
+
+535
+00:35:24,200 --> 00:35:27,120
+not اللي هي نقص اتنين بسيط زي اتنين و y نقص y not
+
+536
+00:35:27,120 --> 00:35:31,040
+اللي هي واحد بساوي سفر دهاللي هو لما نبسطها بيطلع
+
+537
+00:35:31,040 --> 00:35:35,160
+X نقص 2Y بساوي سالب 4 فدي معادلة المماس اللي احنا
+
+538
+00:35:35,160 --> 00:35:38,980
+أمامنا وانتوا في البيت جربوا كيف توجدوها بطريقة
+
+539
+00:35:38,980 --> 00:35:43,360
+السابقة نيجي لعند algebra rules for gradient اللي
+
+540
+00:35:43,360 --> 00:35:46,780
+هي بعض خواص ال gradient خواص ال gradient شبه لخواص
+
+541
+00:35:46,780 --> 00:35:49,660
+مين يا جماعة ال derivative العادية ال gradient ل F
+
+542
+00:35:49,660 --> 00:35:52,580
+زائد G سواء لGradient لF زايد لGradient تنجي لأنه
+
+543
+00:35:52,580 --> 00:35:57,860
+ال gradient عبارة عناللي هي دل F by دل X في تجاه I
+
+544
+00:35:57,860 --> 00:36:00,280
+زي دل F by دل Y في تجاه J يعني نبقى قارعين
+
+545
+00:36:00,280 --> 00:36:03,220
+derivatives فطبيعي يكون خواصها هي خواص الـ
+
+546
+00:36:03,220 --> 00:36:06,240
+derivative العادية والـ partial derivatives فالـ
+
+547
+00:36:06,240 --> 00:36:10,020
+gradient لف نقص G سوى الـ gradient لف نقص G الـ
+
+548
+00:36:10,020 --> 00:36:12,940
+gradient لكيف لف K حيث constant سوى كيف الـ
+
+549
+00:36:12,940 --> 00:36:16,290
+gradient لفالـ F في G بساوية الـ F في الـ Gradient
+
+550
+00:36:16,290 --> 00:36:19,430
+للـ G زائد G في Gradient للـ F والـ Gradient للـ F
+
+551
+00:36:19,430 --> 00:36:23,090
+على G بساوية الـ G Gradient للـ F نقص F Gradient
+
+552
+00:36:23,090 --> 00:36:26,730
+للـ G على G تلت يعني اللي انتوا عارفينها من زمان
+
+553
+00:36:26,730 --> 00:36:31,870
+هذه الخواصة بس خلونا نتحقق عدديا من واحدة تنتير بس
+
+554
+00:36:31,870 --> 00:36:37,460
+عشر سنةللأطمأنة مش أكثر لأن مش مطلوب الورحانة مثال
+
+555
+00:36:37,460 --> 00:36:42,560
+خمسة We illustrate two of the rules with يعني بدنا
+
+556
+00:36:42,560 --> 00:36:47,460
+نوضح قاعدتين من اللي فاتن بوساطة ايش احنا مثلا عدد
+
+557
+00:36:47,460 --> 00:36:51,880
+f of x,y بيساوي x نقص y و g of x,y بيساوي 3y ال
+
+558
+00:36:51,880 --> 00:36:55,340
+gradient اللافظة ما انتوا عارفين دل f by دل x بيه
+
+559
+00:36:55,340 --> 00:37:00,540
+i دل f by دل y هي واحد اللي هي بيصير نقص هذا واحد
+
+560
+00:37:00,540 --> 00:37:08,030
+وهنا نقص واحد بيصير i نقص jDEL F DEL J BY DEL X
+
+561
+00:37:08,030 --> 00:37:12,970
+صفر إذاً Zero مافيش هنا DEL J BY DEL Y تلاتة إذاً
+
+562
+00:37:12,970 --> 00:37:18,230
+الـ Gradient DEL F عبارة عن مين تلاتة Gإذاً غرضنا
+
+563
+00:37:18,230 --> 00:37:21,710
+يا جماعة بس نحقق في هذا المثال المثال del f minus
+
+564
+00:37:21,710 --> 00:37:26,290
+g اي موضح كيف يساوي del f minus del g والـ
+
+565
+00:37:26,290 --> 00:37:30,410
+gradient للـ f ناقص gradient g الـ gradient للـ f
+
+566
+00:37:30,410 --> 00:37:34,590
+minus g بنيجي من اتراح الـ f ناقص g هي x ناقص y
+
+567
+00:37:34,590 --> 00:37:38,650
+ناقص 3 وy بيصير x ناقص 4y الـ gradient لهذه اللي
+
+568
+00:37:38,650 --> 00:37:43,360
+هي del f by del .. del هذا by del x يعني Iلأن
+
+569
+00:37:43,360 --> 00:37:47,240
+تفاضلها بالنسبة لـ X1 وتفاضلها بالنسبة ل Y ناقص 4
+
+570
+00:37:47,240 --> 00:37:53,240
+I ناقص 4J هي في الواقع فعلا I ناقص G ناقص 3J يعني
+
+571
+00:37:53,240 --> 00:37:57,680
+I ناقص 4G يعني الـ Gradient لـ F ناقص Gradient لـ
+
+572
+00:37:57,680 --> 00:38:01,400
+G خلّينا نشوف التانية Gradient للـ F في G نجيب الـ
+
+573
+00:38:01,400 --> 00:38:05,860
+F في G هي الـ F X minus Y في G اللي هي تلاتة Y
+
+574
+00:38:05,860 --> 00:38:08,440
+تصبح تلاتة X في Y
+
+575
+00:38:11,100 --> 00:38:14,640
+نقص تلاتة Y تربيع لأن في تلاتة Y يكون تلاتة Y
+
+576
+00:38:14,640 --> 00:38:19,120
+تربيع ضربت الـ F في G في G يا جماعة طلعت عندي
+
+577
+00:38:19,120 --> 00:38:22,600
+بالشكل هذا الـ Gradient لهذه الدالة الجديدة هو
+
+578
+00:38:22,600 --> 00:38:27,260
+عبارة عن تلاتة Yهذه السفر تلاتة Y في اتجاه الـ I
+
+579
+00:38:27,260 --> 00:38:30,240
+بفضل بالنسبة ل X طبعا قبل ما انتوا عارفين بعدين
+
+580
+00:38:30,240 --> 00:38:33,480
+بفضل بالنسبة ل Y بالنسبة ل Y اللي عبارة عن تلاتة X
+
+581
+00:38:33,480 --> 00:38:37,620
+ناقص ستة Y عشانك تلاتة X ناقص ستة Y في اتجاه الـ G
+
+582
+00:38:37,620 --> 00:38:41,520
+هذه اللي هي ال gradient لل FG نيجي الآن نحسب ال
+
+583
+00:38:41,520 --> 00:38:45,790
+gradient لل Gاللي هي الـ G في الـ gradient للـ F
+
+584
+00:38:45,790 --> 00:38:48,590
+والـ F في الـ gradient للـ G حسب قانونها، بدنا
+
+585
+00:38:48,590 --> 00:38:53,430
+نقول أن هذا بساوي هذا نتحقق منه بس من خلال هذا
+
+586
+00:38:53,430 --> 00:38:57,830
+المثال، هذا اللي هي حاصل ضرب دالتين، هي تفضولين،
+
+587
+00:38:57,830 --> 00:39:01,310
+خلينا نشوف كيف هذا الكلام بساوي اللي فوق من خلال
+
+588
+00:39:01,310 --> 00:39:08,130
+مثلناجي هج ثلاثة جي ثلاثة و اي ثلاثة و اي ثلاثة و
+
+589
+00:39:08,130 --> 00:39:15,500
+اي ثلاثة و اي ثلاثة و اي ثلاثةالـ Gradient للـ G
+
+590
+00:39:15,500 --> 00:39:22,940
+عبارة عن ثلاثة G إذاً الـ Gradient .. آسف فعلا
+
+591
+00:39:22,940 --> 00:39:26,660
+بدأت فائدك بدأنا الـ G .. G اللي عبارة عن جداش
+
+592
+00:39:26,660 --> 00:39:30,940
+عبارة عن تلاتة Y عشان ما خربتش في ال Gradient للـ
+
+593
+00:39:30,940 --> 00:39:36,000
+F ال Gradient للـ F هي I نقص G هي نزلتها زائد الـ
+
+594
+00:39:36,000 --> 00:39:40,140
+F جداش قيمتها X minus Y هيها ضربتها في من؟ في
+
+595
+00:39:40,140 --> 00:39:44,290
+Gradient للـ G Gradient للـ G تلاتة Gالان هذا
+
+596
+00:39:44,290 --> 00:39:49,210
+المقدار جيت بساوي عبارة عن تلاتة Y في I minus G
+
+597
+00:39:49,210 --> 00:39:53,150
+خليته زي ما هو بده يصل للي فوق يا جماعة زائد هذه X
+
+598
+00:39:53,150 --> 00:39:57,310
+دخلت للي جوا بيصير تلاتة X ناقص تلاتة Y الان خلينا
+
+599
+00:39:57,310 --> 00:40:02,390
+نلم هنا على بعض التلاتة Y في ناقص J اللي بيصير
+
+600
+00:40:02,390 --> 00:40:07,210
+ناقص تلاتة YJ لما زيدها على ناقص تلاتة YJ بيصير
+
+601
+00:40:07,210 --> 00:40:11,910
+ناقص تلاتة YJ هيبتبقى هنا تلاتة X في اتجاه الـ J
+
+602
+00:40:11,910 --> 00:40:15,190
+زي ما هي الان اللي ضالت هنا اللي ما عملناهاش هي
+
+603
+00:40:15,190 --> 00:40:18,750
+تلاتة Y في اتجاه الـ I هيها زي ما هي إذا تلاتة Y
+
+604
+00:40:18,750 --> 00:40:22,310
+في اتجاه الـ I هيها زي ما هي ودول الناقص تلاتة في
+
+605
+00:40:22,310 --> 00:40:24,590
+اتجاه الـ J مع الناقص تلاتة في اتجاه الـ J بيصير
+
+606
+00:40:24,590 --> 00:40:28,190
+ناقص 6 مع اتجاه الـ J وهذه تلاتة X زي ما هي بيكون
+
+607
+00:40:28,190 --> 00:40:32,210
+فعلا هذا المقدار بسوء هذا المقدار اللي هي من خلال
+
+608
+00:40:32,210 --> 00:40:36,860
+اللي هو مثلنا اللي أماناالعناء دي اللي حكيناه قبل
+
+609
+00:40:36,860 --> 00:40:39,980
+شوية كله بس بدل functions of two variables خلّيني
+
+610
+00:40:39,980 --> 00:40:43,740
+أحكي functions of three variables Similarly وسهل
+
+611
+00:40:43,740 --> 00:40:50,080
+functions of three variables هي كما يعنيعندي for a
+
+612
+00:40:50,080 --> 00:40:53,220
+differentiable function f of x,y,z and a unit
+
+613
+00:40:53,220 --> 00:40:58,200
+vector u بالسوية u1 في تجارة i و u2 في تجارة j زاد
+
+614
+00:40:58,200 --> 00:41:01,020
+u3 في تجارة k إذا أن الآن الـ function عندنا بدل
+
+615
+00:41:01,020 --> 00:41:03,820
+ما هي two variables هي three و الـ vector طبعا بدل
+
+616
+00:41:03,820 --> 00:41:08,100
+ما يكون اللي هو بدلات i و j هيكون بدلات i,j و k
+
+617
+00:41:08,100 --> 00:41:11,350
+لأنه صار face basedالأن نحن لدينا الـ Gradient للـ
+
+618
+00:41:11,350 --> 00:41:14,830
+F عارفينه دل أف واي دل X في تجاه I زاد دل أف واي
+
+619
+00:41:14,830 --> 00:41:18,330
+دل Y في تجاه J زاد دل أف واي دل Z في تجاه K هذا
+
+620
+00:41:18,330 --> 00:41:21,270
+تعريفه لـ Function of Three Variables الـ
+
+621
+00:41:21,270 --> 00:41:23,810
+Directional Derivative في هذه الحالة هي Gradient
+
+622
+00:41:23,810 --> 00:41:29,260
+للـ F ضد Uدُط الـ U اللي هي عبارة عن هذا دُط ال U
+
+623
+00:41:29,260 --> 00:41:33,340
+بيصير عبارة عن دل F by دل X في U واحد زي دل F by
+
+624
+00:41:33,340 --> 00:41:37,400
+دل Y في U اتنين زي دل F by دل Z في U تلاتة إلى أن
+
+625
+00:41:37,400 --> 00:41:40,720
+نفس ما عملنا قبل بشوية الـ directional derivative
+
+626
+00:41:40,720 --> 00:41:45,440
+can once again be written in the form D U F بساوية
+
+627
+00:41:45,440 --> 00:41:48,520
+ال gradient لل F دُط ال U اللي هي بساوية المجموعة
+
+628
+00:41:48,520 --> 00:41:50,820
+للأول في المجموعة الذاتية في الـ cosine الزاوية
+
+629
+00:41:50,820 --> 00:41:53,650
+اللي بينهموالـ magnitude ده الـ U طبعا هذا بناخده
+
+630
+00:41:53,650 --> 00:41:58,410
+احنا اللي هو direction يعني unit vector بساوي واحد
+
+631
+00:41:58,410 --> 00:42:01,630
+طبعا فبصير ال gradient لف كوسيني ثتا وهذه زي ما
+
+632
+00:42:01,630 --> 00:42:05,770
+قلنا قبل اللي هو most increases rapidly لف in the
+
+633
+00:42:05,770 --> 00:42:09,610
+direction of ال gradient أو most decreases rapidly
+
+634
+00:42:09,610 --> 00:42:12,250
+in the opposite direction اللي هو في longest ال
+
+635
+00:42:12,250 --> 00:42:18,500
+gradientوالعموديات فيه اللي هي بتخلي الـ F0 الـ
+
+636
+00:42:18,500 --> 00:42:21,700
+derivative بالـ 0 نفس اللي حكيناه قبل شوية ما بديش
+
+637
+00:42:21,700 --> 00:42:26,380
+أعيده ولو نهيته يلّا نصلّع النجلة ناخد أخر مثال
+
+638
+00:42:26,380 --> 00:42:30,090
+مثال السادس find the derivativeأف اكس واي اكس
+
+639
+00:42:30,090 --> 00:42:32,470
+تكعيب نقص اكس واي تربية نقص سد عنده النقطة
+
+640
+00:42:32,470 --> 00:42:35,770
+المذكورة بي نوت واحد واحد سفر in the direction of
+
+641
+00:42:35,770 --> 00:42:39,770
+V الأمامي بعدين سألني in what directions does F
+
+642
+00:42:39,770 --> 00:42:42,890
+change most rapidly at B or not يعني بيقوللي في أي
+
+643
+00:42:42,890 --> 00:42:47,890
+direction F increases most rapidly does it change
+
+644
+00:42:47,890 --> 00:42:51,570
+most rapidly يعني سواء increases أو decreases and
+
+645
+00:42:51,570 --> 00:42:55,850
+what are the rates of change in each case ماشي طيب
+
+646
+00:42:55,850 --> 00:42:59,810
+عشان نشوفأول شي نجي للأول عشان نودد الـ Direction
+
+647
+00:42:59,810 --> 00:43:03,230
+الـ derivative لازم نودد الـ Direction في اتجاهه و
+
+648
+00:43:03,230 --> 00:43:07,090
+نودده اللي هو التفاضل الـ Direction of V in the
+
+649
+00:43:07,090 --> 00:43:10,230
+direction of V Direction of V أش بساوي اللي هو الـ
+
+650
+00:43:10,230 --> 00:43:15,710
+V على مجنتيودهالـ V هي مجموعته الجذر التربي إلى 4
+
+651
+00:43:15,710 --> 00:43:24,070
+زي 9 زي 36 يعني 7 يعني بيصير 2 على 7 في اتجاه I زي
+
+652
+00:43:24,070 --> 00:43:32,750
+3 على 7 في اتجاه K زي 6 على 7 في اتجاه Jلأن هي الـ
+
+653
+00:43:32,750 --> 00:43:35,830
+U لأنها نوجد في الـ Gradient لأن هي اللي بيوجد في
+
+654
+00:43:35,830 --> 00:43:37,890
+الـ Directional Derivative الـ Gradient و الـ
+
+655
+00:43:37,890 --> 00:43:41,650
+Direction الـ Gradient إيش بيساوي؟ بنفضل هذه
+
+656
+00:43:41,650 --> 00:43:46,110
+بالنسبة لـ X ثلاثة X تربيع ناقص Y تربيع هي تجاه
+
+657
+00:43:46,110 --> 00:43:52,070
+الـ I ناقص بنفضل ل Y اللي هي اتنين XY لحالها في
+
+658
+00:43:52,070 --> 00:43:56,430
+اتجاه الـ J ونفضل بالنسبة لZ الآن اللي هي في اتجاه
+
+659
+00:43:56,430 --> 00:44:00,870
+الـ K بتطلع ناقص واحدهي الجديدة and so الـ
+
+660
+00:44:00,870 --> 00:44:04,490
+gradient لـ F عند مين عند واحد واحد و سفر هذه
+
+661
+00:44:04,490 --> 00:44:10,530
+بتصير عبارة عن اللي هي عبارة عن تلاتة ناقص واحد
+
+662
+00:44:10,530 --> 00:44:15,490
+اللي هي اتنين اه وهذه ناقص اتنين واحد في واحد في
+
+663
+00:44:15,490 --> 00:44:20,630
+واحد ناقص اتنين وهذه ناقص اللي هي ايش K هي اللي هو
+
+664
+00:44:20,630 --> 00:44:23,810
+ال gradient لـ F عنده النقطة المذكورة اذا then the
+
+665
+00:44:23,810 --> 00:44:26,330
+derivative of F at واحد واحد و Zero in the
+
+666
+00:44:26,330 --> 00:44:30,520
+direction of V isالـ Gradient للـ F في الـ
+
+667
+00:44:30,520 --> 00:44:34,500
+Direction في الـ U يعني الـ Gradient للـ F هي اللي
+
+668
+00:44:34,500 --> 00:44:38,920
+هي الـ U وهي الـ U وهي ال Gradient للـ F هي الـ U
+
+669
+00:44:38,920 --> 00:44:43,040
+وهي ال Gradient للـ F نضربهم dot product 2 ع 7
+
+670
+00:44:43,040 --> 00:44:49,520
+بيصير 4 ع 7 نقص 3 ع 7 في نقص 2 نقص 6 ع 7 و6 ع 7 في
+
+671
+00:44:49,520 --> 00:44:54,880
+نقص 1 اللي هي اسف نقص 3 في نقص 2 بيصير زائد 6 ع 7
+
+672
+00:44:54,880 --> 00:44:59,900
+و6 ع 7 في نقص 1 نقص 6 ع 7هذه تذهب مع هذه وتظهر 4
+
+673
+00:44:59,900 --> 00:45:04,900
+على 7 اللي هي derivative of f in the direction of
+
+674
+00:45:04,900 --> 00:45:09,720
+v طيب لأ ال function increases most rapidly in the
+
+675
+00:45:09,720 --> 00:45:12,820
+direction of ال gradient صح ولا لأ يعني في اتجاه
+
+676
+00:45:12,820 --> 00:45:16,380
+هذا بتظهر اللي هو ايش increases most rapidly اللي
+
+677
+00:45:16,380 --> 00:45:19,140
+هو في اتجاه هذا اتجاهه مين؟
+
+678
+00:45:24,830 --> 00:45:28,330
+على الجدر التربيعي للأربعة زايد الأربعة زايد واحد
+
+679
+00:45:28,330 --> 00:45:31,750
+تلاتة يعني اللي هو بيصير اللي هو function
+
+680
+00:45:31,750 --> 00:45:35,230
+increases most rapidly in the direction vector
+
+681
+00:45:35,230 --> 00:45:40,470
+اللي هو U بتساوي 2 ع 3 I نقص 2 ع 3 J نقص 1 ع 3 K
+
+682
+00:45:40,470 --> 00:45:43,470
+and decreases most rapidly in the opposite
+
+683
+00:45:43,470 --> 00:45:47,080
+direction اللي هو في ناقص اللي هو gradient للـFأو
+
+684
+00:45:47,080 --> 00:45:53,640
+بمعنى آخر في اتجاه ناقص 2 على 3i زائد 2 ع 3j زائد
+
+685
+00:45:53,640 --> 00:45:57,840
+1 ع 3k هذا اللي هو decreases مستقبلي اللي هون
+
+686
+00:45:57,840 --> 00:46:01,920
+بدناه اللي هو القيمة ال directional derivative في
+
+687
+00:46:01,920 --> 00:46:05,430
+كل حالةالـ rates of change اللي هو معناته الـ
+
+688
+00:46:05,430 --> 00:46:08,930
+directional derivative in these directions are الـ
+
+689
+00:46:08,930 --> 00:46:11,270
+derivative اللي طبعا في الحالة الأولى most rapidly
+
+690
+00:46:11,270 --> 00:46:13,570
+بيكون بساوي الـ gradient نفسه اللي هو الجدر
+
+691
+00:46:13,570 --> 00:46:16,690
+التربيعي لأربعة زيادة الأربعة زيادة الواحد هي
+
+692
+00:46:16,690 --> 00:46:19,170
+الأربعة زيادة الأربعة زيادة الواحد اللي هو زي ما
+
+693
+00:46:19,170 --> 00:46:23,550
+قلنا تلاتة و اللي هي في حالة اللي هي ال decreases
+
+694
+00:46:23,550 --> 00:46:26,350
+most rapidly بتكون ال derivative في هذه الحالة
+
+695
+00:46:26,350 --> 00:46:30,120
+سالب تلاتة و بيكون هيك احنا أنهينا المحاضرةوالى
+
+696
+00:46:30,120 --> 00:46:33,780
+لقاء الآخر السلام عليكم ورحمة الله وبركاته
+