Information about Multivariable Calculus - Part 1

  • Title: Introduction

  • Series: Multivariable Calculus

  • YouTube-Title: Multivariable Calculus 1 | Introduction

  • Bright video: https://youtu.be/4QhZTagNq9Q

  • Dark video: https://youtu.be/WQ-0s3KwYCA

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

  • Dark-PDF: Download PDF version of the dark video

  • Print-PDF: Download printable PDF version

  • Thumbnail (bright): Download PNG

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  • Subtitle on GitHub: mc01_sub_eng.srt

  • Timestamps

    00:00 Intro

    00:39 Prerequisites

    02:15 Applications of the course

    02:58 Content of the course

    04:20 Credits

  • Subtitle in English

    1 00:00:00,779 –> 00:00:02,759 Hello and welcome to this

    2 00:00:02,769 –> 00:00:04,590 new series about multi-

    3 00:00:04,869 –> 00:00:06,119 variable calculus.

    4 00:00:07,030 –> 00:00:08,590 This will be a video course

    5 00:00:08,600 –> 00:00:10,039 where I explain things

    6 00:00:10,050 –> 00:00:11,560 concerning functions

    7 00:00:11,569 –> 00:00:13,359 defined on R^n

    8 00:00:14,300 –> 00:00:15,869 For example, we will talk

    9 00:00:15,880 –> 00:00:17,840 a lot about derivatives like

    10 00:00:17,850 –> 00:00:19,159 partial derivatives,

    11 00:00:19,170 –> 00:00:21,079 directional derivatives and

    12 00:00:21,090 –> 00:00:22,430 total derivatives.

    13 00:00:23,200 –> 00:00:24,579 However, before we start

    14 00:00:24,590 –> 00:00:26,139 with this, I really want

    15 00:00:26,149 –> 00:00:27,870 to thank all the nice people

    16 00:00:27,879 –> 00:00:29,649 who make such new video courses

    17 00:00:29,659 –> 00:00:30,940 like this possible.

    18 00:00:31,770 –> 00:00:33,470 And there please don’t forget

    19 00:00:33,479 –> 00:00:35,180 if you support me on Steady,

    20 00:00:35,189 –> 00:00:36,819 you get access to the PDF

    21 00:00:36,830 –> 00:00:38,349 versions and quizzes for

    22 00:00:38,360 –> 00:00:39,470 all the videos.

    23 00:00:40,369 –> 00:00:41,939 Speaking of other videos,

    24 00:00:41,950 –> 00:00:43,639 I can tell you it will be

    25 00:00:43,650 –> 00:00:45,490 helpful that first you watch

    26 00:00:45,500 –> 00:00:47,139 my real analysis course

    27 00:00:47,900 –> 00:00:49,759 simply because there we

    28 00:00:49,770 –> 00:00:51,029 discuss the normal

    29 00:00:51,040 –> 00:00:52,560 calculus in one

    30 00:00:52,569 –> 00:00:53,240 variable.

    31 00:00:54,279 –> 00:00:56,139 In other words, multivariable

    32 00:00:56,150 –> 00:00:57,930 calculus will extend

    33 00:00:57,939 –> 00:00:59,259 this series here.

    34 00:01:00,139 –> 00:01:01,779 However, of course, you don’t

    35 00:01:01,790 –> 00:01:03,349 need to understand everything

    36 00:01:03,360 –> 00:01:05,099 in real analysis to start

    37 00:01:05,110 –> 00:01:06,500 with this multivariable

    38 00:01:06,510 –> 00:01:07,709 calculus course.

    39 00:01:08,400 –> 00:01:09,980 Moreover, soon you will

    40 00:01:09,989 –> 00:01:11,819 notice that we will extend

    41 00:01:11,830 –> 00:01:13,660 a lot of notions from one

    42 00:01:13,669 –> 00:01:15,410 variable here to several

    43 00:01:15,419 –> 00:01:16,480 variables there.

    44 00:01:17,449 –> 00:01:19,190 And exactly for this reason,

    45 00:01:19,260 –> 00:01:20,919 also my linear algebra

    46 00:01:20,930 –> 00:01:22,510 course can help you there.

    47 00:01:23,569 –> 00:01:25,180 This is what you will understand

    48 00:01:25,190 –> 00:01:26,940 immediately when we write

    49 00:01:26,949 –> 00:01:28,529 down functions from

    50 00:01:28,540 –> 00:01:30,300 R^n to R^m.

    51 00:01:31,430 –> 00:01:32,790 In the linear algebra course,

    52 00:01:32,800 –> 00:01:34,510 such functions would

    53 00:01:34,519 –> 00:01:36,139 be linear functions.

    54 00:01:36,870 –> 00:01:38,849 However, here in multivariable

    55 00:01:38,860 –> 00:01:40,739 calculus, we will consider

    56 00:01:40,750 –> 00:01:42,419 a lot of different functions

    57 00:01:42,430 –> 00:01:43,980 from R^n to R^m.

    58 00:01:44,889 –> 00:01:46,250 But still, we will

    59 00:01:46,260 –> 00:01:47,709 ask if we can

    60 00:01:47,720 –> 00:01:49,589 approximate these functions

    61 00:01:49,599 –> 00:01:51,069 with linear functions

    62 00:01:52,110 –> 00:01:53,769 which as you might know from

    63 00:01:53,779 –> 00:01:55,220 the one variable case

    64 00:01:55,230 –> 00:01:56,779 leads to the notion of

    65 00:01:56,790 –> 00:01:57,860 derivatives.

    66 00:01:58,870 –> 00:02:00,699 Moreover, you also know

    67 00:02:00,709 –> 00:02:02,599 such derivatives can help

    68 00:02:02,610 –> 00:02:04,080 finding maxima and

    69 00:02:04,089 –> 00:02:05,720 minima of functions.

    70 00:02:06,910 –> 00:02:08,529 And now with this course,

    71 00:02:08,538 –> 00:02:10,320 we will be able to do this

    72 00:02:10,330 –> 00:02:12,160 for quantities of interest

    73 00:02:12,169 –> 00:02:13,610 that depend on several

    74 00:02:13,619 –> 00:02:14,360 factors.

    75 00:02:15,419 –> 00:02:17,039 And indeed, you will see

    76 00:02:17,050 –> 00:02:18,740 we have a lot of applications

    77 00:02:18,750 –> 00:02:19,320 for this.

    78 00:02:20,190 –> 00:02:21,550 In addition, this course

    79 00:02:21,559 –> 00:02:22,919 now can help you to

    80 00:02:22,929 –> 00:02:24,720 understand my more advanced

    81 00:02:24,729 –> 00:02:25,440 courses.

    82 00:02:26,389 –> 00:02:28,039 Indeed quite fitting after

    83 00:02:28,050 –> 00:02:29,929 this will be my manifolds

    84 00:02:29,940 –> 00:02:30,649 course.

    85 00:02:31,539 –> 00:02:32,970 And also functional

    86 00:02:32,979 –> 00:02:34,589 analysis will generalize

    87 00:02:34,600 –> 00:02:36,199 a lot of topics from here.

    88 00:02:37,059 –> 00:02:38,800 And lastly, I can tell you

    89 00:02:38,809 –> 00:02:40,679 if you know how to deal with

    90 00:02:40,690 –> 00:02:42,320 multivariable calculus,

    91 00:02:42,330 –> 00:02:44,000 you also know how to deal

    92 00:02:44,009 –> 00:02:45,520 with complex analysis.

    93 00:02:46,529 –> 00:02:48,119 So in some sense here, you

    94 00:02:48,130 –> 00:02:49,839 can use a lot of facts for

    95 00:02:49,850 –> 00:02:51,649 functions from R² into

    96 00:02:51,660 –> 00:02:52,309 R².

    97 00:02:53,220 –> 00:02:53,679 OK.

    98 00:02:53,690 –> 00:02:55,100 Now I think we can talk a

    99 00:02:55,110 –> 00:02:56,600 little bit about the topics

    100 00:02:56,610 –> 00:02:58,080 you can expect here.

    101 00:02:58,830 –> 00:03:00,259 First, we start very

    102 00:03:00,270 –> 00:03:02,000 simple and talk about

    103 00:03:02,009 –> 00:03:03,600 continuous functions.

    104 00:03:04,410 –> 00:03:06,160 So we will extend the definition

    105 00:03:06,169 –> 00:03:08,119 of continuity that we

    106 00:03:08,130 –> 00:03:09,949 learned in real analysis.

    107 00:03:10,779 –> 00:03:12,570 Then in a similar sense,

    108 00:03:12,580 –> 00:03:14,380 we will generalize the notion

    109 00:03:14,389 –> 00:03:16,110 of a derivative from one

    110 00:03:16,119 –> 00:03:18,029 variable. However,

    111 00:03:18,039 –> 00:03:19,580 it turns out there are a

    112 00:03:19,589 –> 00:03:21,330 lot of different possibilities

    113 00:03:21,339 –> 00:03:22,020 for that.

    114 00:03:22,029 –> 00:03:23,440 And therefore, we will talk

    115 00:03:23,449 –> 00:03:25,059 about partial derivatives,

    116 00:03:25,070 –> 00:03:26,979 directional derivatives and

    117 00:03:26,990 –> 00:03:28,270 total derivatives.

    118 00:03:29,220 –> 00:03:31,169 Indeed, we will see all

    119 00:03:31,179 –> 00:03:32,619 of these notions will be

    120 00:03:32,630 –> 00:03:33,770 important for us.

    121 00:03:34,550 –> 00:03:36,220 Moreover, by using

    122 00:03:36,229 –> 00:03:37,889 these things here, we can

    123 00:03:37,899 –> 00:03:39,770 generalize an important theorem

    124 00:03:39,779 –> 00:03:41,229 from real analysis,

    125 00:03:41,300 –> 00:03:43,289 namely Taylor’s theorem.

    126 00:03:44,199 –> 00:03:45,679 So I would say this is a

    127 00:03:45,690 –> 00:03:47,229 very important fact

    128 00:03:47,330 –> 00:03:48,899 but also the most

    129 00:03:48,910 –> 00:03:50,369 important fact for multi-

    130 00:03:50,380 –> 00:03:52,149 variable calculus will be

    131 00:03:52,160 –> 00:03:53,580 the implicit function

    132 00:03:53,589 –> 00:03:54,259 theorem.

    133 00:03:55,089 –> 00:03:56,889 In fact, this is a result

    134 00:03:56,899 –> 00:03:58,449 that is used in a lot of

    135 00:03:58,460 –> 00:03:59,369 applications

    136 00:04:00,259 –> 00:04:01,639 and in a similar way, the

    137 00:04:01,649 –> 00:04:03,429 next one here is also used

    138 00:04:03,440 –> 00:04:04,080 a lot,

    139 00:04:04,089 –> 00:04:05,910 it’s the famous method of

    140 00:04:06,410 –> 00:04:07,630 Lagrange multipliers.

    141 00:04:08,399 –> 00:04:08,839 OK.

    142 00:04:08,850 –> 00:04:10,220 There we have it, this is

    143 00:04:10,229 –> 00:04:11,820 the overview of the course

    144 00:04:11,830 –> 00:04:13,190 and we start in the next

    145 00:04:13,199 –> 00:04:15,100 video with continuity.

    146 00:04:16,048 –> 00:04:17,649 Therefore, I would say let’s

    147 00:04:17,660 –> 00:04:19,149 meet there and have a nice

    148 00:04:19,160 –> 00:04:19,589 day.

    149 00:04:19,640 –> 00:04:20,390 Bye.

  • Quiz Content

    Q1: Which video courses should you watch before this one?

    A1: Real Analysis and Linear Algebra

    A2: Real Analysis and Functional Analysis

    A3: Functional Analysis and Linear Algebra

    A4: Manifolds and Complex Analysis

    Q2: What is not a correct definition of $f^\prime(x_0)$ for a function $f: \mathbb{R} \rightarrow \mathbb{R}$?

    A1: $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x - x_0}$.

    A2: $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x_0) - f(x)}{x_0 - x}$.

    A3: $\displaystyle \lim_{z \rightarrow x_0} \frac{f(x_0) - f(z)}{x_0 - z}$.

    A4: $\displaystyle \lim_{x_0 \rightarrow x} \frac{f(x_0) - f(x)}{x - x_0}$.

    A5: $\displaystyle \lim_{n \rightarrow \infty} \frac{f(x_n) - f(x_0)}{x_n - x_0}$ if we get the same value for each sequence $x_n \xrightarrow{n \rightarrow \infty} x_0$.

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