# 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

• 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,

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

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

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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