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Chapter 13 Functions of Several Variables. Copyright © Houghton Mifflin Company. All rights reserved.13-2 Definition of a Function of Two Variables.

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Presentation on theme: "Chapter 13 Functions of Several Variables. Copyright © Houghton Mifflin Company. All rights reserved.13-2 Definition of a Function of Two Variables."— Presentation transcript:

1 Chapter 13 Functions of Several Variables

2 Copyright © Houghton Mifflin Company. All rights reserved.13-2 Definition of a Function of Two Variables

3 Copyright © Houghton Mifflin Company. All rights reserved.13-3 Figure 13.2

4 Copyright © Houghton Mifflin Company. All rights reserved.13-4 Figure 13.5 and Figure 13.6

5 Copyright © Houghton Mifflin Company. All rights reserved.13-5 Figure 13.7 and Figure 13.8 Alfred B. Thomas/Earth Scenes USGS

6 Copyright © Houghton Mifflin Company. All rights reserved.13-6 Figure 13.14

7 Copyright © Houghton Mifflin Company. All rights reserved.13-7 Figure Reprinted with permission. © 1997 Automotive Engineering Magazine. Society of Automotive Engineers, Inc.

8 Copyright © Houghton Mifflin Company. All rights reserved.13-8 Figure 13.17

9 Copyright © Houghton Mifflin Company. All rights reserved.13-9 Rotatable Graphs I

10 Copyright © Houghton Mifflin Company. All rights reserved Rotatable Graphs II

11 Copyright © Houghton Mifflin Company. All rights reserved Rotatable Graphs III

12 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.18

13 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.19

14 Copyright © Houghton Mifflin Company. All rights reserved Definition of the Limit of a Function of Two Variables and Figure 13.20

15 Copyright © Houghton Mifflin Company. All rights reserved Definition of Continuity of a Function of Two Variables

16 Copyright © Houghton Mifflin Company. All rights reserved Theorem 13.1 Continuous Functions of Two Variables

17 Copyright © Houghton Mifflin Company. All rights reserved Figure and Figure 13.25

18 Copyright © Houghton Mifflin Company. All rights reserved Theorem 13.2 Continuity of a Composite Function

19 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.28

20 Copyright © Houghton Mifflin Company. All rights reserved Definition of Continuity of a Function of Three Variables

21 Copyright © Houghton Mifflin Company. All rights reserved Definition of Partial Derivatives of a Function of Two Variables

22 Copyright © Houghton Mifflin Company. All rights reserved Notation for First Partial Derivatives

23 Copyright © Houghton Mifflin Company. All rights reserved Figure and Figure 13.30

24 Copyright © Houghton Mifflin Company. All rights reserved Theorem 13.3 Equality of Mixed Partial Derivatives

25 Copyright © Houghton Mifflin Company. All rights reserved Definition of Total Differential

26 Copyright © Houghton Mifflin Company. All rights reserved Definition of Differentiability

27 Copyright © Houghton Mifflin Company. All rights reserved Theorem 13.4 Sufficient Condition for Differentiability

28 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.35

29 Copyright © Houghton Mifflin Company. All rights reserved Theorem 13.5 Differentiability Implies Continuity

30 Copyright © Houghton Mifflin Company. All rights reserved Theorem 13.6 Chain Rule: One Independent Variable and Figure 13.39

31 Copyright © Houghton Mifflin Company. All rights reserved Theorem 13.7 Chain Rule: Two Independent Variables and Figure 13.41

32 Copyright © Houghton Mifflin Company. All rights reserved Theorem 13.8 Chain Rule: Implicit Differentiation

33 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.42, Figure 13.43, and Figure 13.44

34 Copyright © Houghton Mifflin Company. All rights reserved Definition of Directional Derivative

35 Copyright © Houghton Mifflin Company. All rights reserved Theorem 13.9 Directional Derivative

36 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.45

37 Copyright © Houghton Mifflin Company. All rights reserved Definition of Gradient of a Function of Two Variables and Figure 13.48

38 Copyright © Houghton Mifflin Company. All rights reserved Theorem Alternative Form of the Directional Derivative

39 Copyright © Houghton Mifflin Company. All rights reserved Theorem Properties of the Gradient

40 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.50

41 Copyright © Houghton Mifflin Company. All rights reserved Theorem Gradient Is Normal to Level Curves

42 Copyright © Houghton Mifflin Company. All rights reserved Directional Derivative and Gradient for Three Variables

43 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.56

44 Copyright © Houghton Mifflin Company. All rights reserved Definition of Tangent Plane and Normal Line

45 Copyright © Houghton Mifflin Company. All rights reserved Theorem Equation of Tangent Plane

46 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.61

47 Copyright © Houghton Mifflin Company. All rights reserved Theorem Gradient Is Normal to Level Surfaces

48 Copyright © Houghton Mifflin Company. All rights reserved Figure and Theorem Extreme Value Theorem

49 Copyright © Houghton Mifflin Company. All rights reserved Definition of Relative Extrema and Figure 13.64

50 Copyright © Houghton Mifflin Company. All rights reserved Definition of Critical Point

51 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.65

52 Copyright © Houghton Mifflin Company. All rights reserved Theorem Relative Extrema Occur Only at Critical Points

53 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.68

54 Copyright © Houghton Mifflin Company. All rights reserved Theorem Second Partials Test

55 Copyright © Houghton Mifflin Company. All rights reserved Figure and Figure 13.74

56 Copyright © Houghton Mifflin Company. All rights reserved Figure 13.75

57 Copyright © Houghton Mifflin Company. All rights reserved Theorem Least Squares Regression Line

58 Copyright © Houghton Mifflin Company. All rights reserved Figure and Figure 13.78

59 Copyright © Houghton Mifflin Company. All rights reserved Theorem Lagrange's Theorem

60 Copyright © Houghton Mifflin Company. All rights reserved Method of Lagrange Multipliers


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