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 13.15 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.13-10 Rotatable Graphs II

11. Copyright © Houghton Mifflin Company. All rights reserved.13-11 Rotatable Graphs III

12. Copyright © Houghton Mifflin Company. All rights reserved.13-12 Figure 13.18

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

14. Copyright © Houghton Mifflin Company. All rights reserved.13-14 Definition of the Limit of a Function of Two Variables and Figure 13.20

15. Copyright © Houghton Mifflin Company. All rights reserved.13-15 Definition of Continuity of a Function of Two Variables

16. Copyright © Houghton Mifflin Company. All rights reserved.13-16 Theorem 13.1 Continuous Functions of Two Variables

17. Copyright © Houghton Mifflin Company. All rights reserved.13-17 Figure 13.24 and Figure 13.25

18. Copyright © Houghton Mifflin Company. All rights reserved.13-18 Theorem 13.2 Continuity of a Composite Function

19. Copyright © Houghton Mifflin Company. All rights reserved.13-19 Figure 13.28

20. Copyright © Houghton Mifflin Company. All rights reserved.13-20 Definition of Continuity of a Function of Three Variables

21. Copyright © Houghton Mifflin Company. All rights reserved.13-21 Definition of Partial Derivatives of a Function of Two Variables

22. Copyright © Houghton Mifflin Company. All rights reserved.13-22 Notation for First Partial Derivatives

23. Copyright © Houghton Mifflin Company. All rights reserved.13-23 Figure 13.29 and Figure 13.30

24. Copyright © Houghton Mifflin Company. All rights reserved.13-24 Theorem 13.3 Equality of Mixed Partial Derivatives

25. Copyright © Houghton Mifflin Company. All rights reserved.13-25 Definition of Total Differential

26. Copyright © Houghton Mifflin Company. All rights reserved.13-26 Definition of Differentiability

27. Copyright © Houghton Mifflin Company. All rights reserved.13-27 Theorem 13.4 Sufficient Condition for Differentiability

28. Copyright © Houghton Mifflin Company. All rights reserved.13-28 Figure 13.35

29. Copyright © Houghton Mifflin Company. All rights reserved.13-29 Theorem 13.5 Differentiability Implies Continuity

30. Copyright © Houghton Mifflin Company. All rights reserved.13-30 Theorem 13.6 Chain Rule: One Independent Variable and Figure 13.39

31. Copyright © Houghton Mifflin Company. All rights reserved.13-31 Theorem 13.7 Chain Rule: Two Independent Variables and Figure 13.41

32. Copyright © Houghton Mifflin Company. All rights reserved.13-32 Theorem 13.8 Chain Rule: Implicit Differentiation

33. Copyright © Houghton Mifflin Company. All rights reserved.13-33 Figure 13.42, Figure 13.43, and Figure 13.44

34. Copyright © Houghton Mifflin Company. All rights reserved.13-34 Definition of Directional Derivative

35. Copyright © Houghton Mifflin Company. All rights reserved.13-35 Theorem 13.9 Directional Derivative

36. Copyright © Houghton Mifflin Company. All rights reserved.13-36 Figure 13.45

37. Copyright © Houghton Mifflin Company. All rights reserved.13-37 Definition of Gradient of a Function of Two Variables and Figure 13.48

38. Copyright © Houghton Mifflin Company. All rights reserved.13-38 Theorem 13.10 Alternative Form of the Directional Derivative

39. Copyright © Houghton Mifflin Company. All rights reserved.13-39 Theorem 13.11 Properties of the Gradient

40. Copyright © Houghton Mifflin Company. All rights reserved.13-40 Figure 13.50

41. Copyright © Houghton Mifflin Company. All rights reserved.13-41 Theorem 13.12 Gradient Is Normal to Level Curves

42. Copyright © Houghton Mifflin Company. All rights reserved.13-42 Directional Derivative and Gradient for Three Variables

43. Copyright © Houghton Mifflin Company. All rights reserved.13-43 Figure 13.56

44. Copyright © Houghton Mifflin Company. All rights reserved.13-44 Definition of Tangent Plane and Normal Line

45. Copyright © Houghton Mifflin Company. All rights reserved.13-45 Theorem 13.13 Equation of Tangent Plane

46. Copyright © Houghton Mifflin Company. All rights reserved.13-46 Figure 13.61

47. Copyright © Houghton Mifflin Company. All rights reserved.13-47 Theorem 13.14 Gradient Is Normal to Level Surfaces

48. Copyright © Houghton Mifflin Company. All rights reserved.13-48 Figure 13.63 and Theorem 13.15 Extreme Value Theorem

49. Copyright © Houghton Mifflin Company. All rights reserved.13-49 Definition of Relative Extrema and Figure 13.64

50. Copyright © Houghton Mifflin Company. All rights reserved.13-50 Definition of Critical Point

51. Copyright © Houghton Mifflin Company. All rights reserved.13-51 Figure 13.65

52. Copyright © Houghton Mifflin Company. All rights reserved.13-52 Theorem 13.16 Relative Extrema Occur Only at Critical Points

53. Copyright © Houghton Mifflin Company. All rights reserved.13-53 Figure 13.68

54. Copyright © Houghton Mifflin Company. All rights reserved.13-54 Theorem 13.17 Second Partials Test

55. Copyright © Houghton Mifflin Company. All rights reserved.13-55 Figure 13.73 and Figure 13.74

56. Copyright © Houghton Mifflin Company. All rights reserved.13-56 Figure 13.75

57. Copyright © Houghton Mifflin Company. All rights reserved.13-57 Theorem 13.18 Least Squares Regression Line

58. Copyright © Houghton Mifflin Company. All rights reserved.13-58 Figure 13.77 and Figure 13.78

59. Copyright © Houghton Mifflin Company. All rights reserved.13-59 Theorem 13.19 Lagrange's Theorem

60. Copyright © Houghton Mifflin Company. All rights reserved.13-60 Method of Lagrange Multipliers