Transcript Chapter 13
Chapter 13 Functions of Several Variables Definition of a Function of Two Variables Copyright © Houghton Mifflin Company. All rights reserved. 13-2 Figure 13.2 Copyright © Houghton Mifflin Company. All rights reserved. 13-3 Figure 13.5 and Figure 13.6 Copyright © Houghton Mifflin Company. All rights reserved. 13-4 Figure 13.7 and Figure 13.8 Alfred B. Thomas/Earth Scenes USGS Copyright © Houghton Mifflin Company. All rights reserved. 13-5 Figure 13.14 Copyright © Houghton Mifflin Company. All rights reserved. 13-6 Figure 13.15 Reprinted with permission. © 1997 Automotive Engineering Magazine. Society of Automotive Engineers, Inc. Copyright © Houghton Mifflin Company. All rights reserved. 13-7 Figure 13.17 Copyright © Houghton Mifflin Company. All rights reserved. 13-8 Rotatable Graphs I Copyright © Houghton Mifflin Company. All rights reserved. 13-9 Rotatable Graphs II Copyright © Houghton Mifflin Company. All rights reserved. 13-10 Rotatable Graphs III Copyright © Houghton Mifflin Company. All rights reserved. 13-11 Figure 13.18 Copyright © Houghton Mifflin Company. All rights reserved. 13-12 Figure 13.19 Copyright © Houghton Mifflin Company. All rights reserved. 13-13 Definition of the Limit of a Function of Two Variables and Figure 13.20 Copyright © Houghton Mifflin Company. All rights reserved. 13-14 Definition of Continuity of a Function of Two Variables Copyright © Houghton Mifflin Company. All rights reserved. 13-15 Theorem 13.1 Continuous Functions of Two Variables Copyright © Houghton Mifflin Company. All rights reserved. 13-16 Figure 13.24 and Figure 13.25 Copyright © Houghton Mifflin Company. All rights reserved. 13-17 Theorem 13.2 Continuity of a Composite Function Copyright © Houghton Mifflin Company. All rights reserved. 13-18 Figure 13.28 Copyright © Houghton Mifflin Company. All rights reserved. 13-19 Definition of Continuity of a Function of Three Variables Copyright © Houghton Mifflin Company. All rights reserved. 13-20 Definition of Partial Derivatives of a Function of Two Variables Copyright © Houghton Mifflin Company. All rights reserved. 13-21 Notation for First Partial Derivatives Copyright © Houghton Mifflin Company. All rights reserved. 13-22 Figure 13.29 and Figure 13.30 Copyright © Houghton Mifflin Company. All rights reserved. 13-23 Theorem 13.3 Equality of Mixed Partial Derivatives Copyright © Houghton Mifflin Company. All rights reserved. 13-24 Definition of Total Differential Copyright © Houghton Mifflin Company. All rights reserved. 13-25 Definition of Differentiability Copyright © Houghton Mifflin Company. All rights reserved. 13-26 Theorem 13.4 Sufficient Condition for Differentiability Copyright © Houghton Mifflin Company. All rights reserved. 13-27 Figure 13.35 Copyright © Houghton Mifflin Company. All rights reserved. 13-28 Theorem 13.5 Differentiability Implies Continuity Copyright © Houghton Mifflin Company. All rights reserved. 13-29 Theorem 13.6 Chain Rule: One Independent Variable and Figure 13.39 Copyright © Houghton Mifflin Company. All rights reserved. 13-30 Theorem 13.7 Chain Rule: Two Independent Variables and Figure 13.41 Copyright © Houghton Mifflin Company. All rights reserved. 13-31 Theorem 13.8 Chain Rule: Implicit Differentiation Copyright © Houghton Mifflin Company. All rights reserved. 13-32 Figure 13.42, Figure 13.43, and Figure 13.44 Copyright © Houghton Mifflin Company. All rights reserved. 13-33 Definition of Directional Derivative Copyright © Houghton Mifflin Company. All rights reserved. 13-34 Theorem 13.9 Directional Derivative Copyright © Houghton Mifflin Company. All rights reserved. 13-35 Figure 13.45 Copyright © Houghton Mifflin Company. All rights reserved. 13-36 Definition of Gradient of a Function of Two Variables and Figure 13.48 Copyright © Houghton Mifflin Company. All rights reserved. 13-37 Theorem 13.10 Alternative Form of the Directional Derivative Copyright © Houghton Mifflin Company. All rights reserved. 13-38 Theorem 13.11 Properties of the Gradient Copyright © Houghton Mifflin Company. All rights reserved. 13-39 Figure 13.50 Copyright © Houghton Mifflin Company. All rights reserved. 13-40 Theorem 13.12 Gradient Is Normal to Level Curves Copyright © Houghton Mifflin Company. All rights reserved. 13-41 Directional Derivative and Gradient for Three Variables Copyright © Houghton Mifflin Company. All rights reserved. 13-42 Figure 13.56 Copyright © Houghton Mifflin Company. All rights reserved. 13-43 Definition of Tangent Plane and Normal Line Copyright © Houghton Mifflin Company. All rights reserved. 13-44 Theorem 13.13 Equation of Tangent Plane Copyright © Houghton Mifflin Company. All rights reserved. 13-45 Figure 13.61 Copyright © Houghton Mifflin Company. All rights reserved. 13-46 Theorem 13.14 Gradient Is Normal to Level Surfaces Copyright © Houghton Mifflin Company. All rights reserved. 13-47 Figure 13.63 and Theorem 13.15 Extreme Value Theorem Copyright © Houghton Mifflin Company. All rights reserved. 13-48 Definition of Relative Extrema and Figure 13.64 Copyright © Houghton Mifflin Company. All rights reserved. 13-49 Definition of Critical Point Copyright © Houghton Mifflin Company. All rights reserved. 13-50 Figure 13.65 Copyright © Houghton Mifflin Company. All rights reserved. 13-51 Theorem 13.16 Relative Extrema Occur Only at Critical Points Copyright © Houghton Mifflin Company. All rights reserved. 13-52 Figure 13.68 Copyright © Houghton Mifflin Company. All rights reserved. 13-53 Theorem 13.17 Second Partials Test Copyright © Houghton Mifflin Company. All rights reserved. 13-54 Figure 13.73 and Figure 13.74 Copyright © Houghton Mifflin Company. All rights reserved. 13-55 Figure 13.75 Copyright © Houghton Mifflin Company. All rights reserved. 13-56 Theorem 13.18 Least Squares Regression Line Copyright © Houghton Mifflin Company. All rights reserved. 13-57 Figure 13.77 and Figure 13.78 Copyright © Houghton Mifflin Company. All rights reserved. 13-58 Theorem 13.19 Lagrange's Theorem Copyright © Houghton Mifflin Company. All rights reserved. 13-59 Method of Lagrange Multipliers Copyright © Houghton Mifflin Company. All rights reserved. 13-60