Chapter 15: Gradients; Extreme Values; Differentials
Differentiability and Gradient
a. Definition (15.5.1), p. 862
b. Definition (15.5.2), p. 862
c. Theorem 15.1.3, p. 864
d. Differentiability implies continuity, p. 866
Gradients and Directional Derivatives
a. (15.2.1), p. 869
b. Directional derivative, (15.2.2), p. 871, figure 871
c. Theorem 15.2.4, p. 872
d. Theorem 15.2.5, p. 872
e. (15.2.7), p. 874
The Mean-Value Theorem: Chain Rules
a. Theorem 15.3.1, p. 879
b. Theorem 15.3.2, p. 879
c. Theorem 15.3.3. p. 881
d. Theorem 15.3.4, p. 882
e. (15.3.5), (15.3.6), p. 884
f. (15.3.9), p. 888
Local Extreme Values
Definition 15.5.1, p. 903
b. Theorem 15.5.2, p. 903
c. Critical points, stationary points, saddle points, p. 904, figure 15.5.1-2
d. Second partials test, p. 906
a.
Absolute Extreme Values
a. Definition 15.6.1, p. 911
b. Definition 15.6.2, p. 911
c. Extreme-value theorem, p. 912
Maxima and Minima with Side Conditions
a. Lagrange multiplier, (15.7.2), p. 919, figure 15.7.3
Differentials
a. (15.8.1), (15.8.2), p. 926
b. (15.8.5), p. 928
Reconstructing a Function from its Gradient
a. Theorem 15.9.2, p. 935
The Gradient as a Normal …
a. (15.4.1), p. 893, figure 15.4.1
b. (15.4.2), p. 895, figure 15.4.3
c. (15.4.4), (15.4.5), p. 895, figure 15.4.4
d. (15.4.6), p. 896, figure 15.4.5
e. (15.4.7), (15.4.8), pp. 897, 898
f. (15.4.10), p. 898
g. Surface z=g(x,y), (15.4.11), (15.4.12), p. 900
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Differentiability and Gradient
Definition (15.1.1), p. 862
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Differentiability and Gradient
Definition (15.1.2), p. 862
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Differentiability and Gradient
Theorem 15.1.3, p. 864
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Differentiability and Gradient
Differentiability implies continuity, p. 866
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Gradients and Directional Derivatives
(15.2.1), p. 869
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Gradients and Directional Derivatives
Directional derivative, (15.2.2), p. 871, figure 871
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Gradients and Directional Derivatives
Theorem 15.2.4, p. 872
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Gradients and Directional Derivatives
Theorem 15.2.5, p. 872
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Gradients and Directional Derivatives
(15.2.7), p. 874
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Mean-Value Theorem: Chain Rules
Theorem 15.3.1, p. 879
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Mean-Value Theorem: Chain Rules
Theorem 15.3.2, p. 879
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Mean-Value Theorem: Chain Rules
Theorem 15.3.3, p. 881
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Mean-Value Theorem: Chain Rules
Theorem 15.3.4, p. 882
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Mean-Value Theorem: Chain Rules
(15.3.5), (15.3.6), p. 884
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Mean-Value Theorem: Chain Rules
(15.3.9), p. 888
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Gradient as a Norm…
(15.4.1), p. 893, figure 15.4.1
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Gradient as a Norm…
(15.4.2), p. 895, figure 15.4.3
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Gradient as a Norm…
(15.4.4), (15.4.5), p. 895, figure 15.4.4
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Gradient as a Norm…
(15.4.6), p. 896, figure 15.4.5
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Gradient as a Norm…
(15.4.7), (15.4.8), pp. 897, 898
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Gradient as a Norm…
(15.4.10), p. 898
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
The Gradient as a Norm…
Surface z=g(x,y), (15.4.11), (15.4.12), p. 900
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Local Extreme Values
Definition 15.5.1, p. 903
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Local Extreme Values
Theorem 15.5.2, p. 903
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Local Extreme Values
Critical points, stationary points, saddle points, p. 904, figure 15.5.1-2
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Local Extreme Values
Secondary partials test, p. 906
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Absolute Extreme Values
Definition 15.6.1, p. 911
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Absolute Extreme Values
Definition 15.6.2, p. 911
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Absolute Extreme Values
Extreme-value theorem, p. 912
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Maxima and Minima with Side Conditions
Lagrange multiplier, (15.7.2), p. 919, figure 15.7.3
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Differentials
(15.8.1), (15.8.2), p. 926
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Differentials
(15.8.5), p. 928
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.
Reconstructing a Function from its Gradient
Theorem 15.9.2, p. 935
Main Menu
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2003 © John Wiley & Sons, Inc. All rights reserved.