Page i
Instructor’s Manual with Solutions
for Vector Calculus
Sixth Edition
Jerrold E. Marsden
California Institute of Technology
Anthony Tromba
University of California, Santa Cruz
W.H. Freeman and Co.
New York
, Page ii
ISBN-13: 978-1-4292-5479-3
ISBN-10: 1-4292-5479-3
c 2013, 2004, 1997 by W. H. Freeman and Company
All rights reserved.
Printed in the United States of America
First Printing
W. H. Freeman and Company
41 Madison Avenue
New York, NY 10010
Houndmills, Basingstoke RG21 6XS, England
www.whfreeman.com
, Page iii
Contents
1 The Geometry of Euclidean Space 1
1.1 Vectors in Two and Three-Dimensional Space . . . . . . . 1
1.2 The Inner Product, Length, and Distance . . . . . . . . . 12
1.3 Matrices, Determinants and the Cross Product . . . . . . 19
1.4 Cylindrical and Spherical Coordinates . . . . . . . . . . . 29
1.5 n-dimensional Euclidean Space . . . . . . . . . . . . . . . 32
Review Exercises: Chapter 1 . . . . . . . . . . . . . . . . . . . . . 39
2 Differentiation 41
2.1 Functions, Graphs, and Level Surfaces . . . . . . . . . . . 41
2.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . 54
2.3 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4 Introduction to Paths . . . . . . . . . . . . . . . . . . . . 71
2.5 Properties of the Derivative . . . . . . . . . . . . . . . . . 77
2.6 Gradients and Directional Derivatives . . . . . . . . . . . 86
Review Exercises: Chapter 2 . . . . . . . . . . . . . . . . . . . . . 92
3 Higher-Order Derivatives; Maxima and Minima 95
3.1 Iterated Partial Derivatives . . . . . . . . . . . . . . . . . 95
3.2 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 102
3.3 Extrema of Real Valued Functions . . . . . . . . . . . . . 109
3.4 Constrained Extrema and Lagrange multipliers . . . . . . 118
3.5 The Implicit Function Theorem . . . . . . . . . . . . . . . 130
Review Exercises: Chapter 3 . . . . . . . . . . . . . . . . . . . . . 137
, Page iv
Supplementary Problems and Exam Questions for Chapters 1-3 . 143
4 Vector Valued Functions 151
4.1 Acceleration and Newton’s Second Law . . . . . . . . . . . 151
4.2 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.4 Divergence and Curl . . . . . . . . . . . . . . . . . . . . . 176
Review Exercises: Chapter 4 . . . . . . . . . . . . . . . . . . . . . 180
5 Double and Triple Integrals 185
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.2 The Double Integral over a Rectangle . . . . . . . . . . . . 193
5.3 The Double Integral Over More General Regions . . . . . 200
5.4 Changing the Order of Integration . . . . . . . . . . . . . 205
5.5 The Triple Integral . . . . . . . . . . . . . . . . . . . . . . 210
Review Exercises: Chapter 5 . . . . . . . . . . . . . . . . . . . . . 215
6 The Change of Variables Formula and Applications 217
6.1 The Geometry of Maps from R2 to R2 . . . . . . . . . . . 217
6.2 The Change of Variables Theorem . . . . . . . . . . . . . 221
6.3 Applications of Double and Triple Integrals . . . . . . . . 232
6.4 Improper Integrals (Optional) . . . . . . . . . . . . . . . . 240
Review Exercises: Chapter 6 . . . . . . . . . . . . . . . . . . . . . 244
7 Integrals over Curves and Surfaces 247
7.1 The Path Integral . . . . . . . . . . . . . . . . . . . . . . . 247
7.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.3 Parametrized Surfaces . . . . . . . . . . . . . . . . . . . . 257
7.4 Area of a Surface . . . . . . . . . . . . . . . . . . . . . . . 265
7.5 Integrals of Scalar Functions over Surfaces . . . . . . . . . 270
7.6 Surface Integrals of Vector Functions . . . . . . . . . . . . 276
7.7 Applications: Differential Geometry, Physics, Forms of Life 286
Review Examples: Chapter 7 . . . . . . . . . . . . . . . . . . . . 290
Review Exercises: Chapter 7 . . . . . . . . . . . . . . . . . . . . . 294
8 The Integral Theorems of Vector Analysis 297
8.1 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 297
8.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . 302
8.3 Conservative Fields . . . . . . . . . . . . . . . . . . . . . . 310
8.4 Gauss’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . 315
8.5 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . 322
Review Examples: Chapter 8 . . . . . . . . . . . . . . . . . . . . 327
Review Exercises: Chapter 8 . . . . . . . . . . . . . . . . . . . . . 330
Miscellaneous Problems and Exams 333
Instructor’s Manual with Solutions
for Vector Calculus
Sixth Edition
Jerrold E. Marsden
California Institute of Technology
Anthony Tromba
University of California, Santa Cruz
W.H. Freeman and Co.
New York
, Page ii
ISBN-13: 978-1-4292-5479-3
ISBN-10: 1-4292-5479-3
c 2013, 2004, 1997 by W. H. Freeman and Company
All rights reserved.
Printed in the United States of America
First Printing
W. H. Freeman and Company
41 Madison Avenue
New York, NY 10010
Houndmills, Basingstoke RG21 6XS, England
www.whfreeman.com
, Page iii
Contents
1 The Geometry of Euclidean Space 1
1.1 Vectors in Two and Three-Dimensional Space . . . . . . . 1
1.2 The Inner Product, Length, and Distance . . . . . . . . . 12
1.3 Matrices, Determinants and the Cross Product . . . . . . 19
1.4 Cylindrical and Spherical Coordinates . . . . . . . . . . . 29
1.5 n-dimensional Euclidean Space . . . . . . . . . . . . . . . 32
Review Exercises: Chapter 1 . . . . . . . . . . . . . . . . . . . . . 39
2 Differentiation 41
2.1 Functions, Graphs, and Level Surfaces . . . . . . . . . . . 41
2.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . 54
2.3 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4 Introduction to Paths . . . . . . . . . . . . . . . . . . . . 71
2.5 Properties of the Derivative . . . . . . . . . . . . . . . . . 77
2.6 Gradients and Directional Derivatives . . . . . . . . . . . 86
Review Exercises: Chapter 2 . . . . . . . . . . . . . . . . . . . . . 92
3 Higher-Order Derivatives; Maxima and Minima 95
3.1 Iterated Partial Derivatives . . . . . . . . . . . . . . . . . 95
3.2 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 102
3.3 Extrema of Real Valued Functions . . . . . . . . . . . . . 109
3.4 Constrained Extrema and Lagrange multipliers . . . . . . 118
3.5 The Implicit Function Theorem . . . . . . . . . . . . . . . 130
Review Exercises: Chapter 3 . . . . . . . . . . . . . . . . . . . . . 137
, Page iv
Supplementary Problems and Exam Questions for Chapters 1-3 . 143
4 Vector Valued Functions 151
4.1 Acceleration and Newton’s Second Law . . . . . . . . . . . 151
4.2 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.4 Divergence and Curl . . . . . . . . . . . . . . . . . . . . . 176
Review Exercises: Chapter 4 . . . . . . . . . . . . . . . . . . . . . 180
5 Double and Triple Integrals 185
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.2 The Double Integral over a Rectangle . . . . . . . . . . . . 193
5.3 The Double Integral Over More General Regions . . . . . 200
5.4 Changing the Order of Integration . . . . . . . . . . . . . 205
5.5 The Triple Integral . . . . . . . . . . . . . . . . . . . . . . 210
Review Exercises: Chapter 5 . . . . . . . . . . . . . . . . . . . . . 215
6 The Change of Variables Formula and Applications 217
6.1 The Geometry of Maps from R2 to R2 . . . . . . . . . . . 217
6.2 The Change of Variables Theorem . . . . . . . . . . . . . 221
6.3 Applications of Double and Triple Integrals . . . . . . . . 232
6.4 Improper Integrals (Optional) . . . . . . . . . . . . . . . . 240
Review Exercises: Chapter 6 . . . . . . . . . . . . . . . . . . . . . 244
7 Integrals over Curves and Surfaces 247
7.1 The Path Integral . . . . . . . . . . . . . . . . . . . . . . . 247
7.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.3 Parametrized Surfaces . . . . . . . . . . . . . . . . . . . . 257
7.4 Area of a Surface . . . . . . . . . . . . . . . . . . . . . . . 265
7.5 Integrals of Scalar Functions over Surfaces . . . . . . . . . 270
7.6 Surface Integrals of Vector Functions . . . . . . . . . . . . 276
7.7 Applications: Differential Geometry, Physics, Forms of Life 286
Review Examples: Chapter 7 . . . . . . . . . . . . . . . . . . . . 290
Review Exercises: Chapter 7 . . . . . . . . . . . . . . . . . . . . . 294
8 The Integral Theorems of Vector Analysis 297
8.1 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 297
8.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . 302
8.3 Conservative Fields . . . . . . . . . . . . . . . . . . . . . . 310
8.4 Gauss’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . 315
8.5 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . 322
Review Examples: Chapter 8 . . . . . . . . . . . . . . . . . . . . 327
Review Exercises: Chapter 8 . . . . . . . . . . . . . . . . . . . . . 330
Miscellaneous Problems and Exams 333
