11.1 Contents

Preface  ix

Acknowledgements  xi

Historical Introduction: A Brief Account  xiii

Prerequisites and Notation  xxiii

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  19

1.3

Matrices, Determinants, and the Cross Product  31

1.4

Cylindrical and Spherical Coordinates  52

1.5

n-Dimensional Euclidean Space  60

Review Exercises for Chapter 1  70

2

Differentiation  75

2.1

The Geometry of Real-Valued Functions  76

2.2

Limits and Continuity  88

2.3

Differentiation  105

2.4

Introduction to Paths and Curves  116

2.5

Properties of the Derivative  124

2.6

Gradients and Directional Derivatives  135

Review Exercises for Chapter 2  144

3

Higher-Order Derivatives: Maxima and Minima   149

3.1

Iterated Partial Derivatives  150

3.2

Taylor’s Theorem  158

3.3

Extrema of Real-Valued Functions  166

3.4

Constrained Extrema and Lagrange Multipliers  185

3.5

The Implicit Function Theorem [Optional]  203

Review Exercises for Chapter 3  211

4

Vector-Valued Functions  217

4.1

Acceleration and Newton’s Second Law  217

4.2

Arc Length  228

4.3

Vector Fields  236

4.4

Divergence and Curl  245

Review Exercises for Chapter 4  260

5

Double and Triple Integrals  263

5.1

Introduction  263

5.2

The Double Integral Over a Rectangle  271

5.3

The Double Integral Over More General Regions  283

5.4

Changing the Order of Integration  289

5.5

The Triple Integral  294

Review Exercises for Chapter 5  304

6

The Change of Variables Formula and Applications of Integration  307

6.1

The Geometry of Maps from ℝ2 to ℝ2  308

6.2

The Change of Variables Theorem  314

6.3

Applications  329

6.4

Improper Integrals [Optional]  339

Review Exercises for Chapter 6  347

7

Integrals Over Paths and Surfaces  351

7.1

The Path Integral  351

7.2

Line Integrals  358

7.3

Parametrized Surfaces  375

7.4

Area of a Surface  383

7.5

Integrals of Scalar Functions Over Surfaces  393

7.6

Surface Integrals of Vector Fields  400

7.7

Applications to Differential Geometry, Physics, and Forms of Life  413

Review Exercises for Chapter 7  423

8

The Integral Theorems of Vector Analysis  427

8.1

Green’s Theorem  428

8.2

Stokes’ Theorem  439

8.3

Conservative Fields  453

8.4

Gauss’ Theorem  461

8.5

Differential Forms  476

Review Exercises for Chapter 8  490

Answers to Odd-Numbered Exercises  493

Index  533

Photo Credits  545

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To Jerrold E. Marsden, 1942–2010

Jerry Marsden, Carl F. Braun distinguished Professor at the California Institute of Technology, Fellow of the Royal Society (as was Isaac Newton), and one of the world’s pre-eminent applied mathematicians, passed away on September 21, 2010, while working on the sixth edition of Vector Calculus. Jerry’s interests were unusually broad; his work influenced physicists, engineers, life scientists, and mathematicians across the scientific and engineering spectrum. In addition to his many publications (over 400 archival and conference papers and 21 books) and major scientific prizes, he was a brilliant expositor and teacher. He motivated and encouraged colleagues and students alike, around the world and across an astonishing array of disciplines. He was a wonderful person and a close friend for almost half a century. He will be sorely missed.

—Anthony Tromba

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