Vector Calculus

CHAPTER 1 Vectors, Matrices, and Applications 1

1.1 Vectors 1

1.2 Applications in Geometry and Physics 10

1.3 The Dot Product 20

1.4 Matrices and Determinants 30

1.5 The Cross Product 39

Chapter Review 48

CHAPTER 2 Calculus of Functions of Several Variables 52

2.1 Real-Valued and Vector-Valued Functions of Several Variables 52

2.2 Graph of a Function of Several Variables 62

2.3 Limits and Continuity 76

2.4 Derivatives 93

2.5 Paths and Curves in R2 and R3 112

2.6 Properties of Derivatives 123

2.7 Gradient and Directional Derivative 135

2.8 Cylindrical and Spherical Coordinate Systems 151

Chapter Review 159

CHAPTER 3 Vector-Valued Functions of One Variable 164

3.1 World of Curves 164

3.2 Tangents, Velocity, and Acceleration 181

3.3 Length of a Curve 191

3.4 Acceleration and Curvature 200

3.5 Introduction to Differential Geometry of Curves 209

Chapter Review 215

CHAPTER 4 Scalar and Vector Fields 219

4.1 Higher-Order Partial Derivatives 219

4.2 Taylor’s Formula 230

4.3 Extreme Values of Real-Valued Functions 242

4.4 Optimization with Constraints and Lagrange Multipliers 261

4.5 Flow Lines 272

4.6 Divergence and Curl of a Vector Field 278

4.7 Implicit Function Theorem 292

4.8 Appendix: Some Identities of Vector Calculus 298

Chapter Review 302

CHAPTER 5 Integration Along Paths 306

5.1 Paths and Parametrizations 306

5.2 Path Integrals of Real-Valued Functions 316

5.3 Path Integrals of Vector Fields 325

5.4 Path Integrals Independent of Path 341

Chapter Review 360

CHAPTER 6 Double and Triple Integrals 363

6.1 Double Integrals: Definition and Properties 363

6.2 Double Integrals Over General Regions 375

6.3 Examples and Techniques of Evaluation of Double Integrals 394

6.4 Change of Variables in a Double Integral 401

6.5 Triple Integrals 417

Chapter Review 427

CHAPTER 7 Integration Over Surfaces, Properties, and Applications of Integrals 431

7.1 Parametrized Surfaces 431

7.2 World of Surfaces 448

7.3 Surface Integrals of Real-Valued Functions 462

7.4 Surface Integrals of Vector Fields 474

7.5 Integrals: Properties and Applications 484

Chapter Review 495

CHAPTER 8 Classical Integration Theorems of Vector Calculus 499

8.1 Green’s Theorem 499

8.2 The Divergence Theorem 511

8.3 Stokes’ Theorem 524

8.4 Differential Forms and Classical Integration Theorems 536

8.5 Vector Calculus in Electromagnetism 553

8.6 Vector Calculus in Fluid Flow 566

Chapter Review 576

APPENDIX A Various Results Used in This Book and Proofs of Differentiation Theorems 581

APPENDIX B Answers to Odd-Numbered Exercises 590

Index 615