Preface vii

Features ix

Supplements xv

Acknowledgments xvii

About the Authors xix

11 Vectors 1

Preview 1

11.1 Vectors in the Plane 2

11.2 Vectors in Three-Dimensional Space 12

11.3 The Dot Product and Applications 21

11.4 The Cross Product and Triple Product 32

11.5 Lines and Planes in Space 44

Summary of Key Topics 58

Genesis & Development 62

12 Vector-Valued Functions 65

Preview 65

12.1 Vector-Valued Functions—Limits, Derivatives, and Continuity 66

12.2 Velocity and Acceleration 77

12.3 Tangent Vectors and Arc Length 87

12.4 Curvature 97

12.5 Applications of Vector-Valued Functions to Motion 107

Summary of Key Topics 121

Genesis & Development 125

13 Functions of Several Variables 129

Preview 129

13.1 Functions of Several Variables 130

13.2 Cylinders and Quadric Surfaces 141

13.3 Limits and Continuity 150

13.4 Partial Derivatives 156

13.5 Differentiability and the Chain Rule 166

13.6 Gradients and Directional Derivatives 178

13.7 Tangent Planes 187

13.8 Maximum-Minimum Problems 198

13.9 Lagrange Multipliers 212

Summary of Key Topics 222

Genesis & Development 226

14 Multiple Integrals 231

Preview 231

14.1 Double Integrals over Rectangular Regions 232

14.2 Integration over More General Regions 240

14.3 Calculation of Volumes of Solids 248

14.4 Polar Coordinates 254

14.5 Integrating in Polar Coordinates 263

14.6 Triple Integrals 277

14.7 Physical Applications 283

14.8 Other Coordinate Systems 292

Summary of Key Topics 298

Genesis & Development 304

15 Vector Calculus 307

Preview 307

15.1 Vector Fields 308

15.2 Line Integrals 317

15.3 Conservative Vector Fields and Path-Independence 328

15.4 Divergence, Gradient, and Curl 340

15.5 Green’s Theorem 348

15.6 Surface Integrals 358

15.7 Stokes’s Theorem 369

15.8 Flux and the Divergence Theorem 383

Summary of Key Topics 392

Genesis & Development 396

Appendix Answers to Selected Exercises 399

Index