Multivariable calculus. Concepts and contexts

Front Cover......Page 1
Title Page......Page 12
Copyright......Page 13
Contents......Page 14
Untitled......Page 17
8 Infinite Sequences and Series......Page 29
8.1 Sequences......Page 30
8.2 Series......Page 40
8.3 The Integral and Comparison Tests; Estimating Sums......Page 50
8.4 Other Convergence Tests......Page 59
8.5 Power Series......Page 67
8.6 Representations of Functions as Power Series......Page 72
8.7 Taylor and Maclaurin Series......Page 78
8.8 The Binomial Series......Page 90
8.9 Applications of Taylor Polynomials 8.9......Page 94
Applied Project ■ Radiation from the Stars......Page 103
Review......Page 104
Focus on Problem Solving......Page 107
9 Vectors and the Geometry of Space......Page 109
9.1 Three-Dimensional Coordinate Systems......Page 110
9.2 Vectors......Page 115
9.3 The Dot Product......Page 124
9.4 The Cross Product......Page 130
Discovery Project ■ The Geometry of a Tetrahedron......Page 138
9.5 Equations of Lines and Planes......Page 139
Laboratory Project ■ Putting 3D in Perspective......Page 148
9.6 Functions and Surfaces......Page 149
9.7 Cylindrical and Spherical Coordinates......Page 158
Review......Page 163
Focus on Problem Solving......Page 166
10 Vector Functions......Page 167
10.1 Vector Functions and Space Curves......Page 168
10.2 Derivatives and Integrals of Vector Functions......Page 175
10.3 Arc Length and Curvature......Page 181
10.4 Motion in Space: Velocity and Acceleration......Page 189
Applied Project ■ Kepler’s Laws......Page 200
10.5 Parametric Surfaces......Page 201
Review......Page 207
Focus on Problem Solving......Page 209
11 Partial Derivatives......Page 211
11.1 Functions of Several Variables......Page 212
11.2 Limits and Continuity......Page 223
11.3 Partial Derivatives......Page 229
11.4 Tangent Planes and Linear Approximations......Page 242
11.5 The Chain Rule......Page 253
11.6 Directional Derivatives and the Gradient Vector......Page 261
11.7 Maximum and Minimum Values......Page 274
Applied Project ■ Designing a Dumpster......Page 284
Discovery Project ■ Quadratic Approximations and Critical Points......Page 285
11.8 Lagrange Multipliers......Page 286
Applied Project ■ Rocket Science......Page 293
Applied Project ■ Hydro-Turbine Optimization......Page 294
Review......Page 295
Focus on Problem Solving......Page 299
12 Multiple Integrals......Page 301
12.1 Double Integrals over Rectangles......Page 302
12.2 Iterated Integrals......Page 310
12.3 Double Integrals over General Regions......Page 316
12.4 Double Integrals in Polar Coordinates......Page 324
12.5 Applications of Double Integrals......Page 330
12.6 Surface Area......Page 340
12.7 Triple Integrals......Page 345
Discovery Project ■ Volumes of Hyperspheres......Page 354
12.8 Triple Integrals in Cylindrical and Spherical Coordinates......Page 355
Applied Project ■ Roller Derby......Page 361
12.9 Change of Variables in Multiple Integrals......Page 362
Review......Page 371
Focus on Problem Solving......Page 375
13 Vector Calculus......Page 377
13.1 Vector Fields......Page 378
13.2 Line Integrals......Page 385
13.3 The Fundamental Theorem for Line Integrals......Page 397
13.4 Green’s Theorem......Page 406
13.5 Curl and Divergence......Page 413
13.6 Surface Integrals......Page 421
13.7 Stokes’ Theorem......Page 432
Writing Project ■ Three Men and Two Theorems......Page 438
13.8 The Divergence Theorem......Page 439
13.9 Summary......Page 446
Review......Page 447
Focus on Problem Solving......Page 450
Appendixes......Page 452
D Precise Definitions of Limits......Page 453
E A Few Proofs......Page 454
H Polar Coordinates......Page 457
I Complex Numbers......Page 473
J Answers to Odd-Numbered Exercises......Page 482
Index......Page 504