Multivariable calculus

Cover Page
......Page 1
Half-title 
Page......Page 3
Tilte Page
......Page 5
Copyright Page
......Page 4
ABOUT THIS BOOK: NOTES FOR INSTRUCTORS
......Page 7
ACKNOWLEDGEMENTS
......Page 13
HOW TO USE THIS BOOK: NOTES FOR STUDENTS......Page 15
CONTENTS
......Page 17
What Is Multivariable Calculus?......Page 23
11.1: Sequences and Their Limits......Page 24
11.2: Infinite Series, Convergence, and Divergence......Page 33
11.3: Testing for Convergence; Estimating Limits......Page 44
11.4: Absolute Convergence; Alternating Series......Page 54
11.5: Power Series......Page 61
11.6: Power Series as Functions......Page 68
11.7: Taylor Series......Page 75
Summary......Page 79
Interlude: Fourier series......Page 84
12.1: Three-Dimensional Space......Page 86
12.2: Curves and Parametric Equations......Page 96
12.3: Polar Coordinates and Polar Curves......Page 107
12.4: Vectors......Page 117
12.5: Vector-Valued Functions, Derivatives, and Integrals......Page 126
12.6: Modeling Motion......Page 139
12.7: The Dot Product......Page 146
12.8: Lines and Planes in Three Dimensions......Page 158
12.9: The Cross Product......Page 166
Summary......Page 174
Interlude: Beyond Free Fall......Page 179
13.1: Functions of Several Variables......Page 181
13.2: Partial Derivatives......Page 192
13.3: Linear Approximation in Several Variables......Page 201
13.4: The Gradient and Directional Derivatives......Page 209
13.5: Higher-Order Derivatives and Quadratic Approximation......Page 217
13.6: Maxima, Minima, and Quadratic Approximation......Page 222
13.7: The Chain Rule......Page 232
13.8: Local Linearity: Some Theory of the Derivative......Page 242
Summary......Page 247
14.1: Multiple Integrals and Approximating Sums......Page 253
14.2: Calculating Integrals by Iteration......Page 268
14.3: Integrals over Nonrectangular Regions......Page 275
14.4: Double Integrals in Polar Coordinates......Page 283
14.5: Triple Integrals......Page 291
14.6: More Triple Integrals: Cylindrical and Spherical Coordinates......Page 297
14.7: Multiple Integrals Overviewed; Change of Variables......Page 302
Summary......Page 312
Interlude: Mass and Center of Mass......Page 316
15.1: Linear, Circular, and Combined Motion
......Page 319
Interlude: Cycloids and Epicycloids......Page 325
15.2: New Curves from Old......Page 327
15.3: Curvature......Page 331
15.4: Lagrange Multipliers and Constrained Optimization......Page 335
15.5: Improper Multivariable Integrals
......Page 342
Interlude: Constructing Pedal Curves......Page 348
16.1: Line Integrals......Page 350
16.2: More on Line Integrals; A Fundamental Theorem......Page 358
16.3: Green’s Theorem: Relating Line and Area Integrals......Page 369
16.4: Surfaces and Their Parametrizations......Page 377
16.5: Surface Integrals
......Page 383
16.6: Derivatives and Integrals of Vector Fields......Page 390
16.7: Back to Fundamentals: Stokes’s Theorem and the Divergence Theorem
......Page 395
A: Matrices and Matrix Algebra: A Crash Course......Page 405
B:
 Theory of Multivariable Calculus: Brief Glimpses......Page 414
C:
 Table of Derivatives and Integrals......Page 419
ANSWERS TO SELECTED EXERCISES......Page 423
C......Page 437
I......Page 438
P......Page 439
V......Page 440
W......Page 441