Vector calculus

Front Cover......Page 1
Title Page......Page 2
Opening Quotes......Page 3
Jerrold E. Marsden, 1942-2010......Page 8
Preface......Page 10
Historical Introduction: A Brief Account......Page 14
Prerequisites and Notation......Page 24
CONTENTS (with direct page links)......Page 4
 1.1 Vectors in Two- and Three-Dimensional Space......Page 28
 1.2 The Inner Product, Length, and Distance......Page 46
 1.3 Matrices, Determinants, and the Cross Product......Page 58
 1.4 Cylindrical and Spherical Coordinates......Page 79
 1.5 n-Dimensional Euclidean Space......Page 87
 Review Exercises for Chapter 1......Page 97
2. Differentiation......Page 102
 2.1 The Geometry of Real-Valued Functions......Page 103
 2.2 Limits and Continuity......Page 115
 2.3 Differentiation......Page 132
 2.4 Introduction to Paths and Curves......Page 143
 2.5 Properties of the Derivative......Page 151
 2.6 Gradients and Directional Derivatives......Page 162
 Review Exercises for Chapter 2......Page 171
3. Higher-Order Derivatives: Maxima and Minima......Page 176
 3.1 Iterated Partial Derivatives......Page 177
 3.2 Taylor’s Theorem......Page 185
 3.3 Extrema of Real-Valued Functions......Page 193
 3.4 Constrained Extrema and Lagrange Multipliers......Page 212
 3.5 The Implicit Function Theorem [Optional]......Page 230
 Review Exercises for Chapter 3......Page 238
 4.1 Acceleration and Newton’s Second Law......Page 244
 4.2 Arc Length......Page 255
 4.3 Vector Fields......Page 263
 4.4 Divergence and Curl......Page 272
 Review Exercises for Chapter 4......Page 287
 5.1 Introduction......Page 290
 5.2 The Double Integral Over a Rectangle......Page 298
 5.3 The Double Integral Over More General Regions......Page 310
 5.4 Changing the Order of Integration......Page 316
 5.5 The Triple Integral......Page 321
 Review Exercises for Chapter 5......Page 331
6. The Change of Variables Formula and Applications of Integration......Page 334
 6.1 The Geometry of Maps from R2 to R2......Page 335
 6.2 The Change of Variables Theorem......Page 341
 6.3 Applications......Page 356
 6.4 Improper Integrals [Optional]......Page 366
 Review Exercises for Chapter 6......Page 374
 7.1 The Path Integral......Page 378
 7.2 Line Integrals......Page 385
 7.3 Parametrized Surfaces......Page 402
 7.4 Area of a Surface......Page 410
 7.5 Integrals of Scalar Functions Over Surfaces......Page 420
 7.6 Surface Integrals of Vector Fields......Page 427
 7.7 Applications to Differential Geometry, Physics, and Forms of Life......Page 440
 Review Exercises for Chapter 7......Page 450
8. The Integral Theorems of Vector Analysis......Page 454
 8.1 Green’s Theorem......Page 455
 8.2 Stokes’ Theorem......Page 466
 8.3 Conservative Fields......Page 480
 8.4 Gauss’ Theorem......Page 488
 8.5 Differential Forms......Page 503
 Review Exercises for Chapter 8......Page 517
  1.2......Page 520
  1.3......Page 521
  1.4......Page 522
  1.5......Page 523
  review......Page 524
  2.1......Page 525
  2.2......Page 529
  2.3 - 2.4......Page 530
  2.5......Page 531
  2.6......Page 532
  review......Page 533
  3.1......Page 535
  3.2......Page 536
  3.3......Page 537
  3.5......Page 538
  review......Page 539
  4.1 - 4.2......Page 540
  4.4......Page 542
  review......Page 543
  5.1 - 5.2......Page 544
  5.4......Page 545
  review......Page 546
  6.1......Page 547
  6.2 - 6.3......Page 548
  review......Page 549
  7.2......Page 550
  7.3......Page 551
  7.4 - 7.5......Page 552
  7.7......Page 553
  8.1......Page 554
  8.2......Page 555
  8.4......Page 556
  8.5......Page 558
  review......Page 559
INDEX......Page 560
Photo Credits......Page 572
Table of Derivatives......Page 574
Table of Integrals......Page 575
Symbols Index......Page 579
Additional Content......Page 581
 Contents......Page 582
 Preface......Page 584
  2.7  Some Technical Differentiation Theorems......Page 586
  Suppl. 3.1  The Korteweg–de Vries Equation......Page 602
  Suppl. 3.3  Max Planck and The Principle of Least Action......Page 604
  Suppl. 3.4A  Second Derivative Test: Constrained Extrema......Page 606
  Suppl. 3.4B  Proof of the Implicit Function Theorem......Page 609
  Suppl. 4.1A  Equilibria in Mechanics......Page 612
  Suppl. 4.1B  Rotations and the Sunshine Formula......Page 617
  Suppl. 4.1C  The Principle of Least Action......Page 629
  Suppl. 4.1D  Emmy Noether and Hamilton's Principle......Page 639
  Suppl. 4.4  Flows and the Geometry of the Divergence......Page 646
  Suppl. 5.2  Alternative Definition of the Integral......Page 652
  5.6  Technical Integration Theorems......Page 658
 6. Integrals over Curves and Surfaces......Page 670
  Suppl. 6.2A  A Challenging Example......Page 671
  Suppl. 6.2B  The Gaussian Integral......Page 674
  Suppl. 7.4  The Problem of Plateau......Page 676
  Suppl. 8.2  Reorienting Astronauts......Page 680
  Suppl. 8.3  Exact Differentials......Page 682
  Suppl. 8.5  Green's Functions......Page 685
 Selected Answers and Solutions for Additional Content......Page 706