多元微积分 原书第3版 英文PDF电子书下载

PART ONE Basic Material 1

CHAPTER Ⅰ Vectors 3

1．Definition of Points in Space 3

2．Located Vectors 11

3．Scalar Product 14

4．The Norm of a Vector 17

5．Parametric Lines 32

6．Planes 36

7．The Cross Product 44

CHAPTER Ⅱ Differentiation of Vectors 49

1．Derivative 49

2．Length of Curves 62

CHAPTER Ⅲ Functions of Several Variables 66

1．Graphs and Level Curves 66

2．Partial Derivatives 70

3．Differentiability and Gradient 77

4．Repeated Partial Derivatives 82

CHAPTER Ⅳ The Chain Rule and the Gradient 87

1．The Chain Rule 87

2．Tangent Plane 92

3．Directional Derivative 99

4．Functions Depending only on the Distance from the Origin 103

5．The Law of Conservation of Energy 111

6．Further Technique in Partial Differentiation 114

PART TWO Maxima,Minima,and Taylor's Formula 121

CHAPTER Ⅴ Maximum and Minimum 123

1．Critical Points 123

2．Boundary Points 126

3．Lagrange Multipliers 135

CHAPTER Ⅵ Higher Derivatives 143

1．The First Two Terms in Taylor's Formula 143

2．The Quadratic Term at Critical Points 149

3．Algebraic Study of a Quadratic Form 155

4．Partial Differential Operators 162

5．The General Expression for Taylor's Formula 170

Appendix.Taylor's Formula in One Variable 176

PART THREE Curve Integrals and Double Integrals 181

CHAPTER Ⅶ Potential Functions 183

1．Existence and Uniqueness of Potential Functions 184

2．Local Existence of Potential Functions 188

3．An Important Special Vector Field 194

4．Differentiating Under the Integral 198

5．Proof of the Local Existence Theorem 201

CHAPTER Ⅷ Curve Integrals 206

1．Definition and Evaluation of Curve Integrals 207

2．The Reverse Path 217

3．Curve Integrals When the Vector Field Has a Potential Function 220

4．Dependence of the Integral on the Path 228

CHAPTER Ⅸ Double Integrals 233

1．Double Integrals 233

2．Repeated Integrals 242

3．Polar Coordinates 252

CHAPTER Ⅹ Green's Theorem 269

1．The Standard Version 269

2．The Divergence and the Rotation of a Vector Field 280

PART FOUR Triple and Surface Integrals 291

CHAPTER Ⅺ Triple Integrals 293

1．Triple Integrals 293

2．Cylindrical and Spherical Coordinates 298

3．Center of Mass 313

CHAPTER Ⅻ Surface Integrals 318

1．Parametrization,Tangent Plane,and Normal Vector 318

2．Surface Area 325

3．Surface Integrals 333

4．Curl and Divergence of a Vector Field 342

5．Divergence Theorem in 3-Space 345

6．Stokes' Theorem 355

PART FIVE Mappings,Inverse Mappings,and Change of Variables Formula. 365

CHAPTER ⅩⅢ Matrices 367

1．Matrices 367

2．Multiplication of Matrices 372

CHAPTER ⅩⅣ Linear Mappings 385

1．Mappings 385

2．Linear Mappings 392

3．Geometric Applications 398

4．Composition and Inverse of Mappings 404

CHAPTER ⅩⅤ Determinants 412

1．Determinants of Order 2 412

2．Determinants of Order 3 416

3．Additional Properties of Determinants 420

4．Independence of Vectors 428

5．Determinant of a Product 430

6．Inverse of a Matrix 431

CHAPTER ⅩⅥ Applications to Functions of Several Variables 434

1．The Jacobian Matrix 434

2．Differentiability 438

3．The Chain Rule 440

4．Inverse Mappings 443

5．Implicit Functions 446

6．The Hessian 450

CHAPTER ⅩⅦ The Change of Variables Formula 453

1．Determinants as Area and Volume 453

2．Dilations 463

3．Change of Variables Formula in Two Dimensions 469

4．Application of Green's Formula to the Change of Variables Formula 474

5．Change of Variables Formula in Three Dimensions 478

6．Vector Fields on the Sphere 483

APPENDIX Fourier Series 487

1．General Scalar Products 487

2．Computation of Fourier Series 494