向量微积分、线性代数和微分形式 原书第3版 英文2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

向量微积分、线性代数和微分形式 原书第3版 英文PDF格式电子书版下载

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CHAPTER 0 PRELIMINARIES1

0.0 Introduction1

0.1 Reading mathematics1

0.2 Quantifiers and negation4

0.3 Set theory6

0.4 Functions9

0.5 Real numbers17

0.6 Infinite sets22

0.7 Complex numbers25

CHAPTER 1 VECTORS,MATRICES,AND DERIVATIVES32

1.0 Introduction32

1.1 Introducing the actors：points and vectors33

1.2 Introducing the actors：matrices42

1.3 Matrix multiplication as a linear transformation56

1.4 The geometry of Rn67

1.5 Limits and continuity84

1.6 Four big theorems106

1.7 Derivatives in several variables as linear transformations120

1.8 Rules for computing derivatives140

1.9 The mean value theorem and criteria for differentiability148

1.10 Review exercises for chapter 1155

CHAPTER 2 SOLVING EQUATIONS161

2.0 Introduction161

2.1 The main algorithm：row reduction162

2.2 Solving equations with row reduction168

2.3 Matrix inverses and elementary matrices177

2.4 Linear combinations,span,and linear independence182

2.5 Kernels,images,and the dimension formula195

2.6 Abstract vector spaces211

2.7 Eigenvectors and eigenvalues222

2.8 Newton's method232

2.9 Superconvergence252

2.10 The inverse and implicit function theorems259

2.11 Review exercises for chapter 2278

CHAPTER 3 MANIFOLDS,TAYLOR POLYNOMIALS,QUADRATIC FORMS,AND CURVATURE283

3.0 Introduction283

3.1 Manifolds284

3.2 Tangent spaces306

3.3 Taylor polynomials in several variables314

3.4 Rules for computing Taylor polynomials326

3.5 Quadratic forms334

3.6 Classifying critical points of functions343

3.7 Constrained critical points and Lagrange multipliers350

3.8 Geometry of curves and surfaces368

3.9 Review exercises for chapter 3386

CHAPTER 4 INTEGRATION391

4.0 Introduction391

4.1 Defining the integral392

4.2 Probability and centers of gravity407

4.3 What functions can be integrated?421

4.4 Measure zero428

4.5 Fubini's theorem and iterated integrals436

4.6 Numerical methods of integration448

4.7 Other pavings459

4.8 Determinants461

4.9 Volumes and determinants476

4.10 The change of variables formula483

4.11 Lebesgue integrals495

4.12 Review exercises for chapter 4514

CHAPTER 5 VOLUMES OF MANIFOLDS518

5.0 Introduction518

5.1 Parallelograms and their volumes519

5.2 Parametrizations523

5.3 Computing volumes of manifolds530

5.4 Integration and curvature543

5.5 Fractals and fractional dimension545

5.6 Review exercises for chapter 5547

CHAPTER 6 FORMS AND VECTOR CALCULUS549

6.0 Introduction549

6.1 Forms on Rn550

6.2 Integrating form fields over parametrized domains565

6.3 Orientation of manifolds570

6.4 Integrating forms over oriented manifolds581

6.5 Forms in the language of vector calculus592

6.6 Boundary orientation604

6.7 The exterior derivative617

6.8 Grad,curl,div,and all that624

6.9 Elctromagnetism633

6.10 The generalized Stokes's theorem646

6.11 The integral theorems of vector calculus655

6.12 Potentials663

6.13 Review exercises for chapter 6668

APPENDIX：ANALYSIS673

A.0 Introduction673

A.1 Arithmetic of real numbers673

A.2 Cubic and quartic equations677

A.3 Two results in topology：nested compact sets and Heine-Borel682

A.4 Proof of the chain rule683

A.5 Proof of Kantorovich's theorem686

A.6 Proof of lemma 2.9.5(superconvergence)692

A.7 Proof of differentiability of the inverse function694

A.8 Proof of the implicit function theorem696

A.9 Proving equality of crossed partials700

A.10 Functions with many vanishing partial derivatives701

A.11 Proving rules for Taylor polynomials；big O and little o704

A.12 Taylor's theorem with remainder709

A.13 Proving theorem 3.5.3(completing squares)713

A.14 Geometry of curves and surfaces：proofs714

A.15 Stirling's formula and proof of the central limit theorem720

A.16 Proving Fubini's theorem724

A.17 Justifying the use of other pavings727

A.18 Results concerning the determinant729

A.19 Change of variables formula：a rigorous proof734

A.20 Justifying volume 0740

A.21 Lebesgue measure and proofs for Lebesgue integrals742

A.22 Justifying the change of parametrization760

A.23 Computing the exterior derivative765

A.24 The pullback769

A.25 Proving Stokes's theorem774

BIBLIOGRAPHY788

PHOTO CREDITS790

INDEX792