向量微积分、线性代数和微分形式

Preface

Chapter 0 preliminaries

0.0 introduction

0.1 reading mathematics

0.2 quantifiers and negation

0.3 set theory

0.4 functions

0.5 real numbers

0.6 infinite sets

0.7 complex numbers

Chapter 1 vectors～matrices, and derivatives

1.0 introduction

1.1 introducing the actors: points and vectors

1.2 introducing the actors: matrices

1.3 matrix multiplication as a linear transformation

1.4 the geometry of rn

1.5 limits and continuity

1.6 four big theorems

1.7 derivatives in several variables as lineartransformations

1.8 rules for computing derivatives

1.9 the mean value theorem and criteria for differentiability

1.10 review exercises for Chapter 1

Chapter 2 solving equations

2.0 introduction

2.1 the main algorithm: row reduction

2.2 solving equations with row reduction

2.3 matrix inverses and elementary matrices

2.4 linear combinations, span, and linear independence

2.5 kernels, images, and the dimension formula

2.6 abstract vector spaces

2.7 eigenvectors and eigenvalues

2.8 newton's method

2.9 superconvergence

2.10 the inverse and implicit function theorems

2.11 review exercises for Chapter 2

Chapter 3 manifolds, Taylor polynomials, quadratic forms, and curvature

3.0 introduction

3.1 manifolds

3.2 tangent spaces

3.3 Taylor polynomials in several variables

3.4 rules for computing Taylor polynomials

3.5 quadratic forms

3.6 classifying critical points of fimctions

3.7 constrained critical points and lagrange multipliers

3.8 geometry of curves and surfaces

3.9 review exercises for Chapter 3

Chapter 4 integration

4.0 introduction

4.1 defining the integral

4.2 probability and centers of gravity

4.3 what functions can be integrated?

4.4 measure zero

4.5 fhbini's theorem and iterated integrals

4.6 numerical methods of integration

4.7 other pavings

4.8 determinants

4.9 volumes and determinants

4.10 the change of variables formula

4.11 lebesgue integrals

4.12 review exercises for Chapter 4

Chapter 5 volumes of manifolds

5.0 introduction

5.1 parallelograms and their volumes

5.2 parametrizations

5.3 computing volumes of manifolds

5.4 integration and curvature

5.5 fractals and fractional dimension

5.6 review exercises for Chapter 5

Chapter 6 forms and vector calculus

6.0 introduction

6.1 forms on rn

6.2 integrating form fields over parametrized domains

6.3 orientation of manifolds

6.4 integrating forms over oriented manifolds

6.5 forms in the language of vector calculus

6.6 boundary orientation

6.7 the exterior derivative

6.8 grad, curl, div, and all that

6.9 electromagnetism

6.10 the generalized stokes's theorem

6.11 the integral theorems of vector calculus

6.12 potentials

6.13 review exercises for Chapter 6

Appendix: analysis

A.0 introduction

A.1 arithmetic of real numbers

A.2 cubic and quartic equations

A.3 two results in topology: nested compact sets and heine-borel

A.4 proof of the chain rule

A.5 proof of kantorovich's theorem

A.6 proof of lemma 2.9.5 （superconvergence）

A.7 proof of differentiability of the inverse function

A.8 proof of the implicit function theorem

A.9 proving equality of crossed partials

A.10 functions with many vanishing partial derivatives

A.11 proving rules for Taylor polynomials; big o and little o

A.12 Taylor's theorem with remainder

A.13 proving theorem 3.5.3 （completing squares）

A.14 geometry of curves and surfaces: proofs

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

A.16 proving fubiul's theorem

A.17 justifying the use of other pavings

A.18 results concerning the determinant

A.19 change of variables formula: a rigorous proof

A.20 justifying volume 0

A.21 lebesgue measure and proofs for lebesgue integrals

A.22 justifying the change of parametrization

A.23 computing the exterior derivative

A.24 the pullback

A.25 proving stokes's theorem

bibliography

photo credits

index

《数学经典教材：向量微积分、线性代数和微分形式（第3版）（影印版）》是一部优秀的微积分教材，好评不断。《数学经典教材：向量微积分、线性代数和微分形式（第3版）（影印版）》材料的选择和编排有不同于标准方法的三点：（一）在这个水平的研究中，线性代数是研究多变量微积分的极其方便的环境和语言，非线性更像是一个衍生产品；（二）强调计算有效算法，并且通过这些算术工作来证明定理；（三）运用微分形式推广更高维的积分定理。
　　目次：预备知识；向量、矩阵和导数；解方程；流形、泰勒多项式和二次型、曲率；积分；流形的体积；形式和向量微积分。附录：分析。
　　《数学经典教材：向量微积分、线性代数和微分形式（第3版）（影印版）》读者对象：数学专业的本科生以及想学习微积分知识的广大非专业专业人士。