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內容簡介

　　《數學經典教材：嚮量微積分、綫性代數和微分形式（第3版）（影印版）》是一部優秀的微積分教材，好評不斷。《數學經典教材：嚮量微積分、綫性代數和微分形式（第3版）（影印版）》材料的選擇和編排有不同於標準方法的三點：（一）在這個水平的研究中，綫性代數是研究多變量微積分的極其方便的環境和語言，非綫性更像是一個衍生産品；（二）強調計算有效算法，並且通過這些算術工作來證明定理；（三）運用微分形式推廣更高維的積分定理。
　　目次：預備知識；嚮量、矩陣和導數；解方程；流形、泰勒多項式和二次型、麯率；積分；流形的體積；形式和嚮量微積分。附錄：分析。
　　《數學經典教材：嚮量微積分、綫性代數和微分形式（第3版）（影印版）》讀者對象：數學專業的本科生以及想學習微積分知識的廣大非專業專業人士。

內頁插圖

目錄

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

前言/序言

數學經典教材：嚮量微積分、綫性代數和微分形式（第3版）（影印版） [Vector Calculus,Linear Algebra,and Differential Forms:A Unified Ap 下載 mobi epub pdf txt 電子書

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