Advanced calculus of several variables 多元高等微积分

In this high-level treatment, the author provides a modern conceptual approaeh to multivariable calculus, emphasizing the interplay of geometry and analysis via linear algebra and the approximation of nonlinear mappings by linear ones. At the same time, the book gives equal attention to the elassieal applieations and computational methods responsible for much of the interest and importance of this subject.
Beginning with a discussion of Euclidean space and linear mappings, Professor Edwards (University of Georgia) follows with a thorough and detailed exposition of multivariable differential and integral calculus. Among the topics covered are the basies of single-variable differential calculus generalized to higher dimensions, the use of approximation methods to treat the fundamental existenee theorems of multivariable calculus, iterated integrals and change of variables, improper multiple integrals and a comprehensive discussion, from the viewpoint of differential forms, of the classical material associated with line and surface integrals, Stokes' theorem, and vector analysis. The author closes with a modern treatment of some venerable problems of the calculus of variations.

Preface
ⅠEuclidean Space and Linear Mappings
　1　The Vector Space
　2　Subspaces of
　3　Inner Products and Orthogonality
　4　Linear Mappings and Matrices
　5　The Kernel and Image of a Linear Mapping
　6　Determinants
　7　Limits and Continuity
　8　Elementary Topology of
Ⅱ Multivariable Differential Calculus
　1　Curves in
　2　Directional Derivatives and the Differential
　3　The Chain Rule
　4　Lagrange Multipliers and the Classification of Critical Points　for .Functions of Two Variables
　5　Maxima and Minima, Manifolds, and Lagrange Multipliers
　6　Taylor's Formula for Single-Variable Functions
　7　Taylor's Formula in Several Variables
　8　The Classification of Critical Points
Ⅲ Successive Approximations and Implicit Functions
　1　Newton's Method and Contraction Mappings
　2 The Multivariable Mean Value Theorem
　3 The Inverse and Implicit Mapping Theorems
4 Manifolds in
5 Higher Derivatives
Ⅳ Multiple Integrals
1 Area and the 1-Dimensional Integral
2 Volume and the n-Dimensional Integral
3 Step Functions and Riemann Sums
4 Iterated Integrals and Fubini's Theorem
5 Change of Variables
6 I mproper Integrals and Absolutely Integrable Functions
ⅤLine and Surface Integrals; Differential Forms and Stokes' Theorem
1 Pathlength and Line Integrals
2 Green's Theorem
3 Multilinear Functions and the Area of a Parallelepiped
4 Surface Area
5 Differential Forms
6 Stokes' Theorem
7 The Classical Theorems of Vector Analysis
8 Closed and Exact Forms
ⅥThe Calculus of Variations
1 Normed Vector Spaces and Uniform Convergence
2 Continuous Linear Mappings and Differentials
3 The Simplest Variational Problem
4 The Isoperimetric Problem
5 Multiple Integral Problems
Appendix: The Completeness of
Suggested Reading
Subject Index