內容簡介

The book is suitable for a one-year course at the advanced undergraduate level. By omitting certain chapters, a one semester course can be based on it. For instance, if the students already have a good knowledge of partial differentiation and the elementary topology of E’’, then substantial parts of Chapters 4, 5, 7, and 8 can be covered in a semester. Some knowledge of linear algebra is presumed. However, results from linear algebra are reviewed as needed (in some cases without proof).
 

目錄

chapter 1
euclidean spaces
1.1the real number system
1.2euclidean en
1.3elementary geometry of en
1.4basic topological notions in en
1.5convex sets
chapter 2
elementary topology of en
2.1functions
2.2limits and continuity of transformations
2.3sequences in e”
2.4bolzano-weierstrass theorem
2.5relative neighborhoods, continuous transformations
2.6topological spaces
2.7connectedness
2.8compactness
2.9metric spaces
2.10 spaces of continuous functions
2.11 noneuclidean norms on en

.chapter 3
differentiation of real-valued functions
3.1directional and partial derivatives
3.2linear functions
3.3differentiable functions
3.4functions of class c（q）
3.5relative extrema
*3.6convex and concave functions
chapter 4
vector-valued functions of several variables
4.1linear transformations
4.2affine transformations
4.3differentiable transformations
4.4composition
4.5the inverse function theorem
4,6the implicit function theorem
4.7manifolds
4.8the multiplier rule
chapter 5
integration
5.1intervals
5.2measure
5.3integrals over en
5.4integrals over bounded sets
5.5iterated integrals
5.6integrals of continuous functions
5.7change of measure under affine transformations
5.8transformation of integrals
5.9coordinate systems in en
5.10 measurable sets and functions; further properties
5.11 integrals: general definition, convergence theorems
5.12 differentiation under the integral sign
5.13 lp-spaces
chapter 6
curves and line integrals
6.1derivatives
6.2curves in en
6.3differential i-forms
6.4line integrals
*6.5gradient method
*6.6integrating factors; thermal systems
chapter 7
exterior algebra and differential calculus
7.1covectors and differential forms of degree 2
7.2alternating multilinear functions
7.3muiticovectors
7.4differential forms
7.5multivectors
7.6induced linear transformations
7.7transformation law for differential forms
7.8the adjoint and codifferential
*7.9special results for n = 3
7.10 integrating factors (continued)
chapter 8
integration on manifolds
8.1regular transformations
8.2coordinate systems on manifolds
8.3measure and integration on manifolds
8.4the divergence theorem
8.5fluid flow
8.6orientations
8.7integrals of r-forms
8.8stokes’’s formula
8.9regular transformations on submanifolds
8.10 closed and exact differential forms
8.11 motion of a particle
8.12 motion of several particles
appendix 1
axioms for a vector space
appendix 2
mean value theorem; taylor’’s theorem
appendix 3
review of riemann integration
appendix 4
monotone functions
references
answers to problems
index
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