多元函数

所属分类:数学  
出版时间:2010-9   出版时间:世界图书出版公司   作者:弗莱明   页数:411  
Tag标签:数学  

前言

The purpose of this book is to give a systematic development of differentialand integral calculus for functions of several variables. The traditional topicsfrom advanced calculus are included: maxima and minima, chain rule,implicit function theorem, multiple integrals, divergence and Stokes'stheorems, and so on. However, the treatment differs in several importantrespects from the traditional one. Vector notation is used throughout, andthe distinction is maintained between n-dimensional euclidean space E" andits dual. The elements of the Lebesgue theory of integrals are given. Inplace of the traditional vector analysis in E3, we introduce exterior algebraand the calculus of exterior differential forms. The formulas of vectoranalysis then become special cases of formulas about differential forms andintegrals over manifolds lying in E". The book is suitable for a one-year course at the advanced undergraduatelevel. By omitting certain chapters, a one semester course can be based on it.For instance, if the students already have a good knowledge of partialdifferentiation and the elementary topology of En, then substantial parts ofChapters 4, 5, 7, and 8 can be covered in a semester. Some knowledge oflinear algebra is presumed. However, results from linear algebra are reviewedas needed (in some cases without proof).   A number of changes have been made in the first edition. Many of thesewere suggested by classroom experience. A new Chapter 2 on elementarytopology has been added. Additional physical applications——to thermo-dynamics and classical mechanics——have been added in Chapters 6 and 8.Different proofs, perhaps easier for the beginner, have been given for twomain theorems (the Inverse Function Theorem and the Divergence Theorem.)

内容概要

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).

作者简介

作者:(美国)弗莱明(Wendell Fleming)

书籍目录

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

章节摘录

插图:

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《多元函数(第2版)》由世界图书出版公司出版。

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数学


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