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Mathematical Analysis II, Zorich.pdf
Cover
Universitext
Title: Mathematical Analysis II
Copyright
Prefaces
Preface to the English Edition
Preface to the Fourth Russian Edition
Preface to the Third Russian Edition
Preface to the Second Russian Edition
Preface to the First Russian Edition
Table of Contents
9 *Continuous Mappings (General Theory)
9.1 Metric Spaces
9.1.1 Definition and Examples
9.1.2 Open and Closed Subsets of a Metric Space
9.1.3 Subspaces of a Metric space
9.1.4 The Direct Product of Metric Spaces
9.1.5 Problems and Exercises
9.2 Topological Spaces
9.2.1 Basic Definitions
9.2.2 Subspaces of a Topological Space
9.2.3 The Direct Product of Topological Spaces
9.2.4 Problems and Exercises
9.3 Compact Sets
9.3.1 Definition and General Properties of Compact Sets
9.3.2 Metric Compact Sets
9.3.3 Problems and Exercises
9.4 Connected Topological Spaces
9.4.1 Problems and Exercises
9.5 Complete Metric Spaces
9.5.1 Basic Definitions and Examples
9.5.2 The Completion of a Metric Space
9.5.3 Problems and Exercises
9.6 Continuous Mappings of Topological Spaces
9.6.1 The Limit of a Mapping
9.6.2 Continuous Mappings
9.6.3 Problems and Exercises
9.7 The Contraction Mapping Principle
9.7.1 Problems and Exercises
10 *Differential Calculus from a more General Point of View
10.1 Normed Vector Spaces
10.1.1 Some Examples of Vector Spaces in Analysis
10.1.2 Norms in Vector Spaces
10.1.3 Inner Products in Vector Spaces
10.1.4 Problems and Exercises
10.2 Linear and Multilinear Transformations
10.2.1 Definitions and Examples
10.2.2 The Norm of a Transformation
10.2.3 The Space of Continuous Transformations
10.2.4 Problems and Exercises
10.3 The Differential of a Mapping
10.3.1 Mappings Differentiable at a Point
10.3.2 The General Rules for Differentiation
10.3.3 Some Examples
10.3.4 The Partial Derivatives of a Mapping
10.3.5 Problems and Exercises
10.4 The Finite-increment Theorem and some Examples of its Use
10.4.1 The Finite-increment Theorem
10.4.2 Some Applications of the Finite-increment Theorem
10.4.3 Problems and Exercises
10.5 Higher-order Derivatives
10.5.1 Definition of the nth Differential
10.5.2 Derivative with Respect to a Vector and Computation of the Values of the nth Differential
10.5.3 Symmetry of the Higher-order Differentials
10.5.4 Some Remarks
10.5.5 Problems and Exercises
10.6 Taylor's Formula and the Study of Extrema
10.6.1 Taylor's Formula for Mappings
10.6.2 Methods of Studying Interior Extrema
10.6.3 Some Examples
10.6.4 Problems and Exercises
10.7 The General Implicit Function Theorem
10.7.1 Problems and Exercises
11 Multiple Integrals
11.1 The Riemann Integral over an n-Dimensional Interval
11.1.1 Definition of the Integral
11.1.2 The Lebesgue Criterion for Riemann Integrability
11.1.3 The Darboux Criterion
11.1.4 Problems and Exercises
11.2 The Integral over a Set
11.2.1 Admissible Sets
11.2.2 The Integral over a Set
11.2.3 The Measure (Volume) of an Admissible Set
11.2.4 Problems and Exercises
11.3 General Properties of the Integral
11.3.1 The Integral as a Linear Functional
11.3.2 Additivity of the Integral
11.3.3 Estimates for the Integral
11.3.4 Problems and Exercises
11.4 Reduction of a Multiple Integral to an Iterated Integral
11.4.1 Fubini's Theorem
11.4.2 Some Corollaries
11.4.3 Problems and Exercises
11.5 Change of Variable in a Multiple Integral
11.5.1 Statement of the Problem and Heuristic Derivation of the Change of Variable Formula
11.5.2 Measurable Sets and Smooth Mappings
11.5.3 The One-dimensional Case
11.5.4 The Case of an Elementary Diffeomorphism in IR^n
11.5.5 Composite Mappings and the Formula for Change of Variable
11.5.6 Additivity of the Integral and Completion of the Proof of the Formula for Change of Variable in an Integral
11.5.7 Corollaries and Generalizations of the Formula for Change of Variable in a Multiple Integral
11.5.8 Problems and Exercises
11.6 Improper Multiple Integrals
11.6.1 Basic Definitions
11.6.2 The Comparison Test for Convergence of an Improper Integral
11.6.3 Change of Variable in an Improper Integral
11.6.4 Problems and Exercises
12 Surfaces and Differential Forms in IR^n
12.1 Surfaces in IR^n
12.1.1 Problems and Exercises
12.2 Orientation of a Surface
12.2.1 Problems and Exercises
12.3 The Boundary of a Surface and its Orientation
12.3.1 Surfaces with Boundary
12.3.2 Making the Orientations of a Surface and its Boundary Consistent
12.3.3 Problems and Exercises
12.4 The Area of a Surface in Euclidean Space
12.4.1 Problems and Exercises
12.5 Elementary Facts about Differential Forms
12.5.1 Differential Forms: Definition and Examples
12.5.2 Coordinate Expression of a Differential Form
12.5.3 The Exterior Differential of a Form
12.5.4 Transformation of Vectors and Forms under Mappings
12.5.5 Forms on Surfaces
12.5.6 Problems and Exercises
13 Line and Surface Integrals
13.1 The Integral of a Differential Form
13.1.1 The Original Problems, Suggestive Considerations, Examples
13.1.2 Definition of the Integral of a Form over an Oriented Surface
13.1.3 Problems and Exercises
13.2 The Volume Element. Integrals of First and Second Kind
13.2.1 The Mass of a Lamina
13.2.2 The Area of a Surface as the Integral of a Form
13.2.3 The Volume Element
13.2.4 Expression of the Volume Element in Cartesian Coordinates
13.2.5 Integrals of First and Second Kind
13.2.6 Problems and Exercises
13.3 The Fundamental Integral Formulas of Analysis
13.3.1 Green's Theorem
13.3.2 The Gauss—Ostrogradskii Formula
13.3.3 Stokes' Formula in IR^3
13.3.4 The General Stokes Formula
13.3.5 Problems and Exercises
14 Elements of Vector Analysis and Field Theory
14.1 The Differential Operations of Vector Analysis
14.1.1 Scalar and Vector Fields
14.1.2 Vector Fields and Forms in IR^3
14.1.3 The Differential Operators grad, curl, div, and \nabla
14.1.4 Some Differential Formulas of Vector Analysis
14.1.5 *Vector Operations in Curvilinear Coordinates
14.1.6 Problems and Exercises
14.2 The Integral Formulas of Field Theory
14.2.1 The Classical Integral Formulas in Vector Notation
14.2.2 The Physical Interpretation of div, curl, and grad
14.2.3 Other Integral Formulas
14.2.4 Problems and Exercises
14.3 Potential Fields
14.3.1 The Potential of a Vector Field
14.3.2 Necessary Condition for Existence of a Potential
14.3.3 Criterion for a Field to be Potential
14.3.4 Topological Structure of a Domain and Potentials
14.3.5 Vector Potential. Exact and Closed Forms
14.3.6 Problems and Exercises
14.4 Examples of Applications
14.4.1 The Heat Equation
14.4.2 The Equation of Continuity
14.4.3 The Basic Equations of the Dynamics of Continuous Media
14.4.4 The Wave Equation
14.4.5 Problems and Exercises
15 integration of Differential Forms on Manifolds
15.1 A Brief Review of Linear Algebra
15.1.1 The Algebra of Forms
15.1.2 The Algebra of Skew-symmetric Forms
15.1.3 Linear Mappings of Vector Spaces and the Adjoint Mappings of the Conjugate Spaces
15.1.4 Problems and Exercises
15.2 Manifolds
15.2.1 Definition of a Manifold
15.2.2 Smooth Manifolds and Smooth Mappings
15.2.3 Orientation of a Manifold and its Boundary
15.2.4 Partitions of Unity and the Realization of Manifolds as Surfaces in IR^n
15.2.5 Problems and Exercises
15.3 Differential Forms and Integration on Manifolds
15.3.1 The Tangent Space to a Manifold at a Point
15.3.2 Differential Forms on a Manifold
15.3.3 The Exterior Derivative
15.3.4 The Integral of a Form over a Manifold
15.3.5 Stokes' Formula
15.3.6 Problems and Exercises
15.4 Closed and Exact Forms on Manifolds
15.4.1 Poincare's Theorem
15.4.2 Homology and Cohomology
15.4.3 Problems and Exercises
16 Uniform Convergence and the Basic Operations of Analysis on Series and Families of Functions
16.1 Pointwise and Uniform Convergence
16.1.1 Pointwise Convergence
16.1.2 Statement of the Fundamental Problems
16.1.3 Convergence and Uniform Convergence of a Family of Functions Depending on a Parameter
16.1.4 The Cauchy Criterion for Uniform Convergence
16.1.5 Problems and Exercises
16.2 Uniform Convergence of Series of Functions
16.2.1 Basic Definitions and a Test for Uniform Convergence of a Series
16.2.2 The Weierstrass Μ-test for Uniform Convergence of a Series
16.2.3 The Abel-Dirichlet Test
16.2.4 Problems and Exercises
16.3 Functional Properties of a Limit Function
16.3.1 Specifics of the Problem
16.3.2 Conditions for Two Limiting Passages to Commute
16.3.3 Continuity and Passage to the Limit
16.3.4 Integration and Passage to the Limit
16.3.5 Differentiation and Passage to the Limit
16.3.6 Problems and Exercises
16.4 *Compact and Dense Subsets of the Space of Continuous Functions
16.4.1 The Arzelà—Ascoli Theorem
16.4.2 The Metric Space C(K, Y)
16.4.3 Stone's Theorem
16.4.4 Problems and Exercises
17 Integrals Depending on a Parameter
17.1 Proper Integrals Depending on a Parameter
17.1.1 The Concept of an Integral Depending on a Parameter
17.1.2 Continuity of an Integral Depending on a Parameter
17.1.3 Differentiation of an Integral Depending on a Parameter
17.1.4 Integration of an Integral Depending on a Parameter
17.1.5 Problems and Exercises
17.2 Improper Integrals Depending on a Parameter
17.2.1 Uniform Convergence of an Improper Integral with Respect to a Parameter
17.2.2 Limiting Passage under the Sign of an Improper Integral and Continuity of an Improper Integral Depending on a Parameter
17.2.3 Differentiation of an Improper Integral with Respect to a Parameter
17.2.4 Integration of an Improper Integral with Respect to a Parameter
17.2.5 Problems and Exercises
17.3 The Eulerian Integrals
17.3.1 The Beta Function
17.3.2 The Gamma Function
17.3.3 Connection Between the Beta and Gamma Functions
17.3.4 Examples
17.3.5 Problems and Exercises
17.4 Convolution of Functions and Elementary Facts about Generalized Functions
17.4.1 Convolution in Physical Problems (Introductory Considerations)
17.4.2 General Properties of Convolution
17.4.3 Approximate Identities and the Weierstrass Approximation Theorem
17.4.4 *Elementary Concepts Involving Distributions
17.4.5 Problems and Exercises
17.5 Multiple Integrals Depending on a Parameter
17.5.1 Proper Multiple Integrals Depending on a Parameter
17.5.2 Improper Multiple Integrals Depending on a Parameter
17.5.3 Improper Integrals with a Variable Singularity
17.5.4 *Convolution, the Fundamental Solution, and Generalized Functions in the Multidimensional Case
17.5.5 Problems and Exercises
18 Fourier Series and the Fourier Transform
18.1 Basic General Concepts Connected with Fourier Series
18.1.1 Orthogonal Systems of Functions
18.1.2 Fourier Coefficients and Fourier Series
18.1.3 *An Important Source of Orthogonal Systems of Functions in Analysis
18.1.4 Problems and Exercises
18.2 Trigonometric Fourier Series
18.2.1 Basic Types of Convergence of Classical Fourier Series
18.2.2 Investigation of Pointwise Convergence of a Trigonometric Fourier Series
18.2.3 Smoothness of a Function and the Rate of Decrease of the Fourier Coefficients
18.2.4 Completeness of the Trigonometric System
18.2.5 Problems and Exercises
18.3 The Fourier Transform
18.3.1 Representation of a Function by Means of a Fourier Integral
18.3.2 The Connection of the Differential and Asymptotic Properties of a Function and its Fourier Transform
18.3.3 The Main Structural Properties of the Fourier Transform
18.3.4 Examples of Applications
18.3.5 Problems and Exercises
19 Asymptotic Expansions
19.1 Asymptotic Formulas and Asymptotic Series
19.1.1 Basic Definitions
19.1.2 General Facts about Asymptotic Series
19.1.3 Asymptotic Power Series
19.1.4 Problems and Exercises
19.2 The Asymptotics of Integrals (Laplace's Method)
19.2.1 The Idea of Laplace's Method
19.2.2 The Localization Principle for a Laplace Integral
19.2.3 Canonical Integrals and their Asymptotics
19.2.4 The Principal Term of the Asymptotics of a Laplace Integral
19.2.5 *Asymptotic Expansions of Laplace Integrals
19.2.6 Problems and Exercises
Topics and Questions for Midterm Examinations
1. Series and Integrals Depending on a Parameter
2. Problems Recommended as Midterm Questions
3. Integral Calculus (Several Variables)
4. Problems Recommended for Studying the Midterm Topics
Examination Topics
1. Series and Integrals Depending on a Parameter
2. Integral Calculus (Several Variables)
References
1. Classic Works
1.1. Primary Sources
1.2. Major Comprehensive Expository Works
1.3. Classical courses of analysis from the first half of the twentieth century
2. Textbooks
3. Classroom Materials
4. Further Reading
Index of Basic Notation
Subject Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Z
Name Index
Back Cover
Universitext
Springer
Berlin
Heidelberg
New York
Hong Kong
London
Milan
Paris
Tokyo
Vladimir A. Zorich
Mathematical
Analysis II
sfj Springer
Vladimir A. Zorich
Moscow State University
Department of Mathematics (Mech-Math)
Vorobievy Gory
Moscow 119992
Russia
Translator:
Roger Cooke
Burlington, Vermont
USA
e-mail: cooke@emba.uvm.edu
Title of Russian edition:
Matematicheskij Analiz (Part II, 4th corrected edition, Moscow, 2002)
MCCME (Moscow Center for Continuous Mathematical Education Publ.)
Cataloging-in-Publication Data applied for
A catalog record for this book is available from the Library of Congress.
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): Primary 00A05
Secondary: 26-01, 40-01, 42-01, 54-01, 58-01
ISBN 3-540-40633-6 Springer-Verlag Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data
banks. Duplication of this publication or parts thereof is permitted only under the provisions
of the German Copyright Law of September 9,1965, in its current version, and permission
for use must always be obtained from Springer-Verlag. Violations are liable for prosecution
under the German Copyright Law.
Springer-Verlag is a part of Springer Science+Business Media
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© Springer-Verlag Berlin Heidelberg 2004
Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use.
Cover design: design & production GmbH, Heidelberg
Typeset by the translator using a Springer KTjiX macro package
Printed on acid-free paper
46/3111- 5 4 3 21
SPIN 11605140
Prefaces
Preface to the English Edition
An entire generation of mathematicians has grown up during the time be
tween the appearance of the first edition of this textbook and the publication
of the fourth edition, a translation of which is before you. The book is famil
iar to many people, who either attended the lectures on which it is based or
studied out of it, and who now teach others in universities all over the world.
I am glad that it has become accessible to English-speaking readers.
This textbook consists of two parts. It is aimed primarily at university
students and teachers specializing in mathematics and natural sciences, and
at all those who wish to see both the rigorous mathematical theory and
examples of its effective use in the solution of real problems of natural science.
The textbook exposes classical analysis as it is today, as an integral part
of Mathematics in its interrelations with other modern mathematical courses
such as algebra, differential geometry, differential equations, complex and
functional analysis.
The two chapters with which this second book begins, summarize and
explain in a general form essentially all most important results of the first
volume concerning continuous and differentiable functions, as well as differ
ential calculus. The presence of these two chapters makes the second book
formally independent of the first one. This assumes, however, that the reader
is sufficiently well prepared to get by without introductory considerations of
the first part, which preceded the resulting formalism discussed here. This
second book, containing both the differential calculus in its generalized form
and integral calculus of functions of several variables, developed up to the
general formula of Newton-Leibniz-Stokes, thus acquires a certain unity and
becomes more self-contained.
More complete information on the textbook and some recommendations
for its use in teaching can be found in the translations of the prefaces to the
first and second Russian editions.
Moscow, 2003
V. Zorich
Prefaces
VI
Preface to the Fourth Russian Edition
In the fourth edition all misprints that the author is aware of have been
corrected.
Moscow, 2002
V. Zorich
Preface to the Third Russian Edition
The third edition differs from the second only in local corrections (although
in one case it also involves the correction of a proof) and in the addition of
some problems that seem to me to be useful.
Moscow, 2001
V. Zorich
Preface to the Second Russian Edition
In addition to the correction of all the misprints in the first edition of which
the author is aware, the differences between the second edition and the first
edition of this book are mainly the following. Certain sections on individual
topics - for example, Fourier series and the Fourier transform - have been
recast (for the better, I hope). We have included several new examples of
applications and new substantive problems relating to various parts of the
theory and sometimes significantly extending it. Test questions are given, as
well as questions and problems from the midterm examinations. The list of
further readings has been expanded.
Further information on the material and some characteristics of this sec
ond part of the course are given below in the preface to the first edition.
Moscow, 1998
V. Zorich
Preface to the First Russian Edition
The preface to the first part contained a rather detailed characterization of
the course as a whole, and hence I confine myself here to some remarks on
the content of the second part only.
The basic material of the present volume consists on the one hand of
multiple integrals and line and surface integrals, leading to the generalized
Stokes' formula and some examples of its application, and on the other hand
the machinery of series and integrals depending on a parameter, including
Preface to the First Russian Edition
VII
Fourier series, the Fourier transform, and the presentation of asymptotic
expansions.
Thus, this Part 2 basically conforms to the curriculum of the second year
of study in the mathematics departments of universities.
So as not to impose rigid restrictions on the order of presentation of these
two major topics during the two semesters, I have discussed them practically
independently of each other.
Chapters 9 and 10, with which this book begins, reproduce in compressed
and generalized form, essentially all of the most important results that were
obtained in the first part concerning continuous and differentiable functions.
These chapters are starred and written as an appendix to Part 1. This ap
pendix contains, however, many concepts that play a role in any exposition
of analysis to mathematicians. The presence of these two chapters makes the
second book formally independent of the first, provided the reader is suffi
ciently well prepared to get by without the numerous examples and introduc
tory considerations that, in the first part, preceded the formalism discussed
here.
The main new material in the book, which is devoted to the integral
calculus of several variables, begins in Chapter 11. One who has completed
the first part may begin the second part of the course at this point without
any loss of continuity in the ideas.
The language of differential forms is explained and used in the discussion
of the theory of line and surface integrals. All the basic geometric concepts
and analytic constructions that later form a scale of abstract definitions lead
ing to the generalized Stokes' formula are first introduced by using elementary
material.
Chapter 15 is devoted to a similar summary exposition of the integration
of differential forms on manifolds. I regard this chapter as a very desirable
and systematizing supplement to what was expounded and explained using
specific objects in the mandatory Chapters 11-14.
The section on series and integrals depending on a parameter gives, along
with the traditional material, some elementary information on asymptotic
series and asymptotics of integrals (Chap. 19), since, due to its effectiveness,
the latter is an unquestionably useful piece of analytic machinery.
For convenience in orientation, ancillary material or sections that may be
omitted on a first reading, are starred.
The numbering of the chapters and figures in this book continues the
numbering of the first part.
Biographical information is given here only for those scholars not men
tioned in the first part.
As before, for the convenience of the reader, and to shorten the text, the
end of a proof is denoted by
•. Where convenient, definitions are introduced
by the special symbols := or =: (equality by definition), in which the colon
stands on the side of the object being defined.
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