208 - Analysis for Applied Mathematics Ward Cheney (2001 ISBN 978-0-387-95279-6).pdf

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Graduate Texts in Mathematics
TAKEUTr/ZING. Introduction to
35
ALEXANDERIWERMER. Several Complex
Axiomatic Set Theory. 2nd ed.
Variables and Banach Algebras. 3rd ed.
OXTOBY. Measure and Category. 2nd ed.
36
KELLEy/NMIOA et al. Linear
2
Topological Spaces.
3
SCHAEFER. Topological Vector Spaces.
2nd ed.
37
MONC Mathematical Logic.
HrLTON/STMMBACH. A Course in
38
GAUERT/FRlTZSCHE. Several Complex
4
Homological Algebra. 2nd ed.
Variables.
MAC LANE. Categories fo r the Working
39
ARVESON. An Invitation to C·-Algebras.
5
KEMENY/SNELUKNPP. Denumerable
Mathematician. 2nd ed.
40
6
HUGHES/PIPER. Projective Planes.
Markov Chains. 2nd ed.
7
SERRE. A Course in Arithmetic.
41
ApOSTOL. Modular Functions and Dirichlet
8
TAKEUTI/ZARING. Axiomatic Set Theory.
Series in Number Theory.
9
HUMPHREYS. Introduction to Lie Algebras
2nd ed.
and Representation Theory.
42
SERE. Linear Representations ofFinite
COHEN. A Course in Simple Homotopy
Groups.
10
Theory.
43
GILLMAN/JERISON. Rings of Continuous
II
CONWAY. Functions of One Complex
Functions.
Variable I. 2nd ed.
44
ENDIG. Elementary Algebraic Geometry.
BEALS. Advanced Mathematical Analysis.
4 5
LOEVE. Probability Theory I. 4th ed.
12
13
ANDERSON/FuLLER. Rings and Categories
46
LOEVE. Probability Theory II. 4th ed.
of Modules. 2nd ed.
47
MOISE. Geometric Topology in
14
GOLUBITSKy/G UILLEMIN. Stable Mappings
Dimensions 2 and 3.
and Their Singularities.
48
SACHS/WU. General Relativity fo r
15
BERBERIAN. Lectures in Functional
Mathematicians.
Analysis and Operator Theory.
49
GRUENBERG/WEIR. Linear Geometry.
WINTER. The Stucture of Fields.
2nd ed.
16
17
ROSENBLATT. Random Processes. 2nd ed.
50
EDWDS. Fermat's Last Theorem.
18
HALMOS. Measure Theory.
51
KLINGENBERG. A Course in Diferential
19
HALMOS. A Hilbet Space Problem Book.
Geometry .
2nd ed.
52
HARTSHORNE. Algebraic Geometry.
20
HUSEMOLLER. Fibre Bundles. 3rd ed.
53
MANIN. A Course in Mathematical Logic.
21
HUMPHREYS. Linear Algebraic Groups.
54
GRAVERlWANS. Combinatorics with
22
BARNES/MACK. An Algebraic Introduction
Emphasis on the Theory of Graphs.
to Mathematical Logic.
55
BROWN/PEARCY. Introduction to Operator
23
GREUB. Linear Algebra. 4th ed.
Theory I: Elements of Functional
24
HOLMES. Geometric Functional Analysis
Analysis.
and Its Applications.
56
MASSEY. Algebraic Topology: An
25
HEWITT/STROMBERG. Real and Abstract
Introduction.
Analysis.
57
CROWELL/Fox. Introduction to Knot
26
MANES. Algebraic Theories.
Theory.
27
KELLEY. General Topology.
58
KOBLITZ. p-adic Numbers, p-adic Analysis,
28
ZARISKI/SAMUEL. Commutative Algebra.
and Zeta-Functions. 2nd ed.
VoU.
59
LANG. Cyclotomic Fields.
29
ZARISKI/SAMUEL. Commutative Algebra.
60
ARNOLD. Mathematical Methods in
VoUI.
Classical Mechanics. 2nd ed.
30
JACOBSON. Lectures in Abstract Algebra I.
61
WHITEHED. Elements ofHomotopy
Basic Concepts.
Theory.
31
JACOBSON. Lectures in Abstract Algebra II.
62
KARGAPOLOV/MERLZJKOV. Fundamentals
Linear Algebra.
of the Theory of Groups.
32
JACOBSON. Lectures in Abstract Algebra
63
BOLLOBAS. Graph Theory.
III. Theory of Fields and Galois Theory.
64
EDWDS. Fourier Series. Vol. I. 2nd ed.
33
HIRSCH. Differential Topology.
65
WELLS. Differential Analysis on Complex
34
SPITZER. Principles ofRandom Walk.
Manifolds. 2nd ed.
2nd ed.
(continued after index)
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Ward Cheney
Analysis for Applied
Mathematics
With 27 Illustrations
t Springer
Ward Cheney
Department of Mathematics
University of Texas at Austin
Austin, TX 78712-1082
USA
Editorial Board
S. Axler
Mathematics Department
San Francisco State
University
San Francisco, CA 94132
USA
. .
Gehring
Mathematics Department
East Hall
University of Michigan
Ann Arbor, MI 48109
USA
K.A. Ribet
Mathematics Department
University of California
at Berkeley
Berkeley, CA 94720-3840
USA
Mathematics Subject Classification (2000): 46Bxx, 65L60, 32Wxx, 42B1O
Library of Congress Cataloging-in-Publication Data
Cheney, E. W. (Elliott Ward). 1929-
Analysis for applied mathematics
f
Ward Cheney.
p. cm. - (Graduate texts in mathematics ; 208)
Includes bibliographical references and index.
ISBN 0-387-95279-9 (alk. paper)
I.
Mathematical analysis.
I. Title.
II. Series.
QA300.C4437
2001
515-dc21
2001·1020440
Printed on acid-free paper.
©
2001 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York.
NY 10010. USA). except for brief excerpts in connection with reviews or scholarly analysis. Use
in connection with any form of information storage and retrieval. electronic adaptation, computer
software. or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names. trade names. trademarks. etc., in this publication, even if
the former are not especially identified, is not to be taken as a sign that such names, as understood
by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Production managed by Terry Kornak; manufacturing supervised by Jerome Basma.
Photocomposed from the author's TeX files.
Printed and bound by Maple-Vail Book Manufacturing Group, York, PA.
Printed in the United States of America.
9
8
765
4
321
ISBN 0-387-95279-9
SPIN 10833405
Springer-Verlag New York Berlin Heidelberg
A member of BertelsmannSpringer Science+Business Media GmbH
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Preface
This book evolved from a course at our university for beginning graduate stu­
dents in mathematics-particularly students who intended to specialize in ap­
plied mathematics. The content of the couse made it attractive to other math­
ematics students and to graduate students from other disciplines such as en­
gineering, physics, and computer science. Since the course was designed for
two semesters duration, many topics could be included and dealt with in de­
tail. Chapters 1 through 6 relect roughly the actual nature of the course, as it
was taught over a number of years. The content of the course was dictated by
a syllabus governing our preliminary Ph.D. examinations in the subject of ap­
plied mathematics. That syllabus, in turn, expressed a consensus of the faculty
members involved in the applied mathematics program within our department.
The text in its present manifestation is my interpretation of that syllabus: my
colleagues are blameless for whatever laws are present and fo r any inadvertent
deviations from the syllabus.
The book contains two additional chapters having important material not
included in the course: Chapter 8, on mesure and integration, is for the ben­
eit of reades who want a concise presentation of that subject, and Chapter 7
contains some topics closely allied, but peripheral, to the principal thrust of the
course .
This arrangement of the material deserves some explanation. The ordering
of chapters relects our expectation of our students: If they are unacquainted
with Lebesgue integration (for example), they can nevertheless understand the
examples of Chapter 1 on a supericial level, and at the same time, they can
begin to remedy any deiciencies in their knowledge by a little private study
of Chapter 8. Similar remarks apply to other situations, such as where some
point-set topoloy is involved; Section 7.6 will be helpful here. To summarize:
We encourage students to wade boldly into the course, starting with Chapter 1,
and, where necessary, ill in any gaps in their prior preparation. One advantage
of this strategy is that they will see the necessity for topology, measure theory,
and other topics - thus becoming better motivated to study them. In keeping
with this philosophy, I have not hesitated to make forward references in some
proofs to material coming later in the book. For example, the Banach contraction
mapping theorem is needed at least once prior to the section in Chapter 4 where
it is dealt with at length.
Each of the book's six main topics could certainly be the subject of a year's
course (or a lifetime of study), and many of our students indeed study functional
analysis and other topics of the book in separate courses. \10st of them eventu­
ally or simultaneously take a year-long course in analysis that includes complex
analysis and the theory of mesure and integration. However, the applied math­
ematics course is typically taken in the irst year of graduate study. It seems
to bridge the gap between the undergraduate and graduate curricula in a way
that has been found helpful by many students. In particular, the course and the
v
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