mathematical physics - a modern introduction to its foundations

As the unifying theme of the book, vector spaces demand carefulanalysis, and Part I provides this in the more accessible setting of finite dimension in a language thatis conveniently gen

Mathematical Physics

A Modem Introduction to Its Foundations

With 152 Figures

ODTlJ KU1"UPHANESt

M E T U liBRARY

METULIBRARY 2

~themltlcoJphysicS: Imodem

336417Libraryof Congress Cataloging-in-Publication Data

Hassani, Sadri.

Mathematical physics: a modem introductionits foundations /

Sadri Hassani.

p em.

Includesbibliographical referencesand index.

ISBN 0-387-98579-4 (alk paper)

1 Mathematical physics I Title.

QC20.H394 1998

Printed on acid-freepaper.

QC20 14394

c,2.

© 1999Springer-Verlag New York, Inc.

All rights reserved.This work may not be translatedor 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-inconnection with any form of informationstorage and retrieval, electronicadaptation, computer sctt- ware, or by similar or dissimilarmethodology now known or hereafterdevelopedis forbidden The use of general descriptivenames, trade names, trademarks, etc., in this publication, even if the formerare not especiallyidentified, is not to be taken as a sign that such names, as understoodby the Trade Marks and Merchandise Marks Act, may accordinglybe used freely by anyone.

Productionmanagedby Karina Mikhli;manufacturing supervisedby ThomasKing.

Photocomposed copy preparedfrom the author's TeX files.

Printed and bound by HamiltonPrinting Co., Rensselaer, NY.

Printed in the United Statesof America.

9 8 7 6 5 4 3 (Correctedthird printing, 2002)

ISBN 0-387-98579-4 SPIN 10854281

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A member ofBertelsmannSpringer Science+Business Media GmbH

"Ich kann es nun einmal nicht lassen, in diesem Drama von

Mathematik und Physik -<lie sich im Dunkeln befrnchten,

aber von Angesicht zu Angesicht so geme einander verkennen und verleugnen-die Rolle des (wie ich gentigsam erfuhr, oft unerwiinschten)Botenzu spielen."

Hermann Weyl

Itis said that mathematics is the language of Nature.Ifso, then physics is itspoetry Nature started to whisper into our ears when Egyptians and Babylonianswere compelled to invent and use mathematics in their day-to-day activities Thefaint geomettic and arithmetical pidgin of over four thousand years ago, snitablefor rudimentary conversations with nature as applied to simple landscaping, hasturned into a sophisticated language in which the heart of matter is articulated.The interplay between mathematics and physics needs no emphasis Whatmay need to be emphasized is that mathematics is not merely a tool with which thepresentation of physics is facilitated, butthe only medium in which physics cansurvive Just as language is the means by which humans can express their thoughtsand without which they lose their unique identity, mathematics is the only languagethrough which physics can express itself and without which it loses its identity.And just as language is perfected due to its constant usage, mathematics develops

in the most dramatic way because of its usage in physics The quotation by Weylabove, an approximation to whose translation is"In this drama of mathematics and physics-which fertilize each other in the dark, but which prefer to deny and misconstrue each other face to face-I cannot, however, resist playing the role

of a messenger, albeit, as I have abundantly learned, often an unwelcome one:'

is a perfect description of the natnral intimacy between what mathematicians andphysicists do, and the nnnatnral estrangement between the two camps Some of themost beantifnl mathematics has been motivated by physics (differential eqnations

by Newtonian mechanics, differential geometry by general relativity, and operatortheory by qnantnmmechanics), and some of the most fundamental physics has beenexpressed in the most beantiful poetry of mathematics (mechanics in symplecticgeometry, and fundamental forces in Lie group theory)

I do uot want to give the impression that mathematics and physics cannotdevelop independently On the contrary, it is precisely the independence of eachdiscipline that reinforces not only itself, but the other discipline as well-just as thestndy of the grammar of a language improves its usage and vice versa However,the most effective means by which the two camps can accomplish great success

is throngh an inteuse dialogue Fortnnately, with the advent of gauge and stringtheories ofparticle physics, such a dialogue has been reestablished between physicsand mathematics after a relatively long lull

Level and Philosophy of Presentation

This is a book for physics stndeuts interested in the mathematics they use It

is also a book furmathematics stndeuts who wish to see some of the abstractideas with which they are fantiliar come alive in an applied setting The level ofpreseutation is that of an advanced undergraduate or beginning graduate course (orsequence of courses) traditionally called "Mathematical Methods of Physics" orsome variation thereof Unlike most existing mathematical physics books intendedfor the same audience, which are usually lexicographic collections of facts aboutthe diagonalization of matrices, tensor analysis, Legendre polynomials, contourintegration, etc., with little emphasis on formal and systematic development oftopics, this book attempts to strike a balance between formalism and application,between the abstract and the concrete

I have tried to include as mnch of the essential formalism as is necessary torender the book optimally coherent and self-contained This entails stating andproving a large nnmber of theorems, propositions, lemmas, and corollaries Thebenefit of such an approach is that the stndent will recognize clearly both the powerand the limitation of a mathematical idea used in physics There is a tendency on thepart of the uovice to universalize the mathematical methods and ideas eucountered

in physics courses because the limitations of these methods and ideas are notclearly pointed out

There is a great deal of freedom in the topics and the level of presentation thatinstructors can choose from this book My experience has showu that Parts I,TI,

Ill, Chapter 12, selected sections of Chapter 13, and selected sections or examples

of Chapter 19 (or a large snbset of all this) will be a reasonable course content foradvanced undergraduates.Ifone adds Chapters 14 and 20, as well as selected topicsfrom Chapters 21 and 22, one can design a course snitable for first-year graduate

students By judicious choice of topics from Parts VII and VIII, the instructorcan bring the content of the course to a more modern setting Depending on thesophistication of the students, this can be done either in the first year or the secondyear of graduate school.

Features

To betler understand theorems, propositions, and so forth, students need to seethem in action There are over 350 worked-out examples and over 850 problems(many with detailed hints) in this book, providing a vast arena in which studentscan watch the formalism unfold The philosophy underlying this abundance can

be summarized as''Anexample is worth a thousand words of explanation." Thus,whenever a statement is intrinsically vague or hard to grasp, worked-out examplesand/or problems with hints are provided to clarify it The inclusion of such alarge number of examples is the means by which the balance between formalismand application has been achieved However, although applications are essential

in understanding mathematical physics, they are only one side of the coin Thetheorems, propositions, lemmas, and corollaries, being highly condensed versions

of knowledge, are equally important

A conspicuous feature of the book, which is not emphasized in other rable books, is the attempt to exhibit-as much as.it is useful and applicable-«interrelationships among various topics covered Thus, the underlying theme of avector space (which, in my opinion, is the most primitive concept at this level ofpresentation) recurs throughout the book and alerts the reader to the connectionbetween various seemingly unrelated topics

compa-Another useful feature is the presentation of the historical setting in whichmen and women of mathematics and physics worked I have gone against thetrend of the "ahistoricism" of mathematicians and physicists by summarizing thelife stories of the people behind the ideas Many a time, the anecdotes and thehistorical circumstances in which a mathematical or physical idea takes form can

go a long way toward helping us understand and appreciate the idea, especiallyif

the interaction among-and the contributions of-all those having a share in thecreation of the idea is pointed out, and the historical continuity of the development

of the idea is emphasized

To facilitate reference to them, all mathematical statements (definitions, rems, propositions, lemmas, corollaries, and examples) have been nnmbered con-secutively within each section and are preceded by the section number For exam-

theo-ple, 4.2.9 Definition indicates the ninth mathematical statement (which happens

to be a definition) in Section 4.2 The end of a proof is marked by an empty square

D, and that of an example by a filled square III, placed at the right margin of each.Finally, a comprehensive index, a large number of marginal notes, and manyexplanatory underbraced and overbraced comments in equations facilitate the use

and comprehension of the book.Inthis respect, the book is also nsefnl as a

refer-ence.

Organization and Topical Coverage

Aside from Chapter 0, which is a collection of pnrely mathematical concepts,the book is divided into eight parts Part I, consisting of the first fonr chapters, isdevoted to a thorough study of finite-dimensional vector spaces and linear operatorsdefined on them As the unifying theme of the book, vector spaces demand carefulanalysis, and Part I provides this in the more accessible setting of finite dimension in

a language thatis conveniently generalized to the more relevant infinite dimensions,'the subject of the next part

Following a brief discussion of the technical difficulties associated with finity, Part IT is devoted to the two main infinite-dimensional vector spaces ofmathematical physics: the classical orthogonal polynomials, and Foutier seriesand transform

in-Complex variables appear in Partill.Chapter 9 deals with basic properties ofcomplex functions, complex series, and their convergence Chapter 10 discussesthe calculus of residues and its application to the evaluation of definite integrals.Chapter II deals with more advanced topics such as multivalued functions, analyticcontinuation, and the method of steepest descent

Part IV treats mainly ordinary differential equations Chapter 12 shows howordinary differential equations of second order arise in physical problems, andChapter 13 consists of a formal discussion of these differential equations as well

as methods of solving them numerically Chapter 14 brings in the power of plex analysis to a treatment of the hypergeometric differential equation The lastchapter of this part deals with the solution of differential equations using integraltransforms

com-Part V starts with a formal chapter on the theory of operator and their spectraldecomposition in Chapter 16 Chapter 17 focuses on a specific type of operator,namely the integral operators and their corresponding integral equations The for-malism and applications of Stnrm-Liouville theory appear in Chapters 18 and 19,respectively

The entire Part VI is devoted to a discussion of Green's functions Chapter

20 introduces these functions for ordinary differential equations, while Chapters

21 and 22 discuss the Green's functions in an m-dimensional Euclidean space.Some of the derivations in these last two chapters are new and, as far as I know,unavailable anywhere else

Parts VII andvrncontain a thorough discussion of Lie groups and their plications The concept of group is introduced in Chapter 23 The theory of grouprepresentation, with an eye on its application in quantom mechanics, is discussed

ap-in the next chapter Chapters 25 and 26 concentrate on tensor algebra and ten-,sor analysis on manifolds.In Partvrn,the concepts of group and manifold are

brought together in the coutext of Lie groups Chapter 27 discusses Lie groupsand their algebras as well as their represeutations, with special emphasis on theirapplication in physics Chapter 28 is on differential geometry including a briefintroduction to general relativity Lie's original motivation for constructing thegroups that bear his name is discussed in Chapter 29 in the context of a systematictreatment of differential equations using their symmetry groups The book ends in

a chapter that blends many of the ideas developed throughout the previous parts

in order to treat variational problems and their symmetries.Italso provides a mostfitting example of the claim made at the beginning of this preface and one of themost beautiful results of mathematical physics: Noether's theorem ou the relationbetween symmetries and conservation laws

my son, Dane, and my daughter, Daisy, for the time taken away from them while

I was writing the book, and for their support during the long and arduous writing

process.

Many excellent textbooks, too numerous to cite individually here, have enced the writing of this book The following, however, are noteworthy for boththeir excellence and the amount of their influence:

influ-Birkhoff, G.,andG.-C Rota, Ordinary Differential Equations, 3rd ed., New York,

Hamennesh, M Group Theory and its Application to Physical Problems, Dover,

Simmons, G Calculus Gems, New York, McGraw-Hill, 1992.

History of Mathematics archive at www-groups.dcs.st-and.ac.uk:80.

I wonld greatly appreciate any comments and suggestions for improvements.Although extreme care was taken to correctall the misprints, the mere volume ofthe book makes it very likely that I have missed some (perhaps many) of them Ishall be most grateful to those readers kind enough to bring to my attention anyremaining mistakes, typographical or otherwise Please feel free to contact me.Sadri Hassani

Campus Box 4560Department of PhysicsIllinois State UniversityNormal, IL 61790-4560, USAe-mail: hassani@entropy.phy.i1stu.edu

Itis my pleasure to thankall those readers who pointed out typographical mistakesand suggested a few clarifying changes With the exceptionofa couple that requiredsubstantial revisiou, I have incorporated all the corrections and suggestions in thissecond printing

Mathematics and physics arelike the game of chess (or, for that matter, like anygamej-i-you willleam only by ''playing'' them No amount of reading about thegame will make you a master.Inthis book you will find a large number of examplesand problems Go through as many examples as possible, and try to reproduce them.Pay particular attention to sentences like "The reader may check "or"Itisstraightforwardto show "These are red flags warning you that for a goodunderstanding of the material at hand, yon need to provide the missing steps Theproblems often fill in missing steps as well; and in this respect they are essentialfor a thorough understanding of the book Do not get discouraged if you cannot get

to the solution of a problem at your first attempt Ifyou start from the beginningand think about each problem hard enough, youwill get to the solution, and you

will see that the subsequent problems will not be as difficult

The extensive index makes the specific topics about which you may be terested to leam easily accessible Often the marginal notes will help you easilylocate the index entry you are after

in-I have included a large collection of biographical sketches of mathematicalphysicists of the past These are truly inspiring stories, and I encourage you to readthem They let you see that even under excruciating circumstances, the human mindcan work miracles You will discover how these remarkable individuals overcamethe political, social, and economic conditions of their time to let us get a faintglimpse of the truth They are our true heroes

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

PrefaceNote to the ReaderList of Symbols

o Mathematical Preliminaries

0.3 Metric Spaces 0.4 Cardinality

0.5 Mathematical Induction0.6 P r o b l e m s

1 Vectors and Transformations1.1 Vector Spaces

1.3 Linear Transformations 1.4 Algebras

1.5 Problems

2 Operator Algebra2.1 Algebra«s:(V)

17

19

19

23 32414449

49

2.2 Derivatives of Functions of Operators 2.3 Conjugation of Operators 2.4 Hermitian and Unitary Operators2.5 Projection Operators 2.6 OperatorsinNumerical Analysis

4 Spectral Decomposition4.1 Direct Sums 4.2 InvariantSubspaces 4.3 Eigeuvaluesand Eigenvectors 4.4 Spectral Decomposition4.5 Functions of Operators4.6 Polar Decomposition4.7 Real VectorSpaces4.8 Problems

5 Hilbert Spaces5.1 The Question of Convergence 5.2 The Space of Square-IntegrableFunctions

6 Generalized Functions6.1 ContinuousIndex6.2 Generalized Functions 6.3 Problems

7 Classical Orthogonal Polynomials7.1 General Properties 7.2 Classification

7.3 Recurrence Relations 7.4 Examples of Classical Orthogonal Polynomials

566163677076 82 82 87 89

9193

101

103109109112114117125129130138

143

145145150157

159159165169172

172175176179

10.2 Classification of Isolated Singularities

10.3 Evaluation of Definite Integrals

11 Complex Analysis: AdvancedTopics

ILl Meromorphic Functions

11.2 Multivalued Functions

11.3 Analytic Continuation

1104 The Gamma and Beta Functions .

11.5 Method of Steepest Descent

11.6 Problems

270270273275290

293

293295302309312319

12 Separation of Variables in Spherical Coordinates 32712.1 PDEs of Mathematical Physics 32712.2 Separation oftheAngular Part of the Laplacian 33112.3 Construction of Eigenvalues ofL2 334

1204 Eigenvectors of L2: Spherical Harmonics 33812.5 Problems , 346

13 Second-Order Linear Differential Equations 34813.1 General Properties of ODEs 34913.2 Existence and Uniqneness for First-Order DEs 350

13.10 Problems 394

15.1 Integral Representation of the Hypergeometric Function 43415.2 Integral Representation of the Confiuent Hypergeometric

Function 437

15.5 Problems 445

V Operators on Hilbert Spaces

16 AnInlroduction to Operator Theory16.1 From Abstract to Integral and Differential Operators 16.2 Bounded Operators in Hilbert Spaces

16.3 Spectra of Linear Operators

16.6 Spectrum of Compact Operators16.7 Spectral Theorem for Compact Operators 16.8 Resolvents

16.9 Problems

17 Integral Equations17.1 Classification

449

451451453457458464467473480485488488

17.2 Fredholm Integra!Equations 494

18.1 Unbounded Operators with Compact Resolvent 50718.2 Sturm-Liouville Systems and SOLDEs 51318.3 Other Properties of Sturm-Liouville Systems 51718.4 Problems 522

20 Green's FunctionsinOne Dimension

20.1 Calculationof Some Green's Functions

20.2 Formal Considerations

20.3 Green's Functions for SOLDOs

20.4 EigenfunctionExpansion of Green's Fnnctions

20.5 Problems

21 Multidimensional Green's Functions: Formalism

21.1 Properties of Partial Differential Equations

21.2 MultidimensionalGFs and Delta Functions

22.4 The Fourier TransformTechnique

22.5 The EigenfunctionExpansion Technique

22.6 Problems

551

r-554557565577580

583584592596600603610

613613621626628636641

VII Groups and Manifolds

23 Group Theory

23.1 Groups

23.2 Subgroups 23.3 Group Action 23.4 The Symmetric Groups,

23.5 Problems

24 Group Representation Theory

24.1 Definitions and Examples24.2 Orthogonality Properties .24.3 Analysis of Representations 24.4 Group Algebra .24.5 Relationship of Characters to Those of a Subgroup 24.6 Irreducible Basis Functions

24.7 Tensor Product of Representations 24.8 Representations of the Symmetric Group24.9 Problems

25 Algebra of Tensors

25.1 Multilinear Mappings25.2 Symmetries of Tensors 25.3 Exterior Algebra 25.4 hmer Product Revisited 25.5 The Hodge Star Operator25.6 Problems

26 Analysis of Tensors

26.1 Differentiable Manifolds .26.2 Curves and Tangent Vectors 26.3 Differential of a Map 26.4 Tensor Fields on Manifolds26.5 Exterior Calculus 26.6 Symplectic Geometry26.7 Problems

VIII Lie Groups and Their Applications

27 Lie Groups and Lie Algebras

27.1 Lie Groups and Their Algebras 27.2 An Outline of Lie Algebra Theory 27.3 Representation of Compact Lie Groups

649

651

652656663664669

673 -J

673680685687692695699707723

728

729736739749756758

763

763770776780791801808

813

815

815833845

27.4 Representationof the General Linear Group

27.5 Representationof Lie Algebras

856859876

28.7 Problems 932

29.1 Synunetries of Algebraic Equations 93629.2 Synunetry Groups of Differential Equations 941

30 Calcnlusof Variations, Symmetries, and ConservationLaws 973

30.1 The Calculus of Variations 97330.2 Symmetry Groups of VariationalProblems 98830.3 ConservationLaws and Noether's Theorem 992

30.5 Problems 1000

j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j J

Set of ordered pairs (a, b)witha E A andbE B {(aI, a2, , an)lat E Al

Union, (Intersection)

Ais eqnivalent toB

x is mapped to f(x) via the map f

forall (valnes of)There exists (a valne of)Eqnivalence class to whichabelongsComposition of maps f andg

ifand only ifSet of functions on (a, b)with continnons derivatives np to orderk

Set of complex (or real) n-tnplesSet of polynomials intwith complex coefficientsSet of polynomials intwith real coefficientsSet of polynomials with complex coefficients of degreenor lessSet ofall complex seqnences(atl~1 snch that L:~I latl2< 00Inner prodnct ofla) andIbl

Norm (length) of the vectorla)

Exterior (wedge) product of skew-symmetric tensors A and BSet ofall skew-symmetric tensors of type(p,0) on V

Mathematical Preliminaries

This introductory chapter gathers together some of the most basic tools and notionsthat are used throughout the book Italso introduces some common vocabularyand notations used in modem mathematical physics literature Readers familiar

with such concepts as sets, maps, equivalence relations, and metric spaces may

wish to skip this chapter

0.1 Sets

Modem mathematics starts with the basic (and undefinable) concept of set Wethink of a set as a structureless family, or collection, of objects We speak, forexample, of the set of students in a college, of men in a city, of women workingconcept ofset for a corporation, of vectors in space, of points in a plane, or of events in theelaborated continuum of space-time Each member a of a set A is called an element of that

sel This relation is denoted bya E A (read "a is an element of A" or "abelongs

toA"), and its negation by a ¢ A Sometimes a is called a point of the set Atoemphasize a geometric connotation

A set is usually designated by enumeration of its elements between braces.For example, {2, 4, 6, 8} represents the set consisting of the first four even naturalnumbers; {O, ±I, ±2, ±3, } is the set of all integers; {I,x, x 2 , x 3, }is theset of all nonnegative powers ofx;and {I,i, -1, -i}is the set of the four complexfourth roots of unity.Inmany cases, a set is defined by a (mathematical) statementthat holds for all ofits elements Such a set is generally denoted by{xIP (x)}andread "the set of allx's such that P(x)is true." The foregoing examples of sets can

be written alternatively as follows:

{n In is even and I < n < 9}

{±nInis a natural number}

{y Iy = x" and n is a uatural uumber}

{zIZ4 = I andzis a complex uumber}

x as{x"}~o'This last notation will be used frequeutly iu this book A set with asingle element is called a singleton

Ifa E A whenever aE B,we say thatB is a subset ofA and write B C A or A:J B.IfBe A and A c B,thenA = B.1f Be A and A i'B, thenBis called

a proper subset ofA.The set defined by{ala i'a}iscalled the empty set and

is denoted by 0 Clearly, 0 contains no elements and is a subset of any arbitraryset The collection of all subsets (including 0) of a set A is denoted by 2 A •Thereason for this notation is that the number of subsets of a set containingn elements

is 2" (Problema.I).1fA and B are sets, their union, denoted by A U B, is the set

containing all elements that belong toAorBor both The intersection of the sets

.!\and B,denoted byA n B,is the set containing all elements belonging to both

Aand B.If{B.}.El is a collection of sets,1we denote their union byU.B.andtheir intersection byn.B a-

Inany application of set theory there is an underlying universal set whosesubsets are the objects of study This universal set is usually clear from the context.For exaunple, in the study of the properties of integers, the set of integers, denoted

by Z, is the universal set The set of reals,JR,is the universal set in real analysis,and the set of complex numbers,iC, is the universal set in complex analysis With

a universal set X in mind, one can write X~Ainstead of~ A.The complement

of a setA is denoted by~A and defined as

~A sa {a IaIi!A}.

The complement ofB inA(or their difference) is

A ~B =={ala EA anda Ii!B}.

From two given setsAandB, it is possible to form the Cartesian prodnct ofA

andB,denoted byA x B,which is the set of ordered pairs(a, b),wherea EA

andbE B.This is expressed in set-theoretic notatiou as

A x B ={(a, b)la EA and b e B}.

1HereI is an index set or a counting set-with its typical element denoted byct.In most cases,I is the set of (nonnegative)

it

relation and

equivalence relation

We can generalize this to an arbitrary number of sets.IfAI, Az, , An are sets,

then the Cartesian product of these sets is

Al x Az x x An = {(ai, az, , an)!ai EAd,which is a set of ordered n-tuples.IfAl =Az = =An =A, then we write

An instead of A x A x··· x A, and

An= {(ai, az, , an)Ia; E Aj

The most familiar example of a Cartesian product occurs when A= R ThenJRz is the set of pairs (XI, xz) with XI, xz E JR This is simply the points in theEuclidean plane Similarly, JR3 is the set of triplets (XI, xz,X3),or the points inspace, andJRn = {(XI, Xz, , Xn)!Xi E JRj is the set of real n-tuples.

0.1.1 Equivalence Relations

There are many instances in which the elements of a set are naturally groupedtogether For example, all vector potentials that differ by the gradient of a scalarfunction can be grouped together because they all give the same magnetic field.Similarly, all quantum state functions (of unit "length") that differ by a multi-plicative complex number of unit length can be grouped together because they allrepresent the same physical state The abstraction of these ideas is summarized inthe following definition

0.1.1 Definition Let A be a set A relation on A is a comparison test between ordered pairs ofelements of A If the pair (a, b) E A x A pass this test, we write

at>b and read "a is related to b" An equivalence relation an A is a relation that has the fallowing properties:

af>a V'aEA,

a s-b ~ b s a a.b e A,

a i-b.b» c ==>- a[>c a.b;c E A,

(reflexivity)(symmetry)(transivity)

When at>b, we say that "a is equivalent to b" The set [a] = {bE Albt>aj ofall

equivalence class elements that are equivalent to a is called the equivalence class of a.

The reader may verify the following property of equivalence relations

0.1.2 Proposition. If »isan equivalence relation an A and a, bE A,then either

[a] n[b] =0or [a] =[bl

representative ofan

equivalence class

Therefore,a' E [a] implies that [a'] = [a].Inother words, any element of

an equivalence class can be chosen to be a representative of that class Because

of the symmetry of equivalence relations, sometimes we denote them byc-o

0.1.3 Example LetAbe the set of humanbeings.Leta »bbe interpretedas"ais older

than b." Then clearly,I> is a relation but not an equivalence relation On the other hand, if

we interpretaE>bas"aandbhave the same paternal grandfather," then l> is an equivalence relation, as the reader may check The equivalence class ofa is the set of all grandchildren

ofa's paternalgrandfather

LetV bethe set of vector potentials Write A l> A' if A - A' = Vf for some function

f. The reader may verify that" is an equivalence relation and that [A] is the set of allvector potentials giving rise to the same magnetic field.

Let the underlying set be Z x (Z - {OJ) Say"(a, b)is related to(c, d)"ifad = be.

Then this relation is an equivalence relation Furthermore,[(a, b)]can be identified as the

0.1.4 Definition Let A be a set and {Ra} a collection ofsubsets ofA We say that

partition ofa set {Ra} is a partition of A, or {Ra} partitions A, ifthe Ra' s are disjoint, i.e., have

no elementin common, and Uo;Ba= A.

Now consider the collection {[a]Ia E A} of all equivalence classes ofA.

quotient set These classes are disjoint, aod evidently their union covers all ofA. Therefore,

the collection of equivalence classes of A is a partition of A This collection is

denoted byA/1><1aod is called the quotient set ofAunder the equivalence relation1><1.

0.1.5 Example Let the underlying set belll.3.Define an equivalence relation onlll.3bysaying that PI E lR3 andP2E }R3 are equivalent if they lie on the same line passing through the origin Then ]R3I l><l is the set of all lines in space passing through the origin If we choose the unit vector with positive third coordinate along a given line as the representative

of that line, then]R3Il><l can be identified with the upper unit hemisphere.e ]R3Il><l is calledprojective space the projective space associated with]R3.

On the setIEof integers define a relation by writingm e- nform, n E IEifm - nis divisible by k,wherekis a fixed integer Then e- is not only a relation, but an equivalence relation.Inthis case, we have

Z/"= {[O], [1], ,[k - I]},

as the reader is urged to verify.

For the equivalence relation defined onIEx IEof Example 0.1.3, the setIEx lEIl><l can

0.2 Maps

map, domain,

codomain, image

To communicate between sets, one introduces the concept of a map A mapf

from a set X to a setY,denoted by f :X -> Yor X ~ Y, is a correspondencebetween elements of X aod those ofY in which all the elements of X participate,

Figure 1 The mapf maps all of the set X onto a subset ofY.The shaded areain Yis

f(K),the range off.

and each element of X corresponds to only one element ofY (see Figure 1) If

y E Y is the element that corresponds to x E Xvia the map f, we write

y= f(x) or x f > f(x) orand callf (x) the image of x onder f Thus, by the definition of map, x E X canhave only one image The set X is called the domain, and Y the codomain or

the target space Two maps f :X > Y and g :X >Y are said to be equal if

function f(x) = g(x) for all x E X

0.2.1 Box A map whose codomain is the set ofreal numbers IR or the set ofcomplex numbersiCis commonly called a function.

A special map that applies to all sets A is idA : A >A, called the identity

identity map map ofA, and defined by

we call f(A) = {f(x)lx E A} the image of A Similarly,ifB C f(X), we call

preimage f- 1(B) = {x E Xlf(x) E B) the inverse image, or preimage, of B. In words,

f-1 (B ) consists of all elements in X whose images are in B C Y.1f B consists

of a single elementb, thenr:'(b) = {x E Xlf(x) = b) consists of all elements

of X that are mapped to b.Note that it is possible for many points of X to havethe same image in Y The subset f(X) of the codomain of a map f is called the

range off (see Figure 1)

Figure 2 Thecomposition of two maps is another map.

IfI : X > Y and g : Y > W, then the mapping h : X > W given

by h(x) = g(f(x)) is' called the composition of I and g, and is denoted by

h = g0I (see Figure 2).3Itis easy to verify thatloidx=l=idyol

Ifl(xI) = I(X2) implies that XI = X2,we call I injective, or one-to-one

(denoted I-I) For an injective map only one element of X corresponds to an

element of Y. IfI(X) = Y, the mapping is said to be surjective, oronto A

map that is both injective and surjective is said to be bijective, or to be a one correspondence Two sets that are in one-to-one correspondence, have, by

one-to-definition, the same nnmber of elements IfI :X > Yis a bijection fromXontoY, then for eachy E Y there is one and only one elementX in X for which

I(x) = y.Thus, there is a mapping I-I :Y > X given by I-I(y) = x,where

X is the nniqne element such that I(x) = y This mapping is called the inverse

of I.The inverse ofI is also identified as the map that satisfiesI 0 I-I = idy

and I-I 0 I = idx-For example, one can easily verify that ln-I = exp andexp"! = ln, becauseIn(e X

) = X andelnx = x.

Given a mapI :X >Y,we can define a relationtxlonX by sayingXI txl

X2 if l(xI) = I(X2) The reader may check that this is in fact an equivalence

relation The equivalence classes are subsets of X all of whose elements map to

the same point in Y. In fact, [x] = 1-1(f(X» Corresponding toI, there is amap! : X/txl > Y given by !([x]) = I(x). This map is injective because

if !([XI]) = !([X2]),then l(xI) = I(X2), so XI andX2belong to the sameequivalence class; therefore, [XI] = [X2]. Itfollows that! :X/txl > I(X) isbijective

IfI and g are both bijections with inverses I-I and g -I, respectively,then goI

also has an inverse, and verifying that (g0 f)-I = I-log-I is straightforward

3Note the importance of the order in whichthe composition is written The reverse ordermaynot even exist.

0.2.2 Example As an example of the preirnage of a set, consider the sine and cosine

functions Then it should be clearthat

Similarly, sin- 1[O,!l consists of all the intervals on the x-axis markedbyheavy line

segments in Figure 3, i.e., all the pointswhose sine lies between0 and~

As examples of maps, we consider fonctions1 :lR-+lR stndied in calculus.The Iwo fonctions 1 :lR -+ lR audg : lR-+ (-I,+I) given, respectively, by

1 (x) = x 3audg(x)= taubx are bijective The latter function, by the way, shows

that there are as mauy points in the whole real line as there are in the interval

(-I,+1).Ifwe denote the set of positive real numbers by lR+, then the function

1 : lR-+ lR+ given by I(x) = x 2 is surjective but not injective (bothx aud

~x map tox 2 ).The function g :lR+-+ lR given by the same rule, g(x) = x 2 ,

is injective but not surjective Onthe other haud, h : lR+-+ lR+ again given by

h(x) = x 2is bijective, butu : lR-+ lR given by the same rule is neither injective

nor surjective.

LetM n x ndenote the setofn xnreal matrices Define a function det :M n x n -+

lR by det(A) = det A,where det Ais the determinaut of Afor AE J\1nxn.This tion is clearly surjective (why?) but not injective The set of all matrices whosedeterminaut is 1 is det- I( I ).Such matrices occur frequently in physical applica-

fonc-tions.

Another example of interest is1 :C -+ lRgiven by1(z) = [z] This function

is also neither injective nor swjective Here1-1 ( 1) is the unit circle, the circle

of radius I in the complex plaue

The domain of a map cau be a Cartesiau product of a set, as in1 :X x X -+ Y.

Two specific cases are worthy of mention The first is when Y = R An example

of this case is the dot product onvectors, Thus, if X is the set of vectors in space,

we cau define I(a,b) = a· b.The second case is when Y = X Then 1is

called a binary operation on X, whereby au element in X is associated with two'

elements in X For instance, let X = Z, the set of all integers; then the fonction

I: Z xZ-+ Zdefined by[tm, n) = mnis the binary operation of multiplication

of integers Similarly,g : lR x lR-+ lR given by g(x, y) = x+yis the binaryoperation of addition of real numbers

0.3 Metric Spaces

Although sets are at the root of modem mathematics, they are only of formal audabstract interest by themselves To make sets useful, it is necessary to introducesome structnres on them There are two general procedures for the implementa-tion of such structnres These are the abstractions of the two major brauches ofmathematics-algebra aud aualysis

Figure 3 The unionof all the intervals on the x-axis marked by heavyline segments is

-1[0 1]

We canturna set into an algebraic structure by introducing a binary operation

on it For example, a vector space consists, among other things, of the binaryoperation of vector addition A group is, among other things, a set together with thebinary operation of "multiplication" There are many other examples of algebraicsystems, and they constitute the rich subject of algebra

When analysis, the other branch of mathematics, is abstracted using the concept

of sets, it leads to topology, in which the concept of continuity plays a central role.This is also a rich subject with far-reaching implications and applications We shallnot go into any details of these two areas of mathematics Although some algebraicsystems will be discussed and the ideas of limit and continuity will be used in thesequel, this will be done in an intuitive fashion, by introducing and employing theconcepts when they are needed On the other hand, some general concepts will

be introduced when they require minimum prerequisites One of these is a metricspace:

0.3.1 Definition A metric space is a set X together with a real-valued function

metric space defined d: X x X ~ lRsuch that

(b) d(x, y)= d(y, x).

(c) d(x, y) ::: d(x, z)+d(z, y).

(symmetry)

(the triangle inequality)

It is worthwhile to point out that X is a completely arbitrary set and needs

no other structure.Inthis respect Definition 0.3.1 is very broad and encompassesmany different situations, as the following examples will show Before exantiningthe examples, note that the functionddefined above is the abstraction of the notion

of distance: (a) says that the distance between any two points is always nonnegativeand is zero ouly if the two points coincide;(b) says that the distance between twopoints does not change if the two points are interchanged; (c) states the known fact

that the sum of the lengths of two sides of a triangle is always greater than or equal

to the length of the third side Now consider these examples:

1 Let X= iQI,the set of rational numbers, and defined(x, y) = Ix - yl.

2 Let X= R, and again defined(x, y) = Ix - yl.

3 Let X consist of the points on the surface of a sphere We can define twodistance functions on X Letdt (P, Q) be the length of the chord joiningP and Q on the sphere We can also define another metric, dz(P, Q), as thelength of the arc of the great circle passing through points P and Qon thesurface of the sphere.Itis not hard to convince oneselfthatdl anddasatisfyall the properties of a metric function

4 LeteO[a, b] denote the set of continuous real-valued functions on the closed

interval [a, b] We can define d(f, g) = J:If(x) - g(x)1 dx for f, g E

eO(a, b).

5 Leten(a, b) denote the set of bounded continuons real-valned fnnctions on

the closed interval[a, b].We then define

d(f, g) = max lIf(x) - g(x)IJ

xe[a,b]

for f,g E eB(a, b) This notation says: Take the absolute valne of the

difference inf andg at allxin the interval[a, b]and then pick the maximum

ofall these values

The metric function creates a natural setting in which to test the "closeness"

of points in a metric space One occasionon whichthe ideaof closeness becomes

sequence defined essential is in the study of a seqnence A sequence is a mapping s : N * X from

the set of natural numbers N into the metric space X Such a mapping associateswith a positive integern apoints(n) of the metric space X.Itis customary to write

Sn(orXnto match the symbol X) instead ofs(n) and to enumerate the values of

the function by writing{xnJ~I'

Knowledge of the behavior of a sequence for large values ofn is of fundamental

importance.Inparticular, it is important to know whether a sequence approaches

convergence defined a finite value asn increases.

0.3.2 Box. Suppose thatfor some x andfor any positive real numbere,there exists a natural number N such that dtx-; x) < e whenevern > N Then we

say that the sequence {xn}~l converges to x and writelimn~ood(x n l x) =

Oar d(xno x) *0or simply X n *X.

Itmay not be possible to test directly for the convergence of a given sequencebecause this requires a knowledge of the limit pointx. However, it is possible to

Figure 4 The distance between the elements of a Cauchy sequence gets smaller and smaller.

limm.n >ood(xm, xn) = 0, as shown in Figure 4 We can test directly whether

or not a sequence is Cauchy However, the fact that a sequence is Cauchy doesnot guarantee that it converges For example, let the metric space be the set ofrational numbersIQIwith the metric functiond(x, y) = Ix - YI, and consider thesequence {xn}~l where X n = Lk~l (- li+1

/k Itis clear thatX n is a rational

number for anyn. Also, to show that IX m - X nI4 0 is an exercise in calculus.Thus, the sequence is Cauchy However, it is probably known to the reader that

lim n + ooXn =In2, whichis nota rational number.

A metric space io which every Cauchy sequence converges is called a completemetric space Complete metric spaces playa crucial role in modern analysis Theprecediog example shows thatIQIis not a complete metric space However, ifthelimit poiots ofall Cauchy sequences are added toIQI,the resulting space becomescomplete This complete space is, of course, the real number system R Ittums outthat any iocomplete metric space can be "enlarged" to a complete metric space

is a bijection ontoF nis said to be finite withnelements

Although some steps had been taken before him in the direction of a definitive theory of

sets, the creator of the theoryof sets is considered tobeGeorg Cantor(1845-1918),who

was born in Russia of Danish-Jewish parentage but moved toGermanywithhis parents.

His father urged him to study engineering, and Cantor entered the University of Berlinin

1863 with thatintention.There he came under the influence ofWeierstrassand turned to pure mathematics He became Privatdozent at Halle in 1869 and professorin1879 When

he was twenty-nine he published his first revolutionary paper on the theory of infinite sets intheJournal fUrMathematik.Although some of its propositions were deemed faulty by the

older mathematicians, its overall originality and brilliance attracted attention He continued

to publish papers on the theory of sets and on transfinitenumbers until 1897

One of Cantor's main concerns was to differentiate among

infinitesets by "size" and, like Balzano before him, he decided

that one-to-one correspondence should be the basic principle.

Inhis correspondencewithDedekindin1873,Cantor posed the

question of whether the set of real numbers can be put into

one-to-one correspondence with the integers, and some weeks later

he answered in the negative He gave two proofs The first is more

complicated than the second, which is the one most often used

today In 1874 Cantor occupied himself with the equivalence of

the points of a line and the points ofR"and sought to prove

that a one-to-one correspondence between these two sets was

impossible Three years later he proved that there is such a correspondence He wrote to Dedekind, "I seeitbut I do not believe it." He later showed that given any set, it is always possible to create a new set, the set of subsets of the given set, whose cardinal number is larger than that of the given set.If 1'::0is the given set, then the cardinal number of the set

of subsets is denoted by 2~o Cantor proved that 2~O= C, where c is the cardinal number

of the continuum; i.e., the set of real numbers.

Cantor's work, which resolved age-old problems and reversed much previous thought, could hardly be expected to receive immediate acceptance His ideas on transfinite ordi- nal and cardinal numbers aroused the hostility of the powerful Leopold Kronecker, who attacked Cantor's theory savagely over morethana decade, repeatedly preventing Cantor from obtaining a more prominent appointment in Berlin Though Kronecker died in 1891, his attacks left mathematicians suspicious of Cantor's work Poincare referred to set theory

as an interesting "pathological case." He also predicted that "Later generationswillregard [Cantor's]Mengenlehreas 'a disease from which one has recovered." At one time Cantor suffered a nervous breakdown, but resumed work in 1887.

Many prominent mathematicians, however, were impressed by the uses to which the newtheoryhad alreadybeenpatin analysis,measuretheory,andtopology Hilbert spreadCantor'sideas in Germany, and in 1926 said, "No one shall expel us from the paradise which Cantor created for us." He praised Cantor's transfinite arithmetic as "the most astonishing product

of mathematical thought, one of the most beautiful realizations of human activity in the domain of the purely intelligible." Bertrand Russell described Cantor's work as "probably the greatest of which the age can boast." The subsequent utility of Cantor's work in formalizing mathematics-a movement largely led by Hilbert-seems at odds with Cantor's Platonic view that the greater importance of his work was in its implications for metaphysics and theology That his work could be so seainlessly diverted from the goals intended by its creator is strong testimony to its objectivity and craftsmanship.

of energy levels of the bound states of a hydrogen atom.

Itmay seem surprising that a subset (such as the set of all even numbers)can be put into one-to-one correspondence with the full set (the set of all natural

numbers); however, this is a property shared by all infinite sets In fact, sometimes infinite sets are defined as those sets that are in one-to-one correspondence with at

least one of their proper subsets.Itis also astonishing to discover that there are asmany rational numbers as there are natnral numbers After all, there are infinitelymany rational numbers just in the interval (0, I)-or between any two distinct real

numbers.

Sets that are neither finite nor countably infinite are said to be uncountable Insome sense they are "more infinite" than any countable set Examples of uncount-able sets are the points in the interval(-1,+1),the real numbers, the points in aplane, and the points in space.Itcan be shown that these sets have the same cardinal-ity: There are as many points in three-dimensional space-the whole universe-asthere are in the interval(-I,+1) or in any other finite interval

Cardinality is a very intricate mathematical notion with many surprising results.Consider the interval [0, 1] Remove the open interval(~, ~) from its middle This

means that the points ~ and ~ will not be removed From the remaining portion,

[0, ~] U [~, 1], remove the two middle thirds; the remaining portion will then be

a pattern and discovers a formula, and wants to make sure that the formula holdsfor all natural numbers For this purpose, one uses mathematical induction Theinduction principle essence of mathematical induction is stated as follows:

4 Figure 5 TheCantor set after one, two, three, and four "dissections."

0.5.1 Box Suppose that there is associated with each natural number

(pos-itive integer) n a statement Sn Then Sn is true for every pos(pos-itive integer

provided the following two conditions hold:

I. S, is true.

2 If Sm is true for some given positive integer m,

then Sm+l is also true.

We illustrate the use of mathematical induction by proving the binomial binomial theorem orem:

the-where we have used the notation

( ; ) '" k!(mm~k)!' The mathematical statement Sm is Equation (1) We note that SI is trivially true Now we assume that Sm is true and show that Sm+1 is also true This means starting

with Equation (1) and showing that

Then the induction principle ensures that the statement (equation) holds for all

which the reader can easily verify, we finally obtain

Mathematical induction is also used indefiningquantities involving integers.inductive definitions Such definitions are called inductive definitions For example, inductive definition

is used in defining powers: a l =a and am=am-lao

0.6 Problems

0.1.Show that the number of subsets of a set containingnelements is 2"

tk= n(n+ I).

0.2. Let A, B,and C be sets in a universal setU.Show that

(a) A C BandB C CimpliesA C C

(b) AcBiffAnB=AiffAUB=B.

(c) A c Band B C C implies (A UB) c C

(d) AUB = (A ~ B) U(AnB)U(B ~A).

Hint: To show the equality of two sets, show that each set is a subset of the other

0.3. For eachnE N, let

In= Ixlix - II< n and Ix+II > ~}

FindUnInandnnIn.

0.4.Show thata' E [a] implies that [a']= [a].

0.5 Can you define a binary operatiou of "multiplication" on the set of vectors in

space? What about vectors in the plane?

0.6. Show that (f ag)-1 = g-1 a1-1 whenI and g are both bijections

0.7. Take any two open intervals(a, b)and(c, d),and show that there are as manypoints in the first as there are in the second, regardless of the size of the intervals.Hint: Find a (linear) algebraic relation between points of the two intervals

0.8. Use mathematical induction to derive the Leibnizrulefor differentiating aLeibniz rule product:

intu-in this chapter and much more

2 Kelley,J General Topology,Springer-Verlag, 1985 The introductory ter of this classic reference is a detailed introduction to set theory and map-pings

chap-3 Simmons,G.lntroduction to Topologyand Modern Analysis,Krieger, 1983.The first chapter of this book covers not only set theory and mappings, butalso the Cantor set and the fact that integers are as abundant as rational

numbers.

Finite-Dimensional Vector Spaces

j j j j j j j j j j j j j j

Vectors and Transformations

Two- and three-dimensional vectors-undoubtedly familiar objects to the can easily be generalized to higher dimensions Representing vectors by their

reader-components, one can conceiveof vectors havingN components This is themost

immediate generalization of vectors inthe plane and in space, and such vectorsare called N-dimensionalCartesian vectors Cartesian vectors are limited in two

respects: Their components are real, and their dimensionality is finite Some plicationsinphysics require the removal of one or both of these limitations.Itistherefore convenient to study vectors stripped of any dimensionality or reality ofcomponents Such properties become consequences of more fundamental defini-tions Although we will be concentrating on finite-dimensional vector spaces in

ap-this part of the book, many of the concepts and examples introduced here apply toinfinite-dimensional spaces as well

1.1 Vector Spaces

Let us begin with the definition of an abstract (complex) vector space.'

1.1.1 Definition.A vector spaceVover <Cis a set oj objects denoted by 10), Ib),

vector space defined [z),and so on, called vectors, with the following propertiesr:

1 To every pair ofvectors la) and Ib) inVthere corresponds a vector la)+Ib), also inV,called the sum oj la) and Ib), such that

(a) la)+Ib) = Ib)+la),

1Keepin mindthat C is the set of complexnumbers andRthe set ofreels

2Thebra, ( I, andket, I), notation for vectors, invented by Dirac, is veryuseful whendealingwith complexvectorspaces However, it is somewhat clumsyfor certain topicssuchas nann andmetrics and therefore be abandoned in thosediscussions.

(b) la}+(Ib)+Ie))= (Ia)+Ib))+[c),

(c) There exists a unique vector 10} E V,called the zero vector, such that la}+10) = Ja} for every vector la),

(d) To every vector la}E Vthere corresponds a unique vector - Ja} (also

inV)such thatla)+(-Ia}) = 10)

scalars are numbers

complexVS.real

vector space

concept offield

summarized

2 To every complex number: a-{llso called a scalar-s-and every vector la)

there corresponds a vector a la)inVsuch that

(a) a(f3la})= (af3)la),

(b) J]a) = la)

3 Multiplication involving vectors and scalars is distributive:

(a) a(la)+Ib})= ala)+alb).

(b) (a+13)la} = ala)+f3la).

The vector space defined above is also called a complex vector space It ispossible to replaceiCwith IR-the set of real numbers-s-in which case the resultingspace will be called a real vector space Real and complex numbers are prototypes

of a mathematical structure called field A field is a set of objects with twobinaryoperations called addition and multiplication Each operation distributes with re-spect to the other, and each operation has an identity The identity for addition isdenoted by°and is calledadditive identity The identity for multiplication is de-

noted by I and is calledmultiplicative identity Furthermore, every element has an

additive inverse, and every element except the additive identity has a multiplicative

inverse.

1.1.2.Example SOME VECTOR SPACES

1 1R is a vector space overthefieldof real numbers.

2 C is a vector space overthe fieldof realnumbers.

3 C is a vector space over the complex numbers.

4 LetV= Rand letthefieldof scalars heC Thisisnota vector space, because property

2of Definition1.1.1 is not satisfied: A complexnumber timesa realnumber is not

a realnumber and therefore doesnotbelongto V.

5 The setof "arrows"in theplane(or in space) forms a vector space over1R under the

parallelogram law of additioo of planar (or spatial) vectors

3Complex numbers, particularly when they aretreated asvariables,areusually denoted byz,and we shall adhere to this convention in Part ITI However, in the discussionof vectorspaces, we havefoundit more convenient to use lower case Greek letters to denotecomplexnumbers as scalars.