Mathematics for physics: A guided tour for graduate students

822 Pages·2009·4.2 MB·English

by  Michael Stone

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MathematicsforPhysics
AGuidedTourforGraduateStudents
An engagingly written account of mathematical tools and ideas, this book provides a
graduate-levelintroductiontothemathematicsusedinresearchinphysics.
Thefirsthalfofthebookfocusesonthetraditionalmathematicalmethodsofphysics:
differential and integral equations, Fourier series and the calculus of variations. The
secondhalfcontainsanintroductiontomoreadvancedsubjects, includingdifferential
geometry,topologyandcomplexvariables.
The authors’ exposition avoids excess rigour whilst explaining subtle but impor-
tant points often glossed over in more elementary texts. The topics are illustrated at
everystagebycarefullychosenexamples,exercisesandproblemsdrawnfromrealistic
physicssettings. Thesemakeitusefulbothasatextbookinadvancedcoursesandfor
self-study. Password-protectedsolutionstotheexercisesareavailabletoinstructorsat
www.cambridge.org/9780521854030.
michaelstoneisaProfessorintheDepartmentofPhysicsattheUniversityofIllinois
atUrbana-Champaign.Hehasworkedonquantumfieldtheory,superconductivity,the
quantumHalleffectandquantumcomputing.
paulgoldbartisaProfessorintheDepartmentofPhysicsattheUniversityofIllinois
atUrbana-Champaign,wherehedirectstheInstituteforCondensedMatterTheory.His
researchrangeswidelyoverthefieldofcondensedmatterphysics,includingsoftmatter,
disorderedsystems,nanoscienceandsuperconductivity.

MATHEMATICS FOR PHYSICS
AGuided Tour for Graduate Students
MICHAELSTONE
UniversityofIllinoisatUrbana-Champaign
and
PAULGOLDBART
UniversityofIllinoisatUrbana-Champaign
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521854030
© M. Stone and P. Goldbart 2009
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2009
ISBN-13 978-0-511-59516-5 eBook (EBL)
ISBN-13 978-0-521-85403-0 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
TothememoryofMike’smother,AileenStone:9×9=81.
ToPaul’smotherandfather,CaroleandColinGoldbart.

Contents
Preface pagexi
Acknowledgments xiii
1 Calculusofvariations 1
1.1 Whatisitgoodfor? 1
1.2 Functionals 1
1.3 Lagrangianmechanics 10
1.4 Variableendpoints 27
1.5 Lagrangemultipliers 32
1.6 Maximumorminimum? 36
1.7 Furtherexercisesandproblems 38
2 Functionspaces 50
2.1 Motivation 50
2.2 Normsandinnerproducts 51
2.3 Linearoperatorsanddistributions 66
2.4 Furtherexercisesandproblems 76
3 Linearordinarydifferentialequations 86
3.1 Existenceanduniquenessofsolutions 86
3.2 Normalform 93
3.3 Inhomogeneousequations 94
3.4 Singularpoints 97
3.5 Furtherexercisesandproblems 98
4 Lineardifferentialoperators 101
4.1 Formalvs.concreteoperators 101
4.2 Theadjointoperator 104
4.3 Completenessofeigenfunctions 117
4.4 Furtherexercisesandproblems 132
5 Greenfunctions 140
5.1 Inhomogeneouslinearequations 140
5.2 ConstructingGreenfunctions 141
vii
viii Contents
5.3 ApplicationsofLagrange’sidentity 150
5.4 Eigenfunctionexpansions 153
5.5 AnalyticpropertiesofGreenfunctions 155
5.6 LocalityandtheGelfand–Dikiiequation 165
5.7 Furtherexercisesandproblems 167
6 Partialdifferentialequations 174
6.1 ClassificationofPDEs 174
6.2 Cauchydata 176
6.3 Waveequation 181
6.4 Heatequation 196
6.5 Potentialtheory 201
6.6 Furtherexercisesandproblems 224
7 Themathematicsofrealwaves 231
7.1 Dispersivewaves 231
7.2 Makingwaves 242
7.3 Nonlinearwaves 246
7.4 Solitons 255
7.5 Furtherexercisesandproblems 260
8 Specialfunctions 264
8.1 Curvilinearcoordinates 264
8.2 Sphericalharmonics 270
8.3 Besselfunctions 278
8.4 Singularendpoints 298
8.5 Furtherexercisesandproblems 305
9 Integralequations 311
9.1 Illustrations 311
9.2 Classificationofintegralequations 312
9.3 Integraltransforms 313
9.4 Separablekernels 321
9.5 Singularintegralequations 323
9.6 Wiener–HopfequationsI 327
9.7 Somefunctionalanalysis 332
9.8 Seriessolutions 338
9.9 Furtherexercisesandproblems 342
10 Vectorsandtensors 347
10.1 Covariantandcontravariantvectors 347
10.2 Tensors 350
10.3 Cartesiantensors 362
10.4 Furtherexercisesandproblems 372