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Mathematical methods for physics and engineering PDF

1363 Pages·2006·7.251 MB·English
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This page intentionally left blank Mathematical Methods for Physics and Engineering The third edition of this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics ever likely to be needed for an undergraduate course in any of the physical sciences. As well as lucid descriptions of all the topics covered and many worked examples, it contains more than 800 exercises. A number of additional topics have been included and the text has undergone significantreorganisationinsomeareas.Newstand-alonechapters: • giveasystematicaccountofthe‘specialfunctions’ofphysicalscience • coveranextendedrangeofpracticalapplicationsofcomplexvariablesincluding WKBmethodsandsaddle-pointintegrationtechniques • provideanintroductiontoquantumoperators. Further tabulations, of relevance in statistics and numerical integration, have been added. In this edition, all 400 odd-numbered exercises are provided with complete worked solutions in a separate manual, available to both students and their teachers; these are in addition to the hints and outline answers given in the main text. The even-numbered exercises have no hints, answers or worked solutions and can be used for unaided homework; full solutions to them are availabletoinstructorsonapassword-protectedwebsite. Ken Riley read mathematics at the University of Cambridge and proceeded to a Ph.D. there in theoretical and experimental nuclear physics. He became a researchassociateinelementaryparticlephysicsatBrookhaven,andthen,having taken up a lectureship at the Cavendish Laboratory, Cambridge, continued this researchattheRutherfordLaboratoryandStanford;inparticularhewasinvolved in the experimental discovery of a number of the early baryonic resonances. As well as having been Senior Tutor at Clare College, where he has taught physics andmathematicsforover40years,hehasservedonmanycommitteesconcerned with the teaching and examining of these subjects at all levels of tertiary and undergraduate education. He is also one of the authors of 200 Puzzling Physics Problems. Michael Hobson read natural sciences at the University of Cambridge, spe- cialising in theoretical physics, and remained at the Cavendish Laboratory to completeaPh.D.inthephysicsofstar-formation.AsaresearchfellowatTrinity Hall, Cambridge and subsequently an advanced fellow of the Particle Physics and Astronomy Research Council, he developed an interest in cosmology, and in particular in the study of fluctuations in the cosmic microwave background. Hewasinvolvedinthefirstdetectionofthesefluctuationsusingaground-based interferometer.HeiscurrentlyaUniversityReaderattheCavendishLaboratory, his research interests include both theoretical and observational aspects of cos- mology, and heis the principalauthor ofGeneral Relativity: An Introduction for Physicists.HeisalsoaDirectorofStudiesinNaturalSciencesatTrinityHalland enjoysanactiveroleintheteachingofundergraduatephysicsandmathematics. Stephen Bence obtained both his undergraduate degree in Natural Sciences andhisPh.D.inAstrophysicsfromtheUniversityofCambridge.Hethenbecame a Research Associate with a special interest in star-formation processes and the structure of star-forming regions. In particular, his research concentratedon the physics of jets and outflows from young stars. He has had considerable experi- ence of teaching mathematics and physics to undergraduate and pre-universtiy students. ii Mathematical Methods for Physics and Engineering Third Edition K.F. RILEY, M.P. HOBSON and S.J. BENCE cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridgecb22ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Informationo nthi stitle :www.cambri dge.org/9780521861533 ©K.F.Riley,M.P.HobsonandS.J.Bence2006 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace without the written permission of Cambridge University Press. Firstpublishedinprintformat 2006 isbn-13 978-0-511-16842-0 eBook(EBL) isbn-10 0-511-16842-x eBook(EBL) isbn-13 978-0-521-86153-3 hardback isbn-10 0-521-86153-5 hardback isbn-13 978-0-521-67971-8 paperback isbn-10 0-521-67971-0 paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurls forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Contents Preface to the third edition pagexx Preface to the second edition xxiii Preface to the first edition xxv 1 Preliminary algebra 1 1.1 Simplefunctionsandequations 1 Polynomialequations;factorisation;propertiesofroots 1.2 Trigonometricidentities 10 Singleangle;compoundangles;double-andhalf-angleidentities 1.3 Coordinategeometry 15 1.4 Partialfractions 18 Complicationsandspecialcases 1.5 Binomialexpansion 25 1.6 Propertiesofbinomialcoefficients 27 1.7 Someparticularmethodsofproof 30 Proofbyinduction;proofbycontradiction;necessaryandsufficientconditions 1.8 Exercises 36 1.9 Hintsandanswers 39 2 Preliminary calculus 41 2.1 Differentiation 41 Differentiation from first principles; products; the chain rule; quotients; implicit differentiation;logarithmicdifferentiation; Leibnitz’theorem;special pointsofafunction;curvature;theoremsofdifferentiation v CONTENTS 2.2 Integration 59 Integration from first principles; the inverse of differentiation; by inspec- tion; sinusoidal functions; logarithmic integration; using partial fractions; substitution method; integration by parts; reduction formulae; infinite and improperintegrals;planepolarcoordinates;integralinequalities;applications ofintegration 2.3 Exercises 76 2.4 Hintsandanswers 81 3 Complex numbers and hyperbolic functions 83 3.1 Theneedforcomplexnumbers 83 3.2 Manipulationofcomplexnumbers 85 Addition and subtraction; modulus and argument; multiplication; complex conjugate;division 3.3 Polarrepresentationofcomplexnumbers 92 Multiplicationanddivisioninpolarform 3.4 deMoivre’stheorem 95 trigonometric identities; finding the nth roots of unity; solving polynomial equations 3.5 Complexlogarithmsandcomplexpowers 99 3.6 Applicationstodifferentiationandintegration 101 3.7 Hyperbolicfunctions 102 Definitions; hyperbolic–trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculusofhyperbolicfunctions 3.8 Exercises 109 3.9 Hintsandanswers 113 4 Series and limits 115 4.1 Series 115 4.2 Summationofseries 116 Arithmeticseries;geometricseries;arithmetico-geometricseries;thedifference method;seriesinvolvingnaturalnumbers;transformationofseries 4.3 Convergenceofinfiniteseries 124 Absolute and conditional convergence; series containing only real positive terms;alternatingseriestest 4.4 Operationswithseries 131 4.5 Powerseries 131 Convergenceofpowerseries;operationswithpowerseries 4.6 Taylorseries 136 Taylor’stheorem;approximationerrors;standardMaclaurinseries 4.7 Evaluationoflimits 141 4.8 Exercises 144 4.9 Hintsandanswers 149 vi CONTENTS 5 Partial differentiation 151 5.1 Definitionofthepartialderivative 151 5.2 Thetotaldifferentialandtotalderivative 153 5.3 Exactandinexactdifferentials 155 5.4 Usefultheoremsofpartialdifferentiation 157 5.5 Thechainrule 157 5.6 Changeofvariables 158 5.7 Taylor’stheoremformany-variablefunctions 160 5.8 Stationaryvaluesofmany-variablefunctions 162 5.9 Stationaryvaluesunderconstraints 167 5.10 Envelopes 173 5.11 Thermodynamicrelations 176 5.12 Differentiationofintegrals 178 5.13 Exercises 179 5.14 Hintsandanswers 185 6 Multiple integrals 187 6.1 Doubleintegrals 187 6.2 Tripleintegrals 190 6.3 Applicationsofmultipleintegrals 191 Areasandvolumes;masses,centresofmassandcentroids;Pappus’theorems; momentsofinertia;meanvaluesoffunctions 6.4 Changeofvariablesinmultipleintegrals 199 (cid:1)Change of variables in double integrals; evaluation of the integral I = ∞ e−x2dx; change of variables in triple integrals; general properties of −∞ Jacobians 6.5 Exercises 207 6.6 Hintsandanswers 211 7 Vector algebra 212 7.1 Scalarsandvectors 212 7.2 Additionandsubtractionofvectors 213 7.3 Multiplicationbyascalar 214 7.4 Basisvectorsandcomponents 217 7.5 Magnitudeofavector 218 7.6 Multiplicationofvectors 219 Scalarproduct;vectorproduct;scalartripleproduct;vectortripleproduct vii CONTENTS 7.7 Equationsoflines,planesandspheres 226 7.8 Usingvectorstofinddistances 229 Pointtoline;pointtoplane;linetoline;linetoplane 7.9 Reciprocalvectors 233 7.10 Exercises 234 7.11 Hintsandanswers 240 8 Matrices and vector spaces 241 8.1 Vectorspaces 242 Basisvectors;innerproduct;someusefulinequalities 8.2 Linearoperators 247 8.3 Matrices 249 8.4 Basicmatrixalgebra 250 Matrixaddition;multiplicationbyascalar;matrixmultiplication 8.5 Functionsofmatrices 255 8.6 Thetransposeofamatrix 255 8.7 ThecomplexandHermitianconjugatesofamatrix 256 8.8 Thetraceofamatrix 258 8.9 Thedeterminantofamatrix 259 Propertiesofdeterminants 8.10 Theinverseofamatrix 263 8.11 Therankofamatrix 267 8.12 Specialtypesofsquarematrix 268 Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian andanti-Hermitian;unitary;normal 8.13 Eigenvectorsandeigenvalues 272 Ofanormalmatrix;ofHermitianandanti-Hermitianmatrices;ofaunitary matrix;ofageneralsquarematrix 8.14 Determinationofeigenvaluesandeigenvectors 280 Degenerateeigenvalues 8.15 Changeofbasisandsimilaritytransformations 282 8.16 Diagonalisationofmatrices 285 8.17 QuadraticandHermitianforms 288 Stationarypropertiesoftheeigenvectors;quadraticsurfaces 8.18 Simultaneouslinearequations 292 Range;nullspace;N simultaneouslinearequationsinN unknowns;singular valuedecomposition 8.19 Exercises 307 8.20 Hintsandanswers 314 9 Normal modes 316 9.1 Typicaloscillatorysystems 317 9.2 Symmetryandnormalmodes 322 viii

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