Mathematical Methods for Students of Physics and Related Fields, 2nd ed - Sadri Hassani (Springer, 2009).pdf
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Mathematical Methods for Students of Physics and Related Fields, 2nd ed - Sadri Hassani (Springer, 2009).pdf
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MathematicalMethodsSadriHassaniMathematicalMethodsForStudentsofPhysicsandRelatedFields123SadriHassaniIIlinoisStateUniversityNormal,ILUSAhassanientropy.phy.ilstu.eduISBN:
978-0-387-09503-5e-ISBN:
978-0-387-09504-2LibraryofCongressControlNumber:
2008935523c?
SpringerScience+BusinessMedia,LLC2009Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrights.Printedonacid-freeTomywife,Sarah,andtomychildren,DaneArashandDaisyBitaPrefacetotheSecondEditionInthisnewedition,whichisasubstantiallyrevisedversionoftheoldone,Ihaveaddedfivenewchapters:
VectorsinRelativity(Chapter8),TensorAnalysis(Chapter17),IntegralTransforms(Chapter29),CalculusofVaria-tions(Chapter30),andProbabilityTheory(Chapter32).ThediscussionofvectorsinPartII,especiallytheintroductionoftheinnerproduct,offeredtheopportunitytopresentthespecialtheoryofrelativity,whichunfortunately,inmostundergraduatephysicscurriculareceiveslittleattention.Whilethemainmotivationforthischapterwasvectors,IgrabbedtheopportunitytodeveloptheLorentztransformationandMinkowskidistance,thebedrocksofthespecialtheoryofrelativity,fromfirstprinciples.Theshortsection,VectorsandIndices,attheendofChapter8ofthefirstedition,wastooshorttodemonstratetheimportanceofwhattheindicesarereallyusedfor,tensors.So,Iexpandedthatshortsectionintoasomewhatcomprehensivediscussionoftensors.Chapter17,TensorAnalysis,takesafreshlookatvectortransformationsintroducedintheearlierdiscussionofvectors,andshowsthenecessityofclassifyingthemintothecovariantandcontravariantcategories.Itthenintroducestensorsbasedonandasagen-eralizationofthetransformationpropertiesofcovariantandcontravariantvectors.Inlightofthesetransformationproperties,theKroneckerdelta,in-troducedearlierinthebook,takesonanewlook,andanaturalandextremelyusefulgeneralizationofitisintroducedleadingtotheLevi-Civitasymbol.Adiscussionofconnectionsandmetricsmotivatesafour-dimensionaltreatmentofMaxwellsequationsandamanifestunificationofelectricandmagneticfields.ThechapterendswithRiemanncurvaturetensoranditsplaceinEin-steinsgeneralrelativity.TheFourierseriestreatmentalonedoesnotdojusticetothemanyappli-cationsinwhichaperiodicfunctionsaretoberepresented.Fouriertransformisapowerfultooltorepresentfunctionsinsuchawaythatthesolutiontomany(partial)differentialequationscanbeobtainedelegantlyandsuccinctly.Chapter29,IntegralTransforms,showsthepowerofFouriertransforminmanyillustrationsincludingthecalculationofGreensfunctionsforLaplace,heat,andwavedifferentialoperators.Laplacetransforms,whichareusefulinsolvinginitial-valueproblems,arealsoincluded.viiiPrefacetoSecondEditionTheDiracdeltafunction,aboutwhichthereisacomprehensivediscussioninthebook,allowsaverysmoothtransitionfrommultivariablecalculustotheCalculusofVariations,thesubjectofChapter30.Thischaptertakesanintuitiveapproachtothesubject:
replacethesumbyanintegralandtheKroneckerdeltabytheDiracdeltafunction,andyougetfrommultivariablecalculustothecalculusofvariations!
Well,thetransitionmaynotbeassimpleasthis,buttheheartoftheintuitiveapproachis.OncethetransitionismadeandthemasterEuler-Lagrangeequationisderived,manyexamples,includingsomewithconstraint(whichusetheLagrangemultipliertechnique),andsomefromelectromagnetismandmechanicsarepresented.ProbabilityTheoryisessentialforquantummechanicsandthermody-namics.ThisisthesubjectofChapter32.Startingwiththebasicnotionoftheprobabilityspace,whoseprerequisiteisanunderstandingofelementarysettheory,whichisalsoincluded,thenotionofrandomvariablesanditscon-nectiontoprobabilityisintroduced,averageandvariancearedefined,andbinomial,Poisson,andnormaldistributionsarediscussedinsomedetail.Asidefromtheabovemajorchanges,Ihavealsoincorporatedsomeotherimportantchangesincludingtherearrangementofsomechapters,addingnewsectionsandsubsectionstosomeexistingchapters(forinstance,thedynamicsoffluidsinChapter15),correctingallthemistakes,bothtypographicandconceptual,towhichIhavebeendirectedbymanyreadersofthefirstedition,andaddingmoreproblemsattheendofeachchapter.Stylistically,IthoughtsplittingthesometimesverylongchaptersintosmalleronesandcollectingtherelatedchaptersintoPartsmakethereadingofthetextsmoother.IhopeIwasnotwrong!
Iwouldliketothankthemanyinstructors,students,andgeneralreaderswhocommunicatedtomecomments,suggestions,anderrorstheyfoundinthebook.Amongthose,IespeciallythankDanHollandforthemanydiscussionswehavehadaboutthebook,RafaelBenguriaandGebhardGrublforpointingoutsomeimportanthistoricalandconceptualmistakes,andAliErdemandThomasFergusonforreadingmultiplechaptersofthebook,catchingmanymistakes,andsuggestingwaystoimprovethepresentationofthematerial.JeromeBrozekmeticulouslyanddiligentlyreadmostofthebookandfoundnumerouserrors.Althoughalawyerbyprofession,Mr.Brozek,asahobby,hasakeeninterestinmathematicalphysics.Ithankhimforthisinterestandforputtingittouseonmybook.Lastbutnotleast,Iwanttothankmyfamily,especiallymywifeSarahforherunwaveringsupport.S.H.Normal,ILJanuary,2008PrefaceInnocentlight-mindedmen,whothinkthatastronomycanbelearntbylookingatthestarswithoutknowledgeofmath-ematicswill,inthenextlife,bebirds.Plato,TimaeosThisbookisintendedtohelpbridgethewidegapseparatingthelevelofmath-ematicalsophisticationexpectedofstudentsofintroductoryphysicsfromthatexpectedofstudentsofadvancedcoursesofundergraduatephysicsandengi-neering.Whilenothingbeyondsimplecalculusisrequiredforintroductoryphysicscoursestakenbyphysics,engineering,andchemistrymajors,thenextlevelofcoursesbothinphysicsandengineeringalreadydemandsareadi-nessforsuchintricateandsophisticatedconceptsasdivergence,curl,andStokestheorem.Itistheaimofthisbooktomakethetransitionbetweenthesetwolevelsofexposureassmoothaspossible.LevelandPedagogyIbelievethatthebestpedagogytoteachmathematicstobeginningstudentsofphysicsandengineering(evenmathematics,althoughsomeofmymathe-maticalcolleaguesmaydisagreewithme)istointroduceandusetheconceptsinamultitudeofappliedsettings.Thismethodisnotunliketeachingalan-guagetoachild:
itisbyrepeatedusagebytheparentsortheteacherofthesamewordindifferentcircumstancesthatachildlearnsthemeaningoftheword,andbyrepeatedactive(andsometimeswrong)usageofwordsthatthechildlearnstousetheminasentence.AndwhatbetterplacetousethelanguageofmathematicsthaninNatureitselfinthecontextofphysics.Istartwiththefamiliarnotionof,say,aderivativeoranintegral,butinterpretitentirelyintermsofphysicalideas.Thus,aderivativeisameansbywhichoneobtainsvelocityfrompositionvectorsoraccelerationfromvelocityvectors,andintegralisameansbywhichoneobtainsthegravitationalorelectricfieldofalargenumberofchargedormassiveparticles.Ifconcepts(e.g.,infiniteseries)donotsuccumbeasilytophysicalinterpretation,thenIimmediatelysubjugatethephysicalxPrefacesituationtothemathematicalconcepts(e.g.,multipoleexpansionofelectricpotential).Becauseofmybeliefinthispedagogy,Ihavekeptformalismtoabareminimum.Afterall,achildneedsnoknowledgeoftheformalismofhisorherlanguage(i.e.,grammar)tobeabletoreadandwrite.Similarly,anoviceinphysicsorengineeringneedstoseealotofexamplesinwhichmathematicsisusedtobeableto“speakthelanguage.”AndIhavesparednoefforttoprovidetheseexamplesthroughoutthebook.Ofcourse,formalism,atsomestage,becomesimportant.Justasgrammaristaughtatahigherstageofachildseducation(say,inhighschool),mathematicalformalismistobetaughtatahigherstageofeducationofphysicsandengineeringstudents(possiblyinadvancedundergraduateorgraduateclasses).FeaturesTheuniquefeaturesofthisbook,whichsetitapartfromtheexistingtext-books,aretheinseparabletreatmentsofphysicalandmathematicalconcepts,thelargenumberoforiginalillustrativeexamples,theaccessibilityofthebooktosophomoresandjuniorsinphysicsandengineeringprograms,andthelargenumberofhistoricalnotesonpeopleandideas.Allmathematicalconceptsinthebookareeitherintroducedasanaturaltoolforexpressingsomephysicalconceptor,upontheirintroduction,immediatelyusedinaphysicalsetting.Thus,forexample,differentialequationsarenottreatedassomemathematicalequalitiesseekingsolutions,butratherasastatementaboutthelawsofNature(e.g.,thesecondlawofmotion)whosesolutionsdescribethebehaviorofaphysicalsystem.Almostallexamplesandproblemsinthisbookcomedirectlyfromphysi-calsituationsinmechanics,electromagnetism,and,toalesserextent,quan-tummechanicsandthermodynamics.Althoughtheexamplesaredrawnfromphysics,theyareconceptuallyatsuchanintroductorylevelth
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- Mathematical Methods for Students of Physics and Related Fields 2nd ed Sadri Hassani Springer 2009
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