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1、MethodsofMathematicalPhysicsIAsetoflecturenotesbyMichaelStonePIMANDER-CASAUBONAlexandriaFlorenceLondoniiCopyrightc2001,2002M.Stone.Allrightsreserved.Nopartofthismaterialcanbereproduced,storedortransmittedwithoutthewrittenpermissionoftheauthor.Forinformationcon
2、tact:MichaelStone,LoomisLaboratoryofPhysics,UniversityofIllinois,1110WestGreenStreet,Urbana,IL61801,USA.PrefaceThesenoteswerepreparedforPHYCS-498MMA,afairlytraditionalone-semestermathematicalmethodscourseforbegininggraduatestudentsinphysics.Theemphasisisonlinear
3、operatorsandstressestheanalogybetweensuchoperatorsactingonfunctionspacesandmatricesactingon nitedimen-sionalspaces.Theoperatorlanguagethenprovidesauni edframeworkforinvestigatingordinarydi erentialequations,partialdi erentialequations,andintegralequations.Althou
4、ghthismathematicsisapplicabletoawiderangephysicalphenom-ena,theillustrativeexamplesaremostlydrawnfromclassicalandquantummechanics.Classicalmechanicsisasubjectfamiliartoallphysicsstudentsandthepointbeingillustratedisimmediatelyunderstandablewithoutanyfurtherspeci
5、alizedknowledge.Similarlyallphysicsstudentshavestudiedquantummechanics,andherethematrix/di erential-operatoranalogyliesattheheartofthesubject.Themathematicalprerequisitesforthecourseareasoundgraspofun-dergraduatecalculus(includingthevectorcalculusneededforelectr
6、icityandmagnetismcourses),linearalgebra(themorethebetter),andcompetenceatcomplexarithmetic.Fouriersumsandintegrals,aswellasbasicordinarydi erentialequationtheoryreceiveaquickreview,butitwouldhelpifthereaderhadsomepriorexperiencetobuildon.Contourintegrationisnotr
7、equired.iiiivPREFACEContentsPrefaceiii1CalculusofVariations11.1Whatisitgoodfor?.......................11.2Functionals............................21.2.1TheFunctionalDerivative................21.2.2Examples.........................31.2.3FirstIntegral..............
8、.........81.3LagrangianMechanics......................91.3.1OneDegreeofFreedom..................101.3.2Noether'sTheorem....................141.3.3ManyDegreesofFreedom
