剑桥大学――方法

MethodsPaulMetcalfeSummer1998Thesenotesarem[e-mail]:1.8Date:1999/09/1717:44:23Thefollowingpeoplehavemaintainedthesenotes.–datePaulMetcalfeContentsIntroductionv1Fourierseries11.1Propertiesofsineandcosine......................11.2DefinitionofFourierseries.......................11.2.1Themeaningofgoodbehaviour................31.3ComplexFourierseries.........................31.4Sineandcosineseries..........................41.5Parseval’stheorem...........................42TheWaveEquation52.1Wavesonanelasticstring........................52.2Separationofvariables.........................62.3Oscillationenergy............................72.4Solutionincharacteristicco-ordinates.................82.5Wavereflectionandtransmission....................83Green’sFunctions113.1TheDiracdeltafunction........................113.1.1Representations.........................113.2SecondorderlinearODEs.......................123.3DefinitionofGreen’sfunction.....................123.3.1Definingproperties.......................133.4ConstructingGx:boundaryvalueproblems............133.4.1Derivationofjumpconditions.................143.4.2Example............................143.5ConstructingGx:initialvalueproblems..............143.5.1Example............................154Sturm-LiouvilleTheory174.1Self-adjointformandboundaryvalues.................174.2Eigenfunctionexpansions........................184.2.1Realeigenvalues........................184.2.2Orthogonaleigenfunctions...................184.2.3Completeeigenfunctions....................184.3Example:Legendrepolynomials....................194.4Inhomogeneousboundaryvalueproblem................20iiiivCONTENTS5Applications:Laplace’sEquation215.1Cartesians................................215.2Planepolars...............................235.3Sphericalpolars.............................235.3.1Thefullgloryofsphericalpolars................246CalculusofVariations256.1Theproblem...............................256.2Euler-Lagrangeequations........................256.3Examples................................266.4PrincipleofLeastAction........................276.5Generalisations.............................276.6Integralconstraints...........................287CartesianTensorsinR297.1Tensors?.................................297.2Transformationlaws..........................307.3Tensoralgebra..............................307.4QuotientLaws..............................307.5Isotropictensors.............................317.5.1Sphericallysymmetricintegrals................327.6Symmetricandantisymmetrictensors.................327.7PhysicalApplications..........................33IntroductionThesenotesarebasedonthecourse“Methods”givenbyDr.E.P.ShellardinCam-bridgeintheMichælmasTerm1996.ThesetypesetnotesaretotallyunconnectedwithDr.Shellard.Theyaremorevaguelybasedonthecoursethanmynotesusuallyare,andIhavemainlyusedDr.Shellard’snotestogetasenseoforderingandcontent.Othersetsofnotesareavailablefordifferentcourses.Atthetimeoftypingthesecourseswere:ProbabilityDiscreteMathematicsAnalysisFurtherAnalysisMethodsQuantumMechanicsFluidDynamics1QuadraticMathematicsGeometryDynamicsofD.E.’sFoundationsofQMElectrodynamicsMethodsofMath.PhysFluidDynamics2Waves(etc.)StatisticalPhysicsGeneralRelativityDynamicalSystemsPhysiologicalFluidDynamicsBifurcationsinNonlinearConvectionSlowViscousFlowsTurbulenceandSelf-SimilarityAcousticsNon-NewtonianFluidsSeismicWavesTheymaybedownloadedfrom://@lists.cam.ac.uktogetacopyofthesetsyourequire.vCopyright(c)TheArchimedeans,CambridgeUniversity.Allrightsreserved.Redistributionanduseofthesenotesinelectronicorprintedform,withorwithoutmodification,arepermittedprovidedthatthefollowingconditionsaremet:1.Redistributionsoftheelectronicfilesmustretaintheabovecopyrightnotice,thislistofconditionsandthefollowingdisclaimer.2.Redistributionsinprintedformmustreproducetheabovecopyrightnotice,thislistofconditionsandthefollowingdisclaimer.3.Allmaterialsderivedfromthesenotesmustdisplaythefollowingacknowledge-ment:ThisproductincludesnotesdevelopedbyTheArchimedeans,CambridgeUniversityandtheircontributors.4.NeitherthenameofTheArchimedeansnorthenamesoftheircontributorsmaybeusedtoendorseorpromoteproductsderivedfromthesenotes.5.Neitherthesenotesnoranyderivedproductsmaybesoldonafor-profitbasis,althoughafeemayberequiredforthephysicalactofcopying.6.Youmustcauseanyeditedversionstocarryprominentnoticesstatingthatyoueditedthemandthedateofanychange.THESENOTESAREPROVIDEDBYTHEARCHIMEDEANSANDCONTRIB-UTORS“ASIS”ANDANYEXPRESSORIMPLIEDWARRANTIES,INCLUDING,BUTNOTLIMITEDTO,THEIMPLIEDWARRANTIESOFMERCHANTABIL-ITYANDFITNESSFORAPARTICULARPURPOSEAREDISCLAIMED.INNOEVENTSHALLTHEARCHIMEDEANSORCONTRIBUTORSBELIABLEFORANYDIRECT,INDIRECT,INCIDENTAL,SPECIAL,EXEMPLARY,ORCONSE-QUENTIALDAMAGESHOWEVERCAUSEDANDONANYTHEORYOFLI-ABILITY,WHETHERINCONTRACT,STRICTLIABILITY,ORTORT(