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1、PartialDifferentialEquationsinPhysicsBYARNOLDSOMMEKFELDUnioersityofMunich,GemzanyTranslatedbyERNSTG.STRAUSTheInstifutrforAdoancedStudy,Piinreton,N.J.1949ACAIIEMICPRESSINC.,PUBLISHERSNEWYOKK,N.Y.Copyright1949BYACADEMICPRESSINC.126EAST23~~STREET,NEWYOR
2、K10,N.Y.AllRightsReservedNOPARTOFTHISBOOKMAYBEREPRODUCEDINANYFORM,BYPHOTOSTAT,MICROFILM,ORANYOTHERMEANS,WITHOUTWRITTENPERMISSIONFROMTHEPUBLISHERFirstprinting,1949Secondprintin^,1963PRINTEDINTHEUNITEDSTATESOFAMERICA-THEMURRAYPRINTINGCOMPANYWAKEFIELD,M
3、ABBACHUSETT8FOREWORDThetopicwithwhichIregularlyconcludemysix-termseriesoflecturesinMunichisthepartialdifferentialequationsofphysics.Wedonotreallydealwithmathematicalphysics,butwithphysicalmathe-matics;notwiththemathematicalformulationofphysicalfacts,
4、butwiththephysicalmotivationofmathematicalmethods.Theoft-mentioned“prestabilizedharmony”betweenwhatismathematicallyinterestingandwhatisphysicallyimportantismetateachstepandlendsanesthetic-Ishouldliketosaymetaphysical--attractiontooursubject.Theproble
5、mstobetreatedbelongmainlytotheclassicalmathe-rhaticalliterature,asshownbytheirconnectionwiththenamesofLaplace,Fourier,Green,Gauss,Riemann,andWilliamThomson.Inordertoshowthatthesemethodsareadequatetodealwithactualproblems,wetreatthepropagationofradiow
6、avesinsomedetailinChapterVI.ChapterVdealswiththegeneralmethodofeigenfunctions.Themostspectaculardomainofapplicationofthatmethodiswavemechanics,asweshowherewiththehelpofsomeselected,particularlysimpleexamples.Themathematicallyrigorousfoundationoftheex
7、istenceandthepropertiesofeigenfunctionswiththehelpoftheoremsaboutinte-gralequationscamatbegivenhere;thelatterarementionedonlyocca-sionallyasthecounterpartofthecorrespondingtheoremsondifferentialequations.ChapterIVonBesselfunctionsandsphericalharmonic
8、siscompara-tivelylengthydespiteadevelopmentthatisasconciseaspossible.Forthesakeofbrevitywehaverelegatedsomeproofstotheexercises,aswehavealsodoneinotherchapters.Aspecialsectionisdedicatedtothebeautifulmethodofreciprocalradiiandtothedemonstrationofthef
