topics in mathematical physics - victor palamodov

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1、TopicsinMathematicalPhysicsProf.V.PalamodovSpringsemester2002ContentsChapter1:Di®erentialequationsofMathematicalPhysics1.1Di®erentialequationsofelliptictype1.2Di®usionequations1.3Waveequations1.4Systems1.5Nonlinearequations1.6Hamilton-Jacobitheory1.7Relativistic¯eldtheor

2、y1.8Classi¯cation1.9Initialandboundaryvalueproblems1.10InverseproblemsChapter2:Elementarymethods2.1Changeofvariables2.2Bilinearintegrals2.3Conservationlaws2.4Methodofplanewaves2.5Fouriertransform2.6TheoryofdistributionsChapter3:Fundamentalsolutions3.1Basicde¯nitionandpro

3、perties3.2Fundamentalsolutionsforellipticoperators3.3Moreexamples3.4Hyperbolicpolynomialsandsourcefunctions3.5Wavepropagators3.6Inhomogeneoushyperbolicoperators3.7RieszgroupsChapter4:TheCauchyproblem4.1De¯nitions4.2Cauchyproblemfordistributions4.3HyperbolicCauchyproblem4

4、.4SolutionoftheCauchyproblemforwaveequations4.5Domainofdependence2Chapter5:Helmholtzequationandscattering5.1Time-harmonicwaves5.2Sourcefunctions5.3Radiationconditions5.4Scatteringonobstacle5.5Interferenceanddi®ractionChapter6:Geometryofwaves6.1Wavefronts6.2Hamilton-Jacob

5、itheory6.3Geometryofrays6.4Anintegrablecase6.5Legendretransformationandgeometricduality6.6Ferm¶atprinciple6.7ThemajorHuygensprinciple6.8Geometricaloptics6.9Caustics6.10GeometricalconservationlawChapter7:ThemethodofFourierintegrals7.1Elementsofsymplecticgeometry7.2Generat

6、ingfunctions7.3Fourierintegrals7.4Lagrangedistributions7.5HyperbolicCauchyproblemrevisitedChapter8:Electromagneticwaves8.1Vectoranalysis8.2Maxwellequations8.3Harmonicanalysisofsolutions8.4Cauchyproblem8.5Localconservationlaws3Chapter1Di®erentialequationsofMathematicalPhy

7、sics1.1Di®erentialequationsofelliptictypeLetXbeanEuclideanspaceofdimensionnwithacoordinatesystemx1;:::;xn:²TheLaplaceequationis@2@2:¢u=0;¢=+:::+@x2@x21n¢iscalledtheLaplaceoperator.Asolutioninadomain•½Xiscalledharmonicfunctionin•.Itdescribesastablemembrane,electrostaticor

8、gravity¯eld.²TheHelmholtzequation¡¢2¢+!u=0Forn=1itiscalledtheequationofharmonicoscillator.Asolutionisat