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数学物理的几何方法(英文版)
Thisbookalmstointroducethebeginningorworkingphysicisttoawiderangeofaualytictoolswhichhavetheiror/ginindifferentialgeometryandwhichhaverecentlyfoundincreasinguseintheoreticalphysics.Itisnotuncom-montodayforaphysicistsmathematicaleducationtoignoreallbutthesim-plestgeometricalideas,despitethefactthatyoungphysicistsareencouragedtodevelopmentalpicturesandintuitionappropriatetophysicalphenomena.Thiscuriousneglectofpicturesofonesmathematicaltoolsmaybeseenastheoutcomeofagradualevolutionovermanycenturies.Geometrywascertainlyextremelyimportanttoancientandmedievalnaturalphilosophers;itwasingeometricaltermsthatPtolemy,Copernicus,Kepler,andGalileoallexpressedtheirthinking.ButwhenDescartesintroducedcoordinatesintoEuclideangeometry,heshowedthatthestudyofgeometrycouldberegardedasanappli.cationofalgrebra.Sincethen,the/mportanceofthestudyofgeometryintheeducationofscientistshassteadily
1Somebasicmathematics1.1ThespaceRnanditstopology1.2Mappings1.3Realanalysis1.4Grouptheory1.5Linearalgebra1.6Thealgebraofsquarematrices1.7Bibliography2Dffferentiablemanifoldsandtensors2.1Defmitionofamanifold2.2Thesphereasamanifold2.3Otherexamplesofmanifolds2.4Globalconsiderations2.5Curves2.6FunctionsonM2.7Vectorsandvectorfields2.8Basisvectorsandbasisvectorfields2.9Fiberbundles2.10Examplesoffiberbundles2.11Adeeperlookatfiberbundles2.12Vectorfieldsandintegralcurves2.13Exponentiationoftheoperatord/dZ2.14Liebracketsandnoncoordinatebases2.15Whenisabasisacoordinatebasis?2.16One-forms2.17Examplesofone-forms2.18TheDiracdeltafunction2.19Thegradientandthepictorialrepresentationofaone-form2.20Basisone-formsandcomponentsofone-forms2.21Indexnotation2.22Tensorsandtensorfields2.23Examplesoftensors2.24Componentsoftensorsandtheouterproduct2.25Contraction2.26Basistransformations2.27Tensoroperationsoncomponents2.28Functionsandscalars2.29Themetrictensoronavectorspace2.30Themetrictensorfieldonamanifold2.31Specialrelativity2.32Bibliography3LiederivativesandLiegroups3.1Introduction:howavectorfieldmapsamanifoldintoitself3.2Liedraggingafunction3.3Liedraggingavectorfield3.4Liederivatives3.5Liederivativeofaone-form3.6Submanifolds3.7Frobeniustheorem(vectorfieldversion)3.8ProofofFrobeniustheorem3.9Anexample:thegeneratorsors23.10Invariance3.11Killingvectorfields3.12Killingvectorsandconservedquantitiesinparticledynamics3.13Axialsymmetry3.14AbstractLiegroups3.15ExamplesofLiegroups3.16Liealgebrasandtheirgroups3.17Realizationsandrepresentatidns3.18Sphericalsymmetry,sphericalharmonicsandrepresentationsoftherotationgroup3.19Bibliography4DifferentialformsAThealgebraandintegralcalculusofforms4.1Definitionofvolume-thegeometricalroleofdifferentialforms4.2Notationanddefinitionsforantisymmetrictensors4.3Differentialforms4.4Manipulatingdifferentialforms4.5Restrictionofforms4.6Fieldsofforms4.7Handednessandorientability4.8Volumesandintegrationonorientedmanifolds4.9N-vectors,duals,andthesymbol4.10Tensordensities4.11GeneralizedKroneckerdeltas4.12Determinantsand4.13MetricvolumeelementsBThedifferentialcalculusofformsanditsapplications4.14Theexteriorderivative4.15Notationforderivatives4.16Familiarexamplesofexteriordifferentiation4.17Integrabilityconditionsforpartialdifferentialequations4.18Exactforms4.19Proofofthelocalexactnessofclosedforms4.20Liederivativesofforms4.21Liederivativesandexteriorderivativescommute4.22Stokestheorem4.23Gausstheoremandthedefinitionofdivergence4.24Aglanceatcohomologytheory4.25Differentialformsanddifferentialequations4.26Frobeninstheorem(differentialformsversion)4.27ProofoftheequivalenceofthetwoversionsofFrobeniustheorem4.28Conservationlaws4.29Vectorsphericalharmonics4.30Bibliography5ApplicationsinphysicsAThermodynamics5.1Simplesystems5.2Maxwellandothermathematicalidentities5.3Compositethermodynamicsystems:CaratheodorystheoremBHamilton/anmechanics5.4Hamiltodianvectorfields5.5Canonicaltransformations5.6Mapbetweenvectorsandone-formsprovidedby5.7Poissonbracket5.8Many-particlesystems:symplecticforms5.9Lineardynamicalsystems:thesymplecticinnerproductandconservedquantities5.10FiberbundlestructureoftheHamiltonianequationsCElectromagnetism5.11RewritingMaxwellsequationsusingdifferentialforms5.12Chargeandtopology5.13Thevectorpotential5.14Planewaves:asimpleexampleDDynamicsofaperfectfluid5.15RoleofLiederivatives5.16Thecomovingtime-derivative5.17Equationofmotion5.18ConservationofvorticityECosmology5.19Thecosmologicalprinciple5.20Liealgebraofmaximalsymmetry5.21Themetricofasphericallysymmetricthree-space5.22ConstructionofthesixKillingvectors5.23Open,closed,andflatuniverses5.24Bibliography6ConnectionsforRiemnnnianmanifoldsandgaugetheories6.1Introduction6.2Parallelismoncurvedsurfaces6.3Thecovariantderivative6.4Components:covariantderivativesofthebasis6.5Torsion6.6Geodesics6.7Normalcoordinates6.8Riemanntensor6.9GeometricinterpretationoftheRiemanntensor6.10Flatspaces6.11Compatibilityoftheconnectionwithvolume-measureorthemetric6.12Metricconnections6.13Theaffineconnectionandtheequivalenceprinciple6.14Connectionsandgaugetheories:theexampleofelectromagnetism6.15BibfiographyAppendix:solutionsandhintsforselectedexercisesNotationIndex
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