数学物理的几何方法

1Somebasicmathematics

1.1ThespaceRnanditstopology

1.2Mappings

1.3Realanalysis

1.4Grouptheory

1.5Linearalgebra

1.6Thealgebraofsquarematrices

1.7Bibliography

2Dffferentiablemanifoldsandtensors

2.1Defmitionofamanifold

2.2Thesphereasamanifold

2.3Otherexamplesofmanifolds

2.4Globalconsiderations

2.5Curves

2.6FunctionsonM

2.7Vectorsandvectorfields

2.8Basisvectorsandbasisvectorfields

2.9Fiberbundles

2.10Examplesoffiberbundles

2.11Adeeperlookatfiberbundles

2.12Vectorfieldsandintegralcurves

2.13Exponentiationoftheoperatord/dZ

2.14Liebracketsandnoncoordinatebases

2.15Whenisabasisacoordinatebasis？

2.16One-forms

2.17Examplesofone-forms

2.18TheDiracdeltafunction

2.19Thegradientandthepictorialrepresentationofaone-form

2.20Basisone-formsandcomponentsofone-forms

2.21Indexnotation

2.22Tensorsandtensorfields

2.23Examplesoftensors

2.24Componentsoftensorsandtheouterproduct

2.25Contraction

2.26Basistransformations

2.27Tensoroperationsoncomponents

2.28Functionsandscalars

2.29Themetrictensoronavectorspace

2.30Themetrictensorfieldonamanifold

2.31Specialrelativity

2.32Bibliography

3LiederivativesandLiegroups

3.1Introduction:howavectorfieldmapsamanifoldintoitself

3.2Liedraggingafunction

3.3Liedraggingavectorfield

3.4Liederivatives

3.5Liederivativeofaone-form

3.6Submanifolds

3.7Frobeniustheorem(vectorfieldversion)

3.8ProofofFrobeniustheorem

3.9Anexample:thegeneratorsors2

3.10Invariance

3.11Killingvectorfields

3.12Killingvectorsandconservedquantitiesinparticledynamics

3.13Axialsymmetry

3.14AbstractLiegroups

3.15ExamplesofLiegroups

3.16Liealgebrasandtheirgroups

3.17Realizationsandrepresentatidns

3.18Sphericalsymmetry,sphericalharmonicsandrepresentationsoftherotationgroup

3.19Bibliography

4DifferentialformsAThealgebraandintegralcalculusofforms

4.1Definitionofvolume-thegeometricalroleofdifferentialforms

4.2Notationanddefinitionsforantisymmetrictensors

4.3Differentialforms

4.4Manipulatingdifferentialforms

4.5Restrictionofforms

4.6Fieldsofforms

4.7Handednessandorientability

4.8Volumesandintegrationonorientedmanifolds

4.9N-vectors,duals,andthesymbol

4.10Tensordensities

4.11GeneralizedKroneckerdeltas

4.12Determinantsand

4.13MetricvolumeelementsBThedifferentialcalculusofformsanditsapplications

4.14Theexteriorderivative

4.15Notationforderivatives

4.16Familiarexamplesofexteriordifferentiation

4.17Integrabilityconditionsforpartialdifferentialequations

4.18Exactforms

4.19Proofofthelocalexactnessofclosedforms

4.20Liederivativesofforms

4.21Liederivativesandexteriorderivativescommute

4.22Stokestheorem

4.23Gausstheoremandthedefinitionofdivergence

4.24Aglanceatcohomologytheory

4.25Differentialformsanddifferentialequations

4.26Frobeninstheorem(differentialformsversion)

4.27ProofoftheequivalenceofthetwoversionsofFrobeniustheorem

4.28Conservationlaws

4.29Vectorsphericalharmonics

4.30Bibliography

5ApplicationsinphysicsAThermodynamics

5.1Simplesystems

5.2Maxwellandothermathematicalidentities

5.3Compositethermodynamicsystems:CaratheodorystheoremBHamilton/anmechanics

5.4Hamiltodianvectorfields

5.5Canonicaltransformations

5.6Mapbetweenvectorsandone-formsprovidedby

5.7Poissonbracket

5.8Many-particlesystems:symplecticforms

5.9Lineardynamicalsystems:thesymplecticinnerproductandconservedquantities

5.10FiberbundlestructureoftheHamiltonianequationsCElectromagnetism

5.11RewritingMaxwellsequationsusingdifferentialforms

5.12Chargeandtopology

5.13Thevectorpotential

5.14Planewaves:asimpleexampleDDynamicsofaperfectfluid

5.15RoleofLiederivatives

5.16Thecomovingtime-derivative

5.17Equationofmotion

5.18Conservationofvorticity

ECosmology

5.19Thecosmologicalprinciple

5.20Liealgebraofmaximalsymmetry

5.21Themetricofasphericallysymmetricthree-space

5.22ConstructionofthesixKillingvectors

5.23Open,closed,andflatuniverses

5.24Bibliography

6ConnectionsforRiemnnnianmanifoldsandgaugetheories

6.1Introduction

6.2Parallelismoncurvedsurfaces

6.3Thecovariantderivative

6.4Components:covariantderivativesofthebasis

6.5Torsion

6.6Geodesics

6.7Normalcoordinates

6.8Riemanntensor

6.9GeometricinterpretationoftheRiemanntensor

6.10Flatspaces

6.11Compatibilityoftheconnectionwithvolume-measureorthemetric

6.12Metricconnections

6.13Theaffineconnectionandtheequivalenceprinciple

6.14Connectionsandgaugetheories:theexampleofelectromagnetism

6.15Bibfiography

Appendix:solutionsandhintsforselectedexercises

Notation

Index

　　Thisbookalmstointroducethebeginningorworkingphysicisttoawiderangeofaualytictoolswhichhavetheiror/ginindifferentialgeometryandwhichhaverecentlyfoundincreasinguseintheoreticalphysics.Itisnotuncom-montodayforaphysicistsmathematicaleducationtoignoreallbutthesim-plestgeometricalideas,despitethefactthatyoungphysicistsareencouragedtodevelopmentalpicturesandintuitionappropriatetophysicalphenomena.Thiscuriousneglectofpicturesofonesmathematicaltoolsmaybeseenastheoutcomeofagradualevolutionovermanycenturies.Geometrywascertainlyextremelyimportanttoancientandmedievalnaturalphilosophers;itwasingeometricaltermsthatPtolemy,Copernicus,Kepler,andGalileoallexpressedtheirthinking.ButwhenDescartesintroducedcoordinatesintoEuclideangeometry,heshowedthatthestudyofgeometrycouldberegardedasanappli.cationofalgrebra.