数学物理的几何方法（英文版）

图书标准信息

• 作者 [英]舒茨 著
• 出版社 世界图书出版公司
• 出版时间 2009-06
• 版次 1
• ISBN 9787510004513
• 定价 35.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 250页
• 字数 185千字
• 正文语种 英语

【内容简介】

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

【目录】

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