物理学家用的微分几何和李群
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作者[斯洛伐克]费茨科 著
出版社世界图书出版公司
出版时间2008-11
版次1
装帧平装
货号1-4
上书时间2024-07-12
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图书标准信息
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作者
[斯洛伐克]费茨科 著
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出版社
世界图书出版公司
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出版时间
2008-11
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版次
1
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ISBN
9787506292672
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定价
138.00元
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装帧
平装
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开本
16开
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纸张
胶版纸
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页数
697页
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正文语种
英语
- 【内容简介】
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《物理学家用的微分几何和李群》以一种非正式的形式写作,作者给出了1000多例子重在强调对一般理论的深刻理解。微分几何在现代理论物理和应用数学中扮演着越来越重要的角色。《物理学家用的微分几何和李群》给出了在理论物理和应用数学中很重要的几何知识的引入,包括,流形、张量场、微分形式、联络、辛几何、李群作用、族以及自旋。《物理学家用的微分几何和李群》将要为读者很好的学习拉格郎日现代处理方法、哈密顿力学、电磁、规范场,相对论以及万有引力做充足的准备。《物理学家用的微分几何和李群》很适合作为物理、数学以及工程专业的高年级本科生以及研究生的教程,也是一本很难得自学教程。
- 【目录】
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Preface
Introduction
1Theconceptofamanifold
1.1Topologyandcontinuousmaps
1.2ClassesofsmoothnessofmapsofCartesianspaces
1.3Smoothstructure,smoothmanifold
1.4Smoothmapsofmanifolds
1.5AtechnicaldescriptionofsmoothsurfacesinRn
SummaryofChapter1
2Vectorandtensorfields
2.1CurvesandfunctionsonM
2.2Tangentspace,vectorsandvectorfields
2.3Integralcurvesofavectorfield
2.4Linearalgebraoftensors(multilinearalgebra)
2.5TensorfieldsonM
2.6Metrictensoronamanifold
SummaryofChapter2
3Mappingsoftensorsinducedbymappingsofmanifolds
3.1Mappingsoftensorsandtensorfields
3.2Inducedmetrictensor
SummaryofChapter3
4Liederivative
4.1Localflowofavectorfield
4.2LietransportandLiederivative
4.3PropertiesoftheLiederivative
4.4ExponentoftheLiederivative
4.5Geometricalinterpretationofthecommutator[V,W],non-holonomicframes
4.6Isometriesandconformaltransformations,Killingequations
SummaryofChapter4
5Exterioralgebra
5.1Motivation:volumesofparaUelepipeds
5.2p-formsandexteriorproduct
5.3ExterioralgebraAL*
5.4Interiorproductiv
5.5OrientationinL
5.6DeterminantandgeneralizedKroneckersymbols
5.7Themetricvolumeform
5.8Hodge(duality)operator*
SummaryofChapter5
6Differentialcalculusofforms
6.1Formsonamanifold
6.2Exteriorderivative
6.3Orientability,HodgeoperatorandvolumeformonM
6.4V-valuedforms
SummaryofChapter6
7Integralcalculusofforms
7.1Quantitiesundertheintegralsignregardedasdifferentialforms
7.2Euclideansimplicesandchains
7.3Simplicesandchainsonamanifold
7.4Integralofaformoverachainonamanifold
7.5Stokestheorem
7.6Integraloveradomainonanorientablemanifold
7.7IntegraloveradomainonanorientableRiemannianmanifold
7.8Integralandmapsofmanifolds
SummaryofChapter7
8ParticularcasesandapplicationsofStokestheorem
8.1Elementarysituations
8.2DivergenceofavectorfieldandGausstheorem
8.3CodifferentialandLaPlace-deRhanaoperator
8.4Greenidentities
8.5VectoranalysisinE3
8.6Functionsofcomplexvariables
SummaryofChapter8
9Poincarelemmaandcohomologies
9.1Simpleexamplesofclosednon-exactforms
9.2Constructionofapotentialoncontractiblemanifolds
9.3*CohomologiesanddeRhamcomplex
SummaryofChapter9
10Liegroups:basicfacts
10.1Automorphismsofvariousstructuresandgroups
10.2Liegroups:basicconcepts
SummaryofChapter10
11DifferentialgeometryonLiegroups
11.1Left-invarianttensorfieldsonaLiegroup
11.2LiealgebragofagroupG
11.3One-parametersubgroups
11.4Exponentialmap
11.5DerivedhomomorphismofLiealgebras
11.6InvariantintegralonG
11.7MatrixLiegroups:enjoysimplifications
SummaryofChapter11
12RepresentationsofLiegroupsandLiealgebras
12.1Basicconcepts
12.2Irreducibleandequivalentrepresentations,Schurslemma
12.3Adjointrepresentation,Killing-Cartanmetric
12.4Basicconstructionswithgroups,Liealgebrasandtheirrepresentations
12.5Invarianttensorsandintertwiningoperators
12.6*Liealgebracohomologies
SummaryofChapter12
13ActionsofLiegroupsandLiealgebrasonmanifolds
13.1Actionofagroup,orbitandstabilizer
13.2Thestructureofhomogeneousspaces,G/H
13.3Coveringhomomorphism,coveringsSU(2)→SO(3)andSL(2,C)→L↑+
13.4RepresentationsofGandginthespaceoffunctionsonaG-space,fundamentalfields
13.5RepresentationsofGandginthespaceoftensorfieldsoftypep
SummaryofChapter13
14Hamiltonianmechanicsandsymplecticmanifolds
14.1Poissonandsymplecticstructureonamanifold
14.2Darbouxtheorem,canonicaltransformationsandsymplectomorphisms
14.3Poincare-CartanintegralinvariantsandLiouvillestheorem
14.4Symmetriesandconservationlaws
14.5*Momentmap
14.6*Orbitsofthecoadjointaction
14.7*Symplecticreduction
SummaryofChapter14
15ParalleltransportandlinearconnectiononM
15.1Accelerationandparalleltransport
15.2Paralleltransportandcovariantderivative
15.3Compatibilitywithmetric,RLCconnection
15.4Geodesics
15.5Thecurvaturetensor
15.6ConnectionformsandCartanstructureequations
15.7Geodesicdeviationequation(Jacobisequation)
15.8*Torsion,completeparallelismandfiatconnection
SummaryofChapter15
16Fieldtheoryandthelanguageofforms
16.1DifferentialformsintheMinkowskispaceE13
16.2Maxwellsequationsintermsofdifferentialforms
16.3Gaugetransformations,actionintegral
16.4Energy-momentumtensor,space-timesymmetriesandconservation
lawsduetothem
16.5*Einsteingravitationalfieldequations,HilbertandCartanaction
16.6*Non-linearsigmamodelsandharmonicmaps
SummaryofChapter16
17DifferentialgeometryonTMandT*M
17.1TangentbundleTMandcotangentbundleT*M
17.2Conceptofafiberbundle
17.3ThemapsTfandT*f
17.4Verticalsubspace,verticalvectors
17.5LiftsonTMandT*M
17.6CanonicaltensorfieldsonTMandT*M
17.7Identitiesbetweenthetensorfieldsintroducedhere
SummaryofChapter17
18HamiltonianandLagrangianequations
18.1Second-orderdifferentialequationfields
18.2Euler-Lagrangefield
18.3ConnectionbetweenLagrangianandHamiltonianmechanics,Legendremap
18.4Symmetriesliftedfromthebasemanifold(configurationspace)
18.5Time-dependentHamiltonian,actionintegral
SummaryofChapter18
19Linearconnectionandtheframebundle
19.1Framebundleπ:LM→M
19.2ConnectionformonLM
19.3k-dimensionaldistributionDonamanifold.M
19.4Geometricalinterpretationofaconnectionform:horizontaldistributiononLM
19.5HorizontaldistributiononLMandparalleltransportonM
19.6TensorsonMinthelanguageofLMandtheirparalleltransport
SummaryofChapter19
20ConnectiononaprincipalG-bundle
20.1PrincipalG-bundles
21Gaugetheoriesandconnections
22*SpinorfieldsandtheDiracoperator
AppendixASomerelevantalgebraicstructures
A.ILinearspaces
A.2Associativealgebras
A.3Liealgebras
A.4Modules
A.5Grading
A.6Categoriesandfunctors
AppendixBStarring
Bibliography
Indexof(frequentlyused)symbols
Index
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