物理学家用的微分几何和李群

图书标准信息

• 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
• ISBN 9787506292672
• 定价 138.00元
• 装帧 平装
• 开本 16开
• 纸张 胶版纸
• 页数 697页
• 正文语种 英语

【内容简介】

《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】

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