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流形导论
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作者: [法] 图
出版社: 世界图书出版公司
2015-01
版次: 2
ISBN: 9787510084485
定价: 79.00
装帧: 平装
开本: 24开
纸张: 胶版纸
页数: 410页
正文语种: 英语
分类: 自然科学
3 张插图图片
101人买过
  • 　　Thisisacompletelyrevisededition,withmorethanfiftypagesofnewmaterialscatteredthroughout.Inkeepingwiththeconventionalmeaningofchaptersandsections,Ihavereorgaruzedthebookintotwenty-ninesectionsinsevenchapters.ThemainadditionsareSection200ntheLiederivativeandinteriormultiplication,twointrinsicoperationsonamanifoldtooimportanttoleaveout,newcriteriainSection21fortheboundaryorientation,andanewappendixonquaternionsandthesymplecticgroup.
    　　Apartfromcorrectingerrorsandmisprints,Ihavethoughtthrougheveryproofagain,clarifiedmanypassages,andaddednewexamples,exercises,hints,andsolutions.Intheprocess,everysectionhasbeenrewritten,sometimesquitedrastically.Therevisionsaresoextensivethatitisnotpossibletoenumeratethemallhere.Eachchapternowcomeswithanintroductoryessaygivinganoverviewofwhatistocome.Toprovideatimelineforthedevelopmentofideas,Ihaveindicatedwheneverpossi-blethehistoricaloriginoftheconcepts,andhaveaugmentedthebibliographywithhistoricalreferences. PrefacetotheSecondEdition
    PrefacetotheFirstEdition
    ABriefIntroduction

    Chapter1EuclideanSpaces
    1SmoothFunctionsonaEuclideanSpace
    1.1C∞VersusAnalyticFunctions
    1.2Taylor'sTheoremwithRemainder
    Problems
    2TangentVectorsinRnasDerivations
    2.1TheDirectionalDerivative
    2.2GermsofFunctions
    2.3DerivationsataPoint
    2.4VectorFields
    2.5VectorFieldsasDerivations
    Problems
    3TheExteriorAlgebraofMulticovectors
    3.1DualSpace
    3.2Permutations
    3.3MultilinearFunctions
    3.4ThePermutationActiononMultilinearFunctions
    3.5TheSymmetrizingandAlternatingOperators
    3.6TheTensorProduct
    3.7TheWedgeProduct
    3.8AnticommutativityoftheWedgeProduct
    3.9AssociativityoftheWedgeProduct
    3.10ABasisfork—Covectors
    Problems
    4DifferentialFormsonRn
    4.1Differential1—FormsandtheDifferentialofaFunction
    4.2Differentialk—Forms
    4.3DifferentialFormsasMultilinearFunctionsonVectorFields
    4.4TheExteriorDerivative
    4.5ClosedFormsandExactForms
    4.6ApplicationstoVectorCalculus
    4.7ConventiononSubscriptsandSuperscripts
    Problems

    Chapter2Manifolds
    5Manifolds
    5.1TopologicalManifolds
    5.2CompatibleCharts
    5.3SmoothManifolds
    5.4ExamplesofSmoothManifolds
    Problems
    6SmoothMapsonaManifold
    6.1SmoothFunctionsonaManifold
    6.2SmoothMapsBetweenManifolds
    6.3Diffeomorphisms
    6.4SmoothnessinTermsofComponents
    6.5ExamplesofSmoothMaps
    6.6PartialDerivatives
    6.7TheInverseFunctionTheorem
    Problems
    7Quotients
    7.1TheQuotientTopology
    7.2ContinuityofaMaponaQuotient
    7.3IdentificationofaSubsettoaPoint
    7.4ANecessaryConditionforaHausdorffQuotient
    7.5OpenEquivalenceRelations
    7.6RealProjectiveSpace
    7.7TheStandardC∞AtlasonaRealProjectiveSpace
    Problems

    Chapter3TheTangentSpace
    8TheTangentSpace
    8.1TheTangentSpaceataPoint
    8.2TheDifferentialofaMap
    8.3TheChainRule
    8.4BasesfortheTangentSpaceataPoint
    8.5ALocalExpressionfortheDifferential
    8.6CurvesinaManifold
    8.7ComputingtheDifferentialUsingCurves
    8.8ImmersionsandSubmersions
    8.9Rank，andCriticalandRegularPoints
    Problems
    9Submanifolds
    9.1Submanifolds
    9.2LevelSetsofaFunction
    9.3TheRegularLevelSetTheorem
    9.4ExamplesofRegularSubmanifolds
    Problems
    10CategoriesandFunctors
    10.1Categories
    10.2Functors
    10.3TheDualFunctorandtheMulticovectorFunctor
    Problems
    11TheRankofaSmoothMap
    11.1ConstantRankTheorem
    11.2TheImmersionandSubmersionTheorems
    11.3ImagesofSmoothMaps
    11.4SmoothMapsintoaSubmanifold
    11.5TheTangentPlanetoaSurfaceinR3
    Problems
    12TheTangentBundle
    12.1TheTopologyoftheTangentBundle
    12.2TheManifoldStructureontheTangentBundle
    12.3VectorBundles
    12.4SmoothSections
    12.5SmoothFrames
    Problems
    13BumpFunctionsandPartitionsofUnity
    13.1C∞BumpFunctions
    13.2PartitionsofUnity
    13.3ExistenceofaPartitionofUnity
    Problems
    14VectorFields
    14.1SmoothnessofaVectorField
    14.2IntegralCurves
    14.3LocalFlows
    14.4TheLieBracket
    14.5ThePushforwardofVectorFields
    14.6RelatedVectorFields
    Problems

    Chapter4LieGroupsandLieAlgebras
    15LieGroups
    15.1ExamplesofLieGroups
    15.2LieSubgroups
    15.3TheMatrixExponential
    15.4TheTraceofaMatrix
    15.5TheDifferentialofdetattheIdentity
    Problems
    16LieAlgebras
    16.1TangentSpaceattheIdentityofaLieGroup
    16.2Left—InvariantVectorFieldsonaLieGroup
    16.3TheLieAlgebraofaLieGroup
    16.4TheLieBracketongl（n，R）
    16.5ThePushforwardofLeft—InvariantVectorFields
    16.6TheDifferentialasaLieAlgebraHomomorphism
    Problems

    Chapter5DifferentialForms
    17Differential1—Forms
    17.1TheDifferentialofaFunction
    17.2LocalExpressionforaDifferential1—Form
    17.3TheCotangentBundle
    17.4CharacterizationofC∞l—Forms
    17.5Pullbackofl—Forms
    17.6Restrictionofl—FormstoanImmersedSubmanifold
    Problems
    18Differentialk—Forms
    18.1DifferentialForms
    18.2LocalExpressionforak—Form
    18.3TheBundlePointofView
    18.4Smoothk—Forms
    18.5Pullbackofk—Forms
    18.6TheWedgeProduct
    18.7DifferentialFormsonaCircle
    18.8InvariantFormsonaLieGroup
    Problems
    19TheExteriorDerivative
    19.1ExteriorDerivativeonaCoordinateChart
    19.2LocalOperators
    19.3ExistenceofanExteriorDerivativeonaManifold
    19.4UniquenessoftheExteriorDerivative
    19.5ExteriorDifferentiationUnderaPullback
    19.6Restrictionofk—FormstoaSubmanifold
    19.7ANowhere—Vanishing1—FormontheCircle
    Problems
    20TheLieDerivativeandInteriorMultiplication
    20.1FamiliesofVectorFieldsandDifferentialForms
    20.2TheLieDerivativeofaVectorField
    20.3TheLieDerivativeofaDifferentialForm
    20.4InteriorMultiplication
    20.5PropertiesoftheLieDerivative
    20.6GlobalFormulasfortheLieandExteriorDerivatives
    Problems

    Chapter6Integration
    21Orientations
    21.1OrientationsofaVectorSpace
    21.2Orientationsandn—Covectors
    21.3OrientationsonaManifold
    21.4OrientationsandDifferentialForms
    21.5OrientationsandAtlases
    Problems
    22ManifoldswithBoundary
    22.1SmoothInvarianceofDomaininRn
    22.2ManifoldswithBoundary
    22.3TheBoundaryofaManifoldwithBoundary
    22.4TangentVectors，DifferentialForms，andOrientations
    22.5Outward—PointingVectorFields
    22.6BoundaryOrientation
    Problems
    23IntegrationonManifolds
    23.1TheRiemannIntegralofaFunctiononRn
    23.2IntegrabilityConditions
    23.3TheIntegralofann—FormonRn
    23.4IntegralofaDifferentialFormoveraManifold
    23.5Stokes'sTheorem
    23.6LineIntegralsandGreen'sTheorem
    Problems

    Chapter7DeRhamTheory
    24DeRhamCohomology
    24.1DeRharnCohomology
    24.2ExamplesofdeRhamCohomology
    24.3DiffeomorphismInvariance
    24.4TheRingStructureondeRhamCohomology
    Problems
    25TheLongExactSequenceinCohomology
    25.1ExactSequences
    25.2CohomologyofCochainComplexes
    25.3TheConnectingHomomorphism
    25.4TheZig—ZagLemma
    Problems
    26TheMayer—VietorisSequence
    26.1TheMayer—VietorisSequence
    26.2TheCohomologyoftheCircle
    26.3TheEulerCharacteristic
    Problems
    27HomotopyInvariance
    27.1SmoothHomotopy
    27.2HomotopyType
    27.3DeformationRetractions
    27.4TheHomotopyAxiomfordeRhamCohomology
    Problems
    28ComputationofdeRhamCohomology
    28.1CohomologyVectorSpaceofaTorus
    28.2TheCohomologyRingofaTorus
    28.3TheCohomologyofaSurfaceofGenusg
    Problems
    29ProofofHomotopyInvarianee
    29.1ReductiontoTwoSections
    29.2CochainHomotopies
    29.3DifferentialFormsonM×R
    29.4ACochainHomotopyBetweeni0*andi1*
    29.5VerificationofCochainHomotopy
    Problems

    Appendices
    APoint—SetTopology
    A.1TopologicalSpaces
    A.2SubspaceTopology
    A.3Bases
    A.4FirstandSecondCountability
    A.5SeparationAxioms
    A.6ProductTopology
    A.7Continuity
    A.8Compactness
    A.9BoundednessinRn
    A.10Connectedness
    A.11ConnectedComponents
    A.12Closure
    A.13Convergence
    Problems
    BTheInverseFunctionTheoremonRnandRelatedResults
    B.1TheInverseFunctionTheorem
    B.2TheImplicitFunctionTheorem
    B.3ConstantRankTheorem
    Problems
    CExistenceofaPartitionofUnityinGeneral
    DLinearAlgebra
    D.1QuotientVectorSpaces
    D.2LinearTransformations
    D.3DirectProductandDirectSum
    Problems
    EQuaternionsandtheSymplecticGroup
    E.1RepresentationofLinearMapsbyMatrices
    E.2QuaternionicConjugation
    E.3QuaternionicInnerProduct
  • 内容简介:
    　　Thisisacompletelyrevisededition,withmorethanfiftypagesofnewmaterialscatteredthroughout.Inkeepingwiththeconventionalmeaningofchaptersandsections,Ihavereorgaruzedthebookintotwenty-ninesectionsinsevenchapters.ThemainadditionsareSection200ntheLiederivativeandinteriormultiplication,twointrinsicoperationsonamanifoldtooimportanttoleaveout,newcriteriainSection21fortheboundaryorientation,andanewappendixonquaternionsandthesymplecticgroup.
    　　Apartfromcorrectingerrorsandmisprints,Ihavethoughtthrougheveryproofagain,clarifiedmanypassages,andaddednewexamples,exercises,hints,andsolutions.Intheprocess,everysectionhasbeenrewritten,sometimesquitedrastically.Therevisionsaresoextensivethatitisnotpossibletoenumeratethemallhere.Eachchapternowcomeswithanintroductoryessaygivinganoverviewofwhatistocome.Toprovideatimelineforthedevelopmentofideas,Ihaveindicatedwheneverpossi-blethehistoricaloriginoftheconcepts,andhaveaugmentedthebibliographywithhistoricalreferences.
  • 目录:
    PrefacetotheSecondEdition
    PrefacetotheFirstEdition
    ABriefIntroduction

    Chapter1EuclideanSpaces
    1SmoothFunctionsonaEuclideanSpace
    1.1C∞VersusAnalyticFunctions
    1.2Taylor'sTheoremwithRemainder
    Problems
    2TangentVectorsinRnasDerivations
    2.1TheDirectionalDerivative
    2.2GermsofFunctions
    2.3DerivationsataPoint
    2.4VectorFields
    2.5VectorFieldsasDerivations
    Problems
    3TheExteriorAlgebraofMulticovectors
    3.1DualSpace
    3.2Permutations
    3.3MultilinearFunctions
    3.4ThePermutationActiononMultilinearFunctions
    3.5TheSymmetrizingandAlternatingOperators
    3.6TheTensorProduct
    3.7TheWedgeProduct
    3.8AnticommutativityoftheWedgeProduct
    3.9AssociativityoftheWedgeProduct
    3.10ABasisfork—Covectors
    Problems
    4DifferentialFormsonRn
    4.1Differential1—FormsandtheDifferentialofaFunction
    4.2Differentialk—Forms
    4.3DifferentialFormsasMultilinearFunctionsonVectorFields
    4.4TheExteriorDerivative
    4.5ClosedFormsandExactForms
    4.6ApplicationstoVectorCalculus
    4.7ConventiononSubscriptsandSuperscripts
    Problems

    Chapter2Manifolds
    5Manifolds
    5.1TopologicalManifolds
    5.2CompatibleCharts
    5.3SmoothManifolds
    5.4ExamplesofSmoothManifolds
    Problems
    6SmoothMapsonaManifold
    6.1SmoothFunctionsonaManifold
    6.2SmoothMapsBetweenManifolds
    6.3Diffeomorphisms
    6.4SmoothnessinTermsofComponents
    6.5ExamplesofSmoothMaps
    6.6PartialDerivatives
    6.7TheInverseFunctionTheorem
    Problems
    7Quotients
    7.1TheQuotientTopology
    7.2ContinuityofaMaponaQuotient
    7.3IdentificationofaSubsettoaPoint
    7.4ANecessaryConditionforaHausdorffQuotient
    7.5OpenEquivalenceRelations
    7.6RealProjectiveSpace
    7.7TheStandardC∞AtlasonaRealProjectiveSpace
    Problems

    Chapter3TheTangentSpace
    8TheTangentSpace
    8.1TheTangentSpaceataPoint
    8.2TheDifferentialofaMap
    8.3TheChainRule
    8.4BasesfortheTangentSpaceataPoint
    8.5ALocalExpressionfortheDifferential
    8.6CurvesinaManifold
    8.7ComputingtheDifferentialUsingCurves
    8.8ImmersionsandSubmersions
    8.9Rank，andCriticalandRegularPoints
    Problems
    9Submanifolds
    9.1Submanifolds
    9.2LevelSetsofaFunction
    9.3TheRegularLevelSetTheorem
    9.4ExamplesofRegularSubmanifolds
    Problems
    10CategoriesandFunctors
    10.1Categories
    10.2Functors
    10.3TheDualFunctorandtheMulticovectorFunctor
    Problems
    11TheRankofaSmoothMap
    11.1ConstantRankTheorem
    11.2TheImmersionandSubmersionTheorems
    11.3ImagesofSmoothMaps
    11.4SmoothMapsintoaSubmanifold
    11.5TheTangentPlanetoaSurfaceinR3
    Problems
    12TheTangentBundle
    12.1TheTopologyoftheTangentBundle
    12.2TheManifoldStructureontheTangentBundle
    12.3VectorBundles
    12.4SmoothSections
    12.5SmoothFrames
    Problems
    13BumpFunctionsandPartitionsofUnity
    13.1C∞BumpFunctions
    13.2PartitionsofUnity
    13.3ExistenceofaPartitionofUnity
    Problems
    14VectorFields
    14.1SmoothnessofaVectorField
    14.2IntegralCurves
    14.3LocalFlows
    14.4TheLieBracket
    14.5ThePushforwardofVectorFields
    14.6RelatedVectorFields
    Problems

    Chapter4LieGroupsandLieAlgebras
    15LieGroups
    15.1ExamplesofLieGroups
    15.2LieSubgroups
    15.3TheMatrixExponential
    15.4TheTraceofaMatrix
    15.5TheDifferentialofdetattheIdentity
    Problems
    16LieAlgebras
    16.1TangentSpaceattheIdentityofaLieGroup
    16.2Left—InvariantVectorFieldsonaLieGroup
    16.3TheLieAlgebraofaLieGroup
    16.4TheLieBracketongl（n，R）
    16.5ThePushforwardofLeft—InvariantVectorFields
    16.6TheDifferentialasaLieAlgebraHomomorphism
    Problems

    Chapter5DifferentialForms
    17Differential1—Forms
    17.1TheDifferentialofaFunction
    17.2LocalExpressionforaDifferential1—Form
    17.3TheCotangentBundle
    17.4CharacterizationofC∞l—Forms
    17.5Pullbackofl—Forms
    17.6Restrictionofl—FormstoanImmersedSubmanifold
    Problems
    18Differentialk—Forms
    18.1DifferentialForms
    18.2LocalExpressionforak—Form
    18.3TheBundlePointofView
    18.4Smoothk—Forms
    18.5Pullbackofk—Forms
    18.6TheWedgeProduct
    18.7DifferentialFormsonaCircle
    18.8InvariantFormsonaLieGroup
    Problems
    19TheExteriorDerivative
    19.1ExteriorDerivativeonaCoordinateChart
    19.2LocalOperators
    19.3ExistenceofanExteriorDerivativeonaManifold
    19.4UniquenessoftheExteriorDerivative
    19.5ExteriorDifferentiationUnderaPullback
    19.6Restrictionofk—FormstoaSubmanifold
    19.7ANowhere—Vanishing1—FormontheCircle
    Problems
    20TheLieDerivativeandInteriorMultiplication
    20.1FamiliesofVectorFieldsandDifferentialForms
    20.2TheLieDerivativeofaVectorField
    20.3TheLieDerivativeofaDifferentialForm
    20.4InteriorMultiplication
    20.5PropertiesoftheLieDerivative
    20.6GlobalFormulasfortheLieandExteriorDerivatives
    Problems

    Chapter6Integration
    21Orientations
    21.1OrientationsofaVectorSpace
    21.2Orientationsandn—Covectors
    21.3OrientationsonaManifold
    21.4OrientationsandDifferentialForms
    21.5OrientationsandAtlases
    Problems
    22ManifoldswithBoundary
    22.1SmoothInvarianceofDomaininRn
    22.2ManifoldswithBoundary
    22.3TheBoundaryofaManifoldwithBoundary
    22.4TangentVectors，DifferentialForms，andOrientations
    22.5Outward—PointingVectorFields
    22.6BoundaryOrientation
    Problems
    23IntegrationonManifolds
    23.1TheRiemannIntegralofaFunctiononRn
    23.2IntegrabilityConditions
    23.3TheIntegralofann—FormonRn
    23.4IntegralofaDifferentialFormoveraManifold
    23.5Stokes'sTheorem
    23.6LineIntegralsandGreen'sTheorem
    Problems

    Chapter7DeRhamTheory
    24DeRhamCohomology
    24.1DeRharnCohomology
    24.2ExamplesofdeRhamCohomology
    24.3DiffeomorphismInvariance
    24.4TheRingStructureondeRhamCohomology
    Problems
    25TheLongExactSequenceinCohomology
    25.1ExactSequences
    25.2CohomologyofCochainComplexes
    25.3TheConnectingHomomorphism
    25.4TheZig—ZagLemma
    Problems
    26TheMayer—VietorisSequence
    26.1TheMayer—VietorisSequence
    26.2TheCohomologyoftheCircle
    26.3TheEulerCharacteristic
    Problems
    27HomotopyInvariance
    27.1SmoothHomotopy
    27.2HomotopyType
    27.3DeformationRetractions
    27.4TheHomotopyAxiomfordeRhamCohomology
    Problems
    28ComputationofdeRhamCohomology
    28.1CohomologyVectorSpaceofaTorus
    28.2TheCohomologyRingofaTorus
    28.3TheCohomologyofaSurfaceofGenusg
    Problems
    29ProofofHomotopyInvarianee
    29.1ReductiontoTwoSections
    29.2CochainHomotopies
    29.3DifferentialFormsonM×R
    29.4ACochainHomotopyBetweeni0*andi1*
    29.5VerificationofCochainHomotopy
    Problems

    Appendices
    APoint—SetTopology
    A.1TopologicalSpaces
    A.2SubspaceTopology
    A.3Bases
    A.4FirstandSecondCountability
    A.5SeparationAxioms
    A.6ProductTopology
    A.7Continuity
    A.8Compactness
    A.9BoundednessinRn
    A.10Connectedness
    A.11ConnectedComponents
    A.12Closure
    A.13Convergence
    Problems
    BTheInverseFunctionTheoremonRnandRelatedResults
    B.1TheInverseFunctionTheorem
    B.2TheImplicitFunctionTheorem
    B.3ConstantRankTheorem
    Problems
    CExistenceofaPartitionofUnityinGeneral
    DLinearAlgebra
    D.1QuotientVectorSpaces
    D.2LinearTransformations
    D.3DirectProductandDirectSum
    Problems
    EQuaternionsandtheSymplecticGroup
    E.1RepresentationofLinearMapsbyMatrices
    E.2QuaternionicConjugation
    E.3QuaternionicInnerProduct
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