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[德] 阿莫恩 著 / 世界图书出版公司 / 2012-09 / 平装
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Aswiththefirst,thesecondvolumecontainssubstantiallymorematerialthancanbecoveredinaone-semestercourse.Suchcoursesmayomitmanybeautifulandwell-groundedapplicationswhichconnectbroadlytomanyareasofmathematics.Weofcoursehopethatstudentswillpursuethismaterialindependently;teachersmayfinditusefulforundergraduateseminars.
ForewordChapterⅥIntegralcalculusinonevariable1JumpcontinuousfunctionsStaircaseandjumpcontinuousfunctionsAcharacterizationofjumpcontinuousfunctionsTheBanachspaceofjumpcontinuousfunctions2ContinuousextensionsTheextensionofuniformlycontinuousfunctionsBoundedlinearoperatorsThecontinuousextensionofboundedlinearoperators3TheCauchy-RiemannIntegralTheintegralofstaircasefunctionsTheintegralofjumpcontinuousfunctionsRiemannsums4PropertiesofintegralsIntegrationofsequencesoffunctionsTheorientedintegralPositivityandmonotonyofintegralsComponentwiseintegrationThefirstfundamentaltheoremofcalculusTheindefiniteintegralThemeanvaluetheoremforintegrals5ThetechniqueofintegrationVariablesubstitutionIntegrationbypartsTheintegralsofrationalfunctions6SumsandintegralsTheBernoullinumbersRecursionformulasTheBernoullipolynomialsTheEuler-MaclaurinsumformulaPowersumsAsymptoticequivalenceTheBiemannζfunctionThetrapezoidrule7FourierseriesTheL2scalarproductApproximatinginthequadraticmeanOrthonormalsystemsIntegratingperiodicfunctionsFouriercoefficientsClassicalFourierseriesBessel'sinequalityCompleteorthonormalsystemsPiecewisecontinuouslydifferentiablefunctionsUniformconvergence8ImproperintegralsAdmissiblefunctionsImproperintegralsTheintegralcomparisontestforseriesAbsolutelyconvergentintegralsThemajorantcriterion9ThegammafunctionEuler'sintegralrepresentationThegammafunctiononC\(-N)Gauss'srepresentationformulaThereflectionformulaThelogarithmicconvexityofthegammafunctionStirling'sformulaTheEulerbetaintegralChapterⅦMultivariabledifferentialcalculus1ContinuouslinearmapsThecompletenessof/L(E,F)Finite-dimensionalBanachspacesMatrixrepresentationsTheexponentialmapLineardifferentialequationsGronwall'slemmaThevariationofconstantsformulaDeterminantsandeigenvaluesFundamentalmatricesSecondorderlineardifferentialequationsDifferentiabilityThedefinitionThederivativeDirectionalderivativesPartialderivativesTheJacobimatrixAdifferentiabilitycriterionTheRieszrepresentationtheoremThegradientComplexdifferentiabilityMultivariabledifferentiationrulesLinearityThechainruleTheproductruleThemeanvaluetheoremThedifferentiabilityoflimitsofsequencesoffunctionsNecessaryconditionforlocalextremaMultilinearmapsContinuousmultilinearmapsThecanonicalisomorphismSymmetricmultilinearmapsThederivativeofmultilinearmapsHigherderivativesDefinitionsHigherorderpartialderivativesThechainruleTaylor'sformulaFunctionsofmvariablesSufficientcriterionforlocalextrema6NemytskiioperatorsandthecalculusofvariationsNemytskiioperatorsThecontinuityofNemytskiioperatorsThedifferentiabilityofNemytskiioperatorsThedifferentiabilityofparameter-dependentintegralsVariationalproblemsTheEuler-LagrangeequationClassicalmechanics7InversemapsThederivativeoftheinverseoflinearmapsTheinversefunctiontheoremDiffeomorphismsThesolvabilityofnonlinearsystemsofequations8ImplicitfunctionsDifferentiablemapsonproductspacesTheimplicitfunctiontheoremRegularvaluesOrdinarydifferentialequationsSeparationofvariablesLipschitzcontinuityanduniquenessThePicard-Lindeloftheorem9ManifoldsSubmanifoldsofRnGraphsTheregularvaluetheoremTheimmersiontheoremEmbeddingsLocalchartsandparametrizationsChangeofcharts10TangentsandnormalsThetangentialinRnThetangentialspaceCharacterizationofthetangentialspaceDifferentiablemapsThedifferentialandthegradientNormalsConstrainedextremaApplicationsofLagrangemultipliersChapterⅧLineintegrals1CurvesandtheirlengthsThetotalvariationRectifiablepathsDifferentiablecurvesRectifiablecurves2CurvesinRnUnittangentvectorsParamctrizationbyarclengthOrientedbasesTheFrenetn-frameCurvatureofplanecurvesIdentifyinglinesandcirclesInstantaneouscirclesalongcurvesThevectorproductThecurvatureandtorsionofspacecurves3PfaffformsVectorfieldsandPfaffformsThecanonicalbasisExactformsandgradientfieldsThePoincarelemmaDualoperatorsTransformationrulesModules4LineintegralsThedefinitionElementarypropertiesThefundamentaltheoremoflineintegralsSimplyconnectedsetsThehomotopyinvarianceoflineintegrals5HolomorphicfunctionsComplexlineintegralsHolomorphismTheCauchyintegraltheoremTheorientationofcirclesTheCauchyintegralformulaAnalyticfunctionsLiouville'stheoremTheFresnelintegralThemaximumprincipleHarmonicfunctionsGoursat'stheoremTheWeierstrassconvergencetheorem6MeromorphiefunctionsTheLaurentexpansionRemovablesingularitiesIsolatedsingularitiesSimplepolesThewindingnumberThecontinuityofthewindingnumberThegeneralizedCauchyintegraltheoremTheresiduetheoremFourierintegralsReferencesIndex
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