分析（第2卷）

图书标准信息

• 作者 [德]阿莫恩 著
• 出版社 世界图书出版公司
• 出版时间 2012-09
• 版次 1
• ISBN 9787510047992
• 定价 89.00元
• 装帧 平装
• 开本 16开
• 纸张 其他
• 页数 400页

【内容简介】

　　Aswiththefirst,thesecondvolumecontainssubstantiallymorematerialthancanbecoveredinaone-semestercourse.Suchcoursesmayomitmanybeautifulandwell-groundedapplicationswhichconnectbroadlytomanyareasofmathematics.Weofcoursehopethatstudentswillpursuethismaterialindependently;teachersmayfinditusefulforundergraduateseminars.

【目录】

Foreword
ChapterⅥIntegralcalculusinonevariable
1Jumpcontinuousfunctions
Staircaseandjumpcontinuousfunctions
Acharacterizationofjumpcontinuousfunctions
TheBanachspaceofjumpcontinuousfunctions
2Continuousextensions
Theextensionofuniformlycontinuousfunctions
Boundedlinearoperators
Thecontinuousextensionofboundedlinearoperators
3TheCauchy-RiemannIntegral
Theintegralofstaircasefunctions
Theintegralofjumpcontinuousfunctions
Riemannsums
4Propertiesofintegrals
Integrationofsequencesoffunctions
Theorientedintegral
Positivityandmonotonyofintegrals
Componentwiseintegration
Thefirstfundamentaltheoremofcalculus
Theindefiniteintegral
Themeanvaluetheoremforintegrals
5Thetechniqueofintegration
Variablesubstitution
Integrationbyparts
Theintegralsofrationalfunctions
6Sumsandintegrals
TheBernoullinumbers
Recursionformulas
TheBernoullipolynomials
TheEuler-Maclaurinsumformula
Powersums
Asymptoticequivalence
TheBiemannζfunction
Thetrapezoidrule
7Fourierseries
TheL2scalarproduct
Approximatinginthequadraticmean
Orthonormalsystems
Integratingperiodicfunctions
Fouriercoefficients
ClassicalFourierseries
Bessel'sinequality
Completeorthonormalsystems
Piecewisecontinuouslydifferentiablefunctions
Uniformconvergence
8Improperintegrals
Admissiblefunctions
Improperintegrals
Theintegralcomparisontestforseries
Absolutelyconvergentintegrals
Themajorantcriterion
9Thegammafunction
Euler'sintegralrepresentation
ThegammafunctiononC\（-N）
Gauss'srepresentationformula
Thereflectionformula
Thelogarithmicconvexityofthegammafunction
Stirling'sformula
TheEulerbetaintegral
ChapterⅦMultivariabledifferentialcalculus
1Continuouslinearmaps
Thecompletenessof/L（E,F）
Finite-dimensionalBanachspaces
Matrixrepresentations
Theexponentialmap
Lineardifferentialequations
Gronwall'slemma
Thevariationofconstantsformula
Determinantsandeigenvalues
Fundamentalmatrices
Secondorderlineardifferentialequations
Differentiability
Thedefinition
Thederivative
Directionalderivatives
Partialderivatives
TheJacobimatrix
Adifferentiabilitycriterion
TheRieszrepresentationtheorem
Thegradient
Complexdifferentiability
Multivariabledifferentiationrules
Linearity
Thechainrule
Theproductrule
Themeanvaluetheorem
Thedifferentiabilityoflimitsofsequencesoffunctions
Necessaryconditionforlocalextrema
Multilinearmaps
Continuousmultilinearmaps
Thecanonicalisomorphism
Symmetricmultilinearmaps
Thederivativeofmultilinearmaps
Higherderivatives
Definitions
Higherorderpartialderivatives
Thechainrule
Taylor'sformula
Functionsofmvariables
Sufficientcriterionforlocalextrema
6Nemytskiioperatorsandthecalculusofvariations
Nemytskiioperators
ThecontinuityofNemytskiioperators
ThedifferentiabilityofNemytskiioperators
Thedifferentiabilityofparameter-dependentintegrals
Variationalproblems
TheEuler-Lagrangeequation
Classicalmechanics
7Inversemaps
Thederivativeoftheinverseoflinearmaps
Theinversefunctiontheorem
Diffeomorphisms
Thesolvabilityofnonlinearsystemsofequations
8Implicitfunctions
Differentiablemapsonproductspaces
Theimplicitfunctiontheorem
Regularvalues
Ordinarydifferentialequations
Separationofvariables
Lipschitzcontinuityanduniqueness
ThePicard-Lindeloftheorem
9Manifolds
SubmanifoldsofRn
Graphs
Theregularvaluetheorem
Theimmersiontheorem
Embeddings
Localchartsandparametrizations
Changeofcharts
10Tangentsandnormals
ThetangentialinRn
Thetangentialspace
Characterizationofthetangentialspace
Differentiablemaps
Thedifferentialandthegradient
Normals
Constrainedextrema
ApplicationsofLagrangemultipliers
ChapterⅧLineintegrals
1Curvesandtheirlengths
Thetotalvariation
Rectifiablepaths
Differentiablecurves
Rectifiablecurves
2CurvesinRn
Unittangentvectors
Paramctrizationbyarclength
Orientedbases
TheFrenetn-frame
Curvatureofplanecurves
Identifyinglinesandcircles
Instantaneouscirclesalongcurves
Thevectorproduct
Thecurvatureandtorsionofspacecurves
3Pfaffforms
VectorfieldsandPfaffforms
Thecanonicalbasis
Exactformsandgradientfields
ThePoincarelemma
Dualoperators
Transformationrules
Modules
4Lineintegrals
Thedefinition
Elementaryproperties
Thefundamentaltheoremoflineintegrals
Simplyconnectedsets
Thehomotopyinvarianceoflineintegrals
5Holomorphicfunctions
Complexlineintegrals
Holomorphism
TheCauchyintegraltheorem
Theorientationofcircles
TheCauchyintegralformula
Analyticfunctions
Liouville'stheorem
TheFresnelintegral
Themaximumprinciple
Harmonicfunctions
Goursat'stheorem
TheWeierstrassconvergencetheorem
6Meromorphiefunctions
TheLaurentexpansion
Removablesingularities
Isolatedsingularities
Simplepoles
Thewindingnumber
Thecontinuityofthewindingnumber
ThegeneralizedCauchyintegraltheorem
Theresiduetheorem
Fourierintegrals
References
Index