微积分和数学分析引论．Ⅱ．1

Chapter1FunctionsofSeveralVariablesandTheirDerivatives

1.1PointsandPointsSetsinthePlaneandinSpace

a.Sequencesofpoints.Convergence

b.Setsofpointsintheplane

c.Theboundaryofaset.Closedandopensets

d.Closureassetoflimitpoints

e.Pointsandsetsofpointsinspace

1.2FunctionsofSeveralIndependentVariables

a.Functionsandtheirdomains

b.Thesimplesttypesoffunctions

c.Geometricalrepresentationoffunctions

1.3Continuity

a.Definition

b.Theconceptoflimitofafunctionofseveralvariables

c.Theordertowhichafunctionvanishes

1.4ThePartialDerivativesofaFunction

a.Definition.Geometricalrepresentation

b.Examples

c.Continuityandtheexistenceofpartialderivatives

d.Changeoftheorderofdifferentiation,36

1.5TheDifferentialofaFunctionandItsGeometricalMeaning

a.Theconceptofdifferentiability

b.Directionalderivatives

c.Geometricinterpretationofdifferentiability,Thetangentplane

d.Thetotaldifferentialofafunction

e.Applicationtothecalculusoferrors

1.6FunctionsofFunctions(CompoundFunctions)andtheIntroductionofNewIndependentVariables

a.Compoundfunctions.Thechainrule

b.Examples

c.Changeofindependentvariables

1.7TheMeanValueTheoremandTaylorsTheoremforFunctionsofSeveralVariables

a.Preliminaryremarksaboutapproximationbypolynomials

b.Themeanvaluetheorem

c.Taylorstheoremforseveralindependentvariables

1.8IntegralsofaFunctionDependingonaParameter

a.Examplesanddefinitions

b.Continuityanddifferentiabilityofanintegralwithrespecttotheparameter

c.Interchangeofintegrations.Smoothingoffunctions

1.9DifferentialsandLineIntegrals

a.Lineardifferentialforms

b.Lineintegralsoflineardifferentialforms

c.Dependenceoflineintegralsonendpoints

1.10TheFundamentalTheoremonIntegrabilityofLinearDifferentialForms

a.Integrationoftotaldifferentials

b.Necessaryconditionsforlineintegralstodependonlyontheendpoints

c.Insufficiencyoftheintegrabilityconditions

d.Simplyconnectedsets

e.Thefundamentaltheorem

APPENDIX

……

Chapter2Vectors,Matrices,LinearTransformations

Chapter3DevelopmentsandApplicationsoftheDifferentialCalculus

Chapter4MultipleIntegrals

Chapter5RelationsBetweenSurfaceandVolumeIntegrals

Chapter6DifferentialEquations

Chapter7CalculusofVariations

Chapter8FunctionsofaComplexVariable

ListofBiographicalDates

Index

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　　本书系统地阐述了微积分学的基本理论。在叙述上，作者尽量作到既严谨而又通俗易懂，并指出概念之间的内在联系和直观背景。本书是《微积分和数学分析引论》的第2卷第1册，为单变量情形，它适合于理工科大学师生、数学工作者和工程技术人员。