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1、TomIN.ApostolCALCULUSVOLUMEIIMlultiVariableCalculusandLinearAlgebra,withApplicationstoDifFerentialEquationsandProbabilitySECONDEDITIONJohnWiley&SonsNewYorkLondonSydneyTorontoCONSULTINGEDITORGeorgeSpringer,IndianaUniversityCOPYRIGHT01969BYXEROXCORPORATION.Allrightsreserved
2、.Nopartofthematerialcoveredbythiscopyrightmaybeproducedinanyform,orbyanymeansofreproduction.Previouseditioncopyright01962byXeroxCorporation.LibrarofCongressCatalogCardNumber:67-14605ISBN0471000078PrintedintheUnitedStatesofAmerica.1098765432ToJaneandStephenPREFACEThisbooki
3、sacontinuationoftheauthor’sCalculus,VolumeI,SecondEdition.Thepresentvolumehasbeenwrittenwiththesameunderlyingphilosophythatprevailedinthefirst.Soundtrainingintechniqueiscombinedwithastrongtheoreticaldevelopment.Everyefforthasbeenmadetoconveythespiritofmodernmathematicswit
4、houtundueemphasisonformalization.AsinVolumeI,historicalremarksareincludedtogivethestudentasenseofparticipationintheevolutionofideas.Thesecondvolumeisdividedintothreeparts,entitledLinearAnalysis,NonlinearAna!ysis,andSpecialTopics.ThelasttwochaptersofVolumeIhavebeenrepeated
5、asthefirsttwochaptersofVolumeIIsothatallthematerialonlinearalgebrawillbecompleteinonevolume.Part1containsanintroductiontolinearalgebra,includinglineartransformations,matrices,determinants,eigenvalues,andquadraticforms.Applicationsaregiventoanalysis,inparticulartothestudyo
6、flineardifferentialequations.Systemsofdifferentialequationsaretreatedwiththehelpofmatrixcalculus.ExistenceanduniquenesstheoremsareprovedbyPicard’smethodofsuccessiveapproximations,whichisalsocastinthelanguageofcontractionoperators.Part2discussesthecalculusoffunctionsofseve
7、ralvariables.Differentialcalculusisunifiedandsimplifiedwiththeaidoflinearalgebra.Itincludeschainrulesforscalarandvectorfields,andapplicationstopartialdifferentialequationsandextremumproblems.Integralcalculusincludeslineintegrals,multipleintegrals,andsurfaceintegrals,witha
8、pplicationstovectoranalysis.Herethetreatmentisalongmoreorlessclassicallinesanddoesnotincludeafor
