MathematicalAnalysis:AConciseIntroduction
- 作者
- 出版社 Wiley-Interscience
- 出版时间 2007年11月
- ISBN 9780470107966
- 定价 854.30
内容简介
Aself-containedintroductiontothefundamentalsofmathematicalanalysisMathematicalAnalysis:AConciseIntroductionpresentsthefoundationsofanalysisandillustratesitsroleinmathematics.Byfocusingontheessentials,reinforcinglearningthroughexercises,andfeaturingaunique"learnbydoing"approach,thebookdevelopsthereader'sproofwritingskillsandestablishesfundamentalcomprehensionofanalysisthatisessentialforfurtherexplorationofpureandappliedmathematics.Thisbookisdirectlyapplicabletoareassuchasdifferentialequations,probabilitytheory,numericalanalysis,differentialgeometry,andfunctionalanalysis.MathematicalAnalysisiscomposedofthreeparts:?PartOnepresentstheanalysisoffunctionsofonevariable,includingsequences,continuity,differentiation,Riemannintegration,series,andtheLebesgueintegral.Adetailedexplanationofproofwritingisprovidedwithspecificattentiondevotedtostandardprooftechniques.Tofacilitateanefficienttransitiontomoreabstractsettings,theresultsforsinglevariablefunctionsareprovedusingmethodsthattranslatetometricspaces.?PartTwoexploresthemoreabstractcounterpartsoftheconceptsoutlinedearlierinthetext.Thereaderisintroducedtothefundamentalspacesofanalysis,includingLpspaces,andthebooksuccessfullydetailshowappropriatedefinitionsofintegration,continuity,anddifferentiationleadtoapowerfulandwidelyapplicablefoundationforfurtherstudyofappliedmathematics.Theinterrelationbetweenmeasuretheory,topology,anddifferentiationisthenexaminedintheproofoftheMultidimensionalSubstitutionFormula.Furtherareasofcoverageinthissectionincludemanifolds,Stokes'Theorem,Hilbertspaces,theconvergenceofFourierseries,andRiesz'RepresentationTheorem.?PartThreeprovidesanoverviewofthemotivationsforanalysisaswellasitsapplicationsinvarioussubjects.Aspecialfocusonordinaryandpartialdifferentialequationspresentssometheoreticalandpracticalchallengesthatexistintheseareas.TopicalcoverageincludesNavier-Stokesequationsandthefiniteelementmethod.MathematicalAnalysis:AConciseIntroductionincludesanextensiveindexandover900exercisesranginginlevelofdifficulty,fromconceptualquestionsandadaptationsofproofstoproofswithandwithouthints.Theseopportunitiesforreinforcement,alongwiththeoverallconciseandwell-organizedtreatmentofanalysis,makethisbookessentialforreadersinupper-undergraduateorbeginninggraduatemathematicscourseswhowouldliketobuildasolidfoundationinanalysisforfurtherworkinallanalysis-basedbranchesofmathematics.
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