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1、DepartmentofMathematicalSciencesAdvancedCalculusandAnalysisMA1002IanCrawiiApril13,2000,Version1.3CopyrightÓ2000byIanCrawandtheUniversityofAberdeenAllrightsreserved.Additionalcopiesmaybeobtainedfrom:DepartmentofMathematicalSciencesUniversityofAberdeenAberdeenAB92TYDSN:mth200-101982-8For
2、ewordTheseNotesThenotescontainthematerialthatIusewhenpreparinglecturesforacourseIgavefromthemid1980'suntil1994;inthatsensetheyaremylecturenotes."Lectureswereonceuseful,butnowwhenallcanread,andbooksaresonu-merous,lecturesareunnecessary."SamuelJohnson,1799.Lecturenoteshavebeenaroundforce
3、nturies,eitherinformally,ashandwrittennotes,orformallyastextbooks.Recentlyimprovementsintypesettinghavemadeiteasiertoproducepersonalised"printednotesashere,buttherehasbeennofundamentalchange.Experienceshowsthatveryfewpeopleareabletouselecturenotesasasubstituteforlectures;ifitwereother
4、wise,lecturing,asaprofessionwouldhavediedoutbynow.Thesenoteshavealonghistory;a rstcourseinanalysis"ratherlikethishasbeengivenwithintheMathematicsDepartmentforatleast30years.Duringthattimemanypeoplehavetaughtthecourseandallhavelefttheirmarkonit;clarifyingpointsthathaveproveddicult,sel
5、ectingtheright"examplesandsoon.Icertainlybene tedfromthenotesthatDrStuartDaggerhadwritten,whenItookoverthecoursefromhimandthisversionbuildsonthatfoundation,itslefheavilyinfluencedby(Spivak1967)whichwastherecommendedtextbookformostofthetimethesenoteswereused.ThenotesarewritteninLATEXwhi
6、challowsahigherlevelviewofthetext,andsimpli esthepreparationofsuchthingsastheindexonpage101andnumberedequations.Youwill ndthatmostequationsarenotnumbered,orarenumberedsymbolically.HoweversometimesIwanttoreferbacktoanequation,andinthatcaseitisnumberedwithinthesection.ThusEquation(1.1)re
7、ferstothe rstnumberedequationinChapter1andsoon.AcknowledgementsThesenotes,intheirprintedform,havebeenseenbymanystudentsinAberdeensincetheywere rstwritten.Ithankthose(now)anonymousstudentswhohelpedtoimprovetheirqualitybypointingoutstupidities,repetitionsmisprintsandsoon.Sincethenotesh
