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1、DepartmentofMathematicalSciencesAdvancedCalculusandAnalysisMA1002IanCrawiiApril13,2000,Version1.3CopyrightÓ2000byIanCrawandtheUniversityofAberdeenAllrightsreserved.Additionalcopiesmaybeobtainedfrom:DepartmentofMathematicalSciencesUniversityofAberdeenAberdeenAB92TYDSN:mth200-101982-8ForewordTheseNote
2、sThenotescontainthematerialthatIusewhenpreparinglecturesforacourseIgavefromthemid1980'suntil1994;inthatsensetheyaremylecturenotes."Lectureswereonceuseful,butnowwhenallcanread,andbooksaresonu-merous,lecturesareunnecessary."SamuelJohnson,1799.Lecturenoteshavebeenaroundforcenturies,eitherinformally,ash
3、andwrittennotes,orformallyastextbooks.Recentlyimprovementsintypesettinghavemadeiteasiertoproducepersonalised"printednotesashere,buttherehasbeennofundamentalchange.Experienceshowsthatveryfewpeopleareabletouselecturenotesasasubstituteforlectures;ifitwereotherwise,lecturing,asaprofessionwouldhavediedo
4、utbynow.Thesenoteshavealonghistory;a rstcourseinanalysis"ratherlikethishasbeengivenwithintheMathematicsDepartmentforatleast30years.Duringthattimemanypeoplehavetaughtthecourseandallhavelefttheirmarkonit;clarifyingpointsthathaveproveddicult,selectingtheright"examplesandsoon.Icertainlybene tedfromth
5、enotesthatDrStuartDaggerhadwritten,whenItookoverthecoursefromhimandthisversionbuildsonthatfoundation,itslefheavilyinfluencedby(Spivak1967)whichwastherecommendedtextbookformostofthetimethesenoteswereused.ThenotesarewritteninLATEXwhichallowsahigherlevelviewofthetext,andsimpli esthepreparationofsuchthin
6、gsastheindexonpage101andnumberedequations.Youwill ndthatmostequationsarenotnumbered,orarenumberedsymbolically.HoweversometimesIwanttoreferbacktoanequation,andinthatcaseitisnumberedwithinthesection.ThusEquation(1.1)referstothe rstnumberedequationinChapter1andsoon.AcknowledgementsThesenotes,intheirpri
7、ntedform,havebeenseenbymanystudentsinAberdeensincetheywere rstwritten.Ithankthose(now)anonymousstudentswhohelpedtoimprovetheirqualitybypointingoutstupidities,repetitionsmisprintsandsoon.Sincethenotesh
