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1、DepartmentofMathematicalSciencesAdvancedCalculusandAnalysisMA1002IanCrawiiApril13,2000,Version1.3CopyrightÓ2000byIanCrawandtheUniversityofAberdeenAllrightsreserved.Additionalcopiesmaybeobtainedfrom:DepartmentofMathematicalSciencesUniversityofAberdeenAberdeenAB92TYDSN:mth200-101982-8F
2、orewordTheseNotesThenotescontainthematerialthatIusewhenpreparinglecturesforacourseIgavefromthemid1980'suntil1994;inthatsensetheyaremylecturenotes."Lectureswereonceuseful,butnowwhenallcanread,andbooksaresonu-merous,lecturesareunnecessary."SamuelJohnson,1799.Lecturenoteshavebeenaroundf
3、orcenturies,eitherinformally,ashandwrittennotes,orformallyastextbooks.Recentlyimprovementsintypesettinghavemadeiteasiertoproducepersonalised"printednotesashere,buttherehasbeennofundamentalchange.Experienceshowsthatveryfewpeopleareabletouselecturenotesasasubstituteforlectures;ifitwer
4、eotherwise,lecturing,asaprofessionwouldhavediedoutbynow.Thesenoteshavealonghistory;a rstcourseinanalysis"ratherlikethishasbeengivenwithintheMathematicsDepartmentforatleast30years.Duringthattimemanypeoplehavetaughtthecourseandallhavelefttheirmarkonit;clarifyingpointsthathaveproveddi
5、cult,selectingtheright"examplesandsoon.Icertainlybene tedfromthenotesthatDrStuartDaggerhadwritten,whenItookoverthecoursefromhimandthisversionbuildsonthatfoundation,itslefheavilyinfluencedby(Spivak1967)whichwastherecommendedtextbookformostofthetimethesenoteswereused.Thenotesarewritten
6、inLATEXwhichallowsahigherlevelviewofthetext,andsimpli esthepreparationofsuchthingsastheindexonpage101andnumberedequations.Youwill ndthatmostequationsarenotnumbered,orarenumberedsymbolically.HoweversometimesIwanttoreferbacktoanequation,andinthatcaseitisnumberedwithinthesection.ThusEqu
7、ation(1.1)referstothe rstnumberedequationinChapter1andsoon.AcknowledgementsThesenotes,intheirprintedform,havebeenseenbymanystudentsinAberdeensincetheywere rstwritten.Ithankthose(now)anonymousstudentswhohelpedtoimprovetheirqualitybypointingoutstupidities,repetitionsmisprintsandsoon.Si
8、ncethenotesh
