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1、ADVANCEDCALCULUS1996–1997GilbertWeinsteinOffice:CH493BTel:(205)934-3724(205)934-2154FAX:(205)934-9025Email:weinstei@math.uab.eduOfficehours:Monday1:00pm–2:30pmWednesday8:30am–10:00am3AbouttheCourseWelcometoAdvancedCalculus!Inthiscourse,youwilllearnAnalysis,thatisthetheoryofdifferentiationandintegrationo
2、ffunctions.However,youwillalsolearnsomethingmorefundamentalthanAnalysis.Youwilllearnwhatisamathematicalproof.Youmayhaveseensomeproofsearlier,buthere,youwilllearnhowtowriteyourownproofs.Youwillalsolearnhowtounderstandsomeoneelse’sproof,findaflawinaproof,fixadeficientproofifpossibleanddiscarditifnot.Inot
3、herwordsyouwilllearnthetradeofmathematicalexploration.Mathematicalreasoningtakestime.InCalculus,youexpectedtoreadaprob-lemandimmediatelyknowhowtoproceed.Hereyoumayexpectsomefrustrationandyoushouldplantospendalotoftimethinkingbeforeyouwritedownanything.Analysiswasnotdiscoveredovernight.Ittookcenturi
4、esforthecorrectapproachtoemerge.Youwillhavetogothroughanacceleratedprocessofrediscovery.Twentyhoursofworkaweekoutsideclassisnotanunusualaverageforthiscourse.Thecourseisruninthefollowingway.Inthesenotes,youwillfindDefinitions,Theorems,andExamples.Iwillexplainthedefinitions.Athome,onyourown,youwilltryto
5、provethetheoremsandthestatementsintheexamples.Youwillusenobooksandnohelpfromanyone.Itwillbejustyou,thepencilandthepaper.Everystatementyoumakemustbejustified.Inyourarguments,youmayuseanyresultwhichprecedesinthenotestheitemyouareproving.Youmayusetheseresultseveniftheyhavenotyetbeenproven.However,youma
6、ynotuseresultsfromothersources.Then,inclass,Iwillcallforvolunteerstopresenttheirsolutionsattheboard.Everycorrectproofisworthonepoint.Ifmorethanonepersonvolunteerforanitem,theonewiththefewestpointsiscalledtotheboard,tiestobebrokenbylot.Yourgradewillbedeterminedbythenumberofpointsyouhaveaccumulateddu
7、ringtheterm.Youhavetounderstandtheproofspresentedbyothers.Someoftheideasmaybeusefultoyoulater.Youmustquestionyourpeerswhenyouthinkafaultyargumentisgivenorsomethingisnotentirelycleartoyou.Ifyoudon’t,Imostpro
