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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,thatisthetheoryofdifferen
2、tiationandintegrationoffunctions.However,youwillalsolearnsomethingmorefundamentalthanAnalysis.Youwilllearnwhatisamathematicalproof.Youmayhaveseensomeproofsearlier,buthere,youwilllearnhowtowriteyourownproofs.Youwillalsolearnhowtounderstandsomeoneelse’sproof,findaflawinaproof,fixad
3、eficientproofifpossibleanddiscarditifnot.Inotherwordsyouwilllearnthetradeofmathematicalexploration.Mathematicalreasoningtakestime.InCalculus,youexpectedtoreadaprob-lemandimmediatelyknowhowtoproceed.Hereyoumayexpectsomefrustrationandyoushouldplantospendalotoftimethinkingbeforeyo
4、uwritedownanything.Analysiswasnotdiscoveredovernight.Ittookcenturiesforthecorrectapproachtoemerge.Youwillhavetogothroughanacceleratedprocessofrediscovery.Twentyhoursofworkaweekoutsideclassisnotanunusualaverageforthiscourse.Thecourseisruninthefollowingway.Inthesenotes,youwillfin
5、dDefinitions,Theorems,andExamples.Iwillexplainthedefinitions.Athome,onyourown,youwilltrytoprovethetheoremsandthestatementsintheexamples.Youwillusenobooksandnohelpfromanyone.Itwillbejustyou,thepencilandthepaper.Everystatementyoumakemustbejustified.Inyourarguments,youmayuseanyresul
6、twhichprecedesinthenotestheitemyouareproving.Youmayusetheseresultseveniftheyhavenotyetbeenproven.However,youmaynotuseresultsfromothersources.Then,inclass,Iwillcallforvolunteerstopresenttheirsolutionsattheboard.Everycorrectproofisworthonepoint.Ifmorethanonepersonvolunteerforani
7、tem,theonewiththefewestpointsiscalledtotheboard,tiestobebrokenbylot.Yourgradewillbedeterminedbythenumberofpointsyouhaveaccumulatedduringtheterm.Youhavetounderstandtheproofspresentedbyothers.Someoftheideasmaybeusefultoyoulater.Youmustquestionyourpeerswhenyouthinkafaultyargument
8、isgivenorsomethingisnotentirelycleartoyou.Ifyoudon’t,Imostpro
