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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,thatisthetheoryofdifferentiatio
2、nandintegrationoffunctions.However,youwillalsolearnsomethingmorefundamentalthanAnalysis.Youwilllearnwhatisamathematicalproof.Youmayhaveseensomeproofsearlier,buthere,youwilllearnhowtowriteyourownproofs.Youwillalsolearnhowtounderstandsomeoneelse’sproof,findaflawinaproof,fixadeficientproof
3、ifpossibleanddiscarditifnot.Inotherwordsyouwilllearnthetradeofmathematicalexploration.Mathematicalreasoningtakestime.InCalculus,youexpectedtoreadaprob-lemandimmediatelyknowhowtoproceed.Hereyoumayexpectsomefrustrationandyoushouldplantospendalotoftimethinkingbeforeyouwritedownanything
4、.Analysiswasnotdiscoveredovernight.Ittookcenturiesforthecorrectapproachtoemerge.Youwillhavetogothroughanacceleratedprocessofrediscovery.Twentyhoursofworkaweekoutsideclassisnotanunusualaverageforthiscourse.Thecourseisruninthefollowingway.Inthesenotes,youwillfindDefinitions,Theorems,and
5、Examples.Iwillexplainthedefinitions.Athome,onyourown,youwilltrytoprovethetheoremsandthestatementsintheexamples.Youwillusenobooksandnohelpfromanyone.Itwillbejustyou,thepencilandthepaper.Everystatementyoumakemustbejustified.Inyourarguments,youmayuseanyresultwhichprecedesinthenotestheite
6、myouareproving.Youmayusetheseresultseveniftheyhavenotyetbeenproven.However,youmaynotuseresultsfromothersources.Then,inclass,Iwillcallforvolunteerstopresenttheirsolutionsattheboard.Everycorrectproofisworthonepoint.Ifmorethanonepersonvolunteerforanitem,theonewiththefewestpointsiscalle
7、dtotheboard,tiestobebrokenbylot.Yourgradewillbedeterminedbythenumberofpointsyouhaveaccumulatedduringtheterm.Youhavetounderstandtheproofspresentedbyothers.Someoftheideasmaybeusefultoyoulater.Youmustquestionyourpeerswhenyouthinkafaultyargumentisgivenorsomethingisnotentirelycleartoyou.
8、Ifyoudon’t,Imostpro
