foundations of infinitesimal calculus 2nd ed. - k. stroyan

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1、MathematicalBackground:FoundationsofInnitesimalCalculussecondeditionbyK.D.Stroyanydydyedxxdxdxy=f(x)Figure0.1:AMicroscopicViewoftheTangentCopyrightc1997byAcademicPress,Inc.-Allrightsreserved.TypesetwithAMS-TEXiPrefacetotheMathematicalBackgroundWewantyoutoreasonwithmathematics.Wearenottryingto

2、geteveryonetogiveformalizedproofsinthesenseofcontemporarymathematics;`proof'inthiscoursemeans`convincingargument.'Weexpectyoutousecorrectreasoningandtogivecarefulexpla-nations.Theprojectsbringouttheseissuesinthewaywendbestformoststudents,butthepuremathematicalquestionsalsointerestsomestudents

3、.Thisbookofmathemat-icalbackground"showshowtollinthemathematicaldetailsofthemaintopicsfromthecourse.Theseproofsarecompletelyrigorousinthesenseofmodernmathematics{technicallybulletproof.Wewrotethisbookoffoundationsinparttoprovideaconvenientreferenceforastudentwhomightliketoseethetheorem-proo

4、f"approachtocalculus.Wealsowroteitfortheinterestedinstructor.Inre-thinkingthepresentationofbeginningcalculus,wefoundthatasimplerbasisforthetheorywasbothpossibleanddesirable.Thepointwiseapproachmostbooksgivetothetheoryofderivativesspoilsthesubject.ClearsimpleargumentsliketheproofoftheFundamenta

5、lTheorematthestartofChapter5belowarenotpossibleinthatapproach.Theresultofthepointwiseapproachisthatinstructorsfeeltheyhavetoeitherbedishonestwithstudentsordisclaimgoodintuitiveapproximations.Thisissadbecauseitmakesaclearsubjectseemobscure.Itisalsounnecessary{byandlarge,theintuitiveideasworkpro

6、videdyournotionofderivativeisstrongenough.Thisbookshowshowtobridgethegapbetweenintuitionandtechnicalrigor.Afunctionwithapositivederivativeoughttobeincreasing.Afterall,theslopeispositiveandthegraphissupposedtolooklikeanincreasingstraightline.HowcouldthefunctionNOTbeincreasing?Pointwisederivativ

7、esmakethisbizarrethingpossible-apositivederivative"ofanon-increasingfunction.Ourconclusionissimple.ThatdenitionisWRONGinthesensethatitdoesNOTsupporttheintendedidea.Youmightagreethatthecounterintuitiveconsequencesofpointwisederivatives