Differential Calculus 微分学 数学论文外文翻译

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1、DifferentialCalculus微分学HistoricalIntroduction历史介绍/D/x*p2R6r$]$NewtonandLeibniz,quiteindependentlyofoneanother,werelargelyresponsiblefordevelopingtheideasofintegralcalculustothepointwherehithertoinsurmountableproblemscouldbesolvedbymoreorlessroutinemethods.Thesuccessfulaccomplishmentsofthese

2、menwereprimarilyduetothefactthattheywereabletofusetogethertheintegralcalculuswiththesecondmainbranchofcalculus,differentialcalculus.+V0E$f!k%w6l-{(Thecentralideaofdifferentialcalculusisthenotionofderivative.Liketheintegral,thederivativeoriginatedfromaproblemingeometry—theproblemoffindingthe

3、tangentlineatapointofacurve.Unliletheintegral,however,thederivativeevolvedverylateinthehistoryofmathematics.Theconceptwasnotformulateduntilearlyinthe17thcenturywhentheFrenchmathematicianPierredeFermat,attemptedtodeterminethemaximaandminimaofcertainspecialfunctions.Fermat’sidea,basicallyvery

4、simple,canbeunderstoodifwerefertoacurveandassumethatateachofitspointsthiscurvehasadefinitedirectionthatcanbedescribedbyatangentline.Fermatnoticedthatatcertainpointswherethecurvehasamaximumorminimum,thetangentlinemustbehorizontal.Thustheproblemoflocatingsuchextremevaluesisseentodependontheso

5、lutionofanotherproblem,thatoflocatingthehorizontaltangents.Thisraisesthemoregeneralquestionofdeterminingthedirectionofthetangentlineatanarbitrarypointofthecurve.ItwastheattempttosolvethisgeneralproblemthatledFermattodiscoversomeoftherudimentaryideasunderlyingthenotionofderivative.Atfirstsig

6、htthereseemstobenoconnectionwhateverbetweentheproblemoffindingtheareaofaregionlyingunderacurveandtheproblemoffindingthetangentlineatapointofacurve.Thefirstpersontorealizethatthesetwoseeminglyremoteideasare,infact,ratherintimatelyrelatedappearstohavebeenNewton’steacher,IsaacBarrow(1630-1677)

7、.However,NewtonandLeibnizwerethefirsttounderstandtherealimportanceofthisrelationandtheyexploitedittothefullest,thusinauguratinganunprecedentederainthedevelopmentofmathematics.Althoughthederivativewasoriginallyformulatedtostudytheproblemoftangents,itwasso