微积分.docx
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微积分.docx
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微积分
Historyofcalculus
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Thisisasub-articletoCalculusandHistoryofmathematics.
Calculus,historicallyknownasinfinitesimalcalculus,isamathematicaldisciplinefocusedonlimits,functions,derivatives,integrals,andinfiniteseries.Ideasleadinguptothenotionsoffunction,derivative,andintegralweredevelopedthroughoutthe17thcentury,butthedecisivestepwasmadebyIsaacNewtonandGottfriedLeibniz.PublicationofNewton'smaintreatisestookmanyyears,whereasLeibnizpublishedfirst(Novamethodus,1684)andthewholesubjectwassubsequentlymarredbyaprioritydisputebetweenthetwoinventorsofcalculus.
Contents
[hide]
∙1AncientGreekprecursorsofthecalculus
∙2Medievaldevelopment
∙3Pioneersofmoderncalculus
∙4NewtonandLeibniz
o4.1Newton
o4.2Leibniz
o4.3Legacy
∙5Integrals
∙6Symbolicmethods
∙7Calculusofvariations
∙8Applications
∙9Non-Europeanantecedentsofthecalculus
o9.1Indianmathematics
o9.2Islamicmathematics
∙10Seealso
∙11Notes
∙12Furtherreading
∙13Externallinks
[edit]AncientGreekprecursorsofthecalculus
Greekmathematiciansarecreditedwithasignificantuseofinfinitesimals.Democritusisthefirstpersonrecordedtoconsiderseriouslythedivisionofobjectsintoaninfinitenumberofcross-sections,buthisinabilitytorationalizediscretecross-sectionswithacone'ssmoothslopepreventedhimfromacceptingtheidea.Atapproximatelythesametime,ZenoofEleadiscreditedinfinitesimalsfurtherbyhisarticulationoftheparadoxeswhichtheycreate.
AntiphonandlaterEudoxusaregenerallycreditedwithimplementingthemethodofexhaustion,whichmadeitpossibletocomputetheareaandvolumeofregionsandsolidsbybreakingthemupintoaninfinitenumberofrecognizableshapes.
ArchimedesofSyracusedevelopedthismethodfurther,whilealsoinventingheuristicmethodswhichresemblemoderndayconceptssomewhat.(SeeArchimedes'QuadratureoftheParabola,TheMethod,ArchimedesonSpheres&Cylinders.[1])ItwasnotuntilthetimeofNewtonthatthesemethodswereincorporatedintoageneralframeworkofintegralcalculus.Itshouldnotbethoughtthatinfinitesimalswereputonarigorousfootingduringthistime,however.OnlywhenitwassupplementedbyapropergeometricproofwouldGreekmathematiciansacceptapropositionastrue.
Archimedeswasthefirsttofindthetangenttoacurve,otherthanacircle,inamethodakintodifferentialcalculus.Whilestudyingthespiral,heseparatedapoint'smotionintotwocomponents,oneradialmotioncomponentandonecircularmotioncomponent,andthencontinuedtoaddthetwocomponentmotionstogethertherebyfindingthetangenttothecurve.[2]ThepioneersofthecalculussuchasIsaacBarrowandJohannBernoulliweredilligentstudentsofArchimedes,seeforinstanceC.S.Roero(1983).
[edit]Medievaldevelopment
ThemathematicalstudyofcontinuitywasrevivedinthefourteenthcenturybytheOxfordCalculatorsandFrenchcollaboratorssuchasNicoleOresme.Theyprovedthe"Mertonmeanspeedtheorem":
thatauniformlyacceleratedbodytravelsthesamedistanceasabodywithuniformspeedwhosespeedishalfthefinalvelocityoftheacceleratedbody.[3]
[edit]Pioneersofmoderncalculus
Inthe17thcentury,EuropeanmathematiciansIsaacBarrow,RenéDescartes,PierredeFermat,BlaisePascal,JohnWallisandothersdiscussedtheideaofaderivative.Inparticular,inMethodusaddisquirendammaximametminimaandinDetangentibuslinearumcurvarum,Fermatdevelopedanadequalitymethodfordeterminingmaxima,minima,andtangentstovariouscurvesthatwasequivalenttodifferentiation.[4]IsaacNewtonwouldlaterwritethathisownearlyideasaboutcalculuscamedirectlyfrom"Fermat'swayofdrawingtangents."[5]
Ontheintegralside,Cavalieridevelopedhismethodofindivisiblesinthe1630sand40s,providingamoremodernformoftheancientGreekmethodofexhaustion,[disputed–discuss]andcomputingCavalieri'squadratureformula,theareaunderthecurvesxnofhigherdegree,whichhadpreviouslyonlybeencomputedfortheparabola,byArchimedes.Torricelliextendedthisworktoothercurvessuchasthecycloid,andthentheformulawasgeneralizedtofractionalandnegativepowersbyWallisin1656.Ina1659treatise,Fermatiscreditedwithaningenioustrickforevaluatingtheintegralofanypowerfunctiondirectly.[6]Fermatalsoobtainedatechniqueforfindingthecentersofgravityofvariousplaneandsolidfigures,whichinfluencedfurtherworkinquadrature.JamesGregory,influencedbyFermat'scontributionsbothtotangencyandtoquadrature,wasthenabletoprovearestrictedversionofthesecondfundamentaltheoremofcalculusinthemid-17thcentury.[citationneeded]ThefirstfullproofofthefundamentaltheoremofcalculuswasgivenbyIsaacBarrow.[7]
NewtonandLeibniz,buildingonthiswork,independentlydevelopedthesurroundingtheoryofinfinitesimalcalculusinthelate17thcentury.Also,Leibnizdidagreatdealofworkwithdevelopingconsistentandusefulnotationandconcepts.Newtonprovidedsomeofthemostimportantapplicationstophysics,especiallyofintegralcalculus.
ThefirstproofofRolle'stheoremwasgivenbyMichelRollein1691usingmethodsdevelopedbytheDanishmathematicianJohannvanWaverenHudde.[8]ThemeanvaluetheoreminitsmodernformwasstatedbyBernardBolzanoandAugustin-LouisCauchy(1789–1857)alsoafterthefoundingofmoderncalculus.ImportantcontributionswerealsomadebyBarrow,Huygens,andmanyothers.
[edit]NewtonandLeibniz
IsaacNewton
GottfriedLeibniz
BeforeNewtonandLeibniz,theword“calculus”wasageneraltermusedtorefertoanybodyofmathematics,butinthefollowingyears,"calculus"becameapopulartermforafieldofmathematicsbasedupontheirinsights.[9]ThepurposeofthissectionistoexamineNewtonandLeibniz’sinvestigationsintothedevelopingfieldofinfinitesimalcalculus.Specificimportancewillbeputonthejustificationanddescriptivetermswhichtheyusedinanattempttounderstandcalculusastheythemselvesconceivedit.
Bythemiddleoftheseventeenthcentury,Europeanmathematicshadchangeditsprimaryrepositoryofknowledge.IncomparisontothelastcenturywhichmaintainedHellenisticmathematicsasthestartingpointforresearch,Newton,Leibnizandtheircontemporariesincreasinglylookedtowardstheworksofmoremodernthinkers.[10]Europehadbecomehometoaburgeoningmathematicalcommunityandwiththeadventofenhancedinstitutionalandorganizationalbasesanewleveloforganizationandacademicintegrationwasbeingachieved.Importantly,however,thecommunitylackedformalism;insteaditconsistedofadisorderedmassofvariousmethods,techniques,notations,theories,andparadoxes.
Newtoncametocalculusaspartofhisinvestigationsinphysicsandgeometry.Heviewedcalculusasthescientificdescriptionofthegenerationofmotionandmagnitudes.Incomparison,Leibnizfocusedonthetangentproblemandcametobelievethatcalculuswasametaphysicalexplanationofchange.Thesedifferencesinapproachshouldneitherbeoveremphasizednorunderappreciated.Importantly,thecoreoftheirinsightwastheformalizationoftheinversepropertiesbetweentheintegralandthedifferential.Thisinsighthadbeenanticipatedbytheirpredecessors,buttheywerethefirsttoconceivecalculusasasysteminwhichnewrhetoricanddescriptivetermswerecreated.[11]Theiruniquediscoverieslaynotonlyintheirimagination,butalsointheirabilitytosynthesizetheinsightsaroundthemintoauniversalalgorithmicprocess,therebyforminganewmathematicalsystem.
Seealso:
Leibniz–Newtoncalculuscontroversy
[edit]Newton
NewtoncompletednodefinitivepublicationformalizinghisFluxionalCalculus;rather,manyofhismathematicaldiscoveriesweretransmittedthroughcorrespondence,smallerpapersorasembeddedaspectsinhisotherdefinitivecompilations,suchasthePrincipiaandOpticks.NewtonwouldbeginhismathematicaltrainingasthechosenheirofIsaacBarrowinCambridge.Hisincredibleaptitudewasrecognizedearlyandhequicklylearnedthecurrenttheories.By1664Newtonhadmadehisfirstimportantcontributionbyadvancingthebinomialtheorem,whichhehadextendedtoincludefractionalandnegativeexponents.Newtonsucceededinexpandingtheapplicabilityofthebinomialtheorembyapplyingthealgebraoffinitequantitiesinananalysisofinfiniteseries.Heshowedawillingnesstoviewinfiniteseriesnotonlyasapproximatedevices,butalsoasalternativeformsofexpressingaterm.[12]
ManyofNewton’scriticalinsightsoccurredduringtheplagueyearsof1665-1666[13]whichhelaterdescribedas,“theprimeofmya
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