微积分文档格式.docx

微积分文档格式.docx

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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,René

Descartes,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