微分中值定理与不等式的证明-英文文献

1、重庆三峡学院毕业设计外文文献翻译毕业设计题目微分中值定理与不等式的证明翻译题目牛顿与莱布尼兹创立微积分之解析学院数学与统计学院专业信息与计算科学姓名张冬班级2011级信本班学号201106014111指导教师张春涛Social science album - college journals of philosophy and social sciences社会科学辑-高等专科学校学报哲学与社会科学版NewtonandLeibnizcreatedcalculusresolutionAbstract:Thispapermainlydiscussesthebackgroundofthetimeso

2、fNewtonandLeibniz,andtheirphilosophyofitsfounderiswidelyusedinvariousfieldsofnaturalsciencebasicmathematicaltools-calculus.Keywords:Newton;Leibniz;calculus;philosophicalthoughtToday,calculushasbecomethebasicmathematicaltoolsarewidelyusedinvariousfieldsofnaturalsciences.Engelssaid:inalltheoreticalach

3、ievements,notliketheinventionofthefirsthalfofthe17thcenturycalculusasbeseenasthehighestvictoryforthehumanspirit,somewhere,weseethehumanspiritpureandonlymerit,itisinhere.1(p.244)ThispapertriesfromNewton,Leibnizfoundedthehistoricalbackgroundandphilosophyisseenasthehighestvictoryofthehumanspiritcalculu

4、sitsExpandprofilingFirstly,NewtonsbackgroundofthetimesandphilosophyNewton(IsaacNewton,bornin1642-1727)wasbornin1642inEngland.,in1661,hegotintotheUniversityofCambridge,in1665,Londonwaspopularinplague,Newtonreturnedtothecountryside,alldayhewasthinkingaboutavarietyofissues,usedhiswisdomandseveralyearst

5、heknowledge,inventedthenumberofflowcytometry(calculus),theanalysisofthegravityandlight.2(p.155)Onmay20,1665,Newtonsmanuscripthadtherecordof“thenumberofstreamssurgery”.,otherpapersinthenumberofstreams(differential)andintegration,andsolutionflowequationwithintegraltable.In1669,Newtoncirculatedentitled

6、theuseofaninfinitenumberofequationsanalyticsbookletinhisfriends,here,Newtonwasnotonlyseekingavariableuniversalmethodfortheinstantaneousrateofchangeinanothervariable,butalsoprovedthattheareacanbeobtainedbyseekingtheinverseprocessoftherateofchange.Becausetheareaisalsousedtorepresentthethusobtained.New

7、tonprovedthiscanbeobtainedbytheinverseprocessofseekingtherateofchange(moreprecisely,andthelimitcanbeobtainedbytheanti-differential),thefactisthatwearenowtalkingaboutfundamentaltheoremofcalculus.where,Newtonusedtheinfinitesimalmethod,calledinstantaneousinfinitelysmallincrementofthevariable,instantane

8、ousinfinitesimal,isnotcomponentorinfinitesimal,Newtonobtainedabandontheinstantaneousrateofchange.3(p.199)In1671Newtonmadehisresultsonthestudyofcalculusorganizedintoflowmethodandtheinfiniteseries(1736),where,hebelievedthatthevariablewascontinuousmotiongeneratedvariablecalledflowtherateofchangeofavari

9、ablewascalledthenumberofstreams.Newtonstatedmoreclearlythebasicproblemsofthecalculus:therelationshipbetweenthetwostreamswereknown,foundtherelationshipbetweenthenumberoftheirflow,aswellasitsinverseproblem.Thenumberofstreamslawandinfinitedserieswasamorecompletecalculusbook.Thesecondhalfofthebookpassed

10、by20problemsintroducedanumberofstreamsinfiniteseriesapplicationofthelaw.In1676,Newtonwroteseekingcurvededge-shapedarea(1704),wherethethinkingofNewtonscalculussignificantchanged,hegaveupamicroorinfinitesimal,whileusedtheinitialandThelastratiomethod.In1687Newtonpublisheditslandmarkscientificbooksmathe

11、maticalprinciplesofnaturalphilosophy,thenumberofflowcytometry(Calculus)wasoneofthethreemajorfindings.AsEinsteinsaid:Newtonroadyouhavefoundinyoureraistheonewiththehighestofthinkingandcreativeskillsofpeoplewhofoundtheonlywayyoucreatedtheconcepteventodaystillguidingourthinkinginphysics.4(p.192)Newtonli

12、vedinanageBritainwaschangingtimes,whentheBritishbrakethecivilwar,thebourgeoisieandthearistocracyclasscompromised,madethebourgeoisrevolutioninBritainnothavethoroughness.TheBritishbourgeoissuperstructuretodoeternalargumentforalltheexistenceoftheexploitingclasses,thereforetheabsoluteideologicalbecometh

13、edominantideologyofthedominant,italsoaffectedthenaturalscientistsabsolutemetaphysicalwayofthinking.NewtonsideashavealsobeenabourgeoisrevolutioninBritaindidnotthoroughlyThusNewtonoftencannotfindtheinitialthecausefromthenatureorthethingsofitself,andbymeansoftheexternaldrivingforce.Beforetheageof30,New

14、tondiscoveredcalculus,andestablishedClassicalMechanics.However,hespendedalmostnothingonthestudyofthenaturalsciences.Thiswasincapitalismandtheformationperiod,thebourgeoisiehadtotheologyattacktohelpliberatesciencefromtheology.But,Whenthebourgeoisentrenched,thegradualintensificationoftheclassstruggle,t

15、hebourgeoisiegraduallydeclined,theyseizedawidevarietyofreligiousbeliefsasslaveryideologicalweaponofthepeople.Newtoninfluencedbyitsformerlifeduetotheideologicaltendencyofspontaneousmaterialism,hewasagreatachievement,thelatterhalfwascompletelyaddictedtothestudyoftheology.NewtoninheritedthetraditionofB

16、aconsempiricism,specialattentiontotheroleofexperimentationandinductivereasoning,heassertedthenaturalsciencecanonlyexplaintheworldstartingfromtheempiricalfacts.Thiswasforadvocatingtalkedagainstscholasticism,misusethedivinetodistortthenatureplayedapositiverole.However,Newtonwasrigidlyadheretotheempiri

17、calfacts,one-sidedemphasisontheimportanceofinduction.Emotionalmaterial,allstuckinthephenomenonofthings,relyingsolelyontheinductivemethodwastohavenosystemofuniversalrationalknowledgeto.ontheissueofanalysisandsynthesis,deductiveandinductivemetaphysicsNewtoncaughtinacontradiction.5(p.123)Secondly,Leibn

18、izsbackgroundofthetimesandphilosophyLeibniz(GottfriedWilhelmLeibniz,1646-1716)wasborninGermany.,1672inParis,whereexposuredtotheHuygenssomemathematicalcelebrities,toleadintothefieldofmathematics,thebeginningofthecalculuscreativework2(p.165)In1684,Leibnizpublishedthefirstofficialhistoryofmathematicsca

19、lculusliterature,ademandlimitandtangentmethod.Thisliteratureofsummarizedresultsofthestudyofcalculussince1673,whichdefinedthedifferential,extensivelyusedthedifferentialsymboldx,dy,andalsogivedthesum,commossion,accumulated,andquotientthepoweroftherulesofdifferentiation.Includingtheapplicationofthediff

20、erentialmethodinthetangentline,greatminimaandinflection.Twoyearslater,hepublishedaacalculusthesisesotericgeometryandnotVariablesandtheirinfiniteanalysis,whichforthefirsttimeusingtheintegralsymbol preliminarydiscussesthepoints(orquadrature)problemwithdifferentialquadraturetangentreciprocalproblem.Wel

21、l-knownNewton-LeibnizformulaBAf(x)dx=f(b)-f(a),gaveusanoutlineofthebasicshapeofthecalculusandtheblueprintforthedevelopment.Newtonscalculuswasthedeparturefromthepointofviewofkinematics,LeibnizfromgeometryviewedtoconsiderthecreationofcalculussymbolwasfarbetterthanNewtonssymbol,andeffectivelypromotedth

22、edevelopmentofthecalculusof6(p.120)Newtondiscoveredcalculus(1665-1666)atleastnineyearsearlierthanLeibniz,Leibniz,however,publisheditscalculusarticlethreeyearsearlierthanNewton.AccordingtotheevidencebeprovidedbytheLeibnizhimselfthatheformedin1674,theideasandmethodsofthedifferential.IftheNewtoncalculu

23、swasthedeparturefromthepointofviewofkinematics,Leibnizwasfromthepointofviewofphilosophyandgeometrytoconsider,especiallycloselyrelatedtothedifferentialtriangleandBarrow,LeibnizMartinezcalleditacharacteristictriangle.BarrowsdifferentialtriangleLeibnizhasinspiredthedifferentialtriangle,madehimawareofth

24、etangentlineandquadratureproblemisapairofmutuallyinverseproblem.Leibnizfirstexpresstheinverserelationshipbetweenthedifferentialandintegral.Leibnizsmanyresearchanddevelopmentofideaswareincludedinhundredsofpagesofnoteswrittensince1673,butunpublished.In1673,hesawtheimportanceofthetangenttothecurveofthe

25、positiveandinverseproblemshedoesbelievethattheanti-methodisequivalenttotheareaandvolumebysummingtherequirements.In1684,Leibnizpublishedhisfirstdifferentialcalculuspapersaseekingofmaximum,minimum,andanewmethodoftangent,italsoappliedtofractionalorunreasonableamount,andthisnewapproachwonderfultypeofcal

26、culationmadeapreliminaryfinishinghispreviousstudies,describedthebasicprinciplesofthedifferentialcalculus,infinitelysmalltotheinfinitelysmallincrementofthefunctionwastheindependentvariableOftheresults,andtheincrementofthisfunctioniscalledadifferential,indicatedbytheletterd.1675-1676,fromseekingcurved

27、edgesshapedareaofthedeparturereceivecreditfortheconceptofthetogivecalculustheoremBAF(x)dx=f(b)-f(a).The1686theLeibnizcalculusthesispotentialgeometryandanalysiscannotbeseparatedandunlimited.In1693,hegaveaproofofthetheorem.ThesewerepublishedintheTeacherof.Unifyingthedifferentialandintegralcalculustheo

28、rycanbeestablishedwasanimportantsign.BeforetheendoftheThirtyYearsWar,LeibnizwasbornintheconfrontationbetweentheGermanLutheranprinceswiththeCatholicprinces.Inordertochangethesituationofreligiousstrife,LeibnizdeterminedtofoundanewCatholicandLutherancanfitonthedoctrineoftheentity,tobecomethethetwofacti

29、onsChurchcombinedphilosophicalfoundation.Leibnizsintentionwasimpossibletoachieve,buthelaterwasthereforeproposedtheCartesiantheentitydoctrine-monadology.MonadologywasthecoreofthephilosophyofLeibniz.LeibnizEverythingwasmadeofthelistofthisspiritentity,listneitherasubstancewhichhadacertainquality,itwasa

30、spiritualofLeibnizputitthaninthesoulonlythespiritofthelistwastherealexistenceoftheentity,areinseparablefromthelist,thatwasnotthepointofthepartofthepureentity,Leibnizdeducedaseriesoffeatures:notthepart,itcannotbeinanaturalwaybyacombinationofvariouspartsoftheproduction,orbythedecompositionofthevarious

31、partsoftheeliminationofitsbirthanddeathonlyforthesuddencreationofGodordestruction;notthepart,youcannotimagineanythingcouldgoinsidetocausechanges,listbecameindependentorsomethingcompletelyisolated,amongthelistcannothaveanyrealinteractionorinfluencelistRoomnodifferenceintheamount,butonlyqualitativelyd

32、ifferent.7(p.85)Inshort,thebasicconceptofLeibnizidealism,metaphysics.TheorderoftheuniversewareattributedtoGodspre-determined.Hecertainlymanyinevitabletruthdidntcomefromexperience,hebelievedthatnotonlyknowtheobjectsareposedbythespiritofthelist.Themainbodyofunderstandingcanonlymindasaspiritualentityli

33、st.HisallthedevelopmentandchangeswareattributedtoGodsfront,itactuallynegatedtherealdevelopment,whichwasthenegativesideofhispointofview.Butontheotherhand,Leibnizsphilosophyalsohadpositiveaspects,wasrichinitsphilosophicaldialecticthinking,hecertainlyentitiditselfontheforce,whichwasdynamic,itwasvirtual

34、lycertainideaofmatterandmotioncannotbeseparated,histryingtosolvecannotbeseparatedpointsandcontinuitycontradictionsandproblems,cameintocontactwiththeindividualandthewholeproblemofdiscontinuityandcontinuityoftheunityofopposites,thedialecticofreasonandexperiencetopromotecombinedtomakeacertaincontributi

35、on.Third,Newton,LeibnizfoundedcalculusscomparisonNewtonandLeibniz,withtheirdifferentways,thecreationofcalculus.IfNewtonfinallyconclusedearlierthanLeibniz,LeibnizownconclusionsearlierthanNewton.AlthoughNewtonscalculusapplicationsfarmorethantheworkofLeibniz,stimulateanddecidedalmosttheentire18thcentur

36、yanalysisofdirection,butLeibnizsucceededinestablishingamoreconvenientsystemofsymbolsandcalculationmethods.Twoofthefoundersofthecalculus,aBritishdiscreet,rigorousscholarshipdemeanor,aGermanphilosophyspeculativementality,warmandbold.Duetoerroryinandyangofthebackground,excessivepursuitofrigorousNewtonh

37、adnotyetbeenpublishedhisfindings,Leibnizrobbedaposttoholdtheevent.NewtonandLeibnizsphilosophicaldifferencesofopinionledtothedifferentmethodstheycreatedcalculus.Newtoninsistedmaterialismempiricismwithspecialemphasisonexperimentationandinductivereasoning.Inthestudyofclassicalmechanicslawandthelawofgra

38、vity,heencounteredsomecannotsolvemathproblems,thesemathematicalproblemscannotbesolvedwithEuclideangeometryandalgebrainthe16thcentury,Newtonstartedstudyingnewordercurvature,area,lengthofthecurve,thecenterofgravity,theminimumandmaximumvaluesandotherproblems-flowcountingmethod.Newtonsstudyusedtheinitia

39、lratioandthefinalthanhethoughtthenumberofstreamswasthenewborntheamountoftheinitialratioordisappearancethanthenewborntheamountofinitialratiowastheratioofthenewborninstant,disappearedlastistheamountoftheratioofthedisappearedthemoment.4(p.180)thisinterpretationistoovague,andnotaprecisemathematicalconce

40、pts,butanintuitivedescription.Initialratioandthefinalratioofthephysicalprototypewasamathematicalabstractionoftheinitialvelocityandthefinalvelocity,ineachoftheinstantaneousspeedoftheprocessofmovingtheobjectforpositionisself-evident,Newtonwastheinitialratioandlastdeparturefromtheobjectivefactsthanthei

41、ntuitiveconcept.Sohegavethepointofviewofthelimit.Leibnizscalculuscreationbeginedwithtangentandquadrature,herecognizedthatfromthedifferentialtriangle:Foundthetangentofthecurvedependedontheordinatetheratioofthedifferenceandthedifferencebetweentheabscissa;seekingcurvededgegraphicsareawasdependentonthet

42、heabscissaunlimitedinter-cellandverticalcoordinatesofinfinitelythinrectangle.LeibnizrecognizeSum,differenceoperationisreversible.TheLeibnizinfinitesimalideologicalgavethefundamentaltheoremofcalculusanddevelopintoahigherorderdifferential.TheLeibnizinfinitesimalphases,whichstemsfromhisphilosophymonado

43、logythinking.Leibniz:differentlistitsperceptioninmonadology.Theclaritywasnotthesamefromoneperceptiontoanotherperceptionoftransitionandchange,thedevelopmentwasconstitutedbythelistofthings,fromjuniortoseniorlevelsequence.6(p.91)canbesaid,theLeibnizinfinitesimalphasespreciselycorrespondinglistofthediff

44、erentlevelsofsystemanditsobjectiveidealistphilosophicalsystem.Leibnizmicro-integrationoftheresearchprocess,theprincipleofcontinuitybecomethecornerstoneofitswork,theprincipleofcontinuityisrootedinhisphilosophyunlimitednatureofthinking.NewtonandLeibnizcreatedcalculusthesamepoint:thecreationofanewmathe

45、maticaldisciplinesfromdifferentangles,thecalculushasawiderangeofusesandcanbeappliedtogeneralfunction;algebraicgeometricformsfromthepastfreed;studytheinverserelationshipbetweenthedifferentialandanti-differential.NewtonandLeibnizcreatedatdifferentpointsoftheCalculus:NewtoninheritedBaconsempiricism,asp

46、ecialfavorsummarized.NewtonscalculuswithsignificantlyfromthephysicalmodelfromMechanicsTuotaioftraces,tothemathematicalmodelofthemechanicalmovement,oneofthebasicconcepts,suchastheamountofnascentdisappearance,instantaneous,theinitialratioandthefinalratioofconceptsfromthemechanicalmovementisthemathemat

47、icalabstractionoftheinstantaneousstateofthemechanicalmovement.Hefoundedcalculusisdesignedtosolvespecificproblems,theemphasisistopromoteconcreteresults.Leibnizemphasizesgeneralmethodsandalgorithmscanbeappliedtospecificproblems,inordertoharmonizethevariousissues.Leibnizspentalotoftimeonthechoiceofsymb

48、ols,inventedarichsuggestivesymbolsystem.Hestretchedthesum(and)thefirstletterSexpressedintegraldifferentialofxdx,thissetwasstraightforwardandeasytousesymbolsstillinuse.Newtonscalculuswasanaturalextensionofthepuregeometryconcernwastheapplicationofcalculusinphysics.Experience,specificandcautionarethech

49、aracteristicsofhiswork,thepracticeofthisbinding,sothathedidnotenjoyplaying.Leibnizconcerninthebroadestsenseofthecalculus,andstrivetocreatetheperfectsystemtoestablishthecalculus.Heimaginative,likepromotion,boldandspeculative,sodidnothesitatetoannouncethebirthofanewdiscipline.NewtonandLeibnizwarescientificgiantsoftheirage.Calculushadbeenabletobecomeanindependentdisciplineandarevolutionaryimpactontheentirenaturalsciences,relymainlyontheworkofNewtonandLeibniz.AscanbeseenfromtheprocessofcreationofthecalculusofNewtonandLeibniz:Whenthegiantphilosophicalcontemplationintoscientificconclusions,thedevel