微分中值定理與不等式的證明-英文文獻

1、重慶三峽學院畢業(yè)設計外文文獻翻譯畢業(yè)設計題目微分中值定理與不等式的證明翻譯題目牛頓與萊布尼茲創(chuàng)立微積分之解析學院數(shù)學與統(tǒng)計學院專業(yè)信息與計算科學姓名張冬班級2011級信本班學號201106014111指導教師張春濤Social science album - college journals of philosophy and social sciences社會科學輯-高等?？茖W校學報哲學與社會科學版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