高斯JohannCarlFriedrichGauss.docx
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高斯JohannCarlFriedrichGauss.docx
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高斯JohannCarlFriedrichGauss
JohannCarlFriedrichGauss
Born:
30April1777inBrunswick,DuchyofBrunswick(nowGermany)
Died:
23Feb1855inGöttingen,Hanover(nowGermany)
Attheageofseven,CarlFriedrichGaussstartedelementaryschool,andhispotentialwasnoticedalmostimmediately.Histeacher,Büttner,andhisassistant,MartinBartels,wereamazedwhenGausssummedtheintegersfrom1to100instantlybyspottingthatthesumwas50pairsofnumberseachpairsummingto101.
In1788GaussbeganhiseducationattheGymnasiumwiththehelpofBüttnerandBartels,wherehelearntHighGermanandLatin.AfterreceivingastipendfromtheDukeofBrunswick-Wolfenbüttel,GaussenteredBrunswickCollegiumCarolinumin1792.AttheacademyGaussindependentlydiscoveredBode'slaw,thebinomialtheoremandthearithmetic-geometricmean,aswellasthelawofquadraticreciprocityandtheprimenumbertheorem.
In1795GaussleftBrunswicktostudyatGöttingenUniversity.Gauss'steachertherewasKästner,whomGaussoftenridiculed.HisonlyknownfriendamongstthestudentswasFarkasBolyai.Theymetin1799andcorrespondedwitheachotherformanyyears.
GaussleftGöttingenin1798withoutadiploma,butbythistimehehadmadeoneofhismostimportantdiscoveries-theconstructionofaregular17-gonbyrulerandcompassesThiswasthemostmajoradvanceinthisfieldsincethetimeofGreekmathematicsandwaspublishedasSectionVIIofGauss'sfamouswork,DisquisitionesArithmeticae.
GaussreturnedtoBrunswickwherehereceivedadegreein1799.AftertheDukeofBrunswickhadagreedtocontinueGauss'sstipend,herequestedthatGausssubmitadoctoraldissertationtotheUniversityofHelmstedt.HealreadyknewPfaff,whowaschosentobehisadvisor.Gauss'sdissertationwasadiscussionofthefundamentaltheoremofalgebra.
Withhisstipendtosupporthim,Gaussdidnotneedtofindajobsodevotedhimselftoresearch.HepublishedthebookDisquisitionesArithmeticaeinthesummerof1801.Thereweresevensections,allbutthelastsection,referredtoabove,beingdevotedtonumbertheory.
InJune1801,Zach,anastronomerwhomGausshadcometoknowtwoorthreeyearspreviously,publishedtheorbitalpositionsofCeres,anew"smallplanet"whichwasdiscoveredbyGPiazzi,anItalianastronomeron1January,1801.Unfortunately,Piazzihadonlybeenabletoobserve9degreesofitsorbitbeforeitdisappearedbehindtheSun.Zachpublishedseveralpredictionsofitsposition,includingonebyGausswhichdifferedgreatlyfromtheothers.WhenCereswasrediscoveredbyZachon7December1801itwasalmostexactlywhereGausshadpredicted.Althoughhedidnotdisclosehismethodsatthetime,Gausshadusedhisleastsquaresapproximationmethod.
InJune1802GaussvisitedOlberswhohaddiscoveredPallasinMarchofthatyearandGaussinvestigateditsorbit.OlbersrequestedthatGaussbemadedirectoroftheproposednewobservatoryinGöttingen,butnoactionwastaken.GaussbegancorrespondingwithBessel,whomhedidnotmeetuntil1825,andwithSophieGermain.
GaussmarriedJohannaOstoffon9October,1805.Despitehavingahappypersonallifeforthefirsttime,hisbenefactor,theDukeofBrunswick,waskilledfightingforthePrussianarmy.In1807GaussleftBrunswicktotakeupthepositionofdirectoroftheGöttingenobservatory.
GaussarrivedinGöttingeninlate1807.In1808hisfatherdied,andayearlaterGauss'swifeJohannadiedaftergivingbirthtotheirsecondson,whowastodiesoonafterher.GausswasshatteredandwrotetoOlbersaskinghimtogivehimahomeforafewweeks,
togathernewstrengthinthearmsofyourfriendship-strengthforalifewhichisonlyvaluablebecauseitbelongstomythreesmallchildren.
Gausswasmarriedforasecondtimethenextyear,toMinnathebestfriendofJohanna,andalthoughtheyhadthreechildren,thismarriageseemedtobeoneofconvenienceforGauss.
Gauss'sworkneverseemedtosufferfromhispersonaltragedy.Hepublishedhissecondbook,TheoriamotuscorporumcoelestiuminsectionibusconicisSolemambientium,in1809,amajortwovolumetreatiseonthemotionofcelestialbodies.Inthefirstvolumehediscusseddifferentialequations,conicsectionsandellipticorbits,whileinthesecondvolume,themainpartofthework,heshowedhowtoestimateandthentorefinetheestimationofaplanet'sorbit.Gauss'scontributionstotheoreticalastronomystoppedafter1817,althoughhewentonmakingobservationsuntiltheageof70.
MuchofGauss'stimewasspentonanewobservatory,completedin1816,buthestillfoundthetimetoworkonothersubjects.HispublicationsduringthistimeincludeDisquisitionesgeneralescircaserieminfinitam,arigoroustreatmentofseriesandanintroductionofthehypergeometricfunction,Methodusnovaintegraliumvaloresperapproximationeminveniendi,apracticalessayonapproximateintegration,BestimmungderGenauigkeitderBeobachtungen,adiscussionofstatisticalestimators,andTheoriaattractioniscorporumsphaeroidicorumellipticorumhomogeneorummethodusnovatractata.Thelatterworkwasinspiredbygeodesicproblemsandwasprincipallyconcernedwithpotentialtheory.Infact,Gaussfoundhimselfmoreandmoreinterestedingeodesyinthe1820s.
Gausshadbeenaskedin1818tocarryoutageodesicsurveyofthestateofHanovertolinkupwiththeexistingDanishgrid.Gausswaspleasedtoacceptandtookpersonalchargeofthesurvey,makingmeasurementsduringthedayandreducingthematnight,usinghisextraordinarymentalcapacityforcalculations.HeregularlywrotetoSchumacher,OlbersandBessel,reportingonhisprogressanddiscussingproblems.
Becauseofthesurvey,GaussinventedtheheliotropewhichworkedbyreflectingtheSun'sraysusingadesignofmirrorsandasmalltelescope.However,inaccuratebaselineswereusedforthesurveyandanunsatisfactorynetworkoftriangles.Gaussoftenwonderedifhewouldhavebeenbetteradvisedtohavepursuedsomeotheroccupationbuthepublishedover70papersbetween1820and1830.
In1822GausswontheCopenhagenUniversityPrizewithTheoriaattractionis...togetherwiththeideaofmappingonesurfaceontoanothersothatthetwoaresimilarintheirsmallestparts.Thispaperwaspublishedin1825andledtothemuchlaterpublicationofUntersuchungenüberGegenständederHöherenGeodäsie(1843and1846).ThepaperTheoriacombinationisobservationumerroribusminimisobnoxiae(1823),withitssupplement(1828),wasdevotedtomathematicalstatistics,inparticulartotheleastsquaresmethod.
Fromtheearly1800sGausshadaninterestinthequestionofthepossibleexistenceofanon-Euclideangeometry.HediscussedthistopicatlengthwithFarkasBolyaiandinhiscorrespondencewithGerlingandSchumacher.Inabookreviewin1816hediscussedproofswhichdeducedtheaxiomofparallelsfromtheotherEuclideanaxioms,suggestingthathebelievedintheexistenceofnon-Euclideangeometry,althoughhewasrathervague.GaussconfidedinSchumacher,tellinghimthathebelievedhisreputationwouldsufferifheadmittedinpublicthathebelievedintheexistenceofsuchageometry.
In1831FarkasBolyaisenttoGausshissonJánosBolyai'sworkonthesubject.Gaussreplied
topraiseitwouldmeantopraisemyself.
Again,adecadelater,whenhewasinformedofLobachevsky'sworkonthesubject,hepraisedits"genuinelygeometric"character,whileinalettertoSchumacherin1846,statesthathe
hadthesameconvictionsfor54years
indicatingthathehadknownoftheexistenceofanon-Euclideangeometrysincehewas15yearsofage(thisseemsunlikely).
Gausshadamajorinterestindifferentialgeometry,andpublishedmanypapersonthesubject.Disquisitionesgeneralescircasuperficiescurva(1828)washismostrenownedworkinthisfield.Infact,thispaperrosefromhisgeodesicinterests,butitcontainedsuchgeometricalideasasGaussiancurvature.ThepaperalsoincludesGauss'sfamoustheoremaegregrium:
IfanareainE3canbedeveloped(i.e.mappedisometrically)intoanotherareaofE3,thevaluesoftheGaussiancurvaturesareidenticalincorrespondingpoints.
Theperiod1817-1832wasaparticularlydistressingtimeforGauss.Hetookinhissickmotherin1817,whostayeduntilherdeathin1839,whilehewasarguingwithhiswifeandherfamilyaboutwhethertheyshouldgotoBerlin.HehadbeenofferedapositionatBerlinUniversityandMinnaandherfamilywerekeentomovethere.Gauss,however,neverlikedchangeanddecidedtostayinGöttingen.In1831Gauss'ssecondwifediedafteralongillness.
In1831,WilhelmWeberarrivedinGöttingenasphysicsprofessorfillingTobiasMayer'schair.GausshadknownWebersince1828andsupportedhisappointment.Gausshadworkedonphysicsbefore1831,publishingÜbereinneuesallgemeinesGrundgesetzderMechanik,whichcontainedtheprincipleofleastconstraint,andPrincipiageneraliatheoriaefiguraefluidoruminstatuaequilibriiwhichdiscussedforcesofattraction.ThesepaperswerebasedonGauss'spotentialtheory,whichprovedofgreatimportanceinhisworkonphysics.Helatercametobelievehispotentialtheoryandhismethodofleastsquaresprovidedvitallinksbetweenscienceandnature.
In1832,GaussandWeberbeganinvestigatingthetheoryofterrestrialmagnetismafterAlexandervonHumboldtattemptedtoobtainGauss'sassistanceinmakingagridofmagneticobservationpointsaroundtheEarth.Gausswasexcitedbythisprospectandby1840hehadwrittenthreeimportantpapersonthesubject:
Intensitasvismagneticaeterrestrisadmensuramabsolutamrevocata(1832),AllgemeineTheoriedesErdmagnetismus(1839)and
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