泰勒公式外文翻译Word格式.docx
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泰勒公式外文翻译Word格式.docx
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Theorem1.IfamappingfromaneighborhoodofapointxinanormedspaceXintoanormedspaceYhasderivativesuptoordern-1inclusiveinUandhasann-thorderderivativeatthepointx,then
(1)
as.
Equality
(1)isoneofthevarietiesofTaylor'
sformula,writtenhereforrathergeneralclassesofmappings.
Proof.WeproveTaylor'
sformulabyinduction.
Foritistruebydefinitionof.
Assumeformula
(1)istrueforsome.
Thenbythemean-valuetheorem,formula(12)ofSect.10.5,andtheinductionhypothesis,weobtain.
WeshallnottakethetimeheretodiscussotherversionsofTaylor'
sformula,whicharesometimesquiteuseful.Theywerediscussedearlierindetailfornumericalfunctions.Atthispointweleaveittothereadertoderivethem(see,forexample,Problem1below).
2.MethodsofStudyingInteriorExtrema
UsingTaylor'
sformula,weshallexhibitnecessaryconditionsandalsosufficientconditionsforaninteriorlocalextremumofreal-valuedfunctionsdefinedonanopensubsetofanormedspace.Asweshallsee,theseconditionsareanalogoustothedifferentialconditionsalreadyknowntousforanextremumofareal-valuedfunctionofarealvariable.
Theorem2.Letbeareal-valuedfunctiondefinedonanopensetUinanormedspaceXandhavingcontinuousderivativesuptoorderinclusiveinaneighborhoodofapointandaderivativeoforderkatthepointxitself.
Ifand,thenforxtobeanextremumofthefunctionfitis:
necessarythatkbeevenandthattheformbesemidefinite,
and
sufficientthatthevaluesoftheformontheunitspherebeboundedawayfromzero;
moreover,xisalocalminimumiftheinequalities
holdonthatsphere,andalocalmaximumif
Proof.FortheproofweconsidertheTaylorexpansion
(1)offinaneighborhoodofx.Theassumptionsenableustowrite
whereisareal-valuedfunction,andas.
Wefirstprovethenecessaryconditions.
Since,thereexistsavectoronwhich.Thenforvaluesoftherealparametertsufficientlyclosetozero,
andtheexpressionintheouterparentheseshasthesamesignas.
Forxtobeanextremumitisnecessaryfortheleft-handside(andhencealsotheright-handside)ofthislastequalitytobeofconstantsignwhentchangessign.Butthisispossibleonlyifkiseven.
Thisreasoningshowsthatifxisanextremum,thenthesignofthedifferenceisthesameasthatofforsufficientlysmallt;
henceinthatcasetherecannotbetwovectors,atwhichtheformassumesvalueswithoppositesigns.
Wenowturntotheproofofthesufficiencyconditions.Fordefinitenessweconsiderthecasewhenfor.Then
and,sinceas,thelastterminthisinequalityispositiveforallvectorssufficientlyclosetozero.Thus,forallsuchvectorsh,
thatis,xisastrictlocalminimum.
Thesufficientconditionforastrictlocalmaximumisverifiedsimiliarly.
Remark1.IfthespaceXisfinite-dimensional,theunitspherewithcenterat,beingaclosedboundedsubsetofX,iscompact.Thenthecontinuousfunction(ak-form)hasbothamaximalandaminimalvalueon.Ifthesevaluesareofoppositesign,thenfdoesnothaveanextremumatx.Iftheyarebothofthesamesign,then,aswasshowninTheorem2,thereisanextremum.Inthelattercase,asufficientconditionforanextremumcanobviouslybestatedastheequivalentrequirementthattheformbeeitherpositive-ornegative-definite.
Itwasthisformoftheconditionthatweencounteredinstudyingrealvaluedfunctionson.
Remark2.Aswehaveseenintheexampleoffunctions,thesemi-definitenessoftheformexhibitedinthenecessaryconditionsforanextremumisnotasufficientcriterionforanextremum.
Remark3.Inpractice,whenstudyingextremaofdifferentiablefunctionsonenormallyusesonlythefirstorseconddifferentials.Iftheuniquenessandtypeofextremumareobviousfromthemeaningoftheproblembeingstudied,onecanrestrictattentiontothefirstdifferentialwhenseekinganextremum,simplyfindingthepointxwhere
3.SomeExamples
Example1.Letand.Inotherwords,isacontinuouslydifferentiablereal-valuedfunctiondefinedinandasmoothreal-valuedfunctiondefinedontheclosedinterval.
Considerthefunction
(2)
definedbytherelation
(3)
Thus,
(2)isareal-valuedfunctionaldefinedonthesetoffunctions.
Thebasicvariationalprinciplesconnectedwithmotionareknowninphysicsandmechanics.Accordingtotheseprinciples,theactualmotionsaredistinguishedamongalltheconceivablemotionsinthattheyproceedalongtrajectoriesalongwhichcertainfunctionalshaveanextremum.Questionsconnectedwiththeextremaoffunctionalsarecentralinoptimalcontroltheory.Thus,findingandstudyingtheextremaoffuncti
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