泰勒公式外文翻译.docx

泰勒公式外文翻译.docx

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泰勒公式外文翻译

Taylor'sFormulaandtheStudyofExtrema

1.Taylor'sFormulaforMappings

Theorem1.Ifamappingf:

u「.yfromaneighborhoodu=uxofapointxinanormedspaceXintoanormedspaceYhasderivativesuptoorderr1」nelusiveinUandhasann-thorderderivativefnxatthepointx,then

fxh=fx•f，Xh+fnxhn

（1）

ash—；o.

Equality

（1）isoneofthevarietiesofTaylor'sformula,writtenhereforrathergeneralclassesofmappings.

Proof.WeproveTaylor'sformulabyinduction.

Forn=1itistruebydefinitionoff'x.

Assumeformula

（1）istrueforsomen_i三n.

Thenbythemean-valuetheorem,formula（12）ofSect.10.5,andtheinductionhypothesis,weobtain.

fxh

as—o.

WeshallnottakethetimeheretodiscussotherversionsofTaylor'sformula,whicharesometimesquiteuseful.Theywerediscussedearlierindetailfornumericalfunctions.Atthispointweleaveittothereadertoderivethem（see,forexample,Problem1below）.

2.MethodsofStudyingInteriorExtrema

UsingTaylor'sformula,weshallexhibitnecessaryconditionsandalsosufficientconditionsforaninteriorlocalextremumofreal-valuedfunctionsdefinedonanopensubsetofanormedspace.Asweshallsee,theseconditionsareanalogoustothedifferentialconditionsalreadyknowntousforanextremumofareal-valuedfunctionofarealvariable.

Theorem2.Letf：

u—；rbeareal-valuedfunctiondefinedonanopensetUinanormedspaceXandhavingcontinuousderivativesuptoorderk—1_1inclusiveinaneighborhoodofapointxandaderivativefkxoforderkatthepointxitself.

Iff,x=o/-,fk4x=oandfkx=o,thenforxtobeanextremumofthefunctionfitis:

necessarythatkbeevenandthattheformfkxhkbesemidefinite,and

sufficientthatthevaluesoftheform

xhkontheunitsphereh=1beboundedaway

fromzero;moreover,xisalocalminimumiftheinequalities

fkxhk.0,

holdonthatsphere,andalocalmaximumif

fkXhk=：

：

：

0,

x.The

Proof.FortheproofweconsidertheTaylorexpansion⑴offinaneighborhoodofassumptionsenableustowrite

xh-fx詁fkxhkh

where:

hisareal-valuedfunction,and:

h「0ash>0.Wefirstprovethenecessaryconditions.

Sincefkx=0,thereexistsavectorh0=0onwhichfkxhS=0.Thenforvaluesofthe

realparametertsufficientlyclosetozero,

k

h：

亠很［th。

h0

f（x柚0J-f（x）=£■fCjx牠ktot（th0Jh0k

k!

andtheexpressionintheouterparentheseshasthesamesignasxh0.

Forxtobeanextremumitisnecessaryfortheleft-handside（andhencealsotheright-handside）ofthislastequalitytobeofconstantsignwhentchangessign.Butthisispossibleonlyifkiseven.

Thisreasoningshowsthatifxisanextremum,thenthesignofthediffereneefxth0x

isthesameasthatoffkxhfforsufficientlysmallt;henceinthatcasetherecannotbetwovectorsm,natwhichtheformfkxassumesvalueswithoppositesigns.

Wenowturntotheproofofthesufficiencyconditions.Fordefinitenessweconsiderthecasewhenfkxhk：

：

：

乙>0forh=1.Thenand,since:

-h^-0ash—0,thelastterminthisinequalityispositiveforallvectorsh=0sufficientlyclosetozero.Thus,forallsuchvectorsh,

f（x+h）-f（xfF（Jxhk+口小$

出E（h”

fXh_fX.0,

thatis,xisastrictlocalminimum.

Thesufficientconditionforastrictlocalmaximumisverifiedsimilarly.

Remark1.IfthespaceXisfinite-dimensional,theunitspheresx；1withcenteratx三x,beingaclosedboundedsubsetofX,iscompact.Thenthecontinuousfunctionfkxhk=二讥fxF—hik（ak-form）hasbothamaximalandaminimalvalueonsx；1.Ifthesevaluesareofoppositesign,thenfdoesnothaveanextremumatx.Iftheyarebothofthesamesign,then,aswasshowninTheorem2,thereisanextremum.Inthelattercase,asufficientconditionforanextremumcanobviouslybestatedastheequivalentrequirementthattheformfkxhkbeeitherpositive-ornegative-definite.

ItwasthisformoftheconditionthatweencounteredinstudyingrealvaluedfunctionsonRn.

Remark2.Aswehaveseenintheexampleoffunctions:

Rn—；r,thesemi-definitenessoftheformfkxhkexhibitedinthenecessaryconditionsforanextremumisnotasufficientcriterionforanextremum.

Remark3.Inpractice,whenstudyingextremaofdifferentiablefunctionsonenormallyusesonlythefirstorseconddifferentials.Iftheuniquenessandtypeofextremumareobviousfromthemeaningoftheproblembeingstudied,onecanrestrictattentiontothefirstdifferentialwhenseekinganextremum,simplyfindingthepointxwheref,x=0

3.SomeExamples

Example1.Letl二c1r3；randf^c1a,b；R.Inotherwords,J,u2,u3—lJ,u2,u3

isacontinuouslydifferentiablereal-valuedfunctiondefinedinr3andx—fxasmoothreal-valuedfunctiondefinedontheclosedintervalb,b「r.

Considerthefunction

F:

C1a,b；RLR

（2）

definedbytherelation

f•二C1a,b；R—Ff

b

Lx,fx,f,xd^R（3）

a

Thus,

（2）isareal-valuedfunctionaldefinedonthesetoffunctionsc1a,b；r.

Thebasicvariationalprinciplesconnectedwithmotionareknowninphysicsandmechanics.Accordingtotheseprinciples,theactualmotionsaredistinguishedamongalltheconceivablemotionsinthattheyproceedalongtrajectoriesalongwhichcertainfunctionalshaveanextremum.Questionsconnectedwiththeextremaoffunctionalsarecentralinoptimal

controltheory.Thus,findingandstudyingtheextremaoffunctionalsisaproblemofintrinsicimportanee,andthetheoryassociatedwithitisthesubjectofalargeareaofanalysis-thecalculusofvariations.Wehavealreadydoneafewthingstomakethetransitionfromtheanalysisoftheextremaofnumericalfunctionstotheproblemoffindingandstudyingextremaoffunctionalsseemnaturaltothereader.However,weshallnotgodeeplyintothespecialproblemsofvariationalcalculus,butratherusetheexampleofthefunctional（3）toillustrateonlythegeneralideasofdifferentiationandstudyoflocalextremaconsideredabove.

Weshallshowthatthefunctional（3）isadifferentiatemappingandfinditsdifferential.

Weremarkthatthefunction（3）canberegardedasthecompositionofthemappings

Fifx二Lx,fx,f,x

definedbytheformula

Fi：

c1a,b；R>ca,b；R

followedbythemapping

g：

二Ca,b；R—F2ggxdx：

二R

a

Bypropertiesoftheintegral,themappingf?

isobviouslylinearandcontinuous,sothatitsdifferentiabilityisclear.

WeshallshowthatthemappingF1isalsodifferentiable,andthat

Ff（fh（x）=©L（x,f（x,ffx为（x旷岳L（x,f（x）f[x）

forh.二C1a,b;；R.

Indeed,bythecorollarytothemean-valuetheorem,wecanwriteinthepresentcase

3

L“+sfii,u2+&,u3X1,u2,u3卜迟占L（J,u2,u3

im

-sup：

：

iLuTJiLu1；：

2LuT-「2Lu1；：

3LuT：

-pLu.■:

0-1

（8）

乞30貨；；：

iLu•F-；]Luinjax

i迄,2,311

where

Ifwenowrecallthatthenormfc1ofthefunctionfin

c1a,b；Ris

f

maxfc

u=u1,u2,u3and=卫,「2,「3.

（wherefcisthemaximumabsolutevalueofthefunctionontheclosedintervala,bl）,then,

settingu1=x,u2=fx,u3=f,x,1=o,•2=hx,and3x,weobtainfrominequality（8）,takingaccountoftheuniformcontinuityofthefunctionsjiLu1,u2,u3,^1,2,3,onbounded

subsetsofr,that

max

o<<

Lx,fxhx,f‘xh‘x_Lx,fx,f'x—^Lx,fx,f'xhx「BLx,fx,f'xh‘x*

=ohjashj

ButthismeansthatEq.（7）holds.

Bythechainrulefordifferentiatingacompositefunction,wenowconcludethatthefunctional（3）isindeeddifferentiable,and

b

F'fh二a±Lx,fX,f'Xhx：

：

3Lx,fX,f'Xh'Xdx（9）

Weoftenconsidertherestrictionofthefunctional（3）totheaffinespaceconsistingofthefunctionsf二c1a,b；Rthatassumefixedvaluesfa=a,fb=battheendpointsoftheclosedintervala,bIInthiscase,thefunctionshinthetangentspacerc：

musthavethevaluezeroattheendpointsoftheclosedintervala,blTakingthisfactintoaccount,wemay

integratebypartsin（9）andbringitintotheform

F,f

h=a-2Lx,fX,f,X-：

3Lx,fx,f,xhxdx

（10）

ofcourseundertheassumptionthatandfbelongtothecorrespondingclassc2.

Inparticular,iffisanextremum（extremal）ofsuchafunctional,thenbyTheorem2wehavef,fh=oforeveryfunctionhc1a,b；Rsuchthatha=hb=o.Fromthisandrelation（10）onecaneasilyconclude（seeProblem3below）thatthefunctionfmustsatisfytheequation

（11）

-2Lx,fx,f,x；：

3Lx,fx,f,x=0

Thisisafrequently-encounteredformoftheequationknowninthecalculusofvariationsastheEuler-Lagrangeequation.

Letusnowconsidersomespecificexamples.

Example2.Theshortest-pathproblem

Amongallthecurvesinaplanejoiningtwofixedpoints,findthecurvethathasminimallength.

Theanswerinthiscaseisobvious,anditratherservesasacheckontheformalcomputationswewillbedoinglater.

WeshallassumethatafixedCartesiancoordinatesystemhasbeenchosenintheplane,inwhichthetwopointsare,forexample,o,oand10.Weconfineourselvestojustthecurvesthatarethegraphsoffunctionsf■c10,1；rassumingthevaluezeroatbothendsoftheclosedinterval0,d.Thelengthofsuchacurve

fch（^+（f,2（xdx（12）

dependsonthefunctionfandisafunctionalofthetypeconsideredinExample1.InthiscasethefunctionLhastheform

andthereforethenecessarycondition（11）foranextremalherereducestotheequation

/X

2_0I—**

fromwhichitfollowsthat

一f‘x=常数（13）

\：

1+（f,2（x）

ontheclosedinterval0,11

Sincethefunction一uisnotconstantonanyinterval,Eq.（13）ispossibleonlyif

$14u2

f,x三constona,b].Thusasmoothext