泰勒公式外文翻译Word下载.docx

泰勒公式外文翻译Word下载.docx

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Assumeformula

（1）istrueforsome.n,1,N

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

1,,，，，nnfx，h,fx，fh，，fxhx？

，，，，，，，，,,n!

,

,1n,1，，,,,,n,,,,,，，，，，，，，,fx，h,fx，fxh，，fxhh？

，，，，sup,,，，n,1!

0,,1,,

n,1n,,,,o,hhoh,,,,,,,,,,

WeshallnottakethetimeheretodiscussotherversionsofTaylor'

sformula,whicharesometimesquiteuseful.Theywerediscussedearlierindetailfornumericalfunctions.Atthispointweleaveittothereadertoderivethem（see,forexample,Problem1below）.2.MethodsofStudyingInteriorExtrema

UsingTaylor'

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

f:

U,RTheorem2.Letbeareal-valuedfunctiondefinedonanopensetUina

k,1,1normedspaceXandhavingcontinuousderivativesuptoorderinclusiveina

，，k，，fxx,Uneighborhoodofapointandaderivativeoforderkatthepointxitself.

，，，，,k,1k，，，，，，fx,0,？

fx,0fx,0Ifand,thenforxtobeanextremumofthefunctionfitis:

，，kk，，fxhnecessarythatkbeevenandthattheformbesemidefinite,

and

1

，，kkf，，xhsufficientthatthevaluesoftheformontheunitspherebeboundedawayh,1

fromzero;

moreover,xisalocalminimumiftheinequalities

，，kkf，，xh,,,0,

holdonthatsphere,andalocalmaximumif

Proof.FortheproofweconsidertheTaylorexpansion

（1）offinaneighborhoodofx.The

assumptionsenableustowrite

1k，，kkf，，，，x，h,fx,f，，，，xh，,hhk!

，，，，where,hisareal-valuedfunction,and,h,0as.h,0

Wefirstprovethenecessaryconditions.

，，k，，kkf，，x,0Since,thereexistsavectoronwhich.Thenforvaluesoftheh,0f，，xh,000

realparametertsufficientlyclosetozero,

1kk，，kf，，，，x，th,fx,f，，，，，，xth，,thth0000!

k

1,,k，，kkk，，，，,fxh，,thht,,000!

k,,

，，kkandtheexpressionintheouterparentheseshasthesamesignas.f，，xh0

Forxtobeanextremumitisnecessaryfortheleft-handside（andhencealsotheright-hand

side）ofthislastequalitytobeofconstantsignwhentchangessign.Butthisispossibleonlyifkiseven.

Thisreasoningshowsthatifxisanextremum,thenthesignofthedifference，，，，fx，th,fx0

，，kkisthesameasthatofforsufficientlysmallt;

henceinthatcasetherecannotbetwof，，xh0

，，k，，fxvectorshh,atwhichtheformassumesvalueswithoppositesigns.01

Wenowturntotheproofofthesufficiencyconditions.Fordefinitenessweconsiderthe

，，kk，，fxh,,,0h,1casewhenfor.Then

k1，，kk，，fx，h,f，，x,f，，，，xh，,hhk!

k,,,,1h,,k，，k,,，，，，,fx，,hh,,,,!

kh,,,,,,

1,,k，，,,，,hh,,!

，，,h,0h,0and,sinceas,thelastterminthisinequalityispositiveforallvectors

h,0sufficientlyclosetozero.Thus,forallsuchvectorsh,

2

，，，，fx，h,fx,0

thatis,xisastrictlocalminimum.

Thesufficientconditionforastrictlocalmaximumisverifiedsimiliarly.Remark1.IfthespaceXisfinite-dimensional,theunitspherewithcenterat,，，Sx;

1x,X

beingaclosedboundedsubsetofX,iscompact.Thenthecontinuousfunction

ii，，，，kkk1（ak-form）hasbothamaximalandaminimalvalueon，，.If，，，，Sx;

1fxh,,fxh,？

h？

ii1k

thesevaluesareofoppositesign,thenfdoesnothaveanextremumatx.Iftheyarebothof

thesamesign,then,aswasshowninTheorem2,thereisanextremum.Inthelattercase,asufficientconditionforanextremumcanobviouslybestatedastheequivalentrequirement

，，kkf，，xhthattheformbeeitherpositive-ornegative-definite.

Itwasthisformoftheconditionthatweencounteredinstudyingrealvaluedfunctions

non.R

nf:

R,RRemark2.Aswehaveseenintheexampleoffunctions,thesemi-definiteness

，，kkf，，xhoftheformexhibitedinthenecessaryconditionsforanextremumisnotasufficientcriterionforanextremum.

Remark3.Inpractice,whenstudyingextremaofdifferentiablefunctionsonenormallyusesonlythefirstorseconddifferentials.Iftheuniquenessandtypeofextremumareobviousfromthemeaningoftheproblembeingstudied,onecanrestrictattentiontothefirst

，，fx,0differentialwhenseekinganextremum,simplyfindingthepointxwhere

3.SomeExamples

，，，，131123123，，，，，，，，,,L,CR;

Rf,Ca,b;

Ru,u,u!

Lu,u,uExample1.Letand.Inotherwords,

3，，x!

fxisacontinuouslydifferentiablereal-valuedfunctiondefinedinRandasmooth

,a,b,Rreal-valuedfunctiondefinedontheclosedinterval.

Considerthefunction

，，1，，,,F:

Ca,b;

R,R

（2）

definedbytherelation

，，1，，，，,,f,Ca,b;

R!

Ff

b,，，，，，，,Lx,fx,fxdx,R（3）,a

，，1，，,,Ca,b;

RThus,

（2）isareal-valuedfunctionaldefinedonthesetoffunctions.

Thebasicvariationalprinciplesconnectedwithmotionareknowninphysicsandmechanics.Accordingtotheseprinciples,theactualmotionsaredistinguishedamongalltheconceivablemotionsinthattheyproceedalongtrajectoriesalongwhichcertainfunctionalshaveanextremum.Questionsconnectedwiththeextremaoffunctionalsarecentralinoptimal

3

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

Weshallshowthatthefunctional（3）isadifferentiatemappingandfinditsdifferential.Weremarkthatthefunction（3）canberegardedasthecompositionofthemappings

（4），，F，，，，，，，，fx,Lx,fx,fx1

definedbytheformula

，，1（5）,,,,F:

C，，a,b;

R,C，，a,b;

R1

followedbythemapping

b,,g,C，，，，，，a,b;

Fg,gxdx,R（6）2,a

Bypropertiesoftheintegral,themappingisobviouslylinearandcontinuous,sothatF2

itsdifferentiabilityisclear.

Weshallshowthatthemappingisalsodifferentiable,andthatF1

,,,（7），，，，F，，，，，，，，fhx,,Lx,fx,fxh，，，，，，x，,Lx,fx.fxh，，x231

，，1,,h,C，，a,b;

Rfor.

Indeed,bythecorollarytothemean-valuetheorem,wecanwriteinthepresentcase

3112233123123i，，，，，，Lu，,,u，,,u，,,Lu,u,u,,Lu,u,u,i,,1i

，，，，，，，，，，，，,sup,Lu，,,,,Lu,Lu，,,,,Lu,Lu，,,,,Lu,,11223311,,,01

i，，，，,3max,Lu，,u,,Lu,max,（8）ii0,,,1,1,2,3ii,1,2,3

123123，，，，u,u,u,u,,,,,,,whereand.

,，，1,，，,,Ca,b;

Rfmaxf,fIfwenowrecallthatthenormofthefunctionfinis，，1,,ccc,,

,a,bf（whereisthemaximumabsolutevalueofthefunctionontheclosedinterval）,then,c

23,23,11，，，，，，，，u,fx,,hx,,hxu,fxu,x,,0setting,,,,,and,weobtainfrominequality（8）,

123takingaccountoftheuniformcontinuityofthefunctions，，,onbounded,Lu,u,u,i,1,2,3i

4

3subsetsof,thatR

,,,,,，，，，，，，，，，，，，，，，，，，，，，，，，，，，，，，，maxLx,fx，hx,fx，hx,Lx,fx,fx,,Lx,fx,fxhx,,Lx,fx,fxhx230,x,b

，，as,ohh,0，，1，，1cc

ButthismeansthatEq.（7）holds.

Bythechainrulefordifferentiatingacompositefunction,wenowconcludethatthefunctional（3）isindeeddifferentiable,and

b,,,,，，，，，，F，，，，，，fh,,Lx,fx,fxh，，，，，，x，,Lx,fx,fxh，，xdx（9）23,a

Weoftenconsidertherestrictionofthefunctional（3）totheaffinespaceconsistingofthe

，，1,,f,C，，a,b;

R，，，，functionsthatassumefixedvaluesfa,A,fb,Battheendpointsofthe

，，1,,closedintervala,b.Inthiscase,thefunctionshinthetangentspaceTC,musthavethef

,a,bvaluezeroattheendpointsoftheclosedinterval.Takingthisfactintoaccount,wemay

integratebypartsin（9）andbringitintotheform

bd,,,,,，，，，，，，，，，，，，，，，FfhLx,fx,fxLx,fx,fxhxdx（10）,,,,,,23,dxa,,

，，2CofcourseundertheassumptionthatLandfbelongtothecorrespondingclass.

Inparticular,iffisanextremum（extremal）ofsuchafunctional,thenbyTheorem2wehave

，，,1，，,,Ffh,0h,C，，a,b;

R，，，，ha,hb,0foreveryfunctionsuchthat.Fromthisandrelation（10）

onecaneasilyconclude（seeProblem3below）thatthefunctionfmustsatisfytheequation

d,,，，，，,Lx,f，，，，x,fx,,Lx,f，，，，x,fx,0（11）23dx

Thisisafrequently-encounteredformoftheequationknowninthecalculusofvariationsastheEuler-Lagrangeequation.

Letusnowconsidersomespecificexamples.

Example2.Theshortest-pathproblem

Amongallthecurvesinaplanejoiningtwofixedpoints,findthecurvethathasminimallength.

Theanswerinthiscaseisobvious,anditratherservesasacheckontheformalcomputationswewillbedoinglater.

WeshallassumethatafixedCartesiancoordinatesystemhasbeenchosenintheplane,in

，，0,0，，1,0whichthetwopointsare,forexample,and.Weconfineourselvestojustthe

，，1，，,,f,C0,1;

Rcurvesthatarethegraphsoffunctionsassumingthevaluezeroatbothendsof

,0,1theclosedinterval.Thelengthofsuchacurve

12,，，，，，，Ff,1，fxdx（12）,0

dependsonthefunctionfandisafunctionalofthetypeconsideredinExample1.Inthiscasethefunction