泰勒公式外文翻译Word下载.docx
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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
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