泰勒公式外文翻译

Taylor'sFormulaandtheStudyofExtrema
1.Taylor'sFormulaforMappings
Theorem1. Ifamapping f:U Y fromaneighborhoodU Ux ofapointxina
normedspaceXintoanormedspaceYhasderivativesuptoordern1-inclusiveinUandhas
ann-thorderderivative
fnx
atthepointx,then
fx
h
fxf,xh
1fnxhn
oh
n
(1)
n!
ash 0.
Equality(1)isoneofthevarietiesofTaylor'sformula,writtenhereforrathergeneralclassofmappings.
Proof.WeproveTaylor'sformulabyinduction.
Forn 1 itistruebydefinitionof f,x.
Assumeformula(1)istrueforsomen 1 N.
Thenbythemean-valuetheorem,formula(12)ofSect.10.5,andtheinductionhypothesis,weobtain.
fxh
fx
f,xh
1
fnxhn
n!
supf,x
h
f,x
f,,x
h
1
nxhn1
h
f
0
1
n
1!
o
n1
ho
n
h
h
ash 0.
WeshallnottakethetimeheretodiscussotherversionsofTaylor'sformula,whicharesometimesquiteuful.Theywerediscusdearlierindetailfornumericalfunctions.Atthispointweleaveittothereadertoderivethem(e,forexample,Problem1below).
2.MethodsofStudyingInteriorExtrema
UsingTaylor'sformula,weshallexhibitnecessaryconditionsandalsosufficientconditionsforaninteriorlocalextremumofreal-valuedfunctionsdefinedonanopensubtofanormedspace.Asweshalle,theconditionsareanalogoustothedifferentialconditionsalreadyknowntousforanextremumofareal-valuedfunctionofarealvariable.
Theorem2.
Let
f:U
R
beareal-valuedfunctiondefinedonan
opentU
in
a
normedspaceXandhaving
continuousderivativesuptoorderk1
1inclusive
in
a
neighborhoodofapointx
U
andaderivative
fkxoforderkatthepointxitlf.
Iff,x0,,fk
1x
0
and
fkx0,thenforxtobeanextremumofthefunctionfitis:
fk
xhk
bemidefinite,
necessarythatkbeevenandthattheform
and
sufficientthatthevaluesoftheform fk xhk ontheunitsphere h 1 beboundedaway
fromzero;moreover,xisalocalminimumiftheinequalities
fkxhk
0,
holdonthatsphere,andalocalmaximumif
fkxhk
0,
Proof.FortheproofweconsidertheTaylorexpansion(1)offinaneighborhoodofx.Theassumptionnableustowrite
fx h fx
wherehisareal-valuedfunction,andWefirstprovethenecessaryconditions.

1
fk
xhk
hhk
k!
h
0
ash
0.
Sincefkx0,thereexistsavector
h0
0
onwhichf
k
xh0k
0.Thenforvaluesofthe
realparametertsufficientlyclotozero,
fxth
fx
1f
kxthk
ththk
0
k!
0
0
0
1fkxhk
thh
ktk
k!
0
0
0
k k
andtheexpressionintheouterparentheshasthesamesignfasxh0.
Forxtobeanextremumitisnecessaryfortheleft-handside(andhencealsotheright-handside)ofthislastequalitytobeofconstantsignwhentchangessign.Butthisispossibleonlyifkiven.
Thisreasoningshowsthatifxisanextremum,thenthesignofthedifferencefxth0 fx
isthesameasthatof
fk
xh0k
forsufficientlysmallt;henceinthatcatherecannotbetwo
vectorsh0,h1
atwhichtheform
fkx
assumesvalueswithoppositesigns.
Wenowturntotheproofofthesufficiencyconditions.Fordefinitenessweconsiderthe
cawhenfk
xhk
0
forh
1.Then
fxh
fx
1
fk
xhk
k
hh
k!
k
1fk
x
h
h
hk
k!
h
1
h
hk
k!
and,sinceh0ash0,h0sufficientlyclotozero.

thelastterminthisinequalityispositiveforallvectorsThus,forallsuchvectorsh,
f
xh
f
x
0,
thatis,xisastrictlocalminimum.
Thesufficientconditionforastrictlocalmaximumisverifiedsimiliarly.
Remark1.IfthespaceXisfinite-dimensional,theunitsphere
Sx;1
withcenteratx
X,
being
a
clodbounded
subtofX,iscompact.Then
the
continuous
function
f
kxhk
i1
ikfxhi1
hik
(ak-form)hasbothamaximalandaminimalvalueon
Sx;1
.If
thevaluesareofoppositesign,thenfdoesnothaveanextremumatx.Iftheyarebothofthesamesign,then,aswasshowninTheorem2,thereisanextremum.Inthelatterca,asufficientconditionforanextremumcanobviouslybestatedastheequivalentrequirement
thattheform

f

k

xhk

beeitherpositive-ornegative-definite.
ItwasthisformoftheconditionthatweencounteredinstudyingrealvaluedfunctionsonRn.
Remark2.

Aswehaveenintheexampleoffunctions

:Rn

R,themi-definiteness
oftheform

f

k

xhk

exhibitedinthenecessaryconditionsforanextremumisnotasufficient
criterionforanextremum.
Remark3.Inpractice,whenstudyingextremaofdifferentiablefunctionsonenormallyusonlythefirstorconddifferentials.Iftheuniquenessandtypeofextremumareobviousfromthemeaningoftheproblembeingstudied,onecanrestrictattentiontothefirst
differentialwhenekinganextremum,
simplyfindingthepointxwheref,x
0
3.SomeExamples
Example1.LetLC1
R3;R
and
f
C1a,b;R.Inotherwords,
u1,u2,u3
Lu1,u2,u3
isacontinuouslydifferentiablereal-valuedfunctiondefinedinR3and
x
fx
asmooth
real-valuedfunctiondefinedontheclodintervala,bR.
Considerthefunction
F:C1
a,b;R
R
(2)
definedbytherelation
f
C1
a,b;R
F
f
b
x,f,x
(3)
Lx,f
dx
R
a
Thus,(2)isareal-valuedfunctionaldefinedonthetoffunctionsC1
a,b;R
.
Thebasicvariationalprinciplesconnectedwithmotionareknowninphysicsandmechanics.Accordingtotheprinciples,theactualmotionsaredistinguishedamongalltheconceivablemotionsinthattheyproceedalongtrajectoriesalongwhichcertainfunctionalshaveanextremum.Questionsconnectedwiththeextremaoffunctionalsarecentralinoptimal
controltheory.Thus,findingandstudyingtheextremaoffunctionalsisaproblem
ofintrinsicimportance,andthetheoryassociatedwithitisthesubjectofalargeareaofanalysis-thecalculusofvariations.Wehavealreadydoneafewthingstomakethetransitionfromtheanalysisoftheextremaofnumericalfunctionstotheproblemoffindingandstudyingextremaoffunctionalsemnaturaltothereader.However,weshallnotgodeeplyintothespecialproblemsofvariationalcalculus,butratherutheexampleofthefunctional
toillustrateonlythegeneralideasofdifferentiationandstudyoflocalextremaconsideredabove.
Weshallshowthatthefunctional(3)isadifferentiatemappingandfinditsdifferential.Weremarkthatthefunction(3)canberegardedasthecompositionofthemappings
F1f
x
Lx,f
x,f,
x
(4)
definedbytheformula
F1:C1
a,b;R
Ca,b;R
(5)
followedbythemapping
b
(6)
gCa,b;R
F2
g
gxdxR
a
Bypropertiesoftheintegral,themapping
F2
isobviouslylinearandcontinuous,sothat
itsdifferentiabilityisclear.
WeshallshowthatthemappingF1isalsodifferentiable,andthat
F1,f
hx
2Lx,f
x,f,
xhx
3Lx,f
x.f,
x
h,x
(7)
for
h
C1
a,b;R.
Indeed,bythecorollarytothemean-valuetheorem,wecanwriteintheprentca
3
Lu1
1,u2
2,u3
3
Lu1,u2,u3
iLu1,u2,u3i
i1
sup
1Lu
1Lu1
2Lu
2Lu1
3Lu
3Lu
0
1
3max
iLu
u
iLu
max
i
(8)
0
1
i1,2,3
i
1,2,3
where
u
u1,u2,u3
and
1,2,3.
If
we
now
recallthatthenorm
fc1
ofthe
function
f
inC1a,b;Ris
maxfc,f,
c
(where
f
cisthemaximumabsolutevalueofthefunctionontheclodinterval
a,b),then,
ttingu1
x,u2
fx,
u3
f,x
,
1
0,
2
hx,and3
h,x
,weobtainfrominequality(8),
taking
accountof
theuniform
continuity
ofthefunctions
iLu1,u2,u3,i1,2,3,
onbounded
subtsofR3,that
maxLx,fx hx,f,x h,x Lx,fx,f,x 2Lx,fx,f,xhx 3Lx,fx,f,xh,x
0 xb
ohc1 ashc1 0
ButthismeansthatEq.(7)holds.
Bythechainrulefordifferentiatingacompositefunction,wenowconcludethatthefunctional(3)isindeeddifferentiable,and
F,fh
b
3Lx,fx,f,xh,xdx
(9)
2Lx,fx,f,xhx
a
Weoftenconsidertherestrictionofthefunctional(3)totheaffinespaceconsistingofthefunctionsfC1a,b;RthatassumefixedvaluesfaA,fbBattheendpointsofthe
clodinterval a,b.Inthisca,thefunctionshinthetangentspaceTCf1,musthavethe
valuezeroattheendpointsoftheclodintervala,b.Takingthisfactintoaccount,wemayintegratebypartsin(9)andbringitintotheform
b
2Lx,fx,f,x
d
3Lx,fx,f,xhxdx
(10)
F,fh
a
dx
ofcourundertheassumptionthatLandfbelongtothecorrespondingclassC2.
Inparticular,iffisanextremum(extremal)ofsuchafunctional,thenbyTheorem2wehave
F,fh 0 foreveryfunction h C1 a,b;R suchthatha hb 0.Fromthisandrelation(10)
onecaneasilyconclude(eProblem3below)thatthefunctionfmustsatisfytheequation
2Lx,fx,f,x d 3Lx,fx,f,x 0 (11)
dx
Thisisafrequently-encounteredformoftheequationknowninthecalculusofvariationsastheEuler-Lagrangeequation.
Letusnowconsidersomespecificexamples.
Example2. Theshortest-pathproblem
Amongallthecurvesinaplanejoiningtwofixedpoints,findthecurvethathasminimallength.
Theanswerinthiscaisobvious,anditratherrvesasacheckontheformal
computationswewillbedoinglater.
WeshallassumethatafixedCartesiancoordinatesystemhasbeenchonintheplane,in
whichthetwopointsare,for
example,
0,0
and
1,0.Weconfineourlvestojustthe
curvesthatarethegraphsoffunctions
f
C1
0,1;R
assumingthevaluezeroatbothendsof
theclodinterval0,1.Thelengthofsuchacurve
1
2
(12)
Ff
1f,
xdx
0
dependsonthefunctionfandisafunctionalofthetypeconsideredinExample1.InthiscathefunctionLhastheform
Lu1,u2,u3 1 u32
andthereforethenecessarycondition(11)foranextremalherereducestotheequation
d
f,x
0
dx
f,
2
1
x
fromwhichitfollowsthat
f
,x
常数
(13)
f,2
1
x
ontheclodinterval0,1
Sincethefunction
u
isnotconstantonanyinterval,
Eq.(13)ispossibleonlyif
u2
1
f,x conston a,b.Thusasmoothextremalofthisproblemmustbealinearfunctionwho
graphpassthroughthepoints0,0and1,0.Itfollowsthatfx0,andwearriveattheclodintervalofthelinejoiningthetwogivenpoints.
Example3. Thebrachistochroneproblem
Theclassicalbrachistochroneproblem,podbyJohannBernoulliIin1696,wastofind
theshapeofatrackalongwhichapointmasswouldpassfromaprescribedpoint P to
0
anotherfixedpointP1atalowerlevelundertheactionofgravityintheshortesttime.
Weneglectfriction,ofcour.Inaddition,weshallassumethatthetrivialcainwhichbothpointslieonthesameverticallineixcluded.
Inthevertical
planepassingthroughthepointsP0andP1weintroducearectangular
coordinatesystemsuchthatP0isattheorigin,
thex-axisisdirectedverticallydownward,
andthepoint
P1
haspositivecoordinatesx1,y1
.Weshallfindtheshapeofthetrackamong
thegraphsof
smoothfunctionsdefinedontheclodinterval0,x1andsatisfyingthe
conditionf0
0
,fx1y1.Atthemomentweshallnottaketimetodiscussthisbynomeans
uncontroversialassumption(eProblem4below).
Iftheparticlebeganitsdescentfromthepoint P0 withzerovelocity,thelawofvariation
ofitsvelocityinthecoordinatescanbewrittenas
v 2gx (14)
Recallingthatthedifferentialofthearclengthiscomputedbytheformula
2
ds dx2 dy2 1 f, xdx (15)
wefindthetimeofdescent
Ff

1
2g

x1
1f
,
2
(16)
xdx
x
alongthetrajectorydefinedbythegraphofthefunctiony fx ontheclodinterval0,x1.
Forthefunctional(16)
1
2
3
1u
32
Lu
,u
,u
u1
,
andthereforethecondition(11)foranextremumreducesinthiscatotheequation
d
f,x
0,
dx
f
,
2
x1
x
fromwhichitfollowsthat
f
,x
(17)
cx
2
1 f, x
wherecisanonzeroconstant,sincethepointsarenotbothonthesameverticalline.Takingaccountof(15),wecanrewrite(17)inthef