泰勒公式外文翻译.doc

该【泰勒公式外文翻译 】是由【吴老师】上传分享，文档一共【15】页，该文档可以免费在线阅读，需要了解更多关于【泰勒公式外文翻译 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。1Taylor'sFormulaandtheStudyofExtremaTaylor'-1inclusiveinUandhasann-thorderderivativeatthepointx,then(1)(1)isoneofthevarietiesofTaylor'sformula,'(1)-valuetheorem,formula(12),andtheinductionhypothesis,'sformula,(see,forexample,Problem1below).MethodsofStudyingInteriorExtremaUsingTaylor'sformula,weshallexhibitnecessaryconditionsandalsosufficientconditionsforaninteriorlocalextremumofreal-,theseconditionsareanalogoustothedifferentialconditionsalreadyknowntousforanextremumofareal--,thenforxtobeanextremumofthefunctionfitis:necessarythatkbeevenandthattheformbesemidefinite,and2sufficientthatthevaluesoftheformontheunitspherebeboundedawayfromzero;moreover,xisalocalminimumiftheinequalities,holdonthatsphere,andalocalmaximumif,(1)-valuedfunction,,,-handside(andhencealsotheright-handside),thenthesignofthedifferenceisthesameasthatofforsufficientlysmallt;henceinthatcasetherecannotbetwovectors,,sinceas,,forallsuchvectorsh,4,thatis,-dimensional,theunitspherewithcenterat,beingaclosedboundedsubsetofX,(ak-form),,then,aswasshowninTheorem2,,asufficientconditionforanextremumcanobviouslybestatedastheequivalentrequirementthattheformbeeitherpositive-ornegative-,thesemi-,,onecanrestrictattentiontothefirstdifferentialwhenseekinganextremum,,isacontinuouslydifferentiablereal-valuedfunctiondefinedinandasmoothreal-(2)definedbytherelation(3)Thus,(2)isareal-,,findingandstudyingtheextremaoffunctionalsisaproblem5ofintrinsicimportance,andthetheoryassociatedwithitisthesubjectofalargeareaofanalysis-,weshallnotgodeeplyintothespecialproblemsofvariationalcalculus,butratherusetheexampleofthefunctional(3)(3)(3)positionofthemappings(4)definedbytheformula(5)followedbythemapping(6)Bypropertiesoftheintegral,themappingisobviouslylinearandcontinuous,,andthat(7),bythecorollarytothemean-valuetheorem,wecanwriteinthepresentcase(8)(whereisthemaximumabsolutevalueofthefunctionontheclosedinterval),then,setting,,,,,and,weobtainfrominequality(8),ountoftheuniformcontinuityofthefunctions,onboundedsubsetsof5,thatasButthismeansthatEq.(7),wenowconcludethatthefunctional(3)isindeeddifferentiable,and(9)Weoftenconsidertherestrictionofthefunctional(3)totheaffinespaceconsistingofthefunctionsthatassumefixedvalues,,thefunctionshinthetangentspace,,wemayintegratebypartsin(9)andbringitintotheform(10),iffisanextremum(extremal)ofsuchafunctional,(10)onecaneasilyconclude(seeProblem3below)thatthefunctionfmustsatisfytheequation(11)Thisisafrequently-encounteredformoftheequationknowninthecalculusofvariationsastheEuler--pathproblemAmongallthecurvesinaplanejoiningtwofixedpoints,,,inwhichthetwopointsare,forexample,(12)(11)foranextremalherereducestotheequationfromwhichitfollowsthat(13)ontheclosedintervalSincethefunctionisnotconstantonanyinterval,Eq.(13),,posedbyJohannBernoulliIin1696,,,,thex-axisisdirectedverticallydownward,,.Atthemomentweshallnottaketimetodiscussthisbynomeansuncontroversialassumption(seeProblem4below).Iftheparticlebeganitsdescentfromthepointwithzerovelocity,thelawofvariationofitsvelocityinthesecoordinatescanbewrittenas(14)putedbytheformula(15)8wefindthetimeofdescent(16)(16),andthereforethecondition(11)foranextremumreducesinthiscasetotheequation,fromwhichitfollowsthat(17)wherecisanonzeroconstant,(15),wecanrewrite(17)intheform(18)However,fromthegeometricpointofview,(19)whereistheanglebetweenthetangenttothetrajectoryandthepositivex-.(18)withthesecondequationin(19),wefind(20)Butitfollowsfrom(19)and(20)that,,fromwhichwefind(21)Settingand,wewriterelations(20)and(21)as(22)Since,itfollowsthatonlyfor,.Itfollowsfromtheformofthefunction(22).(21)implies,andwearriveatthesimplerform8(23),whichisascalingcoefficient,mustbechosensothatthecurve(23),asonecanseebysketchingthecurve(23),isbynomeansalwaysunique,andthisshowsthatthenecessarycondition(11),fromphysicalconsiderationsitisclearwhichofthepossiblevaluesoftheparameterashouldbepreferred(andthis,ofcourse,putation).-1阶(包括n-1在内)的导数,而在点x处有n阶导数。,那么当时有(1)等式(1)是各种形式的泰勒公式中的一种,这一次它确实是对非常一般的函数类写出来的公式了。我们用归纳法证明泰勒公式(1)。当时,由的定义,(1)式成立。假设(1)对成立。于是根据有限增量定理,5章中公式(12)和所作的归纳假设,我们得到,当时成立。这里我们不再继续讨论其他的,有时甚至是十分有用的泰勒公式形式。当时,在研究数值函数时,曾详细地讨论过它们。现在我们把它们的结论提供应读者(例如,可参看练****1)。。我们将看到,这些条件类似于我们熟知的实变量的实值函数的极值的微分条件。定理2设是定义在赋范空间X的开集U上的实值函数,且f在某个点的邻域有直到阶.(包括k-1阶在内的)导映射,在点x本身有k阶导映射。如果且,那么为使x是函数f的极值点必要条件是:k是偶数,是半定的。充分条件是:在单位球面上的值不为零;这时,如果在这个球面上10,那么x是严格局部极小点;如果,那么x是严格局部极大点为了证明定理,我们考察函数f在点x邻域内的泰勒展开式。由所作的假设可得,其中是实值函数,而且当时。我们先证必要条件。因为,所以有向量,使。于是,对于充分接近于零的实参量t,括号内的表达式与同号。为使x是极值点,当t变号时最后一个等式的左边(从而右边)必须不改变符号。这只有当k为偶数时才可能。上述讨论说明,如果x是极值点,那么对于充分小的t,差的符号与相同,因而在这种情况下不可能有两个向量,,使在它们上的取值有不同的符号。我们转到极值充分条件的证明。为了确定起见,我们研究,当的情况。这时,又因时,所以不等式的右端对于所有充分接近于零的向量均为正。因而对所有这些向量h,