泰勒公式外文翻译Word下载.docx

1、Assume formula （1） is true for some . n,1,NThen by the mean-value theorem, formula （12） of Sect. 10.5, and the induction hypothesis, we obtain. 1,，nnfx，h,fx，fh，fxhx？，,n!,1n,1，,n,，,fx，h,fx，fxh，fxhh？，sup ,，n,1!,0,1,n,1n,o,hhoh,We shall not take the time here to discuss other versions of Taylors formul

2、a, which are sometimes quite useful. They were discussed earlier in detail for numerical functions. At this point we leave it to the reader to derive them （see, for example, Problem 1 below）. 2. Methods of Studying Interior Extrema Using Taylors formula, we shall exhibit necessary conditions and als

3、o sufficient conditions for an interior local extremum of real-valued functions defined on an open subset of a normed space. As we shall see, these conditions are analogous to the differential conditions already known to us for an extremum of a real-valued function of a real variable. f:U,RTheorem 2

4、. Let be a real-valued function defined on an open set U in a k,1,1normed space X and having continuous derivatives up to order inclusive in a ，k，fxx,Uneighborhood of a point and a derivative of order k at the point x itself. ，,k,1k，fx,0,？,fx,0fx,0If and , then for x to be an extremum of the functio

5、n f it is: ，kk，fxhnecessary that k be even and that the form be semidefinite, and 1 ，kkf，xhsufficient that the values of the form on the unit sphere be bounded away h,1from zero; moreover, x is a local minimum if the inequalities ，kkf，xh,0, hold on that sphere, and a local maximum if Proof. For the

6、proof we consider the Taylor expansion （1） of f in a neighborhood of x. The assumptions enable us to write 1k，kk f，x，h,fx,f，xh，,hhk!，where ,h is a real-valued function, and ,h,0 as. h,0We first prove the necessary conditions. ，k，kkf，x,0Since, there exists a vector on which. Then for values of the h,

7、0f，xh,000real parameter t sufficiently close to zero, 1kk，kf，x，th,fx,f，xth，,thth 0000!k1,k，kkk，,fxh，,thht ,000!k,，kkand the expression in the outer parentheses has the same sign as. f，xh0For x to be an extremum it is necessary for the left-hand side （and hence also the right-hand side） of this last

8、equality to be of constant sign when t changes sign. But this is possible only if k is even. This reasoning shows that if x is an extremum, then the sign of the difference ， fx，th,fx0，kkis the same as that of for sufficiently small t; hence in that case there cannot be two f，xh0，k，fxvectors hh, at w

9、hich the form assumes values with opposite signs. 01We now turn to the proof of the sufficiency conditions. For definiteness we consider the ，kk，fxh,0h,1case when for. Then k1，kk，fx，h,f，x,f，xh，,hh k!k,1h,k，k,，,fx，,hh ,!kh,1,k，,，,hh ,!，,h,0h,0and, since as, the last term in this inequality is positiv

10、e for all vectors h,0sufficiently close to zero. Thus, for all such vectors h, 2 , ，fx，h,fx,0that is, x is a strict local minimum. The sufficient condition for a strict local maximum is verified similiarly. Remark 1. If the space X is finite-dimensional, the unit sphere with center at, ，Sx;1x,Xbeing

11、 a closed bounded subset of X, is compact. Then the continuous function ii，kkk1 （a k-form） has both a maximal and a minimal value on ，. If ，Sx;1fxh,fxh,？,h？ii1kthese values are of opposite sign, then f does not have an extremum at x. If they are both of the same sign, then, as was shown in Theorem 2

12、, there is an extremum. In the latter case, a sufficient condition for an extremum can obviously be stated as the equivalent requirement ，kkf，xhthat the form be either positive- or negative-definite. It was this form of the condition that we encountered in studying realvalued functions non. Rnf:R,RR

13、emark 2. As we have seen in the example of functions, the semi-definiteness ，kkf，xhof the form exhibited in the necessary conditions for an extremum is not a sufficient criterion for an extremum. Remark 3. In practice, when studying extrema of differentiable functions one normally uses only the firs

14、t or second differentials. If the uniqueness and type of extremum are obvious from the meaning of the problem being studied, one can restrict attention to the first ,，fx,0differential when seeking an extremum, simply finding the point x where 3. Some Examples ，131123123，,L,CR;Rf,Ca,b;Ru,u,u!Lu,u,uEx

15、ample 1. Let and . In other words, 3，x!fxis a continuously differentiable real-valued function defined in R and a smooth ,a,b,Rreal-valued function defined on the closed interval . Consider the function ，1，,F:Ca,b;R,R （2） defined by the relation ，1，,f,Ca,b;R!Ff b,，,Lx,fx,fxdx,R （3） ,a，1，,Ca,b;RThus,

16、 （2） is a real-valued functional defined on the set of functions . The basic variational principles connected with motion are known in physics and mechanics. According to these principles, the actual motions are distinguished among all the conceivable motions in that they proceed along trajectories

17、along which certain functionals have an extremum. Questions connected with the extrema of functionals are central in optimal 3 control theory. Thus, finding and studying the extrema of functionals is a problem of intrinsic importance, and the theory associated with it is the subject of a large area

18、of analysis - the calculus of variations. We have already done a few things to make the transition from the analysis of the extrema of numerical functions to the problem of finding and studying extrema of functionals seem natural to the reader. However, we shall not go deeply into the special proble

19、ms of variational calculus, but rather use the example of the functional （3） to illustrate only the general ideas of differentiation and study of local extrema considered above. We shall show that the functional （3） is a differentiate mapping and find its differential. We remark that the function （3

20、） can be regarded as the composition of the mappings , （4） ，F，fx,Lx,fx,fx1defined by the formula ，1 （5） ,F:C，a,b;R,C，a,b;R1followed by the mapping b,g,C，a,b;Fg,gxdx,R （6） 2,aBy properties of the integral, the mapping is obviously linear and continuous, so that F2its differentiability is clear. We sh

21、all show that the mapping is also differentiable, and that F1, （7） ，F，fhx,Lx,fx,fxh，x，,Lx,fx.fxh，x231，1,h,C，a,b;Rfor . Indeed, by the corollary to the mean-value theorem, we can write in the present case 3112233123123i，Lu，,u，,u，,Lu,u,u,Lu,u,u, i,1i，,sup,Lu，,Lu,Lu，,Lu,Lu，,Lu, 11223311,01i，,3max,Lu，,u

22、,Lu,max, （8） ii0,1,1,2,3ii,1,2,3123123，u,u,u,u,where and . ,，1,，,Ca,b;Rfmaxf,fIf we now recall that the norm of the function f in is ，1,ccc,a,bf（where is the maximum absolute value of the function on the closed interval）, then, c23,23,11，u,fx,hx,hxu,fxu,x,0setting, , , , , and, we obtain from inequa

23、lity （8）, 123taking account of the uniform continuity of the functions，, on bounded ,Lu,u,u,i,1,2,3i4 3subsets of , that R,， ，maxLx,fx，hx,fx，hx,Lx,fx,fx,Lx,fx,fxhx,Lx,fx,fxhx230,x,b， as ,ohh,0，1，1ccBut this means that Eq. （7） holds. By the chain rule for differentiating a composite function, we now

24、conclude that the functional （3） is indeed differentiable, and b,，F，fh,Lx,fx,fxh，x，,Lx,fx,fxh，xdx （9） 23,aWe often consider the restriction of the functional （3） to the affine space consisting of the ，1,f,C，a,b;R，functions that assume fixed values fa,A, fb,B at the endpoints of the ，1,closed interva

25、l a,b. In this case, the functions h in the tangent spaceTC, must have the f,a,bvalue zero at the endpoints of the closed interval . Taking this fact into account, we may integrate by parts in （9） and bring it into the form bd,，FfhLx,fx,fxLx,fx,fxhxdx （10） ,23,dxa,，2Cof course under the assumption t

26、hat L and f belong to the corresponding class . In particular, if f is an extremum （extremal） of such a functional, then by Theorem 2 we have ，,1，,Ffh,0h,C，a,b;R，ha,hb,0 for every function such that . From this and relation （10） one can easily conclude （see Problem 3 below） that the function f must

27、satisfy the equation d,，,Lx,f，x,fx,Lx,f，x,fx,0 （11） 23dxThis is a frequently-encountered form of the equation known in the calculus of variations as the Euler-Lagrange equation. Let us now consider some specific examples. Example 2. The shortest-path problem Among all the curves in a plane joining t

28、wo fixed points, find the curve that has minimal length. The answer in this case is obvious, and it rather serves as a check on the formal computations we will be doing later. We shall assume that a fixed Cartesian coordinate system has been chosen in the plane, in ，0,0，1,0which the two points are,

29、for example, and . We confine ourselves to just the ，1，,f,C0,1;Rcurves that are the graphs of functions assuming the value zero at both ends of ,0,1the closed interval . The length of such a curve 12,，Ff,1，fxdx （12） ,0depends on the function f and is a functional of the type considered in Example 1. In this case the function