數(shù)學(xué)及應(yīng)用數(shù)學(xué)專業(yè)外文翻譯--多元函數(shù)的極值

1、外文原文EXTREME VALUES OF FUNCTIONS OF SEVERAL REAL VARIABLES1. Stationary PointsDefinition 1.1 Let and . The point a is said to be: (1) a local maximum iffor all points sufficiently close to ;(2) a local minimum iffor all points sufficiently close to ;(3) a global (or absolute) maximum iffor all points

2、 ;(4) a global (or absolute) minimum iffor all points ;(5) a local or global extremum if it is a local or global maximum or minimum.Definition 1.2 Let and . The point a is said to be critical or stationary point if and a singular point if does not exist at .Fact 1.3 Let and .If has a local or global

3、 extremum at the point , then must be either:(1) a critical point of , or(2) a singular point of , or(3) a boundary point of .Fact 1.4 If is a continuous function on a closed bounded set then is bounded and attains its bounds.Definition 1.5 A critical point which is neither a local maximum nor minim

4、um is called a saddle point.Fact 1.6 A critical point is a saddle point if and only if there are arbitrarily small values of for which takes both positive and negative values.Definition 1.7 If is a function of two variables such that all second order partial derivatives exist at the point , then the

5、 Hessian matrix of at is the matrixwhere the derivatives are evaluated at.If is a function of three variables such that all second order partial derivatives exist at the point , then the Hessian of f at is the matrixwhere the derivatives are evaluated at.Definition 1.8 Let be an matrix and, for each

6、 ,let be the matrix formed from the first rows and columns of .The determinants det(),are called the leading minors of Theorem 1.9(The Leading Minor Test). Suppose that is a sufficiently smooth function of two variables with a critical point atand H the Hessian of at.If , then is:(1) a local maximum

7、 if 0>det(H1) = fxx and 0<det(H)=; (2) a local minimum if 0<det(H1) = fxx and 0<det(H)=; (3) a saddle point if neither of the above hold.where the partial derivatives are evaluated at.Suppose that is a sufficiently smooth function of three variables with a critical point at and Hessian H

8、 at.If , then is:(1) a local maximum if 0>det(H1), 0<det(H2) and 0>det(H3);(2) a local minimum if 0<det(H1), 0<det(H2) and 0>det(H3);(3) a saddle point if neither of the above hold.where the partial derivatives are evaluated at.Key Points.·A continuous function on a closed bou

9、nded set is bounded and achieves its bounds.·To find the extreme values of a function on a closed bounded set it is necessary to consider the value of the function at stationary points(), singular points (does not exist) and boundary points(points on the edge of the set).·Stationary points

10、 can be classified as local maxim , local minim or saddle points.·If The Leading Minor Test 1.9 is not applicable, the stationary point must be classified by directly applying Definition 1.1 and Fact 1.6. For example in the two variable case, if has a stationary point at ,we consider the sign o

11、ffor arbitrarily small, positive and negative values of and (that are not both zero). In each case, if det(H)= 0, then can be either a local extremum or a saddle point.Example. Find and classify the stationary points of the following functions: (1) (2) Solution. (1) ,soijkCritical points occur when

12、,i.e. when(1) (2) (3) Using equations (2) and (3) to eliminate y and z from (1), we see thator ,giving , and .Hence we have three stationary points: , and . Since, and ,the Hessian matrix is At ,which has leading minors >0,And det .By the Leading Minor Test, then, is a local minimum. At ,which ha

13、s leading minors >0,And det .By the Leading Minor Test, then, is also a local minimum.At , the Hessian isSince det, we can apply the leading minor test which tells us that this is a saddle point since the first leading minor is 0. An alternative method is as follows. In this case we consider the

14、value of the expression，for arbitrarily small values of h, k and l. But for very small h, k and l, cubic terms and above are negligible in comparison to quadratic and linear terms, so that.If h, k and l are all positive, . However, if and and ,then .Hence close to ,both increases and decreases, so i

15、s a saddle point.(2) soij.Stationary points occur when ,i.e. at .Let us classify this stationary point without considering the Leading Minor Test (in this case the Hessian has determinant 0 at so the test is not applicable).LetCompleting the square we see that So for any arbitrarily small values of

16、h and k, that are not both 0, and we see that f has a local maximum at .2. Constrained Extrema and Lagrange MultipliersDefinition 2.1 Let f and g be functions of n variables. An extreme value of f(x) subject to the condition g(x) = 0, is called a constrained extreme value and g(x) = 0 is called the

17、constraint.Definition 2.2 If is a function of n variables, the Lagrangian function of f subject to the constraint is the function of n+1 variableswhere is known as the Lagrange multiplier. The Lagrangian function of f subject to the k constraints , is the function with k Lagrange multipliers, Key Po

18、ints.·To find the extreme values of f subject to the constraint g(x) = 0: (1) calculate, remembering that it is a function of the n+1 variables and (2) find values of such that (you do not have to explicitly find the corresponding values of ): (3) evaluate f at these points to find the required

19、 extrema.·Note that the equation is equivalent to the equations,and So, in the two variable case, we have Lagranian function and are solving the equations:, , and .·With more than one constraint we solve the equation.Theorem 2.3 Let and be a point on the curve C, with equation g(x,y) = 0,

20、at which f restricted to C has a local extremum.Suppose that both and have continuous partial derivatives near to and that is not an end point of and that . Then there is some such that is a critical point of the Lagrangian Function.Proof. Sketch only. Since P is not an end point and ,has a tangent

21、at with normal .If is not parallel to at , then it has non-zero projection along this tangent at .But then f increases and decreases away from along ,so is not an extremum. Henceand are parallel and there is some¸such that and the result follows.Example. Find the rectangular box with the larges

22、t volume that fits inside the ellipsoid ,given that it sides are parallel to the axes.Solution. Clearly the box will have the greatest volume if each of its corners touch the ellipse. Let one corner of the box be corner (x, y, z) in the positive octant, then the box has corners (±x,±y,

23、7;z) and its volume is V= 8xyz.We want to maximize V given that . (Note that since the constraint surface is bounded a max/min does exist). The Lagrangian isand this has critical points when , i.e. when (Note that will always be the constraint equation.) As we want to maximize V we can assume that s

24、o that .)Hence, eliminating , we getso that and But then so or ,which implies that and (they are all positive by assumption). So L has only one stationary point (for some value of , which we could work out if we wanted to). Since it is the only stationary point it must the required max and the max v

25、olume is.中文譯文 多元函數(shù)的極值1. 穩(wěn)定點(diǎn)定義1.1 使并且. 對(duì)于任意一點(diǎn)有以下定義: (1)如果對(duì)于所有充分地接近時(shí)，則是一個(gè)局部極大值;(2)如果對(duì)于所有充分地接近時(shí)，則是一個(gè)局部極小值;(3)如果對(duì)于所有點(diǎn)成立，則是一個(gè)全局極大值（或絕對(duì)極大值）;(4) 如果對(duì)于所有點(diǎn)成立，則是一個(gè)全局極小值（或絕對(duì)極小值）; (5) 局部極大（?。┲到y(tǒng)稱為局部極值；全局極大（?。┲到y(tǒng)稱為全局極值.定義 1.2 使并且.對(duì)于任意一點(diǎn)，如果，并且對(duì)于任意奇異點(diǎn)都不存在，則稱是一個(gè)關(guān)鍵點(diǎn)或穩(wěn)定點(diǎn).結(jié)論 1.3 使并且.如果有局部極值或全局極值對(duì)于一點(diǎn), 則 一定是:(1)函數(shù)的一個(gè)關(guān)鍵點(diǎn), 或

26、者(2)函數(shù)的一個(gè)奇異點(diǎn), 或者 (3)定義域的一個(gè)邊界點(diǎn).結(jié)論 1.4 如果函數(shù)是一個(gè)在閉區(qū)間上的連續(xù)函數(shù)，則在區(qū)間上有邊界并且可以取到邊界值.定義 1.5 對(duì)于任一個(gè)關(guān)鍵點(diǎn)，當(dāng)既不是局部極大值也不是局部極小值時(shí)，叫做函數(shù)的鞍點(diǎn).結(jié)論 1.6 對(duì)于一個(gè)關(guān)鍵點(diǎn)是鞍點(diǎn)當(dāng)且僅當(dāng)任意小時(shí)，對(duì)于函數(shù)取正值和負(fù)值.定義 1.7 如果 是二元函數(shù)，并且在點(diǎn)處所有二階偏導(dǎo)數(shù)都存在，則則根據(jù)函數(shù)在點(diǎn)處導(dǎo)數(shù)，有在點(diǎn)處的Hessian矩陣為：.推廣：如果 是三元函數(shù)，并且在點(diǎn)處所有二階偏導(dǎo)數(shù)都存在，則根據(jù)函數(shù)在點(diǎn)處導(dǎo)數(shù)，有在點(diǎn)處的Hessian矩陣為：.定義 1.8 矩陣是 階矩陣，并且對(duì)于每一個(gè)都有,從矩陣中選

27、取左上端的行和列，令其為階的矩陣.則行列式det(),叫做矩陣的順序主子式. 定理 1.9 假如是一個(gè)充分光滑的二元函數(shù)，且在點(diǎn)處穩(wěn)定，其Hessian 矩陣為H .如果，則根據(jù)偏導(dǎo)數(shù)判定點(diǎn)是：(1) 一個(gè)局部極大值點(diǎn)， 如果0>det(H1) = fxx并且0<det(H)=; (2) 一個(gè)局部極小值點(diǎn)， 如果0<det(H1) = fxx并且0<det(H)=; (3) 一個(gè)鞍點(diǎn)，如果點(diǎn)既不是局部極大值點(diǎn)也不是局部極小值點(diǎn).假如是一個(gè)充分光滑的三元函數(shù)，且在點(diǎn)處穩(wěn)定，其Hessian 矩陣為H .如果，則根據(jù)偏導(dǎo)數(shù)判定點(diǎn)是： (1) 一個(gè)局部極大值點(diǎn)， 如果當(dāng)0&g

28、t;det(H1), 0<det(H2) 并且 0>det(H3)時(shí);(2) 一個(gè)局部極小值點(diǎn)， 如果當(dāng)0<det(H1), 0<det(H2) 并且 0>det(H3)時(shí);(3) 一個(gè)鞍點(diǎn)，如果點(diǎn)既不是局部極大值點(diǎn)也不是局部極小值點(diǎn). 在不同的情況下 ,當(dāng)det(H)= 0時(shí), 點(diǎn)是一個(gè)局部極值點(diǎn)，或者是一個(gè)鞍點(diǎn).關(guān)鍵點(diǎn).·在有界閉集上的連續(xù)函數(shù)有邊界，并且可以取到其邊界值.·當(dāng)確定函數(shù)在有界閉集上的極值時(shí)，必須考慮函數(shù)在穩(wěn)定點(diǎn)(即 時(shí)), 奇異點(diǎn) (當(dāng) 不存在時(shí)) 和邊界點(diǎn)(點(diǎn)在集合的邊緣)處的函數(shù)值.·穩(wěn)定點(diǎn)可以分為局部極大值點(diǎn)

29、、局部極小值點(diǎn)或鞍點(diǎn). ·對(duì)于穩(wěn)定點(diǎn)，當(dāng)應(yīng)用定理 1.9 不能分類時(shí)，可依據(jù)定義1.1和結(jié)論1.6對(duì)穩(wěn)定點(diǎn)進(jìn)行直接分類.例如，在二元情況下，如果 在點(diǎn) 處的點(diǎn)是穩(wěn)定點(diǎn)，我們可以考慮函數(shù)的符號(hào)，當(dāng) 和 任意小( 和 可為正值和負(fù)值,但不同時(shí)為0)時(shí). 例. 確定下列函數(shù)的穩(wěn)定點(diǎn)并說明是哪一類點(diǎn): (1) (2) 解. (1) ,soijk當(dāng)時(shí)有穩(wěn)定點(diǎn),也就是說, 當(dāng) (1) (2) (3) 時(shí),將方程(2)和方程(3)帶入到方程(1)可以消去變量y和z, 由此可以得到即,得,和.因此我們可以得到函數(shù)的三個(gè)穩(wěn)定點(diǎn):,和. 又因?yàn)?和,則Hessian矩陣為 在點(diǎn)處, 則順序主子式 >

30、;0,并且行列式.根據(jù)主子式判定方法，則點(diǎn)是一個(gè)局部極小值點(diǎn). 在點(diǎn)處, 則順序主子式 >0,并且行列式.根據(jù)主子式判定方法，則點(diǎn)也是一個(gè)極小值點(diǎn). 在點(diǎn)處，Hessian矩陣為因此det,根據(jù)主子式判定方法，第一主子式為0，由此我們可以知道該點(diǎn)是一個(gè)鞍點(diǎn). 下面是另一種計(jì)算方法，在這種情況下，我們考慮現(xiàn)在下面函數(shù)表達(dá)式，的值，對(duì)于任意h, k和l無限小時(shí). 擔(dān)當(dāng)h, k和l非常小時(shí), 三次及三次以上方程相對(duì)線性二次方程時(shí)可忽略不計(jì),則原方程可為.當(dāng)h, k和l 都為正時(shí),.然而， 當(dāng)、和,則.因此當(dāng)接近時(shí),同時(shí)增加或者同時(shí)減少, 所以 是一個(gè)鞍點(diǎn).(2) soij.當(dāng)時(shí)有穩(wěn)定點(diǎn),也就是說, 當(dāng)在時(shí). 現(xiàn)在我們?cè)诓豢紤]主子式判定方法的情況下為該穩(wěn)定點(diǎn)進(jìn)行分類（因?yàn)樵跁r(shí)Hessian矩陣的行列式為0，所以該判定方法在此刻無法應(yīng)用）.令配成完全平方的形式為所以對(duì)h和k為任意小時(shí)（h和k都不為0），有，因此我們可以確定函數(shù)f 在點(diǎn)處有局部極大值.2. 條件極值和Lagrange乘數(shù)法定義 2.1 函數(shù)f和函數(shù)g都是n元函數(shù).對(duì)于限制在條件g(