1610912322-b551b095a53deaf3d3fbd1ed05ae9b84 (824701), страница 91
Текст из файла (страница 91)
Theorem 6is not applicable, but since f (x, y) = (x 2 − 1)2 + y 4 − 1, it is obvious that thefunction f (x, y) has a strict minimum −1 (even a global minimum) at the points(−1, 0), and (1, 0), while there is no extremum at (0, 0), since for x = 0 and y = 0,we have f (0, y) = y 4 > 0, and for y = 0 and sufficiently small x = 0 we havef (x, 0) = x 4 − 2x 2 < 0.Remark 2 After the quadratic form (8.68) has been obtained, the study of its def7initeness can be carried out using themSylvesteri jcriterion. We recall that by theSylvester criterion, a quadratic form ij =1 aij x x with symmetric matrix⎛a11⎜ ..⎝ .am1···...···⎞a1m.. ⎟. ⎠ammis positive-definite if and only if all its principal minors are positive; the form isnegative-definite if and only if a11 < 0 and the sign of the principal minor reverseseach time its order increases by one.Example 4 Let us find the extrema of the functionf (x, y) = xy ln x 2 + y 2 ,which is defined everywhere in the plane R2 except at the origin.7 J.J.Sylvester (1814–1897) – British mathematician.
His best-known works were on algebra.8.4 Real-valued Functions of Several VariablesSolving the system of equations⎧ 2∂f⎪2⎪⎪⎨ ∂x (x, y) = y ln x + y +⎪⎪ ∂f⎪⎩ (x, y) = x ln x 2 + y 2 +∂y4672x 2 y= 0,x2 + y22xy 2= 0,+ y2x2we find all the critical points of the function11(0, ±1); (±1, 0);±√ ,±√;2e2e11±√ ,∓√.2e2eSince the function is odd with respect to each of its arguments individually, thepoints (0, ±1) and (±1, 0) are obviously not extrema of the function.It is also clear that this function does not change its value when the signs of bothvariables x and y are changed. Thus by studying only one of the remaining criticalpoints, for example, ( √1 , √1 ) we will be able to draw conclusions on the nature2e2eof the others.Since∂ 2f6xy4x 3 y(x,y)=−,∂x 2x 2 + y 2 (x 2 + y 2 )2∂ 2f4x 2 y 2(x, y) = ln x 2 + h2 + 2 − 2,∂x∂y(x + y 2 )2∂ 2f6xy4xy 3(x,y)=−,∂y 2x 2 + y 2 (x 2 + y 2 )2at the point ( √1 , √1 ) the quadratic form ∂ij f (x0 )hi hj has the matrix2e2e2 0,0 2that is, it is positive-definite, and consequently at that point the function has a localminimum111=− .f √ ,√2e2e2eBy the observations made above on the properties of this function, one can conclude immediately that111f −√ ,−√=−2e2e2eis also a local minimum and11111f √ ,−√= f −√ , √=2e2e2e2e2e4688 Differential Calculus in Several Variablesare local maxima of the function.
This, however, could have been verified directly,by checking the definiteness of the corresponding quadratic form. For example, atthe point (− √1 , √1 ) the matrix of the quadratic form (8.68) has the form2e2e−2 0,0 −2from which it is clear that it is negative-definite.Remark 3 It should be kept in mind that we have given necessary conditions (Theorem 5) and sufficient conditions (Theorem 6) for an extremum of a function only atan interior point of its domain of definition.
Thus in seeking the absolute maximumor minimum of a function, it is necessary to examine the boundary points of thedomain of definition along with the critical interior points, since the function mayassume its maximal or minimal value at one of these boundary points.The general principles of studying noninterior extrema will be considered inmore detail later (see the section devoted to extrema with constraint). It is useful tokeep in mind that in searching for minima and maxima one may use certain simpleconsiderations connected with the nature of the problem along with the formal techniques, and sometimes even instead of them.
For example, if a differentiable function being studied in Rm must have a minimum because of the nature of the problemand turns out to be unbounded above, then if the function has only one critical point,one can assert without further investigation that that point is the minimum.Example 5 (Huygens’ problem) On the basis of the laws of conservation of energyand momentum of a closed mechanical system one can show by a simple computation that when two perfectly elastic balls having mass m1 and m2 and initial velocities v1 and v2 collide, their velocities after a central collision (when the velocitiesare directed along the line joining the centers) are determined by the relationsṽ1 =(m1 − m2 )v1 + 2m2 v2,m1 + m2ṽ2 =(m2 − m1 )v2 + 2m1 v1.m1 + m2In particular, if a ball of mass M moving with velocity V strikes a motionless ballof mass m, then the velocity v acquired by the latter can be found from the formulav=2MV,m+M(8.70)from which one can see that if 0 ≤ m ≤ M, then V ≤ v ≤ 2V .How can a significant part of the kinetic energy of a larger mass be communicated to a body of small mass? To do this, for example, one can insert balls withintermediate masses between the balls of small and large mass: m < m1 < m2 <· · · < mn < M.
Let us compute (after Huygens) how the masses m1 , m2 , . . . , mn8.4 Real-valued Functions of Several Variables469should be chosen to that the body m will acquire maximum velocity after successive central collisions.In accordance with formula (8.70) we obtain the following expression for therequired velocity as a function of the variables m1 , m2 , .
. . , mn :v=m1m2mnM·· ··· ··· 2n+1 V .m + m1 m1 + m2mn−1 + mn mn + M(8.71)Thus Huygens’ problem reduces to finding the maximum of the functionf (m1 , . . . , mn ) =m1mnM.· ··· ··m + m1mn−1 + mn mn + MThe system of equations (8.65), which gives the necessary conditions for an interior extremum, reduces to the following system in the present case:⎧m · m2 − m21 = 0,⎪⎪⎪⎪⎨ m1 · m3 − m2 = 0,2..⎪⎪.⎪⎪⎩2mn−1 · M − mn = 0,from which it follows that the numbersm, m1 , . . . , mn , M form a geometric pro√gression with ratio q equal to n+1 M/m.The value of the velocity (8.71) that results from this choice of masses is givenby2q n+1V,(8.72)v=1+qwhich agrees with (8.70) if n = 0.It is clear from physical considerations that formula (8.72) gives the maximalvalue of the function (8.71).
However, this can also be verified formally (withoutinvoking the cumbersome second derivatives. See Problem 9 at the end of this section).We remark that it is clear from (8.72) that if m → 0, then v → 2n+1 V . Thusthe intermediate masses do indeed significantly increase the portion of the kineticenergy of the mass M that is transmitted to the small mass m.8.4.6 Some Geometric Images Connected with Functionsof Several Variablesa.
The Graph of a Function and Curvilinear CoordinatesLet x, y, and z be Cartesian coordinates of a point in R3 and let z = f (x, y) be acontinuous function defined in some domain G of the plane R2 of the variables xand y.4708 Differential Calculus in Several VariablesBy the general definition of the graph of a function, the graph of the functionf : G → R in our case is the set S = {(x, y, z) ∈ R3 | (x, y) ∈ G, z = f (x, y)} inthe space R3 .FIt is obvious that the mapping G → S defined by the relation (x, y) →(x, y, f (x, y)) is a continuous one-to-one mapping of G onto S, by which onecan determine every point of S by exhibiting the point of G corresponding to it, or,what is the same, giving the coordinates (x, y) of this point of G.Thus the pairs of numbers (x, y) ∈ G can be regarded as certain coordinates ofthe points of a set S – the graph of the function z = f (x, y).
Since the points of S aregiven by pairs of numbers, we shall conditionally agree to call S a two-dimensionalsurface in R3 . (The general definition of a surface will be given later.)If we define a path Γ : I → G in G, then a path F ◦ Γ : I → S automaticallyappears on the surface S. If x = x(t) and y = y(t) is a parametric definition ofthe path Γ , then the path F ◦ Γ on S is given by the three functions x = x(t),y = y(t), z = z(t) = f (x(t), y(t)). In particular, if we set x = x0 + t, y = y0 , weobtain a curve x = x0 + t, y = y0 , z = f (x0 + t, y0 ) on the surface S along whichthe coordinate y = y0 of the points of S does not change. Similarly one can exhibit acurve x = x0 , y = y0 + t, z = f (x0 , y0 + t) on S along which the first coordinate x0of the points of S does not change.
By analogy with the planar case these curves onS are naturally called coordinate lines on the surface S. However, in contrast to thecoordinate lines in G ⊂ R2 , which are pieces of straight lines, the coordinate lineson S are in general curves in R3 . For that reason, the coordinates (x, y) of points ofthe surface S are often called curvilinear coordinates on S.Thus the graph of a continuous function z = f (x, y), defined in a domain G ⊂R 2 is a two-dimensional surface S in R3 whose points can be defined by curvilinearcoordinates (x, y) ∈ G.At this point we shall not go into detail on the general definition of a surface,since we are interested only in a special case of a surface – the graph of a function. However, we assume that from the course in analytic geometry the reader iswell acquainted with some important particular surfaces in R3 (such as a plane, anellipsoid, paraboloids, and hyperboloids).b.
The Tangent Plane to the Graph of a FunctionDifferentiability of a function z = f (x, y) at the point (x0 , y0 ) ∈ G means thatf (x, y) = f (x0 , y0 ) + A(x − x0 ) + B(y − y0 ) ++ o (x − x0 )2 + (y − y0 )2 as (x, y) → (x0 , y0 ),(8.73)where A and B are certain constants.In R3 let us consider the planez = z0 + A(x − x0 ) + B(y − y0 ),(8.74)8.4 Real-valued Functions of Several Variables471where z0 = f (x0 , y0 ).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
