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. . , gm ∈ C(I m ; R)such thatm 1 mf x ,...,x =x i gi x 1 , . . . , x mi=1inI m,and in additiongi (0) =∂f(0),∂x ii = 1, . . . , m.8.4 Real-valued Functions of Several Variables4776. Prove the following generalization of Rolle’s theorem for functions of severalvariables.If the function f is continuous in a closed ball B(0; r), equal to zero on theboundary of the ball, and differentiable in the open ball B(0; r), then at least one ofthe points of the open ball is a critical point of the function.7. Verify that the functionf (x, y) = y − x 2 y − 3x 2does not have an extremum at the origin, even though its restriction to each linepassing through the origin has a strict local minimum at that point.8. The method of least squares.
This is one of the commonest methods of processingthe results of observations. It consists of the following. Suppose it is known that thephysical quantities x and y are linearly related:y = ax + b(8.79)or suppose an empirical formula of this type has been constructed on the basis ofexperimental data.Let us assume that n observations have been made, in each of which both x and ywere measured, resulting in n pairs of values x1 , y1 ; . . . ; xn , yn . Since the measurements have errors, even if the relation (8.79) is exact, the equalitiesyk = axk + bmay fail to hold for some of the values of k ∈ {1, .
. . , n}, no matter what the coefficients a and b are.The problem is to determine the unknown coefficients a and b in a reasonableway from these observational results.Basing his argument on analysis of the probability distribution of the magnitudeof observational errors, Gauss established that the most probable values for the coefficients a and b with a given set of observational results should be sought by useof the following least-squares principle:If δk = (axk + b) − yk is the discrepancy in the kth observation, then a and bshould be chosen so that the quantityΔ=nδk2 ,k=1that is, the sum of the squares of the discrepancies, has a minimum.a) Show that the least-squares principle for relation (8.79) leads to the followingsystem of linear equations![xk , xk ]a + [xk , 1]b = [xk , yk ],[1, xk ]a + [1, 1]b = [1, yk ],4788 Differential Calculus in Several VariablesTable 8.1Temperature, °CTemperature, °CFrequency, %0Frequency, %39201365542518210743025415100for determining the coefficients a and b.
Here, following Gauss, we write [xk , xk ] :=x1 x1 + · · · + xn xn , [xk , 1] := x1 · 1 + · · · + xn · 1, [xk , yk ] := x1 y1 + · · · + xn yn , andso forth.b) Write the system of equations for the numbers a1 , . . . , am , b to which theleast-squares principle leads when Eq. (8.79) is replaced by the relationy=mai x i + b,i=1(or, more briefly, y = ai x i + b) between the quantities x 1 , . . . , x m and y.c) How can the method of least squares be used to find empirical formulas ofthe formy = cx1α1 · · · xnαnconnecting physical quantities x1 , .
. . , xm with the quantity y?d) (M. Germain) The frequency R of heart contractions was measured at different temperatures T in several dozen specimens of Nereis diversicolor. The frequencies were expressed in percents relative to the contraction frequency at 15 °C. Theresults are given in Table 8.1.The dependence of R on T appears to be exponential. Assuming R = AebT , findthe values of the constants A and b that best fit the experimental results.9.
a) Show that in Huygens’ problem, studied in Example 5, the function (8.71)tends to zero if at least one of the variables m1 , . . . , mn tends to infinity.b) Show that the function (8.71) has a maximum point in Rn and hence theunique critical point of that function in Rn must be its maximum.c) Show that the quantity v defined by formula (8.72) is monotonically increasing as n increases and find its limit as n → ∞.10.
a) During so-called exterior disk grinding the grinding tool – a rapidly rotatinggrinding disk (with an abrasive rim) that acts as a file – is brought into contact withthe surface of a circular machine part that is rotating slowly compared with the disk(see Fig. 8.3).The disk K is gradually pressed against the machine part D, causing a layer H ofmetal to be removed, reducing the part to the required size and producing a smoothworking surface for the device. In the machine where it will be placed this surfacewill usually be a working surface. In order to extend its working life, the metal of themachine part is subjected to a preliminary annealing to harden the steel.
However,8.4 Real-valued Functions of Several Variables479Fig. 8.3because of the high temperature in the contact zone between the machine part andthe grinding disk, structural changes can (and frequently do) occur in a certain layerΔ of metal in the machine part, resulting in decreased hardness of the steel in thatlayer. The quantity Δ is a monotonic function of the rate s at which the disk isapplied to the machine part, that is, Δ = ϕ(s). It is known that there is a certaincritical rate s0 > 0 at which the relation Δ = 0 still holds, while Δ > 0 whenevers > s0 . For the following discussion it is convenient to introduce the relations = ψ(Δ)inverse to the one just given.
This new relation is defined for Δ > 0.Here ψ is a monotonically increasing function known experimentally, definedfor Δ ≥ 0, and ψ(0) = s0 > 0.The grinding process must be carried out in such a way that there are no structuralchanges in the metal on the surface eventually produced.In terms of rapidity, the optimal grinding mode under these conditions wouldobviously be a set of variations in the rate s of application of the grinding disk forwhichs = ψ(δ),where δ = δ(t) is the thickness of the layer of metal not yet removed up to time t, or,what is the same, the distance from the rim of the disk at time t to the final surfaceof the device being produced. Explain this.b) Find the time needed to remove a layer of thickness H when the rate of application of the disk is optimally adjusted.c) Find the dependence s = s(t) of the rate of application of the disk on timeψin the optimal mode under the condition that the function Δ −→ s is linear: s =s0 + λΔ.Due to the structural properties of certain kinds of grinding lathes, the rate s canundergo only discrete changes.
This poses the problem of optimizing the productivity of the process under the additional condition that only a fixed number n ofswitches in the rate s are allowed. The answers to the following questions give apicture of the optimal mode.4808 Differential Calculus in Several Variables H dδd) What is the geometric interpretation of the grinding time t (H ) = 0 ψ(δ)that you found in part b) for the optimal continuous variation of the rate s?e) What is the geometric interpretation of the time lost in switching from theoptimal continuous mode of variation of s to the time-optimal stepwise mode ofvariation of s?f) Show that the points 0 = xn+1 < xn < · · · < x1 < x0 = H of the closed interval [0, H ] at which the rate should be switched must satisfy the conditions 111−=−(xi )(xi − xi−1 ) (i = 1, .
. . , n)ψ(xi+1 ) ψ(xi )ψand consequently, on the portion from xi to xi+1 , the rate of application of the diskhas the form s = ψ(xi+1 ) (i = 0, . . . , n).g) Show that in the linear case, when ψ(Δ) = s0 + λΔ, the points xi (in part f))on the closed interval [0, H ] are distributed so that the numberss0 s0s0s0< + xn < · · · < + x1 < + Hλλλλform a geometric progression.11.
a) Verify that the tangent to a curve Γ : I → Rm is defined invariantly relativeto the choice of coordinate system in Rm .b) Verify that the tangent plane to the graph S of a function y = f (x 1 , . . . , x m )is defined invariantly relative to the choice of coordinate system in Rm .c) Suppose the set S ⊂ Rm × R1 is the graph of a function y = f (x 1 , . . . , x m )in coordinates (x 1 , .
. . , x m , y) in Rm × R1 and the graph of a function ỹ =f˜(x̃ 1 , . . . , x̃ m ) in coordinates (x̃ 1 , . . . , x̃ m , ỹ) in Rm × R1 . Verify that the tangentplane to S is invariant relative to a linear change of coordinates in Rm × R1 .∂2fd) Verify that the Laplacian Δf = mi=1 ∂x i 2 (x) is defined invariantly relativeto orthogonal coordinate transformations in Rm .8.5 The Implicit Function Theorem8.5.1 Statement of the Problem and Preliminary ConsiderationsIn this section we shall prove the implicit function theorem, which is important bothintrinsically and because of its numerous applications.Let us begin by explaining the problem. Suppose, for example, we have the relationx2 + y2 − 1 = 0(8.80)between the coordinates x, y of points in the plane R2 .
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