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, y n−1 :=⎪⎪⎪⎪1 1m 1n−1 ˜ 1m 1n−1 = 0,⎪⎪⎨. = F x ,...,x ,y ,...,y ,f x ,...,x ,y ,...,y..(8.98)⎪⎪n−11m1n−1⎪⎪x ,...,x ,y ,...,y:=Φ⎪⎪ 1⎩n−1m1n−1=Fx , . . . , x , y , . . . , y , f˜ x 1 , . . . , x m , y 1 , . . . , y n−1 = 0.It is clear that Φ i ∈ C (p) (I˜xm × I˜yn−1 ; R) (i = 1, . . . , n − 1), andΦ i x01 , .
. . , x0m ; y01 , . . . , y0n−1 = 0 (i, . . . , n − 1),since f˜(x01 , . . . , x0m , y01 , . . . , y0n−1 ) = y0n and F i (x0 , y0 ) = 0 (i = 1, . . . , n).By definition of the functions Φ k (k = 1, . . . , n − 1),∂Φ k∂F k ∂F k ∂ f˜=+ n · i∂y i∂y i∂y∂y(i, k = 1, . . . , n − 1).(8.99)Further settingΦ n x 1 , . . . , x m , y1 , . . . , y n−1 :== F n x 1 , . .
. , x m , y 1 , . . . , y n−1 , f˜ x 1 , . . . , x m , y 1 , . . . , y n−1 ,we find by (8.97) that Φ n ≡ 0 in its domain of definition, and therefore∂Φ n ∂F n ∂F n ∂ f˜=+ n · i ≡ 0 (i = 1, . . . , n − 1).∂y i∂y i∂y∂y(8.100)4928 Differential Calculus in Several VariablesTaking account of relations (8.99) and (8.100) and the properties of determinants,we can now observe that the determinant of the matrix (8.94) equals the determinantof the matrix⎛ 1⎞∂ f˜∂F∂F 1 ∂ f˜∂F 1∂F 1∂F 1+···+n ·n ·n11n−1n−1∂y∂y∂y∂y∂y∂y⎜ ∂y...... ⎟..⎜⎟.⎜...
⎟=⎝ n⎠∂ f˜∂F∂F n ∂ f˜∂F n∂F n∂F n+····+·nnnl1n−ln−1∂y∂y∂y∂y∂y∂y∂y⎛ ∂Φ 1⎞11∂F· · · ∂y∂Φn−11∂y n⎜ ∂y.... ⎟..⎜ ...... ⎟⎜⎟= ⎜ n−1⎟.∂Φ n−1∂F n−1 ⎟⎜ ∂Φ···n ⎠n−1∂y⎝ ∂y 1∂y∂F n0···0∂y nBy assumption,∂F n∂y n= 0, and the determinant of the matrix (8.94) is nonzero.Consequently, in some neighborhood of (x01 , . . . , x0m , y01 , . . . y0n−1 ) the determinantof the matrix⎞⎛ ∂Φ 1∂Φ 1···1n−1∂y⎜ ∂y. 1.. ⎟..m 1n−1⎜ ....
⎟⎠ x ,...,x ,y ,...,y⎝n−1∂Φ n−1· · · ∂Φ∂y 1∂y n−1is nonzero.Then by the induction hypothesis there exist an interval I m+n−1 = Ixm × Iyn−1 ⊂I˜xm × I˜yn−1 , which is a neighborhood of (x01 , . . . , x0m , y01 , . . . , y0n−1 ) in Rm−1 , anda mapping f ∈ C (p) (Ixm ; Iyn−1 ) such that the system (8.98) is equivalent on theinterval I m+n−1 = Ixm × Iyn−1 to the relations⎧ 11 1m⎪⎨ y.
= f x , . . . , x ,..⎪⎩ n−1= f n−1 x 1 , . . . , x m .yx ∈ Ixm .(8.101)d) Since Iyn−1 ⊂ I˜yn−1 , and Ixm ⊂ I˜xm , substituting f 1 , . . . , f n−1 from (8.101) inplace of the corresponding variables in the functiony n = f˜ x 1 , . . . , x m , y 1 , . . . , y n−1from (8.97) we obtain a relationyn = f n x1, . .
. , xmbetween y n and (x 1 , . . . , x m ).(8.102)8.5 The Implicit Function Theoreme) We now show that the system⎧ 11 1m⎪⎨ y. = f x , . . . , x ,..⎪⎩ ny = f n x1, . . . , xm ,493x ∈ Ixm ,(8.103)which defines a mapping f ∈ C (p) (Ixm ; Iyn ), where Iyn = Iyn−1 × Iy1 , is equivalent tothe system of equations (8.90) in the neighborhood I m+n = Ixm × Iyn .In fact, inside I˜m+n = (I˜xm × I˜yn−1 ) × Iy1 we began by replacing the last equationof the original system (8.90) with the equality y n = f˜(x, y 1 , .
. . , y n−1 ), which isequivalent to it by virtue of (8.97). From the second system so obtained, we passedto a third system equivalent to it by replacing the variable y n in the first n − 1equations with f˜(x, y 1 , . . . , y n−1 ). We then replaced the first n − 1 equations (8.98)of the third system inside Ixm × Iyn−1 ⊂ I˜xm × I˜yn−1 with relations (8.101), which areequivalent to them. In that way, we obtained a fourth system, after which we passedto the final system (8.103), which is equivalent to it inside Ixm × Iyn−1 × Iy1 = I m+n ,by replacing the variables y 1 , . . . , y n−1 with their expressions (8.101) in the lastequation y n = f˜(x 1 , . .
. , x m , y 1 , . . . , y n−1 ) of the fourth system, obtaining (8.102)as the last equation.f) To complete the proof of the theorem it remains only to verify formula (8.96).Since the systems (8.90) and (8.91) are equivalent in the neighborhood Ixm × Iynof the point (x0 , y0 ), it follows thatF x, f (x) ≡ 0, if x ∈ Ixm .In coordinates this means that in the domain IxmF k x 1 , . . .
, x m , f 1 x 1 , . . . , x m , . . . , f n x 1 , . . . , x m ≡ 0 (k = 1, . . . , n).(8.104)Since f ∈ C (p) (Ixm ; Iyn ) and F ∈ C (p) (U ; Rn ), where p ≥ 1, it follows thatF (·, f (·)) ∈ C (p) (Ixm ; Rn ) and, differentiating the identity (8.104), we obtain∂F k ∂F k ∂f j+·= 0 (k = 1, . . . , n; i = 1, . . . , m).∂x i∂y j ∂x in(8.105)j =1Relations (8.105) are obviously equivalent to the single matrix equalityFx (x, y) + Fy (x, y) · f (x) = 0,in which y = f (x).Taking account of the invertibility of the matrix Fy (x, y) in a neighborhood ofthe point (x0 , y0 ), we find by this equality that*−1 ) *) f (x) = − Fy x, f (x)Fx x, f (x) ,and the theorem is completely proved.4948 Differential Calculus in Several Variables8.5.5 Problems and Exercises1. On the plane R2 with coordinates x and y a curve is defined by the relationF (x, y) = 0, where F ∈ C (2) (R2 , R). Let (x0 , y0 ) be a noncritical point of the function F (x, y) lying on the curve.a) Write the equation of the tangent to this curve at this point (x0 , y0 ).b) Show that if (x0 , y0 ) is a point of inflection of the curve, then the followingequality holds: 2 2Fxx Fy − 2FxyFx Fy + FyyFx (x0 , y0 ) = 0.c) Find a formula for the curvature of the curve at the point (x0 , y0 ).2.
The Legendre transform in m variables. The Legendre transform of x 1 , . . . , x mand the function f (x 1 , . . . , x m ) is the transformation to the new variables ξ1 , . . . , ξmand function f ∗ (ξ1 , . . . , ξm ) defined by the relations⎧∂f 1⎪⎪ξ =x , . . . , x m (i = 1, . . . , m),⎪⎨ i ∂x im⎪∗⎪f(ξ,...,ξ)=ξi x i − f x 1 , . . .
, x m .⎪1m⎩(8.106)i=1a) Give a geometric interpretation of the Legendre transform (8.106) as the transition from the coordinates (x 1 , . . . , x m , f (x 1 , . . . , x m )) of a point on the graph ofthe function f (x) to the parameters (ξ1 , .
. . , ξm , f ∗ (ξ1 , . . . , ξm )) defining the equation of the plane tangent to the graph at that point.b) Show that the Legendre transform is guaranteed to be possible locally if f ∈2f(2)C and det( ∂x∂i ∂xj ) = 0.c) Using the same definition of convexity for a function f (x) = f (x 1 , . . . , x m )as in the one-dimensional case (taking x to be the vector (x 1 , . . . , x m ) ∈ Rm ), showthat the Legendre transform of a convex function is a convex function.d) Show that∗df =mi=1x dξi +imi=1ξi dx − df =imx i dξi ,i=1and deduce from this relation that the Legendre transform is involutive, that is, verifythe equality ∗ ∗f (x) = f (x).e) Taking account of d), write the transform (8.106) in the following form, whichis symmetric in the variables:8.5 The Implicit Function Theorem495⎧m 1⎪∗m⎪⎪ξi x i ,⎨ f (ξ1 , . .
. , ξm ) + f x , . . . , x =⎪⎪∂f 1⎪⎩ ξi =x , . . . , xm ,∂x ii=1(8.107)∂f ∗xi =(ξ1 , . . . , ξm )∂ξior, more briefly, in the formf ∗ (ξ ) + f (x) = ξ x,x = ∇f ∗ (ξ ),ξ = ∇f (x),where∇f (x) =∂f∂f,...,(x),∂x m∂x 1ξ x = ξi x i =m∇f ∗ (ξ ) =∂r ∗∂f ∗,...,(ξ ),∂ξ1∂ξmξi x i .i=1f) The matrix formed from the second-order partial derivatives of a function(and sometimes the determinant of this matrix) is called the Hessian of the functionat a given point.2f∂2f ∗Let dij and dij∗ be the co-factors of the elements ∂x∂i ∂xj and ∂ξi ∂ξj of the Hessians⎛⎜⎜⎝∂2f∂x 1 ∂x 1...···...∂2f∂x 1 ∂x m∂2f∂x m ∂x 1···∂2f∂x m ∂x m...⎞⎛⎟⎟ (x),⎠⎜⎜⎝∂2f ∗∂ξ1 ∂ξ1...···...∂2f ∗∂ξ1 ∂ξm∂2f ∗∂ξm ∂ξ1···∂2f ∗∂ξm ∂ξm...⎞⎟⎟ (ξ )⎠of the functions f (x) and f ∗ (ξ ), and let d and d ∗ be the determinants of thesematrices.
Assuming that d = 0, show that d · d ∗ = 1 and thatdij∗∂ 2f(x)=(ξ ),∂x i ∂x jd∗dij∂ 2f ∗(x).(ξ ) =∂ξi ∂ξjdg) A soap film spanning a wire frame forms a so-called minimal surface, havingminimal area among all the surfaces spanning the contour.If that surface is locally defined as the graph of a function z = f (x, y), it turnsout that the function f must satisfy the following equation for minimal surfaces: − 2fx fy fxy+ 1 + fx2 fyy= 0.1 + fy2 fxxShow that after a Legendre transform is performed this equation is brought intothe form ∗ + 2ξ ηfξ∗η + 1 + ξ 2 fξ∗ξ = 0.1 + η2 fηη4968 Differential Calculus in Several Variables3. Canonical variables and the Hamilton equations.8a) In the calculus of variations and the fundamental principles of classical mechanics the following system of equations, due to Euler and Lagrange, plays animportant role:⎧⎪⎨ ∂L − d ∂L (t, x, v) = 0,∂xdt ∂v(8.108)⎪⎩ v = ẋ(t),where L(t, x, v) is a given function of the variables t, x, v, of which t is usuallytime, x the coordinate, and v the velocity.The system (8.108) consists of two relations in three variables.
Usually we wishto determine x = x(t) and v = v(t) from (8.108), which essentially reduces to determining the relation x = x(t), since v = dxdt .Write the first equation of (8.108) in more detail, expanding the derivative dtdtaking account of the equalities x = x(t) and v = v(t).b) Show that if we change from the coordinates t, x, v, L to the so-called canonical coordinates t, x, p, H by performing the Legendre transform (see Problem 2)⎧⎨ p = ∂L ,∂v⎩H = pv − Lwith respect to the variables v and L to replace them with p and H , then the Euler–Lagrange system (8.108) assumes the symmetric formṗ = −∂H,∂xẋ =∂H,∂p(8.109)in which it is called system of Hamilton equations.c) In the multidimensional case, when L = L(t, x 1 , .
. . , x m , v 1 , . . . , v m ) theEuler–Lagrange system has the form⎧⎪⎨ ∂L − d ∂L (t, x, v) = 0,∂x idt ∂v i(8.110)⎪⎩ iiv = ẋ (t) (i = 1, . . . , m),where for brevity we have set x = (x 1 , . . . , x m ), v = (v 1 , . . . , v m ).By performing a Legendre transform with respect to the variables v 1 , . . . , v m , L,change from the variables t, x 1 , . . . , x m , v 1 , .
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