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. . , v m , L to the canonical variables8 W.R. Hamilton (1805–1865) – famous Irish mathematician and specialist in mechanics. He stateda variational principle (Hamilton’s principle), constructed a phenomenological theory of optic phenomena, and was the creator of quaternions and the founder of vector analysis (in fact, the term“vector” is due to him).8.5 The Implicit Function Theorem497t, x 1 , . . . , x m , p1 , . . . , pm , H and show that in these variables the system (8.110)becomes the following system of Hamilton equations:ṗi = −∂H,∂x iẋ i =∂H∂pi(i = 1, . .
. , m).(8.111)4. The implicit function theorem.The solution of this problem gives another proof of the fundamental theorem of thissection, perhaps less intuitive and constructive than the one given above, but shorter.a) Suppose the hypotheses of the implicit function theorem are satisfied, and letFyi (x, y) =∂F i∂F i,...,(x, y)∂y n∂y 1be the ith row of the matrix Fy (x, y).Show that the determinant of the matrix formed from the vectors Fyi (xi , yi ) isnonzero if all the points (xi , yi ) (i = 1, .
. . , n) lie in some sufficiently small neighborhood U = Ixm × Iyn of (x0 , y0 ).b) Show that, if for x ∈ Ixm there are points y1 , y2 ∈ Iyn such that F (x, y1 ) = 0and F (x, y2 ) = 0, then for each i ∈ {1, . . . , n} there is a point (x, yi ) lying on theclosed interval with endpoints (x, y1 ) and (x, y2 ) such thatFyi (x, yi )(y2 − y1 ) = 0 (i = 1, . .
. , n).Show that this implies that y1 = y2 , that is, if the implicit function f : Ixm → Iynexists, it is unique.c) Show that if the open ball B(y0 ; r) is contained in Iyn , then F (x0 , y) = 0 for,y − y0 ,Rn = r > 0.d) The function ,F (x0 , y),2Rn is continuous and has a positive minimum valueμ on the sphere ,y − y0 ,Rn = r.e) There exists δ > 0 such that for ,x − x0 ,Rm < δ we have===F (x, y)=2 n ≥ 1 μ,R2=2==F (x, y)= n < 1 μ,R2if ,y − y0 ,Rn = r,if y = y0 .f) For any fixed x such that ,x − x0 , < δ the function ,F (x, y),2Rn attains aminimum at some interior point y = f (x) of the open ball ,y −y0 ,Rn ≤ r, and sincethe matrix Fy (x, f (x)) is invertible, it follows that F (x, f (x)) = 0. This establishesthe existence of the implicit function f : B(x0 ; δ) → B(y0 ; r).g) If Δy = f (x + Δx) − f (x), then) *−1 ) *· F̃x Δx,Δy = − F̃y4988 Differential Calculus in Several Variableswhere F̃y is the matrix whose rows are the vectors Fyi (xi , yi ) (i = 1, .
. . , n), (xi , yi )being a point on the closed interval with endpoints (x, y) and (x + Δx, y + Δy).The symbol F̃x has a similar meaning.Show that this relation implies that the function y = f (x) is continuous.h) Show that*−1 ) *) · F̃x x, f (x) .f (x) = − F̃y x, f (x)5. “If f (x, y, z) = 0, then∂z∂y·∂y∂x·∂x∂z= −1.”a) Give a precise meaning to this statement.b) Verify that it holds in the example of Clapeyron’s ideal gas equationP ·V= constTand in the general case of a function of three variables.c) Write the analogous statement for the relation f (x 1 , . . . , x m ) = 0 among mvariables. Verify that it is correct.6.
Show that the roots of the equationzn + c1 zn−1 + · · · + cn = 0are smooth functions of the coefficients, at least when they are all distinct.8.6 Some Corollaries of the Implicit Function Theorem8.6.1 The Inverse Function TheoremDefinition 1 A mapping f : U → V , where U and V are open subsets of Rm , is aC (p) -diffeomorphism or a diffeomorphism of smoothness p (p = 0, 1, .
. .), if1) f ∈ C (p) (U ; V );2) f is a bijection;3) f −1 ∈ C (p) (V ; U ).A C (0) -diffeomorphism is called a homeomorphism.As a rule, in this book we shall consider only the smooth case, that is, the casep ∈ N or p = ∞.The basic idea of the following frequently used theorem is that if the differentialof a mapping is invertible at a point, then the mapping itself is invertible in someneighborhood of the point.8.6 Some Corollaries of the Implicit Function Theorem499Theorem 1 (Inverse function theorem) If a mapping f : G → Rm of a domain G ⊂Rm is such that10 f ∈ C (p) (G; Rm ), p ≥ 1,20 y0 = f (x0 ) at x0 ∈ G,30 f (x0 ) is invertible,then there exists a neighborhood U (x0 ) ⊂ G of x0 and a neighborhood V (y0 ) ofy0 such that f : U (x0 ) → V (y0 ) is a C (p) -diffeomorphism. Moreover, if x ∈ U (x0 )and y = f (x) ∈ V (y0 ), then−1 −1 (y) = f (x) .fProof We rewrite the relation y = f (x) in the formF (x, y) = f (x) − y = 0.(8.112)The function F (x, y) = f (x) − y is defined for x ∈ G and y ∈ Rm , that is it isdefined in the neighborhood G × Rm of the point (x0 , y0 ) ∈ Rm × Rm .We wish to solve Eq.
(8.112) with respect to x in some neighborhood of (x0 , y0 ).By hypotheses 10 , 20 , 30 of the theorem the mapping F (x, y) has the property thatF ∈ C (p) G × Rm ; Rm ,p ≥ 1,F (x0 , y0 ) = 0,Fx (x0 , y0 ) = f (x0 )is invertible.By the implicit function theorem there exist a neighborhood Ix × Iy of (x0 , y0 )and a mapping g ∈ C (p) (Iy ; Ix ) such thatf (x) − y = 0 ⇔ x = g(y)(8.113)for any point (x, y) ∈ Ix × Iy and)*−1 ) *g (y) = − Fx (x, y)Fy (x, y) .In the present caseFx (x, y) = f (x),Fy (x, y) = −E,where E is the identity matrix; therefore−1g (y) = f (x) .(8.114)If we set V = Iy and U = g(V ), relation (8.113) shows that the mappings f :U → V and g : V → U are mutually inverse, that is, g = f −1 on V .5008 Differential Calculus in Several VariablesFig.
8.5Since V = Iy , it follows that V is a neighborhood of y0 . This means that underhypotheses 10 , 20 , and 30 the image y0 = f (x0 ) of x0 ∈ G, which is an interior pointof G, is an interior point of the image f (G) of G. By formula (8.114) the matrixg (y0 ) is invertible. Therefore the mapping g : V → U has properties 10 , 20 , and30 relative to the domain V and the point y0 ∈ V .
Hence by what has already beenproved x0 = g(y0 ) is an interior point of U = g(V ).Since by (8.114) hypotheses 10 , 20 , and 30 obviously hold at any point y ∈ V ,any point x = g(y) is an interior point of U . Thus U is an open (and obviously evenconnected) neighborhood of x0 ∈ Rm .We have now verified that the mapping f : U → V satisfies all the conditions ofDefinition 1 and the assertion of Theorem 1.We shall now give several examples that illustrate Theorem 1.The inverse function theorem is very often used in converting from one coordinate system to another. The simplest version of such a change of coordinates wasstudied in analytic geometry and linear algebra and has the form⎛ 1⎞ ⎛ 1⎞ ⎛ 1⎞1yxa1 · · · am.. ⎟ ⎜ ..
⎟..⎜ .. ⎟ ⎜ ...⎝ . ⎠=⎝ .. ⎠⎝ . ⎠a1mym···mamxmjmor, in compact notation, y j = ai x i . This linear transformation A : Rmx → Ry has−1mmman inverse A : Ry → Rx defined on the entire space Ry if and only if the matrixjj(ai ) is invertible, that is, det(ai ) = 0.The inverse function theorem is a local version of this proposition, based on thefact that in a neighborhood of a point a smooth mapping behaves approximately likeits differential at the point.Example 1 (Polar coordinates) The mapping f : R2+ → R2 of the half-plane R2+ ={(ρ, ϕ) ∈ R2 | ρ ≥ 0} onto the plane R2 defined by the formulax = ρ cos ϕ,y = ρ sin ϕ,is illustrated in Fig.
8.5.(8.115)8.6 Some Corollaries of the Implicit Function Theorem501Fig. 8.6The Jacobian of this mapping, as can be easily computed, is ρ, that is, it isnonzero in a neighborhood of any point (ρ, ϕ), where ρ > 0. Therefore formulas(8.115) are locally invertible and hence locally the numbers ρ and ϕ can be taken asnew coordinates of the point previously determined by the Cartesian coordinates xand y.The coordinates (ρ, ϕ) are a well known system of curvilinear coordinates on theplane – polar coordinates.
Their geometric interpretation is shown in Fig. 8.5. Wenote that by the periodicity of the functions cos ϕ and sin ϕ the mapping (8.115) isonly locally a diffeomorphism when ρ > 0; it is not bijective on the entire plane.That is the reason that the change from Cartesian to polar coordinates always involves a choice of a branch of the argument ϕ (that is, an indication of its range ofvariation).Polar coordinates (ρ, ψ, ϕ) in three-dimensional space R3 are called sphericalcoordinates.
They are connected with Cartesian coordinates by the formulasz = ρ cos ψ,y = ρ sin ψ sin ϕ,(8.116)x = ρ sin ψ cos ϕ.The geometric meaning of the parameters ρ, ψ , and ϕ is shown in Fig. 8.6.The Jacobian of the mapping (8.116) is ρ 2 sin ψ , and so by Theorem 1 the mapping is invertible in a neighborhood of each point (ρ, ψ, ϕ) at which ρ > 0 andsin ψ = 0.The sets where ρ = const, ϕ = const, or ψ = const in (x, y, z)-space obviouslycorrespond to a spherical surface (a sphere of radius ρ), a half-plane passing throughthe z-axis, and the surface of a cone whose axis is the z-axis respectively.Thus in passing from coordinates (x, y, z) to coordinates (ρ, ψ, ϕ), for example,the spherical surface and the conical surface are flattened; they correspond to piecesof the planes ρ = const and ψ = const respectively.
We observed a similar phenomenon in the two-dimensional case, where an arc of a circle in the (x, y)-planecorresponded to a closed interval on the line in the plane with coordinates (ρ, ϕ)(see Fig. 8.5). Please note that this is a local straightening.5028 Differential Calculus in Several VariablesIn the m-dimensional case polar coordinates are introduced by the relationsx 1 = ρ cos ϕ1 ,x 2 = ρ sin ϕ1 cos ϕ2 ,...(8.117)x m−1= ρ sin ϕ1 sin ϕ2 · · · sin ϕm−2 cos ϕm−1 ,xm= ρ sin ϕ1 sin ϕ2 · · · sin ϕm−2 sin ϕm−1 .The Jacobian of this transformation isρ m−1 sinm−2 ϕ1 sinm−3 ϕ2 · · · sin ϕm−2 ,(8.118)and by Theorem 1 it is also locally invertible everywhere where this Jacobian isnonzero.Example 2 (The general idea of local rectification of curves) New coordinates areusually introduced for the purpose of simplifying the analytic expression for theobjects that occur in a problem and making them easier to visualize in the newnotation.Suppose for example, a curve in the plane R2 is defined by the equationF (x, y) = 0.Assume that F is a smooth function, that the point (x0 , y0 ) lies on the curve, that is,F (x0 , y0 ) = 0, and that this point is not a critical point of F .
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