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For example, supposeFy (x, y) = 0.Let us try to choose coordinates ξ, η so that in these coordinates a closed intervalof a coordinate line, for example, the line η = 0, corresponds to an arc of this curve.We setξ = x − x0 ,The Jacobi matrix1Fxη = F (x, y).0Fy(x, y)of this transformation has as its determinant the number Fy (x, y), which by assumption is nonzero at (x0 , y0 ). Then by Theorem 1, this mapping is a diffeomorphism ofa neighborhood of (x0 , y0 ) onto a neighborhood of the point (ξ, η) = (0, 0). Hence,inside this neighborhood, the numbers ξ and η can be taken as new coordinates ofpoints lying in a neighborhood of (x0 , y0 ).

In the new coordinates, the curve obviously has the equation η = 0, and in this sense we have indeed achieved a localrectification of it (see Fig. 8.7).8.6 Some Corollaries of the Implicit Function Theorem503Fig. 8.7Fig. 8.88.6.2 Local Reduction of a Smooth Mapping to Canonical FormIn this subsection we shall consider only one question of this type. To be specific, weshall exhibit a canonical form to which one can locally reduce any smooth mappingof constant rank by means of a suitable choice of coordinates.We recall that the rank of a mapping f : U → Rn of a domain U ⊂ Rm at a pointx ∈ U is the rank of the linear transformation tangent to it at the point, that is, therank of the matrix f (x).

The rank of a mapping at a point is usually denoted rankf (x).Theorem 2 (The rank theorem) Let f : U → Rn be a mapping defined in a neighborhood U ⊂ Rm of a point x0 ∈ Rm . If f ∈ C (p) (U ; Rn ), p ≥ 1, and the mappingf has the same rank k at every point x ∈ U , then there exist neighborhoods O(x0 )of x0 and O(y0 ) of y0 = f (x0 ) and diffeomorphisms u = ϕ(x), v = ψ(y) of thoseneighborhoods, of class C (p) , such that the mapping v = ψ ◦ f ◦ ϕ −1 (u) has thecoordinate representation 1 u , .

. . , uk , . . . , um = u → v = v 1 , . . . , v n = u1 , . . . , uk , 0, . . . , 0(8.119)in the neighborhood O(u0 ) = ϕ(O(x0 )) of u0 = ϕ(x0 ).In other words, the theorem asserts (see Fig. 8.8) that one can choose coordinates(u1 , . . . , um ) in place of (x 1 , . . . , x m ) and (v 1 , . . . , v n ) in place of (y 1 , . . .

, y n ) in5048 Differential Calculus in Several Variablessuch a way that locally the mapping has the form (8.119) in the new coordinates,that is, the canonical form for a linear transformation of rank k.Proof We write the coordinate representationy1 = f 1 x1, . . . , xm ,...yk = f k x1, . . . , xm ,y k+1 = f k+1 x 1 , . . . , x m ,...ny = f n x1, . . . , xm(8.120)of the mapping f : U → Rny , which is defined in a neighborhood of the pointx0 ∈ R mx .

In order to avoid relabeling the coordinates and the neighborhood U , weshall assume that at every point x ∈ U , the principal minor of order k in the upperleft corner of the matrix f (x) is nonzero.Let us consider the mapping defined in a neighborhood U of x0 by the equalitiesu1 = ϕ 1 x 1 , .

. . , x m = f 1 x 1 , . . . , x m ,...uk = ϕ k x 1 , . . . , x m = f k x 1 , . . . , x m ,uk+1 = ϕ k+1 x 1 , . . . , x m = x k+1 ,...mu = ϕm x1, . . . , xm = xm.The Jacobi matrix of this mapping has the form⎛∂f 1∂x 1⎜⎜ .⎜ .⎜ .⎜⎜ ∂f k⎜⎜ ∂x 1⎜ .⎜ .⎜ .⎜⎜⎜⎜⎜⎜⎝···...∂f 1∂x k···...∂f k∂x k0...........................∂f 1∂x k+1...∂f k∂x k+1...···...···...1..0and by assumption its determinant is nonzero in U ..∂f 1∂x m⎞⎟..

⎟⎟. ⎟⎟⎟∂f k ⎟∂x m ⎟.. ⎟⎟,. ⎟⎟⎟0 ⎟⎟⎟⎟⎠1(8.121)8.6 Some Corollaries of the Implicit Function Theorem505By the inverse function theorem, the mapping u = ϕ(x) is a diffeomorphismof smoothness p of some neighborhood Õ(x0 ) ⊂ U of x0 onto a neighborhoodÕ(u0 ) = ϕ(Õ(x0 )) of u0 = ϕ(x0 ).Comparing relations (8.120) and (8.121), we see that the composite function g =f ◦ ϕ −1 : Õ(u0 ) → Rny has the coordinate representationy 1 = f 1 ◦ ϕ −1 u1 , . . . , um = u1 ,...ky = f k ◦ ϕ −1 u1 , . . .

, um = uk ,y k+1 = f k+1 ◦ ϕ −1 u1 , . . . , um = g k+1 u1 , . . . , um ,...y n = f n ◦ ϕ −1 u1 , . . . , um = g n u1 , . . . , um .(8.122)Since the mapping ϕ −1 : Õ(u0 ) → Õ(x0 ) has maximal rank m at each point u ∈Õ(u0 ), and the mapping f : Õ(x0 ) → Rny has rank k at every point x ∈ Õ(x0 ), it follows, as is known from linear algebra, that the matrix g (u) = f (ϕ −1 (u))(ϕ −1 ) (u)has rank k at every point u ∈ Õ(u0 ).Direct computation of the Jacobi matrix of the mapping (8.122) yields⎧⎪⎪⎪1⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪0⎪⎪⎪⎪⎪..⎪⎪.⎪⎪⎨⎪⎪⎪⎪∂g k+1⎪⎪⎪∂u1⎪⎪⎪⎪ ..⎪⎪.⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ∂g n∂u10......1...···...∂g k+1∂uk···∂g n∂uk..............................0.....∂g k+1∂uk+1....···...∂g n∂uk+1···j...⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪∂g k+1 ⎪⎪⎪m∂u ⎪⎪⎪..

⎪⎪⎪. ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪n∂g ⎪⎭.∂umHence at each point u ∈ Õ(u0 ) we obtain ∂g(u) = 0 for i = k + 1, . . . , m;∂uij = k + 1, . . . , n. Assuming that the neighborhood Õ(u0 ) is convex (which canbe achieved by shrinking Õ(u0 ) to a ball with center at u0 , for example), we canconclude from this that the functions g j , j = k + 1, . . . , n, really are independent ofthe variables uk+1 , . . .

, um .5068 Differential Calculus in Several VariablesAfter this decisive observation, we can rewrite the mapping (8.122) asy 1 = u1 ,...y k = uk ,y k+1 = g k+1 u1 , . . . , uk ,...ny = g n u1 , . . . , uk .(8.123)At this point we can exhibit the mapping ψ . We setv 1 = y 1 =: ψ 1 (y),...v k = y k =: ψ k (y),v k+1 = y k+1 − g k+1 y 1 , . . .

, y k =: ψ k+1 (y),...nnn1kv = y − g y , . . . , y =: ψ n (y).(8.124)It is clear from the construction of the functions g j (j = k + 1, . . . , n) that themapping ψ is defined in a neighborhood of y0 and belongs to class C (p) in thatneighborhood.The Jacobi matrix of the mapping (8.124) has the form⎧⎫..⎪⎪⎪⎪10.⎪⎪⎪⎪⎪⎪.⎪⎪..⎪⎪.⎪⎪..0⎪⎪⎪⎪⎪⎪⎪⎪.⎪⎪.⎪⎪01.⎪⎪⎪⎪⎪ .⎪⎪⎪....⎪⎪.....⎪⎪....⎪ .⎪⎪⎪⎨⎬....⎪⎪⎪⎪.k+1k+1⎪⎪..

1⎪⎪⎪− ∂g∂y 1· · · − ∂g∂y k0⎪⎪⎪⎪⎪⎪⎪⎪⎪.....⎪⎪.....⎪⎪..⎪⎪...⎪⎪⎪⎪⎪⎪.⎪⎪⎪⎪..⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪.⎩ ∂g n⎭∂g n.− ∂y 1· · · − ∂y k. 01Its determinant equals 1, and so by Theorem 1 the mapping ψ is a diffeomorphism of smoothness p of some neighborhood Õ(y0 ) of y0 ∈ Rny onto a neighborhood Õ(v0 ) = ψ(Õ(y0 )) of v0 ∈ Rnv .Comparing relations (8.123) and (8.124), we see that in a neighborhood O(u0 ) ⊂Õ(u0 ) of u0 so small that g(O(u0 )) ⊂ Õ(y0 ), the mapping ψ ◦ f ◦ ϕ −1 : O(u0 ) →8.6 Some Corollaries of the Implicit Function Theorem507Rny is a mapping of smoothness p from this neighborhood onto some neighborhoodO(v0 ) ⊂ Õ(v0 ) of v0 ∈ Rnv and that it has the canonical formv 1 = u1 ,...v k = uk ,v k+1 = 0,...(8.125)v n = 0.Setting ϕ −1 (O(u0 )) = O(x0 ) and ψ −1 (O(v0 )) = O(y0 ), we obtain the neighborhoods of x0 and y0 whose existence is asserted in the theorem.

The proof is nowcomplete.Theorem 2, like Theorem 1, is obviously a local version of the correspondingtheorem from linear algebra.In connection with the proof just given of Theorem 2, we make the followingremarks, which will be useful in what follows.Remark 1 If the rank of the mapping f : U → Rn is n at every point of the originalneighborhood U ⊂ Rm , then the point y0 = f (x0 ), where x0 ∈ U , is an interior pointof f (U ), that is, f (U ) contains a neighborhood of this point.Proof Indeed, from what was just proved, the mapping ψ ◦ f ◦ ϕ −1 : O(u0 ) →O(v0 ) has the form 1u , .

. . , un , . . . , um = u → v = v 1 , . . . , v n = u1 , . . . , un ,in this case, and so the image of a neighborhood of u0 = ϕ(x0 ) contains some neighborhood of v0 = ψ ◦ f ◦ ϕ −1 (u0 ).But the mappings ϕ : O(x0 ) → O(u0 ) and ψ : O(y0 ) → O(v0 ) are diffeomorphisms, and therefore they map interior points to interior points.

Writing the originalmapping f as f = ψ −1 ◦ (ψ ◦ f ◦ ϕ −1 ) ◦ ϕ, we conclude that y0 = f (x0 ) is an interior point of the image of a neighborhood of x0 .Remark 2 If the rank of the mapping f : U → Rn is k at every point of a neighborhood U and k < n, then, by Eqs. (8.120), (8.124), and (8.125), in some neighborhood of x0 ∈ U ⊂ Rm the following n − k relations hold; (i = k + 1, .

. . , n).f i x 1, . . . , x m = gi f 1 x 1, . . . , x m , . . . , f k x 1, . . . , x m(8.126)These relations are written under the assumption we have made that the principalminor of order k of the matrix f (x0 ) is nonzero, that is, the rank k is realized on theset of functions f 1 , . . . , f k . Otherwise one may relabel the functions f 1 , . . . , f nand again have this situation.5088 Differential Calculus in Several Variables8.6.3 Functional DependenceDefinition 2 A system of continuous functions f i (x) = f i (x 1 , . . . , x m ) (i = 1,.

. . , n) is functionally independent in a neighborhood of a point x0 = (x01 , . . . , x0m )if for any continuous function F (y) = F (y 1 , . . . , y n ) defined in a neighborhood ofy0 = (y01 , . . . , y0n ) = (f 1 (x0 ), . . . , f n (x0 )) = f (x0 ), the relation F f 1 x1, . . . , xm , . . . , f n x1, . . . , xm ≡ 0is possible at all points of a neighborhood of x0 only when F (y 1 , . . . , y n ) ≡ 0 in aneighborhood of y0 .The linear independence studied in algebra is independence with respect to linearrelationsF y 1 , .

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