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. . , y n = λ1 y 1 + · · · + λn y n .If a system is not functionally independent, it is said to be functionally dependent.When vectors are linearly dependent, one of them obviously is a linear combination of the others. A similar situation holds in the relation of functional dependenceof a system of smooth functions.Proposition 1 If a system f i (x 1 , . . . , x m ) (i = 1, . . . , n) of smooth functions defined on a neighborhood U (x0 ) of the point x0 ∈ Rm is such that the rank of thematrix⎛ 1⎞∂f∂f 1· · · ∂xm1∂x⎜ ... ⎟..⎜ .... ⎟⎝⎠ (x)∂f n∂f n· · · ∂x m∂x 1is equal to the same number k at every point x ∈ U , thena) when k = n, the system is functionally independent in a neighborhood of x0 ;b) when k < n, there exist a neighborhood of x0 and k functions of the system,say f 1 , .
. . , f k such that the other n − k functions can be represented as f i x 1, . . . , x m = gi f 1 x 1, . . . , x m , . . . , f k x 1, . . . , x min this neighborhood, where g i (y 1 , . . . , y k ), (i = k + 1, . . . , n) are smooth functionsdefined in a neighborhood of y0 = (f 1 (x0 ), . .
. , f n (x0 )) and depending only on kcoordinates of the variable point y = (y 1 , . . . , y n ).Proof In fact, if k = n, then by Remark 1 after the rank theorem, the image of aneighborhood of the point x0 under the mapping8.6 Some Corollaries of the Implicit Function Theoremy1 = f 1 x1, . . . , xm ,...yn = f n x1, . . . , xm509(8.127)contains a neighborhood of y0 = f (x0 ).
But then the relation F f 1 x1, . . . , xm , . . . , f n x1, . . . , xm ≡ 0can hold in a neighborhood of x0 only ifF y1, . . . , yn ≡ 0in a neighborhood of y0 . This proves assertion a).If k < n and the rank k of the mapping (8.127) is realized on the functionsf 1 , . . . , f k , then by Remark 2 after the rank theorem, there exists a neighborhood ofy0 = f (x0 ) and n − k functions g i (y) = g i (y 1 , . . . , y k ) (i = k + 1, .
. . , n), definedon that neighborhood, having the same order of smoothness as the functions of theoriginal system, and such that relations (8.126) hold in some neighborhood of x0 .This proves b).yiWe have now shown that if k < n there exist n − k special functions F i (y) =− g i (y 1 , . . . , y k ) (i = k + 1, . . . , n) that establish the relationsF i f 1 (x), .
. . , f k (x), f i (x) ≡ 0 (i = k + 1, . . . , n)between the functions of the system f 1 , . . . , f k , . . . , f n in a neighborhood of thepoint x0 .8.6.4 Local Resolution of a Diffeomorphism into a Compositionof Elementary OnesIn this subsection we shall show how, using the inverse function theorem, one canrepresent a diffeomorphic mapping locally as a composition of diffeomorphisms,each of which changes only one coordinate.Definition 3 A diffeomorphism g : U → Rm of an open set U ⊂ Rm will be calledelementary if its coordinate representation is!y i = x i , i ∈ {1, . .
. , m}, i = j,y j = gj x 1, . . . , x m ,that is, under the diffeomorphism g : U → Rm only one coordinate of the pointbeing mapped is changed.5108 Differential Calculus in Several VariablesProposition 2 If f : G → Rm is a diffeomorphism of an open set G ⊂ Rm , then forany point x0 ∈ G there is a neighborhood of the point in which the representationf = g1 ◦ · · · ◦ gn holds, where g1 , . . . , gn are elementary diffeomorphisms.Proof We shall verify this by induction.If the original mapping f is itself elementary, the proposition holds trivially for it.Assume that the proposition holds for diffeomorphisms that alter at most (k − 1)coordinates, where k − 1 < n. Now consider a diffeomorphism f : G → Rm thatalters k coordinates:y1 = f 1 x1, .
. . , xm ,...yk = f k x1, . . . , xm ,(8.128)y k+1 = x k+1 ,...ym = xm.We have assumed that it is the first k coordinates that are changed, which canbe achieved by linear changes of variable. Hence this assumption causes no loss ingenerality.Since f is a diffeomorphism, its Jacobi matrix f (x) is nondegenerate at eachpoint, for −1 )*−1ff (x) = f (x) .Let us fix x0 ∈ G and compute the determinant of f (x0 ): ∂f 1 1 ∂x . .. k ∂f ∂x 1 . . .···...∂f 1∂x k···...∂f k0...∂x k...........................∂f 1∂x k+1...∂f k∂x k+1...···...···...1..0... . ∂f 1 ∂x 1k∂fm∂x (x ) = ..0 ... k.
∂f∂x 10 1 ∂f 1∂x m···...···.. . (x0 ) = 0.∂f k ∂f 1∂x k∂x kThus one of the minors of order k − 1 of this last determinant must be nonzero.Again, for simplicity of notation, we shall assume that the principal minor of orderk − 1 is nonzero. Now consider the auxiliary mapping g : G → Rm defined by the8.6 Some Corollaries of the Implicit Function Theorem511equalitiesu1 = f 1 x 1 , . .
. , x m ,...k−1= f k−1 x 1 , . . . , x m ,u(8.129)uk = x k ,...um = x m .Since the Jacobian ∂f 11 1· · · ∂x∂fk−1 ∂x ..... .... k−1k−1 ∂f· · · ∂f ∂x 1k−1∂x ..... .. ..0........................∂f 1∂x k...∂f k−1∂x k...···...···...1...0.. . 1 ∂f ∂x 1k−1∂f .∂x m (x ) = .... 0 ∂f k−1. ∂x 10 1 ∂f 1∂x m···...···.. . (x0 ) = 0∂f k−1 ∂f 1∂x k−1∂x k−1of the mapping g : G → Rm is nonzero at x0 ∈ G, the mapping g is a diffeomorphism in some neighborhood of x0 .Then, in some neighborhood of u0 = g(x0 ) the mapping inverse to g, x =g −1 (u), is defined, making it possible to introduce new coordinates (u1 , . .
. , um )in a neighborhood of x0 .Let h = f ◦ g −1 . In other words, the mapping y = h(u) is the mapping (8.128)y = f (x) written in u-coordinates. The mapping h, being the composition of diffeomorphisms, is a diffeomorphism of some neighborhood of u0 . Its coordinateexpression obviously has the formy 1 = f 1 ◦ g −1 (u) = u1 ,...y k−1 = f k−1 ◦ g −1 (u) = uk−1 ,y k = f k ◦ g −1 (u),y k+1 = uk+1 ,...y m = um ,that is, h is an elementary diffeomorphism.5128 Differential Calculus in Several VariablesBut f = h ◦ g, and by the induction hypothesis the mapping g defined by (8.129)can be resolved into a composition of elementary diffeomorphisms.
Thus, the diffeomorphism f , which alters k coordinates, can also be resolved into a compositionof elementary diffeomorphisms in a neighborhood of x0 , which completes the induction.8.6.5 Morse’s LemmaThis same circle of ideas contains an intrinsically beautiful lemma of Morse9 on thelocal reduction of smooth real-valued functions to canonical form in a neighborhoodof a nondegenerate critical point. This lemma is also important in applications.Definition 4 Let x0 be a critical point of the function f ∈ C (2) (U ; R) defined in aneighborhood U of this point.The critical point x0 is a nondegenerate critical point of f if the Hessian of the2ffunction at that point (that is, the matrix ∂x∂i ∂xj (x0 ) formed from the second-orderpartial derivatives) has a nonzero determinant.If x0 is a critical point of the function, that is, f (x0 ) = 0, then by Taylor’s formulaf (x) − f (x0 ) =1 ∂ 2fj(x0 ) x i − x0i x j − x0 + o ,x − x0 ,2 .
(8.130)ij2!∂x ∂xi,jMorse’s lemma asserts that one can make a local change of coordinates x = g(y)such that the function will have the form2 2 2 2(f ◦ g)(y) − f (x0 ) = − y 1 − · · · − y k + y k+1 + · · · + y mwhen expressed in y-coordinates.If the remainder term o(,x − x0 ,2 ) were not present on the right-hand side ofEq. (8.130), that is, the difference f (x) − f (x0 ) were a simple quadratic form, then,a is known from algebra, it could be brought into the indicated canonical form bya linear transformation.
Thus the assertion we are about to prove is a local versionof the theorem on reduction of a quadratic form to canonical form. The proof willuse the idea of the proof of this algebraic theorem. We shall also rely on the inversefunction theorem and the following proposition.Hadamard’s lemma Let f : U → R be a function of class C (p) (U ; R), p ≥ 1,defined in a convex neighborhood U of the point 0 = (0, . . . , 0) ∈ Rm and such that9 H.C.M. Morse (1892–1977) – American mathematician; his main work was devoted to the application of topological methods in various areas of analysis.8.6 Some Corollaries of the Implicit Function Theorem513f (0) = 0. Then there exist functions gi ∈ C (p−1) (U ; R) (i = 1, .
. . , m) such that theequalitym f x1, . . . , xm =x i gi x 1 , . . . , x m(8.131)i=1holds in U , and gi (0) =∂f(0).∂x iProof Equality (8.131) is essentially another useful expression for Taylor’s formulawith the integral form of the remainder term. It follows from the equalitiesf x1, .
. . , xm =01df (tx 1 , . . . , tx m )dt =xidtmi=101∂f 1tx , . . . , tx m dt,i∂xif we setgi x 1 , . . . , x m =01∂f 1tx , . . . , tx m dti∂x(i = 1, . . . , m).∂fThe fact that gi (0) = ∂xi (0) (i = 1, . . . , m) is obvious, and it is also not difficultto verify that gi ∈ C (p−1) (U ; R). However, we shall not undertake the verificationjust now, since we shall later give a general rule for differentiating an integral depending on a parameter, from which the property we need for the functions gi willfollow immediately.Thus, up to this verification, Hadamard’s formula (8.131) is proved.Morse’s lemma If f : G → R is a function of class C (3) (G; R) defined on an openset G ⊂ Rm and x0 ∈ G is a nondegenerate critical point of that function, then thereexists a diffeomorphism g : V → U of some neighborhood of the origin 0 in Rmonto a neighborhood U of x0 such that) 2 2 * ) k+1 2 2 *+ y+ · · · + ym(f ◦ g)(y) = f (x0 ) − y 1 + · · · + y kfor all y ∈ V .Proof By linear changes of variable we can reduce the problem to the case whenx0 = 0 and f (x0 ) = 0, and from now on we shall assume that these conditions hold.Since x0 = 0 is a critical point of f , we have gi (0) = 0 in formula (8.131) (i = 1,.
. . , m). Then, also by Hadamard’s lemma,m gi x 1 , . . . , x m =x j hij x 1 , . . . , x m ,j =15148 Differential Calculus in Several Variableswhere hij are smooth functions in a neighborhood of 0 and consequentlym f x1, . . . , xm =x i x j hij x 1 , . . . , x m .(8.132)i,j =1By making the substitution h̃ij = 12 (hij + hj i ) if necessary, we can assume thathij = hj i . We remark also that, by the uniqueness of the Taylor expansion, the con2ftinuity of the functions hij implies that hij (0) = ∂x∂i ∂xj (0) and hence the matrix(hij (0)) is nondegenerate.The function f has now been written in a manner that resembles a quadraticform, and we wish, so to speak, to reduce it to diagonal form.As in the classical case, we proceed by induction.Assume that there exist coordinates u1 , .
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