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. . , um in a neighborhood U1 of 0 ∈ Rm ,that is, a diffeomorphism x = ϕ(u), such thatm2 2ui uj Hij u1 , . . . , um(f ◦ ϕ)(u) = ± u1 ± · · · ± ur−1 +(8.133)i,j =rin the coordinates u1 , . . . , um , where r ≥ 1 and Hij = Hj i .We observe that relation (8.133) holds for r = 1, as one can see from (8.132),where Hij = hij .i jBy the hypothesis of the lemma the quadratic form mi,j =1 x x hij (0) is nondegenerate, that is, det(hij (0)) = 0. The change of variable x = ϕ(u) is carried outby a diffeomorphism, so that det ϕ (0) = 0.
But then the matrix of the quadratic12r−12i jform ±(u ) ±· · ·±(u ) + mi,j =r u u Hij (0) obtained from the matrix (hij (0))through right-multiplication by the matrix ϕ (0) and left-multiplication by the transpose of ϕ (0) is also nondegenerate. Consequently, at least one of the numbersHij (0)(i, j = r, . . . , m) is nonzero.
By a linear change of variable we can bring thei jform mi,j =r u u Hij (0) to diagonal form, and so we may assume that Hrr (0) = 0in Eq. (8.133). By the continuity of the functions Hij (u) the inequality Hrr (u) = 0will also hold in some neighborhoodof u = 0.√Let us set ψ(u1 , . . . , um ) = |Hrr (u)|. Then the function ψ belongs to the classC (1) (U2 ; R) in some neighborhood U2 ⊂ U1 of u = 0.
We now change to coordinates (v 1 , . . . , v m ) by the formulasv i = ui ,i = r, ui Hir (u1 , . . . , um ) .v r = ψ u1 , . . . , um ur +Hrr (u1 , . . . , um )(8.134)i>rThe Jacobian of the transformation (8.134) at u = 0 is obviously equal to ψ(0),that is, it is nonzero. Then by the inverse function theorem we can assert that insome neighborhood U3 ⊂ U2 of u = 0 the mapping v = ψ(u) defined by (8.134)8.6 Some Corollaries of the Implicit Function Theorem515is a diffeomorphism of class C (1) (U3 ; Rm ) and therefore the variables (v 1 , .
. . , v m )can indeed serve as coordinates of points in U3 .We now separate off in Eq. (8.133) all termsmur uj Hrj u1 , . . . , um ,ur ur Hrr u1 , . . . , um + 2(8.135)j =r+1containing ur . In the expression (8.135) for the sum of these terms we have used thefact that Hij = Hj i .Comparing (8.134) and (8.135), we see that we can rewrite (8.135) in the form 21 i±v r v r −u Hir u1 , .
. . , um.Hrri>rThe ambiguous sign ± appears in front of v r v r because Hrr = ±(ψ)2 , the positive sign being taken if Hrr > 0 and the negative sign if Hrr < 0.Thus, after the substitution v = ψ(u), the expression (8.133) becomes the equalityr) i 2 * i jf ◦ ϕ ◦ ψ −1 (v) =± v+v v H̃ij v 1 , . . .
, v m ,i=1i,j >rwhere H̃ij are new smooth functions that are symmetric with respect to the indicesi and j . The mapping ϕ ◦ ψ −1 is a diffeomorphism. Thus the induction from r − 1to r is now complete, and Morse’s lemma is proved.8.6.6 Problems and Exercises1. Compute the Jacobian of the change of variable (8.118) from polar coordinatesto Cartesian coordinates in Rm .2. a) Let x0 be a noncritical point of a smooth function F : U → R defined ina neighborhood U of x0 = (x01 , . .
. , x0m ) ∈ Rm . Show that in some neighborhoodŨ ⊂ U of x0 one can introduce curvilinear coordinates (ξ 1 , . . . , ξ m ) such that theset of points defined by the condition F (x) = F (x0 ) will be given by the equationξ m = 0 in these new coordinates.b) Let ϕ, ψ ∈ C (k) (D; R), and suppose that (ϕ(x) = 0) ⇒ (ψ(x) = 0) in thedomain D. Show that if grad ϕ = 0, then there is a decomposition ψ = θ · ϕ in D,where θ ∈ C (k−1) (D; R).3. Let f : R2 → R2 be a smooth mapping satisfying the Cauchy–Riemann equations∂f 1 ∂f 2=,∂x 1 ∂x 2∂f 1∂f 2=−.∂x 2∂x 15168 Differential Calculus in Several Variablesa) Show that the Jacobian of such a mapping is zero at a point if and only iff (x) is the zero matrix at that point.b) Show that if f (x) = 0, then the inverse f −1 to the mapping f is defined ina neighborhood of f and also satisfies the Cauchy–Riemann equations.4.
Functional dependence (direct proof).a) Show that the functions π i (x) = x i (i = 1, . . . , m), regarded as functions ofthe point x = (x 1 , . . . , x m ) ∈ Rm , form an independent system of functions in aneighborhood of any point of Rm .b) Show that, for any function f ∈ C(Rm ; R) the system π 1 , . . . , π m , f is functionally dependent.c) If the system of smooth functions f 1 , . .
. , f k , k < m, is such that the rankof the mapping f = (f 1 , . . . , f k ) equals k at a point x0 = (x01 , . . . , x0m ) ∈ Rm , thenin some neighborhood of this point one can complete it to an independent systemf 1 , . . . , f m consisting of m smooth functions.d) If the systemξ i = f i x 1 , . . .
, x m (i = 1, . . . , m)of smooth functions is such that the mapping f = (f 1 , . . . , f m ) has rank m at thepoint x0 = (x01 , . . . , x0m ), then the variables (ξ 1 , . . . , ξ m ) can be used as curvilinearcoordinates in some neighborhood U (x0 ) of x0 , and any function ϕ : U (x0 ) → Rcan be written as ϕ(x) = F (f 1 (x), . . . , f m (x)), where F = ϕ ◦ f −1 .e) The rank of the mapping provided by a system of smooth functions is alsocalled the rank of the system. Show that if the rank of a system of smooth functionsf i (x 1 , . . . , x m ) (i = 1, . .
. , k) is k and the rank of the system f 1 , . . . , f m , ϕ is alsok at some point x0 ∈ Rm , then ϕ(x) = F (f 1 (x), . . . , f k (x)) in a neighborhood ofthe point.Hint: Use c) and d) and show thatF f 1, . . . , f m = F f 1, . . . , f k .5. Show that the rank of a smooth mapping f : Rm → Rn is a lower semicontinuousfunction, that is rank f (x) ≥ rank f (x0 ) in a neighborhood of a point x0 ∈ Rm .6. a) Give a direct proof of Morse’s lemma for functions f : R → R.b) Determine whether Morse’s lemma is applicable at the origin to the followingfunctions:f (x) = x 3 ;1f (x) = x sin ;xf (x, y) = x 3 − 3xy 2 ;12f (x) = e−1/x sin2 ;xf (x, y) = x 2 .c) Show that nondegenerate critical points of a function f ∈ C (3) (Rm ; R) areisolated: each of them has a neighborhood in which it is the only critical point of f .d) Show that the number k of negative squares in the canonical representationof a function in the neighborhood of a nondegenerate critical point is independent8.7 Surfaces in Rn and Constrained Extrema517of the reduction method, that is, independent of the coordinate system in which thefunction has canonical form.
This number is called the index of the critical point.8.7 Surfaces in Rn and the Theory of Extrema with ConstraintTo acquire an informal understanding of the theory of extrema with constraint,which is important in applications, it is useful to have some elementary information on surfaces (manifolds) in Rn .8.7.1 k-Dimensional Surfaces in RnGeneralizing the concept of a law of motion of a point mass x = x(t), we havepreviously introduced the concept of a path in Rn as a continuous mapping Γ : I →Rn of an interval I ⊂ R.
The degree of smoothness of the path was defined a thedegree of smoothness of this mapping. The support Γ (I ) ⊂ Rn of a path can be arather peculiar set in Rn , which it would be a great stretch to call a curve in someinstances. For example, the support of a path might be a single point.Similarly, a continuous or smooth mapping f : I k → Rn of a k-dimensional interval I k ⊂ Rk , called a singular k-cell in Rn , may have as its image f (I k ) not atall what one would like to call a k-dimensional surface in Rn . For example, it mightagain be simply a point.In order for a smooth mapping f : G → Rn of a domain G ⊂ Rk to define ak-dimensional geometric figure in Rn whose points are described by k independentparameters (t 1 , . . . , t k ) ∈ G, it suffices, as we know from the preceding section, torequire that the rank of the mapping f : G → Rn be k at each point t ∈ G (naturally,k ≤ n).
In that case the mapping f : G → f (G) is locally one-to-one (that is, in aneighborhood of each point t ∈ G).Indeed, suppose rank f (t0 ) = k and this rank is realized, for example, on the firstk of the n functions⎧ 11 1k⎪⎨ .x = f t , . . . , t ,..(8.136)⎪⎩ nx = f n t 1, . . . , t kthat define the coordinate expressions for the mapping f : G → Rn .Then, by the inverse function theorem the variables t 1 , . . .
, t k can be expressedin terms of x 1 , . . . , x k in some neighborhood U (t0 ) of t0 . It follows that the setf (U (t0 )) can be written asx k+1 = ϕ k+1 x 1 , . . . , x k , . . . , x n = ϕ n x 1 , . . . , x k(that is, it projects in a one-to-one manner onto the coordinate plane of x 1 , . . . , x k ),and therefore the mapping f : U (t0 ) → f (U (t0 )) is indeed one-to-one.5188 Differential Calculus in Several VariablesFig. 8.9However, even the simple example of a smooth one-dimensional path (Fig. 8.9)makes it clear that the local injectivity of the mapping f : G → Rn from the parameter domain G into Rn is by no means necessarily a global injectivity.
The trajectory may have multiple self-intersections, so that if we wish to define a smoothk-dimensional surface in Rn and picture it as a set that has the structure of a slightlydeformed piece of a k-dimensional plane (a k-dimensional subspace of Rn ) neareach of its points, it is not enough merely to map a canonical piece G ⊂ Rk of ak-dimensional plane in a regular manner into Rn . It is also necessary to be sure thatit happens to be globally imbedded in this space.Definition 1 We shall call a set S ⊂ Rn a k-dimensional smooth surface in Rn (or ak-dimensional submanifold of Rn ) if for every point x0 ∈ S there exist a neighborhood U (x0 ) in Rn and a diffeomorphism ϕ : U (x0 ) → I n of this neighborhood ontothe standard n-dimensional cube I n = {t ∈ Rn | |t i | < 1, i = 1, .
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