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. . , n} of the spaceRn under which the image of the set S ∩ U (x0 ) is the portion of the k-dimensionalplane in Rn defined by the relations t k+1 = 0, . . . , t n = 0 lying inside I n (Fig. 8.10).We shall measure the degree of smoothness of the surface S by the degree ofsmoothness of the diffeomorphism ϕ.If we regard the variables t 1 , . . . , t n as new coordinates in a neighborhood ofU (x0 ), Definition 1 can be rewritten briefly as follows: the set S ⊂ Rn is a k-Fig. 8.108.7 Surfaces in Rn and Constrained Extrema519dimensional surface (k-dimensional submanifold) in Rn if for every point x0 ∈ Sthere is a neighborhood U (x0 ) and coordinates t 1 , .

. . , t n in U (x0 ) such that in thesecoordinates the set S ∩ U (x0 ) is defined by the relationst k+1 = · · · = t n = 0.The role of the standard n-dimensional cube in Definition 1 is rather artificialand approximately the same as the role of the standard size and shape of a page ina geographical atlas. The canonical location of the interval in the coordinate systemt 1 , . . . , t n is also a matter of convention and nothing more, since any cube in Rn canalways be transformed into the standard n-dimensional cube by an additional lineardiffeomorphism.We shall often use this remark when abbreviating the verification that a set S ⊂Rn is a surface in Rn .Let us consider some examples.Example 1 The space Rn itself is an n-dimensional surface of class C (∞) .

As themapping ϕ : Rn → I n here, one can take, for example, the mappingξi =2arctan x iπ(i = 1, . . . , n).Example 2 The mapping constructed in Example 1 also establishes that the subspace of the vector space Rn defined by the conditions x k+1 = · · · = x n = 0 is ak-dimensional surface in Rn (or a k-dimensional submanifold of Rn ).Example 3 The set in Rn defined by the system of relations⎧ 1 11 k1k+1 + · · · + a 1 x n = 0,⎪n⎨ a. 1 x + · · · + ak x + ak+1 x..⎪⎩ n−k 1n−k k+1a1 x + · · · + akn−k x k + ak+1x+ · · · + ann−k x n = 0,provided this system has rank n − k, is a k-dimensional submanifold of Rn .Indeed, suppose for example that the determinant 1 ak+1 · · ·an1 ... ..

.. .. a n−k · · · a n−k k+1is nonzero. Then the linear transformationt 1 = x1,...t k = xk ,n5208 Differential Calculus in Several Variablest k+1 = a11 x 1 + · · · + an1 x n ,...t n = a1n−k x 1 + · · · + ann−k x n ,is obviously nondegenerate. In the coordinates t 1 , . . . , t n the set is defined by theconditions t k+1 = · · · = t n = 0, already considered in Example 2.Example 4 The graph of a smooth function x n = f (x 1 , . . .

, x n−1 ) defined in a domain G ⊂ Rn−1 is a smooth (n − 1)-dimensional surface in Rn .Indeed, setting!t i = x i (i = 1, . . . , n − 1),t n = x n − f x 1 , . . . , x n−1 ,we obtain a coordinate system in which the graph of the function has the equationt n = 0.Example 5 The circle x 2 + y 2 = 1 in R2 is a one-dimensional submanifold of R2 , asis established by the locally invertible conversion to polar coordinates (ρ, ϕ) studiedin the preceding section. In these coordinates the circle has equation ρ = 1.Example 6 This example is a generalization of Example 3 and at the same time, ascan be seen from Definition 1, gives a general form for the coordinate expression ofsubmanifolds of Rn .Let F i (x 1 , .

. . , x n ) (i = 1, . . . , n − k) be a system of smooth functions of rankn − k. We shall show that the relations⎧ 1 1x , . . . , x k , x k+1 , . . . , x n = 0,⎪⎨F..(8.137).⎪⎩ n−k 1kk+1nx ,...,x ,x ,...,x = 0Fdefine a k-dimensional submanifold S in Rn .Suppose the condition ∂F 1∂F 1 ∂x k+1 · · ·∂x n ..... .... ∂F n−kn−k∂F k+1 · · ·∂x n∂x (x0 ) = 0(8.138)holds at a point x0 ∈ S. Then by the inverse function theorem the transformation!t i = x i (i = 1, . .

. , k),t i = F i−k x 1 , . . . , x n (i = k + 1, . . . , n)is a diffeomorphism of a neighborhood of this point.8.7 Surfaces in Rn and Constrained Extrema521In the new coordinates t 1 , . . . , t n the original system will have the form t k+1 =· · · = t n = 0; thus, S is a k-dimensional smooth surface in Rn .Example 7 The set E of points of the plane R2 satisfying the equation x 2 − y 2 = 0consists of two lines that intersect at the origin. This set is not a one-dimensionalsubmanifold of R2 (verify this!) precisely because of this point of intersection.If the origin 0 ∈ R2 is removed from E, then the set E\0 will now obviouslysatisfy Definition 1.

We remark that the set E\0 is not connected. It consists of fourpairwise disjoint rays.Thus a k-dimensional surface in Rn satisfying Definition 1 may happen to be adisconnected subset consisting of several connected components (and these components are connected k-dimensional surfaces). A surface in Rn is often taken to meana connected k-dimensional surface. Just now we shall be interested in the problemof finding extrema of functions defined on surfaces. These are local problems, andtherefore connectivity will not manifest itself in them.Example 8 If a smooth mapping f : G → Rn of the domain G ⊂ Rn defined incoordinate form by (8.136) has rank k at the point t0 ∈ G, then there exists a neighborhood U (t0 ) ⊂ G of this point<b>Текст обрезан, так как является слишком большим</b>.

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