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Unformatted text preview: Implicit function theorem 1 Chapter 6 Implicit function theorem Chapter 5 has introduced us to the concept of manifolds of dimension m contained in R n . In the present chapter we are going to give the exact definition of such manifolds and also discuss the crucial theorem of the beginnings of this subject. The name of this theorem is the title of this chapter. We definitely want to maintain the following point of view. An m-dimensional manifold M R n is an object which exists and has various geometric and calculus properties which are inherent to the manifold, and which should not depend on the particular mathematical formulation we use in describing the manifold. Since our goal is to do lots of calculus on M , we need to have formulas we can use in order to do this sort of work. In the very discussion of these methods we shall gain a clear and precise understanding of what a manifold actually is. We have already done this sort of work in the preceding chapter in the case of hypermani- folds. There we discussed the intrinsic gradient and the fact that the tangent space at a point of such a manifold has dimension n- 1 etc. We also discussed the version of the implicit function theorem that we needed for the discussion of hypermanifolds. We noticed at that time that we were really always working with only the local description of M , and that we didnt particularly care whether we were able to describe M with formulas that held for the entire manifold. The same will be true in the present chapter. A. Implicit presentation We go back to the idea of implicitly defined manifolds introduced briefly on p. 526. That is, we are going to have n- m real-valued functions on R n , and the manifold will be (locally) the intersection of level sets of these functions. Let us denote k = n- m ; this integer is frequently called the codimension of M . We are assuming 1 m n- 1, so that 1 k n- 1. (Hypermanifolds correspond to k = 1.) Let us call the constraint functions g 1 ,...,g k . Then we are proposing that M be described locally as the set of all points x R n satisfying the equations g 1 ( x ) = 0 , . . . g k ( x ) = 0 . Of course, since we want to do calculus we assume at least that each g i C 1 . But more is required. We must also have functions which are independent from one another. For instance, 2 Chapter 6 it would be ridiculous to think that the two equations ( x 2 + y 2 + z 2 = 1 , 2 x 2 + 2 y 2 + 2 z 2 = 2 , define a 1-dimensional manifold in R 3 ! But more than the qualitative idea of independence is required. Even in the case n = 2 and k = 1 we have seen that a single C 1 equation g ( x 1 ,x 2 ) = 0 does not necessarily give us a true manifold; see p. 52. We have learned in general that significant treatment of hypermanifolds (the case k = 1) requires that the gradient of the defining function be nonzero on M . The proper extension of this to our more general situation is that the gradients...
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- Fall '08