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Math Help - Prove solution exists

  1. #1
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    Prove solution exists

    Let  f: \mathbb{R}^{k+n} \rightarrow \mathbb{R}^n be of class  C^1 ; suppose that f(a) = 0 and that Df(a) has rank n. Show that if c is a point of  \mathbb{R}^n sufficiently close to 0, then the equation f(x) = c has a solution.

    I'm pretty sure that I have to use the implicit function theorem here, but I'm not sure how to proceed. Any ideas?
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  2. #2
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    Quote Originally Posted by JG89 View Post
    Let  f: \mathbb{R}^{k+n} \rightarrow \mathbb{R}^n be of class  C^1 ; suppose that f(a) = 0 and that Df(a) has rank n. Show that if c is a point of  \mathbb{R}^n sufficiently close to 0, then the equation f(x) = c has a solution.

    I'm pretty sure that I have to use the implicit function theorem here, but I'm not sure how to proceed. Any ideas?

    As you can see in "statement of the theorem" in http://en.wikipedia.org/wiki/Implicit_function_theorem" , or

    perhaps a little clearer in Implicit Function Theorem -- from Wolfram MathWorld ,

    this is almost the exact wording of the implicit function theorem, so I don't understand what is it

    that you cannot understand here?

    Perhaps it is just a matter of writing a=(x_1,...,x_n,x_{n+1},...,x_{n+m})\in\mathbb{R}^{  n+m} so that it'll look clearer...?

    Tonio
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  3. #3
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    Let  a = (a_1, a_2) The theorem says that since f(a) = 0 and  \frac{\partial f}{\partial y} (a) has rank n (this follows from Df(a) having rank n), then there is a neighborhood B of  a_1 in  \mathbb{R}^k and a  C^r map  g: B \rightarrow \mathbb{R}^n such that  g(a_1) = a_2 and f(x, g(x)) = 0 for all x in B.

    How does this show that if c is a point very close to 0 ( but not quite 0), that there is a solution to f(x) = c?
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