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Math Help - Problem involving implicit function theorem.

  1. #1
    Senior Member Pinkk's Avatar
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    Problem involving implicit function theorem.

    Suppose F(x,y) is a C^{1} function such that F(0,0) = 0. What conditions on F will guarantee that the equation F(F(x,y),y) = 0 can be solved for y as a C^{1} function of x near (0,0).

    So obviously \partial_{2}F(0,0) \ne 0 is one condition just by what the hypothesis of the implicit function theorem uses (I think this part is obvious, but I could be wrong and if I am, please explain why this is part of a condition that will guarantee what we want). However, another condition that is needed that will guarantee what we want is \partial_{1}F(0,0) \ne -1, and I just don't see how to arrive at that conclusion. Any help would be appreciated.
    Last edited by Pinkk; October 18th 2010 at 07:16 PM.
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  2. #2
    Senior Member Pinkk's Avatar
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    Will it have something to with being able to solve the equation x = F(x,y), and if so, how exactly can we arrive to the conclusion that \partial_{1}F(0,0) \ne 1 and \partial_{2}F(0,0) \ne 0 will guarantee we get want we desire? If not, still the same question, how do we arrive that those two conditions will guarantee what we want?
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  3. #3
    Senior Member Pinkk's Avatar
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    Sorry for the multiple posts but I really want to know how to solve this since it's practice for my upcoming midterm. Here is what I have so far.

    If \partial_{y}F(F(0,0),0) = \partial_{y}F(0,0) \ne 0, then we can solve the equation F(F(x,y), y) = 0 for y as a C^{1} function of F(x,y) near (F(0,0),0) = (0,0). Now, we have to show that we can find a function g that is C^{1} near (0,0) such that g(x) = F(x,y)...

    ...and that is where I get stuck. I think this is the right approach but if it is, how do I proceed, and if not, what do I do?
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