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Inverse/Implicit Function Theorems

  1. Jan 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Let B=B(0,r) be an open ball of radius r centered at the origin in R^n. Suppose U is an open subset of R^n containing the closed ball of radius r centered at the origin, f is a function from U to R^n that is differentiable, f(0) = 0, and ||Df(x) - I|| <= s < 1 for all x in the open ball. Prove that if ||y|| < r(1-s), then there is an x in the open ball such that f(x) = y.

    2. Relevant equations

    I'm pretty sure that this problem uses the inverse and implicit function theorems.

    3. The attempt at a solution

    I'm not sure how to start this problem, and I don't really have any idea what to do with the I that is being subtracted from the derivative matrix. Can somebody please get me pointed in the right direction?
  2. jcsd
  3. Jan 29, 2009 #2


    Staff: Mentor

    I think you're probably on the right track with the implicit function theorem. I can't offer any more help than that, but in checking "Vector Calculus" (Marsden & Tromba) last night, their section on implicit function theorem looked an awful lot like your problem. The proof used the mean value theorem, which might be where your Df(x) comes into play. I'm assuming that Df(x) is the matrix of partials of fi with respect to xj.
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