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Kate2010
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Homework Statement
Let f [tex]\in[/tex] C1(Rn) be a function such that f(0) = 0 and [tex]\delta[/tex]1f(0) is nonzero. ([tex]\delta[/tex]1 means partial derivative with respect to x1)
Show that there exist neighbourhoods U and V of x=0[tex]\in[/tex] Rn and a diffeomorphism g:U->V such that f(g(x)) = x1 for all x = (x1,...,xn) [tex]\in[/tex] U.
Homework Equations
Inverse function theorem:
Let S[tex]\subseteq[/tex]Rn be open, f [tex]\in[/tex]C1(S,Rn) and x0[tex]\in[/tex]S. If Df(x0) is invertible then there exists an open neighbourhood U of x0 such that f(U) is open and f: U -> f(U) is a diffeomorphism. Also Df-1(f(x0)) = (D(f(x0))-1.
The Attempt at a Solution
I have a hint that says to apply the inverse function theorem to some function F:Rn -> Rn.
But, I really have no idea what this should be. I think it should involve f but I'm not sure how. Also I noticed g(x) = f-1(x1) if f-1 exists but then I'm not sure this makes sense as f is defined on Rn not R. So any help with finding this F would be great!