1

Basic question.

I started reading Ordinary Diff. Equations by V. I. Arnold and am a little confused about one of the exercises: proving a diffeomorphism from U to V (in the context of the text, this is defined as a one-to-one invertibly differentiable mapping) can only exist given $dimU=dimV$. It's been a while since I've taken multivariable calculus. How would I use the implicit function theorem here?

Looking online I found several proofs which use (simple-looking) tools from topology, but this is my first encounter with manifolds and the such and so I couldn't make much sense of them.

Full context: http://puu.sh/28E5l (it is Problem 2)

Thanks for your help!

devr
  • 11
  • Presumably, if we assume say U=$R^2$, V=$R$, we could use the implicit function theorem to say that $f(x,y)=f(x_0,y_0)$ defines y as a function of x near some $(x_0,y_0)\in R^2$. This would be in contradiction to the injectivity. (And in the case where $V=R^2, U=R$ we could look at $f^{-1}$). This is surprisingly simple... is this it? – devr Feb 26 '13 at 13:07
  • Yes, it's simple once you have the implicit function theorem. The theorem tells you exactly how preimages of points look like, so you can see that are not single points. –  Feb 26 '13 at 13:58

0 Answers0