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!