# Find a diffeomorphism between two sets

I'm having a problem with this multivariate calculus problem. Here it is: find a diffeomorphism between set $S = \{ (x,y) \in \mathbb{R}^2: 2<x^2+y^2<20, x^2<y<3x^2 \}$ and some open cube $P \subset \mathbb{R}^2$.

Is it even possible. $S$ is not connected while cube $P$ is and every diffeomorphism is a homeomorphism...?

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What maps have you thought of? – rschwieb Jan 17 '13 at 15:25
Have you drawn a picture of $S$? – Michael Albanese Jan 17 '13 at 15:34
You can use functions similar to ones defining the boundary of this set. In particular, notice that the collection of circle arcs (with varying radius) and parts of parabola graphs form curvilinear coordinate lines on this set. But first of all draw the picture as Michael suggests! – Marek Jan 17 '13 at 16:22
Is it even possible. $S$ is not connected while cube $P$ is and every diffeomorphism is a homeomorphism...? – hnCas Jan 17 '13 at 18:16