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Can anyone come up with a diffeomorphism of $\mathbb{R}^{2}$ onto the unit disc $\left \{ (x,y): x^{2}+y^{2}< 1 \right \}$?

I tried the following example: $$F(x,y)=(\frac{x}{\sqrt{1+x^{2}+y^{2}}},\frac{y}{\sqrt{1+x^{2}+y^{2}}})$$, but I am not sure if this works out. In addition, it is very hard to prove that $Jf(x)\neq 0$.

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You just need to produce an inverse, as in M. B.'s answer, but I don't think that calculating the partial derivatives of this is so bad. $\arctan$ is something you might look into as well. – Dylan Moreland Mar 8 '12 at 3:20
up vote 4 down vote accepted

That looks a like nice attempt. Show that it is smooth with smooth inverse given by $$(x,y) \to \left (\frac{x}{\sqrt{1-x^2-y^2}}, \frac{y}{\sqrt{1-x^2-y^2}}\right).$$

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