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 $Jf(x)\neq 0$.