Let $h:]0,+\infty[ \times ]-1,1[ \to \mathbb{R}\times ]0,+\infty[$ be defined by :
$h((x,y)) = (u,v) = (xy, x\sqrt{1-y^2})$
I need to show that h is bijective.
What I did:
I showed that the jacobian of $h$ in a point $(a,b) \in ]0,+\infty[ \times ]-1,1[$ is :
$ DH_{(x,y)}(a,b)=-a(\frac{2b^2}{\sqrt{1-y^2}}+\sqrt{1-b^2}) <0 \neq 0 $
Is that enough to say that h is bijective or it is only injective?
Is there a need to show that $h^{-1}(u,v) \in ]0,+\infty[ \times ]-1,1[$ ?
Many thanks!
EDIT:
I calculated the inverse:
$h((x,y)) = (u,v) \Leftrightarrow \left\{ \begin{array}{@{}l} y=\frac{u}{\sqrt{u^2+v^2}}\\ x=\sqrt{u^2+v^2} \end{array} \right.$
Now I guess that means h is bijective ?
Thank you!