Is $f$ a diffeomorphism

Question

$f:\Bbb{R}^{2}\to \Bbb{R}^{2}$ be the function $f(x,y)=(e^{x}cosy,e^{x}siny)$. If $f$ is diffemorphism, then i need to show that $f$ is bijective and $f$, $f^{-1}$ are smooth. I know that the Jacobian matrix is $e^{2x}\gt 0$ so that $Df(x,y)$ is invertible for all $(x,y)\in\Bbb{R}^{2}$. But how to show f is bijective and smooth?

Answer 1

It is not injective (and therefore not bijective), since $f(0,2\pi)=f(0,0)=(1,0)$.

However, since $(\forall (x,y)\in\mathbb R^2):f'(x,y)\neq0$, $f$ is a local diffeomorphism, by the inverse function theorem.