Where does $f(r,\varphi,\theta)=(r\sin{\theta}\cos{\varphi},r\sin{\theta}\sin{\varphi},r\cos{\theta})$ have a locally differentiable inversion?

Question

$$f(r,\varphi,\theta)=(r\sin{\theta}\cos{\varphi},r\sin{\theta}\sin{\varphi},r\cos{\theta})$$

$$f:(0,\infty)\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^3$$

How can I find out on which points $f$ is a local Diffeomorphism and for which is it not?

Answer 1

Hint: If you know Inverse Function Theorem then it is enough to check where rank$[Df(p)] = 3$.