I have a question regarding the following task:
Examine whether the mapping $g: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, defined by $$(r,\phi,\theta) \mapsto (r \sin(\theta) \cos(\phi), r \sin(\phi) \sin(\theta), r \cos(\theta)),$$ is a diffeomorphism. Determine, for each point $p \in \mathbb{R}^3$, whether the mapping is a local diffeomorphism at that point. If so, specify open neighborhoods of $p$ and $g(p)$ between which $g$ defines a diffeomorphism.
I have already shown that $g$ is a local diffeomorphism for $(r, \phi, \theta)$ with $r \neq 0$ and $\theta \neq \pi k$, where $k$ is an integer. I only have a question regarding the choice of open neighborhoods. For the neighborhood of $(r, \phi, \theta)$, I have chosen $U:=U_{|r|}(r, \phi, \theta)$, and for $g(r, \phi, \theta)$, I have chosen $V := g(U_{|r|}(r, \phi, \theta))$.
Are these neighborhoods chosen correctly? I am unsure because the neighborhoods need to be chosen in a three-dimensional space.