For the question below, I have read the wolfram article which restricts all three points based on the representation s cartesian coordinates. For example, $r\in[0,\infty]$. But how would I then find express all points so that the space near them is invertible?
Near which points $(r, \phi, \theta)$ is the following spherical coordinate transformation in space invertible?
$x=r\cos\phi \sin\theta$
$y=r\sin\phi \sin\theta$$z=r\cos\theta$
The Jacobian-determinant of the transformation is $\sin(\theta)r^2$. If this is non-zero the transformation is invertible in a neighbourhood of that point. Thus the only possible exceptions are points that solve $\sin(\theta)r^2=0$. And thus $r=0$ or $\theta= \pi p$, $p\in \mathbb Z$ (regardless of the values of the remaining coordinates) are the exceptions - and indeed the transformation isn't invertible at those points. To see this note that $(r,\theta, \phi)=(0,\theta, \phi)$ all maps to $(0,0,0)$, $(r,\theta, \phi)=(r,2\pi p,\phi)$ all maps to $(0,0,r)$, $(r,\theta, \phi)=(r,\pi ,\phi)=(r,3\pi,\phi)$ all maps to $(0,0,-r)$.
EDIT:
The determinant is the determinant of: $$ \begin{pmatrix} \partial_r x & \partial_\theta x & \partial_\phi x\\ \partial_r y & \partial_\theta y & \partial_\phi y\\ \partial_r z & \partial_\theta z & \partial_\phi z \end{pmatrix} $$