To compare and contrast: the rectangular, cylindrical and spherical coordinates systems.

Given:
Point is represented as ${\displaystyle (r,\theta ,z).}$
Proof:
A point in cylindrical coordinate is represented by ${\displaystyle (r,\theta .z).}$
${\displaystyle (r,\theta )}$ are polar coordinates
z is the usual - coordinates
The rectangular (x, y, z) and the spherical coordinates ${\displaystyle (r,\theta ,z)}$ are related as
${\displaystyle x=r\cos \theta }$
${\displaystyle y=r\sin \theta }$
${\displaystyle z=z}$
These are used to convert from cylindrical to rectangular coordinates.
Rectangular to cylindrical coordinates:
${\displaystyle r^2=x^2+y^2}$
${\displaystyle \tan \theta =\frac{y}{x}}$
${\displaystyle z=z}$
A point in spherical coordinate is represented by ${\displaystyle (\rho ,\theta ,\phi ).}$
The rectangular (x, y, z) and the spherical coordinates ${\displaystyle (\rho ,\theta ,\phi )}$ are related as
${\displaystyle x=\rho \sin \phi \cos \theta }$
${\displaystyle y=\rho \sin \phi \sin \theta }$
${\displaystyle z=\rho \cos \phi }$
These are used to convert from spherical to rectangular coordinates.
Rectangular to spherical coordinates:
${\displaystyle {\rho }^2=x^2+y^2+z^2}$
${\displaystyle \tan \theta =\frac{y}{x}}$
${\displaystyle \phi ={\cos }^{-1}\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)}$
Spherical to rectangular coordinates:
${\displaystyle r=\rho \sin \phi }$
${\displaystyle \theta =\phi }$
${\displaystyle z=\rho \cos \phi }$
Cylindricalto spherical coordinates:
${\displaystyle {\rho }^2=\sqrt{r^2+z^2}}$
${\displaystyle \theta =\theta }$
${\displaystyle \phi ={\cos }^{-1}\left(\frac{z}{\sqrt{r^2+z^2}}\right)}$