The Student[VectorCalculus] package supports the following coordinate systems:

In two dimensions - cartesian and polar.

In three dimensions - cartesian, cylindrical, and spherical.

The conversions from the various coordinate systems to Cartesian (rectangular) coordinates in two-dimensional space

are given by:

cartesian:

$x=u$ 

$y=v$ 

polar:

$x=u\cos \left(v\right)$ 

$y=u\sin \left(v\right)$ 

The conversions from the various coordinate systems to Cartesian coordinates in three-dimensional space

are given as follows.

cartesian:

$x=u$

$y=v$

$z=w$

cylindrical:

$x=u\cos \left(v\right)$ 

$y=u\sin \left(v\right)$

$z=w$

spherical:

$x=u\cos \left(w\right)\sin \left(v\right)$

$y=u\sin \left(w\right)\sin \left(v\right)$

$z=u\cos \left(v\right)$

Moon, P., and Spencer, D.E. Field Theory Handbook. 2d ed. Berlin: Springer-Verlag, 1971.

Spiegel, Murray R.  Mathematical Handbook of Formulas and Tables. New York:  McGraw Hill Book Company, 1968, pp. 126-130.