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 2-space

are given by:

Cartesian:

polar:

The conversions from the various coordinate systems to Cartesian coordinates in 3-space

are given as follows.

Cartesian:

cylindrical:

spherical:

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.