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Consider a cartesian, a cylindrical, and a spherical coordinate system, related as shown in Figure 1.

**Figure 1:** Standard relations between cartesian, cylindrical, and
spherical coordinate systems. The
origin is the same for all three. The positive *z*-axes of the
cartesian and cylindrical systems coincide with the positive polar
axis of the spherical system. The initial rays of the cylindrical and
spherical systems coincide with the positive *x*-axis of the cartesian
system, and the rays =90° coincide with the positive
*y*-axis.

Then the cartesian coordinates (*x*,*y*,*z*), the cylindrical coordinates
(*r*,,*z*), and the spherical coordinates (,,) of
a point are related as follows:

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.