# Spherical coordinates

The numbers which are related to the Cartesian coordinates by the formulas

where , , .

Figure: s086660a

The coordinate surfaces are (see Fig.): concentric spheres with centre ; half-planes that pass through the axis ; circular cones with vertex and axis . The system of spherical coordinates is orthogonal.

The Lamé coefficients are

The element of surface area is

The volume element is

The basic operations of vector calculus are

The numbers , called generalized spherical coordinates, are related to the Cartesian coordinates by the formulas

where , , , , . The coordinate surface are: ellipsoids , half-planes and elliptical cones .

#### Comments

If the surface has been given by , then the element of surface area can be written as:

A general method to transform vector functions when new coordinates are introduced is, e.g., given in [a1].

#### References

[a1] | D.E. Rutherford, "Vector methods" , Oliver & Boyd (1949) |

[a2] | M.R. Spiegel, "Vector analysis and an introduction to tensor analysis" , McGraw-Hill (1959) |

[a3] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) pp. 11; 258 |

**How to Cite This Entry:**

Spherical coordinates. D.D. Sokolov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Spherical_coordinates&oldid=18298