Spherical coordinates

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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 .


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].


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098