The numbers which are related to the Cartesian coordinates by the formulas
where , , .
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].