Orthogonal Coordinate
Systems
Note: This is a practical guide to orthogonal coordinate systems rather than a rigorous derivation. Note further that this is a work in progress and may never be completed. Please let me know if you can fill out part of the tables that I have left blank.
Here we consider 13 orthogonal coordinate systems and list their line differentials, area differentials, volume differentials, gradients, divergences, curls, Laplacians, and transformations to the other systems. By an “orthogonal” system we mean that all three axes are perpendicular to each other. Suppose for instance, that we have three axes labeled u1, u2, and u3. (In a spherical coordinate system, we would have u1 = r, u2 = q, and u3 = f.) A coordinate system is orthogonal if
In addition to the axes, parameters h1, h2,
and h3 are useful in deriving the desired quantities. For the spherical system, we have 
, 
, and 
.
We begin by
considering two vectors, 
and 
, in any orthogonal system.
It is always true that
where 
is the angle between
the two vectors, and 
is the vector that is
perpendicular to the two vectors following a right-hand rule. The fact that these quantities are invariant
means that other relationships hold such as
It also means that the dot product is always formed by multiplying the corresponding components of the vectors and adding, and that the cross product is found by placing the coordinates in a matrix and taking the determinate.
The differential line, area, and volume elements are always given by
Line
Differential: 
Area
Differential: 
where ds takes on one of three values depending on the axes to which the area is being projected:
Volume
Differential: 
The gradient, divergence, curl, and Laplacian are always found by
Gradient: 
Divergence: 
Curl: 
Laplacian: 
where
and 
is a scalar. The coordinate transformations depend on the
definitions of the coordinate systems.
The tables
that follow explicitly list the key quantities: conversion to and from
rectangular coordinates; line, surface, and volume differentials; and gradient,
divergence, curl, and Laplacian. To
perform the derivation, we let 
be a vector that
represents coordinates in the rectangular system, and let 
be a vector that
represents the coordinates in any other system. 
is found by the
transformation
where 
is a three-by-three
matrix. Reversing the transformation,
we obtain
We assume a vector, 
, where u1, u2, and u3
are chosen as appropriate to the coordinate system under consideration. Finally, we assume a scalar function, 
, of u1, u2, and u3.
