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 u_{1}, u_{2}, and u_{3}. (In a spherical coordinate system, we would have u_{1} = r, u_{2} = q, and u_{3} = f.) A coordinate system is orthogonal if
_{}
_{}
_{}
In addition to the axes, parameters h_{1}, h_{2}, and h_{3} 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 righthand 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 threebythree matrix. Reversing the transformation, we obtain
_{}
We assume a vector, _{}, where u_{1}, u_{2}, and u_{3} are chosen as appropriate to the coordinate system under consideration. Finally, we assume a scalar function, _{}, of u_{1}, u_{2}, and u_{3}.
Coordinate System 
_{} 
_{} _{} 
_{} _{} 
Conversion from Rectangular Coordinates 
Conversion to Rectangular Coordinates 
Alternative Conversion to Rectangular Coordinates 
Rectangular 
_{} 
_{} 
_{} 
_{} 
_{} 

Cylindrical 
_{} 
_{} 
_{} 
_{} 
_{} 

Spherical 
_{} 
_{} 
_{} 
_{} 
_{} 

Parabolic Cylindrical 
_{} 
_{} 
_{} 
_{} 
_{} 

Paraboloidal 
_{} 
_{} 
_{} 
_{} 
_{} 

Elliptic Cylindrical 
_{} 
_{} 
_{} 
_{} 
_{} 

Prolate Spheroidal 
_{} 
_{} 
_{} 
_{} 
_{} 

Oblate Spheroidal 
_{} 
_{} 
_{} 
_{} 
_{} 

Bipolar 
_{} 
_{} 
_{} 
_{} 
_{} 

Toroidal 
_{} 
_{} 
_{} 
_{} 
_{} 

Conical 
_{} 
_{} 
_{} 

_{} 

Confocal Ellipsoidal 
_{} 
_{} 
_{} 

_{} 
_{} 
Confocal Paraboloidal 
_{} 
_{} 
_{} 

_{} 
_{} 
Coordinate System 
_{} 
_{} 
Differentials 
Rectangular 
_{} 
_{} 
_{} 
Cylindrical 
_{} 
_{} 
_{} 
Spherical 
_{} 
_{} 
_{} 
Parabolic Cylindrical 
_{} 
_{} 
_{} 
Paraboloidal 
_{} 
_{} 
_{} 
Elliptic Cylindrical 
_{} 
_{} 
_{} 
Prolate Spheroidal 
_{} 
_{} 
_{} 
Oblate Spheroidal 
_{} 
_{} 
_{} 
Bipolar 
_{} 
_{} 
_{} 
Toroidal 
_{} 
_{} 
_{} 
Conical 
_{} 
_{} 
_{} 
Confocal Ellipsoidal 
_{} 
_{} 
_{} 
Confocal Paraboloidal 
_{} 
_{} 
_{} 
Coordinate System 
_{} 
_{} 
Vector Operations 
Rectangular 
_{} 
_{} 
_{} 
Cylindrical 
_{} 
_{} 
_{} 
Spherical 
_{} 
_{} 
_{} 
Parabolic Cylindrical 
_{} 
_{} 
_{} 
Paraboloidal 
_{} 
_{} 
_{} 
Elliptic Cylindrical 
_{} 
_{} 
_{} 
Prolate Spheroidal 
_{} 
_{} 
_{} 
Oblate Spheroidal 
_{} 
_{} 
_{} 
Bipolar 
_{} 
_{} 
_{} 
Toroidal 
_{} 
_{} 
_{} 
Conical 
_{} 
_{} 
_{} 
Confocal Ellipsoidal 
_{} 
_{} 
_{} 
Confocal Paraboloidal 
_{} 
_{} 
_{} 