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.


 

 

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