Ellipsoidal coordinates

Scale factors and differential operators

For brevity in the equations below, we introduce a function

${\displaystyle S(\sigma )\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(a^2+\sigma \right)\left(b^2+\sigma \right)\left(c^2+\sigma \right)}$

where ${\displaystyle \sigma }$ can represent any of the three variables ${\displaystyle (\lambda ,\mu ,\nu )}$. Using this function, the scale factors can be written

${\displaystyle h_{\lambda }=\frac{1}{2}\sqrt{\frac{\left(\lambda -\mu \right)\left(\lambda -\nu \right)}{S(\lambda )}}}$

${\displaystyle h_{\mu }=\frac{1}{2}\sqrt{\frac{\left(\mu -\lambda \right)\left(\mu -\nu \right)}{S(\mu )}}}$

${\displaystyle h_{\nu }=\frac{1}{2}\sqrt{\frac{\left(\nu -\lambda \right)\left(\nu -\mu \right)}{S(\nu )}}}$

Hence, the infinitesimal volume element equals

${\displaystyle dV=\frac{\left(\lambda -\mu \right)\left(\lambda -\nu \right)\left(\mu -\nu \right)}{8\sqrt{-S(\lambda )S(\mu )S(\nu )}}d\lambda d\mu d\nu }$

and the Laplacian is defined by

${\displaystyle \begin{array}{cc}{\mathrm{\nabla }}^2\mathrm{\Phi }=& \frac{4\sqrt{S(\lambda )}}{\left(\lambda -\mu \right)\left(\lambda -\nu \right)}\frac{\mathrm{\partial }}{\mathrm{\partial }\lambda }\left[\sqrt{S(\lambda )}\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }\lambda }\right]\\ & +\frac{4\sqrt{S(\mu )}}{\left(\mu -\lambda \right)\left(\mu -\nu \right)}\frac{\mathrm{\partial }}{\mathrm{\partial }\mu }\left[\sqrt{S(\mu )}\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }\mu }\right]\\ & +\frac{4\sqrt{S(\nu )}}{\left(\nu -\lambda \right)\left(\nu -\mu \right)}\frac{\mathrm{\partial }}{\mathrm{\partial }\nu }\left[\sqrt{S(\nu )}\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }\nu }\right]\end{array}}$

Other differential operators such as ${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}}$ and ${\displaystyle \mathrm{\nabla }\times \mathbf{F}}$ can be expressed in the coordinates ${\displaystyle (\lambda ,\mu ,\nu )}$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

Angular parametrization

An alternative (but non-orthogonal) parametrization exists that closely follows the angular parametrization of spherical coordinates:

${\displaystyle x=as\sin \theta \cos \varphi ,}$

${\displaystyle y=bs\sin \theta \sin \varphi ,}$

${\displaystyle z=cs\cos \theta .}$

Here, ${\displaystyle s>0}$ parametrizes the concentric ellipsoids around the origin and ${\displaystyle \theta \in [0,\pi ]}$ and ${\displaystyle \varphi \in [0,2\pi ]}$ are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element is

${\displaystyle dxdydz=abc{s}^2\sin \theta dsd\theta d\varphi .}$