Ellipsoidal Coordinates - Scale Factors and Differential Operators

Scale Factors and Differential Operators

For brevity in the equations below, we introduce a function

S(\sigma) \ \stackrel{\mathrm{def}}{=}\ \left( a^{2} + \sigma \right) \left( b^{2} + \sigma \right) \left( c^{2} + \sigma \right)

where can represent any of the three variables . Using this function, the scale factors can be written

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

Hence, the infinitesimal volume element equals

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

\nabla^{2} \Phi = \frac{4\sqrt{S(\lambda)}}{\left( \lambda - \mu \right) \left( \lambda - \nu\right)} \frac{\partial}{\partial \lambda} \left \ + \
\frac{4\sqrt{S(\mu)}}{\left( \mu - \lambda \right) \left( \mu - \nu\right)} \frac{\partial}{\partial \mu} \left \ + \ \frac{4\sqrt{S(\nu)}}{\left( \nu - \lambda \right) \left( \nu - \mu\right)} \frac{\partial}{\partial \nu} \left

Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.

Read more about this topic:  Ellipsoidal Coordinates

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