Elliptic Coordinate System - Scale Factors

Scale Factors

In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates are equal to

h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu} = a\sqrt{\cosh^{2}\mu - \cos^{2}\nu}.

Using the double argument identities for hyperbolic functions, the scale factors can be equivalently expressed as

h_{\mu} = h_{\nu} = a\sqrt{\frac{1}{2} (\cosh2\mu - \cos2\nu}).

Consequently, an infinitesimal element of area equals

dA = a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu = a^{2} \left( \cosh^{2}\mu - \cos^{2}\nu \right) d\mu d\nu = \frac{a^{2}}{4} \left( \cosh 2 \mu - \cos 2\nu \right) d\mu d\nu

and the Laplacian reads

\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right) = \frac{1}{a^{2} \left( \cosh^{2}\mu - \cos^{2}\nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right).

An alternative expression for the Laplacian, again using the double argument identities for hyperbolic functions is

\nabla^{2} \Phi = \frac{4}{a^{2} \left( \cosh 2 \mu - \cos 2 \nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right)

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:  Elliptic Coordinate System

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