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

Famous quotes containing the words scale and/or factors:

    “I find it interesting that the meanest life, the poorest existence, is attributed to God’s will, but as human beings become more affluent, as their living standard and style begin to ascend the material scale, God descends the scale of responsibility at a commensurate speed.”
    —Maya Angelou (b. 1928)

    “I always knew I wanted to be somebody. I think that’s where it begins. People decide, “I want to be somebody. I want to make a contribution. I want to leave my mark here.” Then different factors contribute to how you will do that.”
    —Faith Ringgold (b. 1934)