Parabolic Coordinates - Two-dimensional Scale Factors

Two-dimensional Scale Factors

The scale factors for the parabolic coordinates are equal

h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}}

Hence, the infinitesimal element of area is

dA = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau

and the Laplacian equals

\nabla^{2} \Phi = \frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} \Phi}{\partial \sigma^{2}} + \frac{\partial^{2} \Phi}{\partial \tau^{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:  Parabolic Coordinates

Famous quotes containing the words scale and/or factors:

    “The Humanity of men and women is inversely proportional to their Numbers. A Crowd is no more human than an Avalanche or a Whirlwind. A rabble of men and women stands lower in the scale of moral and intellectual being than a herd of Swine or of Jackals.”
    —Aldous Huxley (1894–1963)

    “The economic dependence of woman and her apparently indestructible illusion that marriage will release her from loneliness and work and worry are potent factors in immunizing her from common sense in dealing with men at work.”
    —Mary Barnett Gilson (1877–?)