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:

    “Another armored animal—scale
    lapping scale with spruce-cone regularity until they
    form the uninterrupted central
    tail-row!”
    —Marianne Moore (1887–1972)

    “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)