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

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