Parabolic Coordinates - Three-dimensional Scale Factors

Three-dimensional Scale Factors

The three dimensional scale factors are:

It is seen that The scale factors and are the same as in the two-dimensional case. The infinitesimal volume element is then

dV = h_\sigma h_\tau h_\varphi\, d\sigma\,d\tau\,d\varphi = \sigma\tau \left( \sigma^{2} + \tau^{2} \right)\,d\sigma\,d\tau\,d\varphi

and the Laplacian is given by

\nabla^2 \Phi = \frac{1}{\sigma^{2} + \tau^{2}} \left[ \frac{1}{\sigma} \frac{\partial}{\partial \sigma} \left( \sigma \frac{\partial \Phi}{\partial \sigma} \right) + \frac{1}{\tau} \frac{\partial}{\partial \tau} \left( \tau \frac{\partial \Phi}{\partial \tau} \right)\right] + \frac{1}{\sigma^2\tau^2}\frac{\partial^2 \Phi}{\partial \varphi^2}

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.

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