Editing Parabolic coordinates (section)

==Three-dimensional scale factors==

The three dimensional scale factors are:

:<math>h_{\sigma} = \sqrt{\sigma^2+\tau^2}</math>
:<math>h_{\tau}   = \sqrt{\sigma^2+\tau^2}</math>
:<math>h_{\varphi} = \sigma\tau</math>

It is seen that the scale factors <math>h_{\sigma}</math> and <math>h_{\tau}</math> are the same as in the two-dimensional case. The infinitesimal volume element is then

:<math>
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
</math>

and the Laplacian is given by

:<math>
\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}
</math>

Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> 
and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau, \phi)</math> by substituting 
the scale factors into the general formulae 
found in [[orthogonal coordinates]].