Bipolar Coordinates - Scale Factors

Scale Factors

The scale factors for the bipolar coordinates (σ, τ) are equal

h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos\sigma}

Thus, the infinitesimal area element equals

dA = \frac{a^2}{\left( \cosh \tau - \cos\sigma \right)^2} \, d\sigma\, d\tau

and the Laplacian is given by

\nabla^2 \Phi = \frac{1}{a^2} \left( \cosh \tau - \cos\sigma \right)^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 (σ, τ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

Read more about this topic:  Bipolar Coordinates

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