Editing Bipolar coordinates (section)

== Scale factors ==
To obtain the scale factors for bipolar coordinates, we take the differential of the equation for <math> x + iy </math>, which gives
:<math>
dx + i\, dy = \frac{-ia}{\sin^2\bigl(\tfrac{1}{2}(\sigma + i \tau)\bigr)}(d\sigma +i\,d\tau).
</math>
Multiplying this equation with its [[complex conjugate]] yields
:<math>
(dx)^2 + (dy)^2 = \frac{a^2}{\bigl[2\sin\tfrac{1}{2}\bigl(\sigma + i\tau\bigr) \sin\tfrac{1}{2}\bigl(\sigma - i\tau\bigr)\bigr]^2} \bigl((d\sigma)^2 + (d\tau)^2\bigr).
</math>
Employing the trigonometric identities for products of sines and cosines, we obtain
:<math>
2\sin\tfrac{1}{2}\bigl(\sigma + i\tau\bigr) \sin\tfrac{1}{2}\bigl(\sigma - i\tau\bigr)
 = \cos\sigma - \cosh\tau,
</math>
from which it follows that
:<math>
(dx)^2 + (dy)^2 = \frac{a^2}{(\cosh \tau - \cos\sigma)^2} \bigl((d\sigma)^2 + (d\tau)^2\bigr).
</math>

Hence the scale factors for ''σ'' and ''τ'' are equal, and given by
:<math>
h_\sigma = h_\tau = \frac{a}{\cosh \tau - \cos\sigma}.
</math>

Many results now follow in quick succession from the general formulae for [[orthogonal coordinates]].
Thus, the [[infinitesimal]] area element equals

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

and the [[Laplacian]] is given by 

:<math>
\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).
</math>

Expressions for <math>\nabla f</math>, <math>\nabla \cdot \mathbf{F}</math>, and <math>\nabla \times \mathbf{F}</math> can be expressed obtained by substituting the scale factors into the general formulae found in [[orthogonal coordinates]].