Elliptic cylindrical coordinates

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular ${\displaystyle z}$-direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$ are generally taken to be fixed at ${\displaystyle -a}$ and ${\displaystyle +a}$, respectively, on the ${\displaystyle x}$-axis of the Cartesian coordinate system.

The most common definition of elliptic cylindrical coordinates ${\displaystyle (\mu ,\nu ,z)}$ is

${\displaystyle x=a\ \cosh \mu \ \cos \nu }$

${\displaystyle y=a\ \sinh \mu \ \sin \nu }$

${\displaystyle z=z}$

where ${\displaystyle \mu }$ is a nonnegative real number and ${\displaystyle \nu \in [0,2\pi ]}$.

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

${\displaystyle {\frac {x^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {y^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1}$

shows that curves of constant ${\displaystyle \mu }$ form ellipses, whereas the hyperbolic trigonometric identity

${\displaystyle {\frac {x^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {y^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1}$

shows that curves of constant ${\displaystyle \nu }$ form hyperbolae.

The scale factors for the elliptic cylindrical coordinates ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are equal

${\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}}$

whereas the remaining scale factor ${\displaystyle h_{z}=1}$. Consequently, an infinitesimal volume element equals

${\displaystyle dV=a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)d\mu d\nu dz}$

and the Laplacian equals

${\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right)+{\frac {\partial ^{2}\Phi }{\partial z^{2}}}}$

Other differential operators such as ${\displaystyle \nabla \cdot \mathbf {F} }$ and ${\displaystyle \nabla \times \mathbf {F} }$ can be expressed in the coordinates ${\displaystyle (\mu ,\nu ,z)}$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

An alternative and geometrically intuitive set of elliptic coordinates ${\displaystyle (\sigma ,\tau ,z)}$ are sometimes used, where ${\displaystyle \sigma =\cosh \mu }$ and ${\displaystyle \tau =\cos \nu }$. Hence, the curves of constant ${\displaystyle \sigma }$ are ellipses, whereas the curves of constant ${\displaystyle \tau }$ are hyperbolae. The coordinate ${\displaystyle \tau }$ must belong to the interval [-1, 1], whereas the ${\displaystyle \sigma }$ coordinate must be greater than or equal to one.

The coordinates