Parabolic coordinates

Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas.

Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges.

Two-dimensional parabolic coordinates

Two-dimensional parabolic coordinates $(\sigma ,\tau )$ are defined by the equations, in terms of cartesian coordinates:

$x=\sigma \tau$ $y={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)$ The curves of constant $\sigma$ form confocal parabolae

$2y={\frac {x^{2}}{\sigma ^{2}}}-\sigma ^{2}$ that open upwards (i.e., towards $+y$ ), whereas the curves of constant $\tau$ form confocal parabolae

$2y=-{\frac {x^{2}}{\tau ^{2}}}+\tau ^{2}$ that open downwards (i.e., towards $-y$ ). The foci of all these parabolae are located at the origin.

Two-dimensional scale factors

The scale factors for the parabolic coordinates $(\sigma ,\tau )$ are equal

$h_{\sigma }=h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2}}}$ Hence, the infinitesimal element of area is

$dA=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau$ and the Laplacian equals

$\nabla ^{2}\Phi ={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}\Phi }{\partial \sigma ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \tau ^{2}}}\right)$ Other differential operators such as $\nabla \cdot \mathbf {F}$ and $\nabla \times \mathbf {F}$ can be expressed in the coordinates $(\sigma ,\tau )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

Three-dimensional parabolic coordinates

The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional orthogonal coordinates. The parabolic cylindrical coordinates are produced by projecting in the $z$ -direction. Rotation about the symmetry axis of the parabolae produces a set of confocal paraboloids, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:

$x=\sigma \tau \cos \varphi$ $y=\sigma \tau \sin \varphi$ $z={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)$ where the parabolae are now aligned with the $z$ -axis, about which the rotation was carried out. Hence, the azimuthal angle $\phi$ is defined

$\tan \varphi ={\frac {y}{x}}$ The surfaces of constant $\sigma$ form confocal paraboloids

$2z={\frac {x^{2}+y^{2}}{\sigma ^{2}}}-\sigma ^{2}$ that open upwards (i.e., towards $+z$ ) whereas the surfaces of constant $\tau$ form confocal paraboloids

$2z=-{\frac {x^{2}+y^{2}}{\tau ^{2}}}+\tau ^{2}$ that open downwards (i.e., towards $-z$ ). The foci of all these paraboloids are located at the origin.

The Riemannian metric tensor associated with this coordinate system is

$g_{ij}={\begin{bmatrix}\sigma ^{2}+\tau ^{2}&0&0\\0&\sigma ^{2}+\tau ^{2}&0\\0&0&\sigma ^{2}\tau ^{2}\end{bmatrix}}$ Three-dimensional scale factors

The three dimensional scale factors are:

$h_{\sigma }={\sqrt {\sigma ^{2}+\tau ^{2}}}$ $h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2}}}$ $h_{\varphi }=\sigma \tau$ It is seen that the scale factors $h_{\sigma }$ and $h_{\tau }$ 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 $\nabla \cdot \mathbf {F}$ and $\nabla \times \mathbf {F}$ can be expressed in the coordinates $(\sigma ,\tau ,\phi )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.