Parabolic coordinates

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.

The Cartesian coordinates x and y can be converted to parabolic coordinates by: : \sigma = \operatorname{sign}(x)\sqrt{\sqrt{x^{2} +y^{2}}-y}

: \tau = \sqrt{\sqrt{x^{2} +y^{2}}+y}

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 \varphi 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.

Bibliography

• Same as Morse & Feshbach (1953), substituting u**k for ξk.