Paraboloidal coordinates

Basic formulas

The Cartesian coordinates (x, y, z) can be produced from the ellipsoidal coordinates (\mu, \nu, \lambda) by the equations{{Citation

: x^{2} = \frac{4}{b-c}(\mu - b)(b - \nu)(b - \lambda)

: y^{2} = \frac{4}{b-c}(\mu - c)(c - \nu)(\lambda - c)

: z = \mu + \nu + \lambda - b - c

with

: \mu b \lambda c \nu 0

Consequently, surfaces of constant \mu are downward opening elliptic paraboloids:

: \frac{x^2}{\mu - b} + \frac{y^2}{\mu - c} = -4 (z - \mu)

Similarly, surfaces of constant \nu are upward opening elliptic paraboloids,

: \frac{x^2}{b - \nu} + \frac{y^2}{c - \nu} = 4 (z - \nu)

whereas surfaces of constant \lambda are hyperbolic paraboloids:

: \frac{x^2}{b - \lambda} - \frac{y^2}{\lambda - c} = 4 (z - \lambda)

Scale factors

The scale factors for the paraboloidal coordinates (\mu, \nu, \lambda) are

: h_{\mu} = \left[ \frac{\left(\mu - \nu \right) \left(\mu - \lambda \right)}{ \left(\mu - b \right) \left(\mu - c \right)} \right]^{1/2}

: h_{\nu} = \left[\frac{\left(\mu - \nu \right) \left( \lambda - \nu \right)}{ \left(b - \nu \right) \left(c - \nu \right)} \right]^{1/2}

: h_{\lambda} = \left[\frac{\left(\lambda - \nu \right) \left(\mu - \lambda \right)}{ \left(b - \lambda \right) \left(\lambda - c \right)} \right]^{1/2}

Hence, the infinitesimal volume element is

: dV = \frac{(\mu - \nu)(\mu - \lambda)(\lambda - \nu)}{\left[(\mu - b)(\mu - c)(b - \nu)(c - \nu)(b - \lambda)(\lambda - c) \right]^{1/2}} \ d\lambda d\mu d\nu

Differential operators

Common differential operators can be expressed in the coordinates (\mu, \nu, \lambda) by substituting the scale factors into the general formulas for these operators, which are applicable to any three-dimensional orthogonal coordinates. For instance, the gradient operator is

: \nabla = \left[\frac{ \left(\mu - b \right) \left(\mu - c \right)}{\left(\mu - \nu \right) \left(\mu - \lambda \right)} \right]^{1/2} \mathbf{e}{\mu} \frac{\partial}{\partial \mu} + \left[\frac{\left(b - \nu \right) \left(c - \nu \right)}{\left(\mu - \nu \right) \left( \lambda - \nu \right)} \right]^{1/2} \mathbf{e}{\nu} \frac{\partial}{\partial \nu} + \left[\frac{\left(b - \lambda \right) \left(\lambda - c \right)}{\left(\lambda - \nu \right) \left(\mu - \lambda \right)} \right]^{1/2} \mathbf{e}_{\lambda} \frac{\partial}{\partial \lambda}

and the Laplacian is

: \begin{align} \nabla^2 = &\left[\frac{ \left(\mu - b \right) \left(\mu - c \right)}{\left(\mu - \nu \right) \left(\mu - \lambda \right)} \right]^{1/2} \frac{\partial}{\partial \mu} \left[(\mu - b)^{1/2}(\mu - c)^{1/2} \frac{\partial}{\partial \mu} \right] \ &+ \left[\frac{\left(b - \nu \right) \left(c - \nu \right)}{\left(\mu - \nu \right) \left( \lambda - \nu \right)} \right]^{1/2} \frac{\partial}{\partial \nu} \left[ (b - \nu)^{1/2}(c - \nu)^{1/2}\frac{\partial}{\partial \nu} \right] \ &+ \left[\frac{\left(b - \lambda \right) \left(\lambda - c \right)}{\left(\lambda - \nu \right) \left(\mu - \lambda \right)} \right]^{1/2} \frac{\partial}{\partial \lambda} \left[(b - \lambda)^{1/2}(\lambda -c)^{1/2} \frac{\partial}{\partial \lambda} \right] \end{align}

Applications

Paraboloidal coordinates can be useful for solving certain partial differential equations. For instance, the Laplace equation and Helmholtz equation are both separable in paraboloidal coordinates. Hence, the coordinates can be used to solve these equations in geometries with paraboloidal symmetry, i.e. with boundary conditions specified on sections of paraboloids.

The Helmholtz equation is (\nabla^2 + k^2) \psi = 0. Taking \psi = M(\mu)N(\nu)\Lambda(\lambda), the separated equations are

: \begin{align} &(\mu - b)(\mu - c) \frac{d^2M}{d\mu^2} + \frac{1}{2}\left[2 \mu - (b + c) \right] \frac{dM}{d\mu} + \left[k^2 \mu^2 + \alpha_3 \mu - \alpha_2\right] M = 0 \ &(b - \nu)(c - \nu) \frac{d^2N}{d\nu^2} + \frac{1}{2}\left[2 \nu - (b + c) \right] \frac{dN}{d\nu} + \left[k^2 \nu^2 + \alpha_3 \nu - \alpha_2\right] N = 0 \ &(b - \lambda)(\lambda - c) \frac{d^2\Lambda}{d\lambda^2} - \frac{1}{2}\left[2 \lambda - (b + c) \right] \frac{d\Lambda}{d\lambda} - \left[k^2 \lambda^2 + \alpha_3 \lambda - \alpha_2\right] \Lambda = 0 \ \end{align}

where \alpha_2 and \alpha_3 are the two separation constants. Similarly, the separated equations for the Laplace equation can be obtained by setting k = 0 in the above.

Each of the separated equations can be cast in the form of the Baer equation. Direct solution of the equations is difficult, however, in part because the separation constants \alpha_2 and \alpha_3 appear simultaneously in all three equations.

Following the above approach, paraboloidal coordinates have been used to solve for the electric field surrounding a conducting paraboloid.{{Citation

References

Bibliography

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

References

1. Willatzen and Yoon (2011), p. 219
2. Willatzen and Yoon (2011), p. 227