Parabolic cylindrical coordinates

Coordinate surfaces of parabolic cylindrical coordinates. The red parabolic cylinder corresponds to σ=2, whereas the yellow parabolic cylinder corresponds to τ=1. The blue plane corresponds to z=2. These surfaces intersect at the point P (shown as a black sphere), which has Cartesian coordinates roughly (2, −1.5, 2).

In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular $z$-direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges.

Basic definition

Parabolic coordinate system showing curves of constant σ and τ the horizontal and vertical axes are the x and y coordinates respectively. These coordinates are projected along the z-axis, and so this diagram will hold for any value of the z coordinate.

The parabolic cylindrical coordinates Template:Math are defined in terms of the Cartesian coordinates Template:Math by:

$\begin{array}{cc}x& =\sigma \tau \\ y& =\frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =z\end{array}$

The surfaces of constant Template:Math form confocal parabolic cylinders

$2y=\frac{x^2}{{\sigma }^2}-{\sigma }^2$

that open towards Template:Math, whereas the surfaces of constant Template:Math form confocal parabolic cylinders

$2y=-\frac{x^2}{{\tau }^2}+{\tau }^2$

that open in the opposite direction, i.e., towards Template:Math. The foci of all these parabolic cylinders are located along the line defined by Template:Math. The radius Template:Math has a simple formula as well

$r=\sqrt{x^2+y^2}=\frac{1}{2}({\sigma }^2+{\tau }^2)$

that proves useful in solving the Hamilton–Jacobi equation in parabolic coordinates for the inverse-square central force problem of mechanics; for further details, see the Laplace–Runge–Lenz vector article.

Scale factors

The scale factors for the parabolic cylindrical coordinates Template:Math and Template:Math are:

$\begin{array}{cc}{h}_{\sigma }& =h_{\tau }=\sqrt{{\sigma }^2+{\tau }^2}\\ h_z& =1\end{array}$

Differential elements

The infinitesimal element of volume is

$dV=h_{\sigma }{h}_{\tau }{h}_{z}d\sigma d\tau dz=({\sigma }^2+{\tau }^2)d\sigma d\tau dz$

The differential displacement is given by:

$d𝐥=\sqrt{{\sigma }^2+{\tau }^2}d\sigma \widehat{𝝈}+\sqrt{{\sigma }^2+{\tau }^2}d\tau \widehat{𝝉}+dz\widehat{𝐳}$

The differential normal area is given by:

$d𝐒=\sqrt{{\sigma }^2+{\tau }^2}d\tau dz\widehat{𝝈}+\sqrt{{\sigma }^2+{\tau }^2}d\sigma dz\widehat{𝝉}+({\sigma }^2+{\tau }^2)d\sigma d\tau \widehat{𝐳}$

Del

Let Template:Math be a scalar field. The gradient is given by

$\mathrm{\nabla }f=\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \sigma }\widehat{𝝈}+\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \tau }\widehat{𝝉}+\frac{\partial f}{\partial z}\widehat{𝐳}$

The Laplacian is given by

${\mathrm{\nabla }}^{2}f=\frac{1}{{\sigma }^2+{\tau }^2}(\frac{{\partial }^{2}f}{\partial {\sigma }^2}+\frac{{\partial }^{2}f}{\partial {\tau }^2})+\frac{{\partial }^{2}f}{\partial z^2}$

Let Template:Math be a vector field of the form:

$𝐀=A_{\sigma }\widehat{𝝈}+A_{\tau }\widehat{𝝉}+A_z\widehat{𝐳}$

The divergence is given by

$\mathrm{\nabla }\cdot 𝐀=\frac{1}{{\sigma }^2+{\tau }^2}(\frac{\partial (\sqrt{{\sigma }^2+{\tau }^2}{A}_{\sigma })}{\partial \sigma }+\frac{\partial (\sqrt{{\sigma }^2+{\tau }^2}{A}_{\tau })}{\partial \tau })+\frac{\partial A_z}{\partial z}$

The curl is given by

$\mathrm{\nabla }\times 𝐀=(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \tau }-\frac{\partial A_{\tau }}{\partial z})\widehat{𝝈}-(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \sigma }-\frac{\partial A_{\sigma }}{\partial z})\widehat{𝝉}+\frac{1}{{\sigma }^2+{\tau }^2}(\frac{\partial \left(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\tau }\right)}{\partial \sigma }-\frac{\partial \left(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\sigma }\right)}{\partial \tau })\widehat{𝐳}$

Other differential operators can be expressed in the coordinates Template:Math by substituting the scale factors into the general formulae found in orthogonal coordinates.

Relationship to other coordinate systems

Relationship to cylindrical coordinates Template:Math:

$\begin{array}{cc}\rho \cos \phi & =\sigma \tau \\ \rho \sin \phi & =\frac{1}{2}({\tau }^2-{\sigma }^2)\\ z& =z\end{array}$

Parabolic unit vectors expressed in terms of Cartesian unit vectors:

$\begin{array}{cc}\widehat{𝝈}& =\frac{\tau \widehat{𝐱}-\sigma \widehat{𝐲}}{\sqrt{{\tau }^2+{\sigma }^2}}\\ \widehat{𝝉}& =\frac{\sigma \widehat{𝐱}+\tau \widehat{𝐲}}{\sqrt{{\tau }^2+{\sigma }^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$

Parabolic cylinder harmonics

Since all of the surfaces of constant Template:Math, Template:Math and Template:Math are conicoids, Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written:

$V=S(\sigma )T(\tau )Z(z)$

and Laplace's equation, divided by Template:Math, is written:

$\frac{1}{{\sigma }^2+{\tau }^2}[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}]+\frac{\ddot{Z}}{Z}=0$

Since the Template:Math equation is separate from the rest, we may write

$\frac{\ddot{Z}}{Z}=-m^2$

where Template:Math is constant. Template:Math has the solution:

$Z_m(z)=A_1{e}^{imz}+A_2{e}^{-imz}$

Substituting Template:Math for $\ddot{Z}/Z$, Laplace's equation may now be written:

$[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}]=m^2({\sigma }^2+{\tau }^2)$

We may now separate the Template:Math and Template:Math functions and introduce another constant Template:Math to obtain:

$\ddot{S}-(m^2{\sigma }^2+n^2)S=0$

$\ddot{T}-(m^2{\tau }^2-n^2)T=0$

The solutions to these equations are the parabolic cylinder functions

$S_{mn}(\sigma )=A_3{y}_1(n^2/2m,\sigma \sqrt{2m})+A_4{y}_2(n^2/2m,\sigma \sqrt{2m})$

$T_{mn}(\tau )=A_5{y}_1(n^2/2m,i\tau \sqrt{2m})+A_6{y}_2(n^2/2m,i\tau \sqrt{2m})$

The parabolic cylinder harmonics for Template:Math are now the product of the solutions. The combination will reduce the number of constants and the general solution to Laplace's equation may be written:

$V(\sigma ,\tau ,z)=\sum _{m,n}{A}_{mn}{S}_{mn}{T}_{mn}{Z}_m$

Applications

The classic applications of parabolic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which such coordinates allow a separation of variables. A typical example would be the electric field surrounding a flat semi-infinite conducting plate.

See also

• Parabolic coordinates
• Orthogonal coordinate system
• Curvilinear coordinates

Bibliography

• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
• Template:Cite book

External links

• MathWorld description of parabolic cylindrical coordinates