Parabolic cylindrical coordinates

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

The parabolic cylindrical coordinates $(σ, τ, z)$ are defined in terms of the Cartesian coordinates $(x, y, z)$ by:

{\begin{aligned}x&=\sigma \tau \\y&={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}

The surfaces of constant $σ$ form confocal parabolic cylinders

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

that open towards $+ y$ , whereas the surfaces of constant $τ$ form confocal parabolic cylinders

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

that open in the opposite direction, i.e., towards $- y$ . The foci of all these parabolic cylinders are located along the line defined by $x = y = 0$ . The radius $r$ has a simple formula as well

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

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 $σ$ and $τ$ are:

{\begin{aligned}h_{\sigma }&=h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2}}}\\h_{z}&=1\end{aligned}}

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:

d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,{\boldsymbol {\hat {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,{\boldsymbol {\hat {\tau }}}+dz\,\mathbf {\hat {z}}

The differential normal area is given by:

d\mathbf {S} ={\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,dz{\boldsymbol {\hat {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,dz{\boldsymbol {\hat {\tau }}}+\left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau \mathbf {\hat {z}}

Del

Let $f$ be a scalar field. The gradient is given by

\nabla f={\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \sigma }{\boldsymbol {\hat {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \tau }{\boldsymbol {\hat {\tau }}}+{\partial f \over \partial z}\mathbf {\hat {z}}

The Laplacian is given by

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

Let $A$ be a vector field of the form:

\mathbf {A} =A_{\sigma }{\boldsymbol {\hat {\sigma }}}+A_{\tau }{\boldsymbol {\hat {\tau }}}+A_{z}\mathbf {\hat {z}}

The divergence is given by

\nabla \cdot \mathbf {A} ={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }) \over \partial \sigma }+{\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }) \over \partial \tau }\right)+{\partial A_{z} \over \partial z}

The curl is given by

\nabla \times \mathbf {A} =\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \tau }}-{\frac {\partial A_{\tau }}{\partial z}}\right){\boldsymbol {\hat {\sigma }}}-\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \sigma }}-{\frac {\partial A_{\sigma }}{\partial z}}\right){\boldsymbol {\hat {\tau }}}+{\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\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 }}\right)\mathbf {\hat {z}}

Other differential operators can be expressed in the coordinates $(σ, τ)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

Relationship to other coordinate systems

Relationship to cylindrical coordinates $(ρ, φ, z)$ :

{\begin{aligned}\rho \cos \varphi &=\sigma \tau \\\rho \sin \varphi &={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}

Parabolic unit vectors expressed in terms of Cartesian unit vectors:

{\begin{aligned}{\boldsymbol {\hat {\sigma }}}&={\frac {\tau {\hat {\mathbf {x} }}-\sigma {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\boldsymbol {\hat {\tau }}}&={\frac {\sigma {\hat {\mathbf {x} }}+\tau {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}

Parabolic cylinder harmonics

Since all of the surfaces of constant $σ$ , $τ$ and $z$ 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 $V$ , is written:

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

Since the $Z$ equation is separate from the rest, we may write

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

where $m$ is constant. $Z (z)$ has the solution:

Z_{m}(z)=A_{1}\,e^{imz}+A_{2}\,e^{-imz}

Substituting $- m 2$ for ${\ddot {Z}}/Z$ , Laplace's equation may now be written:

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

We may now separate the $S$ and $T$ functions and introduce another constant $n 2$ 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 $(m, n)$ 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.

Bibliography

Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 658. ISBN 0-07-043316-X. LCCN 52011515.
Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 186–187. LCCN 55010911.
Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 181. LCCN 59014456. ASIN B0000CKZX7.
Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. p. 96. LCCN 67025285.
Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9. Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
Moon P, Spencer DE (1988). "Parabolic-Cylinder Coordinates (μ, ν, z)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 21–24 (Table 1.04). ISBN 978-0-387-18430-2.

External links

MathWorld description of parabolic cylindrical coordinates

Orthogonal coordinate systems
Two dimensional	Cartesian Polar (Log-polar) Parabolic Bipolar Elliptic
Three dimensional	Cartesian Cylindrical Spherical Parabolic Paraboloidal Oblate spheroidal Prolate spheroidal Ellipsoidal Elliptic cylindrical Toroidal Bispherical Bipolar cylindrical Conical 6-sphere

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