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Parabolic Cylindrical Coordinates

\begin{figure}\begin{center}\BoxedEPSF{curv_coords_Parabolic.epsf scaled 1100}\end{center}\end{figure}

A system of Curvilinear Coordinates. There are several different conventions for the orientation and designation of these coordinates. Arfken (1970) defines coordinates $(\xi, \eta, z)$ such that

$\displaystyle x$ $\textstyle =$ $\displaystyle \xi\eta$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(\eta^2-\xi^2)$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle z.$ (3)

In this work, following Morse and Feshbach (1953), the coordinates $(u, v, z)$ are used instead. In this convention, the traces of the coordinate surfaces of the $xy$-Plane are confocal Parabolas with a common axis. The $u$ curves open into the Negative x-Axis; the $v$ curves open into the Positive x-Axis. The $u$ and $v$ curves intersect along the y-Axis.
$\displaystyle x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(u^2-v^2)$ (4)
$\displaystyle y$ $\textstyle =$ $\displaystyle uv$ (5)
$\displaystyle z$ $\textstyle =$ $\displaystyle z,$ (6)

where $u \in [0,\infty)$, $v \in [0, \infty)$, and $z \in (-\infty, \infty)$. The Scale Factors are
$\displaystyle h_1$ $\textstyle =$ $\displaystyle \sqrt{u^2+v^2}$ (7)
$\displaystyle h_2$ $\textstyle =$ $\displaystyle \sqrt{u^2+v^2}$ (8)
$\displaystyle h_3$ $\textstyle =$ $\displaystyle 1.$ (9)

Laplace's Equation is
\nabla^2 f = {1\over u^2 +v^2}\left({{\partial^2 f\over \par...
...over \partial v^2 }}\right)+ {\partial^2 f\over \partial z^2}.
\end{displaymath} (10)

The Helmholtz Differential Equation is Separable in parabolic cylindrical coordinates.

See also Confocal Paraboloidal Coordinates, Helmholtz Differential Equation--Parabolic Cylindrical Coordinates, Parabolic Coordinates


Arfken, G. ``Parabolic Cylinder Coordinates ($\xi$, $\eta$, $z$).'' §2.8 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, p. 97, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 658, 1953.

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© 1996-9 Eric W. Weisstein