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

are used instead. In this convention, the traces of the coordinate surfaces of the

-Plane are confocal Parabolas with a common axis. The

curves open into the Negative x-Axis; the

curves open into the Positive x-Axis. The

and

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

$\begin{displaymath} \nabla^2 f = {1\over u^2 +v^2}\left({{\partial^2 f\over \par... ...over \partial v^2 }}\right)+ {\partial^2 f\over \partial z^2}. \end{displaymath}$

(10)

The Helmholtz Differential Equation is Separable in parabolic cylindrical coordinates.

References

Arfken, G. ``Parabolic Cylinder Coordinates ( $\xi$ , $\eta$ , ).'' §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.