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Parabolic Coordinates

\begin{figure}\begin{center}\BoxedEPSF{ParabolicCoordinates.epsf scaled 800}\end{center}\end{figure}

A system of Curvilinear Coordinates in which two sets of coordinate surfaces are obtained by revolving the parabolas of Parabolic Cylindrical Coordinates about the x-Axis, which is then relabeled the z-Axis. There are several notational conventions. Whereas $(u, v, \theta)$ is used in this work, Arfken (1970) uses $(\xi, \eta, \varphi)$.

The equations for the parabolic coordinates are

$\displaystyle x$ $\textstyle =$ $\displaystyle uv\cos\theta$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle uv\sin\theta$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(u^2-v^2),$ (3)

where $u \in [0,\infty)$, $v \in [0, \infty)$, and $\theta \in [0, 2\pi)$. To solve for $u$, $v$, and $\theta$, examine
$\displaystyle x^2+y^2+z^2$ $\textstyle =$ $\displaystyle u^2v^2+{\textstyle{1\over 4}}(u^4-2u^2v^2+v^4)$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}(u^4+2u^2v^2+v^4)$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}(u^2+v^2)^2,$ (4)

\sqrt{x^2+y^2+z^2} = {\textstyle{1\over 2}}(u^2+v^2)
\end{displaymath} (5)

\sqrt{x^2+y^2+z^2}+z = u^2
\end{displaymath} (6)

\sqrt{x^2+y^2+z^2}-z = v^2.
\end{displaymath} (7)

We therefore have
$\displaystyle u$ $\textstyle =$ $\displaystyle \sqrt{\sqrt{x^2+y^2+z^2}+z}$ (8)
$\displaystyle v$ $\textstyle =$ $\displaystyle \sqrt{\sqrt{x^2+y^2+z^2}-z}$ (9)
$\displaystyle \theta$ $\textstyle =$ $\displaystyle \tan^{-1}\left({y\over x}\right).$ (10)

The Scale Factors are
$\displaystyle h_u$ $\textstyle =$ $\displaystyle \sqrt{u^2+v^2}$ (11)
$\displaystyle h_v$ $\textstyle =$ $\displaystyle \sqrt{u^2+v^2}$ (12)
$\displaystyle h_\theta$ $\textstyle =$ $\displaystyle uv.$ (13)

The Line Element is
\end{displaymath} (14)

and the Volume Element is
\end{displaymath} (15)

The Laplacian is

$\displaystyle \nabla^2 f$ $\textstyle =$ $\displaystyle {1\over uv(u^2+v^2)}\left[{{\partial\over\partial u}\left({uv {\p...
...partial v}}\right)}\right]+ {1\over u^2 v^2} {\partial^2f\over\partial\theta^2}$  
  $\textstyle =$ $\displaystyle {1\over u^2+v^2}\left[{{1\over u}{\partial\over\partial u}\left({...
...partial v}}\right)}\right]+ {1\over u^2 v^2} {\partial^2f\over\partial\theta^2}$  
  $\textstyle =$ $\displaystyle {1\over u^2+v^2}\left({{1\over u}{\partial f\over\partial u}+{\pa...
...f\over\partial v^2}}\right)+{1\over u^2v^2}{\partial^2 f\over\partial\theta^2}.$ (16)

The Helmholtz Differential Equation is Separable in parabolic coordinates.

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


Arfken, G. ``Parabolic Coordinates ($\xi$, $\eta$, $\phi$).'' §2.12 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109-112, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 660, 1953.

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© 1996-9 Eric W. Weisstein