Parabolic Coordinates

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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

, 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)

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

(5)

and

$\begin{displaymath} \sqrt{x^2+y^2+z^2}+z = u^2 \end{displaymath}$

(6)

$\begin{displaymath} \sqrt{x^2+y^2+z^2}-z = v^2. \end{displaymath}$

(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

$\begin{displaymath} ds^2=(u^2+v^2)(du^2+dv^2)+u^2v^2\,d\theta^2, \end{displaymath}$

(14)

and the Volume Element is

$\begin{displaymath} dV=uv(u^2+v^2)\,du\,dv\,d\theta. \end{displaymath}$

(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.

References

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.