# 拋物柱面坐標系

## 基本定義

$x = \sigma \tau$
$y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right)$
$z = z$

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

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

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

## 標度因子

$h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}}$
$h_{z}=1$

$dV = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz$
$\nabla^{2} \Phi = \frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} \Phi}{\partial \sigma^{2}} + \frac{\partial^{2} \Phi}{\partial \tau^{2}} \right) + \frac{\partial^{2} \Phi}{\partial z^{2}}$

## 參考文獻

• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 186–187.
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 181. ASIN B0000CKZX7.
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: p. 96.
• Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. ISBN 0-86720-293-9. Same as Morse & Feshbach (1953), substituting uk for ξk
• Moon P, Spencer DE. 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. 1988: pp. 21–24 (Table 1.04). ISBN 978-0387184302.