# 橢圓柱坐標系

## 基本定義

$x = a \ \cosh \mu \ \cos \nu$
$y = a \ \sinh \mu \ \sin \nu$
$z = z$

$\mu$ 的等值曲線形成了橢圓，而 $\nu$ 的等值曲線則形成了雙曲線

$\frac{x^{2}}{a^{2} \cosh^{2} \mu} + \frac{y^{2}}{a^{2} \sinh^{2} \mu} = \cos^{2} \nu + \sin^{2} \nu = 1$
$\frac{x^{2}}{a^{2} \cos^{2} \nu} - \frac{y^{2}}{a^{2} \sin^{2} \nu} = \cosh^{2} \mu - \sinh^{2} \mu = 1$

## 標度因子

$h_{\mu}=h_{\nu}=a\sqrt{\sinh^{2}\mu+\sin^{2}\nu}$
$h_{z}=1$

$dV = a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu dz$
$\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right) + \frac{\partial^{2} \Phi}{\partial z^{2}}$

## 第二種定義

$d_{1}+d_{2}=2a\sigma$
$d_{1} - d_{2}=2a\tau$

$d_{1}=a(\sigma+\tau)$
$d_{2}=a(\sigma - \tau)$

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

## 第二種標度因子

$h_{\sigma} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{\sigma^{2} - 1}}$
$h_{\tau} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{1 - \tau^{2}}}$
$h_{z}=1$

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

## 參考文獻

• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 657. ISBN 0-07-043316-X.
• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 182–183.
• Korn GA. Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 179.
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: p. 97.
• Moon P, Spencer DE. Elliptic-Cylinder Coordinates (η, ψ, z). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer-Verlag. 1988: pp. 17–20 (Table 1.03). ISBN 978-0387184302.