# 扁球面坐標系

## 第一種表述

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

### 坐標曲面

$\mu$坐標曲面是扁球面 ：

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

$\nu$坐標曲面是半個單葉旋轉雙曲面 ：

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

$\phi$坐標曲面是個半平面 ：

$x\sin\phi - y\cos\phi=0$

### 逆變換

$\tan \phi = \frac{y}{x}$

$d_{1}^{2} = (\sqrt{x^{2}+y^{2}} + a)^{2} + z^{2}$
$d_{2}^{2} = (\sqrt{x^{2}+y^{2}} - a)^{2} + z^{2}$

$\cosh \mu = \frac{d_{1} + d_{2}}{2a}$
$\cos \nu = \frac{d_{1} - d_{2}}{2a}$

### 標度因子

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

$h_{\phi} = a \cosh\mu \ \cos\nu$

$dV = a^{3} \cosh\mu \ \cos\nu \ \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu d\phi$
$\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)} \left[ \frac{1}{\cosh \mu} \frac{\partial}{\partial \mu} \left( \cosh \mu \frac{\partial \Phi}{\partial \mu} \right) + \frac{1}{\cos \nu} \frac{\partial}{\partial \nu} \left(\cos \nu \frac{\partial \Phi}{\partial \nu} \right) \right] + \frac{1}{a^{2} \left( \cosh^{2}\mu\cos^{2}\nu \right)} \frac{\partial^{2} \Phi}{\partial \phi^{2}}$

## 第二種表述

$x = a\sqrt{(1+\zeta^2)(1-\xi^2)}\,\cos \phi$
$y = a\sqrt{(1+\zeta^2)(1-\xi^2)}\,\sin \phi$
$z = a \zeta \xi$

### 標度因子

$h_{\zeta} = a\sqrt{\frac{\zeta^2 + \xi^2}{1+\zeta^2}}$
$h_{\xi} = a\sqrt{\frac{\zeta^2 + \xi^2}{1 - \xi^2}}$
$h_{\phi} = a\sqrt{(1+\zeta^2)(1 - \xi^2)}$

$dV = a^{3} (\zeta^2+\xi^2)\,d\zeta\,d\xi\,d\phi$
$\nabla^{2} V = \frac{1}{a^2 \left( \zeta^2 + \xi^2 \right)} \left\{ \frac{\partial}{\partial \zeta} \left[ \left(1+\zeta^2\right) \frac{\partial V}{\partial \zeta} \right] + \frac{\partial}{\partial \xi} \left[ \left( 1 - \xi^2 \right) \frac{\partial V}{\partial \xi} \right] \right\} + \frac{1}{a^2 \left( 1+\zeta^2 \right) \left( 1 - \xi^{2} \right)} \frac{\partial^2 V}{\partial \phi^{2}}$

## 第三種表述

$\sigma=\cosh\mu$
$\tau=\cos \nu$
$\phi=\phi$

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

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

### 坐標曲面

$\sigma$坐標曲面是扁球面 ：

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

$\tau$坐標曲面是單葉旋轉雙曲面 ：

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

$\phi$坐標曲面是半個平面 ：

$x\sin\phi - y\cos\phi=0$

### 標度因子

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

$dV = a^{3} \sigma \tau \frac{\sigma^{2} + \tau^{2}}{\sqrt{\left( \sigma^{2} + 1 \right) \left( 1 - \tau^{2} \right)}} d\sigma d\tau d\phi$
$\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sigma^{2} + \tau^{2} \right)} \left\{ \frac{\partial}{\partial \sigma} \left[ \left( \sigma^{2} + 1 \right) \frac{\partial \Phi}{\partial \sigma} \right] + \frac{\partial}{\partial \tau} \left[ \left( 1 - \tau^{2} \right) \frac{\partial \Phi}{\partial \tau} \right] \right\} + \frac{1}{a^{2} \left( \sigma^{2} + 1 \right) \left( 1 - \tau^{2} \right)} \frac{\partial^{2} \Phi}{\partial \phi^{2}}$

## 參考文獻

1. ^ Smythe, 1968。
2. ^ Abramowitz and Stegun, p. 752。

## 參考目錄

### 不按照命名常規

• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 662. 採用$\xi_1=a\sinh\mu$$\xi_2=\sin\nu$$\xi_3=\cos\phi$
• Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 115. ISBN 0-86720-293-9. 如同Morse & Feshbach (1953)，採用$u_k$來替代$\xi_k$
• Smythe, WR. Static and Dynamic Electricity 3rd ed. New York: McGraw-Hill. 1968.
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: p. 98. 採用混合坐標$\xi=a\sinh\mu$$\eta=\sin\nu$$\phi=\phi$

### 按照命名常規

• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 177.採用第一種表述$(\mu,\ \nu,\ \phi)$，又加介紹了簡併的第三種表述$(\sigma,\ \tau,\ \phi)$
• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: p. 182. 如同Korn and Korn (1961)，但採用餘緯度$\theta=90^{\circ} - \nu$來替代緯度$\nu$
• Moon PH, Spencer DE. Oblate spheroidal coordinates (η, θ, ψ)//Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer Verlag. 1988: pp. 31–34 (Table 1.07). ISBN 0-387-02732-7. Moon and Spencer採用餘緯度常規$\theta=90^{\circ} - \nu$，又改名$\phi$$\psi$

### 特異命名常規

• Landau LD, Lifshitz EM, Pitaevskii LP. Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) 2nd edition. New York: Pergamon Press. 1984: pp. 19–29. ISBN 978-0750626347.視扁球面坐標系為橢球坐標系的極限。採用第二種表述。