# 長球面坐標系

## 基本定義

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

### 坐標曲面

$\mu$ 坐標曲面長球面

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

$\nu$ 坐標曲面是半個旋轉雙葉雙曲面

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

$\nu<\pi/2$ 時，坐標曲面在 xy-平面以上；當 $\nu>\pi/2$ 時，坐標曲面在 xy-平面以下。

$\phi$ 坐標曲面是個半平面 ：

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

### 標度因子

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

$h_{\phi} = a \sinh\mu \ \sin\nu$

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

## 第二種表述

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

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

### 坐標曲面

$\sigma$ 坐標曲面長球面

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

$\tau$ 坐標曲面是半個旋轉雙曲面

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

$\tau>0$ 時，坐標曲面在 xy-平面以上；當 $\tau<0$ 時，坐標曲面在 xy-平面以下。

$\phi$ 坐標曲面是個半平面 ：

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

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

### 標度因子

$h_{\sigma} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{\sigma^{2} - 1}}$
$h_{\tau} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{1 - \tau^{2}}}$
$h_{\phi} = a \sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}$

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

## 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. 661. 採用 $\xi_1=a\cosh\mu$$\xi_2=\sin\nu$$\xi_3=\cos\phi$
• Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. 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. 97. 採用混合坐標 $\xi=\cosh\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. 180–182. 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. 28–30 (Table 1.06). 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. 視長球面坐標系為橢球坐標系的極限。