# 長球面坐標系

## 基本定義

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

### 坐標曲面

${\displaystyle \mu }$  坐標曲面長球面

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

${\displaystyle \nu }$  坐標曲面是半個旋轉雙葉雙曲面

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

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

${\displaystyle \phi }$  坐標曲面是個半平面 ：

${\displaystyle x\sin \phi -y\cos \phi =0}$

### 標度因子

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

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

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

## 第二種表述

${\displaystyle \sigma =\cosh \mu }$
${\displaystyle \tau =\cos \nu }$
${\displaystyle \phi =\phi }$

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

### 坐標曲面

${\displaystyle \sigma }$  坐標曲面長球面

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

${\displaystyle \tau }$  坐標曲面是半個旋轉雙曲面

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

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

${\displaystyle \phi }$  坐標曲面是個半平面 ：

${\displaystyle x\sin \phi -y\cos \phi =0}$

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

### 標度因子

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

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

## 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 ${\displaystyle \xi _{1}=a\cosh \mu }$ ${\displaystyle \xi _{2}=\sin \nu }$ ${\displaystyle \xi _{3}=\cos \phi }$
• 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 ${\displaystyle u_{k}}$  來替代 ${\displaystyle \xi _{k}}$
• 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 ${\displaystyle \xi =\cosh \mu }$ ${\displaystyle \eta =\sin \nu }$ ${\displaystyle \phi =\phi }$

### 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 ${\displaystyle (\mu ,\ \nu ,\ \phi )}$  ，又加介紹了簡併的第三種表述 ${\displaystyle (\sigma ,\ \tau ,\ \phi )}$
• 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 ${\displaystyle \theta =90^{\circ }-\nu }$  來替代緯度 ${\displaystyle \nu }$
• 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 ${\displaystyle \theta =90^{\circ }-\nu }$  ，又改名 ${\displaystyle \phi }$ ${\displaystyle \psi }$

### 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。