# 扁球面坐標系

## 第一種表述

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

### 坐標曲面

${\displaystyle \mu }$坐標曲面是扁球面 ：

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

${\displaystyle \nu }$坐標曲面是半個單葉旋轉雙曲面 ：

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

${\displaystyle \phi }$坐標曲面是個半平面 ：

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

### 逆變換

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

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

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

### 標度因子

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

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

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

## 第二種表述

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

### 標度因子

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

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

## 第三種表述

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

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

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

### 坐標曲面

${\displaystyle \sigma }$坐標曲面是扁球面 ：

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

${\displaystyle \tau }$坐標曲面是單葉旋轉雙曲面 ：

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

${\displaystyle \phi }$坐標曲面是半個平面 ：

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

### 標度因子

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

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

## 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${\displaystyle \xi _{1}=a\sinh \mu }$${\displaystyle \xi _{2}=\sin \nu }$${\displaystyle \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${\displaystyle u_{k}}$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. 98. 採用混合坐標${\displaystyle \xi =a\sinh \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. 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. 31–34 (Table 1.07). 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. 視扁球面坐標系為橢球坐標系的極限。採用第二種表述。