# 雙球坐標系

## 基本定義

${\displaystyle x=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\cos \phi }$
${\displaystyle y=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\sin \phi }$
${\displaystyle z=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}}$

${\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}$

### 坐標曲面

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

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

### 逆變換

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

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

${\displaystyle \tau }$${\displaystyle d_{1}}$${\displaystyle d_{2}}$ 的比例的自然對數

${\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}}$

${\displaystyle \cos \sigma ={\frac {d_{1}^{2}+d_{2}^{2}-4a^{2}}{2d_{1}d_{2}}}}$

### 標度因子

${\displaystyle h_{\sigma }=h_{\tau }={\frac {a}{\cosh \tau -\cos \sigma }}}$

${\displaystyle h_{\phi }={\frac {a\sin \sigma }{\cosh \tau -\cos \sigma }}}$

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

## 參考目錄

• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 665–666.
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 182.
• Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 113. ISBN 0-86720-293-9.
• Moon PH, Spencer DE. Toroidal Coordinates (η, θ, ψ). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions 2nd ed., 3rd revised printing. New York: Springer Verlag. 1988: pp. 110–112 (Section IV, E4Ry). ISBN 0-387-02732-7.