# 雙球坐標系

## 基本定義

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

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

### 坐標曲面

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

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

### 逆變換

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

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

$\tau$$d_{1}$$d_{2}$ 的比例的自然對數

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

$\cos \sigma =\frac{d_1^2 + d_2^2 - 4a^2}{2 d_1 d_2}$

### 標度因子

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

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

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

## 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.