# 橢球坐標系

## 基本公式

$x^{2} =\frac{(a^{2}+\lambda) (a^{2}+\mu)( a^{2}+\nu)}{(a^{2} - b^{2}) (a^{2} - c^{2})}$
$y^{2} = \frac{( b^{2} + \lambda)( b^{2} + \mu)( b^{2} + \nu )}{( b^{2} - a^{2})( b^{2} - c^{2})}$
$z^{2} = \frac{( c^{2} + \lambda)( c^{2} + \mu)( c^{2} + \nu)}{( c^{2} - b^{2} ) ( c^{2} - a^{2})}$

$\lambda > - c^{2} > \mu > - b^{2} > \nu > - a^{2}$

## 坐標曲面

$\lambda$-坐標曲面是橢球面 ：

$\frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} + \frac{z^{2}}{c^{2} + \lambda} = 1$

$\mu$-坐標曲面是單葉雙曲面 (hyperboloid of one sheet) ：

$\frac{x^{2}}{a^{2} + \mu} + \frac{y^{2}}{b^{2} + \mu} + \frac{z^{2}}{c^{2} + \mu} = 1$

$\nu$-坐標曲面是双葉雙曲面 (hyperboloid of two sheet) ：

$\frac{x^{2}}{a^{2} + \nu} + \frac{y^{2}}{b^{2} + \nu} + \frac{z^{2}}{c^{2} + \nu} = 1$

## 標度因子

$S(\sigma) \ \stackrel{\mathrm{def}}{=}\ ( a^{2} + \sigma)( b^{2} + \sigma ) ( c^{2} + \sigma)$

$h_{\lambda} = \frac{1}{2} \sqrt{\frac{( \lambda - \mu)( \lambda - \nu)}{S(\lambda)}}$
$h_{\mu} = \frac{1}{2} \sqrt{\frac{( \mu - \lambda)( \mu - \nu)}{S(\mu)}}$
$h_{\nu} = \frac{1}{2} \sqrt{\frac{( \nu - \lambda)( \nu - \mu)}{S(\nu)}}$

$dV = \frac{( \lambda - \mu)( \lambda - \nu)( \mu - \nu)}{8\sqrt{-S(\lambda) S(\mu) S(\nu)}} \ d\lambda d\mu d\nu$
$\nabla^{2} \Phi = \frac{4\sqrt{S(\lambda)}}{\left( \lambda - \mu \right) \left( \lambda - \nu\right)} \frac{\partial}{\partial \lambda} \left[ \sqrt{S(\lambda)} \frac{\partial \Phi}{\partial \lambda} \right] \ + \ \frac{4\sqrt{S(\mu)}}{\left( \mu - \lambda \right) \left( \mu - \nu\right)} \frac{\partial}{\partial \mu} \left[ \sqrt{S(\mu)} \frac{\partial \Phi}{\partial \mu} \right]$
$+ \ \frac{4\sqrt{S(\nu)}}{\left( \nu - \lambda \right) \left( \nu - \mu\right)} \frac{\partial}{\partial \nu} \left[ \sqrt{S(\nu)} \frac{\partial \Phi}{\partial \nu} \right]$

## 參考目錄

• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 663.
• Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. ISBN 0-86720-293-9.
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: pp. 101–102.
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 176.
• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956年: pp. 178–180.
• Moon PH, Spencer DE. Ellipsoidal Coordinates (η, θ, λ). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer Verlag. 1988: pp. 40–44 (Table 1.10). ISBN 0-387-02732-7.