# 椭球坐标系

λ=-0.1 红色椭球, μ=-0.5 蓝色单叶双曲面, ν=-0.8 品红色双叶双曲面.

## 基本公式

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

${\displaystyle -\lambda

## 坐标曲面

${\displaystyle \lambda }$-坐标曲面是椭球面 ：

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

${\displaystyle \mu }$-坐标曲面是单叶双曲面 (hyperboloid of one sheet) ：

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

${\displaystyle \nu }$-坐标曲面是双叶双曲面 (hyperboloid of two sheet) ：

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

## 标度因子

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

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

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

## 参考目录

• 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: 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.