椭球坐标系

椭球坐标 ${\displaystyle (\lambda ,\ \mu ,\ \nu )}$  以直角坐标 ${\displaystyle (x,\ y,\ z)}$  定义为：

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

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

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

其中，椭球坐标遵守以下限制：

${\displaystyle -\lambda <c^{2}<-\mu <b^{2}<-\nu <a^{2}}$  。

${\displaystyle \lambda }$ -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) ：

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

为了简化标度因子的计算，设定函数

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

其中，参数 ${\displaystyle \sigma }$  可以代表任何一个椭球坐标 ${\displaystyle (\lambda ,\ \mu ,\ \nu )}$  。

椭球坐标的标度因子分别为

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

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

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

无穷小体积元素等于

${\displaystyle dV={\frac {(\lambda -\mu )(\lambda -\nu )(\mu -\nu )}{8{\sqrt {-S(\lambda )S(\mu )S(\nu )}}}}\ d\lambda d\mu d\nu }$  。

拉普拉斯算子是

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

${\displaystyle +\ {\frac {4{\sqrt {S(\nu )}}}{\left(\nu -\lambda \right)\left(\nu -\mu \right)}}{\frac {\partial }{\partial \nu }}\left[{\sqrt {S(\nu )}}{\frac {\partial \Phi }{\partial \nu }}\right]}$  。

其它微分算子，例如 ${\displaystyle \nabla \cdot \mathbf {F} }$  与 ${\displaystyle \nabla \times \mathbf {F} }$  ，都可以用椭球坐标表达，只需要将标度因子代入正交坐标条目内对应的一般公式。

• 椭球
• 类球面

• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 663.  引文格式1维护：冗余文本 (link)
• Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. ISBN 0-86720-293-9.  引文格式1维护：冗余文本 (link)
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: pp. 101–102.  引文格式1维护：冗余文本 (link)
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 176.  引文格式1维护：冗余文本 (link)
• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 178–180.  引文格式1维护：冗余文本 (link)
• 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.  引文格式1维护：冗余文本 (link)