橢球坐標系

橢球坐標系（英語：Ellipsoidal coordinates）是一種三維正交坐標系，是橢圓坐標系的推廣。與大多數的三維正交坐標系的生成方法不同，橢球坐標系不是由任何二維正交坐標系延伸或旋轉生成的。

橢球坐標 $(\lambda ,\ \mu ,\ \nu )$ 以直角坐標 $(x,\ y,\ z)$ 定義為：

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})}}

；

其中，橢球坐標遵守以下限制：

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

。

$\lambda$ -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

。

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

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

。

為了簡化標度因子的計算，設定函數

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

；

其中，參數 $\sigma$ 可以代表任何一個橢球坐標 $(\lambda ,\ \mu ,\ \nu )$ 。

橢球坐標的標度因子分別為

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 )}}}

。

無窮小體積元素等於

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]

。

其它微分算子，例如 $\nabla \cdot \mathbf {F}$ 與 $\nabla \times \mathbf {F}$ ，都可以用橢球坐標表達，只需要將標度因子代入正交坐標條目內對應的一般公式。

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.

基本公式