查看“橢球坐標系”的源代码

[[File:Quadriken-konfok.svg|350px|thumb|椭球坐标系: a=1,b=0.8,c=0.6, <br />
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.]]
'''橢球坐標系'''（{{lang-en|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。

==基本公式==
橢球坐標 <math>(\lambda,\ \mu,\ \nu)</math> 以[[直角坐標]] <math>(x,\ y,\ z)</math> 定義為：
:<math>x^{2} =\frac{(a^{2}+\lambda) (a^{2}+\mu)( a^{2}+\nu)}{(a^{2} - b^{2}) (a^{2} - c^{2})}</math> 、
:<math>y^{2} = \frac{( b^{2} + \lambda)( b^{2} + \mu)( b^{2} + \nu )}{( b^{2} - a^{2})( b^{2} - c^{2})}</math> 、
:<math>z^{2} = \frac{( c^{2} + \lambda)( c^{2} + \mu)( c^{2} + \nu)}{( c^{2} - b^{2}  ) ( c^{2} - a^{2})}</math> ；

其中，橢球坐標遵守以下限制：
:<math>- \lambda < c^{2} < - \mu < b^{2} < -\nu < a^{2}</math> 。

==坐標曲面==
[[File:Ellipsoid-kl.svg|thumb|350px|椭球上的与双曲面相交的曲线，a=1, b=0.8, c=0.6.]]
<math>\lambda</math>-坐標曲面是橢球面 ：
:<math>\frac{x^{2}}{a^{2} + \lambda} +  \frac{y^{2}}{b^{2} + \lambda} + \frac{z^{2}}{c^{2} + \lambda} = 1</math> 。

<math>\mu</math>-坐標曲面是[[單葉雙曲面]] ({{lang|en|hyperboloid of one sheet}}) ：
:<math>\frac{x^{2}}{a^{2} + \mu} +  \frac{y^{2}}{b^{2} + \mu} + \frac{z^{2}}{c^{2} + \mu} = 1</math> 。

<math>\nu</math>-坐標曲面是[[双葉雙曲面]] ({{lang|en|hyperboloid of two sheet}}) ：
:<math>\frac{x^{2}}{a^{2} + \nu} +  \frac{y^{2}}{b^{2} + \nu} + \frac{z^{2}}{c^{2} + \nu} = 1</math> 。

==標度因子==
為了簡化標度因子的計算，設定函數
:<math>S(\sigma) \ \stackrel{\mathrm{def}}{=}\  ( a^{2} + \sigma)( b^{2} + \sigma ) ( c^{2} + \sigma)</math> ；

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

橢球坐標的標度因子分別為
:<math>h_{\lambda} = \frac{1}{2} \sqrt{\frac{( \lambda - \mu)( \lambda - \nu)}{S(\lambda)}}</math> 、
:<math>h_{\mu} = \frac{1}{2} \sqrt{\frac{( \mu - \lambda)( \mu - \nu)}{S(\mu)}}</math> 、
:<math>h_{\nu} = \frac{1}{2} \sqrt{\frac{( \nu - \lambda)( \nu - \mu)}{S(\nu)}}</math> 。

無窮小體積元素等於
:<math>dV = \frac{( \lambda - \mu)( \lambda - \nu)( \mu - \nu)}{8\sqrt{-S(\lambda) S(\mu) S(\nu)}} \  d\lambda d\mu d\nu</math> 。

[[拉普拉斯算子]]是 
:<math>
\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]
</math>
:::<math> 
+ \  \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]
</math> 。

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

==參閱==
*[[橢球]]
*[[類球面]]

{{正交坐標系}}

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

[[Category:坐標系|T]]