橢球坐標系

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橢圓坐標系

橢球坐標系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[编辑]

橢球坐標 (\lambda,\ \mu,\ \nu)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}

坐標曲面[编辑]

\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[编辑]

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

拉普拉斯算子


\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}\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。

參閱[编辑]

參考目錄[编辑]

  • 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.