拋物面坐標系

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拋物面坐標系是一種三維正交坐標系,是二維拋物線坐標系的推廣。與大多數的三維正交坐標系的生成方法不同,拋物面坐標系不是由任何二維正交坐標系延伸或旋轉生成的。

三维抛物面坐标系的坐标表面

基本公式[编辑]

直角坐標 (x,\ y,\ z) 變換至拋物面坐標 ( \lambda,\ \mu,\ \nu )

x^{2} = \frac{\left( A - \lambda \right) \left( A - \mu \right) \left( A - \nu \right)}{B - A}
y^{2} = \frac{\left( B - \lambda \right) \left( B - \mu \right) \left( B - \nu \right)}{A - B}
z = \frac{1}{2} \left( A + B - \lambda - \mu -\nu \right)

其中,拋物面坐標遵守以下限制:

\lambda < B < \mu < A < \nu

坐標曲面[编辑]

\lambda-坐標曲面是橢圓拋物面 (elliptic paraboloid) :

\frac{x^{2}}{\lambda - A}+\frac{y^{2}}{\lambda - B}=2z+\lambda

\mu-坐標曲面是雙曲拋物面

\frac{x^{2}}{\mu - A}+\frac{y^{2}}{\mu - B}=2z+\mu

\nu-坐標曲面也是橢圓拋物面 :

\frac{x^{2}}{\nu - A} +  \frac{y^{2}}{\nu - B}  = 2z + \nu

標度因子[编辑]

拋物面坐標的標度因子分別為

h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right)}{ \left( A - \lambda \right) \left( B - \lambda \right)}}
h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \mu \right) \left( \lambda - \mu \right)}{ \left( A - \mu \right) \left( B - \mu \right)}}
h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \nu \right) \left( \mu - \nu \right)}{ \left( A - \nu \right) \left( B - \nu \right)}}

無窮小體積元素等於

dV = \frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right) \left( \nu - \mu\right)}{8\sqrt{\left( A - \lambda \right) \left( B - \lambda \right) \left( A - \mu \right) \left( \mu - B \right) \left( \nu - A \right) \left( \nu  - B \right) }} \  d\lambda d\mu d\nu

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]

  • Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 184–185. 
  • Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 180. ASIN B0000CKZX7. 
  • Arfken G. Mathematical Methods for Physicists 2nd ed. Orlando, FL: Academic Press. 1970: pp. 119–120. 
  • Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: p. 98. 
  • Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. ISBN 0-86720-293-9.  Same as Morse & Feshbach (1953), 代替 uk 為 ξk.
  • Moon P, Spencer DE. Paraboloidal Coordinates (μ,\ ν,\ λ). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer-Verlag. 1988: pp. 44–48 (Table 1.11). ISBN 978-0387184302. 

外部連結[编辑]