查看“︁拋物面坐標系”︁的源代码

'''拋物面坐標系'''（{{lang-en|Paraboloidal coordinates}}）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。
[[File:Parabolic coordinates 3D.png|thumb|right|200px|三维抛物面坐标系的[[坐标表面]]。]]

==基本公式==
從[[直角坐標]] <math>(x,\ y,\ z)</math> 變換至拋物面坐標 <math>( \lambda,\ \mu,\ \nu )</math> ：
:<math>x^{2} = \frac{\left( A - \lambda \right) \left( A - \mu \right) \left( A - \nu \right)}{B - A}</math> 、
:<math>y^{2} = \frac{\left( B - \lambda \right) \left( B - \mu \right) \left( B - \nu \right)}{A - B}</math> 、
:<math>z = \frac{1}{2} \left( A + B - \lambda - \mu -\nu \right)</math> ；

其中，拋物面坐標遵守以下限制：
:<math>\lambda < B < \mu < A < \nu</math> 。

==坐標曲面==
<math>\lambda</math>-坐標曲面是[[橢圓拋物面]] ({{lang|en|elliptic paraboloid}}) ：
:<math>\frac{x^{2}}{\lambda - A}+\frac{y^{2}}{\lambda - B}=2z+\lambda</math> 。

<math>\mu</math>-坐標曲面是[[雙曲拋物面]] ：
:<math>\frac{x^{2}}{\mu - A}+\frac{y^{2}}{\mu - B}=2z+\mu</math> 。

<math>\nu</math>-坐標曲面也是橢圓拋物面 ：
:<math>\frac{x^{2}}{\nu - A} +  \frac{y^{2}}{\nu - B}  = 2z + \nu</math> 。

==標度因子==
拋物面坐標的標度因子分別為
:<math>h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right)}{ \left( A - \lambda \right) \left( B - \lambda \right)}}</math> 、

:<math>h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \mu \right) \left( \lambda - \mu \right)}{ \left( A - \mu \right) \left( B - \mu \right)}}</math> 、

:<math>h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \nu \right) \left( \mu - \nu \right)}{ \left( A - \nu \right) \left( B - \nu \right)}}</math> 。

無窮小體積元素等於
:<math>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
</math> 。

其它微分算子，例如 <math>\nabla \cdot \mathbf{F}</math> 、<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==
{{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 | id = ISBN 0-07-043316-X| pages = p. 664}}

* {{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. 184–185 }}

* {{cite book | author = Korn GA, Korn TM |date = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id =ASIN B0000CKZX7 | pages = p. 180}}

* {{cite book | author = Arfken G | date = 1970 | title = Mathematical Methods for Physicists | edition = 2nd ed. | publisher = Academic Press | location = Orlando, FL | pages = pp. 119–120}}

* {{cite book | author = Sauer R, Szabó I | date = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | pages = p. 98}}

* {{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}}  Same as Morse & Feshbach (1953), 代替 ''u''<sub>''k''</sub> 為 ξ<sub>''k''</sub>.  

* {{cite book | author = Moon P, Spencer DE | date = 1988 | chapter = Paraboloidal Coordinates (μ,\ ν,\ 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 | pages = pp. 44–48 (Table 1.11) | isbn = 978-0387184302}}

==外部連結==
{{Portal|数学}}
* [http://mathworld.wolfram.com/ConfocalParaboloidalCoordinates.html MathWorld 的拋物面坐標系] {{Wayback|url=http://mathworld.wolfram.com/ConfocalParaboloidalCoordinates.html |date=20210309100132 }} 

[[Category:坐標系|P]]