# 拋物面坐標系

## 基本公式

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

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

## 坐標曲面

$\lambda$-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

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

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

## 參考目錄

• 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.