抛物面坐标系

基本公式

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

${\displaystyle \lambda

坐标曲面

${\displaystyle \lambda }$-坐标曲面是椭圆抛物面 (elliptic paraboloid) ：

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

${\displaystyle \mu }$-坐标曲面是双曲抛物面

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

${\displaystyle \nu }$-坐标曲面也是椭圆抛物面 ：

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

标度因子

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

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

参考目录

• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 664. ISBN 0-07-043316-X.
• 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 为 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.