# 拋物柱面坐標系

## 基本定義

${\displaystyle x=\sigma \tau }$
${\displaystyle y={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)}$
${\displaystyle z=z}$

${\displaystyle 2y={\frac {x^{2}}{\sigma ^{2}}}-\sigma ^{2}}$

${\displaystyle 2y=-{\frac {x^{2}}{\tau ^{2}}}+\tau ^{2}}$

${\displaystyle r={\sqrt {x^{2}+y^{2}}}={\frac {1}{2}}\left(\sigma ^{2}+\tau ^{2}\right)}$

## 標度因子

${\displaystyle h_{\sigma }=h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2}}}}$
${\displaystyle h_{z}=1}$

${\displaystyle dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz}$
${\displaystyle \nabla ^{2}\Phi ={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}\Phi }{\partial \sigma ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}\Phi }{\partial z^{2}}}}$

## 參考文獻

• Philip M. Morse, Herman Feshbach. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 658. ISBN 0-07-043316-X.
• Henry Margenau, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 186–187.
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 181. ASIN B0000CKZX7.
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: p. 96.
• Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. ISBN 0-86720-293-9. Same as Morse & Feshbach (1953), substituting uk for ξk
• Moon P, Spencer DE. Parabolic-Cylinder Coordinates (μ, ν, z). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer-Verlag. 1988: pp. 21–24 (Table 1.04). ISBN 978-0387184302.

## 外部連結

• 數學世界的拋物柱面坐標系頁面