抛物线坐标系

（重定向自拋物線坐標系）

二维抛物线坐标系

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

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

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

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

二维标度因子

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

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

三维抛物线坐标系

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

${\displaystyle \tan \phi ={\frac {y}{x}},\qquad 0\leq \phi \leq 2\pi }$

${\displaystyle \sigma ={\sqrt {-z+{\sqrt {x^{2}+y^{2}+z^{2}}}}}}$
${\displaystyle \tau ={\sqrt {z+{\sqrt {x^{2}+y^{2}+z^{2}}}}}}$
${\displaystyle \phi =\tan ^{-1}{\frac {y}{x}}}$

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

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

三维标度因子

${\displaystyle h_{\sigma }={\sqrt {\sigma ^{2}+\tau ^{2}}}}$
${\displaystyle h_{\tau }={\sqrt {\sigma ^{2}+\tau ^{2}}}}$
${\displaystyle h_{\phi }=\sigma \tau \,}$

${\displaystyle dV=h_{\sigma }h_{\tau }h_{\phi }=\sigma \tau \left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau \,d\phi }$
${\displaystyle \nabla ^{2}\Phi ={\frac {1}{\sigma ^{2}+\tau ^{2}}}\left[{\frac {1}{\sigma }}{\frac {\partial }{\partial \sigma }}\left(\sigma {\frac {\partial \Phi }{\partial \sigma }}\right)+{\frac {1}{\tau }}{\frac {\partial }{\partial \tau }}\left(\tau {\frac {\partial \Phi }{\partial \tau }}\right)\right]+{\frac {1}{\sigma ^{2}\tau ^{2}}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}$

第二种表述

${\displaystyle \eta ={-z+{\sqrt {x^{2}+y^{2}+z^{2}}}}}$
${\displaystyle \xi ={z+{\sqrt {x^{2}+y^{2}+z^{2}}}}}$
${\displaystyle \phi =\arctan {y \over x}}$

${\displaystyle \eta =-z+{\sqrt {x^{2}+z^{2}}}}$
${\displaystyle \xi =z+{\sqrt {x^{2}+z^{2}}}}$

${\displaystyle \left.z\right|_{\eta =c}={x^{2} \over 2c}-{c \over 2}}$

${\displaystyle \left.z\right|_{\xi =b}={b \over 2}-{x^{2} \over 2b}}$

${\displaystyle {x^{2} \over 2c}-{c \over 2}={b \over 2}-{x^{2} \over 2b}}$

${\displaystyle x={\sqrt {bc}}}$

${\displaystyle z_{c}={bc \over 2c}-{c \over 2}={b-c \over 2}}$

${\displaystyle {\frac {dz_{c}}{dx}}={\frac {x}{c}}={\sqrt {\frac {b}{c}}}=s_{c}}$

${\displaystyle {dz_{b} \over dx}=-{x \over b}=-{\sqrt {c \over b}}=s_{b}}$

${\displaystyle s_{c}s_{b}=-1}$

${\displaystyle x={\sqrt {\eta \xi }}\ \cos \phi }$
${\displaystyle y={\sqrt {\eta \xi }}\ \sin \phi }$
${\displaystyle z={\frac {1}{2}}(\xi -\eta )}$

参考文献

• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 660. ISBN 0-07-043316-X.
• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 185–186.
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 180.
• 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.
• Moon P, Spencer DE. Parabolic Coordinates (μ, ν, ψ). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer-Verlag. 1988: pp. 34–36 (Table 1.08). ISBN 978-0387184302.
1. ^ Menzel, Donald H. Mathematical Physics. United States of America: Dover Publications. 1961: pp. 139. ISBN 978-0486600567 （英语）.