圓錐坐標系

维基百科,自由的百科全书
跳转至: 导航搜索
圓錐坐標系的幾個坐標曲面。紅色圓球面的 r=2 。錐軸為 z-軸的藍色圓錐面的 \mu=\cosh(1) 。錐軸為 x-軸的黃色圓錐面 \nu^2= 2/3 。z-軸是垂直的,以白色表示。 x-軸以綠色表示。三個坐標曲面相交於點 P (以黑色的圓球表示),直角坐標大約為 (1.26,\  - 0.78,\  1.34)

圓錐坐標系是一種三維正交坐標系。它的三個坐標曲面分別為同心圓球面,錐軸為 x-軸的圓錐面,錐軸為 z-軸的圓錐面。

基本定義[编辑]

圓錐坐標 (r,\ \mu,\ \nu) 通常定義為

x = \frac{r\mu\nu}{bc}
y = \frac{r}{b} \sqrt{\frac{\left( \mu^{2} - b^{2} \right) \left( \nu^{2} - b^{2} \right)}{\left( b^{2} - c^{2} \right)} }
z = \frac{r}{c} \sqrt{\frac{\left( \mu^{2} - c^{2} \right) \left( \nu^{2} - c^{2} \right)}{\left( c^{2} - b^{2} \right)} }

其中,0\le r < \inf \nu^{2} < c^{2} < \mu^{2} < b^{2}

坐標曲面[编辑]

r 坐標曲面是圓心在原點的圓球面:

x^{2} + y^{2} + z^{2} = r^{2}

\mu\nu 坐標曲面是兩個相交的圓錐面:

\frac{x^{2}}{\mu^{2}} + \frac{y^{2}}{\mu^{2} - b^{2}} + \frac{z^{2}}{\mu^{2} - c^{2}} = 0
\frac{x^{2}}{\nu^{2}} + \frac{y^{2}}{\nu^{2} - b^{2}} + \frac{z^{2}}{\nu^{2} - c^{2}} = 0

標度因子[编辑]

如同球坐標系,徑向坐標 r 的標度因子是

h_{r} = 1

坐標\mu\nu 的標度因子分別是

h_{\mu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \mu^{2} \right) \left( \mu^{2} - c^{2} \right)}}
h_{\nu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \nu^{2} \right) \left( c^{2} - \nu^{2} \right)}}

參閱[编辑]

參考目錄[编辑]

  • Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 659. ISBN 0-07-043316-X. 
  • Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 183–184. 
  • Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 179. ASIN B0000CKZX7. 
  • Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: pp. 991–100. 
  • Arfken G. Mathematical Methods for Physicists 2nd ed. Orlando, FL: Academic Press. 1970: pp. 118–119. ASIN B000MBRNX4. 
  • Moon P, Spencer DE. Conical Coordinates (r, θ, λ)//Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer-Verlag. 1988: pp. 37–40 (Table 1.09). ISBN 978-0387184302.