# 圓錐坐標系

## 基本定義

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

## 坐標曲面

${\displaystyle r}$ 坐標曲面是圓心在原點的圓球面：

${\displaystyle x^{2}+y^{2}+z^{2}=r^{2}}$

${\displaystyle \mu }$${\displaystyle \nu }$ 坐標曲面是兩個相交的圓錐面：

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

### 標度因子

${\displaystyle h_{r}=1}$

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

## 參考目錄

• 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.