橢球坐標系

橢球坐標系（英語：Ellipsoidal coordinates）是一種三維正交坐標系，是橢圓坐標系的推廣。與大多數的三維正交坐標系的生成方法不同，橢球坐標系不是由任何二維正交坐標系延伸或旋轉生成的。 橢球坐標 ( λ ,   μ ,   ν ) {\displaystyle (\lambda ,\ \mu ,\ \nu )} 以直角坐標 ( x ,   y ,   z ) {\displaystyle (x,\ y,\ z)} 定義為： x 2 = ( a 2 + λ ) ( a 2 + μ ) ( a 2 + ν ) ( a 2 − b 2 ) ( a 2 − c 2 ) {\displaystyle x^{2}={\frac {(a^{2}+\lambda )(a^{2}+\mu )(a^{2}+\nu )}{(a^{2}-b^{2})(a^{2}-c^{2})}}} 、 y 2 = ( b 2 + λ ) ( b 2 + μ ) ( b 2 + ν ) ( b 2 − a 2 ) ( b 2 − c 2 ) {\displaystyle y^{2}={\frac {(b^{2}+\lambda )(b^{2}+\mu )(b^{2}+\nu )}{(b^{2}-a^{2})(b^{2}-c^{2})}}} 、 z 2 = ( c 2 + λ ) ( c 2 + μ ) ( c 2 + ν ) ( c 2 − b 2 ) ( c 2 − a 2 ) {\displaystyle z^{2}={\frac {(c^{2}+\lambda )(c^{2}+\mu )(c^{2}+\nu )}{(c^{2}-b^{2})(c^{2}-a^{2})}}} ； 其中，橢球坐標遵守以下限制： − λ < c 2 < − μ < b 2 < − ν < a 2 {\displaystyle -\lambda <c^{2}<-\mu <b^{2}<-\nu <a^{2}} 。 λ {\displaystyle \lambda } -坐標曲面是橢球面 ： x 2 a 2 + λ + y 2 b 2 + λ + z 2 c 2 + λ = 1 {\displaystyle {\frac {x^{2}}{a^{2}+\lambda }}+{\frac {y^{2}}{b^{2}+\lambda }}+{\frac {z^{2}}{c^{2}+\lambda }}=1} 。 μ {\displaystyle \mu } -坐標曲面是單葉雙曲面 (hyperboloid of one sheet) ： x 2 a 2 + μ + y 2 b 2 + μ + z 2 c 2 + μ = 1 {\displaystyle {\frac {x^{2}}{a^{2}+\mu }}+{\frac {y^{2}}{b^{2}+\mu }}+{\frac {z^{2}}{c^{2}+\mu }}=1} 。 ν {\displaystyle \nu } -坐標曲面是双葉雙曲面 (hyperboloid of two sheet) ： x 2 a 2 + ν + y 2 b 2 + ν + z 2 c 2 + ν = 1 {\displaystyle {\frac {x^{2}}{a^{2}+\nu }}+{\frac {y^{2}}{b^{2}+\nu }}+{\frac {z^{2}}{c^{2}+\nu }}=1} 。 為了簡化標度因子的計算，設定函數 S ( σ )   = d e f   ( a 2 + σ ) ( b 2 + σ ) ( c 2 + σ ) {\displaystyle S(\sigma )\ {\stackrel {\mathrm {def} }{=}}\ (a^{2}+\sigma )(b^{2}+\sigma )(c^{2}+\sigma )} ； 其中，參數 σ {\displaystyle \sigma } 可以代表任何一個橢球坐標 ( λ ,   μ ,   ν ) {\displaystyle (\lambda ,\ \mu ,\ \nu )} 。 橢球坐標的標度因子分別為 h λ = 1 2 ( λ − μ ) ( λ − ν ) S ( λ ) {\displaystyle h_{\lambda }={\frac {1}{2}}{\sqrt {\frac {(\lambda -\mu )(\lambda -\nu )}{S(\lambda )}}}} 、 h μ = 1 2 ( μ − λ ) ( μ − ν ) S ( μ ) {\displaystyle h_{\mu }={\frac {1}{2}}{\sqrt {\frac {(\mu -\lambda )(\mu -\nu )}{S(\mu )}}}} 、 h ν = 1 2 ( ν − λ ) ( ν − μ ) S ( ν ) {\displaystyle h_{\nu }={\frac {1}{2}}{\sqrt {\frac {(\nu -\lambda )(\nu -\mu )}{S(\nu )}}}} 。 無窮小體積元素等於 d V = ( λ − μ ) ( λ − ν ) ( μ − ν ) 8 − S ( λ ) S ( μ ) S ( ν )   d λ d μ d ν {\displaystyle dV={\frac {(\lambda -\mu )(\lambda -\nu )(\mu -\nu )}{8{\sqrt {-S(\lambda )S(\mu )S(\nu )}}}}\ d\lambda d\mu d\nu } 。 拉普拉斯算子是 ∇ 2 Φ = 4 S ( λ ) ( λ − μ ) ( λ − ν ) ∂ ∂ λ [ S ( λ ) ∂ Φ ∂ λ ]   +   4 S ( μ ) ( μ − λ ) ( μ − ν ) ∂ ∂ μ [ S ( μ ) ∂ Φ ∂ μ ] {\displaystyle \nabla ^{2}\Phi ={\frac {4{\sqrt {S(\lambda )}}}{\left(\lambda -\mu \right)\left(\lambda -\nu \right)}}{\frac {\partial }{\partial \lambda }}\left[{\sqrt {S(\lambda )}}{\frac {\partial \Phi }{\partial \lambda }}\right]\ +\ {\frac {4{\sqrt {S(\mu )}}}{\left(\mu -\lambda \right)\left(\mu -\nu \right)}}{\frac {\partial }{\partial \mu }}\left[{\sqrt {S(\mu )}}{\frac {\partial \Phi }{\partial \mu }}\right]} +   4 S ( ν ) ( ν − λ ) ( ν − μ ) ∂ ∂ ν [ S ( ν ) ∂ Φ ∂ ν ] {\displaystyle +\ {\frac {4{\sqrt {S(\nu )}}}{\left(\nu -\lambda \right)\left(\nu -\mu \right)}}{\frac {\partial }{\partial \nu }}\left[{\sqrt {S(\nu )}}{\frac {\partial \Phi }{\partial \nu }}\right]} 。 其它微分算子，例如 ∇ ⋅ F {\displaystyle \nabla \cdot \mathbf {F} } 與 ∇ × F {\displaystyle \nabla \times \mathbf {F} } ，都可以用橢球坐標表達，只需要將標度因子代入正交坐標條目內對應的一般公式。 橢球 類球面 Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 663.  引文格式1维护：冗余文本 (link) Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. ISBN 0-86720-293-9.  引文格式1维护：冗余文本 (link) Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: pp. 101–102.  引文格式1维护：冗余文本 (link) Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 176.  引文格式1维护：冗余文本 (link) Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 178–180.  引文格式1维护：冗余文本 (link) Moon PH, Spencer DE. Ellipsoidal Coordinates (η, 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. 40–44 (Table 1.10). ISBN 0-387-02732-7.  引文格式1维护：冗余文本 (link)