Mixing
橢球坐標系
橢圓坐標系
橢球坐標系（英语：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。
目录
1 基本公式
2 坐標曲面
3 標度因子
4 參閱
5 參考目錄
基本公式
橢球坐標 (λ, μ, ν){displaystyle (lambda , mu , nu )}
x2=(a2+λ)(a2+μ)(a2+ν)(a2−b2)(a2−c2){displaystyle x^{2}={frac {(a^{2}+lambda )(a^{2}+mu )(a^{2}+nu )}{(a^{2}-b^{2})(a^{2}-c^{2})}}}、
y2=(b2+λ)(b2+μ)(b2+ν)(b2−a2)(b2−c2){displaystyle y^{2}={frac {(b^{2}+lambda )(b^{2}+mu )(b^{2}+nu )}{(b^{2}-a^{2})(b^{2}-c^{2})}}}、
z2=(c2+λ)(c2+μ)(c2+ν)(c2−b2)(c2−a2){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>−c2>μ>−b2>ν>−a2{displaystyle lambda >-c^{2}>mu >-b^{2}>nu >-a^{2}}。
坐標曲面
λ{displaystyle lambda }
x2a2+λ+y2b2+λ+z2c2+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 }
x2a2+μ+y2b2+μ+z2c2+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 }
x2a2+ν+y2b2+ν+z2c2+ν=1{displaystyle {frac {x^{2}}{a^{2}+nu }}+{frac {y^{2}}{b^{2}+nu }}+{frac {z^{2}}{c^{2}+nu }}=1}。
標度因子
為了簡化標度因子的計算，設定函數
S(σ) =def (a2+σ)(b2+σ)(c2+σ){displaystyle S(sigma ) {stackrel {mathrm {def} }{=}} (a^{2}+sigma )(b^{2}+sigma )(c^{2}+sigma )}；
其中，參數 σ{displaystyle sigma }
橢球坐標的標度因子分別為
hλ=12(λ−μ)(λ−ν)S(λ){displaystyle h_{lambda }={frac {1}{2}}{sqrt {frac {(lambda -mu )(lambda -nu )}{S(lambda )}}}}、
hμ=12(μ−λ)(μ−ν)S(μ){displaystyle h_{mu }={frac {1}{2}}{sqrt {frac {(mu -lambda )(mu -nu )}{S(mu )}}}}、
hν=12(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
dV=(λ−μ)(λ−ν)(μ−ν)8−S(λ)S(μ)S(ν) dλdμdν{displaystyle dV={frac {(lambda -mu )(lambda -nu )(mu -nu )}{8{sqrt {-S(lambda )S(mu )S(nu )}}}} dlambda dmu dnu }。
拉普拉斯算子是
∇2Φ=4S(λ)(λ−μ)(λ−ν)∂∂λ[S(λ)∂Φ∂λ] + 4S(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]}
+ 4S(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]}。
![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]](https://wikimedia.org/api/rest_v1/media/math/render/svg/998c61bdad85baca9a1232ac126fff1b6c7ee1a2)
其它微分算子，例如 ∇⋅F{displaystyle nabla cdot mathbf {F} }
參閱
- 橢球
- 類球面
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參考目錄
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)
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