橢球坐標系: a=1,b=0.8,c=0.6, 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 )}
直角坐標
(
x
,
y
,
z
)
{\displaystyle (x,\ y,\ z)}
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}}
橢球上的與雙曲面相交的曲線，a=1, b=0.8, c=0.6.
λ
{\displaystyle \lambda }
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 }
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 }
雙葉雙曲面 (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 }
(
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} }
∇
×
F
{\displaystyle \nabla \times \mathbf {F} }
正交坐標 條目內對應的一般公式。
Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 663. Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. ISBN 0-86720-293-9 Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: pp. 101–102. Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 176. Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 178–180. 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 
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/998c61bdad85baca9a1232ac126fff1b6c7ee1a2)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8efed646e4a9013b1e5bc0161d543c6c58a892)
