Izdužene sferoidne koordinate u trodimenzionalnom prostoru predstavljaju ortogonalni koordinatni sistem nastao rotacijom sferoida oko velike osi. Rotacijom oko manje osi dobijaju se spljoštene sferoidne koordinate. Izdužene sferoidne koordinate koriste se da se reše različite parcijalne diferencijalne jednačine, u kojima granični uslovi odgovaraju izduženom sferoidu sa dva fokusa na velikoj osi. Jedan od realnih primera je elektron u elektromagnetnom polju dva pozitivno nabijena jezgra, kao što je slučaj u jonizovanom molekulu vodonika
H
2
+
{\displaystyle H_{2}^{+}}
Najčešća definicija izduženih sferoidnih koordinata
(
μ
,
ν
,
ϕ
)
{\displaystyle (\mu ,\nu ,\phi )}
x
=
a
sinh
μ
sin
ν
cos
ϕ
{\displaystyle x=a\ \sinh \mu \ \sin \nu \ \cos \phi }
y
=
a
sinh
μ
sin
ν
sin
ϕ
{\displaystyle y=a\ \sinh \mu \ \sin \nu \ \sin \phi }
z
=
a
cosh
μ
cos
ν
{\displaystyle z=a\ \cosh \mu \ \cos \nu }
gde je
μ
{\displaystyle \mu }
ν
∈
[
0
,
π
]
{\displaystyle \nu \in [0,\pi ]}
ϕ
{\displaystyle \phi }
[
0
,
2
π
)
{\displaystyle [0,2\pi )}
Kvadrirajući gornje izraze dobija se:
z
2
a
2
cosh
2
μ
+
x
2
+
y
2
a
2
sinh
2
μ
=
cos
2
ν
+
sin
2
ν
=
1
{\displaystyle {\frac {z^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {x^{2}+y^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1}
što pokazuje da površi konstantnoga
μ
{\displaystyle \mu }
elipse , koje se rotiraju oko osi, koje spajaju njihove fokuse . Na sličan način dobija se i sledeća relacija:
z
2
a
2
cos
2
ν
−
x
2
+
y
2
a
2
sin
2
ν
=
cosh
2
μ
−
sinh
2
μ
=
1
{\displaystyle {\frac {z^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {x^{2}+y^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1}
iz koje se vidi da površi konstantnoga
ν
{\displaystyle \nu }
Lameovi koeficijenti skaliranja [ uredi | uredi izvor ] Lameovi koeficijenti skaliranja za eliptične koordinate
(
μ
,
ν
)
{\displaystyle (\mu ,\nu )}
h
μ
=
h
ν
=
a
sinh
2
μ
+
sin
2
ν
{\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}}
a azimutalni Lameov koeficijent je:
h
ϕ
=
a
sinh
μ
sin
ν
{\displaystyle h_{\phi }=a\sinh \mu \ \sin \nu }
Infinitezimalni element zapremine je:
d
V
=
a
3
sinh
μ
sin
ν
(
sinh
2
μ
+
sin
2
ν
)
d
μ
d
ν
d
ϕ
{\displaystyle dV=a^{3}\sinh \mu \ \sin \nu \ \left(\sinh ^{2}\mu +\sin ^{2}\nu \right)d\mu d\nu d\phi }
a Laplasijan je:
∇
2
Φ
=
1
a
2
(
sinh
2
μ
+
sin
2
ν
)
[
∂
2
Φ
∂
μ
2
+
∂
2
Φ
∂
ν
2
+
coth
μ
∂
Φ
∂
μ
+
cot
ν
∂
Φ
∂
ν
]
+
1
a
2
sinh
2
μ
sin
2
ν
∂
2
Φ
∂
ϕ
2
{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left[{\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}+\coth \mu {\frac {\partial \Phi }{\partial \mu }}+\cot \nu {\frac {\partial \Phi }{\partial \nu }}\right]+{\frac {1}{a^{2}\sinh ^{2}\mu \sin ^{2}\nu }}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}
Postoji alternativna definicija preko tri koordinate
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\tau ,\phi )}
σ
=
cosh
μ
{\displaystyle \sigma =\cosh \mu }
τ
=
cos
ν
{\displaystyle \tau =\cos \nu }
Onda dobijamo:
x
=
a
(
σ
2
−
1
)
(
1
−
τ
2
)
cos
ϕ
{\displaystyle x=a{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}\cos \phi }
y
=
a
(
σ
2
−
1
)
(
1
−
τ
2
)
sin
ϕ
{\displaystyle y=a{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}\sin \phi }
z
=
a
σ
τ
{\displaystyle z=a\ \sigma \ \tau }
Alternativni Lameovi koeficijenti [ uredi | uredi izvor ] Lameovi koeficijenti za
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\tau ,\phi )}
h
σ
=
a
σ
2
−
τ
2
σ
2
−
1
{\displaystyle h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}}}
h
τ
=
a
σ
2
−
τ
2
1
−
τ
2
{\displaystyle h_{\tau }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{1-\tau ^{2}}}}}
h
ϕ
=
a
(
σ
2
−
1
)
(
1
−
τ
2
)
{\displaystyle h_{\phi }=a{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}}
Infinitezimalni element zapremine je:
d
V
=
a
3
(
σ
2
−
τ
2
)
d
σ
d
τ
d
ϕ
{\displaystyle dV=a^{3}\left(\sigma ^{2}-\tau ^{2}\right)d\sigma d\tau d\phi }
a Laplasijan je:
∇
2
Φ
=
1
a
2
(
σ
2
−
τ
2
)
{
∂
∂
σ
[
(
σ
2
−
1
)
∂
Φ
∂
σ
]
+
∂
∂
τ
[
(
1
−
τ
2
)
∂
Φ
∂
τ
]
}
+
1
a
2
(
σ
2
−
1
)
(
1
−
τ
2
)
∂
2
Φ
∂
ϕ
2
{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left\{{\frac {\partial }{\partial \sigma }}\left[\left(\sigma ^{2}-1\right){\frac {\partial \Phi }{\partial \sigma }}\right]+{\frac {\partial }{\partial \tau }}\left[\left(1-\tau ^{2}\right){\frac {\partial \Phi }{\partial \tau }}\right]\right\}+{\frac {1}{a^{2}\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}
Divergencija je:
div
A
(
σ
,
τ
,
ϕ
)
=
1
a
(
σ
2
−
τ
2
)
∂
∂
σ
[
A
σ
(
σ
2
−
τ
2
)
(
σ
2
−
1
)
]
+
{\displaystyle \operatorname {div} \mathbf {A} (\sigma ,\tau ,\phi )={\frac {1}{a(\sigma ^{2}-\tau ^{2})}}{\frac {\partial }{\partial \sigma }}\left[A_{\sigma }{\sqrt {(\sigma ^{2}-\tau ^{2})(\sigma ^{2}-1)}}\right]+}
1
a
(
σ
2
−
τ
2
)
∂
∂
τ
[
A
τ
(
σ
2
−
τ
2
)
(
1
−
τ
2
)
]
+
1
a
(
σ
2
−
1
)
(
1
−
τ
2
)
∂
∂
ϕ
[
A
ϕ
]
{\displaystyle {\frac {1}{a(\sigma ^{2}-\tau ^{2})}}{\frac {\partial }{\partial \tau }}\left[A_{\tau }{\sqrt {(\sigma ^{2}-\tau ^{2})(1-\tau ^{2})}}\right]+{\frac {1}{a{\sqrt {(\sigma ^{2}-1)(1-\tau ^{2})}}}}{\frac {\partial }{\partial \phi }}{\Big [}A_{\phi }{\Big ]}}
Izduženi sferoidni koordinatni sistem Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers , McGraw-Hill.
Abramowitz, Milton; Stegun, Irene A., eds. (1965), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover. ISBN 978-0-486-61272-0 .
Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN 978-0-07-043316-8 
![{\displaystyle \nu \in [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afa60490b32a53df8fe49e7bd023f1adf3e61f05)
![{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left[{\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}+\coth \mu {\frac {\partial \Phi }{\partial \mu }}+\cot \nu {\frac {\partial \Phi }{\partial \nu }}\right]+{\frac {1}{a^{2}\sinh ^{2}\mu \sin ^{2}\nu }}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48caeab189e1f1d9da9233fb21d53bb92851f8fc)
![{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left\{{\frac {\partial }{\partial \sigma }}\left[\left(\sigma ^{2}-1\right){\frac {\partial \Phi }{\partial \sigma }}\right]+{\frac {\partial }{\partial \tau }}\left[\left(1-\tau ^{2}\right){\frac {\partial \Phi }{\partial \tau }}\right]\right\}+{\frac {1}{a^{2}\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f15e727f9d0e40ee12fa68cb58a07f2011f82707)
![{\displaystyle \operatorname {div} \mathbf {A} (\sigma ,\tau ,\phi )={\frac {1}{a(\sigma ^{2}-\tau ^{2})}}{\frac {\partial }{\partial \sigma }}\left[A_{\sigma }{\sqrt {(\sigma ^{2}-\tau ^{2})(\sigma ^{2}-1)}}\right]+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbc19e4a6bdf3e3202eee7b479894bb8ca261e6)
![{\displaystyle {\frac {1}{a(\sigma ^{2}-\tau ^{2})}}{\frac {\partial }{\partial \tau }}\left[A_{\tau }{\sqrt {(\sigma ^{2}-\tau ^{2})(1-\tau ^{2})}}\right]+{\frac {1}{a{\sqrt {(\sigma ^{2}-1)(1-\tau ^{2})}}}}{\frac {\partial }{\partial \phi }}{\Big [}A_{\phi }{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9612a7367490aa53fa3dfc1bad027b852bbcf32)