Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system
(
λ
,
μ
,
ν
)
{\displaystyle (\lambda ,\mu ,\nu )}
that generalizes the two-dimensional elliptic coordinate system . Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces , the ellipsoidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
The Cartesian coordinates
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
can be produced from the ellipsoidal coordinates
(
λ
,
μ
,
ν
)
{\displaystyle (\lambda ,\mu ,\nu )}
by the equations
x
2
=
(
a
2
+
λ
)
(
a
2
+
μ
)
(
a
2
+
ν
)
(
a
2
−
b
2
)
(
a
2
−
c
2
)
{\displaystyle x^{2}={\frac {\left(a^{2}+\lambda \right)\left(a^{2}+\mu \right)\left(a^{2}+\nu \right)}{\left(a^{2}-b^{2}\right)\left(a^{2}-c^{2}\right)}}}
y
2
=
(
b
2
+
λ
)
(
b
2
+
μ
)
(
b
2
+
ν
)
(
b
2
−
a
2
)
(
b
2
−
c
2
)
{\displaystyle y^{2}={\frac {\left(b^{2}+\lambda \right)\left(b^{2}+\mu \right)\left(b^{2}+\nu \right)}{\left(b^{2}-a^{2}\right)\left(b^{2}-c^{2}\right)}}}
z
2
=
(
c
2
+
λ
)
(
c
2
+
μ
)
(
c
2
+
ν
)
(
c
2
−
b
2
)
(
c
2
−
a
2
)
{\displaystyle z^{2}={\frac {\left(c^{2}+\lambda \right)\left(c^{2}+\mu \right)\left(c^{2}+\nu \right)}{\left(c^{2}-b^{2}\right)\left(c^{2}-a^{2}\right)}}}
where the following limits apply to the coordinates
−
λ
<
c
2
<
−
μ
<
b
2
<
−
ν
<
a
2
.
{\displaystyle -\lambda <c^{2}<-\mu <b^{2}<-\nu <a^{2}.}
Consequently, surfaces of constant
λ
{\displaystyle \lambda }
are ellipsoids
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,}
whereas surfaces of constant
μ
{\displaystyle \mu }
are hyperboloids 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,}
because the last term in the lhs is negative, and surfaces of constant
ν
{\displaystyle \nu }
are hyperboloids of two sheets
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}
because the last two terms in the lhs are negative.
Scale factors and differential operators
For brevity in the equations below, we introduce a function
S
(
σ
)
=
d
e
f
(
a
2
+
σ
)
(
b
2
+
σ
)
(
c
2
+
σ
)
{\displaystyle S(\sigma )\ {\stackrel {\mathrm {def} }{=}}\ \left(a^{2}+\sigma \right)\left(b^{2}+\sigma \right)\left(c^{2}+\sigma \right)}
where
σ
{\displaystyle \sigma }
can represent any of the three variables
(
λ
,
μ
,
ν
)
{\displaystyle (\lambda ,\mu ,\nu )}
.
Using this function, the scale factors can be written
h
λ
=
1
2
(
λ
−
μ
)
(
λ
−
ν
)
S
(
λ
)
{\displaystyle h_{\lambda }={\frac {1}{2}}{\sqrt {\frac {\left(\lambda -\mu \right)\left(\lambda -\nu \right)}{S(\lambda )}}}}
h
μ
=
1
2
(
μ
−
λ
)
(
μ
−
ν
)
S
(
μ
)
{\displaystyle h_{\mu }={\frac {1}{2}}{\sqrt {\frac {\left(\mu -\lambda \right)\left(\mu -\nu \right)}{S(\mu )}}}}
h
ν
=
1
2
(
ν
−
λ
)
(
ν
−
μ
)
S
(
ν
)
{\displaystyle h_{\nu }={\frac {1}{2}}{\sqrt {\frac {\left(\nu -\lambda \right)\left(\nu -\mu \right)}{S(\nu )}}}}
Hence, the infinitesimal volume element equals
d
V
=
(
λ
−
μ
)
(
λ
−
ν
)
(
μ
−
ν
)
8
−
S
(
λ
)
S
(
μ
)
S
(
ν
)
d
λ
d
μ
d
ν
{\displaystyle dV={\frac {\left(\lambda -\mu \right)\left(\lambda -\nu \right)\left(\mu -\nu \right)}{8{\sqrt {-S(\lambda )S(\mu )S(\nu )}}}}\ d\lambda d\mu d\nu }
and the Laplacian is defined by
∇
2
Φ
=
4
S
(
λ
)
(
λ
−
μ
)
(
λ
−
ν
)
∂
∂
λ
[
S
(
λ
)
∂
Φ
∂
λ
]
+
{\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]\ +\ }
4
S
(
μ
)
(
μ
−
λ
)
(
μ
−
ν
)
∂
∂
μ
[
S
(
μ
)
∂
Φ
∂
μ
]
+
4
S
(
ν
)
(
ν
−
λ
)
(
ν
−
μ
)
∂
∂
ν
[
S
(
ν
)
∂
Φ
∂
ν
]
{\displaystyle {\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]\ +\ {\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]}
Other differential operators such as
∇
⋅
F
{\displaystyle \nabla \cdot \mathbf {F} }
and
∇
×
F
{\displaystyle \nabla \times \mathbf {F} }
can be expressed in the coordinates
(
λ
,
μ
,
ν
)
{\displaystyle (\lambda ,\mu ,\nu )}
by substituting
the scale factors into the general formulae
found in orthogonal coordinates .
See also
Focaloid (shell given by two coordinate surfaces)
References
Bibliography
Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I . New York: McGraw-Hill. p. 663.
Zwillinger D (1992). Handbook of Integration . Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9 .
Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs . New York: Springer Verlag. pp. 101–102. LCCN 67025285 .
Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers . New York: McGraw-Hill. p. 176. LCCN 59014456 .
Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry . New York: D. van Nostrand. pp. 178–180. LCCN 55010911 .
Moon PH, Spencer DE (1988). "Ellipsoidal Coordinates (η, θ, λ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd, 3rd print ed.). New York: Springer Verlag. pp. 40–44 (Table 1.10). ISBN 0-387-02732-7 .
Unusual convention
Landau LD, Lifshitz EM, Pitaevskii LP (1984). Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics ) (2nd ed.). New York: Pergamon Press. pp. 19–29. ISBN 978-0-7506-2634-7 . Uses (ξ, η, ζ) coordinates that have the units of distance squared.
External links
Two dimensional Three dimensional