Paraboloidal coordinates are a three-dimensional orthogonal coordinate system
(
λ
,
μ
,
ν
)
that generalizes the two-dimensional parabolic coordinate system. Similar to the related ellipsoidal coordinates, the paraboloidal coordinate system has orthogonal quadratic coordinate surfaces that are not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
The Cartesian coordinates
(
x
,
y
,
z
)
can be produced from the ellipsoidal coordinates
(
λ
,
μ
,
ν
)
by the equations
x
2
=
(
A
−
λ
)
(
A
−
μ
)
(
A
−
ν
)
B
−
A
y
2
=
(
B
−
λ
)
(
B
−
μ
)
(
B
−
ν
)
A
−
B
z
=
1
2
(
A
+
B
−
λ
−
μ
−
ν
)
where the following limits apply to the coordinates
λ
<
B
<
μ
<
A
<
ν
Consequently, surfaces of constant
λ
are elliptic paraboloids
x
2
λ
−
A
+
y
2
λ
−
B
=
2
z
+
λ
and surfaces of constant
ν
are likewise
x
2
ν
−
A
+
y
2
ν
−
B
=
2
z
+
ν
whereas surfaces of constant
μ
are hyperbolic paraboloids
x
2
μ
−
A
+
y
2
μ
−
B
=
2
z
+
μ
The scale factors for the paraboloidal coordinates
(
λ
,
μ
,
ν
)
are
h
λ
=
1
2
(
μ
−
λ
)
(
ν
−
λ
)
(
A
−
λ
)
(
B
−
λ
)
h
μ
=
1
2
(
ν
−
μ
)
(
λ
−
μ
)
(
A
−
μ
)
(
B
−
μ
)
h
ν
=
1
2
(
λ
−
ν
)
(
μ
−
ν
)
(
A
−
ν
)
(
B
−
ν
)
Hence, the infinitesimal volume element equals
d
V
=
(
μ
−
λ
)
(
ν
−
λ
)
(
ν
−
μ
)
8
(
A
−
λ
)
(
B
−
λ
)
(
A
−
μ
)
(
μ
−
B
)
(
ν
−
A
)
(
ν
−
B
)
d
λ
d
μ
d
ν
Differential operators such as
∇
⋅
F
and
∇
×
F
can be expressed in the coordinates
(
λ
,
μ
,
ν
)
by substituting the scale factors into the general formulae found in orthogonal coordinates.
