Parabolic coordinates

[Image:Parabolic coords.svg|thumb|right|384px|In green, confocal parabolae opening upwards, In red, confocal parabolae opening downwards, ]
Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas.
Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges.

Two-dimensional parabolic coordinates

Two-dimensional parabolic coordinates are defined by the equations, in terms of Cartesian coordinates:
The curves of constant form confocal parabolae
that open upwards, whereas the curves of constant form confocal parabolae
that open downwards. The foci of all these parabolae are located at the origin.
The Cartesian coordinates and can be converted to parabolic coordinates by:

Two-dimensional scale factors

The scale factors for the parabolic coordinates are equal
Hence, the infinitesimal element of area is
and the Laplacian equals
Other differential operators such as
and can be expressed in the coordinates by substituting
the scale factors into the general formulae
found in orthogonal coordinates.

Three-dimensional parabolic coordinates

[Image:Parabolic coordinates 3D.png|thumb|right|300px|Coordinate surfaces of the three-dimensional parabolic coordinates. The red paraboloid corresponds to τ=2, the blue paraboloid corresponds to σ=1, and the yellow half-plane corresponds to φ=−60°. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.0, −1.732, 1.5).]
The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional orthogonal coordinates. The parabolic cylindrical coordinates are produced by projecting in the -direction.
Rotation about the symmetry axis of the parabolae produces a set of
confocal paraboloids, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:
where the parabolae are now aligned with the -axis,
about which the rotation was carried out. Hence, the azimuthal angle is defined
The surfaces of constant form confocal paraboloids
that open upwards whereas the surfaces of constant form confocal paraboloids
that open downwards. The foci of all these paraboloids are located at the origin.
The Riemannian metric tensor associated with this coordinate system is

Three-dimensional scale factors

The three dimensional scale factors are:
It is seen that the scale factors and are the same as in the two-dimensional case. The infinitesimal volume element is then
and the Laplacian is given by
Other differential operators such as
and can be expressed in the coordinates by substituting
the scale factors into the general formulae
found in orthogonal coordinates.