Parabolic cylindrical coordinates

Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges.

The radius r has a simple formula as well that proves useful in solving the Hamilton–Jacobi equation in parabolic coordinates for the inverse-square central force problem of mechanics; for further details, see the Laplace–Runge–Lenz vector article.

The gradient is given by The Laplacian is given by Let A be a vector field of the form: The divergence is given by The curl is given by Other differential operators can be expressed in the coordinates (σ, τ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

Relationship to cylindrical coordinates (ρ, φ, z): Parabolic unit vectors expressed in terms of Cartesian unit vectors: Since all of the surfaces of constant σ, τ and z are conicoids, Laplace's equation is separable in parabolic cylindrical coordinates.

, Laplace's equation may now be written: We may now separate the S and T functions and introduce another constant n2 to obtain: The solutions to these equations are the parabolic cylinder functions The parabolic cylinder harmonics for (m, n) are now the product of the solutions.

Coordinate surfaces of parabolic cylindrical coordinates. The red parabolic cylinder corresponds to σ=2, whereas the yellow parabolic cylinder corresponds to τ=1. The blue plane corresponds to z =2. These surfaces intersect at the point P (shown as a black sphere), which has Cartesian coordinates roughly (2, -1.5, 2).
Parabolic coordinate system showing curves of constant σ and τ the horizontal and vertical axes are the x and y coordinates respectively. These coordinates are projected along the z-axis, and so this diagram will hold for any value of the z coordinate.