Parabolic cylindrical coordinates

In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular -direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges. The parabolic cylindrical coordinates (σ, τ, z) are defined in terms of the Cartesian coordinates (x, y, z) by: The surfaces of constant σ form confocal parabolic cylinders that open towards +y, whereas the surfaces of constant τ form confocal parabolic cylinders that open in the opposite direction, i.e., towards −y. The foci of all these parabolic cylinders are located along the line defined by x = y = 0. 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 scale factors for the parabolic cylindrical coordinates σ and τ are: The infinitesimal element of volume is The differential displacement is given by: The differential normal area is given by: Let f be a scalar field. 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. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written: and Laplace's equation, divided by V, is written: Since the Z equation is separate from the rest, we may write where m is constant.