that generalize two-dimensional parabolic coordinates.
They possess elliptic paraboloids as one-coordinate surfaces.
As such, they should be distinguished from parabolic cylindrical coordinates and parabolic rotational coordinates, both of which are also generalizations of two-dimensional parabolic coordinates.
The coordinate surfaces of the former are parabolic cylinders, and the coordinate surfaces of the latter are circular paraboloids.
Differently from cylindrical and rotational parabolic coordinates, but similarly to the related ellipsoidal coordinates, the coordinate surfaces of the paraboloidal coordinate system are not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
are downward opening elliptic paraboloids: Similarly, surfaces of constant
are upward opening elliptic paraboloids, whereas surfaces of constant
are[2] Hence, the infinitesimal volume element is Common differential operators can be expressed in the coordinates
by substituting the scale factors into the general formulas for these operators, which are applicable to any three-dimensional orthogonal coordinates.
For instance, the gradient operator is and the Laplacian is Paraboloidal coordinates can be useful for solving certain partial differential equations.
For instance, the Laplace equation and Helmholtz equation are both separable in paraboloidal coordinates.
Hence, the coordinates can be used to solve these equations in geometries with paraboloidal symmetry, i.e. with boundary conditions specified on sections of paraboloids.
Direct solution of the equations is difficult, however, in part because the separation constants
Following the above approach, paraboloidal coordinates have been used to solve for the electric field surrounding a conducting paraboloid.