Paraboloidal coordinates
that generalize two-dimensional parabolic coordinates.They possess elliptic paraboloids as one-coordinate surfaces.The coordinate surfaces of the former are parabolic cylinders, and the coordinate surfaces of the latter are circular paraboloids.are downward opening elliptic paraboloids: Similarly, surfaces of constantare upward opening elliptic paraboloids, whereas surfaces of constantare[2] Hence, the infinitesimal volume element is Common differential operators can be expressed in the coordinatesby 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.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 constantsFollowing the above approach, paraboloidal coordinates have been used to solve for the electric field surrounding a conducting paraboloid.