Paraboloidal coordinates are three-dimensional
orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
that generalize two-dimensional
parabolic coordinates. They possess elliptic
paraboloids as one-coordinate surfaces. As such, they should be distinguished from
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 z-direction. Hence, the coordinate surfa ...
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
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system (\lambda, \mu, \nu) that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic ...
, the coordinate surfaces of the paraboloidal coordinate system are ''not'' produced by rotating or projecting any two-dimensional orthogonal coordinate system.
Basic formulas
The Cartesian coordinates
can be produced from the ellipsoidal coordinates
by the equations
:
:
:
with
:
Consequently, surfaces of constant
are downward opening elliptic paraboloids:
:
Similarly, surfaces of constant
are ''upward'' opening elliptic paraboloids,
:
whereas surfaces of constant
are hyperbolic paraboloids:
:
Scale factors
The scale factors for the paraboloidal coordinates
are
:
:
:
Hence, the infinitesimal volume element is
:
Differential operators
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
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...
is
:
and the
Laplacian is
:
Applications
Paraboloidal coordinates can be useful for solving certain
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
. For instance, the
Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \n ...
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.
The Helmholtz equation is
. Taking
, the separated equations are
[Willatzen and Yoon (2011), p. 227]
:
where
and
are the two separation constants. Similarly, the separated equations for the Laplace equation can be obtained by setting
in the above.
Each of the separated equations can be cast in the form of the
Baer equation. Direct solution of the equations is difficult, however, in part because the separation constants
and
appear simultaneously in all three equations.
Following the above approach, paraboloidal coordinates have been used to solve for the
electric field surrounding a
conducting paraboloid.
References
Bibliography
*
*
*
*
*
*
* Same as Morse & Feshbach (1953), substituting ''u''
''k'' for ξ
''k''.
*
External links
MathWorld description of confocal paraboloidal coordinates
{{Orthogonal coordinate systems
Three-dimensional coordinate systems
Orthogonal coordinate systems