![Parabolic cylindrical coordinates](https://upload.wikimedia.org/wikipedia/commons/8/85/Parabolic_cylindrical_coordinates.png)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, parabolic cylindrical coordinates are a three-dimensional
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
that results from projecting the two-dimensional
parabolic coordinate system in the
perpendicular
-direction. Hence, the
coordinate surfaces
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
are
confocal In geometry, confocal means having the same foci: confocal conic sections.
* For an optical cavity consisting of two mirrors, confocal means that they share their foci. If they are identical mirrors, their radius of curvature, ''R''mirror, equals ' ...
parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
of edges.
Basic definition
![Parabolic coords](https://upload.wikimedia.org/wikipedia/commons/9/90/Parabolic_coords.svg)
The parabolic cylindrical coordinates are defined in terms of the
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
by:
:
The surfaces of constant form confocal parabolic cylinders
:
that open towards , whereas the surfaces of constant form confocal parabolic cylinders
:
that open in the opposite direction, i.e., towards . The foci of all these parabolic cylinders are located along the line defined by . The radius has a simple formula as well
:
that proves useful in solving the
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechan ...
in parabolic coordinates for the
inverse-square
In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understo ...
central force
In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force.
: \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat
where \vec F is the force, F is a vecto ...
problem of
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
; for further details, see the
Laplace–Runge–Lenz vector
In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For ...
article.
Scale factors
The scale factors for the parabolic cylindrical coordinates and are:
:
Differential elements
The infinitesimal element of volume is
:
The differential displacement is given by:
:
The differential normal area is given by:
:
Del
Let be a scalar field. The
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
is given by
:
The
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
is given by
:
Let be a vector field of the form:
:
The
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
is given by
:
The
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
is given by
:
Other differential operators can be expressed in the coordinates by substituting the scale factors into the general formulae found in
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 ...
.
Relationship to other coordinate systems
Relationship to
cylindrical coordinate
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
s :
:
Parabolic unit vectors expressed in terms of Cartesian unit vectors:
:
Parabolic cylinder harmonics
Since all of the surfaces of constant , and are
conicoid
In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the ''apex'' or ''vertex'' — and any point of some fixed space curve — the ''dire ...
s, Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of the
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
, a separated solution to Laplace's equation may be written:
:
and Laplace's equation, divided by , is written:
:
Since the equation is separate from the rest, we may write
:
where is constant. has the solution:
:
Substituting for
, Laplace's equation may now be written:
:
We may now separate the and functions and introduce another constant to obtain:
:
:
The solutions to these equations are the
parabolic cylinder functions
:
:
The parabolic cylinder harmonics for are now the product of the solutions. The combination will reduce the number of constants and the general solution to Laplace's equation may be written:
:
Applications
The classic applications of parabolic cylindrical coordinates are in solving
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 ...
, e.g.,
Laplace's 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 = \nab ...
or the
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
, for which such coordinates allow a
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
. A typical example would be the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
surrounding a flat semi-infinite conducting plate.
See also
*
Parabolic coordinates
*
Orthogonal coordinate system
*
Curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
Bibliography
*
*
*
*
* Same as Morse & Feshbach (1953), substituting ''u''
''k'' for ξ
''k''.
*
External links
MathWorld description of parabolic cylindrical coordinates
{{Orthogonal coordinate systems
Three-dimensional coordinate systems
Orthogonal coordinate systems