Editing Parabolic coordinates

[[Image:Parabolic coords.svg|thumb|right|384px|In green, confocal parabolae opening upwards, <math>2y = \frac {x^2}{\sigma^2}-\sigma^2</math> In red, confocal parabolae opening downwards, <math>2y =-\frac{x^2}{\tau^2}+\tau^2</math>]]
'''Parabolic coordinates''' are a two-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] in which the [[Coordinate system#Coordinate line|coordinate lines]] are [[confocal]] [[parabola]]s. [[Parabolic cylindrical coordinates|A three-dimensional version]] of parabolic coordinates is obtained by rotating the two-dimensional [[coordinate system|system]] about the symmetry axis of the parabolas.  

Parabolic coordinates have found many applications, e.g., the treatment of the [[Stark effect]] and the [[potential theory]] of the edges.

== Two-dimensional parabolic coordinates ==

Two-dimensional parabolic coordinates <math>(\sigma, \tau)</math> are defined by the equations, in terms of Cartesian coordinates:

:<math>
x = \sigma \tau
</math>

:<math>
y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right)
</math>

The curves of constant <math>\sigma</math> form confocal parabolae

:<math>
2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2}
</math>

that open upwards (i.e., towards <math>+y</math>), whereas the curves of constant <math>\tau</math> form confocal parabolae

:<math>
2y = -\frac{x^{2}}{\tau^{2}} + \tau^{2}
</math>

that open downwards (i.e., towards <math>-y</math>).  The foci of all these parabolae are located at the origin.

The Cartesian coordinates <math>x</math> and <math>y</math> can be converted to parabolic coordinates by:
:<math>
\sigma = \operatorname{sign}(x)\sqrt{\sqrt{x^{2} +y^{2}}-y}
</math>

:<math>
\tau = \sqrt{\sqrt{x^{2} +y^{2}}+y}
</math>

==Two-dimensional scale factors==

The scale factors for the parabolic coordinates <math>(\sigma, \tau)</math> are equal

:<math>
h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}}
</math>

Hence, the infinitesimal element of area is

:<math>
dA = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau
</math>

and the [[Laplacian]] equals

:<math>
\nabla^{2} \Phi = \frac{1}{\sigma^{2} + \tau^{2}} 
\left(  \frac{\partial^{2} \Phi}{\partial \sigma^{2}} + 
\frac{\partial^{2} \Phi}{\partial \tau^{2}} \right)
</math>

Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> 
and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau)</math> by substituting 
the scale factors into the general formulae 
found in [[orthogonal coordinates]].

==Three-dimensional parabolic coordinates==

[[Image:Parabolic coordinates 3D.png|thumb|right|300px|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of the three-dimensional parabolic coordinates.  The red paraboloid corresponds to τ=2, the blue paraboloid corresponds to σ=1, and the yellow half-plane corresponds to φ=−60°.  The three surfaces intersect at the point '''P''' (shown as a black sphere) with [[Cartesian coordinate system|Cartesian coordinates]] roughly (1.0, −1.732, 1.5).]]

The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional [[orthogonal coordinates]]. The [[parabolic cylindrical coordinates]] are produced by projecting in the <math>z</math>-direction.
Rotation about the symmetry axis of the parabolae produces a set of 
confocal paraboloids, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:

:<math>
x = \sigma \tau \cos \varphi
</math>

:<math>
y = \sigma \tau \sin \varphi
</math>

:<math>
z = \frac{1}{2} \left(\tau^{2} - \sigma^{2} \right)
</math>

where the parabolae are now aligned with the <math>z</math>-axis,
about which the rotation was carried out.  Hence, the azimuthal angle <math>\varphi</math> is defined

:<math>
\tan \varphi = \frac{y}{x}
</math>

The surfaces of constant <math>\sigma</math> form confocal paraboloids

:<math>
2z = \frac{x^{2} + y^{2}}{\sigma^{2}} - \sigma^{2}
</math>

that open upwards (i.e., towards <math>+z</math>) whereas the surfaces of constant <math>\tau</math> form confocal paraboloids 

:<math>
2z = -\frac{x^{2} + y^{2}}{\tau^{2}} + \tau^{2}
</math>

that open downwards (i.e., towards <math>-z</math>).  The foci of all these paraboloids are located at the origin.

The [[Riemannian manifold|Riemannian]] [[metric tensor]] associated with this coordinate system is

:<math> g_{ij} = \begin{bmatrix} \sigma^2+\tau^2 & 0 & 0\\0 & \sigma^2+\tau^2 & 0\\0 & 0  & \sigma^2\tau^2 \end{bmatrix} </math>

==Three-dimensional scale factors==

The three dimensional scale factors are:

:<math>h_{\sigma} = \sqrt{\sigma^2+\tau^2}</math>
:<math>h_{\tau}   = \sqrt{\sigma^2+\tau^2}</math>
:<math>h_{\varphi} = \sigma\tau</math>

It is seen that the scale factors <math>h_{\sigma}</math> and <math>h_{\tau}</math> are the same as in the two-dimensional case. The infinitesimal volume element is then

:<math>
dV = h_\sigma h_\tau h_\varphi\, d\sigma\,d\tau\,d\varphi = \sigma\tau \left( \sigma^{2} + \tau^{2} \right)\,d\sigma\,d\tau\,d\varphi
</math>

and the Laplacian is given by

:<math>
\nabla^2 \Phi = \frac{1}{\sigma^{2} + \tau^{2}} 
\left[
\frac{1}{\sigma} \frac{\partial}{\partial \sigma} 
\left( \sigma \frac{\partial \Phi}{\partial \sigma} \right) +
\frac{1}{\tau} \frac{\partial}{\partial \tau} 
\left( \tau \frac{\partial \Phi}{\partial \tau} \right)\right] +
\frac{1}{\sigma^2\tau^2}\frac{\partial^2 \Phi}{\partial \varphi^2}
</math>

Other differential operators such as <math>\nabla \cdot \mathbf{F}</math> 
and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau, \phi)</math> by substituting 
the scale factors into the general formulae 
found in [[orthogonal coordinates]].

== See also ==

* [[Parabolic cylindrical coordinates]]
* [[Orthogonal coordinate system]]
* [[Curvilinear coordinates]]

==Bibliography==
*{{cite book | author = [[Philip M. Morse|Morse PM]], [[Herman Feshbach|Feshbach H]] | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | isbn = 0-07-043316-X|lccn=52011515 | pages = 660}}
*{{cite book | author = [[Henry Margenau|Margenau H]], Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | url = https://archive.org/details/mathematicsofphy0002marg| url-access = registration| publisher = D. van Nostrand | location = New York | pages = [https://archive.org/details/mathematicsofphy0002marg/page/185 185–186] | lccn = 55010911 }}
*{{cite book | author = Korn GA, [[Theresa M. Korn|Korn TM]] |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | pages = 180 | lccn = 59014456}}
*{{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | pages = 96 | lccn = 67025285}}  
*{{cite book | author = Zwillinger D | year = 1992 | title = Handbook of Integration | publisher = Jones and Bartlett | location = Boston, MA | isbn = 0-86720-293-9 | pages = 114}}  Same as Morse & Feshbach (1953), substituting ''u''<sub>''k''</sub> for ξ<sub>''k''</sub>.
*{{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Parabolic Coordinates (μ, ν, ψ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer-Verlag | location = New York | pages = 34–36 (Table 1.08) | isbn = 978-0-387-18430-2}}

==External links==
* {{springer|title=Parabolic coordinates|id=p/p071170}}
*[http://mathworld.wolfram.com/ParabolicCoordinates.html MathWorld description of parabolic coordinates]

{{Orthogonal coordinate systems}}
[[Category:Orthogonal coordinate systems]]