parabolic coordinates

Two-dimensional parabolic coordinates $(\backslash sigma,\; \backslash tau)$ are defined by the equations, in terms of Cartesian coordinates:

:$$

x = \sigma \tau

:$$

y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right)

The curves of constant $\backslash sigma$ form confocal parabolae

:$$

2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2}

that open upwards (i.e., towards $+y$), whereas the curves of constant $\backslash tau$ form confocal parabolae

:$$

2y = -\frac{x^{2}}{\tau^{2}} + \tau^{2}

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

The Cartesian coordinates $x$ and $y$ can be converted to parabolic coordinates by:

:$$

\sigma = \operatorname{sign}(x)\sqrt{\sqrt{x^{2} +y^{2}}-y}

:$$

\tau = \sqrt{\sqrt{x^{2} +y^{2}}+y}

The scale factors for the parabolic coordinates $(\backslash sigma,\; \backslash tau)$ are equal

:$$

h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}}

Hence, the infinitesimal element of area is

:$$

dA = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau

and the Laplacian equals

:$$

\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)

Other differential operators such as $\backslash nabla\; \backslash cdot\; \backslash mathbf\{F\}$

and $\backslash nabla\; \backslash times\; \backslash mathbf\{F\}$ can be expressed in the coordinates $(\backslash sigma,\; \backslash tau)$ by substituting

the scale factors into the general formulae

found in orthogonal coordinates.

Image:Parabolic coordinates 3D.png 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 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 $z$-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:

:$$

x = \sigma \tau \cos \varphi

:$$

y = \sigma \tau \sin \varphi

:$$

z = \frac{1}{2} \left(\tau^{2} - \sigma^{2} \right)

where the parabolae are now aligned with the $z$-axis,

about which the rotation was carried out. Hence, the azimuthal angle $\backslash varphi$ is defined

:$$

\tan \varphi = \frac{y}{x}

The surfaces of constant $\backslash sigma$ form confocal paraboloids

:$$

2z = \frac{x^{2} + y^{2}}{\sigma^{2}} - \sigma^{2}

that open upwards (i.e., towards $+z$) whereas the surfaces of constant $\backslash tau$ form confocal paraboloids

:$$

2z = -\frac{x^{2} + y^{2}}{\tau^{2}} + \tau^{2}

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

The Riemannian metric tensor associated with this coordinate system is

:

The three dimensional scale factors are:

:$h\_\{\backslash sigma\}\; =\; \backslash sqrt\{\backslash sigma^2+\backslash tau^2\}$

:$h\_\{\backslash tau\}\; =\; \backslash sqrt\{\backslash sigma^2+\backslash tau^2\}$

:$h\_\{\backslash varphi\}\; =\; \backslash sigma\backslash tau$

It is seen that the scale factors $h\_\{\backslash sigma\}$ and $h\_\{\backslash tau\}$ are the same as in the two-dimensional case. The infinitesimal volume element is then

:$$

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

and the Laplacian is given by

:$$

\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}

Other differential operators such as $\backslash nabla\; \backslash cdot\; \backslash mathbf\{F\}$

and $\backslash nabla\; \backslash times\; \backslash mathbf\{F\}$ can be expressed in the coordinates $(\backslash sigma,\; \backslash tau,\; \backslash phi)$ by substituting

the scale factors into the general formulae

found in orthogonal coordinates.

• Parabolic cylindrical coordinates
• Orthogonal coordinate system
• Curvilinear coordinates

• {{cite book | author = Morse PM, 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 = 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, 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_k for ξ_k.
• {{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}}

• {{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