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| [[File:Parabolic cylindrical coordinates.png|thumb|right|350px|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of parabolic cylindrical coordinates. The red parabolic cylinder corresponds to σ=2, whereas the yellow parabolic cylinder corresponds to τ=1. The blue plane corresponds to ''z''=2. These surfaces intersect at the point '''P''' (shown as a black sphere), which has [[Cartesian coordinate system|Cartesian coordinates]] roughly (2, -1.5, 2).]]
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| In [[mathematics]], '''parabolic cylindrical coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] that results from projecting the two-dimensional [[parabolic coordinates|parabolic coordinate system]] in the
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| perpendicular <math>z</math>-direction. Hence, the [[Coordinate system#Coordinate surface|coordinate surfaces]] are [[confocal]] [[parabola|parabolic]] cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the [[potential theory]] of edges.
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| ==Basic definition==
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| [[File:Parabolic coords.svg|thumb|right|250px|Parabolic coordinate system showing curves of constant σ and τ the horizontal and vertical axes are the x and y coordinates respectively. These coordinates are projected along the z-axis, and so this diagram will hold for any value of the z coordinate.]]
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| The parabolic cylindrical coordinates <math>(\sigma, \tau, z)</math> are defined in terms of the [[Cartesian coordinates]] ''(x,y,z)'' by:
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| :<math>x = \sigma \tau\,</math>
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| :<math>y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right)</math>
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| :<math>z = z\,</math>
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| The surfaces of constant <math>\sigma</math> form confocal parabolic cylinders
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| :<math> | |
| 2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2}
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| </math>
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| that open towards <math>+y</math>, whereas the surfaces of constant <math>\tau</math> form confocal parabolic cylinders
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| :<math>
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| 2y = -\frac{x^{2}}{\tau^{2}} + \tau^{2}
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| </math>
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| that open in the opposite direction, i.e., towards <math>-y</math>. The foci of all these parabolic cylinders are located along the line defined by <math>x=y=0</math>. The radius ''r'' has a simple formula as well
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| :<math>
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| r = \sqrt{x^{2} + y^{2}} = \frac{1}{2} \left( \sigma^{2} + \tau^{2} \right)
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| </math>
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| that proves useful in solving the [[Hamilton-Jacobi equation]] in parabolic coordinates for the [[inverse-square law|inverse-square]] [[central force]] problem of [[mechanics]]; for further details, see the [[Laplace–Runge–Lenz vector]] article.
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| ==Scale factors==
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| The scale factors for the parabolic cylindrical coordinates <math>\sigma</math> and <math>\tau</math> are:
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| :<math>
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| h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}}
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| </math>
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| :<math>h_{z}=1\,</math>
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| The infinitesimal element of volume is
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| :<math>
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| dV = h_\sigma h_\tau h_z=\left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz
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| </math>
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| and the Laplacian equals
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| :<math>
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| \nabla^{2} \Phi = \frac{1}{\sigma^{2} + \tau^{2}}
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| \left( \frac{\partial^{2} \Phi}{\partial \sigma^{2}} +
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| \frac{\partial^{2} \Phi}{\partial \tau^{2}} \right) +
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| \frac{\partial^{2} \Phi}{\partial z^{2}}
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| </math>
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| Other differential operators such as <math>\nabla \cdot \mathbf{F}</math>
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| and <math>\nabla \times \mathbf{F}</math> can be expressed in the coordinates <math>(\sigma, \tau)</math> by substituting
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| the scale factors into the general formulae
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| found in [[orthogonal coordinates]].
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| == Parabolic cylinder harmonics ==
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| Since all of the surfaces of constant σ, τ and ''z'' are conicoid, Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of the [[separation of variables]], a separated solution to Laplace's equation may be written:
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| :<math>V=S(\sigma)\,T(\tau)\,Z(z)</math>
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| and Laplace's equation, divided by ''V'' , is written: | |
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| :<math>\frac{1}{\sigma^2+\tau^2}
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| \left[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}\right]+\frac{\ddot{Z}}{Z}=0
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| </math>
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| Since the ''Z'' equation is separate from the rest, we may write
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| :<math>\frac{\ddot{Z}}{Z}=-m^2</math>
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| where ''m'' is constant. ''Z(z)'' has the solution:
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| :<math>Z_m(z)=A_1\,e^{imz}+A_2\,e^{-imz}\,</math>
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| Substituting <math>-m^2</math> for <math>\ddot{Z}/Z</math> , Laplace's equation may now be written:
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| :<math>\left[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}\right]=m^2(\sigma^2+\tau^2)</math>
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| We may now separate the ''S'' and ''T'' functions and introduce another constant <math>n^2</math> to obtain:
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| :<math>\ddot{S} - (m^2\sigma^2+n^2)S=0</math>
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| :<math>\ddot{T} - (m^2\tau^2 -n^2)T=0</math>
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| The solutions to these equations are the [[parabolic cylinder functions]]
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| :<math>S_{mn}(\sigma) = A_3\,y_1(n^2/2m,\sigma\sqrt{2m}) + A_4\,y_2(n^2/2m,\sigma\sqrt{2m})</math>
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| :<math>T_{mn}(\tau) = A_5\,y_1(n^2/2m,i\tau \sqrt{2m}) + A_6\,y_2(n^2/2m,i\tau \sqrt{2m})</math>
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| The parabolic cylinder harmonics for (m,n) 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:
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| :<math>V(\sigma,\tau,z)=\sum_{m,n} A_{mn} S_{mn} T_{mn} Z_m\,</math>
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| ==Applications==
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| The classic applications of parabolic cylindrical coordinates are in solving [[partial differential equations]],
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| e.g., [[Laplace's equation]] or the [[Helmholtz equation]], for which such coordinates allow a
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| [[separation of variables]]. A typical example would be the [[electric field]] surrounding a
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| flat semi-infinite conducting plate.
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| ==Bibliography==
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| *{{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 | page = 658}}
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| *{{cite book | author = [[Henry Margenau|Margenau H]], Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | publisher = D. van Nostrand | location = New York | pages = 186–187 | lccn = 5510911 }}
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| *{{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 | page = 181 | lccn = 5914456}}
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| *{{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | page = 96 | lccn = 6725285}}
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| *{{cite book | author = Zwillinger D | year = 1992 | title = Handbook of Integration | publisher = Jones and Bartlett | location = Boston, MA | isbn = 0-86720-293-9 | page = 114}} Same as Morse & Feshbach (1953), substituting ''u''<sub>''k''</sub> for ξ<sub>''k''</sub>.
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| *{{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Parabolic-Cylinder Coordinates (μ, ν, z) | 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 = 21–24 (Table 1.04) | isbn = 978-0-387-18430-2}}
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| ==External links==
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| *[http://mathworld.wolfram.com/ParabolicCylindricalCoordinates.html MathWorld description of parabolic cylindrical coordinates]
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| {{Orthogonal coordinate systems}}
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| [[Category:Coordinate systems]]
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Emilia Shryock is my title but you can call me anything you like. South Dakota is where me and my spouse reside. One of over the counter std test (Keep Reading) very best issues in the globe for me is to do aerobics and now I'm attempting to make cash with it. Hiring is her day occupation now but she's always needed her personal company.