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| [[File:Bipolar cylindrical coordinates.png|thumb|350px|right|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of the bipolar cylindrical coordinates. The yellow crescent corresponds to σ, whereas the red tube corresponds to τ and the blue plane corresponds to ''z''=1. The three surfaces intersect at the point '''P''' (shown as a black sphere).]]
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| '''Bipolar cylindrical coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] that results from projecting the two-dimensional [[bipolar coordinates|bipolar coordinate system]] in the
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| perpendicular <math>z</math>-direction. The two lines of [[Focus (geometry)|foci]]
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| <math>F_{1}</math> and <math>F_{2}</math> of the projected [[Apollonian circles]] are generally taken to be
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| defined by <math>x=-a</math> and <math>x=+a</math>, respectively, (and by <math>y=0</math>) in the [[Cartesian coordinate system]].
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| The term "bipolar" is often used to describe other curves having two singular points (foci), such as [[ellipse]]s, [[hyperbola]]s, and [[Cassini oval]]s. However, the term ''bipolar coordinates'' is never used to describe coordinates associated with those curves, e.g., [[elliptic coordinates]].
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| ==Basic definition==
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| The most common definition of bipolar cylindrical coordinates <math>(\sigma, \tau, z)</math> is
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| :<math>
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| x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}
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| </math>
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| :<math>
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| y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma}
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| </math>
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| :<math>
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| z = \ z
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| </math>
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| where the <math>\sigma</math> coordinate of a point <math>P</math>
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| equals the angle <math>F_{1} P F_{2}</math> and the
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| <math>\tau</math> coordinate equals the [[natural logarithm]] of the ratio of the distances <math>d_{1}</math> and <math>d_{2}</math> to the focal lines
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| :<math>
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| \tau = \ln \frac{d_{1}}{d_{2}}
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| </math>
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| (Recall that the focal lines <math>F_{1}</math> and <math>F_{2}</math> are located at <math>x=-a</math> and <math>x=+a</math>, respectively.)
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| Surfaces of constant <math>\sigma</math> correspond to cylinders of different radii
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| :<math>
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| x^{2} +
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| \left( y - a \cot \sigma \right)^{2} = \frac{a^{2}}{\sin^{2} \sigma}
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| </math>
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| that all pass through the focal lines and are not concentric. The surfaces of constant <math>\tau</math> are non-intersecting cylinders of different radii
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| :<math>
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| y^{2} +
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| \left( x - a \coth \tau \right)^{2} = \frac{a^{2}}{\sinh^{2} \tau}
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| </math>
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| that surround the focal lines but again are not concentric. The focal lines and all these cylinders are parallel to the <math>z</math>-axis (the direction of projection). In the <math>z=0</math> plane, the centers of the constant-<math>\sigma</math> and constant-<math>\tau</math> cylinders lie on the <math>y</math> and <math>x</math> axes, respectively.
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|
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| ==Scale factors==
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| The scale factors for the bipolar coordinates <math>\sigma</math> and <math>\tau</math> are equal
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| :<math> | |
| h_{\sigma} = h_{\tau} = \frac{a}{\cosh \tau - \cos\sigma}
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| </math>
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| whereas the remaining scale factor <math>h_{z}=1</math>.
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| Thus, the infinitesimal volume element equals
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| :<math>
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| dV = \frac{a^{2}}{\left( \cosh \tau - \cos\sigma \right)^{2}} d\sigma d\tau dz
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| </math>
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| and the Laplacian is given by
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| :<math>
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| \nabla^{2} \Phi =
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| \frac{1}{a^{2}} \left( \cosh \tau - \cos\sigma \right)^{2}
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| \left(
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| \frac{\partial^{2} \Phi}{\partial \sigma^{2}} +
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| \frac{\partial^{2} \Phi}{\partial \tau^{2}}
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| \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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| ==Applications==
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| The classic applications of bipolar coordinates are in solving [[partial differential equations]],
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| e.g., [[Laplace's equation]] or the [[Helmholtz equation]], for which bipolar coordinates allow a
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| [[separation of variables]]. A typical example would be the [[electric field]] surrounding two
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| parallel cylindrical conductors.
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| ==Bibliography==
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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 = 187–190 | 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 = 182 | lccn = 5914456}}
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| *{{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Conical Coordinates (r, θ, λ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer-Verlag | location = New York | isbn = 978-0-387-18430-2 | nopp = true | page = unknown}}
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| ==External links==
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| *[http://mathworld.wolfram.com/BipolarCylindricalCoordinates.html MathWorld description of bipolar 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 contact me anything you like. Hiring has been my occupation for some time but I've currently utilized for an additional one. South Dakota is where I've always been residing. Doing ceramics is what my family members and I appreciate.
My web-site ... bikedance.com