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| [[File:Prolate spheroidal coordinates.png|thumb|right|325px|The three [[Coordinate system#Coordinate surface|coordinate surfaces]] of prolate spheroidal coordinates. The red prolate spheroid (stretched sphere) corresponds to μ=1, and the blue two-sheet [[hyperboloid]] corresponds to ν=45°. The yellow half-plane corresponds to φ=-60°, which is measured relative to the ''x''-axis (highlighted in green). The black sphere represents the intersection point of the three surfaces, which has [[Cartesian coordinate system|Cartesian coordinates]] of roughly (0.831, -1.439, 2.182).]]
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| '''Prolate spheroidal coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] that results from rotating the two-dimensional [[elliptic coordinates|elliptic coordinate system]] about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces [[oblate spheroidal coordinates]]. Prolate spheroidal coordinates can also be considered as a [[limiting case]] of [[ellipsoidal coordinates]] in which the two smallest [[ellipsoid|principal axes]] are equal in length.
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| Prolate spheroidal coordinates can be used to solve various [[partial differential equation]]s in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the ''z''-axis. One example is solving for the [[wavefunction]] of an [[electron]] moving in the [[electromagnetic field]] of two positively charged [[atomic nucleus|nuclei]], as in the [[hydrogen molecular ion]], H<sub>2</sub><sup>+</sup>. Another example is solving for the [[electric field]] generated by two small [[electrode]] tips. Other limiting cases include areas generated by a line segment (μ=0) or a line with a missing segment (ν=0).
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| ==Definition==
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| [[File:ProlateSpheroidCoord.png|thumb|360 px|right|Prolate spheroidal coordinates μ and ν for ''a=1''. The lines of equal values of μ and ν are shown on the x-z plane, i.e. for φ=0. The surfaces of constant μ and ν are obtained by rotation about the z axis, so that the diagram is valid for any plane containing the z axis: i.e. for any φ.]]
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| The most common definition of prolate spheroidal coordinates <math>(\mu, \nu, \phi)</math> is
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| :<math>
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| x = a \ \sinh \mu \ \sin \nu \ \cos \phi
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| </math>
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| :<math>
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| y = a \ \sinh \mu \ \sin \nu \ \sin \phi
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| </math>
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| :<math>
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| z = a \ \cosh \mu \ \cos \nu
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| </math>
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| where <math>\mu</math> is a nonnegative real number and <math>\nu \in [0, \pi]</math>. The azimuthal angle <math>\phi</math> belongs to the interval <math>[0, 2\pi]</math>.
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| The trigonometric identity
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| :<math>
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| \frac{z^{2}}{a^{2} \cosh^{2} \mu} + \frac{x^{2} + y^{2}}{a^{2} \sinh^{2} \mu} = \cos^{2} \nu + \sin^{2} \nu = 1
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| </math>
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| shows that surfaces of constant <math>\mu</math> form [[prolate]] [[spheroids]], since they are [[ellipse]]s rotated about the axis
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| joining their foci. Similarly, the hyperbolic trigonometric identity
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| :<math>
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| \frac{z^{2}}{a^{2} \cos^{2} \nu} - \frac{x^{2} + y^{2}}{a^{2} \sin^{2} \nu} = \cosh^{2} \mu - \sinh^{2} \mu = 1
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| </math>
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| shows that surfaces of constant <math>\nu</math> form
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| [[hyperboloid]]s of revolution.
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| ==Scale factors==
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| The scale factors for the elliptic coordinates <math>(\mu, \nu)</math> are equal
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| :<math>
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| h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu}
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| </math>
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| whereas the azimuthal scale factor equals
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| :<math>
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| h_{\phi} = a \sinh\mu \ \sin\nu
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| </math>
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| Consequently, an infinitesimal volume element equals
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| :<math>
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| dV = a^{3} \sinh\mu \ \sin\nu \
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| \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu d\phi
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| </math>
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| and the Laplacian can be written
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| :<math>
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| \nabla^{2} \Phi =
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| \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)}
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| \left[
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| \frac{\partial^{2} \Phi}{\partial \mu^{2}} +
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| \frac{\partial^{2} \Phi}{\partial \nu^{2}} +
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| \coth \mu \frac{\partial \Phi}{\partial \mu} +
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| \cot \nu \frac{\partial \Phi}{\partial \nu}
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| \right] +
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| \frac{1}{a^{2} \sinh^{2}\mu \sin^{2}\nu}
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| \frac{\partial^{2} \Phi}{\partial \phi^{2}}
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| </math>
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| 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>(\mu, \nu, \phi)</math> by substituting the scale factors into the general formulae found in [[orthogonal coordinates]].
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| ==Alternative definition==
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| [[File:Prolate spheroidal coordinates degenerate.png|thumb|right|250px|In principle, a definition of prolate spheroidal coordinates could be degenerate. In other words, a single set of coordinates might correspond to two points in [[Cartesian coordinate system|Cartesian coordinates]]; this is illustrated here with two black spheres, one on each sheet of the hyperboloid and located at (''x'', ''y'', ±''z''). However, neither of the definitions presented here are degenerate.]]
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| An alternative and geometrically intuitive set of prolate spheroidal coordinates <math>(\sigma, \tau, \phi)</math> are sometimes used,
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| where <math>\sigma = \cosh \mu</math> and <math>\tau = \cos \nu</math>. Hence, the curves of constant <math>\sigma</math> are prolate spheroids, whereas the curves of constant <math>\tau</math> are hyperboloids of revolution. The coordinate <math>\tau</math> belongs to the interval [-1, 1], whereas the <math>\sigma</math> coordinate must be greater than or equal to one.
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|
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| The coordinates <math>\sigma</math> and <math>\tau</math> have a simple relation to the distances to the foci <math>F_{1}</math> and <math>F_{2}</math>. For any point in the plane, the ''sum'' <math>d_{1}+d_{2}</math> of its distances to the foci equals <math>2a\sigma</math>, whereas their ''difference'' <math>d_{1}-d_{2}</math> equals <math>2a\tau</math>. Thus, the distance to <math>F_{1}</math> is <math>a(\sigma+\tau)</math>, whereas the distance to <math>F_{2}</math> is <math>a(\sigma-\tau)</math>. (Recall that <math>F_{1}</math> and <math>F_{2}</math> are located at <math>z=-a</math> and <math>z=+a</math>, respectively.) This gives the following expressions for <math>\sigma</math>, <math>\tau</math>, and <math>\phi</math>:
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| :<math>
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| \sigma = \frac{1}{2a}\left(\sqrt{x^2+y^2+(z+a)^2}+\sqrt{x^2+y^2+(z-a)^2}\right)
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| </math>
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| :<math>
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| \tau = \frac{1}{2a}\left(\sqrt{x^2+y^2+(z+a)^2}-\sqrt{x^2+y^2+(z-a)^2}\right)
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| </math>
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| :<math>
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| \phi = \arctan\left(\frac{y}{x}\right)
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| </math>
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| Unlike the analogous [[oblate spheroidal coordinates]], the prolate spheroid coordinates (σ, τ, φ) are ''not'' degenerate; in other words, there is a [[1-to-1 mapping|unique, reversible correspondence]] between them and the [[Cartesian coordinates]]
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| :<math>
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| x = a \sqrt{\left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right)} \cos \phi
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| </math>
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| :<math>
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| y = a \sqrt{\left( \sigma^{2} - 1 \right) \left(1 - \tau^{2} \right)} \sin \phi
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| </math>
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| :<math>
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| z = a\ \sigma\ \tau
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| </math>
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| ==Alternative scale factors==
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| The scale factors for the alternative elliptic coordinates <math>(\sigma, \tau, \phi)</math> are
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| :<math>
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| h_{\sigma} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{\sigma^{2} - 1}}
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| </math>
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| :<math>
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| h_{\tau} = a\sqrt{\frac{\sigma^{2} - \tau^{2}}{1 - \tau^{2}}}
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| </math>
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| while the azimuthal scale factor is now
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| :<math>
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| h_{\phi} = a \sqrt{\left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}
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| </math> | |
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| Hence, the infinitesimal volume element becomes
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| :<math> | |
| dV = a^{3} \left( \sigma^{2} - \tau^{2} \right) d\sigma d\tau d\phi
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| </math>
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| and the Laplacian equals
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| :<math> | |
| \nabla^{2} \Phi =
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| \frac{1}{a^{2} \left( \sigma^{2} - \tau^{2} \right)}
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| \left\{
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| \frac{\partial}{\partial \sigma} \left[
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| \left( \sigma^{2} - 1 \right) \frac{\partial \Phi}{\partial \sigma}
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| \right] +
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| \frac{\partial}{\partial \tau} \left[
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| \left( 1 - \tau^{2} \right) \frac{\partial \Phi}{\partial \tau}
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| \right]
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| \right\}
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| + \frac{1}{a^{2} \left( \sigma^{2} - 1 \right) \left( 1 - \tau^{2} \right)}
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| \frac{\partial^{2} \Phi}{\partial \phi^{2}}
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| </math>
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| 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]].
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| As is the case with [[spherical coordinates]], Laplace's equation may be solved by the method of [[separation of variables]] to yield solutions in the form of '''prolate spheroidal harmonics''', which are convenient to use when boundary conditions are defined on a surface with a constant prolate spheroidal coordinate (See Smythe, 1968).
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| ==References==
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| {{reflist}}
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| ==Bibliography==
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| ===No angles convention===
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| *{{cite book | author = Morse PM, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | page = 661}} Uses ξ<sub>1</sub> = a cosh μ, ξ<sub>2</sub> = sin ν, and ξ<sub>3</sub> = cos φ.
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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 | last = Smythe | first = WR| title = Static and Dynamic Electricity |edition = 3rd | publisher = McGraw-Hill | location = New York | year = 1968}}
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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 = 97 | lccn = 6725285}} Uses coordinates ξ = cosh μ, η = sin ν, and φ.
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| ===Angle convention===
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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 | page = 177 | lccn = 5914456}} Korn and Korn use the (μ, ν, φ) coordinates, but also introduce the degenerate (σ, τ, φ) coordinates.
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| *{{cite book | author = Margenau H, Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | publisher = D. van Nostrand | location = New York| pages = 180–182 | lccn = 5510911 }} Similar to Korn and Korn (1961), but uses [[colatitude]] θ = 90° - ν instead of [[latitude]] ν.
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| *{{cite book | author = Moon PH, Spencer DE | year = 1988 | chapter = Prolate Spheroidal Coordinates (η, θ, 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 | isbn = 0-387-02732-7 | pages = 28–30 (Table 1.06)}} Moon and Spencer use the colatitude convention θ = 90° - ν, and rename φ as ψ.
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| ===Unusual convention===
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| *{{cite book | author = Landau LD, Lifshitz EM, Pitaevskii LP | year = 1984 | title = Electrodynamics of Continuous Media (Volume 8 of the [[Course of Theoretical Physics]]) | edition = 2nd | publisher = Pergamon Press | location = New York | isbn = 978-0-7506-2634-7 | pages = 19–29 }} Treats the prolate spheroidal coordinates as a limiting case of the general [[ellipsoidal coordinates]]. Uses (ξ, η, ζ) coordinates that have the units of distance squared.
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| ==External links==
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| *[http://mathworld.wolfram.com/ProlateSpheroidalCoordinates.html MathWorld description of prolate spheroidal coordinates]
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| {{Orthogonal coordinate systems}}
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| [[Category:Coordinate systems]]
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