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| '''Ellipsoidal coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] <math>(\lambda, \mu, \nu)</math> that generalizes the two-dimensional [[elliptic coordinates|elliptic coordinate system]]. Unlike most three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]]s that feature [[quadratic function|quadratic]] [[Coordinate system#Coordinate surface|coordinate surfaces]], the ellipsoidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
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| ==Basic formulae==
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| The Cartesian coordinates <math>(x, y, z)</math> can be produced from the ellipsoidal coordinates
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| <math>( \lambda, \mu, \nu )</math> by the equations
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| :<math>
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| x^{2} = \frac{\left( a^{2} + \lambda \right) \left( a^{2} + \mu \right) \left( a^{2} + \nu \right)}{\left( a^{2} - b^{2} \right) \left( a^{2} - c^{2} \right)}
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| </math>
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| :<math>
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| y^{2} = \frac{\left( b^{2} + \lambda \right) \left( b^{2} + \mu \right) \left( b^{2} + \nu \right)}{\left( b^{2} - a^{2} \right) \left( b^{2} - c^{2} \right)}
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| </math>
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| :<math>
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| z^{2} = \frac{\left( c^{2} + \lambda \right) \left( c^{2} + \mu \right) \left( c^{2} + \nu \right)}{\left( c^{2} - b^{2} \right) \left( c^{2} - a^{2} \right)}
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| </math>
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| where the following limits apply to the coordinates
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| :<math>
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| - \lambda < c^{2} < - \mu < b^{2} < -\nu < a^{2}.
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| </math>
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| Consequently, surfaces of constant <math>\lambda</math> are [[ellipsoid]]s
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| :<math>
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| \frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} + \frac{z^{2}}{c^{2} + \lambda} = 1,
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| </math>
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| whereas surfaces of constant <math>\mu</math> are [[hyperboloid]]s of one sheet
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| :<math>
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| \frac{x^{2}}{a^{2} + \mu} + \frac{y^{2}}{b^{2} + \mu} + \frac{z^{2}}{c^{2} + \mu} = 1,
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| </math>
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| because the last term in the lhs is negative, and surfaces of constant <math>\nu</math> are [[hyperboloid]]s of two sheets
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| :<math>
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| \frac{x^{2}}{a^{2} + \nu} + \frac{y^{2}}{b^{2} + \nu} + \frac{z^{2}}{c^{2} + \nu} = 1
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| </math>
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| because the last two terms in the lhs are negative.
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| ==Scale factors and differential operators==
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| For brevity in the equations below, we introduce a function
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| :<math>
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| S(\sigma) \ \stackrel{\mathrm{def}}{=}\ \left( a^{2} + \sigma \right) \left( b^{2} + \sigma \right) \left( c^{2} + \sigma \right)
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| </math>
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| where <math>\sigma</math> can represent any of the three variables <math>(\lambda, \mu, \nu )</math>. | |
| Using this function, the scale factors can be written
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|
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| :<math>
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| h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \mu \right) \left( \lambda - \nu\right)}{S(\lambda)}}
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| </math> | |
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| :<math>
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| h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda\right) \left( \mu - \nu\right)}{S(\mu)}}
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| </math>
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| :<math>
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| h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \lambda\right) \left( \nu - \mu\right)}{S(\nu)}}
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| </math>
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| Hence, the infinitesimal volume element equals
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| :<math>
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| dV = \frac{\left( \lambda - \mu \right) \left( \lambda - \nu \right) \left( \mu - \nu\right)}{8\sqrt{-S(\lambda) S(\mu) S(\nu)}} \ d\lambda d\mu d\nu
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| </math>
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| and the [[Laplacian]] is defined by
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| :<math> | |
| \nabla^{2} \Phi =
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| \frac{4\sqrt{S(\lambda)}}{\left( \lambda - \mu \right) \left( \lambda - \nu\right)}
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| \frac{\partial}{\partial \lambda} \left[ \sqrt{S(\lambda)} \frac{\partial \Phi}{\partial \lambda} \right] \ + \
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| </math>
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| :::::<math>
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| \frac{4\sqrt{S(\mu)}}{\left( \mu - \lambda \right) \left( \mu - \nu\right)}
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| \frac{\partial}{\partial \mu} \left[ \sqrt{S(\mu)} \frac{\partial \Phi}{\partial \mu} \right] \ + \
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| \frac{4\sqrt{S(\nu)}}{\left( \nu - \lambda \right) \left( \nu - \mu\right)}
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| \frac{\partial}{\partial \nu} \left[ \sqrt{S(\nu)} \frac{\partial \Phi}{\partial \nu} \right]
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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>(\lambda, \mu, \nu)</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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| ==See also==
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| * [[Focaloid]] (shell given by two coordinate surfaces)
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| ==References==
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| {{reflist}}
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| ==Bibliography==
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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 = 663}}
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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}}
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| *{{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | pages = 101–102 | lccn = 6725285}}
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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 = 176 | lccn = 5914456}}
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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 = 178–180 | lccn = 5510911 }}
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| *{{cite book | author = Moon PH, Spencer DE | year = 1988 | chapter = Ellipsoidal Coordinates (η, 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 = 0-387-02732-7 | pages = 40–44 (Table 1.10)}}
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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 }} 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/ConfocalEllipsoidalCoordinates.html MathWorld description of confocal ellipsoidal coordinates]
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| {{Orthogonal coordinate systems}}
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| [[Category:Coordinate systems]]
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The author is recognized by the title of Figures Wunder. What I love doing is to collect badges but I've been using on new things recently. Hiring is her working day job now but she's always wanted her own business. Minnesota is where he's been residing for many years.
Feel free to visit my web-site http://calvaryhill.net/