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| '''Paraboloidal coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] <math>(\lambda, \mu, \nu)</math> that generalizes the two-dimensional [[parabolic coordinates|parabolic coordinate system]]. Similar to the related [[ellipsoidal coordinates]], the paraboloidal coordinate system has [[orthogonal coordinates|orthogonal]] [[quadratic]] [[Coordinate system#Coordinate surface|coordinate surfaces]] that are ''not'' produced by rotating or projecting any two-dimensional orthogonal coordinate system. | | Irwin Butts is what my wife loves to contact me although I don't really like being known as like that. Years in the past we moved to North Dakota. One of the [http://www.Health.com/health/condition-article/0,,20188476,00.html std testing] at home things she enjoys most is to [http://www.pinellashealth.com/STD/TypesofSTD.pdf study comics] and she'll be starting something else alongside with it. I utilized to be unemployed but now I am a librarian and the salary has [http://www.hotporn123.com/user/KGottscha over the counter std test] been truly satisfying.<br><br>over the counter [http://xrambo.com/user/RKDBG std testing at home] test Here is my web page: [https://alphacomms.Zendesk.com/entries/53455304-Curing-Your-Candida-Albicans-How-To-Make-It-Happen-Easily at home std test] |
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| [[File:Parabolic coordinates 3D.png|thumb|right|300px|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of the three-dimensional paraboloidal coordinates.]]
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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 - \lambda \right) \left( A - \mu \right) \left( A - \nu \right)}{B - A}
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| </math>
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| :<math> | |
| y^{2} = \frac{\left( B - \lambda \right) \left( B - \mu \right) \left( B - \nu \right)}{A - B}
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| </math>
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| :<math>
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| z =
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| \frac{1}{2} \left( A + B - \lambda - \mu -\nu \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 < B < \mu < A < \nu
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| </math>
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| Consequently, surfaces of constant <math>\lambda</math> are elliptic [[paraboloid]]s
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| :<math>
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| \frac{x^{2}}{\lambda - A} + \frac{y^{2}}{\lambda - B} = 2z + \lambda
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| </math>
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| and surfaces of constant <math>\nu</math> are likewise | |
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| :<math>
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| \frac{x^{2}}{\nu - A} + \frac{y^{2}}{\nu - B} = 2z + \nu
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| </math>
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| whereas surfaces of constant <math>\mu</math> are hyperbolic [[paraboloid]]s
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| :<math> | |
| \frac{x^{2}}{\mu - A} + \frac{y^{2}}{\mu - B} = 2z + \mu
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| </math>
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| ==Scale factors==
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| The scale factors for the paraboloidal coordinates <math>(\lambda, \mu, \nu )</math> are
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|
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| :<math>
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| h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right)}{ \left( A - \lambda \right) \left( B - \lambda \right)}}
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| </math>
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| :<math>
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| h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \mu \right) \left( \lambda - \mu \right)}{ \left( A - \mu \right) \left( B - \mu \right)}}
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| </math>
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| :<math>
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| h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \nu \right) \left( \mu - \nu \right)}{ \left( A - \nu \right) \left( B - \nu \right)}}
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| </math> | |
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| Hence, the infinitesimal volume element equals
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| :<math> | |
| dV = \frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right) \left( \nu - \mu\right)}{8\sqrt{\left( A - \lambda \right) \left( B - \lambda \right) \left( A - \mu \right) \left( \mu - B \right) \left( \nu - A \right) \left( \nu - B \right) }} \ d\lambda d\mu d\nu
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| </math>
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| 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 the scale factors into the general formulae found in [[orthogonal coordinates]].
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| ==References==
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| {{reflist}}
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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 = 664}}
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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 = 184–185 | 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 = 180 | lccn = 5914456}}
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| *{{cite book | author = Arfken G | year = 1970 | title = Mathematical Methods for Physicists | edition = 2nd | publisher = Academic Press | location = Orlando, FL | pages = 119–120}}
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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 = 98 | 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 = Paraboloidal Coordinates (μ, ν, 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 | pages = 44–48 (Table 1.11) | isbn = 978-0-387-18430-2}}
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| ==External links==
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| *[http://mathworld.wolfram.com/ConfocalParaboloidalCoordinates.html MathWorld description of confocal paraboloidal coordinates]
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| {{Orthogonal coordinate systems}}
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| [[Category:Coordinate systems]]
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