Bispherical coordinates: Difference between revisions

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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.
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[[File:Parabolic coordinates 3D.png|thumb|right|300px|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of the three-dimensional paraboloidal coordinates.]]
 
==Basic formulae==
 
The Cartesian coordinates <math>(x, y, z)</math> can be produced from the ellipsoidal coordinates
<math>( \lambda, \mu, \nu )</math> by the equations
 
:<math>
x^{2} = \frac{\left( A - \lambda \right) \left( A - \mu \right) \left( A - \nu \right)}{B - A}
</math>
 
:<math>
y^{2} = \frac{\left( B - \lambda \right) \left( B - \mu \right) \left( B - \nu \right)}{A - B}
</math>
 
:<math>
z =
\frac{1}{2} \left( A + B - \lambda - \mu -\nu \right)
</math>
 
where the following limits apply to the coordinates
 
:<math>
\lambda < B < \mu < A < \nu
</math>
 
Consequently, surfaces of constant <math>\lambda</math> are elliptic [[paraboloid]]s
 
:<math>
\frac{x^{2}}{\lambda - A} +  \frac{y^{2}}{\lambda - B}  = 2z + \lambda
</math>
 
and surfaces of constant <math>\nu</math> are likewise
 
:<math>
\frac{x^{2}}{\nu - A} +  \frac{y^{2}}{\nu - B} = 2z + \nu
</math>
 
whereas surfaces of constant <math>\mu</math> are hyperbolic [[paraboloid]]s
 
:<math>
\frac{x^{2}}{\mu - A} +  \frac{y^{2}}{\mu - B} = 2z + \mu
</math>
 
==Scale factors==
 
The scale factors for the paraboloidal coordinates <math>(\lambda, \mu, \nu )</math> are
:<math>
h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda \right) \left( \nu - \lambda \right)}{ \left( A - \lambda \right) \left( B - \lambda \right)}}
</math>
 
:<math>
h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \mu \right) \left( \lambda - \mu \right)}{ \left( A - \mu \right) \left( B - \mu \right)}}
</math>
 
:<math>
h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \nu \right) \left( \mu - \nu \right)}{ \left( A - \nu \right) \left( B - \nu \right)}}
</math>
 
Hence, the infinitesimal volume element equals
 
:<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
</math>
 
Differential operators such as <math>\nabla \cdot \mathbf{F}</math>
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]].
 
==References==
{{reflist}}
 
==Bibliography==
*{{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}}
*{{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&ndash;185 | lccn = 5510911 }}
*{{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}}
*{{cite book | author = Arfken G | year = 1970 | title = Mathematical Methods for Physicists | edition = 2nd | publisher = Academic Press | location = Orlando, FL | pages = 119&ndash;120}}
*{{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | page = 98 | lccn = 6725285}}
*{{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>. 
*{{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&ndash;48 (Table 1.11) | isbn = 978-0-387-18430-2}}
 
==External links==
*[http://mathworld.wolfram.com/ConfocalParaboloidalCoordinates.html MathWorld description of confocal paraboloidal coordinates]
 
{{Orthogonal coordinate systems}}
 
[[Category:Coordinate systems]]

Latest revision as of 04:33, 25 August 2014

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