Bispherical coordinates: Difference between revisions

Revision as of 19:32, 14 December 2013

Paraboloidal coordinates are a three-dimensional orthogonal coordinate system $(\lambda ,\mu ,\nu )$ that generalizes the two-dimensional parabolic coordinate system. Similar to the related ellipsoidal coordinates, the paraboloidal coordinate system has orthogonal quadratic coordinate surfaces that are not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

Basic formulae

The Cartesian coordinates $(x,y,z)$ can be produced from the ellipsoidal coordinates $(\lambda ,\mu ,\nu )$ by the equations

x^{2}={\frac {\left(A-\lambda \right)\left(A-\mu \right)\left(A-\nu \right)}{B-A}}

y^{2}={\frac {\left(B-\lambda \right)\left(B-\mu \right)\left(B-\nu \right)}{A-B}}

z={\frac {1}{2}}\left(A+B-\lambda -\mu -\nu \right)

where the following limits apply to the coordinates

\lambda <B<\mu <A<\nu

Consequently, surfaces of constant $\lambda$ are elliptic paraboloids

{\frac {x^{2}}{\lambda -A}}+{\frac {y^{2}}{\lambda -B}}=2z+\lambda

and surfaces of constant $\nu$ are likewise

{\frac {x^{2}}{\nu -A}}+{\frac {y^{2}}{\nu -B}}=2z+\nu

whereas surfaces of constant $\mu$ are hyperbolic paraboloids

{\frac {x^{2}}{\mu -A}}+{\frac {y^{2}}{\mu -B}}=2z+\mu

Scale factors

The scale factors for the paraboloidal coordinates $(\lambda ,\mu ,\nu )$ are

h_{\lambda }={\frac {1}{2}}{\sqrt {\frac {\left(\mu -\lambda \right)\left(\nu -\lambda \right)}{\left(A-\lambda \right)\left(B-\lambda \right)}}}

h_{\mu }={\frac {1}{2}}{\sqrt {\frac {\left(\nu -\mu \right)\left(\lambda -\mu \right)}{\left(A-\mu \right)\left(B-\mu \right)}}}

h_{\nu }={\frac {1}{2}}{\sqrt {\frac {\left(\lambda -\nu \right)\left(\mu -\nu \right)}{\left(A-\nu \right)\left(B-\nu \right)}}}

Hence, the infinitesimal volume element equals

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

Differential operators such as $\nabla \cdot \mathbf {F}$ and $\nabla \times \mathbf {F}$ can be expressed in the coordinates $(\lambda ,\mu ,\nu )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

References

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Bibliography

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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External links

MathWorld description of confocal paraboloidal coordinates

Template:Orthogonal coordinate systems

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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.
+[[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]]

Bispherical coordinates: Difference between revisions

Revision as of 19:32, 14 December 2013

Contents

Basic formulae

Scale factors

References

Bibliography

External links

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Bispherical coordinates: Difference between revisions

Revision as of 19:32, 14 December 2013

Basic formulae

Scale factors

References

Bibliography

External links

Navigation menu

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