Elliptic cylindrical coordinates: Difference between revisions

Revision as of 14:26, 15 March 2013

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system $(λ, μ, ν)$ that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is 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 $(λ, μ, ν)$ by the equations

x^{2} = \frac{(a^{2} + λ) (a^{2} + μ) (a^{2} + ν)}{(a^{2} - b^{2}) (a^{2} - c^{2})}

y^{2} = \frac{(b^{2} + λ) (b^{2} + μ) (b^{2} + ν)}{(b^{2} - a^{2}) (b^{2} - c^{2})}

z^{2} = \frac{(c^{2} + λ) (c^{2} + μ) (c^{2} + ν)}{(c^{2} - b^{2}) (c^{2} - a^{2})}

where the following limits apply to the coordinates

- λ < c^{2} < - μ < b^{2} < - ν < a^{2} .

Consequently, surfaces of constant $λ$ are ellipsoids

\frac{x^{2}}{a^{2} + λ} + \frac{y^{2}}{b^{2} + λ} + \frac{z^{2}}{c^{2} + λ} = 1,

whereas surfaces of constant $μ$ are hyperboloids of one sheet

\frac{x^{2}}{a^{2} + μ} + \frac{y^{2}}{b^{2} + μ} + \frac{z^{2}}{c^{2} + μ} = 1,

because the last term in the lhs is negative, and surfaces of constant $ν$ are hyperboloids of two sheets

\frac{x^{2}}{a^{2} + ν} + \frac{y^{2}}{b^{2} + ν} + \frac{z^{2}}{c^{2} + ν} = 1

because the last two terms in the lhs are negative.

Scale factors and differential operators

For brevity in the equations below, we introduce a function

S (σ) \overset{d e f}{=} (a^{2} + σ) (b^{2} + σ) (c^{2} + σ)

where $σ$ can represent any of the three variables $(λ, μ, ν)$ . Using this function, the scale factors can be written

h_{λ} = \frac{1}{2} \sqrt{\frac{(λ - μ) (λ - ν)}{S (λ)}}

h_{μ} = \frac{1}{2} \sqrt{\frac{(μ - λ) (μ - ν)}{S (μ)}}

h_{ν} = \frac{1}{2} \sqrt{\frac{(ν - λ) (ν - μ)}{S (ν)}}

Hence, the infinitesimal volume element equals

d V = \frac{(λ - μ) (λ - ν) (μ - ν)}{8 \sqrt{- S (λ) S (μ) S (ν)}} d λ d μ d ν

and the Laplacian is defined by

\nabla^{2} Φ = \frac{4 \sqrt{S (λ)}}{(λ - μ) (λ - ν)} \frac{\partial}{\partial λ} [\sqrt{S (λ)} \frac{\partial Φ}{\partial λ}] +

\frac{4 \sqrt{S (μ)}}{(μ - λ) (μ - ν)} \frac{\partial}{\partial μ} [\sqrt{S (μ)} \frac{\partial Φ}{\partial μ}] + \frac{4 \sqrt{S (ν)}}{(ν - λ) (ν - μ)} \frac{\partial}{\partial ν} [\sqrt{S (ν)} \frac{\partial Φ}{\partial ν}]

Other differential operators such as $\nabla \cdot F$ and $\nabla \times F$ can be expressed in the coordinates $(λ, μ, ν)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Bibliography

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
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

Unusual convention

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 Uses (ξ, η, ζ) coordinates that have the units of distance squared.

External links

MathWorld description of confocal ellipsoidal coordinates

Template:Orthogonal coordinate systems

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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.
+==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^{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)}
+</math>
+:<math>
+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)}
+</math>
+:<math>
+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)}
+</math>
+where the following limits apply to the coordinates
+:<math>
+- \lambda < c^{2} < - \mu < b^{2} < -\nu < a^{2}.
+</math>
+Consequently, surfaces of constant <math>\lambda</math> are [[ellipsoid]]s
+:<math>
+\frac{x^{2}}{a^{2} + \lambda} +  \frac{y^{2}}{b^{2} + \lambda} + \frac{z^{2}}{c^{2} + \lambda} = 1,
+</math>
+whereas surfaces of constant <math>\mu</math> are [[hyperboloid]]s of one sheet
+:<math>
+\frac{x^{2}}{a^{2} + \mu} +  \frac{y^{2}}{b^{2} + \mu} + \frac{z^{2}}{c^{2} + \mu} = 1,
+</math>
+because the last term in the lhs is negative, and surfaces of constant <math>\nu</math> are [[hyperboloid]]s of two sheets
+:<math>
+\frac{x^{2}}{a^{2} + \nu} +  \frac{y^{2}}{b^{2} + \nu} + \frac{z^{2}}{c^{2} + \nu} = 1
+</math>
+because the last two terms in the lhs are negative.
+==Scale factors and differential operators==
+For brevity in the equations below, we introduce a function
+:<math>
+S(\sigma) \ \stackrel{\mathrm{def}}{=}\  \left( a^{2} + \sigma \right) \left( b^{2} + \sigma \right) \left( c^{2} + \sigma \right)
+</math>
+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
+:<math>
+h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \mu \right) \left( \lambda - \nu\right)}{S(\lambda)}}
+</math>
+:<math>
+h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda\right) \left( \mu - \nu\right)}{S(\mu)}}
+</math>
+:<math>
+h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \lambda\right) \left( \nu - \mu\right)}{S(\nu)}}
+</math>
+Hence, the infinitesimal volume element equals
+:<math>
+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
+</math>
+and the [[Laplacian]] is defined by
+:<math>
+\nabla^{2} \Phi =
+\frac{4\sqrt{S(\lambda)}}{\left( \lambda - \mu \right) \left( \lambda - \nu\right)}
+\frac{\partial}{\partial \lambda} \left[ \sqrt{S(\lambda)} \frac{\partial \Phi}{\partial \lambda} \right] \  +  \
+</math>
+:::::<math>
+\frac{4\sqrt{S(\mu)}}{\left( \mu - \lambda \right) \left( \mu - \nu\right)}
+\frac{\partial}{\partial \mu} \left[ \sqrt{S(\mu)} \frac{\partial \Phi}{\partial \mu} \right] \  + \
+\frac{4\sqrt{S(\nu)}}{\left( \nu - \lambda \right) \left( \nu - \mu\right)}
+\frac{\partial}{\partial \nu} \left[ \sqrt{S(\nu)} \frac{\partial \Phi}{\partial \nu} \right]
+</math>
+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>(\lambda, \mu, \nu)</math> by substituting
+the scale factors into the general formulae
+found in [[orthogonal coordinates]].
+==See also==
+* [[Focaloid]] (shell given by two coordinate surfaces)
+==References==
+{{reflist}}
+==Bibliography==
+*{{cite book | author = Morse PM, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | page = 663}}
+*{{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}}
+*{{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | pages = 101&ndash;102 | lccn = 6725285}}
+*{{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}}
+*{{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&ndash;180 | lccn = 5510911 }}
+*{{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&ndash;44 (Table 1.10)}}
+===Unusual convention===
+*{{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&ndash;29 }}  Uses (ξ, η, ζ) coordinates that have the units of distance squared.
+==External links==
+*[http://mathworld.wolfram.com/ConfocalEllipsoidalCoordinates.html MathWorld description of confocal ellipsoidal coordinates]
+{{Orthogonal coordinate systems}}
+[[Category:Coordinate systems]]

Elliptic cylindrical coordinates: Difference between revisions

Revision as of 14:26, 15 March 2013

Contents

Basic formulae

Scale factors and differential operators

See also

References

Bibliography

Unusual convention

External links

Navigation menu

Elliptic cylindrical coordinates: Difference between revisions

Revision as of 14:26, 15 March 2013

Basic formulae

Scale factors and differential operators

See also

References

Bibliography

Unusual convention

External links

Navigation menu

Search