Parabolic cylindrical coordinates: Difference between revisions
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[[File:Conical coordinates.png|thumb|380px|right|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of the conical coordinates. The constants ''b'' and ''c'' were chosen as 1 and 2, respectively. The red sphere represents ''r''=2, the blue elliptic cone aligned with the vertical ''z''-axis represents μ=cosh(1) and the yellow elliptic cone aligned with the (green) ''x''-axis corresponds to ν<sup>2</sup> = 2/3. The three surfaces intersect at the point '''P''' (shown as a black sphere) with [[Cartesian coordinate system|Cartesian coordinates]] roughly (1.26, -0.78, 1.34). The elliptic cones intersect the sphere in taco-shaped curves.]] | |||
'''Conical coordinates''' are a three-dimensional [[orthogonal coordinates|orthogonal]] [[coordinate system]] consisting of | |||
concentric spheres (described by their radius <math>r</math>) and by two families of perpendicular cones, aligned along the ''z''- and ''x''-axes, respectively. | |||
==Basic definitions== | |||
The conical coordinates <math>(r, \mu, \nu)</math> are defined by | |||
:<math> | |||
x = \frac{r\mu\nu}{bc} | |||
</math> | |||
:<math> | |||
y = \frac{r}{b} \sqrt{\frac{\left( \mu^{2} - b^{2} \right) \left( \nu^{2} - b^{2} \right)}{\left( b^{2} - c^{2} \right)} } | |||
</math> | |||
:<math> | |||
z = \frac{r}{c} \sqrt{\frac{\left( \mu^{2} - c^{2} \right) \left( \nu^{2} - c^{2} \right)}{\left( c^{2} - b^{2} \right)} } | |||
</math> | |||
with the following limitations on the coordinates | |||
:<math> | |||
\nu^{2} < c^{2} < \mu^{2} < b^{2} | |||
</math> | |||
Surfaces of constant <math>r</math> are spheres of that radius centered on the origin | |||
:<math> | |||
x^{2} + y^{2} + z^{2} = r^{2} | |||
</math> | |||
whereas surfaces of constant <math>\mu</math> and <math>\nu</math> are mutually perpendicular cones | |||
:<math> | |||
\frac{x^{2}}{\mu^{2}} + \frac{y^{2}}{\mu^{2} - b^{2}} + \frac{z^{2}}{\mu^{2} - c^{2}} = 0 | |||
</math> | |||
:<math> | |||
\frac{x^{2}}{\nu^{2}} + \frac{y^{2}}{\nu^{2} - b^{2}} + \frac{z^{2}}{\nu^{2} - c^{2}} = 0 | |||
</math> | |||
In this coordinate system, both [[Laplace's equation]] and the [[Helmholtz equation]] are separable. | |||
==Scale factors== | |||
The scale factor for the radius <math>r</math> is one (<math>h_{r} = 1</math>), as in [[spherical coordinates]]. The scale factors for the two conical coordinates are | |||
:<math> | |||
h_{\mu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \mu^{2} \right) \left( \mu^{2} - c^{2} \right)}} | |||
</math> | |||
:<math> | |||
h_{\nu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \nu^{2} \right) \left( c^{2} - \nu^{2} \right)}} | |||
</math> | |||
==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 = 659}} | |||
*{{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 = 183–184 | 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 = 179 | lccn = 5914456}} | |||
*{{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | pages = 991–100 | lccn = 6725285}} | |||
*{{cite book | author = Arfken G | year = 1970 | title = Mathematical Methods for Physicists | edition = 2nd | publisher = Academic Press | location = Orlando, FL | pages = 118–119 | id = ASIN B000MBRNX4}} | |||
*{{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Conical Coordinates (r, 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 | pages = 37–40 (Table 1.09) | isbn = 978-0-387-18430-2}} | |||
==External links== | |||
*[http://mathworld.wolfram.com/ConicalCoordinates.html MathWorld description of conical coordinates] | |||
{{Orthogonal coordinate systems}} | |||
[[Category:Coordinate systems]] |
Revision as of 05:24, 1 March 2013
Conical coordinates are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius ) and by two families of perpendicular cones, aligned along the z- and x-axes, respectively.
Basic definitions
The conical coordinates are defined by
with the following limitations on the coordinates
Surfaces of constant are spheres of that radius centered on the origin
whereas surfaces of constant and are mutually perpendicular cones
In this coordinate system, both Laplace's equation and the Helmholtz equation are separable.
Scale factors
The scale factor for the radius is one (), as in spherical coordinates. The scale factors for the two conical coordinates are
References
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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