Bispherical coordinates - Revision history

en>Jonesey95: Fixing citation DOI or LCCN parameter error using AutoEd

2014-08-25T03:33:49Z

Fixing citation DOI or LCCN parameter error using AutoEd

← Older revision		Revision as of 05:33, 25 August 2014
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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]

	~~[[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 = 184–185 \| 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 = 119–120}}
	*{{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 = 44–48 (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]~~]

en>LilHelpa: Typo fixing and general fixes using AWB

2013-12-14T17:32:49Z

Typo fixing and general fixes using AWB

← Older revision		Revision as of 19:32, 14 December 2013
Line 1:		Line 1:
	~~The name of~~ the ~~author is~~ [~~http~~://~~www.Rxlist.com~~/~~collection~~-of-~~images~~/~~genital_warts_hpv_picture~~/~~pictures.htm Figures] but it's not~~ the ~~most masucline name out there. North Dakota is her beginning location but she will have over~~ the ~~counter~~ [~~http~~://~~youulike.com~~/~~blogs~~/~~114396~~/~~390615~~/~~items~~-to-~~know~~-~~when~~-~~confronted~~-~~wi std home test] test to move one working day or an additional. One of~~ the ~~very very best issues~~ in ~~over over~~ the ~~counter~~ [~~https://healthcoachmarketing~~.~~zendesk~~.~~com/entries/53672184~~-~~Candida~~-~~Tips~~-~~And~~-~~Cures~~-~~That~~-~~May~~-~~Meet~~-~~Your~~-~~Needs std testing at home] test the counter std test~~ ([http://~~www~~.~~pornextras~~.~~info~~/~~blog/187567 please click the next page~~]~~) globe for him is to collect badges but he is struggling to discover time for it. For years he's been working as a meter reader and it's something he truly appreciate.~~		'''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 = 184–185 \| 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 = 119–120}}
			*{{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 = 44–48 (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]]

en>AManWithNoPlan: | Assisted by Citation bot r394

2012-04-03T18:09:17Z

| Assisted by Citation bot r394

New page

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