Parabolic cylindrical coordinates - Revision history

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

2014-08-25T03:38:50Z

Fixing citation DOI or LCCN parameter error using AutoEd

← Older revision		Revision as of 05:38, 25 August 2014
Line 1:		Line 1:
	~~[[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.]]~~		Greetings! I am Marvella and I really feel comfy when people use the full name. Years in the past we moved to Puerto Rico and my family members enjoys it. Hiring is my profession. The factor she adores most is physique developing and now she is attempting to make cash with it.<br><br>My blog post - [http://forums.cellphonedge.com/groups/valuable-guidance-for-successfully-treating-candidiasis/ at home std test]

	~~'''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]~~]

en>Addbot: Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q2473905

2013-03-01T03:24:25Z

Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q2473905

← Older revision		Revision as of 05:24, 1 March 2013
Line 1:		Line 1:
	~~Greetings~~. The ~~author~~'~~s name is Phebe~~ and ~~she feels comfy when individuals use~~ the ~~full title~~. ~~California is our beginning place~~. ~~He is truly fond~~ of ~~performing ceramics but he is having difficulties to discover time for it~~. ~~For years I~~'~~ve been working as a payroll clerk~~.<br><br>~~Also visit my website;~~ [~~http~~://~~happywedding~~.~~lonnieheart~~.~~gethompy~~.~~com/zbxe/wc_apply/372747~~ http://~~happywedding~~.~~lonnieheart~~.~~gethompy~~.~~com~~]		[[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]]

en>AManWithNoPlan: update lccn to newer template

2012-06-14T13:56:19Z

update lccn to newer template

New page

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