Particle shower - Revision history

en>Headbomb: Various citation cleanup + AWB fixes, typo(s) fixed: Moliere → Molière (2) using AWB

2014-08-23T22:48:22Z

Various citation cleanup + AWB fixes, typo(s) fixed: Moliere → Molière (2) using AWB

← Older revision		Revision as of 00:48, 24 August 2014
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	~~Alyson Meagher~~ is ~~the title her parents gave her~~ but she ~~doesn~~'t [http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ telephone psychic] like when individuals use psychics online ([http://www.khuplaza.com/dent/14869889 http://www.khuplaza.com/dent/14869889]) her ~~complete title~~. ~~Ohio is where~~ my ~~house~~ is ~~but my spouse wants us~~ to ~~transfer. She is truly fond of caving~~ but ~~she doesn~~'t ~~have the time recently. Since I was eighteen I've been operating as~~ a ~~bookkeeper but soon my wife and I will begin our personal company~~.<br><br>~~Here is~~ my ~~page - psychic phone (~~[http://www.~~octionx~~.~~sinfauganda~~.Co.~~ug/node~~/~~22469 click through the next internet site~~])		The writer is called Irwin. Years ago we moved to North Dakota. She is a librarian but she's usually needed her own company. The favorite pastime for my children and me is to play baseball but I haven't made a dime with it.<br><br>Also visit my weblog :: [http://www.articlecube.com/profile/Nydia-Lyons/532548 http://www.articlecube.com/]

78.48.73.195: let's please be exact... this error/inaccuracy caused me quite some trouble

2014-02-26T20:30:57Z

let's please be exact... this error/inaccuracy caused me quite some trouble

← Older revision		Revision as of 22:30, 26 February 2014
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	~~In [[mathematics]],~~ the '~~''biharmonic equation''' is a fourth-order~~ [~~[partial differential equation]] which arises in areas of [[continuum mechanics]], including [[linear elasticity]] theory and the solution of [[Stokes flow]]s~~. ~~It is written as~~		Alyson Meagher is the title her parents gave her but she doesn't [http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ telephone psychic] like when individuals use psychics online ([http://www.khuplaza.com/dent/14869889 http://www.khuplaza.com/dent/14869889]) her complete title. Ohio is where my house is but my spouse wants us to transfer. She is truly fond of caving but she doesn't have the time recently. Since I was eighteen I've been operating as a bookkeeper but soon my wife and I will begin our personal company.<br><br>Here is my page - psychic phone ([http://www.octionx.sinfauganda.Co.ug/node/22469 click through the next internet site])

	~~:<math>\nabla^4\varphi=0<~~/~~math>~~

	or

	~~:<math>\nabla^2\nabla^2\varphi=0<~~/~~math>~~

	or

	~~:<math>\Delta^2\varphi=0<~~/~~math>~~

	~~where <math>\nabla^4<~~/~~math> is the fourth power of the [[del~~]~~] operator and the square of the~~ [~~[laplacian]] operator <math>\nabla^2<~~/~~math> (or <math>\Delta<~~/~~math>), and it is known as the '''biharmonic operator''' or the '''bilaplacian operator'''~~.

	~~For example, in three dimensional [[cartesian coordinates]] the biharmonic equation has the form~~
	~~: <math>~~
	~~{\partial^4 \varphi\over \partial x^4 } +~~
	~~{\partial^4 \varphi\over \partial y^4 } +~~
	~~{\partial^4 \varphi\over \partial z^4 }+~~
	~~2{\partial^4 \varphi\over \partial x^2\partial y^2}+~~
	~~2{\partial^4 \varphi\over \partial y^2\partial z^2}+~~
	~~2{\partial^4 \varphi\over \partial x^2\partial z^2} = 0~~.
	</~~math>~~

	~~As another example, in ''n''-dimensional [[Euclidean space]],~~

	:~~<math>\nabla^4 \left({1\over r}\right)= {3(15-8n+n^2)\over r^5}<~~/~~math>~~

	~~where~~

	~~:<math>r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}~~.</~~math>~~

	~~which, for ''n=3 and n=5'' only, becomes the biharmonic equation~~.

	~~A solution to the biharmonic equation~~ is ~~called a '''biharmonic function'''. Any [[harmonic function]]~~ is ~~biharmonic,~~ but ~~the converse is not always true~~.

	~~In two-dimensional [[polar coordinates]], the biharmonic equation~~ is

	~~:<math>~~
	~~\frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial}{\partial r} \left(\frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial \varphi}{\partial r}\right)\right)\right)~~
	~~+ \frac{2}{r^2} \frac{\partial^4 \varphi}{\partial \theta^2 \partial r^2}~~
	~~+ \frac{1}{r^4} \frac{\partial^4 \varphi}{\partial \theta^4}~~
	~~- \frac{2}{r^3} \frac{\partial^3 \varphi}{\partial \theta^2 \partial r}~~
	~~+ \frac{4}{r^4} \frac{\partial^2 \varphi}{\partial \theta^2} = 0~~
	~~</math>~~

	~~which can be solved by separation~~ of ~~variables. The result is~~ the ~~[[Michell solution]]~~.

	~~==2 dimensional space==~~
	~~The general solution to the 2 dimensional case is~~
	~~:<math>~~
	~~x v(x,y) - y u(x,y) + w(x,y)~~
	~~</math>~~
	~~where <math>u(x,y)</math>, <math>v(x,y)</math> and <math>w(x,y)</math> are [[harmonic functions]]~~ and <~~math~~>~~v(x,y)~~<~~/math~~> is ~~a [[harmonic conjugate]] of <math>u~~(~~x,y)</math>.~~

	~~Just as~~ [[harmonic functions]] in 2 variables are closely related to complex [[analytic functions]], so are biharmonic functions in 2 variables. The general form of a biharmonic function in 2 variables can also be written as
	:~~<math>~~
	~~\operatorname{Im}(\bar{z}f(z) + g(z))~~
	</~~math>~~
	~~where <math>f(z)<~~/~~math> and <math>g(z)</math> are [[analytic functions]].~~

	~~==See also==~~

	* [[Harmonic function]]

	~~== References ==~~

	* Eric W Weisstein, ''CRC Concise Encyclopedia of Mathematics'', CRC Press, 2002. ~~ISBN 1-58488-347-2~~.
	* S I Hayek, ''Advanced Mathematical Methods in Science and Engineering'', Marcel Dekker, 2000. ~~ISBN 0-8247-0466-5~~.
	* {{cite book \| author=J P Den Hartog \| title=Advanced Strength of Materials \| publisher=Courier Dover Publications \| year=Jul 1, 1987 \| isbn= 0-486-65407-9}}

	~~==External links==~~
	* {{MathWorld \| urlname=BiharmonicEquation \| title=Biharmonic Equation}}
	* {{MathWorld \| urlname=BiharmonicOperator \| title=Biharmonic Operator}}

	~~[[Category:Elliptic partial differential equations]~~]

en>NickPenguin: no consensus to merge

2013-12-21T18:56:19Z

no consensus to merge

← Older revision		Revision as of 20:56, 21 December 2013
Line 1:		Line 1:
	~~The writer~~ is ~~known~~ as ~~Irwin Wunder but it's [http~~://~~tti-ttw.com/index.php?do~~=/~~profile-2200~~/~~info/ home std test~~] ~~not at home std test std home test~~ the ~~most masucline name out there. The preferred pastime for my children~~ and me is to [~~http~~://~~www~~.~~health.com~~/~~health/condition-article/0~~,,~~20188476,00~~.~~html perform baseball~~] but ~~I haven't produced a dime with it~~. ~~For a while I've been in South Dakota and my mothers and fathers reside nearby. Since she was eighteen at home std testing she's been working as a~~ [~~http~~:/~~/www~~.~~Britishskinfoundation~~.~~org.uk~~/~~SkinInformation~~/~~AtoZofSkindisease~~/~~HerpesSimplex.aspx receptionist~~] ~~but her~~ [~~https:~~/~~/hcbsales~~.~~com/node/3739 std testing at home~~] ~~marketing by no means arrives~~.<br><br>~~Stop by my site ... over the counter std test~~ (~~[http:~~//~~www~~.~~webmdbook~~.~~com/index~~.~~php?do~~=~~/profile~~-~~11685/info/ Read Webpage~~])		In [[mathematics]], the '''biharmonic equation''' is a fourth-order [[partial differential equation]] which arises in areas of [[continuum mechanics]], including [[linear elasticity]] theory and the solution of [[Stokes flow]]s. It is written as

			:<math>\nabla^4\varphi=0</math>

			or

			:<math>\nabla^2\nabla^2\varphi=0</math>

			or

			:<math>\Delta^2\varphi=0</math>

			where <math>\nabla^4</math> is the fourth power of the [[del]] operator and the square of the [[laplacian]] operator <math>\nabla^2</math> (or <math>\Delta</math>), and it is known as the '''biharmonic operator''' or the '''bilaplacian operator'''.

			For example, in three dimensional [[cartesian coordinates]] the biharmonic equation has the form
			: <math>
			{\partial^4 \varphi\over \partial x^4 } +
			{\partial^4 \varphi\over \partial y^4 } +
			{\partial^4 \varphi\over \partial z^4 }+
			2{\partial^4 \varphi\over \partial x^2\partial y^2}+
			2{\partial^4 \varphi\over \partial y^2\partial z^2}+
			2{\partial^4 \varphi\over \partial x^2\partial z^2} = 0.
			</math>

			As another example, in ''n''-dimensional [[Euclidean space]],

			:<math>\nabla^4 \left({1\over r}\right)= {3(15-8n+n^2)\over r^5}</math>

			where

			:<math>r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.</math>

			which, for ''n=3 and n=5'' only, becomes the biharmonic equation.

			A solution to the biharmonic equation is called a '''biharmonic function'''. Any [[harmonic function]] is biharmonic, but the converse is not always true.

			In two-dimensional [[polar coordinates]], the biharmonic equation is

			:<math>
			\frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial}{\partial r} \left(\frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial \varphi}{\partial r}\right)\right)\right)
			+ \frac{2}{r^2} \frac{\partial^4 \varphi}{\partial \theta^2 \partial r^2}
			+ \frac{1}{r^4} \frac{\partial^4 \varphi}{\partial \theta^4}
			- \frac{2}{r^3} \frac{\partial^3 \varphi}{\partial \theta^2 \partial r}
			+ \frac{4}{r^4} \frac{\partial^2 \varphi}{\partial \theta^2} = 0
			</math>

			which can be solved by separation of variables. The result is the [[Michell solution]].

			==2 dimensional space==
			The general solution to the 2 dimensional case is
			:<math>
			x v(x,y) - y u(x,y) + w(x,y)
			</math>
			where <math>u(x,y)</math>, <math>v(x,y)</math> and <math>w(x,y)</math> are [[harmonic functions]] and <math>v(x,y)</math> is a [[harmonic conjugate]] of <math>u(x,y)</math>.

			Just as [[harmonic functions]] in 2 variables are closely related to complex [[analytic functions]], so are biharmonic functions in 2 variables. The general form of a biharmonic function in 2 variables can also be written as
			:<math>
			\operatorname{Im}(\bar{z}f(z) + g(z))
			</math>
			where <math>f(z)</math> and <math>g(z)</math> are [[analytic functions]].

			==See also==

			* [[Harmonic function]]

			== References ==

			* Eric W Weisstein, ''CRC Concise Encyclopedia of Mathematics'', CRC Press, 2002. ISBN 1-58488-347-2.
			* S I Hayek, ''Advanced Mathematical Methods in Science and Engineering'', Marcel Dekker, 2000. ISBN 0-8247-0466-5.
			* {{cite book \| author=J P Den Hartog \| title=Advanced Strength of Materials \| publisher=Courier Dover Publications \| year=Jul 1, 1987 \| isbn= 0-486-65407-9}}

			==External links==
			* {{MathWorld \| urlname=BiharmonicEquation \| title=Biharmonic Equation}}
			* {{MathWorld \| urlname=BiharmonicOperator \| title=Biharmonic Operator}}

			[[Category:Elliptic partial differential equations]]

en>Obondu: /* See also */

2012-08-24T14:21:05Z