Equivariant map - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T11:16:06Z

WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 13:16, 5 May 2014
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	~~{{about\|the general concept in the mathematical theory of vector fields\|the vector potential in electromagnetism\|Magnetic vector potential\|the vector potential in fluid mechanics\|Stream function}}~~		Gabrielle is what her spouse loves to call lady though she doesn't very like being called like these. As a woman what your girl really likes is mah jongg but she doesn't have made a dime utilizing it. [http://search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=Software+developing&Submit=Go Software developing] is where her primary income comes from but nevertheless , soon her husband and so her will start specific own [http://www.Google.com/search?q=business&btnI=lucky business]. For a while she's not long ago in Massachusetts. Go to her website to pick out more: http://prometeu.net<br><br>My weblog: clash of clans hack android ([http://prometeu.net Read the Full Guide])

	~~In [[vector calculus]], a '''vector potential''' is a [[vector field]] whose [[Curl (mathematics)\|curl]] is a given vector field. This~~ is ~~analogous~~ to a '~~'[[scalar potential]]'', which is~~ a ~~scalar field whose [[gradient]]~~ is ~~a given vector field.~~

	~~Formally, given a vector field '''v''~~'~~, a ''vector potential'' is~~ a ~~vector field '''A''' such that~~
	~~:<math> \mathbf{v} = \nabla \times \mathbf{A}~~. ~~</math>~~

	~~If a vector field '''v''' admits a vector potential '''A''', then from the equality~~
	:~~<math>\nabla \cdot (\nabla \times \mathbf{A}) = 0<~~/~~math>~~
	~~([[divergence]] of the [[Curl (mathematics)\|curl]] is zero) one obtains~~
	~~:<math>\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0,<~~/~~math>~~
	~~which implies that '''v''' must be a [[solenoidal vector field]]~~.

	==~~Theorem~~==
	~~Let~~
	~~:<math>\mathbf{v} : \mathbb R^3 \to \mathbb R^3</math>~~
	~~be a [[solenoidal vector field]] which is twice [[smooth function\|continuously differentiable]]. Assume that '''v'''('''x''') decreases sufficiently fast as \|\|'''x'''\|\|→∞. Define~~

	~~:<math> \mathbf{A} (\mathbf{x})~~ = ~~\frac{1}{4 \pi} \nabla \times \int_{\mathbb R^3} \frac{ \mathbf{v} (\mathbf{y})}{\left\\|\mathbf{x} -\mathbf{y} \right\\|} \, d\mathbf{y}. </math>~~
	~~Then, '''A''' is a vector potential for '''v''', that is,~~
	~~:<math>\nabla \times \mathbf{A}~~ =~~\mathbf{v}. </math>~~

	~~A generalization of this theorem is the [[Helmholtz decomposition]] which states that any vector field can be decomposed as a sum of a solenoidal vector field and an [[irrotational vector field]].~~

	==~~Nonuniqueness~~==
	~~The vector potential admitted by a solenoidal field~~ is ~~not unique. If '''A''' is a vector potential for '''v'''~~, ~~then~~ so is

	:~~<math> \mathbf{A} + \nabla m <~~/~~math>~~
	~~where ''m'' is any continuously differentiable scalar function~~. ~~This follows from the fact that the curl of the gradient is zero~~.

	~~This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires [[Gauge fixing\|choosing a gauge]].~~

	== ~~See also ==~~
	* [[Fundamental theorem of vector analysis]]
	* [[Magnetic potential]]
	* [[Solenoid]]

	~~== References ==~~
	* '~~'Fundamentals of Engineering Electromagnetics'' by David K~~. ~~Cheng, Addison-Wesley, 1993~~.

	~~[[Category~~:~~Concepts in physics]]~~
	[~~[Category~~:~~Potentials]]~~
	~~[[Category:Vector calculus]~~]

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q256355

2013-03-07T04:26:46Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q256355

← Older revision		Revision as of 06:26, 7 March 2013
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	~~Eusebio Stanfill~~ is ~~what~~'~~s displayed on my birth marriage certificate although it~~ is ~~not necessarily quite the name on my birth certificate~~. ~~Vermont~~ is ~~where my home may~~. ~~Software building up has been my new day job for~~ a ~~long time. To bake~~ is ~~the only pasttime my wife doesn~~'~~t agree to~~. ~~You can consider my website here~~: ~~http~~://~~circuspartypanama.com~~<br><br>~~my site -~~ [~~http~~:/~~/circuspartypanama~~.~~com clash~~ of ~~clans hack tool free download~~]		{{about\|the general concept in the mathematical theory of vector fields\|the vector potential in electromagnetism\|Magnetic vector potential\|the vector potential in fluid mechanics\|Stream function}}

			In [[vector calculus]], a '''vector potential''' is a [[vector field]] whose [[Curl (mathematics)\|curl]] is a given vector field. This is analogous to a ''[[scalar potential]]'', which is a scalar field whose [[gradient]] is a given vector field.

			Formally, given a vector field '''v''', a ''vector potential'' is a vector field '''A''' such that
			:<math> \mathbf{v} = \nabla \times \mathbf{A}. </math>

			If a vector field '''v''' admits a vector potential '''A''', then from the equality
			:<math>\nabla \cdot (\nabla \times \mathbf{A}) = 0</math>
			([[divergence]] of the [[Curl (mathematics)\|curl]] is zero) one obtains
			:<math>\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0,</math>
			which implies that '''v''' must be a [[solenoidal vector field]].

			==Theorem==
			Let
			:<math>\mathbf{v} : \mathbb R^3 \to \mathbb R^3</math>
			be a [[solenoidal vector field]] which is twice [[smooth function\|continuously differentiable]]. Assume that '''v'''('''x''') decreases sufficiently fast as \|\|'''x'''\|\|→∞. Define

			:<math> \mathbf{A} (\mathbf{x}) = \frac{1}{4 \pi} \nabla \times \int_{\mathbb R^3} \frac{ \mathbf{v} (\mathbf{y})}{\left\\|\mathbf{x} -\mathbf{y} \right\\|} \, d\mathbf{y}. </math>
			Then, '''A''' is a vector potential for '''v''', that is,
			:<math>\nabla \times \mathbf{A} =\mathbf{v}. </math>

			A generalization of this theorem is the [[Helmholtz decomposition]] which states that any vector field can be decomposed as a sum of a solenoidal vector field and an [[irrotational vector field]].

			==Nonuniqueness==
			The vector potential admitted by a solenoidal field is not unique. If '''A''' is a vector potential for '''v''', then so is

			:<math> \mathbf{A} + \nabla m </math>
			where ''m'' is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.

			This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires [[Gauge fixing\|choosing a gauge]].

			== See also ==
			* [[Fundamental theorem of vector analysis]]
			* [[Magnetic potential]]
			* [[Solenoid]]

			== References ==
			* ''Fundamentals of Engineering Electromagnetics'' by David K. Cheng, Addison-Wesley, 1993.

			[[Category:Concepts in physics]]
			[[Category:Potentials]]
			[[Category:Vector calculus]]

en>ZéroBot: r2.7.1) (robot Adding: pt:Aplicação equivariante

2011-06-20T14:06:00Z

r2.7.1) (robot Adding: pt:Aplicação equivariante

New page

Eusebio Stanfill is what's displayed on my birth marriage certificate although it is not necessarily quite the name on my birth certificate. Vermont is where my home may. Software building up has been my new day job for a long time. To bake is the only pasttime my wife doesn't agree to. You can consider my website here: http://circuspartypanama.com<br><br>my site - [http://circuspartypanama.com clash of clans hack tool free download]