Rankine vortex - Revision history

89.110.14.92 at 10:48, 9 November 2014

2014-11-09T10:48:35Z

← Older revision		Revision as of 12:48, 9 November 2014
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	~~In [[mathematics]], a differentiable manifold <math>\scriptstyle M</math> of dimension ''n''~~ is called '''parallelizable''' <ref>{{citation \| last1=Bishop\|first1=R.L.\|last2=Goldberg\|first2=S.I.\|title=Tensor Analysis on Manifolds\|year=1968\|page=160}}</ref> if there exist [[Smooth function\|smooth]] [[vector field]]s		The person who wrote the article is known as Jayson Hirano and he completely digs that title. Distributing production is how he tends to make a residing. For years he's been living in Alaska and he doesn't strategy on changing it. Playing badminton is a thing that he is completely addicted to.<br><br>my web site ... love psychic readings ([http://si.dgmensa.org/xe/index.php?document_srl=48014&mid=c0102 My Web Page])

	~~:<math>\{V_1, \dots,V_n\}</math>~~

	~~on the manifold, such~~ that ~~at any point <math>\scriptstyle p</math> of <math>\scriptstyle M</math> the [[tangent vector]]s~~

	~~:<math>\{V_i(p)\}_{i\in \{1,\dots,n\}}</math>~~

	provide a [[basis of a vector space\|basis]] of the [[tangent space]] at <math>\scriptstyle p</math>. Equivalently, the [[tangent bundle]] is a [[trivial bundle]],<ref>{{citation \| last1=Milnor\|first1=J.W.\|last2=Stasheff\|first2=J.D.\|title~~=Characteristic Classes\|year=1974\|page=15}}</ref> so that the associated [[principal bundle]] of [[frame bundle\|linear frames]] has a section on <math>\scriptstyle M</math>.~~

	A particular choice of such a basis of vector fields on <math>\scriptstyle M</math> is called a '''[[Parallelization (mathematics)\|parallelization]]''' (or an '''absolute parallelism''') of <math>\scriptstyle M</math>.

	~~==Examples==~~
	*An example with ''n'' = 1 is ~~the [[circle]]: we can take ''V''<sub>1</sub>~~ to ~~be the unit tangent vector field, say pointing in the anti-clockwise direction. The [[torus]] of dimension ''n'' is also parallelizable, as can be seen by expressing it as~~ a ~~[[cartesian product]] of circles~~. For ~~example, take~~ '~~'n'' = 2,~~ and construct a torus from a square of [[graph paper]] with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, any [[Lie group]] ''G'' is parallelizable, since a basis for the tangent space at the [[identity element]] can be moved around by the action of the translation group of ''G'' on ~~''G'' (any translation~~ is a ~~diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in ''G'').~~

	*A classical problem was to determine which of the [[sphere]]s ''S''<sup>''n''</sup> are parallelizable. The zero-dimensional case ''S''<sup>0</sup> is trivially parallelizable. The case ''S''<sup>1</sup> is the circle, which is parallelizable as has already been explained. The [[hairy ball theorem]] shows that ~~''S''<sup>2</sup>~~ is ~~not parallelizable~~. ~~However ''S''~~<~~sup~~>3<~~/sup~~> ~~is parallelizable, since it is the Lie group [[SU(2)]]~~. ~~The only other parallelizable sphere is ''S''<sup>7</sup>; this was proved in 1958, by [[Michel Kervaire]], and by [[Raoul Bott]] and [[John Milnor]], in independent work~~. The parallelizable spheres correspond precisely to elements of unit norm in the [[normed division algebra]]s of the real numbers, complex numbers, [[quaternion]]s, and [[octonion]]s, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires [[algebraic topology]].

	*The product of parallelizable [[manifold]]s is parallelizable.

	*Any [[Orientability\|orientable]] [[3-manifold\|three-dimensional manifold]] is parallelizable.

	~~==Remarks==~~
	*The term '''''framed manifold''''' (~~occasionally '''''rigged manifold''''') is most usually applied to an embedded manifold with a given trivialisation of the~~ [~~[normal bundle]], and also for an abstract (i~~.e. ~~non-embedded) manifold with a given stable trivialisation of the [[tangent bundle]]~~.

	==~~See also==~~
	* [[Chart (topology)]]
	* [[Differentiable manifold]]
	* [[Frame bundle]]
	* [[Kervaire invariant]]
	* [[Orthonormal frame bundle]]
	* [[Principal bundle]]
	* [[Connection (mathematics)]]
	* [[G-structure]]

	~~==Notes==~~
	~~{{Reflist}}~~

	~~== References ==~~
	* {{citation \| last1=Bishop\|first1=R.L.\|last2=Goldberg\|first2=S.I. \| title = Tensor Analysis on Manifolds\| publisher=The Macmillan Company \| year=1968\|edition=First Dover 1980\|isbn=0-486-64039-6}}
	* {{citation \| last1=Milnor\|first1=J.W.\|last2=Stasheff\|first2=J.D. \|author2-link=Jim Stasheff\| title = Characteristic Classes\| publisher=Princeton University Press \| year=1974}}

	~~[[Category:Differential topology]]~~
	~~[[Category:Fiber bundles]]~~
	~~[[Category:Vector bundles]]~~
	~~[[Category:Manifolds]]~~

en>Briardew: Moved some content around, removed some bozo's comment about tornadoes with no supporting reference, and fixed the broken link

2013-06-19T23:06:45Z

Moved some content around, removed some bozo's comment about tornadoes with no supporting reference, and fixed the broken link

← Older revision		Revision as of 01:06, 20 June 2013
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	Hi there. ~~Allow me begin by introducing~~ the ~~author~~, ~~her name~~ is ~~Sophia~~. For ~~years she~~'~~s been living in Kentucky but her spouse wants them~~ to ~~move~~. ~~Distributing manufacturing~~ is ~~exactly where my primary earnings arrives from~~ and it's ~~something I truly appreciate~~. ~~Doing ballet~~ is ~~something she would never give up~~.<br><br>~~Here~~ is ~~my homepage ::~~ [~~http:~~//1.~~234~~.36.~~240/fxac/m001_2/7330 free psychic~~]		In [[mathematics]], a differentiable manifold <math>\scriptstyle M</math> of dimension ''n'' is called '''parallelizable''' <ref>{{citation \| last1=Bishop\|first1=R.L.\|last2=Goldberg\|first2=S.I.\|title=Tensor Analysis on Manifolds\|year=1968\|page=160}}</ref> if there exist [[Smooth function\|smooth]] [[vector field]]s

			:<math>\{V_1, \dots,V_n\}</math>

			on the manifold, such that at any point <math>\scriptstyle p</math> of <math>\scriptstyle M</math> the [[tangent vector]]s

			:<math>\{V_i(p)\}_{i\in \{1,\dots,n\}}</math>

			provide a [[basis of a vector space\|basis]] of the [[tangent space]] at <math>\scriptstyle p</math>. Equivalently, the [[tangent bundle]] is a [[trivial bundle]],<ref>{{citation \| last1=Milnor\|first1=J.W.\|last2=Stasheff\|first2=J.D.\|title=Characteristic Classes\|year=1974\|page=15}}</ref> so that the associated [[principal bundle]] of [[frame bundle\|linear frames]] has a section on <math>\scriptstyle M</math>.

			A particular choice of such a basis of vector fields on <math>\scriptstyle M</math> is called a '''[[Parallelization (mathematics)\|parallelization]]''' (or an '''absolute parallelism''') of <math>\scriptstyle M</math>.

			==Examples==
			*An example with ''n'' = 1 is the [[circle]]: we can take ''V''<sub>1</sub> to be the unit tangent vector field, say pointing in the anti-clockwise direction. The [[torus]] of dimension ''n'' is also parallelizable, as can be seen by expressing it as a [[cartesian product]] of circles. For example, take ''n'' = 2, and construct a torus from a square of [[graph paper]] with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, any [[Lie group]] ''G'' is parallelizable, since a basis for the tangent space at the [[identity element]] can be moved around by the action of the translation group of ''G'' on ''G'' (any translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in ''G'').

			*A classical problem was to determine which of the [[sphere]]s ''S''<sup>''n''</sup> are parallelizable. The zero-dimensional case ''S''<sup>0</sup> is trivially parallelizable. The case ''S''<sup>1</sup> is the circle, which is parallelizable as has already been explained. The [[hairy ball theorem]] shows that ''S''<sup>2</sup> is not parallelizable. However ''S''<sup>3</sup> is parallelizable, since it is the Lie group [[SU(2)]]. The only other parallelizable sphere is ''S''<sup>7</sup>; this was proved in 1958, by [[Michel Kervaire]], and by [[Raoul Bott]] and [[John Milnor]], in independent work. The parallelizable spheres correspond precisely to elements of unit norm in the [[normed division algebra]]s of the real numbers, complex numbers, [[quaternion]]s, and [[octonion]]s, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires [[algebraic topology]].

			*The product of parallelizable [[manifold]]s is parallelizable.

			*Any [[Orientability\|orientable]] [[3-manifold\|three-dimensional manifold]] is parallelizable.

			==Remarks==
			*The term '''''framed manifold''''' (occasionally '''''rigged manifold''''') is most usually applied to an embedded manifold with a given trivialisation of the [[normal bundle]], and also for an abstract (i.e. non-embedded) manifold with a given stable trivialisation of the [[tangent bundle]].

			==See also==
			* [[Chart (topology)]]
			* [[Differentiable manifold]]
			* [[Frame bundle]]
			* [[Kervaire invariant]]
			* [[Orthonormal frame bundle]]
			* [[Principal bundle]]
			* [[Connection (mathematics)]]
			* [[G-structure]]

			==Notes==
			{{Reflist}}

			== References ==
			* {{citation \| last1=Bishop\|first1=R.L.\|last2=Goldberg\|first2=S.I. \| title = Tensor Analysis on Manifolds\| publisher=The Macmillan Company \| year=1968\|edition=First Dover 1980\|isbn=0-486-64039-6}}
			* {{citation \| last1=Milnor\|first1=J.W.\|last2=Stasheff\|first2=J.D. \|author2-link=Jim Stasheff\| title = Characteristic Classes\| publisher=Princeton University Press \| year=1974}}

			[[Category:Differential topology]]
			[[Category:Fiber bundles]]
			[[Category:Vector bundles]]
			[[Category:Manifolds]]

en>FrescoBot: Bot: link syntax/spacing

2012-06-23T10:33:15Z

Bot: link syntax/spacing

New page

Hi there. Allow me begin by introducing the author, her name is Sophia. For years she's been living in Kentucky but her spouse wants them to move. Distributing manufacturing is exactly where my primary earnings arrives from and it's something I truly appreciate. Doing ballet is something she would never give up.<br><br>Here is my homepage :: [http://1.234.36.240/fxac/m001_2/7330 free psychic]