Material derivative - Revision history

en>Xezbeth: Unlinked: Continuum

2015-01-12T08:53:55Z

← Older revision		Revision as of 10:53, 12 January 2015
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	~~{{Unreferenced\|date=December 2009}}~~

	~~In [[mathematics]]~~, ~~the '''fundamental class''' is a [[homology (mathematics)\|homology]] class [''M''] associated~~ to ~~an [[oriented]] [[manifold]] ''M'', which corresponds~~ to ~~"the whole manifold",~~ and ~~pairing~~ with ~~which corresponds~~ to ~~"integrating over~~ the ~~manifold". Intuitively~~, the ~~fundamental class can be thought~~ of as ~~the sum of the (top-dimensional) simplices of~~ a ~~suitable triangulation~~ of ~~the manifold~~.

	~~==Definition==~~
	~~===Closed, orientable===~~
	~~When ''M''~~ is ~~a [[connected space\|connected]] [[orientable]] [[closed manifold]] of dimension ''n'', the top homology group is [[infinite cyclic]]:~~ <~~math~~>~~H_n(M,\mathbf{Z}) \cong \mathbf{Z}~~<~~/math~~>~~, and an orientation is a choice~~ of ~~generator, a choice of isomorphism <math>\mathbf{Z} \to H_n(M,\mathbf{Z})<~~/~~math>~~. ~~The generator is called~~ the '~~''fundamental class'''~~.

	If ~~''M'' is disconnected (but still orientable),~~ a ~~fundamental class is a fundamental class for each connected component (corresponding to an orientation for each component).~~

	~~It represents,~~ in ~~a sense~~, ~~''integration over M'', and in relation with [[de Rham cohomology]] it is exactly~~ that~~; namely for ''M'' a smooth manifold, an~~ [~~[differential form\|''n''-form~~]~~] ω can be paired with the fundamental class as~~

	:<~~math~~>~~\langle\omega, [M]\rangle = \int_M \omega~~<~~/math~~>

	~~to get a real number, which is~~ the ~~integral~~ of ~~ω over ''M'',~~ and ~~depends only on the cohomology class~~ of ω.

	~~===Non-orientable===~~
	~~If ''M'' is not orientable, one cannot define~~ a ~~fundamental class, or more precisely, one cannot define a fundamental class ''over <math>\mathbf{Z}</math>'' (or over <math>\mathbf{R}</math>), as <math>H_n(M;\mathbf{Z})=0</math>,~~ and ~~indeed, one cannot integrate differential ''n''-forms over non-orientable manifolds~~.

	~~However, every closed manifold is~~ <~~math~~>~~\mathbf{Z}/2~~<~~/math~~>~~-orientable~~, and
	~~<math>H_n(M;\mathbf{Z}/2)=\mathbf{Z}/2</math> (for ''M'' connected)~~. ~~Thus every closed manifold~~ is ~~<math>\mathbf{Z}/2</math>-oriented (not just orient''able'': there is no ambiguity in choice~~ of ~~orientation)~~, ~~and has a <math>\mathbf{Z}/2</math>-fundamental class~~.

	This ~~<math>\mathbf{Z}/2</math>-fundamental class is used in defining [[Stiefel–Whitney numbers]].~~

	~~===With boundary===~~
	~~If ''M'' is~~ a ~~compact orientable manifold with boundary~~, ~~then~~ the ~~top relative homology group is again infinite cyclic~~ <~~math~~>~~H_n(M,\partial M)\cong \mathbf{Z}~~<~~/math~~>, ~~and as~~ with ~~closed manifolds~~, a ~~choice~~ of ~~isomorphism is a fundamental class~~.

	~~==Poincaré duality==~~
	~~{{Main\|Poincaré duality}}~~
	~~{{Expand section\|date=December 2008}}~~
	~~Under [[Poincaré duality]], the fundamental class~~ is ~~dual~~ to ~~the bottom class of~~ a ~~connected manifold (a generator of~~ <~~math~~>~~H_0~~<~~/math~~>): in the ~~closed case~~, ~~Poincaré duality is the statement that the [[cap product]]~~ with the ~~fundamental class yields an isomorphism <math>H_* (M, R) \~~to ~~H^{n~~-*}(M, R)</math>.

	~~See also [[Twisted Poincaré duality]]~~

	~~==Applications==~~
	~~{{Expand section\|date=December 2008}}~~
	~~In the [[Bruhat decomposition]]~~ of the ~~[[flag variety]]~~ of ~~a [[Lie group]],~~ the ~~fundamental class corresponds~~ to the ~~top-dimension [[Schubert cell]]~~, or ~~equivalently the [[longest element of~~ a ~~Coxeter group]].~~

	~~==See also==~~
	*[[Longest element of a ~~Coxeter group]]~~
	*[[Poincaré duality]]

	~~== External links ==~~
	*[http://www.map.mpim-bonn.mpg.de/Fundamental_class Fundamental class] at the ~~Manifold Atlas.~~
	* The Encyclopedia of ~~Mathematics article on [http://www~~.~~encyclopediaofmath.org/index.php/Fundamental_class the fundamental class].~~

	~~{{DEFAULTSORT:Fundamental Class}}~~
	~~[[Category:Algebraic topology]]~~

108.40.64.124 at 16:09, 22 January 2014

2014-01-22T16:09:49Z

← Older revision		Revision as of 18:09, 22 January 2014
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	~~Prior to invest loads of money on things like controls actually memory cards, appear~~ the net for a secondhand edition. Occasionally a store will probably getting out of used-game hardware, which could be quite affordable. Make sure you look which has a web-based seller's feedback to be able to the purchase so this whether you are having what you covered.<br><br>		{{For\|the fundamental class in class field theory\|class formation}}
			{{Unreferenced\|date=December 2009}}

	~~Backside line~~ is, ~~this is actually worth exploring if more powerful~~ and ~~healthier strategy games~~, ~~especially to tell~~ the ~~truth keen on Clash~~ of ~~Clans~~. ~~Want realize what opinions you possess~~, ~~when you do.~~<br><br>~~A personalized little ones who enjoy video games~~, ~~then conscious how challenging it really~~ is to ~~pull them out from the t. v.. Their eye can [http:~~/~~/en~~.~~Search.wordpress.com/?q=prove+stuck prove stuck] towards~~ the ~~keep tabs on for hours as they will play their preferred games~~. If ~~you want aid regulating your children~~'~~s clash of clans Hack time~~, ~~then your pursuing article has some tips~~ for ~~you~~.<br><br>~~Normally clash~~ of ~~clans identify tool no survey plots believe in among their people~~. ~~Society~~ is ~~probably definitely powered by look pressure~~, one ~~of the most powerful forces during the planet. Whenever long as peer power utilizes its power to gain good~~, ~~clash of clans hack tool no critique will have its house in community.~~<br><br>~~And additionally the game is their mobile edition~~, ~~it is performing not lack substance~~ as ~~many mobile games. So~~, ~~defragging the process registry will boost its system overall performance which will a fantastic extent~~. ~~I usually get at all from 4000 to 5~~,~~000 m~~ - ~~Points from a day~~ (~~$4 to $5~~ for ~~Amazon~~. ~~Fruit showed off the massively anticipated i~~ - ~~Business phone 5 for the to start with time in San Francisco on Wednesday morning~~ (~~September 12, 2012). Should you have any kind of concerns concerning~~ in ~~which and the best way to work with how to hack clash~~ of ~~clans ([http://circuspartypanama.com click the next post]~~), ~~it is possible to call us at the webpage. Is actually an~~ a ~~huge demand during some i~~ - ~~Phone line 4 application not exclusive promoting business but definitely helps users to make extra money~~.<br><br>~~Your war abject~~ is ~~agnate~~ in ~~your approved village, except that your battle abject will not gain resources~~. ~~Barrio in your warfare abject simpley can~~'~~t be anon improved possibly rearranged~~, ~~as it by it's self mimics this adjustment to accomplished completed advancement certifications of your apple inside alertness day. Struggle bases additionally never charge to take their pieces rearmed~~, ~~defenses reloaded perhaps characters healed~~, as ~~these kinds~~ of ~~products are consistently ready~~. ~~The association alcazar with your war abject penalty be abounding alone~~ to the ~~one in your whole village.~~<br><br>~~To master game play throughout~~ the ~~shooter video games~~, ~~learn about your weapons~~. See ~~everything there is realize about each and all~~ of the ~~weapon style in~~ the ~~overall game. Each weapon excels in~~ the ~~certain ways~~, ~~but lies short in others~~. ~~When you know i would say~~ the ~~pluses and minuses~~ of ~~each weapon, you may easily use them to full advantage~~.		In [[mathematics]], the '''fundamental class''' is a [[homology (mathematics)\|homology]] class [''M''] associated to an [[oriented]] [[manifold]] ''M'', which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold". Intuitively, the fundamental class can be thought of as the sum of the (top-dimensional) simplices of a suitable triangulation of the manifold.

			==Definition==
			===Closed, orientable===
			When ''M'' is a [[connected space\|connected]] [[orientable]] [[closed manifold]] of dimension ''n'', the top homology group is [[infinite cyclic]]: <math>H_n(M,\mathbf{Z}) \cong \mathbf{Z}</math>, and an orientation is a choice of generator, a choice of isomorphism <math>\mathbf{Z} \to H_n(M,\mathbf{Z})</math>. The generator is called the '''fundamental class'''.

			If ''M'' is disconnected (but still orientable), a fundamental class is a fundamental class for each connected component (corresponding to an orientation for each component).

			It represents, in a sense, ''integration over M'', and in relation with [[de Rham cohomology]] it is exactly that; namely for ''M'' a smooth manifold, an [[differential form\|''n''-form]] ω can be paired with the fundamental class as

			:<math>\langle\omega, [M]\rangle = \int_M \omega</math>

			to get a real number, which is the integral of ω over ''M'', and depends only on the cohomology class of ω.

			===Non-orientable===
			If ''M'' is not orientable, one cannot define a fundamental class, or more precisely, one cannot define a fundamental class ''over <math>\mathbf{Z}</math>'' (or over <math>\mathbf{R}</math>), as <math>H_n(M;\mathbf{Z})=0</math>, and indeed, one cannot integrate differential ''n''-forms over non-orientable manifolds.

			However, every closed manifold is <math>\mathbf{Z}/2</math>-orientable, and
			<math>H_n(M;\mathbf{Z}/2)=\mathbf{Z}/2</math> (for ''M'' connected). Thus every closed manifold is <math>\mathbf{Z}/2</math>-oriented (not just orient''able'': there is no ambiguity in choice of orientation), and has a <math>\mathbf{Z}/2</math>-fundamental class.

			This <math>\mathbf{Z}/2</math>-fundamental class is used in defining [[Stiefel–Whitney numbers]].

			===With boundary===
			If ''M'' is a compact orientable manifold with boundary, then the top relative homology group is again infinite cyclic <math>H_n(M,\partial M)\cong \mathbf{Z}</math>, and as with closed manifolds, a choice of isomorphism is a fundamental class.

			==Poincaré duality==
			{{Main\|Poincaré duality}}
			{{Expand section\|date=December 2008}}
			Under [[Poincaré duality]], the fundamental class is dual to the bottom class of a connected manifold (a generator of <math>H_0</math>): in the closed case, Poincaré duality is the statement that the [[cap product]] with the fundamental class yields an isomorphism <math>H_* (M, R) \to H^{n-*}(M, R)</math>.

			See also [[Twisted Poincaré duality]]

			==Applications==
			{{Expand section\|date=December 2008}}
			In the [[Bruhat decomposition]] of the [[flag variety]] of a [[Lie group]], the fundamental class corresponds to the top-dimension [[Schubert cell]], or equivalently the [[longest element of a Coxeter group]].

			==See also==
			*[[Longest element of a Coxeter group]]
			*[[Poincaré duality]]

			== External links ==
			*[http://www.map.mpim-bonn.mpg.de/Fundamental_class Fundamental class] at the Manifold Atlas.
			* The Encyclopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Fundamental_class the fundamental class].

			{{DEFAULTSORT:Fundamental Class}}
			[[Category:Algebraic topology]]

en>Giftlite: /* Orthogonal coordinates */ mv :

2012-05-29T17:08:01Z

Orthogonal coordinates: mv :

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