en>Headbomb: Various citation cleanup + AWB fixes, added orphan tag using AWB

2014-08-24T16:07:07Z

Various citation cleanup + AWB fixes, added orphan tag using AWB

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	~~In [[differential topology]],~~ a ~~mathematical discipline,~~ and ~~more specifically in [[Morse theory]], a '''gradient-like vector field''' is a generalization of [[gradient vector field]].~~		I'm a 34 years old and study at the college (Industrial and Labor Relations).<br>In my spare time I try to teach myself Italian. I have been there and look forward to returning sometime near future. I like to read, preferably on my ebook reader. I like to watch Supernatural and Sons of Anarchy as well as documentaries about nature. I love Running.<br><br>My web site: [http://www.sensitiv.dk/sys6_site/modules/import_template/showme.php?a%5B%5D=%3Ca+href%3Dhttp%3A%2F%2Fhowtohackassasinscreedpirates.wordpress.com%2F%3Eassasins+creed+pirates+hack%3C%2Fa%3E assasin's creed pirates hack tokens free android hack]

	~~The primary motivation is as a technical tool in the construction of [[Morse function]]s, to show that one can construct a function whose critical points are~~ at ~~distinct levels. One first constructs a Morse function, then uses gradient-like vector fields to move around~~ the ~~critical points, yielding a different Morse function.~~

	~~== Definition ==~~
	~~Given a [[Morse function]] ''f'' on a manifold ''M,'' a gradient-like vector field ''X'' for the function ''f'' is, informally:~~
	* away from critical points, ''X'' points "in the same direction as" the [[gradient]] of ''f,'' and
	* near a critical point (in the neighborhood of a critical point)~~, it equals the gradient of ''f,'' when ''f'' is written in standard form given in the [[Morse lemma]]s~~.
	~~Formally:~~<~~ref~~>~~[http://books.google~~.~~com/books?id=TtKyqozvgIwC&pg=PA63 p. 63]</ref>~~
	* away from critical points, <math>X \cdot f > 0,</math>
	* around every critical point there ~~is a neighborhood on which ''f'' is given as in the Morse lemmas:~~
	~~:<math>f(x) = f(b) - x_1^2 - \cdots - x_{\alpha}^2 + x_{\alpha +1}^2 + \cdots + x_n^2 </math>~~
	and ~~on which ''X'' equals the gradient of ''f~~.''

	~~== Dynamical system ==~~
	~~The associated [[dynamical system]] of a gradient-~~like ~~vector field~~, ~~a '''gradient-~~like ~~dynamical system''', is a special case~~ of ~~a [[Morse–Smale system]]~~.

	~~==References==~~
	~~{{reflist}}~~
	~~{{refbegin}}~~
	* An introduction to Morse theory, Yukio Matsumoto, 2002, Section 2.3: ~~Gradient-like vector fields,~~ [http://~~books~~.~~google~~.~~com~~/~~books~~?id=~~TtKyqozvgIwC&pg=PA56 p. 56–69]~~
	* [http://hilbertthm90.wordpress.com~~/2009/09/25/gradient-like-vector-fields-exist/ Gradient-Like Vector Fields Exist], September 25, 2009~~
	~~{{refend}}~~

	~~[[Category:Morse theory]]~~
	~~[[Category:Differential topology]~~]


	~~{{differential-geometry-stub}}~~
	~~{{topology-stub}}~~

128.214.198.130 at 11:22, 11 August 2011

2011-08-11T11:22:05Z

New page

In [[differential topology]], a mathematical discipline, and more specifically in [[Morse theory]], a '''gradient-like vector field''' is a generalization of [[gradient vector field]].

The primary motivation is as a technical tool in the construction of [[Morse function]]s, to show that one can construct a function whose critical points are at distinct levels. One first constructs a Morse function, then uses gradient-like vector fields to move around the critical points, yielding a different Morse function.

== Definition ==
Given a [[Morse function]] ''f'' on a manifold ''M,'' a gradient-like vector field ''X'' for the function ''f'' is, informally:
* away from critical points, ''X'' points "in the same direction as" the [[gradient]] of ''f,'' and
* near a critical point (in the neighborhood of a critical point), it equals the gradient of ''f,'' when ''f'' is written in standard form given in the [[Morse lemma]]s.
Formally:<ref>[http://books.google.com/books?id=TtKyqozvgIwC&pg=PA63 p. 63]</ref>
* away from critical points, <math>X \cdot f > 0,</math>
* around every critical point there is a neighborhood on which ''f'' is given as in the Morse lemmas:
:<math>f(x) = f(b) - x_1^2 - \cdots - x_{\alpha}^2 + x_{\alpha +1}^2 + \cdots + x_n^2 </math>
and on which ''X'' equals the gradient of ''f.''

== Dynamical system ==
The associated [[dynamical system]] of a gradient-like vector field, a '''gradient-like dynamical system''', is a special case of a [[Morse–Smale system]].

==References==
{{reflist}}
{{refbegin}}
* An introduction to Morse theory, Yukio Matsumoto, 2002, Section 2.3: Gradient-like vector fields, [http://books.google.com/books?id=TtKyqozvgIwC&pg=PA56 p. 56–69]
* [http://hilbertthm90.wordpress.com/2009/09/25/gradient-like-vector-fields-exist/ Gradient-Like Vector Fields Exist], September 25, 2009
{{refend}}

[[Category:Morse theory]]
[[Category:Differential topology]]

{{differential-geometry-stub}}
{{topology-stub}}

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en>Headbomb: Various citation cleanup + AWB fixes, added orphan tag using AWB

128.214.198.130 at 11:22, 11 August 2011