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2011-08-11T11:22:05Z

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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]].

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]]

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