Gradient-like vector field

<h2 id="definition">Definition</h2>
Given a <a href="/facts/Morse_function/bJRqH8BQ">Morse function</a> f on a manifold M, a gradient-like vector field X for the function f is, informally:

<ul><li>away from critical points, X points "in the same direction as" the <a href="/facts/Gradient/TsR7uukj">gradient</a> of f, and</li>
<li>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 <a href="/facts/Morse_lemma/bJRqH8BQ">Morse lemmas</a>.</li></ul>
Formally:<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<ul><li>away from critical points, 
 
 
 
 X
 ⋅
 f
 >
 0
 ,
 
 
 {\displaystyle X\cdot f>0,}
 
</li>
<li>around every critical point there is a neighborhood on which f is given as in the Morse lemmas:</li></ul>

f
        (
        x
        )
        =
        f
        (
        b
        )
        −
        
          x
          
            1
          
          
            2
          
        
        −
        ⋯
        −
        
          x
          
            α
          
          
            2
          
        
        +
        
          x
          
            α
            +
            1
          
          
            2
          
        
        +
        ⋯
        +
        
          x
          
            n
          
          
            2
          
        
      
    
    {\displaystyle f(x)=f(b)-x_{1}^{2}-\cdots -x_{\alpha }^{2}+x_{\alpha +1}^{2}+\cdots +x_{n}^{2}}

and on which X equals the gradient of f.

<h2 id="dynamical-system">Dynamical system</h2>
The associated <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> of a gradient-like vector field, a gradient-like dynamical system, is a special case of a <a href="/facts/Morse%25E2%2580%2593Smale_system/VyScnCjV">Morse–Smale system</a>.

<ul><li>An introduction to Morse theory, Yukio Matsumoto, 2002, Section 2.3: Gradient-like vector fields, <a href="https://books.google.com/books?id=TtKyqozvgIwC&pg=PA56">p. 56–69</a></li>
<li><a href="http://hilbertthm90.wordpress.com/2009/09/25/gradient-like-vector-fields-exist/">Gradient-Like Vector Fields Exist</a> <a href="https://web.archive.org/web/20120326013348/http://hilbertthm90.wordpress.com/2009/09/25/gradient-like-vector-fields-exist/">Archived</a> 2012-03-26 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>, September 25, 2009</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">p. 63 <a href="https://books.google.com/books?id=TtKyqozvgIwC&pg=PA63" target="_blank">https://books.google.com/books?id=TtKyqozvgIwC&pg=PA63</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Gradient-like vector field open-in-new

Gradient-like vector field