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One of the basic characteristics of an isolated [[Singular point|singular point]] of a vector field. Let a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s0856001.png" /> be defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s0856002.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s0856003.png" /> be a sphere of small radius surrounding a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s0856004.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s0856005.png" />. The degree of the mapping (cf. [[Degree of a mapping|Degree of a mapping]])
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s0856006.png" /></td> </tr></table>
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is then called the index, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s0856007.png" />, of the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s0856008.png" /> of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s0856009.png" />, i.e.
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One of the basic characteristics of an isolated [[Singular point|singular point]] of a vector field. Let a vector field  $  X $
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be defined on  $  \mathbf R  ^ {n} $,  
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and let  $  Q $
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be a sphere of small radius surrounding a singular point $  x _ {0} $
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such that  $  X \mid  _ {Q} \neq 0 $.  
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The degree of the mapping (cf. [[Degree of a mapping|Degree of a mapping]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s08560010.png" /></td> </tr></table>
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$$
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f: Q  \rightarrow  S  ^ {n-} 1 ,\ \
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f( x)  = X(
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\frac{z)}{\| X( x) \| }
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,
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$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s08560011.png" /> is non-degenerate, then
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is then called the index,  $  \mathop{\rm ind} _ {x _ {0}  } ( X) $,
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of the singular point  $  x _ {0} $
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of the vector field  $  X $,
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i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085600/s08560012.png" /></td> </tr></table>
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$$
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\mathop{\rm ind} _ {x _ {0}  } ( X)  =   \mathop{\rm deg}  f _ {x _ {0}  } .
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$$
  
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If  $  x _ {0} $
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is non-degenerate, then
  
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$$
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\mathop{\rm ind} _ {x _ {0}  } ( X)  =  \mathop{\rm sign}  \mathop{\rm det}  \left \|
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 +
\frac{\partial  X  ^ {j} }{\partial  x  ^ {i} }
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\right \| .
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$$
  
 
====Comments====
 
====Comments====

Revision as of 08:14, 6 June 2020


One of the basic characteristics of an isolated singular point of a vector field. Let a vector field $ X $ be defined on $ \mathbf R ^ {n} $, and let $ Q $ be a sphere of small radius surrounding a singular point $ x _ {0} $ such that $ X \mid _ {Q} \neq 0 $. The degree of the mapping (cf. Degree of a mapping)

$$ f: Q \rightarrow S ^ {n-} 1 ,\ \ f( x) = X( \frac{z)}{\| X( x) \| } , $$

is then called the index, $ \mathop{\rm ind} _ {x _ {0} } ( X) $, of the singular point $ x _ {0} $ of the vector field $ X $, i.e.

$$ \mathop{\rm ind} _ {x _ {0} } ( X) = \mathop{\rm deg} f _ {x _ {0} } . $$

If $ x _ {0} $ is non-degenerate, then

$$ \mathop{\rm ind} _ {x _ {0} } ( X) = \mathop{\rm sign} \mathop{\rm det} \left \| \frac{\partial X ^ {j} }{\partial x ^ {i} } \right \| . $$

Comments

See also Poincaré theorem; Rotation of a vector field.

References

[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] J.A. Thorpe, "Elementary topics in differential geometry" , Springer (1979)
[a3] C. Conley, E. Zehnder, "Morse type index theory for flows and periodic solutions of Hamiltonian equations" Comm. Pure Appl. Math. , 37 (1984) pp. 207–253
[a4] K.P. Rybakovskii, "The homotopy index and partial differential equations" , Springer (1987) (Translated from Russian)
How to Cite This Entry:
Singular point, index of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_point,_index_of_a&oldid=48721
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article