Singular point, index of a

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)