# Singular point, index of a

From Encyclopedia of Mathematics

One of the basic characteristics of an isolated singular point of a vector field. Let a vector field be defined on , and let be a sphere of small radius surrounding a singular point such that . The degree of the mapping (cf. Degree of a mapping)

is then called the index, , of the singular point of the vector field , i.e.

If is non-degenerate, then

#### 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=11420

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article