# 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

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