1

How can I prove that the index of a smooth vector field is well-defined? All I know that it is locally constant.

Definition. Given an open set $U\subset\Bbb{R^m}$, and a smooth vector field $v:U\to\Bbb{R^m}$ with an isolated zero $z\in U$, the function $\overline{v}:v(x)/\|v(x)\|$ maps a small sphere centred at $z$ into the unit sphere. The degree of this mapping is called the index of the vector field $v$ at the zero $z$, denoted by $i(z)$.

algebraically_speaking
  • 11
  • Well, what is your definition of the index of a vector field? – Pax Nov 04 '14 at 05:53
  • Given an open set $U\subset\Bbb{R^m}$, and a smooth vector field $v:U\to\Bbb{R^m}$ with an isolated zero $z\in U$, the function $\overline{v}:v(x)/|v(x)|$ maps a small sphere centred at $z$ into the unit sphere. The degree of this mapping is called the index of the vector field $v$ at the zero $z$, denoted by $i(z)$. – algebraically_speaking Nov 04 '14 at 06:05

0 Answers0