I'm having troubles to understand an equivalent definition, or characterization of the degree of mapping as an index of a vector field :
Consider $X \in C^\infty(\mathbb R^2, \mathbb R^2)$, $\, X(x)\neq 0, \, \forall x \in \mathbb S^1$. Define $F:\mathbb S^1 \rightarrow \mathbb S^1$ as $F(x) := \frac{X(x)}{||X(x)||}.$ Then $$\deg(F) = I_X(0),$$ where $I_X(x)$ is the index of the vector field $X$ at the point $x$, see definition below.
How does this make sense?
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Definition. Let $\xi \in C(U,\mathbb R^2), \, U \subset \mathbb R^2 \, $open, a continuous vector field. Let all singularities (zeros) of $\xi$ be isolated. Let $x \in U$, $\,r > 0$ small enough, such that $$\{y \in \mathbb R^2 : 0 < |y - x| \leq r\} \subset U$$ and doesn't contain any singularities of $\xi$. Let $\varphi: [0,2\pi] \rightarrow \mathbb R$ a continuous polar function for the vector field $t \mapsto \xi(c(t))$ along the circle $c: t \mapsto x + r (\cos t, \sin t)$. Then $$ I(x) := I_\xi(x) = \frac{1}{2 \pi} \big(\varphi (2\pi) - \varphi(0)\big), $$ and is independent of the coices made.
Any help is as always greatly appreciated!