Consider a vector field $F(x,y):\mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose $F(x_0)=0$ and det $nabla F(x_0) \neq 0$, prove that the index (winding number) of the vector field F at $x_0$ is 1 or -1 depending on whether det $\nabla F(x_0) > 0$ or $<0$.
I know the winding number can be defined as $$\frac{1}{2 \pi} \int_0^1 \nabla \theta$$ where $\theta$ is the angle between the x axis and the curve, but I am struggling on how to proceed with the integral.