The above figure is from Milnor's "Topology from the differentiable viewpoint." He just wrote that they are examples of index -1,0,1,2 vector fields. However, I had a little bit confusing about this conclusion; could someone explain how he can conclude the index of vector fields? Thank you in advance.
The index of $v$ is the degree of $S^r\rightarrow S^1$ defined by $g(x)={{v(x)}\over{\|v(x)\|}}$, where $S^r$ is a small sphere around the singularity that we suppose to be at $0$ here.
the case $i=-1$ follows from the fact that if you follow the flow around $S^r$ you make one turn around the singularity in the opposite orientation of $S^r$
The case $i=0$ follows from the fact that $g$ is not surjective, since we do not turn around the singularity for example the image of $g$ does not contain an element orthogonal to the horizontal axis.
the case $i=1$ follows from the fact that if you follow the flow around $S^r$ you make one turn around the singularity.
the case $i=2$ follows from the fact that if you follow the flow around $S^r$ you make two turn around the singularity.