Let $D$ be the disk with two disks cut out, e.g. let $D$ be the set of points $(x, y)$ satisfying all of the following: $x^2 + y^2 \le 16$, $(x-2)^2+y^2 \ge 1$, $(x+2)^2+y^2 \ge 1$. Show that any vector field that is defined on D and points towards the interior of D has at least one equilibrium point.
I have no idea how to prove this proposition. We are learning the index of vector field. It seems that the proof needs the theorem that "the index of the vector field tangent to a simple closed curve (and not vanishing on this curve) equals 1". However, I don't know how to use this.