 Solution

Along a path avoiding the origin, polar coordinates givewhere the angle is followed continuously along the path. For a closed path its total change is an integer number of turns, soFor a simple counterclockwise loop, the value is $2\pi$ if the origin lies inside and zero if it lies outside. Clockwise orientation reverses the nonzero sign. The winding number formula also handles repeated or self-intersecting loops. A curve passing through the origin is outside the ordinary line integral's domain.