Along a path avoiding the origin, polar coordinates give

F \cdot d r = \frac{- y d x + x d y}{x ^{2} + y ^{2}} = d θ,

where the angle is followed continuously along the path. For a closed path its total change is an integer number of turns, so

\oint_{C} F \cdot d r = 2 π wind (C, 0) .

For a simple counterclockwise loop, the value is

2 π

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.