Please refer to image below (plot link),
Consider the vector field F.
It is conservative - thus, the circulation for all simple closed curves (except those through the origin) is zero. Yet, visually, this appears absurd; consider a curve C that is a circle of radius 4 centered at the origin. If F is a force field, one definition of circulation is that it is the work done on a particle as it traverses a said path. An infinitesimal increment of such work is defined as the dot product of the vector field and the vector tangent to the curve at each point, times the arc length infinitesimal: (F dot T)ds - that is, work done ALONG the curve.
Orienting the curve counterclockwise using a parametrization, it is clear that the vector field is always positively aligned with the tangent vector of the curve - and thus, NO NEGATIVE WORK IS DONE anywhere along the curve. Despite this, the circulation around the circle is [zero][3]. By the same integral, the work done in the [second quadrant][4] is negative.
What am I missing?