Stokes Theorem 2D Circulation

What you are seeing: a vector field plotted as arrows with a rectangular region. The circulation around the rectangle equals the surface integral of the curl: $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_R (\partial Q / \partial x - \partial P / \partial y) dA$ . For the unit-curl field $\mathbf{F} = (-y/2, x/2)$ the right-hand side is just the area; for the shear $\mathbf{F} = (y, 0)$ it is the negative of the area; conservative fields give zero.

Figure 1. Vector field arrows and Stokes-theorem circulation rectangle.

fieldunit

rect width2.00

rect height1.50

WHAT TO TRY

Drag and resize the loop in a curl=1 field: the circulation around it always equals the enclosed area, so the circulation-versus-area plot is a straight line of slope equal to the curl.
Switch the field to curl=-1 or curl=0: the circulation flips sign or vanishes, even though the loop is unchanged. The line integral around the boundary measures the curl inside, that is Stokes theorem.
Stretch the rectangle taller or wider at fixed area: the circulation stays put, because Stokes cares only about the enclosed curl flux, not the shape of the loop.