Stokes theorem in 2D (Green's theorem)

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.