Line integrals: path dependence and Stokes

What you are seeing: a vector field in the plane, and two paths from $A = (-1, 0)$ to $B = (1, 0)$: the straight line (orange) and the upper semicircular arc (cyan). The line integral $\int_A^B \mathbf{F} \cdot d\mathbf{r}$ is computed on each path using Simpson quadrature. For a conservative field the two values coincide; for a non-conservative field they differ, and the closed-loop integral around the semicircle equals the curl integrated over the enclosed area (Stokes' theorem).

The selector switches between two conservative fields ($\mathbf{F} = (2xy, x^2)$ and $\mathbf{F} = (x, y)$) and two non-conservative fields ($\mathbf{F} = (-y, x)$ with curl 2, $\mathbf{F} = (y, 0)$ with curl $-1$). The readout reports both path integrals and their difference (the closed-loop value).