 Solution

For a continuously differentiable vector field $\mathbf{F}$ near a piecewise smooth oriented surface $S$, Stokes theorem statesThe circulation around the whole boundary equals the flux of the curl through the surface. The unit normal vector and boundary direction must be compatible with the right-hand rule: curling the right-hand fingers in the boundary direction points the thumb along the normal. Reversing both orientations reverses both integrals.