 Stokes theorem on a cut cylindrical surface

Stokes theorem applies to an oriented surface with several boundary components. Cutting a cylindrical side along one generator produces a parameter rectangle: the two copies of the cut are traversed in opposite directions and their line integrals cancel for a single-valued smooth vector field. The upper and lower rims retain opposite directions when viewed from the same side. This proves why cylindrical topology does not require a modification of Stokes theorem; it requires inclusion of the entire oriented boundary.