= 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.
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