Suppose I have a surface $S$ with no boundary e.g. a sphere. Can I still use Stokes' theorem? E.g. $$\int_S \nabla \times F\cdot dS= \int _{\partial \Sigma} F\cdot dr=0 \quad (\text{presumably?})$$
I guess this is instead saying "$\partial \Sigma=\emptyset$" not "$\partial\Sigma$ does not exist".