Let us consider the standard version of Stokes' Theorem, $$\iint_S\nabla\times\mathbf{F}\cdot d\mathbf{\sigma}=\oint_C\mathbf{F}\cdot d\mathbf{s}$$ where $S$ is a smooth orientable surface bounded by a connected, smooth simple closed curve $C$. Can one apply this to, say, the punctured torus? (A picture shown below.) The standard proof of Stokes' Theorem uses Green's theorem on the parametrization of the surface, so as long as we choose the parametrization to have a simply connected domain, Stokes' Theorem seems to hold.

A punctured torus

  • $\begingroup$ Can you state clearly which integral you are considering, under what assumption ? And it depends if you can extend your differential form to the whole torus. If you can't, you'll have to substract a path integral around the removed point. $\endgroup$ – reuns Nov 29 '16 at 5:27
  • $\begingroup$ The last bullet point on page 2 of this might answer your question. $\endgroup$ – Mattos Nov 29 '16 at 5:35

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