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I'm working on this exercise from an old Spivak Differential Geometry book which I must be misunderstanding. The question reads:

Let $M$ and $N$ be compact $n$-dimensional manifolds with boundary and $M \subset N \setminus \partial N$. For any closed $(n-1)$-form $\omega$ on $N$ show $\int_{\partial M} \omega=\int_{\partial N} \omega$.

Both integrals seem to me to be trivially equal as by Stoke's theorem $\int_{\partial M} \omega=\int_M d\omega =\int_M 0=0$? Am I missing something here? Any external verification? I feel like I'm taking crazy pills, but this book has so many typos...

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  • $\begingroup$ I feel fairly confident that this is a trivial exercise as written. I'm now more interested in what Spivak might have really had in mind. Can you think of a more interesting question with minimal changes to the one above? For example, if $\omega$ is an $(n-1)$-form on $N$ but only closed over $N \setminus M$ then we still have $$\int_{\partial M} \omega=\int_{\partial M}\omega+\int_{N \setminus M} d\omega=\int_{\partial M + \partial N -\partial M} \omega=\int_{\partial N} \omega$$ Any other ideas? $\endgroup$ – spitespike Mar 26 '13 at 2:42
  • $\begingroup$ Yes, I believe that's what he must have had in mind. Think of the generalizations of Gauss's Law. $\endgroup$ – Ted Shifrin Apr 29 '13 at 22:24
  • $\begingroup$ What generalization do you have in mind? $\endgroup$ – spitespike May 1 '13 at 1:03

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