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I see in my course the following theorem:

If $\omega$ is an exact k-form over an oriented compact manifold M of dimension $k$, then $\int_M \omega=0$.

I don't have a proof of this theorem and I only know it's an application of stokes theorem.

Is this theorem correct?

Thank you for any help.

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  • $\begingroup$ stokes theorem... $\endgroup$ – Tsemo Aristide Jan 9 at 18:28
  • $\begingroup$ Why don't you just use that $\omega$ is exact and then apply stokes theorem on you own? $\endgroup$ – klirk Jan 9 at 18:28
  • $\begingroup$ Because it's not as easy as it seems. Apply stokes theorem on wich manifold(boundary, interior)? is the integral 0 because there is no boundary, or is it because $\omega$ is closed then $d\omega=0$? $\endgroup$ – PerelMan Jan 9 at 18:33
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$$ \text{With }\omega = d\eta, \qquad \int_M \omega = \int_M d\eta = \int_{\partial M} \eta = 0 \quad \text{since }M \text{ has no boundary.}$$

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