# integral of a k-form over an oriented compact manifold

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.

• stokes theorem... – Tsemo Aristide Jan 9 at 18:28
• Why don't you just use that $\omega$ is exact and then apply stokes theorem on you own? – klirk Jan 9 at 18:28
• 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 $d\omega=0$? – PerelMan Jan 9 at 18:33

$$\text{With }\omega = d\eta, \qquad \int_M \omega = \int_M d\eta = \int_{\partial M} \eta = 0 \quad \text{since }M \text{ has no boundary.}$$