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If $\omega$ is an $n$-form on a compact $n$-dimensional manifold $M$ without boundary, then $\omega $ is exact if and only if $\int\limits_{M}{\omega }=0$.

Maybe there are two ways - use de Rham theory, and another way is to prove this directly.I don't know both. Help!

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    You want to add that $M$ is oriented (otherwise the integral doesn't make sense) and connected (otherwise the statement isn't true.)2012-02-19

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