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I am trying to read through J. Simmons, Minimal Varieties in Riemannian Manifolds, and in the proof of Proposition 1.2.2 he calls "Stokes theorem" to the following result: $$ \int_M\delta\theta=\int_{\partial M}\star\theta $$ where $\theta$ is a 1-form on $M$, $\delta$ is the codifferential and $\star$ is the Hodge star operator.

I looked for "stokes codifferential" on the internet. In google or in MO it didn't give anything. On MSE there is this unanswered question.

Is this result explained in some book? I don't do differential geometry and I don't know the literature.

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    $\begingroup$ Note that in this case $\delta \theta$ is the function $-*d*\theta$. When one says "integrate a function over M", they mean the n-form $fd\text{vol}_n = *f$. So the left-hand-side really means $\int -d*\theta$. The given statement is Stokes' theorem applied to the 1-form $*\theta$, except that (unless I have made a sign error) there is a missing minus sign somewhere. $\endgroup$
    – mme
    Commented Apr 4, 2023 at 23:50
  • $\begingroup$ @nme okay, that explains it then, thx. $\endgroup$
    – Elías Guisado Villalgordo
    Commented Apr 4, 2023 at 23:55

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