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I am reading a paper which uses closed forms [tex] \omega [/tex] on a p-dimensional closed submanifold [tex] \Sigma[/tex] of a larger manifold [tex]M[/tex]. When we integrate [tex]\omega[/tex] we get a number

[tex] Q(\Sigma) =\int _{\Sigma}\omega [/tex] which, in principle, depends on the choice of [tex]\Sigma [/tex] but because [tex] \omega [/tex]is closed, [tex] Q(\Sigma)[/tex] is said to be unchanged by continuous deformations of [tex]\Sigma[/tex]. The converse is supposed to be true as well.

Now, I understand this as a generalization of Gauss' law in electrostatics (three dimensions) but I only remember that for this particular case, the demonstration I saw was purely geometrical. I know that this has to use the fact that [tex] \Sigma [/tex] does not have a boundary (closed submanifold). How exactly is this proved?

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# Integration of closed forms

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