I'm looking at prop 19.5 of Taylor's PDE book.(adsbygoogle = window.adsbygoogle || []).push({});

The theorem is:

If M is a compact, connected, oriented manifold of dimension n, and a is an n-form, then a=dB where B is an n-1 form iff the ∫a over M is 0.

I'm trying to understand why a=dB implies ∫a = 0.

If M has no boundary, than this follows from Stokes theorem.

However, if M has a boundary, then it seems like this is a counterexample:

a = dx^dy, B=xdy, M=unit square in R^2

Here, ∫a = 1, and a=dB.

The general definitions of compact manifold I've found don't assume no boundary.

What am I missing?

thanks

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# Compact manifold question

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