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redrzewski
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I'm looking at prop 19.5 of Taylor's PDE book.
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
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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