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Why is Stokes theorem not applicable / not used correctly in this case?

The expression in a chart for a differential 1-form (using Einstein conv.) looks like: $$\omega = \omega_idx^i$$ $$\implies d\omega = d\omega_i \wedge dx^i = \partial_j\omega_idx^j\wedge dx^i.$$ For $...
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Integral of a volume form and stokes theorem

It turns out there are many possible choices of $g$ and $M$. Here's a way to construct one with $g=1$ using a partition of unity. For convenience, I'll use $n$ for the dimension of $\partial A$. It ...
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