Why the volume form in Sphere is locally exact but not globally exact? here the integral is integral $$\int_{S^n}w$$ with $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots dx_{n+1}. $$
2026-04-15 12:58:48.1776257928
Locally exact vs globally exact
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Well, if it were globally exact, because $\partial S^n = \emptyset$, we'd have to have $\int_{S^n}\omega = 0$ by Stokes's Theorem.