Let $S$ be the cylinder $0\leq z\leq1, x^2+y^2=1$ in $\mathbb{R^3}$ oriented with the outward normal. Let $D_0=${$(x,y,0)|x^2+y^2\leq1$} oriented with the downward normal and $D_1=${$(x,y,1)|x^2+y^2\leq1$} oriented with the upward normal. Let $\omega=xdx\wedge dy+ydx\wedge dz+zdx\wedge dy$. Is there any relationship between $\int_S \omega$, $\int_{D_0} \omega$ and $\int_{D_1} \omega$?
2026-04-08 04:12:51.1775621571
Integral of a 2-form over a cylinder
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