Closed one-forms on the two-sphere are exact
= Closed one-forms on the two-sphere are exact
{title2=$H^1_{\mathrm{dR}}(S^2)=0$}
Remove the north pole and south pole separately. Each resulting chart is diffeomorphic to the plane, so the <Poincaré lemma> supplies local primitives of a closed one-form. Their difference has zero derivative on the connected overlap and is constant. Adjust one primitive by that constant and glue. This proves exactness without assuming a prior computation of <de Rham cohomology>.