= Green-theorem proof of exactness on a disc
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On a star-shaped planar domain containing zero, let $\eta$ be a smooth <closed differential one-form> and set $f(x)=\int_{[0,x]}\eta$. <Green theorem> on the triangle with vertices $0,x,x+h$ gives $f(x+h)-f(x)=\int_{[x,x+h]}\eta$. Dividing by a displacement and taking its <limit> proves $df=\eta$. Thus every such form is an <exact differential form>, a degree-one version of the <Poincare lemma> which uses only the planar integral theorem.
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