Here's Rudin's full proof of poincare lemma Rudin's proof on Poincare Lemma.
I just have one question, why do we need $E$ to be convex?
2026-03-31 10:39:04.1774953544
Question about Rudin's proof of poincare lemma
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To guarantee that $f$ does not depend on $x_j$ for $j>p$. Being convex, you can put a path along the axis of each variable where the derivative is zero, so $f$ is constant in that variable (that the derivative is zero at a single point would not be enough for this) .