the Poincaré lemma
/ pwan-kah-RAY /
The Poincaré lemma is the precise statement of the slogan 'closed means locally exact'. It says that if a differential form has zero exterior derivative (it is closed), then on any small enough, nicely shaped patch you can always find a form whose derivative it is (it is exact there). The whole subtlety of cohomology is that 'locally' cannot in general be upgraded to 'globally' — but locally, there is never an obstruction.
Precisely: on a contractible open set U (for instance any ball, or any star-shaped region in R^n), every closed k-form with k >= 1 is exact. Equivalently, for k >= 1 the de Rham cohomology H^k_dR(U) = 0 when U is contractible. The proof is constructive: one builds an explicit homotopy operator h such that omega = d(h omega) + h(d omega); when d omega = 0 this gives omega = d(h omega), so h omega is the desired primitive. The operator h is an integration along the contracting radial lines, assembled using the interior product, and it is exactly the form-level shadow of the contraction of U to a point.
This lemma is the local foundation of de Rham theory: it guarantees that the only thing a closed form can detect is global topology, because locally it is always exact and hence invisible. It is what makes the de Rham complex a fine resolution and powers the comparison with other cohomology theories. A careful restriction: the lemma fails in degree 0 (a closed 0-form is a locally constant function, which is 'exact' only in the trivial sense), and crucially it needs the domain to be contractible — on an annulus or a punctured plane there ARE closed forms that are not exact, and that failure is exactly the point where cohomology is born.
On R^2 (which is contractible) every closed 1-form is exact. Take omega = (2xy + 1) dx + x^2 dy; check it is closed: d omega = (2x dx^dy) wait compute partial_x(x^2) - partial_y(2xy + 1) = 2x - 2x = 0, so d omega = 0. The lemma promises a potential f; integrating, f = x^2 y + x works, and indeed df = (2xy + 1) dx + x^2 dy = omega. By contrast, on R^2 minus the origin (not contractible) the closed form d theta has no global potential — same local recipe, but the hole blocks assembling one primitive over the whole space.
On any ball the local potential always exists; the obstruction to a global one is exactly the topology a hole introduces.
The lemma is local (or about contractible domains); it is NOT a claim that closed implies exact globally — believing that is the single most common error here. The same form d theta is closed everywhere but exact only on simply-shaped pieces, never on the full punctured plane.