Poincare lemma
/ pwan-kah-RAY /
The Poincare lemma is the theorem that rescues the dream that 'closed implies exact'. We know every exact form is closed; the lemma says that locally the converse also holds. On any region that is contractible — one you can shrink continuously to a single point, like a ball, a star-shaped region, or all of R^n — every closed form is exact. So in such a region, d omega = 0 guarantees you can find an eta with omega = d eta.
It is most familiar in disguise. For a 1-form in the plane, the Poincare lemma says that if Q_x = P_y on a simply connected region, then P dx + Q dy has a potential function f with df = omega — the standard fact that a curl-free field on a nice domain is a gradient. The lemma also tells you how to build the potential: integrate the form along straight rays from the contraction point. This explicit homotopy formula is constructive, so the lemma is not just an existence statement; it hands you the antiderivative.
The crucial fine print is the word 'locally', or equivalently 'on a contractible region'. The lemma fails the moment the domain has a hole, and that failure is the whole content of de Rham cohomology. The angle form (-y dx + x dy)/(x^2 + y^2) is closed everywhere on the punctured plane, yet has no global potential, because the punctured plane is not contractible. So the Poincare lemma is best read as a precise statement of when topology does not get in the way — and a flag for exactly when it does.
On all of R^2 take omega = (3x^2 + y) dx + (x + 2y) dy. Check closed: P_y = 1 = Q_x, so d omega = 0. The plane is contractible, so the Poincare lemma promises a potential. Integrate: f = x^3 + xy + y^2 works, since df = (3x^2 + y) dx + (x + 2y) dy = omega.
On a contractible domain, passing the closedness test guarantees a potential — and you can build it by integrating.
The hypothesis is contractibility, which is stronger than the domain merely being connected. For 1-forms the working condition is usually stated as simply connected (no loops that fail to shrink); for higher-degree forms you need higher contractibility, and a domain can be simply connected yet still trap a 2-form.