closed form

A differential form is called closed when its exterior derivative vanishes: d omega = 0. The name is a relative of 'closed loop' — nothing is changing as you apply d, so the form has no internal twist or source that d can detect. For a 1-form in the plane, P dx + Q dy is closed exactly when Q_x = P_y, the equal-mixed-partials condition you may remember as the test for a conservative field.

Closed is the weaker, more local cousin of exact. Every exact form is automatically closed, because d squared equals zero: if omega = d eta then d omega = d(d eta) = 0. The reverse — is every closed form exact? — is the interesting question, and the answer depends on the shape of the space. Locally (on any small ball, or any region with no holes) the Poincare lemma says yes: closed implies exact. Globally it can fail, and that failure is precisely how forms detect the topology of the region.

The classic cautionary tale is omega = (-y dx + x dy)/(x^2 + y^2) on the plane minus the origin. A direct computation shows d omega = 0, so omega is closed; yet its integral around a loop circling the origin is 2 pi, not zero, which an exact form could never produce. So omega is closed but not exact. That single mismatch is the seed of de Rham cohomology, and in physics it is exactly the angle differential whose nonzero loop integral signals the hole at the origin — the same structure behind winding numbers, the Aharonov-Bohm phase, and circulation that cannot be undone.