closed and exact forms
These two words sort differential forms by their relationship with the exterior derivative d. A form is closed if d kills it — its exterior derivative is zero. A form is exact if it is itself the d of something — it is a derivative. The physics picture: a closed 1-form is like a force field with zero curl (locally conservative), and an exact 1-form is like one that is genuinely the gradient of a potential. The interesting question is when 'locally conservative' fails to be 'globally a gradient'.
Precisely: a k-form omega is closed if d omega = 0, and exact if omega = d eta for some (k-1)-form eta. Because d^2 = 0, every exact form is automatically closed: d(d eta) = 0. So the exact forms sit inside the closed forms. The whole question of de Rham cohomology is the converse: is every closed form exact? The answer is yes locally (the Poincaré lemma) but can fail globally, and the failure — the quotient (closed forms)/(exact forms) in each degree — is exactly H^k_dR(M). In the language of cochain complexes, closed = cocycle (in the kernel of d), exact = coboundary (in the image of d).
This is the heart of the field. Whether a closed form is exact is a topological question: the form d theta on the punctured plane (the angle form) is closed but not exact precisely because the plane has a hole, and its failure to be exact detects the hole. A standard caution: 'closed' and 'exact' are degree-dependent and global notions. A form can be exact on a small ball but not on the whole manifold; exactness is never a pointwise property, since it asks for a single global primitive eta defined everywhere at once.
On the punctured plane R^2 minus the origin, the 1-form omega = (-y dx + x dy)/(x^2 + y^2) is closed (you can check d omega = 0 directly) but NOT exact: its integral around the unit circle is 2 pi, not zero, whereas every exact form integrates to zero around any loop. Locally it equals d(arctan(y/x)) = d theta, but the angle function theta cannot be defined continuously on the whole punctured plane — the hole obstructs a global primitive. This single form generates H^1_dR(R^2 minus a point) = R.
The angle form is the textbook closed-but-not-exact form; its nonzero loop integral measures the hole it cannot see across.
Closed does not imply exact — that gap is the entire content of cohomology. Conversely, being exact is strictly stronger than being closed. And note d theta is a genuine global 1-form on the punctured plane even though theta is not a global function; do not read 'd of something' as 'globally exact'.