exact form
A form is exact if it is the exterior derivative of some other form: omega is exact when there is an eta with omega = d eta. The form eta is called a potential or primitive for omega. This is the form-language version of having an antiderivative — exact is to a form what 'has a potential function' is to a vector field.
Why care? Because exactness is what makes integration easy and path-independent. If omega = d eta then the generalized Stokes' theorem turns the integral of omega over a region into the integral of eta over the boundary alone, and for a 1-form that is the fundamental theorem of calculus: the integral of df along a curve from A to B is just f(B) - f(A), depending only on the endpoints. An exact 1-form integrates to zero around any closed loop, which is exactly the statement that a force field with a potential does no net work on a round trip — a conservative field. So 'exact' is the form-theoretic name for 'conservative'.
Every exact form is closed (because d squared is zero), so exactness is the stronger condition. Whether a given closed form is actually exact depends on the topology of the domain: on a region with no holes, the Poincare lemma guarantees closed implies exact, and you can build the potential by an explicit integration along radial paths. On a region with holes, some closed forms refuse to be exact, and the count of those stubborn forms is what de Rham cohomology measures. Finding a potential, in practice, is the same antiderivative-hunting you did in first-year calculus, now organized by degree.
The 1-form omega = y dx + x dy is exact: it equals d(xy), since d(xy) = y dx + x dy. So its line integral from (0, 0) to (3, 2) is just xy evaluated at the endpoints, 3*2 - 0 = 6, no matter what path you take. The potential xy plays the role of an antiderivative.
An exact 1-form integrates to a difference of endpoint values — the fundamental theorem of calculus in form language.
A potential, when it exists, is never unique: you can add any closed form to eta without changing d eta. For a 1-form this is the familiar freedom of adding a constant to a potential function; in gauge theory it is the gauge freedom itself.