de Rham cohomology
/ duh-RAHM /
We have two facts that beg to be compared: every exact form is closed, but not every closed form is exact. The gap between them — the closed forms that are not exact — turns out not to be an accident but a measurement of the holes in the space. De Rham cohomology is the bookkeeping that turns 'closed-but-not-exact' into a precise count of those holes, using only calculus on the space.
The construction is a clean piece of algebra. At each degree k, consider the closed k-forms (the ones d kills) and quotient out by the exact k-forms (the ones that are d of something). Because d squared is zero, every exact form is closed, so this quotient makes sense; the result is the k-th de Rham cohomology, and its dimension is a number called the k-th Betti number. If the quotient is trivial (every closed form is exact), the space has no k-dimensional holes. If it is nonzero, each independent leftover closed-but-not-exact form points at a distinct hole. On the punctured plane the angle form (-y dx + x dy)/(x^2 + y^2) is the leftover class in degree 1, and it detects the single hole at the origin — the first Betti number is one.
The astonishing theorem of de Rham is that this object, defined purely by differentiating and integrating smooth forms, computes exactly the same numbers as topology computes by counting holes with triangles and chains. Calculus on a smooth space sees its global shape. This is why a closed form whose loop integral is nonzero is a genuine obstruction you cannot massage away, why electromagnetism on a space with holes has physical consequences (flux quantization, the Aharonov-Bohm effect), and why de Rham cohomology is the bridge from the local language of differential forms to the global language of topology.
On the circle (a 1-dimensional loop), the form d(theta) — the angle differential — is closed but not exact, because theta is not a single-valued function (it jumps by 2 pi each lap). It cannot be written as df for any honest f on the circle, so it represents a nonzero class: the first de Rham cohomology of the circle is one-dimensional, detecting the loop.
The angle form on the circle is closed but not exact; its leftover cohomology class is the loop the circle wraps.
De Rham cohomology measures holes of the domain, not flaws of the form. Calling d(theta) 'closed but not exact' is a statement about the circle's shape; on a contractible piece of the circle the same form does have a potential. The obstruction lives in the space, and the form merely reveals it.