Closed and exact forms

Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $\omega$ be a differential k-form on $U$ with $C^1$ coefficients, where $k \ge 1$. The form $\omega$ is closed if $d\omega = 0$, where $d$ is the exterior derivative. The form $\omega$ is exact if $\omega = d\eta$ for some $(k - 1)$-form $\eta$ on $U$.

Remarks

Each exact form $d\eta$ with $\eta$ of class $C^2$ is closed, because d d = 0. A one-form $\sum_i f_i \, dx_i$ is closed if and only if $\partial_j f_i = \partial_i f_j$ for all $i, j$. It is exact if and only if $(f_1, \dots, f_n)$ is a conservative field. In $\mathbb{R}^3$, closed means that the curl is zero. The angle form $d\theta = \frac{-y \, dx + x \, dy}{x^2 + y^2}$ on $\mathbb{R}^2 \setminus \{0\}$ is closed but not exact. Its integral around the unit circle is $2\pi$, not $0$. On a convex open set, each closed form is exact. This is the Poincaré lemma.

Explanations

L1 human

Exact forms are derivatives of other forms, and closed forms only pass the local test for this: their derivative is zero. The gap between the two notions is a measure of the holes of the domain.