What is closed form? — Mathematics

Definition 21.13 University Mathematics — Year 3 · Chapter 21 — Differential Forms and Stokes’ Theorem

$\omega \in \Omega^k(U)$ is closed if $\dd\omega = 0$ , exact if $\omega = \dd\eta$ for some $\eta \in \Omega^{k-1}(U)$ (a primitive of $\omega$ ). Exact $\Rightarrow$ closed by $\dd^2 = 0$ ; the converse is a question about the shape of $U$ .

Examples

Example 21.14 (The angular form)

On $U = \R^2 \setminus \{0\}$ ,

\omega_\theta = \frac{x\,\dd y - y\,\dd x}{x^2 + y^2}

is closed (direct computation: Exercise 21.4) but not exact: its integral along the unit circle is $2\pi \neq 0$ , while integrals of exact forms along closed curves vanish (Proposition 21.28). Locally, $\omega_\theta = \dd\theta$ for any smooth determination $\theta$ of the polar angle — whence the name and the obstruction: no such determination exists on all of $U$ . This single form runs the winding number (Section 21.6) and, through it, the residue theorem of Chapter 17.

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