closed form কী?

Example 21.14 (The angular form)

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

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.