What is Exact and closed forms?

Example 20.7 (Reconstructing a potential)

Let $\omega = y\,\eu^{xy}\,\dd x + (x\,\eu^{xy} + 2y)\,\dd y$ on $\R^2$. It is closed: both cross derivatives equal $\eu^{xy}(1 + xy)$. To find a potential, integrate $P$ in $x$ at fixed $y$:

then adjust $c$ by matching $f_y$: $x\,\eu^{xy} + c'(y) = x\,\eu^{xy} + 2y$ gives $c(y) = y^2$. So $f(x,y) = \eu^{xy} + y^2$, and for any piecewise $\mathcal C^1$ arc from $(0,0)$ to $(1,1)$,

independently of the path — the two-step recipe (integrate in $x$, correct in $y$) is the practical converse of Theorem 20.6 on domains where closed forms are exact.

Example 20.8 (Closed does not imply exact)

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

is closed (direct computation: both $P_y$ and $Q_x$ equal $\frac{y^2 - x^2}{(x^2+y^2)^2}$), but its integral along the unit circle is $2\pi \neq 0$ (same computation as Example 20.3, divided by $1$): $\omega$ is not exact on $U$. Locally, $\omega = \dd\theta$ for a determination $\theta$ of the polar angle; the failure is global — the angle cannot be defined continuously around the puncture. On domains without holes the pathology disappears: on a star-shaped open set, every closed $\mathcal{C}^1$ form is exact (Poincaré’s lemma, Exercise 20.8).