I've seen in many references the statement that "a volume form on a compact manifold is not exact". This is often proved from the Stokes theorem as seen for instance in this question: Volume form on a compact manifold is not exact.
However, this proof supposes the manifold has empty boundary and I think this fails for manifolds with boundary. Consider the closed disk in $\mathbb{R}$, $$D = \{(x,y) \in \mathbb{R}^2 \,:\, x^2 + y^2 \leq 1\}.$$ This is a compact manifold with boundary, namely the unit circle $\mathbb{S}^1$. Any volume form for $\mathbb{R}^2$ can be restricted to a volume form on the disk. For example, $\omega = dx \wedge dy$ can be restricted to a volume form on the disk and $\omega$, being a 2-form on $\mathbb{R}^2$, is exact (the 1-form $\alpha = \frac{1}{2}(x dy - y dx)$ has $\omega$ as its exterior derivative). I fail to see why $\omega|_{D}$ is not exact as well.
By this token, shouldn't the statement be "a volume form on a compact manifold without boundary is not exact"?