This is the first part of the proof that my differential forms textbook (Differential Forms by Guillemin and Haine) gives for the Poincaré Lemma for Rectangles. Everything up to the last math line $\left(\int \frac{\partial f_i}{\partial x_i}dx_i\right)$ I understand. However, I have a few questions on it. First, why does it become $f(x)$ and not $f_i(x)$? This could be a typo but I want to make sure I'm not misunderstanding something. Second, why is $f_i$ supported on $U$? Nowhere before this point have I seen mention of what $U$ is. Finally, and this is the more important question, the idea is that since $\mbox{supp}(\mu) \subset \mbox{int}(Q)$ then the end points $a_i, b_i \notin \mbox{supp}(\mu)$ and thus $f_i(a_i) = f_i(b_i) = 0$? This isn't explained super well so I wanted to make sure. I would appreciate any help!