How can I show that for an open subset $U \subset \mathbb{R}^n $ and $F \in C_c^1(U;\mathbb{R}^n)$ $\int_U \mathbb{div}Fd\mathcal{L}^n=0$ ?
I know that for every function $f \in C^1(\mathbb{R}^n;\mathbb{R})$, if $f$ and $\partial_if$ are integrable, $\int_U \partial_ifd\mathcal{L}^n=0$.
Why do I need that the function $F$ has compact support ?