Divergence in $\mathbb{R}^n$

Question

I have a homework problem and I don't have any idea how to solve it. I want to use the divergence theorem, but I am working in $\mathbb{R}^n$ that is not bounded. So, I don't think I can use it. The problem states the following:

Let $F$ be a $C^1$ field in $\mathbb{R^n}$ with compact support. Prove that $$ \int_{\mathbb{R^n}} \mbox{div} F \, dx = 0$$

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Thanks in advace for any advice.

Edit: I also know that if $F=(f_1,\dots,f_n)$ has compact support and the partials derivatives $(f_i)_{x_j}$ exists, then each $(f_i)_{x_j}$ has also compact support.

Answer 1

Use the fact that $F$ has compact support. Let $U\subset \mathbb{R}^n$ be such that $\operatorname{supp}F \subsetneq U$ (such a $U$ exists since $\operatorname{supp} F$ is compact, hence bounded). Then $F$ is zero on the boundary of $U$, as are it's derivatives. Futhermore, $F$ is zero outside of $U$, so we can restrict integration from $\mathbb{R}^n$ to $U$. Now apply the divergence theorem:

$$\int\limits_{\mathbb{R}^n} \nabla F = \int\limits_U \nabla F = \int \limits_{\partial U} F \cdot n = 0$$