I have a doubt about divergence theorem in particolar about the hipothesis on the vector field.
Divergence theorem If $\Omega \subset \mathbb{R}^N$, open and bounded, with $C^1$ boundary and $F \in C^1(\bar{\Omega},\mathbb{R})$ then $$\int_{\Omega} div(F)=\int_{\partial \Omega} <F,\nu>dH_N$$
But in some versions of the theorem I saw the hypothesis $F \in C^1_c(\mathbb{R}^N)$, where $C^1_c(\mathbb{R}^N)$ is the space of continous and differentiable function with compact support.
Question: does $F \in C^1(\bar{\Omega},\mathbb{R})$, whit $\Omega$ open and bounded implies $F \in C^1_c(\mathbb{R}^N)$?
My attempt: By definition: \begin{align*} supp(f)&:=\overline{\left \{ x \in \Omega: F(x) \neq 0 \right \}} \\ &=\overline{\Omega \setminus \left \{ x \in \Omega: F(x)=0 \right \}}\\&=\overline{\Omega \setminus F^{-1}(\left\{0 \right \})} \end{align*} But $\Omega \setminus F^{-1}(\left\{0 \right \})$ is a subset of $\Omega$, and hence is bounded, since $\Omega$ is bounded; so $supp(f)=\overline{\Omega \setminus F^{-1}(\left\{0 \right \})}$ is a closed and bounded hence compact.
Is it correct? Thanks