If $\varphi\in \mathcal{C}^1_c(A)$ and $G\in \mathcal{C}^1(A,\mathbb{R}^n)$, then $\int_A \text{div} (\varphi G(x))\ dx=0$. ($K$ is compact support)
Proof:
$I=\int_B \text{div} (\varphi \ G(x))\ dx=\int_{\partial B}\langle\varphi G(x),N_{\partial B}\rangle \ dH^{n-1}=\int_{\partial B}\varphi\langle G(x),N_{\partial B}\rangle \ dH^{n-1}=0$
I don't understand why we need a transition from Lebesgue measure to the Hausdorff one. Could you please explain? Thanks