On the existence of distributional potentials

We present proofs for the existence of distributional potentials $F\in{\mathcal D}'(\Omega)$ for distributional vector fields $G\in{\mathcal D}'(\Omega)^n$, i.e. $\operatorname{grad} F=G$, where $\Omega$ is an open subset of ${\mathbb R}^n$. A key ingredient of our treatment is the use of the Bogovskii formula, assigning vector fields $v\in{\mathcal D}(\Omega)^n$ with $\operatorname{div} v=\varphi$ to functions $\varphi\in{\mathcal D}(\Omega)$ with $\int \varphi(x)\,\mathrm{d}x=0$.