Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be an integrable function. Then, I wonder if there exists any integrable $F : \mathbb{R}^n \to \mathbb{R}$ such that \begin{equation} \nabla F = f \end{equation} where $\nabla F$ is understood as the weak gradient on $\mathbb{R}^n$.
For $n=1$, it seems quite straightforward by integration on the real line. However, I do not see how I can extend to $n>1$. Could anyone help me?