Conservative Fields and Denseness

Question

Let $\Omega\subset\mathbb{R}^3$ be a open domain and $F=(F_1,F_2,F_3):\Omega\rightarrow\mathbb{R}^3$ a continuous field. Suppose that does not exist $u:\Omega\rightarrow\mathbb{R}$ with $\nabla u=F$. Is it possible to find $G=(G_1,G_2,G_3):\Omega\rightarrow\mathbb{R}^3$ close to $F$ in the $L^2$ norm, or in the $\sup$ norm such that $G=\nabla v$ for some $v$.