Non-zero divergence of a vector field

Question

It is known that if a vector field is divergence free, thus for sure it is the curl of a suitable vector field.

My question is: if a vector field is not divergence free, one can aspect that it is anyway curl of a vector field?

Thank you.

Answer 1

The answer to your question is a trivial no, simply by taking the contrapositive of the following obvious theorem, which I state in excruciating detail so there is no ambiguity:

Let $U\subset\Bbb{R}^3$ be an open set, $G:U\to\Bbb{R}^3$ be a twice Frechet-differentiable vector field, and define $F:U\to\Bbb{R}^3$, $F:=\text{curl}(G)$. Then, $\text{div}(F):U\to\Bbb{R}$ is the zero function.

The proof is by equality of mixed partials (which holds due to twice Frechet-differentiability of $G$).