Can every differentiable scalar function be written as a divergence of some vector field?

Question

My question is simple: can every differentiable function $f$ defined on a bounded, connected subset of $\mathbb{R}^3$ be written as a divergence of some vector field ? That is, given the vector field $\mathbf{F}$, you can write:$$f=\nabla\cdot\mathbf{F}$$ Is this always possible? How to prove it?

Answer 1

$$f(x,y,z)=\left(\frac{1}{3}\int_0^xf(x,y,z)dx\right)^{'}_x+\left(\frac{1}{3}\int_0^yf(x,y,z)dy\right)^{'}_y+\left(\frac{1}{3}\int_0^zf(x,y,z)dz\right)^{'}_z.$$

Answer 2

Sure. For example, supposing the intersection of the domain with every vertical line is an interval containing zero, you could just take

$$ \mathbf F(x,y,z) = (0, 0, \int_0^z f(x,y,t)\,dt) $$

For a domain of a more complex shape it could be more tedious to patch a solution together from smaller ones, but the existence of an $\mathbf F$ isn't really at risk.