What is the most compact representation of the general solution to $$ \text{div}(f)=\frac{\partial f_1}{\partial x_1}+\frac{\partial f_2}{\partial x_2}+\frac{\partial f_3}{\partial x_3}=0 $$ for a $f:\mathbb{R}^3\to\mathbb{R}$ that goes to zero exponentially as $|\vec{x}|\to\infty$.
My first approach was the Helmholtz theorem with a potential that satisfies the Laplace equation. Edit: Using the Helmholtz theorem, $f=-\nabla \Phi + \nabla\times f$, one can write $$ \nabla\cdot f = -\nabla^2 \Phi + \nabla\cdot \nabla\times f=-\Delta \Phi =0 $$
The Poincar'e Lemma, as in Spivak's Calculus on Manifolds, includes the fact that for smooth functions ( $x$ and $f(x)$ in $R^3$, domain of $f$ a star-shaped set) you have $$ f(x) = {\rm curl\,}\Big( \int_0^1 f(tx)\times tx\,dt \Big) +\int_0^1 t^2x{\rm \, div\,}f(tx)\,dt. $$ (I have changed the notation to express it for vector fields; Spivak has it for differential forms.)
When the divergence of $f$ is zero in a star-shaped set there are many functions $g$ such that $f = {\rm curl\,} g$; the first integral above is one of them.
It looks like we can't guarantee the exponential decay: Suppose $|f(x)| \le ce^{-|x|}$. Then $$ \bigg| \int_0^1 f(tx)\times tx\,dt \bigg| \le \int_0^1 ce^{-t|x|}t|x|\,dt $$ $$ =\bigg[ -c\big(t+\frac{1}{|x|}\big)e^{-t|x|} \bigg]_0^1 = c\Big(-\big(1+\frac{1}{|x|}\big)e^{-|x|}+\frac{1}{|x|} \Big). $$