In this question, asked by @Emilio Pisanty, he says that
"...the polarization can be split into a curl-free component, which is the gradient of something, and a divergence-free component, which is the curl of something else"
I wonder whether this is true for any vector field or is it some peculiarity about the electric field. I suspect the former. If yes, what is a proof?
The result is true for any twice-continuously-differentiable vector field on a bounded domain in $\mathbb{R}^3$, or for any twice-continuously-differentiable vector field on an unbounded domain in $\mathbb{R}^3$ that vanishes faster than $1/r$ at infinity. This is known as Helmholtz's theorem (the process itself is called Helmholtz decomposition), or the fundamental theorem of vector calculus. Proofs of this are readily available, for example, here: https://en.wikipedia.org/wiki/Helmholtz_decomposition
