Electromagnetic Identity - Seems easy, ridiculously hard

Question

$\newcommand{\b}{\textbf}$I'm trying to prove the following identity. Assume we are in a source free region where we have chosen the radiation gauge ($\phi=0$ and $\nabla\cdot \b{A}=0$). Assuming the fields have a finite extension (i.e. die off at infinity), show that

$$\int_{\mathbb{R}^3}\left(\b{E}\cdot\nabla\right)\left(\b{r}\times\b{A}\right) \,d^3\b{r}=\b{0}$$

I know this is true because I am proving a known identity for the electromagnetic angular momentum, written on this wiki page. (Proving the equivalence of those first two definitions). Every step I have taken so far is fine, and I have everything lined up except for that integral above. I feel like I've tried massaging it with pretty much every vector calculus identity, every variant of Stokes's theorem and the divergence theorem, etc. I've tried decomposing it with levi-civita symbols only to go in a circle. I've tried doing gymnastics with feynman subscript notation and even tried proving it indirectly, all without success.

I can easily include my derivation by the way. I've only left it out not to put off any potential helpers.

EDIT: Correction, *"Seems easy, ridiculously hard for my non-insightful self"

Answer 1

It's actually pretty simple. Let's use index notation: the integrand is

$$ E_l \partial_l \left(\varepsilon_{ijk} r_j A_k\right) = \partial_l (E_l \varepsilon_{ijk} r_j A_k) - (\partial_l E_l) \varepsilon_{ijk} r_j A_k $$

The first term is a divergence so it vanishes by Gauss' law (assuming everything goes to zero sufficiently fast), and the last is zero because $\nabla \cdot \mathbf{E} = 0$.