It can be found: $$F_l(r) = -\dfrac{1}{4\,\pi}\nabla\int_V \dfrac{\nabla'\cdot F(r')}{|r-r'|}\,\mathrm{d^3}r' + \dfrac{1}{4\,\pi}\nabla\int_{S}\dfrac{F(r')}{|r-r'|}\cdot \mathrm{\hat{n}\,d^2}r'$$
which is the longitudinal component of the Helmholtz decomposition
Now why the second term reduces to $\mathbf{0}$ if $F(r)$ is regular at infinity as well as the surface sprawls to infinity.
I can only image the numerator will remain constant but the denominator will go down to $\mathbf{0}$