First of all, I am not sure of the terminology here, I am interested in the function $$F(x)=x|x|^{-d},x\in \mathbb{R}^d\setminus \{0\}$$ in dimension $d\geq 2$. I read somewhere that this is called the gravitational force (and I agree in dimension 3, but I am not sure for other dimensions), and elsewhere that this comes from the Coulomb potential.
Anyway, the divergence of this force is $-c_d\delta_0$ in the distributional sense (for some constant $c_d$), i.e. for any sufficiently smooth domain $D\subset \mathbb{R}^d$ not having $0$ in the boudnary and with normal vector $n_D$ on the boundary, $$ \int_{\partial D}n_D(x)\cdot F(x)S(dx)=-c_d1_{0\in D}. $$
First question: does anyone knows a good keyword to look upon this function and this kind of facts in the literature? Or a textbook?
Second question: consider now the force of mass spread uniformly over some smooth set $A\subset \mathbb{R}^d$: $$ F_A(x)=\int F(y-x)1_A(y)dy. $$ I'm pretty sure this is well defined, at least outside $\partial A$. Has the divergence of $F_A$ also a good interpretation in the distributional sense? Ultimately I would like to replace $1_A(y)dy$ by some finite measure $\mu(dy)$...