My question occured reading this paper in the first example of section 6.
Let $f:\Bbb{R}^3 \rightarrow \Bbb{R}^3$, $f(x)=|x|$ a vector field. Now divide $\Bbb{R}^3$ up into disjoint unit cubes centered at the points of $\Bbb{Z}^3$. Put the flow between the adjacent points $u,v \in \Bbb{Z}^3$ to the integral of $\nabla f(x) * (u-v)$ across the common face of the cubes centered at $u,v$.
Now the author says that the magnitd of the flow is of order $\frac{1}{|x|²}$. Why is this?
I have tried to solve the flow integral but there came up the problem that in cartesian coordinates it's hard to deal with the functions, and in polar coordinates it's hard to find a parametrization for the cube faces. Is there an easy way to approximate it?