 $\psi \in { C^0 _c ({{{\bf}R}}^d \times r { {\bf}{S}})}{}$ $$\begin{aligned} { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \psi (x, \omega) { \left \langle { \mathrm{div}_x}(fv) \right \rangle }} & = \int _{v \neq 0} { \psi \left ( x, r \frac{v}{|v|}\right )}{ \mathrm{div}_x}