 \frac{r}{|v|} { \left ( I - \frac{v \otimes v}{|v|^0} \right ) }a \cdot (\nabla _v \varphi ) \left ( x, r{ \frac{v}{|v|}}\right ) f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t \\ & = { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\