} \varphi \left ( x, r{ \frac{v}{|v|}}\right ) f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t \\ & = \int _{{{{\bf}R}}_+} \eta ^{\;\prime} (t) \int _{|v| = r} \varphi \left ( x, r{ \frac{v}{|v|}}\right ) f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t \\ & = \int _{{{{\bf}R}}_+} \eta ^{\;\prime} (t) \int _{|v| = r} \varphi \left ( x, 