= Commuting velocity derivative for kinetic transport
{title2=$Z_j=\partial_{v_j}+t\sum_i(\partial_{v_j}a_i)\partial_{x_i}$}
The differential operators $Z_j$ commute with $D=\partial_t+a(v)\cdot\nabla_x$. Indeed $[D,\partial_{v_j}]=-\sum_i(\partial_{v_j}a_i)\partial_{x_i}$, canceled by differentiating the explicit time factor in $Z_j$. Hence $Z_jf$ satisfies the same <transport equation> as $f$, and obeys the same <L2 norm> conservation when boundary fluxes vanish. For the standard <Jacobian matrix> $(Da)_{ij}=\partial_{v_j}a_i$, the vector of these derivatives is $\nabla_vf+t(Da)^T\nabla_xf$.
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