= Solution
Let $n(\mathbf x)=\int f\,d^3v$ and $\overline A=n^{-1}\int Af\,d^3v$. Assume the <one-particle distribution function> and its weighted tails decay enough that all required <velocity> integrals converge and boundary terms at infinite <velocity> vanish. Multiply the steady <Collisionless Boltzmann equation> by $v_j$ and integrate over <velocity>. In <Cartesian coordinates>, spatial differentiation commutes with this integration, while <integration by parts> gives
$$
\int v_j\frac{\partial f}{\partial v_i}d^3v=-\delta_{ij}n.
$$
Consequently the three force-balance equations are
$$
\boxed{\sum_{i=1}^3\frac{\partial}{\partial x_i}\left(n\overline{v_iv_j}\right)=-n\frac{\partial\Phi}{\partial x_j},\qquad j=1,2,3.}
$$
Explicitly the $x$ component is $\partial_x(n\overline{v_x^2})+\partial_y(n\overline{v_xv_y})+\partial_z(n\overline{v_xv_z})=-n\Phi_x$, with cyclic counterparts for $y,z$. For equal stellar <masses>, multiplying by the common <mass> replaces $n$ by the <mass density> $\rho$; for multiple stellar <masses>, use a mass-weighted <one-particle distribution function> or derive a separate equation for each tracer population.
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