= Stellar velocity moment
{title2=$\overline A=\nu^{-1}\int A(\mathbf v)f\,d^3v$}
For a <galactic distribution function> $f(\mathbf x,\mathbf v)$ with tracer <number density> $\nu=\int f\,d^3v$, a velocity moment is the weighted <integral> of a velocity monomial. The first moment gives the mean <velocity> $\overline v_i$; the second moment satisfies $\overline{v_iv_j}=\overline v_i\overline v_j+\sigma_{ij}^2$, separating ordered motion from the random <velocity dispersion> tensor. Multiplying the <Collisionless Boltzmann equation> by $v_i$ and integrating yields the <Jeans equation>, with vanishing velocity-space boundary terms when the distribution decays sufficiently rapidly.
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