Velocity moment (source code)

= Velocity moment
{title2=$M_{i_1\cdots i_r}=\int v_{i_1}\cdots v_{i_r}f\,d^3v$}

Integrating a <phase-space distribution function> against products of particle <velocity> components produces its velocity moments. The zeroth gives number density, the first gives number flux, and, after multiplying by particle <mass>, the second gives <momentum flux>. Replacing $\mathbf v$ by $\mathbf v-\mathbf u$, where $\mathbf u$ is the <mean velocity>, defines central velocity moments; the second is the <pressure tensor> divided by particle <mass>.