text: The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy. For a fluid consisting of only one kind of particle, the number density  is given by
  The average value of any function  is
  Since the conservation equations involve tensors, the Einstein summation convention will be used where repeated indices in a product indicate summation over those indices. Thus  and , where  is the particle velocity vector. Define  as some function of momentum  only, which is conserved in a collision. Assume also that the force  is a function of position only, and that f is zero for . Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as
  where the last term is zero, since  is conserved in a collision. The values of  correspond to moments of velocity  (and momentum , as they are linearly dependent).