Microscopic local conservation laws for classical fluids

The local densities and current densities of conserved quantities are expressed in a tutorial scope, for a fluid under arbitrary off-equilibrium conditions, in terms of the two-particle potential W(r) and of the one- and two-particle densities in phase space f and f2. When f and f2 vary significantly over the range of W(r), the density and current density of energy are not defined in a unique fashion, so that conservation of energy can be implemented locally in many different ways. Owing to Galilean invariance, the stress tensor and the heat flux are defined even far from the hydrodynamic regime.