  Using an interpolant form for the gradient of a function of position, we
write an integral version of the conservation equations for a fluid. In the
appropriate limit, these become the usual conservation laws of mass, momentum
and energy. We also discuss the special cases of the Navier-Stokes equations
for viscous flow and the Fourier law for thermal conduction in the presence of
hydrodynamic fluctuations. By means of a discretization procedure, we show how
these equations can give rise to the so-called "particle dynamics" of Smoothed
Particle Hydrodynamics and Dissipative Particle Dynamics.
