The starting point for both theories are the microscopic conservation laws that are obeyed by the particles under observation. These are typically conservation of mass, charge, momentum and energy. For each microscopic conservation law, there is a corresponding macroscopic continuity equation, which basically states that within a volume of a fluid, the influx of each conserved quantity must be balanced by a corresponding outflow of that quantity. By taking these fluid volumes sufficiently small that the macroscopic variables are constant across the volume, yet still large enough for the atomic nature not to manifest itself (coarse graining), these continuity equations may be cast in differential form.
For example, the mass continuity equation is
where 
is the mass density, 
the streaming
velocity of the fluid, S the source term, and t is time
[de Groot and Mazur
[1962]]
.
Similarly, the momentum continuity
equation may be written:
where 
is the pressure tensor
, 
is the acceleration of the fluid particles due to an
external field and 
is the total derivative given by
.
