How do we characterize fluid flow? We could track the positions of
all of the particles in the fluid, by solving the equations of motion
for the particles. While this might be feasible for a small
number of particles, it is clearly absurd for any macroscopic system.
Instead, we seek a reduced description in terms of a small number
of macroscopic variables, which are functions of the fixed position
and the time t, such as the local density
, the local fluid velocity 
,
the local pressure 
, and the local temperature
. We will stick with the more convenient,
and conventional, Eulerian coordinate system, in which
is a coordinate fixed in space. One could also choose
a coordinate system fixed to the particles, the Lagrangian
system.
By throwing away most of the microscopic information, we hope to obtain a simpler, and in most cases, more useful, macroscopic description. However, such a simplification comes with a price: we will often need to introduce phenomenological parameters (such as the viscosity) whose values are not known a priori, but which must be taken from experiment. Exactly how we achieve this reduction from microscopic variables to macroscopic variables is the subject of kinetic theory; the reduced description is often called a continuum theory. Kinetic theory is extremely subtle (and interesting), and vexed the great physicist Boltzmann throughout his career. We won't have much to say about it in this course. However, I should point out that the macroscopic description must eventually break down when we begin probing length scales comparable to microscopic lengths. The relevant microscopic length is the mean free path of the particles in the fluid, which is loosely defined as the average distance a particle moves before colliding with another particle. For example, if we try to understand sound propagation starting from a macroscopic point of view, our analysis will certainly break down once the wavelength of sound is comparable to the mean free path. We see immediately that a continuum theory needs to be used with caution for a very tenuous, or rarefied, gas.