  The divergence theorem of Gauss plays a central role in the derivation of the
governing differential equations in fluid dynamics, electrodynamics,
gravitational fields, and optics. One is often interested in an evolution
equation for the large scale quantities without resolving the details of the
small scales. As a result, there has been a significant effort in developing
time-averaged and spatially-filtered equations for large scale dynamics from
the fully resolved governing differential equations. One should realize that by
starting from these fully-resolved equations (e.g. the Euler or Navier-Stokes
equations) to derive an averaged evolution equation one has already taken the
limit of the wave-numbers approaching infinity with no regards to our
observational abilities at such a limit. As a result, obtaining the evolution
equations for large scale quantities (low wave-numbers) by an averaging or
filtering process is done after the fact. This could explain many of the
theoretical and computational difficulties with the Euler or Navier-Stokes
equations. Here, a rather different approach is proposed. The averaging process
in implemented before the derivation of the differential form of the transport
equations. A new observable divergence concept is defined based on fluxes
calculated from observable quantities at a desired averaging scale, $\alpha$.
An observable divergence theorem is then proved and applied in the derivation
of the observable and regularized transport equations. We further show that the
application of the observable divergence theorem to incompressible flows
results in a formal derivation of the inviscid Leray turbulence model first
proposed in 1934. It is argued that such a methodology in deriving fluid
evolution equations removes many of the theoretical and computational
difficulties in multi-scale problems such as turbulence and shocks.
