ABSTRACT
Let us recall that the natural notion of mass ow or \ ux" _M() of uid particles across some smooth surface , from west-to-east. The ow forced by a velocity eld x(x; t), imparted to particles located at point x in time t, can be mathematically formulated by an analysis-synthesis two-stage process, resulting in the following denitory ux formula for uids with mass density and surfaces with a continuous eld of unit normals, which have a west-to east-orientation, thus yielding
Fwe() = ux = _M() =
Z
v nwed: (I.1)
The vector eld v acts as a \vector density" for this \ ux" F . There are many vector elds describing physical vector magnitudes, for
which corresponding \ uxes" can be analogously dened. Starting with some planar surface , with a unit normal nwe and oriented
from west to east, consider a uniform vector eld j = Jnwe. The ux Fwe() of j across from west to east is naturally dened as
Fwe() = J(u nwe) Z
d =
Z
(Ju) nwed; (I.2)
and the corresponding vector ux density will be j = Ju. When the eld j is nonuniform, both in size and direction, and is a smooth
orientable surface, then the consideration of as the limit of polyhedral surfaces with increasingly small plane surface elements, lead us to dene the ux as:
Fwe() =
Z
j nwe d: (I.3)
