If the vector 
is defined at each point x, y, z in a
given region, then we say that a field of exists. An
example is the field of gravity or the field of magnetic force on
the earth. Another example is the mean field of fluid velocity
in the ocean. In order to uniquely determine the field
of the vector , we must specify three functions of
space:
Nominally, we can state
which implies three functions of space.
The divergence of such a vector field (Div ) is
defined as
the term on the right being an abbreviation of
At this stage the quantity is purely a methematical
definition devoid of physical menaing.
In Art. 1.07 the divergence of the velocity
field is discussed; when so applied the divergence takes on a definite
physical significance. It should be noted at this point, however,
that Div is a scalar.