Lecture51 : Applications of Divergence theorem [Section 51.2]
where is the volume of the sphere If we let in the above equation, as , we have
(79)
Note that, since and are independent of the coordinate system, and the surface integral is a limit of approximating sums, is independent of the coordinate system.
51.2.7
Physical interpretation of divergence:
Recall that, the integral
gives the total mass of the fluid that flows across a surface per unit time, where being the density and the velocity of the fluid. We can also interpret it as the total mass of the fluid that flows from inside of to outside if is the outward unit normal. Thus
is the volume of the sphere 
If we let 
in the above equation, as 
, we have 
(79) 
and 
are independent of the coordinate system, and the surface integral is a limit of approximating sums, 
is independent of the coordinate system. 
per unit time, where 
being the density and 
the velocity of the fluid. We can also interpret it as the total mass of the fluid that flows from inside of 
to outside 
if 
is the outward unit normal. Thus 
