Computations in continuum mechanics often require that the regular time derivation operator 
is replaced by the substantive derivative operator
This can be seen as follows.
Consider a bug that is moving through a volume where there is some scalar, 
e.g. pressure, that varies with time and position: 
If the bug during the time interval from 
to 
moves from 
to 
then the bug experiences a change  in the scalar value,
(the total differential). If the bug is moving with a velocity
the change in particle position is 
and we may write
where  is the gradient of the scalar field p. So:
If the bug is just moving with the flow, the same formula applies, but now the velocity vector,v, is that of the flow, u.
The last parenthesized expression is the substantive derivative of the scalar pressure.
Since the pressure p in this computation is an arbitrary scalar field, we may abstract it and write the substantive derivative operator as