The time derivative appearing in equation (A.1) expresses how the velocity of a specific particle changes. Therefore, what appears in equation (A.3) means the same, that is, the position of the gas element concerning in equation (A.3) moves and the positions at and are generally different. However, considering the velocity field in the space, the time derivative of the velocity should be calculated staying at a fixed point .
These two time derivative are different each other and should be distinguished.
The former time derivative is called Lagrangian time derivative and
is expressed using .
On the other hand, the latter is called Eulerian time derivative and
is expressed using
These two are related with each other.
Consider a function whose independent variables are time and
position , that is
, using the Lagrangian time derivative of ,
represents the the difference of from focusing on a specific
fluid element, whose positions are different owing to its motion.
The element at the position of
at the epoch moves
in time span of .
Thus the difference is expressed as
Applying the above expression on equation of motion based on the Lagrangian derivative (A.3),
we obtain the Eulerian equation motion: