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
.
The difference
, 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
to
in time span of .
Thus the difference is expressed as

where we used the Taylor expansion of . The difference corresponding to the Eulerian derivative is written down as

and this is equal to the second term of the rhs of equation (A.4). Comparing equations (A.4) and (A.5), the Lagrangian derivative contains an extra term besides the Eulerian derivative. That is, the Lagrangian derivative is expressed by the Eulerian derivative as

Applying the above expression on equation of motion based on the Lagrangian derivative (A.3),
we obtain the Eulerian equation motion: