With moving grids the time derivatives must also be transformed. Assume a
time dependent function 
in the time dependent domain 
. Then, for the time derivative 
in
domain 
we get (3.3-56) from the chain rule.
Here, the time derivative on the left side is at a
fixed position 
in the computational space, i.e. at a given grid
point. The time derivative at the right side is at a
fixed position 
in the physical space, i.e. the that appears in the governing physical equations. The quantity
is the speed of the grid points.
The last term in (3.3-56) resembles a convective term and
accounts for the motion of the grid. Hereafter we will use 
for the
speed of the grid points.
With the time derivatives transformed, only time derivatives at fixed points in the computational domain will appear in the equations. Therefore, all computations can be done on the fixed grid in the domain without interpolation even though the grid points are in motion in the physical space.
The expression (3.3-56) for the transformation of the time
derivatives was obtained by a chain rule expansion. To get a conservative
form for the transformed time derivatives we apply the divergence theorem to
an area element 
at time 
and to the same element at time
, 
. Then making up a balance equation and proceeding to the
limit 
, we obtain (3.3-57).
The structure of (3.3-57) is easily grasped. Since 
is a
measure for the area, the first term 
covers
the change of the amount of quantity 
within a cell, whereas the second
term covers the flux 
through the moving cell boundaries.
Again, it is straightforward to proof that (3.3-56) and (3.3-57) are mathematically identical.
