We started by splitting-up the time-dependent region under consideration into infinitely many infinitesimal volumes, and looking at how each infinitesimal volume changes in an infinitesimal time [Transparancy 1].
Of course, a rectangular volume doesn't necessarily stay rectangular with time, so we must calculate its volume using the triple scalar product [Transparancy 2]. This turns out to be quite complicated (the textbook leaves out the details), and so we used Maple to do the calculation. Here is the Maple code which was used:
with(linalg):
v:=vector([v1(x,y,z),v2(x,y,z),v3(x,y,z)]);
v := [v1(x, y, z), v2(x, y, z), v3(x, y, z)]
i:=vector([1,0,0]);
j:=vector([0,1,0]);
k:=vector([0,0,1]);
i := [1, 0, 0]
j := [0, 1, 0]
k := [0, 0, 1]
> A:=evalm(dx*i + map(diff,v,x)*dx*dt);
[ /d \ /d \
A := [dx + dx dt |-- v1(x, y, z)|, dx dt |-- v2(x, y, z)|,
[ \dx / \dx /
/d \]
dx dt |-- v3(x, y, z)|]
\dx /]
B:=evalm(dy*j + map(diff,v,y)*dy*dt);
[ /d \ /d \
B := [dy dt |-- v1(x, y, z)|, dy + dy dt |-- v2(x, y, z)|,
[ \dy / \dy /
/d \]
dy dt |-- v3(x, y, z)|]
\dy /]
C:=evalm(dz*k + map(diff,v,z)*dz*dt);
[ /d \ /d \
C := [dz dt |-- v1(x, y, z)|, dz dt |-- v2(x, y, z)|,
[ \dz / \dz /
/d \]
dz + dz dt |-- v3(x, y, z)|]
\dz /]
Vol:=dotprod(A,crossprod(B,C));
This yields a big mess, which I will leave out, for the sake of
conserving space. Simplifying the mess:
Vol:=simplify(Vol);
This yields an even bigger mess, which again I leave out, for the
sake of conserving space. It has terms involving dt,
dt^2, and dt^3. Noting that dt is an
infinitesimal, we can ignore the dt^2 and dt^3
terms:
Voldt:=subs({dt^2=0,dt^3=0},Vol);
/d \
Voldt := dx dy dz + dx dy dz dt |-- v3(x, y, z)|
\dz /
/d \ /d \
+ dx dy dt |-- v2(x, y, z)| dz + dx dt |-- v1(x, y, z)| dy dz
\dy / \dx /
This can be written quite simply in terms of the divergence of v:
diverge(v,[x,y,z]);
/d \ /d \ /d \
|-- v1(x, y, z)| + |-- v2(x, y, z)| + |-- v3(x, y, z)|
\dx / \dy / \dz /
An expression for the time derivative of the change of volume is
thereby obtained, and the "proof" is completed [Transparancy
3].
(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 70 and 95 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 240 K, which takes a bit longer to download.)
