9—Vector Calculus 1 255tells me about the rotation of the fluid in the immediate neighborhood of that point. If I place a tiny paddlewheel
in the fluid, how will it rotate?
2 ~ω=curl of~v (6)
The factor of 2 is for later convenience.
The fourth way that the sphere can change after expansion and rotation is that it can change its shape.
In a very small time interval, the sphere can slightly distort into an ellipsoid. This will lead to the mathematical
concept of thestrain. This is important in the subject of elasticity and viscosity, but I’ll put it aside for now
save for one observation: how much information is needed to describe whatever it is? The sphere changes to
an ellipsoid, and the first question is: what is the longest axis and how much stretch occurs along it — that’s
the three components of a vector. After that what is the shortest axis and how much contraction occurs along
it? That’s one more vector, but you need its magnitude and only one component to define its direction because
it’s perpendicular to the long axis. After this there’s nothing left. The direction of the third axis is determined
and so is its length if you assume that the total volume hasn’t changed. You can assume that is so because the
question of volume change is already handled by the divergence; you don’t need it here too. The total number
of components needed for this object is2 + 3 = 5. It comes under the heading of tensors.
9.3 Computing the divergence
Now how do you calculate these? I’ll start with the simplest, the divergence, and compute the time derivative of
a volume from the velocity field. To do this, go back to the definition of derivative:
dV
dt= lim
∆t→ 0V(t+ ∆t)−V(t)
∆t(7)
~v∆t
Pick an arbitrary surface to start with and see how the volume changes as the fluid moves, carrying the
surface with it. In time∆ta point on the surface will move by a distance~v∆tand it will carry with it a piece
of neighboring area∆A. This area sweeps out a volume. This piece of volume is not∆Atimesv∆tbecause