Mathematical Tools for Physics

9—Vector Calculus 1 254

How can I get some simpler picture? Do it in the same spirit that you introduce the derivative: Concentrate
on a little piece of the picture. Inject only a little bit of dye and wait only a little time. To make it explicit,
assume that the initial volume of dye forms a sphere of (small) volumeV and let the fluid move for a little time.

1. In a small time∆tthe center of the sphere will move.


2. The sphere can expand or contract, changing its volume.


3. The sphere can rotate.


4. The sphere can distort.

Div, Curl, Strain
The first one, the motion of the center, tells you about the velocity at the center of the sphere. It’s like knowing the
value of a function at a point, and that tells you nothing about the behavior of the function in the neighborhood
of the point.
The second one, the volume, gives new information. You can simply take the time derivativedV/dtto see
if the fluid is expanding or contracting; just check the sign and determine if it’s positive or negative. But how
big is it? That’s not yet in a useful form because the size of this derivative will depend on how much the original
volume is. If you put in twice as much dye, each part of the volume will change and there will be twice as much
rate of change in the total volume. If I divide the time derivative by the volume itself this effect will cancel.
Finally, to get the effect at one point I have to take the limit as the volume approaches a point. This defines a
kind of derivative of the velocity field called the divergence.

lim

V→ 0

1

V

dV

dt

=divergence of~v (5)

This doesn’t tell you how to compute it any more than saying that the derivative is the slope tells you how to
compute an ordinary* derivative. I’ll have to work that out.
But first look at the third way that the sphere can move: rotation. Again, if you take a large object it will
distort a lot and it will be hard to define a single rotation for it. Take a very small sphere instead. The time
derivative of this rotation is its angular velocity, the vector~ω. In the limit as the sphere approaches a point, this

• Can youstartfrom the definition of the derivative as a slope, use it directly, and compute the derivative of
  x^2 with respect tox, getting 2 x?