About the divergence of vector fields

Question

I've been explained that a vector field, when seen as "arrows" in the plane, has 0 divergence when its magnitude doesn't change, i.e. when the "arrows" keep same length. But the following examples puzzle me:

$F(x)=x/|x|$ has always norm 1 but its divergence is not 0

$F(x)=x/|x|^2$ has not constant norm but its divergence is 0

Is there some contradiction or do I have a wrong/incomplete picture?

Answer 1

Divergence has nothing (little?) to do with norms of the vectors.

Think instead of of drawing a closed region $\Omega$ in the plane, and the arrows as measuring the velocity of material flowing through the plane.

The region that you drew encloses some amount of material. As the material flows, that amount changes: in particular, it is common sense that the rate at which the amount of enclosed material changes is equal to the rate at which material is crossing the boundary of the region.

You can think of the divergence at a point as measuring the rate at which the density of material is decreasing at that point. The rate at which the total amount of material in all of $\Omega$ is changing is then

$$\int_\Omega \operatorname{div} v$$

and the rate at which material is passing through the boundary of $\Omega$ is $$\int_{\partial \Omega} v \cdot n$$ and the fact that these two must be equal is exactly the divergence theorem.

Answer 2

Here is a way to visualize the divergence of vector fields, following the answer given by user7530.

1. Launch the excellent applet VfaII by Matthias Kawski.

2. Select the tab DEs/flows

3. Enter your field equation: x/sqrt(x^2+y^2) and y/sqrt(x^2+y^2) for the first field; x/(x^2+y^2) and y/(x^2+y^2) for the second.

4. Dragging the mouse, draw a small rectangle somewhere in the plot, preferably near the origin.

5. Watch the rectangle flow. The shape will become distorted in either case, but the thing to watch is the area. Under the flow of the first field the area grows. Under the flow of the second field it stays the same.

Interpretation: both fields stretch the rectangle in the directions of polar $\theta$-coordinate. But the second field compensates for this stretch by shrinking the rectangle in the radial direction. The first one does not.