What is the meaning of evaluating the divergence at a _point_?

Solution 1:

Divergence doesn't need a direction because it's the average outflow of bananas in every direction around a point $p$. You draw a closed curve around $p$, add up the total number of bananas passing through the curve (positive for outgoing bananas, negative for incoming) and then divide by the area enclosed by the curve. And then the divergence is what happens to that quantity when the curve is very small, enclosing $p$ and not much else.

It takes into account flows into and out of $p$ from all directions, and averages them together.

You can see this from (one possible) definition of the divergence:

$$\mathrm{div}\ {\bf F} = \lim_{S\to\{p\}}{1\over V(S)}\oint_S {{\bf F}\cdot d{\bf n}}$$

Notice the $V(S)$ in the denominator, where we're dividing by the area enclosed by the curve $S$; the integral adds up the total net banana flow, and dividing the total net flow by $V(S)$ finds the average net flow per unit area. (If we were working in three dimensions instead of two, $V(S)$ would be the volume enclosed by the surface $S$.)

${\bf F}\cdot d{\bf n}$ projects each bit of the total flow onto a small normal vector, so that we only add up the contribution of the flow inwards or outwards, and disregard the part of the flow that is parallel to the curve $S$.

Solution 2:

Consider the vector field $x^{\prime}=2xy-6x^2$ and $y^{\prime}=x^2-2y$. Divergence at the origin is $-2$ and it is not a sink. SO, divergence can REALLY only be interpreted as an average. BUT, I totally agree that some books are very lazy with this detail and actually call equilibrium points as source or sink depending on the sign of the divergence only.