what is divergence of a vector field
Draw a tiny circle around a point. If more flows out through its rim than flows in, that point is spouting something. That is divergence: the net amount leaking out of a point. If outflow wins it is a source, if inflow wins it is a sink, and if they are equal it is 0 and the flow simply passes through. As a formula, it is how much the horizontal component spreads eastward (∂u/∂x) plus how much the vertical component spreads northward (∂v/∂y). In engineering, this one number pins down where things leak, where they gather, and where they are compressed.
Drag the probe over the field and eight sample arrows are placed around a dashed ring at that spot. Where the arrows reach outward, the flow is leaking away; where they gather inward, it is being drawn in. Near the origin they spread in every direction, the divergence is positive, and the ring glows as a gold source. Drag far out and the arrows bend inward, the divergence turns negative, and the ring becomes a blue sink. Somewhere in between, on a ring of radius about 2.45, inflow and outflow balance exactly and the divergence is 0. The divergence at the current spot is shown below.
Press the buttons and three fields take turns. In the source, every arrow reaches outward from the origin and the divergence is +1.2, positive; in the sink, they all gather inward and it is −1.2, negative. In the swirl, the arrows circle around the rim, as much flows out as flows in, and the divergence is exactly 0. The key point is that however strong the rotation, it has nothing to do with divergence. Divergence looks only at spreading or gathering, not at turning, and the large number below, changing along with its sign, makes that plain.
Divergence, too, is the sum of two pieces. Drag the probe and two bars appear: how much the horizontal component spreads eastward (∂u/∂x, green) and how much the vertical component spreads northward (∂v/∂y, orange). Adding the two bars gives the divergence; at the origin both are about +1, so the sum is +2, a strong source. Drag outward and one axis at a time bends negative so the sum shrinks, and far out both are negative and it becomes a sink. The single line "∂u/∂x + ∂v/∂y = ∇·F" comes alive in numbers.
The real definition of divergence is the flow passing through a rim: the flux. Drag the center of the circle and move the radius slider, and arrows show how much flow leaks out along the outward normal at each point of the rim. Adding up the whole rim gives the total flux Φ, and dividing it by the circle's area gives Φ/area, the average divergence inside. The smaller you make the circle, the more Φ/area converges to the single value of ∇·F at the center, and the last two of the three readouts below grow closer. So divergence is a quantity you can grasp in a picture: the outflow per unit area across a very small rim.
Now look at divergence not at one point but over the whole field. Move the slider s and the field changes smoothly as F=[s·x − x³/9, s·y − y³/9]. The background color is a map of the divergence: gold is source, positive; blue is sink, negative; and the boundary between them is the ring of zero divergence. Raise s and the spreading at the origin grows stronger, the gold region widens, and the zero ring is pushed outward. Lower s and the gold shrinks while the blue sink fills in toward the center. The radius of that ring is shown below. A single handle moves the whole boundary between source and sink.