Divergence Is the Net Outflow
Picture drawing a tiny circle around a point. If more flow leaves across that boundary than enters it, the point is pumping something out. That is divergence: the net amount leaking out of a point. More out than in makes it a source; more in than out makes it a sink; equal amounts give zero, so the flow just passes through. As a formula it is how much the across part spreads going east (∂u/∂x) plus how much the up part spreads going north (∂v/∂y). In engineering it is the one number that pins down where something leaks, gathers, or gets compressed.
Drag the probe over the field. A small ring of eight sample arrows wraps around wherever it sits. When these arrows splay outward the flow is leaking out there; when they gather inward it is being drawn in. Place it near the origin and the arrows spread every way, so ∇·F reads positive and it shows up as a gold source. Drag it far out and the arrows curl inward, ∇·F turns negative, and it becomes a blue sink. Somewhere between, on a ring of radius about 2.45, the inflow and outflow exactly balance and ∇·F falls to zero.
Switch among three fields with the buttons and read off the sign. The source has every arrow reaching outward from the origin, so ∇·F is +1.2, positive. The sink has them all gathering inward, so it is −1.2, negative. The swirl has every arrow turning around the loop, yet as much comes in as goes out, so ∇·F is exactly zero. Here is the key: no matter how hard it spins, that has nothing to do with divergence. Divergence asks not "does it turn" but only "does it spread or gather." Watch the big readout flip with its sign.
Divergence is, in the end, a sum of two pieces. Drag the probe and the spread of the across part going east (∂u/∂x, green) and the spread of the up part going north (∂v/∂y, orange) each rise as a bar. Add the two bars and you have ∇·F itself. 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, the sum shrinks, and far out both go negative and it becomes a sink. Watch the single line "∂u/∂x + ∂v/∂y = ∇·F" come alive in live numbers.
The real definition of divergence is the flow crossing a boundary, the flux. Drag the circle's center and move the radius slider. All around the rim, arrows show how much flow leaks out along the outward normal. Sum the whole rim and you get the total flux Φ; divide that by the circle's area and Φ/area is the average divergence inside. The smaller you shrink the circle, the more Φ/area converges to the single value of ∇·F at the center. So divergence is a quantity you can picture: the outflow per unit area of a vanishingly small boundary.
Now see divergence not at one point but across the whole field. Move the slider s and the field shifts smoothly to F=[s·x − x³/9, s·y − y³/9]. The background color is the map of ∇·F itself. Gold is a source (positive), blue is a sink (negative), and the boundary between them is the ring of zero divergence. Turn s up and the spreading at the origin strengthens, the gold region widens, and the zero ring is pushed outward. Turn s down and the gold shrinks while the blue sink floods inward. Watch one handle move the entire boundary between source and sink.