Curl Is the Local Swirl
Imagine floating a tiny paddlewheel in moving water, pinned to one spot so it can't drift away. If the wheel spins in place, there's curl there. Curl measures exactly that: the strength of the local swirl. Spinning counter-clockwise counts as positive, clockwise as negative. Where divergence asked "is it spreading out?", curl asks "is it going around?" Two faces of the same field. This one quantity governs eddies in a fluid, spinning machinery, and the very test for whether a field is conservative.
Here's a field with two vortices. Drag the probe. A small curved arrow appears at the probe — that's which way the paddlewheel turns at that point. Over the left vortex the arrow curves counter-clockwise and glows gold (curl positive); over the right one it curves clockwise and turns blue (negative). Halfway between the two vortices the curl is nearly zero, so the arrow shrinks. The longer and thicker the curved arrow, the stronger the swirl at that spot.
The sign of curl is simply the direction of spin. Toggle through three fields. F=[−y, x] rotates counter-clockwise as a solid whirl, so ∇×F=+2; F=[y, −x] is clockwise, so −2. The third, F=[y, 0], is a shear flow that only slides sideways, yet even though every arrow looks like a straight parallel line, its curl is −1, not zero. When the upper row moves faster than the lower, a paddlewheel caught between them turns. The key point: arrows looking straight does not mean there's no curl. The circulation summed around a fixed circle confirms the same sign.
Curl, too, is the difference of two pieces. The first piece, ∂v/∂x, is how much the up-velocity changes as you head east; the second, −∂u/∂y, is how much the across-velocity changes as you head north. Drag the probe and two bars grow separately: green is ∂v/∂x, orange is −∂u/∂y. Their sum is the curl ∇×F, exactly as the formula ∂v/∂x − ∂u/∂y = ∇×F below the picture says. Even one bar alone makes curl; when the two add with the same sign, the swirl peaks like the core of a vortex. When opposite signs cancel, the curl drops toward zero.
Curl is a quantity at a single point — so how do you measure it? Draw a small loop and add up how much the flow goes around its edge (the circulation). Drag the loop's center, and shrink its radius with the slider. The raw circulation grows as the loop gets bigger, but the circulation divided by the loop's area (circulation/area) settles onto one value as you shrink it. That value is exactly the curl at the center. This is the true definition of curl: the swirl per unit area as the loop collapses to a point. Place it over the left vortex and it converges to a positive number; over the right, a negative one.
Now here's a map that paints the curl itself in color. Gold is counter-clockwise rotation (positive), blue is clockwise (negative), and faint arrows show the flow. The slider pulls the two vortices apart. Bring them close and the two patches of curl eat into each other in the middle, so the rotation blurs; pull them far and the gold and blue patches separate cleanly. Gather the curl at every single point and you get a curl map, where you see at a glance which parts of the field turn counter-clockwise and which turn clockwise. The next lesson moves to the line integral, measuring the work done following these arrows.