what is curl of a vector field
Pin a tiny paddlewheel in a stream so it cannot drift. If it spins where it sits, that point has curl. Curl is the strength of that spin in place. Counter-clockwise is positive and clockwise is negative. Divergence asked how much leaks out of a point. Curl asks whether the same field turns there. Eddies in a fluid, spinning machinery, and the test for a conservative field all hang on this one number. The paddlewheel in the figures below is that point's curl.
One field holds two vortices. Drag the probe and a small curved arrow appears. That arrow is the way the paddlewheel turns at the point. Over the left vortex the arrow bends counter-clockwise and turns gold. Over the right one it bends clockwise and turns blue. Halfway between them the curl is nearly 0, so the arrow shrinks. A longer, thicker curve means a stronger swirl. Where the arrow shrinks until it disappears, the curl is gone.
The sign of the curl is the direction of the spin. The buttons switch among three fields. F=[−y, x] is a solid counter-clockwise whirl, so ∇×F=+2, and F=[y, −x] is clockwise, so −2. The third, F=[y, 0], is a shear that only slides sideways. The arrows look like parallel straight lines, yet the curl is −1, not 0. If the upper row moves faster than the lower, a paddlewheel caught between them turns. Straight arrows do not mean there is no curl. The circulation added around the fixed circle carries the same sign.
Curl is also a difference of two pieces. The first, ∂v/∂x, is how much the vertical speed changes as you go east. The second, −∂u/∂y, is how much the horizontal speed changes as you go north. Drag the probe and two bars grow on their own. Green is ∂v/∂x and orange is −∂u/∂y. Their sum is the curl ∇×F, exactly the formula under the picture, ∂v/∂x − ∂u/∂y = ∇×F. One bar alone is enough to make curl. When both bars share a sign, the curl peaks, as at the core of a vortex. When the signs cancel, the curl falls toward 0.
Curl is a quantity at one point, and the way to measure it is to draw a small loop. Add up how much the flow turns along the rim and you have the circulation. Drag the center of the loop and shrink the radius with the slider. The circulation itself grows as the loop grows, but circulation divided by the loop's area settles toward one value as the loop shrinks. That value is the curl at the center. The curl is the turning per unit area once the loop has collapsed to a point. Under the picture the circulation, the circulation per area, and the curl at the center sit in a column, and as the radius falls the middle reading approaches the last one. Over the left vortex it settles positive. Over the right one it settles negative. The faint arrows behind the loop are the field, and the short arrows on the rim are only the part of the flow that runs along the loop.
The last picture paints the curl itself. Gold is counter-clockwise, so positive. Blue is clockwise, so negative. Faint arrows show the flow. The slider pulls the two vortices apart. Bring them close and the two patches of curl wear each other down in the middle, so the rotation blurs. Pull them apart and the gold patch and the blue patch separate cleanly. Gather the curl at every point and the map shows which parts of the field turn counter-clockwise and which turn clockwise. The work accumulated along these arrows continues in the lesson on the line integral.