curl of gradient divergence of curl
Divergence asks whether anything spreads, and curl asks whether anything turns. Between the two there are two promises that never break. A field made from a gradient has curl exactly 0 no matter how you twist it, and the formula is ∇×(∇φ)=0. A field made from a curl has divergence exactly 0 no matter how you stir it, and the formula is ∇·(∇×A)=0. A gradient can climb but does not know how to turn, and a swirl can turn but does not know how to leak. The surprise is the converse. Any field splits into the sum of a pure source piece and a pure swirl piece, and that is the Helmholtz decomposition. It splits one field into two characters, so you can pull apart where the flow begins and where it spins. A source comes from the slope of a height, and a swirl comes from a grain that circles. Stack both on one field and the promises still hold.
There is a scalar bump φ, and gold gradient arrows lie over it. Every arrow points uphill, toward the top of the bump. Raise and lower the bump height with the slider and the arrows grow and shrink and sway, while the curl color map beside them stays one neutral color from start to finish. However you shake it, ∇×(∇φ) does not budge from 0. A gradient field has only the one sense of uphill, so a circling part cannot arise in the first place. It can climb, and it does not know how to turn. Neutral means it does not turn, and however tall you make the bump that color stays as it is.
Now the other side. A swirl field is built from a stream function ψ. The horizontal part is ∂ψ/∂y, and the vertical part is −∂ψ/∂x, and built this way the blue arrows circle along the rim. Raise the strength of ψ with the slider and the swirl grows and fades and sways, while the divergence color map beside it stays that same one neutral color. The divergence does not move from 0. A field shaped from a stream function leaves exactly as much as it enters, so it neither leaks nor gathers at any point. It can turn, and it does not know how to leak. The 3-dimensional identity ∇·(∇×A)=0 shows up in 2D like this. However fierce the blue arrows become, the amount that leaks does not grow, and what enters still leaves.
Now look at a general field, the two added together. Buttons move among three views. First the whole field has arrows that surge one way and circle at the same time, so at a glance you cannot tell source from swirl. Press the source part and only the gradient part remains, a pure flow reaching out from the bump, and the note turns gold. Press the swirl part and only the stream-function part remains, a grain that spins, and the note turns blue. The same field has been split into two characters. That any complicated field splits as a source piece plus a swirl piece is the heart of the Helmholtz decomposition. Each time you move among the buttons the same arrows show a different character, and the two grains you added appear apart.
Pull off only the source piece. The arrows are the gold gradient part, and the background color is its divergence. You already saw that the curl is 0, and the divergence is not 0. The inside of the bump is dyed a strong color, showing that every source and sink of the flow gathers here. Drag the probe and wherever you place it the curl readout stays near 0 and does not turn, while the divergence readout splits clearly into positive and negative by place. The part of one field where flow is born and where it vanishes is carried whole by this source piece. There is no swirl in it at all. Positive is where flow is born and negative is where flow vanishes. Both live only inside this piece.
Now pull off only the swirl piece. The arrows are the blue stream-function part, and the background color is its curl. You already saw that the divergence is 0, and the curl is not 0. The two swirl places are dyed a strong color, and every turning part of the field gathers there. Drag the probe and wherever you place it the divergence readout stays near 0 and does not leak, while the curl readout splits by place into gold for counterclockwise and blue for clockwise. One field is two pieces: the source piece holds every divergence, and the swirl piece holds every curl. The two faces, divergence and curl, join into one field here and split apart again. Gold and blue differ only in the way of turning. Neither side has any leak.