Stokes' Theorem

What to observe

1. Click a patch. The panel on the right opens it up and asks one question on each edge: does the field push me along, or hold me back? Add the four and you have that patch's spin. Divide by its area and you have the curl poking through it, the only part of the curl that patch can feel.
2. Tick cancel the shared edges. Every inner edge belongs to two patches that walk it in opposite directions, so it is added once and subtracted once. The whole mesh dissolves and leaves one bold loop: the edge of the surface. That collapse is the theorem.
3. Now the punchline. Switch between dome, bowl and crumpled. Every patch changes, the normals point all over the place, and the total does not move by a hair, because all three surfaces share the same edge.
4. The number on the left never moves at all, not when you bend the surface and not when you chop it finer, because it is a walk around the one curve all of them hang from. The sum on the right creeps up to meet it as the patches get smaller and cover the surface more honestly.
5. Choose sideways spin, where the curl points along y and never up. A flat surface feels nothing at all. Bend it into a dome and patches light up teal on one side and orange on the other, yet they still add to zero. Only the component through the patch counts, and only the edge decides the total.
6. Choose source: the flow blasts outward, nothing anywhere spins, every patch reads zero and so does the walk around the edge. Spreading is not spinning, and curl is blind to it.