Calculus
Green's Theorem
Take any wobbly loop, chop the region inside it into little squares, and give each square its own tiny walk around its rim. Neighbours share every inner edge and walk it in opposite directions, so all of it cancels. What survives is one walk around the outside, and it equals all the spin inside.
a few big oneshundreds of tiny ones
Walk around the loop—
Spin added over the squares—
Square you picked—
…its curl—
Gap between the two—
Click any square to open it up on the right. Both numbers are measured from the field by brute force, never assumed equal.
What to observe
- Click a square. The panel on the right opens it up and puts an arrow on each of its four edges: teal where the field pushes you along the way you are walking, orange where it holds you back. Add the four and you have that square's circulation. Divide by its area and you have curl, which is all curl ever was.
- Every circle in the region is one of those little walks, drawn as the spin it adds up to. Teal turns anticlockwise, orange turns clockwise, and the size is how hard.
- Now tick cancel the shared edges. Every inner edge is walked twice, once by each of the two squares that share it, and in opposite directions, so it is added once and subtracted once. Watch the whole grid dissolve and leave a single bold loop: the rim, the only edges with nobody on the far side.
- That is the whole theorem, and you can watch it arrive. The number on the left never moves, because it is a walk around the real loop and knows nothing about your grid. Drag squares and the sum on the right climbs to meet it: squares can only approximate a curved region, and the finer they are the less of it they miss.
- Try source: the flow blasts outward, every square is flat and the rim walk is zero. Spreading is not spinning. Then try two-sided, where teal and orange squares fight each other and only their signed total survives to match the rim.
Shortcuts: space run/pause · s step · r reset · f fullscreen