Calculus
Stokes' Theorem
Stretch any surface across a loop and chop it into patches. Neighbours walk every shared edge both ways, so the sum collapses onto the one curve around the edge. Lay it flat and this is Green's theorem; bend it and the answer refuses to move.
spins one wayspins the otherthe patch you picked
Click any patch to open it up · in space, drag to orbit
Patch you picked—
—
View
Surface across the same loop
Field
a few big oneshundreds of tiny ones
walk around the edge—
spin added over the patches—
gap between the two—
What to observe
- In the Flat (Green's theorem) view the surface is just the region inside the loop: Green's theorem is Stokes' theorem lying flat. Click a patch: its four edge walks add up to its spin, and that divided by its area is the curl.
- Tick cancel the shared edges. Every inner edge is walked twice, in opposite directions, so the mesh dissolves and only the edge of the surface is left.
- Go In space and switch between dome, bowl and crumpled. Every patch changes, yet the total stays put, because all three hang from the same edge.
- On Two-sided teal and orange patches fight each other. Only their signed total survives, and it still matches the walk around the edge.
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