Stretch any surface you like across the same wire loop. The spin caught by all of them is identical.
The loop never moves. Change the cap and only the last number changes — the first two stay locked together.
Swap the cap. The two numbers never stop agreeing.
Bend a loop of wire and dip it in a field. Walk once around the wire, adding up how much the field pushes you along as you go: that's the circulation. Now stretch a soap film across the same wire and ask how much curl pokes through the film.
Stokes' theorem says those two numbers are the same. Not approximately, and not just for the obvious flat film — for any surface with that wire as its edge. Swap the flat disk for a dome, a cone, a deep bowl, a wobbly sheet: the areas differ wildly, the tinting redistributes completely, and the total never moves.
It is the same bookkeeping argument as Green's theorem, which is the flat case of exactly this. Chop the film into tiny cells; each cell's own little circulation is curl · n times its area; adjacent cells share edges and walk them in opposite directions, so everything interior cancels. All that survives is the boundary — and the boundary is the wire, which never changed. The film in between was never really the point.
One consequence worth noticing: pick the gradient field and every number drops to zero. A field that is the gradient of something has no curl at all, so it can have no circulation around any closed loop — which is precisely what it means for work to be path-independent.