Divergence & Curl

Stand a tiny paddlewheel in a flowing field. Two things can happen: the flow can stream away from you in all directions (a source) or rush in toward you (a sink) — that's divergence. Or the flow can twist the paddlewheel around — that's curl. Every vector field, anywhere, is doing some mix of these two. Divergence and curl are the two fundamental derivatives that decode what a field does.