Vector calculus · divergence & curl

Does the field spread, or does it spin?

Two numbers describe what a vector field is doing at any point. Divergence asks whether more flows out of a tiny region than flows in — is this a source or a drain? Curl asks whether the field would spin a little paddlewheel dropped into it. Drag the probe around: the ring breathes with the divergence, the paddlewheel turns with the curl.

At the probe

∇·F divergence—

∇×F curl (z)—

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Field

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Legend

F(x,y) — the vector field

flux ring — grows if ∇·F > 0, shrinks if < 0

paddlewheel — spins with the curl

For a 2-D field F = (P, Q), divergence is ∇·F = ∂P/∂x + ∂Q/∂y and the (scalar) curl is ∇×F = ∂Q/∂x − ∂P/∂y. Divergence is flux density — a positive value is a source spraying outward, negative is a sink draining in, zero means whatever flows in flows back out (incompressible). Curl is circulation density — it's how fast an infinitesimal paddlewheel would spin, and notice the shear field has curl without any obvious swirl: uneven flow alone twists the wheel. These two operators are the whole vocabulary of vector calculus: Gauss's divergence theorem ties total flux to sources inside, Stokes' theorem ties circulation to curl through a surface, and together they're how Maxwell's equations, fluid flow, and electromagnetism are written.