Calculus

Divergence & Curl

Two numbers at every point of a flow. Drop in a material patch and its area grows or shrinks with the divergence; drop in a paddle wheel and it spins with the curl.

the patch swellsthe patch is squeezed

Drag the probe anywhere in the field

Divergence at the probe—

—

Probe

Vector field

Strength 1.00

Flow speed 1.0×

Probe radius ε 0.18

Show tracer streaksReference wheel at the centre

Values

divergence ∇ · F—

flux ÷ area—

patch area now—

curl (∇ × F)_z—

circulation ÷ area—

wheel ω = ½ curl—

wheel ÷ reference—

What to observe

Put the material patch on Source: more flow leaves it than enters, so it swells and ∇ · F > 0. Drag the strength negative and the same field becomes a sink that squeezes it.
On Saddle the patch stretches one way and squeezes the other, yet its area stays at 1.00×. Heavy deformation, zero divergence.
Switch to the paddle wheel on Rotation: it turns at ω = ½ curl and wheel ÷ reference reads 1.00× wherever you drag it. On Source it never turns: spreading is not spinning.
On Shear every arrow is horizontal, yet the wheel spins one way above the centreline and the other way below. Curl is local twist, not curved paths.

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