Divergence and Curl

Calculus 3 · Axiom Academy

Two derivatives of a vector field: , a scalar that measures spreading, and , a vector that measures spin. 1. Divergence: The Net Flow Out of a Point The divergence of a field measures how strongly it is spreading out at a point. Surround the point with a tiny box and ask: does more flow out than in? Divide that net outward flow by the box's volume and shrink the box — the limit is the divergence, one number (a scalar) at each point. A scalar — net outflow per unit volume The source field in the picture has divergence +2 2. Curl: The Local Spin of the Field The curl of measures how much the field rotates near a point — drop a tiny paddle wheel in and watch. Unlike divergence, curl is itself a vector : its direction is the axis the wheel spins about (right-hand rule), and its length is twice the local spin rate. It is read off a determinant. The i, j, k determinant — expand it for the three components Component form (what the determinant expands to) It carries a direction (the spin axis, by the right-hand rule) and a length (twice the angular speed of the flow). Expanding the i, j, k determinant gives all three components at once, each a difference of two partials. In the paddle wheel spins; the curl points straight out of the plane. In the source the wheel is pushed outward but never turns: curl is zero. The rotation field has — it points out of the plane, and the wheel turns counter-clockwise. The source has : pushed outward, but never spinning.

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