Computing div F and curl F for F = <x²y, y²z, z²x>

Calculus 3 · Axiom Academy

One vector field, two operators — add the matching partials for the divergence, expand the determinant for the curl. Find the divergence and the curl of the vector field . Nice work — you took both derivatives of a vector field: one scalar and one vector. Divergence dots into : add , giving the scalar . Curl crosses into through the determinant, giving the vector . Check: — the divergence of any curl is zero, and here 0+0+0=0 confirms it. Divergence measures how much this field spreads at a point; curl measures how much it spins — the two quantities behind Green's, Stokes', and the Divergence Theorem.

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