= Solution
For the <curl of a cross product>, the Cartesian contraction is
$$
[\nabla\times(\mathbf w\times\mathbf u)]_i=\epsilon_{ijk}\epsilon_{k\ell m}\partial_j(w_\ell u_m)=\partial_j(w_i u_j-w_j u_i).
$$
The <product rule> then gives
$$
\boxed{\nabla\times(\mathbf w\times\mathbf u)=(\mathbf u\cdot\nabla)\mathbf w+(\nabla\cdot\mathbf u)\mathbf w-(\nabla\cdot\mathbf w)\mathbf u-(\mathbf w\cdot\nabla)\mathbf u.}
$$
For the fluid equation, replace the advective term by $\mathbf w\times\mathbf u+\nabla(|\mathbf u|^2/2)$ using the preceding identity. Take its curl. Both gradient terms have zero curl, while $\nabla\cdot\mathbf w=\nabla\cdot(\nabla\times\mathbf u)=0$ and incompressibility gives $\nabla\cdot\mathbf u=0$. The <vorticity equation> is consequently
$$
\boxed{\partial_t\mathbf w+(\mathbf u\cdot\nabla)\mathbf w=(\mathbf w\cdot\nabla)\mathbf u.}
$$
The left side is the <material derivative> of <vorticity>; the right is <vortex stretching>. It follows here because the given force per mass is a gradient, in addition to the flow being inviscid and incompressible.
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