= Solution
Let $\boldsymbol\omega=\nabla\times\mathbf u$ be the <vorticity>. In component notation, contracting two <Levi-Civita symbols> gives
$$
(\mathbf u\times\boldsymbol\omega)_i=\varepsilon_{ijk}u_j\varepsilon_{klm}\partial_lu_m=u_j\partial_i u_j-u_j\partial_j u_i.
$$
The first term is $\partial_i(|\mathbf u|^2/2)$ and the second is the $i$th component of the <convective acceleration>. Rearranging proves \b[the velocity identity]
$$
\boxed{(\mathbf u\cdot\nabla)\mathbf u=\nabla\left(\tfrac12|\mathbf u|^2\right)-\mathbf u\times\boldsymbol\omega.}
$$
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