Solution
= Solution
Use $w_j=\epsilon_{j\ell m}\partial_\ell u_m$ and contract the <Levi-Civita symbols>:
$$
(\mathbf w\times\mathbf u)_i=\epsilon_{ijk}\epsilon_{j\ell m}(\partial_\ell u_m)u_k=u_\ell\partial_\ell u_i-u_m\partial_i u_m=u_\ell\partial_\ell u_i-\partial_i\frac{|\mathbf u|^2}{2}.
$$
This proves the <convective acceleration identity>, with the sign set by the displayed order $\mathbf w\times\mathbf u$.