= Solution
An <axial vector> transforms under an orthogonal matrix $Q$ as
$$
\boxed{a'_i=(\det Q)Q_{ij}a_j.}
$$
For a proper rotation, $\det Q=1$, so this is the usual <polar vector> rule. Under an orientation-reversing change, the extra sign distinguishes the two. In particular, spatial inversion leaves an axial vector unchanged while reversing a polar vector.
Let $C_i=(\nabla\times u)_i=\epsilon_{ijk}\partial_j u_k$. The determinant identity for the <Levi-Civita symbol> is
$$
\epsilon_{ijk}Q_{jp}Q_{kq}=(\det Q)Q_{ir}\epsilon_{rpq}.
$$
It follows by multiplying the three-index determinant identity by $Q_{ir}$ and using orthogonality. The derivative transformation from part (ii) then gives
$$
\boxed{C'_i=\epsilon_{ijk}Q_{jp}Q_{kq}\partial_pu_q=(\det Q)Q_{ir}C_r.}
$$
Hence the <curl> of a polar vector field is an axial vector field. The Levi-Civita determinant factor is exactly the extra sign required under reflections.
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