Let $u_1,~u_2,~u_3$ be components of the position vector $\vec{u}$:
$$\vec{u}=u_1 \vec{e}_{u_1}+u_2 \vec{e}_{u_2}+u_3 \vec{e}_{u_3}$$
in an ortho-normal coordinate system (not necessarily Cartesian) spun by the ortho-normal basis $\{\vec{e}_{u_1},\vec{e}_{u_2},\vec{e}_{u_3}\}$.
Prove:
$$\nabla\cdot\left(\nabla u_1\times\nabla u_2\right)=0$$
I am mostly interested in the general non-Cartesian case.