= Solution
Apply the ordinary product rule to the components of the <curl>. For example, the first component is
$$
[\nabla\times(\phi\mathbf F)]_x=\partial_y(\phi F_z)-\partial_z(\phi F_y)=\phi(\partial_yF_z-\partial_zF_y)+\phi_yF_z-\phi_zF_y.
$$
The last two terms are the first component of $\nabla\phi\times\mathbf F$. The other components follow cyclically, so
$$
\boxed{\nabla\times(\phi\mathbf F)=\phi\nabla\times\mathbf F+\nabla\phi\times\mathbf F=\phi\nabla\times\mathbf F-\mathbf F\times\nabla\phi.}
$$
The sign in the last form follows from antisymmetry of the <cross product>, not from a different curl convention. Continuous first derivatives suffice for this identity.
Back to article page
