Gradient of a dot product ($\nabla (F\cdot G)=(F\cdot \nabla )G+(G\cdot \nabla )F+F\times \mathrm{curl}G+G\times \mathrm{curl}F$)

For differentiable vector fields, expand the cross products using the contraction of two Levi-Civita symbols. The directional-derivative terms cancel the corresponding terms in the two cross products, leaving ${\partial }_i(F_j{G}_j)$. This proves the displayed gradient identity by the product rule.