Dot-product gradient identity ($\nabla (F\cdot G)=(F\cdot \nabla )G+(G\cdot \nabla )F+F\times (\nabla \times G)+G\times (\nabla \times F)$)

For continuously differentiable vector fields, the gradient of their dot product expands as displayed. The contraction of two Levi-Civita symbols gives $[F\times (\nabla \times G){]}_i=F_j{\partial }_i{G}_j-F_j{\partial }_j{G}_i$; adding the directional derivative cancels the second term. Interchanging the fields and using the product rule proves the identity. It does not require either field to be irrotational.