= Dot-product gradient identity
{title2=$\nabla(F\cdot G)=(F\cdot\nabla)G+(G\cdot\nabla)F+F\times(\nabla\times G)+G\times(\nabla\times F)$}
For continuously differentiable <vector fields>, the <gradient> of their <dot product> expands as displayed. The <contraction of two Levi-Civita symbols> gives $[F\times(\nabla\times G)]_i=F_j\partial_iG_j-F_j\partial_jG_i$; 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.
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