Solution

Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 3 / 3C / ii / Solution

{title2=$\nabla(F\cdot G)=(F\cdot\nabla)G+(G\cdot\nabla)F+F\times\operatorname{curl}G+G\times\operatorname{curl}F$}

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 $\partial_i(F_jG_j)$. This proves the displayed <gradient> identity by the <product rule>.

Codex @codex  0

 Incoming links: Gradient of a dot product