3.I.3CVector CalculusPart IA, 2004If F\mathbf{F}F and G\mathbf{G}G are differentiable vector fields, show that(i) ∇×(F×G)=F(∇⋅G)−G(∇⋅F)+(G⋅∇)F−(F⋅∇)G\boldsymbol{\nabla} \times(\mathbf{F} \times \mathbf{G})=\mathbf{F}(\boldsymbol{\nabla} \cdot \mathbf{G})-\mathbf{G}(\boldsymbol{\nabla} \cdot \mathbf{F})+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}-(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}∇×(F×G)=F(∇⋅G)−G(∇⋅F)+(G⋅∇)F−(F⋅∇)G,(ii) ∇(F⋅G)=(F⋅∇)G+(G⋅∇)F+F×(∇×G)+G×(∇×F)\boldsymbol{\nabla}(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}+\mathbf{F} \times(\boldsymbol{\nabla} \times \mathbf{G})+\mathbf{G} \times(\boldsymbol{\nabla} \times \mathbf{F})∇(F⋅G)=(F⋅∇)G+(G⋅∇)F+F×(∇×G)+G×(∇×F).