Vector Calculus | Part IA, 2003

(a) Prove the identity

(FG)=(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})

(b) If E\mathbf{E} is an irrotational vector field (×E=0(\boldsymbol{\nabla} \times \mathbf{E}=\mathbf{0} everywhere )), prove that there exists a scalar potential ϕ(x)\phi(\mathbf{x}) such that E=ϕ\mathbf{E}=-\boldsymbol{\nabla} \phi.

Show that

(2xy2zex2z,2yex2z,x2y2ex2z)\left(2 x y^{2} z e^{-x^{2} z},-2 y e^{-x^{2} z}, x^{2} y^{2} e^{-x^{2} z}\right)

is irrotational, and determine the corresponding potential ϕ\phi.

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