(a) Prove the identity

$$\boldsymbol{\nabla}(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}+\mathbf{F} \times(\boldsymbol{\nabla} \times \mathbf{G})+\mathbf{G} \times(\boldsymbol{\nabla} \times \mathbf{F})$$

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

Show that

$$\left(2 x y^{2} z e^{-x^{2} z},-2 y e^{-x^{2} z}, x^{2} y^{2} e^{-x^{2} z}\right)$$

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