Let $\mathbf{F}=\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{x})$, where $\mathbf{x}$ is the position vector and $\boldsymbol{\omega}$ is a uniform vector field.

(i) Use the divergence theorem to evaluate the surface integral $\int_{S} \mathbf{F} \cdot d \mathbf{S}$, where $S$ is the closed surface of the cube with vertices $(\pm 1, \pm 1, \pm 1)$.

(ii) Show that $\boldsymbol{\nabla} \times \mathbf{F}=0$. Show further that the scalar field $\phi$ given by

$$\phi=\frac{1}{2}(\boldsymbol{\omega} \cdot \mathbf{x})^{2}-\frac{1}{2}(\boldsymbol{\omega} \cdot \boldsymbol{\omega})(\mathbf{x} \cdot \mathbf{x})$$

satisfies $\mathbf{F}=\boldsymbol{\nabla} \phi$. Describe geometrically the surfaces of constant $\phi$.