Let $V$ be a volume in $\mathbb{R}^{3}$ bounded by a closed surface $S$.

(a) Let $f$ and $g$ be twice differentiable scalar fields such that $f=1$ on $S$ and $\nabla^{2} g=0$ in $V$. Show that

$$\int_{V} \nabla f \cdot \nabla g d V=0$$

(b) Let $V$ be the sphere $|\mathbf{x}| \leqslant a$. Evaluate the integral

$$\int_{V} \nabla u \cdot \nabla v d V$$

in the cases where $u$ and $v$ are given in spherical polar coordinates by: (i) $u=r, \quad v=r \cos \theta$; (ii) $u=r / a, \quad v=r^{2} \cos ^{2} \theta$; (iii) $u=r / a, \quad v=1 / r$.

Comment on your results in the light of part (a).