# 3.II.11A

Prove that

$\nabla \times(\mathbf{a} \times \mathbf{b})=\mathbf{a} \nabla \cdot \mathbf{b}-\mathbf{b} \nabla \cdot \mathbf{a}+(\mathbf{b} \cdot \nabla) \mathbf{a}-(\mathbf{a} \cdot \nabla) \mathbf{b}$

$S$ is an open orientable surface in $\mathbb{R}^{3}$ with unit normal $\mathbf{n}$, and $\mathbf{v}(\mathbf{x})$ is any continuously differentiable vector field such that $\mathbf{n} \cdot \mathbf{v}=0$ on $S$. Let $\mathbf{m}$ be a continuously differentiable unit vector field which coincides with $\mathbf{n}$ on $S$. By applying Stokes' theorem to $\mathbf{m} \times \mathbf{v}$, show that

$\int_{S}\left(\delta_{i j}-n_{i} n_{j}\right) \frac{\partial v_{i}}{\partial x_{j}} d S=\oint_{C} \mathbf{u} \cdot \mathbf{v} d s$

where $s$ denotes arc-length along the boundary $C$ of $S$, and $\mathbf{u}$ is such that $\mathbf{u} d s=d \mathbf{s} \times \mathbf{n}$. Verify this result by taking $\mathbf{v}=\mathbf{r}$, and $S$ to be the disc $|\mathbf{r}| \leqslant R$ in the $z=0$ plane.