= Solution
Expanding with the Levi-Civita symbol and the product rule gives
$$
\nabla\times(a\times b)=a\,\nabla\cdot b-b\,\nabla\cdot a
+(b\cdot\nabla)a-(a\cdot\nabla)b.
$$
The <Stokes theorem> states
$$
\int_S(\nabla\times F)\cdot n\,dS=\oint_C F\cdot dx,
$$
where the boundary orientation follows the right-hand rule about $n$.
Apply it to $F=m\times v$. On $S$, $m=n$ and $v\cdot n=0$. Substitution of the <vector> identity and contraction with $n$ leaves
$$
(\delta_{ij}-n_in_j)\partial_jv_i.
$$
The boundary integrand satisfies $(n\times v)\cdot dx=v\cdot(dx\times n)$, proving the formula. The <vector> $dx\times n$ is the outward co-normal in the tangent plane of $S$.
For the hemisphere, $n=e_r$ and $v=r\sin\theta e_\theta$. The projected divergence is $2\cos\theta$, so the left side is
$$
\int_0^{2\pi}\int_0^{\pi/2}2\cos\theta R^2\sin\theta\,d\theta d\phi=2\pi R^2.
$$
On the positively oriented equator, $dx=Re_\phi d\phi$, hence $dx\times n=Re_\theta d\phi$ and the boundary <integral> is also $2\pi R^2$.
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