Evaluate the following integrals

$\displaystyle I_1 = \oint \vec{r} (\vec{a} \cdot \vec{n}) dS$,

$\displaystyle I_2 = \oint (\vec{a} \cdot \vec{r})\vec{n} dS$,

where $\displaystyle \vec{a}$ is a constant vector, and $\displaystyle \vec{n}$ is an unit vector normal to the closed surface $\displaystyle S$.

I'm not sure whether the Stokes's theorem can be used in this case. For instance, if we write the integrals in the component notation

$\displaystyle I_1 = \oint x_i (a_j dS_j)$,

$\displaystyle I_2 = \oint (a_j x_j)dS_i$,

then the Stokes' theorem would suggest the supstitution $\displaystyle dS_i \to dV \partial_i$ where $\displaystyle S=\partial(V)$. This results in

$\displaystyle I_1 = \int (a_j \partial_j) x_i dV=\vec{a} V$,

$\displaystyle I_2 = \int \partial_i (a_j x_j)dV=\vec{a} V$.

Is this correct?