To evaluate $\int \int_{S} \hat{n}×(\bar{a} × \bar{r}) dS$

Question

There's this surface integral question which I can't make sense of.

Given $\bar{a}$, a constant vector and $V$ is the volume enclosed by surface $S$, then $\int \int_{S} \hat{n}×(\bar{a} × \bar{r}) dS$ is equal to..

We usually integrate $\bar{F}\cdot n$ in surface integral which becomes a scalar function, but here integrand is vector.

Any help or hint. Thanks.

Answer 1

According to (1) in this answer the surface integral equals $$ \int_V\nabla\times(\mathbf{a}\times\mathbf{r})\,dV\,. $$ The proof is simple and essentially a componentwise application of Gauss' theorem. Using $$ \nabla\times(\mathbf{a}\times\mathbf{r})=2\,\mathbf{a} $$ there is not much we have to do. The surface integral equals $2\,\mathbf{a}\,{\rm vol}(V)\,.$