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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.

ogirkar
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    "but here integrand is vector" Integrate each component separately. Use the triple product expansion: https://en.wikipedia.org/wiki/Cross_product#Triple_product_expansion – Robert Z Jul 25 '23 at 08:13

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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 $2\,\mathbf{a}\,{\rm vol}(V)\,.$

Kurt G.
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