= Solution
Apply the <product rule> to the <tensor field>, without assuming any symmetry of $\Sigma$:
$$
\partial_j(r_i\partial_k\Sigma_{kj})=\delta_{ij}\partial_k\Sigma_{kj}+r_i\partial_j\partial_k\Sigma_{kj}=\partial_k\Sigma_{ki}+r_i\partial_j\partial_k\Sigma_{kj}.
$$
Integrating, the <divergence theorem> converts each of the two <divergences> into a boundary term:
$$
\boxed{\int_Vr_i\partial_j\partial_k\Sigma_{kj}\,dV=\int_Sr_i\partial_k\Sigma_{kj}\,dS_j-\int_S\Sigma_{ki}\,dS_k.}
$$
The repeated indices are summed according to <Einstein summation convention>, while $i$ labels the component of the resulting <vector> identity.
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