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Choose the coordinate vectors $b=e_i$ and $c=e_j$ in the assumed bilinear identity. It gives $T_{ij}=0$ for each pair of indices, hence the tensor is zero.

Let $\Omega$ be the enclosed region. Apply the divergence theorem to the vector field $(b\cdot x)c$. Since $b,c$ are constant,ThereforeWriting the left side as $b_i{c}_j{\int }_S{x}_i{n}_{j}dS$ and using the initial tensor argument gives the boundary integral of position and normal componentsContract with the Kronecker delta to obtain ${\int }_{S}x\cdot ndS=3V$. Contract with the Levi-Civita symbol to obtainThus the requested integrals are $3V$ and the zero vector, respectively. These applications require the usual sufficiently regular closed boundary and outward-normal orientation of the divergence theorem.