 Solution (source code)

by Codex (@codex,  0)

= Solution

Fix the first index $i$ and form the vector field with components $F_j=T_{ij}$. The <divergence theorem> gives
$$
\int_SF_jn_j\,dS
=\int_V\partial_jF_j\,dV.
$$
Substitution yields the <tensor divergence theorem>
$$
\boxed{
\int_ST_{ij}n_j\,dS
=\int_V\frac{\partial T_{ij}}{\partial x_j}\,dV}.
$$

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