The major vector calculus theorem applied to convert the volume integral of the tensor's divergence into the final surface integral form is the Divergence Theorem.
Also known as Gauss's Theorem, this theorem is critical for the conversion because it relates the integral of the divergence of a vector or tensor field over a volume to the flux of that field through the surface bounding the volume. In the context of the magnetic force derivation, it allows the volume integral of $\partial_j T^{i j}$ to be replaced by a surface integral of $T^{i j} d S_j$, confirming that the total force can be determined entirely by the field's stresses on the surface.
<aside> ❓