When researching a derivation of the angular momentum balance equations, I came across these excellent online notes from a course on continuum mechanics. While this is helpful, I would appreciate a fuller derivation, with clarification as to where each term in the expression in the first yellow box came from, as well as an explanation as to how 'using the equilibrium equations...' produces the simplification $(\sigma_{12}-\sigma_{21})\Delta v=0$.
As a side note, why is it necessary to consider a rectangular volume in this example and not a cube as was the case in the earlier derivation of the linear momentum balance (see hyperlink)?
