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 Feb 5 comment How the torque/moment-of-force can be mathematically defined? I was saying that $\mathbb{R}^3$ simplify (eccessively) things, because there is an obvious isomorphism between different copies of $\mathbb{R}^3$. Using arbitrary 3-dimensional vector spaces does not provide such an obvious isomorphism, unless you choose a basis for each of them. Also, I would not say that they are different copies of the same space, but different spaces. Feb 5 comment How the torque/moment-of-force can be mathematically defined? This is interesting, also if I would have used general vector spaces and not $\mathbb{R}^3$, which already suggest an isomorphism. You say "... there is little mathematical reason...", probably it is better to say that there is little physical reason, but a big mathematical one. Feb 5 comment How the torque/moment-of-force can be mathematically defined? You all put too much attention on dimensions, I shouldn't have talked about that. Still from a mathematical point of view the problem exists, given that, as you admit, the object belong to different spaces (or sets). Well, right, the problem arises also for simpler formulas with scalar instead of vectors. Feb 5 comment Can $U_{ij}$ or $v_{ij}$ in continuum mechanics be negative? @tpg2114: there is not other restriction that I know of. Also, the difference from this matrix and the diagonal one is a simple change of basis, and one can always choose the reference frame at will. Feb 4 comment How the torque/moment-of-force can be mathematically defined? @MyUserIsThis: this is not mathematically satisfactory. They do belong to the same vector space $\mathbb{R}^3$, but they cannot be added, it is contraddictory. May 8 comment Modulus of action-reaction forces @Qmechanics: hand-written notes of a professor at my University.