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However, my professor said that we need the symmetric part of ##L##, and not only ##L##, because

**otherwise we loose frame indifference**. Unfortunately, he didn't prove this. I'd like to get some feedback on the following argument which should prove the bold sentence.

Assume we have a rotation tensor ##Q(t)## and let ##B_t## and ##B_t^{*}## two configurations related by a rigid motion ##x^{*}=Q(t)x + c(t)##. Then we have, for ##L^{*}=\nabla_{x^*}v## the following relation ##L^{*}=QLQ^T + \dot{Q}Q^T##. Let's focus on the second addendum of ##T##, i.e. ##2 \mu D##. If we were only considering ##S=2 \mu L## instead of the symmetric part of ##L##, we would obtain ##S^{*}=2 \mu L^*=(2 \mu) (QLQ^T + \dot{Q}Q^T) = QSQ^T + 2 \mu \dot{Q}Q^T## and from here we can see that we are not satisfying the requirement for frame indifference.

As another matter of fact, it's clear that if ##S=2 \mu L##, then we don't have symmetry anymore, because of that ##\dot{Q}Q^T## term. However, this is only an algebraic point of view, which has nothing to do with the frame-indifference requirement.

Do you think that this may be what the professor had in mind? Or maybe is there some more physical interpretation?