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Please, is the stress tensor (elasticity theory) actually a pseudo tensor? It seems to me it must change its sign when coordinate system changes its orientation.

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We must integrate something over a small two dimensional surface to obtain the force vector which is applied to this surface. Thus this "something" must be a tensor of the form $\sum_{j<k}\omega^i_{jk}dx^j\wedge dx^k$. All indexes run 1,2,3

The tensor $\omega^i_{jk}$ is skew symmetric in the subscripts ,therefore it has 9 independent components. So it corresponds to a 3*3 matrix. This matrix is a stress tensor. But the correspondence can be only axial. There is no pure tensor correspondence that takes three indexed tensor to two indexed one.

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