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If we have a real vector space $E$ and some inner product $g$, then we can always project any 2nd order tensor onto three subspaces invariant under automorphisms of $E$.

These projections represent its trace, its antisymmetric part and its traceless symmetric part. For a generic tensor, this decomposition looks like

$$T_{ij}= \frac{1}{3}T^k{}_k\space g_{ij}+T_{[ij]}+\left(T_{(ij)} - \frac{1}{3}T^k{}_k\space g_{ij}\right).$$

If we do this to the covariant derivative of a vector field $\left.\vec{v}\right|_P\in T_P\mathbb{R}^3$, we find

$$v_{i;j} = \nabla_{j\space}v_i = \frac{1}{3}g_{ij}(\text{div}\space \vec{v}) + \frac{1}{2}\varepsilon_{ijk}g^{k\ell}(\text{curl}\space\vec{v})_\ell + \frac{1}{2}(\text{shear}\space\vec{v})_{ij}.$$

Now, we can also find the divergence and the curl of a 1st order tensor field using the exterior derivative and the Hodge star thingie (which incorporates the dependence on the metric of these):

$$\text{div}=\star\space\text{d}\space\star,\space\space\space\space\space\text{curl}=\star\space\text{d}$$

This makes sense because the divergence is a scalar field and the curl is a (pseudo)vector field, and both are differential forms (antisymmetric covariant tensor fields) when they have their indices lowered. The shear, though, is a symmetric 2nd order tensor field, so it couldn't possibly be a differential 2-form(... couldn't it?)

My questions are:

  • Is everything I just said true?
  • If so, how comes the flux and circulation of a vector field around a point are related to differential forms, but not its shear?
  • Does this have something to do with the fact that the shear isn't usually taught in undergrad physics, math and engineering courses? (Besides the fact that, you know, it's a 2nd order tensor and nobody wants to deal with those if it's not necessary.)

EDIT: any insight on the shear would be helpful. Knowing its relation to things like Stokes' theorem and theorems alike, or the Helmholtz decomposition theorem (something to be answered here, maybe: Why is shear missing in Helmholtz theorem?) would be nice, even if it doesn't completely answer my original ---and maybe too broad--- question.

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    $\begingroup$ I don't think I can really give you an answer, but out of curiosity: where did you find that definition of shear? I don't know what to look for in the wiki. $\endgroup$ Jul 11, 2020 at 1:27
  • $\begingroup$ @JackozeeHakkiuz there's this post here in math.se in which this is discussed math.stackexchange.com/questions/1196427/what-does-shear-mean but it is indeed something barely spoken about in the wiki, in books and almost anywhere else. Were it not, I wouldn't be asking such a broad question xD $\endgroup$
    – TeicDaun
    Jul 11, 2020 at 9:34

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