I can think of the stiffness tensor, that is a 4th order tensor and relates stress and strain in a continuum medium. (and contains 3^4=81) elements.
 For engineering applications the matrix notation is adopted though, transforming it into a symmetric 2nd order tensor. This is done mainly for visual and calculation convenience (apart from energy and thermodynamics considerations).

Tensors are great objects and they are ultimately adequate in the formulation of physical laws (because of their transformation properties) and I think we loose a lot whole of analytical expressiveness because we don't use them more.

I suppose they are highly unpopular (because they are harder to understand), but I would argue for the research need to create tensor abstractions under a c++ framework (this would be a very very interesting task indeed). A quick search over the Internet reveals only very weak implementations compared to what exists for matrices and vectors.

Best
Nasos


> To: ublas@lists.boost.org
> From: matwey.kornilov@gmail.com
> Date: Fri, 2 Apr 2010 21:47:57 +0400
> Subject: Re: [ublas] multiarray subarray and matrix
>
>
> It is interesting why no linear algebra library implements tensors with rank
> greater than 2? At the same time I can't give a wide-known example of high-
> rank tensor.
>
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> ublas@lists.boost.org
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> Sent to: nasos_i@hotmail.com

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