Ryan Krauss wrote:

We can work out the notations details and make sure it agrees with other SfePy docs for consistency.

matrices denoted by [], vectors by {}

Yes.

But I think a matrix is a second order tensor and a vector is a first order tensor so that the first equation should be:

1/2 \int E_{ij} \varepsilon_{i} \varepsilon_{j}

or something like that - at least for the 1D problem.

Am I thinking correctly?

There are two possible approaches: engineering (using just vectors and matrices) and "mathematical". The engineering approach employs usually symmetry, consider (in a pseudo-latex):

e_ij = 1/2 (du_i/dx_j + du_j/dx_i) - a symmetric second order tensor can be stored as {e} = [e_11, e_22, e_33, e_12, e_13, e_23] - a vector.

I would prefer working with proper tensors in the docs. The storage is just an implementation detail (yes, in sfepy we use this storage too).

r.