Tensor Equations for Anisotropic Materials

[KayDee01]

Hi,
I am in the middle of revising for and a classical electromagnetism exam, and I've hit a wall when it comes to tensor equations.

I know that for anisotropic materials: $J=σE$ and $E=ρJ$
And that in component form the first equation can be written as $J_i = σ_{ij} E_j$

What I'm wondering is, does $J_i=σ_{ii}E_i$ make mathematical sense and if so does it make physical sense when applied to anisotropic materials. The second part of my question is, if this does make sense, is the final value of $J_i$ written as: $J_i=σ_{ij} E_j+σ_{ii}E_i$ or is there another way to combine the two values of $J_i$?

 

[clamtrox]

The component form equation means that $$J_i = \sum_j \sigma_{ij} E_j$$, the notation is known as Einstein summing convention. You don't bother to write down the summation sign, but just implicitly assume that the indices repeated on one side of the equation are summed over.