I think solid state continuum mechanics are also the optimal place to introduce tensors. For some reason, the first tensors many physics students encounter are very abstract. It would be like if the first vectors you encountered were quantum mechanical states. Stress and strain are, in my opinion, the ideal "prototypical rank-2 tensors", and it's useful to spend time really elaborating what that means, the same way we teach students to think of vectors as "things that look like displacement/velocity".
That’s an interesting idea. The best class involving tensors I ever took was an introductory course on differential geometry, and I still think the coordinate-free approach of thinking of tensors as multilinear functions from some number of vectors to some other number of vectors (or a scalar) is great. Everything else just involves picking coordinates and figuring out where the numbers to :)
But I probably like abstractions like this more than most people.
Oddly, undergraduate physics also seems to be missing another, arguably even more fundamental, tensor: the moment of inertia. You can get quite far (in three dimensions, and only in three dimensions) by thinking of rotation as a vector. (Or a quaternion if that floats your boat.) But you can’t get very far by pretending that the moment of inertia is a scalar, and you get very confused very quickly if you treat it as three scalars in the magical coordinate system in which you can write it like that.
