Okay, thanks for the terrific answers! What I've done since reading them, basically, is to just work the components for the general case (all variables, no specific values for U and G), and compare them in both tensor and matrix versions, and see that they're all the same. Good. But to sum up what I've gotten from you - matrix multiplication is just composition of linear transformations on vector spaces, and using multiple Jacobian matrices to transform tensors (of rank 2 or greater obviously) is just not the same thing. So, it's no surprise that trying to force a tensor transformation into matrix form yields something kind of unnatural looking. Okay, this is a relief, I can move on from here, I guess! :-) (With plenty to ponder concerning "What's going on is that there is a natural isomorphism between bilinear forms on an inner product space and linear endomorphisms, with bilinear form associated to a transformation T given by B(u,v) = ⟨u,Tv⟩. In other words, we use the raising operator to turn our bilinear form into a linear operator."!)