The paper doesn't really explain everything you need to be able to read the notation, but, being a tensor diagram aficionado, I have to say I found the way they

The paper doesn't really explain everything you need to be able to read the notation, but, being a tensor diagram aficionado, I have to say I found the way they encoded divergence and curl very exciting: it is all consistent and helpful for doing calculations. The loop-around-a-tensor notation is calculating the covariant derivative. For example, if you have a vector field, the covariant derivative is a "matrix field," where at every point in space you have a matrix that can take in a vector there and spit out the component-wise directional derivative. The divergence graphical notation (connecting the two wires together) is equivalent to taking a trace of this, meaning you create a scalar field where at every point you take the trace of the matrix from the covariant derivative. I guess all I'm saying is that the notation for divergence and curl are consequences to a couple of building blocks that have more generality. I'll also add that, like any language, it takes some getting used to. > Why not just represent divergence and curl for what they are? I'm curious how you would represent them for what they are. Maybe you have something in mind that would make calculations easier?