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The conclusions section has a nice short history of some graphical notations, noting "graphical notations of tensor algebra have a history spanning over a century."

Equation 18 in their paper is in Penrose's original paper ("Applications of negative dimensional tensors"). Fun fact: if you take a planar graph, all of whose vertices are degree-3, then interpret them as triple-products/determinants/cross-products, then the absolute value of the resulting scalar is proportional to the number of proper 4-colorings of the regions complementary to the graph. A reiteration of this is in Bar-Natan's paper "Lie algebras and the four color theorem."

However, their contribution seems to be notation for dealing with 3-dimensional derivatives (gradient, curl, and divergence), which are special due to Hodge duality and the existence of a triple product. Equation 30 implying curl of gradient and divergence of curl both being zero is pretty nice.

I think the authors are correct that the specialization of the notation for 3D vector calculus had not been written up yet.




Yes, your fun fact is in Baez's TWF week 92 (1996):

http://math.ucr.edu/home/baez/week92.html

and the core proposition goes back to Penrose.


I see I was rather oblique, but I brought up the fun fact because that equation is how Penrose proved it (to account for signs). By the way, the TWF article is citing the Bar-Natan paper.




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