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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 centu
by kmill 7y ago
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.
- mikhailfranco 7y agoYes, your fun fact is in Baez's TWF week 92 (1996): http://math.ucr.edu/home/baez/week92.html http://math.ucr.edu/home/baez/week92.html and the core proposition goes back to Penrose.
- kmill 7y agoI 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.