Turning the mathematics of vector calculus into simple pictures

These methods were invented decades ago by Penrose (1971) [2], as a way of visualizing and improving on Einstein&#x27;s summation convention [1]. Similar &quot;string diagrams&quot; were popularized for various categories by John Baez in &quot;This Week&#x27;s Finds&quot; (TWF) [3], with extensive applications to QM by Coecke [4]. There are other modern examples all over the interweb [5] ...<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Einstein_notation" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Einstein_notation</a><p>[2] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Penrose_graphical_notation" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Penrose_graphical_notation</a><p>[3] <a href="http:&#x2F;&#x2F;math.ucr.edu&#x2F;home&#x2F;baez&#x2F;week79.html" rel="nofollow">http:&#x2F;&#x2F;math.ucr.edu&#x2F;home&#x2F;baez&#x2F;week79.html</a> <i>ff</i><p>[4] <a href="https:&#x2F;&#x2F;www.cs.ox.ac.uk&#x2F;ss2014&#x2F;programme&#x2F;Bob.pdf" rel="nofollow">https:&#x2F;&#x2F;www.cs.ox.ac.uk&#x2F;ss2014&#x2F;programme&#x2F;Bob.pdf</a><p>[5] <a href="https:&#x2F;&#x2F;graphicallinearalgebra.net&#x2F;" rel="nofollow">https:&#x2F;&#x2F;graphicallinearalgebra.net&#x2F;</a>

I’ve seen a lot of people enamored with graphical notations like this one, but nobody ever using them for anything nontrivial. In general, people just use them to laboriously reproduce a few results from undergrad courses, nothing more.<p>If ordinary notation is like C, these graphical notations are like code golf languages, a neat gimmick that performs excellently at the few things you hardcoded it to do, and not much else.

Ref of the original paper <a href="https:&#x2F;&#x2F;arxiv.org&#x2F;abs&#x2F;1911.00892" rel="nofollow">https:&#x2F;&#x2F;arxiv.org&#x2F;abs&#x2F;1911.00892</a>. The biggest issue for me is they don&#x27;t have proper upper&#x2F;lower indices. A lower index is a leg pointing down, while an upper index is a leg pointing up. \delta should be \delta_i^j. The upper&#x2F;lower I like to think of as extra type information. You can pair covectors and vectors. The other issue is with the dot product of course, I would write a_i b^i.

Not enough pictures in this article

This is all really confusing. Why not just represent divergence and curl for what they are? Sources&#x2F;sinks, and circulations. As someone who had lots of courses in electromagnetics, I see no basis for physical reality in these diagrams.