
Turning the mathematics of vector calculus into simple pictures - respinal
https://www.technologyreview.com/s/614704/how-to-turn-the-complex-mathematics-of-vector-calculus-into-simple-pictures/
======
mikhailfranco
These methods were invented decades ago by Penrose (1971) [2], as a way of
visualizing and improving on Einstein's summation convention [1]. Similar
"string diagrams" were popularized for various categories by John Baez in
"This Week's Finds" (TWF) [3], with extensive applications to QM by Coecke
[4]. There are other modern examples all over the interweb [5] ...

[1]
[https://en.wikipedia.org/wiki/Einstein_notation](https://en.wikipedia.org/wiki/Einstein_notation)

[2]
[https://en.wikipedia.org/wiki/Penrose_graphical_notation](https://en.wikipedia.org/wiki/Penrose_graphical_notation)

[3]
[http://math.ucr.edu/home/baez/week79.html](http://math.ucr.edu/home/baez/week79.html)
_ff_

[4]
[https://www.cs.ox.ac.uk/ss2014/programme/Bob.pdf](https://www.cs.ox.ac.uk/ss2014/programme/Bob.pdf)

[5] [https://graphicallinearalgebra.net/](https://graphicallinearalgebra.net/)

~~~
kmill
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
Yes, 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
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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knzhou
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.

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.

~~~
wodenokoto
A Nobel Price winner seems to disagree with you.

From the article:

> the American physicist Frank Wilcjek, who worked with Feynman in the 1980s,
> once wrote: “The calculations that eventually got me a Nobel Prize in 2004
> would have been literally unthinkable without Feynman diagrams.“

~~~
knzhou
Diagrammatic intuition is good. I use Feynman diagrams every day. But nobody
computes _with_ Feynman diagrams, they use them as an intuition aid to tell
them _what_ to compute, and that computation is done using standard notation.

~~~
wodenokoto
> ... and that computation is done using standard notation.

I really thought you would have ended that sentence with "Matlab" or "python",
not "standard notation"

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foxes
Ref of the original paper
[https://arxiv.org/abs/1911.00892](https://arxiv.org/abs/1911.00892). The
biggest issue for me is they don't have proper upper/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/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.

~~~
kmill
They're sneaking the Euclidean metric tensor in everywhere, which means
there's no distinction between upper and lower indices. I agree it's
confusing, and I'd rather they kept tensor orientations intact.

The dot product is double contraction with the Euclidean metric; your notation
is contraction of a vector and a dual vector.

------
slantaclaus
Not enough pictures in this article

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

~~~
bollu
While this is true, note that this special relationship between gradient,
divergence, and curl only exists in 3D, due to the fact that 2 + 1 = 3, and
the presence of the [hodge
star]([https://en.wikipedia.org/wiki/Hodge_star_operator](https://en.wikipedia.org/wiki/Hodge_star_operator))
operator that allows us to convert "areas" into "normal vectors to areas". A
curl is really a "infinitesimal area", which we then choose to interpret as "a
normal vector".

This is captured by the [De Rham
cohomology]([https://en.wikipedia.org/wiki/De_Rham_cohomology](https://en.wikipedia.org/wiki/De_Rham_cohomology)),
whose calculations are sometimes simplified by using the [Penrose
notation]([https://en.wikipedia.org/wiki/Penrose_graphical_notation](https://en.wikipedia.org/wiki/Penrose_graphical_notation))
as outlined above.

Having used the notation, I don't think it helps for vector calculus. And I
100% agree with you that we should teach the intuition for div, grad, and curl
first, but I wanted to bring to the attention of anyone who's never seen the
"fully general" form of these objects as to what it looks like.

