4 ms·

Turning the mathematics of vector calculus into simple pictures

139 points by respinal 7y ago

mikhailfranco 7y ago
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/
- fantispug 7y ago
  They do mention Penrose notation in the linked arxiv paper[1], as well as Cvitanović's Birdtracks[2] which let you do calculations on representations of groups. They claim they're making it more accessible with pedagogical examples (and innovations in representing differential vector calculus). I still prefer index notation. Their notation looks pretty intricate and I think it would be really hard to do long calculations in it. It's also not clear to me how you'd generalise it to non-commutative vector algebras like in quantum mechanics. [1] https://arxiv.org/pdf/1911.00892.pdf https://arxiv.org/pdf/1911.00892.pdf [2] http://birdtracks.eu/ http://birdtracks.eu/
- 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 ago
    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 7y ago
      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.
- eternalban 7y ago
  Penrose's paper: https://mathstat.dal.ca/~selinger/papers/papers/graphical-bib/public/Penrose-applications-of-negative-dimensional-tensors.pdf https://mathstat.dal.ca/~selinger/papers/papers/graphical-bi...
madengr 7y ago
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 7y ago
  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.
- kmill 7y ago
  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?
knzhou 7y ago
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 7y ago
  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.“
  - improbable22 7y ago
    Sure, but that's an odd reading of GP's "graphical notations like this one". Feynman diagrams are very useful, and made respectable lots of other graphical notations. There's a nice book on this: https://www.press.uchicago.edu/ucp/books/book/chicago/D/bo3534300.html https://www.press.uchicago.edu/ucp/books/book/chicago/D/bo35... But diagrams as a replacement for writing out tensor indices, like in TFA, which was what I presume GP meant... these have not proven widely useful. I guess people doing tensor network things use them.
  - knzhou 7y ago
    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 7y ago
      > ... and that computation is done using standard notation. I really thought you would have ended that sentence with "Matlab" or "python", not "standard notation"
foxes 7y ago
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 7y ago
  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 7y ago
Not enough pictures in this article

139 points by respinal 7y ago

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/

fantispug 7y ago
They do mention Penrose notation in the linked arxiv paper[1], as well as Cvitanović's Birdtracks[2] which let you do calculations on representations of groups. They claim they're making it more accessible with pedagogical examples (and innovations in representing differential vector calculus). I still prefer index notation. Their notation looks pretty intricate and I think it would be really hard to do long calculations in it. It's also not clear to me how you'd generalise it to non-commutative vector algebras like in quantum mechanics. [1] https://arxiv.org/pdf/1911.00892.pdf https://arxiv.org/pdf/1911.00892.pdf [2] http://birdtracks.eu/ http://birdtracks.eu/
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 ago
  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 7y ago
    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.
eternalban 7y ago
Penrose's paper: https://mathstat.dal.ca/~selinger/papers/papers/graphical-bib/public/Penrose-applications-of-negative-dimensional-tensors.pdf https://mathstat.dal.ca/~selinger/papers/papers/graphical-bi...

madengr 7y ago

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 7y ago
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.
kmill 7y ago
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?

knzhou 7y ago

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 7y ago
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.“
- improbable22 7y ago
  Sure, but that's an odd reading of GP's "graphical notations like this one". Feynman diagrams are very useful, and made respectable lots of other graphical notations. There's a nice book on this: https://www.press.uchicago.edu/ucp/books/book/chicago/D/bo3534300.html https://www.press.uchicago.edu/ucp/books/book/chicago/D/bo35... But diagrams as a replacement for writing out tensor indices, like in TFA, which was what I presume GP meant... these have not proven widely useful. I guess people doing tensor network things use them.
- knzhou 7y ago
  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 7y ago
    > ... and that computation is done using standard notation. I really thought you would have ended that sentence with "Matlab" or "python", not "standard notation"

foxes 7y ago

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 7y ago
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 7y ago

Not enough pictures in this article