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Stream: learning: questions

Topic: string diagrams for vector calculus

John Baez (Oct 06 2025 at 08:41):

Somewhere I've seen a paper on how to use string diagrams for vector calculus identities, but I'm not finding it - perhaps because the authors were not knowledgeable about the string diagram literature and didn't say "string diagram".

Arula Ratnukar is an impressive young neurobiologist who is studying Lagrangian mechanics just to improve her ability to write SF stories. She had a test on vector calculus and did this nice calculation.

This would be more pleasant with string diagrams - at least for people who understand string diagrams.

Peva Blanchard (Oct 06 2025 at 09:01):

It is not a paper, but maybe graphicallinearalgebra.net.

edit: actually no, the website doesn't seem to mention vector calculus.

Oscar Cunningham (Oct 06 2025 at 09:07):

The Graphical Linear Algebra site works in the monoidal category with $\oplus$ as the monoidal product, which is not at all what we want here. We want strings next to each other to represent the tensor product $\otimes$.

I think the search we want to do is for 'vector calculus' along with 'Penrose graphical notation'. Indeed the Wikipedia page for Penrose's notation has a picture that essentially explains the whole thing:
image.png

Oscar Cunningham (Oct 06 2025 at 09:10):

Here's a paper which is probably the one you're thinking of: Vectors and tensors analysis by Penrose’s graphical notation

Oscar Cunningham (Oct 06 2025 at 09:15):

On the other hand, I kinda hate these stupid 'bubbles' representing the derivative. I think you should be able to do the whole thing without them. Write $C(\mathbb{R}^3)$ for the real vector space of sufficiently nice functions on $\mathbb{R}^3$. Then have one kind of string in your diagram for $\mathbb{R}^3$, and one for $C(\mathbb{R}^3)$. You can think of a vector field as an element of $C(\mathbb{R}^3)\otimes\mathbb{R}^3$ and represent it as a box in the string diagram with one of each kind of leg. Then the derivative is just a linear map $C(\mathbb{R}^3)\to C(\mathbb{R}^3)\otimes\mathbb{R}^3$, and everything becomes normal string diagram manipulations with no bubbles.

Oscar Cunningham (Oct 06 2025 at 09:57):

Here's a first attempt at that:
IMG_20251006_104225487_DOC3.jpg

Oscar Cunningham (Oct 06 2025 at 12:00):

Another paper: Boosting Vector Differential Calculus with the Graphical Notation

John Baez (Oct 06 2025 at 13:10):

Thanks, everyone! That last paper, "Boosting Vector Differential Calculus with the Graphical Notation", is the one I was trying to remember! In equation (28) it completes a string diagram proof of the identity Arula Ratkunar was proving,

$\nabla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} \,+\, (\mathbf{B} \cdot \nabla)\mathbf{A} +$
$\mathbf{A} {\times} (\nabla {\times} \mathbf{B}) \,+\, \mathbf{B} {\times} (\nabla {\times} \mathbf{A})$