I’m working on a PhD adjacent to computational differential geometry, and while I’ve made a lot of progress on the computation, I still don’t have much intuition for k-vectors and k-forms. I love coming across articles like this that help me build intuition. The article for which this is the second part was really helpful, but I’m going to have to come back to this second part a few times to fully grasp it. I also loved this quote from one of the articles listed as a source (but with a broken link, I found it at http://yaroslavvb.com/papers/notes/piponi-on.pdf): “Think of a vector as a pin, and a one-form as an onion. You evaluate a one-form on a vector by counting how many onion layers it goes through.”
> I still don’t have much intuition for k-vectors and k-forms
The references that you cite are great. Also Tristan Needham's book "Visual Differential Geometry and Forms", or the classic "Gravitation" where this visual language was fist exposed.
Besides this continuous interpretation, you may also enjoy further insight via the discrete case, called "discrete exterior calculus", or "discrete differential geometry" or even "graph signal processing".
The idea is the following. The base space is a graph, consisting of vertices, edges joining pairs of vertices, triangles joining triplets of edges, and so on.
Now, k-forms on the graph are, by definition, functions defined over the k-cliques of the graph. Thus:
0-forms are functions defined on the vertices
1-forms are functions defined on the edges
2-forms are functions defined on the triangles
...
k-forms are functions defined on the (k+1)-cliques (i.e. complete subgraphs of k+1 vertices)
Now the exterior derivative of a k-form is the (k+1)-form obtained by computing differences over each k-subclique.
For example, if f is a "scalar-field" or 0-form, its differential df is the 1-form defined by (df)(a,b)=f(b)-f(a) for each edge (a,b). If "F" is a 1-form, then dF is the 2-form given by (dF)(a,b,c)=F(a,b)+F(b,c)-F(c,a). And so on. Thus, it is clear intuitively that ddf=0, because all the edge differences cancel out when you traverse the edges of a triangle.
If your graph has a notion of duality (a mapping between k-cliques and (n-k)-cliques, for example when the graph is planar), then you can define the laplacian Δ=dd and all its associated goodies.
Furthermore, you can easily define the integral of a k-form over a k-chain (a subset of k-cliques), and the boundary of a k-chain as the (k-1)-chain obtained by differentiating its indicator function. The the discrete version of Stokes theorem becomes an algebraic triviality.
I recall feeling immense pleasure when I first learned about all this. Hoping you feel something similar!
The same could be said when they are written using Gibbs notation with div and curl. Surprisingly, when Einstein wrote his paper on special relativity, Gibbs vector notation had not yet been widely adopted, and Einstein himself wrote out Maxwell's equations in component form.
In part yes, but these symbols are not contrived aliases that merely wrap the underlying complexity. They are well defined geometrical onjects individually, can be used separately in other contexts etc.
That such a frugal abstraction is possible is by no means guaranteed. Other physical systems are not as elegantly expressed using differential forms.
I’d like to recommend Prof Chern’s course on Discrete Differential Geometry https://cseweb.ucsd.edu/~alchern/teaching/cse270_wi24/. The lecture slide on Exterior Calculus includes many good illustrations of differential forms, and there are more cool figures on related topics. There is also a pdf textbook neatly formatted if you want to dive deeper.
I think this is a fancy way of saying, "the visualization of a vector field". Differential forms are a generalization of a vector field, most of the images in the post that are visualizations of differential forms are images of vector fields. Using vector field instead of differential form, makes the title accessible to people who've taken multivariable calculus, which is a much larger group of people than people that know what a differential form is.
I'm sure there's some interesting stuff in the difference between a differential form and a vector field that the author is trying to get at, it's just interesting that all the images are of vector fields.
> it's just interesting that all the images are of vector fields.
Because they are! You can indeed "flatten" all these forms into vector and scalar fields, and you lose some information but the data is the same. It's like forgetting the types of objects in a programming language and considering only their representation as byte arrays in memory.
When you do multivariable calculus, you soon realize that there are two different kinds of vector fields: those that are "gradients" or "rates" and those that are "speeds", or "displacements" or "flows". They have different units, like 1/length or length/time. Notice that if you change the units of length from meters to centimeters, the arrows that represent gradients become shorter, and the arrows that represent speeds become longer (assuming that you keep the same data).
Likewise, there are two different kinds of scalar fields: potentials, temperatures, etc, that are invariant to unit changes; and densities that change with the cube of the unit scaling.
You don't need differential forms to understand any of this. But differential forms provide a nice formalization where all these objects are of different types, and it explains what differential operators can you apply to objects of one type to obtain another, and so on.
Vectors are rates. Covectors are gradients.
-- Luc Florack
I’m working on a PhD adjacent to computational differential geometry, and while I’ve made a lot of progress on the computation, I still don’t have much intuition for k-vectors and k-forms. I love coming across articles like this that help me build intuition. The article for which this is the second part was really helpful, but I’m going to have to come back to this second part a few times to fully grasp it. I also loved this quote from one of the articles listed as a source (but with a broken link, I found it at http://yaroslavvb.com/papers/notes/piponi-on.pdf): “Think of a vector as a pin, and a one-form as an onion. You evaluate a one-form on a vector by counting how many onion layers it goes through.”
Edit: This one also looks good: https://math.uchicago.edu/~may/REU2018/REUPapers/Bixler.pdf
> I still don’t have much intuition for k-vectors and k-forms
The references that you cite are great. Also Tristan Needham's book "Visual Differential Geometry and Forms", or the classic "Gravitation" where this visual language was fist exposed.
Besides this continuous interpretation, you may also enjoy further insight via the discrete case, called "discrete exterior calculus", or "discrete differential geometry" or even "graph signal processing".
The idea is the following. The base space is a graph, consisting of vertices, edges joining pairs of vertices, triangles joining triplets of edges, and so on.
Now, k-forms on the graph are, by definition, functions defined over the k-cliques of the graph. Thus:
0-forms are functions defined on the vertices
1-forms are functions defined on the edges
2-forms are functions defined on the triangles
...
k-forms are functions defined on the (k+1)-cliques (i.e. complete subgraphs of k+1 vertices)
Now the exterior derivative of a k-form is the (k+1)-form obtained by computing differences over each k-subclique.
For example, if f is a "scalar-field" or 0-form, its differential df is the 1-form defined by (df)(a,b)=f(b)-f(a) for each edge (a,b). If "F" is a 1-form, then dF is the 2-form given by (dF)(a,b,c)=F(a,b)+F(b,c)-F(c,a). And so on. Thus, it is clear intuitively that ddf=0, because all the edge differences cancel out when you traverse the edges of a triangle.
If your graph has a notion of duality (a mapping between k-cliques and (n-k)-cliques, for example when the graph is planar), then you can define the laplacian Δ=dd and all its associated goodies.
Furthermore, you can easily define the integral of a k-form over a k-chain (a subset of k-cliques), and the boundary of a k-chain as the (k-1)-chain obtained by differentiating its indicator function. The the discrete version of Stokes theorem becomes an algebraic triviality.
I recall feeling immense pleasure when I first learned about all this. Hoping you feel something similar!
In some domains differential forms are mind-mindbogglingly expressive. E.g. Maxwell's equations boil down to [1]:
dF = 0, d*F = J
You literally could not use fewer letters. But the corresponding visualizations are not particularly easy or illuminating.
[1] https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...
Sure, but that is because there's a lot of context that is tucked away inside the formalism and corresponding syntax.
The same could be said when they are written using Gibbs notation with div and curl. Surprisingly, when Einstein wrote his paper on special relativity, Gibbs vector notation had not yet been widely adopted, and Einstein himself wrote out Maxwell's equations in component form.
In part yes, but these symbols are not contrived aliases that merely wrap the underlying complexity. They are well defined geometrical onjects individually, can be used separately in other contexts etc.
That such a frugal abstraction is possible is by no means guaranteed. Other physical systems are not as elegantly expressed using differential forms.
I’d like to recommend Prof Chern’s course on Discrete Differential Geometry https://cseweb.ucsd.edu/~alchern/teaching/cse270_wi24/. The lecture slide on Exterior Calculus includes many good illustrations of differential forms, and there are more cool figures on related topics. There is also a pdf textbook neatly formatted if you want to dive deeper.
Web Archive link: https://web.archive.org/web/20240921132410/https://web.tecni...
I think this is a fancy way of saying, "the visualization of a vector field". Differential forms are a generalization of a vector field, most of the images in the post that are visualizations of differential forms are images of vector fields. Using vector field instead of differential form, makes the title accessible to people who've taken multivariable calculus, which is a much larger group of people than people that know what a differential form is.
I'm sure there's some interesting stuff in the difference between a differential form and a vector field that the author is trying to get at, it's just interesting that all the images are of vector fields.
> it's just interesting that all the images are of vector fields.
Because they are! You can indeed "flatten" all these forms into vector and scalar fields, and you lose some information but the data is the same. It's like forgetting the types of objects in a programming language and considering only their representation as byte arrays in memory.
When you do multivariable calculus, you soon realize that there are two different kinds of vector fields: those that are "gradients" or "rates" and those that are "speeds", or "displacements" or "flows". They have different units, like 1/length or length/time. Notice that if you change the units of length from meters to centimeters, the arrows that represent gradients become shorter, and the arrows that represent speeds become longer (assuming that you keep the same data).
Likewise, there are two different kinds of scalar fields: potentials, temperatures, etc, that are invariant to unit changes; and densities that change with the cube of the unit scaling.
You don't need differential forms to understand any of this. But differential forms provide a nice formalization where all these objects are of different types, and it explains what differential operators can you apply to objects of one type to obtain another, and so on.
Backup https://web.archive.org/web/20240213003201/https://web.tecni...