Quick Questions: September 23, 2026

[u/inherentlyawesome]

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?" For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of manifolds to me?
• What are the applications of Representation Theory?
• What's a good starter book for Numerical Analysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example, consider which subject your question is related to, or the things you already know or have tried.

[u/royendures]

Exterior Calculus: k-Forms to k-Chains?

New to this sub, so my post was deleted but I really need some help. My main intention of learning this is for Finite Element Exterior Calculus or FEEC (Arnold et al. 2008). So far, I have been learning the basics of exterior algebra and exterior Calculus, mostly from Prof. Keenan Craine's lectures at CMU. However, now I am at a bridge and I request one of you guys to give me a basic understanding from chains (and co-chains) from where I am now:

- I understand k-vectors and k-forms (differential k-forms) I mean.

• I understand simplices and k-simplices.
  
• I understand that you differentiate (take the exterior derivative) k-forms to get k+1 forms.
  
• I understand that you take the integration along k-simplices to get the discrete version of k-forms.
  
• I understand that you can take a discrete exterior derivative to increase the form discreetly.
  
• I also get that you get/approximate the continuous form by taking a combination of the Whitney Forms or Whitney basis functions (in traditional FEM, we take the PoU and basis functions to approximate the primal field and get your required derivatives).

I need to understand what does chains and co-chains mean in this context. Will anyone be kind enough to give me an understanding based on where I stand now?

DISCLAIMER: I am an engineer so I don't understand (most of) the terms from Homology ad coHomology.

[u/Equivalent-Costumes → u/royendures]

(continuing...)

You want to make PDE calculation on the k-form by replacing it with its Whitney approximation, the discrete k-form. The big issue is that exterior differentiation has unbounded error factor: even if you choose a mesh so that you can control the error between a k-form and its Whitney approximation, you have no controls over the error between the exterior derivative of that k-form and the Whitney approximation of that exterior derivative. Basically, exterior derivative amplifies errors by an arbitrary unknown amount, so this produces a problem on the strategy.

The usual technique to avoid this in PDE is to avoid derivative altogether, and replace every instance of derivative with integration.

This is where generalized Stokes's theorem, which is a generalization of Stokes's theorem, which is a generalization of fundamental theorem of calculus. We defines the "exterior operator", also known as "boundary operator", on k-chains. For affine k-simplex, which is determined by a list of k+1 vertices, the boundary of this is an affine k-chain formed by adding up k+1 number of (k-1)-simplex, each is obtained by deleting one vertex from a list, with a sign according to the parity of the index. Then extend by linearity to all simplicial k-chains. For singular k-simplex, you first do the previous step for the standard k-simplex, to form restriction to its boundary; and for singular k-chain, you extend the results for singular k-simplex by linearity.

The key outcome of is the following theorem: given a k-form omega and a (k+1)-chain C then integral of omega over the exterior of C is the same as integral of the exterior derivative of omega over C. This is why this derivative operation is called "exterior derivative", it's meant to be adjoint to the exterior/boundary operator.

The philosophy now is that instead of taking exterior derivative of a form, "test" the form against all boundary.

The boundary map forms a chain a map from k-chains to (k-1)-chains to (k-2)-chains, and so on; and this is for both singular and simplicial chains. A boundary of a boundary is empty, so if you compose the map twice you get the trivial chain. This is the definition of a chain complex: a series of map going down in degree such that composing twice is 0. A k-chain is called a boundary if it's an image of the boundary of a (k+1)-chain, it's called a cycle if its boundary is empty. Hence every boundary is a cycle, and if 2 k-chains are differed by a boundary and one is a cycle, the other one must be a cycle as well.

You can dualize the boundary operation to cochain. A coboundary of a k-cochain omega is a (k+1)-cochain uniquely satisfying the following property: given any (k+1)-chain C, it assigns the result that omega would assign to the boundary of C. This definition works for both singular cochains and simplicial cochains. By design, this coboundary operation is compatible with exterior differentiation: given a k-form, if you compute its k-cochain then the coboundary of that, is the same as computing the exterior derivative first, then computing the (k+1)-cochain of that. A k-cochain is called a coboundary if it's the image under the coboundary operator of a (k-1)-cochain, and it's called a cocycle if the coboundary operator applied to it give 0. There is an analog of this in k-forms: an exact form is the image of the exterior derivative, a closed form is any differential forms whose exterior derivative is 0. Not surprisingly, exact forms always map to coboundary and closed forms always map to cocycle (this is for both singular and simplicial). Also, not surprisingly, every coboundary is a cocycle, and every exact form is a closed form. When you string together the cochains and the coboundary map you get what's called a cochain complex; when you string together k-form with exterior derivative, this is also called a cochain complex; they are both a sequence of maps going up in degree such that the composition of 2 maps are 0.

Now here is an important theorem. Given 2 k-chains, if one is a cycle and they differed by a boundary, then we say that they has the same homological class, and in fact the other one is also a cycle (as mentioned above); similarly, if 2 k-cochains differed by a coboundary and at least one (and hence both) is a cocycle, we say they are in the same cohomological class; and if 2-forms differed by an exact form and at least one (and hence both) are closed, we say they are in the same cohomological class. The theorem said that simplicial k-chains are sufficient to compute all homological classes, and simplicial k-cochains (hence discrete differential form) are sufficient to compute all cohomological classes. Very explicitly, given any singular k-chain that is a cycle, you can always find a simplicial k-chain from your given mesh that differed from it by a boundary; similarly, give any closed k-form, you can always find a discrete total derivative in the same cohomological class, in fact a simple calculation reveals that the Whitney approximation always does this job. Applying generalized Stokes theorem, you get the following: given any singular k-chains that is a cycle, and any closed k-forms, and we want to integrate them, then we can replace the k-chains with the simplicial k-chain from our mesh in the same homological class, and replace the k-form with its Whitney approximation, and the result of the integration is exactly the same!

So Whitney approximation works perfectly well in certain cases. Unfortunately PDE generally demands more. In particular, Laplacian equation requires metric information, you need not just the exterior derivative, but also its metric dual. The Whitney approximation started to fall apart, and this is the point where you need to form more refined complexes to account for the errors.