# How should I conceptualize differential forms as a generalization of scalar and vector fields?

I'm reading Munkres' "Analysis on Manifolds" at the moment, and I'm having a bit of trouble understanding what exactly differential forms are.

I understand that the point of integrating forms over manifolds is to generalize the concept of integrating functions over intervals/curves/surfaces, as you do in a standard multivariable calculus course.

Manifolds are much easier to generalize from surfaces to me. Taking out all the details, we're just considering subsets of $$\mathbb{R}^n$$ embedded in $$\mathbb{R}^m$$. Surfaces, but in more dimensions (to keep it brief).

Differential forms are MUCH harder to visualize for me. Perhaps it's because I don't tend to think of things from a linear algebra perspective, but the jump from a scalar field (a 0-form) or a vector field (a 1-form) to a $$k$$-form is WAY more difficult.

I just think of a scalar field as, for example, measuring mass or assigning certain parts of $$\mathbb{R}^n$$ as more "weighted" as other parts. I think of a vector field as, concretely, assigning every point in $$\mathbb{R}^n$$ to a vector, and a lot of times imagining it as "fluid flow" is a helpful way for students to visualize this stuff and what integrals may represent.

What is a $$2$$-form though? I understand that an example of one is the curl of a vector field. Is there something concrete that I can point to or think of to help me visualize what a $$k$$-form may represent? And, for the record, an integral? An integral of a scalar field just sums up over some subset of $$\mathbb{R}^n$$, everything where the $$0$$-form "weighs" it, in a sense. And an integral over a $$1$$-form, or a vector field, calculates this, where a greater magnitude signifies more "fluid flow", and takes direction into account. I think I'm missing the linear algebra portion of the definition of a $$k$$-form, but seeing the definitions with multi-indeces and sums just isn't useful to me.

Can someone shed some light on some physical interpretation of $$k$$-forms, and how I can think of generalizing scalar and vector fields in this fashion?

Thanks.

• You might find some of my lectures on YouTube helpful, especially for concrete calculations and applications. That said, my favorite answer to your question — aside from the standard physics applications you basically know — would be that the curvature $2$-form of a surface, when you integrate over a piece of the surface, tells you how much net curvature it has. You can make sense of this for higher-dimensional manifolds, integrating appropriate $2$-forms over little pieces of $2$-dimensional submanifolds. This is not a flux integral. Dec 23, 2020 at 6:43
• @TedShifrin I am actually watching your lecture series RIGHT NOW, and I can say you are fantastic, and thank you for having these videos. You go in a quite different order than Munkres does, and I'm a little confused as to where to start. It looks like you do a lot of multivariable calculus that I've already done (multiple integration, "regular" surface integration" as part of your 3510 videos. You also do a lot of linear algebra where Munkres does none. From someone who's done and comfortable with the "standard" multivariable calculus, but not as good with LA, where do you suppose I start? Dec 23, 2020 at 6:48
• I suppose, in particular, UG does multivariable calculus with differential forms as part of their "multivariable calculus" and does all of multi-variable calculus in two semesters. Our university does multi-variable analysis up to de Rham cohomology, Poincare Lemma, and an intro to Riemannian geometry. The way the undergrad multivariable is in 2 sections as opposed to the graduate-level analysis perspective is confusing to me. Do you feel the 3510 videos are a good supplement to Munkres' book, and can you help me figure out which portions of your content you think would be applicable? Dec 23, 2020 at 6:55
• My course was for students who knew neither linear algebra nor multivariable calculus. Start with the differential forms material, and go back and fill in linear algebra as needed. Other than the notions of linearity and multilinearity, you shouldn't need a lot of the rest. Do determinants carefully, if course. My treatment is less abstract than Munkres or Spivak — no tensors, just everything in terms of determinants. Dec 23, 2020 at 6:56
• Oh, and just a handful of the best students took my course. Your course, like many in Europe, is very sophisticated and may not have enough concrete computation for my taste. You'll find a few lectures on the topological stuff, although I did not define deRham cohomology for first- and second-year students. Dec 23, 2020 at 6:59

If you're looking for a bit of geometric intuition, I've found it useful to think of 2-forms in terms of bivectors (or multivectors more generally): if one thinks of vectors in $$TM$$ as "infinitesimal oriented lengths", then bivectors, which are elements of $$\Lambda^2TM$$ can be thought of as "infinitesimal oriented areas" and so on. The wedge product fits nicely into this picture: for $$u,v\in T_pM$$ the element $$u\wedge v\in\Lambda^2TM$$ corresponds to the parallelogram spanned by $$u$$ and $$v$$ (with an induced orientation). Two different pairs of vectors have the same wedge product precisely if the paralellograms have the same area and are coplanar (in an oriented sense). A bit of algebra/geometry can show that this equivalence is related to the antisymmetry of the wedge product.
Differential forms, however, are not multivectors, they are instead dual to multivectors. That is, a $$k$$-form is a linear map from the space of $$k$$-vectors to the real numbers. They can be defined other ways (such as alternating linear maps on $$TM$$), but these definitions end up being isomorphic.
Integration fits nicely into this geometric picture: for an oriented $$k$$-dimensional (sub)manifold $$N$$ and a $$k$$-form $$\omega$$, one can think of $$\int_N\omega$$ as breaking up $$N$$ into a bunch of small $$k$$-dimensional volumes (expressed as multivectors), evaluating $$\omega$$ on these multivectors, and then summing. This is more or less analogous to Riemann integration in calculus.