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

Question

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.

Answer 1

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.

Ultimately, this picture is a schematic, but it can be made quite precise at the cost of a lot more algebra and abstraction.