What is the motivation for differential forms?

Question

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their properties, defined forms, learned of their pullbacks and the properties of these pullbacks, and defined the differential operator while learning some of its properties. I am currently reading about exact/closed forms in the build up to a certain "Poincare Lemma".

While the theory all seems to be fitting together (albeit with a bit of effort), there has been a nagging question. What is the motivation here? It has been my experience that many mathematical constructions (that I have encountered at least) are done with the goal of better understanding something. I feel like this thing is missing from my understanding of differential forms. Any insight will be appreciated.

Answer 1

Spivak seems to spend more time developing intuition in his book A Comprehensive Introduction to Differential Geometry, volume 1. On p. 111, in chapter 4, he writes:

Classical differential geometers (and classical analysts) did not hesitate to talk about "infinitely small" changes $dx^i$ of the coordinates $x^i$, just as Leibnitz had. No one wanted to admit that this was nonsense, because true results were obtained when these infinitely small quantities were divided into each other (provided one did it in the right way).

Eventually it was realized that the closest one can come to describing an infinitely small change is to describe a direction in which this change is supposed to occur, i.e., a tangent vector. Since $df$ is supposed to be the infinitesimal change of $f$ under an infinitesimal change of the point, $df$ must be a function of this change, which means that $df$ should be a function on tangent vectors. The $dx^i$ themselves then metamorphosed into functions, and it became clear that they must be distinguished from the tangent vectors $\partial/\partial x^i$.

Once this realization came, it was only a matter of making new definitions, which preserved the old notation, and waiting for everybody to catch up. In short, all classical notions involving infinitely small quantitites became functions on tangent vectors, like $df$, except for quotients of infinitely small quantities, which became tangent vectors, like $dc/dt$.

Answer 2

The most obvious uses of differential forms are related to integration. They are the language in which we express Stokes' theorem, for instance: whenever you have a compact, orientable manifold $M^n$ with boundary, the integral of a $(n-1)$-form $\omega$ over $\partial M$ equals the integral of $\mathrm{d}\omega$ over $M$ (in particular, the integral of an exact form over a closed manifold is always zero, as is the integral of a closed form over the boundary).

That is not all, of course. For example, closed/exact forms you mentioned give the de Rham cohomology, an important topological invariant. There's more, but for that you'll have to dig in a bit deeper.

Answer 3

Let's stick to 1-forms on $M$ for simplicity. If you are already convinced of the importance of vector fields on $M$, I have good news for you. Vector fields and 1-forms are dual objects in a suitable sense, but there is one good reason to work with 1-forms rather than vector fields. Namely, if one has a smooth map between manifolds $N\to M$, one can pull-back a differential form from $M$ back to $N$, but one cannot in general push forward a vector field on $N$ to a vector field on $M$. This is certainly not the whole story, but perhaps a beginning. What this shows is that there is a suitable functoriality for 1-forms that's not there for vector fields. That's why it is sometimes more useful to work with 1-forms rather than vector fields, even though the latter are more accessible intuitively.

Answer 4

The other answers are talking about manifolds, but even if you stick to $ \mathbb R ^ n $, there is a motivation from curiosity. There are all these things that we integrate in multivariable calculus and vector calculus: $ f ( x , y ) \, \mathrm đ A $ (area integral), $ f ( x , y , z ) \, \mathrm đ V $ (volume integral), $ \mathbf F ( x , y ) \cdot \mathrm d \mathbf r $ or $ \mathbf F ( x , y , z ) \cdot \mathrm d \mathbf r $ (line integral), $ \mathbf F ( x , y , z ) \cdot \mathrm đ \mathbf S $ (surface flux integral). If it's just notation, why is the notation so suggestive? If these are real things, then what are they? They're differential forms!

If you stick to $ \mathbb R ^ n $ for $ n \leq 3 $, then these can all be treated ad hoc, and we don't need the theory of differential forms to tie them all together. But even then, you might notice the similarities between the Fundamental Theorem of Calculus, Green's Theorem, Stokes's Curl Theorem, and Gauss's Divergence Theorem; and of course these are all special cases of a single generalized Stokes Theorem about exterior differential forms.

Then if you want to make sense of integration in $ \mathbb R ^ n $ for $ n \geq 4 $, it's clear that the differential forms are the basic concept, and we can associate only some of them (those of rank $ 0 $, $ 1 $, $ n - 1 $, and $ n $) with scalar fields or vector fields.