What is a differential form?

can someone please informally (but intuitively) explain what "differential form" mean? I know that there is (of course) some formalism behind it - definition and possible operations with differential forms, but what is the motivation of introducing and using this object (differential form)? I have heard that they somehow generalize integration, are used for integration of manifolds and can evaluate k-dimensional integrals in n-dimensional space ($k \leq n$), but is it really true and is it the main motivation of introducing this object into mathematics? Thank you for explanation

To talk about differential forms, first we need to talk about manifolds and vector fields. Informally speaking, a manifold is any space which is locally Euclidean. That is, the area around every point in a manifold "looks like" Euclidean space, but the space as a whole may not be Euclidean. Examples include spheres and tori. More complex examples are varied and interesting, but are difficult to define in an informal setting.

A smooth manifold is a manifold where the Euclidean regions around each point are in some sense "compatible." This means that if the Euclidean regions of two points overlap, I can both Euclidean coordinate systems in that overlap region, and transfer from one to the other in an infinitely differentiable way.

A smooth function on a smooth manifold is a function whose range is the real numbers, and which is infinitely differentiable with respect Euclidean coordinate systems in the Euclidean regions around points in the manifold. The set of all smooth functions on a manifold $M$ is called $C^\infty(M)$

If things are moving a little fast for you, you may some up the last three paragraphs as, "we have spaces that up close look like Euclidean space, and functions on them that are in some sense differentiable."

Now we will talk about a vector at a point on a smooth manifold. This definition is probably going to sound really strange, but it really is the simplest way to define vectors on smooth manifolds. A vector $v$ at a point $x$ in a smooth manifold $M$ is any function whose domain is $C^\infty(M)$ and whose range is $\mathbb R$, and which satisfies the following three properties:

• $v(f+g)=v(f)+v(g)$
• $v(\lambda f)=\lambda v(f)$
• $v(fg)=v(f)g(x)+f(x)v(g)$

Where $f$ and $g$ are smooth functions on $M$ and $\lambda$ is a real number (notice something that looks like the product rule). What does this mean, and how does it relate to vectors as we are used to seeing them? We are used to seeing vectors defined by a collection of components. But the problem with that is that those coordinates depend on the coordinate system we choose to use. The definition I just gave does not. But if you like coordinates, do not worry; we can transfer between these two definitions. If your coordinates are $(x_1,\cdots,x_n)$ and your vector expressed in Euclidean coordinates at a point $x$ is $v=(v_1,\cdots,v_n)$, we can write the vector as an object of the form I defined above by writing $$v=v_1\frac{\partial}{\partial x_1}+\cdots+v_n\frac{\partial}{\partial x_n}.$$ It can act on a function $f$ by differentiation: $$v(f)=v_1\frac{\partial f}{\partial x_1}(x)+\cdots+v_n\frac{\partial f}{\partial x_n}(x).$$ It is easy to observe that this satisfies each of the three properties above.

A smooth vector field on a smooth manifold is a collection of vectors on a manifold, one at each point, which vary is a smooth (differentiable) way. In other words, it is a function $X$ whose domain and range are $C^\infty(M)$ such that

• $X(f+g)=X(f)+X(g)$
• $X(\lambda f)=\lambda X(f)$
• $X(fg)=X(f)g+fX(g)$

If it is easier for you, it is okay to imagine a vector as a little arrow lying tangent to some surface, and to imagine a vector field as a bunch of such arrows covering the manifold. This is a natural image to think of, but it turns out to be rather unhelpful in practice. But as this is an "informal discussion," go ahead.

We are finally ready to define a differential form.

A differential $k$-form on an $n$ dimensional smooth manifold $M$ is any multilinear function $\omega$ which takes as input $k$ smooth vector fields on $M$, $X_1,\cdots,X_k$ and outputs a scalar function on $M$ so that $$\omega(X_1,\cdots,X_i,\cdots,X_j,\cdots,X_k)=-\omega(X_1,\cdots,X_j,\cdots,X_i,\cdots,X_k).$$ The latter property is called antisymmetry.

So, what is the motivation behind such an object? As far as I know, the most important application of differential forms is, by far, integration on manifolds. There may have been some other reason for their initial discovery and definition, but this is what they are used for. When you think of integration, you think of calculating area and volume. In an $n$-dimensional manifold, we may want to measure the volume or area of any submanifold of $n$ with any dimension less than or equal to $n$. Given a coordinate system on $M$, a differential $k$-form tells us how to measure $k$-dimensional volume according to that coordinate system. Assuming you have taken a multivariable calculus course, you probably remeber seeing a picture of a spherical coordinate volume element.

Pictures like this one also give us some idea about how differential forms work. The picture labels infinitesimal changes in the $\theta$, $\phi$ and $r$ directions, and shows how we can calculate the infinitesimal volume swept out by these changes. Instead, we could consider three vector fields which at each point have vectors tangent to the $\theta$, $\phi$ and $r$ directions respectively. A differential 3-form would combine these three vector fields into the same volume element.

Why the antisymmetry? The antisymmetry allows us to consider orientations. Again, if you have studied multivariable calculus, you know that when integrating over a surface in 3 dimensional space, it is usually important to note which direction normal vectors to the surface point. But if our surface sits in a space of 4 dimensions or more, there is not a unique normal direction at each point, so we instead use the order of our coordinates to determine orientation. If we switch two coordinates we switch orientations. We will get the same result from integration except the sign will be reversed.

I hope I answered some of your questions.

Unfortunately, there is a ton of formalism behind differential forms, and an "intuitive" explanation of them for a non-specialist is probably too much to ask. I myself have spent a reasonable amount of time thinking about what differential forms "are," trying to derive them from first principles, or trying to understand them as generalizations of this-or-that object in elementary calculus.

At this point, however, I think that the best way to approach the daunting concept of differential forms is to realize that differential forms are defined to be the thing that makes Stokes' Theorem true. In other words, you can approach understanding forms in two different ways:

1. You can try to understand differential forms first, and interpret Stokes' Theorem as a profound result about forms, or
2. You can realize that the Fundamental Theorem of Calculus, Green's Theorem, Divergence Theorem, Kelvin-Stokes, etc. are all tantalizingly similar, and you can then try to understand why forms are the correct abstraction that makes all of these theorems special cases of the same general Stokes' Theorem.

I think that the second approach is more enlightening, and leads to less banging your head against the wall trying to understand "why" the definitions are what they are. So I would advise first thoroughly understanding all of the theorems I listed above, and then, once you've convinced yourself that they are all "the same" in some mysterious way you can't quite put your finger on, try picking up a book on differential forms that doesn't start with manifolds as a prerequisite, e.g. Spivak's Calculus on Manifolds. Understanding determinants very well probably won't hurt either.