Necessity of Differential Forms

Question

All the undergraduate and graduate texts on analysis introduce Differential and integral calculus (I will assume this introduction of basic calculus/analysis).

Among them, some books also introduce differential forms. I then understand that knowledge of differential forms is not too necessary in (Real/complex) analysis.

On the other hand, the books, where differential forms are included, do not give any motivation for their consideration in the subsection in which they introduce it. They start the Definition like ...

...... this expression is called $1$-form; ......this expression is called 2 form .....

It is not mentioned in any book, for what purpose it is getting introduced? This really bothers me and gives a feeling like it is memorizing or copy-pasting from some old books, the definitions of differential forms and bombarding it on readers brains! Even in many lectures, I heard that the concept of Differential forms is introduced just as a memorized definition and start games with it! No book explains what is their necessity in analysis?

I believe that almost all the mathematical concepts and especially differential forms have been introduced concerning at least some elementary problem or I feel using differential forms one can interpret some mathematical contexts in a better frame.

My question is

For the study of which elementary problems in analysis, differential forms are necessary?

Answer 1

If you want to extend calculus to the setting of manifolds, then you need differential forms. They are the natural kind of object to integrate over a manifold.

The way that integration works is that you chop up the region that you're integrating over into tiny pieces, compute the contribution of each piece, then add up all those individual contributions.

So how do we integrate over a $k$ dimensional manifold? Chop it up into tiny pieces, in such a way that the $i$th piece is approximately a parallelopiped spanned by tangent vectors $v^i_1,\ldots,v^i_k$. The contribution of the $i$th piece can be viewed as being a function of these $k$ vectors. Thus, to compute the contribution of each piece of the manifold, what we need is a gadget that will assign to each point $p$ on our manifold a real-valued function $f_p(v_1,\ldots,v_k)$. You can argue that $f_p$ should be alternating and multilinear, because chopping up the manifold more finely should not change the value of the integral (and because degenerate parallelopipeds should contribute $0$). We have now discovered the concept of a differential form.

Answer 2

Disclaimer: this is far from a complete answer and I'm hardly an expert on the topic

First one should say that differential forms certainly are not necessary for any problem the same way vector space are not necessary for any problem. They simply are a language which makes working with some problems easier.

I can give a few examples of "elementary" situations where differential forms can be used or are often used:

• Stoke's theorem can be stated using differential forms generalizing several theorems from "classical" vector calculus (Gauß' theorem, Green's theorem, the classical Stoke's theorem, ...).
• Brouwer's fixed-point theorem and the Hairy Ball theorem have quite elegant proofs using differential forms.
• The Bochner-Martinelli integral formula generalizes Cauchy's integral formula in higher dimensions and uses differential forms to integrate over the $2n-1$ dimensional boundary of an open set in $\mathbb{C}^n$.
• Maxwell's equations in electrodynamics can be written down in a very natural way using differential forms.

You may not find these applications elementary and I guess they are not completely elementary. That is probably why differential forms are not usually taught in intrductory courses.

Answer 3

I want to restate the answer I heard most frequently, which is that differential forms are a "coordinate-free" approach to calculus. I am far from an expert on them but having just taken my first course which really fleshed-out differential forms, I think what I say could be useful at this level.

Often, they do feel very throw-away in undergraduate texts, but this is largely due to the settings we do calculus on being very fixed at this point - the real numbers and, at a more advanced level, the complex numbers. At no point in our undergraduate career are we particularly pushed to consider what sort of things unify these settings or even how to generalize these ideas to other contexts. In fact, the idea of "other contexts" to consider calculus in is very amorphous at this stage, as it isn't clear what else exists besides these spaces!

Differential forms provide a very nice way of understanding what is "really going on" when we do calculus without needing to know where we're doing it exactly. We can extend the ideas of calculus we are familiar with to manifolds in general through differential forms.

However, if you are familiar with manifolds, you know we are looking at something which is locally Euclidean, so it still is not obvious to see why we need to change our language - if all of our problems are like problems in the reals, why change how we talk about them?

Well, for one, we don't want to have to concern ourselves so specifically with charts and atlases (beyond their existence) all the times to state even basic theorems, but this may still be unsatisfying. The answer I found most satisfying personally is that, in doing away with all these specific references to how our surface looks like the reals, differential forms are in some sense the "simplest" way to understand what's going on. They illustrate that coordinates are not in and of themselves what are important to what we're doing, but that what we're dealing with is actually more general and applicable than our normal calculus in $\mathbb{R}^n$ would have us think.

The most frequently cited example of this is Stokes' theorem, which, without the language of differential forms, is almost meaningless to state, but which generalizes nicely the Fundamental Theorem of Calculus and contains all the major theorems of vector calculus (Stokes', Green's, Gauss') within it.

Lastly, I know you did not ask for this explicitly, but if you are interested in learning more about differential forms Loring Tu's An Introduction to Manifolds may be worth investigating. It introduces these concepts very nicely in the environment of the reals before extending to manifolds and showing why the language of differential forms is needed to understand calculus on manifolds.

Answer 4

The main reasons for using differential forms come not from analysis, but from geometry. This is differential geometry and so still touches on analysis, so differential forms can have analytic properties, such as being measurable, Lipschitz-continuous, analytic (in the power-series sense), etc. But that's not why people care about them.

When I've seen them discussed in analysis textbooks, it's usually towards the end as optional material. I think the attitude is that you might be familiar with them (from differential geometry) but perhaps not know precise definitions or proofs of theorems about them, and so the analysis textbook will fill in the gaps. Since one purpose of an analysis course is to treat the topics in a calculus course rigorously, this topic should also be covered.

(Most calculus courses don't mention differential forms explicitly, but they are there under the hood. When you take the differential of $ x ^ 2 $ and get $ 2 x \, \mathrm d x $, that's a differential form; when you integrate a vector field $ \mathbf F ( x , y , z ) = M \mathbf i + N \mathbf j + P \mathbf k $ across a surface, the integrand $ \mathbf F ( x , y , z ) \cdot \mathrm đ \mathbf S $ is the differential form $ M \, \mathrm d y \wedge \mathrm d z - N \, \mathrm d x \wedge \mathrm d z + P \, \mathrm d x \wedge \mathrm d y $, etc. If you want a rigorous proof of Stokes's Theorem etc, then it's probably easier to prove the generalized Stokes Theorem for differential forms and treat everything as a special case of that.)

None of this is central to analysis; the analysis is just in support of the geometry. So from the perspective of analysis, it's an optional topic. If you ever want to study differential geometry, however, then you'll definitely need it. (But you can keep the analysis to a minimum if you assume that everything is infinitely differentiable, which is what rigorous geometry books usually do.)

Answer 5

The introduction of differential forms (:= d.f.) in a Calculus/Analysis lesson -not only for mathematicians but for physicists too- is absolutely necessary; provided that there is no Calculus/Analysis lesson without expressions such $\frac {dy}{dx}$ or $f(x)dx$, we cannot avoid d.f.

Teachers teaching without d.f. often answer the absolutely reasonable question "what is $dx$?" saying "it is just something formal" or "it is an extra small/infinitesimal quantity". Of course both answers are nonsense (the first is much worse), precisely because there is no reference to d.f.

Two basic examples.

1. Let $f\in D(I\subseteq \Bbb R, \Bbb R)$ a differentiable real function on an open interval $I$ and $x\in I$, such that $f'(x)\neq 0$. Then its differential at $x$ for a change $Δx$ is defined as

$$df(x, Δx)\equiv dy(x, Δx)=f'(x)Δx.$$

If our function is the identity $id_I:I\to \Bbb R:id_I(x)=x$, then $d(id_I)(x,Δx)\equiv dx(x, Δx)=Δx$, so the above expression can be written simply

$$dy=f'(x)dx.$$ It is immediate that both $dy, dx$ are linear functions $\Bbb R\to \Bbb R$ (they are of the form $v=au$; we also see $dy$ as a function of $dx$ only). These differentials constitute the basis of 1-forms $\Bbb R^2\to (\Bbb R^2)^*$, which means that every such 1-form $ω$ can be written as $ω=a_1dx+a_2dy.$

2. When we see the volume integral $\int_Af(x,y,z)dxdydz\equiv\int_A f(x,y,z)dV$ we have to ask about the integrand. Is there a relationship between $dxdydz$ and $dV$? They say that "there is just a formal identity between them". The answer is that $dxdydz$ is a 3-form called volume element and it is defined as follows:

Volume element $v$ of $\Bbb R^3$ is a 3-form defined via $v(e_1,e_2,e_3)=1$, where $\{e_i\}_{i=1,2,3}$ is the usual orthonormal basis of $\Bbb R^3$. We can proove easily that this form can be written as $v=dxdydz$ and that it gives the volume of solid made of three linear independent vectors $u_i$, i.e. $$\rm vol(u_1,u_2,u_3)=v(u_1,u_2,u_3).$$ Provided that d.f. is "something that can be integrated", the meaning of the above integral is complete.