Ytukyg

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?

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.

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 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.