Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an introduction to exterior calculus, describing (for example) differential operators on curved surfaces in terms of the exterior derivative and the Hodge star.
I teach the class this way because that's how I myself learned differential geometry, but am starting to wonder how much value this approach really has for the students. Exterior calculus takes considerable effort to absorb, and students are often left wondering why we don't use simpler, more conventional language to discuss the same ideas.
Take the gradient, for example. One could say that the gradient of a scalar field $\phi$ on a curved surface is the unique vector field $\nabla\phi$ satisfying
$$ \langle \nabla\phi, u \rangle = D_u \phi, $$
where $\langle \cdot, \cdot \rangle$ is the Riemannian metric and $D_{\cdot}$ is the directional derivative. From there, it takes additional work to introduce the exterior derivative $d$ and the sharp operator $\sharp$ so that one can write the ever so slightly more succinct relationship
$$ \nabla \phi = (d\phi)^\sharp. $$
But what is really gained by having this slightly more succinct expression?
My question is an earnest one. I actually want to know, especially from people who do surface theory, why they bother to use exterior calculus and not a more elementary language. Or to put it another way:
- Are there clear examples where it's beneficial to use exterior calculus, as opposed to simpler language (e.g., writing everything explicitly in terms of the inner product)?
- How pervasive is exterior calculus in surface theory? E.g., roughly what percent of new results about surfaces would one estimate are expressed in the language of differential forms?
- Likewise, how much are new results being written in a more classical language?
- What did the historical development look like? I.e., why has exterior calculus become more (or less) popular over time?
In other words, since this is not my main area, I am trying to get some broad cultural knowledge about how pervasive this tool is, and how likely it is to benefit my students.
Thanks!
Best Answer
Well, there's really not a whole lot more to say beyond what Deane already wrote. He certainly hit the main points, but maybe I can expand a bit on what he wrote and comment on my own experience over the years both learning and teaching differential geometry.
Maybe I should say a little bit about my own education: Originally, I learned differential geometry from O'Neill's Elementary Differential Geometry, which introduces differential forms and makes very good use of them. I went on to read a series of papers in differential geometry that used differential forms extensively, particularly many beautiful papers of S.-s. Chern and his students, and his postdoctoral advisor, the great Élie Cartan. So differential forms became very natural to me, but when I started teaching differential geometry, I found that they were always a little bit of a barrier to students, who had to take some time getting used to them. I experimented with teaching the curves and surfaces course without them, using, for example, do Carmo's wonderful book Differential Geometry of Curves and Surfaces, which avoids differential forms in favor of a classical vector calculus in local coordinates approach and manages to cover a lot of great material from the classical literature. Ultimately, though, I became convinced that this was not an efficient way to go, and have reverted to teaching the curves and surfaces course using differential forms, but spending a little extra time at the beginning to redevelop vector calculus using forms, which has its own benefits.
Your example using differential forms to get a minor improvement in notation does not give any indication of the efficiency of actually using differential forms (mainly because you haven't really used the exterior derivative). Here's a better example: The construction of the Gauss curvature invariant of a metric $g$ on a surface. Locally write $g = {\omega_1}^2 +{\omega_2}^2$ where $\omega_1$ and $\omega_2$ are a ($g$-orthonormal) basis of $1$-forms. There is then a unique $1$-form $\phi$ that satisfies $\mathrm{d}\omega_1 = -\phi\wedge\omega_2$ and $\mathrm{d}\omega_2 = \phi\wedge\omega_1$, and there is a unique function $K$ such that $\mathrm{d}\phi = K\ \omega_1\wedge\omega_2$. Then $K$ clearly depends on $2$ derivatives of the coframing $\omega_i$, but it turns out to depend only $g$ and not on the choice of coframing. Here is why: If $g = {\bar\omega_1}^2 +{\bar\omega_2}^2$, then $\bar\omega_1 = \cos\theta\ \omega_1 + \sin\theta\ \omega_2$ and $\bar\omega_2 = \pm(-\sin\theta\ \omega_1 + \cos\theta\ \omega_2)$ for some function $\theta$ on the domain of the coframing. Then a simple computation yields $\bar\phi = \pm(\phi+\mathrm{d}\theta )$, so $$\bar K\ \bar\omega_1\wedge\bar\omega_2 = \mathrm{d}\bar\phi = \pm \mathrm{d}\phi = \pm K\ \omega_1\wedge\omega_2 = K\ \bar\omega_1\wedge\bar\omega_2\,, $$ so $\bar K = K$ depends only on the metric $g$. (It's hard to imagine any construction of $K$ and proof that it is invariantly defined that starts with writing $$ g = E(x,y)\ \mathrm{d}x^2 + 2F(x,y)\ \mathrm{d}x\mathrm{d}y + G(x,y)\ \mathrm{d}y^2 $$ and goes through the process of defining the Christoffel symbols and then $K(x,y)$ and then proving that the result doesn't depend on the choice of coordinates that isn't considerably longer than this.)
Moreover, you get easy proofs of fundamental results: For example: If $K$ vanishes identically, then $\mathrm{d}\phi = 0$, so, locally $\phi = -\mathrm{d}\theta$ for some function $\theta$. Using this to define $\bar\omega_i$ as above, we get $\mathrm{d}\bar\omega_i = 0$, so $\bar\omega_i = \mathrm{d}x_i$ for some local functions $x_1$ and $x_2$, so $g = {\mathrm{d}x_1}^2 + {\mathrm{d}x_2}^2$, i.e., $g$ is locally flat. Similarly, there are easy proofs that $K\equiv 1$ implies that $g$ is locally isometric to the unit $2$-sphere, and lots of other such results.
Of course, as Deane pointed out, it was Cartan who advocated and popularized the use of differential forms in surface theory and throughout differential geometry. Beginning in the 1890s and continuing through the 1940s, he wrote many highly influential papers and books using differential forms that effectively demonstrated their efficacy. The number of results that he was able to derive using them and the efficiency with which he did it is still astonishing today, and it made an enormous impression on the differential geometers of the time. Throughout the 20th century, his followers continued to develop and apply the techniques in higher dimensions, but also in classical surface theory. Many results from the classical literature were rewritten in forms language and extended. A typical (and beautiful) example is the paper by Chern and Terng, An analogue of Bäcklund's theorem in affine geometry (Rocky Mountain Journal of Mathematics (1980), 105–124, in which they give a much shorter proof of Bäcklund's theorem (about Euclidean surfaces with $K=-1$) using differential forms and discover a corresponding version in affine geometry. (One can give lots of examples of this kind, of course.)
I have no way of reliably estimating the percentage of current research papers about curves and surfaces that use differential forms, but it is fairly high, simply because the language is very efficient; I can say that 100% of my own papers, even the curves and surfaces papers, use differential forms. I can't even imagine translating some of them out into classical vector calculus notation without them becoming much longer and essential unintelligible.
The great efficiency of using differential forms accounts for their popularity; it's a pervasive language throughout differential geometry now. The resistance to introducing differential forms is that there seems to be an extra level of abstraction beyond ordinary vector calculus, but part of that is caused by the common feeling that you have to introduce tensors and all kinds of abstract concepts in order to define them. However, that's not really so if they are introduced the right way. Once students understand that $1$-forms measure velocities, $2$-forms measure areas, $3$-forms measure volumes, etc., they learn the rules pretty quickly and come to appreciate that the exterior algebra keeps track of what are otherwise messy formulas involving determinants and cross-products, etc. Anyone who has struggled to memorize the formulas for div, grad, and curl in cylindrical and spherical coordinates can appreciate the simplicity of the exterior derivative.