Saturday, May 14, 2005


When I first heard the word tensor toward the end of high school, I liked the sound of it. But it took me until a couple years into graduate school to get to my current understanding.

It's not so uncommon for students to come into college with a bit of formal mathematical coursework beyond basic calculus. In my case, I had a second semester calculus course and some differential equations. The math department at UMCP, in a bid to catch early the interest of potential students, invites incoming freshmen with such background to take an introductory real analysis sequence. I took it. It was a shock.

I was good at math in high school, which really meant that I was good at setting up and executing various sorts of calculations. I'd had some experience with proofs, particularly in my high school geometry class, but they were mostly very simple thing. This was the first time I really engaged in the sort of rigorous reading, writing, and exploration that characterizes the subject. Usually, I might say I learned about proofs, and leave it at that; but rigorous mathematical reasoning is a skill that extends beyond the ability to turn coffee into proofs, and just as it bothers me when people think mathematicians do nothing but write page after page of formulae, so it bothers me when (more sophisticated) folks think mathematical writing consists of nothing but theorems, lemmas, and proofs connected haphazardly by a few conjunctions and a definition or two.

At the end of the year, we covered some theorems about multi-dimensional integration from a geometric point of view. In particular, we discussed the generalized version of Stoke's theorem, which meant that we had to learn about differential forms. It was my first introduction to tensors, though we didn't actually say tensor in class, and I didn't really realize what we were working with until rather later.

I was able to do the calculations, and to prove simple theorems, but I don't think any of us had a real feel for differential forms. The intuitive understanding came later.

In my senior year, I took a graduate Riemannian geometry course. It was another shock. I wasn't ready for it, and at the time it seemed like I was catching only a very little bit of the material. It was by no means the fault of Schwartz, who was an inspired lecturer, but that was the only math course I ever took in which I felt completely lost by the semester's close. We learned about tensors and tensor bundles in that class, but what little luck I had understanding the idea really was a result of learning about tensor products in the graduate algebra sequence (which I took at the same time as the geometry class). I remember talking about it to one of the algebraists in the department, who said Well, I know what the algebraic definition of a tensor is, but I've always wished I understood how the physicists think of them. Indeed.

William James said we learn to swim in the winter and to skate in the summer I may not have understood what was going on in my Riemannian geometry class, but I did remember it, and over time I started to digest it a little better. I took a course in differentiable manifolds -- usually a prerequisite for a course on Riemannian geometry, but I've always taken prerequisites rather loosely -- and things made a lot of sense; combined with a course on nonlinear finite elements for continuum mechanical problems and a lot of outside reading, I got a context in which all the things I didn't understand when I took the Riemannian geometry course suddenly made sense.

In all, I have seen four distinct presentations of tensors: a purely algebraic presentation, as suitable for groups or rings as for linear spaces; a presentation based on Kronecker products, in which everything is reduced to matrices; the geometric view, which effectively takes the algebraic notion of a tensor product over vector spaces and combines it with the techniques of analysis; and the indicial presentation which seems common in engineering. In addition, I've seen tensor analysis presented in indicial form with only first and second-order tensors on three dimensional spaces; it always seemed to me that such a presentation was subsumed by a good elementary course on linear algebra or matrix algebra, but I kept my mouth shut. At this point, they all seem interchangeable to me, and even fairly intuitive -- but it took a while for me to feel that way.

I still like the sound of tensor, even if it no longer seems like such a mysterious thing. My last desktop computer -- now running on nine years old, and mostly left to sit while I use my much-faster five-year-old laptop -- was called Vector, another mathematical word that I think sounds good. After a year or two with Vector, I thought I would upgrade, and call the new box Tensor. At the time, I secretly hoped that I would actually have a good grip on what tensors are before I named a computer after them. When I got the laptop, it somehow seemed more fitting to call it Mongoose; but now, using Mongoose to type a page full of tensor calculus for part of my thesis while Vector sits quietly by my knee, I wonder -- maybe it's time to get a new machine that I can name Tensor?

  • Currently drinking: Coffee