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.
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
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
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
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
- Currently drinking: Coffee