As I sometimes used to tell my students, the way we present mathematics in the classroom is a lie. Because every teacher who is any good at all tries to present it in a way that makes it seems as simple as possible.

One wants to solve problem A, or perhaps prove a theorem. After thinking about this for a while, one realizes that fact B might relevant. And then one realizes that this is a little like fact C. And with a little more thought, one gets to D, and then one has the answer to the problem.

This is not the way real mathematical research works, even when it comes to something as simple as figuring out a new way to present a standard topic in class.

The way mathematical thinking is really done can be illustrated by a story told about the mathematician John von Neumann. The mathematician I heard this from told this as a true story, but I don't know whether it's true of not.

Anyway, as the story goes, when von Neumann first came to Princeton to be a member of the Institute for Advanced Study, he hired a maid to take care of his home, as was customary for Princeton professors in those days.

One day, someone asked the maid what it was like working for this famous mathematician.

"Oh, he's just fine," the maid answered. "Very nice. It's just that he's a little strange."

"Strange in what way?"

"Well, all day he just sits at his desk with a pencil and a big stack of paper and all day he scribbles, scribbles, scribbles. And then at the end of the day, he takes all these sheets of paper that he's covered with his scribblings and ... he wads all the papers up into a big ball and throws them all away."

This is in fact the way all mathematicians work. Mathematical thinking is not a neat orderly process. One tries all sorts of things, most of which don't work, and some of which, one eventually realizes, are totally stupid. One spends days chasing down a blind alley and finding nothing except things one already knew from the beginning.

And like as not, when one finally reaches the end and manages to solve the problem or prove the theorem, one will beat one's head with one's fists, asking, "How could I have been so stupid? The answer was right there staring me in the face the whole time."

Real mathematical thinking is something that one never shows to the world at large. Even if still one has all the notes, one can no longer remember how to explain what one was trying to do. And no one would be willing to listen to all that, or read it, in any case.

But in this article, you have a somewhat cleaned-up version of some of my thinking. For years I stared at the formulas for divergence and curl, and saw the concepts explained from a number of different points of view. But I could never manage to see them in a way that really made sense or to explain them in a way that I thought would be satisfactory to students.

Finally, I thought that maybe I had some answers. And so I started writing up what I had figured out, which was certainly simple enough. And then, after writing it up, I realized that there were other things that probably ought to be considered.

I had a final version, I thought, that explained about as much as could be explained. It was eight pages long. And then I realized that what I had figured out for curl was just plain wrong. Irretrievably incorrect. And so I needed to go back to the beginning and try again.

And then I started thinking that the idea of eigenvalues might hold the key I really needed. Looking at the eigenvectors and eigenvalues associated with various two-dimensional vector fields seemed rather enlightening, and I spent considerable time creating pictures (using the "picture" capability in LaTeX) to illustrate these various examples. But in all honesty, I could not say for sure that this was getting me closer to an understanding of divergence, and it seemed clear that it was not very relevent to the concept of curl. (In the ideal case, eigenvalues depend on the diagonal entries of a matrix; curl depends on the off-diagonal entries.)

I wound up with another finished version of an article, or so I thought. I think this one was fourteen or fifteen pages long.

I put this on on my web site, then went on a trip for a month. And on the airliner, and in the city where I was visiting, I had some more thoughts. And I came home a wrote still a new final version of my article. This one, I think, was almost thirty pages.

And at this point, I thought I had about as good an explanation as one could give, especially for divergence.

I still didn't have a good answer to the question of what the significance of divergence and curl was in the three-dimensional case. I was pretty sure that the answer in two dimensions was not the correct answer for three dimensions, but I had a hard time finding good examples to show this.

And I was really frustrated that I couldn't find a good intuitive meaning for the direction of the curl vector.

My feeling was that even though examples showed that the fact that a vector field had a non-zero curl did not necessarily indicate that the vector field was a rotation, still it ought to indicate that it was somehow like a rotation, and the direction of the curl vector ought to indicate the axis around which the rotation took place.

And then by some miracle --- and I no longer remember quite how it happened --- I realized that curl could be related to skew-symmetric matrices, and I vaguely remembered having once heard (at the time, I didn't pay much attention) that skew symmetric matrices could be used to represent rotations. If I'd known more about the use of matrix-theoretic methods in mechanics, I would have realized this much earlier. Anyway, by using the decomposition of a matrix into a symmetric and a skew-symmetric part, I now had an explanation for what curl is that actually made sense.

Because of the way my thinking about divergence and culr had first started, I had been especially concerned from the beginning with the special case of a vector field with constant magnitude. But it seemed to me now that this might be one of those cases where an answer is hard to find because one is really not asking the right question. Certainly in one respect looking at field with constant magnitude is not very natural, since such fields rarely occur naturally in applications.

At this point, my life wss getting complicated because I was beginning the rather lengthy process of selling my apartment and leaving Honolulu forever. In any case, it was now about the fourth time when I had reached the point where I was certain that I had discovered all there was to discover about divergence and curl and that the article was now complete. (Of course in retrospect, I realize that I would have saved an enormous amount of effort by spending more time in the library seeing what I could find in books.)

What happened then was what often happens when doing a mathematical investigation. Even though I was determined to not put any more effort into the article, I couldn't stop my mind from going back over what I had done. And it occured to me that in the special case when a vector field is the gradient field of some function, one could indeed find a very specific meaning for the divergence of a vector field with constant magnitude by using a little theory from the differential geometry of surfaces. And I wrote this up, after I had moved away from Honolulu, using a friend's computer at a time when I was living in a small corner of her apartment.

And then I went off traveling, to San Francisco and to Europe. Because of the changes in the Mathematrics Department computer system at the University of Hawaii, I was no longer able to make any revisions to my divergence-curl article at all.

But then a rather disgusting thing happened. I started becoming more and more aware that the special case (when the vector field is a gradient of a function) was not so relevant after all, and the the crucial consideration was not a matter of the differential geometry or surfaces at all.

In fact, I realized that I now had the final answer for the significance of divergence, and that this answer came directly from the basic formula, and that in fact most of the my article was actually irrelevant.

This, in my experience, is a sort of thing that often occurs in mathematical research. A big part of the education of a mathematician consists of learning a lot of very fancy and high powered theorems, and also seeing how these theorems can be used to obtain other results. And so it is natural that when one works on a new problem (i.e. tries to find and prove a new theorem), one thinks in terms of using all the fancy and high-powered theorems that one knows. In fact, the only reasonable way of working on any mathematical problem is to throw at it every conceivable thing that one knows. And sometimes this works. My own mathematical reputation, for instance, was primarily established by solving a problem by means of a theorem that most of the other people who had worked on that problem were unaware of.

But in the majority of cases, in my experience, when one finally reaches the point where one completely understands a problem and is able to formulate and prove a theorem, all the high-powered stuff one attempted to use on it turns out to be irrelevant, and the solution is simply a matter of straightforward application of basic principles.

When I reached this point, the only thing I could was to write up this new insight and email it to myself, waiting for the moment when I would not be working from internet cafés and could download the secure shell software I needed to add this to the article on my web site.

But my mind still didn't stop. If divergence could be completely explained starting only from the basic formula, wasn't it possible that the same thing could be done for curl? Because divergence and curl, although so opposite to each other, are very much like two sides of the same coin.

So here I am now, once again sitting in a friend's apartment, this time in Amsterdam, holding a keyboard (a QWERTY keyboard, by some miracle!) and a Logitech mouse on my lap as I update the article to create what I hope will now be the final version.

Unfortunately, the explanation for curl didn't turn out as satisfactory as that for divergence, but I'm satisfied that it's the best that can be achieved.

For those readers whose only interest is in the advertised result of the article, namely an understanding of divergence and curl, by the time one reaches the end it may indeed seem that the bulk of my article should be wadded up into a big ball and thrown into the trash. However I believe that most of the ideas developed, even when they turned out not to lead to the desired goal, are legitimately interesting on their own.

Furthermore, I have had the opportunity to do here what one is never allowed when writing an article for publication, namely to record the actual process of the investigation that led to a mathematical result. (Of course it is highly unlikely that anything here is actually new. But it was certainly new for me. And the time I spent in the library suggests that this information is not easy to find in standard books. )