Yesterday...
How to begin? "Yesterday I learned a new mathematical operation"? "Yesterday I was taught something that rocked my knowledge of vector calculus to its core"? Or maybe "Yesterday I learned of mathematical quarks".
I'm sorry, it's a bit hard for me to explain, but yesterday I did have a very profound revelation in math as it relates to the commutativity of the dot product when one of the "vectors" is the del operator.
Oh dear, this is just getting worse, isn't it? Let me start by explaining that we have homework due in Electromagnetism. Most of the problems were fairly straightforward, if a little long and time-consuming. But there were two product identities that simply rebuffed all efforts to prove. This was the more downright bewildering because usually such proofs are very simple: just write out every single term explicitly, and cancel until you're left with zero on both sides of the equals sign. But these two repeatedly withstood every attempt, by me and several other people as well. My only thought was that I must be doing something wrong, but what? As far as I could tell, all the terms were correct, but in each proof I'd end up with 20 or more terms that wouldn't cancel.
But yesterday...
I and a classmate were working on the homework, and having finished the rest of the problems, were getting increasingly frustrated with our lack of progress, or indeed the lack of any conceivable way to
make progress. I should mention that we were working in the main hallway in Wentworth Hall, the main Astronomy/Physics/Chemistry building on campus. We were venting our frustration by reviewing everything we knew about vectors and their operations from the beginning, our conversation going something like
"But that's a vector!"
"Right!"
"And if you dot a vector and the del operator, you get..."
"A scalar!"
"Right!"
"And if you multiply a vector by a scalar, you get-"
"-a vector."
At which point Dr. Heacox, one of our professors from a different class happened to walk by and overhear us. With the desperation of two drowning sailors, we turned to him and poured out our troubles, about how the identities wouldn't work, and we
must be doing something wrong, but what was it, and did he have any idea what was the problem was? To his everlasting credit, he stopped and came over to look, and in about 30 seconds had identified the problem:
dotting the del operator with a vector is not commutative.
That's a rather complex statement, so I will explain. To be
commutative in math means that you can do things in any order and it won't affect the results. For instance, addition is commutative: \(2+3=3+2\). So is multiplication: \(5*7=7*5\). But subtraction and division are not. \(3-4\ne 4-3\), and \(8/2\ne2/8\). Knowing whether an operation is commutative or not is very important.
You can think of vectors as essentially numbers with direction. For instance, velocity is a vector. You can say things like "50 mph to the east", or "2 feet per minute straight up". Speed, in contrast, is a
scalar, a number with no direction. Saying "50 mph" gives me no information about what direction this speed is directed towards. Many of the numbers we deal with in everyday life are scalars (like temperature, price, and speed). Others are vectors, like forces and accelerations (I almost said weight was a scalar, until I remembered that it
does have a direction: towards the center of the Earth).
Anyway, the point of this is that the dot product (one of the two ways to multiply vectors) is commutative: \(A\cdot B=B\cdot A\). However, it is
not commutative when one of the two "vectors" is the del operator. I put vectors is quotes, because the del operator is just that: an operator, not a vector (some operators you may be more familiar with are \(+, -, \times,\) and / or \(\div\)). You are allowed to treat it like a vector for purposes of notational simplification, and so I (naively) assumed that it did indeed behave like a vector at all times. But what I learned, and what really rocked me to my core, is that \(\nabla\cdot A\) does not equal \(A\cdot\nabla\) (the del operator is the upside-down equilateral triangle. The actual symbol is called a “
nabla”.). The quantity \(\nabla\cdot A\) is a very common mathematical and physical quantity, called the
divergence of A, since it measures how much A is spreading out or contracting in. But \(A\cdot\nabla\) is another beast entirely. In fact, it is without physical meaning until multiplied by another vector. That's why I whimsically began this long rant by saying I had discovered a mathematical quark.
Quarks, in the Standard Model of physics (the most successful theory of matter that we have to date), are what make up certain sub-atomic particles like protons and neutrons. Both protons and neutrons are made up of three quarks, and when one of those quarks changes its type (from being, say, a 'down' quark to an 'up' quark) you can have a neutron change into a proton, or vice versa (extremely rare).
The catch is, quarks have
never been observed by themselves in any experiment, so the best we can say, since they seem to fit all the data, is that quarks can never exist freely in nature. Much like the little mathematical beastie \(A\cdot\nabla\) which has no meaning until multiplied by a vector.
"Why is this such a big deal?" , you may well ask. I think the answer, for me, is a combination of the fact that I had never heard of this before -- had seen nary a mention of it in all the math classes I've taken so far. I had no inkling of its existence, yet there it was just waiting to be the solution to my problem. Also, the fact that I was rather...worked up when I learned it. I'd already spent over 4 hours over several days trying to get the one identity to work, and it was driving me crazy when all of a sudden...the answer just fell into my lap, as it were.
After dispensing this astonishing piece of knowledge, Dr. Heacox amiably bid us good day and went off to wherever he was going, while my classmate and I sat and tried to decide whether to laugh long and loud at our stupidity or go over and kick a wall, repeatedly. Unable to make up our minds, we settled for finishing the homework, which went over completely without incident after that (although it still took us an additional 2 hours).
So, that was that. Not especially life-changing to most of you I'm sure, but it was a very moving experience for me, as you can see by the length of this post. You can now go about your life with some interesting information about vectors (perhaps tomorrow I'll tell you more about the other way to multiply vectors, the cross product), while I ... will be taking Dr. Heacox cookies tomorrow.
