Vectors

This first paragraph... you're either going to know what I'm talking about or you won't have the slightest idea. I didn't study tensors in college (or anywhere else except on my own time) and I suspect that they're still a graduate level subject. But they're not that complicated. Mathematicians use collections of values that can be handled as a single entity. They're called "matrices". A tensor is basically a multidimensional matrix of measurements. A zero order tensor is a scalar. A first order tensor is a vector. A second order tensor is a two dimensional matrix. A third order tensor is a three dimensional matrix, and so on.

The important thing about these things is that they are collections of many values but they can be handled as a single unit. It's like...you can buy a dozen eggs at the grocery store and bring them home separately - that's 24 trips to and from the store. Or you can put them into a carton and put the carton in a bag, and bring them home all at once. Just forget "eggs" and think "carton" until you get ready to make omelettes.

Tensors are advanced math and people generally think of advanced math as a kind of puzzle for folks that like that sort of thing. Well, abstract math is (sort of). But tensors are not abstract math - they're the other kind - practical math. In school, you study practical math. For most of us it goes like this: elementary arithmetic, algebra, geometry, trigonometry and precalculus (or college algebra), and then, in a college science curriculum there are differential calculus, integral calculus, and maaaaybe differential equations.

None of that involves tensors. You'll probably study some linear algebra ( that's matrices) and, in physics, you'll learn about vectors. But there's a lot (!) Beyond that….statistics, calculus of variations, numerical analysis, discrete mathematics…

The point is that all this stuff isn't abstract nonsense. It's all...and here's the kicker...it's all quick and easy labor saving devices. That's right. Advanced mathematics is there to make life easier for people that need to do certain jobs.

Take matrices from example. Let's say you had to figure out three values and you have three equations that contain them. Say, you overheard three people talking. One said, "I have 25, 7, and 43, so I have 490." Another said, "I have 13, 9, and 17, so I have 228." The third said, "I have 3, 23, and 37, so I have 488." You might reasonably think that they're talking about numbers of things with three different values, and you guess that they're naming the things in increasing value so you set up the following equations:

25X1+7X2+43X3=490

14X1+9X3+17X3=228

3X1+23X2+37X3=488

There are several ways to solve for the values of the three variables that make them all true at the same time. One is called elimination and it looks like this.

I counted about 27 (grueling) operations there. Here's the "advanced math" matrix method.

See, you can treat matrices like individual numbers so, once I had the coefficients of the variables packed safely away into one matrix (A) and the numbers on the right side of the equations packed into another (B), I just inverted A and multiplied it by B, two easy operations on the spreadsheet, and I had my answers.

It's pretty clear that the three were talking about money - pennies, nickels, and dimes.

It's a lot easier working with matrices than with individual numbers.

By the way, scalars are just individual numbers. A vector is a row (or column) of numbers. A two dimensional matrix has rows and columns, like the ones I used above. You can have a stack of two dimensional matrices to form a three dimensional matrix, and you can keep going adding more and more dimensions until your brain explodes.

When you're talking about tensors, you're usually talking about measured values and things can get pretty deep, but I won't here.

Many of the values that physicists work with have two parts, so they pack well into two valued vectors. Think back to all the things I measured in the playground.

I started at the trailhead and walked to the playground measuring the distance using AllTrails. It was 0.3 mile. That would not be enough for a physicist, though. They would also want to know the direction. My direction was almost due west or pi radians from an east-west line. The vector would be (0.3,3.15). It could also be represented by an arrow pointing west with a scaled length representing 0.3 miles. The vector would represent my displacement. Distance is a scalar; displacement is a vector.

The funny thing about displacement is that it's the distance traveled from start to finish. On a loop hike, displacement would be exactly 0, since, all told, I would have gone nowhere.

Speed is also a scalar. I walked about 1.8 mph. But physicists talk about velocity, which is a vector consisting of speed and direction. I walked 1.8 mph west.

Acceleration is also a vector consisting of change of speed and change of direction. When I was spinning the phone on the cord, I tried to keep the speed constant but there was still acceleration because the direction of the motion was constantly changing. (Actually, since I didn't do a very good job keeping the speed constant, both were changing.)

Most of the vectors in "undergraduate" physics are two-valued. More advanced physics and engineering get to use larger vectors because they're dealing with three dimensions (space) and four dimensions (relativistic space-time). But our universe is growing and some physicists think that we need ten, eleven, or, maybe, an infinite number of dimensions to describe it.

Are big vectors a problem? Well, they're hard to visualize but statisticians, social scientists, researchers, economists...they've had to deal with big vectors for a long time because every case in a dataset is a vector that might be described by two, fifteen, or thousands of values. Just think about what you look like in a census report: age, ethnicity, residence, number of people in your family, whether you're the head of your family - all values in a huge vector.

We will be talking about other quantities in the future: force, work, energy, magnetic fields. If you've never seen vectors in action, you'll get to see how they work.