A Leap Day 2016 Mathematics A To Z: Jacobian
I don’t believe I got any requests for a mathematics term starting ‘J’. I’m as surprised as you. Well, maybe less surprised. I’ve looked at the alphabetical index for Wolfram MathWorld and noticed its relative poverty for ‘J’. It’s not as bad as ‘X’ or ‘Y’, though. But it gives me room to pick a word of my own.
The Jacobian is named for Carl Gustav Jacob Jacobi, who lived in the first half of the 19th century. He’s renowned for work in mechanics, the study of mathematically modeling physics. He’s also renowned for matrices, rectangular grids of numbers which represent problems. There’s more, of course, but those are the points that bring me to the Jacobian I mean to talk about. There are other things named for Jacobi, including other things named “Jacobian”. But I mean to limit the focus to two, related, things.
I discussed mappings some while describing homomorphisms and isomorphisms. A mapping’s a relationship matching things in one set, a domain, to things in a set, the range. The domain and the range can be anything at all. They can even be the same thing, if you like.
A very common domain is … space. Like, the thing you move around in. It’s a region full of points that are all some distance and some direction from one another. There’s almost always assumed to be multiple directions possible. We often call this “Euclidean space”. It’s the space that works like we expect for normal geometry. We might start with a two- or three-dimensional space. But it’s often convenient, especially for physics problems, to work with more dimensions. Four-dimensions. Six-dimensions. Incredibly huge numbers of dimensions. Honest, this often helps. It’s just harder to sketch out.
So we might for a problem need, say, 12-dimensional space. We can describe a point in that with an ordered set of twelve coordinates. Each describes how far you are from some standard reference point known as The Origin. If it doesn’t matter how many dimensions we’re working with, we call it an N-dimensional space. Or we use another letter if N is committed to something or other.
This is our stage. We are going to be interested in some N-dimensional Euclidean space. Let’s pretend N is 2; then our stage looks like the screen you’re reading now. We don’t need to pretend N is larger yet.
Our player is a mapping. It matches things in our N-dimensional space back to the same N-dimensional space. For example, maybe we have a mapping that takes the point with coordinates (3, 1) to the point (-3, -1). And it takes the point with coordinates (5.5, -2) to the point (-5.5, 2). And it takes the point with coordinates (-6, -π) to the point (6, π). You get the pattern. If we start from the point with coordinates (x, y) for some real numbers x and y, then the mapping gives us the point with coordinates (-x, -y).
One more step and then the play begins. Let’s not just think about a single point. Think about a whole region. If we look at the mapping of every point in that whole region, we get out … probably, some new region. We call this the “image” of the original region. With the mapping from the paragraph above, it’s easy to say what the image of a region is. It’ll look like the reflection in a corner mirror of the original region.
What if the mapping’s more complicated? What if we had a mapping that described how something was reflected in a cylindrical mirror? Or a mapping that describes how the points would move if they represent points of water flowing around a drain? — And that last explains why Jacobians appear in mathematical physics.
Many physics problems can be understood as describing how points that describe the system move in time. The dynamics of a system can be understood by how moving in time changes a region of starting conditions. A system might keep a region pretty much unchanged. Maybe it makes the region move, but it doesn’t change size or shape much. Or a system might change the region impressively. It might keep the area about the same, but stretch it out and fold it back, the way one might knead cookie dough.
The Jacobian, the one I’m interested in here, is a way of measuring these changes. The Jacobian matrix describes, for each point in the original domain, how a tiny change in one coordinate causes a change in the mapping’s coordinates. So if we have a mapping from an N-dimensional space to an N-dimensional space, there are going to be N times N values at work. Each one represents a different piece. How much does a tiny change in the first coordinate of the original point change the first coordinate of the mapping of the point? How much does a tiny change in the first coordinate of the original point change the second coordinate of the mapping of the the point? How much does a tiny change in the first coordinate of the original point change the third coordinate of the mapping of the point? … how much does a tiny change in the second coordinate of the original point change the first coordinate of the mapping of the point? And on and on and now you know why mathematics majors are trained on Jacobians with two-by-two and three-by-three matrices. We do maybe a couple four-by-four matrices to remind us that we are born to suffer. We never actually work out bigger matrices. Life is just too short.
(I’ve been talking, by the way, about the mapping of an N-dimensional space to an N-dimensional space. This is because we’re about to get to something that requires it. But we can write a matrix like this for a mapping of an N-dimensional space to an M-dimensional space, a different-sized space. It has uses. Let’s not worry about that.)
If you have a square matrix, one that has as many rows as columns, then you can calculate something named the determinant. This involves a lot of work. It takes even more work the bigger the matrix is. This is why mathematics majors learn to calculate determinants on two-by-two and three-by-three matrices. We do a couple four-by-four matrices and maybe one five-by-five to again remind us about suffering.
Anyway, by calculating the determinant of a Jacobian matrix, we get the Jacobian determinant. Finally we have something simple. The Jacobian determinant says how the area of a region changes in the mapping. Suppose the Jacobian determinant at a point is 2. Then a small region containing that point has an image with twice the original area. Suppose the Jacobian determinant is 0.8. Then a small region containing that point has an image with area 0.8 times the original area. Suppose the Jacobian determinant is -1. Then —
Well, what would you imagine?
If the Jacobian determinant is -1, then a small region around that point gets mapped to something with the same area. What changes is called the handedness. The mapping doesn’t just stretch or squash the region, but it also flips it along at least one dimension. The Jacobian determinant can tell us that.
So the Jacobian matrix, and the Jacobian determinant, are ways to describe how mappings change areas. Mathematicians will often call either of them just “the Jacobian”. We trust context to make clear what we mean. Either one is a way of describing how mappings change space: how they expand or contract, how they rotate, how they reflect spaces. Some fields of mathematics, including a surprising amount of the study of physics, are about studying how space changes.