The Jacobian Determinant

In this video, I want to talk about something called the Jacobian determinant. It's more or less just what it sounds like: it's the determinant of the Jacobian matrix that I've been talking to you the last couple of videos about.

Before we jump into it, I just want to give a quick review of how you think about the determinant itself, just in an ordinary linear algebra context. So, if I'm taking the determinant of some kind of matrix, let's say

3 0
1 2

Something like this. To compute the determinant, you take these diagonal terms here. So you take 3 multiplied by that 2, and then you subtract off the other diagonal, subtract off 1 multiplied by 0. In this case, that evaluates to 6.

But there is, of course, much more than just a computation going on here. There's a really nice geometric intuition. Namely, if we think of this matrix as a linear transformation, as something that's going to take this first basis vector over to the coordinates (3, 0) and that second basis vector over to the coordinates (1, 2), you know, thinking about the columns, you can think of the determinant as measuring how much this transformation stretches or squishes space.

In particular, you'll notice how I have this yellow region highlighted. This region starts off as the unit square—a square with side lengths one—so its area is one. There's nothing special about this particular region; it's just nice as a canonical shape with an area of one so that we can compare it to what happens after the transformation. Ask how much does that area get stretched out? The answer is it gets stretched out by a factor of the determinant. That's kind of what the determinant means: that all areas, if you were to draw any kind of shape, not just that one square, are going to get stretched out by a factor of six.

We can actually verify, looking at this parallelogram that the square turned into, it has a base of three, and then the height is two. Three times two is six, and that has everything to do with the fact that this three showed up here and this two showed up there.

Now, let's think about what this might mean in the context of what I've been describing in the last couple of videos. If you'll remember, we had a multivariable function, something that you can write out as f1 with two inputs, and then the second component f2 also with two inputs. The function that I was looking at, that we were kind of analyzing to learn about the Jacobian, had the first component

x + sin(y)

and the second component

y + sin(x)

The idea was that this function is not at all linear; it's going to make everything very curvy and complicated. However, if we zoom in around a particular region—which is what this outer yellow box represents—zooming in, it will look like a linear transformation. In fact, I can kind of play this forward, and we see that even though everything is crazy inside that zoomed-in version, things loosely look like a linear function.

You'll notice I have this inner yellow box highlighted, and this yellow box inside corresponds to the unit square that I was showing in the last animation. Again, it's just a placeholder as something to watch to see how much the area of any kind of blob in that region gets stretched.

In this particular case, when you play out the animation, areas don't really change that much; they get stretched out a little bit, but it's not that dramatic. So, if we know the matrix that describes the transformation that this looks like zoomed in, the determinant of that matrix will tell us the factor by which areas tend to get stretched out.

In particular, you can think of this little yellow box and the factor by which it gets stretched. As a reminder, the matrix describing that zoomed-in transformation is the Jacobian. It is this thing that kind of holds all of the partial differential information. You take the partial derivative of f with respect to x—sorry, the partial derivative of f1 of that first component—and then the partial derivative of the second component with respect to x.

Then, on the other column, we have the partial derivative of that first component with respect to y, and the partial derivative of that second component with respect to y. If you let's see, and we'll close this off, close off this matrix, and if you evaluate each one of these partial derivatives at a particular point, at whatever point we happen to zoom in on—in this case, it was (-2, 1)—once you plug that into all of these, you get some matrix that's just full of numbers.

What turns out to be a very useful thing later on in multivariable calculus concepts is to take the determinant of that matrix, to kind of analyze how much space is getting stretched or squished in that region. So in the last video, we worked this out for the specific example here where that top left function turned out just to be the constant function 1, right? Because we were taking the partial derivative of this guy with respect to x, and that was 1. Likewise, in the bottom right, that was also a constant function of 1. The others were cosine functions. This one was cosine(x) because we were taking the partial derivative of this second component here with respect to x, and then the top right of our matrix was cosine(y).

These are in general functions of x and y, because you know you're going to plug in whatever the input point you're zooming in on. When we're thinking about the determinant here, let's just go ahead and take the determinant in this form, in the form as a function. So, I'm going to ask about the determinant of this matrix, or maybe you think of it as a matrix-valued function.

In this case, we do the same thing. I mean, procedurally, you know how to take a determinant: we take these diagonals, so that's just going to be 1 times 1, and then we subtract off the product of the other diagonal, subtract off cosine(x) multiplied by cosine(y). As an example, let's plug in this point here that we're zooming in on (-2, 1).

So, I'm going to plug in x = -2 and y = 1. When you plug in cosine(-2), that's going to come out to be approximately -0.42, and when you plug in cosine(1) in this case, that's going to come out to be about 0.54. When we multiply those, when we take 1 minus the product of those, it's going to be about -0.227. That's all stuff that you can plug into your calculator if you want, and what that means is that the total determinant evaluated at that point—the Jacobian determinant at the point (-2, 1)—is about 1.0, sorry, 1.227.

So, that's telling you that areas tend to get stretched out by this factor around that point. That kind of lines up with what we see: we see that areas get stretched out maybe a little bit, but not that much. Right? It's only by a factor of about 1.2.

Now let's contrast this. If instead we zoom in at the point where x = 0 and y = 1, so I'm going to go over here, and all I'm going to change—I'm going to change that x = 0 and y will still equal 1. What that means is that cosine(x)—instead of being -0.42—well, what's cosine(0)? That's actually precisely equal to 1, right? We don't have to approximate on this one, which means when we multiply them 1 times 0.54, that's going to now be about 0.54.

So this one, once we actually perform the subtraction instead, when you take 1 - 0.54, that's going to give us 0.46. So even before watching, because this determinant of the Jacobian around the point (0, 1) is less than 1, this is telling us we should expect areas to get squashed down. Precisely, they should be squashed by a factor of 0.46.

And let's see if this looks right. Right? We're looking at the zoomed-in version around that point, and areas should tend to contract around that. Indeed, they do. You see it got squished down, it looks like, by a fair bit. From our calculation, we can conclude that they got scaled down precisely by a factor of 0.46. That's what the determinant means.

So, like I said, this is actually a very nice notion throughout multivariable calculus: you look at a tiny little local neighborhood around a point, and if you just want to get a general feel for does this function, as a transformation, tend to stretch out that region or to squish it together. You know how much do areas change in that little neighborhood? That's exactly what this Jacobian determinant is, you know, built to solve.

So, with that, I'll see you guys next video.