The Jacobian Determinant

- [Voiceover] In this video, I want to talk about something called the Jacobian determinant. And 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 videos about. And 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, three, zero, one, two, something like this, to compute the determinant, you take these diagonal terms here, so you take three multiplied by that two, and then you subtract off the other diagonal, subtract off one multiplied by zero. And in this case, that evaluates to six. 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, three, zero, one, two, as a linear transformation, as something that's gonna take this first basis vector over to the coordinates three, zero, and that second basis vector over to the coordinates one, two, you know, thinking about the columns, you can think of the determinant as measuring how much this transformation stretches or squishes space. And in particular, you'll notice how I have this yellow region highlighted, and this region starts off as the unit square, a square with side lengths one so its area is one. And 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? And the answer is, it gets stretched out by a factor of the determinant. That's kind of what the determinant means, is that all areas, if you were to draw up any kind of shape, not just that one square, are gonna get stretched out by a factor of six. And 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. And 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. So now, let's think about what this might mean in the context of what I've been describing in the last couple videos. And if you'll remember, we had a multivariable function, something that you can write out as f one with two inputs and then the second component, f two, also with two inputs. And the function that I was looking at, that we were kind of analyzing to learn about the Jacobian, had the first component, x plus sine of y, x plus sine y, and the second component was y plus the sine of x. And the idea was that this function is not at all linear. It's gonna 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. And 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. And 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. So, 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. And in particular, you can think of this little yellow box and the factor by which it gets stretched. And 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, partial of f one of that first component, and then the partial derivative of the second component, with respect to x, and 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. And if you... Let's see, I'm gonna 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 negative two, one, once you plug that into all of these, you get some matrix that's just full of numbers. And what turns out to be a very useful thing later on in multivariable calc 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 this specific example here, where that top left function turned out just to be the constant function, one, right, because we were taking the partial derivative of this guy with respect to x and that was one. And likewise, in the bottom right, that was also a constant function of one. And then 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 of y. And these are, in general, functions of x and y because you know, you're gonna plug in whatever the input point you're zooming in on. And 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. And 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 gonna be one times one, and then we subtract off the product of the other diagonal, subtract off cosine of x multiplied by cosine of y. And as an example, let's plug in this point here that we're zooming in on, negative two, one. So I'm going to plug in x is equal to negative two, and y is equal to one. And when you plug in cosine of negative two, that's gonna come out to be approximately negative 0.42. And when you plug in cosine of y, cosine of one in this case, that's gonna come out to be about 0.54. And when we multiply those, when we take one minus the product of those, it's gonna be about negative 0.227. And 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 negative two, one, is about 1.227. So that's telling you that areas tend to get stretched out by this factor around that point. And 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. And now, let's contrast this. If instead we zoom in at the point where x is equal to zero and y is equal to one, so I'm gonna go over here and all I'm gonna change, all I'm gonna change is that x is equal to zero and y will still equal one, and what that means is that cosine of x, instead of being negative 0.42, well what's cosine of zero, that's actually precisely equal to one, right? We don't have to approximate on this one, which means when we multiply them, one times 0.54, well that, that's gonna now be about 0.54, right? So this one, once we actually perform the subtraction, instead when you take one minus 0.54, that's gonna give us 0.46. So even before watching, because this determinant of the Jacobian around the point zero, one is less than one, this is telling us we should expect areas to get squished down. Precisely, they should be squished 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. And indeed, they do. You see it got squished down, it looks like by a fair bit, and 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, is that 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, 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.