Video transcript
- [Voiceover] In thisvideo, I want to talk about something called the Jacobian determinant. And it's, more or less,just what it sounds like. It's the determinantof 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 aboutthe determinant itself, just in an ordinarylinear algebra context. So if I'm taking the determinantof 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 subtractoff 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 acomputation going on here. There's a really nice geometric intuition. Namely, if we think of thismatrix, three, zero, one, two, as a linear transformation, as something that's gonnatake 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 thedeterminant as measuring how much this transformationstretches or squishes space. And in particular, you'll notice how I have thisyellow region highlighted, and this region startsoff as the unit square, a square with side lengthsone so its area is one. And there's nothing specialabout 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 thatarea get stretched out? And the answer is, it gets stretched out by afactor of the determinant. That's kind of what the determinant means, is that all areas, if you wereto draw up any kind of shape, not just that one square, are gonna get stretchedout by a factor of six. And we can actually verify, looking at this parallelogramthat 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 everythingto do with the fact that this three showed up hereand this two showed up there. So now, let's think aboutwhat this might mean in the context of whatI've been describing in the last couple videos. And if you'll remember, wehad 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 analyzingto learn about the Jacobian, had the first component, x plus sine of y, x plus sine y, and the second componentwas y plus the sine of x. And the idea was that thisfunction is not at all linear. It's gonna make everythingvery curvy and complicated. However, if we zoom inaround a particular region, which is what this outer yellowbox represents, zooming in, it will look like a linear transformation. In fact, I can kind of play this forward, and we see that eventhough everything is crazy, inside that zoomed in version, things loosely looklike a linear function. And you'll notice I have thisinner yellow box highlighted, and this yellow box insidecorresponds to the unit square that I was showing in the last animation. And again, it's just aplaceholder as something to watch to see how much the area ofany 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 thatdescribes the transformation that this looks like zoomed in, the determinant of thatmatrix will tell us the factor by which areastend to get stretched out. And in particular, you canthink of this little yellow box and the factor by which it gets stretched. And as a reminder, the matrix describing thatzoomed in transformation is the Jacobian. It is this thing that kind of holds all of the partialdifferential information. You take the partial derivative of f, with respect to x, sorry, partial of f oneof that first component, and then the partial derivativeof the second component, with respect to x, and then on the other column, we have the partial derivativeof that first component, with respect to y, and the partial derivativeof 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 oneof 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'sjust full of numbers. And what turns out tobe a very useful thing later on in multivariable calc concepts, is to take the determinant of that matrix, to kind of analyze how muchspace 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 functionturned out just to be the constant function, one, right, because we were taking thepartial 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 takingthe partial derivative of this second componenthere 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 pointyou're zooming in on. And when we're thinkingabout the determinant here, let's just go ahead andtake the determinant in this form, in the form as a function. So I'm going to ask about thedeterminant of this matrix, or maybe you think of it asa matrix-valued function. And in this case, we do the same thing. I mean, procedurally, you knowhow to take a determinant. We take these diagonals, so that's just gonna be one times one, and then we subtract off theproduct of the other diagonal, subtract off cosine of x multiplied by cosine of y. And as an example, let'splug in this point here that we're zooming inon, negative two, one. So I'm going to plug in xis equal to negative two, and y is equal to one. And when you plug incosine of negative two, that's gonna come out to beapproximately 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 minusthe product of those, it's gonna be about negative 0.227. And that's all stuff that youcan plug into your calculator if you want. And what that means isthat 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 thatareas tend to get stretched out by this factor around that point. And that kind of linesup with what we see. We see that areas get stretchedout 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 thepoint where x is equal to zero and y is equal to one, so I'm gonna go over hereand all I'm gonna change, all I'm gonna change isthat 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 preciselyequal to one, right? We don't have to approximate on this one, which means when we multiplythem, one times 0.54, well that, that's gonnanow be about 0.54, right? So this one, once we actuallyperform 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 pointzero, one is less than one, this is telling us we should expect areas to get squished down. Precisely, they should besquished by a factor of 0.46. And let's see if this looks right, right? We're looking at the zoomedin version around that point, and areas should tendto 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 theygot scaled down precisely by a factor of 0.46. That's what the determinant means. So like I said, this isactually a very nice notion throughout multivariable calculus, is that you look at a tinylittle local neighborhood around a point, and if you just want toget a general feel for, does this function, as a transformation, tend to stretch out that regionor to squish it together, how much do areas change inthat little neighborhood, that's exactly what thisJacobian determinant is, you know, built to solve. So with that, I'll seeyou guys next video.
