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Current time:0:00Total duration:8:53

- [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.