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OK.
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So far, we've learned how to do double integrals
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in terms of x, y coordinates, also how to switch to polar
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coordinates.
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But marginally, there's a lot of different changes
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of variables that you might want to do.
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So today, we're going to see how to change variables.
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If you want how to do substitutions in double integrals.
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OK.
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So let me start with a simple example.
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Let's say that we want to find the area of an ellipse
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with semi-axis A and D. So that means an ellipse is just
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like a squished circle.
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And so there's A and there's B. And the equation of that ellipse
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is x over A squared plus y over B squared equals 1.
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Well, that's the curve.
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And the inside region is where this is less than 1.
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So it's just like a circle that where you have rescaled x
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and y differently.
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So let's say we want to find the area of that.
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I mean, maybe you know what the area is.
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But let's do it as double integral.
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So if you find that the area is too easy,
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you can integrate any function of the ellipse if you prefer.
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But let's do it just with the area.
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So we know that we want to integrate just the area element,
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let's say the x dy over the region inside the ellipse
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that's x over A squared plus y over B squared less than 1.
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Now we can try to set this up in terms of x and y
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coordinates, you know, set up the bounds by solving
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for x as a function of y if we do it this order.
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And well, do the usual stuff.
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But it doesn't look very pleasant.
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And it's certainly not the best way to do it.
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If this were a circle, we would switch to polar coordinates.
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Well, we can't quite do that yet.
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But you know, an ellipse is just a squished circle.
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So maybe we want to actually first rescale x and y
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by a and b to reduce to the case of a circle.
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So to do that, what we'd like to do is set x over a to be u
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and y over B, B, V.
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So we have two new variables, u and v.
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And we'll try to redo our integral in terms of u and v.
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So how do we do the substitution?
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So in terms of u and v, the condition, the region
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that we are integrating on, will become u squared plus v
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squared less than 1, which is arguably
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nicer than the ellipse.
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That's what's what we're doing it.
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But we need to know what to do with dx and dy.
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Well, here the answer is pretty easy,
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because we just change x and y separately.
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We do two independent substitutions.
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So if we set u equals x over a, that means du is 1 over a dx.
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And here, dv is 1 over b dy.
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So it's very tempting to write.
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And here, we actually, we can write.
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In this particular case, that du dv is 1 over a, b dx dy.
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So let me rewrite that.
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So I get du dv equals 1 over a, b dx dy.
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Or equivalently, dx dy is a, b times du dv.
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So in my double integral, I'm going to write a, b, du dv.
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OK, so now my double integral becomes, well, the double integral
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of a constant in terms of u and v.
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So I can take the constant out.
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I will get a, b times the double integral of u squared plus v squared
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less than 1 of du dv.
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And that is an integral that we know how to do.
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Well, because it's the area, it's just the area of a unit circle.
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So we can just say, oh, this is a, b times the area of the unit
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disk, which we know to be pi.
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If somehow you had some function to integrate,
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then you could have switched to polar coordinates,
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setting u equals R cos theta, v equals R sine theta,
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and then doing it in polar coordinates.
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So here, the substitution worked pretty easily.
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The question is, if we do a change of variables
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where the relation between x and y and u and v is more
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complicated, what can we do?
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Can we still do this?
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Or do we have to be more careful?
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And actually, we have to be more careful.
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So that's what we're going to see next.
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But any question about this first?
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No?
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OK.
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OK, so see, the general problem when we try to do this
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is to figure out what is the scale factor,
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what's the relation between dx, dy and du, dv.
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So we need to find the scaling factor.
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So we need to find dx, dy, that's just du, dv.
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So let's do another example that's still pretty easy,
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but a little bit less easy.
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So let's say that for some reason,
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we want to set, we want to do the change of variables,
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u equals x minus 2y, and v equals x plus y.
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Why would we want to do that?
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Well, that might be to simplify the integrant
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because we're integrating a function that happens to be
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actually involving these guys, rather than x and y.
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Or it might be to simplify the bounds,
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because maybe we're integrating over a region,
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whose equation in xy coordinates is very hard to write down,
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but it becomes much easier in terms of u and v.
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And then the bounds will be much easier to set up with u and v.
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Anyway, so whatever reason might be,
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but typically it would be to simplify the integrant of the bounds.
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Well, how do we convert dx, dy to du, dv?
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So we want to understand what's the relation between,
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let's call, dA, the area element in xy coordinates.
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So dA is dx dy, maybe dy dx, depending on the order.
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And the area element in uv coordinates,
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let me call that dA prime just to make it look different.
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So that would just be du, dv, or dv, du,
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depending on which order I will want to set it up in.
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So to find this relation, it's probably
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best to draw a picture to see what happens.
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Let's consider a small piece of the xy plane with area delta A
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corresponding to just a box with sides delta y and delta x.
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And let's try to figure out what it will look like in terms
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of u and v.
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And then we'll say, well, when we integrate,
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we're really summing the value of a function
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over a lot of small boxes times their area.
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But the problem is that the area of a box in here
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is not the same as the area of a box in uv coordinates.
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There maybe it will look like.
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Actually, if you see that these are linear changes of variables,
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you know that a rectangle will become a parallelogram
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after the change of variables.
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So the area of a parallelogram delta A prime,
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well, we have to figure out how they're related so that we
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can decide what conversion factor, what's the exchange rate
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between these two currencies for area?
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Any questions at this point?
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No, still with me?
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Mostly.
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I see a lot of tired faces.
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Yes?
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So what is the delta A prime parallelogram?
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Why is delta A prime a parallelogram?
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That's a very good question.
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Well, see, here, if I look at this side of the rectangle,
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say there's vertical side, means I'm
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going to increase y keeping x the same.
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If I look at the formulas for u and v,
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they're linear formulas in terms of x and y.
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So if I just increase y, see that u is going to decrease
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at rate of 2, this is going to increase at rate of 1
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at constant rates.
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And it doesn't matter whether I was looking at this side
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or at that side.
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So basically, straight lines become straight lines.
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And if they're parallel, they stay parallel.
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So if you just look at what the transformation from x, y,
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to u, v does, it does this kind of thing.
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Actually, this transformation here,
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you can express by your matrix.
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And remember, we've seen what matrices do to pictures,
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but just take straight lines to straight lines.
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They keep the notion of being parallel,
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but of course, they mess up lengths, angles, and all that.
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So let's see.
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So let's try to figure out what is the array of this guy.
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Well, in fact, what I've been saying
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about this transformation being linear and transforming
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all of the vertical lines in the same way,
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all the horizontal lines in the same way,
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tells me also I should have a constant scaling factor.
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Because how much I've scaled my rectangle
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doesn't depend on where my rectangle is.
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If I move my rectangle to somewhere else,
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I have a rectangle of the same size, same shape.
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It will become a parallelogram of the same size,
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same shape somewhere else.
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So in fact, I can just take the simplest rectangle I can think of
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and see how its array changes.
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And if you don't believe me, then try with any of our rectangle,
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you will see it works exactly the same way.
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So I claim that the area scaling factor here, in this case,
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does not depend on the choice of the rectangle.
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And I should say that because we're actually doing a linear
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change of variables.
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So somehow the exchange rate between uv and xy
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is going to be the same everywhere.
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So let's try to see what happens to the simplest rectangle
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I can think of, namely just the unit square.
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And if you don't trust me, then while I'm doing this one,
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do it with a different rectangle, do the same calculation
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and see that you will get the same conduction ratio.
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So let's say that I take the unit square.
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So some of it has size that goes from 0 to 1,
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both in x and y directions.
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And let's try to figure out what it looks like on the other side.
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So here, the area is 1.
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Let's try to draw it in terms of u and d coordinates.
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OK?
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So here we have x equals 0, y equals 0.
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Well, that tells us u and v are going to be 0.
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Next, let's look at this corner.
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Well, in xy coordinates, this is 1, 0.
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If you plug x equals 1, y equals 0,
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you get u equals 3, v equals 1.
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So that goes somewhere here.
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And so this edge of the square will become this line here.
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Next, let's look at that point.
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So that point here was 0, 1.
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If I plug x equals 0, y equals 1, I will get negative 2 and 1.
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So this edge goes here.
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Then if you put x equals 1, y equals 1,
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you'll get u equals 1, v equals 2.
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So I want 1, 2.
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And these edges will go to these edges here.
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And you see, it does look like a parallelogram.
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So now, what's the area of this parallelogram?
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Well, we can get that by taking the determinant
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of these two vectors.
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So one of them is 3, 1.
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And the other one is minus 2, 1.
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That will be 3 plus 2.
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That's 5.
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This parallelogram is apparently 5 times the size of this square.
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Here, it looks like it's less because I've somehow changed
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my scale.
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I mean, my unit length is smaller here than here.
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So that it should be a lot bigger than that.
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If you do the same calculations not with 0 and 1,
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but with x and x plus delta x and so on,
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you will still find that the area has been multiplied by 5.
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So that tells us, actually, for any over rectangle,
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the area is also multiplied by 5.
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So that tells us that the area element in UV coordinate
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is worth 5 times more than the area element in x, y coordinates.
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So that means the UDV is worth 5 times the x, dy.
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What's so funny?
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What?
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Oh.
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Oh.
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OK.
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I like tango.
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OK.
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Is it OK now?
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Did I miss parallel?
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Or a lot?
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It's really hard to see when you're that close up.
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It's much easier for me to distance.
254
00:17:56.640 --> 00:17:59.640
OK.
255
00:17:59.640 --> 00:18:03.480
So yeah, so we've said our transformation
256
00:18:03.480 --> 00:18:07.320
multiplies the area by 5.
257
00:18:07.320 --> 00:18:09.440
And so the UDV is 5 times the x, dy.
258
00:18:09.440 --> 00:18:14.680
So if I'm integrating some function, the x, dy,
259
00:18:14.680 --> 00:18:17.000
then when I switch to UV coordinates,
260
00:18:17.000 --> 00:18:21.320
I will have to replace that by 1.5 du, dy.
261
00:18:21.320 --> 00:18:24.400
OK.
262
00:18:24.400 --> 00:18:27.640
And of course, I would also, here my function
263
00:18:27.640 --> 00:18:29.440
would probably involve x and y.
264
00:18:29.440 --> 00:18:32.760
I will replace them by u and v's.
265
00:18:32.760 --> 00:18:36.080
And the bounds, well, the shape of my region in the x, y
266
00:18:36.080 --> 00:18:39.040
coordinates, I will have to switch to some shape in the UV
267
00:18:39.040 --> 00:18:40.400
coordinates.
268
00:18:40.400 --> 00:18:43.400
And that's also something that might be easy or might be
269
00:18:43.400 --> 00:18:47.000
tricky depending on what region we're looking at.
270
00:18:47.000 --> 00:18:49.040
So usually we'll do changes of variables
271
00:18:49.040 --> 00:18:51.480
to actually simplify the region.
272
00:18:51.480 --> 00:18:53.200
So that it becomes easier to set up the bounds.
273
00:18:58.240 --> 00:19:05.760
So anyway, so this is kind of an illustration of a general case
274
00:19:05.760 --> 00:19:07.120
and why is that?
275
00:19:07.120 --> 00:19:08.680
Well, here it looks very easy.
276
00:19:08.680 --> 00:19:12.320
We're just using linear formulas and somehow a relation
277
00:19:12.320 --> 00:19:15.840
between the x, dy and du, dv is the same everywhere.
278
00:19:15.840 --> 00:19:18.400
If you take actually more complicated changes of variables,
279
00:19:18.400 --> 00:19:19.280
that's not true.
280
00:19:19.280 --> 00:19:24.480
Because usually you will expect that there's some places
281
00:19:24.480 --> 00:19:27.040
where the rescaling is maybe enlarging things,
282
00:19:27.040 --> 00:19:29.720
and some other places where things are shrunk.
283
00:19:29.720 --> 00:19:33.120
So certainly the x-change rate between du, dv and the x, dy
284
00:19:33.120 --> 00:19:35.000
will fluctuate from point to point.
285
00:19:35.000 --> 00:19:37.480
It's same as if you're trying to change dollars to euros.
286
00:19:37.480 --> 00:19:38.680
It depends on where you do it.
287
00:19:38.680 --> 00:19:43.240
You will get a better rate or a worse one.
288
00:19:43.240 --> 00:19:48.080
So we have to, of course, we'll get a formula where actually
289
00:19:48.080 --> 00:19:52.960
this scaling factor will depend on x and y or on u and v.
290
00:19:52.960 --> 00:19:57.160
But if you fix a point, then we have linear approximation.
291
00:19:57.160 --> 00:20:00.520
And linear approximation tells us, oh, we can do as if our function
292
00:20:00.520 --> 00:20:02.600
is just a linear function of x and y.
293
00:20:02.600 --> 00:20:06.240
So then we can do it the same way we did here.
294
00:20:06.240 --> 00:20:08.680
So let's try to think about that.
295
00:20:18.120 --> 00:20:23.560
So in the general case, well, that means we'll replace x and y
296
00:20:23.560 --> 00:20:27.280
by nu coordinates u and v. And u and v will be some functions
297
00:20:27.280 --> 00:20:29.920
of x and y.
298
00:20:29.920 --> 00:20:35.880
So we'll have an approximation formula, which tells us
299
00:20:35.880 --> 00:20:40.080
that the change in u, if I change x on y a little bit,
300
00:20:40.080 --> 00:20:42.920
will be roughly u sub x times change in x
301
00:20:42.920 --> 00:20:46.120
plus u sub y times change in y.
302
00:20:46.120 --> 00:20:52.880
And the change in v will be roughly v sub x delta x
303
00:20:52.880 --> 00:20:57.360
plus v sub y delta y.
304
00:20:57.360 --> 00:21:01.120
Or the other way to say it, if you want in matrix form,
305
00:21:01.120 --> 00:21:11.760
is delta u delta v is approximately equal to u sub x, u sub y
306
00:21:11.760 --> 00:21:17.200
v sub x, v sub y times delta x and delta y.
307
00:21:22.440 --> 00:21:28.520
So if we look at that, what it tells us, in fact,
308
00:21:28.520 --> 00:21:36.800
is that if we take a small rectangle in x, y coordinates,
309
00:21:36.800 --> 00:21:42.120
so that means we have a certain point x, y.
310
00:21:42.120 --> 00:21:44.800
And then we have a certain width.
311
00:21:44.800 --> 00:21:46.160
This is going to be too small either.
312
00:21:46.160 --> 00:21:56.120
So I have my width delta x.
313
00:21:56.120 --> 00:21:59.240
I have my height delta y.
314
00:21:59.240 --> 00:22:05.000
This is going to correspond to a small uv parallelogram.
315
00:22:05.000 --> 00:22:19.240
And what the shape and the size of that parallelogram
316
00:22:19.240 --> 00:22:22.760
are depend on the partial derivatives of u and v.
317
00:22:22.760 --> 00:22:26.520
So in particular, they depend on at which point we are.
318
00:22:26.520 --> 00:22:30.000
But still, at a given point, it's a bit like that.
319
00:22:30.000 --> 00:22:38.520
And so if we do the same argument as before,
320
00:22:38.520 --> 00:22:41.520
what we'll see is that the scaling factor
321
00:22:41.520 --> 00:22:45.480
is actually the determinant of this transformation.
322
00:22:45.480 --> 00:22:50.520
So that's one thing that maybe we did not emphasize enough
323
00:22:50.520 --> 00:22:53.640
when we did matrices at the beginning of the semester.
324
00:22:53.640 --> 00:22:57.000
But when you have a linear transformation between variables,
325
00:22:57.000 --> 00:22:58.920
the determinant of that transformation
326
00:22:58.920 --> 00:23:03.440
represents how it scales areas.
327
00:23:03.440 --> 00:23:05.640
So one way to think about it is just
328
00:23:05.640 --> 00:23:09.000
to try and see what happens.
329
00:23:09.000 --> 00:23:13.360
Take this side, this side, in x, y coordinates corresponds
330
00:23:13.360 --> 00:23:15.960
to delta x and 0.
331
00:23:15.960 --> 00:23:18.280
And now if you take the image of that,
332
00:23:18.280 --> 00:23:21.320
if you see what happens to delta u and delta v,
333
00:23:21.320 --> 00:23:23.720
that will be basically u sub x delta x.
334
00:23:23.720 --> 00:23:28.320
And v sub x delta x, there's no delta y.
335
00:23:28.320 --> 00:23:31.560
For the other side, so maybe I should do it actually.
336
00:23:34.800 --> 00:23:41.560
So if we move in the x, y coordinates by delta x and 0,
337
00:23:41.560 --> 00:23:48.200
then delta u and delta v will be u sub x,
338
00:23:48.200 --> 00:23:54.480
be approximately u sub x delta x, and v sub x delta x.
339
00:23:54.480 --> 00:24:04.640
And if you move in the other direction along the other side
340
00:24:04.640 --> 00:24:09.080
of your rectangle, 0 and delta y, then the change in u
341
00:24:09.080 --> 00:24:13.240
and the change in v will correspond to, well,
342
00:24:13.240 --> 00:24:14.120
how does u change?
343
00:24:14.120 --> 00:24:19.920
That's u sub y delta y, and v changes by v sub y delta y.
344
00:24:19.920 --> 00:24:22.800
And so now if you take the determinant of these two vectors.
345
00:24:22.800 --> 00:24:27.160
So these are the sides of your parallelogram up here.
346
00:24:27.160 --> 00:24:30.720
And if you take these sides to get the area of the parallelogram,
347
00:24:30.720 --> 00:24:33.160
you will need to take the determinant.
348
00:24:33.160 --> 00:24:37.800
And the determinant of this matrix times delta x times delta y.
349
00:24:44.840 --> 00:24:48.320
So the area in u v coordinates will be the determinant
350
00:24:48.320 --> 00:24:53.200
of a matrix times delta x delta y.
351
00:24:53.200 --> 00:25:03.760
And so what I'm trying to say is that when you have a general change of variables,
352
00:25:03.760 --> 00:25:07.760
the u dv versus dx dy is given by the determinant
353
00:25:07.760 --> 00:25:11.040
of this matrix of partial derivatives.
354
00:25:11.040 --> 00:25:13.000
It doesn't matter in which order you've added.
355
00:25:13.000 --> 00:25:16.520
I mean, you can put, you know, in rows or in columns.
356
00:25:16.520 --> 00:25:19.560
If you transpose a matrix that doesn't change for the determinant,
357
00:25:19.560 --> 00:25:49.520
you know, any sensible matrix that you can write will have the correct determinant.
358
00:25:49.520 --> 00:26:07.440
So what we need to know is the following thing.
359
00:26:07.440 --> 00:26:14.160
So we define something called the Jacobian of a change of variables.
360
00:26:14.160 --> 00:26:23.720
And if you have a different function that you can use a letter j or maybe a more useful notation,
361
00:26:23.720 --> 00:26:25.880
is partial of u v over partial of xy.
362
00:26:25.880 --> 00:26:27.040
But it's a very strange notation.
363
00:26:27.040 --> 00:26:32.800
I mean, that doesn't mean that we're actually taking the partial derivative of anything.
364
00:26:32.800 --> 00:26:38.640
It's just notation to remind us that this has to do with a ratio between the u dv and the
365
00:26:38.640 --> 00:26:45.640
So, it's the determinant of a matrix, use of x, use of y,
366
00:26:45.640 --> 00:26:50.640
this of x, this of y.
367
00:26:50.640 --> 00:26:55.640
The matrix that I had up there.
368
00:26:55.640 --> 00:27:01.640
And what we need to know is that the u dv,
369
00:27:01.640 --> 00:27:08.640
is equal to the absolute value of j dx dy.
370
00:27:08.640 --> 00:27:12.640
Or if you prefer to see it in the easier-to-remembered version,
371
00:27:12.640 --> 00:27:22.640
it's absolute value of d of uv over partial xy dx dy.
372
00:27:22.640 --> 00:27:28.640
So, this is just what you need to remember.
373
00:27:28.640 --> 00:27:34.640
And it says that the RR in uv coordinates is worth,
374
00:27:34.640 --> 00:27:38.640
well, the ratio to the RR in the x-ray coordinates
375
00:27:38.640 --> 00:27:42.640
is given by this Jacobian determinant.
376
00:27:42.640 --> 00:27:44.640
Except one small thing.
377
00:27:44.640 --> 00:27:49.640
It's given by the absolute value of this guy.
378
00:27:49.640 --> 00:27:51.640
So, what's going on here?
379
00:27:51.640 --> 00:27:55.640
What's going on here is when we are saying the determinant
380
00:27:55.640 --> 00:27:58.640
of the transformation tells us how the RR is multiplied.
381
00:27:58.640 --> 00:28:00.640
There's a small catch.
382
00:28:00.640 --> 00:28:03.640
Remember, the determinants are equal to RRs up to sine.
383
00:28:03.640 --> 00:28:07.640
Sometimes the determinant is negative because of reversing
384
00:28:07.640 --> 00:28:09.640
the orientation of things.
385
00:28:09.640 --> 00:28:10.640
But the RR is still the same.
386
00:28:10.640 --> 00:28:13.640
RR is always positive.
387
00:28:13.640 --> 00:28:17.640
So, the RR elements are actually related by the absolute value
388
00:28:17.640 --> 00:28:18.640
of this guy.
389
00:28:18.640 --> 00:28:21.640
So, if you find, you know, minus 10 as your answer,
390
00:28:21.640 --> 00:28:29.640
then the uv is still 10 times dx dy.
391
00:28:29.640 --> 00:28:30.640
Okay.
392
00:28:30.640 --> 00:28:32.640
So, yeah, I mean, I didn't put it all together
393
00:28:32.640 --> 00:28:35.640
because then you would have two sets of vertical bars.
394
00:28:35.640 --> 00:28:37.640
This is a vertical bar for absolute value.
395
00:28:37.640 --> 00:28:39.640
This is a vertical bar for determinant.
396
00:28:39.640 --> 00:28:41.640
They're not the same.
397
00:28:41.640 --> 00:28:45.640
That's the one thing to remember.
398
00:28:45.640 --> 00:28:50.640
Okay. Any questions about this?
399
00:28:50.640 --> 00:28:53.640
No.
400
00:28:53.640 --> 00:29:00.640
Okay.
401
00:29:00.640 --> 00:29:08.640
So, actually, let's do a first example of that.
402
00:29:08.640 --> 00:29:13.640
Let's check what we had for polar coordinates.
403
00:29:13.640 --> 00:29:16.640
You know, last time I told you, well, if we have dx dy,
404
00:29:16.640 --> 00:29:19.640
we should switch it to RDRD theta.
405
00:29:19.640 --> 00:29:22.640
And we had some argument for that by looking at the RR
406
00:29:22.640 --> 00:29:26.640
of a small piece of, you know, a small circular sector.
407
00:29:26.640 --> 00:29:33.640
But, you know, let's check again using this new method.
408
00:29:33.640 --> 00:29:38.640
So, in polar coordinates, I'm setting x equals R cosine theta.
409
00:29:38.640 --> 00:29:43.640
y equals R sine theta.
410
00:29:43.640 --> 00:29:48.640
So, the Jacobian for this change of variables.
411
00:29:48.640 --> 00:29:53.640
So, let's say I'm trying to find the partial derivatives
412
00:29:53.640 --> 00:29:57.640
of xy with respect to R theta.
413
00:29:57.640 --> 00:30:01.640
Well, what is, okay, let me actually write them here again,
414
00:30:01.640 --> 00:30:07.640
just for you.
415
00:30:07.640 --> 00:30:09.640
And so, what does that become?
416
00:30:09.640 --> 00:30:16.640
Partial x of a partial R is just cosine theta.
417
00:30:16.640 --> 00:30:24.640
Partial x of a partial theta is negative R sine theta.
418
00:30:24.640 --> 00:30:27.640
Sorry, I guess I'm going to run out of space here.
419
00:30:27.640 --> 00:30:30.640
So, let me do it underneath.
420
00:30:30.640 --> 00:30:33.640
So, we say x sub R is cosine theta.
421
00:30:33.640 --> 00:30:36.640
x sub theta is negative R sine theta.
422
00:30:36.640 --> 00:30:40.640
y sub R is sine.
423
00:30:40.640 --> 00:30:45.640
y sub theta is R cosine.
424
00:30:45.640 --> 00:30:52.640
And now, if we compute this determinant,
425
00:30:52.640 --> 00:30:58.640
we'll get R cosine square theta plus R sine square theta.
426
00:30:58.640 --> 00:31:01.640
And that simplifies to R.
427
00:31:01.640 --> 00:31:07.640
So, the xdy is, well, absolute value of R,
428
00:31:07.640 --> 00:31:09.640
DRD theta.
429
00:31:09.640 --> 00:31:12.640
But remember that R is always positive.
430
00:31:12.640 --> 00:31:19.640
So, it's RDRD theta.
431
00:31:19.640 --> 00:31:20.640
Okay.
432
00:31:20.640 --> 00:31:24.640
So, that's another way to justify how we did double integrals
433
00:31:24.640 --> 00:31:30.640
in polar coordinates.
434
00:31:30.640 --> 00:31:34.640
Any questions on that?
435
00:31:34.640 --> 00:31:37.640
Well.
436
00:31:37.640 --> 00:31:40.640
Yeah, okay.
437
00:31:40.640 --> 00:31:47.640
Are we switching from polar to xy to xy to polar?
438
00:31:47.640 --> 00:31:52.640
Yeah, so this one seems to be switching.
439
00:31:52.640 --> 00:31:55.640
Well, it depends what you do.
440
00:31:55.640 --> 00:32:00.640
Actually, it has an important thing that I didn't quite say.
441
00:32:00.640 --> 00:32:04.640
So, I said, you know, we're going to switch from xy to uv.
442
00:32:04.640 --> 00:32:07.640
We can also switch from uv to xy.
443
00:32:07.640 --> 00:32:10.640
And, you know, this conversion ratio, the Jacobian,
444
00:32:10.640 --> 00:32:12.640
works both ways.
445
00:32:12.640 --> 00:32:16.640
Once you have found, you know, the ratio between duv and dxdy,
446
00:32:16.640 --> 00:32:20.640
then it works one way or it works the other way.
447
00:32:20.640 --> 00:32:23.640
I mean, here, of course, we get the answer in terms of R.
448
00:32:23.640 --> 00:32:28.640
So, this would let us, you know, switch from xy to R theta.
449
00:32:28.640 --> 00:32:32.640
But we could also switch from R theta to xy,
450
00:32:32.640 --> 00:32:36.640
just we'd invite DRD theta equals one over R dxdy.
451
00:32:36.640 --> 00:32:39.640
So, we have, of course, two replace R by its formula
452
00:32:39.640 --> 00:32:41.640
and xy coordinates.
453
00:32:41.640 --> 00:32:42.640
Usually, we don't do that.
454
00:32:42.640 --> 00:32:46.640
Usually, we actually start with xy and switch to polar.
455
00:32:46.640 --> 00:32:50.640
But, so, in general, you know, when you have this formula relating
456
00:32:50.640 --> 00:32:54.640
duv with dxdy, you can use it both ways.
457
00:32:54.640 --> 00:32:59.640
I have to switch from duv to dxdy or the other way around.
458
00:32:59.640 --> 00:33:02.640
And the thing that I'm not telling you,
459
00:33:02.640 --> 00:33:05.640
but now I should probably tell you, is, you know,
460
00:33:05.640 --> 00:33:08.640
that there's a lot of variables in the formula that we can
461
00:33:08.640 --> 00:33:10.640
change in terms of uv.
462
00:33:10.640 --> 00:33:14.640
I can change the formula that we can change in terms of uv
463
00:33:14.640 --> 00:33:17.640
and the other variable, which is, you know,
464
00:33:17.640 --> 00:33:21.640
if you have a formula that you can change in terms of uv,
465
00:33:21.640 --> 00:33:24.640
because if I solve for xy in terms of uv,
466
00:33:24.640 --> 00:33:28.640
instead of uv in terms of xy, then I can compute two different
467
00:33:28.640 --> 00:33:35.640
formulas that you might get are consistent.
468
00:33:58.640 --> 00:34:08.640
Okay, so, useful remark.
469
00:34:08.640 --> 00:34:20.640
You know, so say that you can compute both
470
00:34:20.640 --> 00:34:24.640
these guys, well, then actually their product will just be one.
471
00:34:24.640 --> 00:34:26.640
So, you know, they're the inverse of each other.
472
00:34:26.640 --> 00:34:29.640
So, it doesn't matter which one you compute.
473
00:34:29.640 --> 00:34:32.640
You can compute whichever one is the easiest to compute,
474
00:34:32.640 --> 00:34:46.640
no matter which one of the two you need.
475
00:34:46.640 --> 00:34:49.640
And one way to see that is that, in fact,
476
00:34:49.640 --> 00:34:52.640
you know, we're looking at the determinants of these matrices
477
00:34:52.640 --> 00:34:55.640
that tell us the relation between the variables.
478
00:34:55.640 --> 00:34:58.640
And then it tells you how delta u delta v relates to delta x delta y.
479
00:34:58.640 --> 00:35:00.640
The other one does the opposite thing.
480
00:35:00.640 --> 00:35:02.640
It means that the inverse matrices.
481
00:35:02.640 --> 00:35:07.640
And the determinant of the inverse matrix is the inverse of the determinant.
482
00:35:07.640 --> 00:35:11.640
So, they're really interchangeable.
483
00:35:11.640 --> 00:35:16.640
I mean, you can just, you know, compute whichever one is the easiest.
484
00:35:16.640 --> 00:35:20.640
So, here, if you wanted, you know, the other theta in terms of dx dy,
485
00:35:20.640 --> 00:35:23.640
it's easier to do this and then move the r over there,
486
00:35:23.640 --> 00:35:27.640
than to first solve for r and theta as functions of x and y,
487
00:35:27.640 --> 00:35:30.640
and then do the entire thing again.
488
00:35:30.640 --> 00:35:32.640
But you can do it if you want.
489
00:35:32.640 --> 00:35:41.640
I mean, it works.
490
00:35:41.640 --> 00:35:43.640
Oh, yeah. The other useful remark.
491
00:35:43.640 --> 00:35:46.640
So, I mentioned it, but let me emphasize again.
492
00:35:46.640 --> 00:35:49.640
So, now, the ratio between du, du, dx, dy,
493
00:35:49.640 --> 00:35:51.640
it's not a constant anymore.
494
00:35:51.640 --> 00:35:55.640
Over there, it used to be 5, but now it's become c r or anything.
495
00:35:55.640 --> 00:35:58.640
I mean, in general, it would be a function that depends on the variables.
496
00:35:58.640 --> 00:36:00.640
So, it's not true that you can just say,
497
00:36:00.640 --> 00:36:06.640
oh, I'll put a constant times du dv. Yes?
498
00:36:06.640 --> 00:36:15.640
It would still work the same.
499
00:36:15.640 --> 00:36:19.640
You know, you could imagine drawing a picture where r and theta are the Cartesian coordinates,
500
00:36:19.640 --> 00:36:22.640
and your picture would be completely messed up.
501
00:36:22.640 --> 00:36:25.640
It would be a very strange thing to do to try to draw, you know,
502
00:36:25.640 --> 00:36:28.640
I'm going to do it, but don't take notes on that.
503
00:36:28.640 --> 00:36:30.640
You know, you could try to draw a picture like that,
504
00:36:30.640 --> 00:36:33.640
and then, you know, a circle would start looking like, you know,
505
00:36:33.640 --> 00:36:36.640
a disk would look like that, and it would be very counter-intuitive.
506
00:36:36.640 --> 00:36:39.640
But you could do it, and that would be equivalent to what we did
507
00:36:39.640 --> 00:36:42.640
with the previous change of variables.
508
00:36:42.640 --> 00:36:46.640
So, you know, in this case, certainly you would never draw a picture like that.
509
00:36:46.640 --> 00:36:53.640
But you could do it.
510
00:36:53.640 --> 00:36:58.640
Okay.
511
00:36:58.640 --> 00:37:02.640
Okay, so now let's do a complete example to see how things fit together, you know,
512
00:37:02.640 --> 00:37:06.640
how we do everything.
513
00:37:06.640 --> 00:37:09.640
So, let's say that we want to compute.
514
00:37:09.640 --> 00:37:12.640
So, I have to warn you, it's going to be a very silly example.
515
00:37:12.640 --> 00:37:15.640
It's an example where it's much easier to compute things without the change of variables,
516
00:37:15.640 --> 00:37:24.640
but you know, it's good practice in the sense that we're going to make it so complicated,
517
00:37:24.640 --> 00:37:29.640
that if we can do this one, then we can do them all.
518
00:37:29.640 --> 00:37:34.640
So, let's say that we want to compute this, and of course it's very easy to compute it directly.
519
00:37:34.640 --> 00:37:39.640
But let's say that for some evil reason,
520
00:37:39.640 --> 00:37:48.640
we want to do that by changing variables to u equals x, and v equals xy.
521
00:37:48.640 --> 00:37:54.640
Okay, that's a very strange idea, but let's do it anyway.
522
00:37:54.640 --> 00:37:58.640
I mean, normally, you know, you would only do this kind of substitution
523
00:37:58.640 --> 00:38:01.640
if either it simplifies a lot of the function you're integrating,
524
00:38:01.640 --> 00:38:05.640
or it simplifies a lot of the region on which you're integrating.
525
00:38:05.640 --> 00:38:11.640
And here, neither happens. So, but anyway,
526
00:38:11.640 --> 00:38:16.640
so the first thing we have to do here is figure out what we're going to be integrating.
527
00:38:16.640 --> 00:38:22.640
Okay, so to do that, first we should figure out what the xdy will be coming in terms of u and v.
528
00:38:22.640 --> 00:38:25.640
So that's what we've just seen using the Jacobian.
529
00:38:25.640 --> 00:38:31.640
Okay, so the first thing to do is find the area element.
530
00:38:31.640 --> 00:38:35.640
And for that, we use the Jacobian. So, well, let's see.
531
00:38:35.640 --> 00:38:40.640
The one that we can do easily is partial of u and v with respect to x and y.
532
00:38:40.640 --> 00:38:43.640
I mean, the other one is not very hot because here you can solve easily,
533
00:38:43.640 --> 00:38:49.640
but the one that's given to you is partial of u and v with respect to x and y.
534
00:38:49.640 --> 00:38:52.640
So partial u, partial x is one.
535
00:38:52.640 --> 00:38:55.640
Partial u, partial y is zero.
536
00:38:55.640 --> 00:39:02.640
Partial v, partial x is y and partial v, partial y is x.
537
00:39:02.640 --> 00:39:07.640
Okay, so it's just x.
538
00:39:07.640 --> 00:39:18.640
So that means that du, du, du is x, dx, dy.
539
00:39:18.640 --> 00:39:23.640
Well, it would be absolute value of x, but x is positive in our region.
540
00:39:23.640 --> 00:39:30.640
So, at least we don't have to worry about that.
541
00:39:30.640 --> 00:39:37.640
Okay.
542
00:39:37.640 --> 00:39:45.640
So now that we have that, we can try to look at the integrand
543
00:39:45.640 --> 00:39:50.640
in terms of u and v.
544
00:39:50.640 --> 00:39:56.640
Okay, so we were integrating x squared y dx dy.
545
00:39:56.640 --> 00:39:59.640
So, let's switch it.
546
00:39:59.640 --> 00:40:09.640
Well, let's first switch the dx dy that becomes one of our x, du, du, du.
547
00:40:09.640 --> 00:40:14.640
So that's actually x, y, du, du, du.
548
00:40:14.640 --> 00:40:17.640
And what is x, y in terms of u and v?
549
00:40:17.640 --> 00:40:20.640
Well, here at least we have a little bit of luck.
550
00:40:20.640 --> 00:40:22.640
No x, y is just v.
551
00:40:22.640 --> 00:40:26.640
So that's v, du, du.
552
00:40:26.640 --> 00:40:30.640
So in fact, what will be computing is a double integral
553
00:40:30.640 --> 00:40:39.640
over some mysterious region of v, du, du.
554
00:40:39.640 --> 00:40:40.640
Okay.
555
00:40:40.640 --> 00:40:44.640
Now, last but not least, we'll have to find what other bounds
556
00:40:44.640 --> 00:40:47.640
for u and v in the new integral.
557
00:40:47.640 --> 00:41:12.640
So that we know how to evaluate this.
558
00:41:12.640 --> 00:41:16.640
In fact, well, we could do it du, du, du, du, du.
559
00:41:16.640 --> 00:41:19.640
We don't know yet.
560
00:41:19.640 --> 00:41:21.640
Oh, amazing.
561
00:41:21.640 --> 00:41:30.640
It went all the way down this time.
562
00:41:30.640 --> 00:41:32.640
Okay.
563
00:41:32.640 --> 00:41:38.640
So, it could be du.
564
00:41:38.640 --> 00:41:43.640
If that's easier.
565
00:41:43.640 --> 00:41:46.640
So let's try to find the bounds.
566
00:41:46.640 --> 00:41:51.640
That's, in this case, that's the hardest part.
567
00:41:51.640 --> 00:41:52.640
Okay.
568
00:41:52.640 --> 00:41:58.640
So let me draw a picture in x, y coordinates.
569
00:41:58.640 --> 00:42:04.640
And try to understand things using that.
570
00:42:04.640 --> 00:42:08.640
So x and y go from 0 to 1, the region that we want to integrate
571
00:42:08.640 --> 00:42:11.640
over was just this square.
572
00:42:11.640 --> 00:42:16.640
Let's try to figure out how u and v vary in there.
573
00:42:16.640 --> 00:42:20.640
So let's say that we are going to do,
574
00:42:20.640 --> 00:42:23.640
let's say we're going to do it du, du, du.
575
00:42:23.640 --> 00:42:24.640
Okay.
576
00:42:24.640 --> 00:42:33.640
So what we want to understand is how u and v vary in here.
577
00:42:33.640 --> 00:42:36.640
So what's going to happen?
578
00:42:36.640 --> 00:42:40.640
So the way we can think about it is we try to figure out how we are
579
00:42:40.640 --> 00:42:42.640
slicing our region.
580
00:42:42.640 --> 00:42:45.640
Okay. So here we are integrating first of our u.
581
00:42:45.640 --> 00:42:48.640
That means we start by keeping u constant.
582
00:42:48.640 --> 00:42:55.640
No, by keeping v constant as u changes.
583
00:42:55.640 --> 00:42:56.640
Okay.
584
00:42:56.640 --> 00:43:02.640
So u changes as v is constant.
585
00:43:02.640 --> 00:43:05.640
What does it mean that I'm keeping v constant?
586
00:43:05.640 --> 00:43:06.640
Well, what is v?
587
00:43:06.640 --> 00:43:08.640
v is xy.
588
00:43:08.640 --> 00:43:12.640
So that means I keep xy equals constant.
589
00:43:12.640 --> 00:43:15.640
What does the curve xy equals constant look like?
590
00:43:15.640 --> 00:43:17.640
Well, it's just a hyperbola.
591
00:43:17.640 --> 00:43:18.640
Okay.
592
00:43:18.640 --> 00:43:21.640
y equals constant of our x.
593
00:43:21.640 --> 00:43:31.640
So if I look at the various values of v that I can take,
594
00:43:31.640 --> 00:43:34.640
for each value of v, if I fix a value of v,
595
00:43:34.640 --> 00:43:38.640
I will be moving on one of these red curves.
596
00:43:38.640 --> 00:43:39.640
Okay.
597
00:43:39.640 --> 00:43:42.640
And u, well, u is the same thing as x.
598
00:43:42.640 --> 00:43:44.640
So that means u will increase.
599
00:43:44.640 --> 00:43:50.640
You know, here maybe it will be point one and it will increase to all the way to one here.
600
00:43:50.640 --> 00:43:58.640
Okay. So we are just traveling on each of these slices.
601
00:43:58.640 --> 00:44:03.640
Now, so the question we must answer here is for a given value of v,
602
00:44:03.640 --> 00:44:05.640
what are the bounds for u?
603
00:44:05.640 --> 00:44:10.640
So I'm traveling on my curve v equals constant and trying to figure out,
604
00:44:10.640 --> 00:44:12.640
when do I enter my region?
605
00:44:12.640 --> 00:44:14.640
When do I leave it?
606
00:44:14.640 --> 00:44:17.640
Well, I enter it when I go through this side.
607
00:44:17.640 --> 00:44:23.640
So the question is, what's the value of u here?
608
00:44:23.640 --> 00:44:27.640
Well, we don't know that very easily until we look at this formulas.
609
00:44:27.640 --> 00:44:33.640
So u equals x, okay, but we don't know what x is at that point.
610
00:44:33.640 --> 00:44:37.640
Well, u equals x and v equals xy.
611
00:44:37.640 --> 00:44:39.640
What do we know here?
612
00:44:39.640 --> 00:44:43.640
Well, we don't know x, but we know y, certainly.
613
00:44:43.640 --> 00:44:44.640
Okay.
614
00:44:44.640 --> 00:44:48.640
So let's forget about trying to find u.
615
00:44:48.640 --> 00:44:51.640
And let's say for now, we know y equals 1.
616
00:44:51.640 --> 00:44:55.640
Well, if we set y equals 1,
617
00:44:55.640 --> 00:44:59.640
that tells us that u and v are both equal to x.
618
00:44:59.640 --> 00:45:10.640
So in terms of u and v, the equation of this in u v coordinates is u equals v.
619
00:45:10.640 --> 00:45:11.640
Okay.
620
00:45:11.640 --> 00:45:15.640
I mean, the other way to do it is say that you know you want y equals 1.
621
00:45:15.640 --> 00:45:17.640
You want to know what is y in terms of u and v.
622
00:45:17.640 --> 00:45:18.640
Well, it's easy.
623
00:45:18.640 --> 00:45:23.640
y is v over u.
624
00:45:23.640 --> 00:45:27.640
So let me actually add an extra step in case that's.
625
00:45:27.640 --> 00:45:32.640
So we know that y is v over u equals 1.
626
00:45:32.640 --> 00:45:36.640
So that means u equals v is my equation.
627
00:45:36.640 --> 00:45:37.640
Okay.
628
00:45:37.640 --> 00:45:42.640
So when I'm here, when I'm entering my regime,
629
00:45:42.640 --> 00:45:50.640
the value of u at this point is just v, u equals v.
630
00:45:50.640 --> 00:45:52.640
That's the hard part.
631
00:45:52.640 --> 00:45:55.640
Now we need to figure out, you know, so we start that u equals v.
632
00:45:55.640 --> 00:45:58.640
u increases, increases, increases.
633
00:45:58.640 --> 00:45:59.640
Where does it exit?
634
00:45:59.640 --> 00:46:01.640
The exit's when we are here.
635
00:46:01.640 --> 00:46:03.640
What's the value of u here?
636
00:46:03.640 --> 00:46:04.640
1.
637
00:46:04.640 --> 00:46:05.640
That one is easier, right?
638
00:46:05.640 --> 00:46:08.640
This side here.
639
00:46:08.640 --> 00:46:10.640
So this side here is x equals 1.
640
00:46:10.640 --> 00:46:12.640
That means u equals 1.
641
00:46:12.640 --> 00:46:16.640
So we start that u equals 1.
642
00:46:16.640 --> 00:46:20.640
Now, we've done the inner integral.
643
00:46:20.640 --> 00:46:23.640
What about the outer?
644
00:46:23.640 --> 00:46:31.640
So we have to figure out what is the first and what is the last value of v that we want to consider.
645
00:46:31.640 --> 00:46:35.640
Well, if you look at all these hyperbolas, x, y equals constant.
646
00:46:35.640 --> 00:46:39.640
What's the smallest value of x, y that we'll ever want to look at in here?
647
00:46:39.640 --> 00:46:40.640
0.
648
00:46:40.640 --> 00:46:41.640
Okay.
649
00:46:41.640 --> 00:46:44.640
Let me actually, where's my yellow chalk?
650
00:46:44.640 --> 00:46:45.640
Is it?
651
00:46:45.640 --> 00:46:48.640
No.
652
00:46:48.640 --> 00:46:56.640
So this one here, that's actually v equals 0.
653
00:46:56.640 --> 00:46:59.640
So we'll start that v equals 0.
654
00:46:59.640 --> 00:47:02.640
And what's the last hyperbola we want to look at?
655
00:47:02.640 --> 00:47:05.640
Well, it's the one that's right there in the corner.
656
00:47:05.640 --> 00:47:07.640
It's this one here.
657
00:47:07.640 --> 00:47:09.640
And that's v equals 1.
658
00:47:09.640 --> 00:47:14.640
So v goes from 0 to 1.
659
00:47:14.640 --> 00:47:15.640
Okay.
660
00:47:15.640 --> 00:47:18.640
So we can compute this.
661
00:47:18.640 --> 00:47:24.640
It's not particularly easier than that one, but it's not harder either.
662
00:47:24.640 --> 00:47:26.640
How else could we have gotten these bounds?
663
00:47:26.640 --> 00:47:28.640
Because that was quite evil.
664
00:47:28.640 --> 00:47:31.640
So I would like to recommend that you give it a try this way.
665
00:47:31.640 --> 00:47:38.640
In case it works well, just try to picture what are the slices in terms of human v.
666
00:47:38.640 --> 00:47:41.640
And how you travel on them, where you enter, where you leave,
667
00:47:41.640 --> 00:47:43.640
staying in the x, y picture.
668
00:47:43.640 --> 00:47:45.640
So it doesn't work well.
669
00:47:45.640 --> 00:47:50.640
Another way is to draw a picture in the uv pic coordinates.
670
00:47:50.640 --> 00:47:57.640
Oops.
671
00:47:57.640 --> 00:48:01.640
Switch to a uv picture.
672
00:48:01.640 --> 00:48:03.640
So what do I mean by that?
673
00:48:03.640 --> 00:48:09.640
Well, we had here a picture in x, y coordinates, where we had our sides.
674
00:48:09.640 --> 00:48:15.640
And we're going to try to draw what it looks like in terms of u and v.
675
00:48:15.640 --> 00:48:17.640
So here we say this is x equals 1.
676
00:48:17.640 --> 00:48:20.640
That becomes u equals 1.
677
00:48:20.640 --> 00:48:24.640
So we'll draw u equals 1.
678
00:48:24.640 --> 00:48:27.640
This side we say is y equals 1.
679
00:48:27.640 --> 00:48:29.640
That becomes u equals v.
680
00:48:29.640 --> 00:48:32.640
That's what we've done over there.
681
00:48:32.640 --> 00:48:37.640
Okay. So u equals v.
682
00:48:37.640 --> 00:48:39.640
Now we have the two other sides to deal with.
683
00:48:39.640 --> 00:48:41.640
Well, let's look at this one first.
684
00:48:41.640 --> 00:48:43.640
So that was x equals 0.
685
00:48:43.640 --> 00:48:45.640
What happens when x equals 0?
686
00:48:45.640 --> 00:48:47.640
Well, both u and v are 0.
687
00:48:47.640 --> 00:48:51.640
So this side actually gets squished in the change of variables.
688
00:48:51.640 --> 00:48:57.640
It's a bit strange, but it's a bit the same thing as when you switch to polar coordinates at the origin.
689
00:48:57.640 --> 00:49:00.640
r is 0, but theta can be anything.
690
00:49:00.640 --> 00:49:03.640
Sometimes it's not always one point is one point.
691
00:49:03.640 --> 00:49:07.640
So anyway, this is the origin.
692
00:49:07.640 --> 00:49:10.640
And then the last side, y equals 0.
693
00:49:10.640 --> 00:49:14.640
And x varies just becomes v equals 0.
694
00:49:14.640 --> 00:49:20.640
So somehow, in the change of variables, this square becomes this triangle.
695
00:49:20.640 --> 00:49:29.640
And now, if we want to integrate du dv, it means we're going to slice by v equals constant.
696
00:49:29.640 --> 00:49:33.640
So we're going to integrate other slices like this.
697
00:49:33.640 --> 00:49:37.640
And you see for each value of v, we go from u equals v to u equals 1.
698
00:49:37.640 --> 00:49:40.640
And v goes from 0 to 1.
699
00:49:40.640 --> 00:49:44.640
So you get the same bounce just by drawing a different picture.
700
00:49:44.640 --> 00:49:47.640
So it's up to you to decide whether you prefer to, you know,
701
00:49:47.640 --> 00:49:51.640
think on this picture or draw that one instead.
702
00:49:51.640 --> 00:50:01.640
It depends on which problem you're doing.