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And if you let z increase, keep it constant, but let it increase by 1 to negative 1, we get a new plane still parallel to the xy-plane. And it's got a, but its distance from the xy-plane is negative 1. And the rest of these guys, they're all still values, constant values of z. Now in terms of our graph, what that means is that these represent constant values of the graph itself. These represent constant values for the function itself. So because we always represent the output of the function as the height off of the xy-plane, these represent constant values for the output. So what that's going to look like, so what we do is we say, where do these slices cut into the graph?
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Contour plots Multivariable calculus Khan Academy.mp3
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Now in terms of our graph, what that means is that these represent constant values of the graph itself. These represent constant values for the function itself. So because we always represent the output of the function as the height off of the xy-plane, these represent constant values for the output. So what that's going to look like, so what we do is we say, where do these slices cut into the graph? So I'm going to draw on all of the points where those slices cut into the graph. And these are called contour lines. We're still in three dimensions, so we're not done yet.
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Contour plots Multivariable calculus Khan Academy.mp3
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So what that's going to look like, so what we do is we say, where do these slices cut into the graph? So I'm going to draw on all of the points where those slices cut into the graph. And these are called contour lines. We're still in three dimensions, so we're not done yet. So what I'm going to do is take all these contour lines, and I'm going to squish them down onto the xy-plane. So what that means, each of them has some kind of z component at the moment, and we're just going to chop it down and squish them all nice and flat onto the xy-plane. And now we have something two-dimensional, and it still represents some of the outputs of our function.
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Contour plots Multivariable calculus Khan Academy.mp3
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We're still in three dimensions, so we're not done yet. So what I'm going to do is take all these contour lines, and I'm going to squish them down onto the xy-plane. So what that means, each of them has some kind of z component at the moment, and we're just going to chop it down and squish them all nice and flat onto the xy-plane. And now we have something two-dimensional, and it still represents some of the outputs of our function. Not all of them, it's not perfect, but it does give a very good idea. So I'm going to switch over to a two-dimensional graph here. And this is that same function that we were just looking at.
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Contour plots Multivariable calculus Khan Academy.mp3
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And now we have something two-dimensional, and it still represents some of the outputs of our function. Not all of them, it's not perfect, but it does give a very good idea. So I'm going to switch over to a two-dimensional graph here. And this is that same function that we were just looking at. Let's actually move it a little bit more central here. So this is the same function that we were just looking at, but each of these lines represents a constant output of the function. So it's important to realize we're still representing a function that has a two-dimensional input and a one-dimensional output.
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Contour plots Multivariable calculus Khan Academy.mp3
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And this is that same function that we were just looking at. Let's actually move it a little bit more central here. So this is the same function that we were just looking at, but each of these lines represents a constant output of the function. So it's important to realize we're still representing a function that has a two-dimensional input and a one-dimensional output. It's just that we're looking in the input space of that function as a whole. So this is still f of xy, and then some expression of those guys. But this line might represent the constant value of f when all of the values where it outputs 3.
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Contour plots Multivariable calculus Khan Academy.mp3
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So it's important to realize we're still representing a function that has a two-dimensional input and a one-dimensional output. It's just that we're looking in the input space of that function as a whole. So this is still f of xy, and then some expression of those guys. But this line might represent the constant value of f when all of the values where it outputs 3. Over here, this also, both of these circles together give you all the values where f outputs 3. This one over here will tell you where it outputs 2. And you can't know this just looking at the contour plot.
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Contour plots Multivariable calculus Khan Academy.mp3
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But this line might represent the constant value of f when all of the values where it outputs 3. Over here, this also, both of these circles together give you all the values where f outputs 3. This one over here will tell you where it outputs 2. And you can't know this just looking at the contour plot. So typically, if someone's drawing it, if it matters that you know the specific values, they'll mark it somehow. They'll let you know what value each line corresponds to. But as soon as you know that this line corresponds to 0, it tells you that every possible input point that sits somewhere on this line will evaluate to 0 when you pump it through the function.
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Contour plots Multivariable calculus Khan Academy.mp3
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And you can't know this just looking at the contour plot. So typically, if someone's drawing it, if it matters that you know the specific values, they'll mark it somehow. They'll let you know what value each line corresponds to. But as soon as you know that this line corresponds to 0, it tells you that every possible input point that sits somewhere on this line will evaluate to 0 when you pump it through the function. And this actually gives a very good feel for the shape of things. If you like thinking in terms of graphs, you can kind of imagine how these circles and everything would pop out of the page. You can also look.
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Contour plots Multivariable calculus Khan Academy.mp3
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But as soon as you know that this line corresponds to 0, it tells you that every possible input point that sits somewhere on this line will evaluate to 0 when you pump it through the function. And this actually gives a very good feel for the shape of things. If you like thinking in terms of graphs, you can kind of imagine how these circles and everything would pop out of the page. You can also look. Notice how the lines are really close together over here, very, very close together, but they're a little more spaced over here. How do you interpret that? Well, over here, this means it takes a very, very small step to increase the value of the function by 1.
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Contour plots Multivariable calculus Khan Academy.mp3
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You can also look. Notice how the lines are really close together over here, very, very close together, but they're a little more spaced over here. How do you interpret that? Well, over here, this means it takes a very, very small step to increase the value of the function by 1. Very small step, and it increases by 1. But over here, it takes a much larger step to increase the function by the same value. So over here, this kind of means steepness.
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Contour plots Multivariable calculus Khan Academy.mp3
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Well, over here, this means it takes a very, very small step to increase the value of the function by 1. Very small step, and it increases by 1. But over here, it takes a much larger step to increase the function by the same value. So over here, this kind of means steepness. If you see a very short distance between contour lines, it's going to be very steep. But over here, it's much more shallow. And you can do things like this to kind of get a better feel for the function as a whole.
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Contour plots Multivariable calculus Khan Academy.mp3
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So over here, this kind of means steepness. If you see a very short distance between contour lines, it's going to be very steep. But over here, it's much more shallow. And you can do things like this to kind of get a better feel for the function as a whole. The idea of a whole bunch of concentric circles usually corresponds to a maximum or a minimum. And you end up seeing these a lot. Another common thing people will do with contour plots as they represent them is color them.
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Contour plots Multivariable calculus Khan Academy.mp3
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And you can do things like this to kind of get a better feel for the function as a whole. The idea of a whole bunch of concentric circles usually corresponds to a maximum or a minimum. And you end up seeing these a lot. Another common thing people will do with contour plots as they represent them is color them. So what that might look like is here, where warmer colors like orange correspond to high values, and cooler colors like blue correspond to low values. And the contour lines end up going along the division between red and green here, between light green and green. And that's another way where colors tell you the output, and then the contour lines themselves can be thought of as the borders between different colors.
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Contour plots Multivariable calculus Khan Academy.mp3
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And here, I want to talk about second partial derivatives. So I'm going to write some kind of multivariable function, let's say it's, I don't know, sine of x times y squared. Sine of x multiplied by y squared. And if you take the partial derivative, you have two options, given that there's two variables, you can go one way and say, what's the partial derivative, partial derivative of f with respect to x, and what you do for that, x looks like a variable, as far as this direction is concerned, y looks like a constant, so you differentiate this by saying the derivative of sine of x is cosine x, you know, you're differentiating with respect to x, and then that looks like it's multiplied by a constant, so you just continue multiplying that by a constant. But you could also go another direction, you could also say, you know, what's the partial derivative with respect to y, and in that case, you're considering y to be the variable, so here it looks at y, and says y squared looks like a variable, x looks like a constant, sine of x then just looks like sine of a constant, which is a constant, so that'll be that constant, sine of x, multiplied by the derivative of y squared, which is gonna be two times y. Two times y. And these are what you might call first partial derivatives, and there's some alternate notation here, df dy, you could also say f and then a little subscript y, and over here, similarly, you'd say f with a little subscript x.
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Symmetry of second partial derivatives.mp3
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And if you take the partial derivative, you have two options, given that there's two variables, you can go one way and say, what's the partial derivative, partial derivative of f with respect to x, and what you do for that, x looks like a variable, as far as this direction is concerned, y looks like a constant, so you differentiate this by saying the derivative of sine of x is cosine x, you know, you're differentiating with respect to x, and then that looks like it's multiplied by a constant, so you just continue multiplying that by a constant. But you could also go another direction, you could also say, you know, what's the partial derivative with respect to y, and in that case, you're considering y to be the variable, so here it looks at y, and says y squared looks like a variable, x looks like a constant, sine of x then just looks like sine of a constant, which is a constant, so that'll be that constant, sine of x, multiplied by the derivative of y squared, which is gonna be two times y. Two times y. And these are what you might call first partial derivatives, and there's some alternate notation here, df dy, you could also say f and then a little subscript y, and over here, similarly, you'd say f with a little subscript x. Now each of these two functions, these two partial derivatives that you get, are also multivariable functions, they take in two variables and they output a scalar, so we can do something very similar, where here, you might then apply the partial derivative with respect to x to that partial derivative of your original function with respect to x, right? It's just like a second derivative in ordinary calculus, but this time, we're doing it partial, so when you do it with respect to x, cosine x looks like cosine of a variable, the derivative of which is negative sine times that variable, and y squared here just looks like a constant, so it just stays constant at y squared, and similarly, you could go down a different branch of options here, and say what if you did your partial derivative with respect to y of that whole function, which itself is a partial derivative with respect to x? And if you did that, then y squared now looks like the variable, so you're gonna take the derivative of that, which is two y, and then what's in front of it just looks like a constant as far as the variable y is concerned, so that stays as cosine of x.
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Symmetry of second partial derivatives.mp3
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And these are what you might call first partial derivatives, and there's some alternate notation here, df dy, you could also say f and then a little subscript y, and over here, similarly, you'd say f with a little subscript x. Now each of these two functions, these two partial derivatives that you get, are also multivariable functions, they take in two variables and they output a scalar, so we can do something very similar, where here, you might then apply the partial derivative with respect to x to that partial derivative of your original function with respect to x, right? It's just like a second derivative in ordinary calculus, but this time, we're doing it partial, so when you do it with respect to x, cosine x looks like cosine of a variable, the derivative of which is negative sine times that variable, and y squared here just looks like a constant, so it just stays constant at y squared, and similarly, you could go down a different branch of options here, and say what if you did your partial derivative with respect to y of that whole function, which itself is a partial derivative with respect to x? And if you did that, then y squared now looks like the variable, so you're gonna take the derivative of that, which is two y, and then what's in front of it just looks like a constant as far as the variable y is concerned, so that stays as cosine of x. And the notation here, first of all, just as in single variable calculus, it's common to kind of do a abusive notation with this kind of thing and write partial squared of f divided by partial x squared. And this always, I don't know, when I first learned about these things, they always threw me off because here, this Leibniz notation, you have the great intuition of nudging the x and nudging the f, but you kind of lose that when you do this, but it makes sense if you think of this partial partial x as being an operator and you're just applying it twice. And over here, the way that that would look, it's a little bit funny, because you still have that partial squared f on top, but then on the bottom, you write partial y, partial x.
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Symmetry of second partial derivatives.mp3
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And if you did that, then y squared now looks like the variable, so you're gonna take the derivative of that, which is two y, and then what's in front of it just looks like a constant as far as the variable y is concerned, so that stays as cosine of x. And the notation here, first of all, just as in single variable calculus, it's common to kind of do a abusive notation with this kind of thing and write partial squared of f divided by partial x squared. And this always, I don't know, when I first learned about these things, they always threw me off because here, this Leibniz notation, you have the great intuition of nudging the x and nudging the f, but you kind of lose that when you do this, but it makes sense if you think of this partial partial x as being an operator and you're just applying it twice. And over here, the way that that would look, it's a little bit funny, because you still have that partial squared f on top, but then on the bottom, you write partial y, partial x. And I'm putting them in these order just because it's as if I wrote it that way. This reflects the fact that first I did the x derivative, then I did the y derivative. And you could do this on this side also.
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Symmetry of second partial derivatives.mp3
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And over here, the way that that would look, it's a little bit funny, because you still have that partial squared f on top, but then on the bottom, you write partial y, partial x. And I'm putting them in these order just because it's as if I wrote it that way. This reflects the fact that first I did the x derivative, then I did the y derivative. And you could do this on this side also. And this might feel tedious, but it's actually kind of worth doing for a result that we end up seeing here that I find a little bit surprising, actually. So here, if we go down the path of doing, in this case, like a partial derivative with respect to x, and you're thinking of this as being applied to your original partial derivative with respect to y, it looks here, it says sine of x looks like a variable, two y looks like a constant, so what we end up getting is derivative of sine of x, cosine x, multiplied by that two y. And a pretty cool thing worth pointing out here that maybe you take it for granted, maybe you think it's as surprising as I did when I first saw it, both of these turn out to be equal, right, even though it was a very different way that we got there, right?
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Symmetry of second partial derivatives.mp3
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And you could do this on this side also. And this might feel tedious, but it's actually kind of worth doing for a result that we end up seeing here that I find a little bit surprising, actually. So here, if we go down the path of doing, in this case, like a partial derivative with respect to x, and you're thinking of this as being applied to your original partial derivative with respect to y, it looks here, it says sine of x looks like a variable, two y looks like a constant, so what we end up getting is derivative of sine of x, cosine x, multiplied by that two y. And a pretty cool thing worth pointing out here that maybe you take it for granted, maybe you think it's as surprising as I did when I first saw it, both of these turn out to be equal, right, even though it was a very different way that we got there, right? You first take the partial derivative with respect to x, and you get cosine xy squared, which looks very different from sine x two y, and then when you take the derivative with respect to y, you know, you get a certain value, and when you go down the other path, you also get that same value. And maybe the way that you'd write this is that you'd say, let me just copy this guy over here, and what you might say is that the partial derivative of f, when you do it the other way around, when instead of doing x and then y, you do y and then x, partial x, that these guys are equal to each other, and that's a pretty cool result. And maybe in this case, given that the original function just looks like the product of two things, you can kind of reason through why it's the case, but what's surprising is that this turns out to be true for, I mean, not all functions, there's actually a certain criterion.
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Symmetry of second partial derivatives.mp3
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And a pretty cool thing worth pointing out here that maybe you take it for granted, maybe you think it's as surprising as I did when I first saw it, both of these turn out to be equal, right, even though it was a very different way that we got there, right? You first take the partial derivative with respect to x, and you get cosine xy squared, which looks very different from sine x two y, and then when you take the derivative with respect to y, you know, you get a certain value, and when you go down the other path, you also get that same value. And maybe the way that you'd write this is that you'd say, let me just copy this guy over here, and what you might say is that the partial derivative of f, when you do it the other way around, when instead of doing x and then y, you do y and then x, partial x, that these guys are equal to each other, and that's a pretty cool result. And maybe in this case, given that the original function just looks like the product of two things, you can kind of reason through why it's the case, but what's surprising is that this turns out to be true for, I mean, not all functions, there's actually a certain criterion. There's a special theorem, it's called Schwarz's theorem, where if the second partial derivatives of your function are continuous at the relevant point, that's the circumstance for this being true, but for all intents and purposes, the kind of functions you can expect to run into, this is the case. This order of partial derivatives doesn't matter truth turns out to hold, which is actually pretty cool. And I'd encourage you to play around with some other functions, just come up with any multivariable function, maybe a little bit more complicated than just multiplying two separate things there, and see that it's true, and maybe try to convince yourself why it's true in certain cases.
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Symmetry of second partial derivatives.mp3
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And maybe in this case, given that the original function just looks like the product of two things, you can kind of reason through why it's the case, but what's surprising is that this turns out to be true for, I mean, not all functions, there's actually a certain criterion. There's a special theorem, it's called Schwarz's theorem, where if the second partial derivatives of your function are continuous at the relevant point, that's the circumstance for this being true, but for all intents and purposes, the kind of functions you can expect to run into, this is the case. This order of partial derivatives doesn't matter truth turns out to hold, which is actually pretty cool. And I'd encourage you to play around with some other functions, just come up with any multivariable function, maybe a little bit more complicated than just multiplying two separate things there, and see that it's true, and maybe try to convince yourself why it's true in certain cases. I think that would actually be a really good exercise. And just before I go, one thing I should probably mention, a bit of notation that people will commonly use. With this second partial derivative, sometimes instead of saying partial squared F, partial X squared, they'll just write it as partial and then XX.
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Symmetry of second partial derivatives.mp3
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And I'd encourage you to play around with some other functions, just come up with any multivariable function, maybe a little bit more complicated than just multiplying two separate things there, and see that it's true, and maybe try to convince yourself why it's true in certain cases. I think that would actually be a really good exercise. And just before I go, one thing I should probably mention, a bit of notation that people will commonly use. With this second partial derivative, sometimes instead of saying partial squared F, partial X squared, they'll just write it as partial and then XX. And over here, this would be partial, let's see, first you did it with X, then Y, so over here you do it first X and then Y, kind of the order of these reverses, because you're reading left to right, but when you do it with this, you're kind of reading right to left for how you multiply it in. Which would mean that this guy, let's see, this guy over here, you know, he would be partial, first you did the Y, and then you did the X. So those two, those two guys are just different notations for the same thing.
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Symmetry of second partial derivatives.mp3
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Alright everyone, we've gotten to one of my all-time favorite multivariable calculus topics, divergence, and in the next few videos I'm going to describe what it is mathematically and how you compute it and all of that, but here I just want to give a very visual understanding of what it is that it's trying to represent. So I've got a human pictured in front of us, a vector field, and I've said before that a pretty neat way to understand vector fields is to think of them as representing a fluid flow, and what I mean by that is you can think of every single point in space as a particle, maybe like an air particle or a water particle, something to that effect, and since what a vector field does is it associates each point in space with some kind of vector, and remember I mean whenever we represent vector fields we only show a small subset of all of those vectors, but in principle you should be thinking of every one of those infinitely many points in space being associated with one of these vectors, and the fact that they're kind of smoothly changing as you traverse across space means that showing this very small, you know, finite subsample of those infinitely many vectors still gives a pretty good feel for what's going on. So if we have these fluid particles and you, you know, have a vector assigned to each one, kind of a natural thought you might have is to say what would happen if you let things progress over time where at any given instant the velocity of one of these particles is given by that vector connected to it, and as it moves, you know, it'll be touching a different vector so its velocity might turn, it might go in a different direction, and for each one it'll kind of traverse some path as determined by the vectors that it's touching as it goes, and when you think of all of them doing this at once it'll feel like a certain, a certain fluid flow, and for this, for this you don't actually have to imagine, I went ahead and put together an animation for you. So we'll put some water molecules, or dots, to represent a small sample of the water molecules throughout space here, and then I'm just gonna let it play where each one moves along the vector that it's closest to, and I'll just let it play forward here where each one is flowing along the vector that's touching the point where it is in that moment. So for example if we were to, you know, go back and maybe focus our attention on just one vector like this guy, one particle excuse me, he's attached to this vector so he'll be moving in that direction, but just for an instant because after he moves a little he'll be attached to a different vector. So if you kind of let it play and follow that particular dot after a little bit you'll find him, you know, elsewhere. I think, I think this is the one right, and now he's gonna be moving along this vector or whatever one is really attached to him.
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Divergence intuition, part 1.mp3
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So we'll put some water molecules, or dots, to represent a small sample of the water molecules throughout space here, and then I'm just gonna let it play where each one moves along the vector that it's closest to, and I'll just let it play forward here where each one is flowing along the vector that's touching the point where it is in that moment. So for example if we were to, you know, go back and maybe focus our attention on just one vector like this guy, one particle excuse me, he's attached to this vector so he'll be moving in that direction, but just for an instant because after he moves a little he'll be attached to a different vector. So if you kind of let it play and follow that particular dot after a little bit you'll find him, you know, elsewhere. I think, I think this is the one right, and now he's gonna be moving along this vector or whatever one is really attached to him. And thinking about all of the particles all at once doing this gives a good sort of global view of the vector field, and if you're studying math you might start to ask some natural questions about the nature of that fluid flow. Like for example you might wonder if you were to just look in a certain region and count the number of water molecules that are inside that region, does that count of yours change as you play this, this animation, as you let this flow over time? And in this particular example you can look and it doesn't look like the count changes, certainly not by much, it's not increasing over time or decreasing over time, and in a little bit if I gave you the function that determines this vector field you will be able to tell me why it's the case that the number of molecules in that region doesn't tend to change.
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Divergence intuition, part 1.mp3
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I think, I think this is the one right, and now he's gonna be moving along this vector or whatever one is really attached to him. And thinking about all of the particles all at once doing this gives a good sort of global view of the vector field, and if you're studying math you might start to ask some natural questions about the nature of that fluid flow. Like for example you might wonder if you were to just look in a certain region and count the number of water molecules that are inside that region, does that count of yours change as you play this, this animation, as you let this flow over time? And in this particular example you can look and it doesn't look like the count changes, certainly not by much, it's not increasing over time or decreasing over time, and in a little bit if I gave you the function that determines this vector field you will be able to tell me why it's the case that the number of molecules in that region doesn't tend to change. But if you were to look at another example, like a guy that looks like this, and if I were to say I want you to focus on what happens around the origin, in that little region around the origin, and you can probably predict how once I start playing it, once I put some water molecules in there and let them flow along the vectors that they flow along, the density inside that region around the origin decreases. So we put a whole bunch of vectors there, and I'll just play it for a quick instant, just kind of let it jump for an instant, and one thing that characterizes this field around the origin is that decrease in density, and what you might say if you want it to be suggestive of the operation that I'm leading to here, is that the water molecules tend to diverge away from the origin, so the kind of divergence of the vector field near that origin is positive, and you'll see what I mean mathematically by that in the next couple videos, but if we were to flip all of these vectors, right, and we were to flip them around, now if I were to ask about the density in that same region around the origin, you can probably see how it's going to increase, and when I play that fluid flow over just a short spurt of time, the density in that region increases, so these don't diverge away, they converge towards the origin, and that fact actually has some mathematical significance for the function representing this vector field around that point, and even if the vector field doesn't represent fluid flow, if it represents like a magnetic field or an electric field or things like that, there's a certain meaning to this idea of diverging away from a point or converging to a point, and another way that people sometimes think about this, if you look at that same kind of outward flowing vector field, is rather than thinking of a decrease in density, imagining that the fluid would have to constantly be repopulated around that point, so you're really thinking of the origin as a source of fluid, and if I had animated this better, a whole bunch of other points should be sources of fluid so that the density doesn't decrease everywhere, but the idea is that points of positive divergence, where things are diverging away, would have to have a source of that fluid in order to kind of keep things sustaining, and conversely, if you were to look at that kind of inward flow, or what you might call negative divergence example, and you were to play it, but it were to go continuously, you'd have to think of that center point as a sink, where all the fluid just sort of flows away, and that's actually a technical term. People will say the vector field has a sink at such and such point, or the electromagnetic field has a sink at such and such point, and that often has a certain significance, and if we go back to that original example here, where there is no change in fluid density, what you might notice, this feels a lot more like actual water than the other ones, because there is no change in density there, and if you can find a way to mathematically describe that lack of a change in density, that's a pretty good way to model water flow, and again, even if it's not water flow, but it's something like the electromagnetic field, there's often a significance to this no changing in density idea, so with that, I think I've jabbered on enough about the visuals of it, and in the next video, I'll tell you what divergence is mathematically, how you compute it, go through a couple examples, things like that.
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Divergence intuition, part 1.mp3
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So what I'd like to do here is talk about curvature. Curvature. And I've drawn on the xy plane here a certain curve, so this is our x-axis, this is our y-axis, and this is a curve running through space. And I'd like you to imagine that this is a road of some kind and you're driving on it, and you're at a certain point. So let's say this point right here. And if you imagine what it feels like to drive along this road and where you need to have your steering wheel, you're turning it a little bit to the right, not a lot because it's kind of a gentle curve at this point, you're not curving a lot, but the steering wheel isn't straight, you are still turning on the road. And now, imagine that your steering wheel's stuck, that it's not gonna move, and however you're turning it, you're stuck in that situation.
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Curvature intuition.mp3
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And I'd like you to imagine that this is a road of some kind and you're driving on it, and you're at a certain point. So let's say this point right here. And if you imagine what it feels like to drive along this road and where you need to have your steering wheel, you're turning it a little bit to the right, not a lot because it's kind of a gentle curve at this point, you're not curving a lot, but the steering wheel isn't straight, you are still turning on the road. And now, imagine that your steering wheel's stuck, that it's not gonna move, and however you're turning it, you're stuck in that situation. What's gonna happen, hopefully you're on an open field or something, because your car's gonna trace out some kind of circle. Your steering wheel can't do anything different, you're just turning at a certain rate, and that's gonna have you tracing out some giant circle. And this depends on where you are, right?
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Curvature intuition.mp3
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And now, imagine that your steering wheel's stuck, that it's not gonna move, and however you're turning it, you're stuck in that situation. What's gonna happen, hopefully you're on an open field or something, because your car's gonna trace out some kind of circle. Your steering wheel can't do anything different, you're just turning at a certain rate, and that's gonna have you tracing out some giant circle. And this depends on where you are, right? If you had been at a different point on the curve, where the curve was rotating a lot, let's say you were back a little bit towards the start, at the start here, you have to turn the steering wheel to the right, but you're turning it much more sharply to stay on this part of the curve than you were to be on this relatively straight part. And the circle that you draw as a result is much smaller. And this turns out to be a pretty nice way to think about a measure for just how much the curve actually curves.
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Curvature intuition.mp3
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And this depends on where you are, right? If you had been at a different point on the curve, where the curve was rotating a lot, let's say you were back a little bit towards the start, at the start here, you have to turn the steering wheel to the right, but you're turning it much more sharply to stay on this part of the curve than you were to be on this relatively straight part. And the circle that you draw as a result is much smaller. And this turns out to be a pretty nice way to think about a measure for just how much the curve actually curves. And one way you could do this is you can think, okay, what is the radius of that circle, the circle that you would trace out if your steering wheel locked at any given point? And if you kind of follow the point along different parts of the curve and see, oh, what's the different circle that my car would trace out if it was stuck at that point? You get circles of varying different radii, right?
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Curvature intuition.mp3
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And this turns out to be a pretty nice way to think about a measure for just how much the curve actually curves. And one way you could do this is you can think, okay, what is the radius of that circle, the circle that you would trace out if your steering wheel locked at any given point? And if you kind of follow the point along different parts of the curve and see, oh, what's the different circle that my car would trace out if it was stuck at that point? You get circles of varying different radii, right? And this radius actually has a very special name. I'll call this R. This is called the radius of curvature. And you can kind of see how this is a good way to describe how much you're turning, radius of curvature.
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Curvature intuition.mp3
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You get circles of varying different radii, right? And this radius actually has a very special name. I'll call this R. This is called the radius of curvature. And you can kind of see how this is a good way to describe how much you're turning, radius of curvature. You may have heard with a car descriptions of the turning radius. You know, if you have a car with a very good turning radius, it's very small, because what that means is if you turned it all the way, you could trace out just a very small circle. But a car with a bad turning radius, you know, you don't turn very much at all, so you'd have to trace out a much larger circle.
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Curvature intuition.mp3
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And you can kind of see how this is a good way to describe how much you're turning, radius of curvature. You may have heard with a car descriptions of the turning radius. You know, if you have a car with a very good turning radius, it's very small, because what that means is if you turned it all the way, you could trace out just a very small circle. But a car with a bad turning radius, you know, you don't turn very much at all, so you'd have to trace out a much larger circle. And curvature itself isn't this R. It's not the radius of curvature. But what it is is it's the reciprocal of that, one over R. And there's a special symbol for it. It's kind of a K, and I'm not sure in handwriting how I'm gonna distinguish it from an actual K. Maybe give it a little curly there.
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Curvature intuition.mp3
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But a car with a bad turning radius, you know, you don't turn very much at all, so you'd have to trace out a much larger circle. And curvature itself isn't this R. It's not the radius of curvature. But what it is is it's the reciprocal of that, one over R. And there's a special symbol for it. It's kind of a K, and I'm not sure in handwriting how I'm gonna distinguish it from an actual K. Maybe give it a little curly there. It's the Greek letter kappa, and this is curvature. And I want you to think for a second why we would take one over R. R is a perfectly fine description of how much the road curves. But why is it that you would think one divided by R instead of R itself?
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Curvature intuition.mp3
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It's kind of a K, and I'm not sure in handwriting how I'm gonna distinguish it from an actual K. Maybe give it a little curly there. It's the Greek letter kappa, and this is curvature. And I want you to think for a second why we would take one over R. R is a perfectly fine description of how much the road curves. But why is it that you would think one divided by R instead of R itself? And the reason, basically, is you want curvature to be a measure of how much it curves in the sense that more sharp turns should give you a higher number. So if you're at a point where you're turning the steering wheel a lot, you want that to result in a much higher number. But radius of curvature will be really small when you're turning it a lot.
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Curvature intuition.mp3
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But why is it that you would think one divided by R instead of R itself? And the reason, basically, is you want curvature to be a measure of how much it curves in the sense that more sharp turns should give you a higher number. So if you're at a point where you're turning the steering wheel a lot, you want that to result in a much higher number. But radius of curvature will be really small when you're turning it a lot. But if you're at a point that's basically like a straight road, you know, there's some slight curve to it, but it's basically a straight road, you want the curvature to be a very small number. But in this case, the radius of curvature is very large. So it's pretty helpful to just have one divided by R as the measure of how much the road is turning.
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Curvature intuition.mp3
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But radius of curvature will be really small when you're turning it a lot. But if you're at a point that's basically like a straight road, you know, there's some slight curve to it, but it's basically a straight road, you want the curvature to be a very small number. But in this case, the radius of curvature is very large. So it's pretty helpful to just have one divided by R as the measure of how much the road is turning. And in the next video, I'm gonna go ahead and start describing a little bit more mathematically how we capture this value. Because as a loose description, if you're just kind of drawing pictures, it's perfectly fine to say, oh, yeah, yeah, you imagine a circle that's kind of closely hugging the curve. It's what your steering wheel would do if you were locked.
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Curvature intuition.mp3
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So it's pretty helpful to just have one divided by R as the measure of how much the road is turning. And in the next video, I'm gonna go ahead and start describing a little bit more mathematically how we capture this value. Because as a loose description, if you're just kind of drawing pictures, it's perfectly fine to say, oh, yeah, yeah, you imagine a circle that's kind of closely hugging the curve. It's what your steering wheel would do if you were locked. But in math, we will describe this curve parametrically. It'll be the output of a certain vector-valued function. And I wanna know how you can capture this idea, this one over R curvature idea in a certain formula.
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Curvature intuition.mp3
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So I want to introduce you to a different way. Some people might have taught this first, but the way I taught it in the first double integral video is kind of the way that I always think about when I do the problems. But sometimes it's more useful to think about the way I'm about to show you. And maybe you won't see the difference. Or maybe you say, oh, Sal, those are just the exact same things. Someone actually emailed me and told me that I should make it so I could scroll things. And I said, oh, that's not too hard to do.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And maybe you won't see the difference. Or maybe you say, oh, Sal, those are just the exact same things. Someone actually emailed me and told me that I should make it so I could scroll things. And I said, oh, that's not too hard to do. So I just did that. I scrolled my drawing. But anyway, let's say we have a surface in three dimensions.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And I said, oh, that's not too hard to do. So I just did that. I scrolled my drawing. But anyway, let's say we have a surface in three dimensions. It's a function of x and y. You give me a coordinate down here, and I'll tell you how high the surface is at that point. And we want to figure out the volume under that surface.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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But anyway, let's say we have a surface in three dimensions. It's a function of x and y. You give me a coordinate down here, and I'll tell you how high the surface is at that point. And we want to figure out the volume under that surface. So we can very easily figure out the volume of a very small column underneath this surface. So this whole area, or this whole volume, is what we're trying to figure out, right? Between the dotted lines.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And we want to figure out the volume under that surface. So we can very easily figure out the volume of a very small column underneath this surface. So this whole area, or this whole volume, is what we're trying to figure out, right? Between the dotted lines. I think you can see it. You have some experience visualizing this right now. So let's say that I have a little area here.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Between the dotted lines. I think you can see it. You have some experience visualizing this right now. So let's say that I have a little area here. We could call that dA. Let me see if I can draw this. Let's say we have a little area down here.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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So let's say that I have a little area here. We could call that dA. Let me see if I can draw this. Let's say we have a little area down here. A little square, the xy-plane. And it's, depending on how you view it, this side of it is dx. Its length is dx.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Let's say we have a little area down here. A little square, the xy-plane. And it's, depending on how you view it, this side of it is dx. Its length is dx. And its height, you could say, on that side is dy. Because it's a little small change in y there, and it's a little small change in x here. And its area, the area of this little square, is going to be dx times dy.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Its length is dx. And its height, you could say, on that side is dy. Because it's a little small change in y there, and it's a little small change in x here. And its area, the area of this little square, is going to be dx times dy. And then if we wanted to figure out the volume of the solid between this little area and the surface, we could just multiply this area times the function, right? Because the height at this point is going to be the value of the function, roughly at this point. This is going to be an approximation, and then we're going to take an infinite sum.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And its area, the area of this little square, is going to be dx times dy. And then if we wanted to figure out the volume of the solid between this little area and the surface, we could just multiply this area times the function, right? Because the height at this point is going to be the value of the function, roughly at this point. This is going to be an approximation, and then we're going to take an infinite sum. I think you know where this is going. But let me do that. Let me at least draw the little column that I want to show you.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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This is going to be an approximation, and then we're going to take an infinite sum. I think you know where this is going. But let me do that. Let me at least draw the little column that I want to show you. So that's one end of it. That's another end of it. That's the front end of it.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Let me at least draw the little column that I want to show you. So that's one end of it. That's another end of it. That's the front end of it. That's the other end of it. So we have a little figure that looks something like that. A little column.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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That's the front end of it. That's the other end of it. So we have a little figure that looks something like that. A little column. Right? It intersects the top of the surface. And the volume of this column, not too difficult.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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A little column. Right? It intersects the top of the surface. And the volume of this column, not too difficult. It's going to be this little area down here, which is, we could call that da. Sometimes written like that. da.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And the volume of this column, not too difficult. It's going to be this little area down here, which is, we could call that da. Sometimes written like that. da. It's a little bit small area. And we're going to multiply that area times the height of this column, and that's the function at that point. So it's f of x and y.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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da. It's a little bit small area. And we're going to multiply that area times the height of this column, and that's the function at that point. So it's f of x and y. And of course, we could have also written it as, this da is just dx times dy, or dy times dx. I'm going to write it in every different way. So we could also have written this as f of xy times dx times dy, and of course, since multiplication is associative, I could have also written it as f of xy times dy dx.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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So it's f of x and y. And of course, we could have also written it as, this da is just dx times dy, or dy times dx. I'm going to write it in every different way. So we could also have written this as f of xy times dx times dy, and of course, since multiplication is associative, I could have also written it as f of xy times dy dx. These are all equivalent, and these all represent the volume of this column that's between this little area here and the surface. So now, if we wanted to figure out the volume of the entire surface, there's a couple of things we could do. We could add up all the volumes in the x direction, because between the lower x bound and the upper x bound, and then we'd have kind of a thin sheet, although it'll already have some depth, because we're not adding up just dx's, there's also a dy back there.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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So we could also have written this as f of xy times dx times dy, and of course, since multiplication is associative, I could have also written it as f of xy times dy dx. These are all equivalent, and these all represent the volume of this column that's between this little area here and the surface. So now, if we wanted to figure out the volume of the entire surface, there's a couple of things we could do. We could add up all the volumes in the x direction, because between the lower x bound and the upper x bound, and then we'd have kind of a thin sheet, although it'll already have some depth, because we're not adding up just dx's, there's also a dy back there. So we would have a volume of a figure that would extend from the lower x all the way to the upper x, go back dy, and come back here, if we wanted to sum up all the dx's. And if we wanted to do that, which expression would we use? Well, we would be summing with respect to x first, so we could use this expression, right?
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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We could add up all the volumes in the x direction, because between the lower x bound and the upper x bound, and then we'd have kind of a thin sheet, although it'll already have some depth, because we're not adding up just dx's, there's also a dy back there. So we would have a volume of a figure that would extend from the lower x all the way to the upper x, go back dy, and come back here, if we wanted to sum up all the dx's. And if we wanted to do that, which expression would we use? Well, we would be summing with respect to x first, so we could use this expression, right? And actually, we could write it here, but it just becomes confusing if we're summing with respect to x, but we have the dy written here first. It's really not incorrect, but it just becomes a little ambiguous, are we summing with respect to x or y? But here, we could say, OK, if we want to sum up all the dx's first, let's do that.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Well, we would be summing with respect to x first, so we could use this expression, right? And actually, we could write it here, but it just becomes confusing if we're summing with respect to x, but we have the dy written here first. It's really not incorrect, but it just becomes a little ambiguous, are we summing with respect to x or y? But here, we could say, OK, if we want to sum up all the dx's first, let's do that. We're taking the sum with respect to x. And I'm going to write down the actual, normally you just write numbers here, but I'm going to say, well, the lower bound here is x is equal to a, and the upper bound here is x is equal to b. And that'll give us the volume of, you could imagine, a sheet with depth, right?
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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But here, we could say, OK, if we want to sum up all the dx's first, let's do that. We're taking the sum with respect to x. And I'm going to write down the actual, normally you just write numbers here, but I'm going to say, well, the lower bound here is x is equal to a, and the upper bound here is x is equal to b. And that'll give us the volume of, you could imagine, a sheet with depth, right? The sheet is going to be parallel to the x-axis, right? And then once we have that sheet, my video, I think that's the newspaper people trying to sell me something. Anyway, so once we have this sheet, I'll try to draw it here, too, I don't want to get this picture too muddied up.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And that'll give us the volume of, you could imagine, a sheet with depth, right? The sheet is going to be parallel to the x-axis, right? And then once we have that sheet, my video, I think that's the newspaper people trying to sell me something. Anyway, so once we have this sheet, I'll try to draw it here, too, I don't want to get this picture too muddied up. But once we have that sheet, then we could integrate those, we could add up the dy's, right? Because this width right here is still dy, we could add up all the different dy's, and we would have the volume of the whole figure. So once we take this sum, then we could take this sum, where y is going from its bottom, which we said was c, from y is equal to c, to y's upper bound, to y is equal to d. Fair enough.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Anyway, so once we have this sheet, I'll try to draw it here, too, I don't want to get this picture too muddied up. But once we have that sheet, then we could integrate those, we could add up the dy's, right? Because this width right here is still dy, we could add up all the different dy's, and we would have the volume of the whole figure. So once we take this sum, then we could take this sum, where y is going from its bottom, which we said was c, from y is equal to c, to y's upper bound, to y is equal to d. Fair enough. And then once we evaluate this whole thing, we have the volume of this solid, or the volume under the surface. Now we could have gone the other way. I know this gets a little bit messy, but I think you get what I'm saying.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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So once we take this sum, then we could take this sum, where y is going from its bottom, which we said was c, from y is equal to c, to y's upper bound, to y is equal to d. Fair enough. And then once we evaluate this whole thing, we have the volume of this solid, or the volume under the surface. Now we could have gone the other way. I know this gets a little bit messy, but I think you get what I'm saying. Let's start with that little dA we had originally. Let's start with that little dA. Instead of going this way, instead of summing up the dx's and getting this sheet, let's sum up the dy's first.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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I know this gets a little bit messy, but I think you get what I'm saying. Let's start with that little dA we had originally. Let's start with that little dA. Instead of going this way, instead of summing up the dx's and getting this sheet, let's sum up the dy's first. So we could take, we're summing in the y direction first, so we would get a sheet that's parallel to the y-axis now, so the top of the sheet would look something like that. So if we're summing the dy's first, we would take the sum, we would take the integral with respect to y, and the lower bound would be y is equal to c, and the upper bound is y is equal to d. And then we would have that sheet with a little depth. The depth is dx.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Instead of going this way, instead of summing up the dx's and getting this sheet, let's sum up the dy's first. So we could take, we're summing in the y direction first, so we would get a sheet that's parallel to the y-axis now, so the top of the sheet would look something like that. So if we're summing the dy's first, we would take the sum, we would take the integral with respect to y, and the lower bound would be y is equal to c, and the upper bound is y is equal to d. And then we would have that sheet with a little depth. The depth is dx. And then we could take the sum of all of those. Sorry, my throat is dry. I just had a bunch of almonds to get power to be able to record these videos.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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The depth is dx. And then we could take the sum of all of those. Sorry, my throat is dry. I just had a bunch of almonds to get power to be able to record these videos. But once I have one of these sheets, and if I want to sum up all of the x's, then I could take the infinite sum of infinitely small columns, or in this view, sheets, infinitely small depths, and the lower bound is x is equal to a, and the upper bound is x is equal to b. And once again, I would have the volume of the figure. And all I showed you here is that there's two ways of doing the order of integration.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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I just had a bunch of almonds to get power to be able to record these videos. But once I have one of these sheets, and if I want to sum up all of the x's, then I could take the infinite sum of infinitely small columns, or in this view, sheets, infinitely small depths, and the lower bound is x is equal to a, and the upper bound is x is equal to b. And once again, I would have the volume of the figure. And all I showed you here is that there's two ways of doing the order of integration. Now, another way of saying this, if this little original square was da, and this is a shorthand that you'll see all the time, especially in physics textbooks, is that we are integrating along the domain. Because the xy-plane here is our domain. So we're going to do a double integral, a two-dimensional integral.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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And all I showed you here is that there's two ways of doing the order of integration. Now, another way of saying this, if this little original square was da, and this is a shorthand that you'll see all the time, especially in physics textbooks, is that we are integrating along the domain. Because the xy-plane here is our domain. So we're going to do a double integral, a two-dimensional integral. We're saying that the domain here is two-dimensional. And we're going to take that over f of x and y times da. The reason why I want to show you this is, you see this in physics books all the time.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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So we're going to do a double integral, a two-dimensional integral. We're saying that the domain here is two-dimensional. And we're going to take that over f of x and y times da. The reason why I want to show you this is, you see this in physics books all the time. And I don't think it's a great thing to do, because it is a shorthand, and maybe it looks simpler. But for me, whenever I see something that I don't know how to compute, or that's not obvious for me to know how to compute, it actually is more confusing. So I wanted to just show you that what you see in this physics book, when someone writes this, it's the exact same thing as this or this.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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The reason why I want to show you this is, you see this in physics books all the time. And I don't think it's a great thing to do, because it is a shorthand, and maybe it looks simpler. But for me, whenever I see something that I don't know how to compute, or that's not obvious for me to know how to compute, it actually is more confusing. So I wanted to just show you that what you see in this physics book, when someone writes this, it's the exact same thing as this or this. The da could either be dx times dy, or it could either be dy times dx. And when they do this double integral over domain, that's the same thing as just adding up all of these squares. Where we do it here, we're very ordered about it, right?
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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So I wanted to just show you that what you see in this physics book, when someone writes this, it's the exact same thing as this or this. The da could either be dx times dy, or it could either be dy times dx. And when they do this double integral over domain, that's the same thing as just adding up all of these squares. Where we do it here, we're very ordered about it, right? We go in the x direction, and then we add all of those up in the y direction, and we get the entire volume. Or we could go the other way around. When we say that we're just taking the double integral, first of all, that tells us we're doing it in two dimensions over domain, it leaves it a little bit ambiguous in terms of how we're going to sum up all of the da's.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Where we do it here, we're very ordered about it, right? We go in the x direction, and then we add all of those up in the y direction, and we get the entire volume. Or we could go the other way around. When we say that we're just taking the double integral, first of all, that tells us we're doing it in two dimensions over domain, it leaves it a little bit ambiguous in terms of how we're going to sum up all of the da's. And they do it intentionally in physics books, because you don't have to do it using Cartesian coordinates, using x's and y's. You could do it in polar coordinates. You could do it a ton of different ways.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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When we say that we're just taking the double integral, first of all, that tells us we're doing it in two dimensions over domain, it leaves it a little bit ambiguous in terms of how we're going to sum up all of the da's. And they do it intentionally in physics books, because you don't have to do it using Cartesian coordinates, using x's and y's. You could do it in polar coordinates. You could do it a ton of different ways. But I just wanted to show you that this is another way of having an intuition of the volume under a surface. And these are the exact same thing as this type of notation that you might see in a physics book. Sometimes they won't write a domain, sometimes they'd write over a surface.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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You could do it a ton of different ways. But I just wanted to show you that this is another way of having an intuition of the volume under a surface. And these are the exact same thing as this type of notation that you might see in a physics book. Sometimes they won't write a domain, sometimes they'd write over a surface. And we'll later do those integrals. Here the surface is easy. It's a flat plane, but sometimes it'll end up being a curve or something like that.
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Double integrals 4 Double and triple integrals Multivariable Calculus Khan Academy.mp3
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Let's see if we can use our knowledge of Green's theorem to solve some actual line integrals. And actually, before I show an example, I want to make one clarification on Green's theorem. All of the examples that I did is I had a region like this. And the inside of the region was to the left of what we traversed. So all my examples, I went counterclockwise. And so our region was to the left of, if you imagine walking along the path in that direction, it was always to our left. And that's the situation which Green's theorem would apply.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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And the inside of the region was to the left of what we traversed. So all my examples, I went counterclockwise. And so our region was to the left of, if you imagine walking along the path in that direction, it was always to our left. And that's the situation which Green's theorem would apply. So if you were to take a line integral along this path, the closed line integral, maybe we could even specify it like that. You'll see that in some textbooks. Along the curve C of F dot dr, this is what equals the double integral over this region r of the partial of q with respect to x minus the partial of p with respect to y d area.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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And that's the situation which Green's theorem would apply. So if you were to take a line integral along this path, the closed line integral, maybe we could even specify it like that. You'll see that in some textbooks. Along the curve C of F dot dr, this is what equals the double integral over this region r of the partial of q with respect to x minus the partial of p with respect to y d area. And just as a reminder, this q and p are coming from the components of F. F, in this situation, F would be written as F of xy is equal to p of xy times the i component plus q of xy times the j component. So this is a situation where the inside of the region is to the left of the direction that we're taking the path. If it was in the reverse, then we would put a minus sign right here.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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Along the curve C of F dot dr, this is what equals the double integral over this region r of the partial of q with respect to x minus the partial of p with respect to y d area. And just as a reminder, this q and p are coming from the components of F. F, in this situation, F would be written as F of xy is equal to p of xy times the i component plus q of xy times the j component. So this is a situation where the inside of the region is to the left of the direction that we're taking the path. If it was in the reverse, then we would put a minus sign right here. If this arrow went the other way, we'd put a minus sign. And we can do that because we know that when we're taking line integrals through vector fields, if we reverse the direction, it becomes the negative of that. We showed that, I think, four or five videos ago.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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If it was in the reverse, then we would put a minus sign right here. If this arrow went the other way, we'd put a minus sign. And we can do that because we know that when we're taking line integrals through vector fields, if we reverse the direction, it becomes the negative of that. We showed that, I think, four or five videos ago. With that said, it was convenient to write Green's theorem up here. Let's actually solve a problem. Let's say I have the line integral.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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We showed that, I think, four or five videos ago. With that said, it was convenient to write Green's theorem up here. Let's actually solve a problem. Let's say I have the line integral. Let's say we're over a curve. I'll define the curve in a second. But let's say that the integral we're trying to solve is x squared minus y squared dx plus 2xy dy.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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Let's say I have the line integral. Let's say we're over a curve. I'll define the curve in a second. But let's say that the integral we're trying to solve is x squared minus y squared dx plus 2xy dy. And then our curve, they're giving us the boundary. The boundary is the region. I'll do it in a different color.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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But let's say that the integral we're trying to solve is x squared minus y squared dx plus 2xy dy. And then our curve, they're giving us the boundary. The boundary is the region. I'll do it in a different color. So the curve is boundary of the region given by all of the points x, y such that x is greater than or equal to 0, less than or equal to 1. And then y is greater than or equal to 2x squared and less than or equal to 2x. So let's draw this region that we're dealing with right now.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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I'll do it in a different color. So the curve is boundary of the region given by all of the points x, y such that x is greater than or equal to 0, less than or equal to 1. And then y is greater than or equal to 2x squared and less than or equal to 2x. So let's draw this region that we're dealing with right now. So let me draw my y-axis and then my x-axis right there. And let's see, x goes from 0 to 1. So if we make, that's obviously 0.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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So let's draw this region that we're dealing with right now. So let me draw my y-axis and then my x-axis right there. And let's see, x goes from 0 to 1. So if we make, that's obviously 0. Let's say that that is x is equal to 1. So that's all the x values. And y varies.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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So if we make, that's obviously 0. Let's say that that is x is equal to 1. So that's all the x values. And y varies. It's above 2x squared and below 2x. Normally, if you get to large enough numbers, 2x squared is larger. But if you're below 1, this is actually going to be smaller than that.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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And y varies. It's above 2x squared and below 2x. Normally, if you get to large enough numbers, 2x squared is larger. But if you're below 1, this is actually going to be smaller than that. So the upper boundary is 2x. So it's 1, 2. This is a line y is equal to 2x.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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But if you're below 1, this is actually going to be smaller than that. So the upper boundary is 2x. So it's 1, 2. This is a line y is equal to 2x. So that is the line y is. Let me draw a straighter line than that. The line y is equal to 2x looks something like that.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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This is a line y is equal to 2x. So that is the line y is. Let me draw a straighter line than that. The line y is equal to 2x looks something like that. That right there is y is equal to 2x. And then this bottom curve, maybe I'll do that in yellow. And then the bottom curve right here is y is going to be greater than 2x squared.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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The line y is equal to 2x looks something like that. That right there is y is equal to 2x. And then this bottom curve, maybe I'll do that in yellow. And then the bottom curve right here is y is going to be greater than 2x squared. It might look something like this. And of course, the region that they're talking about is this. But we're saying that the curve is the boundary of this region.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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And then the bottom curve right here is y is going to be greater than 2x squared. It might look something like this. And of course, the region that they're talking about is this. But we're saying that the curve is the boundary of this region. And we're going to go in a counterclockwise direction. I have to specify that. Counterclockwise direction.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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But we're saying that the curve is the boundary of this region. And we're going to go in a counterclockwise direction. I have to specify that. Counterclockwise direction. So our curve, we could start at any point, really. But we're going to go like that and then get to that point and then come back down along that top curve just like that. And so this met the condition that the inside of the region is always going to be our left.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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Counterclockwise direction. So our curve, we could start at any point, really. But we're going to go like that and then get to that point and then come back down along that top curve just like that. And so this met the condition that the inside of the region is always going to be our left. So we can just do the straight up Green's theorem. We don't have to do the negative of it. And let's define our region.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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And so this met the condition that the inside of the region is always going to be our left. So we can just do the straight up Green's theorem. We don't have to do the negative of it. And let's define our region. So our region, let's just do our region. This integral right here is going to go. I'll just do it the way y goes from y.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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And let's define our region. So our region, let's just do our region. This integral right here is going to go. I'll just do it the way y goes from y. Let me do it. y varies from y is equal to 2x squared to y is equal to 2x. And so maybe we'll integrate with respect to y first.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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I'll just do it the way y goes from y. Let me do it. y varies from y is equal to 2x squared to y is equal to 2x. And so maybe we'll integrate with respect to y first. And then x, I'll do the outside, the boundary of x goes from 0 to 1. So they set us up well to do an indefinite integral. And now we just have to figure out what goes over here.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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And so maybe we'll integrate with respect to y first. And then x, I'll do the outside, the boundary of x goes from 0 to 1. So they set us up well to do an indefinite integral. And now we just have to figure out what goes over here. Green's theorem. This right here is our f. It would look like this in this situation. f of x, y is going to be equal to x squared minus y squared i plus 2xyj.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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And now we just have to figure out what goes over here. Green's theorem. This right here is our f. It would look like this in this situation. f of x, y is going to be equal to x squared minus y squared i plus 2xyj. We've seen this in multiple videos. You take the dot product of this with dr, you're going to get this thing right here. So this expression right here is our p of x, y.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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f of x, y is going to be equal to x squared minus y squared i plus 2xyj. We've seen this in multiple videos. You take the dot product of this with dr, you're going to get this thing right here. So this expression right here is our p of x, y. And this expression right here is our q of x, y. So inside of here, we're just going to apply Green's theorem straight up. So the partial of q with respect to x.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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So this expression right here is our p of x, y. And this expression right here is our q of x, y. So inside of here, we're just going to apply Green's theorem straight up. So the partial of q with respect to x. The partial of q with respect to x. So take the derivative of this with respect to x, we're just going to end up with a 2y. And then from that, we're going to subtract the partial of p with respect to y.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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So the partial of q with respect to x. The partial of q with respect to x. So take the derivative of this with respect to x, we're just going to end up with a 2y. And then from that, we're going to subtract the partial of p with respect to y. So if you take the derivative of this with respect to y, that becomes 0. And then here, you have the derivative with respect to y here is minus 2y. So you have a minus 2y just like that.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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And then from that, we're going to subtract the partial of p with respect to y. So if you take the derivative of this with respect to y, that becomes 0. And then here, you have the derivative with respect to y here is minus 2y. So you have a minus 2y just like that. And so this simplifies to 2y minus minus 2y. That's 2y plus 2y. I'm just subtracting a negative.
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Green's theorem example 1 Multivariable Calculus Khan Academy.mp3
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