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Let me draw one of those rectangles where you could view. And there's different ways to do this, but this is just a review. Where you could review, that's maybe one of the rectangles. The area of the rectangle is just base times height, right? We're going to make these rectangles really skinny and just sum up an infini... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
The area of the rectangle is just base times height, right? We're going to make these rectangles really skinny and just sum up an infinite number of them. So we want to make them infinitely small, but let's just call the base of this rectangle dx. And then the height of this rectangle is going to be f of x at that poin... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And then the height of this rectangle is going to be f of x at that point. It's going to be f of, if this is x naught or whatever, you could just call it f of x. That's the height of that rectangle. And if we wanted to take the sum of all of these rectangles, there's just going to be a bunch of them. One there, one the... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And if we wanted to take the sum of all of these rectangles, there's just going to be a bunch of them. One there, one there. Then we'll get the area. And if we have infinite number of these rectangles and they're infinitely skinny, we have exactly the area under that curve. That's the intuition behind the definite inte... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And if we have infinite number of these rectangles and they're infinitely skinny, we have exactly the area under that curve. That's the intuition behind the definite integral. And the way we write that, it's the definite integral. We're going to take the sums of these rectangles from x is equal to a to x is equal to b.... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
We're going to take the sums of these rectangles from x is equal to a to x is equal to b. And the sum, or the areas that we're summing up, are going to be the height is f of x and the width is d of x. It's going to be f of x times d of x. This is equal to the area under the curve, f of x, y is equal to f of x, from x i... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
This is equal to the area under the curve, f of x, y is equal to f of x, from x is equal to a to x is equal to b. And that's just a little bit of review. But hopefully you'll now see the parallel of how we extend this to taking the volume under a surface. First of all, what is a surface? If we're thinking in three dime... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
First of all, what is a surface? If we're thinking in three dimensions, a surface is going to be a function of x and y. So we can write a surface as, instead of saying that y is a function of x, we can write a surface as z is equal to a function of x and y. So you can kind of view it as the domain. The domain is all of... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So you can kind of view it as the domain. The domain is all of the set of valid things that you can input into a function. So now, before our domain was just, at least for most of what we dealt with, was just the x-axis, or kind of the real number line in the x direction. Now, our domain is the xy-plane. We can give it... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Now, our domain is the xy-plane. We can give it any x and any y, and we'll just deal with reals right now, but I don't want to get too technical. And then it'll pop out another number, and if we wanted to graph it, it'll be our height. And so that could be the height of a surface. So let me just show you what a surface... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And so that could be the height of a surface. So let me just show you what a surface looks like, in case you don't remember, and we'll actually figure out the volume under this surface. So this is a surface. I'll tell you its function in a second, but it's pretty neat to look at. But as you can see, it's a surface. It'... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
I'll tell you its function in a second, but it's pretty neat to look at. But as you can see, it's a surface. It's like a piece of paper that's bent. Let me rotate it to its traditional form. So this is the x direction, this is the y direction, and the height is a function of where we are in the xy-plane. So how do we f... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Let me rotate it to its traditional form. So this is the x direction, this is the y direction, and the height is a function of where we are in the xy-plane. So how do we figure out the volume under a surface like this? How do we figure out the volume? It seems like a bit of a stretch, given what we've learned from this... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
How do we figure out the volume? It seems like a bit of a stretch, given what we've learned from this. But what if, and I'm just going to draw an abstract surface here, let me draw some axes. Let's say that's my x-axis, that's my y-axis, that's my z-axis. I practice these videos ahead of time, so I'm often wondering wh... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Let's say that's my x-axis, that's my y-axis, that's my z-axis. I practice these videos ahead of time, so I'm often wondering what I'm about to draw. So that's x, that's y, and that's z. And let's say I have some surface. I'll just draw something. I don't know what it is. Some surface. | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And let's say I have some surface. I'll just draw something. I don't know what it is. Some surface. This is our surface. z is a function of x and y. So if you give me a coordinate in the xy-plane, say here, I'll put it into the function and it'll give us a z value. | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Some surface. This is our surface. z is a function of x and y. So if you give me a coordinate in the xy-plane, say here, I'll put it into the function and it'll give us a z value. And I'll plot it there and it'll be a point on the surface. So what we want to figure out is the volume under the surface, and we have to sp... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So if you give me a coordinate in the xy-plane, say here, I'll put it into the function and it'll give us a z value. And I'll plot it there and it'll be a point on the surface. So what we want to figure out is the volume under the surface, and we have to specify bounds. From here we said x is equal to a to x is equal t... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
From here we said x is equal to a to x is equal to b. So let's make a square bound first, because this keeps it a lot simpler. So let's say that the region of the x and y region of this part of the surface under which we want to calculate the volume, let's say the shadow, if the sun was right above the surface, the sha... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Let me try my best to draw this neatly. So this is what we're going to try to figure out the volume of. So if we wanted to draw it in the xy-plane, you can kind of view it as the projection of the surface in the xy-plane or the shadow of the surface in the xy-plane. What are the bounds? You can almost view it as what a... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
What are the bounds? You can almost view it as what are the bounds of the domain? Let's say that this right here, that's 0, 0 in the xy-plane. Let's say that this is y is equal to a. That's this line right here, y is equal to a. And let's say that this line right here is x is equal to b. I hope you get that. This is th... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Let's say that this is y is equal to a. That's this line right here, y is equal to a. And let's say that this line right here is x is equal to b. I hope you get that. This is the xy-plane. If we have a constant x, it would be a line like that, a constant y, a line like that, and then we have the area in between it. So ... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
This is the xy-plane. If we have a constant x, it would be a line like that, a constant y, a line like that, and then we have the area in between it. So how do we figure out the volume under this? Well, if I just wanted to figure out the area of, let's just say, this sliver. Let's say we had a constant y. So let's say ... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Well, if I just wanted to figure out the area of, let's just say, this sliver. Let's say we had a constant y. So let's say I had some sliver. I don't want to confuse you. Let's say that I have some constant y. I just want to give you the intuition. Let's say, I don't know what that is, it's an arbitrary y. But for some... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
I don't want to confuse you. Let's say that I have some constant y. I just want to give you the intuition. Let's say, I don't know what that is, it's an arbitrary y. But for some constant y, what if I could just figure out the area under the curve there? How would I figure out just the area under that curve? It'll be a... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
But for some constant y, what if I could just figure out the area under the curve there? How would I figure out just the area under that curve? It'll be a function of which y I pick, right? Because if I pick a y here, it'll be a different area. If I pick a y there, it'll be a different area. But I could view this now a... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Because if I pick a y here, it'll be a different area. If I pick a y there, it'll be a different area. But I could view this now as a very similar problem to this one up here. I could have my dx's. Let me pick a vibrant color so you can see it. Let's say that's a dx, right? That's a change in x. | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
I could have my dx's. Let me pick a vibrant color so you can see it. Let's say that's a dx, right? That's a change in x. And then the height, this height, is going to be a function of the x I have and the y I've picked. Although I'm assuming to some degree that that's a constant y. So what would be the area of this she... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
That's a change in x. And then the height, this height, is going to be a function of the x I have and the y I've picked. Although I'm assuming to some degree that that's a constant y. So what would be the area of this sheet of paper? It's kind of a constant y. It's a sheet of paper within this volume. You can kind of v... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So what would be the area of this sheet of paper? It's kind of a constant y. It's a sheet of paper within this volume. You can kind of view it. Well, we said the height of each of these rectangles is f of xy. That's the height. It depends which x and y we pick down here. | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
You can kind of view it. Well, we said the height of each of these rectangles is f of xy. That's the height. It depends which x and y we pick down here. And then its width is going to be dx. And then if we integrated it from x is equal to 0, which was back here, all the way to x is equal to b, what would it look like? ... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
It depends which x and y we pick down here. And then its width is going to be dx. And then if we integrated it from x is equal to 0, which was back here, all the way to x is equal to b, what would it look like? It would look like that. x is going from 0 to b. Fair enough. And this would actually give us a function of y... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
It would look like that. x is going from 0 to b. Fair enough. And this would actually give us a function of y. This would give us an expression so that if I would know the area of this kind of sliver of the volume for any given value of y. If you give me a y, I can tell you the area of the sliver that corresponds to th... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And this would actually give us a function of y. This would give us an expression so that if I would know the area of this kind of sliver of the volume for any given value of y. If you give me a y, I can tell you the area of the sliver that corresponds to that y. Now what could I do? If I know the area of any given sli... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Now what could I do? If I know the area of any given sliver, what if I multiplied the area of that sliver times dy? This is a dy. Let me do it in a vibrant color. So dy, a very small change in y. If I multiply this area times a small dy, then all of a sudden I have a sliver of volume. Hopefully that makes some sense. | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Let me do it in a vibrant color. So dy, a very small change in y. If I multiply this area times a small dy, then all of a sudden I have a sliver of volume. Hopefully that makes some sense. I'm making that little cut that I took the area of, I'm making it three dimensional. So what would be the volume of that sliver? Th... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Hopefully that makes some sense. I'm making that little cut that I took the area of, I'm making it three dimensional. So what would be the volume of that sliver? The volume of that sliver will be this function of y times dy or this whole thing times dy. So it would be the integral from 0 to b of f of x, y dx. That give... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
The volume of that sliver will be this function of y times dy or this whole thing times dy. So it would be the integral from 0 to b of f of x, y dx. That gives us the area of this blue sheet. Now if I multiply this whole thing times dy, I get this volume. It gets some depth. This little area that I'm shading right here... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Now if I multiply this whole thing times dy, I get this volume. It gets some depth. This little area that I'm shading right here gives depth of that sheet. Now if I added all of those sheets that now have depth, if I took the infinite sum, so if I took the integral of this from my lower y bound, from 0 to my upper y bo... | Double integral 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
You produce some little trinket that people enjoy buying. And the main costs that you have are labor, you know, the workers that you have creating these, and steel. And let's just say that your labor costs are $20 per hour, $20 each hour, and then your steel costs are $2,000. $2,000, keep the numbers kind of related to... | Lagrange multiplier example, part 1.mp3 |
$2,000, keep the numbers kind of related to each other, $2,000 for every ton of steel. And then you've had your analysts work a little bit on trying to model the revenues you can make with your widgets as a function of hours of labor and tons of steel. Now let's say the revenue model that they've come up with, the reve... | Lagrange multiplier example, part 1.mp3 |
If you put in a given amount of labor and a given amount of steel, this is about how much money you're gonna expect to earn. And of course you wanna earn as much as you can, but let's say you actually have a budget for how much you're able to spend on all these things, and your budget, the budget is $20,000. You're wil... | Lagrange multiplier example, part 1.mp3 |
Now this is exactly the kind of problem that the Lagrange multiplier technique is made for. We're trying to maximize some kind of function and we have a constraint. Now right now the constraint isn't written as a formula, but we can pretty easily write it as a formula because what makes up our budget? Well, it's gonna ... | Lagrange multiplier example, part 1.mp3 |
Well, it's gonna be the number of hours of labor multiplied by 20, so that's gonna be $20 per hour multiplied by the number of hours you put in plus $2,000 per tons of steel times the tons of steel that you put in. So the constraint is basically that you have to have these values equal $20,000. I mean, you could say le... | Lagrange multiplier example, part 1.mp3 |
You could say you're not willing to go any more than that, but intuitively and in reality, it's gonna be the case that in order to maximize your revenues, you should squeeze every dollar that you have available and actually hit this constraint. So this right here is the constraint of our problem. And let's go ahead and... | Lagrange multiplier example, part 1.mp3 |
And I'm gonna call it G of HS, which is gonna be that guy. And now if you'll remember in the last few videos, the way we visualize something like this is to think about the set of all possible inputs. So in this case, you might be thinking about the HS plane, you know, the number of hours of labor on one axis, the numb... | Lagrange multiplier example, part 1.mp3 |
And this constraint, well, in this case, it's a linear function. So this constraint is gonna give us some kind of line that tells us which pairs of S and H are gonna achieve that constraint. And then the revenue function that we're dealing with will have certain contours. You know, maybe revenues of $10,000 have a cert... | Lagrange multiplier example, part 1.mp3 |
You know, maybe revenues of $10,000 have a certain contour that looks like this, and revenues of $100,000 have a certain contour that looks like this. But what we want is to find which value is barely touching the constraint curve, just tangent to it at a given point, because that's gonna be the contour line where if y... | Lagrange multiplier example, part 1.mp3 |
And the way to think about finding that tangency is to consider the vector perpendicular to the tangent line to the curve at that point, which fortunately is represented by, let's see, let me make some room for myself here, represented by the gradient of our R function, the function whose contours this is, the revenue.... | Lagrange multiplier example, part 1.mp3 |
It's called the Lagrange multiplier. So let's go ahead and start working it out. Let's first compute the gradient of R. So the gradient of R is gonna be the partial derivative of R with respect to its first variable, which is H. So partial derivative with respect to H. And the second component is its partial derivative... | Lagrange multiplier example, part 1.mp3 |
So that'll be 100 times 2 3rds times H, H to the power of, well, we've gotta subtract one from 2 3rds when we bring it down, so that'll be negative 1 3rd, multiplied by S to the 1 3rd. And then the second component here, the partial derivative with respect to S, is gonna be 100 times, well, now by treating S as the var... | Lagrange multiplier example, part 1.mp3 |
So when we take the gradient of G, which is its partial derivative with respect to H, partial H, and its partial derivative with respect to S, partial S, well, the partial with respect to H is just 20. The function looks like 20 times H plus something that's a constant. So that ends up being 20, and then the partial wi... | Lagrange multiplier example, part 1.mp3 |
So that's great. And this means when we set the gradient of R equal to the gradient of G, the pair of equations that we get, and let me just write it all out again, is we have this top one, which I'll call 2 3rds times, and let's go ahead and do a little simplifying while I'm rewriting things here. So H to the 1 3rd is... | Lagrange multiplier example, part 1.mp3 |
So all of this, that first component, is being set equal to the first component of the gradient of G, which is 20 times lambda, times this Lagrange multiplier. Because we're not setting the gradients equal to each other, we're just setting them proportional to each other. So that's the first equation. And then the seco... | Lagrange multiplier example, part 1.mp3 |
In the last couple videos, I showed how you can take a function, just a function with two inputs, and find the tangent plane to its graph. And the way that you think about this, you first find a point, some kind of input point, which is, I'll just write abstractly as x-naught and y-naught, and then you see where that p... | Local linearization.mp3 |
This will go under the name local linearization. Local linearization. It's kind of a long word. Zation. And what this basically means, the word local means you're looking at a specific input point. So in this case, it's at a specific input point, x-naught, y-naught. And the idea of a linearization, a linearization mean... | Local linearization.mp3 |
Zation. And what this basically means, the word local means you're looking at a specific input point. So in this case, it's at a specific input point, x-naught, y-naught. And the idea of a linearization, a linearization means you're approximating the function with something simpler, with something that's actually linea... | Local linearization.mp3 |
And the idea of a linearization, a linearization means you're approximating the function with something simpler, with something that's actually linear. And I'll tell you what I mean by linear in just a moment. But the whole idea here is that we don't really care about, you know, tangent planes in an abstract 3D space t... | Local linearization.mp3 |
The whole reason for doing this is that this is a really good way to approximate a function, which is potentially a very complicated function, with something that's much easier, something that has constant partial derivatives. Now my goal of this video is going to be to show how we write this local linearization here i... | Local linearization.mp3 |
So just to remind us of where we were and what we got to in the last couple videos, I'll write it a little bit more abstractly this time, rather than a specific example. The way you do this local linearization is first you find the partial derivative of f with respect to x, which I'll write with the subscript notation.... | Local linearization.mp3 |
You evaluate it at the point about which you're approximating. And then you multiply that by x minus that constant. So the only variable right here, everything is a constant, but the only variable part is that x. And then we add to that, basically doing the same thing with y. You take the partial derivative with respec... | Local linearization.mp3 |
And then we add to that, basically doing the same thing with y. You take the partial derivative with respect to y, you evaluate it at the input point, the point about which you are linearizing, and then you multiply it by y minus y sub o. And then to this entire thing, because you want to make sure that when you evalua... | Local linearization.mp3 |
This way you can just think about adding whatever the function itself evaluates to at that point. And this will ensure that your linearization actually equals the function itself at the local point, because hopefully if you're approximating it near a point, then at that point it's actually equal. So what do I mean by t... | Local linearization.mp3 |
The word linear has a very precise formulation, especially in the context of linear algebra, and admittedly, this is not actually a linear function in the technical sense. But loosely what it means, and the reason people call it linear, is that this x term here, this variable term, doesn't have anything fancy going on ... | Local linearization.mp3 |
And similarly, this y term, it's just being multiplied by a constant. It's not squared, there's no square root, it's not in an exponent or anything like that. And although there is a more technical meaning of the word linear, this is all it really means in this context. This is all you need to think about. Each variabl... | Local linearization.mp3 |
This is all you need to think about. Each variable is just multiplied by a constant. Now you might see this in a more complicated form, or what's at first a more complicated form, using vectors. So first of all, let's think about how we would start describing everything going on here with vectors. So the input, rather ... | Local linearization.mp3 |
So first of all, let's think about how we would start describing everything going on here with vectors. So the input, rather than talking about the input as being a pair of points, what I want to say is that there's some vector, some vector that has these as its components, and we just want to capture that all, and I w... | Local linearization.mp3 |
So I'll just kind of try to emphasize bold-faced x equals this vector, and that's confusing, because x is already one of the input variables that's just a number. But I'll try to emphasize it, just making it bold. You'll see this in writing a lot. x is this input vector. And then similarly, the specified input about wh... | Local linearization.mp3 |
x is this input vector. And then similarly, the specified input about which we are approximating, you would call, let's say I'll make it a nice bold-faced x, not. We'll do that not to just kind of indicate that it's a constant of some kind. And what that is, it's a vector containing the two numbers x-not, y-not. So thi... | Local linearization.mp3 |
And what that is, it's a vector containing the two numbers x-not, y-not. So this is just our starting to write things in a more vectorized way, and the convenience here is that then if you're dealing with a function with three input variables, or four, or 100, you could still just write it as this bold-faced x with the... | Local linearization.mp3 |
We can start thinking of this as a dot product, actually. So let me first just kind of move this guy out of the way and give ourselves some room. So he's gonna just go up there, just the same guy. And now I want to think about writing this other term here as a dot product. And what that looks like is we have the two pa... | Local linearization.mp3 |
And now I want to think about writing this other term here as a dot product. And what that looks like is we have the two partial derivatives, f sub x and f sub y, indicating the partial derivatives with respect to x and y, and each one of them is evaluated, let's see, I'll do it evaluating at our bold-faced x-not. And ... | Local linearization.mp3 |
So really, you're thinking about this as being a vector that contains two different variables, you're just packing it into a single symbol. And the dot product here is against, you know, the first component is x minus x-not, so I'd write that as x minus x-not, the number, and then similarly, y minus, let's see, I'll do... | Local linearization.mp3 |
And if that feels unfamiliar, maybe go back and check out the videos on the gradient, but this whole vector is basically just saying take the gradient and evaluate it at that vector input, you know, x-not. And then the second component here, that's telling you you've got x and y minus x-not, y-not. So what you're basic... | Local linearization.mp3 |
So this right here, this is just vector terms where you're thinking of this as being a vector with two components, and this one is a vector with two components. But if your function happened to be something more complicated with, you know, 100 input variables, this would be the same thing you write down. You would just... | Local linearization.mp3 |
And this is what a linear term looks like in vector terminology, because this dot product is telling you that all of the components of that bold-faced x vector, the, that expands into, you know, not bold-faced x, y, z, whatever else it expands to, all of those are just being multiplied by some kind of constant. So we t... | Local linearization.mp3 |
So you would take that as the linear term. And now, I kind of like to add it on to the front, actually, where you think about taking the function itself and evaluating it at that constant input, x-not. Because that way, you can kind of think this is your constant term, this is your constant term, and then the rest of t... | Local linearization.mp3 |
The rest of your stuff is your linear. Because later on, if we start adding other terms, like a quadratic term or more complicated things, you can kind of keep adding them on the end. So this right here is the expression that you will often see for the local linearization. And the only place where the actual variable s... | Local linearization.mp3 |
And the only place where the actual variable shows up, the variable vector, is right here, is this guy. Because, you know, when you evaluate the function f at a specified input, that's just a constant. When you evaluate the gradient at that input, it's just a constant. And we're subtracting off that specified input, th... | Local linearization.mp3 |
And we're subtracting off that specified input, that's just a constant. So this is the only place where your variable shows up. So once all is said and done, and once you do your computations, this is a very simple function. And the important part is maybe this is much simpler than the function f itself, which allows y... | Local linearization.mp3 |
And the important part is maybe this is much simpler than the function f itself, which allows you to, you know, maybe compute something more quickly if you're writing a program that needs to, you know, deal with some kind of complicated function, but runtime is an issue. Or maybe it's a function that you never knew in ... | Local linearization.mp3 |
So that's the plane that kind of slants down, slants down like that. It's the intersection of that plane and the cylinder, actually I shouldn't even call it a cylinder because if you just have x squared plus y squared is equal to one, it would essentially be like a pole, an infinite pole that keeps going up forever and... | Stokes example part 1 Multivariable Calculus Khan Academy.mp3 |
But we've sliced that pole with y plus z is equal to two to get and where they intersect, we get our path C. We also have this vector field defined in this way and we're asked to evaluate the line integral of F dot dr over this path with this orientation. And we could directly solve this line integral. We've done it be... | Stokes example part 1 Multivariable Calculus Khan Academy.mp3 |
But this is about Stokes' Theorem. So let's see if we can apply Stokes' Theorem to this circumstance. So Stokes' Theorem tells us that this is going to be the same thing as the surface integral over a piecewise smooth surface bounded by this boundary right over here. And actually we'll pick the simplest of all the, of ... | Stokes example part 1 Multivariable Calculus Khan Academy.mp3 |
And actually we'll pick the simplest of all the, of all of the surfaces bounded by this. I'll pick the portion of the plane that is bounded by C. So S is going to be the portion, portion of the plane, of the plane Y plus Z is equal to two bounded, bounded by C. So it's gonna be the surface integral over this. We could ... | Stokes example part 1 Multivariable Calculus Khan Academy.mp3 |
We have information on it. It's a nice flat, it's at an incline, but it's a nice flat surface that we're gonna be able to operate on analytically. And so we're gonna take the surface integral of the curl, the curl of our vector field dotted with, and we could say dot ds if we think of the vector differential, or we cou... | Stokes example part 1 Multivariable Calculus Khan Academy.mp3 |
And we've talked already about orientation, so we wanna make sure that we get our orientation right. We are traversing it in this direction, and if we think about the little man analogy, if when he's walking in this direction, if he's walking with his head upwards, then the surface will be to his left, and so we want t... | Stokes example part 1 Multivariable Calculus Khan Academy.mp3 |
The other analogy, if you're twisting a bottle cap like this then the bottle cap will move in that direction, in the direction of this normal vector. So we've set it all up. Now all we have to do is evaluate, now we have to do is evaluate this integral. To evaluate it, we have to do a couple of things. We one, have to ... | Stokes example part 1 Multivariable Calculus Khan Academy.mp3 |
All right. So in the last video, I set up the scaffolding for the quadratic approximation, which I'm calling Q, of a function, an arbitrary two-variable function, which I'm calling F. And the form that we have right now looks like quite a lot, actually. We have six different terms. Now, the first three were just basica... | Quadratic approximation formula, part 2.mp3 |
Now, the first three were just basically stolen from the local linearization formula and written in their full abstractness. It almost makes it seem a little bit more complicated than it is. And then these next three terms are basically the quadratic parts. We have what is basically x squared. We take it as x minus x-n... | Quadratic approximation formula, part 2.mp3 |
We have what is basically x squared. We take it as x minus x-naught squared so that we don't mess with anything previously once we plug in x equals x-naught. But basically, we think of this as x-squared. And then this here is basically x times y, but of course, we're matching each one of them with the corresponding x-n... | Quadratic approximation formula, part 2.mp3 |
And then this here is basically x times y, but of course, we're matching each one of them with the corresponding x-naught, y-naught. And then this term is the y-squared. And the question at hand is how do we fill in these constants, the coefficients in front of each one of these quadratic terms to make it so that this ... | Quadratic approximation formula, part 2.mp3 |
And I showed that in the very first video, kind of what that hugging means. Now, in formulas, the goal here, I should probably state what it is that we want is for the second partial derivatives of q. So for example, if we take the partial derivative with respect to x twice in a row, we want it to be the case that if y... | Quadratic approximation formula, part 2.mp3 |
And of course, we want this to be true not just with the second partial derivative with respect to x twice in a row, but if we did it with the other ones. Like for example, let's say we took the partial derivative first with respect to x, and then with respect to y. This is called the mixed partial derivative. We want ... | Quadratic approximation formula, part 2.mp3 |
We want it to be the case that when we evaluate that at the point of interest, it's the same as taking the mixed partial derivative of f, with respect to x, and then with respect to y, and we evaluate it at that same point. At that same point. And remember, for almost all functions that you deal with, when you take thi... | Quadratic approximation formula, part 2.mp3 |
You could take it first with respect to x, then y, or you could do it first with respect to y, and then with respect to x. Usually, these guys are equal. There are some functions for which this isn't true, but we're gonna basically assume that we're dealing with functions where this is. So, that's the only mixed partia... | Quadratic approximation formula, part 2.mp3 |
So, that's the only mixed partial derivative that we have to take into account. And I'll just kinda get rid of that guy there. And then of course, the final one, final one, just to have it on record here, is that we want the partial derivative when we take it with respect to y two times in a row, and we evaluate that a... | Quadratic approximation formula, part 2.mp3 |
There's kind of a lot of, this is, there's a lot of writing that goes on with these things, and that's just kind of par for the course when it comes to multivariable calculus, but you take the partial derivative with respect to y at both of them, and you want it to be the same value at this point. So, even though there... | Quadratic approximation formula, part 2.mp3 |
So, let's go ahead and do it. This first, this partial derivative with respect to x twice, what we'll do is I'll take one of those out and think partial derivative with respect to x, and then on the inside, I'm gonna put what the partial derivative of this entire expression with respect to x is. Well, we just take it o... | Quadratic approximation formula, part 2.mp3 |
This first term here is a constant, so that goes to zero. The second term here actually has the variable x in it. And when we take its partial derivative, since this is a linear term, it's just gonna be that constant sitting in front of it. So, it'll be that constant, which is the value of the partial derivative of f w... | Quadratic approximation formula, part 2.mp3 |
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