Datasets:
DarthJudie
/
MultiCalc

Dataset card Data Studio Files
xet
Community
Sentence
stringlengths
131
8.39k
video_title
stringlengths
12
104
And let me just kind of clean it up at least a little bit here. So what we're gonna do is we're gonna extend this, and I'm gonna change its name because I don't want it to be a linear function anymore. What I want is for this to be a quadratic function, so instead I'm gonna call it q of x, y. And now we're gonna add some terms, and what I could do, what I could do is add a constant times x squared, since that's something we're allowed, plus some kind of constant times x, y, and then another constant times y squared. But the problem there is that if I just add these as they are then it might mess things up when I plug in x-naught and y-naught, right? Well, it was very important that when you plug in those values that you get the original value of the function and that the partial derivatives of the approximation also match that of the function. And that could mess things up because once you start plugging in x-naught and y-naught over here, that might actually mess with the value.
Quadratic approximation formula, part 1.mp3
And now we're gonna add some terms, and what I could do, what I could do is add a constant times x squared, since that's something we're allowed, plus some kind of constant times x, y, and then another constant times y squared. But the problem there is that if I just add these as they are then it might mess things up when I plug in x-naught and y-naught, right? Well, it was very important that when you plug in those values that you get the original value of the function and that the partial derivatives of the approximation also match that of the function. And that could mess things up because once you start plugging in x-naught and y-naught over here, that might actually mess with the value. So we're basically gonna do the same thing we did with the linearization where we put in, every time we have an x, we kind of attach it. We say x minus x-naught, just to make sure that we don't mess things up when we plug in x-naught. So instead, instead of what I had written there, what we're gonna add as our quadratic approximation is some kind of constant, and we'll fill in that constant in a moment, times x minus x-naught squared.
Quadratic approximation formula, part 1.mp3
And that could mess things up because once you start plugging in x-naught and y-naught over here, that might actually mess with the value. So we're basically gonna do the same thing we did with the linearization where we put in, every time we have an x, we kind of attach it. We say x minus x-naught, just to make sure that we don't mess things up when we plug in x-naught. So instead, instead of what I had written there, what we're gonna add as our quadratic approximation is some kind of constant, and we'll fill in that constant in a moment, times x minus x-naught squared. And then we're gonna add another constant multiplied by x minus x-naught times y minus y-naught. And then that times yet another constant, which I'll call c, multiplied by y minus y-naught squared. All right, this is quite a lot going on.
Quadratic approximation formula, part 1.mp3
So instead, instead of what I had written there, what we're gonna add as our quadratic approximation is some kind of constant, and we'll fill in that constant in a moment, times x minus x-naught squared. And then we're gonna add another constant multiplied by x minus x-naught times y minus y-naught. And then that times yet another constant, which I'll call c, multiplied by y minus y-naught squared. All right, this is quite a lot going on. This is a heck of a function. And these are three different constants that we're gonna try to fill in to figure out what they should be to most closely approximate the original function f. Now the important part of making this x minus x-naught and y minus y-naught is that when we plug in here, when we plug in x-naught for our variable x, and when we plug in y-naught for our variable y, all of this stuff is just gonna go to zero, and it's gonna cancel out. And moreover, when we take the partial derivatives, all of it's gonna go to zero as well.
Quadratic approximation formula, part 1.mp3
All right, this is quite a lot going on. This is a heck of a function. And these are three different constants that we're gonna try to fill in to figure out what they should be to most closely approximate the original function f. Now the important part of making this x minus x-naught and y minus y-naught is that when we plug in here, when we plug in x-naught for our variable x, and when we plug in y-naught for our variable y, all of this stuff is just gonna go to zero, and it's gonna cancel out. And moreover, when we take the partial derivatives, all of it's gonna go to zero as well. And we'll see that in a moment. Maybe I'll just actually show that right now. Or rather, I think I'll call the video done here, and then start talking about how we fill in these constants in the next video.
Quadratic approximation formula, part 1.mp3
Let's say y is greater than or equal to 0, and is less than or equal to 4. And then let's say that z is greater than or equal to 0, and is less than or equal to 2. Now I know, using basic geometry, you could figure out, just multiply the width times the height times the depth, and you'd have the volume. But I want to do this example just so that you get used to what a triple integral looks like, how it relates to a double integral, and then later in the next video we could do something slightly more complicated. So let's just draw this volume. So this is my x-axis. This is my z-axis.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
But I want to do this example just so that you get used to what a triple integral looks like, how it relates to a double integral, and then later in the next video we could do something slightly more complicated. So let's just draw this volume. So this is my x-axis. This is my z-axis. This is the y, x, y, z. OK, so x is between 0 and 3, so that's x is equal to 0. This is x is equal to, say, 1, 2, 3. y is between 0 and 4, 1, 2, 3, 4. So the xy-plane, it looks something like this.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
This is my z-axis. This is the y, x, y, z. OK, so x is between 0 and 3, so that's x is equal to 0. This is x is equal to, say, 1, 2, 3. y is between 0 and 4, 1, 2, 3, 4. So the xy-plane, it looks something like this. The kind of base of our cube will look something like this. And then z is between 0 and 2, so 0 is the xy-plane. And then 1, 2.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
So the xy-plane, it looks something like this. The kind of base of our cube will look something like this. And then z is between 0 and 2, so 0 is the xy-plane. And then 1, 2. So this would be the top part, and I'll maybe do that in a slightly different color. So this is along the xz-axis. You'd have a boundary here, and then it would come in like this.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And then 1, 2. So this would be the top part, and I'll maybe do that in a slightly different color. So this is along the xz-axis. You'd have a boundary here, and then it would come in like this. You'd have a boundary here, come in like that, boundary there, and then like that. So we want to figure out the volume of this cube. And you could do it.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
You'd have a boundary here, and then it would come in like this. You'd have a boundary here, come in like that, boundary there, and then like that. So we want to figure out the volume of this cube. And you could do it. You could say, well, the depth is 3, the width is 4, so this area is 12 times the height. 12 times 2 is 24. You could say it's 24 cubic units, whatever units we're doing.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And you could do it. You could say, well, the depth is 3, the width is 4, so this area is 12 times the height. 12 times 2 is 24. You could say it's 24 cubic units, whatever units we're doing. But let's do it as a triple integral. So what does a triple integral mean? Well, what we could do is we could take the volume of a very small, I don't want to say area, of a very small volume.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
You could say it's 24 cubic units, whatever units we're doing. But let's do it as a triple integral. So what does a triple integral mean? Well, what we could do is we could take the volume of a very small, I don't want to say area, of a very small volume. So let's say I wanted to take the volume of a small cube someplace in the volume under question. And it'll start to make more sense, or it starts to become a lot more useful when we have variable boundaries and surfaces and curves as boundaries. But let's say we want to figure out the volume of this little small cube here.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
Well, what we could do is we could take the volume of a very small, I don't want to say area, of a very small volume. So let's say I wanted to take the volume of a small cube someplace in the volume under question. And it'll start to make more sense, or it starts to become a lot more useful when we have variable boundaries and surfaces and curves as boundaries. But let's say we want to figure out the volume of this little small cube here. That's my cube. It's someplace in this larger cube, this larger rectangle, cubic rectangle, whatever you want to call it. So what's the volume of that cube?
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
But let's say we want to figure out the volume of this little small cube here. That's my cube. It's someplace in this larger cube, this larger rectangle, cubic rectangle, whatever you want to call it. So what's the volume of that cube? Let's say that its width is dy, so that length right there is dy. Its height is dz, right? The way I drew it, z is up and down.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
So what's the volume of that cube? Let's say that its width is dy, so that length right there is dy. Its height is dz, right? The way I drew it, z is up and down. And then its depth is dx. This is dx. So this is dx, this is dz, this is dy.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
The way I drew it, z is up and down. And then its depth is dx. This is dx. So this is dx, this is dz, this is dy. So you could say that a small volume within this larger volume, you could call that dv, which is kind of the volume differential. And that would be equal to, you could say, it's just the width times the length times the height. dy, dx times dy times dz.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
So this is dx, this is dz, this is dy. So you could say that a small volume within this larger volume, you could call that dv, which is kind of the volume differential. And that would be equal to, you could say, it's just the width times the length times the height. dy, dx times dy times dz. And you could switch the orders of these, right? Because multiplication is associative and order doesn't matter and all of that. But anyway, what can you do with it here?
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
dy, dx times dy times dz. And you could switch the orders of these, right? Because multiplication is associative and order doesn't matter and all of that. But anyway, what can you do with it here? Well, we can take the integral. All integrals help us do is it helps us take infinite sums of infinitely small distances, like a dz or a dx or a dy, et cetera. So what we could do is we could take this cube and first add it up in, let's say, the z direction.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
But anyway, what can you do with it here? Well, we can take the integral. All integrals help us do is it helps us take infinite sums of infinitely small distances, like a dz or a dx or a dy, et cetera. So what we could do is we could take this cube and first add it up in, let's say, the z direction. So we could take that cube and then add it along the up and down axis, the z axis, so that we get the volume of a column. So what would that look like? Well, since we're going up and down, we're taking the sum in the z direction, we'd have an integral.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
So what we could do is we could take this cube and first add it up in, let's say, the z direction. So we could take that cube and then add it along the up and down axis, the z axis, so that we get the volume of a column. So what would that look like? Well, since we're going up and down, we're taking the sum in the z direction, we'd have an integral. And then what's the lowest z value? Well, it's z is equal to 0. And what's the upper bound?
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
Well, since we're going up and down, we're taking the sum in the z direction, we'd have an integral. And then what's the lowest z value? Well, it's z is equal to 0. And what's the upper bound? Like if you were to just keep adding these cubes and keep going up, you'd run into the upper bound. And what's the upper bound? It's z is equal to 2.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And what's the upper bound? Like if you were to just keep adding these cubes and keep going up, you'd run into the upper bound. And what's the upper bound? It's z is equal to 2. And of course, you would take the sum of these dv's. I'll write dz first, just so it reminds us that we're going to take the integral with respect to z first. And let's say we'll do y next.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
It's z is equal to 2. And of course, you would take the sum of these dv's. I'll write dz first, just so it reminds us that we're going to take the integral with respect to z first. And let's say we'll do y next. And then we'll do x. So this integral, this value, as I have written it, will figure out the volume of a column given any x and y. It'll be a function of x and y, but since we're dealing with all constant here, it's actually going to be a constant value.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And let's say we'll do y next. And then we'll do x. So this integral, this value, as I have written it, will figure out the volume of a column given any x and y. It'll be a function of x and y, but since we're dealing with all constant here, it's actually going to be a constant value. It'll be the constant value of the volume of one of these columns. And so essentially, it'll be 2 times dy dx. Because the height of one of these columns is 2.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
It'll be a function of x and y, but since we're dealing with all constant here, it's actually going to be a constant value. It'll be the constant value of the volume of one of these columns. And so essentially, it'll be 2 times dy dx. Because the height of one of these columns is 2. And then its width and its depth is dy and dx. So then if we wanted to figure out the entire volume, so what we did just now is we figured out the height of a column. So then we could take those columns and sum them in the y direction, right?
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
Because the height of one of these columns is 2. And then its width and its depth is dy and dx. So then if we wanted to figure out the entire volume, so what we did just now is we figured out the height of a column. So then we could take those columns and sum them in the y direction, right? So for summing in the y direction, we could just take another integral of this, sum in the y direction. And y goes from 0 to what? y goes from 0 to 4.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
So then we could take those columns and sum them in the y direction, right? So for summing in the y direction, we could just take another integral of this, sum in the y direction. And y goes from 0 to what? y goes from 0 to 4. I wrote this integral a little bit too far to the left. It looks strange. But I think you get the idea.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
y goes from 0 to 4. I wrote this integral a little bit too far to the left. It looks strange. But I think you get the idea. y is equal to 4. y is equal to 0 to y is equal to 4. And then that'll give us the volume of kind of a sheet that is parallel to the zy plane. And then all we have left to do is add up a bunch of those sheets in the x direction.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
But I think you get the idea. y is equal to 4. y is equal to 0 to y is equal to 4. And then that'll give us the volume of kind of a sheet that is parallel to the zy plane. And then all we have left to do is add up a bunch of those sheets in the x direction. We'll have the volume of our entire figure. So to add up those sheets, we would have to sum in the x direction, and we'd go from x is equal to 0 to x is equal to 3. And to evaluate this, it's actually fairly straightforward.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And then all we have left to do is add up a bunch of those sheets in the x direction. We'll have the volume of our entire figure. So to add up those sheets, we would have to sum in the x direction, and we'd go from x is equal to 0 to x is equal to 3. And to evaluate this, it's actually fairly straightforward. So first we're taking the integral with respect to z. Well, we don't have anything written under here, but we can just assume that there's a 1, right? Because dz times dy times dx is the same thing as 1 times dz times dy dx.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And to evaluate this, it's actually fairly straightforward. So first we're taking the integral with respect to z. Well, we don't have anything written under here, but we can just assume that there's a 1, right? Because dz times dy times dx is the same thing as 1 times dz times dy dx. So what's the value of this integral? Well, the antiderivative of 1 with respect to z is just z, right, because the derivative of z is 1. And you evaluate that from 2 to 0.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
Because dz times dy times dx is the same thing as 1 times dz times dy dx. So what's the value of this integral? Well, the antiderivative of 1 with respect to z is just z, right, because the derivative of z is 1. And you evaluate that from 2 to 0. So then you're left with 2 minus 0. So you're just left with 2. You're going to take the integral of that from y is equal to 0 to y is equal to 4 dy.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And you evaluate that from 2 to 0. So then you're left with 2 minus 0. So you're just left with 2. You're going to take the integral of that from y is equal to 0 to y is equal to 4 dy. And then you have the x from x is equal to 0 to x is equal to 3 dx. And notice, when we just took the integral with respect to z, we ended up with a double integral. And this double integral is the exact integral we would have done in the previous videos on the double integral where you would have just said, well, z is a function of x and y.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
You're going to take the integral of that from y is equal to 0 to y is equal to 4 dy. And then you have the x from x is equal to 0 to x is equal to 3 dx. And notice, when we just took the integral with respect to z, we ended up with a double integral. And this double integral is the exact integral we would have done in the previous videos on the double integral where you would have just said, well, z is a function of x and y. So you could have written z is a function of x and y is always equal to 2. It's a constant function. It's independent of x and y.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And this double integral is the exact integral we would have done in the previous videos on the double integral where you would have just said, well, z is a function of x and y. So you could have written z is a function of x and y is always equal to 2. It's a constant function. It's independent of x and y. But if you had defined z in this way and you wanted to figure out the volume under this surface where the surface is z is equal to 2, we would have ended up with this. So you see that what we're doing with the triple integral, it's really, really nothing different. And you might be wondering, well, why are we doing it at all?
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
It's independent of x and y. But if you had defined z in this way and you wanted to figure out the volume under this surface where the surface is z is equal to 2, we would have ended up with this. So you see that what we're doing with the triple integral, it's really, really nothing different. And you might be wondering, well, why are we doing it at all? And I'll show you that in a second. But anyway, to evaluate this, you can take the antiderivative of this with respect to y. You get 2y.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And you might be wondering, well, why are we doing it at all? And I'll show you that in a second. But anyway, to evaluate this, you can take the antiderivative of this with respect to y. You get 2y. Let me scroll down a little bit. You get 2y, evaluating that at 4 and 0. And then so you get 2 times 4, so it's 8 minus 0.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
You get 2y. Let me scroll down a little bit. You get 2y, evaluating that at 4 and 0. And then so you get 2 times 4, so it's 8 minus 0. And then you integrate that with respect to x from 0 to 3. So that's 8x from 0 to 3. So that'll be equal to 24 units cubed.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And then so you get 2 times 4, so it's 8 minus 0. And then you integrate that with respect to x from 0 to 3. So that's 8x from 0 to 3. So that'll be equal to 24 units cubed. So I know the obvious question is, what is this good for? Well, when you have a kind of a constant value within the volume, you're right. You could have just done a double integral.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
So that'll be equal to 24 units cubed. So I know the obvious question is, what is this good for? Well, when you have a kind of a constant value within the volume, you're right. You could have just done a double integral. But what if I were told you, our goal is not to figure out the volume of this figure. Our goal is to figure out the mass of this figure. And even more, this volume, this area of space or whatever, its mass is not uniform.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
You could have just done a double integral. But what if I were told you, our goal is not to figure out the volume of this figure. Our goal is to figure out the mass of this figure. And even more, this volume, this area of space or whatever, its mass is not uniform. If its mass was uniform, you could just multiply its uniform density times its volume, and you would get its mass. But let's say the density changes. It could be a volume of some gas, or it could be even some material with different compounds in it.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And even more, this volume, this area of space or whatever, its mass is not uniform. If its mass was uniform, you could just multiply its uniform density times its volume, and you would get its mass. But let's say the density changes. It could be a volume of some gas, or it could be even some material with different compounds in it. So let's say that its density is a variable function of x, y, and z. So let's say that the density, this row, this thing that looks like a p is what you normally use in physics for density. So its density is a function of x, y, and z.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
It could be a volume of some gas, or it could be even some material with different compounds in it. So let's say that its density is a variable function of x, y, and z. So let's say that the density, this row, this thing that looks like a p is what you normally use in physics for density. So its density is a function of x, y, and z. Let's just to make it simple, let's make it x times y times z. So if we wanted to figure out the mass of any small volume, it would be that volume times the density. Because the units of density are like kilograms per meter cubed.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
So its density is a function of x, y, and z. Let's just to make it simple, let's make it x times y times z. So if we wanted to figure out the mass of any small volume, it would be that volume times the density. Because the units of density are like kilograms per meter cubed. So if you multiply it times meter cubed, you get kilograms. So we could say that the mass, I'll make up notation, d mass. This isn't a function.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
Because the units of density are like kilograms per meter cubed. So if you multiply it times meter cubed, you get kilograms. So we could say that the mass, I'll make up notation, d mass. This isn't a function. Well, I don't want to write it in parentheses because it makes it look like a function. So a differential of mass, or a very small mass, is going to equal the density at that point, which would be x, y, z, times the volume of that small mass. And that volume of that small mass we could write as dv.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
This isn't a function. Well, I don't want to write it in parentheses because it makes it look like a function. So a differential of mass, or a very small mass, is going to equal the density at that point, which would be x, y, z, times the volume of that small mass. And that volume of that small mass we could write as dv. And we know that dv is the same thing as the width times the height times the depth. dv doesn't always have to be dx times dy times dz. If we're doing other coordinates, if we're doing polar coordinates, it could be something slightly different.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
And that volume of that small mass we could write as dv. And we know that dv is the same thing as the width times the height times the depth. dv doesn't always have to be dx times dy times dz. If we're doing other coordinates, if we're doing polar coordinates, it could be something slightly different. And we'll do that eventually. But if we want to figure out the mass, since we're using rectangular coordinates, it would be the density function at that point times our differential of volume. So times dx, dy, dz.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
If we're doing other coordinates, if we're doing polar coordinates, it could be something slightly different. And we'll do that eventually. But if we want to figure out the mass, since we're using rectangular coordinates, it would be the density function at that point times our differential of volume. So times dx, dy, dz. And of course, we can change the order here. So when you want to figure out the volume, when you want to figure out the mass, which I will do in the next video, we essentially will have to integrate this function as opposed to just 1 over z, y, and x. And I'm going to do that in the next video.
Triple integrals 1 Double and triple integrals Multivariable Calculus Khan Academy.mp3
So in the last video, I talked about curvature and the radius of curvature, and I described it purely geometrically, where I'm saying you imagine driving along a certain road, your steering wheel locks, and you're wondering what the radius of the circle that you draw with your car, you know, through whatever surrounding fields there are on the road as a result. And the special symbol that we have for this, for this idea of curvature, is a little kappa, and that's gonna be one divided by the radius. And the reason we do that is basically you want a large kappa, a large curvature, to correspond with a sharp turn. So sharp turn, small radius, large curvature. But the question is, how do we describe this in a more mathematical way? So I'm gonna go ahead and clear up, get rid of the circle itself and all of that radius, and just be looking at the curve itself in its own right. The way you typically describe a curve like this is parametrically.
Curvature formula, part 1.mp3
So sharp turn, small radius, large curvature. But the question is, how do we describe this in a more mathematical way? So I'm gonna go ahead and clear up, get rid of the circle itself and all of that radius, and just be looking at the curve itself in its own right. The way you typically describe a curve like this is parametrically. So you'll have some kind of vector-valued function, s, that takes in a single parameter, t, and then it's gonna output the x and y coordinates, each as functions of t. And so this will be the x coordinate, the y coordinate. And in this specific case, I'll just tell you the curve that I drew happens to be parameterized by one minus the sine of t as the x-component function. Actually, no, it's t minus sine of t, and the bottom part is one minus cosine of t. That's the curve that I drew.
Curvature formula, part 1.mp3
The way you typically describe a curve like this is parametrically. So you'll have some kind of vector-valued function, s, that takes in a single parameter, t, and then it's gonna output the x and y coordinates, each as functions of t. And so this will be the x coordinate, the y coordinate. And in this specific case, I'll just tell you the curve that I drew happens to be parameterized by one minus the sine of t as the x-component function. Actually, no, it's t minus sine of t, and the bottom part is one minus cosine of t. That's the curve that I drew. And the way that you're thinking about this is for each value t, you get a certain vector that puts you at a point on the curve. And as t changes, the vector you get changes, but the tip of that vector kind of traces out the curve as a whole, right? You know, maybe.
Curvature formula, part 1.mp3
Actually, no, it's t minus sine of t, and the bottom part is one minus cosine of t. That's the curve that I drew. And the way that you're thinking about this is for each value t, you get a certain vector that puts you at a point on the curve. And as t changes, the vector you get changes, but the tip of that vector kind of traces out the curve as a whole, right? You know, maybe. And you can imagine just the vector drawing the curve as t varies. And the thought behind making curvature mathematical, here I'll kind of clear up some room for myself, is that you take the tangent vectors here, so you might imagine like a unit tangent vector at every given point, and you're wondering how quickly those guys change direction, right? So with the little schematic that I have drawn here, I might call this guy, I'm just gonna call this guy t1, for like the first tangent vector, t2, t3, and I haven't specified where they start or anything, I just want to give a feel for you've got various different tangent vectors, and I'm just gonna say that all of them, each one of those t sub somethings, has a unit length.
Curvature formula, part 1.mp3
You know, maybe. And you can imagine just the vector drawing the curve as t varies. And the thought behind making curvature mathematical, here I'll kind of clear up some room for myself, is that you take the tangent vectors here, so you might imagine like a unit tangent vector at every given point, and you're wondering how quickly those guys change direction, right? So with the little schematic that I have drawn here, I might call this guy, I'm just gonna call this guy t1, for like the first tangent vector, t2, t3, and I haven't specified where they start or anything, I just want to give a feel for you've got various different tangent vectors, and I'm just gonna say that all of them, each one of those t sub somethings, has a unit length. And the idea of curvature is to look at how quickly that unit tangent vector changes directions. So you know, you might imagine just a completely different space, so rather than rooting each vector on the curve, let's see what it would look like if you just kind of write each vector in its own right off in some other spot. So this guy here would be t1, and then t2 points a little bit down, and then t3 points kind of much more down.
Curvature formula, part 1.mp3
So with the little schematic that I have drawn here, I might call this guy, I'm just gonna call this guy t1, for like the first tangent vector, t2, t3, and I haven't specified where they start or anything, I just want to give a feel for you've got various different tangent vectors, and I'm just gonna say that all of them, each one of those t sub somethings, has a unit length. And the idea of curvature is to look at how quickly that unit tangent vector changes directions. So you know, you might imagine just a completely different space, so rather than rooting each vector on the curve, let's see what it would look like if you just kind of write each vector in its own right off in some other spot. So this guy here would be t1, and then t2 points a little bit down, and then t3 points kind of much more down. So all of these, this would be t1, that guy is t2, and these are the same vectors, I'm just kind of drawing them all rooted at the same spot, so it's a little easier to see how they turn. And you want to say, okay, how much do you change as you go from t1 to t2, is that a large angle change? And as you go from t2 to t3, is that a large change as well?
Curvature formula, part 1.mp3
So this guy here would be t1, and then t2 points a little bit down, and then t3 points kind of much more down. So all of these, this would be t1, that guy is t2, and these are the same vectors, I'm just kind of drawing them all rooted at the same spot, so it's a little easier to see how they turn. And you want to say, okay, how much do you change as you go from t1 to t2, is that a large angle change? And as you go from t2 to t3, is that a large change as well? And you can kind of see how if you have a curve, and let's say it's, if you have a curve that curves quite a bit, you know, it's doing something like that, then the unit vector, the unit tangent vector at this point, changes quite rapidly over a short distance to be something almost 90 degrees different. Whereas if you take the unit vector here and see how much does it change as you go from this point over to this point, it doesn't really change that much. So the thought behind curvature is we're going to take the rate of change of that unit tangent vector, so the rate of change of t, and I'm going to let capital T be a function that tells you whatever the unit tangent vector at each point is, and I'm not going to take the rate of change in terms of, you know, the parameter, little t, which is what we use to parametrize the curve, because it shouldn't matter how you parametrize the curve.
Curvature formula, part 1.mp3
And as you go from t2 to t3, is that a large change as well? And you can kind of see how if you have a curve, and let's say it's, if you have a curve that curves quite a bit, you know, it's doing something like that, then the unit vector, the unit tangent vector at this point, changes quite rapidly over a short distance to be something almost 90 degrees different. Whereas if you take the unit vector here and see how much does it change as you go from this point over to this point, it doesn't really change that much. So the thought behind curvature is we're going to take the rate of change of that unit tangent vector, so the rate of change of t, and I'm going to let capital T be a function that tells you whatever the unit tangent vector at each point is, and I'm not going to take the rate of change in terms of, you know, the parameter, little t, which is what we use to parametrize the curve, because it shouldn't matter how you parametrize the curve. Maybe you're driving along it really quickly or really slowly. Instead, what you want to take is the rate of change with respect to arc length, arc length. And I'm using the variable s here to denote arc length.
Curvature formula, part 1.mp3
So the thought behind curvature is we're going to take the rate of change of that unit tangent vector, so the rate of change of t, and I'm going to let capital T be a function that tells you whatever the unit tangent vector at each point is, and I'm not going to take the rate of change in terms of, you know, the parameter, little t, which is what we use to parametrize the curve, because it shouldn't matter how you parametrize the curve. Maybe you're driving along it really quickly or really slowly. Instead, what you want to take is the rate of change with respect to arc length, arc length. And I'm using the variable s here to denote arc length. And what I mean by that is if you take just a tiny little step here, the distance of that step, the actual distance in the xy plane, you'd consider to be the arc length. And if you imagine it being really, really small, you're considering that a ds, a tiny change in the arc length. So this is the quantity that we care about.
Curvature formula, part 1.mp3
And I'm using the variable s here to denote arc length. And what I mean by that is if you take just a tiny little step here, the distance of that step, the actual distance in the xy plane, you'd consider to be the arc length. And if you imagine it being really, really small, you're considering that a ds, a tiny change in the arc length. So this is the quantity that we care about. How much does that unit tangent vector change with respect to a tiny change in the arc length? You know, as we travel along, let's say it was a distance, of like 0.5, right? You want to know, did this unit vector change a lot or a little bit?
Curvature formula, part 1.mp3
So this is the quantity that we care about. How much does that unit tangent vector change with respect to a tiny change in the arc length? You know, as we travel along, let's say it was a distance, of like 0.5, right? You want to know, did this unit vector change a lot or a little bit? But I should add something here. It's this tiny change in the vector, that would tell you what the vector connecting their two tips is. So this would be a vector-valued quantity, and curvature itself should just be a number.
Curvature formula, part 1.mp3
You want to know, did this unit vector change a lot or a little bit? But I should add something here. It's this tiny change in the vector, that would tell you what the vector connecting their two tips is. So this would be a vector-valued quantity, and curvature itself should just be a number. So what we really care about is the size of this guy. So what we really care about, the size of this, which is a vector-valued quantity, and that'll be an indication of just how much the curve curves. But if on a sharper-turned curve, you go over that same distance, and then suddenly the change in the tangent vectors goes by quite a bit, that would be telling you it's a high curvature.
Curvature formula, part 1.mp3
So this would be a vector-valued quantity, and curvature itself should just be a number. So what we really care about is the size of this guy. So what we really care about, the size of this, which is a vector-valued quantity, and that'll be an indication of just how much the curve curves. But if on a sharper-turned curve, you go over that same distance, and then suddenly the change in the tangent vectors goes by quite a bit, that would be telling you it's a high curvature. And in the next video, I'm gonna talk through what that looks like, how you think about this tangent vector function, this unit tangent vector function, and what it looks like to take the derivative of that with respect to arc length. It can get a little convoluted, in terms of the symbols involved. And the constant picture you should have in the back of your mind is that circle, that circle that's hugging the curve very closely at a certain point.
Curvature formula, part 1.mp3
So I thought I'd make a little video here to spell out exactly how it is that we describe points and vectors in three dimensions. And before we do that, I think it'll be valuable if we start off by describing points and vectors in two dimensions. And I'm assuming if you're learning about multivariable calculus that a lot of you have already learned about this, and you might be saying, what's the point? I already know how to represent points and vectors in two dimensions. But there's a huge value in analogy here, because as soon as you start to compare two dimensions and three dimensions, you start to see patterns for how it could extend to other dimensions that you can't necessarily visualize, or when it might be useful to think about one dimension versus another. So in two dimensions, if you have some kind of point just off sitting there, we typically represent it, you've got an x-axis and a y-axis that are perpendicular to each other. And we represent this number with a pair, sorry, we represent this point with a pair of numbers.
Representing points in 3d Multivariable calculus Khan Academy.mp3
I already know how to represent points and vectors in two dimensions. But there's a huge value in analogy here, because as soon as you start to compare two dimensions and three dimensions, you start to see patterns for how it could extend to other dimensions that you can't necessarily visualize, or when it might be useful to think about one dimension versus another. So in two dimensions, if you have some kind of point just off sitting there, we typically represent it, you've got an x-axis and a y-axis that are perpendicular to each other. And we represent this number with a pair, sorry, we represent this point with a pair of numbers. So in this case, I don't know, it might be something like one, three. And what that would represent is it's saying that you have to move a distance of one along the x-axis, and then a distance of three up along the y-axis. So you know this, let's say that's the distance of one, that's the distance of three, it might not be exactly that the way I drew it, but let's say that those are the coordinates.
Representing points in 3d Multivariable calculus Khan Academy.mp3
And we represent this number with a pair, sorry, we represent this point with a pair of numbers. So in this case, I don't know, it might be something like one, three. And what that would represent is it's saying that you have to move a distance of one along the x-axis, and then a distance of three up along the y-axis. So you know this, let's say that's the distance of one, that's the distance of three, it might not be exactly that the way I drew it, but let's say that those are the coordinates. What this means is every point in two-dimensional space can be given a pair of numbers like this, and you think of them as instructions, where it's kind of telling you how far to walk in one way, how far to walk in another. But you can also think of the reverse, right? Every time that you have a pair of things, you know that you should be thinking two-dimensionally.
Representing points in 3d Multivariable calculus Khan Academy.mp3
So you know this, let's say that's the distance of one, that's the distance of three, it might not be exactly that the way I drew it, but let's say that those are the coordinates. What this means is every point in two-dimensional space can be given a pair of numbers like this, and you think of them as instructions, where it's kind of telling you how far to walk in one way, how far to walk in another. But you can also think of the reverse, right? Every time that you have a pair of things, you know that you should be thinking two-dimensionally. And that's actually a surprisingly powerful idea that I don't think I appreciated for a long time, how there's this back and forth between pairs of numbers and points in space, and it lets you visualize things you didn't think you could visualize, or it lets you understand things that are inherently visual just by kind of going back and forth. And in three dimensions, there's a similar dichotomy, but between triplets of points and points in three-dimensional space. So let me just plop down a point in three-dimensional space here.
Representing points in 3d Multivariable calculus Khan Academy.mp3
Every time that you have a pair of things, you know that you should be thinking two-dimensionally. And that's actually a surprisingly powerful idea that I don't think I appreciated for a long time, how there's this back and forth between pairs of numbers and points in space, and it lets you visualize things you didn't think you could visualize, or it lets you understand things that are inherently visual just by kind of going back and forth. And in three dimensions, there's a similar dichotomy, but between triplets of points and points in three-dimensional space. So let me just plop down a point in three-dimensional space here. And it's hard to get a feel for exactly where it is until you move things around. This is one thing that makes three dimensions hard, is you can't really draw it without moving it around or showing a difference in perspective in various ways. But we describe points like this, again, with a set of coordinates, but this time it's a triplet.
Representing points in 3d Multivariable calculus Khan Academy.mp3
So let me just plop down a point in three-dimensional space here. And it's hard to get a feel for exactly where it is until you move things around. This is one thing that makes three dimensions hard, is you can't really draw it without moving it around or showing a difference in perspective in various ways. But we describe points like this, again, with a set of coordinates, but this time it's a triplet. And this particular point, I happen to know, is one, two, five. And what those numbers are telling you is how far to move parallel to each axis. So just like with two dimensions, we have an x-axis and a y-axis, but now there's a third axis that's perpendicular to both of them and moves us into that third dimension, the z-axis.
Representing points in 3d Multivariable calculus Khan Academy.mp3
But we describe points like this, again, with a set of coordinates, but this time it's a triplet. And this particular point, I happen to know, is one, two, five. And what those numbers are telling you is how far to move parallel to each axis. So just like with two dimensions, we have an x-axis and a y-axis, but now there's a third axis that's perpendicular to both of them and moves us into that third dimension, the z-axis. And the first number in our coordinate is gonna tell us how far, whoop, can't move those guys, how far we need to move in the x direction as our first step. The second number, two in this case, tells us how far we have to move parallel to the y-axis for our second step. And then the third number tells us how far up we have to go to get to that point.
Representing points in 3d Multivariable calculus Khan Academy.mp3
So just like with two dimensions, we have an x-axis and a y-axis, but now there's a third axis that's perpendicular to both of them and moves us into that third dimension, the z-axis. And the first number in our coordinate is gonna tell us how far, whoop, can't move those guys, how far we need to move in the x direction as our first step. The second number, two in this case, tells us how far we have to move parallel to the y-axis for our second step. And then the third number tells us how far up we have to go to get to that point. And you can do this for any point in three-dimensional space, right? Any point that you have, you can give the instructions for how to move along the x and then how to move parallel to the y and how to move parallel to the z to get to that point, which means there's this back and forth between triplets of numbers and points in 3D. So whenever you come across a triplet of things, and you'll see this in the next video when we start talking about three-dimensional graphs, you'll know, just by virtue of the fact that it's a triplet, ah, yes, I should be thinking in three dimensions somehow.
Representing points in 3d Multivariable calculus Khan Academy.mp3
And then the third number tells us how far up we have to go to get to that point. And you can do this for any point in three-dimensional space, right? Any point that you have, you can give the instructions for how to move along the x and then how to move parallel to the y and how to move parallel to the z to get to that point, which means there's this back and forth between triplets of numbers and points in 3D. So whenever you come across a triplet of things, and you'll see this in the next video when we start talking about three-dimensional graphs, you'll know, just by virtue of the fact that it's a triplet, ah, yes, I should be thinking in three dimensions somehow. Just in the same way that whenever you have pairs, you should be thinking, ah, this is a very two-dimensional thing. So there's another context, though, where pairs of numbers come up. And that would be vectors.
Representing points in 3d Multivariable calculus Khan Academy.mp3
So whenever you come across a triplet of things, and you'll see this in the next video when we start talking about three-dimensional graphs, you'll know, just by virtue of the fact that it's a triplet, ah, yes, I should be thinking in three dimensions somehow. Just in the same way that whenever you have pairs, you should be thinking, ah, this is a very two-dimensional thing. So there's another context, though, where pairs of numbers come up. And that would be vectors. So a vector you might represent, you know, you typically represent it with an arrow. Whoa. Ah, help, help.
Representing points in 3d Multivariable calculus Khan Academy.mp3
And that would be vectors. So a vector you might represent, you know, you typically represent it with an arrow. Whoa. Ah, help, help. So vectors, so vectors we typically represent with some kind of arrow. Let's make this arrow a nice color. An arrow.
Representing points in 3d Multivariable calculus Khan Academy.mp3
Ah, help, help. So vectors, so vectors we typically represent with some kind of arrow. Let's make this arrow a nice color. An arrow. And if it's a vector from the origin to a simple point, the coordinates of that vector are just the same as those of the point. And the convention is to write those coordinates in a column. You know, it's not set in stone, but typically if you see numbers in a column, you should be thinking about it as a vector, some kind of arrow.
Representing points in 3d Multivariable calculus Khan Academy.mp3
An arrow. And if it's a vector from the origin to a simple point, the coordinates of that vector are just the same as those of the point. And the convention is to write those coordinates in a column. You know, it's not set in stone, but typically if you see numbers in a column, you should be thinking about it as a vector, some kind of arrow. And if it's a pair with parentheses around it, you just think about it as a point. And even though both of these are ways of representing the same pair of numbers, the main difference is that a vector, you could have started it at any point in space. It didn't have to start in the origin.
Representing points in 3d Multivariable calculus Khan Academy.mp3
You know, it's not set in stone, but typically if you see numbers in a column, you should be thinking about it as a vector, some kind of arrow. And if it's a pair with parentheses around it, you just think about it as a point. And even though both of these are ways of representing the same pair of numbers, the main difference is that a vector, you could have started it at any point in space. It didn't have to start in the origin. So if we have that same guy, but you know, if he starts here and he still has a rightward component of one and an upward component of three, we think of that as the same vector. And typically these are representing motion of some kind, whereas points are just representing like actual points in space. And the other big thing that you can do is you can add vectors together.
Representing points in 3d Multivariable calculus Khan Academy.mp3
It didn't have to start in the origin. So if we have that same guy, but you know, if he starts here and he still has a rightward component of one and an upward component of three, we think of that as the same vector. And typically these are representing motion of some kind, whereas points are just representing like actual points in space. And the other big thing that you can do is you can add vectors together. So you know, if you had another, let's say you have another vector that has a large x component, but a small negative y component like this guy. And what that means is you can kind of add by imagining that that second vector started at the tip of the first one. And then however you get from the origin to the new tip there, that's gonna be the resulting vector.
Representing points in 3d Multivariable calculus Khan Academy.mp3
And the other big thing that you can do is you can add vectors together. So you know, if you had another, let's say you have another vector that has a large x component, but a small negative y component like this guy. And what that means is you can kind of add by imagining that that second vector started at the tip of the first one. And then however you get from the origin to the new tip there, that's gonna be the resulting vector. So I'd say this is the sum of those two vectors. And you can't really do that with points as much. In order to think about adding points, you end up thinking about them as vectors.
Representing points in 3d Multivariable calculus Khan Academy.mp3
And then however you get from the origin to the new tip there, that's gonna be the resulting vector. So I'd say this is the sum of those two vectors. And you can't really do that with points as much. In order to think about adding points, you end up thinking about them as vectors. And the same goes with three dimensions. For a given point, if you draw an arrow from the origin up to that point, this arrow would be represented with that same triplet of numbers, but you typically do it in a column. I call this a column vector.
Representing points in 3d Multivariable calculus Khan Academy.mp3
In order to think about adding points, you end up thinking about them as vectors. And the same goes with three dimensions. For a given point, if you draw an arrow from the origin up to that point, this arrow would be represented with that same triplet of numbers, but you typically do it in a column. I call this a column vector. It's not three, that's five. And the difference between the point and the arrow is you can think of the arrow or the vector as starting anywhere in space. It doesn't really matter as long as it's got those same components for how far does it move parallel to the x, how far does it move parallel to the y-axis, and how far does it move parallel to the z-axis.
Representing points in 3d Multivariable calculus Khan Academy.mp3
And the one I have in mind has a special name. It's a helix. And the first two components kind of make it look like a circle. It's gonna be cosine of t for the x component, sine of t for the y component, but this is three-dimensional, and it makes it a little different from a circle. I'm gonna have the last component be t divided by five. And what this looks like, we can visualize it pretty well. I'm gonna go on over to the graph of it here.
Curvature of a helix, part 1.mp3
It's gonna be cosine of t for the x component, sine of t for the y component, but this is three-dimensional, and it makes it a little different from a circle. I'm gonna have the last component be t divided by five. And what this looks like, we can visualize it pretty well. I'm gonna go on over to the graph of it here. So this shape is called a helix, and you can sort of see how, looking from the xy plane's perspective, it looks as if it's gonna draw a circle. And really, these lines should all line up when you're facing it, but it's due to the perspective where things farther away look smaller. But it would just be drawing a circle.
Curvature of a helix, part 1.mp3
I'm gonna go on over to the graph of it here. So this shape is called a helix, and you can sort of see how, looking from the xy plane's perspective, it looks as if it's gonna draw a circle. And really, these lines should all line up when you're facing it, but it's due to the perspective where things farther away look smaller. But it would just be drawing a circle. But then the z component, because z increases while your parameter t increases, you're kind of rising as if it's a spiral staircase. And now, before we compute curvature, to know what we're really going for, what this represents, you kind of imagine yourself, you know, maybe this isn't a road, but it's like a space freeway, right? And you're driving your spaceship along it, and you imagine that you get stuck at some point, or maybe not that you get stuck, but all of your instruments lock, and your steering wheel locks, or your joystick, or however you're steering, it all just locks up.
Curvature of a helix, part 1.mp3
But it would just be drawing a circle. But then the z component, because z increases while your parameter t increases, you're kind of rising as if it's a spiral staircase. And now, before we compute curvature, to know what we're really going for, what this represents, you kind of imagine yourself, you know, maybe this isn't a road, but it's like a space freeway, right? And you're driving your spaceship along it, and you imagine that you get stuck at some point, or maybe not that you get stuck, but all of your instruments lock, and your steering wheel locks, or your joystick, or however you're steering, it all just locks up. And you're gonna trace out a certain circle in space, right? And that circle might look something like this. So if you were turning however you were on the helix, but then you can't do anything different, you might trace out a giant circle.
Curvature of a helix, part 1.mp3
And you're driving your spaceship along it, and you imagine that you get stuck at some point, or maybe not that you get stuck, but all of your instruments lock, and your steering wheel locks, or your joystick, or however you're steering, it all just locks up. And you're gonna trace out a certain circle in space, right? And that circle might look something like this. So if you were turning however you were on the helix, but then you can't do anything different, you might trace out a giant circle. And what we care about is the radius of that circle, and if you take one divided by the radius of that circle you trace out, that's gonna be the curvature. That's gonna be the little kappa curvature. And of course, the way that we compute it, we don't directly talk about that circle at all, but it's actually a good thing to keep in the back of your mind.
Curvature of a helix, part 1.mp3
So if you were turning however you were on the helix, but then you can't do anything different, you might trace out a giant circle. And what we care about is the radius of that circle, and if you take one divided by the radius of that circle you trace out, that's gonna be the curvature. That's gonna be the little kappa curvature. And of course, the way that we compute it, we don't directly talk about that circle at all, but it's actually a good thing to keep in the back of your mind. The way that we compute it is to first find a unit tangent vector function with the same parameter. And what that means, you know, if you imagine your helix kind of spiraling through three-dimensional space, man, I am not as good an artist as the computer is when it comes to drawing a helix. But the unit tangent vector function would be something that gives you a tangent vector at every given point, you know, kind of the direction that you on your spaceship are traveling.
Curvature of a helix, part 1.mp3
And of course, the way that we compute it, we don't directly talk about that circle at all, but it's actually a good thing to keep in the back of your mind. The way that we compute it is to first find a unit tangent vector function with the same parameter. And what that means, you know, if you imagine your helix kind of spiraling through three-dimensional space, man, I am not as good an artist as the computer is when it comes to drawing a helix. But the unit tangent vector function would be something that gives you a tangent vector at every given point, you know, kind of the direction that you on your spaceship are traveling. And to do that, you take the derivative of your parameterization, that derivative, which is gonna give you a tangent vector, but it might not be a unit tangent vector, so you divide it by its own magnitude, and that'll give you a unit tangent vector. And then ultimately, the goal that we're shooting for is gonna be to find the derivative of this tangent vector function with respect to the arc length. So as a first step, we'll start by finding a derivative of our parameterization function.
Curvature of a helix, part 1.mp3
But the unit tangent vector function would be something that gives you a tangent vector at every given point, you know, kind of the direction that you on your spaceship are traveling. And to do that, you take the derivative of your parameterization, that derivative, which is gonna give you a tangent vector, but it might not be a unit tangent vector, so you divide it by its own magnitude, and that'll give you a unit tangent vector. And then ultimately, the goal that we're shooting for is gonna be to find the derivative of this tangent vector function with respect to the arc length. So as a first step, we'll start by finding a derivative of our parameterization function. So when we take that derivative, luckily there's not a lot of new things going on. Let's see, derivative, so S prime. From single-variable calculus, we just take the derivative of each component.
Curvature of a helix, part 1.mp3
So as a first step, we'll start by finding a derivative of our parameterization function. So when we take that derivative, luckily there's not a lot of new things going on. Let's see, derivative, so S prime. From single-variable calculus, we just take the derivative of each component. So cosine goes to negative sine, negative sine of t. Sine, its derivative is cosine, cosine of t. And then the derivative of t divided by five is just a constant, that's just one over five. Boy, it is hard to say the word derivative over and over. Say it five times.
Curvature of a helix, part 1.mp3
From single-variable calculus, we just take the derivative of each component. So cosine goes to negative sine, negative sine of t. Sine, its derivative is cosine, cosine of t. And then the derivative of t divided by five is just a constant, that's just one over five. Boy, it is hard to say the word derivative over and over. Say it five times. Okay, so that's S prime of t. And now what we need to do, we need to find the magnitude of S prime of t. So what that involves, as we're taking the magnitude, S prime of t as a vector, we take the square root of the sum of the squares of each of its components. So sine, negative sine squared, just looks like sine squared, sine squared of t, cosine squared, cosine squared of t, and then 1 5th squared, and that's just 1 25th. And you might notice I use a lot of these sine-cosine pairs in examples, partly because they draw circles, and lots of things are fun that involve drawing circles, but also because it has a tendency to let things simplify, especially if you're taking a magnitude, because sine squared plus cosine squared just equals one.
Curvature of a helix, part 1.mp3
Say it five times. Okay, so that's S prime of t. And now what we need to do, we need to find the magnitude of S prime of t. So what that involves, as we're taking the magnitude, S prime of t as a vector, we take the square root of the sum of the squares of each of its components. So sine, negative sine squared, just looks like sine squared, sine squared of t, cosine squared, cosine squared of t, and then 1 5th squared, and that's just 1 25th. And you might notice I use a lot of these sine-cosine pairs in examples, partly because they draw circles, and lots of things are fun that involve drawing circles, but also because it has a tendency to let things simplify, especially if you're taking a magnitude, because sine squared plus cosine squared just equals one. So this entire formula, this entire formula, boils down to the square root of one plus one divided by 25. And for this, you might kind of be thinking off to the side that that's 25 over 25 plus one over 25. So making even more room here, what that equals is the square root of 26 divided by 25.
Curvature of a helix, part 1.mp3
And you might notice I use a lot of these sine-cosine pairs in examples, partly because they draw circles, and lots of things are fun that involve drawing circles, but also because it has a tendency to let things simplify, especially if you're taking a magnitude, because sine squared plus cosine squared just equals one. So this entire formula, this entire formula, boils down to the square root of one plus one divided by 25. And for this, you might kind of be thinking off to the side that that's 25 over 25 plus one over 25. So making even more room here, what that equals is the square root of 26 divided by 25. And just because 25 is already a square, and it kind of might make things look nice, I'm gonna write this as the square root of 26 divided by five, that square root of 25. So this whole thing is the magnitude of our derivative, right? And we think to ourselves, it's quite lucky that this came out to be a constant, because as we saw with the more general formula, it's often pretty nasty, and it can get pretty bad, but in this case, it's just a constant, which is nice, because as we go up, and we start to think about what our unit tangent vector function for the helix should be, we're just gonna take the derivative function and divide each term by that magnitude, right?
Curvature of a helix, part 1.mp3
So making even more room here, what that equals is the square root of 26 divided by 25. And just because 25 is already a square, and it kind of might make things look nice, I'm gonna write this as the square root of 26 divided by five, that square root of 25. So this whole thing is the magnitude of our derivative, right? And we think to ourselves, it's quite lucky that this came out to be a constant, because as we saw with the more general formula, it's often pretty nasty, and it can get pretty bad, but in this case, it's just a constant, which is nice, because as we go up, and we start to think about what our unit tangent vector function for the helix should be, we're just gonna take the derivative function and divide each term by that magnitude, right? So it's gonna look almost identical. It's gonna be negative sine of t, except now we're dividing by that magnitude, and that magnitude, of course, is root 26 over five. So we go up here, and we say we're dividing this by root 26 over five, that whole quantity, and then similarly, y component is cosine of t divided by the quantity of the root, 26 divided by five, and the last part is 1 5th, 1 5th, I'll put that in parentheses, divided by that same amount, root 26 over five.
Curvature of a helix, part 1.mp3
In the last couple videos, I've been talking about curl, where if we have a two-dimensional vector field V defined with component functions P and Q, I've said that the 2D curl of that function V gives you a new function that also takes in x and y as inputs, and its formula is the partial derivative of Q with respect to x minus the partial derivative of P with respect to y. And my hope is that this is more than just a formula, and that you can understand how this represents fluid rotation in two dimensions. But what I want to do here is show how the original intuition I gave for this formula might be a little oversimplified. Because, for example, if we look at this, the partial Q partial x component, I said that you can imagine that Q at some point starting off a little bit negative, so the y component of the output is a little negative, then as you move positively in the x direction, it goes to being 0, and then it goes to being a little bit positive. And with this particular picture, it's hopefully a little bit clear why this can correspond to counterclockwise rotation in the fluid. But this is only a very specific circumstance for what partial Q partial x being positive could look like. You know, it might also look like Q starting off a little bit positive, and then as you move in the x direction, it becomes even more positive, and then even more positive.
2d curl nuance.mp3
Because, for example, if we look at this, the partial Q partial x component, I said that you can imagine that Q at some point starting off a little bit negative, so the y component of the output is a little negative, then as you move positively in the x direction, it goes to being 0, and then it goes to being a little bit positive. And with this particular picture, it's hopefully a little bit clear why this can correspond to counterclockwise rotation in the fluid. But this is only a very specific circumstance for what partial Q partial x being positive could look like. You know, it might also look like Q starting off a little bit positive, and then as you move in the x direction, it becomes even more positive, and then even more positive. And according to the formula, this should contribute as much to positive curl as this very clear-cut kind of whirlpool example. And to illustrate what this might look like, if we take a look at this vector field here, if we look in the center, this is kind of the clear-cut whirlpool counterclockwise rotation example. And if we play the fluid flow, the fluid does indeed rotate counterclockwise in the region.
2d curl nuance.mp3
You know, it might also look like Q starting off a little bit positive, and then as you move in the x direction, it becomes even more positive, and then even more positive. And according to the formula, this should contribute as much to positive curl as this very clear-cut kind of whirlpool example. And to illustrate what this might look like, if we take a look at this vector field here, if we look in the center, this is kind of the clear-cut whirlpool counterclockwise rotation example. And if we play the fluid flow, the fluid does indeed rotate counterclockwise in the region. But contrast that with what goes on over here on the right. This doesn't look like rotation in that sense at all. Instead, the fluid particles are just kind of rushing up through it.
2d curl nuance.mp3
And if we play the fluid flow, the fluid does indeed rotate counterclockwise in the region. But contrast that with what goes on over here on the right. This doesn't look like rotation in that sense at all. Instead, the fluid particles are just kind of rushing up through it. But in fact, the curl in this region is going to be just as strong as it is over here. And I'll show that with the formula, and kind of computing it through in just a moment. But the image that you might have in your mind is to imagine a paddle wheel of sorts, where let's say it's got arms kind of like that, and then you hold down with your thumb that middle portion.
2d curl nuance.mp3
Instead, the fluid particles are just kind of rushing up through it. But in fact, the curl in this region is going to be just as strong as it is over here. And I'll show that with the formula, and kind of computing it through in just a moment. But the image that you might have in your mind is to imagine a paddle wheel of sorts, where let's say it's got arms kind of like that, and then you hold down with your thumb that middle portion. So even though the paddle wheel left to its own devices would just kind of fly up, I want to say, let's say you're holding that down with your thumb, but it's free to rotate. Then the vectors on its left are pointing up, but less strongly than the vectors on its right, which are even greater. So if you imagine that setup, and you kind of have your paddle wheel there, then when you play the fluid rotation, holding your thumb down, but letting the paddle wheel rotate freely, it's also going to rotate just as it would over here in the easier to see whirlpool example.
2d curl nuance.mp3
But the image that you might have in your mind is to imagine a paddle wheel of sorts, where let's say it's got arms kind of like that, and then you hold down with your thumb that middle portion. So even though the paddle wheel left to its own devices would just kind of fly up, I want to say, let's say you're holding that down with your thumb, but it's free to rotate. Then the vectors on its left are pointing up, but less strongly than the vectors on its right, which are even greater. So if you imagine that setup, and you kind of have your paddle wheel there, then when you play the fluid rotation, holding your thumb down, but letting the paddle wheel rotate freely, it's also going to rotate just as it would over here in the easier to see whirlpool example. And in terms of the formula, this is because a situation like this one here, where q goes from being negative to zero to positive, should be treated just the same as a situation like this, as far as 2D curl is concerned, because this term in the 2D curl formula is going to come out the same for either one of these. And it's worth pointing out, by the way, curl isn't something that mathematicians and physicists came across trying to understand fluid flow. Instead, they found this term as being significant in various other formulas and circumstances, and I think electromagnetism might be where it originally came about.
2d curl nuance.mp3
So if you imagine that setup, and you kind of have your paddle wheel there, then when you play the fluid rotation, holding your thumb down, but letting the paddle wheel rotate freely, it's also going to rotate just as it would over here in the easier to see whirlpool example. And in terms of the formula, this is because a situation like this one here, where q goes from being negative to zero to positive, should be treated just the same as a situation like this, as far as 2D curl is concerned, because this term in the 2D curl formula is going to come out the same for either one of these. And it's worth pointing out, by the way, curl isn't something that mathematicians and physicists came across trying to understand fluid flow. Instead, they found this term as being significant in various other formulas and circumstances, and I think electromagnetism might be where it originally came about. But then in trying to understand this formula, they realized that you can give a fluid flow interpretation that gives a very deep understanding of what's going on beyond just the symbols themselves. So let me go ahead and walk through this example in terms of the formula representing the vector field. It's a relatively straightforward formula, actually.
2d curl nuance.mp3