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So if, for example, let's say you did all of your calculations, and it turned out that lambda star was equal to 2.3. Previously, that just seemed like some dummy number that you ignore, and you just look at whatever the associated values here are. But if you plug this in the computer and you see lambda star equals 2.3,... | Meaning of Lagrange multiplier.mp3 |
So you increase your budget by just a little bit, a little dB. Then the ratio of the change in the maximizing revenue to that dB is about 2.3. So what that would mean is increasing your budget by $1 is gonna increase m star over here. It would mean that m star increases by about $2.30 for every dollar that you increase... | Meaning of Lagrange multiplier.mp3 |
It would mean that m star increases by about $2.30 for every dollar that you increase your budget. And that's information you'd wanna know, right? If you see that this lambda star is a number bigger than one, you'd say, hey, maybe we should increase our budget. We increase it from $10,000 to 10,001, and we're making mo... | Meaning of Lagrange multiplier.mp3 |
We increase it from $10,000 to 10,001, and we're making more money. So maybe as long as lambda star is greater than one, you should keep doing whatever it takes to increase that budget. So this fact is quite surprising, I think, and it seems like it totally comes out of nowhere. So what I'm gonna do in the next video i... | Meaning of Lagrange multiplier.mp3 |
Some type of substance. It gives us the mass density at any point in three dimensions. And let's say we have another function. This is a scalar function. It just gives us a number for any point in 3D. And then let's say we have another function, v, which is a vector function. It gives us a vector for any point in three... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
This is a scalar function. It just gives us a number for any point in 3D. And then let's say we have another function, v, which is a vector function. It gives us a vector for any point in three dimensions, and this right over here tells us the velocity of that same fluid or gas or whatever we're talking about. Now let'... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
It gives us a vector for any point in three dimensions, and this right over here tells us the velocity of that same fluid or gas or whatever we're talking about. Now let's imagine another function. This might all look a little bit familiar, because we went through a very similar exercise in two dimensions when we talke... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
Now we're just extending it to three dimensions. Let's say we have a function f, and it is equal to the product of rho and v. So for any point in x, y, z, this will give us a vector, and then we'll multiply it times this scalar right over here for that same point in three dimensions. So it's equal to rho times v. Let m... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
Rho times v. And there's a couple of ways you could conceptualize this. You could view this as it obviously maintains the direction of the velocity, but now its magnitude, one way to think about it, is kind of the momentum density. If that doesn't make too much sense, you don't have to worry too much about it. Hopefull... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
Hopefully, as we use these two functions and we think a little bit more about them relative to a surface, it'll make a little bit more conceptual sense. Now, what I want to do is think about what it means given this function f to evaluate the surface integral over some surface. So we're going to evaluate over some surf... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
We're going to evaluate f dot n, where n is the unit normal vector at each point on that surface, dS, d capital S, d surface. So let's think about what this is saying. So first let me draw my axes. So I have my z-axis. This could be my x-axis. And let's say that this right over here is my y-axis. And let's say my surfa... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
So I have my z-axis. This could be my x-axis. And let's say that this right over here is my y-axis. And let's say my surface, I'll use that same color, my surface looks something like that. So that is my surface. That is the surface in question. That is S. Now let's think about the units, and hopefully that'll give us ... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
And let's say my surface, I'll use that same color, my surface looks something like that. So that is my surface. That is the surface in question. That is S. Now let's think about the units, and hopefully that'll give us a conceptual understanding of what this thing right over here is measuring. It's completely analogou... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
That is S. Now let's think about the units, and hopefully that'll give us a conceptual understanding of what this thing right over here is measuring. It's completely analogous to what we did in the two-dimensional case with line integrals. So we have a dS. A dS is a little chunk of area of that surface. So that is dS. ... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
A dS is a little chunk of area of that surface. So that is dS. So this is going to be area. And if we want to pick particular units, this could be square meters. And I think when you do particular units, it starts to make a little bit more concrete sense. Now, the normal vector at that dS, the normal vector is going to... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
And if we want to pick particular units, this could be square meters. And I think when you do particular units, it starts to make a little bit more concrete sense. Now, the normal vector at that dS, the normal vector is going to point right out of it. It's literally normal to that plane. It's literally normal to that p... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
It's literally normal to that plane. It's literally normal to that plane. It has a magnitude 1. So that is our unit normal vector. And F is defined throughout this three-dimensional space. You give me any x, y, z, I'll know its mass density, I'll know its velocity, and I'll get some F. I'll get some F at any point in t... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
So that is our unit normal vector. And F is defined throughout this three-dimensional space. You give me any x, y, z, I'll know its mass density, I'll know its velocity, and I'll get some F. I'll get some F at any point in three-dimensional space, including on the surface, including right over here. So right over here,... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
So right over here, F might look something like this. So that is F right at that point. So what does all this mean? Well, when you take the dot product of two vectors, it's essentially saying how much do they go together. And since N is a unit vector, since it has a magnitude 1, this is essentially saying what is the m... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
Well, when you take the dot product of two vectors, it's essentially saying how much do they go together. And since N is a unit vector, since it has a magnitude 1, this is essentially saying what is the magnitude of the component of F that's going in the direction of N? Or what is the magnitude of the component of F th... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
How much of F is normal to the surface? So the component of F that is normal to the surface might look something like that. And this right over here will essentially just give the magnitude of that. And it's just going to keep the units of F. And right over here just specifies the direction. It has no units associated ... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
And it's just going to keep the units of F. And right over here just specifies the direction. It has no units associated with it. It's dimensionless. F's units are going to be units of mass density. So it could be, let's say, it could be kilogram per meter cubed. Well, that's actually just the rho part. So it's mass de... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
F's units are going to be units of mass density. So it could be, let's say, it could be kilogram per meter cubed. Well, that's actually just the rho part. So it's mass density times velocity times meters per second. Let me write in those colors just so we have clear what's happening here. So the units of F are going to... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
So it's mass density times velocity times meters per second. Let me write in those colors just so we have clear what's happening here. So the units of F are going to be the units of rho, which are going to be kilogram per cubic meter. That's mass density. Times the units of V, which is meters per second. And we're goin... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
That's mass density. Times the units of V, which is meters per second. And we're going to multiply that times meters squared. So what you have is you have a meter and then a meter squared in the numerator. That's meters cubed in the numerator. And meters cubed in the denominator. That cancels out. | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
So what you have is you have a meter and then a meter squared in the numerator. That's meters cubed in the numerator. And meters cubed in the denominator. That cancels out. And so the units that we get for this are kilogram per second. And so the way to conceptualize it, given how we've defined F, what we say F represe... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
That cancels out. And so the units that we get for this are kilogram per second. And so the way to conceptualize it, given how we've defined F, what we say F represents, the way to conceptualize this, this is saying how much mass, how much mass given this mass density, this velocity, is going directly out of this littl... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
And then if we were to add up all the ds's, and this is what essentially that surface integral is, we're essentially saying how much mass in kilograms per second, that's what we picked, how much mass is traveling across this surface at any given moment in time. And this is really the same idea we do with the line integ... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
So this is the flux through a 2D surface. And this isn't like some crazy abstract thing. I mean, you could imagine something like water vapor in your bathroom. Water vapor in your bathroom, and I like to imagine that because that's actually visible, especially when sunlight is shining through it. And we've all seen wat... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
Water vapor in your bathroom, and I like to imagine that because that's actually visible, especially when sunlight is shining through it. And we've all seen water vapor in our bathroom when you have a ray of sunlight, and you can see how the particles are traveling, and you see they have a certain density at different ... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
Maybe you have a window, and so if the surface was the window, and the window, let's say the window is open, so it's kind of a, there's nothing physical there, it's just kind of a rectangular surface that things can pass freely through. If an F was essentially the mass density of the water vapor times the velocity of t... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
I'm conceptualizing it, this is kind of a river, obviously this would be the surface that we normally see, but obviously it has some depth, it's three-dimensional in nature. And so we would know the density, maybe it's constant, you know the density and you know the velocity at any point, that's what F gives us. So tha... | Conceptual understanding of flux in three dimensions Multivariable Calculus Khan Academy.mp3 |
We have our usual setup here for this constrained optimization situation. We have a function we want to maximize, which I'm thinking of as revenues for some company, a constraint, which I'm thinking of as some kind of budget for that company, and as you know if you've gotten to this video, one way to solve this constra... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
And the way to start writing all of that in formulas would be to make explicit the fact that if you consider this value, you know, the $10,000 that is your budget, which I'm calling b, a variable, and not a constant, then you have to acknowledge that h star and s star are dependent on b, right? It's a very implicit rel... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So that's kind of a mouthful, it takes a lot just to even phrase what's going on, but in the context of an economic example, it has a very clear, precise meaning, which is if you increase your budget by a dollar, right, if you increase it from $10,000 to $10,001, you're wondering for that tiny change in budget, that ti... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
This just seems like it comes out of nowhere. Well, there are a couple clever observations that go into proving this. The first, the first is to notice what happens if we evaluate this Lagrangian function itself at this critical point, when you input h star, s star, and lambda star. And remember, the way that these guy... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
And remember, the way that these guys are defined is that you look at all of the values where the gradient of the Lagrangian equals the zero vector, and then if you get multiple options, you know, sometimes when you set the gradient equal to zero, you get multiple solutions, and whichever one maximizes r, that is h sta... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
And we subtract off lambda star times b of h star, s star, minus the constant that is your budget, you know, something you might think of as $10,000, whatever you set little b equal to. Okay, Grant, you might say, why does this tell us anything? You're just plugging in stars instead of the usual variables. But the key ... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
But the key is that if you plug in h star and s star, this value has to equal zero because h star and s star have to satisfy the constraint. Remember, one of the cool parts about this Lagrangian function as a whole is that when you take its partial derivative with respect to lambda, all that's left is this constraint f... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
And if you remember from the Lagrangian video, all that really boils down to is the fact that the constraint holds, right? Which would be your budget achieves $10,000. When you plug in the appropriate h star and s star to this value, you're hitting this constrained amount of money that you can spend. So by virtue of ho... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So by virtue of how h star and s star are defined, the fact that they are solutions to the constrained optimization problem means this whole portion goes to zero. So we can just kind of cancel all that out. And what's left, what's left here is the maximum possible revenue, right? So evidently, when you evaluate the Lag... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So evidently, when you evaluate the Lagrangian at this critical point, at h star, s star, and lambda star, it equals m star. It equals the maximum possible value for the function you're trying to maximize. So ultimately what we want is to understand how that maximum value changes when you consider it a function of the ... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So evidently, what we can look for is to just ask how the Lagrangian changes as you consider it a function of the budget. Now this is an interesting thing to observe, because if we just look up at the definition of the Lagrangian, if you just look at this formula, if I told you to take the derivative of this with respe... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
You would notice that this goes to zero, it doesn't have a little b. This would also go to zero. And all you'd be left with would be negative lambda times negative b, and the derivative of that with respect to b would be lambda. So you might say, oh yeah, of course, of course, this, the derivative of that Lagrangian wi... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So you might say, oh yeah, of course, of course, this, the derivative of that Lagrangian with respect to b, once we work it all out, the only term that was left there was the lambda. And that's compelling, but ultimately, it's not entirely right. That overlooks the fact that L is not actually defined as a function of b... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
When we defined the Lagrangian, we were considering b to be a constant. So if you really want to consider this to be a function that involves b, the way we should write it, and I'll go ahead and erase this guy, the way we should write this Lagrangian is to say, you're a function of h star, which itself is dependent on ... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
And then we can consider it as a fourth variable, so we're adding on yet another variable to this function, the value of b itself. The value of b itself here. So now, when we want to know what is the value of the Lagrangian at the critical point, h star, s star, lambda star, as a function of b, so that can be kind of c... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
It only depends on b, but it kind of goes through a four variable function. And so just to make it explicit, this would equal the value of r as a function of h star and s star, and each one of those is a function of little b, right? So this term is saying what's your revenue evaluated on the maximizing h and s for the ... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So what you subtract off minus lambda star at b of h star and s star, but each of these guys is also a function of little b, little b, minus little b, right? So you have this large, kind of complicated multivariable function, it's defined in terms of h stars and s stars, which are themselves very implicit, right? We ju... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
And from here, if we want to evaluate the derivative of L, we want to evaluate the derivative of this Lagrangian with respect to little b, which is really the only thing it depends on, it's just via all of these other variables, we use the multivariable chain rule. And at this point, if you don't know the multivariable... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So we'll start with the partial derivative of L with respect to h star, with respect to h star, and we're going to multiply that by the derivative of h star with respect to b. And this might seem like a very hard thing to think about, like how do we know how h star changes as b changes? But don't worry about it, you'll... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
And then we add in partial derivative of L with respect to that second variable, s star, with respect to whatever the second variable is, multiplied by the derivative of s star with respect to b. You can see how you really need to know what the multivariable chain rule is, right? This would all seem kind of out of the ... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So what we now add in is the partial derivative of L with respect to that lambda star, with respect to lambda star, multiplied by the derivative of lambda star with respect to little b. And then finally, finally, we take the partial derivative of this Lagrangian with respect to that little b, which we're now considerin... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So now, if you're thinking that this is going to be horrifying to compute, I can understand where you're coming from. You have to know the derivative of lambda star with respect to b, you have to somehow intimately be familiar with how this lambda star changes as you change b, and like I said, that's such an implicit r... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So somehow you're supposed to know how that changes when you slightly alter b over here. Well, you don't really have to worry about things, because by definition h star, s star, and lambda star are whatever values make the gradient of L equal to 0. But if you think about that, what does it mean for the gradient of L to... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
Well, what it means is that when you take its derivative with respect to that first variable, h star, it equals 0. When you take its derivative with respect to the second variable, that equals 0 as well. And with respect to this third variable, that's going to equal 0. By definition, h star, s star, and lambda star are... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
By definition, h star, s star, and lambda star are whatever values make it the case so that when you plug them in, the partial derivative of the Lagrangian with respect to any one of those variables equals 0. So we don't even have to worry about most of this equation. The only part that matters here is the partial deri... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
What is the rate of change of a variable with respect to itself? It's 1. It is 1. So all of this stuff, this entire multivariable chain rule boils down to a single innocent-looking factor which is the partial derivative of L partial derivative of L with respect to little b. And now, there's something very subtle here, ... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
So all of this stuff, this entire multivariable chain rule boils down to a single innocent-looking factor which is the partial derivative of L partial derivative of L with respect to little b. And now, there's something very subtle here, right? Because this might seem obvious. I'm saying the derivative of L with respec... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
I'm saying the derivative of L with respect to b equals the derivative of L with respect to b. But maybe I should give a different notation here, right? Because here when I'm taking the derivative, really I'm considering L as a single variable function, right? I'm considering not what happens as you can freely change a... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
I'm considering not what happens as you can freely change all four of these variables. Three of them are locked into place by b. So maybe I should really give that a different name. I should call that L star. L star is a single variable function. Whereas this L is a multivariable function. This is the function where yo... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
I should call that L star. L star is a single variable function. Whereas this L is a multivariable function. This is the function where you can freely change the values of h and s and lambda and b as you put them in. So if we kind of scroll up to look at its definition, which I've written all over I guess here, let me ... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
This is the function where you can freely change the values of h and s and lambda and b as you put them in. So if we kind of scroll up to look at its definition, which I've written all over I guess here, let me actually rewrite its definition, right? I think that'll be useful. I'm going to rewrite that L, if I consider... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
I'm going to rewrite that L, if I consider it as a four variable function of h s lambda and b that what that equals is r evaluated at h and s minus lambda multiplied by this constraint function b evaluated at h and s minus little b. And this is now when I'm considering little b to be a variable. So this is the Lagrangi... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
Whereas the thing up here that I'm considering a single variable function has three of its inputs locked into place. So effectively it's just a single variable function with respect to b. So it's actually quite miraculous that the single variable derivative of that L here, I should L star with respect to b ends up bein... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
Usually in any usual circumstance all of these other terms would have come into play somehow. But what's special here is that by the definition of this L star the specific way in which these h star, s star, and lambda stars are locked into place happens to be one in which all of these partial derivatives go to zero. So... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
Well this R has no b's in it so don't need to care about that. This term over here it's partial derivative is negative one, right, just because there's a b here and that's multiplied by the constant lambda so that all just equals lambda. But if we're in the situation where lambda is locked into place as a function of l... | Proof for the meaning of Lagrange multipliers Multivariable Calculus Khan Academy.mp3 |
Hey everyone. So in the last video I was talking about divergence and kind of laying down the intuition that we need for it. Where you're imagining a vector field as representing some kind of fluid flow where particles move according to the vector that they're attached to in that point in time. And as they move to a di... | Divergence intuition, part 2.mp3 |
And as they move to a different point, the vector they're attached to is different, so their velocity changes in some way. And the key question that we want to think about is if you have a given point somewhere in space, does fluid tend to flow towards that point or does it tend to flow more away from it? Does it diver... | Divergence intuition, part 2.mp3 |
And what I want to do here is start kind of closing our grasp on that intuition a little bit more tightly as if we are trying to discover the formula for divergence ourselves. Because ultimately that's what I'm gonna get to, a formula for divergence. But I want it to be something that's not just plopped down in front o... | Divergence intuition, part 2.mp3 |
So a vector field like the one I have pictured above is given as a function, a multivariable function, with a two-dimensional input, since it's a two-dimensional vector field, and then some kind of two-dimensional output. And it's common to, whoop, it's common to write p and q as the functions for the components of the... | Divergence intuition, part 2.mp3 |
And the divergence is kind of like a derivative, where you might denote it by just div, and in the same way that your derivative, you have this operator, and what it does is it takes in a function, and what you get is a whole new function. This div operator, you think of as taking in a vector field of some kind, and yo... | Divergence intuition, part 2.mp3 |
It'll be something that just takes in points in space and outputs a number. Because what you're thinking, the thing that it's trying to do is take in some specific point with x, y coordinates and just give you a single number to tell you, hey, does fluid tend to diverge away from it? How much? Or does it tend to flow t... | Divergence intuition, part 2.mp3 |
Or does it tend to flow towards it, and how much? So this is the kind of the form of the thing that we're going for. So here what we're gonna do is just start thinking about cases where this divergence is positive or negative or zero and what that should look like. So for example, let's say we want cases where the dive... | Divergence intuition, part 2.mp3 |
So for example, let's say we want cases where the divergence of our vector field at a specific point, at a specific point x, y, is positive. What might that look like? So one case would be where your point, nothing is happening at that point, the vector attached to it is zero, and everyone around it is going away. This... | Divergence intuition, part 2.mp3 |
This is kind of the extreme example of positive divergence. And I animated this in the last video where we have all of the vectors pointing away from the origin, and if you look at a region around that origin, all the fluid particles kind of go out of that region. And that's the quintessential positive divergence examp... | Divergence intuition, part 2.mp3 |
But it doesn't have to look like that. It actually, I mean, you could have something where there is a little bit of movement at your point, and then maybe there's movement towards it as well from one side, and vectors are kind of going towards it, but they're going away from it even more rapidly on the other side. So i... | Divergence intuition, part 2.mp3 |
So these are the kind of situations you might see for positive divergence. Now negative divergence, negative divergence, let's think about what examples of that might look like. Divergence of V at a given point, and really it's something that takes in all points of the plane, but we're just looking at specific points. ... | Divergence intuition, part 2.mp3 |
So if the divergence is negative, well, the quintessential example here is that nothing happens at your point, but all of the vectors around it are kind of flowing in towards it. And this is the thing where I animated where we took this and we flipped all of the vectors and said, ah, there, if you start playing the flu... | Divergence intuition, part 2.mp3 |
But again, this isn't the only example that you might have. You could have a little bit of activity at your point itself, and maybe it is the case that things do flow away from it a little bit as you're going away. And some of the fluid particles are going away, and it's just the case that the fluid particles flowing i... | Divergence intuition, part 2.mp3 |
Because then if you're looking at any kind of region around your point, you say fluid particles are coming in quite rapidly, a lot of particles per time, but they're not leaving too rapidly around the other end. So kind of loosely, intuitively, this is what a negative divergence case might look like. And finally, anoth... | Divergence intuition, part 2.mp3 |
Right, if it's just absolutely zero. And one thing this could look like is, you know, you have something going on, but nothing really changes, and all of the fluid just kind of flows in and it flows out, and on the whole it balances. You know, if you take any kind of region, the amount flowing in is balanced with the a... | Divergence intuition, part 2.mp3 |
So somehow our whole function, our function takes things from this two-dimensional space and plugs it onto this output. T, you're thinking of as just another number line up here, so t, and then you've got two separate functions here, you know, x of t and y of t, x of t and y of t, each of which take that same value for... | Multivariable chain rule intuition.mp3 |
And in this way, you're thinking of it as just a single variable function that goes from t and ultimately outputs f, it's just that there's multidimensional stuff happening in between. And now if we start thinking about the derivative of it, what does that mean? What does that mean for the conception of the picture tha... | Multivariable chain rule intuition.mp3 |
Well, that bottom part, that dt, you're thinking of as a tiny change to t, right? So you're thinking of it as kind of a nudge. I'll draw it as a sizable line here for like moving from some original input over, but you might in principle think of it as a very, very tiny nudge in t. And over here, you'd say, well, that's... | Multivariable chain rule intuition.mp3 |
Again, imagine this is a very small nudge, I'm gonna give it some size here just so I can write into it. And then whatever that nudge in the output space, and it's a nudge in some direction, that's gonna correspond to some change in f, some change based on, you know, based on the differential properties of the multivar... | Multivariable chain rule intuition.mp3 |
But you can actually reason about what these should be because it's not just an arbitrary change in x or an arbitrary change in y, it's the one that was caused by dt. So if I go over here, I might say that dx is caused by that dt, and the whole meaning of the derivative, the whole meaning of the single variable derivat... | Multivariable chain rule intuition.mp3 |
But really, you're saying there's a tiny nudge in t and that results in a change in x, and this derivative is what tells you the ratio between those sizes. And similarly, similarly, that change in y here, that change in y is gonna be somehow proportional to the change in t, and that proportion is given by the derivativ... | Multivariable chain rule intuition.mp3 |
I'll describe this in a much more formal way that's a little bit more airtight than the kind of hand-waving, nudging around, but the intuition you get from just writing this as a fraction is basically the scaffolding for that formal argument, so it's a fine thing to do. I don't think mathematicians are shaking their he... | Multivariable chain rule intuition.mp3 |
You could kind of think of it like this partial x is canceling out with that dx if you wanted, or you could just say, this is a tiny nudge in x, this is gonna result in some change in f, I'm not sure what, but the meaning of the derivative is the ratio between those two, and that's what lets you figure it out. And simi... | Multivariable chain rule intuition.mp3 |
So let's say you have yourself some kind of multivariable function, and this time let's say it's got some very high dimensional input. So x1, x2, on and on and on and on, up to x sub n for some large number n. In the last couple videos I told you about the Laplacian operator, which is a way of taking in your scalar val... | Explicit Laplacian formula.mp3 |
And what I want to show you here is another formula that you might commonly see for this Laplacian. So first let's kind of abstractly write out what the gradient of f will look like. So we start by taking this del operator, which is gonna be a vector full of partial differential operators, partial with respect to x1, p... | Explicit Laplacian formula.mp3 |
You take that whole thing, and then you just kind of imagine multiplying it by your function. So what you end up getting is all the different partial derivatives of f, right? It's partial of f with respect to the first variable, and then kind of on and on and on, up until you get the partial derivative of f with respec... | Explicit Laplacian formula.mp3 |
What you end up getting is, well you start by multiplying the first components, which involves taking the partial derivative with respect to x1, that first variable, of the partial derivative of f with respect to that same variable. So it looks like the second partial derivative of f with respect to that first variable... | Explicit Laplacian formula.mp3 |
And then you imagine kind of adding what the product of these next two items will be, and for very similar reasons, that's gonna look like the second partial derivative of f with respect to that second variable, partial x2 squared. And you do this to all of them, and you're adding them all up until you find yourself do... | Explicit Laplacian formula.mp3 |
So people will say the Laplacian of your function f is equal to, and then using sigma notation, you'd say the sum from i is equal to one up to, you know, one, two, three, up to n. So the sum from that up to n of your second partial derivatives, partial squared of f with respect to that ith variable. So, you know, if yo... | Explicit Laplacian formula.mp3 |
Personally, I always like to think about it as taking the divergence of the gradient of f, because you're thinking about the gradient field and the divergence of that kind of corresponds to maxima and minima of your original function, which is what I talked about in the initial intuition of Laplacian video. But this fo... | Explicit Laplacian formula.mp3 |
We can now express this as a double integral over the domain of the parameters that we care about. And we're going to do that in this video. And then in the next series of videos, we'll do the same thing for this expression. We're actually going to do that using Green's Theorem. What we're going to do is we're going to... | Stokes' theorem proof part 3 Multivariable Calculus Khan Academy.mp3 |
We're actually going to do that using Green's Theorem. What we're going to do is we're going to see we get the same expressions, which will show us that Stokes' Theorem is true, at least for this special class of surfaces that we are studying right here. But they're pretty general. Now let's now try to do that. So our ... | Stokes' theorem proof part 3 Multivariable Calculus Khan Academy.mp3 |
Now let's now try to do that. So our surface integral, I'm just going to rewrite it down here. It's the surface integral over our surface of the curl of f. Let me go a little bit lower. So we have our surface integral of the curl of f dot ds. Well, we've already figured out what our curl of f is here two videos ago. An... | Stokes' theorem proof part 3 Multivariable Calculus Khan Academy.mp3 |
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