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And then we're multiplying that times du dv. Now du and dv are just scalar quantities. They're infinitesimally small, but for the sake of this argument, you can just view that they're not vectors. They're just scalar quantities. And so you can essentially include them. If you have the cross product, if you have a cross b times some scalar value, x, you could rewrite this as x times a cross b. Or you could write this as a cross x times b because x is just a scalar value.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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They're just scalar quantities. And so you can essentially include them. If you have the cross product, if you have a cross b times some scalar value, x, you could rewrite this as x times a cross b. Or you could write this as a cross x times b because x is just a scalar value. It's just a number. So we could do the same thing over here. We can rewrite all of this business as, and I'm going to group the du where we have the partial with respect to u in the denominator, and I'll do the same thing with the v's.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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Or you could write this as a cross x times b because x is just a scalar value. It's just a number. So we could do the same thing over here. We can rewrite all of this business as, and I'm going to group the du where we have the partial with respect to u in the denominator, and I'll do the same thing with the v's. And so you'll get the partial of r with respect to u times du times that scalar. So that'll give us a vector. And we're going to cross that with the partial of r with respect to v dv.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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We can rewrite all of this business as, and I'm going to group the du where we have the partial with respect to u in the denominator, and I'll do the same thing with the v's. And so you'll get the partial of r with respect to u times du times that scalar. So that'll give us a vector. And we're going to cross that with the partial of r with respect to v dv. Now, these might look notationally like two different things, but that just comes from the necessity of when we take partial derivative to say, oh, no, this vector function is defined, it's a function of multiple variables, and this is taking a derivative with respect to only one of them. So this is how much does our vector change when you have a very small change in u. But this is also an infinitesimally small change in u over here.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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And we're going to cross that with the partial of r with respect to v dv. Now, these might look notationally like two different things, but that just comes from the necessity of when we take partial derivative to say, oh, no, this vector function is defined, it's a function of multiple variables, and this is taking a derivative with respect to only one of them. So this is how much does our vector change when you have a very small change in u. But this is also an infinitesimally small change in u over here. We're just using slightly different notation. So for the sake of this, this is a little bit loosey-goosey mathematics, but it'll hopefully give you the intuition for why this thing can be written in a different way. These are essentially the same quantity.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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But this is also an infinitesimally small change in u over here. We're just using slightly different notation. So for the sake of this, this is a little bit loosey-goosey mathematics, but it'll hopefully give you the intuition for why this thing can be written in a different way. These are essentially the same quantity. So if you divide by something and multiply by something, you can cancel them out. If you divide by something and multiply by something, you can cancel them out. And all you're left with then is the differential of r, and since we lost the information that it's in the u direction, I'll write it here, the differential of r in the u direction.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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These are essentially the same quantity. So if you divide by something and multiply by something, you can cancel them out. If you divide by something and multiply by something, you can cancel them out. And all you're left with then is the differential of r, and since we lost the information that it's in the u direction, I'll write it here, the differential of r in the u direction. I don't want to get the notation confused. This is just the differential. This is just how much r changed.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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And all you're left with then is the differential of r, and since we lost the information that it's in the u direction, I'll write it here, the differential of r in the u direction. I don't want to get the notation confused. This is just the differential. This is just how much r changed. This is not the partial derivative of r with respect to u. This right over here is how much does r change given per unit change, per small change in u. This just says a differential in the direction of, I guess, as u changes, this is how much that infinitely small change that just r changes.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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This is just how much r changed. This is not the partial derivative of r with respect to u. This right over here is how much does r change given per unit change, per small change in u. This just says a differential in the direction of, I guess, as u changes, this is how much that infinitely small change that just r changes. This isn't change in r with respect to change in u. We're going to cross that with the partial of r in the v direction. Now this right over here, let's just conceptualize this.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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This just says a differential in the direction of, I guess, as u changes, this is how much that infinitely small change that just r changes. This isn't change in r with respect to change in u. We're going to cross that with the partial of r in the v direction. Now this right over here, let's just conceptualize this. This goes back to our original visions of what a surface integral was all about. If we're on a surface, and I'll draw a surface, right? Let me draw another surface.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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Now this right over here, let's just conceptualize this. This goes back to our original visions of what a surface integral was all about. If we're on a surface, and I'll draw a surface, right? Let me draw another surface. I won't use the one that I had already drawn on. If we draw a surface, and for a very small change in u, and we're not going to think about the rate. We're thinking about the change in r. You're going in that direction.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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Let me draw another surface. I won't use the one that I had already drawn on. If we draw a surface, and for a very small change in u, and we're not going to think about the rate. We're thinking about the change in r. You're going in that direction. If that thing looks like this, this is actually a distance moved on the surface. Because it's not, remember, this isn't the derivative. This is the differential.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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We're thinking about the change in r. You're going in that direction. If that thing looks like this, this is actually a distance moved on the surface. Because it's not, remember, this isn't the derivative. This is the differential. It's just a small change along the surface. That's that over there. This is a small change when you change v. It's also a change along the surface.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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This is the differential. It's just a small change along the surface. That's that over there. This is a small change when you change v. It's also a change along the surface. If you take the cross product of these two things, you get a vector that is orthogonal. You get a vector that is normal to the surface. Its magnitude, and we saw this when we first learned about cross products, its magnitude is equal to the area that is defined by these two vectors.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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This is a small change when you change v. It's also a change along the surface. If you take the cross product of these two things, you get a vector that is orthogonal. You get a vector that is normal to the surface. Its magnitude, and we saw this when we first learned about cross products, its magnitude is equal to the area that is defined by these two vectors. Its magnitude is equal to area. In a lot of ways, you can really think of it as a unit normal vector times ds. The way that we would, I guess, notationally do this is we can call this, because this is kind of a ds, but it's a vector version of the ds.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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Its magnitude, and we saw this when we first learned about cross products, its magnitude is equal to the area that is defined by these two vectors. Its magnitude is equal to area. In a lot of ways, you can really think of it as a unit normal vector times ds. The way that we would, I guess, notationally do this is we can call this, because this is kind of a ds, but it's a vector version of the ds. Over here, this is just an area right over here. This is just a scalar value. Now, we have a vector that points normally from the surface, but its magnitude is the same thing as that ds that we were just talking about.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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The way that we would, I guess, notationally do this is we can call this, because this is kind of a ds, but it's a vector version of the ds. Over here, this is just an area right over here. This is just a scalar value. Now, we have a vector that points normally from the surface, but its magnitude is the same thing as that ds that we were just talking about. We can call this thing right over here, we can call this ds. The key difference here is this is a vector now. We'll call it ds with a little vector over it to know that this is this thing.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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Now, we have a vector that points normally from the surface, but its magnitude is the same thing as that ds that we were just talking about. We can call this thing right over here, we can call this ds. The key difference here is this is a vector now. We'll call it ds with a little vector over it to know that this is this thing. This isn't the scalar ds that is just concerned with the area. When you view things this way, we just saw that this entire thing simplifies to ds. Then our whole surface integral can be rewritten.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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We'll call it ds with a little vector over it to know that this is this thing. This isn't the scalar ds that is just concerned with the area. When you view things this way, we just saw that this entire thing simplifies to ds. Then our whole surface integral can be rewritten. Instead of writing it like this, we can write it as the integral, or the surface integral, those integral signs were too fancy, the surface integral of f dot, instead of saying a normal vector times a scalar quantity, a little chunk of area on the surface, we can now just call that the vector differential ds. I want to make it clear, these are two different things. This is a vector.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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Then our whole surface integral can be rewritten. Instead of writing it like this, we can write it as the integral, or the surface integral, those integral signs were too fancy, the surface integral of f dot, instead of saying a normal vector times a scalar quantity, a little chunk of area on the surface, we can now just call that the vector differential ds. I want to make it clear, these are two different things. This is a vector. This is essentially what we're calling it. This right over here is a scalar times a normal vector. These are three different ways of really representing the same thing.
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Vector representation of a surface integral Multivariable Calculus Khan Academy.mp3
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So in the last couple videos, I talked about the multivariable chain rule, which I have written up here, and if you haven't seen those, go take a look. And here I want to write it out in vector notation, and this helps us generalize it a little bit when the intermediary space is a little bit higher dimensional. So instead of writing x of t and y of t as separate functions and just trying to emphasize, oh, they have the same input space, and whatever x takes in, that's the same number y takes in, it's better and a little bit cleaner if we say there's a vector-valued function that takes in a single number t, and then it outputs some kind of vector. In this case, you could say the components of v are x of t and y of t, and that's fine. But I want to talk about what this looks like if we start writing everything in vector notation. And just since we see dx dt here and dy dt here, you might start thinking, oh, well, we should take the derivative of that vector-valued function. The derivative of v with respect to t, and when we compute this, it's nothing more than just taking the derivatives of each component.
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Vector form of the multivariable chain rule.mp3
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In this case, you could say the components of v are x of t and y of t, and that's fine. But I want to talk about what this looks like if we start writing everything in vector notation. And just since we see dx dt here and dy dt here, you might start thinking, oh, well, we should take the derivative of that vector-valued function. The derivative of v with respect to t, and when we compute this, it's nothing more than just taking the derivatives of each component. So in this case, the derivative of x, so you'd write dx dt, and the derivative of y, dy dt. This is the vector-valued derivative. And now you might start to notice something here.
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Vector form of the multivariable chain rule.mp3
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The derivative of v with respect to t, and when we compute this, it's nothing more than just taking the derivatives of each component. So in this case, the derivative of x, so you'd write dx dt, and the derivative of y, dy dt. This is the vector-valued derivative. And now you might start to notice something here. Okay, so we've got one of those components multiplied by a certain value, and another component multiplied by a certain value. You might recognize this as a dot product. This would be the dot product between the vector that contains the derivatives, the partial derivatives, partial of f with respect to y, partial of f with respect to x. Oh, whoops, don't know why I wrote it that way.
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Vector form of the multivariable chain rule.mp3
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And now you might start to notice something here. Okay, so we've got one of those components multiplied by a certain value, and another component multiplied by a certain value. You might recognize this as a dot product. This would be the dot product between the vector that contains the derivatives, the partial derivatives, partial of f with respect to y, partial of f with respect to x. Oh, whoops, don't know why I wrote it that way. So up here, that's with respect to x, and then here to y. So this whole thing, we're taking the dot product with the vector that contains ordinary derivative, dx dt, and ordinary derivative, dy dt. And of course, both of these are special vectors.
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Vector form of the multivariable chain rule.mp3
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This would be the dot product between the vector that contains the derivatives, the partial derivatives, partial of f with respect to y, partial of f with respect to x. Oh, whoops, don't know why I wrote it that way. So up here, that's with respect to x, and then here to y. So this whole thing, we're taking the dot product with the vector that contains ordinary derivative, dx dt, and ordinary derivative, dy dt. And of course, both of these are special vectors. They're not just random. The left one, that's the gradient of f, gradient of f. And the right vector here, that's what we just wrote. That's the derivative of v with respect to t. Just for being quick, I'm gonna write that as v prime of t. That's saying completely the same thing as dv dt.
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Vector form of the multivariable chain rule.mp3
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And of course, both of these are special vectors. They're not just random. The left one, that's the gradient of f, gradient of f. And the right vector here, that's what we just wrote. That's the derivative of v with respect to t. Just for being quick, I'm gonna write that as v prime of t. That's saying completely the same thing as dv dt. And this right here is another way to write the multivariable chain rule. And maybe if you were being a little bit more exact, you would emphasize that when you take the gradient of f, the thing that you input into it is the output of that vector-valued function. You know, you're throwing in x of t and y of t. So you might emphasize that you take in that as an input, and then you multiply it by the derivative, the vector-valued derivative of v of t. And when I say multiply, I mean dot product.
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Vector form of the multivariable chain rule.mp3
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That's the derivative of v with respect to t. Just for being quick, I'm gonna write that as v prime of t. That's saying completely the same thing as dv dt. And this right here is another way to write the multivariable chain rule. And maybe if you were being a little bit more exact, you would emphasize that when you take the gradient of f, the thing that you input into it is the output of that vector-valued function. You know, you're throwing in x of t and y of t. So you might emphasize that you take in that as an input, and then you multiply it by the derivative, the vector-valued derivative of v of t. And when I say multiply, I mean dot product. These are vectors, and you're taking the dot product. And this should seem very familiar to the single-variable chain rule. And just to remind us, I'll throw it up here.
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Vector form of the multivariable chain rule.mp3
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You know, you're throwing in x of t and y of t. So you might emphasize that you take in that as an input, and then you multiply it by the derivative, the vector-valued derivative of v of t. And when I say multiply, I mean dot product. These are vectors, and you're taking the dot product. And this should seem very familiar to the single-variable chain rule. And just to remind us, I'll throw it up here. If you take the derivative of, you know, composition of two single-variable functions, f and g, you take the derivative of the outside, f prime, and throw in g, throw in what was the interior function, and you multiply it by the derivative of that interior function, g prime of t. And this is super helpful in single-variable calculus for computing a lot of derivatives. And over here, it has a very similar form, right? The gradient, which really serves the function of the true extension of the derivative for multivariable functions, for scalar-valued multivariable functions at least.
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Vector form of the multivariable chain rule.mp3
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And just to remind us, I'll throw it up here. If you take the derivative of, you know, composition of two single-variable functions, f and g, you take the derivative of the outside, f prime, and throw in g, throw in what was the interior function, and you multiply it by the derivative of that interior function, g prime of t. And this is super helpful in single-variable calculus for computing a lot of derivatives. And over here, it has a very similar form, right? The gradient, which really serves the function of the true extension of the derivative for multivariable functions, for scalar-valued multivariable functions at least. You take that derivative and throw in the inner function, which just happens to be a vector-valued function, but you throw it in there, and then you multiply it by the derivative of that. But multiplying vectors in this context means taking the dot product of the two. And this could mean, if you have a function with a whole bunch of different variables, so let's say you have, you know, some f of x, or not f of x, but f of, like, x1 and x2, and it takes in a whole bunch of different variables, and it goes out to x100.
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Vector form of the multivariable chain rule.mp3
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The gradient, which really serves the function of the true extension of the derivative for multivariable functions, for scalar-valued multivariable functions at least. You take that derivative and throw in the inner function, which just happens to be a vector-valued function, but you throw it in there, and then you multiply it by the derivative of that. But multiplying vectors in this context means taking the dot product of the two. And this could mean, if you have a function with a whole bunch of different variables, so let's say you have, you know, some f of x, or not f of x, but f of, like, x1 and x2, and it takes in a whole bunch of different variables, and it goes out to x100. And then what you throw into it is a vector-valued function that's something that's vector-valued, takes in a single variable, and, you know, in order to be able to compose them, it's gonna have a whole bunch of intermediary functions. And I'll, you know, you could write it as x1, x2, x3, all the way up to x100. And these are all functions at this point.
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Vector form of the multivariable chain rule.mp3
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And this could mean, if you have a function with a whole bunch of different variables, so let's say you have, you know, some f of x, or not f of x, but f of, like, x1 and x2, and it takes in a whole bunch of different variables, and it goes out to x100. And then what you throw into it is a vector-valued function that's something that's vector-valued, takes in a single variable, and, you know, in order to be able to compose them, it's gonna have a whole bunch of intermediary functions. And I'll, you know, you could write it as x1, x2, x3, all the way up to x100. And these are all functions at this point. These are component functions of your vector-valued v. This expression still makes sense, right? You can still take the gradient of f. It's gonna have 100 components. You can plug in any vector, any set of 100 different numbers, and in particular, the output of a vector-valued function with 100 different components is gonna work.
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Vector form of the multivariable chain rule.mp3
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And these are all functions at this point. These are component functions of your vector-valued v. This expression still makes sense, right? You can still take the gradient of f. It's gonna have 100 components. You can plug in any vector, any set of 100 different numbers, and in particular, the output of a vector-valued function with 100 different components is gonna work. And then you take the dot product with the derivative of this. And that's the more general version of the multivariable chain rule. And another cool way about writing it like this is you can interpret it in terms of the directional derivative.
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Vector form of the multivariable chain rule.mp3
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You can plug in any vector, any set of 100 different numbers, and in particular, the output of a vector-valued function with 100 different components is gonna work. And then you take the dot product with the derivative of this. And that's the more general version of the multivariable chain rule. And another cool way about writing it like this is you can interpret it in terms of the directional derivative. And I think I'll do that in the next video. I won't do that here. So a certain way to interpret this with the directional derivative.
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Vector form of the multivariable chain rule.mp3
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So I'm gonna go ahead and go through a specific example here. And just to remind you of kind of what all these terms are, how there's actually kind of a pattern to what's going on, this here represents, you could think of it as the constant term, where this is just gonna evaluate to some kind of number. These two terms are what you might call the linear term. Linear. Because if you actually look, the only places where the variable x and y comes up is here, where it's just being multiplied by a constant, and here, where it's just being multiplied by a constant. So it's just variables times constants in there. And then all of this stuff at the end, which is kind of the whole essence of a quadratic approximation, where you start to have things, like you get an x squared, and you get like x gets to be multiplied by y.
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Quadratic approximation example.mp3
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Linear. Because if you actually look, the only places where the variable x and y comes up is here, where it's just being multiplied by a constant, and here, where it's just being multiplied by a constant. So it's just variables times constants in there. And then all of this stuff at the end, which is kind of the whole essence of a quadratic approximation, where you start to have things, like you get an x squared, and you get like x gets to be multiplied by y. All of this stuff is the quadratic term. And though it seems like a lot now, you'll see in the context of an actual example, it's not necessarily as bad as it seems. So let's say we're looking at the function f of x, y, and let's say it's gonna be e to the x divided by two multiplied by sine of y.
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Quadratic approximation example.mp3
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And then all of this stuff at the end, which is kind of the whole essence of a quadratic approximation, where you start to have things, like you get an x squared, and you get like x gets to be multiplied by y. All of this stuff is the quadratic term. And though it seems like a lot now, you'll see in the context of an actual example, it's not necessarily as bad as it seems. So let's say we're looking at the function f of x, y, and let's say it's gonna be e to the x divided by two multiplied by sine of y. This is our multivariable function. And let's say we want to approximate this near some kind of point. And I'm gonna choose a point that's, you know, something that we can actually evaluate these at.
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Quadratic approximation example.mp3
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So let's say we're looking at the function f of x, y, and let's say it's gonna be e to the x divided by two multiplied by sine of y. This is our multivariable function. And let's say we want to approximate this near some kind of point. And I'm gonna choose a point that's, you know, something that we can actually evaluate these at. So like x, it would be convenient if that was zero. And then y, I'll go with pi halves, because that's something where I'll know how to evaluate sine, and where I'll know how to evaluate its derivatives, things like that. So we're trying to approximate this function near this point.
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Quadratic approximation example.mp3
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And I'm gonna choose a point that's, you know, something that we can actually evaluate these at. So like x, it would be convenient if that was zero. And then y, I'll go with pi halves, because that's something where I'll know how to evaluate sine, and where I'll know how to evaluate its derivatives, things like that. So we're trying to approximate this function near this point. Now, first things first, we're just gonna need to get all of the different partial derivatives and second partial derivatives. We know we're gonna need them, so let's just kind of start working it through and figuring out what all of them are. So, let's start with the partial derivative with respect to x.
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Quadratic approximation example.mp3
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So we're trying to approximate this function near this point. Now, first things first, we're just gonna need to get all of the different partial derivatives and second partial derivatives. We know we're gonna need them, so let's just kind of start working it through and figuring out what all of them are. So, let's start with the partial derivative with respect to x. So this is also a function of x, y. And we look up at the original function, the only place where x shows up is in this e to the x over two. The derivative of that is 1 1 2, we bring down that 1 1 2 times e to the x over two.
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Quadratic approximation example.mp3
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So, let's start with the partial derivative with respect to x. So this is also a function of x, y. And we look up at the original function, the only place where x shows up is in this e to the x over two. The derivative of that is 1 1 2, we bring down that 1 1 2 times e to the x over two. And this is being multiplied by something that looks like a constant, as far as x is concerned, sine of y. Now when we do the partial derivative with respect to y, what we get, this first part just looks like a constant, so we kind of keep that constant there, as far as y is concerned. And the derivative of sine is cosine.
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Quadratic approximation example.mp3
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The derivative of that is 1 1 2, we bring down that 1 1 2 times e to the x over two. And this is being multiplied by something that looks like a constant, as far as x is concerned, sine of y. Now when we do the partial derivative with respect to y, what we get, this first part just looks like a constant, so we kind of keep that constant there, as far as y is concerned. And the derivative of sine is cosine. Cosine of y. And then now, let's start taking second partial derivatives. So I'll start by doing the one where we take the partial derivative with respect to x twice.
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Quadratic approximation example.mp3
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And the derivative of sine is cosine. Cosine of y. And then now, let's start taking second partial derivatives. So I'll start by doing the one where we take the partial derivative with respect to x twice. Now here, I'll actually do this, I'll do this in a different color. How about, let's do like yellow. Just to make clear which ones are the second partial derivatives.
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Quadratic approximation example.mp3
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So I'll start by doing the one where we take the partial derivative with respect to x twice. Now here, I'll actually do this, I'll do this in a different color. How about, let's do like yellow. Just to make clear which ones are the second partial derivatives. So, partial with respect to x twice, also a function of x, y, like all of these guys. And so let's look up at the original partial derivative with respect to x, and we're now gonna take its derivative, again with respect to x. This is the only place where x shows up, that 1 1 2 kind of comes down again, so now it's gonna be 1 1 4 times e to the x over two.
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Quadratic approximation example.mp3
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Just to make clear which ones are the second partial derivatives. So, partial with respect to x twice, also a function of x, y, like all of these guys. And so let's look up at the original partial derivative with respect to x, and we're now gonna take its derivative, again with respect to x. This is the only place where x shows up, that 1 1 2 kind of comes down again, so now it's gonna be 1 1 4 times e to the x over two. And we just keep that sine of y, cause it looks like we're just multiplying by a constant. Sine of y. Next, we'll do the mixed partial derivative, where you do first with respect to x, then with respect to y, or you could do it the other way, because with almost all functions, it kind of doesn't matter which order you take the two.
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Quadratic approximation example.mp3
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This is the only place where x shows up, that 1 1 2 kind of comes down again, so now it's gonna be 1 1 4 times e to the x over two. And we just keep that sine of y, cause it looks like we're just multiplying by a constant. Sine of y. Next, we'll do the mixed partial derivative, where you do first with respect to x, then with respect to y, or you could do it the other way, because with almost all functions, it kind of doesn't matter which order you take the two. So I'll go ahead and just look at the one that was with respect to x, and now let's think of its derivative with respect to y. This whole 1 1 2 e to the x halves, looks like a constant, derivative of sine of y is cosine y. So we take that constant, the 1 1 2 e to the x halves, and then we multiply it by the derivative of sine of y, which is cosine of y.
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Quadratic approximation example.mp3
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Next, we'll do the mixed partial derivative, where you do first with respect to x, then with respect to y, or you could do it the other way, because with almost all functions, it kind of doesn't matter which order you take the two. So I'll go ahead and just look at the one that was with respect to x, and now let's think of its derivative with respect to y. This whole 1 1 2 e to the x halves, looks like a constant, derivative of sine of y is cosine y. So we take that constant, the 1 1 2 e to the x halves, and then we multiply it by the derivative of sine of y, which is cosine of y. And then finally, we take the second derivative, second partial derivative with respect to y twice in a row. So f with respect to y twice in a row. And for this one, let's take a look at the partial derivative with respect to y.
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Quadratic approximation example.mp3
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So we take that constant, the 1 1 2 e to the x halves, and then we multiply it by the derivative of sine of y, which is cosine of y. And then finally, we take the second derivative, second partial derivative with respect to y twice in a row. So f with respect to y twice in a row. And for this one, let's take a look at the partial derivative with respect to y. This part is the only part where y shows up, derivative of cosine is negative sine, and then e to the x halves just still looks like a constant. So we bring that negative out front, that constant e to the x halves, and it was negative sine, so that negative went out front. Sine of y.
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Quadratic approximation example.mp3
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And for this one, let's take a look at the partial derivative with respect to y. This part is the only part where y shows up, derivative of cosine is negative sine, and then e to the x halves just still looks like a constant. So we bring that negative out front, that constant e to the x halves, and it was negative sine, so that negative went out front. Sine of y. So that's all of the partial differential information that we're gonna need. And now we know we're gonna need to evaluate all of these guys, all of these partial derivatives at the specific point, because if we go up and look at the original function that we have, we're gonna need to evaluate f at this point, both of the partial derivatives at this point, the second partial derivatives. Oh, I'm realizing actually that I made a little bit of a mistake here.
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Quadratic approximation example.mp3
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Sine of y. So that's all of the partial differential information that we're gonna need. And now we know we're gonna need to evaluate all of these guys, all of these partial derivatives at the specific point, because if we go up and look at the original function that we have, we're gonna need to evaluate f at this point, both of the partial derivatives at this point, the second partial derivatives. Oh, I'm realizing actually that I made a little bit of a mistake here. This should be 1 1 2 out in front of each of these guys. That should be plus 1 1 2 of this second partial derivative and 1 1 2 of this second partial derivative. The mixed partial derivative, it's still one, but these guys should have a 1 1 2.
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Quadratic approximation example.mp3
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Oh, I'm realizing actually that I made a little bit of a mistake here. This should be 1 1 2 out in front of each of these guys. That should be plus 1 1 2 of this second partial derivative and 1 1 2 of this second partial derivative. The mixed partial derivative, it's still one, but these guys should have a 1 1 2. That was a mistake on my part. In either case, though, we're gonna need to evaluate all of these guys. So if we go back down, let's just start plugging in the point zero and pi halves to each one of these.
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Quadratic approximation example.mp3
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The mixed partial derivative, it's still one, but these guys should have a 1 1 2. That was a mistake on my part. In either case, though, we're gonna need to evaluate all of these guys. So if we go back down, let's just start plugging in the point zero and pi halves to each one of these. So the function itself, when we plug in zero, e to the zero is one, and sine of pi halves, sine of pi halves is also one, so this entire thing just comes to one. If we do this for the next one, again, e to the zero is gonna be one, sine of y is also gonna be one, but now we have that 1 1 2 sitting there, so that'll end up as 1 1 2. If we look at the partial derivative with respect to y, cosine of pi halves is zero, so this entire thing is gonna be zero.
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Quadratic approximation example.mp3
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So if we go back down, let's just start plugging in the point zero and pi halves to each one of these. So the function itself, when we plug in zero, e to the zero is one, and sine of pi halves, sine of pi halves is also one, so this entire thing just comes to one. If we do this for the next one, again, e to the zero is gonna be one, sine of y is also gonna be one, but now we have that 1 1 2 sitting there, so that'll end up as 1 1 2. If we look at the partial derivative with respect to y, cosine of pi halves is zero, so this entire thing is gonna be zero. Moving right along, if we do, let's take a look at the second partial derivative with respect to x. Again, e to the zero will be one, and sine of pi halves will be one, so this ends up just being that 1 1 4. The mixed partial derivative here, if we have 1 1 2 by the pattern starting to continue, we've got the one.
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Quadratic approximation example.mp3
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If we look at the partial derivative with respect to y, cosine of pi halves is zero, so this entire thing is gonna be zero. Moving right along, if we do, let's take a look at the second partial derivative with respect to x. Again, e to the zero will be one, and sine of pi halves will be one, so this ends up just being that 1 1 4. The mixed partial derivative here, if we have 1 1 2 by the pattern starting to continue, we've got the one. This one's actually zero, so cosine of pi halves is zero, so the whole thing will be zero. And then the last one, it'll be negative one times that one again, for sine of pi halves is one. So all of that just comes out to be negative one.
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Quadratic approximation example.mp3
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The mixed partial derivative here, if we have 1 1 2 by the pattern starting to continue, we've got the one. This one's actually zero, so cosine of pi halves is zero, so the whole thing will be zero. And then the last one, it'll be negative one times that one again, for sine of pi halves is one. So all of that just comes out to be negative one. So this is, I mean, I kind of chose a convenient example, right, where all the derivatives look very similar to the thing itself, which is actually pretty common, so we get to leverage a lot of the work that we did earlier. So now we have these six different constants, and I kind of can't keep them all on the screen at the same time, but we've got the six different constants, so now we just plug each one of these in to the quadratic approximation. So if we make our quadratic approximation of our function, the first term is that constant term, so we take a look up, and we say, where does f of x, y go at this point, and it'll just be one.
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Quadratic approximation example.mp3
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So all of that just comes out to be negative one. So this is, I mean, I kind of chose a convenient example, right, where all the derivatives look very similar to the thing itself, which is actually pretty common, so we get to leverage a lot of the work that we did earlier. So now we have these six different constants, and I kind of can't keep them all on the screen at the same time, but we've got the six different constants, so now we just plug each one of these in to the quadratic approximation. So if we make our quadratic approximation of our function, the first term is that constant term, so we take a look up, and we say, where does f of x, y go at this point, and it'll just be one. I'm gonna have to do a lot of scrolling back and forth here, there's a lot of text to deal with. And the next thing is gonna be something times x minus zero, the kind of x-coordinate of our specified point, and that something is the first derivative with respect to x, so that's gonna be 1 1⁄2. So come back down here, we've got 1 1⁄2.
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Quadratic approximation example.mp3
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So if we make our quadratic approximation of our function, the first term is that constant term, so we take a look up, and we say, where does f of x, y go at this point, and it'll just be one. I'm gonna have to do a lot of scrolling back and forth here, there's a lot of text to deal with. And the next thing is gonna be something times x minus zero, the kind of x-coordinate of our specified point, and that something is the first derivative with respect to x, so that's gonna be 1 1⁄2. So come back down here, we've got 1 1⁄2. And then similarly, we're gonna have something multiplied by y minus the y-coordinate of the point about which we are approximating, and for that, we take a look at the partial derivative with respect to y, which was just zero, so that's pretty convenient. That's just gonna end up being zero. And then for the second partial derivative terms, maybe I'll actually be able to keep it on the same screen here, we're gonna have something multiplied by x minus its coordinate squared, and that something is whatever the partial derivative with respect to x twice is, which is 1⁄4.
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Quadratic approximation example.mp3
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So come back down here, we've got 1 1⁄2. And then similarly, we're gonna have something multiplied by y minus the y-coordinate of the point about which we are approximating, and for that, we take a look at the partial derivative with respect to y, which was just zero, so that's pretty convenient. That's just gonna end up being zero. And then for the second partial derivative terms, maybe I'll actually be able to keep it on the same screen here, we're gonna have something multiplied by x minus its coordinate squared, and that something is whatever the partial derivative with respect to x twice is, which is 1⁄4. So we go ahead and plug in 1⁄4, and then for the mixed partial derivative, we'll put it down here, it'll be something multiplied by x minus its constant, and then y minus that pi halves, and that something is the mixed partial derivative, which in this case is zero. Oh, and I'm realizing I made the same mistake again, it's not 1⁄4, it's 1⁄2. For the same reason that I made a mistake up here earlier where it's actually 1⁄2 multiplied by this second partial derivative, and 1⁄2 by the second partial derivative there.
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Quadratic approximation example.mp3
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And then for the second partial derivative terms, maybe I'll actually be able to keep it on the same screen here, we're gonna have something multiplied by x minus its coordinate squared, and that something is whatever the partial derivative with respect to x twice is, which is 1⁄4. So we go ahead and plug in 1⁄4, and then for the mixed partial derivative, we'll put it down here, it'll be something multiplied by x minus its constant, and then y minus that pi halves, and that something is the mixed partial derivative, which in this case is zero. Oh, and I'm realizing I made the same mistake again, it's not 1⁄4, it's 1⁄2. For the same reason that I made a mistake up here earlier where it's actually 1⁄2 multiplied by this second partial derivative, and 1⁄2 by the second partial derivative there. I guess I keep forgetting that. Good lesson, I suppose, that that's an easy thing to forget if you find yourself computing one of these, where I'll put it in here, multiply that guy by 1⁄2. It's similar to a Taylor expansion in single variable calculus, where you kind of have to remember what's what that squared term would be, has a 1⁄2 associated with it.
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Quadratic approximation example.mp3
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For the same reason that I made a mistake up here earlier where it's actually 1⁄2 multiplied by this second partial derivative, and 1⁄2 by the second partial derivative there. I guess I keep forgetting that. Good lesson, I suppose, that that's an easy thing to forget if you find yourself computing one of these, where I'll put it in here, multiply that guy by 1⁄2. It's similar to a Taylor expansion in single variable calculus, where you kind of have to remember what's what that squared term would be, has a 1⁄2 associated with it. So for that same reason, now we're gonna have, and this time I won't forget it, will be 1⁄2 multiplied by something multiplied by the y minus pi halves minus that y coordinate of the point we're approximating near. And this time that something is negative one. So we can kind of plug in here negative one.
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Quadratic approximation example.mp3
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It's similar to a Taylor expansion in single variable calculus, where you kind of have to remember what's what that squared term would be, has a 1⁄2 associated with it. So for that same reason, now we're gonna have, and this time I won't forget it, will be 1⁄2 multiplied by something multiplied by the y minus pi halves minus that y coordinate of the point we're approximating near. And this time that something is negative one. So we can kind of plug in here negative one. And now this is something we can simplify quite a bit because that one stays there, 1⁄2 of x minus zero, that's just x halves. This whole part cancels out to zero, so there's nothing there. Over here we have 1⁄2 times 1⁄4, 1⁄8 times x squared.
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Quadratic approximation example.mp3
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So we can kind of plug in here negative one. And now this is something we can simplify quite a bit because that one stays there, 1⁄2 of x minus zero, that's just x halves. This whole part cancels out to zero, so there's nothing there. Over here we have 1⁄2 times 1⁄4, 1⁄8 times x squared. So that's x squared divided by eight. This mixed partial derivative term is zero, so that's pretty nice. And then this last term here is just negative 1⁄2.
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Quadratic approximation example.mp3
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Over here we have 1⁄2 times 1⁄4, 1⁄8 times x squared. So that's x squared divided by eight. This mixed partial derivative term is zero, so that's pretty nice. And then this last term here is just negative 1⁄2. So let's see, I'll write it down as negative 1⁄2 by y minus pi halves squared. By y minus pi halves squared. So that is the quadratic approximation.
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Quadratic approximation example.mp3
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And then this last term here is just negative 1⁄2. So let's see, I'll write it down as negative 1⁄2 by y minus pi halves squared. By y minus pi halves squared. So that is the quadratic approximation. And you can see this actually feels like a quadratic function. We've got up to x squared and up to y squared. And there's a sense in which this is a simpler function.
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Quadratic approximation example.mp3
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So that is the quadratic approximation. And you can see this actually feels like a quadratic function. We've got up to x squared and up to y squared. And there's a sense in which this is a simpler function. I mean, it looks like it's got more terms than the original one, which was e to the x halves sine of y. But if it's a computer that needs to compute these things, for example, it's much easier to deal with polynomials. That's a faster thing to do.
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Quadratic approximation example.mp3
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And there's a sense in which this is a simpler function. I mean, it looks like it's got more terms than the original one, which was e to the x halves sine of y. But if it's a computer that needs to compute these things, for example, it's much easier to deal with polynomials. That's a faster thing to do. Also, for theoretical purposes, it can be nice to deal with just a quadratic polynomial to make conclusions about things. We'll see that in the context of something called the second partial derivative test. But just to get a feel for what this means, let's pull up the graph of the relevant functions.
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Quadratic approximation example.mp3
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That's a faster thing to do. Also, for theoretical purposes, it can be nice to deal with just a quadratic polynomial to make conclusions about things. We'll see that in the context of something called the second partial derivative test. But just to get a feel for what this means, let's pull up the graph of the relevant functions. So this here is the graph of the original function, e to the x halves times sine of y. And the point that we're approximating near was where x equals zero. So let's see how we get oriented.
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Quadratic approximation example.mp3
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But just to get a feel for what this means, let's pull up the graph of the relevant functions. So this here is the graph of the original function, e to the x halves times sine of y. And the point that we're approximating near was where x equals zero. So let's see how we get oriented. X is equal to zero, and then y is equal to pi halves. So this is the point we're approximating near. And the quadratic approximation, when you plug everything in, has a graph that looks like this white surface here.
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Quadratic approximation example.mp3
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So let's see how we get oriented. X is equal to zero, and then y is equal to pi halves. So this is the point we're approximating near. And the quadratic approximation, when you plug everything in, has a graph that looks like this white surface here. So if I get rid of that original graph, this is how we're approximating the function near that point and that does a pretty good job, right? Because even as you step pretty far away from that point, it's pretty closely hugging the original surface. If you go very far away, you know, it certainly doesn't get the oscillating nature of that sine component.
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Quadratic approximation example.mp3
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And the quadratic approximation, when you plug everything in, has a graph that looks like this white surface here. So if I get rid of that original graph, this is how we're approximating the function near that point and that does a pretty good job, right? Because even as you step pretty far away from that point, it's pretty closely hugging the original surface. If you go very far away, you know, it certainly doesn't get the oscillating nature of that sine component. And the exponential component grows faster than the quadratic one. But nearby, this actually gives a very good feel for the shape of the graph. And again, later on, we'll see how this is a pretty useful theoretical tool for drawing conclusions about qualitative features of the shape of the graph.
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Quadratic approximation example.mp3
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If you go very far away, you know, it certainly doesn't get the oscillating nature of that sine component. And the exponential component grows faster than the quadratic one. But nearby, this actually gives a very good feel for the shape of the graph. And again, later on, we'll see how this is a pretty useful theoretical tool for drawing conclusions about qualitative features of the shape of the graph. The fact that this looks kind of like a saddle is gonna end up being kind of important in certain contexts. But before we get to any of that, in the next couple of videos, I'm gonna talk about a simpler or rather a more generalizable form of writing down this quadratic approximation using vector notation. Because right now we're just limited to, you know, two variables.
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Quadratic approximation example.mp3
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And again, later on, we'll see how this is a pretty useful theoretical tool for drawing conclusions about qualitative features of the shape of the graph. The fact that this looks kind of like a saddle is gonna end up being kind of important in certain contexts. But before we get to any of that, in the next couple of videos, I'm gonna talk about a simpler or rather a more generalizable form of writing down this quadratic approximation using vector notation. Because right now we're just limited to, you know, two variables. And you can imagine how monstrous this might look if you were dealing even just with a three variable function, right? Where, think of all the different possible second partial derivatives of a three variable function or a four variable function. It would quickly get out of hand.
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Quadratic approximation example.mp3
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So where we left off, we had this two dimensional vector field V, and I have it pictured here as kind of a yellow vector field, and I just stuck it in three dimensions in kind of an awkward way where I put it on the xy plane and said, pretend this is in three dimensions. And then when you describe the rotation around each point, what we were familiar with is 2D curl, that's where you get this vector field. It's not quite a 3D vector field because you're only assigning points on the xy plane to three dimensional vectors, rather than every point in space to a vector, but we're getting there. So here, let's actually extend this to a fully three dimensional vector field. And first of all, let me just kind of clear up the board from the computations that we did in the last part. And as I do that, kind of start thinking about how you might want to extend the vector field that I have here that's pretty much two dimensional into three dimensions. And one idea you might have, so kind of get rid of the circles and the plane, is to take this vector field and then just kind of copy it into different slices.
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3d curl intuition, part 2.mp3
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So here, let's actually extend this to a fully three dimensional vector field. And first of all, let me just kind of clear up the board from the computations that we did in the last part. And as I do that, kind of start thinking about how you might want to extend the vector field that I have here that's pretty much two dimensional into three dimensions. And one idea you might have, so kind of get rid of the circles and the plane, is to take this vector field and then just kind of copy it into different slices. So, you might get something kind of like this. And I've drawn each slice a little bit sparser than the original one, so technically, that original one, if you look on the xy plane, I've pictured many more vectors, but it's really the same vector field. And all I've done here is said that every slice in space, just copy that same vector field.
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3d curl intuition, part 2.mp3
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And one idea you might have, so kind of get rid of the circles and the plane, is to take this vector field and then just kind of copy it into different slices. So, you might get something kind of like this. And I've drawn each slice a little bit sparser than the original one, so technically, that original one, if you look on the xy plane, I've pictured many more vectors, but it's really the same vector field. And all I've done here is said that every slice in space, just copy that same vector field. So if you look from above, you can maybe see how really it's just the same vector field kind of copied a bunch. And if you look at each slice, in the same way that on the xy plane, you've got this vector field sitting on a slice, every other part of space will have that. And even though there's only, what, like six or seven slices displayed here, in principle, you're thinking that every one of those infinitely many slices of space has a copy of this vector field.
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3d curl intuition, part 2.mp3
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And all I've done here is said that every slice in space, just copy that same vector field. So if you look from above, you can maybe see how really it's just the same vector field kind of copied a bunch. And if you look at each slice, in the same way that on the xy plane, you've got this vector field sitting on a slice, every other part of space will have that. And even though there's only, what, like six or seven slices displayed here, in principle, you're thinking that every one of those infinitely many slices of space has a copy of this vector field. And in a formula, what does that mean? Well, what it means is we're taking not just x and y as input points, but we're gonna start taking z in as well. So if I go, I'm gonna say that z is an input point as well.
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3d curl intuition, part 2.mp3
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And even though there's only, what, like six or seven slices displayed here, in principle, you're thinking that every one of those infinitely many slices of space has a copy of this vector field. And in a formula, what does that mean? Well, what it means is we're taking not just x and y as input points, but we're gonna start taking z in as well. So if I go, I'm gonna say that z is an input point as well. And I wanna be considering these as vectors in three dimensions, so rather than just saying that it's got x and y components, I'm gonna pretend like it has a z component, it has a z component that just happens to be zero for this case. And the fact that you have a z in the input, but the output doesn't depend on the z, corresponds to the fact that all the slices are the same. As you change the z direction, the vectors won't change at all, they're just carbon copies of each other.
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3d curl intuition, part 2.mp3
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So if I go, I'm gonna say that z is an input point as well. And I wanna be considering these as vectors in three dimensions, so rather than just saying that it's got x and y components, I'm gonna pretend like it has a z component, it has a z component that just happens to be zero for this case. And the fact that you have a z in the input, but the output doesn't depend on the z, corresponds to the fact that all the slices are the same. As you change the z direction, the vectors won't change at all, they're just carbon copies of each other. And the fact that this output has a z component, but it just happens to be zero, is what corresponds to the fact that it's very flat looking. You know, none of them point up or down in the z direction, they're all purely x and y. So as three-dimensional vector fields go, this one is only barely a three-dimensional vector field, it's kind of phoning it in as far as three-dimensional vector fields are concerned.
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3d curl intuition, part 2.mp3
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As you change the z direction, the vectors won't change at all, they're just carbon copies of each other. And the fact that this output has a z component, but it just happens to be zero, is what corresponds to the fact that it's very flat looking. You know, none of them point up or down in the z direction, they're all purely x and y. So as three-dimensional vector fields go, this one is only barely a three-dimensional vector field, it's kind of phoning it in as far as three-dimensional vector fields are concerned. But it'll be quite good for our example here. Because now, if we start thinking of this as representing a three-dimensional fluid flow, so now rather than just kind of the fluid flow like the one I have pictured over here, where you've got water molecules moving in two dimensions and it's very easy to understand clockwise rotation, counterclockwise rotation, things like that, whereas over here, it's a very kind of chaotic three-dimensional fluid flow. But because it's so flat, if you view it from above, it's still loosely the same, just kind of counterclockwise over here on the right, and clockwise up there above.
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3d curl intuition, part 2.mp3
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So as three-dimensional vector fields go, this one is only barely a three-dimensional vector field, it's kind of phoning it in as far as three-dimensional vector fields are concerned. But it'll be quite good for our example here. Because now, if we start thinking of this as representing a three-dimensional fluid flow, so now rather than just kind of the fluid flow like the one I have pictured over here, where you've got water molecules moving in two dimensions and it's very easy to understand clockwise rotation, counterclockwise rotation, things like that, whereas over here, it's a very kind of chaotic three-dimensional fluid flow. But because it's so flat, if you view it from above, it's still loosely the same, just kind of counterclockwise over here on the right, and clockwise up there above. So if I were to draw like a column, you could think of this column as being, having a tornado of fluid flow, right? Where everything is kind of rotating together in that same direction. So if you were to assign a vector to each point in space to describe the kind of rotation happening around each one of those points in space, you would expect that those inside this column, inside this sort of counterclockwise rotating tornado, and I say counterclockwise, but if we viewed it from below, it would look clockwise.
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3d curl intuition, part 2.mp3
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But because it's so flat, if you view it from above, it's still loosely the same, just kind of counterclockwise over here on the right, and clockwise up there above. So if I were to draw like a column, you could think of this column as being, having a tornado of fluid flow, right? Where everything is kind of rotating together in that same direction. So if you were to assign a vector to each point in space to describe the kind of rotation happening around each one of those points in space, you would expect that those inside this column, inside this sort of counterclockwise rotating tornado, and I say counterclockwise, but if we viewed it from below, it would look clockwise. So that's the tricky part about three dimensions and why we need to describe it with vectors. But you would expect these using your right-hand rule, where you curl the fingers of your right hand around the direction of rotation here, you would expect vectors that point up in the z direction, the positive z direction. And if I do that, if I show what all of the rotation vectors look like, you'll get this.
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3d curl intuition, part 2.mp3
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So if you were to assign a vector to each point in space to describe the kind of rotation happening around each one of those points in space, you would expect that those inside this column, inside this sort of counterclockwise rotating tornado, and I say counterclockwise, but if we viewed it from below, it would look clockwise. So that's the tricky part about three dimensions and why we need to describe it with vectors. But you would expect these using your right-hand rule, where you curl the fingers of your right hand around the direction of rotation here, you would expect vectors that point up in the z direction, the positive z direction. And if I do that, if I show what all of the rotation vectors look like, you'll get this. And maybe this is kind of a mess because there's a lot of things on the screen at this point. So for the moment, I'll kind of remove that original vector field and remove the xy plane, and just kind of focus on this new vector field that I have pictured here. Inside that column, where we had that tornado of rotation I was describing, all of the vectors point in the positive z direction.
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3d curl intuition, part 2.mp3
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And if I do that, if I show what all of the rotation vectors look like, you'll get this. And maybe this is kind of a mess because there's a lot of things on the screen at this point. So for the moment, I'll kind of remove that original vector field and remove the xy plane, and just kind of focus on this new vector field that I have pictured here. Inside that column, where we had that tornado of rotation I was describing, all of the vectors point in the positive z direction. But if we were to view it elsewhere, like over in this region, those are pointing in the negative z direction. And if you stick your thumb in the direction of all of these vectors in the negative z direction, that tells you the direction of, that tells you how the fluid, maybe you're thinking of it as air kind of rushing about the room, how that fluid rotates in three dimensions. So what curl is gonna do, here I'll kind of clear things up, I have the formula from last time that hopefully hasn't looked too in the way while I've been doing this, but it described curl for a two-dimensional vector field.
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3d curl intuition, part 2.mp3
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Inside that column, where we had that tornado of rotation I was describing, all of the vectors point in the positive z direction. But if we were to view it elsewhere, like over in this region, those are pointing in the negative z direction. And if you stick your thumb in the direction of all of these vectors in the negative z direction, that tells you the direction of, that tells you how the fluid, maybe you're thinking of it as air kind of rushing about the room, how that fluid rotates in three dimensions. So what curl is gonna do, here I'll kind of clear things up, I have the formula from last time that hopefully hasn't looked too in the way while I've been doing this, but it described curl for a two-dimensional vector field. If we imagine this not just taking x and y as its inputs, because it's a three-dimensional vector field, but if we imagine it taking x, y, and z, so it's a proper three-dimensional vector field, the output is gonna tell you at every point in space what the rotation that corresponds to that point is. And in the next video, I'm gonna give you the formula and tell you how you actually compute this curl given an arbitrary function, but for right now, we're just getting the visual intuition, we're just trying to understand what it is that curl's going to represent. And in this vector field, this one that was just kind of copies of a 2D one put above, it's almost contrived because all of the rotation happens in these perfect, perfect tornado-like patterns that doesn't really change as you move up and down in the x, y direction.
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3d curl intuition, part 2.mp3
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So what curl is gonna do, here I'll kind of clear things up, I have the formula from last time that hopefully hasn't looked too in the way while I've been doing this, but it described curl for a two-dimensional vector field. If we imagine this not just taking x and y as its inputs, because it's a three-dimensional vector field, but if we imagine it taking x, y, and z, so it's a proper three-dimensional vector field, the output is gonna tell you at every point in space what the rotation that corresponds to that point is. And in the next video, I'm gonna give you the formula and tell you how you actually compute this curl given an arbitrary function, but for right now, we're just getting the visual intuition, we're just trying to understand what it is that curl's going to represent. And in this vector field, this one that was just kind of copies of a 2D one put above, it's almost contrived because all of the rotation happens in these perfect, perfect tornado-like patterns that doesn't really change as you move up and down in the x, y direction. But more generally, you might have a more complicated-looking vector field. So I'll go ahead and kind of finally erase this since it's been a little bit in the way for a while, and erase this guy too. And if you think about just arbitrary three-dimensional vector fields, like let's say this one that I have here, so I don't know about you, but for me, it's really hard to think about the fluid flow associated with this.
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3d curl intuition, part 2.mp3
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And in this vector field, this one that was just kind of copies of a 2D one put above, it's almost contrived because all of the rotation happens in these perfect, perfect tornado-like patterns that doesn't really change as you move up and down in the x, y direction. But more generally, you might have a more complicated-looking vector field. So I'll go ahead and kind of finally erase this since it's been a little bit in the way for a while, and erase this guy too. And if you think about just arbitrary three-dimensional vector fields, like let's say this one that I have here, so I don't know about you, but for me, it's really hard to think about the fluid flow associated with this. I have a vague notion in my mind that okay, like fluid is flowing out from this corner and kind of flowing in here, but it's very hard to think about it all at once. And certainly if you start talking about rotation, it's hard to look at a given point and say, oh yeah, there's gonna be a general fluid rotation in some certain way, and I can give you the vector for that. But as a more loose and vague idea, I can say, okay, given that there's some kind of crazy air current fluid flow happening around here, I can maybe understand that at a specific point, you're gonna have some kind of rotation.
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3d curl intuition, part 2.mp3
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And if you think about just arbitrary three-dimensional vector fields, like let's say this one that I have here, so I don't know about you, but for me, it's really hard to think about the fluid flow associated with this. I have a vague notion in my mind that okay, like fluid is flowing out from this corner and kind of flowing in here, but it's very hard to think about it all at once. And certainly if you start talking about rotation, it's hard to look at a given point and say, oh yeah, there's gonna be a general fluid rotation in some certain way, and I can give you the vector for that. But as a more loose and vague idea, I can say, okay, given that there's some kind of crazy air current fluid flow happening around here, I can maybe understand that at a specific point, you're gonna have some kind of rotation. And here, I'll picture it as if there's like a ball or a globe sitting there in space, and maybe you imagine your new vector field and saying what kind of rotation is it gonna induce in that ball that's just floating there in space. So maybe you're imagining this as like a tennis ball that you're sort of holding in place in space using magnets or magic or something like that, and you're letting the wind sort of freely rotate it, and you're wondering what direction it tends to rotate. And then when it does, and once you have this rotation, you can describe that 3D rotation with some kind of vector.
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3d curl intuition, part 2.mp3
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But as a more loose and vague idea, I can say, okay, given that there's some kind of crazy air current fluid flow happening around here, I can maybe understand that at a specific point, you're gonna have some kind of rotation. And here, I'll picture it as if there's like a ball or a globe sitting there in space, and maybe you imagine your new vector field and saying what kind of rotation is it gonna induce in that ball that's just floating there in space. So maybe you're imagining this as like a tennis ball that you're sort of holding in place in space using magnets or magic or something like that, and you're letting the wind sort of freely rotate it, and you're wondering what direction it tends to rotate. And then when it does, and once you have this rotation, you can describe that 3D rotation with some kind of vector. And in this case, it would be a vector that points out in that direction because we're kind of curling our fingers, curling our right hand fingers over in that direction. And if you don't understand how we describe 3D rotation with a vector, I have a video on that. Maybe go back and check out that video.
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3d curl intuition, part 2.mp3
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And then when it does, and once you have this rotation, you can describe that 3D rotation with some kind of vector. And in this case, it would be a vector that points out in that direction because we're kind of curling our fingers, curling our right hand fingers over in that direction. And if you don't understand how we describe 3D rotation with a vector, I have a video on that. Maybe go back and check out that video. But the idea here is that when you have some sort of crazy fluid flow that's induced by some sort of vector field, and you do this at every point and say, hey, what's the rotation at every single point? That's gonna give you the curl. That is what the curl of a three-dimensional vector field is trying to represent.
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3d curl intuition, part 2.mp3
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Maybe go back and check out that video. But the idea here is that when you have some sort of crazy fluid flow that's induced by some sort of vector field, and you do this at every point and say, hey, what's the rotation at every single point? That's gonna give you the curl. That is what the curl of a three-dimensional vector field is trying to represent. And if this feels confusing, if this feels like something that's hard to wrap your mind around, don't worry, we've all been there. 3D curl is one of the most complicated things in multivariable calculus that we have to describe. But I think the key to understanding it is to just kind of patiently think through and take the time to think about what 2D curl is and start thinking about how you extend that to three dimensions and slowly say, okay, okay, I kind of get it, and tornadoes of rotation, that sort of makes sense.
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3d curl intuition, part 2.mp3
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That is what the curl of a three-dimensional vector field is trying to represent. And if this feels confusing, if this feels like something that's hard to wrap your mind around, don't worry, we've all been there. 3D curl is one of the most complicated things in multivariable calculus that we have to describe. But I think the key to understanding it is to just kind of patiently think through and take the time to think about what 2D curl is and start thinking about how you extend that to three dimensions and slowly say, okay, okay, I kind of get it, and tornadoes of rotation, that sort of makes sense. And if you understand how to represent three-dimensional rotation around a single point with a vector, then understanding three-dimensional curl comes down to thinking about doing that at every single point in space according to whatever rotation the wind flow around that point would induce. But like I said, it is complicated, and it's okay if it doesn't sink the first time. It certainly took me a while to really wrap my head around this 3D curl idea.
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3d curl intuition, part 2.mp3
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And we ended up seeing how specifying that orientation comes down to certain partial derivative information. And first let me just kind of repeat what the conclusion was, but I'll put it in more abstract terms since I did it in a very specific example last time. So basically, if you want some kind of function which gives you a plane that passes through a certain point, well first let's say what that point is, right? Let's say the point was x-naught, y-naught, and z-naught. So these are just constant values, and this is my way of abstractly describing a single point in space, using x-naught to represent a constant x value, y-naught to represent constant y value, that kind of thing. So what it is, is it's gonna be some sort of other constant, a, multiplied by x minus x-naught. So this white x here is the variable, and then x-naught is just a constant.
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Computing a tangent plane.mp3
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Let's say the point was x-naught, y-naught, and z-naught. So these are just constant values, and this is my way of abstractly describing a single point in space, using x-naught to represent a constant x value, y-naught to represent constant y value, that kind of thing. So what it is, is it's gonna be some sort of other constant, a, multiplied by x minus x-naught. So this white x here is the variable, and then x-naught is just a constant. Let me go ahead and make that parentheses there. Okay, so then we add to that b multiplied by, and then b is just some other constant, just like a is some other constant, multiplied by y minus y-naught. And then all of that you add z-naught.
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Computing a tangent plane.mp3
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So this white x here is the variable, and then x-naught is just a constant. Let me go ahead and make that parentheses there. Okay, so then we add to that b multiplied by, and then b is just some other constant, just like a is some other constant, multiplied by y minus y-naught. And then all of that you add z-naught. And now if you just presented this as it is, it's kind of a lot, right? There's five different constants going on. But really what this is saying is you want something where the partial derivative with respect to x is just some constant, and you wanna be able to specify what that constant is.
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Computing a tangent plane.mp3
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And then all of that you add z-naught. And now if you just presented this as it is, it's kind of a lot, right? There's five different constants going on. But really what this is saying is you want something where the partial derivative with respect to x is just some constant, and you wanna be able to specify what that constant is. And similarly, the partial derivative with respect to y is another constant. And you just wanna ensure that this passes through this point, x-naught, y-naught, z-naught. And if you imagine plugging in x, the variable, equals x-naught, the constant, this part goes to zero.
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Computing a tangent plane.mp3
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But really what this is saying is you want something where the partial derivative with respect to x is just some constant, and you wanna be able to specify what that constant is. And similarly, the partial derivative with respect to y is another constant. And you just wanna ensure that this passes through this point, x-naught, y-naught, z-naught. And if you imagine plugging in x, the variable, equals x-naught, the constant, this part goes to zero. Similarly, plugging in y-naught, the constant, makes this part go to zero. So this is a way of specifying that when you evaluate the function at x-naught, y-naught, it equals z-naught. And that's what makes sure that the graph passes through that point.
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Computing a tangent plane.mp3
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And if you imagine plugging in x, the variable, equals x-naught, the constant, this part goes to zero. Similarly, plugging in y-naught, the constant, makes this part go to zero. So this is a way of specifying that when you evaluate the function at x-naught, y-naught, it equals z-naught. And that's what makes sure that the graph passes through that point. So with that said, let's start thinking about how you confine the tangent plane to a graph. And first of all, let's think about what that point is, how you specify such a point. Instead of specifying any three numbers in space, because you have to make sure the point is somewhere on the graph, you instead only specify two.
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Computing a tangent plane.mp3
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And that's what makes sure that the graph passes through that point. So with that said, let's start thinking about how you confine the tangent plane to a graph. And first of all, let's think about what that point is, how you specify such a point. Instead of specifying any three numbers in space, because you have to make sure the point is somewhere on the graph, you instead only specify two. You're basically gonna say what's the x-coordinate, and in this case, let's say the x-coordinate was like one, that'd be like one, and then the y-coordinate, which looks about like negative two. To make it easier, I'm just gonna say, let's say it is negative two. Then the z-coordinate is specified, because this is a graph.
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Computing a tangent plane.mp3
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Instead of specifying any three numbers in space, because you have to make sure the point is somewhere on the graph, you instead only specify two. You're basically gonna say what's the x-coordinate, and in this case, let's say the x-coordinate was like one, that'd be like one, and then the y-coordinate, which looks about like negative two. To make it easier, I'm just gonna say, let's say it is negative two. Then the z-coordinate is specified, because this is a graph. The z-coordinate is forced to be whatever the output of the function is at one, negative two. So this is gonna be whatever the output of our function is at one, negative two. And f here, f is gonna be whatever function gives us this graph.
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Computing a tangent plane.mp3
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Then the z-coordinate is specified, because this is a graph. The z-coordinate is forced to be whatever the output of the function is at one, negative two. So this is gonna be whatever the output of our function is at one, negative two. And f here, f is gonna be whatever function gives us this graph. So maybe I should write down the actual function that I'm using for this graph. In this case, f, which is a function of x and y, is equal to three minus 1 3rd of x squared minus y squared. Okay, so this is the function that we're using, and you evaluate it at that point, and this will give you your point in three-dimensional space that our linear function, that our tangent plane has to pass through.
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Computing a tangent plane.mp3
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And f here, f is gonna be whatever function gives us this graph. So maybe I should write down the actual function that I'm using for this graph. In this case, f, which is a function of x and y, is equal to three minus 1 3rd of x squared minus y squared. Okay, so this is the function that we're using, and you evaluate it at that point, and this will give you your point in three-dimensional space that our linear function, that our tangent plane has to pass through. So we can start writing out our function, right? We can say, okay, so our linear function has a function of x and y. It's gotta make sure it goes through that one and that negative two.
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Computing a tangent plane.mp3
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