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Studying for a test? Prepare with these 13 lessons on Derivatives of multivariable functions.
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Video transskription
- [Voiceover] So where we left off, I 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 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, we'll kind of get rid of the circles and the plane, is to take this vector field and then just kind of copy it to different slices. So, you might get something kind of like this. And then, 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 at 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 it each slice, you know in the same way that in the XY plane, you've got this vector field sitting on the slice, every other part of space will have that. And even though there's only what, like six or seven slices displayed here, in principal 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 what it means is that 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 want to be considering these as vectors in three dimensions, so rather than just saying that that it's got X and Y components, I'm gonna pretend like 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 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, counter clockwise 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 counter clockwise over here on the right, and clockwise like 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 it's, 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 counter clockwise rotating tornado, and I say counter clockwise, 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 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 have that tornado 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 it 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, that described curl for a two dimensional vector field if we imagine that's 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 is going to represent. And in this vector field, this one that was just kind of copies of a 2D put above, it's almost contrived because all of the rotation happens in these perfect tornado-like patters that doesn't really change as you move up and down in the XY direction, but more generally you might have a more complicated looking vector field, so I'll go ahead and 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, and here I'll picture it as if there's like a ball or a globe sitting there in space, and maybe you're imagining your new vector field and saying what kind of rotation is it kind of induce in that ball that's just floating there in space? 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 when 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. And if this feels confusing, if this feels like something that's hard to wrap your mind around, don't worry we've 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 instead of thinking about how you extend that to three dimensions and slowly say, okay, okay, I kind of get it, tornados 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. Like I said, it is complicated, and it's okay if it doesn't sink the first time, it certainly took me awhile to really wrap my head around this 3D curl idea, and with that I'll see you next video.