3d curl intuition, part 2

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. 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."

When you describe the rotation around each point, what we were familiar with as 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. First of all, let me just kind of clear up the board from the computations that we did in the last part. 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.

One idea you might have, so we 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. You might get something kind of like this. 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.

All I've done here is that at every slice in space, just copy that same vector field. If you look from above, you can maybe see how really it's just the same vector field, kind of copied a bunch. 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.

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. 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 going to start taking Z in as well. So if I go, I'm going to 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 it's got X and Y components, I'm going to pretend like it has a Z component. It has a Z component that just happens to be zero for this case. 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.

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, rather than just kind of the fluid flow like the one I have pictured over here where you’ve got, you know, water molecules moving in two dimensions, it's very easy to understand, you know, 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 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. 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 are 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.

If you stick your thumb in the direction of all of these vectors in the negative Z direction, that tells you the direction—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 going to 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. The 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 going to tell you at every point in space what the rotation that corresponds to that point is.

In the next video, I'm going to 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 one put above—it’s almost contrived because all of the rotation happens in these perfect tornado-like patterns that don’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 kind of finally erase this since it’s been a little bit in the way for a while and erase this guy too. 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. Certainly, if you start talking about rotation, it's hard to look at a given point and say, "Oh yeah, there’s going to 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 going to have some kind of rotation. 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 field is saying, "What kind of rotation is it going to 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. You’re wondering what direction it tends to rotate. 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.

We're kind of curling our fingers, curling our right-hand fingers over in that direction. 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 going to give you the curl.

That is what the curl of a three-dimensional vector field is trying to represent. 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, I kind of get it." Tornadoes of rotation—that sort of makes sense. 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 in the first time. It certainly took me a while to really wrap my head around this 3D curl idea. And with that, I'll see you in the next video.