3d curl formula, part 1

So I've spent a couple of videos laying down the foundation for what three-dimensional curl is trying to represent, and here I'm going to go ahead and talk about how you actually compute it.

So, 3D curl is the kind of thing that you take with regards to a three-dimensional vector field; something that takes in a three-dimensional point as its input and then it's gonna output a three-dimensional vector. It's common to write the component functions as um p, q, and r. So each one of these is a scalar-valued function that takes in a three-dimensional point and just outputs a number.

So, it'll be that same 3D point with the coordinates x, y, and z. When you have a three-dimensional vector field like this, the image you might have in mind would be something like this where every point in three-dimensional space has a vector attached to it. And, you know, when you actually look at it, there's quite a lot going on.

But in principle, all that's really happening is that each point in space is associated with a vector, and the point in space is the input and the vector is the output, and you're just kind of gluing them together. Naturally, because between the three dimensions of the input and the three dimensions of the output, we have six dimensions going on, the picture that you're looking at becomes quite messy.

So, the question is, how do you compute this curl value that I've been talking about? Curl of your vector-valued function. Just as a quick reminder, what this is supposed to be is you're going to have some kind of fluid flow induced by this vector field, where you're imagining like air flowing along each vector. What you want is a function that tells you at any given point what is the rotation induced by that fluid flow around that point.

Because rotation is described with a three-dimensional vector, you're expecting this to be vector-valued. It'll be something that equals a vector output. If that doesn't make sense, if that doesn't quite jive, maybe go check out the video on how to represent three-dimensional rotation with a vector.

So, what you have here is going to be something that takes as its input x, y, and z; it takes a three-dimensional point, and what it outputs is a vector describing rotation. There's actually another notation that's quite helpful when it comes to computing this, where you take nabla, that upside-down triangle we used in divergence and gradient, and you imagine taking the cross product between that and your vector v.

As a reminder, this nabla you imagine it as if it's a vector containing partial differential operators. That's the kind of thing where when you say it out loud, it sounds kind of fancy: a vector full of partial differential operators. But all it really means is I'm just going to write a bunch of symbols.

This like partial partial x is something that wants to take in a function, a multivariable function, and tell you its partial derivative. You know, strictly speaking, this doesn't really make sense; like, hey, how can a vector contain these partial differential operators? But as a series of symbolic movements, it's actually quite helpful.

Because when you're multiplying these guys by a thing, it's not really multiplication. You're really going to be giving it some kind of multivariable function like p, q, or r, the component functions of our vector field, and evaluating it.

So, just as a warm-up for how to do this, let's see what this looks like in the case of two dimensions, where we already kind of know the formula for two-dimensional curl. So, what that would look like is you have a smaller, more two-dimensional just partial partial x, partial partial y, you know, del operator, and you're going to take the cross product between that and a two-dimensional vector that's just the component functions p and q.

In this case, p and q would be just functions of x and y. So, I'm kind of overloading notation right over here. I have a two-dimensional vector field that I'm saying p and q are scalar-valued functions with a two-dimensional input, but over here, I'm also using p and q to represent ones with a three-dimensional input.

So, you should think of these as separate, but it's common to use the same names, and this is just kind of going to illustrate the broader, more complicated point. So, when you compute something like this, the cross product, you typically think of it as taking these diagonal components and multiplying them.

So, that would be your partial partial x quote unquote multiplied with q, which means you're taking the partial derivative of q with respect to x, and then you subtract off this diagonal component here, which is um, let's see, so partial partial… oh sorry, this should be a y. This should be partial partial y.

So, okay, sorry about that. That’s the partial with respect to y of p, and that's what you're subtracting off. So, partial partial y of p, so just the partial derivative of that p function with respect to y. Hopefully, this is something you recognize. This is the two-dimensional curl, and it's something we got an intuition for. I want it to be more than just a formula, but hopefully this is kind of reassuring that when you take that del operator, that nabla symbol, and take the cross product with the vector-valued function itself, it gives you a sense of curl.

Now, when we do this in the three-dimensional case, we're going to take a three-dimensional cross product between this three-dimensional vector-ish thing and this three-dimensional function. That would be a good time, by the way, if you're not terribly comfortable with the cross product of how to compute it or how to interpret it and things like that, that would probably be a good time to go find the videos that Sal does on this and build up that intuition for what a cross product actually is and how to compute it.

Because at this point, I'm going to assume that you know how to compute it, because we're doing it in kind of an absurd context of partial differential operators and functions. So, it's important to have that foundation.

The way you compute a thing like this is you construct a determinant. So, I'm going to go down here, determinant of a certain three by three matrix. The top row of that is all of the unit vectors in various directions of three-dimensional space. So these i, j, and k guys: i represents the unit vector in the x direction.

So that would be i is equal to, you know, x component is one, but then the other components are zero. Similarly, j and k represent the unit vectors in the y and z direction. Again, if that doesn't quite make sense, uh, why I'm putting them up there or what we're about to do, maybe check out that cross product video.

So, we put those in the top row as vectors, and this is kind of the trick to computing the cross product. Because again, it's like, what does it mean to put a vector inside a matrix? But it's a notational trick. Then we're going to take the first vector that we're doing the cross product with and put its components in the next row.

So what that would look like is the next row has a partial with respect to x, and oh sorry again, I keep messing up here. That's an x. You do whatever the first component is first, then the second component second, and the third component: the z partial partial z. I don't know why I'm making that little mistake.

Then for the last row, you put in the second vector, which in this case is vector-valued function p, q, and r. P, which is a multivariable function, q, and r. It's worth stepping back and looking at this. This is kind of an absurd thing that usually when we talk about matrices and taking the determinant, all of the components are numbers because you're multiplying numbers together.

But here, we've got like a notational trick layered on top of a notational trick, so that one of the rows is vectors, one of the rows is like partial differential operators, and then the last one, each one of these is a multi-variable function. So, it seems like this absurd, convoluted, as far away from a matrix full of numbers thing as you can get.

But it's actually very helpful for computation because if you go through the process of computing this determinant and saying what could that mean, the thing that pops out is going to be the formula for three-dimensional curl. At the risk of having a video that runs too long, I'll call things in here, but continue going through that operation in the next video. See you then!