3d curl formula, part 2

So I'm explaining the formula for three-dimensional curl, and where we left off, we have this determinant of a 3x3 matrix, which looks absurd because none of the individual components are actual numbers. But nevertheless, I'm about to show how, when you kind of go through the motions of taking a determinant, you get a vector-valued function that corresponds to the curl.

Let me show you what I mean by that. If you're computing the determinant of the guy that we have pictured there in the upper right, you start by taking this upper left component and then multiplying it by the determinant of the submatrix. The submatrix whose rows are not the row of I and whose columns are not the row of I.

What that looks like over here is we're going to take that unit vector I and then multiply it by a certain little determinant. What this subd determinant involves is multiplying this partial with respect to y by R, which means taking the partial derivative with respect to y of the multivariable function R and then subtracting off the partial derivative with respect to z of Q. So we're subtracting off the partial derivative with respect to z of the multivariable function Q.

And then, so that's the first thing that we do. As a second part, we take this J and we're going to subtract off, so you're kind of thinking plus minus plus for the elements in this top row. So we're going to subtract off J multiplied by another subd determinant, and then this one is going to involve this column that it's not part of and this column that it's not part of. You imagine those guys as a 2x2 matrix, and its determinant involves taking the partial derivative with respect to x of R. That's kind of the diagonal, partial with respect to x of R, and then subtracting off the partial derivative with respect to z of P, so partial with respect to z of P.

And then, that's just two out of three of the things we need to do for our overall determinant because the last part we're going to add. We're going to add that top right component K multiplied by the submatrix whose columns involve the column it's not part of and whose rows involve the rows that it's not part of. K multiplied by the determinant of this guy is going to be, let's see, partial with respect to x of Q, so that's partial with respect to x of Q minus partial with respect to y of P, so the partial derivative with respect to y of the multivariable function P.

And that entire expression is the three-dimensional curl of the function whose components are P, Q, and R. So here we have our vector-valued function V whose components are P, Q, and R. When you go through this whole process of imagining the cross product between the Del operator, this nabla symbol, and the vector output P, Q, and R, what you get is this whole expression.

Here we're writing it with IJK notation. If you were writing it as a column vector, I guess I didn't erase some of these guys, but if you were writing this as a column vector, it would look like saying the curl of your vector-valued function V as a function of x, y, and z is equal to. And then what I'd put in for this first component would be what's up there. So that would be your partial with respect to y of R minus the partial of Q with respect to z, so partial of Q with respect to z.

I won't copy it down for all of the other ones, but in principle, you know, whatever this J component is, and I guess we're subtracting it, so you'd subtract there. You'd copy that as the next component, and then over here, but often times when you're computing curl, you kind of switch to using this J, JK notation. My personal preference, I typically default to column vectors, and other people will write in terms of I, J, and K. It doesn't really matter as long as you know how to go back and forth between the two.

One really quick thing that I want to highlight before doing an example of this is that the K component here, the Z component of the output is exactly the two-dimensional curl formula. If you kind of look back to the videos on 2D curl and what its formula is, that is what we have here. In fact, all the other components kind of look like mirrors of that, but you're using slightly different operators and slightly different functions.

If you think about rotation that happens purely in the XY plane, just two-dimensional rotation, and how in three dimensions that's described with a vector in the K direction. Again, if that doesn't quite seem clear, maybe look back at the video on describing rotation with a three-dimensional vector and the right-hand rule. But vectors pointing in the pure Z direction describe rotation in the XY plane.

What's happening with these other guys is kind of similar. Right rotation that happens purely in the XZ plane is going to correspond with a rotation vector in the Y direction, the direction perpendicular to the X. Let's see, so the XZ plane over here. Similarly, this first component kind of tells you all the rotation happening in the YZ plane. The vectors in the I direction, the X direction of the output, correspond to rotation in that plane.

Now, when you compute it, you're not always thinking about, oh, you know, this corresponds to rotation in that plane and this corresponds to rotation in that plane. You're just kind of computing it to get a formula out. But I think it's kind of nice to recognize that all the intuition that we put into the two-dimensional curl does show up here.

Another thing I want to emphasize is this is not a formula to be memorized. I would not, if I were you, try to sit down and memorize this long expression. The only thing that you need to remember, the only thing, is that curl is represented as this Del cross V, this nabla symbol cross product with the vector-valued function V. Because from there, whatever your components are, you can kind of go through the process that I just did. The more you do it, the quicker it becomes.

It's kind of long, but it doesn't take that long, and it's certainly much more fault-tolerant than trying to remember something that has as many moving parts as the formula that you see here. In the next video, I'll go through an actual example of that. I'll have functions for P, Q, and R, and walk through that process in a more concrete context. I'll see you then.