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- [Voiceover] So I've spent a couple 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.
Three-d curl is the kind of thing that you take
with regards to a three-dimensional vector field.
So something that takes in a three-dimensional point
as its input, and then it's going to output
a three-dimensional vector.
It's common to write the component functions as P, Q, and R.
Each one of these is a scale or valued function
that takes in a three-dimensional point,
and just outputs a number.
So, it'll be that same three-d point
with the coordinates x, y, and z.
So 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.
The point in space is the input
and the vector is the output.
You're just gluing them together.
Naturally, 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 value 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
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,
and there's actually another notation
that's quite, quite helpful when it comes to computing this.
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 kinda 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 partial partial x is something that wants
to take in a multi-variable function,
and tell you its partial derivative.
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 multi-variable 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 know the formula
for two-dimensional curl.
What that would look like, is you have a smaller,
more two-dimensional, just partial partial x,
partial partial y, del operator.
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 y are scale or value 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.
This is going to illustrate
the broader, more complicated point.
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,
"multiplied" with Q, which really means you're taking
the partial derivative of Q with respect to x.
Then you subtract off this diagonal component here,
oh sorry, this should be a y.
This should be partial partial y.
Sorry about that.
You need partial partial y of P,
and that's what you're subtracting off.
So partial partial y of P, just the partial derivative
of that P function with respect to y.
Hopefully this is something you recognize.
This is the two-dimensional curl.
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 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 vectorish thing
and this three-dimensional function.
If you're not terribly comfortable with the cross-product,
how to compute it or how to interpret it
and things like that,
now 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.
I'm going to go down here.
Determinant of a certain 3x3 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,
x component is one but then the other components are zero.
Then similarly, J and K represent the unit vectors
in the y and z direction, and again,
if that doesn't quite make sense,
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 rows as vectors.
This is the trick to computing the cross-product,
because again, 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.
What that would look like, is the next row has a
partial partial y,
no sorry, God I keep messing up here,
that's an x, you do whatever the first component is first,
and then the second component second,
and the third component, the z, partial z.
Don't know why I'm making that little mistake.
For the last row, you put in the second vector,
which is in this case, is vector value function P, Q, and R.
P, which is a multi-variable function, Q, and R.
First, it's worth stepping back and looking at this.
This is kind of an absurd thing.
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 a notational trick layered
on top of a notational trick, so that one of the rows
is vectors, one of the rows
is 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.
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 and end here,
but continue going through that operation in the next video.