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Current time:0:00Total duration:5:43

- [Voiceover] So after
introducing the idea of fluid rotation in a
vector field like this, let's start tightening up
our grasp on this intuition to get something that we can
actually apply formulas to. So a vector field like
the one that I had there, that's two-dimensional,
is given by a function that has a two-dimensional input and a two-dimensional output. And it's common to write the
components of that output as the functions p and q. So, each one of those p and q, takes in two different
variables as it's input. P and q. And what I want to do here, is
talk about this idea of curl and you might write it down as just curl, curl of v, the vector field. Which takes in the same inputs
that the vector field does. And because this is the
two-dimensional example, I might write, just to distinguish it from
three-dimensional curl, which is something we'll get
later on, two d curl of v. So you're kind of thinking of
this as a differential thing, in the same way that you have, you know, a derivative, dx is gonna take
in some kind of a function. And you give it a function and
it gives you a new function, the derivative. Here, you think of this 2d
curl, as like an operator, you give it a function,
a vector field function, and it gives you another function, which in this case will be scalar valued. And the reason it's scalar valued, is because at every given point, you want it to give you a number. So if I look back at the vector field, that I have here, we want, that at a point like this, where there's a lot of
counter-clockwise rotation happening around it, for the curl function to return a positive number. But at a point like
this, where there's some, where there's clockwise
rotation happening around it, you want the curl to
return a negative number. So, let's start thinking
about what that should mean. And a good way to understand this two-dimensional curl function and start to get a feel for it, is to imagine the
quintessential 2d curl scenario. Well let's say you have a point and this here's going to be our point, xy, sitting of somewhere in space. And let's say there's no
vector attached to it, as in the values, p and
q, and x and y, are zero. And then let's say that
to the right of it, you have a vector pointing straight up. Above it, in the vector field, you have a vector pointing
straight to the left, to it's left, you have one
pointing straight down, and below it, you have one
pointing straight to the right. So in terms of the
functions, what that means, is this vector, to it's right, whatever point it's evaluated at, that's gonna be q is greater than zero. So this function q, that
corresponds to the y component, the up and down component of each vector, when you evaluate it at this point, to the right of our xy point, q's gonna be greater than zero. Where as if you evaluate
it to the left over here, q would be less than zero, less than zero, in our kind of, perfect curl
will be positive example. And then these bottom guys, if you start thinking
about what this means for, you'd have a rightward vector below, and a leftward vector above, the one below it, whatever
point you're evaluating that at, p, which gives us the kind of, left right component of these vectors, since it's the first
component of the output, would have to be positive. And then above it, above it here, when you evaluate p at that point, would have to be negative. Where as p, if you did it on
the left and right points, would be equal to zero because
there's no x component. And similarly q, if you did it
on the top and bottom points, since there's no up and down
component of those vectors, would also be zero. So this is just the, the very specific, almost contrived scenario
that I'm looking at. And I want to say, hey if this
should have positive curl, maybe if we look at the information, the partial derivative
information to be specific, about p and q, in a scenario like this, it'll give us a way to
quantify the idea of curl. And first let's look at p. So p starts positive, and as y increases, as the y value of our input increases, it goes from being positive
to zero, to negative. So we would expect, that
the partial derivative of p, with respect to y, so as we change that y component,
moving up in the plane, and look at the x
component of the vectors, that should be negative. That should be negative in circumstances where we want positive curl. So all of this we're looking at cases, you know the quintessential
case where curl is positive. So evidently, this is a fact, that corresponds to positive curl. Where as q, let's take a look at q. It starts negative,
when you're at the left. And then becomes zero,
then it becomes positive. So here, as x increases, q increases. So we're expecting that a
partial derivative of q, with respect to x, should be positive. Or at the very least,
the situations where, the partial derivative of q
with respect to x is positive, corresponds to positive
two-dimensional curl. And in fact, it turns out, these guys tell us all you need to know. We can say as a formula, that the 2d curl, 2d curl,
of our vector field v, as a function of x and y, is equal to the partial
derivative of q with respect to x. Partial derivative of
q, with respect to x, and then I'm gonna subtract
off the partial of p, with respect to y. Because I want, when this is negative, for that to correspond
with more positive 2d curl. So I'm gonna subtract off, partial of p, with respect to y. And this right here, is the formula for two-dimensional curl. Which basically, you can
think of it as a measure, at any given point you're asking, how much does the surrounding
information to that point, look like this set-up, like this perfect
counter-clockwise rotation set-up? And the more it looks like this set-up, the more this value will be positive. And if it was the opposite of this, if each of the vectors was turned around and you have clockwise rotation, each of these values
will become the negative of what it had been before. So 2d curl would end up being negative. And in the next video,
I'll show some examples of what it looks like to use this formula.