2d curl formula

WEBVTT
Kind: captions
Language: en

00:00:00.544 --> 00:00:01.821
- [Voiceover] So after
introducing the idea

00:00:01.821 --> 00:00:04.294
of fluid rotation in a
vector field like this,

00:00:04.294 --> 00:00:06.499
let's start tightening up
our grasp on this intuition

00:00:06.499 --> 00:00:09.460
to get something that we can
actually apply formulas to.

00:00:09.460 --> 00:00:11.062
So a vector field like
the one that I had there,

00:00:11.062 --> 00:00:13.523
that's two-dimensional,
is given by a function

00:00:13.523 --> 00:00:15.114
that has a two-dimensional input

00:00:15.114 --> 00:00:17.297
and a two-dimensional output.

00:00:17.297 --> 00:00:19.340
And it's common to write the
components of that output

00:00:19.340 --> 00:00:22.300
as the functions p and q.

00:00:22.300 --> 00:00:24.123
So, each one of those p and q,

00:00:24.123 --> 00:00:27.722
takes in two different
variables as it's input.

00:00:27.722 --> 00:00:29.963
P and q.

00:00:29.963 --> 00:00:33.121
And what I want to do here, is
talk about this idea of curl

00:00:33.121 --> 00:00:37.566
and you might write it down as just curl,

00:00:37.566 --> 00:00:40.096
curl of v, the vector field.

00:00:40.096 --> 00:00:43.161
Which takes in the same inputs
that the vector field does.

00:00:43.161 --> 00:00:45.240
And because this is the
two-dimensional example,

00:00:45.240 --> 00:00:46.343
I might write,

00:00:46.343 --> 00:00:48.119
just to distinguish it from
three-dimensional curl,

00:00:48.119 --> 00:00:51.242
which is something we'll get
later on, two d curl of v.

00:00:51.242 --> 00:00:53.866
So you're kind of thinking of
this as a differential thing,

00:00:53.866 --> 00:00:55.526
in the same way that you have, you know,

00:00:55.526 --> 00:00:59.764
a derivative, dx is gonna take
in some kind of a function.

00:00:59.764 --> 00:01:01.703
And you give it a function and
it gives you a new function,

00:01:01.703 --> 00:01:02.620
the derivative.

00:01:02.620 --> 00:01:05.360
Here, you think of this 2d
curl, as like an operator,

00:01:05.360 --> 00:01:07.438
you give it a function,
a vector field function,

00:01:07.438 --> 00:01:09.365
and it gives you another function,

00:01:09.365 --> 00:01:12.604
which in this case will be scalar valued.

00:01:12.604 --> 00:01:13.626
And the reason it's scalar valued,

00:01:13.626 --> 00:01:15.263
is because at every given point,

00:01:15.263 --> 00:01:16.645
you want it to give you a number.

00:01:16.645 --> 00:01:18.340
So if I look back at the vector field,

00:01:18.340 --> 00:01:19.582
that I have here,

00:01:19.582 --> 00:01:21.277
we want, that at a point like this,

00:01:21.277 --> 00:01:23.146
where there's a lot of
counter-clockwise rotation

00:01:23.146 --> 00:01:24.865
happening around it, for the curl function

00:01:24.865 --> 00:01:26.803
to return a positive number.

00:01:26.803 --> 00:01:28.545
But at a point like
this, where there's some,

00:01:28.545 --> 00:01:31.599
where there's clockwise
rotation happening around it,

00:01:31.599 --> 00:01:35.082
you want the curl to
return a negative number.

00:01:35.082 --> 00:01:37.659
So, let's start thinking
about what that should mean.

00:01:37.659 --> 00:01:38.959
And a good way to understand this

00:01:38.959 --> 00:01:40.085
two-dimensional curl function

00:01:40.085 --> 00:01:41.502
and start to get a feel for it,

00:01:41.502 --> 00:01:45.159
is to imagine the
quintessential 2d curl scenario.

00:01:45.159 --> 00:01:46.262
Well let's say you have a point

00:01:46.262 --> 00:01:48.444
and this here's going to be our point, xy,

00:01:48.444 --> 00:01:49.802
sitting of somewhere in space.

00:01:49.802 --> 00:01:51.846
And let's say there's no
vector attached to it,

00:01:51.846 --> 00:01:55.619
as in the values, p and
q, and x and y, are zero.

00:01:55.619 --> 00:01:58.266
And then let's say that
to the right of it,

00:01:58.266 --> 00:02:00.298
you have a vector pointing straight up.

00:02:00.298 --> 00:02:02.260
Above it, in the vector field,

00:02:02.260 --> 00:02:04.257
you have a vector pointing
straight to the left,

00:02:04.257 --> 00:02:07.067
to it's left, you have one
pointing straight down,

00:02:07.067 --> 00:02:10.457
and below it, you have one
pointing straight to the right.

00:02:10.457 --> 00:02:11.966
So in terms of the
functions, what that means,

00:02:11.966 --> 00:02:13.638
is this vector, to it's right,

00:02:13.638 --> 00:02:15.461
whatever point it's evaluated at,

00:02:15.461 --> 00:02:18.839
that's gonna be q is greater than zero.

00:02:18.839 --> 00:02:21.638
So this function q, that
corresponds to the y component,

00:02:21.638 --> 00:02:23.646
the up and down component of each vector,

00:02:23.646 --> 00:02:25.202
when you evaluate it at this point,

00:02:25.202 --> 00:02:26.723
to the right of our xy point,

00:02:26.723 --> 00:02:28.859
q's gonna be greater than zero.

00:02:28.859 --> 00:02:31.262
Where as if you evaluate
it to the left over here,

00:02:31.262 --> 00:02:35.116
q would be less than zero, less than zero,

00:02:35.116 --> 00:02:38.879
in our kind of, perfect curl
will be positive example.

00:02:38.879 --> 00:02:40.086
And then these bottom guys,

00:02:40.086 --> 00:02:42.025
if you start thinking
about what this means for,

00:02:42.025 --> 00:02:43.604
you'd have a rightward vector below,

00:02:43.604 --> 00:02:45.508
and a leftward vector above,

00:02:45.508 --> 00:02:48.318
the one below it, whatever
point you're evaluating that at,

00:02:48.318 --> 00:02:50.524
p, which gives us the kind of,

00:02:50.524 --> 00:02:52.021
left right component of these vectors,

00:02:52.021 --> 00:02:54.157
since it's the first
component of the output,

00:02:54.157 --> 00:02:56.108
would have to be positive.

00:02:56.108 --> 00:02:58.580
And then above it, above it here,

00:02:58.580 --> 00:03:00.821
when you evaluate p at that point,

00:03:00.821 --> 00:03:02.702
would have to be negative.

00:03:02.702 --> 00:03:04.827
Where as p, if you did it on
the left and right points,

00:03:04.827 --> 00:03:07.467
would be equal to zero because
there's no x component.

00:03:07.467 --> 00:03:09.878
And similarly q, if you did it
on the top and bottom points,

00:03:09.878 --> 00:03:12.316
since there's no up and down
component of those vectors,

00:03:12.316 --> 00:03:13.593
would also be zero.

00:03:13.593 --> 00:03:15.636
So this is just the, the very specific,

00:03:15.636 --> 00:03:18.156
almost contrived scenario
that I'm looking at.

00:03:18.156 --> 00:03:21.232
And I want to say, hey if this
should have positive curl,

00:03:21.232 --> 00:03:23.752
maybe if we look at the information,

00:03:23.752 --> 00:03:25.748
the partial derivative
information to be specific,

00:03:25.748 --> 00:03:28.454
about p and q, in a scenario like this,

00:03:28.454 --> 00:03:31.716
it'll give us a way to
quantify the idea of curl.

00:03:31.716 --> 00:03:33.052
And first let's look at p.

00:03:33.052 --> 00:03:35.734
So p starts positive, and as y increases,

00:03:35.734 --> 00:03:38.218
as the y value of our input increases,

00:03:38.218 --> 00:03:41.712
it goes from being positive
to zero, to negative.

00:03:41.712 --> 00:03:44.232
So we would expect, that
the partial derivative of p,

00:03:44.232 --> 00:03:46.136
with respect to y,

00:03:46.136 --> 00:03:48.957
so as we change that y component,
moving up in the plane,

00:03:48.957 --> 00:03:52.150
and look at the x
component of the vectors,

00:03:52.150 --> 00:03:54.669
that should be negative.

00:03:54.669 --> 00:03:56.016
That should be negative in circumstances

00:03:56.016 --> 00:03:57.618
where we want positive curl.

00:03:57.618 --> 00:03:59.289
So all of this we're looking at cases,

00:03:59.289 --> 00:04:01.913
you know the quintessential
case where curl is positive.

00:04:01.913 --> 00:04:03.632
So evidently, this is a fact,

00:04:03.632 --> 00:04:05.455
that corresponds to positive curl.

00:04:05.455 --> 00:04:07.556
Where as q, let's take a look at q.

00:04:07.556 --> 00:04:09.576
It starts negative,
when you're at the left.

00:04:09.576 --> 00:04:12.514
And then becomes zero,
then it becomes positive.

00:04:12.514 --> 00:04:16.019
So here, as x increases, q increases.

00:04:16.019 --> 00:04:18.353
So we're expecting that a
partial derivative of q,

00:04:18.353 --> 00:04:22.730
with respect to x, should be positive.

00:04:22.730 --> 00:04:24.797
Or at the very least,
the situations where,

00:04:24.797 --> 00:04:27.188
the partial derivative of q
with respect to x is positive,

00:04:27.188 --> 00:04:31.113
corresponds to positive
two-dimensional curl.

00:04:31.113 --> 00:04:32.657
And in fact, it turns out,

00:04:32.657 --> 00:04:34.956
these guys tell us all you need to know.

00:04:34.956 --> 00:04:36.418
We can say as a formula,

00:04:36.418 --> 00:04:41.418
that the 2d curl, 2d curl,
of our vector field v,

00:04:41.980 --> 00:04:46.738
as a function of x and y,

00:04:46.738 --> 00:04:50.721
is equal to the partial
derivative of q with respect to x.

00:04:50.721 --> 00:04:54.378
Partial derivative of
q, with respect to x,

00:04:54.378 --> 00:04:56.804
and then I'm gonna subtract
off the partial of p,

00:04:56.804 --> 00:04:58.116
with respect to y.

00:04:58.116 --> 00:04:59.567
Because I want, when this is negative,

00:04:59.567 --> 00:05:02.377
for that to correspond
with more positive 2d curl.

00:05:02.377 --> 00:05:05.697
So I'm gonna subtract off, partial of p,

00:05:05.697 --> 00:05:08.867
with respect to y.

00:05:08.867 --> 00:05:10.144
And this right here,

00:05:10.144 --> 00:05:13.046
is the formula for two-dimensional curl.

00:05:13.046 --> 00:05:14.845
Which basically, you can
think of it as a measure,

00:05:14.845 --> 00:05:16.041
at any given point you're asking,

00:05:16.041 --> 00:05:18.839
how much does the surrounding
information to that point,

00:05:18.839 --> 00:05:20.604
look like this set-up,

00:05:20.604 --> 00:05:23.680
like this perfect
counter-clockwise rotation set-up?

00:05:23.680 --> 00:05:25.097
And the more it looks like this set-up,

00:05:25.097 --> 00:05:27.802
the more this value will be positive.

00:05:27.802 --> 00:05:29.044
And if it was the opposite of this,

00:05:29.044 --> 00:05:30.716
if each of the vectors was turned around

00:05:30.716 --> 00:05:32.016
and you have clockwise rotation,

00:05:32.016 --> 00:05:34.304
each of these values
will become the negative

00:05:34.304 --> 00:05:35.744
of what it had been before.

00:05:35.744 --> 00:05:38.379
So 2d curl would end up being negative.

00:05:38.379 --> 00:05:39.877
And in the next video,
I'll show some examples

00:05:39.877 --> 00:05:42.913
of what it looks like to use this formula.


3d curl intuition, part 1

WEBVTT
Kind: captions
Language: en

00:00:00.345 --> 00:00:01.185
- [Voiceover] Hello, everyone.

00:00:01.185 --> 00:00:03.706
So I'm gonna start talking
about three-dimensional curl.

00:00:03.706 --> 00:00:05.083
And to do that, I'm gonna start off

00:00:05.083 --> 00:00:07.123
by taking the two-dimensional example

00:00:07.123 --> 00:00:08.036
that I very first used

00:00:08.036 --> 00:00:09.889
when I was introducing the intuition.

00:00:09.889 --> 00:00:11.221
You know I talked about fluid flow

00:00:11.221 --> 00:00:12.588
and I animated it here

00:00:12.588 --> 00:00:14.205
where with this particular vector field

00:00:14.205 --> 00:00:16.990
you see a certain counter-clockwise
rotation on the right

00:00:16.990 --> 00:00:18.978
and a clockwise rotation on the top.

00:00:18.978 --> 00:00:20.361
So I'm gonna take that vector field

00:00:20.361 --> 00:00:22.998
which hopefully we have a
little bit of an intuition for

00:00:22.998 --> 00:00:25.639
and I'm gonna plop it into
three-dimensional space

00:00:25.639 --> 00:00:27.436
on the x-y plane.

00:00:27.436 --> 00:00:29.491
So if I just take that whole vector field

00:00:29.491 --> 00:00:31.416
and I copy it onto the x-y plane

00:00:31.416 --> 00:00:32.617
here's how it looks.

00:00:32.617 --> 00:00:34.767
And the vector spacing might
be a little bit different,

00:00:34.767 --> 00:00:36.608
the choice of what
subsample of points to use

00:00:36.608 --> 00:00:38.536
in displaying vectors might
be a little bit different

00:00:38.536 --> 00:00:40.124
but this is the same field,

00:00:40.124 --> 00:00:42.148
and it's actually worth writing out

00:00:42.148 --> 00:00:44.853
how it's defined in two-dimensions.

00:00:44.853 --> 00:00:47.119
This guy is a function of x and y,

00:00:47.119 --> 00:00:51.503
so it's a vector field function of x and y

00:00:51.503 --> 00:00:53.387
and what it's components are.

00:00:53.387 --> 00:00:57.554
Now the first component
is y cubed minus nine y

00:00:58.838 --> 00:01:03.005
and then the next one
is x cubed minus nine x.

00:01:05.056 --> 00:01:06.545
So now if I look at this guy

00:01:06.545 --> 00:01:07.770
and I say let's start thinking about

00:01:07.770 --> 00:01:09.890
the fluid rotation associated with it,

00:01:09.890 --> 00:01:11.597
because it's in three-dimensions

00:01:11.597 --> 00:01:13.599
it's natural to describe that rotation,

00:01:13.599 --> 00:01:15.493
not just with a number at each point,

00:01:15.493 --> 00:01:16.956
you know with a scale or value

00:01:16.956 --> 00:01:18.677
like the two-dimensional curl gives us,

00:01:18.677 --> 00:01:21.241
but instead to assign
a vector to each one.

00:01:21.241 --> 00:01:22.131
And when you do that,

00:01:22.131 --> 00:01:24.611
when you associate a
vector to each different

00:01:24.611 --> 00:01:26.682
point in space according
to the fluid rotation

00:01:26.682 --> 00:01:28.034
that would be happening there,

00:01:28.034 --> 00:01:30.347
you get something that looks like this.

00:01:30.347 --> 00:01:31.421
Now this is kind of complicated

00:01:31.421 --> 00:01:33.621
because there's two different
vector fields going on.

00:01:33.621 --> 00:01:34.603
One of them, all the vectors

00:01:34.603 --> 00:01:36.474
are perpendicular to the x-y plane.

00:01:36.474 --> 00:01:37.988
So let's just kind of
take it piece by piece

00:01:37.988 --> 00:01:39.803
and see if we can understand it.

00:01:39.803 --> 00:01:43.412
So I've got four different
circled regions here

00:01:43.412 --> 00:01:45.012
and one of them is this one on the right

00:01:45.012 --> 00:01:47.674
where there's counter-clockwise
rotation happening.

00:01:47.674 --> 00:01:49.605
And if you think to your right hand rule,

00:01:49.605 --> 00:01:50.438
I'm gonna go ahead and bring in

00:01:50.438 --> 00:01:52.426
the picture of the right hand rule here,

00:01:52.426 --> 00:01:54.437
where you imagine curling your fingers

00:01:54.437 --> 00:01:56.675
around that direction of rotation,

00:01:56.675 --> 00:01:58.182
in the fingers of your right hand,

00:01:58.182 --> 00:01:59.666
then you stick out your thumb,

00:01:59.666 --> 00:02:01.505
the direction your thumb is pointing

00:02:01.505 --> 00:02:03.068
would be the direction of vectors

00:02:03.068 --> 00:02:05.364
that describe that rotation.

00:02:05.364 --> 00:02:08.324
So if we do that here and if we

00:02:08.324 --> 00:02:10.209
you know, you imagine curling
your right fingers around

00:02:10.209 --> 00:02:11.637
there and sticking up your thumb

00:02:11.637 --> 00:02:12.839
you're gonna get vectors that point

00:02:12.839 --> 00:02:14.691
in the positive z-direction.

00:02:14.691 --> 00:02:16.755
So this is why in that
region you have vectors

00:02:16.755 --> 00:02:19.206
pointing up, positively
in the z-direction,

00:02:19.206 --> 00:02:21.959
they're telling you that
as you view this x-y plane

00:02:21.959 --> 00:02:24.466
from above there's
counter-clockwise rotation.

00:02:24.466 --> 00:02:27.480
But then what about at a different point?

00:02:27.480 --> 00:02:28.571
What about up here at the top

00:02:28.571 --> 00:02:30.133
where you have clockwise rotation?

00:02:30.133 --> 00:02:31.230
Now there, if you imagine

00:02:31.230 --> 00:02:33.170
taking the fingers of your right hand

00:02:33.170 --> 00:02:36.061
and curling them around
that direction of rotation

00:02:36.061 --> 00:02:38.019
your thumb is gonna be
pointing straight down.

00:02:38.019 --> 00:02:40.103
It'll be kind of in the
negative z-direction,

00:02:40.103 --> 00:02:42.679
and we see that with
this vector field here.

00:02:42.679 --> 00:02:44.797
Where below that circle, below that point,

00:02:44.797 --> 00:02:47.567
you have vectors pointing
straight down indicating that

00:02:47.567 --> 00:02:50.398
that's the direction of
rotation in that region.

00:02:50.398 --> 00:02:51.713
So if you do this at every single point

00:02:51.713 --> 00:02:53.121
and you kind of get an understanding

00:02:53.121 --> 00:02:54.521
of what the rotation is at every point

00:02:54.521 --> 00:02:55.484
and assign a vector

00:02:55.484 --> 00:02:57.355
this is the field that you're gonna get.

00:02:57.355 --> 00:02:58.639
And let's go ahead and describe that

00:02:58.639 --> 00:03:00.841
with an actual, with a function.

00:03:00.841 --> 00:03:03.354
Because we know how to compute
the two-D curl at this point.

00:03:03.354 --> 00:03:04.917
You see if this whole thing,

00:03:04.917 --> 00:03:08.288
if we give names to the two
different component functions

00:03:08.288 --> 00:03:10.788
as P and Q

00:03:11.688 --> 00:03:14.667
then the curl, the two-D curl of this guy,

00:03:14.667 --> 00:03:17.583
two-D curl

00:03:17.583 --> 00:03:21.073
of the vector field v

00:03:21.073 --> 00:03:23.406
has a function of x and y

00:03:24.892 --> 00:03:26.768
what it equals is the partial derivative

00:03:26.768 --> 00:03:29.582
of that second component
with respect to x.

00:03:29.582 --> 00:03:33.833
So the partial of q with respect to x

00:03:33.833 --> 00:03:36.194
minus the partial derivative
of that first component

00:03:36.194 --> 00:03:37.642
with respect to y.

00:03:37.642 --> 00:03:42.387
So minus partial of p with respect to y.

00:03:42.387 --> 00:03:44.200
And what we get when we do that,

00:03:44.200 --> 00:03:46.640
the partial of q with respect to x

00:03:46.640 --> 00:03:48.200
So we take the partial derivative of this

00:03:48.200 --> 00:03:49.171
with respect to x,

00:03:49.171 --> 00:03:50.453
that just looks like a derivative

00:03:50.453 --> 00:03:52.278
since there's only x's in there,

00:03:52.278 --> 00:03:55.778
and you get three x squared minus nine.

00:03:57.681 --> 00:03:58.789
And I actually did this,

00:03:58.789 --> 00:04:01.110
there's another video where
this is the example that I do,

00:04:01.110 --> 00:04:05.100
when you take this second
derivative of p with respect to y,

00:04:05.100 --> 00:04:06.681
you're taking the
derivative of this top part

00:04:06.681 --> 00:04:08.262
with respect to y

00:04:08.262 --> 00:04:12.429
and that's three y squared minus nine.

00:04:13.343 --> 00:04:14.372
And what you can do is you can say,

00:04:14.372 --> 00:04:16.218
this minus nine cancels out with that,

00:04:16.218 --> 00:04:17.720
you know minus minus nine.

00:04:17.720 --> 00:04:19.689
So these guys cancel,

00:04:19.689 --> 00:04:22.621
and what you ultimately get is

00:04:22.621 --> 00:04:27.314
three x squared plus three y squared.

00:04:27.314 --> 00:04:28.946
Now what does this mean
for the vector field

00:04:28.946 --> 00:04:29.779
that we see here?

00:04:29.779 --> 00:04:31.938
Because this, this is a
scale of valued quantity

00:04:31.938 --> 00:04:34.270
and yet the vector field that I'm showing,

00:04:34.270 --> 00:04:36.719
with all these blue vectors
indicating rotation,

00:04:36.719 --> 00:04:38.574
these are vectors.

00:04:38.574 --> 00:04:40.537
And because the rotation is happening

00:04:40.537 --> 00:04:42.320
purely in the x-y plane,

00:04:42.320 --> 00:04:44.427
which is perpendicular to the z-axis,

00:04:44.427 --> 00:04:48.219
all of these vectors
purely have a z-component.

00:04:48.219 --> 00:04:50.304
So what you might say is that the curl,

00:04:50.304 --> 00:04:51.438
and I'm not gonna say two-D curl,

00:04:51.438 --> 00:04:54.849
but actual curl of v

00:04:54.849 --> 00:04:59.050
of v as a function of x and y

00:04:59.050 --> 00:05:00.581
is a, it's not a scale or value,

00:05:00.581 --> 00:05:02.764
but it's a vector and it's
gonna be a vector that describes

00:05:02.764 --> 00:05:05.355
these blue ones in the pure z-direction.

00:05:05.355 --> 00:05:06.871
And since they're in the pure z-direction

00:05:06.871 --> 00:05:08.568
the x and y components are zero.

00:05:08.568 --> 00:05:12.113
But that last component is
the formula that we found

00:05:12.113 --> 00:05:15.810
that describes the magnitude of the curl.

00:05:15.810 --> 00:05:20.259
Three x squared minus three y squared,

00:05:20.259 --> 00:05:23.909
and this, let's see kinda
running out of room,

00:05:23.909 --> 00:05:26.098
this here you can think
of as kind of a prototype

00:05:26.098 --> 00:05:27.407
to three-dimensional curl.

00:05:27.407 --> 00:05:29.286
Because really, this vector field v,

00:05:29.286 --> 00:05:31.885
is not quite a three-dimensional
vector field is it?

00:05:31.885 --> 00:05:33.560
It only lives in the x-y plane,

00:05:33.560 --> 00:05:35.796
it only take in x and y's input points.

00:05:35.796 --> 00:05:38.272
So what we wanna do is
start extending this to say,

00:05:38.272 --> 00:05:40.287
how can you make this look
like a three-dimensional

00:05:40.287 --> 00:05:43.721
vector field and still kind
of understand the rotation

00:05:43.721 --> 00:05:46.680
as a three-dimensional vector quantity?

00:05:46.680 --> 00:05:49.224
And that's what I'm gonna
continue doing in the next video.


2d curl intuition

WEBVTT
Kind: captions
Language: en

00:00:00.379 --> 00:00:01.295
- [Voiceover] Hello everyone.

00:00:01.295 --> 00:00:03.061
So I'm gonna start talking about curl.

00:00:03.061 --> 00:00:06.256
Curl is one of those very
cool vector calculus concepts,

00:00:06.256 --> 00:00:08.067
and you'll be pretty happy
that you've learned it

00:00:08.067 --> 00:00:09.654
once you have, if for no other reason

00:00:09.654 --> 00:00:11.487
because it's kind of
artistically pleasing.

00:00:11.487 --> 00:00:12.738
And, there's two different versions,

00:00:12.738 --> 00:00:15.420
there's a two-dimensional curl
and a three-dimensional curl.

00:00:15.420 --> 00:00:16.739
And naturally enough, I'll start talking

00:00:16.739 --> 00:00:17.945
about the two-dimensional version

00:00:17.945 --> 00:00:20.382
and kind of build our
way up to the 3D one.

00:00:20.382 --> 00:00:21.768
And in this particular video,

00:00:21.768 --> 00:00:23.197
I just want to lay down the intuition

00:00:23.197 --> 00:00:25.410
for what's visually going on.

00:00:25.410 --> 00:00:28.293
And, curl has to do with the
fluid flow interpretation

00:00:28.293 --> 00:00:29.589
of vector fields.

00:00:29.589 --> 00:00:30.661
Now this is something
that I've talked about

00:00:30.661 --> 00:00:31.533
in other videos,

00:00:31.533 --> 00:00:33.946
especially the ones on
divergents if you watch that,

00:00:33.946 --> 00:00:35.912
but just as a reminder,
you kind of imagine

00:00:35.912 --> 00:00:38.125
that each point in space is a particle,

00:00:38.125 --> 00:00:40.293
like an air molecule or a water molecule.

00:00:40.293 --> 00:00:41.836
And since what a vector field does

00:00:41.836 --> 00:00:45.010
is associate each point in
space with some kind of vector,

00:00:45.010 --> 00:00:47.513
now remember we don't always
draw every single vector,

00:00:47.513 --> 00:00:49.055
we just draw a small sub-sample,

00:00:49.055 --> 00:00:51.446
but in principle, every
single point in space

00:00:51.446 --> 00:00:52.955
has a vector attached to it.

00:00:52.955 --> 00:00:54.547
You can think of each particle,

00:00:54.547 --> 00:00:56.826
each one of these water
molecules or air molecules,

00:00:56.826 --> 00:01:01.096
as moving over time in such a
way that the velocity vector

00:01:01.096 --> 00:01:03.531
of its movement at any given point in time

00:01:03.531 --> 00:01:05.297
is the vector that it's attached to.

00:01:05.297 --> 00:01:06.951
So as it moves to a
different location in space

00:01:06.951 --> 00:01:08.671
and that velocity vector changes,

00:01:08.671 --> 00:01:10.750
it might be turning or
it might be accelerating,

00:01:10.750 --> 00:01:12.226
and that velocity might change.

00:01:12.226 --> 00:01:16.393
And you end up kind of a
trajectory for your point.

00:01:17.231 --> 00:01:19.511
And since every single
point is moving in this way,

00:01:19.511 --> 00:01:21.277
you can start thinking about a flow,

00:01:21.277 --> 00:01:23.333
kind of a global view of the vector field.

00:01:23.333 --> 00:01:24.696
And for this particular example,

00:01:24.696 --> 00:01:26.752
this particular vector
field that I have pictured,

00:01:26.752 --> 00:01:28.630
I'm gonna go ahead and put a blue dot

00:01:28.630 --> 00:01:30.594
at various points in space,

00:01:30.594 --> 00:01:32.180
and, each one of these you can think of

00:01:32.180 --> 00:01:34.304
as representing a water
molecule or something,

00:01:34.304 --> 00:01:36.539
and I'm just gonna let it play.

00:01:36.539 --> 00:01:39.355
And at any given moment,
if you look at the movement

00:01:39.355 --> 00:01:41.008
of one of these blue dots,

00:01:41.008 --> 00:01:43.155
it's moving along the
vector that it's attached to

00:01:43.155 --> 00:01:45.545
at that point, or if that
vector's not pictured,

00:01:45.545 --> 00:01:47.110
you know the vector that
would be attached to it

00:01:47.110 --> 00:01:48.697
at that point.

00:01:48.697 --> 00:01:50.686
And as we get kind of a
feel for what's going on

00:01:50.686 --> 00:01:52.094
in this entire flow,

00:01:52.094 --> 00:01:55.827
I want you to notice a
couple of particular regions.

00:01:55.827 --> 00:01:57.555
First, let's take a look at this region

00:01:57.555 --> 00:01:59.365
over here on the right.

00:01:59.365 --> 00:02:00.660
Kind of around here.

00:02:00.660 --> 00:02:03.432
And just kind of concentrate
on what's going on there.

00:02:03.432 --> 00:02:04.706
And I'll go ahead

00:02:04.706 --> 00:02:07.263
and start playing the animation over here.

00:02:07.263 --> 00:02:09.074
And what's most noticeable
about this region

00:02:09.074 --> 00:02:11.778
is that there's counterclockwise rotation.

00:02:11.778 --> 00:02:14.259
And this corresponds to an
idea that the vector field

00:02:14.259 --> 00:02:16.874
has a curl here, and
I'll go very specifically

00:02:16.874 --> 00:02:18.729
into what curl means, but just right now

00:02:18.729 --> 00:02:20.629
you should have the idea that in a region

00:02:20.629 --> 00:02:22.573
where there's counterclockwise rotation,

00:02:22.573 --> 00:02:24.808
we want to say the curl is positive.

00:02:24.808 --> 00:02:27.512
Whereas, if you look at a
region that also has rotation,

00:02:27.512 --> 00:02:29.657
but clockwise, going the other way,

00:02:29.657 --> 00:02:32.026
we think of that as being negative curl.

00:02:32.026 --> 00:02:33.948
Here I'll start it over here.

00:02:33.948 --> 00:02:35.401
And in contrast, if you look at a place

00:02:35.401 --> 00:02:38.195
where there's no rotation,
where like at the center here,

00:02:38.195 --> 00:02:40.474
you have some points coming
in from the top right

00:02:40.474 --> 00:02:42.329
and from the bottom
left, and then going out

00:02:42.329 --> 00:02:43.380
from the other corners.

00:02:43.380 --> 00:02:44.677
But there's no net rotation.

00:02:44.677 --> 00:02:48.945
If you were to just put like a
twig somewhere in this water,

00:02:48.945 --> 00:02:50.934
it wouldn't really be rotating.

00:02:50.934 --> 00:02:52.499
These are regions where you think of them

00:02:52.499 --> 00:02:54.666
as having zero curl.

00:02:54.666 --> 00:02:57.483
So with that as a general idea,
clockwise rotation regions

00:02:57.483 --> 00:02:59.025
correspond to positive curl.

00:02:59.025 --> 00:03:01.125
Counterclockwise rotation regions

00:03:01.125 --> 00:03:02.309
correspond to negative curl,

00:03:02.309 --> 00:03:04.523
and then no rotation
corresponds to zero curl.

00:03:04.523 --> 00:03:06.691
In the next video I'm
gonna start going through

00:03:06.691 --> 00:03:08.545
what this means in terms
of the underlying function

00:03:08.545 --> 00:03:11.093
defining the vector field
and how we can start looking

00:03:11.093 --> 00:03:14.177
at the partial differential
information of that function

00:03:14.177 --> 00:03:17.284
to quantify this intuition
of fluid rotation.

00:03:17.284 --> 00:03:19.809
And what's neat is that it's
not just about fluid rotation.

00:03:19.809 --> 00:03:22.760
If you have vector fields in other context

00:03:22.760 --> 00:03:25.017
and you just imagine that
they represent a fluid,

00:03:25.017 --> 00:03:26.201
even though they don't,

00:03:26.201 --> 00:03:28.123
this idea rotation and curling

00:03:28.123 --> 00:03:29.844
actually has certain importance

00:03:29.844 --> 00:03:32.526
in ways that you totally wouldn't expect.

00:03:32.526 --> 00:03:34.671
The gradient turns out
to relate to the curl,

00:03:34.671 --> 00:03:36.929
even though you wouldn't
necessarily think the grading

00:03:36.929 --> 00:03:38.695
has something to do with fluid rotation.

00:03:38.695 --> 00:03:41.320
In electromagnetism, this
idea of fluid rotation

00:03:41.320 --> 00:03:42.415
has a certain importance,

00:03:42.415 --> 00:03:44.493
even though fluids
aren't actually involved.

00:03:44.493 --> 00:03:47.041
So, it's more general than
just the representation

00:03:47.041 --> 00:03:49.432
that we have here, but
it's a very strong visual

00:03:49.432 --> 00:03:53.078
to have in your mind as
you study vector fields.


2d curl example

WEBVTT
Kind: captions
Language: en

00:00:00.832 --> 00:00:02.573
- [Voiceover] So let's compute
the two dimensional curl

00:00:02.573 --> 00:00:03.793
of a vector field.

00:00:03.793 --> 00:00:07.491
The one I have in mind will
have an x component of,

00:00:07.491 --> 00:00:08.931
let's see not nine.

00:00:08.931 --> 00:00:11.431
But y cubed minus nine times y

00:00:12.705 --> 00:00:16.153
and then the y component will be x cubed

00:00:16.153 --> 00:00:18.253
minus nine times x.

00:00:18.253 --> 00:00:20.193
You can kind see I'm just
a sucker for symmetry

00:00:20.193 --> 00:00:21.690
when I choose examples.

00:00:21.690 --> 00:00:23.166
When I showed in the last video

00:00:23.166 --> 00:00:24.872
how the two dimensional curl,

00:00:24.872 --> 00:00:27.372
the 2D curl of a vector field,

00:00:28.221 --> 00:00:33.115
of a vector field v which
is a function of x and y,

00:00:33.115 --> 00:00:36.365
is equal to the partial derivative of q,

00:00:36.365 --> 00:00:39.698
that second component, with respect to x

00:00:41.113 --> 00:00:45.838
minus the partial derivative
of p that first component,

00:00:45.838 --> 00:00:47.755
with respect to y.

00:00:47.755 --> 00:00:49.520
And I went through the
reasoning for why this is true

00:00:49.520 --> 00:00:51.760
but just real quick kinda
the in the nutshell here,

00:00:51.760 --> 00:00:54.640
this partial q, partial x
is because of as you move

00:00:54.640 --> 00:00:58.088
from left to right vectors
tend to go from having a small

00:00:58.088 --> 00:01:01.420
or even negative y component
to a positive y component,

00:01:01.420 --> 00:01:04.230
that corresponds to
counter clockwise rotation.

00:01:04.230 --> 00:01:07.911
And similarly this dp,
dy is because if vectors

00:01:07.911 --> 00:01:11.764
as you move up and down as you
kind of increase the y value

00:01:11.764 --> 00:01:13.761
go from being positive to zero to negative

00:01:13.761 --> 00:01:16.269
or if they're decreasing
that also corresponds

00:01:16.269 --> 00:01:17.720
to counter clockwise rotation.

00:01:17.720 --> 00:01:19.810
So taking the negative of that
will tell you whether or not

00:01:19.810 --> 00:01:22.549
changes in the y direction
around your point

00:01:22.549 --> 00:01:24.790
correspond with counter
clockwise rotation.

00:01:24.790 --> 00:01:27.189
So in this particular case
when we start evaluating that,

00:01:27.189 --> 00:01:30.103
we start by looking at partial
of q with respect to x.

00:01:30.103 --> 00:01:32.159
So we're looking at the second
component and taking its

00:01:32.159 --> 00:01:34.840
partial derivative with
respect to x and in this case,

00:01:34.840 --> 00:01:37.080
nothing but x's show up so
it's just like taking its

00:01:37.080 --> 00:01:41.247
derivative and you get
three x squared minus nine.

00:01:45.138 --> 00:01:48.761
Three x squared minus nine,
and that's the first part.

00:01:48.761 --> 00:01:51.929
Then we subtract off whatever
the partial derivative of p

00:01:51.929 --> 00:01:54.589
with respect to y is, so we
go up here and it's entirely

00:01:54.589 --> 00:01:57.155
in terms of y and trying to
do the symmetry we're just

00:01:57.155 --> 00:02:00.255
taking the same calculation,
three y squared that derivative

00:02:00.255 --> 00:02:02.088
of y cubed minus nine.

00:02:04.492 --> 00:02:08.285
So this right here is
our two dimensional curl.

00:02:08.285 --> 00:02:10.444
And lets go ahead and
interpret what this means.

00:02:10.444 --> 00:02:12.267
And in fact this vector
field that I showed you

00:02:12.267 --> 00:02:14.869
is exactly the one that
I used when I was kind of

00:02:14.869 --> 00:02:17.527
animating the intuition
behind curl to start off with,

00:02:17.527 --> 00:02:19.512
where I had these specific
parts where there is

00:02:19.512 --> 00:02:22.124
positive curl here and
here but negative curl

00:02:22.124 --> 00:02:24.772
up in these clockwise rotating areas.

00:02:24.772 --> 00:02:27.186
So we can actually see
why that's the case here

00:02:27.186 --> 00:02:29.530
and why I chose this specific
function for something

00:02:29.530 --> 00:02:31.750
that'll have lots of good curl examples.

00:02:31.750 --> 00:02:34.551
Cause if we look over in that
region where there should be

00:02:34.551 --> 00:02:37.966
positive curl, that's
where x is equal to three

00:02:37.966 --> 00:02:40.067
and y is equal to zero.

00:02:40.067 --> 00:02:44.422
So I go over here and say
if x is equal to three,

00:02:44.422 --> 00:02:47.922
and y is equal to zero, this whole formula

00:02:48.983 --> 00:02:51.388
becomes let's see, three
times three squared so,

00:02:51.388 --> 00:02:55.079
three times three squared minus nine,

00:02:55.079 --> 00:02:58.899
minus nine and then minus the
quantity now we're plugging

00:02:58.899 --> 00:03:03.299
in y here so that's three
times y squared is just zero

00:03:03.299 --> 00:03:06.863
cause y is equal to zero, minus nine.

00:03:06.863 --> 00:03:09.790
Minus nine and so this part is 27,

00:03:09.790 --> 00:03:14.468
that's three times nine is
27 minus nine gives us 18.

00:03:14.468 --> 00:03:16.720
And then we're subtracting
off a negative nine

00:03:16.720 --> 00:03:19.564
so that's actually plus nine
so this whole thing is 27,

00:03:19.564 --> 00:03:22.943
it's actually quite positive,
so this is a positive number

00:03:22.943 --> 00:03:25.289
and that's why when we go
over here and we're looking

00:03:25.289 --> 00:03:28.318
at the fluid flow, you have
a counter clockwise rotation

00:03:28.318 --> 00:03:29.568
in that region.

00:03:30.420 --> 00:03:33.102
Whereas, let's say we did
all of this but instead of

00:03:33.102 --> 00:03:36.655
x equals three and y
equals zero, we looked at

00:03:36.655 --> 00:03:40.238
x is equal to zero and
y is equal to three.

00:03:42.276 --> 00:03:43.609
So in that case,

00:03:45.583 --> 00:03:48.102
we would instead, so x
equals zero, y equals three,

00:03:48.102 --> 00:03:49.866
let's take a look at where that is.

00:03:49.866 --> 00:03:53.221
X is zero, and then y the tick
mark's here are each one half

00:03:53.221 --> 00:03:56.643
so y equals three is
right here, it's in that

00:03:56.643 --> 00:03:59.840
clockwise rotation area
so if I kind of play this,

00:03:59.840 --> 00:04:01.859
we got the clockwise
rotation, we're expecting

00:04:01.859 --> 00:04:03.010
a negative value.

00:04:03.010 --> 00:04:04.744
Now let's see if that's what we get.

00:04:04.744 --> 00:04:06.625
We go over here and I'm gonna evaluate

00:04:06.625 --> 00:04:08.540
this whole function again.

00:04:08.540 --> 00:04:11.930
How about plugging in
zero for x so this is

00:04:11.930 --> 00:04:15.180
three times zero times zero minus nine.

00:04:17.341 --> 00:04:20.080
And then we're subtracting off

00:04:20.080 --> 00:04:24.620
three times y squared so that's
three times three squared.

00:04:24.620 --> 00:04:26.703
Three squared minus nine.

00:04:27.848 --> 00:04:30.286
And this whole part is zero
minus nine so that becomes

00:04:30.286 --> 00:04:34.815
negative nine and over
here we're subtracting off

00:04:34.815 --> 00:04:38.982
27 minus nine which is 18
so we're subtracting off 18,

00:04:40.341 --> 00:04:44.229
so the whole thing equals negative 27.

00:04:44.229 --> 00:04:47.608
So maybe I should say
that equals negative 27.

00:04:47.608 --> 00:04:50.742
So because this is negative
that's what corresponds

00:04:50.742 --> 00:04:54.051
to the clockwise rotation that we have

00:04:54.051 --> 00:04:55.572
going on in that region.

00:04:55.572 --> 00:04:57.186
And if you went and you plugged in a bunch

00:04:57.186 --> 00:04:59.044
of different points like
you could perhaps see

00:04:59.044 --> 00:05:01.586
how if you plug in zero
for x and zero for y,

00:05:01.586 --> 00:05:04.210
those nines cancel out
which is why over here

00:05:04.210 --> 00:05:06.428
there's no general
rotation around the origin

00:05:06.428 --> 00:05:08.343
when x and y are both equal to zero.

00:05:08.343 --> 00:05:09.934
And you can understand
that every single point

00:05:09.934 --> 00:05:12.314
and the general rotation
around every single point

00:05:12.314 --> 00:05:16.427
just by taking this formula
that we found for 2D curl,

00:05:16.427 --> 00:05:19.233
and plugging in the
corresponding values of x and y.

00:05:19.233 --> 00:05:21.987
So it's actually a very powerful
tool cause you would think

00:05:21.987 --> 00:05:24.184
that's a very complicated
thing to figure out right,

00:05:24.184 --> 00:05:26.908
that if I give you this
pretty complicated fluid flow

00:05:26.908 --> 00:05:29.787
and say hey I want you to
figure out a number that'll

00:05:29.787 --> 00:05:32.445
tell me the general direction
and strength of rotation

00:05:32.445 --> 00:05:35.220
around each point, that's a
lot of information so it's nice

00:05:35.220 --> 00:05:38.311
just to have a small compact formula.


2d curl nuance

WEBVTT
Kind: captions
Language: en

00:00:00.764 --> 00:00:01.673
- [Voiceover] In the last couple videos

00:00:01.673 --> 00:00:03.877
I've been talking about
curl where if we have

00:00:03.877 --> 00:00:05.702
a two dimensional vector field V

00:00:05.702 --> 00:00:08.224
defined with component functions P and Q,

00:00:08.224 --> 00:00:11.014
I've said that the 2D
curl of that function V

00:00:11.014 --> 00:00:14.453
gives you a new function that
also takes in x and y's inputs

00:00:14.453 --> 00:00:16.834
and it's formula is the
partial derivative of Q

00:00:16.834 --> 00:00:18.075
with respect to x

00:00:18.075 --> 00:00:21.101
minus the partial derivative
of P with respect to y.

00:00:21.101 --> 00:00:23.316
And my hope is that this
is more than just a formula

00:00:23.316 --> 00:00:24.507
and that you can understand

00:00:24.507 --> 00:00:28.237
how this represents fluid
rotation in two dimensions

00:00:28.237 --> 00:00:29.429
but what I want to do here is show

00:00:29.429 --> 00:00:31.811
how the original intuition
I gave for this formula

00:00:31.811 --> 00:00:33.493
might be a little oversimplified

00:00:33.493 --> 00:00:35.207
because for example, if we look at this,

00:00:35.207 --> 00:00:38.140
the partial Q, partial x component,

00:00:38.140 --> 00:00:40.220
I said that you can imagine
that Q at some point

00:00:40.220 --> 00:00:42.884
starting off a little bit
negative so the y component

00:00:42.884 --> 00:00:44.483
of the output is a little negative,

00:00:44.483 --> 00:00:46.740
then as you move positively
in the x direction,

00:00:46.740 --> 00:00:48.548
it goes to being zero

00:00:48.548 --> 00:00:50.648
and then it goes to being
a little bit positive,

00:00:50.648 --> 00:00:52.006
and with this particular picture,

00:00:52.006 --> 00:00:53.887
it's hopefully a little bit
clear why this can correspond

00:00:53.887 --> 00:00:56.479
to counter-clockwise
rotation in the fluid,

00:00:56.479 --> 00:00:58.621
but this is only a very
specific circumstance

00:00:58.621 --> 00:01:00.734
for what partial Q,
partial x being positive

00:01:00.734 --> 00:01:01.567
could look like.

00:01:01.567 --> 00:01:05.099
It might also look like Q's
starting a little bit positive

00:01:05.099 --> 00:01:06.907
and then as you move in the x direction,

00:01:06.907 --> 00:01:10.428
it becomes even more positive
and then even more positive.

00:01:10.428 --> 00:01:12.654
And according to the formula,
this should contribute

00:01:12.654 --> 00:01:14.461
as much to positive curl

00:01:14.461 --> 00:01:17.867
as this very clear-cut whirlpool example,

00:01:17.867 --> 00:01:19.769
and to illustrate what
this might look like,

00:01:19.769 --> 00:01:22.266
if we take a look at
this vector field here,

00:01:22.266 --> 00:01:23.782
if we look at the center, this is kinda

00:01:23.782 --> 00:01:28.379
the clear-cut whirlpool
counter-clockwise rotation example.

00:01:28.379 --> 00:01:30.929
And if we play the fluid
flow, the fluid does indeed

00:01:30.929 --> 00:01:33.175
rotate counter-clockwise in the region.

00:01:33.175 --> 00:01:36.467
But contrast that with what
goes on over here on the right.

00:01:36.467 --> 00:01:40.051
This doesn't look like
rotation in that sense at all.

00:01:40.051 --> 00:01:42.009
Instead the fluid
particles are just kind of

00:01:42.009 --> 00:01:44.157
rushing up through it.

00:01:44.157 --> 00:01:47.049
But in fact, the curl in
this region is going to be

00:01:47.049 --> 00:01:48.838
just as strong as it is over here,

00:01:48.838 --> 00:01:50.071
and I'll show that with a formula

00:01:50.071 --> 00:01:51.952
and kind of computing it
through in just a moment,

00:01:51.952 --> 00:01:54.293
but the image that you
might have in your mind

00:01:54.293 --> 00:01:55.891
is to imagine a paddle wheel of sorts

00:01:55.891 --> 00:01:58.138
where let's say it's got
arms kind of like that

00:01:58.138 --> 00:02:01.544
and then you hold down with
your thumb that middle portion,

00:02:01.544 --> 00:02:03.477
so even though the paddle
wheel left to its own devices

00:02:03.477 --> 00:02:05.417
would just fly up, I want to say,

00:02:05.417 --> 00:02:07.135
let's say you're holding
that down with your thumb

00:02:07.135 --> 00:02:08.733
but it's free to rotate.

00:02:08.733 --> 00:02:11.732
Then the vectors on its
left are pointing up

00:02:11.732 --> 00:02:14.156
but less strongly than
the vectors on its right

00:02:14.156 --> 00:02:17.072
which are even greater, so
if you imagine that setup

00:02:17.072 --> 00:02:18.994
and you have your paddle wheel there,

00:02:18.994 --> 00:02:21.042
then when you play the fluid rotation,

00:02:21.042 --> 00:02:22.192
holding your thumb down

00:02:22.192 --> 00:02:24.900
but letting the paddle
wheel rotate freely,

00:02:24.900 --> 00:02:26.716
it's also going to rotate just as it would

00:02:26.716 --> 00:02:30.799
over here in the easier
to see whirlpool example.

00:02:32.546 --> 00:02:34.312
And in terms of the formula,

00:02:34.312 --> 00:02:37.269
this is because a situation
like this one here,

00:02:37.269 --> 00:02:40.328
where Q goes from being
negative to zero to positive,

00:02:40.328 --> 00:02:42.776
should be treated just the
same as a situation like this

00:02:42.776 --> 00:02:44.709
as far as 2d curls is concerned,

00:02:44.709 --> 00:02:47.342
because this term in the 2d curl formula

00:02:47.342 --> 00:02:50.550
is going to come out the
same for either one of these.

00:02:50.550 --> 00:02:51.897
And it's worth pointing out, by the way,

00:02:51.897 --> 00:02:54.238
curl isn't something that
mathematicians and physicists

00:02:54.238 --> 00:02:57.028
came across trying to
understand fluid flow.

00:02:57.028 --> 00:03:00.445
Instead, they found this
term as being significant

00:03:00.445 --> 00:03:02.545
and various other
formulas and circumstances

00:03:02.545 --> 00:03:04.653
and I think electromagnetism
might be where

00:03:04.653 --> 00:03:06.651
it originally came about.

00:03:06.651 --> 00:03:08.801
But then in trying to
understand this formula,

00:03:08.801 --> 00:03:11.392
they realized that you can give
a fluid flow interpretation

00:03:11.392 --> 00:03:13.858
that gives a very deep
understanding of what's going on

00:03:13.858 --> 00:03:15.959
beyond just the symbols themselves.

00:03:15.959 --> 00:03:19.176
So let me go ahead and
walk through this example

00:03:19.176 --> 00:03:22.008
in terms of the formula
representing the vector field.

00:03:22.008 --> 00:03:26.175
It's a relatively
straightforward formula actually.

00:03:27.724 --> 00:03:28.641
So P and Q,

00:03:30.614 --> 00:03:33.582
that x component is going to be negative y

00:03:33.582 --> 00:03:37.072
and the y component, Q, is equal to x.

00:03:37.072 --> 00:03:38.869
So when we apply our 2d curl formula

00:03:38.869 --> 00:03:41.335
and apply the partial Q with respect to x,

00:03:41.335 --> 00:03:43.968
so partial of this second
component with respect to x

00:03:43.968 --> 00:03:44.968
is just one,

00:03:45.985 --> 00:03:49.767
and then we subtract off the
partial of P with respect to y

00:03:49.767 --> 00:03:51.491
which up here is negative one

00:03:51.491 --> 00:03:54.730
because P is just equal to negative y.

00:03:54.730 --> 00:03:57.422
So the 2d curl is equal to two

00:03:57.422 --> 00:03:59.512
and in particular, it's a constant two

00:03:59.512 --> 00:04:02.561
that doesn't depend on x and
y, which is pretty unusual,

00:04:02.561 --> 00:04:04.977
most times that you apply
2d curl to a vector field

00:04:04.977 --> 00:04:07.181
you're going to get some
kind of function of x and y.

00:04:07.181 --> 00:04:08.780
But the fact that this is constant,

00:04:08.780 --> 00:04:11.162
tells us that when we look
over at this fluid flow,

00:04:11.162 --> 00:04:14.652
the sense in which curl,
the formula for curl

00:04:14.652 --> 00:04:17.202
wants to say that rotation
happens around the center

00:04:17.202 --> 00:04:19.344
is just as strong as
it's supposed to happen

00:04:19.344 --> 00:04:20.984
over here on the right,

00:04:20.984 --> 00:04:22.834
or anywhere on the plane for that matter.

00:04:22.834 --> 00:04:23.667
So if we're playing this

00:04:23.667 --> 00:04:25.948
and if you imagine you have
the paddle wheel in the center,

00:04:25.948 --> 00:04:28.257
evidently it would be
rotating just as quickly

00:04:28.257 --> 00:04:30.260
as the paddle on the right
even though it might,

00:04:30.260 --> 00:04:32.041
I don't know, to me, that
feels a little unintuitive

00:04:32.041 --> 00:04:33.524
cause the one on the right I'm thinking,

00:04:33.524 --> 00:04:36.186
okay, you know, maybe there's
a little bit more torque

00:04:36.186 --> 00:04:38.066
on the right side than
there is on the left

00:04:38.066 --> 00:04:39.832
and that's kind of a
counter-balancing act,

00:04:39.832 --> 00:04:41.823
but the idea that that's actually the same

00:04:41.823 --> 00:04:45.182
as the very clear-cut, I see
the counter-clockwise rotation

00:04:45.182 --> 00:04:47.466
with my eyes, example in the center,

00:04:47.466 --> 00:04:49.483
does seem a little unusual.

00:04:49.483 --> 00:04:51.636
But I think it's important to understand

00:04:51.636 --> 00:04:53.402
what else two dimensional
curl can look like

00:04:53.402 --> 00:04:57.069
and what else this formula
might be representing.

00:04:57.069 --> 00:04:59.440
So with that, I'll see you next video.


Describing rotation in 3d with a vector

WEBVTT
Kind: captions
Language: en

00:00:01.241 --> 00:00:02.386
- [Voiceover] How do you describe rotation

00:00:02.386 --> 00:00:04.076
in three dimensions?

00:00:04.076 --> 00:00:06.266
So for example, I have here a globe

00:00:06.266 --> 00:00:07.915
and it's rotating in some way

00:00:07.915 --> 00:00:09.866
and there's a certain
direction that it's rotating

00:00:09.866 --> 00:00:11.747
and a speed with which it's rotating.

00:00:11.747 --> 00:00:14.277
And the question is how could
you give me some numerical

00:00:14.277 --> 00:00:17.694
information that perfectly
describes that rotation?

00:00:17.694 --> 00:00:20.199
So you give me some numbers,
and I can tell you the speed

00:00:20.199 --> 00:00:21.871
and the direction and
everything associated

00:00:21.871 --> 00:00:23.708
with this rotation.

00:00:23.708 --> 00:00:24.999
But before talking about that,

00:00:24.999 --> 00:00:26.654
let's remind ourselves
of how we talked about

00:00:26.654 --> 00:00:28.209
two dimensional rotation.

00:00:28.209 --> 00:00:30.033
So I have here a little pi creature,

00:00:30.033 --> 00:00:32.510
and I set him to start rotating about

00:00:32.510 --> 00:00:34.154
and the way that we can describe this,

00:00:34.154 --> 00:00:36.833
we pretty much need to
just give a rate to it.

00:00:36.833 --> 00:00:41.000
And you might give that rate
as a number of rotations

00:00:42.711 --> 00:00:47.233
per second, some unit of
time. So rotations per second.

00:00:47.233 --> 00:00:49.943
And in this case, I think
I programmed him so that

00:00:49.943 --> 00:00:53.194
he's going to do one rotation
for every five seconds.

00:00:53.194 --> 00:00:56.244
So his rotational rate would be 0.2.

00:00:56.244 --> 00:00:57.676
But that's a little bit ambiguous

00:00:57.676 --> 00:00:59.557
because if you just say,
"Hey, this little pi creature

00:00:59.557 --> 00:01:02.111
is rotating at 0.2 rotations per second,"

00:01:02.111 --> 00:01:04.131
someone could say, "Well, is it clockwise

00:01:04.131 --> 00:01:05.705
or counterclockwise?"

00:01:05.705 --> 00:01:06.964
So there's some ambiguity.

00:01:06.964 --> 00:01:08.473
And the convention that
people have adopted

00:01:08.473 --> 00:01:10.911
is to say, "Well, if I
give you a positive number,

00:01:10.911 --> 00:01:13.279
if the number is positive,
then that's going to tell you

00:01:13.279 --> 00:01:17.268
that the nature of the
rotation is counterclockwise,

00:01:17.268 --> 00:01:18.620
but if I give you a negative number,

00:01:18.620 --> 00:01:20.896
if instead you see something
that's a negative number

00:01:20.896 --> 00:01:22.776
of rotations per second,
that would be rotation

00:01:22.776 --> 00:01:25.007
the other way, going clockwise."

00:01:25.007 --> 00:01:26.910
And that's the convention.

00:01:26.910 --> 00:01:28.744
That's just what people have decided on.

00:01:28.744 --> 00:01:31.524
And with this it's very nice
because given a single number,

00:01:31.524 --> 00:01:33.434
just one number, and it could
be positive or negative,

00:01:33.434 --> 00:01:36.499
you can perfectly describe
two dimensional rotation.

00:01:36.499 --> 00:01:38.032
And there's a minor nuance here,

00:01:38.032 --> 00:01:40.513
usually in physics and
math, we don't actually use

00:01:40.513 --> 00:01:43.512
rotations per unit second but
instead you describe things

00:01:43.512 --> 00:01:47.679
in terms of the number of
radians per unit second.

00:01:49.005 --> 00:01:51.476
And just as a quick
reminder of what that means,

00:01:51.476 --> 00:01:53.589
if you imagine some kind of circle,

00:01:53.589 --> 00:01:55.726
and it could be any circle,
the size doesn't really matter,

00:01:55.726 --> 00:01:59.076
and if you draw the radius to
that and then ask the question

00:01:59.076 --> 00:02:01.530
how far along the circumference
would I have to go

00:02:01.530 --> 00:02:04.595
such that the arc length,
that sort of sub-portion

00:02:04.595 --> 00:02:07.452
of the circumference, is
exactly as long as the radius?

00:02:07.452 --> 00:02:11.003
So if this was R, you'd want
to know how far you have to go

00:02:11.003 --> 00:02:14.024
before that arc length is also R.

00:02:14.024 --> 00:02:16.826
And then that, that angle,
that amount of turning

00:02:16.826 --> 00:02:19.270
that you can do, determines one radian.

00:02:19.270 --> 00:02:21.244
And because there's exactly two pi radians

00:02:21.244 --> 00:02:24.502
for every rotation, to
convert between rotations

00:02:24.502 --> 00:02:25.912
per unit second and
radians per unit second,

00:02:25.912 --> 00:02:28.579
you just multiply this guy by 2π

00:02:29.450 --> 00:02:33.279
so it would be whatever the
number you have there times 2π.

00:02:33.279 --> 00:02:34.874
And the specific numbers
aren't too important.

00:02:34.874 --> 00:02:38.247
The main upshot here is
that with a single number,

00:02:38.247 --> 00:02:40.192
positive or negative, you
can perfectly describe

00:02:40.192 --> 00:02:42.026
two dimensional rotation.

00:02:42.026 --> 00:02:43.811
But if we look over here at the 3D case,

00:02:43.811 --> 00:02:45.184
there's actually more information

00:02:45.184 --> 00:02:47.514
than just one number that
we're going to need to know.

00:02:47.514 --> 00:02:49.745
First of all, you want to know the axis

00:02:49.745 --> 00:02:52.197
around which it's rotating,
so the line that you can draw

00:02:52.197 --> 00:02:55.387
such that all rotation
happens around that line.

00:02:55.387 --> 00:02:57.351
And then you want to
describe the actual rate

00:02:57.351 --> 00:02:59.267
at which it's going. You
know, is it slow rotation

00:02:59.267 --> 00:03:00.254
or is it fast?

00:03:00.254 --> 00:03:03.743
So you need to know a direction
along with a magnitude.

00:03:03.743 --> 00:03:06.260
And you might say to yourself,
"Hey, direction? Magnitude?

00:03:06.260 --> 00:03:07.981
Sounds like we could use a vector."

00:03:07.981 --> 00:03:09.333
And in fact, that's what we do.

00:03:09.333 --> 00:03:11.755
We use some kind of vector whose length

00:03:11.755 --> 00:03:14.047
is going to correspond to the
rate at which it's rotating,

00:03:14.047 --> 00:03:15.741
typically in radians per second,

00:03:15.741 --> 00:03:17.367
it's called the angular velocity.

00:03:17.367 --> 00:03:21.999
And then the direction describes
the axis of rotation itself

00:03:21.999 --> 00:03:24.612
But similar to how in
two dimensions there was

00:03:24.612 --> 00:03:27.508
an ambiguity between clockwise
and counterclockwise,

00:03:27.508 --> 00:03:29.142
if this was the only convention we had,

00:03:29.142 --> 00:03:31.252
it would be ambiguous whether
you should use this vector

00:03:31.252 --> 00:03:35.040
or if you should use one pointing
in the opposite direction.

00:03:35.040 --> 00:03:36.965
And the way I've chosen to
draw these guys, by the way,

00:03:36.965 --> 00:03:38.311
it doesn't matter where they are,

00:03:38.311 --> 00:03:40.848
remember a vector it just has
a magnitude and a direction

00:03:40.848 --> 00:03:42.514
and you can put it anywhere in space.

00:03:42.514 --> 00:03:43.745
I figured it was natural enough

00:03:43.745 --> 00:03:45.301
to just kind of put them around the poles

00:03:45.301 --> 00:03:46.508
just so that you could see them

00:03:46.508 --> 00:03:48.505
on the axis of rotation itself.

00:03:48.505 --> 00:03:51.082
So the question is,
what vector do you use?

00:03:51.082 --> 00:03:52.963
Do you use the one
pointing in this direction?

00:03:52.963 --> 00:03:54.960
Or do you use this green one pointing

00:03:54.960 --> 00:03:56.237
in the opposite direction?

00:03:56.237 --> 00:03:57.630
And for this, we have a convention

00:03:57.630 --> 00:03:59.325
known as the right-hand rule.

00:03:59.325 --> 00:04:01.048
So I'll go ahead and
bring in a picture here

00:04:01.048 --> 00:04:03.964
to illustrate the right-hand rule.

00:04:03.964 --> 00:04:06.005
What you imagine doing
is taking the fingers

00:04:06.005 --> 00:04:08.271
of your right hand and curling them around

00:04:08.271 --> 00:04:10.014
in the direction of rotation.

00:04:10.014 --> 00:04:11.516
And what I mean by that is
the tips of your fingers

00:04:11.516 --> 00:04:13.513
will be pointing the direction

00:04:13.513 --> 00:04:15.745
that the surface of the sphere would move.

00:04:15.745 --> 00:04:17.259
Then when you stick out your thumb,

00:04:17.259 --> 00:04:20.008
that's the direction that
is the choice of vector

00:04:20.008 --> 00:04:22.003
which should describe that rotation.

00:04:22.003 --> 00:04:23.757
So in the specific example we have here,

00:04:23.757 --> 00:04:25.506
when you stick out your right thumb,

00:04:25.506 --> 00:04:28.722
that corresponds to the white
vector, not the green one.

00:04:28.722 --> 00:04:30.765
But if you did things
the other way around,

00:04:30.765 --> 00:04:34.932
whoops, move this a little
bit. Get him to stay in place.

00:04:37.406 --> 00:04:39.008
If you move things the other way around,

00:04:39.008 --> 00:04:41.129
such that the rotation were going kind of

00:04:41.129 --> 00:04:42.538
in the opposite direction,

00:04:42.538 --> 00:04:45.417
then when you imagine curling
the fingers of your right hand

00:04:45.417 --> 00:04:47.855
around that direction, your
thumb is going to point

00:04:47.855 --> 00:04:49.341
according to the green vector.

00:04:49.341 --> 00:04:53.477
But with the original rotation
that I started illustrating,

00:04:53.477 --> 00:04:54.594
it's the white vector,

00:04:54.594 --> 00:04:56.260
the white vector is the one to go with.

00:04:56.260 --> 00:04:57.498
And this is actually pretty cool, right?

00:04:57.498 --> 00:04:59.001
Because you're packing
a lot of information

00:04:59.001 --> 00:05:00.208
into that vector.

00:05:00.208 --> 00:05:02.007
It tells you what the axis is.

00:05:02.007 --> 00:05:05.250
It tells you the speed of
rotation via its magnitude.

00:05:05.250 --> 00:05:08.637
And then the choice of which
direction along the axis

00:05:08.637 --> 00:05:10.346
tells you whether the
globe is going one way

00:05:10.346 --> 00:05:11.516
or if it's going the other.

00:05:11.516 --> 00:05:12.816
So with just three numbers,

00:05:12.816 --> 00:05:15.510
the three dimensional
coordinates of this vector,

00:05:15.510 --> 00:05:17.739
you can perfectly describe any one given

00:05:17.739 --> 00:05:19.620
three dimensional rotation.

00:05:19.620 --> 00:05:22.267
And the reason I'm talking
about this, by the way,

00:05:22.267 --> 00:05:24.265
in a series of videos about curl,

00:05:24.265 --> 00:05:26.521
is because what I'm about to talk about

00:05:26.521 --> 00:05:29.867
is three dimensional curl
which relates to fluid flow

00:05:29.867 --> 00:05:32.506
in three dimensions and
how that induces a rotation

00:05:32.506 --> 00:05:34.272
at every single point in space.

00:05:34.272 --> 00:05:35.920
And what's going to happen
is you're going to associate

00:05:35.920 --> 00:05:38.366
a vector with every single point in space

00:05:38.366 --> 00:05:41.435
to answer the question
what rotation at that point

00:05:41.435 --> 00:05:43.769
is induced by the certain fluid flow?

00:05:43.769 --> 00:05:45.371
And I'm getting a little
bit ahead of myself here.

00:05:45.371 --> 00:05:46.810
For right now you just need to focus

00:05:46.810 --> 00:05:48.738
on a single point of rotation

00:05:48.738 --> 00:05:51.269
and a single vector corresponding to that.

00:05:51.269 --> 00:05:53.459
But it's important to kind
of get your head around

00:05:53.459 --> 00:05:55.889
how exactly we represent
this rotation with a vector

00:05:55.889 --> 00:06:00.464
before moving on to the notably
more cognitively intensive

00:06:00.464 --> 00:06:02.786
subject of three dimensional curl.

00:06:02.786 --> 00:06:05.710
So with that, I will see you next video.


3d curl formula, part 1

WEBVTT
Kind: captions
Language: en

00:00:00.355 --> 00:00:01.355
- [Voiceover] So I've
spent a couple videos

00:00:01.355 --> 00:00:03.574
laying down the foundation
for what three-dimensional

00:00:03.574 --> 00:00:05.308
curl is trying to represent, and here

00:00:05.308 --> 00:00:07.010
I'm going to go ahead and talk about

00:00:07.010 --> 00:00:08.492
how you actually compute it.

00:00:08.492 --> 00:00:11.199
Three-d curl is the kind
of thing that you take

00:00:11.199 --> 00:00:13.594
with regards to a
three-dimensional vector field.

00:00:13.594 --> 00:00:15.646
So something that takes in
a three-dimensional point

00:00:15.646 --> 00:00:17.999
as its input, and then
it's going to output

00:00:17.999 --> 00:00:19.794
a three-dimensional vector.

00:00:19.794 --> 00:00:24.352
It's common to write the component
functions as P, Q, and R.

00:00:24.352 --> 00:00:26.426
Each one of these is a
scale or valued function

00:00:26.426 --> 00:00:28.527
that takes in a three-dimensional point,

00:00:28.527 --> 00:00:30.265
and just outputs a number.

00:00:30.265 --> 00:00:31.711
So, it'll be that same three-d point

00:00:31.711 --> 00:00:34.461
with the coordinates x, y, and z.

00:00:38.025 --> 00:00:40.242
So when you have a three-dimensional
vector field like this,

00:00:40.242 --> 00:00:41.657
the image you might have in mind would be

00:00:41.657 --> 00:00:44.824
something like this, where every point

00:00:45.817 --> 00:00:49.123
in three-dimensional space
has a vector attached to it.

00:00:49.123 --> 00:00:50.671
And you know, when you
actually look at it,

00:00:50.671 --> 00:00:52.999
there's quite a lot going
on, but in principle,

00:00:52.999 --> 00:00:55.730
all that's really happening
is that each point in space

00:00:55.730 --> 00:00:57.757
is associated with a vector.

00:00:57.757 --> 00:00:59.140
The point in space is the input

00:00:59.140 --> 00:01:00.662
and the vector is the output.

00:01:00.662 --> 00:01:03.107
You're just gluing them together.

00:01:03.107 --> 00:01:05.613
Naturally, between the three
dimensions of the input

00:01:05.613 --> 00:01:06.705
and the three dimensions of the output,

00:01:06.705 --> 00:01:08.634
we have six dimensions going on,

00:01:08.634 --> 00:01:11.915
the picture that you're
looking at becomes quite messy.

00:01:11.915 --> 00:01:15.763
So, the question is, how do
you compute this curl value

00:01:15.763 --> 00:01:17.417
that I've been talking about.

00:01:17.417 --> 00:01:20.806
Curl of your vector value function.

00:01:20.806 --> 00:01:23.515
Just as a quick reminder,
what this is supposed to be,

00:01:23.515 --> 00:01:25.648
is you're going to have
some kind of fluid flow

00:01:25.648 --> 00:01:28.732
induced by this vector
field, where you're imagining

00:01:28.732 --> 00:01:30.651
air flowing along each vector.

00:01:30.651 --> 00:01:32.241
What you want is a function that tells you

00:01:32.241 --> 00:01:34.835
at any given point, what is the rotation

00:01:34.835 --> 00:01:37.821
induced by that fluid
flow around that point.

00:01:37.821 --> 00:01:39.525
Because rotation is described

00:01:39.525 --> 00:01:41.242
with a three-dimensional vector,

00:01:41.242 --> 00:01:43.619
you're expecting this to be vector-valued.

00:01:43.619 --> 00:01:46.253
It'll be something that
equals a vector output.

00:01:46.253 --> 00:01:49.344
If that doesn't make sense,
if that doesn't quite jive,

00:01:49.344 --> 00:01:51.153
maybe go check out the
video on how to represent

00:01:51.153 --> 00:01:53.798
three-dimensional rotation with a vector.

00:01:53.798 --> 00:01:55.237
So what you have here
is going to be something

00:01:55.237 --> 00:01:57.798
that takes as its input, x, y, and z.

00:01:57.798 --> 00:01:59.489
It takes a three-dimensional point,

00:01:59.489 --> 00:02:02.415
and what it outputs is a
vector describing rotation,

00:02:02.415 --> 00:02:03.764
and there's actually another notation

00:02:03.764 --> 00:02:06.854
that's quite, quite helpful
when it comes to computing this.

00:02:06.854 --> 00:02:08.932
You take nabla, that
upside-down triangle we used

00:02:08.932 --> 00:02:11.132
in divergence and
gradient, and you imagine

00:02:11.132 --> 00:02:15.955
taking the cross-product
between that and your vector V.

00:02:15.955 --> 00:02:17.955
As a reminder, this nabla, you imagine it

00:02:17.955 --> 00:02:19.937
as if it's a vector containing

00:02:19.937 --> 00:02:21.858
partial differential operators.

00:02:21.858 --> 00:02:23.605
That's the kind of thing where,
when you say it out loud,

00:02:23.605 --> 00:02:25.187
it sounds kinda fancy, a vector full

00:02:25.187 --> 00:02:26.901
of partial differential operators,

00:02:26.901 --> 00:02:28.322
but all it really means is

00:02:28.322 --> 00:02:30.702
I'm just going to write
a bunch of symbols.

00:02:30.702 --> 00:02:33.144
This partial partial x
is something that wants

00:02:33.144 --> 00:02:35.618
to take in a multi-variable function,

00:02:35.618 --> 00:02:39.424
and tell you its partial derivative.

00:02:39.424 --> 00:02:41.187
Strictly speaking, this
doesn't really make sense,

00:02:41.187 --> 00:02:42.435
like, hey, how can a vector contain

00:02:42.435 --> 00:02:44.248
these partial differential operators?

00:02:44.248 --> 00:02:46.415
But as a series of symbolic movements,

00:02:46.415 --> 00:02:47.783
it's actually quite helpful,

00:02:47.783 --> 00:02:51.000
because when you're multiplying
these guys by a thing,

00:02:51.000 --> 00:02:52.597
it's not really multiplication.

00:02:52.597 --> 00:02:53.506
You're really going to be giving it

00:02:53.506 --> 00:02:56.736
some kind of multi-variable
function, like P, Q, or R,

00:02:56.736 --> 00:02:58.702
the component functions
of our vector field,

00:02:58.702 --> 00:03:01.031
and evaluating it.

00:03:01.031 --> 00:03:02.787
So just as a warm-up for how to do this,

00:03:02.787 --> 00:03:04.104
let's see what this looks like in the case

00:03:04.104 --> 00:03:06.719
of two dimensions, where
we already know the formula

00:03:06.719 --> 00:03:08.767
for two-dimensional curl.

00:03:08.767 --> 00:03:11.414
What that would look like,
is you have a smaller,

00:03:11.414 --> 00:03:14.392
more two-dimensional,
just partial partial x,

00:03:14.392 --> 00:03:17.059
partial partial y, del operator.

00:03:19.176 --> 00:03:20.934
You're going to take the
cross-product between that

00:03:20.934 --> 00:03:22.489
and a two-dimensional vector

00:03:22.489 --> 00:03:25.518
that's just the component
functions P and Q.

00:03:25.518 --> 00:03:29.417
In this case, P and Q would
be just functions of x and y.

00:03:29.417 --> 00:03:31.150
So I'm kind of overloading notation right,

00:03:31.150 --> 00:03:33.410
over here I have a
two-dimensional vector field

00:03:33.410 --> 00:03:35.270
that I'm saying, P and y
are scale or value functions

00:03:35.270 --> 00:03:36.702
with a two-dimensional
input, but over here

00:03:36.702 --> 00:03:39.071
I'm also using P and Q to represent ones

00:03:39.071 --> 00:03:41.001
with a three-dimensional input.

00:03:41.001 --> 00:03:42.400
So you should think of these as separate,

00:03:42.400 --> 00:03:44.295
but it's common to use the same names.

00:03:44.295 --> 00:03:45.825
This is going to illustrate

00:03:45.825 --> 00:03:48.602
the broader, more complicated point.

00:03:48.602 --> 00:03:51.783
When you compute something
like this, the cross-product,

00:03:51.783 --> 00:03:52.969
you typically think of it as taking

00:03:52.969 --> 00:03:55.600
these diagonal components
and multiplying them,

00:03:55.600 --> 00:03:58.085
so that would be your partial partial x,

00:03:58.085 --> 00:04:01.267
"multiplied" with Q, which
really means you're taking

00:04:01.267 --> 00:04:04.017
the partial derivative
of Q with respect to x.

00:04:04.017 --> 00:04:08.184
Then you subtract off this
diagonal component here,

00:04:10.750 --> 00:04:11.682
oh sorry, this should be a y.

00:04:11.682 --> 00:04:13.802
This should be partial partial y.

00:04:13.802 --> 00:04:14.635
Sorry about that.

00:04:14.635 --> 00:04:16.315
You need partial partial y of P,

00:04:16.315 --> 00:04:18.021
and that's what you're subtracting off.

00:04:18.021 --> 00:04:21.555
So partial partial y of P,
just the partial derivative

00:04:21.555 --> 00:04:24.265
of that P function with respect to y.

00:04:24.265 --> 00:04:25.640
Hopefully this is something you recognize.

00:04:25.640 --> 00:04:28.265
This is the two-dimensional curl.

00:04:28.265 --> 00:04:29.790
It's something we got an intuition for,

00:04:29.790 --> 00:04:32.082
I want it to be more than just a formula,

00:04:32.082 --> 00:04:33.366
but hopefully this is kind of reassuring

00:04:33.366 --> 00:04:36.899
that when you take that del
operator, that nabla symbol,

00:04:36.899 --> 00:04:40.199
and cross-product with the
vector valued function itself,

00:04:40.199 --> 00:04:41.767
it gives you a sense of curl.

00:04:41.767 --> 00:04:44.274
Now when we do this in the
three-dimensional case,

00:04:44.274 --> 00:04:46.099
we're going to take a
three-dimensional cross-product

00:04:46.099 --> 00:04:48.922
between this three-dimensional
vectorish thing

00:04:48.922 --> 00:04:52.091
and this three-dimensional function.

00:04:52.091 --> 00:04:54.239
If you're not terribly comfortable
with the cross-product,

00:04:54.239 --> 00:04:56.225
how to compute it or how to interpret it

00:04:56.225 --> 00:04:57.466
and things like that,

00:04:57.466 --> 00:04:59.336
now would probably be a good
time to go find the videos

00:04:59.336 --> 00:05:02.345
that Sal does on this and
build up that intuition

00:05:02.345 --> 00:05:06.199
for what a cross-product actually
is and how to compute it.

00:05:06.199 --> 00:05:08.665
Because at this point, I'm
going to assume that you

00:05:08.665 --> 00:05:11.157
know how to compute it
because we're doing it in

00:05:11.157 --> 00:05:14.332
kind of an absurd context of
partial differential operators

00:05:14.332 --> 00:05:17.681
and functions, so it's important
to have that foundation.

00:05:17.681 --> 00:05:19.205
The way you compute a thing like this,

00:05:19.205 --> 00:05:21.406
is you construct a determinant.

00:05:21.406 --> 00:05:23.733
I'm going to go down here.

00:05:23.733 --> 00:05:26.733
Determinant of a certain 3x3 matrix.

00:05:29.898 --> 00:05:33.424
The top row of that is
all of the unit vectors

00:05:33.424 --> 00:05:35.966
in various directions of
three-dimensional space.

00:05:35.966 --> 00:05:39.094
So these I, J, and K guys,
I represents the unit vector

00:05:39.094 --> 00:05:43.744
in the x direction, so that
would be I is equal to,

00:05:43.744 --> 00:05:47.432
x component is one but then
the other components are zero.

00:05:47.432 --> 00:05:50.059
Then similarly, J and K
represent the unit vectors

00:05:50.059 --> 00:05:52.342
in the y and z direction, and again,

00:05:52.342 --> 00:05:54.200
if that doesn't quite make sense,

00:05:54.200 --> 00:05:56.257
why I'm putting them up there
or what we're about to do,

00:05:56.257 --> 00:05:58.625
maybe check out that cross-product video.

00:05:58.625 --> 00:06:00.460
So we put those in the
top rows as vectors.

00:06:00.460 --> 00:06:02.858
This is the trick to
computing the cross-product,

00:06:02.858 --> 00:06:05.173
because again, what does
it mean to put a vector

00:06:05.173 --> 00:06:09.094
inside a matrix, but
it's a notational trick.

00:06:09.094 --> 00:06:10.791
Then we're going to take the first vector

00:06:10.791 --> 00:06:12.393
that we're doing the cross-product with,

00:06:12.393 --> 00:06:15.700
and put its components in the next row.

00:06:15.700 --> 00:06:18.765
What that would look like,
is the next row has a

00:06:18.765 --> 00:06:20.265
partial partial y,

00:06:23.243 --> 00:06:25.324
no sorry, God I keep messing up here,

00:06:25.324 --> 00:06:28.181
that's an x, you do whatever
the first component is first,

00:06:28.181 --> 00:06:30.266
and then the second component second,

00:06:30.266 --> 00:06:33.925
and the third component, the z, partial z.

00:06:33.925 --> 00:06:36.937
Don't know why I'm making
that little mistake.

00:06:36.937 --> 00:06:39.750
For the last row, you
put in the second vector,

00:06:39.750 --> 00:06:43.917
which is in this case, is vector
value function P, Q, and R.

00:06:44.897 --> 00:06:48.897
P, which is a multi-variable
function, Q, and R.

00:06:50.495 --> 00:06:53.251
First, it's worth stepping
back and looking at this.

00:06:53.251 --> 00:06:55.305
This is kind of an absurd thing.

00:06:55.305 --> 00:06:56.979
Usually when we talk about matrices

00:06:56.979 --> 00:06:58.188
and taking the determinant,

00:06:58.188 --> 00:06:59.605
all of the components are numbers

00:06:59.605 --> 00:07:01.756
because you're multiplying
numbers together.

00:07:01.756 --> 00:07:04.025
But here, we've got a
notational trick layered

00:07:04.025 --> 00:07:06.956
on top of a notational trick,
so that one of the rows

00:07:06.956 --> 00:07:08.705
is vectors, one of the rows

00:07:08.705 --> 00:07:10.852
is partial differential operators,

00:07:10.852 --> 00:07:11.805
and then the last one,

00:07:11.805 --> 00:07:14.156
each one of these is a
multi-variable function.

00:07:14.156 --> 00:07:16.609
So it seems like this absurd, convoluted,

00:07:16.609 --> 00:07:19.090
as far away from a matrix
full of numbers thing

00:07:19.090 --> 00:07:21.439
as you can get, but it's actually

00:07:21.439 --> 00:07:23.522
very helpful for computation.

00:07:23.522 --> 00:07:25.974
If you go through the process
of computing this determinant

00:07:25.974 --> 00:07:27.722
and saying, what could that mean,

00:07:27.722 --> 00:07:30.062
the thing that pops out
is going to be the formula

00:07:30.062 --> 00:07:31.698
for three-dimensional curl.

00:07:31.698 --> 00:07:33.910
At the risk of having a
video that runs too long,

00:07:33.910 --> 00:07:35.422
I'll call things and end here,

00:07:35.422 --> 00:07:39.589
but continue going through that
operation in the next video.


3d curl computation example

WEBVTT
Kind: captions
Language: en

00:00:00.256 --> 00:00:01.417
- [Voiceover] So let's
go ahead and work through

00:00:01.417 --> 00:00:03.275
an actual curl computation.

00:00:03.275 --> 00:00:05.574
Let's say our vector valued function V,

00:00:05.574 --> 00:00:07.013
which is a function of X, Y, and Z,

00:00:07.013 --> 00:00:08.848
this is gonna be three-dimensional,

00:00:08.848 --> 00:00:11.309
is defined by the functions, I don't know,

00:00:11.309 --> 00:00:13.979
let's say the first
component is X times Y.

00:00:13.979 --> 00:00:15.893
The second one is

00:00:15.893 --> 00:00:16.893
cosine of Z,

00:00:19.223 --> 00:00:23.223
and then the last component
is Z squared plus Y.

00:00:24.261 --> 00:00:25.655
Let's say.

00:00:25.655 --> 00:00:27.280
So if you take this guy,

00:00:27.280 --> 00:00:29.530
how do you compute the curl

00:00:31.074 --> 00:00:33.817
of that vector valued function?

00:00:33.817 --> 00:00:35.767
So what you do, as I
mentioned in the last video,

00:00:35.767 --> 00:00:38.693
is you imagine taking this del operator

00:00:38.693 --> 00:00:40.853
and taking the cross product between that

00:00:40.853 --> 00:00:42.501
and your vector valued function.

00:00:42.501 --> 00:00:44.405
And what that means when you expand it

00:00:44.405 --> 00:00:46.867
is that del operator,
you just kinda fill it

00:00:46.867 --> 00:00:49.514
with partial differential
operators, you could say,

00:00:49.514 --> 00:00:51.511
but really it's just the
symbol partial partial X,

00:00:51.511 --> 00:00:53.011
partial partial Y,

00:00:54.459 --> 00:00:56.446
partial partial Z.

00:00:56.446 --> 00:00:57.979
And these are things that are just waiting

00:00:57.979 --> 00:00:59.628
to take in some kind of function.

00:00:59.628 --> 00:01:01.810
So we're gonna take the
cross product between that

00:01:01.810 --> 00:01:04.109
and the function that
we have defined here.

00:01:04.109 --> 00:01:07.442
So I'm gonna just actually copy it over,

00:01:09.960 --> 00:01:11.460
copy it over here.

00:01:12.329 --> 00:01:14.288
And a little residue.

00:01:14.288 --> 00:01:16.030
And to compute this cross product,

00:01:16.030 --> 00:01:18.003
we take a certain determinant.

00:01:18.003 --> 00:01:20.070
So I'm gonna write over here determinant,

00:01:20.070 --> 00:01:22.160
and it's gonna be of a
three-by-three matrix,

00:01:22.160 --> 00:01:24.412
but really it's kinda like
a quote-unquote matrix,

00:01:24.412 --> 00:01:26.572
because each component
has something funky.

00:01:26.572 --> 00:01:28.192
So the top row, just like we would have

00:01:28.192 --> 00:01:30.746
with any other cross
product that we're computing

00:01:30.746 --> 00:01:33.625
is gonna have i, j, and
k, these unit vectors

00:01:33.625 --> 00:01:35.135
in three-dimensional space.

00:01:35.135 --> 00:01:36.504
And the second row here is gonna have

00:01:36.504 --> 00:01:38.548
all of these partial
differential operators,

00:01:38.548 --> 00:01:41.381
since that's the first
vector in our cross product.

00:01:41.381 --> 00:01:43.714
So that's partial partial X,

00:01:44.840 --> 00:01:47.009
partial partial Y,

00:01:47.009 --> 00:01:48.565
and again, all of these are just kind of

00:01:48.565 --> 00:01:50.027
waiting to be given a function

00:01:50.027 --> 00:01:52.558
that they can take the derivative of.

00:01:52.558 --> 00:01:53.789
And then that third row is gonna be

00:01:53.789 --> 00:01:54.950
the functions that we have.

00:01:54.950 --> 00:01:57.783
So the first component here is XY,

00:01:59.107 --> 00:02:02.032
the second component is cosine of Z,

00:02:02.032 --> 00:02:03.032
cosine of Z,

00:02:04.749 --> 00:02:07.930
and then that final component
is Z squared plus Y.

00:02:07.930 --> 00:02:08.763
Z squared

00:02:09.602 --> 00:02:10.786
plus Y.

00:02:10.786 --> 00:02:12.226
Now let's all get some room here,

00:02:12.226 --> 00:02:14.524
maybe make it more visible.

00:02:14.524 --> 00:02:16.664
So this is the determinant
we need to compute.

00:02:16.664 --> 00:02:19.233
And this is gonna be broken
up into three different parts.

00:02:19.233 --> 00:02:20.933
The first one,

00:02:20.933 --> 00:02:23.016
we take this top part, i,

00:02:24.187 --> 00:02:28.371
and multiply it by the
determinant of this sub-matrix.

00:02:28.371 --> 00:02:31.319
So when we do that, this sub-determinant,

00:02:31.319 --> 00:02:33.781
we're taking partial
derivative with respect to Y

00:02:33.781 --> 00:02:35.638
of Z squared plus Y.

00:02:35.638 --> 00:02:37.101
Now as far as Y is concerned,

00:02:37.101 --> 00:02:38.239
Z looks like a constant.

00:02:38.239 --> 00:02:39.818
So Z squared is a constant,

00:02:39.818 --> 00:02:42.813
and the partial derivative of
this entire guy is just one.

00:02:42.813 --> 00:02:45.019
So that'll look like one.

00:02:45.019 --> 00:02:47.202
And then we're subtracting
off the partial derivative

00:02:47.202 --> 00:02:49.779
with respect to Z of cosine of Z,

00:02:49.779 --> 00:02:52.124
and that just looks the
same as a derivative

00:02:52.124 --> 00:02:54.725
of cosine Z, which is negative sine,

00:02:54.725 --> 00:02:57.381
so that's negative sine

00:02:57.381 --> 00:02:58.214
of Z.

00:03:00.330 --> 00:03:01.932
So that's the first part.

00:03:01.932 --> 00:03:04.463
And then as the next
part, we're gonna take j,

00:03:04.463 --> 00:03:06.739
but we're subtracting, 'cause
you're always kinda thinking

00:03:06.739 --> 00:03:10.454
plus, minus, plus, when you're
doing these determinants.

00:03:10.454 --> 00:03:13.171
So we're gonna subtract off j,

00:03:13.171 --> 00:03:15.307
multiplied by its own
little sub-determinant,

00:03:15.307 --> 00:03:18.140
and this time the
sub-determinant is gonna involve

00:03:18.140 --> 00:03:20.462
the two columns that it's not part of.

00:03:20.462 --> 00:03:21.832
So you're imagining this first column

00:03:21.832 --> 00:03:24.177
and this second column as
being part of a matrix.

00:03:24.177 --> 00:03:25.942
So the first thing you do is you take

00:03:25.942 --> 00:03:27.846
this partial derivative with respect to X

00:03:27.846 --> 00:03:29.309
of Z squared plus Y.

00:03:29.309 --> 00:03:30.516
Well no Xs show up there, right?

00:03:30.516 --> 00:03:31.909
That's Z squared and Y.

00:03:31.909 --> 00:03:34.742
Each look like constants
as far as X is concerned.

00:03:34.742 --> 00:03:35.992
So that's zero.

00:03:37.041 --> 00:03:39.061
Then we take the partial with respect to Z

00:03:39.061 --> 00:03:40.454
of X times Y.

00:03:40.454 --> 00:03:42.196
And again, there's no
Z that shows up there,

00:03:42.196 --> 00:03:46.677
so that's also zero, so we're
kinda subtracting off zero.

00:03:46.677 --> 00:03:49.556
And then finally, we're
adding this last component.

00:03:49.556 --> 00:03:52.575
We're gonna add that last component, k,

00:03:52.575 --> 00:03:55.454
multiplied by the determinant
of this sub-matrix

00:03:55.454 --> 00:03:57.474
of the columns that it's not part of.

00:03:57.474 --> 00:03:58.798
So this involves partial derivative

00:03:58.798 --> 00:04:01.497
with respect to X of cosine Z.

00:04:01.497 --> 00:04:04.748
Well no Xs show up there,
so that's just zero.

00:04:04.748 --> 00:04:06.583
So that's just a zero.

00:04:06.583 --> 00:04:08.301
And then we're subtracting off the partial

00:04:08.301 --> 00:04:11.064
with respect to Y of X times Y.

00:04:11.064 --> 00:04:13.456
Well X looks like a constant,
Y looks like the variable,

00:04:13.456 --> 00:04:15.870
so that partial derivative is just X.

00:04:15.870 --> 00:04:18.120
So we're subtracting off X.

00:04:19.888 --> 00:04:21.885
Which means if we simplify this,

00:04:21.885 --> 00:04:24.671
so the curl of our vector field,

00:04:24.671 --> 00:04:27.922
curl of our vector field as a whole,

00:04:27.922 --> 00:04:30.589
as this function of X, Y, and Z,

00:04:31.846 --> 00:04:33.285
is equal to,

00:04:33.285 --> 00:04:35.886
and that first component, the i component,

00:04:35.886 --> 00:04:38.138
we've got one minus negative sine of Z,

00:04:38.138 --> 00:04:39.903
so minus minus sine of Z.

00:04:39.903 --> 00:04:41.153
That's one plus

00:04:42.722 --> 00:04:43.555
sine of Z.

00:04:45.462 --> 00:04:47.505
And then the j component,
we're subtracting off,

00:04:47.505 --> 00:04:48.852
but it's zero.

00:04:48.852 --> 00:04:49.952
Usually if you're subtracting off,

00:04:49.952 --> 00:04:51.926
you'd have to make sure
to remember to flip those,

00:04:51.926 --> 00:04:55.107
but both of those are zero, so
the entire j component here,

00:04:55.107 --> 00:04:58.218
or the Y component of the output, is zero.

00:04:58.218 --> 00:05:02.385
And then finally, the k
component is zero minus X,

00:05:03.605 --> 00:05:06.938
so that entire thing is just negative X.

00:05:08.644 --> 00:05:09.759
And that's the curl of the function.

00:05:09.759 --> 00:05:11.245
And in general, that's how you do it.

00:05:11.245 --> 00:05:13.938
You would take a look at the way

00:05:13.938 --> 00:05:16.562
that your function is defined
in each component there,

00:05:16.562 --> 00:05:18.095
and imaging taking the cross product

00:05:18.095 --> 00:05:21.508
between this del symbol,
this partial partial X,

00:05:21.508 --> 00:05:23.412
partial partial Y, partial partial Z,

00:05:23.412 --> 00:05:24.420
and you take the cross product

00:05:24.420 --> 00:05:26.184
between that and your function,

00:05:26.184 --> 00:05:29.389
and it involves taking six
different partial derivatives.

00:05:29.389 --> 00:05:31.525
And mainly it's a matter of bookkeeping

00:05:31.525 --> 00:05:32.895
to make sure you do it right.

00:05:32.895 --> 00:05:34.828
And you'll end up with
something like this.