Curvature of a helix, part 1

WEBVTT
Kind: captions
Language: en

00:00:00.238 --> 00:00:01.843
- [Voiceover] So let's
compute the curvature of a

00:00:01.843 --> 00:00:04.527
three dimensional parametric curve

00:00:04.527 --> 00:00:07.460
and the one I have in
mind has a special name.

00:00:07.460 --> 00:00:10.744
It's a helix and the first two components

00:00:10.744 --> 00:00:13.036
kind of make it look like a circle.

00:00:13.036 --> 00:00:16.122
It's going to be cosine
of t for the x component,

00:00:16.122 --> 00:00:18.019
sine of t for the y component

00:00:18.019 --> 00:00:20.266
but this is three dimensional,
I know it makes it

00:00:20.266 --> 00:00:21.790
a little different from a circle.

00:00:21.790 --> 00:00:23.017
I'm going to have the last component

00:00:23.017 --> 00:00:24.767
be t divided by five.

00:00:26.693 --> 00:00:28.694
And what this looks like we
can visualize it pretty well.

00:00:28.694 --> 00:00:31.306
I'm going to go on over
to the graph of it here.

00:00:31.306 --> 00:00:34.610
So this shape is called a
helix and you can see how

00:00:34.610 --> 00:00:37.176
looking from the x y planes perspective

00:00:37.176 --> 00:00:38.973
it looks as if it's going to draw a circle

00:00:38.973 --> 00:00:40.623
and really these lines should all line up

00:00:40.623 --> 00:00:42.781
when you're facing it but
it's due to the perspective

00:00:42.781 --> 00:00:44.766
where things farther away look smaller.

00:00:44.766 --> 00:00:46.485
But it would just be drawing a circle,

00:00:46.485 --> 00:00:49.382
but then the z component
because z increases

00:00:49.382 --> 00:00:51.744
while your parameter t
increases, you're kind

00:00:51.744 --> 00:00:54.650
of rising, as if it's a spiral staircase.

00:00:54.650 --> 00:00:56.572
And now, before we compute curvature,

00:00:56.572 --> 00:00:59.928
to know what we're really going
for, what this represents,

00:00:59.928 --> 00:01:03.168
you kind of imagine yourself
, maybe this isn't a road,

00:01:03.168 --> 00:01:06.305
but it's like a space
freeway, and you're driving

00:01:06.305 --> 00:01:09.045
your spaceship along it and
you imagine that you get stuck

00:01:09.045 --> 00:01:12.016
at some point, or maybe
not that you get stuck,

00:01:12.016 --> 00:01:13.405
but that all of your instruments lock.

00:01:13.405 --> 00:01:14.915
And your steering wheel
locks or your joystick

00:01:14.915 --> 00:01:17.014
or however you're steering,
it all just locks up

00:01:17.014 --> 00:01:21.561
and you're gonna trace out
a certain circle in space.

00:01:21.561 --> 00:01:24.505
And that circle might
look something like this.

00:01:24.505 --> 00:01:26.435
So if you were turning
however you are on the helix,

00:01:26.435 --> 00:01:28.090
but then you can't do anything different,

00:01:28.090 --> 00:01:29.940
you might trace out a giant circle.

00:01:29.940 --> 00:01:33.360
And what we care about is
the radius of that circle.

00:01:33.360 --> 00:01:37.478
And if you take one divided
by the radius of that circle

00:01:37.478 --> 00:01:40.298
you trace out that's
going to be the curvature.

00:01:40.298 --> 00:01:42.166
That's going to be the
little kappa curvature.

00:01:42.166 --> 00:01:43.817
And of course the way that we compute it,

00:01:43.817 --> 00:01:45.495
we don't directly talk
about that circle at all,

00:01:45.495 --> 00:01:46.530
but it's actually a good thing

00:01:46.530 --> 00:01:48.278
to keep in the back of your mind.

00:01:48.278 --> 00:01:51.613
The way that we compute
it is to first find

00:01:51.613 --> 00:01:55.168
the unit tangent vector
function with the same parameter

00:01:55.168 --> 00:01:57.993
and what that means if you
imagine your helix spiraling

00:01:57.993 --> 00:02:00.253
through three dimensional space.

00:02:00.253 --> 00:02:02.746
Man, I am not as good an
artist as the computer is

00:02:02.746 --> 00:02:05.517
when it comes to drawing a helix.

00:02:05.517 --> 00:02:07.340
But the unit tangent vector function

00:02:07.340 --> 00:02:09.059
would be something that gives you

00:02:09.059 --> 00:02:11.959
a tangent vector at every given point,

00:02:11.959 --> 00:02:13.417
you know kind of the direction that you

00:02:13.417 --> 00:02:15.470
on your space ship are travelling.

00:02:16.826 --> 00:02:19.441
And to do that you take the derivative of

00:02:19.441 --> 00:02:21.245
your parameterization.

00:02:21.245 --> 00:02:22.872
That derivative, which
is going to give you

00:02:22.872 --> 00:02:25.322
a tangent vector, but it might
not be a unit tangent vector,

00:02:25.322 --> 00:02:27.785
so you divide it by its own magnitude

00:02:31.458 --> 00:02:33.727
And that'll give you
a unit tangent vector.

00:02:33.727 --> 00:02:36.256
And then ultimately, the
goal that we're shooting for

00:02:36.256 --> 00:02:38.052
is gonna be to find the derivative of this

00:02:38.052 --> 00:02:41.984
tangent vector function with
respect to the arc length.

00:02:41.984 --> 00:02:44.484
So as a first step, we'll start
by finding a derivative of

00:02:44.484 --> 00:02:46.915
our paremetarization function.

00:02:46.915 --> 00:02:48.944
So when we take that
derivative, luckily there's not

00:02:48.944 --> 00:02:51.926
a lot of new things
going on for derivatives.

00:02:51.926 --> 00:02:55.293
So S prime from single variable calculus,

00:02:55.293 --> 00:02:57.138
we just take the derivative
of each component,

00:02:57.138 --> 00:03:01.241
so cosine goes to negative sine of t.

00:03:02.477 --> 00:03:06.094
Sine, it's derivative is cosine of t

00:03:07.009 --> 00:03:10.152
and then the derivative
of t divided by five

00:03:10.152 --> 00:03:11.163
is just a constant.

00:03:11.163 --> 00:03:12.583
That's just one over five.

00:03:12.583 --> 00:03:15.456
Boy, it is hard to say the
word derivative over and over.

00:03:15.456 --> 00:03:17.559
Say it five times.

00:03:17.559 --> 00:03:21.019
So that's S prime of t,
and now what we need to do

00:03:21.019 --> 00:03:24.542
we need to find the
magnitude of S prime of t.

00:03:24.542 --> 00:03:28.192
So what that involves is
we're taking the magnitude

00:03:28.192 --> 00:03:30.805
S prime of t as a vector.

00:03:32.022 --> 00:03:33.282
We take the square root

00:03:33.282 --> 00:03:35.633
of the sum of the squares
of each of its components.

00:03:35.633 --> 00:03:39.265
So, sine, negative sine squared

00:03:39.265 --> 00:03:41.517
just looks like sine squared.

00:03:41.517 --> 00:03:42.900
Sine squared of t.

00:03:43.861 --> 00:03:44.791
Cosine squared.

00:03:46.436 --> 00:03:50.102
Cosine squared of t.

00:03:50.102 --> 00:03:51.755
And then one fifth squared

00:03:51.755 --> 00:03:54.412
and that's just one twenty fifth.

00:03:56.326 --> 00:03:59.897
You might notice I use a lot
of these sine cosine pairs

00:03:59.897 --> 00:04:01.783
in examples, partly
because they draw circles

00:04:01.783 --> 00:04:03.943
and lots of things are fun
that involve drawing circles.

00:04:03.943 --> 00:04:06.994
But also because it has a
tendency to let things simplify

00:04:06.994 --> 00:04:08.553
especially if you are taking a magnitude

00:04:08.553 --> 00:04:10.153
because sine squared plus cosine squared

00:04:10.153 --> 00:04:12.270
just equals one.

00:04:12.270 --> 00:04:17.067
So this entire formula boils down

00:04:17.067 --> 00:04:21.434
to the square root of one
plus one divided by 25.

00:04:22.680 --> 00:04:25.467
And for this you might be
thinking off to the side that

00:04:25.467 --> 00:04:30.467
that's 25 over 25 plus one over 25

00:04:31.544 --> 00:04:34.981
So making even more room
here, what that equals

00:04:36.227 --> 00:04:40.274
is the square root of 26 divided by 25.

00:04:43.259 --> 00:04:45.742
And just because 25 is
already a square and it might

00:04:45.742 --> 00:04:47.983
make things look nice I am
going to write this as the

00:04:47.983 --> 00:04:51.500
square root of 26 divided by 5,

00:04:51.500 --> 00:04:53.164
the square root of 25.

00:04:53.164 --> 00:04:57.601
This whole thing is the
magnitude of our derivative.

00:04:59.190 --> 00:05:00.919
And we think to ourselves,
it's quite lucky

00:05:00.919 --> 00:05:02.475
that this came out to be a constant

00:05:02.475 --> 00:05:04.982
because, as we saw with
the more general formula,

00:05:04.982 --> 00:05:07.352
it's often pretty nasty
and it can get pretty bad.

00:05:07.352 --> 00:05:09.313
But in this case, it's just a constant,

00:05:09.313 --> 00:05:12.090
which is nice because
as we go up and we start

00:05:12.090 --> 00:05:15.547
to think about what our
unit tangent vector function

00:05:15.547 --> 00:05:18.898
for the helix should be, we're just going

00:05:18.898 --> 00:05:20.553
to take the derivative function

00:05:20.553 --> 00:05:23.744
and divide each term by that magnitude.

00:05:23.744 --> 00:05:25.529
So it's going to look almost identical.

00:05:25.529 --> 00:05:28.406
It's going to be negative sine of t.

00:05:28.406 --> 00:05:31.173
Except now we're dividing
by that magnitude

00:05:31.173 --> 00:05:32.294
and that magnitude of course

00:05:32.294 --> 00:05:34.024
is root 26 over five,

00:05:35.016 --> 00:05:38.131
So we go up here and we
say we're dividing this

00:05:38.131 --> 00:05:42.938
by root 26 over five.

00:05:42.938 --> 00:05:44.304
That whole quantity.

00:05:45.604 --> 00:05:49.942
And then similarly y
component is cosine of t

00:05:49.942 --> 00:05:53.609
divided by the quantity of the root

00:05:53.609 --> 00:05:56.163
26 divided by 5.

00:05:57.443 --> 00:06:02.120
The last part is one fifth,
I'll put that in parenthesis,

00:06:02.120 --> 00:06:05.177
divided by that same amount.

00:06:05.177 --> 00:06:06.674
root 26 over five.

00:06:06.674 --> 00:06:08.243
So we're just taking the whole vector

00:06:08.243 --> 00:06:10.243
and we're dividing it by the magnitude

00:06:10.243 --> 00:06:12.345
that it has and we're lucky, again,

00:06:12.345 --> 00:06:14.401
that even though this vector
is a function, and it can

00:06:14.401 --> 00:06:16.627
depend on t, the magnitude doesn't.

00:06:16.627 --> 00:06:19.079
So the unit tangent vector function we get

00:06:19.079 --> 00:06:21.821
as a result is relatively simple.

00:06:21.821 --> 00:06:24.829
And I'll call it a day
here and then continue on

00:06:24.829 --> 00:06:27.003
on with the same line of
reasoning in the next video.


Multivariable chain rule and directional derivatives

WEBVTT
Kind: captions
Language: en

00:00:00.665 --> 00:00:02.331
- [Voiceover] So, in the last video,

00:00:02.331 --> 00:00:04.038
I introduced the vector form of

00:00:04.038 --> 00:00:05.745
the multivariable chain rule

00:00:05.745 --> 00:00:07.045
and just to remind ourselves,

00:00:07.045 --> 00:00:09.361
I'm saying you have
some kind of function f,

00:00:09.361 --> 00:00:10.661
and in this case I said it comes from

00:00:10.661 --> 00:00:13.831
a 100 dimensional space,
so you might imagine--

00:00:13.831 --> 00:00:16.066
Well, I can't imagine a
100 dimensional space,

00:00:16.066 --> 00:00:17.397
but in principle, you're just thinking

00:00:17.397 --> 00:00:20.437
of some area that's 100 dimensions,

00:00:20.437 --> 00:00:21.810
it can be two if you wanted to think

00:00:21.810 --> 00:00:23.557
more concretely in two dimensions.

00:00:23.557 --> 00:00:25.751
And it's a scalar valued function

00:00:25.751 --> 00:00:28.027
so it just outputs to a number line,

00:00:28.027 --> 00:00:29.733
some kind of number line that I'll think

00:00:29.733 --> 00:00:31.566
of as f as its output.

00:00:33.690 --> 00:00:35.608
And what we're gonna do is we compose it

00:00:35.608 --> 00:00:36.990
with a vector valued function

00:00:36.990 --> 00:00:40.087
so some function that
takes in a single number t

00:00:40.087 --> 00:00:43.279
and then outputs into that
super high dimensional space.

00:00:43.279 --> 00:00:45.839
So you're thinking, you go
from the single variable t

00:00:45.839 --> 00:00:47.952
to some very high dimensional space

00:00:47.952 --> 00:00:49.944
that we think of as full of vectors,

00:00:49.944 --> 00:00:53.750
and then you take from that
over to a single variable,

00:00:53.750 --> 00:00:55.908
over to a number.

00:00:55.908 --> 00:00:57.452
And you know, the way you'd write that out

00:00:57.452 --> 00:01:00.581
is you'd say f composed
with the output of v,

00:01:00.581 --> 00:01:02.748
so f composed with v of t,

00:01:04.544 --> 00:01:07.570
and what we're interested in
doing is taking its derivative.

00:01:07.570 --> 00:01:11.715
So the derivative of that composition is--

00:01:11.715 --> 00:01:13.788
and I told you and we
kind of walked through

00:01:13.788 --> 00:01:16.591
where this come from, the gradient of f,

00:01:16.591 --> 00:01:20.758
evaluated at v of t, evaluated
at your original output,

00:01:22.186 --> 00:01:24.872
that product, with the derivative of v,

00:01:24.872 --> 00:01:27.963
the vectorized derivative,
and what that means, you know,

00:01:27.963 --> 00:01:31.202
for v, you're just taking the
derivative of every component.

00:01:31.202 --> 00:01:33.722
So when you take this and
you take the derivative

00:01:33.722 --> 00:01:37.054
with respect to t, all that means is

00:01:37.054 --> 00:01:40.955
that each component, you're
taking the derivative of it.

00:01:40.955 --> 00:01:44.246
The x1 dt, the x2 dt, on and on until d

00:01:44.246 --> 00:01:47.579
and then the one hundredth component dt.

00:01:50.666 --> 00:01:52.007
So this was the vectorized form

00:01:52.007 --> 00:01:54.649
of the multivariable chain rule.

00:01:54.649 --> 00:01:56.315
And what I wanna do here is show

00:01:56.315 --> 00:02:00.541
how this looks a lot like
a directional derivative.

00:02:00.541 --> 00:02:01.800
And if you haven't watched the video

00:02:01.800 --> 00:02:03.995
on the directional derivative,
maybe go back, take a look,

00:02:03.995 --> 00:02:06.392
kind of remind yourself, but in principle,

00:02:06.392 --> 00:02:08.668
you say, if you're in
the input space of f,

00:02:08.668 --> 00:02:13.260
and you nudge yourself
along some kind of vector v,

00:02:13.260 --> 00:02:14.722
and maybe just because I'm using v there,

00:02:14.722 --> 00:02:17.567
I'll instead say some kind of vector w.

00:02:17.567 --> 00:02:19.273
So not a function, just a vector.

00:02:19.273 --> 00:02:21.508
And you're wondering, hey,
how much does that result

00:02:21.508 --> 00:02:24.149
in a change to the output of f,

00:02:24.149 --> 00:02:26.100
that's answered by the
directional derivative

00:02:26.100 --> 00:02:28.985
and you'd write directional derivative

00:02:28.985 --> 00:02:32.033
in the direction of w of f,

00:02:32.033 --> 00:02:33.821
the directional derivative of f,

00:02:33.821 --> 00:02:36.584
and I should say at some
point, some input point,

00:02:36.584 --> 00:02:38.981
p for that input point and
it's a vector in this case,

00:02:38.981 --> 00:02:41.544
like a 100 dimensional vector.

00:02:41.544 --> 00:02:43.077
And the way you evaluate it,

00:02:43.077 --> 00:02:45.556
is you take the gradient of f,

00:02:45.556 --> 00:02:48.278
this is why we use the nabla
notation in the first place,

00:02:48.278 --> 00:02:50.229
it's an indicative of how we compute it,

00:02:50.229 --> 00:02:53.967
the gradient of f evaluated
at that same input point,

00:02:53.967 --> 00:02:55.836
the same input vector p.

00:02:55.836 --> 00:02:58.374
So here, just to be clear,
you'd be thinking of

00:02:58.374 --> 00:03:02.370
whatever vector to your
input point, that's p.

00:03:02.370 --> 00:03:03.873
But then the nudge, the nudge away

00:03:03.873 --> 00:03:06.515
from that input point is w.

00:03:06.515 --> 00:03:08.587
And you take the dot product

00:03:08.587 --> 00:03:11.800
between that and the vector itself,

00:03:11.800 --> 00:03:14.917
the vector that represents
your nudge direction.

00:03:14.917 --> 00:03:15.771
But that looks a lot like

00:03:15.771 --> 00:03:17.518
the multivariable chain rule up here,

00:03:17.518 --> 00:03:20.647
except instead of w, you're
taking the derivative,

00:03:20.647 --> 00:03:22.679
the vector value derivative of v,

00:03:22.679 --> 00:03:24.466
so this whole thing you could say

00:03:24.466 --> 00:03:27.480
is the directional derivative
in the direction of

00:03:27.480 --> 00:03:31.406
the derivative of t, and
that's kind of confusing.

00:03:31.406 --> 00:03:34.585
Directional derivative in the
direction of a derivative,

00:03:34.585 --> 00:03:38.790
of f, and what point are you taking this,

00:03:38.790 --> 00:03:41.223
at what point are you taking
this directional derivative?

00:03:41.223 --> 00:03:44.636
Well, it's wherever the output of v is.

00:03:44.636 --> 00:03:47.608
So this is very compact,
it's saying quite a bit here.

00:03:47.608 --> 00:03:49.910
But a way that you could
be thinking about this,

00:03:49.910 --> 00:03:53.493
is v of t, so I'm gonna
kind of erase here.

00:03:56.917 --> 00:03:59.584
V of t has you zooming all about

00:04:01.685 --> 00:04:03.473
and as you shift t, it kind of moves you

00:04:03.473 --> 00:04:05.708
through this space in some way.

00:04:05.708 --> 00:04:07.333
And each one of these output points here

00:04:07.333 --> 00:04:11.000
represents the vector,
v of t at some point,

00:04:13.307 --> 00:04:16.517
the derivative of that, what
does this derivative represent?

00:04:16.517 --> 00:04:18.874
That's the tangent vector to that motion,

00:04:18.874 --> 00:04:21.202
you know, so you're zipping
about through that space,

00:04:21.202 --> 00:04:22.644
the tangent vector to your motion,

00:04:22.644 --> 00:04:25.727
that's how we interpret v prime of t,

00:04:29.633 --> 00:04:32.787
the derivative of v with respect to t.

00:04:32.787 --> 00:04:34.225
I mean why should that make sense?

00:04:34.225 --> 00:04:36.135
Why should the directional derivative

00:04:36.135 --> 00:04:38.451
in the direction of v prime of t,

00:04:38.451 --> 00:04:41.865
this change to the
intermediary function v,

00:04:41.865 --> 00:04:45.115
have anything to do with the
multivariable chain rule?

00:04:45.115 --> 00:04:46.985
Well, remember what
we're asking when we say

00:04:46.985 --> 00:04:50.308
dt of this composition is we're saying

00:04:50.308 --> 00:04:54.177
we take a tiny nudge to t,
so that tiny change here,

00:04:54.177 --> 00:04:56.313
in the value t, and we're wondering

00:04:56.313 --> 00:04:59.401
what changed that result
in after the composition?

00:04:59.401 --> 00:05:02.490
Well, at a given point,
that tiny nudge in t

00:05:02.490 --> 00:05:06.187
causes a change in the
direction of v prime of t.

00:05:06.187 --> 00:05:07.284
That's kind of the whole meaning

00:05:07.284 --> 00:05:08.991
of this vector value derivative.

00:05:08.991 --> 00:05:11.957
You change t by a little bit,
and that's gonna tell you

00:05:11.957 --> 00:05:14.274
how you move in the output space.

00:05:14.274 --> 00:05:16.346
But then you say, "Okay,
so I've moved a little bit

00:05:16.346 --> 00:05:18.621
"in this intermediary
100 dimensional space,

00:05:18.621 --> 00:05:21.263
"how does that influence the output of f

00:05:21.263 --> 00:05:22.685
"based on the behavior of just

00:05:22.685 --> 00:05:24.432
"the multivariable function f?"

00:05:24.432 --> 00:05:27.317
Well, that's what the
directional derivative is asking.

00:05:27.317 --> 00:05:30.324
It says you take a nudge in
the direction of some vector,

00:05:30.324 --> 00:05:32.641
in this case, I wrote
v prime of t over here.

00:05:32.641 --> 00:05:34.826
More generally, you
could say any vector w,

00:05:34.826 --> 00:05:36.492
you take a nudge in that direction.

00:05:36.492 --> 00:05:38.077
And more importantly, you know,

00:05:38.077 --> 00:05:40.271
the size of v prime of t matters here.

00:05:40.271 --> 00:05:42.059
If you're moving really
quickly, you would expect

00:05:42.059 --> 00:05:44.009
that change to be larger, so the fact

00:05:44.009 --> 00:05:47.473
that v prime of t would
be larger is helpful.

00:05:47.473 --> 00:05:49.405
And the directional
derivative is telling you

00:05:49.405 --> 00:05:52.575
the size of the change in f as a ratio

00:05:52.575 --> 00:05:56.557
of the proportion of
that directional vector

00:05:56.557 --> 00:05:58.385
that you went along. Right?

00:05:58.385 --> 00:06:00.661
You could--another notation
for the directional derivative

00:06:00.661 --> 00:06:03.755
is to say partial f, and then partial

00:06:03.755 --> 00:06:05.732
whatever that vector is.

00:06:05.732 --> 00:06:08.007
Basically saying you
take size of that nudge

00:06:08.007 --> 00:06:11.636
along that vector as a
proportion of the vector itself,

00:06:11.636 --> 00:06:13.960
and then you consider
the change to the output

00:06:13.960 --> 00:06:15.376
and you're taking the ratio.

00:06:15.376 --> 00:06:17.530
So I think this is a very
beautiful way of understanding

00:06:17.530 --> 00:06:20.050
the multivariable chain rule.

00:06:20.050 --> 00:06:21.756
Cause it give this image of, you know,

00:06:21.756 --> 00:06:23.260
you're thinking of v of
t, and you're thinking

00:06:23.260 --> 00:06:24.722
of zipping along in some way,

00:06:24.722 --> 00:06:26.917
and the direction and
value of your velocity

00:06:26.917 --> 00:06:28.948
as you zip along is what determines

00:06:28.948 --> 00:06:31.590
the change in the output
of the function f.

00:06:31.590 --> 00:06:33.459
So hopefully, that helps
give a better understanding

00:06:33.459 --> 00:06:35.247
both of the directional derivative

00:06:35.247 --> 00:06:37.319
and of the multivariable chain rule.

00:06:37.319 --> 00:06:40.540
It's one of those nice
little interpretations.


Curvature of a helix, part 2

WEBVTT
Kind: captions
Language: en

00:00:00.798 --> 00:00:02.015
- [Voiceover] So, where we left off,

00:00:02.015 --> 00:00:04.581
we were looking at this,
this parametric function

00:00:04.581 --> 00:00:06.933
for a three dimensional
curve, and what it draws,

00:00:06.933 --> 00:00:10.451
I showed you, was a helix
in three dimensional space.

00:00:10.451 --> 00:00:12.378
And, we're trying to find it's curvature,

00:00:12.378 --> 00:00:14.572
which...the way you think about that,

00:00:14.572 --> 00:00:15.977
you have a circle...

00:00:15.977 --> 00:00:17.800
You're thinking of the
circle that most closely

00:00:17.800 --> 00:00:19.681
hugs that curve, or you could imagine

00:00:19.681 --> 00:00:22.734
you're flying a spaceship
and all your instruments

00:00:22.734 --> 00:00:25.080
lock up when you're turning,
and you're wondering

00:00:25.080 --> 00:00:26.774
what circle in space
you're going to trace out,

00:00:26.774 --> 00:00:29.932
and you're looking for one
divided by the radius of that.

00:00:29.932 --> 00:00:32.417
Or if you prefer my,
you know, kindergartner

00:00:32.417 --> 00:00:34.576
drawing skills, you could
just look at the helix

00:00:34.576 --> 00:00:36.917
over here while we work it through.

00:00:36.917 --> 00:00:39.584
And, the part where we left off,

00:00:40.686 --> 00:00:42.150
we had this tangent vector function.

00:00:42.150 --> 00:00:45.830
This unit tangent vector
function for our curve.

00:00:45.830 --> 00:00:47.989
So at every given value, T, whatever point

00:00:47.989 --> 00:00:50.032
that corresponds to on the curve,

00:00:50.032 --> 00:00:52.772
this function is going
to give us the vector

00:00:52.772 --> 00:00:55.872
that is of unit length
and tangent to the curve.

00:00:55.872 --> 00:00:58.287
And the ultimate goal,
for curvature, is to find

00:00:58.287 --> 00:01:02.664
the derivative of that
unit tangent vector,

00:01:02.664 --> 00:01:04.952
with respect to arclength.

00:01:04.952 --> 00:01:07.227
And specifically we want it's magnitude.

00:01:07.227 --> 00:01:09.804
And what that typically
requires, and I talked

00:01:09.804 --> 00:01:11.604
about this in the videos on the formula,

00:01:11.604 --> 00:01:14.553
is you take it's derivative, with respect

00:01:14.553 --> 00:01:16.410
to the parameter, T, because

00:01:16.410 --> 00:01:17.966
that's the thing you can actually do,

00:01:17.966 --> 00:01:20.671
and that might not correspond
to unit length, right?

00:01:20.671 --> 00:01:22.668
If you nudge the parameter,
T, that might not

00:01:22.668 --> 00:01:26.070
nudge you a corresponding
length on the curve.

00:01:26.070 --> 00:01:29.344
But you correct that by
dividing out, by the derivative

00:01:29.344 --> 00:01:33.872
of the parameterization
function, with respect to T.

00:01:33.872 --> 00:01:35.670
And that's actually arclength,

00:01:35.670 --> 00:01:39.837
this magnitude of the derivative
of the parameterization

00:01:40.901 --> 00:01:42.328
with respect to T.

00:01:42.328 --> 00:01:44.023
Boy, is that a hard word.

00:01:44.023 --> 00:01:46.786
So, with our given
function, let's go ahead

00:01:46.786 --> 00:01:48.121
and start doing that.

00:01:48.121 --> 00:01:50.060
And I think first, given
that we have this kind of

00:01:50.060 --> 00:01:52.103
fractions and fractions
form, I'm just going

00:01:52.103 --> 00:01:55.307
to start by writing it a
little bit more simply.

00:01:55.307 --> 00:01:57.026
So our unit tangent vector function,

00:01:57.026 --> 00:01:59.661
that first component,
where we're dividing by

00:01:59.661 --> 00:02:02.041
the square root of 26 divided by five,

00:02:02.041 --> 00:02:04.038
instead I'm just going to write that as

00:02:04.038 --> 00:02:07.075
negative five times the sine of t.

00:02:07.075 --> 00:02:10.337
So, I'm just kind of moving
that five up into the numerator,

00:02:10.337 --> 00:02:13.855
divided by the root of 26,
and I'll do the same thing

00:02:13.855 --> 00:02:17.755
for cosine, where we move that
five up into the numerator.

00:02:17.755 --> 00:02:21.755
Cosine of t, divided by
the root of 26, and then

00:02:23.991 --> 00:02:26.754
the last part, one fifth
divided by square root

00:02:26.754 --> 00:02:30.097
of 26 fifths, gives us just one divided

00:02:30.097 --> 00:02:31.847
by square root of 26.

00:02:34.509 --> 00:02:38.236
Alright, so the first step in
using our curvature formula

00:02:38.236 --> 00:02:41.255
is going to be to take the
derivative of this guy, right?

00:02:41.255 --> 00:02:44.888
We need the derivative of
the tangent vector function.

00:02:44.888 --> 00:02:46.595
So, we go ahead and start doing that.

00:02:46.595 --> 00:02:50.762
We see that d big T, d little
t, so tangent vector function

00:02:51.727 --> 00:02:54.188
parameter, is equal to, and
we just take the derivative

00:02:54.188 --> 00:02:56.011
of each component.

00:02:56.011 --> 00:03:00.178
So, negative five sine goes
to negative five cosine.

00:03:03.395 --> 00:03:07.389
And we divide out by the
constant, square root of 26.

00:03:07.389 --> 00:03:10.593
Similarly, five cosine
goes to negative five sine,

00:03:10.593 --> 00:03:14.831
since the derivative of
cosine is negative sine.

00:03:14.831 --> 00:03:18.788
Negative five sine divided
by 26, or square root of 26.

00:03:18.788 --> 00:03:21.474
And the last component is just a constant,

00:03:21.474 --> 00:03:25.026
so derivative is nothing, it's zero.

00:03:25.026 --> 00:03:27.282
Next step, we're going
to take the magnitude

00:03:27.282 --> 00:03:28.833
of that vector that we just found.

00:03:28.833 --> 00:03:30.969
So we're trying to find the
magnitude of the derivative

00:03:30.969 --> 00:03:33.210
of the tangent vector function.

00:03:33.210 --> 00:03:34.928
So we say alright.

00:03:34.928 --> 00:03:37.028
Magnitude of what we just found,

00:03:37.028 --> 00:03:39.695
d big T, d little t, involves...

00:03:41.732 --> 00:03:45.331
so this magnitude will
be the square root of,

00:03:45.331 --> 00:03:47.815
make the little tic there,
square root of the sum

00:03:47.815 --> 00:03:49.208
of the squares of these guys.

00:03:49.208 --> 00:03:53.794
So the square of that first
component is going to be 25,

00:03:53.794 --> 00:03:56.394
multiply by cosine squared,

00:03:56.394 --> 00:03:58.218
cosine squared of t,

00:03:58.218 --> 00:04:01.097
all divided by 26, right, because the

00:04:01.097 --> 00:04:04.264
square of the square root of 26 is 26.

00:04:05.137 --> 00:04:08.970
And then we add to that
25 times sine squared,

00:04:11.527 --> 00:04:14.694
sine squared of t, also divided by 26.

00:04:18.087 --> 00:04:21.350
And from that, we can factor out,

00:04:21.350 --> 00:04:24.933
factor out 25 over 26,
inside that radical,

00:04:26.640 --> 00:04:29.424
because both terms involve
multiplying by 25 and

00:04:29.424 --> 00:04:32.106
dividing by 26, and what
we're left with is a

00:04:32.106 --> 00:04:34.381
nice, and friendly cosine/sine pair.

00:04:34.381 --> 00:04:36.901
The reason we love
things involving circles.

00:04:36.901 --> 00:04:37.923
This always happens.

00:04:37.923 --> 00:04:39.385
Nice cancellation.

00:04:39.385 --> 00:04:42.133
This just becomes one, so what we're left

00:04:42.133 --> 00:04:45.300
with on the whole, is root 25 over 26.

00:04:48.019 --> 00:04:49.064
Pretty nice.

00:04:49.064 --> 00:04:51.038
Root 25 over 26.

00:04:51.038 --> 00:04:53.221
So for a curvature equation, we go up

00:04:53.221 --> 00:04:55.183
and we can start plugging that in.

00:04:55.183 --> 00:04:58.193
So we just found that
numerator, and found that it was

00:04:58.193 --> 00:05:01.276
the square root of 25, divided by 26.

00:05:03.885 --> 00:05:06.395
The entire thing, 25 over 26.

00:05:06.395 --> 00:05:09.354
And we already found the magnitude

00:05:09.354 --> 00:05:10.688
of the derivative itself.

00:05:10.688 --> 00:05:12.407
That's one of the things we needed to do

00:05:12.407 --> 00:05:13.533
to find the tangent vector.

00:05:13.533 --> 00:05:16.950
That's where this 26 over five came from.

00:05:17.910 --> 00:05:20.314
I deleted it from where we
did last video to make room,

00:05:20.314 --> 00:05:22.089
but if you look at the
last video, you can see

00:05:22.089 --> 00:05:25.468
where we got the square
root of 26 divided by five.

00:05:25.468 --> 00:05:28.208
And I'll actually write
that as square root of

00:05:28.208 --> 00:05:32.125
26 divided by 25, just
that's how we originally

00:05:33.966 --> 00:05:37.113
found it, I'm just putting the
five back under the radical.

00:05:37.113 --> 00:05:39.225
And it's tempting, if you
aren't looking too closely,

00:05:39.225 --> 00:05:41.095
to think these guys cancel
out, but it's actually

00:05:41.095 --> 00:05:42.766
they're the opposite of each other, right?

00:05:42.766 --> 00:05:46.134
One is 25 over 26 the other is 26 over 25.

00:05:46.134 --> 00:05:48.549
So if we put everything
under the radical there

00:05:48.549 --> 00:05:51.706
it's going to say equals the square root.

00:05:51.706 --> 00:05:55.873
And we're going to have 25
over 26, divided by 26 over 25

00:05:58.812 --> 00:06:02.527
and when we flip that bottom and multiply,

00:06:02.527 --> 00:06:06.527
what we get is 25 squared
divided by 26 squared.

00:06:10.248 --> 00:06:14.056
And the root of that whole
thing, just gives us 25

00:06:14.056 --> 00:06:16.973
over 26, and that is our curvature.

00:06:18.386 --> 00:06:20.232
That right there, is the answer.

00:06:20.232 --> 00:06:21.567
That is the curvature.

00:06:21.567 --> 00:06:23.007
So it's a little bit greater than one.

00:06:23.007 --> 00:06:24.156
No, no, sorry.

00:06:24.156 --> 00:06:25.679
So, it's a little bit less than one,

00:06:25.679 --> 00:06:27.233
which means that you're
curving a little bit less

00:06:27.233 --> 00:06:30.716
than you would if it was
a circle with radius one.

00:06:30.716 --> 00:06:33.154
Which kind of makes sense if
we look at the image here,

00:06:33.154 --> 00:06:36.149
because if the helix were
completely flattened out, right,

00:06:36.149 --> 00:06:38.333
if you imagine squishing
this down onto the xy plane,

00:06:38.333 --> 00:06:41.107
you'd just be going around
a circle with radius one.

00:06:41.107 --> 00:06:42.874
But by kind of pulling that spring and

00:06:42.874 --> 00:06:44.973
pulling it so that there's a z component,

00:06:44.973 --> 00:06:47.074
you're making it a
little bit more straight.

00:06:47.074 --> 00:06:49.187
So, the curvature should
go down a little bit,

00:06:49.187 --> 00:06:51.057
because it's becoming a
little bit more straight.

00:06:51.057 --> 00:06:52.810
The radius of curvature will go up.

00:06:52.810 --> 00:06:55.771
So that's the curvature of a helix,

00:06:55.771 --> 00:06:58.174
and that's a pretty good
example of how you can find

00:06:58.174 --> 00:07:00.391
the curvature by walking through directly,

00:07:00.391 --> 00:07:03.050
the idea of finding dt/ds.

00:07:03.050 --> 00:07:05.174
And you know, getting
that unit tangent vector.

00:07:05.174 --> 00:07:07.276
Getting the little unit arclength.

00:07:07.276 --> 00:07:10.933
And in the next example, I
think I'll go through one

00:07:10.933 --> 00:07:12.755
where you just use the formula.

00:07:12.755 --> 00:07:14.648
Where it's something a
little bit more complicated

00:07:14.648 --> 00:07:17.214
than thinking about this and
you turn to the formula itself.

00:07:17.214 --> 00:07:18.366
And I'll see you then.


More formal treatment of multivariable chain rule

WEBVTT
Kind: captions
Language: en

00:00:01.833 --> 00:00:02.793
- [Voiceover] Hello everyone.

00:00:02.793 --> 00:00:04.881
So this is what I might
call a more optional video.

00:00:04.881 --> 00:00:06.891
In the last couple videos, I talked about

00:00:06.891 --> 00:00:08.601
this multivariable chain rule,

00:00:08.601 --> 00:00:10.805
and I give some justification.

00:00:10.805 --> 00:00:11.779
And it might have been considered

00:00:11.779 --> 00:00:13.477
a little bit hand-wavy by some.

00:00:13.477 --> 00:00:14.768
I was doing a lot of things

00:00:14.768 --> 00:00:17.040
that looked kind of
like taking a derivative

00:00:17.040 --> 00:00:19.430
with respect to t, and
then multiplying that

00:00:19.430 --> 00:00:21.193
by an infinitesimal quantity, dt,

00:00:21.193 --> 00:00:23.092
and thinking of canceling those out.

00:00:23.092 --> 00:00:24.705
And some people might say, "Ah!

00:00:24.705 --> 00:00:25.755
"But this isn't really a fraction.

00:00:25.755 --> 00:00:28.152
"That's a derivative, that's
a differential operator,

00:00:28.152 --> 00:00:29.931
"and you're treating it incorrectly."

00:00:29.931 --> 00:00:31.740
And while that's true,

00:00:31.740 --> 00:00:33.781
the intuitions underlying a lot of this

00:00:33.781 --> 00:00:35.710
actually matches with the
formal argument pretty well.

00:00:35.710 --> 00:00:37.572
So what I wanna do here is just talk about

00:00:37.572 --> 00:00:40.155
what the formal argument behind

00:00:41.806 --> 00:00:42.937
the multivariable chain rule is,

00:00:42.937 --> 00:00:46.345
and just to remind ourselves
of the setup of where we are.

00:00:46.345 --> 00:00:48.515
You're thinking of v as
a vector-valued function,

00:00:48.515 --> 00:00:51.377
so this is something that
takes as an input, t,

00:00:51.377 --> 00:00:52.774
that lives on a number line,

00:00:52.774 --> 00:00:56.941
and then v maps this to some
kind of high-dimensional space.

00:00:59.250 --> 00:01:00.697
In the simplest case, you
might just think of that

00:01:00.697 --> 00:01:01.860
as a two-dimensional space,

00:01:01.860 --> 00:01:04.012
maybe it's three-dimensional space.

00:01:04.012 --> 00:01:05.454
Or it could be 100-dimensional.

00:01:05.454 --> 00:01:06.917
You don't have to literally
be visualizing it.

00:01:06.917 --> 00:01:09.381
And then f, our function f,

00:01:09.381 --> 00:01:11.621
somehow takes that 100-dimensional space,

00:01:11.621 --> 00:01:13.333
or two-dimensional, or three-dimensional,

00:01:13.333 --> 00:01:14.584
whatever it is,

00:01:14.584 --> 00:01:16.665
and then maps it onto the number line.

00:01:16.665 --> 00:01:19.264
So the overall effect of
the composition function

00:01:19.264 --> 00:01:21.884
is to just take a real
number to a real number

00:01:21.884 --> 00:01:23.164
so it's a single-variable function.

00:01:23.164 --> 00:01:25.616
So that's where we're taking
this ordinary derivative,

00:01:25.616 --> 00:01:27.754
rather than a partial
derivative, or gradient,

00:01:27.754 --> 00:01:29.236
or anything like that.

00:01:29.236 --> 00:01:31.374
But because it goes through
a multi-dimensional space,

00:01:31.374 --> 00:01:34.337
and you have this intermediary,
multivariable nature to it,

00:01:34.337 --> 00:01:36.014
that's why you have a gradient,

00:01:36.014 --> 00:01:37.221
and a vector-value derivative.

00:01:37.221 --> 00:01:39.339
With the formal argument,
the first thing you might do

00:01:39.339 --> 00:01:42.890
is just write out the formal
definition of a derivative.

00:01:42.890 --> 00:01:44.580
And in this case,

00:01:44.580 --> 00:01:45.893
it's a limit.

00:01:45.893 --> 00:01:48.145
Definitions of derivatives
are always gonna be

00:01:48.145 --> 00:01:50.577
some kind of limit as a
variable goes to zero.

00:01:50.577 --> 00:01:53.831
And here, you're loosely
thinking about h as being dt.

00:01:53.831 --> 00:01:55.270
And you could write delta t,

00:01:55.270 --> 00:01:58.350
but it's common to use h
just because that can be used

00:01:58.350 --> 00:01:59.919
for whatever your
differential quantity is.

00:01:59.919 --> 00:02:01.527
So that's on the denominator,

00:02:01.527 --> 00:02:02.725
'cause you're thinking of it as dt.

00:02:02.725 --> 00:02:05.774
And the top is whatever the change

00:02:05.774 --> 00:02:07.347
to this whole function is

00:02:07.347 --> 00:02:09.770
when you nudge that input by t.

00:02:09.770 --> 00:02:13.631
And what I mean by that
is you'll take f of v,

00:02:13.631 --> 00:02:16.131
not of t, but of t plus h,

00:02:16.131 --> 00:02:18.545
that kind of nudged output value,

00:02:18.545 --> 00:02:20.826
and you're wondering how different that is

00:02:20.826 --> 00:02:24.576
from f of v of t, the
original value, v of t.

00:02:27.237 --> 00:02:28.656
So this is what happens
when you just apply

00:02:28.656 --> 00:02:30.485
the formal definition of the derivative,

00:02:30.485 --> 00:02:34.529
the ordinary derivative, to
your composition function.

00:02:34.529 --> 00:02:36.188
And now, what do you do

00:02:36.188 --> 00:02:38.816
as you're trying to reason
about what this should equal?

00:02:38.816 --> 00:02:40.003
And a good place to start, actually,

00:02:40.003 --> 00:02:41.957
is to look back to the
intuition that I was giving

00:02:41.957 --> 00:02:44.187
for the multivariable chain
rule in the first place.

00:02:44.187 --> 00:02:47.376
You imagine nudging your input by some dt,

00:02:47.376 --> 00:02:49.235
some tiny change, and I was saying,

00:02:49.235 --> 00:02:50.443
oh, so that causes a change

00:02:50.443 --> 00:02:53.128
in the intermediary space

00:02:53.128 --> 00:02:55.295
of some kind of, you know,

00:02:56.315 --> 00:02:58.479
you could call it dv,
a change in the vector.

00:02:58.479 --> 00:02:59.356
And the way that you're thinking that,

00:02:59.356 --> 00:03:00.873
that as you take the
vector value derivative

00:03:00.873 --> 00:03:05.136
and multiply it by dt, it's
the proportionality constant

00:03:05.136 --> 00:03:07.866
between the size of your nudge
and the resulting vector.

00:03:07.866 --> 00:03:10.689
And loosely, you might imagine
those dt's crossing out

00:03:10.689 --> 00:03:12.146
as if they were fractions.

00:03:12.146 --> 00:03:13.379
It doesn't really matter.

00:03:13.379 --> 00:03:14.899
And then you say, "What does this change?"

00:03:14.899 --> 00:03:17.506
This change by a dv cause for f,

00:03:17.506 --> 00:03:20.662
and by definition, the resulting nudge

00:03:20.662 --> 00:03:23.030
to the output space of f

00:03:23.030 --> 00:03:25.525
is the directional
derivative in the direction

00:03:25.525 --> 00:03:29.211
of whatever your vector
nudge is of the function f.

00:03:29.211 --> 00:03:30.794
So this is the loose intuition,

00:03:30.794 --> 00:03:33.162
and where does that
carry over to formality?

00:03:33.162 --> 00:03:36.093
You say, "Well, in this
intermediary space,

00:03:36.093 --> 00:03:38.304
"we have to deal with the
vector value derivative of v."

00:03:38.304 --> 00:03:39.873
So it might be a good thing to just

00:03:39.873 --> 00:03:41.878
write down that definition, right?

00:03:41.878 --> 00:03:42.900
Write down the fact

00:03:42.900 --> 00:03:47.369
that the definition for the
vector value derivative of v,

00:03:47.369 --> 00:03:48.910
again, it looks almost identical.

00:03:48.910 --> 00:03:50.304
All these derivative definitions

00:03:50.304 --> 00:03:52.704
really do look kind of the same

00:03:52.704 --> 00:03:56.243
'cause what you're doing
is you're taking the limit

00:03:56.243 --> 00:03:57.852
as h goes to zero,

00:03:57.852 --> 00:04:00.974
h we're still thinking of as being dt.

00:04:00.974 --> 00:04:02.298
So that kind of sits on the bottom.

00:04:02.298 --> 00:04:05.095
But here you're just wondering
how your vector changes.

00:04:05.095 --> 00:04:07.561
And the difference, even though
we're kind of writing this

00:04:07.561 --> 00:04:09.867
the same way, and it looks
almost identical notationally,

00:04:09.867 --> 00:04:14.204
what's on the numerator
here, this v of t plus h,

00:04:14.204 --> 00:04:16.568
and this v of t, these are vectors.

00:04:16.568 --> 00:04:18.509
So this is kind of a
vector minus a vector.

00:04:18.509 --> 00:04:19.554
When you take the limit,

00:04:19.554 --> 00:04:21.159
you're getting a limiting vector,

00:04:21.159 --> 00:04:22.899
something in your high-dimensional space.

00:04:22.899 --> 00:04:24.229
It's not just a number.

00:04:24.229 --> 00:04:25.878
And now, another way to write this,

00:04:25.878 --> 00:04:27.571
one that's more helpful,

00:04:27.571 --> 00:04:29.271
more conducive to manipulation,

00:04:29.271 --> 00:04:32.080
is to say not that it equals
the limit of this value,

00:04:32.080 --> 00:04:36.479
and I'm gonna go ahead and
just copy this value here,

00:04:36.479 --> 00:04:38.810
kind of down here, and say,

00:04:38.810 --> 00:04:40.176
the value of our derivative

00:04:40.176 --> 00:04:43.509
actually equals this, subject
to some kind of error,

00:04:43.509 --> 00:04:45.527
which I'll just write as E of h,

00:04:45.527 --> 00:04:47.830
like an error function of h.

00:04:47.830 --> 00:04:48.894
And what you should be thinking is that

00:04:48.894 --> 00:04:51.877
that error function goes
to zero as h goes to zero.

00:04:51.877 --> 00:04:53.641
This is just writing
things so that we're able

00:04:53.641 --> 00:04:55.858
to manipulate it a little bit more easily.

00:04:55.858 --> 00:04:58.669
So I'll give ourselves some room here.

00:04:58.669 --> 00:05:00.324
And what you can do with this

00:05:00.324 --> 00:05:02.211
is multiply all sides by h.

00:05:02.211 --> 00:05:05.720
So this is our vector value derivative,

00:05:05.720 --> 00:05:07.220
just rewriting it.

00:05:08.177 --> 00:05:10.381
Multiply it by h.

00:05:10.381 --> 00:05:11.929
And you're thinking of this h as a dt,

00:05:11.929 --> 00:05:13.021
so maybe in the back of your mind,

00:05:13.021 --> 00:05:15.207
you're kind of thinking of
canceling this dt with the h.

00:05:15.207 --> 00:05:18.736
And what it equals is this
top, this numerator here,

00:05:18.736 --> 00:05:20.653
which was v of t plus h

00:05:23.407 --> 00:05:24.490
minus v of t.

00:05:26.103 --> 00:05:27.355
And in the back of your mind,

00:05:27.355 --> 00:05:30.710
you might be thinking,
this whole thing represents

00:05:30.710 --> 00:05:32.856
dv, a change in v.

00:05:32.856 --> 00:05:35.126
So the idea of canceling
out that dt with the h

00:05:35.126 --> 00:05:36.456
really does kind of come through here.

00:05:36.456 --> 00:05:37.904
But the difference between the more

00:05:37.904 --> 00:05:39.874
hand-waving argument before
of canceling those out

00:05:39.874 --> 00:05:41.341
and what we're doing here

00:05:41.341 --> 00:05:45.693
is now we're accounting
for that error function.

00:05:45.693 --> 00:05:47.214
In this case it's now multiplied by h

00:05:47.214 --> 00:05:51.007
'cause everything was
multiplied by h error function.

00:05:51.007 --> 00:05:53.857
And there's actually another
way that I'm gonna write this.

00:05:53.857 --> 00:05:56.981
There's a very useful
convention in analysis

00:05:56.981 --> 00:05:58.751
where I'll take something like this

00:05:58.751 --> 00:05:59.739
and instead I'll write it

00:05:59.739 --> 00:06:01.156
as little o of h.

00:06:02.508 --> 00:06:04.189
And this isn't literally a function.

00:06:04.189 --> 00:06:06.544
It's just a stand-in to
say whatever this is,

00:06:06.544 --> 00:06:07.832
whatever function that represents,

00:06:07.832 --> 00:06:11.884
it satisfies the property that
when we take that function

00:06:11.884 --> 00:06:13.540
and divide it by h,

00:06:13.540 --> 00:06:17.367
that will go to zero as
h goes to zero, right?

00:06:17.367 --> 00:06:18.842
Which is true here because
you imagine taking this

00:06:18.842 --> 00:06:20.597
and dividing by h, and that would be,

00:06:20.597 --> 00:06:23.154
this h cancels out and you
just have your error function

00:06:23.154 --> 00:06:24.205
is gonna go to zero.

00:06:24.205 --> 00:06:27.663
So now what I do is I use
this entire expression

00:06:27.663 --> 00:06:29.996
to write this v of t plus h.

00:06:31.134 --> 00:06:32.110
And the reason I wanna do that

00:06:32.110 --> 00:06:33.886
if we kind of scroll back up

00:06:33.886 --> 00:06:36.700
is because we see v of t plus h showing up

00:06:36.700 --> 00:06:38.739
in the original definition we care about.

00:06:38.739 --> 00:06:42.051
So this is just a way of
starting to get a grapple on that

00:06:42.051 --> 00:06:44.778
a little bit more firmly.

00:06:44.778 --> 00:06:48.945
So what I'd write, I'd say
that that v of t plus h,

00:06:49.898 --> 00:06:53.172
v of t plus h, that nudged output value,

00:06:53.172 --> 00:06:56.823
is equal to the original
value that I have, v of t

00:06:56.823 --> 00:07:00.257
plus, and it's gonna be
plus this derivative term,

00:07:00.257 --> 00:07:01.966
and you can kind of think that it's almost

00:07:01.966 --> 00:07:04.218
like a Taylor polynomial,

00:07:04.218 --> 00:07:06.497
where this is our first order term.

00:07:06.497 --> 00:07:08.854
We're evaluating it at whatever that t is,

00:07:08.854 --> 00:07:11.229
but we're multiplying it
by the value of that nudge,

00:07:11.229 --> 00:07:12.370
that linear term.

00:07:12.370 --> 00:07:16.930
And then the rest of the stuff
is just some little o of h.

00:07:16.930 --> 00:07:18.564
And maybe you'd say,
"Shouldn't you be subtracting

00:07:18.564 --> 00:07:19.602
"off that little o of h?"

00:07:19.602 --> 00:07:20.869
And it's not an actual function.

00:07:20.869 --> 00:07:22.749
It just represents anything that shrinks.

00:07:22.749 --> 00:07:25.215
And maybe I should say
it's the absolute value,

00:07:25.215 --> 00:07:27.485
like the magnitude, 'cause in this case,

00:07:27.485 --> 00:07:29.324
this is a vector-valued quantity.

00:07:29.324 --> 00:07:30.739
You know, that error is a vector.

00:07:30.739 --> 00:07:33.056
So it's the size of that vector

00:07:33.056 --> 00:07:36.223
divided by the size of h goes to zero.

00:07:37.584 --> 00:07:39.313
So this is the main tool that
we're gonna end up using.

00:07:39.313 --> 00:07:41.646
This is the way to represent

00:07:42.573 --> 00:07:44.366
v of t plus h.

00:07:44.366 --> 00:07:47.065
And now if we go back up
to the original definition

00:07:47.065 --> 00:07:48.714
of the vector value derivative,

00:07:48.714 --> 00:07:52.025
and I'll go ahead and copy that,

00:07:52.025 --> 00:07:54.275
go ahead and copy that guy.

00:07:57.870 --> 00:07:59.620
Little bit of debris.

00:08:04.303 --> 00:08:06.150
So copy that original definition

00:08:06.150 --> 00:08:10.045
for the ordinary derivative
of the composition function,

00:08:10.045 --> 00:08:11.650
and now when I write things in according

00:08:11.650 --> 00:08:14.307
to all the manipulations that we just did,

00:08:14.307 --> 00:08:17.320
this is really, it's still a limit,

00:08:17.320 --> 00:08:19.153
'cause h goes to zero,

00:08:20.137 --> 00:08:22.892
but what we put on the inside here

00:08:22.892 --> 00:08:23.975
is it's f of,

00:08:24.863 --> 00:08:27.443
now instead of writing v of t plus h,

00:08:27.443 --> 00:08:29.596
I'm gonna use everything
that I did up there.

00:08:29.596 --> 00:08:32.064
It's the value of v of t

00:08:32.064 --> 00:08:33.647
plus the derivative

00:08:36.674 --> 00:08:39.473
at our point times the size of h.

00:08:39.473 --> 00:08:40.970
So again, it's kind of
like a Taylor polynomial.

00:08:40.970 --> 00:08:42.390
This is your linear term,

00:08:42.390 --> 00:08:44.613
and then it's plus something
that we don't care about,

00:08:44.613 --> 00:08:46.145
something that's gonna get really small

00:08:46.145 --> 00:08:47.151
as h goes small,

00:08:47.151 --> 00:08:49.995
and really small in comparison
to h, more importantly.

00:08:49.995 --> 00:08:52.292
And from that you subtract off

00:08:52.292 --> 00:08:53.292
f of v of t.

00:08:56.817 --> 00:08:57.650
Kind of running off the edge.

00:08:57.650 --> 00:08:59.159
I always keep running off the edge.

00:08:59.159 --> 00:09:01.360
And all of that is divided by h.

00:09:01.360 --> 00:09:03.752
Now, the point here is

00:09:03.752 --> 00:09:05.801
when you look at this limit,
because we're taking it

00:09:05.801 --> 00:09:07.366
as h goes to zero,

00:09:07.366 --> 00:09:10.987
we'll basically be able to
ignore this o of h component

00:09:10.987 --> 00:09:12.451
because as h goes to zero,

00:09:12.451 --> 00:09:14.793
this gets very, very
small in comparison to h.

00:09:14.793 --> 00:09:17.969
So everything that's on the inside here

00:09:17.969 --> 00:09:20.246
is basically just the v of t

00:09:20.246 --> 00:09:22.746
plus this vector value, right?

00:09:24.390 --> 00:09:26.887
And this is h times some kind of vector.

00:09:26.887 --> 00:09:28.288
But if you think back, I made a video

00:09:28.288 --> 00:09:31.576
on the formal definition of
the directional derivative.

00:09:31.576 --> 00:09:33.013
And if you remembered,
or if you kind of go back

00:09:33.013 --> 00:09:36.658
and take a look now, this is
exactly the formal definition

00:09:36.658 --> 00:09:38.284
of the directional derivative.

00:09:38.284 --> 00:09:40.076
We're taking h to go to zero,

00:09:40.076 --> 00:09:41.808
the thing we're multiplying it by

00:09:41.808 --> 00:09:43.252
is a certain vector quantity.

00:09:43.252 --> 00:09:46.994
That vector is the nudge
to your original value,

00:09:46.994 --> 00:09:49.276
and then we're dividing everything by h.

00:09:49.276 --> 00:09:51.266
So by definition, this entire thing

00:09:51.266 --> 00:09:55.276
is the directional derivative
in the direction of

00:09:55.276 --> 00:09:58.345
the derivative of the function of t.

00:09:58.345 --> 00:10:00.382
I'm writing v prime t
instead of getting the whole

00:10:00.382 --> 00:10:01.882
dv, dt down there.

00:10:02.767 --> 00:10:05.274
All of that of f evaluated at where?

00:10:05.274 --> 00:10:06.395
Well, the place that we're starting

00:10:06.395 --> 00:10:09.145
is just v of t, so that's v of t.

00:10:12.703 --> 00:10:14.332
And that's it, that's the answer.

00:10:14.332 --> 00:10:16.266
'Cause when you evaluate
the directional derivative,

00:10:16.266 --> 00:10:19.943
the way that you do that,
you take the gradient of f,

00:10:19.943 --> 00:10:22.321
evaluate it at whatever
point you're starting at,

00:10:22.321 --> 00:10:25.099
in this case it's the output of v of t,

00:10:25.099 --> 00:10:27.223
and you take the dot product between that

00:10:27.223 --> 00:10:30.178
and the vector value derivative.

00:10:30.178 --> 00:10:31.447
Well, I mean (chuckles),

00:10:31.447 --> 00:10:33.508
the dot product between that
and whatever your vector is,

00:10:33.508 --> 00:10:35.258
which, in this case, is
the vector-value derivative

00:10:35.258 --> 00:10:38.455
of v, and that's the
multivariable chain rule.

00:10:38.455 --> 00:10:41.453
And if you look back through
the line of reasoning,

00:10:41.453 --> 00:10:43.200
it all really did match

00:10:43.200 --> 00:10:46.241
the thoughts of kind of nudging, nudging,

00:10:46.241 --> 00:10:47.851
and seeing how that nudged, right?

00:10:47.851 --> 00:10:49.808
Because the reason we thought to use

00:10:49.808 --> 00:10:50.891
the vector-value derivative

00:10:50.891 --> 00:10:52.688
was because of that intuition.

00:10:52.688 --> 00:10:54.809
And the reason for all the
manipulation that I did

00:10:54.809 --> 00:10:59.022
is just because I wanted
to be able to express

00:10:59.022 --> 00:11:01.579
what a nudge to the input of v looks like.

00:11:01.579 --> 00:11:03.112
And what that looks like
is the original value

00:11:03.112 --> 00:11:05.758
plus a certain vector here.

00:11:05.758 --> 00:11:08.431
This was the resulting nudge
in the intermediary space.

00:11:08.431 --> 00:11:10.098
I wanted to express that in a formal way.

00:11:10.098 --> 00:11:12.151
And sure, we have this kind of o of h term

00:11:12.151 --> 00:11:15.685
that expresses something
that shrinks really fast,

00:11:15.685 --> 00:11:17.531
but once you express it like that,

00:11:17.531 --> 00:11:20.656
you just end up plopping out

00:11:20.656 --> 00:11:23.928
the definition of the
directional derivative.

00:11:23.928 --> 00:11:26.621
So I hope that gives kind
of a satisfying reason

00:11:26.621 --> 00:11:29.811
for those of you who are a
little bit more rigor-inclined

00:11:29.811 --> 00:11:32.593
for why the multivariable
chain rule works.

00:11:32.593 --> 00:11:35.071
I should also maybe mention
there's a more general

00:11:35.071 --> 00:11:37.358
multivariable chain rule
for vector-valued functions.

00:11:37.358 --> 00:11:38.560
I'll get to that at another point

00:11:38.560 --> 00:11:40.056
when I talk about the connections

00:11:40.056 --> 00:11:43.139
between multivariable
calculus and linear algebra.

00:11:43.139 --> 00:11:47.008
But for now, that's pretty
much all you need to know

00:11:47.008 --> 00:11:48.256
on the multivariable chain rule

00:11:48.256 --> 00:11:50.710
when the ultimate composition is,

00:11:50.710 --> 00:11:53.192
you know, just a real
number to a real number.

00:11:53.192 --> 00:11:54.706
And I'll see you next video.


Vector form of the multivariable chain rule

WEBVTT
Kind: captions
Language: en

00:00:00.364 --> 00:00:01.688
- [Voiceover] So in the
last couple of videos,

00:00:01.688 --> 00:00:03.523
I talked about the
multi-variable chain rule,

00:00:03.523 --> 00:00:04.751
which I have up here, and if you

00:00:04.751 --> 00:00:06.628
haven't seen those go take a look.

00:00:06.628 --> 00:00:09.063
Here and I want to write
it out in vector notation,

00:00:09.063 --> 00:00:11.466
and this helps us generalize
it a little bit when

00:00:11.466 --> 00:00:15.260
the intermediary space is a
little bit higher dimensional.

00:00:15.260 --> 00:00:17.944
So, instead of writing X of T and

00:00:17.944 --> 00:00:21.211
Y of T as separate functions,
and just trying to emphasize

00:00:21.211 --> 00:00:23.015
"oh they have the same input space, and

00:00:23.015 --> 00:00:25.869
whatever X takes in that's
the same number Y takes in."

00:00:25.869 --> 00:00:28.444
It's better and a little bit cleaner

00:00:28.444 --> 00:00:32.037
if we say there's a vector valued function

00:00:32.037 --> 00:00:33.737
that takes in a single number "T,"

00:00:33.737 --> 00:00:36.808
then it outputs some kind of vector.

00:00:36.808 --> 00:00:38.228
In this case you could say the components

00:00:38.228 --> 00:00:41.895
of V are X of T and Y
of T, and that's fine.

00:00:46.160 --> 00:00:48.459
But I want to talk about
what this looks like

00:00:48.459 --> 00:00:52.372
if we start writing
everything in vector notation,

00:00:52.372 --> 00:00:56.566
and just since we see
DX/DT here and DY/DT here,

00:00:56.566 --> 00:00:59.072
you might start thinking,
"oh we should take

00:00:59.072 --> 00:01:03.664
the derivative of that
vector valued function."

00:01:03.664 --> 00:01:07.441
The derivative of V, with respect to T,

00:01:07.441 --> 00:01:09.120
and when we compute this
it's nothing more than

00:01:09.120 --> 00:01:11.603
taking the derivatives of each component.

00:01:11.603 --> 00:01:13.785
So in this case, the derivative of X,

00:01:13.785 --> 00:01:17.952
so you'd write DX/DT,
and the derivative of Y,

00:01:19.251 --> 00:01:23.418
DY/DT. This is the
vector value derivative.

00:01:25.182 --> 00:01:26.940
And now you might start
to notice something here.

00:01:26.940 --> 00:01:29.258
Okay so we've got one of
those components multiplied

00:01:29.258 --> 00:01:31.318
by a certain value and
another component multiplied

00:01:31.318 --> 00:01:33.355
by a certain value, you might

00:01:33.355 --> 00:01:36.022
recognize this as a dot product.

00:01:37.528 --> 00:01:39.129
This would be the dot product between the

00:01:39.129 --> 00:01:41.005
vector that contains the derivatives,

00:01:41.005 --> 00:01:45.172
the partial derivatives,
partial of F with respect to Y,

00:01:46.065 --> 00:01:49.355
partial of F with respect to X, oh,

00:01:49.355 --> 00:01:52.766
whoops, don't know why
I wrote it that way,

00:01:52.766 --> 00:01:56.933
but up here that's with respect
to X, and then here to Y.

00:01:59.750 --> 00:02:03.261
So this whole thing, we're
taking the dot product

00:02:03.261 --> 00:02:07.428
with the vector that contains
ordinary derivative DX/DT

00:02:08.294 --> 00:02:11.711
and ordinary derivative DY/DT.

00:02:15.072 --> 00:02:16.653
And of course both of
these are special vectors,

00:02:16.653 --> 00:02:19.667
they're not just random, the left one,

00:02:19.667 --> 00:02:24.496
that's the gradient of F,
and the right vector here

00:02:24.496 --> 00:02:25.755
that's what we just wrote that's

00:02:25.755 --> 00:02:28.217
the derivative of V with respect to T,

00:02:28.217 --> 00:02:32.423
just for being quick I'm gonna
write that as V prime of T.

00:02:32.423 --> 00:02:35.371
That's saying completely
the same thing as VDVT,

00:02:35.371 --> 00:02:37.712
and this right here is
another way to write

00:02:37.712 --> 00:02:39.700
the multi-variable chain rule,

00:02:39.700 --> 00:02:41.349
and maybe if you were being
a little bit more exact

00:02:41.349 --> 00:02:44.481
you would emphasize that when
you take the gradient of F

00:02:44.481 --> 00:02:47.253
the thing that you input
into it is the output

00:02:47.253 --> 00:02:48.771
of that vector valued function,

00:02:48.771 --> 00:02:51.379
you know you're throwing
in X of T and Y of T,

00:02:51.379 --> 00:02:53.895
so you might emphasize
that you take in that

00:02:53.895 --> 00:02:57.054
as an input, and then
you multiply it by the

00:02:57.054 --> 00:03:01.221
derivative, the vector
valued derivative of V of T.

00:03:03.266 --> 00:03:05.325
And when I say multiply,
I mean dot product, right,

00:03:05.325 --> 00:03:07.317
these are vectors and you're
taking the dot product,

00:03:07.317 --> 00:03:09.646
it should seem very familiar to, you know,

00:03:09.646 --> 00:03:11.790
the single-variable chain rule.

00:03:11.790 --> 00:03:13.823
And just to remind us
I'll throw it up here,

00:03:13.823 --> 00:03:18.264
if you take the derivative
of composition of

00:03:18.264 --> 00:03:21.347
two single-variable functions F of G,

00:03:23.872 --> 00:03:26.625
you take the derivative
of the outside F prime,

00:03:26.625 --> 00:03:30.792
and throw in G, throw in what
was the interior function,

00:03:31.923 --> 00:03:32.778
and you multiply it by the

00:03:32.778 --> 00:03:36.568
derivative of that interior
function, G prime of T.

00:03:36.568 --> 00:03:38.331
And this is super helpful
in single-variable calculus

00:03:38.331 --> 00:03:40.437
for computing a lot of derivatives,

00:03:40.437 --> 00:03:42.579
and over here it has a
very similar form right?

00:03:42.579 --> 00:03:45.029
The gradient which really
serves the function

00:03:45.029 --> 00:03:47.926
of the true extension
of the derivative for

00:03:47.926 --> 00:03:50.736
multi-variable functions for
scalar valued multi-variable

00:03:50.736 --> 00:03:53.543
functions at least. You
take that derivative

00:03:53.543 --> 00:03:56.428
and throw in the inner function,

00:03:56.428 --> 00:03:58.253
which just happens to be
a vector valued function.

00:03:58.253 --> 00:04:00.222
You throw it in there,
and then you multiply it

00:04:00.222 --> 00:04:02.526
by the derivative of that,
but multiplying vectors

00:04:02.526 --> 00:04:06.487
in this context means taking
the dot product of the two,

00:04:06.487 --> 00:04:07.550
and this could mean if you have a function

00:04:07.550 --> 00:04:09.191
with a whole bunch of different variables,

00:04:09.191 --> 00:04:13.024
let's say you have some
F of X, or not F of X,

00:04:14.487 --> 00:04:17.154
F of X1 and X2 and it takes in a

00:04:19.244 --> 00:04:23.411
whole bunch of variables
that it goes out to X100.

00:04:24.449 --> 00:04:28.222
And then what you throw into
it is the vector value function

00:04:28.222 --> 00:04:32.305
that's vector valued,
takes in a single variable,

00:04:34.086 --> 00:04:36.343
and in order to be able to compose them

00:04:36.343 --> 00:04:39.359
it's gonna have a whole bunch
of intermediary functions,

00:04:39.359 --> 00:04:43.312
and you can write it as X1, X2,
X3, all the way up to X100 ,

00:04:43.312 --> 00:04:45.289
and these are all functions at this point.

00:04:45.289 --> 00:04:49.036
These are component functions
of your vector valued V.

00:04:49.036 --> 00:04:51.175
This expression still makes sense, right?

00:04:51.175 --> 00:04:53.130
You can still take the gradient of F,

00:04:53.130 --> 00:04:55.099
it's gonna have 100
components, you can plug

00:04:55.099 --> 00:04:58.876
in any vector, any set
of 100 different numbers,

00:04:58.876 --> 00:05:00.819
and in particular the output of a

00:05:00.819 --> 00:05:03.348
vector valued function with
100 different components

00:05:03.348 --> 00:05:05.153
is gonna work, and then
you take the dot product

00:05:05.153 --> 00:05:07.801
with the derivative of
this. That's the more

00:05:07.801 --> 00:05:11.284
general version of the
multi-variable chain rule,

00:05:11.284 --> 00:05:13.005
and then the cool way
about writing it like this,

00:05:13.005 --> 00:05:16.902
you can interpret it in terms
of the directional derivative,

00:05:16.902 --> 00:05:19.012
and I think I'll do
that in the next video,

00:05:19.012 --> 00:05:22.019
so, that's a certain way to interpret

00:05:22.019 --> 00:05:24.309
this with a directional derivative.


Multivariable chain rule intuition

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.073
- [Voiceover] So in the last
video, I introduced this

00:00:02.073 --> 00:00:04.796
multi-variable chain
rule and here I want to

00:00:04.796 --> 00:00:07.863
explain a loose intuition
for why it's true,

00:00:07.863 --> 00:00:10.148
why you would expect
something like this to happen.

00:00:10.148 --> 00:00:12.339
So the way you think about
an expression like this,

00:00:12.339 --> 00:00:14.189
you have this multi-variable function

00:00:14.189 --> 00:00:16.275
f of xy and you're plugging things in,

00:00:16.275 --> 00:00:18.096
but just that function itself, you'll be

00:00:18.096 --> 00:00:21.845
thinking of taking a two dimensional space

00:00:21.845 --> 00:00:24.718
you know here's our xy plane,

00:00:24.718 --> 00:00:28.371
and then mapping it to, you
know, just a real number line

00:00:28.371 --> 00:00:30.400
and I'll think of this
as f, as the output.

00:00:30.400 --> 00:00:34.429
So somehow our whole
function takes things from

00:00:34.429 --> 00:00:37.755
this two dimensional space
and plugs it onto this output.

00:00:37.755 --> 00:00:39.296
T you're thinking of just another

00:00:39.296 --> 00:00:41.129
number line up here, so t, and then

00:00:41.129 --> 00:00:44.023
you've got separate functions here,

00:00:44.023 --> 00:00:46.255
you know x of t and y of t.

00:00:46.255 --> 00:00:47.755
X of t and y of t.

00:00:50.893 --> 00:00:53.828
Each of which take that same value for

00:00:53.828 --> 00:00:55.374
a specific input, you know it's not that

00:00:55.374 --> 00:00:56.556
they're acting on different inputs,

00:00:56.556 --> 00:00:59.076
x of some other input t
and y of some other input,

00:00:59.076 --> 00:01:01.860
it's the same one and then they move that

00:01:01.860 --> 00:01:03.948
somewhere to this output space

00:01:03.948 --> 00:01:06.448
which itself get's moved over.

00:01:07.710 --> 00:01:09.048
And in this way you're thinking of it

00:01:09.048 --> 00:01:10.714
as just a single variable function

00:01:10.714 --> 00:01:13.026
that goes from t and ultimately outputs f

00:01:13.026 --> 00:01:14.704
it's just that there's
multi-dimensional stuff

00:01:14.704 --> 00:01:16.932
happening in between and now if we start

00:01:16.932 --> 00:01:19.180
thinking about the derivative of it -

00:01:19.180 --> 00:01:21.833
what does that mean, what
does that mean for the

00:01:21.833 --> 00:01:25.172
conception of the picture
that we have going on here?

00:01:25.172 --> 00:01:27.459
Well, that bottom part, that dt

00:01:27.459 --> 00:01:30.394
you're thinking of as a
tiny change to t, right?

00:01:30.394 --> 00:01:32.196
So you're thinking of
it as kind of a nudge,

00:01:32.196 --> 00:01:34.090
I'll draw it as a sizable line here

00:01:34.090 --> 00:01:36.543
for like moving from
some original input over,

00:01:36.543 --> 00:01:38.179
but you might in
principal think of it as a

00:01:38.179 --> 00:01:40.625
very, very tiny nudge in t.

00:01:40.625 --> 00:01:42.831
And over here you'd say
well, that's gonna move

00:01:42.831 --> 00:01:45.959
your intermediary output in the xy plane

00:01:45.959 --> 00:01:48.873
to, you know maybe it'll
move it in some amount,

00:01:48.873 --> 00:01:51.138
again imagine this is a very small nudge,

00:01:51.138 --> 00:01:52.137
I'm going to give it some size here

00:01:52.137 --> 00:01:53.875
just so I can write into it and

00:01:53.875 --> 00:01:55.948
then whatever that nudge
in the output space

00:01:55.948 --> 00:01:57.889
right, it's a nudge in some direction

00:01:57.889 --> 00:02:01.501
that's going to correspond
to some change in f.

00:02:01.501 --> 00:02:05.501
Some change based on the
differential properties

00:02:07.295 --> 00:02:09.895
of the multi-variable function itself.

00:02:09.895 --> 00:02:13.583
And if we think about this, this change

00:02:13.583 --> 00:02:16.343
you might break it into components

00:02:16.343 --> 00:02:20.010
and say this shift here
has some kind of dx,

00:02:21.553 --> 00:02:23.279
some kind of shift in the x direction

00:02:23.279 --> 00:02:26.423
and some kind of dy, some
shift in the y direction.

00:02:26.423 --> 00:02:28.906
But you can actually reason
about what these should be

00:02:28.906 --> 00:02:30.349
coz it's not just an arbitrary change in x

00:02:30.349 --> 00:02:31.721
or an arbitrary change in y,

00:02:31.721 --> 00:02:34.975
it's the one that was caused by dt.

00:02:34.975 --> 00:02:38.721
So if I go over here, I might say that dx

00:02:38.721 --> 00:02:42.904
is caused by that dt and the whole meaning

00:02:42.904 --> 00:02:45.131
of the derivative, the whole meaning

00:02:45.131 --> 00:02:47.864
of the single variable derivative

00:02:47.864 --> 00:02:50.614
would be that when we take dx dt,

00:02:51.900 --> 00:02:54.006
this is the factor that
tells us, you know,

00:02:54.006 --> 00:02:56.250
a tiny nudge in t, how
much does that change

00:02:56.250 --> 00:02:58.450
the x component and if you want you could

00:02:58.450 --> 00:03:00.702
think of this as kind of
cancelling out the dts

00:03:00.702 --> 00:03:02.973
and you're just left with
x, but really you're saying

00:03:02.973 --> 00:03:05.313
there's a tiny nudge in
t and that results in a

00:03:05.313 --> 00:03:07.596
change in x and this derivative is what

00:03:07.596 --> 00:03:10.148
tells you the ratio between those sizes.

00:03:10.148 --> 00:03:13.231
And similarly, that change in y here,

00:03:16.538 --> 00:03:19.404
that change in y is gonna be somehow

00:03:19.404 --> 00:03:21.987
proportional to the change in t

00:03:22.884 --> 00:03:24.784
and that proportion is given by the

00:03:24.784 --> 00:03:26.599
derivative of y with respect to t

00:03:26.599 --> 00:03:29.307
that's the whole point of the derivative,

00:03:29.307 --> 00:03:31.930
no no, with respect to t and again

00:03:31.930 --> 00:03:33.489
you can kind of think of it as if

00:03:33.489 --> 00:03:34.977
you're cancelling out the ts and

00:03:34.977 --> 00:03:37.126
this is why the fractional writing,

00:03:37.126 --> 00:03:39.553
this Leibniz notation is
actually pretty helpful.

00:03:39.553 --> 00:03:42.313
You know, people will say,
oh mathematicians would

00:03:42.313 --> 00:03:43.928
like, share their heads at the idea of

00:03:43.928 --> 00:03:47.059
treating these like
fractions, but not only is it

00:03:47.059 --> 00:03:49.232
a useful thing to do coz it is a

00:03:49.232 --> 00:03:51.764
helpful mnemonic, it's
reflective of what you're

00:03:51.764 --> 00:03:54.064
gonna do when you make
a very formal argument.

00:03:54.064 --> 00:03:56.118
And I think I'll do that in
one of the following videos,

00:03:56.118 --> 00:03:59.520
I'll describe this in a
very, a much more formal way

00:03:59.520 --> 00:04:01.442
that's a little bit more airtight than the

00:04:01.442 --> 00:04:03.656
kind of hand-waving nudging around.

00:04:03.656 --> 00:04:05.472
But the intuition you
get from just writing

00:04:05.472 --> 00:04:08.241
this is a fraction is
basically the scaffolding

00:04:08.241 --> 00:04:09.700
for that formal argument, so it's a

00:04:09.700 --> 00:04:12.144
fine thing to do, I don't
think mathematicians

00:04:12.144 --> 00:04:14.254
are shaking their heads every time that a

00:04:14.254 --> 00:04:16.286
student or a teacher does this.

00:04:16.286 --> 00:04:18.286
But anyway, so this is kind of gives you

00:04:18.286 --> 00:04:20.798
what that dx is, what that dy is

00:04:20.798 --> 00:04:22.319
and then over here if you're saying

00:04:22.319 --> 00:04:25.840
how much does that change
the ultimate output of the f?

00:04:25.840 --> 00:04:29.262
You could say, well, your
nudge of size dx over here,

00:04:29.262 --> 00:04:31.926
you're wondering how much
that changes the output of f,

00:04:31.926 --> 00:04:34.347
that's the meaning of the
partial derivative, right.

00:04:34.347 --> 00:04:35.972
If we say we have the partial derivative

00:04:35.972 --> 00:04:38.889
with respect to x, what that means,

00:04:39.782 --> 00:04:42.298
is that if you take a tiny nudge of size x

00:04:42.298 --> 00:04:44.792
this is giving you the ratio between that

00:04:44.792 --> 00:04:48.317
and the ultimate change to
the output that you want.

00:04:48.317 --> 00:04:50.278
You could think of it like this partial x

00:04:50.278 --> 00:04:52.701
is cancelling out with
that dx if you wanted

00:04:52.701 --> 00:04:55.534
or you could just say,
this is a tiny nudge in x,

00:04:55.534 --> 00:04:57.318
this is going to result
in some change in f -

00:04:57.318 --> 00:04:59.206
I'm not sure what - but the meaning of

00:04:59.206 --> 00:05:02.965
the derivative is the
ratio between those two

00:05:02.965 --> 00:05:04.655
and that's what lets you figure it out.

00:05:04.655 --> 00:05:08.414
And similarly, you might
call this the change in f

00:05:08.414 --> 00:05:10.747
caused by x, like, due to x.

00:05:13.137 --> 00:05:16.046
Due to, I should say to dx.

00:05:16.046 --> 00:05:18.470
But that's not the only thing
changing the value of f right?

00:05:18.470 --> 00:05:19.961
That's not the only change happening

00:05:19.961 --> 00:05:23.747
in the input space, you also
have another change in f

00:05:23.747 --> 00:05:26.914
and this one I might say is due to dy.

00:05:30.966 --> 00:05:34.762
Due to that tiny shift in
y and what that's gonna be

00:05:34.762 --> 00:05:36.254
we know it's going to
be proportional to that

00:05:36.254 --> 00:05:40.422
tiny shift in y and the
proportionality constant -

00:05:40.422 --> 00:05:43.638
this is the meaning of
the partial derivative,

00:05:43.638 --> 00:05:46.433
that when you nudge y in some way it

00:05:46.433 --> 00:05:48.582
results in some kind of
nudge in f and the ratio

00:05:48.582 --> 00:05:51.438
between those two is what
the derivative gives.

00:05:51.438 --> 00:05:53.396
So ultimately, if you
put this all together

00:05:53.396 --> 00:05:54.984
what you'd say is there's
two different things

00:05:54.984 --> 00:05:57.255
causing an ultimate change to f.

00:05:57.255 --> 00:05:58.727
So if you put these together, and you

00:05:58.727 --> 00:06:01.363
want to know what the
total change in f is -

00:06:01.363 --> 00:06:02.927
so I might go over here and say

00:06:02.927 --> 00:06:06.923
the total change in f,
one of them is caused

00:06:06.923 --> 00:06:10.778
by partial f, partial x
- and I can multiply it

00:06:10.778 --> 00:06:13.119
by dx here, but really, we know that dx,

00:06:13.119 --> 00:06:16.327
the change there was in turn caused by dt

00:06:16.327 --> 00:06:19.532
so that in turn is caused by the change

00:06:19.532 --> 00:06:22.684
in the x component that was due to dt.

00:06:22.684 --> 00:06:25.184
That was of course of size dt.

00:06:26.051 --> 00:06:27.439
And then for similar
reasons, the other way

00:06:27.439 --> 00:06:30.439
that this changes in the y direction

00:06:32.472 --> 00:06:35.389
is a partial of f with respect to y

00:06:36.274 --> 00:06:38.997
but what caused that initial shift in y,

00:06:38.997 --> 00:06:43.164
you'd say that was a shift
in y that was due to t,

00:06:44.683 --> 00:06:48.158
and that size is dy dt times
dt, you could think of it.

00:06:48.158 --> 00:06:51.192
So slight nudge in t causes a change in y,

00:06:51.192 --> 00:06:53.229
that change in y causes the change in f

00:06:53.229 --> 00:06:55.035
and when you add those two together that's

00:06:55.035 --> 00:06:57.016
everything that's going
on, that's everything

00:06:57.016 --> 00:06:59.398
that influences the ultimate change in f.

00:06:59.398 --> 00:07:00.908
So then if you take this whole expression

00:07:00.908 --> 00:07:03.107
and you divide everything out by dt

00:07:03.107 --> 00:07:06.774
so you know, kind of
erase it from this side

00:07:08.216 --> 00:07:10.992
and put it over here, dt,

00:07:10.992 --> 00:07:12.920
this is your multi-variable chain rule,

00:07:12.920 --> 00:07:15.064
and of course I've just
written the same thing again

00:07:15.064 --> 00:07:17.144
but hopefully this gives
a little bit on intuition

00:07:17.144 --> 00:07:19.275
for how you're composing different nudges

00:07:19.275 --> 00:07:22.565
and why you wanna think about it that way.

00:07:22.565 --> 00:07:23.772
Of course, you can see this, and you see

00:07:23.772 --> 00:07:26.592
the partial f kind of
cancels out with that dx

00:07:26.592 --> 00:07:29.234
and this partial y kind of
cancels out with that dy

00:07:29.234 --> 00:07:31.346
and you're left with
the two different things

00:07:31.346 --> 00:07:32.970
that constitute a change in x,

00:07:32.970 --> 00:07:34.913
you know this one is only
partially the change in f,

00:07:34.913 --> 00:07:36.818
this is also partially the change in f,

00:07:36.818 --> 00:07:39.497
but together they give
the ultimate change in f

00:07:39.497 --> 00:07:41.033
and I think that gives
a very strong reason,

00:07:41.033 --> 00:07:45.200
if you break it down like
that, why this should be true.


Multivariable chain rule

WEBVTT
Kind: captions
Language: en

00:00:01.641 --> 00:00:02.614
- [Voiceover] So I've written here

00:00:02.614 --> 00:00:04.211
three different functions.

00:00:04.211 --> 00:00:06.397
The first on is a multivariable function,

00:00:06.397 --> 00:00:09.211
it has a two variable input, x, y,

00:00:09.211 --> 00:00:10.907
and a single variable output,

00:00:10.907 --> 00:00:12.226
that's x squared times y,

00:00:12.226 --> 00:00:13.460
that's just a number,

00:00:13.460 --> 00:00:16.248
and then the other two
functions are each just regular

00:00:16.248 --> 00:00:18.208
old single variable functions.

00:00:18.208 --> 00:00:20.358
And what I want to do is start thinking

00:00:20.358 --> 00:00:22.353
about the composition of them.

00:00:22.353 --> 00:00:24.445
So, I'm going to take,
as the first component,

00:00:24.445 --> 00:00:28.056
the value of the function x of t,

00:00:28.056 --> 00:00:29.797
so you pump t through that,

00:00:29.797 --> 00:00:33.046
and then you make that
the first component of f.

00:00:33.046 --> 00:00:34.260
And the second component

00:00:34.260 --> 00:00:38.120
will be the value of the function y of t.

00:00:38.120 --> 00:00:39.800
So, the image that you
might have in your head

00:00:39.800 --> 00:00:40.823
for something like this

00:00:40.823 --> 00:00:42.713
is you can think of t

00:00:42.713 --> 00:00:45.677
as just living on a
number line of some kind,

00:00:45.677 --> 00:00:47.233
then you have x and y,

00:00:47.233 --> 00:00:48.703
which is just the plane,

00:00:48.703 --> 00:00:50.550
so that will be, you know,

00:00:50.550 --> 00:00:52.210
your x-coordinate, your y-coordinate,

00:00:52.210 --> 00:00:53.653
two-dimensional space,

00:00:53.653 --> 00:00:55.018
and then you have your output,

00:00:55.018 --> 00:00:57.171
which is just whatever the value of f is.

00:00:57.171 --> 00:00:58.555
And for this whole function,

00:00:59.558 --> 00:01:01.095
for this whole composition of functions,

00:01:01.095 --> 00:01:03.510
you're thinking of xt, yt,

00:01:03.510 --> 00:01:05.091
as taking a single point in t,

00:01:05.091 --> 00:01:06.216
and kind of moving it over

00:01:06.216 --> 00:01:07.922
to two-dimensional space somewhere,

00:01:07.922 --> 00:01:10.797
and then from there, our
multivariable function

00:01:10.797 --> 00:01:11.942
takes that back down.

00:01:11.942 --> 00:01:14.110
So, this is just the
single variable function,

00:01:14.110 --> 00:01:17.897
nothing too fancy going on
in terms of where you start

00:01:17.897 --> 00:01:18.837
and where you end up,

00:01:18.837 --> 00:01:20.951
it's just what's happening in the middle.

00:01:20.951 --> 00:01:23.657
And what I want to know is
what's the derivative of this

00:01:23.657 --> 00:01:25.104
function.

00:01:25.104 --> 00:01:27.363
If I take this, and it's
just an ordinary derivative,

00:01:27.363 --> 00:01:28.327
not a partial derivative,

00:01:28.327 --> 00:01:30.376
because this is just a
single variable function,

00:01:30.376 --> 00:01:32.454
one variable input, one variable output,

00:01:32.454 --> 00:01:34.075
how do you take it's derivative?

00:01:34.075 --> 00:01:35.973
And there's a special rule for this,

00:01:35.973 --> 00:01:36.851
it's called the chain rule,

00:01:36.851 --> 00:01:38.553
the multivariable chain rule,

00:01:38.553 --> 00:01:40.561
but you don't actually need it.

00:01:40.561 --> 00:01:42.917
So, let's actually walk through this,

00:01:42.917 --> 00:01:44.375
showing that you don't need it.

00:01:44.375 --> 00:01:45.516
It's not that you'll never need it,

00:01:45.516 --> 00:01:47.060
it's just for computations like this

00:01:47.060 --> 00:01:48.509
you could go without it.

00:01:48.509 --> 00:01:49.941
It's a very useful theoretical tool,

00:01:49.941 --> 00:01:52.115
a very useful model to have in mind

00:01:52.115 --> 00:01:54.989
for what function composition looks like

00:01:54.989 --> 00:01:58.951
and implies for derivatives
in the multivariable world.

00:01:58.951 --> 00:02:01.096
So, let's just start
plugging things in here.

00:02:01.096 --> 00:02:03.649
If I have f(x) and y(t),

00:02:03.649 --> 00:02:05.978
the first thing I might do is write

00:02:05.978 --> 00:02:06.658
okay, f,

00:02:06.658 --> 00:02:10.453
and instead of x of t,
just write in cosine of t,

00:02:10.453 --> 00:02:14.234
since that's the function
that I have for x of t,

00:02:14.234 --> 00:02:16.281
and then y we replace that with sine of t,

00:02:17.721 --> 00:02:18.564
sine of t,

00:02:19.651 --> 00:02:22.302
and of course I'm hoping to
take the derivative of this.

00:02:24.477 --> 00:02:26.693
And then from there, we can
go to the definition of f,

00:02:26.693 --> 00:02:29.053
f of xy equals f squared times y,

00:02:29.053 --> 00:02:31.539
which means we take that
first component squared.

00:02:31.539 --> 00:02:35.421
So we'll take that first
component, cosine of t,

00:02:36.442 --> 00:02:38.010
and then square it,

00:02:40.311 --> 00:02:41.633
square that guy,

00:02:41.633 --> 00:02:44.564
and then we'll multiply it
by the second component,

00:02:44.564 --> 00:02:47.919
sine of t, sine of t,

00:02:47.919 --> 00:02:49.763
and again we're just
taking this derivative.

00:02:51.977 --> 00:02:52.772
And you might be wondering,

00:02:52.772 --> 00:02:53.563
okay, why am I doing this,

00:02:53.563 --> 00:02:56.371
you're just showing me how
to take a first derivative,

00:02:56.371 --> 00:02:58.031
an ordinary derivative?

00:02:58.031 --> 00:03:00.227
But the pattern that
we'll see is gonna lead us

00:03:00.227 --> 00:03:01.770
to the multivariable chain rule.

00:03:01.770 --> 00:03:03.989
And it's actually kind of
surprising when you see it

00:03:03.989 --> 00:03:04.646
in this context,

00:03:04.646 --> 00:03:07.325
because it pops out in a way
that you might not expect

00:03:07.325 --> 00:03:09.215
things to pop out.

00:03:09.215 --> 00:03:11.192
So, continuing our chugging along,

00:03:11.192 --> 00:03:12.678
when you take the derivative of this,

00:03:12.678 --> 00:03:14.210
you do the product rule,

00:03:14.210 --> 00:03:16.723
left d right, plus right d left,

00:03:16.723 --> 00:03:19.686
so in this case, the left
is cosine squared of t,

00:03:19.686 --> 00:03:21.832
we just leave that as it is,

00:03:21.832 --> 00:03:23.243
cosine squared of t,

00:03:24.427 --> 00:03:27.336
and multiply it by the
derivative of the right, d right,

00:03:27.336 --> 00:03:29.323
so that's going to be cosine of t,

00:03:30.703 --> 00:03:32.768
cosine of t,

00:03:32.768 --> 00:03:34.396
and then we add to that right,

00:03:34.396 --> 00:03:38.689
which is, keep that right side unchanged,

00:03:38.689 --> 00:03:41.127
multiply it by the derivative of the left,

00:03:41.127 --> 00:03:42.383
and for that we use the chain rule,

00:03:42.383 --> 00:03:44.088
the single variable chain rule,

00:03:44.088 --> 00:03:46.335
where you think of taking the
derivative of the outside,

00:03:46.335 --> 00:03:48.877
so you plug two down,

00:03:48.877 --> 00:03:50.568
like you're taking the
derivative of two x,

00:03:50.568 --> 00:03:53.736
but you're just writing
in cosine, instead of x.

00:03:53.736 --> 00:03:54.845
Cosine t,

00:03:55.867 --> 00:04:00.174
and then you multiply that by
the derivative of the inside,

00:04:00.174 --> 00:04:01.388
that's a tongue twister,

00:04:02.318 --> 00:04:04.575
which is negative sine of t,

00:04:04.575 --> 00:04:05.874
negative sine of t.

00:04:05.874 --> 00:04:08.002
And I'm afraid I'm gonna
run off the edge here,

00:04:08.002 --> 00:04:11.460
certainly with the many many
parentheses that I need.

00:04:11.460 --> 00:04:13.121
I'll go ahead and rewrite this though.

00:04:13.121 --> 00:04:15.040
I'm gonna rewrite it anyway because

00:04:15.040 --> 00:04:19.545
there's a certain pattern
that I hope to make clear.

00:04:19.545 --> 00:04:21.316
So, let me just rewrite this side,

00:04:26.706 --> 00:04:28.997
let's copy that down here,

00:04:28.997 --> 00:04:30.882
I just want to rewrite this guy.

00:04:31.902 --> 00:04:32.944
You might be wondering why,

00:04:32.944 --> 00:04:34.287
but it'll become clear in just a moment

00:04:34.287 --> 00:04:36.205
why I want to do this.

00:04:36.205 --> 00:04:38.576
So, in this case, I'm gonna write this as

00:04:39.656 --> 00:04:42.052
two times cosine of t,

00:04:44.212 --> 00:04:45.342
times sine of t,

00:04:47.602 --> 00:04:50.529
then all of them multiplied
by negative sine of t,

00:04:50.529 --> 00:04:51.931
negative sine of t.

00:04:55.555 --> 00:04:56.687
So this is the derivative,

00:04:56.687 --> 00:04:59.785
this is the derivative of
the composition of functions

00:04:59.785 --> 00:05:01.793
that ultimately was a
single variable function,

00:05:01.793 --> 00:05:04.589
but it kind of wind through
two different variables.

00:05:04.589 --> 00:05:06.232
And I just want to make an observation

00:05:06.232 --> 00:05:08.789
in terms of the partial derivatives of f.

00:05:08.789 --> 00:05:11.936
So, let me just make a copy of this guy,

00:05:11.936 --> 00:05:14.120
give ourselves a little
bit of room down here,

00:05:16.061 --> 00:05:17.227
paste that over here.

00:05:18.411 --> 00:05:20.357
So let's look at the
partial derivatives of f

00:05:20.357 --> 00:05:22.139
for a second here.

00:05:22.139 --> 00:05:25.210
So, if I took the partial
derivative with respect to x,

00:05:25.210 --> 00:05:27.749
partial x,

00:05:27.749 --> 00:05:30.077
which means y is treated as a constant.

00:05:30.077 --> 00:05:32.925
So I take the derivative
of x squared to get two x,

00:05:32.925 --> 00:05:34.691
and then multiply it by that constant,

00:05:34.691 --> 00:05:36.496
which is just y,

00:05:36.496 --> 00:05:38.595
and if I also do it with respect to y,

00:05:38.595 --> 00:05:40.267
get all of them in there.

00:05:41.532 --> 00:05:42.857
So, now y looks like a variable,

00:05:42.857 --> 00:05:44.409
x looks like a constant,

00:05:44.409 --> 00:05:46.494
so x squared also looks like a constant,

00:05:46.494 --> 00:05:48.124
constant times a variable,

00:05:48.124 --> 00:05:50.483
the derivative is just that constant.

00:05:51.830 --> 00:05:55.486
These two, their pattern comes
up in the ultimate result

00:05:55.486 --> 00:05:56.344
that we got.

00:05:56.344 --> 00:05:58.074
And this is the whole
reason that I rewrote it.

00:05:58.074 --> 00:05:59.874
If you look at this two x y,

00:05:59.874 --> 00:06:01.528
you can see that over here,

00:06:01.528 --> 00:06:03.413
where cosine corresponds to x,

00:06:03.413 --> 00:06:06.443
sine corresponds to y, based
on our original functions,

00:06:06.443 --> 00:06:08.928
and an x squared here

00:06:08.928 --> 00:06:12.592
corresponds with squaring
the x that we put in there.

00:06:12.592 --> 00:06:15.569
Then if we take the derivative
of our two intermediary

00:06:15.569 --> 00:06:16.462
functions,

00:06:16.462 --> 00:06:19.640
the ordinary derivative
of x, with respect to t,

00:06:19.640 --> 00:06:21.659
that's derivative of cosine,

00:06:21.659 --> 00:06:24.943
negative sine of t,

00:06:25.827 --> 00:06:27.854
and then similarly derivative of y,

00:06:27.854 --> 00:06:31.021
just the ordinary derivative,
no partials going on here,

00:06:31.021 --> 00:06:33.007
with respect to t,

00:06:33.007 --> 00:06:34.256
that's equal to cosine,

00:06:34.256 --> 00:06:35.860
derivative of sine is cosine.

00:06:36.957 --> 00:06:38.195
And these guys show up, right,

00:06:38.195 --> 00:06:40.866
you see negative sine over here,

00:06:40.866 --> 00:06:43.357
and you see cosine show up over here.

00:06:44.395 --> 00:06:45.403
And we can generalize this,

00:06:45.403 --> 00:06:46.617
we can write it down and say

00:06:46.617 --> 00:06:48.852
at least for this specific example,

00:06:48.852 --> 00:06:51.419
it looks like the derivative
of the composition is

00:06:51.419 --> 00:06:53.898
this part, which is the partial of f

00:06:53.898 --> 00:06:55.861
with respect to y,

00:06:55.861 --> 00:06:57.305
right, that's kind of
what it looks like here,

00:06:57.305 --> 00:06:59.651
once we've plugged in the
intermediary functions,

00:07:00.689 --> 00:07:03.957
multiply it by this guy, was
the ordinary derivative of y,

00:07:03.957 --> 00:07:05.483
with respect to t.

00:07:05.483 --> 00:07:08.293
So, that was the ordinary derivative of y,

00:07:08.293 --> 00:07:09.779
with respect to t.

00:07:09.779 --> 00:07:11.015
And then very similarly,

00:07:11.015 --> 00:07:13.493
this guy was the partial of f,

00:07:13.493 --> 00:07:16.153
with respect to x, partial x,

00:07:16.153 --> 00:07:19.108
and we're multiplying it
by the ordinary derivative

00:07:19.108 --> 00:07:20.141
of x of t.

00:07:20.141 --> 00:07:23.682
So, over here, x of t, with respect to t.

00:07:24.994 --> 00:07:27.753
And of course, when I write
this partial f, partial y,

00:07:27.753 --> 00:07:30.664
what I really mean is
you plug in for x and y,

00:07:30.664 --> 00:07:32.366
the two coordinate functions,

00:07:32.366 --> 00:07:34.406
x of t, y of t.

00:07:34.406 --> 00:07:36.255
So, if I say partial
f, partial y over here,

00:07:36.255 --> 00:07:37.241
what I really mean is

00:07:37.241 --> 00:07:39.234
you take that x squared
and then you plug in

00:07:39.234 --> 00:07:42.908
x of t squared to get cosine squared.

00:07:42.908 --> 00:07:43.977
And same deal over here,

00:07:43.977 --> 00:07:45.273
you're always plugging things in,

00:07:45.273 --> 00:07:47.554
so you ultimately have a function of t.

00:07:47.554 --> 00:07:48.985
But this right here has a name,

00:07:48.985 --> 00:07:51.047
this is the multivariable chain rule.

00:07:51.047 --> 00:07:52.175
And it's important enough,

00:07:52.175 --> 00:07:56.052
I'll just write it out
all on it's own here.

00:07:56.052 --> 00:07:59.413
If we take the ordinary
derivative, with respect to t,

00:07:59.413 --> 00:08:01.945
of a composition of a
multivariable function,

00:08:01.945 --> 00:08:03.519
in this case just two variables,

00:08:03.519 --> 00:08:05.586
x of t, y of t,

00:08:05.586 --> 00:08:08.207
where we're plugging in
two intermediary functions,

00:08:08.207 --> 00:08:09.721
x of t, y of t,

00:08:09.721 --> 00:08:11.957
each of which just single variable,

00:08:11.957 --> 00:08:15.191
the result is that we take
the partial derivative,

00:08:17.821 --> 00:08:19.331
with respect to x,

00:08:21.055 --> 00:08:22.785
and we multiply it by

00:08:26.669 --> 00:08:29.189
the derivative of x with respect to t,

00:08:30.750 --> 00:08:32.986
and then we add to that

00:08:32.986 --> 00:08:35.762
the partial derivative with respect to y,

00:08:37.768 --> 00:08:41.336
multiplied by the derivative
of y with respect to t.

00:08:43.129 --> 00:08:45.737
So, this entire expression here

00:08:45.737 --> 00:08:47.596
is what you might call the simple version

00:08:47.596 --> 00:08:49.338
of the multivariable chain rule.

00:08:50.398 --> 00:08:52.410
There's a more general version,

00:08:52.410 --> 00:08:54.059
and we'll kind of build up to it,

00:08:54.059 --> 00:08:56.062
but this is the simplest
example you can think of,

00:08:56.062 --> 00:08:58.761
where you start with one dimension,

00:08:58.761 --> 00:09:00.993
and then you move over
to two dimension somehow,

00:09:00.993 --> 00:09:03.419
and then you move from those
two dimensions down to one.

00:09:05.401 --> 00:09:06.445
So, this is that,

00:09:06.445 --> 00:09:08.542
and in the next video I'm gonna talk about

00:09:08.542 --> 00:09:10.274
the intuition for why this is true.

00:09:10.274 --> 00:09:12.196
You know, here I just went
through an example and showed

00:09:12.196 --> 00:09:13.646
oh but it just happens to be true,

00:09:13.646 --> 00:09:14.963
it fills this pattern.

00:09:14.963 --> 00:09:16.991
But there's a very nice line of reasoning

00:09:16.991 --> 00:09:18.935
for where this comes about,

00:09:18.935 --> 00:09:20.999
and I'll also talk about
a more generalized form,

00:09:20.999 --> 00:09:22.456
where you'll see it.

00:09:22.456 --> 00:09:23.926
We start using vector notation,

00:09:23.926 --> 00:09:26.402
it makes things look very clean,

00:09:26.402 --> 00:09:28.936
and I might even get around
to a more formal argument

00:09:28.936 --> 00:09:30.254
for why this is true.

00:09:30.254 --> 00:09:31.676
So, we'll see in next video.