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so -- OK, so remember last
time,

00:00:25.000 --> 00:00:32.000
on Tuesday we learned about the
chain rule,

00:00:32.000 --> 00:00:39.000
and so for example we saw that
if we have a function that

00:00:39.000 --> 00:00:44.000
depends,
sorry, on three variables,

00:00:44.000 --> 00:00:50.000
x,y,z,
that x,y,z themselves depend on

00:00:50.000 --> 00:00:54.000
some variable,
t,

00:00:54.000 --> 00:01:06.000
then you can find a formula for
df/dt by writing down wx/dx dt

00:01:06.000 --> 00:01:12.000
wy dy/dt wz dz/dt.
And, the meaning of that

00:01:12.000 --> 00:01:17.000
formula is that while the change
in w is caused by changes in x,

00:01:17.000 --> 00:01:21.000
y, and z, x,
y, and z change at rates dx/dt,

00:01:21.000 --> 00:01:25.000
dy/dt, dz/dt.
And, this causes a function to

00:01:25.000 --> 00:01:31.000
change accordingly using,
well, the partial derivatives

00:01:31.000 --> 00:01:37.000
tell you how sensitive w is to
changes in each variable.

00:01:37.000 --> 00:01:45.000
OK, so, we are going to just
rewrite this in a new notation.

00:01:45.000 --> 00:01:52.000
So, I'm going to rewrite this
in a more concise form as

00:01:52.000 --> 00:01:59.000
gradient of w dot product with
velocity vector dr/dt.

00:01:59.000 --> 00:02:04.000
So, the gradient of w is a
vector formed by putting

00:02:04.000 --> 00:02:08.000
together all of the partial
derivatives.

00:02:08.000 --> 00:02:12.000
OK, so it's the vector whose
components are the partials.

00:02:12.000 --> 00:02:15.000
And, of course,
it's a vector that depends on

00:02:15.000 --> 00:02:19.000
x, y, and z, right?
These guys depend on x, y, z.

00:02:19.000 --> 00:02:22.000
So, it's actually one vector
for each point,

00:02:22.000 --> 00:02:31.000
x, y, z.
You can talk about the gradient

00:02:31.000 --> 00:02:39.000
of w at some point,
x, y, z.

00:02:39.000 --> 00:02:41.000
So, at each point,
it gives you a vector.

00:02:41.000 --> 00:02:47.000
That actually is what we will
call later a vector field.

00:02:47.000 --> 00:02:59.000
We'll get back to that later.
And, dr/dt is just the velocity

00:02:59.000 --> 00:03:07.000
vector dx/dt,
dy/dt, dz/dt.

00:03:07.000 --> 00:03:14.000
OK, so the new definition for
today is the definition of the

00:03:14.000 --> 00:03:18.000
gradient vector.
And, our goal will be to

00:03:18.000 --> 00:03:21.000
understand a bit better,
what does this vector mean?

00:03:21.000 --> 00:03:24.000
What does it measure?
And, what can we do with it?

00:03:24.000 --> 00:03:29.000
But, you see that in terms of
information content,

00:03:29.000 --> 00:03:33.000
it's really the same
information that's already in

00:03:33.000 --> 00:03:38.000
the partial derivatives,
or in the differential.

00:03:38.000 --> 00:03:43.000
So, yes, and I should say,
of course you can also use the

00:03:43.000 --> 00:03:49.000
gradient and other things like
approximation formulas and so

00:03:49.000 --> 00:03:52.000
on.
And so far, it's just notation.

00:03:52.000 --> 00:03:57.000
It's a way to rewrite things.
But, so here's the first cool

00:03:57.000 --> 00:04:03.000
property of the gradient.
So, I claim that the gradient

00:04:03.000 --> 00:04:11.000
vector is perpendicular to the
level surface corresponding to

00:04:11.000 --> 00:04:18.000
setting the function,
w, equal to a constant.

00:04:18.000 --> 00:04:22.000
OK, so if I draw a contour plot
of my function,

00:04:22.000 --> 00:04:28.000
so, actually forget about z
because I want to draw a two

00:04:28.000 --> 00:04:32.000
variable contour plot.
So, say I have a function of

00:04:32.000 --> 00:04:35.000
two variables,
x and y, then maybe it has some

00:04:35.000 --> 00:04:38.000
contour plot.
And, I'm saying if I take the

00:04:38.000 --> 00:04:42.000
gradient of a function at this
point, (x,y).

00:04:42.000 --> 00:04:46.000
So, I will have a vector.
Well, if I draw that vector on

00:04:46.000 --> 00:04:51.000
top of a contour plot,
it's going to end up being

00:04:51.000 --> 00:04:54.000
perpendicular to the level
curve.

00:04:54.000 --> 00:04:57.000
Same thing if I have a function
of three variables.

00:04:57.000 --> 00:04:59.000
Then, I can try to draw its
contour plot.

00:04:59.000 --> 00:05:03.000
Of course, I can't really do it
because the contour plot would

00:05:03.000 --> 00:05:05.000
be living in space with x,
y, and z.

00:05:05.000 --> 00:05:09.000
But, it would be a bunch of
level faces, and the gradient

00:05:09.000 --> 00:05:11.000
vector would be a vector in
space.

00:05:11.000 --> 00:05:15.000
That vector is perpendicular to
the level faces.

00:05:15.000 --> 00:05:24.000
So, let's try to see that on a
couple of examples.

00:05:24.000 --> 00:05:32.000
So, let's do a first example.
What's the easiest case?

00:05:32.000 --> 00:05:36.000
Let's take a linear function of
x, y, and z.

00:05:36.000 --> 00:05:42.000
So, I will take w equals a1
times x plus a2 times y plus a3

00:05:42.000 --> 00:05:47.000
times z.
Well, so, what's the gradient

00:05:47.000 --> 00:05:53.000
of this function?
Well, the first component will

00:05:53.000 --> 00:05:58.000
be a1.
That's partial w partial x.

00:05:58.000 --> 00:06:03.000
Then, a2, that's partial w
partial y, and a3,

00:06:03.000 --> 00:06:15.000
partial w partial z.
Now, what is the levels of this?

00:06:15.000 --> 00:06:22.000
Well, if I set w equal to some
constant, c, that means I look

00:06:22.000 --> 00:06:27.000
at the points where a1x a2y a3z
equals c.

00:06:27.000 --> 00:06:30.000
What kind of service is that?
It's a plane.

00:06:30.000 --> 00:06:39.000
And, we know how to find a
normal vector to this plane just

00:06:39.000 --> 00:06:48.000
by looking at the coefficients.
So, it's a plane with a normal

00:06:48.000 --> 00:06:51.000
vector exactly this gradient.
And, in fact,

00:06:51.000 --> 00:06:55.000
in a way, this is the only case
you need to check because of

00:06:55.000 --> 00:06:58.000
linear approximations.
If you replace a function by

00:06:58.000 --> 00:07:02.000
its linear approximation,
that means you will replace the

00:07:02.000 --> 00:07:04.000
level surfaces by their tension
planes.

00:07:04.000 --> 00:07:08.000
And then, you'll actually end
up in this situation.

00:07:08.000 --> 00:07:09.000
But maybe that's not very
convincing.

00:07:09.000 --> 00:07:25.000
So, let's do another example.
So, let's do a second example.

00:07:25.000 --> 00:07:28.000
Let's say we look at the
function x^2 y^2.

00:07:28.000 --> 00:07:32.000
OK, so now it's a function of
just two variables because that

00:07:32.000 --> 00:07:36.000
way we'll be able to actually
draw a picture for you.

00:07:36.000 --> 00:07:40.000
OK, so what are the level sets
of this function?

00:07:40.000 --> 00:07:44.000
Well, they're going to be
circles, right?

00:07:44.000 --> 00:07:54.000
w equals c is a circle,
x^2 y^2 = c.

00:07:54.000 --> 00:07:58.000
So, I should say,
maybe, sorry,

00:07:58.000 --> 00:08:08.000
the level curve is a circle.
So, the contour plot looks

00:08:08.000 --> 00:08:16.000
something like that.
Now, what's the gradient vector?

00:08:16.000 --> 00:08:20.000
Well, the gradient of this
function, so,

00:08:20.000 --> 00:08:26.000
partial w partial x is 2x.
And partial w partial y is 2y.

00:08:26.000 --> 00:08:31.000
So, let's say I take a point,
x comma y, and I try to draw my

00:08:31.000 --> 00:08:34.000
gradient vector.
So, here at x,

00:08:34.000 --> 00:08:38.000
y, so, I have to draw the
vector, <2x,

00:08:38.000 --> 00:08:41.000
2y>.
What does it look like?

00:08:41.000 --> 00:08:42.000
Well, it's going in that
direction.

00:08:42.000 --> 00:08:49.000
It's parallel to the position
vector for this point.

00:08:49.000 --> 00:08:51.000
It's actually twice the
position vector.

00:08:51.000 --> 00:08:55.000
So, I guess it goes more or
less like this.

00:08:55.000 --> 00:09:01.000
What's interesting,
too, is it is perpendicular to

00:09:01.000 --> 00:09:04.000
this circle.
OK, so it's a general feature.

00:09:04.000 --> 00:09:10.000
Actually, let me show you more
examples, oops,

00:09:10.000 --> 00:09:16.000
not the one I want.
So, I don't know if you can see

00:09:16.000 --> 00:09:19.000
it so well.
Well, hopefully you can.

00:09:19.000 --> 00:09:22.000
So, here I have a contour plot
of a function,

00:09:22.000 --> 00:09:25.000
and I have a blue vector.
That's the gradient vector at

00:09:25.000 --> 00:09:28.000
the pink point on the plot.
So, you can see,

00:09:28.000 --> 00:09:32.000
I can move the pink point,
and the gradient vector,

00:09:32.000 --> 00:09:37.000
of course, changes because the
gradient depends on x and y.

00:09:37.000 --> 00:09:42.000
But, what doesn't change is
that it's always perpendicular

00:09:42.000 --> 00:09:46.000
to the level curves.
Anywhere I am,

00:09:46.000 --> 00:09:53.000
my gradient stays perpendicular
to the level curve.

00:09:53.000 --> 00:09:57.000
OK, is that convincing?
Is that visible for people who

00:09:57.000 --> 00:10:05.000
can't see blue?
OK, so, OK, so we have a lot of

00:10:05.000 --> 00:10:16.000
evidence, but let's try to prove
the theorem because it will be

00:10:16.000 --> 00:10:22.000
interesting.
So, first of all,

00:10:22.000 --> 00:10:30.000
sorry, any questions about the
statement, the example,

00:10:30.000 --> 00:10:34.000
anything, yes?
Ah, very good question.

00:10:34.000 --> 00:10:37.000
Does the gradient vector,
why is the gradient vector

00:10:37.000 --> 00:10:40.000
perpendicular in one direction
rather than the other?

00:10:40.000 --> 00:10:43.000
So, we'll see the answer to
that in a few minutes.

00:10:43.000 --> 00:10:46.000
But let me just tell you
immediately, to the side,

00:10:46.000 --> 00:10:50.000
which side it's pointing to,
it's always pointing towards

00:10:50.000 --> 00:10:54.000
higher values of a function.
OK, and we'll see in that maybe

00:10:54.000 --> 00:11:03.000
about half an hour.
So, well, let me say actually

00:11:03.000 --> 00:11:13.000
points towards higher values of
w.

00:11:13.000 --> 00:11:24.000
OK, any other questions?
I don't see any questions.

00:11:24.000 --> 00:11:28.000
OK, so let's try to prove this
theorem, at least this part of

00:11:28.000 --> 00:11:30.000
the theorem.
We're not going to prove that

00:11:30.000 --> 00:11:38.000
just yet.
That will come in a while.

00:11:38.000 --> 00:11:44.000
So, well, maybe we want to
understand first what happens if

00:11:44.000 --> 00:11:48.000
we move inside the level curve,
OK?

00:11:48.000 --> 00:11:52.000
So, let's imagine that we are
taking a moving point that stays

00:11:52.000 --> 00:11:55.000
on the level curve or on the
level surface.

00:11:55.000 --> 00:12:00.000
And then, we know,
well, what happens is that the

00:12:00.000 --> 00:12:03.000
function stays constant.
But, we can also know how

00:12:03.000 --> 00:12:07.000
quickly the function changes
using the chain rule up there.

00:12:07.000 --> 00:12:11.000
So, maybe the chain rule will
actually be the key to

00:12:11.000 --> 00:12:15.000
understanding how the gradient
vector and the motion on the

00:12:15.000 --> 00:12:23.000
level service relate.
So, let's take a curve,

00:12:23.000 --> 00:12:31.000
r equals r of t,
that stays inside,

00:12:31.000 --> 00:12:42.000
well, maybe I should say on the
level surface,

00:12:42.000 --> 00:12:48.000
w equals c.
So, let's think about what that

00:12:48.000 --> 00:12:51.000
means.
So, just to get you used to

00:12:51.000 --> 00:12:55.000
this idea, I'm going to draw a
level surface of a function of

00:12:55.000 --> 00:12:59.000
three variables.
OK, so it's a surface given by

00:12:59.000 --> 00:13:03.000
the equation w of x,
y, z equals some constant,

00:13:03.000 --> 00:13:07.000
c.
And, so now I'm going to have a

00:13:07.000 --> 00:13:11.000
point on that,
and it's going to move on that

00:13:11.000 --> 00:13:15.000
surface.
So, I will have some parametric

00:13:15.000 --> 00:13:19.000
curve that lives on this
surface.

00:13:19.000 --> 00:13:25.000
So, the question is,
what's going to happen at any

00:13:25.000 --> 00:13:29.000
given time?
Well, the first observation is

00:13:29.000 --> 00:13:32.000
that the velocity vector,
what can I say about the

00:13:32.000 --> 00:13:37.000
velocity vector of this motion?
It's going to be tangent to the

00:13:37.000 --> 00:13:39.000
level surface,
right?

00:13:39.000 --> 00:13:42.000
If I move on a surface,
then at any point,

00:13:42.000 --> 00:13:45.000
my velocity is tangent to the
curve.

00:13:45.000 --> 00:13:49.000
But, if it's tangent to the
curve, then it's also tangent to

00:13:49.000 --> 00:13:53.000
the surface because the curve is
inside the surface.

00:13:53.000 --> 00:13:56.000
So, OK, it's getting a bit
cluttered.

00:13:56.000 --> 00:13:58.000
Maybe I should draw a bigger
picture.

00:13:58.000 --> 00:14:06.000
Let me do that right away here.
So, I have my level surface,

00:14:06.000 --> 00:14:11.000
w equals c.
I have a curve on that,

00:14:11.000 --> 00:14:19.000
and at some point,
I'm going to have a certain

00:14:19.000 --> 00:14:28.000
velocity.
So, the claim is that the

00:14:28.000 --> 00:14:40.000
velocity, v,
equals dr/dt is tangent -- --

00:14:40.000 --> 00:14:48.000
to the level,
w equals c because it's tangent

00:14:48.000 --> 00:14:50.000
to the curve,
and the curve is inside the

00:14:50.000 --> 00:14:52.000
level,
OK?

00:14:52.000 --> 00:14:55.000
Now, what else can we say?
Well, we have,

00:14:55.000 --> 00:15:03.000
the chain rule will tell us how
the value of w changes.

00:15:03.000 --> 00:15:12.000
So, by the chain rule,
we have dw/dt.

00:15:12.000 --> 00:15:20.000
So, the rate of change of the
value of w as I move along this

00:15:20.000 --> 00:15:28.000
curve is given by the dot
product between the gradient and

00:15:28.000 --> 00:15:34.000
the velocity vector.
And, so, well,

00:15:34.000 --> 00:15:43.000
maybe I can rewrite it as w dot
v, and that should be,

00:15:43.000 --> 00:15:50.000
well, what should it be?
What happens to the value of w

00:15:50.000 --> 00:15:54.000
as t changes?
Well, it stays constant because

00:15:54.000 --> 00:15:58.000
we are moving on a curve.
That curve might be

00:15:58.000 --> 00:16:02.000
complicated, but it stays always
on the level,

00:16:02.000 --> 00:16:08.000
w equals c.
So, it's zero because w of t

00:16:08.000 --> 00:16:18.000
equals c, which is a constant.
OK, is that convincing?

00:16:18.000 --> 00:16:21.000
OK, so now if we have a dot
product that's zero,

00:16:21.000 --> 00:16:25.000
that tells us that these two
guys are perpendicular.

00:16:25.000 --> 00:16:37.000
So -- So if the gradient vector
is perpendicular to v,

00:16:37.000 --> 00:16:44.000
OK, that's a good start.
We know that the gradient is

00:16:44.000 --> 00:16:48.000
perpendicular to this vector
tangent that's tangent to the

00:16:48.000 --> 00:16:51.000
level surface.
What about other vectors

00:16:51.000 --> 00:16:55.000
tangent to the level surface?
Well, in fact,

00:16:55.000 --> 00:17:00.000
I could use any curve drawn on
the level of w equals c.

00:17:00.000 --> 00:17:03.000
So, I could move,
really, any way I wanted on

00:17:03.000 --> 00:17:06.000
that surface.
In particular,

00:17:06.000 --> 00:17:11.000
I claim that I could have
chosen my velocity vector to be

00:17:11.000 --> 00:17:15.000
any vector tangent to the
surface.

00:17:15.000 --> 00:17:22.000
OK, so let's write this.
So this is true for any curve,

00:17:22.000 --> 00:17:30.000
or, I'll say for any motion on
the level surface,

00:17:30.000 --> 00:17:40.000
w equals c.
So that means v can be any

00:17:40.000 --> 00:17:53.000
vector tangent to the surface
tangent to the level.

00:17:53.000 --> 00:18:01.000
See, for example,
OK, let me draw one more

00:18:01.000 --> 00:18:06.000
picture.
OK, so I have my level surface.

00:18:06.000 --> 00:18:09.000
So, I'm drawing more and more
levels, and they never quite

00:18:09.000 --> 00:18:12.000
look the same.
But I have a point.

00:18:12.000 --> 00:18:16.000
And, at this point,
I have the tangent plane to the

00:18:16.000 --> 00:18:24.000
level surface.
OK, so this is tangent plane to

00:18:24.000 --> 00:18:30.000
the level.
Then, if I choose any vector in

00:18:30.000 --> 00:18:35.000
that tangent plane.
Let's say I choose the one that

00:18:35.000 --> 00:18:39.000
goes in that direction.
Then, I can actually find a

00:18:39.000 --> 00:18:42.000
curve that goes in that
direction, and stays on the

00:18:42.000 --> 00:18:45.000
level.
So, here, that would be a curve

00:18:45.000 --> 00:18:50.000
that somehow goes from the right
to the left, and of course it

00:18:50.000 --> 00:18:53.000
has to end up going up or
something like that.

00:18:53.000 --> 00:19:05.000
OK, so given any vector tangent
-- -- let's call that vector v

00:19:05.000 --> 00:19:14.000
tangent to the level,
we get that the gradient is

00:19:14.000 --> 00:19:20.000
perpendicular to v.
So, if the gradient is

00:19:20.000 --> 00:19:24.000
perpendicular to this vector
tangent to this curve,

00:19:24.000 --> 00:19:28.000
but also to any vector,
I can draw that tangent to my

00:19:28.000 --> 00:19:29.000
surface.
So, what does that mean?

00:19:29.000 --> 00:19:34.000
Well, that means the gradient
is actually perpendicular to the

00:19:34.000 --> 00:19:38.000
tangent plane or to the surface
at this point.

00:19:38.000 --> 00:19:43.000
So, the gradient is
perpendicular.

00:20:02.000 --> 00:20:04.000
And, well, here,
I've illustrated things with a

00:20:04.000 --> 00:20:06.000
three-dimensional example,
but really it works the same if

00:20:06.000 --> 00:20:10.000
you have only two variables.
Then you have a level curve

00:20:10.000 --> 00:20:13.000
that has a tangent line,
and the gradient is

00:20:13.000 --> 00:20:23.000
perpendicular to that line.
OK, any questions?

00:20:23.000 --> 00:20:36.000
No?
OK, so, let's see.

00:20:36.000 --> 00:20:39.000
That's actually pretty neat
because there is a nice

00:20:39.000 --> 00:20:43.000
application of this,
which is to try to figure out,

00:20:43.000 --> 00:20:44.000
now we know,
actually, how to find the

00:20:44.000 --> 00:20:46.000
tangent plane to anything,
pretty much.

00:21:13.000 --> 00:21:19.000
OK, so let's see.
So, let's say that,

00:21:19.000 --> 00:21:27.000
for example,
I want to find -- -- the

00:21:27.000 --> 00:21:42.000
tangent plane -- -- to the
surface with equation,

00:21:42.000 --> 00:21:50.000
let's say, x^2 y^2-z^2 = 4 at
the point (2,1,

00:21:50.000 --> 00:22:01.000
1).
Let me write that.

00:22:01.000 --> 00:22:06.000
So, how do we do that?
Well, one way that we already

00:22:06.000 --> 00:22:09.000
know,
if we solve this for z,

00:22:09.000 --> 00:22:12.000
so we can write z equals a
function of x and y,

00:22:12.000 --> 00:22:16.000
then we know tangent plane
approximation for the graph of a

00:22:16.000 --> 00:22:19.000
function,
z equals some function of x and

00:22:19.000 --> 00:22:21.000
y.
But, that doesn't look like

00:22:21.000 --> 00:22:24.000
it's the best way to do it.
OK, the best way to it,

00:22:24.000 --> 00:22:27.000
now that we have the gradient
vector, is actually to directly

00:22:27.000 --> 00:22:30.000
say, oh, we know the normal
vector to this plane.

00:22:30.000 --> 00:22:35.000
The normal vector will just be
the gradient.

00:22:35.000 --> 00:22:40.000
Oh, I think I have a cool
picture to show.

00:22:40.000 --> 00:22:42.000
OK, so that's what it looks
like.

00:22:42.000 --> 00:22:49.000
OK, so here you have the
surface x2 y2-z2 equals four.

00:22:49.000 --> 00:22:52.000
That's called a hyperboloid
because it looks like when you

00:22:52.000 --> 00:22:55.000
get when you spin a hyperbola
around an axis.

00:22:55.000 --> 00:23:00.000
And, here's a tangent plane at
the given point.

00:23:00.000 --> 00:23:02.000
So, it doesn't look very
tangent because it crosses the

00:23:02.000 --> 00:23:04.000
surface.
But, it's really,

00:23:04.000 --> 00:23:08.000
if you think about it,
you will see it's really the

00:23:08.000 --> 00:23:12.000
plane that's approximating the
surface in the best way that you

00:23:12.000 --> 00:23:18.000
can at this given point.
It is really the tangent plane.

00:23:18.000 --> 00:23:27.000
So, how do we find this plane?
Well, you can plot it on a

00:23:27.000 --> 00:23:30.000
computer.
That's not exactly how you

00:23:30.000 --> 00:23:33.000
would look for it in the first
place.

00:23:33.000 --> 00:23:38.000
So, the way to do it is that we
compute the gradient.

00:23:38.000 --> 00:23:43.000
So, a gradient of what?
Well, a gradient of this

00:23:43.000 --> 00:23:49.000
function.
OK, so I should say,

00:23:49.000 --> 00:23:56.000
this is the level set,
w equals four,

00:23:56.000 --> 00:24:07.000
where w equals x^2 y^2 - z^2.
And so, we know that the

00:24:07.000 --> 00:24:14.000
gradient of this,
well, what is it?

00:24:14.000 --> 00:24:22.000
2x, then 2y,
and then negative 2z.

00:24:22.000 --> 00:24:27.000
So, at this given point,
I guess we are at x equals two.

00:24:27.000 --> 00:24:29.000
So, that's four.
And then, y and z are one.

00:24:29.000 --> 00:24:37.000
So, two, negative two.
OK, and that's going to be the

00:24:37.000 --> 00:24:44.000
normal vector to the surface or
to the tangent plane.

00:24:44.000 --> 00:24:47.000
That's one way to define the
tangent plane.

00:24:47.000 --> 00:24:50.000
All right, it has the same
normal vector as the surface.

00:24:50.000 --> 00:24:52.000
That's one way to define the
normal vector to the surface,

00:24:52.000 --> 00:24:56.000
if you prefer.
Being perpendicular to the

00:24:56.000 --> 00:25:01.000
surface means that you are
perpendicular to its tangent

00:25:01.000 --> 00:25:05.000
plane.
OK, so the equation is,

00:25:05.000 --> 00:25:12.000
well, 4x 2y-2z equals
something, where something is,

00:25:12.000 --> 00:25:19.000
well, we should just plug in
that point.

00:25:19.000 --> 00:25:26.000
We'll get eight plus two minus
two looks like we'll get eight.

00:25:26.000 --> 00:25:29.000
And, of course,
we could simplify dividing

00:25:29.000 --> 00:25:32.000
everything by two,
but it's not very important

00:25:32.000 --> 00:25:34.000
here.
OK, so now if you have a

00:25:34.000 --> 00:25:36.000
surface given by an evil
equation,

00:25:36.000 --> 00:25:40.000
and a point on the surface,
well, you know how to find the

00:25:40.000 --> 00:25:44.000
tangent plane to the surface at
that point.

00:25:44.000 --> 00:25:52.000
OK, any questions?
No.

00:25:52.000 --> 00:26:00.000
OK, let me give just another
reason why, another way that we

00:26:00.000 --> 00:26:04.000
could have seen this.
So, I claim,

00:26:04.000 --> 00:26:07.000
in fact, we could have done
this without the gradient,

00:26:07.000 --> 00:26:09.000
or using the gradient in a
somehow disguised way.

00:26:09.000 --> 00:26:18.000
So, here's another way.
So, the other way to do it

00:26:18.000 --> 00:26:22.000
would be to start with a
differential,

00:26:22.000 --> 00:26:26.000
OK?
dw, while it's pretty much the

00:26:26.000 --> 00:26:31.000
same content,
but let me write it as a

00:26:31.000 --> 00:26:35.000
differential,
dw is 2xdx 2ydy-2zdz.

00:26:35.000 --> 00:26:44.000
So, at a given point,
at (2,1, 1),

00:26:44.000 --> 00:26:52.000
this is 4dx 2dy-2dz.
Now, if we want to change this

00:26:52.000 --> 00:26:56.000
into an approximation formula,
we can.

00:26:56.000 --> 00:27:07.000
We know that the change in w is
approximately equal to 4 delta x

00:27:07.000 --> 00:27:15.000
2 delta y - 2 delta z.
OK, so when do we stay on the

00:27:15.000 --> 00:27:19.000
level surface?
Well, we stay on the level

00:27:19.000 --> 00:27:24.000
surface when w doesn't change,
so, when this becomes zero,

00:27:24.000 --> 00:27:25.000
OK?
Now, what does this

00:27:25.000 --> 00:27:28.000
approximation sign mean?
Well, it means for small

00:27:28.000 --> 00:27:31.000
changes in x,
y, z, this guy will be close to

00:27:31.000 --> 00:27:33.000
that guy.
It also means something else.

00:27:33.000 --> 00:27:36.000
Remember, these approximation
formulas, they are linear

00:27:36.000 --> 00:27:39.000
approximations.
They mean that we replace the

00:27:39.000 --> 00:27:43.000
function, actually,
by some closest linear formula

00:27:43.000 --> 00:27:47.000
that will be nearby.
And so, in particular,

00:27:47.000 --> 00:27:52.000
if we set this equal to zero
instead of approximately zero,

00:27:52.000 --> 00:27:56.000
it means we'll actually be
moving on the tangent plane to

00:27:56.000 --> 00:27:59.000
the level set.
If you want strict equalities

00:27:59.000 --> 00:28:03.000
in approximations means that we
replace the function by its

00:28:03.000 --> 00:28:04.000
tangent approximation.

00:28:37.000 --> 00:28:44.000
So -- [APPLAUSE] OK,
so the level corresponds to

00:28:44.000 --> 00:28:53.000
delta w equals zero,
and its tangent plane

00:28:53.000 --> 00:29:03.000
corresponds to four delta x plus
two delta y minus two delta z

00:29:03.000 --> 00:29:08.000
equals zero.
That's what I'm trying to say,

00:29:08.000 --> 00:29:10.000
basically.
And, what's delta x?

00:29:10.000 --> 00:29:12.000
Well, that means it's the
change in x.

00:29:12.000 --> 00:29:15.000
So, what's the change in x here?
That means, well,

00:29:15.000 --> 00:29:19.000
we started with x equals two,
and we moved to some other

00:29:19.000 --> 00:29:25.000
value, x.
So, that's actually x- 2, right?

00:29:25.000 --> 00:29:28.000
That's how much x has changed
compared to 2.

00:29:28.000 --> 00:29:37.000
And, two times (y - 1) minus
two times z - 1 = 0.

00:29:37.000 --> 00:29:42.000
That's the equation of a
tangent plane.

00:29:42.000 --> 00:29:46.000
It's the same equation as the
one over there.

00:29:46.000 --> 00:29:48.000
These are just two different
methods to get it.

00:29:48.000 --> 00:29:52.000
OK, so this one explains to you
what's going on in terms of

00:29:52.000 --> 00:29:57.000
approximation formulas.
This one goes right away,

00:29:57.000 --> 00:30:02.000
by using the gradient factor.
So, in a way,

00:30:02.000 --> 00:30:06.000
with this one,
you don't have to think nearly

00:30:06.000 --> 00:30:11.000
as much.
But, you can use either one.

00:30:11.000 --> 00:30:17.000
OK, questions?
No?

00:30:17.000 --> 00:30:23.000
OK, so let's move on to new
topic, which is another

00:30:23.000 --> 00:30:30.000
application of a gradient
vector, and that is directional

00:30:30.000 --> 00:30:32.000
derivatives.

00:30:44.000 --> 00:30:52.000
OK, so let's say that we have a
function of two variables,

00:30:52.000 --> 00:30:56.000
x and y.
Well, we know how to compute

00:30:56.000 --> 00:31:02.000
partial w over partial x or
partial w over partial y,

00:31:02.000 --> 00:31:07.000
which measure how w changes if
I move in the direction of the x

00:31:07.000 --> 00:31:10.000
axis or in the direction of the
y axis.

00:31:10.000 --> 00:31:13.000
So, what about moving in other
directions?

00:31:13.000 --> 00:31:16.000
Well, of course,
we've seen other approximation

00:31:16.000 --> 00:31:18.000
formulas and so on.
But, we can still ask,

00:31:18.000 --> 00:31:21.000
is there a derivative in every
direction?

00:31:21.000 --> 00:31:25.000
And that's basically,
yes, that's the directional

00:31:25.000 --> 00:31:30.000
derivative.
OK, so these are derivatives in

00:31:30.000 --> 00:31:40.000
the direction of I hat or j hat,
the vectors that go along the x

00:31:40.000 --> 00:31:50.000
or the y axis.
So, what if we move in another

00:31:50.000 --> 00:32:01.000
direction, let's say,
the direction of some unit

00:32:01.000 --> 00:32:09.000
vector, let's call it u .
OK, so if I give you a unit

00:32:09.000 --> 00:32:13.000
vector, you can ask yourself,
if I move in the direction,

00:32:13.000 --> 00:32:16.000
how quickly will my function
change?

00:32:16.000 --> 00:32:29.000
So -- So, let's look at the
straight trajectory.

00:32:29.000 --> 00:32:34.000
What this should mean is I
start at some value,

00:32:34.000 --> 00:32:37.000
x, y, and there I have my
vector u.

00:32:37.000 --> 00:32:41.000
And, I'm going to move in a
straight line in the direction

00:32:41.000 --> 00:32:46.000
of u.
And, I have the graph of my

00:32:46.000 --> 00:32:54.000
function -- -- and I'm asking
myself how quickly does the

00:32:54.000 --> 00:33:02.000
value change when I move on the
graph in that direction?

00:33:02.000 --> 00:33:10.000
OK, so let's look at a straight
line trajectory So,

00:33:10.000 --> 00:33:18.000
we have a position vector,
r, that will depend on some

00:33:18.000 --> 00:33:26.000
parameter which I will call s.
You'll see why very soon,

00:33:26.000 --> 00:33:30.000
in such a way that the
derivative is this given unit

00:33:30.000 --> 00:33:33.000
vector u hat.
So, why do I use s for my

00:33:33.000 --> 00:33:36.000
parameter rather than t.
Well, it's a convention.

00:33:36.000 --> 00:33:41.000
I'm moving at unit speed along
this line.

00:33:41.000 --> 00:33:45.000
So that means that actually,
I'm parameterizing things by

00:33:45.000 --> 00:33:48.000
the distance that I've traveled
along a curve,

00:33:48.000 --> 00:33:54.000
sorry, along this line.
So, here it's called s in the

00:33:54.000 --> 00:33:59.000
sense of arc length.
Actually, it's not really an

00:33:59.000 --> 00:34:06.000
arc because it's a straight
line, so it's the distance along

00:34:06.000 --> 00:34:09.000
the line.
OK, so because we are

00:34:09.000 --> 00:34:15.000
parameterizing by distance,
we are just using s as a

00:34:15.000 --> 00:34:21.000
convention just to distinguish
it from other situations.

00:34:21.000 --> 00:34:27.000
And, so, now,
the question will be,

00:34:27.000 --> 00:34:32.000
what is dw/ds?
What's the rate of change of w

00:34:32.000 --> 00:34:36.000
when I move like that?
Well, of course we know the

00:34:36.000 --> 00:34:40.000
answer because that's a special
case of the chain rule.

00:34:40.000 --> 00:34:44.000
So, that's how we will actually
compute it.

00:34:44.000 --> 00:34:49.000
But, in terms of what it means,
it really means we are asking

00:34:49.000 --> 00:34:51.000
ourselves,
we start at a point and we

00:34:51.000 --> 00:34:54.000
change the variables in a
certain direction,

00:34:54.000 --> 00:34:57.000
which is not necessarily the x
or the y direction,

00:34:57.000 --> 00:35:01.000
but really any direction.
And then, what's the derivative

00:35:01.000 --> 00:35:07.000
in that direction?
OK, does that make sense as a

00:35:07.000 --> 00:35:08.000
concept?
Kind of?

00:35:08.000 --> 00:35:10.000
I see some faces that are not
completely convinced.

00:35:10.000 --> 00:35:14.000
So, maybe you should show more
pictures.

00:35:14.000 --> 00:35:21.000
Well, let me first write down a
bit more and show you something.

00:35:40.000 --> 00:35:45.000
So I just want to give you the
actual definition.

00:35:45.000 --> 00:35:50.000
Sorry, first of all in case you
wonder what this is all about,

00:35:50.000 --> 00:35:55.000
so let's say the components of
our unit vector are two numbers,

00:35:55.000 --> 00:36:00.000
a and b.
Then, it means we'll move along

00:36:00.000 --> 00:36:05.000
the line x of s equals some
initial value,

00:36:05.000 --> 00:36:09.000
the point where we are actually
at the directional derivative

00:36:09.000 --> 00:36:13.000
plus s times a,
or I meant to say plus a times

00:36:13.000 --> 00:36:19.000
s.
And, y of s equals y0 bs.

00:36:19.000 --> 00:36:38.000
And then, we plug that into w.
And then we take the derivative.

00:36:38.000 --> 00:36:45.000
So, we have a notation for that
which is going to be dw/ds with

00:36:45.000 --> 00:36:53.000
a subscript in the direction of
u to indicate in which direction

00:36:53.000 --> 00:37:03.000
we are actually going to move.
And, that's called the

00:37:03.000 --> 00:37:17.000
directional derivative -- -- in
the direction of u.

00:37:17.000 --> 00:37:28.000
OK, so, let's see what it means
geometrically.

00:37:28.000 --> 00:37:33.000
So, remember,
we've seen things about partial

00:37:33.000 --> 00:37:36.000
derivatives,
and we see that the partial

00:37:36.000 --> 00:37:41.000
derivatives are the slopes of
slices of the graph by vertical

00:37:41.000 --> 00:37:45.000
planes that are parallel to the
x or the y directions.

00:37:45.000 --> 00:37:48.000
OK, so, if I have a point,
at any point,

00:37:48.000 --> 00:37:52.000
I can slice the graph of my
function by two planes,

00:37:52.000 --> 00:37:57.000
one that's going along the x,
one along the y direction.

00:37:57.000 --> 00:38:02.000
And then, I can look at the
slices of the graph.

00:38:02.000 --> 00:38:04.000
Let me see if I can use that
thing.

00:38:04.000 --> 00:38:07.000
So, we can look at the slices
of the graph that are drawn

00:38:07.000 --> 00:38:10.000
here.
In fact, we look at the tangent

00:38:10.000 --> 00:38:14.000
lines to the slices,
and we look at the slope and

00:38:14.000 --> 00:38:17.000
that gives us the partial
derivatives in case you are on

00:38:17.000 --> 00:38:21.000
that side and want to see also
the pointer that was here.

00:38:21.000 --> 00:38:26.000
So, now, similarly,
the directional derivative

00:38:26.000 --> 00:38:31.000
means, actually,
we'll be slicing our graph by

00:38:31.000 --> 00:38:37.000
the vertical plane.
It's not really colorful,

00:38:37.000 --> 00:38:43.000
something more colorful.
We'll be slicing things by a

00:38:43.000 --> 00:38:46.000
plane that is now in the
direction of this vector,

00:38:46.000 --> 00:38:51.000
u, and we'll be looking at the
slope of the slice of the graph.

00:38:51.000 --> 00:38:57.000
So, what that looks like here,
so that's the same applet the

00:38:57.000 --> 00:39:03.000
way that you've used on your
problem set in case you are

00:39:03.000 --> 00:39:08.000
wondering.
So, now, I'm picking a point on

00:39:08.000 --> 00:39:12.000
the contour plot.
And, at that point,

00:39:12.000 --> 00:39:15.000
I slice the graph.
So, here I'm starting by

00:39:15.000 --> 00:39:17.000
slicing in the direction of the
x axis.

00:39:17.000 --> 00:39:20.000
So, in fact,
what I'm measuring here by the

00:39:20.000 --> 00:39:24.000
slope of the slice is the
partial in the x direction.

00:39:24.000 --> 00:39:28.000
It's really partial f partial
x, which is also the directional

00:39:28.000 --> 00:39:31.000
derivative in the direction of
i.

00:39:31.000 --> 00:39:37.000
And now, if I rotate the slice,
then I have all of these

00:39:37.000 --> 00:39:40.000
planes.
So, you see at the bottom left,

00:39:40.000 --> 00:39:42.000
I have the direction in which
I'm going.

00:39:42.000 --> 00:39:44.000
There's this,
like, rotating line that tells

00:39:44.000 --> 00:39:47.000
you in which direction I'm going
to be moving.

00:39:47.000 --> 00:39:49.000
And for each direction,
I have a plane.

00:39:49.000 --> 00:39:52.000
And, when I slice by that
plane, I will get,

00:39:52.000 --> 00:39:56.000
so I have this direction here
going maybe to the southwest.

00:39:56.000 --> 00:40:00.000
So, that gives me a slice of my
graph by a vertical plane,

00:40:00.000 --> 00:40:03.000
and the slice has a certain
slope.

00:40:03.000 --> 00:40:08.000
And, the slope is going to be
the directional derivative in

00:40:08.000 --> 00:40:14.000
that direction.
OK, I think that's as graphic

00:40:14.000 --> 00:40:22.000
as I can get.
OK, any questions about that?

00:40:22.000 --> 00:40:33.000
No?
OK, so let's see how we compute

00:40:33.000 --> 00:40:41.000
that guy.
So, let me just write again

00:40:41.000 --> 00:40:49.000
just in case you want to,
in case you didn't hear me it's

00:40:49.000 --> 00:40:58.000
the slope of the slice of the
graph by a vertical plane -- --

00:40:58.000 --> 00:41:03.000
that contains the given
direction,

00:41:03.000 --> 00:41:06.000
that's parallel to the
direction, u.

00:41:06.000 --> 00:41:11.000
So, how do we compute it?
Well, we can use the chain rule.

00:41:11.000 --> 00:41:22.000
The chain rule implies that
dw/ds is actually the gradient

00:41:22.000 --> 00:41:31.000
of w dot product with the
velocity vector dr/ds.

00:41:31.000 --> 00:41:35.000
But, remember we say that we
are going to be moving at unit

00:41:35.000 --> 00:41:39.000
speed in the direction of u.
So, in fact,

00:41:39.000 --> 00:41:50.000
that's just gradient w dot
product with the unit vector u.

00:41:50.000 --> 00:41:57.000
OK, so the formula that we
remember is really dw/ds in the

00:41:57.000 --> 00:42:03.000
direction of u is gradient w dot
product of u.

00:42:03.000 --> 00:42:13.000
And, maybe I should also say in
words, this is the component of

00:42:13.000 --> 00:42:19.000
the gradient in the direction of
u.

00:42:19.000 --> 00:42:21.000
And, maybe that makes more
sense.

00:42:21.000 --> 00:42:25.000
So, for example,
the directional derivative in

00:42:25.000 --> 00:42:29.000
the direction of I hat is the
component along the x axes.

00:42:29.000 --> 00:42:32.000
That's the same as,
indeed, the partial derivatives

00:42:32.000 --> 00:42:40.000
in the x direction.
Things make sense.

00:42:40.000 --> 00:42:50.000
dw/ds in the direction of I hat
is, sorry, gradient w dot I hat,

00:42:50.000 --> 00:42:59.000
which is wx,maybe I should
write, partial w of partial x.

00:42:59.000 --> 00:43:09.000
OK, now, so that's basically
what we need to know to compute

00:43:09.000 --> 00:43:12.000
these guys.
So now, let's go back to the

00:43:12.000 --> 00:43:16.000
gradient and see what this tells
us about the gradient.

00:43:42.000 --> 00:43:51.000
[APPLAUSE]
I see you guys are having fun.

00:43:51.000 --> 00:43:54.000
OK, OK, let's do a little bit
of geometry here.

00:43:54.000 --> 00:44:00.000
That should calm you down.
So, we said dw/ds in the

00:44:00.000 --> 00:44:04.000
direction of u is gradient w dot
u.

00:44:04.000 --> 00:44:11.000
That's the same as the length
of gradient w times the length

00:44:11.000 --> 00:44:15.000
of u.
Well, that happens to be one

00:44:15.000 --> 00:44:23.000
because we are taking the unit
vector times the cosine of the

00:44:23.000 --> 00:44:30.000
angle between the gradient and
the given unit vector,

00:44:30.000 --> 00:44:36.000
u, so, have this angle, theta.
OK, that's another way of

00:44:36.000 --> 00:44:39.000
saying we are taking the
component of a gradient in the

00:44:39.000 --> 00:44:43.000
direction of u.
But now, what does that tell us?

00:44:43.000 --> 00:44:46.000
Well,
let's try to figure out in

00:44:46.000 --> 00:44:50.000
which directions w changes the
fastest,

00:44:50.000 --> 00:44:54.000
in which direction it increases
the most or decreases the most,

00:44:54.000 --> 00:45:03.000
or doesn't actually change.
So, when is this going to be

00:45:03.000 --> 00:45:05.000
the largest?
If I fix a point,

00:45:05.000 --> 00:45:09.000
if I set a point,
then the gradient vector at

00:45:09.000 --> 00:45:12.000
that point is given to me.
But, the question is,

00:45:12.000 --> 00:45:15.000
in which direction does it
change the most quickly?

00:45:15.000 --> 00:45:19.000
Well, what I can change is the
direction, and this will be the

00:45:19.000 --> 00:45:25.000
largest when the cosine is one.
So, this is largest when the

00:45:25.000 --> 00:45:33.000
cosine of the angle is one.
That means the angle is zero.

00:45:33.000 --> 00:45:40.000
That means u is actually in the
direction of the gradient.

00:45:40.000 --> 00:45:42.000
OK, so that's a new way to
think about the direction of a

00:45:42.000 --> 00:45:47.000
gradient.
The gradient is the direction

00:45:47.000 --> 00:45:57.000
in which the function increases
the most quickly at that point.

00:45:57.000 --> 00:46:08.000
So, the direction of gradient w
is the direction of fastest

00:46:08.000 --> 00:46:15.000
increase of w at the given
point.

00:46:15.000 --> 00:46:24.000
And, what is the magnitude of w?
Well, it's actually the

00:46:24.000 --> 00:46:33.000
directional derivative in that
direction.

00:46:33.000 --> 00:46:37.000
OK, so if I go in that
direction, which gives me the

00:46:37.000 --> 00:46:40.000
fastest increase,
then the corresponding slope

00:46:40.000 --> 00:46:44.000
will be the length of the
gradient.

00:46:44.000 --> 00:46:51.000
And, with the direction of the
fastest decrease?

00:46:51.000 --> 00:46:53.000
It's going in the opposite
direction, right?

00:46:53.000 --> 00:46:55.000
I mean, if you are on a
mountain, and you know that you

00:46:55.000 --> 00:46:57.000
are facing the mountain,
that's the direction of fastest

00:46:57.000 --> 00:46:59.000
increase.
The direction of fastest

00:46:59.000 --> 00:47:01.000
decrease is behind you straight
down.

00:47:01.000 --> 00:47:11.000
OK, so, the minimal value of
dw/ds is achieved when cosine of

00:47:11.000 --> 00:47:18.000
theta is minus one.
That means theta equals 180�.

00:47:18.000 --> 00:47:27.000
That means u is in the
direction of minus the gradient.

00:47:27.000 --> 00:47:30.000
It points opposite to the
gradient.

00:47:30.000 --> 00:47:43.000
And, finally,
when do we have dw/ds equals

00:47:43.000 --> 00:47:48.000
zero?
So, in which direction does the

00:47:48.000 --> 00:47:52.000
function not change?
Well, we have two answers to

00:47:52.000 --> 00:47:54.000
that.
One is to just use the formula.

00:47:54.000 --> 00:47:58.000
So, that's one cosine theta
equals zero.

00:47:58.000 --> 00:48:03.000
That means theta equals 90 degrees.
That means that u is

00:48:03.000 --> 00:48:08.000
perpendicular to the gradient.
The other way to think about

00:48:08.000 --> 00:48:11.000
it, the direction in which the
value doesn't change is a

00:48:11.000 --> 00:48:14.000
direction that's tangent to the
level surface.

00:48:14.000 --> 00:48:18.000
If we are not changing a,
it means we are moving along

00:48:18.000 --> 00:48:24.000
the level.
And, that's the same thing --

00:48:24.000 --> 00:48:30.000
-- as being tangent to the
level.

00:48:30.000 --> 00:48:36.000
So, let me just show that on
the picture here.

00:48:36.000 --> 00:48:39.000
So, if actually show you the
gradient, you can't really see

00:48:39.000 --> 00:48:41.000
it here.
I need to move it a bit.

00:48:41.000 --> 00:48:44.000
So, the gradient here is
pointing straight up at the

00:48:44.000 --> 00:48:50.000
point that I have chosen.
Now, if I choose a slice that's

00:48:50.000 --> 00:48:52.000
perpendicular,
and a direction that's

00:48:52.000 --> 00:48:55.000
perpendicular to the gradient,
so that's actually tangent to

00:48:55.000 --> 00:48:57.000
the level curve,
then you see that my slice is

00:48:57.000 --> 00:49:00.000
flat.
I don't actually have any slop.

00:49:00.000 --> 00:49:04.000
The directional derivative in a
direction that's perpendicular

00:49:04.000 --> 00:49:06.000
to the gradient is basically
zero.

00:49:06.000 --> 00:49:08.000
Now, if I rotate,
then the slope sort of

00:49:08.000 --> 00:49:11.000
increases, increases,
increases, and it becomes the

00:49:11.000 --> 00:49:14.000
largest when I'm going in the
direction of a gradient.

00:49:14.000 --> 00:49:17.000
So, here, I have,
actually, a pretty big slope.

00:49:17.000 --> 00:49:20.000
And now, if I keep rotating,
then the slope will decrease

00:49:20.000 --> 00:49:22.000
again.
Then it becomes zero when I

00:49:22.000 --> 00:49:25.000
perpendicular,
and then it becomes negative.

00:49:25.000 --> 00:49:29.000
It's the most negative when I
pointing away from the gradient

00:49:29.000 --> 00:49:33.000
and then becomes zero again when
I'm back perpendicular.

00:49:33.000 --> 00:49:38.000
OK, so for example,
if I give you a contour plot,

00:49:38.000 --> 00:49:41.000
and I ask you to draw the
direction of the gradient

00:49:41.000 --> 00:49:43.000
vector,
well, at this point,

00:49:43.000 --> 00:49:46.000
for example,
you would look at the picture.

00:49:46.000 --> 00:49:49.000
The gradient vector would be
going perpendicular to the

00:49:49.000 --> 00:49:52.000
level.
And, it would be going towards

00:49:52.000 --> 00:49:55.000
higher values of a function.
I don't know if you can see the

00:49:55.000 --> 00:49:57.000
labels, but the thing in the
middle is a minimum.

00:49:57.000 --> 00:50:03.000
So, it will actually be
pointing in this kind of

00:50:03.000 --> 00:50:08.000
direction.
OK, so that's it for today.