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so -- OK, so remember last
time,

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00:00:25,000 --> 00:00:32,000
on Tuesday we learned about the
chain rule,

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00:00:32,000 --> 00:00:39,000
and so for example we saw that
if we have a function that

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00:00:39,000 --> 00:00:44,000
depends,
sorry, on three variables,

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00:00:44,000 --> 00:00:50,000
x,y,z,
that x,y,z themselves depend on

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00:00:50,000 --> 00:00:54,000
some variable,
t,

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00:00:54,000 --> 00:01:06,000
then you can find a formula for
df/dt by writing down wx/dx dt

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00:01:06,000 --> 00:01:12,000
wy dy/dt wz dz/dt.
And, the meaning of that

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formula is that while the change
in w is caused by changes in x,

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00:01:17,000 --> 00:01:21,000
y, and z, x,
y, and z change at rates dx/dt,

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00:01:21,000 --> 00:01:25,000
dy/dt, dz/dt.
And, this causes a function to

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00:01:25,000 --> 00:01:31,000
change accordingly using,
well, the partial derivatives

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00:01:31,000 --> 00:01:37,000
tell you how sensitive w is to
changes in each variable.

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00:01:37,000 --> 00:01:45,000
OK, so, we are going to just
rewrite this in a new notation.

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00:01:45,000 --> 00:01:52,000
So, I'm going to rewrite this
in a more concise form as

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00:01:52,000 --> 00:01:59,000
gradient of w dot product with
velocity vector dr/dt.

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00:01:59,000 --> 00:02:04,000
So, the gradient of w is a
vector formed by putting

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00:02:04,000 --> 00:02:08,000
together all of the partial
derivatives.

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00:02:08,000 --> 00:02:12,000
OK, so it's the vector whose
components are the partials.

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00:02:12,000 --> 00:02:15,000
And, of course,
it's a vector that depends on

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00:02:15,000 --> 00:02:19,000
x, y, and z, right?
These guys depend on x, y, z.

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00:02:19,000 --> 00:02:22,000
So, it's actually one vector
for each point,

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00:02:22,000 --> 00:02:31,000
x, y, z.
You can talk about the gradient

30
00:02:31,000 --> 00:02:39,000
of w at some point,
x, y, z.

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00:02:39,000 --> 00:02:41,000
So, at each point,
it gives you a vector.

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00:02:41,000 --> 00:02:47,000
That actually is what we will
call later a vector field.

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00:02:47,000 --> 00:02:59,000
We'll get back to that later.
And, dr/dt is just the velocity

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00:02:59,000 --> 00:03:07,000
vector dx/dt,
dy/dt, dz/dt.

35
00:03:07,000 --> 00:03:14,000
OK, so the new definition for
today is the definition of the

36
00:03:14,000 --> 00:03:18,000
gradient vector.
And, our goal will be to

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00:03:18,000 --> 00:03:21,000
understand a bit better,
what does this vector mean?

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00:03:21,000 --> 00:03:24,000
What does it measure?
And, what can we do with it?

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00:03:24,000 --> 00:03:29,000
But, you see that in terms of
information content,

40
00:03:29,000 --> 00:03:33,000
it's really the same
information that's already in

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00:03:33,000 --> 00:03:38,000
the partial derivatives,
or in the differential.

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00:03:38,000 --> 00:03:43,000
So, yes, and I should say,
of course you can also use the

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00:03:43,000 --> 00:03:49,000
gradient and other things like
approximation formulas and so

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00:03:49,000 --> 00:03:52,000
on.
And so far, it's just notation.

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00:03:52,000 --> 00:03:57,000
It's a way to rewrite things.
But, so here's the first cool

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00:03:57,000 --> 00:04:03,000
property of the gradient.
So, I claim that the gradient

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00:04:03,000 --> 00:04:11,000
vector is perpendicular to the
level surface corresponding to

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00:04:11,000 --> 00:04:18,000
setting the function,
w, equal to a constant.

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00:04:18,000 --> 00:04:22,000
OK, so if I draw a contour plot
of my function,

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00:04:22,000 --> 00:04:28,000
so, actually forget about z
because I want to draw a two

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00:04:28,000 --> 00:04:32,000
variable contour plot.
So, say I have a function of

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00:04:32,000 --> 00:04:35,000
two variables,
x and y, then maybe it has some

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00:04:35,000 --> 00:04:38,000
contour plot.
And, I'm saying if I take the

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00:04:38,000 --> 00:04:42,000
gradient of a function at this
point, (x,y).

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00:04:42,000 --> 00:04:46,000
So, I will have a vector.
Well, if I draw that vector on

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00:04:46,000 --> 00:04:51,000
top of a contour plot,
it's going to end up being

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00:04:51,000 --> 00:04:54,000
perpendicular to the level
curve.

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00:04:54,000 --> 00:04:57,000
Same thing if I have a function
of three variables.

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00:04:57,000 --> 00:04:59,000
Then, I can try to draw its
contour plot.

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00:04:59,000 --> 00:05:03,000
Of course, I can't really do it
because the contour plot would

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00:05:03,000 --> 00:05:05,000
be living in space with x,
y, and z.

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00:05:05,000 --> 00:05:09,000
But, it would be a bunch of
level faces, and the gradient

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00:05:09,000 --> 00:05:11,000
vector would be a vector in
space.

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00:05:11,000 --> 00:05:15,000
That vector is perpendicular to
the level faces.

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00:05:15,000 --> 00:05:24,000
So, let's try to see that on a
couple of examples.

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00:05:24,000 --> 00:05:32,000
So, let's do a first example.
What's the easiest case?

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00:05:32,000 --> 00:05:36,000
Let's take a linear function of
x, y, and z.

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00:05:36,000 --> 00:05:42,000
So, I will take w equals a1
times x plus a2 times y plus a3

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00:05:42,000 --> 00:05:47,000
times z.
Well, so, what's the gradient

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00:05:47,000 --> 00:05:53,000
of this function?
Well, the first component will

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00:05:53,000 --> 00:05:58,000
be a1.
That's partial w partial x.

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00:05:58,000 --> 00:06:03,000
Then, a2, that's partial w
partial y, and a3,

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00:06:03,000 --> 00:06:15,000
partial w partial z.
Now, what is the levels of this?

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00:06:15,000 --> 00:06:22,000
Well, if I set w equal to some
constant, c, that means I look

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00:06:22,000 --> 00:06:27,000
at the points where a1x a2y a3z
equals c.

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00:06:27,000 --> 00:06:30,000
What kind of service is that?
It's a plane.

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00:06:30,000 --> 00:06:39,000
And, we know how to find a
normal vector to this plane just

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00:06:39,000 --> 00:06:48,000
by looking at the coefficients.
So, it's a plane with a normal

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00:06:48,000 --> 00:06:51,000
vector exactly this gradient.
And, in fact,

80
00:06:51,000 --> 00:06:55,000
in a way, this is the only case
you need to check because of

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00:06:55,000 --> 00:06:58,000
linear approximations.
If you replace a function by

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00:06:58,000 --> 00:07:02,000
its linear approximation,
that means you will replace the

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00:07:02,000 --> 00:07:04,000
level surfaces by their tension
planes.

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00:07:04,000 --> 00:07:08,000
And then, you'll actually end
up in this situation.

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00:07:08,000 --> 00:07:09,000
But maybe that's not very
convincing.

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00:07:09,000 --> 00:07:25,000
So, let's do another example.
So, let's do a second example.

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00:07:25,000 --> 00:07:28,000
Let's say we look at the
function x^2 y^2.

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00:07:28,000 --> 00:07:32,000
OK, so now it's a function of
just two variables because that

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00:07:32,000 --> 00:07:36,000
way we'll be able to actually
draw a picture for you.

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00:07:36,000 --> 00:07:40,000
OK, so what are the level sets
of this function?

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00:07:40,000 --> 00:07:44,000
Well, they're going to be
circles, right?

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00:07:44,000 --> 00:07:54,000
w equals c is a circle,
x^2 y^2 = c.

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00:07:54,000 --> 00:07:58,000
So, I should say,
maybe, sorry,

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00:07:58,000 --> 00:08:08,000
the level curve is a circle.
So, the contour plot looks

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00:08:08,000 --> 00:08:16,000
something like that.
Now, what's the gradient vector?

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00:08:16,000 --> 00:08:20,000
Well, the gradient of this
function, so,

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00:08:20,000 --> 00:08:26,000
partial w partial x is 2x.
And partial w partial y is 2y.

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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

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00:08:31,000 --> 00:08:34,000
gradient vector.
So, here at x,

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00:08:34,000 --> 00:08:38,000
y, so, I have to draw the
vector, <2x,

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00:08:38,000 --> 00:08:41,000
2y>.
What does it look like?

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00:08:41,000 --> 00:08:42,000
Well, it's going in that
direction.

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00:08:42,000 --> 00:08:49,000
It's parallel to the position
vector for this point.

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00:08:49,000 --> 00:08:51,000
It's actually twice the
position vector.

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00:08:51,000 --> 00:08:55,000
So, I guess it goes more or
less like this.

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00:08:55,000 --> 00:09:01,000
What's interesting,
too, is it is perpendicular to

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00:09:01,000 --> 00:09:04,000
this circle.
OK, so it's a general feature.

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00:09:04,000 --> 00:09:10,000
Actually, let me show you more
examples, oops,

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00:09:10,000 --> 00:09:16,000
not the one I want.
So, I don't know if you can see

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00:09:16,000 --> 00:09:19,000
it so well.
Well, hopefully you can.

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00:09:19,000 --> 00:09:22,000
So, here I have a contour plot
of a function,

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00:09:22,000 --> 00:09:25,000
and I have a blue vector.
That's the gradient vector at

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00:09:25,000 --> 00:09:28,000
the pink point on the plot.
So, you can see,

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00:09:28,000 --> 00:09:32,000
I can move the pink point,
and the gradient vector,

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00:09:32,000 --> 00:09:37,000
of course, changes because the
gradient depends on x and y.

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00:09:37,000 --> 00:09:42,000
But, what doesn't change is
that it's always perpendicular

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00:09:42,000 --> 00:09:46,000
to the level curves.
Anywhere I am,

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00:09:46,000 --> 00:09:53,000
my gradient stays perpendicular
to the level curve.

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00:09:53,000 --> 00:09:57,000
OK, is that convincing?
Is that visible for people who

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can't see blue?
OK, so, OK, so we have a lot of

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00:10:05,000 --> 00:10:16,000
evidence, but let's try to prove
the theorem because it will be

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00:10:16,000 --> 00:10:22,000
interesting.
So, first of all,

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00:10:22,000 --> 00:10:30,000
sorry, any questions about the
statement, the example,

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00:10:30,000 --> 00:10:34,000
anything, yes?
Ah, very good question.

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00:10:34,000 --> 00:10:37,000
Does the gradient vector,
why is the gradient vector

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00:10:37,000 --> 00:10:40,000
perpendicular in one direction
rather than the other?

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00:10:40,000 --> 00:10:43,000
So, we'll see the answer to
that in a few minutes.

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00:10:43,000 --> 00:10:46,000
But let me just tell you
immediately, to the side,

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00:10:46,000 --> 00:10:50,000
which side it's pointing to,
it's always pointing towards

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00:10:50,000 --> 00:10:54,000
higher values of a function.
OK, and we'll see in that maybe

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00:10:54,000 --> 00:11:03,000
about half an hour.
So, well, let me say actually

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00:11:03,000 --> 00:11:13,000
points towards higher values of
w.

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00:11:13,000 --> 00:11:24,000
OK, any other questions?
I don't see any questions.

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00:11:24,000 --> 00:11:28,000
OK, so let's try to prove this
theorem, at least this part of

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00:11:28,000 --> 00:11:30,000
the theorem.
We're not going to prove that

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00:11:30,000 --> 00:11:38,000
just yet.
That will come in a while.

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00:11:38,000 --> 00:11:44,000
So, well, maybe we want to
understand first what happens if

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00:11:44,000 --> 00:11:48,000
we move inside the level curve,
OK?

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00:11:48,000 --> 00:11:52,000
So, let's imagine that we are
taking a moving point that stays

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00:11:52,000 --> 00:11:55,000
on the level curve or on the
level surface.

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00:11:55,000 --> 00:12:00,000
And then, we know,
well, what happens is that the

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00:12:00,000 --> 00:12:03,000
function stays constant.
But, we can also know how

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00:12:03,000 --> 00:12:07,000
quickly the function changes
using the chain rule up there.

144
00:12:07,000 --> 00:12:11,000
So, maybe the chain rule will
actually be the key to

145
00:12:11,000 --> 00:12:15,000
understanding how the gradient
vector and the motion on the

146
00:12:15,000 --> 00:12:23,000
level service relate.
So, let's take a curve,

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00:12:23,000 --> 00:12:31,000
r equals r of t,
that stays inside,

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00:12:31,000 --> 00:12:42,000
well, maybe I should say on the
level surface,

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00:12:42,000 --> 00:12:48,000
w equals c.
So, let's think about what that

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00:12:48,000 --> 00:12:51,000
means.
So, just to get you used to

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00:12:51,000 --> 00:12:55,000
this idea, I'm going to draw a
level surface of a function of

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00:12:55,000 --> 00:12:59,000
three variables.
OK, so it's a surface given by

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00:12:59,000 --> 00:13:03,000
the equation w of x,
y, z equals some constant,

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00:13:03,000 --> 00:13:07,000
c.
And, so now I'm going to have a

155
00:13:07,000 --> 00:13:11,000
point on that,
and it's going to move on that

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00:13:11,000 --> 00:13:15,000
surface.
So, I will have some parametric

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00:13:15,000 --> 00:13:19,000
curve that lives on this
surface.

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00:13:19,000 --> 00:13:25,000
So, the question is,
what's going to happen at any

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00:13:25,000 --> 00:13:29,000
given time?
Well, the first observation is

160
00:13:29,000 --> 00:13:32,000
that the velocity vector,
what can I say about the

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00:13:32,000 --> 00:13:37,000
velocity vector of this motion?
It's going to be tangent to the

162
00:13:37,000 --> 00:13:39,000
level surface,
right?

163
00:13:39,000 --> 00:13:42,000
If I move on a surface,
then at any point,

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00:13:42,000 --> 00:13:45,000
my velocity is tangent to the
curve.

165
00:13:45,000 --> 00:13:49,000
But, if it's tangent to the
curve, then it's also tangent to

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00:13:49,000 --> 00:13:53,000
the surface because the curve is
inside the surface.

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00:13:53,000 --> 00:13:56,000
So, OK, it's getting a bit
cluttered.

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00:13:56,000 --> 00:13:58,000
Maybe I should draw a bigger
picture.

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00:13:58,000 --> 00:14:06,000
Let me do that right away here.
So, I have my level surface,

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00:14:06,000 --> 00:14:11,000
w equals c.
I have a curve on that,

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00:14:11,000 --> 00:14:19,000
and at some point,
I'm going to have a certain

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00:14:19,000 --> 00:14:28,000
velocity.
So, the claim is that the

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00:14:28,000 --> 00:14:40,000
velocity, v,
equals dr/dt is tangent -- --

174
00:14:40,000 --> 00:14:48,000
to the level,
w equals c because it's tangent

175
00:14:48,000 --> 00:14:50,000
to the curve,
and the curve is inside the

176
00:14:50,000 --> 00:14:52,000
level,
OK?

177
00:14:52,000 --> 00:14:55,000
Now, what else can we say?
Well, we have,

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00:14:55,000 --> 00:15:03,000
the chain rule will tell us how
the value of w changes.

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00:15:03,000 --> 00:15:12,000
So, by the chain rule,
we have dw/dt.

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00:15:12,000 --> 00:15:20,000
So, the rate of change of the
value of w as I move along this

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00:15:20,000 --> 00:15:28,000
curve is given by the dot
product between the gradient and

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the velocity vector.
And, so, well,

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00:15:34,000 --> 00:15:43,000
maybe I can rewrite it as w dot
v, and that should be,

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00:15:43,000 --> 00:15:50,000
well, what should it be?
What happens to the value of w

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00:15:50,000 --> 00:15:54,000
as t changes?
Well, it stays constant because

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00:15:54,000 --> 00:15:58,000
we are moving on a curve.
That curve might be

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00:15:58,000 --> 00:16:02,000
complicated, but it stays always
on the level,

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00:16:02,000 --> 00:16:08,000
w equals c.
So, it's zero because w of t

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00:16:08,000 --> 00:16:18,000
equals c, which is a constant.
OK, is that convincing?

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00:16:18,000 --> 00:16:21,000
OK, so now if we have a dot
product that's zero,

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00:16:21,000 --> 00:16:25,000
that tells us that these two
guys are perpendicular.

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00:16:25,000 --> 00:16:37,000
So -- So if the gradient vector
is perpendicular to v,

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00:16:37,000 --> 00:16:44,000
OK, that's a good start.
We know that the gradient is

194
00:16:44,000 --> 00:16:48,000
perpendicular to this vector
tangent that's tangent to the

195
00:16:48,000 --> 00:16:51,000
level surface.
What about other vectors

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00:16:51,000 --> 00:16:55,000
tangent to the level surface?
Well, in fact,

197
00:16:55,000 --> 00:17:00,000
I could use any curve drawn on
the level of w equals c.

198
00:17:00,000 --> 00:17:03,000
So, I could move,
really, any way I wanted on

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00:17:03,000 --> 00:17:06,000
that surface.
In particular,

200
00:17:06,000 --> 00:17:11,000
I claim that I could have
chosen my velocity vector to be

201
00:17:11,000 --> 00:17:15,000
any vector tangent to the
surface.

202
00:17:15,000 --> 00:17:22,000
OK, so let's write this.
So this is true for any curve,

203
00:17:22,000 --> 00:17:30,000
or, I'll say for any motion on
the level surface,

204
00:17:30,000 --> 00:17:40,000
w equals c.
So that means v can be any

205
00:17:40,000 --> 00:17:53,000
vector tangent to the surface
tangent to the level.

206
00:17:53,000 --> 00:18:01,000
See, for example,
OK, let me draw one more

207
00:18:01,000 --> 00:18:06,000
picture.
OK, so I have my level surface.

208
00:18:06,000 --> 00:18:09,000
So, I'm drawing more and more
levels, and they never quite

209
00:18:09,000 --> 00:18:12,000
look the same.
But I have a point.

210
00:18:12,000 --> 00:18:16,000
And, at this point,
I have the tangent plane to the

211
00:18:16,000 --> 00:18:24,000
level surface.
OK, so this is tangent plane to

212
00:18:24,000 --> 00:18:30,000
the level.
Then, if I choose any vector in

213
00:18:30,000 --> 00:18:35,000
that tangent plane.
Let's say I choose the one that

214
00:18:35,000 --> 00:18:39,000
goes in that direction.
Then, I can actually find a

215
00:18:39,000 --> 00:18:42,000
curve that goes in that
direction, and stays on the

216
00:18:42,000 --> 00:18:45,000
level.
So, here, that would be a curve

217
00:18:45,000 --> 00:18:50,000
that somehow goes from the right
to the left, and of course it

218
00:18:50,000 --> 00:18:53,000
has to end up going up or
something like that.

219
00:18:53,000 --> 00:19:05,000
OK, so given any vector tangent
-- -- let's call that vector v

220
00:19:05,000 --> 00:19:14,000
tangent to the level,
we get that the gradient is

221
00:19:14,000 --> 00:19:20,000
perpendicular to v.
So, if the gradient is

222
00:19:20,000 --> 00:19:24,000
perpendicular to this vector
tangent to this curve,

223
00:19:24,000 --> 00:19:28,000
but also to any vector,
I can draw that tangent to my

224
00:19:28,000 --> 00:19:29,000
surface.
So, what does that mean?

225
00:19:29,000 --> 00:19:34,000
Well, that means the gradient
is actually perpendicular to the

226
00:19:34,000 --> 00:19:38,000
tangent plane or to the surface
at this point.

227
00:19:38,000 --> 00:19:43,000
So, the gradient is
perpendicular.

228
00:20:02,000 --> 00:20:04,000
And, well, here,
I've illustrated things with a

229
00:20:04,000 --> 00:20:06,000
three-dimensional example,
but really it works the same if

230
00:20:06,000 --> 00:20:10,000
you have only two variables.
Then you have a level curve

231
00:20:10,000 --> 00:20:13,000
that has a tangent line,
and the gradient is

232
00:20:13,000 --> 00:20:23,000
perpendicular to that line.
OK, any questions?

233
00:20:23,000 --> 00:20:36,000
No?
OK, so, let's see.

234
00:20:36,000 --> 00:20:39,000
That's actually pretty neat
because there is a nice

235
00:20:39,000 --> 00:20:43,000
application of this,
which is to try to figure out,

236
00:20:43,000 --> 00:20:44,000
now we know,
actually, how to find the

237
00:20:44,000 --> 00:20:46,000
tangent plane to anything,
pretty much.

238
00:21:13,000 --> 00:21:19,000
OK, so let's see.
So, let's say that,

239
00:21:19,000 --> 00:21:27,000
for example,
I want to find -- -- the

240
00:21:27,000 --> 00:21:42,000
tangent plane -- -- to the
surface with equation,

241
00:21:42,000 --> 00:21:50,000
let's say, x^2 y^2-z^2 = 4 at
the point (2,1,

242
00:21:50,000 --> 00:22:01,000
1).
Let me write that.

243
00:22:01,000 --> 00:22:06,000
So, how do we do that?
Well, one way that we already

244
00:22:06,000 --> 00:22:09,000
know,
if we solve this for z,

245
00:22:09,000 --> 00:22:12,000
so we can write z equals a
function of x and y,

246
00:22:12,000 --> 00:22:16,000
then we know tangent plane
approximation for the graph of a

247
00:22:16,000 --> 00:22:19,000
function,
z equals some function of x and

248
00:22:19,000 --> 00:22:21,000
y.
But, that doesn't look like

249
00:22:21,000 --> 00:22:24,000
it's the best way to do it.
OK, the best way to it,

250
00:22:24,000 --> 00:22:27,000
now that we have the gradient
vector, is actually to directly

251
00:22:27,000 --> 00:22:30,000
say, oh, we know the normal
vector to this plane.

252
00:22:30,000 --> 00:22:35,000
The normal vector will just be
the gradient.

253
00:22:35,000 --> 00:22:40,000
Oh, I think I have a cool
picture to show.

254
00:22:40,000 --> 00:22:42,000
OK, so that's what it looks
like.

255
00:22:42,000 --> 00:22:49,000
OK, so here you have the
surface x2 y2-z2 equals four.

256
00:22:49,000 --> 00:22:52,000
That's called a hyperboloid
because it looks like when you

257
00:22:52,000 --> 00:22:55,000
get when you spin a hyperbola
around an axis.

258
00:22:55,000 --> 00:23:00,000
And, here's a tangent plane at
the given point.

259
00:23:00,000 --> 00:23:02,000
So, it doesn't look very
tangent because it crosses the

260
00:23:02,000 --> 00:23:04,000
surface.
But, it's really,

261
00:23:04,000 --> 00:23:08,000
if you think about it,
you will see it's really the

262
00:23:08,000 --> 00:23:12,000
plane that's approximating the
surface in the best way that you

263
00:23:12,000 --> 00:23:18,000
can at this given point.
It is really the tangent plane.

264
00:23:18,000 --> 00:23:27,000
So, how do we find this plane?
Well, you can plot it on a

265
00:23:27,000 --> 00:23:30,000
computer.
That's not exactly how you

266
00:23:30,000 --> 00:23:33,000
would look for it in the first
place.

267
00:23:33,000 --> 00:23:38,000
So, the way to do it is that we
compute the gradient.

268
00:23:38,000 --> 00:23:43,000
So, a gradient of what?
Well, a gradient of this

269
00:23:43,000 --> 00:23:49,000
function.
OK, so I should say,

270
00:23:49,000 --> 00:23:56,000
this is the level set,
w equals four,

271
00:23:56,000 --> 00:24:07,000
where w equals x^2 y^2 - z^2.
And so, we know that the

272
00:24:07,000 --> 00:24:14,000
gradient of this,
well, what is it?

273
00:24:14,000 --> 00:24:22,000
2x, then 2y,
and then negative 2z.

274
00:24:22,000 --> 00:24:27,000
So, at this given point,
I guess we are at x equals two.

275
00:24:27,000 --> 00:24:29,000
So, that's four.
And then, y and z are one.

276
00:24:29,000 --> 00:24:37,000
So, two, negative two.
OK, and that's going to be the

277
00:24:37,000 --> 00:24:44,000
normal vector to the surface or
to the tangent plane.

278
00:24:44,000 --> 00:24:47,000
That's one way to define the
tangent plane.

279
00:24:47,000 --> 00:24:50,000
All right, it has the same
normal vector as the surface.

280
00:24:50,000 --> 00:24:52,000
That's one way to define the
normal vector to the surface,

281
00:24:52,000 --> 00:24:56,000
if you prefer.
Being perpendicular to the

282
00:24:56,000 --> 00:25:01,000
surface means that you are
perpendicular to its tangent

283
00:25:01,000 --> 00:25:05,000
plane.
OK, so the equation is,

284
00:25:05,000 --> 00:25:12,000
well, 4x 2y-2z equals
something, where something is,

285
00:25:12,000 --> 00:25:19,000
well, we should just plug in
that point.

286
00:25:19,000 --> 00:25:26,000
We'll get eight plus two minus
two looks like we'll get eight.

287
00:25:26,000 --> 00:25:29,000
And, of course,
we could simplify dividing

288
00:25:29,000 --> 00:25:32,000
everything by two,
but it's not very important

289
00:25:32,000 --> 00:25:34,000
here.
OK, so now if you have a

290
00:25:34,000 --> 00:25:36,000
surface given by an evil
equation,

291
00:25:36,000 --> 00:25:40,000
and a point on the surface,
well, you know how to find the

292
00:25:40,000 --> 00:25:44,000
tangent plane to the surface at
that point.

293
00:25:44,000 --> 00:25:52,000
OK, any questions?
No.

294
00:25:52,000 --> 00:26:00,000
OK, let me give just another
reason why, another way that we

295
00:26:00,000 --> 00:26:04,000
could have seen this.
So, I claim,

296
00:26:04,000 --> 00:26:07,000
in fact, we could have done
this without the gradient,

297
00:26:07,000 --> 00:26:09,000
or using the gradient in a
somehow disguised way.

298
00:26:09,000 --> 00:26:18,000
So, here's another way.
So, the other way to do it

299
00:26:18,000 --> 00:26:22,000
would be to start with a
differential,

300
00:26:22,000 --> 00:26:26,000
OK?
dw, while it's pretty much the

301
00:26:26,000 --> 00:26:31,000
same content,
but let me write it as a

302
00:26:31,000 --> 00:26:35,000
differential,
dw is 2xdx 2ydy-2zdz.

303
00:26:35,000 --> 00:26:44,000
So, at a given point,
at (2,1, 1),

304
00:26:44,000 --> 00:26:52,000
this is 4dx 2dy-2dz.
Now, if we want to change this

305
00:26:52,000 --> 00:26:56,000
into an approximation formula,
we can.

306
00:26:56,000 --> 00:27:07,000
We know that the change in w is
approximately equal to 4 delta x

307
00:27:07,000 --> 00:27:15,000
2 delta y - 2 delta z.
OK, so when do we stay on the

308
00:27:15,000 --> 00:27:19,000
level surface?
Well, we stay on the level

309
00:27:19,000 --> 00:27:24,000
surface when w doesn't change,
so, when this becomes zero,

310
00:27:24,000 --> 00:27:25,000
OK?
Now, what does this

311
00:27:25,000 --> 00:27:28,000
approximation sign mean?
Well, it means for small

312
00:27:28,000 --> 00:27:31,000
changes in x,
y, z, this guy will be close to

313
00:27:31,000 --> 00:27:33,000
that guy.
It also means something else.

314
00:27:33,000 --> 00:27:36,000
Remember, these approximation
formulas, they are linear

315
00:27:36,000 --> 00:27:39,000
approximations.
They mean that we replace the

316
00:27:39,000 --> 00:27:43,000
function, actually,
by some closest linear formula

317
00:27:43,000 --> 00:27:47,000
that will be nearby.
And so, in particular,

318
00:27:47,000 --> 00:27:52,000
if we set this equal to zero
instead of approximately zero,

319
00:27:52,000 --> 00:27:56,000
it means we'll actually be
moving on the tangent plane to

320
00:27:56,000 --> 00:27:59,000
the level set.
If you want strict equalities

321
00:27:59,000 --> 00:28:03,000
in approximations means that we
replace the function by its

322
00:28:03,000 --> 00:28:04,000
tangent approximation.

323
00:28:37,000 --> 00:28:44,000
So -- [APPLAUSE] OK,
so the level corresponds to

324
00:28:44,000 --> 00:28:53,000
delta w equals zero,
and its tangent plane

325
00:28:53,000 --> 00:29:03,000
corresponds to four delta x plus
two delta y minus two delta z

326
00:29:03,000 --> 00:29:08,000
equals zero.
That's what I'm trying to say,

327
00:29:08,000 --> 00:29:10,000
basically.
And, what's delta x?

328
00:29:10,000 --> 00:29:12,000
Well, that means it's the
change in x.

329
00:29:12,000 --> 00:29:15,000
So, what's the change in x here?
That means, well,

330
00:29:15,000 --> 00:29:19,000
we started with x equals two,
and we moved to some other

331
00:29:19,000 --> 00:29:25,000
value, x.
So, that's actually x- 2, right?

332
00:29:25,000 --> 00:29:28,000
That's how much x has changed
compared to 2.

333
00:29:28,000 --> 00:29:37,000
And, two times (y - 1) minus
two times z - 1 = 0.

334
00:29:37,000 --> 00:29:42,000
That's the equation of a
tangent plane.

335
00:29:42,000 --> 00:29:46,000
It's the same equation as the
one over there.

336
00:29:46,000 --> 00:29:48,000
These are just two different
methods to get it.

337
00:29:48,000 --> 00:29:52,000
OK, so this one explains to you
what's going on in terms of

338
00:29:52,000 --> 00:29:57,000
approximation formulas.
This one goes right away,

339
00:29:57,000 --> 00:30:02,000
by using the gradient factor.
So, in a way,

340
00:30:02,000 --> 00:30:06,000
with this one,
you don't have to think nearly

341
00:30:06,000 --> 00:30:11,000
as much.
But, you can use either one.

342
00:30:11,000 --> 00:30:17,000
OK, questions?
No?

343
00:30:17,000 --> 00:30:23,000
OK, so let's move on to new
topic, which is another

344
00:30:23,000 --> 00:30:30,000
application of a gradient
vector, and that is directional

345
00:30:30,000 --> 00:30:32,000
derivatives.

346
00:30:44,000 --> 00:30:52,000
OK, so let's say that we have a
function of two variables,

347
00:30:52,000 --> 00:30:56,000
x and y.
Well, we know how to compute

348
00:30:56,000 --> 00:31:02,000
partial w over partial x or
partial w over partial y,

349
00:31:02,000 --> 00:31:07,000
which measure how w changes if
I move in the direction of the x

350
00:31:07,000 --> 00:31:10,000
axis or in the direction of the
y axis.

351
00:31:10,000 --> 00:31:13,000
So, what about moving in other
directions?

352
00:31:13,000 --> 00:31:16,000
Well, of course,
we've seen other approximation

353
00:31:16,000 --> 00:31:18,000
formulas and so on.
But, we can still ask,

354
00:31:18,000 --> 00:31:21,000
is there a derivative in every
direction?

355
00:31:21,000 --> 00:31:25,000
And that's basically,
yes, that's the directional

356
00:31:25,000 --> 00:31:30,000
derivative.
OK, so these are derivatives in

357
00:31:30,000 --> 00:31:40,000
the direction of I hat or j hat,
the vectors that go along the x

358
00:31:40,000 --> 00:31:50,000
or the y axis.
So, what if we move in another

359
00:31:50,000 --> 00:32:01,000
direction, let's say,
the direction of some unit

360
00:32:01,000 --> 00:32:09,000
vector, let's call it u .
OK, so if I give you a unit

361
00:32:09,000 --> 00:32:13,000
vector, you can ask yourself,
if I move in the direction,

362
00:32:13,000 --> 00:32:16,000
how quickly will my function
change?

363
00:32:16,000 --> 00:32:29,000
So -- So, let's look at the
straight trajectory.

364
00:32:29,000 --> 00:32:34,000
What this should mean is I
start at some value,

365
00:32:34,000 --> 00:32:37,000
x, y, and there I have my
vector u.

366
00:32:37,000 --> 00:32:41,000
And, I'm going to move in a
straight line in the direction

367
00:32:41,000 --> 00:32:46,000
of u.
And, I have the graph of my

368
00:32:46,000 --> 00:32:54,000
function -- -- and I'm asking
myself how quickly does the

369
00:32:54,000 --> 00:33:02,000
value change when I move on the
graph in that direction?

370
00:33:02,000 --> 00:33:10,000
OK, so let's look at a straight
line trajectory So,

371
00:33:10,000 --> 00:33:18,000
we have a position vector,
r, that will depend on some

372
00:33:18,000 --> 00:33:26,000
parameter which I will call s.
You'll see why very soon,

373
00:33:26,000 --> 00:33:30,000
in such a way that the
derivative is this given unit

374
00:33:30,000 --> 00:33:33,000
vector u hat.
So, why do I use s for my

375
00:33:33,000 --> 00:33:36,000
parameter rather than t.
Well, it's a convention.

376
00:33:36,000 --> 00:33:41,000
I'm moving at unit speed along
this line.

377
00:33:41,000 --> 00:33:45,000
So that means that actually,
I'm parameterizing things by

378
00:33:45,000 --> 00:33:48,000
the distance that I've traveled
along a curve,

379
00:33:48,000 --> 00:33:54,000
sorry, along this line.
So, here it's called s in the

380
00:33:54,000 --> 00:33:59,000
sense of arc length.
Actually, it's not really an

381
00:33:59,000 --> 00:34:06,000
arc because it's a straight
line, so it's the distance along

382
00:34:06,000 --> 00:34:09,000
the line.
OK, so because we are

383
00:34:09,000 --> 00:34:15,000
parameterizing by distance,
we are just using s as a

384
00:34:15,000 --> 00:34:21,000
convention just to distinguish
it from other situations.

385
00:34:21,000 --> 00:34:27,000
And, so, now,
the question will be,

386
00:34:27,000 --> 00:34:32,000
what is dw/ds?
What's the rate of change of w

387
00:34:32,000 --> 00:34:36,000
when I move like that?
Well, of course we know the

388
00:34:36,000 --> 00:34:40,000
answer because that's a special
case of the chain rule.

389
00:34:40,000 --> 00:34:44,000
So, that's how we will actually
compute it.

390
00:34:44,000 --> 00:34:49,000
But, in terms of what it means,
it really means we are asking

391
00:34:49,000 --> 00:34:51,000
ourselves,
we start at a point and we

392
00:34:51,000 --> 00:34:54,000
change the variables in a
certain direction,

393
00:34:54,000 --> 00:34:57,000
which is not necessarily the x
or the y direction,

394
00:34:57,000 --> 00:35:01,000
but really any direction.
And then, what's the derivative

395
00:35:01,000 --> 00:35:07,000
in that direction?
OK, does that make sense as a

396
00:35:07,000 --> 00:35:08,000
concept?
Kind of?

397
00:35:08,000 --> 00:35:10,000
I see some faces that are not
completely convinced.

398
00:35:10,000 --> 00:35:14,000
So, maybe you should show more
pictures.

399
00:35:14,000 --> 00:35:21,000
Well, let me first write down a
bit more and show you something.

400
00:35:40,000 --> 00:35:45,000
So I just want to give you the
actual definition.

401
00:35:45,000 --> 00:35:50,000
Sorry, first of all in case you
wonder what this is all about,

402
00:35:50,000 --> 00:35:55,000
so let's say the components of
our unit vector are two numbers,

403
00:35:55,000 --> 00:36:00,000
a and b.
Then, it means we'll move along

404
00:36:00,000 --> 00:36:05,000
the line x of s equals some
initial value,

405
00:36:05,000 --> 00:36:09,000
the point where we are actually
at the directional derivative

406
00:36:09,000 --> 00:36:13,000
plus s times a,
or I meant to say plus a times

407
00:36:13,000 --> 00:36:19,000
s.
And, y of s equals y0 bs.

408
00:36:19,000 --> 00:36:38,000
And then, we plug that into w.
And then we take the derivative.

409
00:36:38,000 --> 00:36:45,000
So, we have a notation for that
which is going to be dw/ds with

410
00:36:45,000 --> 00:36:53,000
a subscript in the direction of
u to indicate in which direction

411
00:36:53,000 --> 00:37:03,000
we are actually going to move.
And, that's called the

412
00:37:03,000 --> 00:37:17,000
directional derivative -- -- in
the direction of u.

413
00:37:17,000 --> 00:37:28,000
OK, so, let's see what it means
geometrically.

414
00:37:28,000 --> 00:37:33,000
So, remember,
we've seen things about partial

415
00:37:33,000 --> 00:37:36,000
derivatives,
and we see that the partial

416
00:37:36,000 --> 00:37:41,000
derivatives are the slopes of
slices of the graph by vertical

417
00:37:41,000 --> 00:37:45,000
planes that are parallel to the
x or the y directions.

418
00:37:45,000 --> 00:37:48,000
OK, so, if I have a point,
at any point,

419
00:37:48,000 --> 00:37:52,000
I can slice the graph of my
function by two planes,

420
00:37:52,000 --> 00:37:57,000
one that's going along the x,
one along the y direction.

421
00:37:57,000 --> 00:38:02,000
And then, I can look at the
slices of the graph.

422
00:38:02,000 --> 00:38:04,000
Let me see if I can use that
thing.

423
00:38:04,000 --> 00:38:07,000
So, we can look at the slices
of the graph that are drawn

424
00:38:07,000 --> 00:38:10,000
here.
In fact, we look at the tangent

425
00:38:10,000 --> 00:38:14,000
lines to the slices,
and we look at the slope and

426
00:38:14,000 --> 00:38:17,000
that gives us the partial
derivatives in case you are on

427
00:38:17,000 --> 00:38:21,000
that side and want to see also
the pointer that was here.

428
00:38:21,000 --> 00:38:26,000
So, now, similarly,
the directional derivative

429
00:38:26,000 --> 00:38:31,000
means, actually,
we'll be slicing our graph by

430
00:38:31,000 --> 00:38:37,000
the vertical plane.
It's not really colorful,

431
00:38:37,000 --> 00:38:43,000
something more colorful.
We'll be slicing things by a

432
00:38:43,000 --> 00:38:46,000
plane that is now in the
direction of this vector,

433
00:38:46,000 --> 00:38:51,000
u, and we'll be looking at the
slope of the slice of the graph.

434
00:38:51,000 --> 00:38:57,000
So, what that looks like here,
so that's the same applet the

435
00:38:57,000 --> 00:39:03,000
way that you've used on your
problem set in case you are

436
00:39:03,000 --> 00:39:08,000
wondering.
So, now, I'm picking a point on

437
00:39:08,000 --> 00:39:12,000
the contour plot.
And, at that point,

438
00:39:12,000 --> 00:39:15,000
I slice the graph.
So, here I'm starting by

439
00:39:15,000 --> 00:39:17,000
slicing in the direction of the
x axis.

440
00:39:17,000 --> 00:39:20,000
So, in fact,
what I'm measuring here by the

441
00:39:20,000 --> 00:39:24,000
slope of the slice is the
partial in the x direction.

442
00:39:24,000 --> 00:39:28,000
It's really partial f partial
x, which is also the directional

443
00:39:28,000 --> 00:39:31,000
derivative in the direction of
i.

444
00:39:31,000 --> 00:39:37,000
And now, if I rotate the slice,
then I have all of these

445
00:39:37,000 --> 00:39:40,000
planes.
So, you see at the bottom left,

446
00:39:40,000 --> 00:39:42,000
I have the direction in which
I'm going.

447
00:39:42,000 --> 00:39:44,000
There's this,
like, rotating line that tells

448
00:39:44,000 --> 00:39:47,000
you in which direction I'm going
to be moving.

449
00:39:47,000 --> 00:39:49,000
And for each direction,
I have a plane.

450
00:39:49,000 --> 00:39:52,000
And, when I slice by that
plane, I will get,

451
00:39:52,000 --> 00:39:56,000
so I have this direction here
going maybe to the southwest.

452
00:39:56,000 --> 00:40:00,000
So, that gives me a slice of my
graph by a vertical plane,

453
00:40:00,000 --> 00:40:03,000
and the slice has a certain
slope.

454
00:40:03,000 --> 00:40:08,000
And, the slope is going to be
the directional derivative in

455
00:40:08,000 --> 00:40:14,000
that direction.
OK, I think that's as graphic

456
00:40:14,000 --> 00:40:22,000
as I can get.
OK, any questions about that?

457
00:40:22,000 --> 00:40:33,000
No?
OK, so let's see how we compute

458
00:40:33,000 --> 00:40:41,000
that guy.
So, let me just write again

459
00:40:41,000 --> 00:40:49,000
just in case you want to,
in case you didn't hear me it's

460
00:40:49,000 --> 00:40:58,000
the slope of the slice of the
graph by a vertical plane -- --

461
00:40:58,000 --> 00:41:03,000
that contains the given
direction,

462
00:41:03,000 --> 00:41:06,000
that's parallel to the
direction, u.

463
00:41:06,000 --> 00:41:11,000
So, how do we compute it?
Well, we can use the chain rule.

464
00:41:11,000 --> 00:41:22,000
The chain rule implies that
dw/ds is actually the gradient

465
00:41:22,000 --> 00:41:31,000
of w dot product with the
velocity vector dr/ds.

466
00:41:31,000 --> 00:41:35,000
But, remember we say that we
are going to be moving at unit

467
00:41:35,000 --> 00:41:39,000
speed in the direction of u.
So, in fact,

468
00:41:39,000 --> 00:41:50,000
that's just gradient w dot
product with the unit vector u.

469
00:41:50,000 --> 00:41:57,000
OK, so the formula that we
remember is really dw/ds in the

470
00:41:57,000 --> 00:42:03,000
direction of u is gradient w dot
product of u.

471
00:42:03,000 --> 00:42:13,000
And, maybe I should also say in
words, this is the component of

472
00:42:13,000 --> 00:42:19,000
the gradient in the direction of
u.

473
00:42:19,000 --> 00:42:21,000
And, maybe that makes more
sense.

474
00:42:21,000 --> 00:42:25,000
So, for example,
the directional derivative in

475
00:42:25,000 --> 00:42:29,000
the direction of I hat is the
component along the x axes.

476
00:42:29,000 --> 00:42:32,000
That's the same as,
indeed, the partial derivatives

477
00:42:32,000 --> 00:42:40,000
in the x direction.
Things make sense.

478
00:42:40,000 --> 00:42:50,000
dw/ds in the direction of I hat
is, sorry, gradient w dot I hat,

479
00:42:50,000 --> 00:42:59,000
which is wx,maybe I should
write, partial w of partial x.

480
00:42:59,000 --> 00:43:09,000
OK, now, so that's basically
what we need to know to compute

481
00:43:09,000 --> 00:43:12,000
these guys.
So now, let's go back to the

482
00:43:12,000 --> 00:43:16,000
gradient and see what this tells
us about the gradient.

483
00:43:42,000 --> 00:43:51,000
[APPLAUSE]
I see you guys are having fun.

484
00:43:51,000 --> 00:43:54,000
OK, OK, let's do a little bit
of geometry here.

485
00:43:54,000 --> 00:44:00,000
That should calm you down.
So, we said dw/ds in the

486
00:44:00,000 --> 00:44:04,000
direction of u is gradient w dot
u.

487
00:44:04,000 --> 00:44:11,000
That's the same as the length
of gradient w times the length

488
00:44:11,000 --> 00:44:15,000
of u.
Well, that happens to be one

489
00:44:15,000 --> 00:44:23,000
because we are taking the unit
vector times the cosine of the

490
00:44:23,000 --> 00:44:30,000
angle between the gradient and
the given unit vector,

491
00:44:30,000 --> 00:44:36,000
u, so, have this angle, theta.
OK, that's another way of

492
00:44:36,000 --> 00:44:39,000
saying we are taking the
component of a gradient in the

493
00:44:39,000 --> 00:44:43,000
direction of u.
But now, what does that tell us?

494
00:44:43,000 --> 00:44:46,000
Well,
let's try to figure out in

495
00:44:46,000 --> 00:44:50,000
which directions w changes the
fastest,

496
00:44:50,000 --> 00:44:54,000
in which direction it increases
the most or decreases the most,

497
00:44:54,000 --> 00:45:03,000
or doesn't actually change.
So, when is this going to be

498
00:45:03,000 --> 00:45:05,000
the largest?
If I fix a point,

499
00:45:05,000 --> 00:45:09,000
if I set a point,
then the gradient vector at

500
00:45:09,000 --> 00:45:12,000
that point is given to me.
But, the question is,

501
00:45:12,000 --> 00:45:15,000
in which direction does it
change the most quickly?

502
00:45:15,000 --> 00:45:19,000
Well, what I can change is the
direction, and this will be the

503
00:45:19,000 --> 00:45:25,000
largest when the cosine is one.
So, this is largest when the

504
00:45:25,000 --> 00:45:33,000
cosine of the angle is one.
That means the angle is zero.

505
00:45:33,000 --> 00:45:40,000
That means u is actually in the
direction of the gradient.

506
00:45:40,000 --> 00:45:42,000
OK, so that's a new way to
think about the direction of a

507
00:45:42,000 --> 00:45:47,000
gradient.
The gradient is the direction

508
00:45:47,000 --> 00:45:57,000
in which the function increases
the most quickly at that point.

509
00:45:57,000 --> 00:46:08,000
So, the direction of gradient w
is the direction of fastest

510
00:46:08,000 --> 00:46:15,000
increase of w at the given
point.

511
00:46:15,000 --> 00:46:24,000
And, what is the magnitude of w?
Well, it's actually the

512
00:46:24,000 --> 00:46:33,000
directional derivative in that
direction.

513
00:46:33,000 --> 00:46:37,000
OK, so if I go in that
direction, which gives me the

514
00:46:37,000 --> 00:46:40,000
fastest increase,
then the corresponding slope

515
00:46:40,000 --> 00:46:44,000
will be the length of the
gradient.

516
00:46:44,000 --> 00:46:51,000
And, with the direction of the
fastest decrease?

517
00:46:51,000 --> 00:46:53,000
It's going in the opposite
direction, right?

518
00:46:53,000 --> 00:46:55,000
I mean, if you are on a
mountain, and you know that you

519
00:46:55,000 --> 00:46:57,000
are facing the mountain,
that's the direction of fastest

520
00:46:57,000 --> 00:46:59,000
increase.
The direction of fastest

521
00:46:59,000 --> 00:47:01,000
decrease is behind you straight
down.

522
00:47:01,000 --> 00:47:11,000
OK, so, the minimal value of
dw/ds is achieved when cosine of

523
00:47:11,000 --> 00:47:18,000
theta is minus one.
That means theta equals 180�.

524
00:47:18,000 --> 00:47:27,000
That means u is in the
direction of minus the gradient.

525
00:47:27,000 --> 00:47:30,000
It points opposite to the
gradient.

526
00:47:30,000 --> 00:47:43,000
And, finally,
when do we have dw/ds equals

527
00:47:43,000 --> 00:47:48,000
zero?
So, in which direction does the

528
00:47:48,000 --> 00:47:52,000
function not change?
Well, we have two answers to

529
00:47:52,000 --> 00:47:54,000
that.
One is to just use the formula.

530
00:47:54,000 --> 00:47:58,000
So, that's one cosine theta
equals zero.

531
00:47:58,000 --> 00:48:03,000
That means theta equals 90 degrees.
That means that u is

532
00:48:03,000 --> 00:48:08,000
perpendicular to the gradient.
The other way to think about

533
00:48:08,000 --> 00:48:11,000
it, the direction in which the
value doesn't change is a

534
00:48:11,000 --> 00:48:14,000
direction that's tangent to the
level surface.

535
00:48:14,000 --> 00:48:18,000
If we are not changing a,
it means we are moving along

536
00:48:18,000 --> 00:48:24,000
the level.
And, that's the same thing --

537
00:48:24,000 --> 00:48:30,000
-- as being tangent to the
level.

538
00:48:30,000 --> 00:48:36,000
So, let me just show that on
the picture here.

539
00:48:36,000 --> 00:48:39,000
So, if actually show you the
gradient, you can't really see

540
00:48:39,000 --> 00:48:41,000
it here.
I need to move it a bit.

541
00:48:41,000 --> 00:48:44,000
So, the gradient here is
pointing straight up at the

542
00:48:44,000 --> 00:48:50,000
point that I have chosen.
Now, if I choose a slice that's

543
00:48:50,000 --> 00:48:52,000
perpendicular,
and a direction that's

544
00:48:52,000 --> 00:48:55,000
perpendicular to the gradient,
so that's actually tangent to

545
00:48:55,000 --> 00:48:57,000
the level curve,
then you see that my slice is

546
00:48:57,000 --> 00:49:00,000
flat.
I don't actually have any slop.

547
00:49:00,000 --> 00:49:04,000
The directional derivative in a
direction that's perpendicular

548
00:49:04,000 --> 00:49:06,000
to the gradient is basically
zero.

549
00:49:06,000 --> 00:49:08,000
Now, if I rotate,
then the slope sort of

550
00:49:08,000 --> 00:49:11,000
increases, increases,
increases, and it becomes the

551
00:49:11,000 --> 00:49:14,000
largest when I'm going in the
direction of a gradient.

552
00:49:14,000 --> 00:49:17,000
So, here, I have,
actually, a pretty big slope.

553
00:49:17,000 --> 00:49:20,000
And now, if I keep rotating,
then the slope will decrease

554
00:49:20,000 --> 00:49:22,000
again.
Then it becomes zero when I

555
00:49:22,000 --> 00:49:25,000
perpendicular,
and then it becomes negative.

556
00:49:25,000 --> 00:49:29,000
It's the most negative when I
pointing away from the gradient

557
00:49:29,000 --> 00:49:33,000
and then becomes zero again when
I'm back perpendicular.

558
00:49:33,000 --> 00:49:38,000
OK, so for example,
if I give you a contour plot,

559
00:49:38,000 --> 00:49:41,000
and I ask you to draw the
direction of the gradient

560
00:49:41,000 --> 00:49:43,000
vector,
well, at this point,

561
00:49:43,000 --> 00:49:46,000
for example,
you would look at the picture.

562
00:49:46,000 --> 00:49:49,000
The gradient vector would be
going perpendicular to the

563
00:49:49,000 --> 00:49:52,000
level.
And, it would be going towards

564
00:49:52,000 --> 00:49:55,000
higher values of a function.
I don't know if you can see the

565
00:49:55,000 --> 00:49:57,000
labels, but the thing in the
middle is a minimum.

566
00:49:57,000 --> 00:50:03,000
So, it will actually be
pointing in this kind of

567
00:50:03,000 --> 00:50:08,000
direction.
OK, so that's it for today.