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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.