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Last time we saw things about
gradients and directional
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00:00:29,000 --> 00:00:32,000
derivatives.
Before that we studied how to
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00:00:32,000 --> 00:00:37,000
look for minima and maxima of
functions of several variables.
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00:00:37,000 --> 00:00:41,000
And today we are going to look
again at min/max problems but in
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00:00:41,000 --> 00:00:45,000
a different setting,
namely, one for variables that
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are not independent.
And so what we will see is you
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may have heard of Lagrange
multipliers.
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00:00:52,000 --> 00:00:59,000
And this is the one point in
the term when I can shine with
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00:00:59,000 --> 00:01:05,000
my French accent and say
Lagrange's name properly.
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00:01:05,000 --> 00:01:08,000
OK.
What are Lagrange multipliers
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00:01:08,000 --> 00:01:13,000
about?
Well, the goal is to minimize
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00:01:13,000 --> 00:01:19,000
or maximize a function of
several variables.
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00:01:19,000 --> 00:01:22,000
Let's say, for example,
f of x, y, z,
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but where these variables are
no longer independent.
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00:01:41,000 --> 00:01:43,000
They are not independent.
That means that there is a
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00:01:43,000 --> 00:01:47,000
relation between them.
The relation is maybe some
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00:01:47,000 --> 00:01:52,000
equation of the form g of x,
y, z equals some constant.
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00:01:52,000 --> 00:01:57,000
You take the relation between
x, y, z, you call that g and
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00:01:57,000 --> 00:02:02,000
that gives you the constraint.
And your goal is to minimize f
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00:02:02,000 --> 00:02:05,000
only of those values of x,
y, z that satisfy the
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00:02:05,000 --> 00:02:07,000
constraint.
What is one way to do that?
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00:02:07,000 --> 00:02:10,000
Well, one to do that,
if the constraint is very
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00:02:10,000 --> 00:02:14,000
simple, we can maybe solve for
one of the variables.
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00:02:14,000 --> 00:02:17,000
Maybe we can solve this
equation for one of the
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00:02:17,000 --> 00:02:21,000
variables, plug it back into f,
and then we have a usual
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00:02:21,000 --> 00:02:25,000
min/max problem that we have
seen how to do.
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00:02:25,000 --> 00:02:28,000
The problem is sometimes you
cannot actually solve for x,
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00:02:28,000 --> 00:02:31,000
y, z in here because this
condition is too complicated and
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00:02:31,000 --> 00:02:38,000
then we need a new method.
That is what we are going to do.
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00:02:38,000 --> 00:02:41,000
Why would we care about that?
Well, one example is actually
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00:02:41,000 --> 00:02:43,000
in physics.
Maybe you have seen in
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00:02:43,000 --> 00:02:47,000
thermodynamics that you study
quantities about gases,
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00:02:47,000 --> 00:02:50,000
and those quantities that
involve pressure,
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00:02:50,000 --> 00:02:53,000
volume and temperature.
And pressure,
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00:02:53,000 --> 00:02:56,000
volume and temperature are not
independent of each other.
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00:02:56,000 --> 00:02:59,000
I mean you know probably the
equation PV = NRT.
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00:02:59,000 --> 00:03:01,000
And, of course,
there you could actually solve
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00:03:01,000 --> 00:03:03,000
to express things in terms of
one or the other.
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00:03:03,000 --> 00:03:07,000
But sometimes it is more
convenient to keep all three
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00:03:07,000 --> 00:03:09,000
variables but treat them as
constrained.
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00:03:09,000 --> 00:03:19,000
It is just an example of a
situation where you might want
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00:03:19,000 --> 00:03:24,000
to do this.
Anyway, we will look mostly at
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particular examples,
but just to point out that this
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00:03:28,000 --> 00:03:32,000
is useful when you study guesses
in physics.
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The first observation is we
cannot use our usual method of
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00:03:35,000 --> 00:03:36,000
looking for critical points of
f.
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00:03:36,000 --> 00:03:40,000
Because critical points of f
typically will not satisfy this
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00:03:40,000 --> 00:03:43,000
condition and so won't be good
solutions.
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We need something else.
Let's look at an example,
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00:03:49,000 --> 00:03:53,000
and we will see how that leads
us to the method.
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00:03:53,000 --> 00:04:03,000
For example,
let's say that I want to find
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00:04:03,000 --> 00:04:17,000
the point closest to the origin
-- -- on the hyperbola xy equals
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3 in the plane.
That means I have this
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00:04:23,000 --> 00:04:26,000
hyperbola, and I am asking
myself what is the point on it
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that is the closest to the
origin?
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00:04:29,000 --> 00:04:31,000
I mean we can solve this by
elementary geometry,
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00:04:31,000 --> 00:04:34,000
we don't need actually Lagrange
multipliers,
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00:04:34,000 --> 00:04:38,000
but we are going to do it with
Lagrange multipliers because it
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00:04:38,000 --> 00:04:41,000
is a pretty good example.
What does it mean?
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00:04:41,000 --> 00:04:47,000
Well, it means that we want to
minimize distance to the origin.
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00:04:47,000 --> 00:04:49,000
What is the distance to the
origin?
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00:04:49,000 --> 00:04:53,000
If I have a point,
at coordinates (x,
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00:04:53,000 --> 00:04:58,000
y) and then the distance to the
origin is square root of x
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00:04:58,000 --> 00:05:02,000
squared plus y squared.
Well, do we really want to
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00:05:02,000 --> 00:05:05,000
minimize that or can we minimize
something easier?
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00:05:05,000 --> 00:05:06,000
Yeah.
Maybe we can minimize the
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00:05:06,000 --> 00:05:14,000
square of a distance.
Let's forget this guy and
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00:05:14,000 --> 00:05:23,000
instead -- Actually,
we will minimize f of x,
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00:05:23,000 --> 00:05:27,000
y equals x squared plus y
squared,
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00:05:27,000 --> 00:05:39,000
that looks better,
subject to the constraint xy =
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3.
And so we will call this thing
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00:05:44,000 --> 00:05:50,000
g of x, y to illustrate the
general method.
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00:05:50,000 --> 00:05:58,000
Let's look at a picture.
Here you can see in yellow the
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00:05:58,000 --> 00:06:02,000
hyperbola xy equals three.
And we are going to look for
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00:06:02,000 --> 00:06:05,000
the points that are the closest
to the origin.
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00:06:05,000 --> 00:06:08,000
What can we do?
Well, for example,
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00:06:08,000 --> 00:06:13,000
we can plot the function x
squared plus y squared,
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00:06:13,000 --> 00:06:17,000
function f.
That is the contour plot of f
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00:06:17,000 --> 00:06:21,000
with a hyperbola on top of it.
Now let's see what we can do
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00:06:21,000 --> 00:06:25,000
with that.
Well, let's ask ourselves,
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00:06:25,000 --> 00:06:30,000
for example,
if I look at points where f
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00:06:30,000 --> 00:06:34,000
equals 20 now.
I think I am at 20 but you
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00:06:34,000 --> 00:06:37,000
cannot really see it.
That is a circle with a point
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00:06:37,000 --> 00:06:41,000
whose distant square is 20.
Well, can I find a solution if
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00:06:41,000 --> 00:06:44,000
I am on the hyperbola?
Yes, there are four points at
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00:06:44,000 --> 00:06:46,000
this distance.
Can I do better?
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00:06:46,000 --> 00:06:49,000
Well, let's decrease for
distance.
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Yes, we can still find points
on the hyperbola and so on.
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00:06:52,000 --> 00:06:56,000
Except if we go too low then
there are no points on this
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00:06:56,000 --> 00:07:00,000
circle anymore in the hyperbola.
If we decrease the value of f
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00:07:00,000 --> 00:07:03,000
that we want to look at that
will somehow limit value beyond
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00:07:03,000 --> 00:07:07,000
which we cannot go,
and that is the minimum of f.
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00:07:07,000 --> 00:07:13,000
We are trying to look for the
smallest value of f that will
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00:07:13,000 --> 00:07:17,000
actually be realized on the
hyperbola.
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00:07:17,000 --> 00:07:20,000
When does that happen?
Well, I have to backtrack a
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00:07:20,000 --> 00:07:23,000
little bit.
It seems like the limiting case
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00:07:23,000 --> 00:07:26,000
is basically here.
It is when the circle is
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00:07:26,000 --> 00:07:31,000
tangent to the hyperbola.
That is the smallest circle
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00:07:31,000 --> 00:07:37,000
that will hit the hyperbola.
If I take a larger value of f,
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00:07:37,000 --> 00:07:39,000
I will have solutions.
If I take a smaller value of f,
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00:07:39,000 --> 00:07:41,000
I will not have any solutions
anymore.
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So, that is the situation that
we want to solve for.
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00:07:49,000 --> 00:07:54,000
How do we find that minimum?
Well, a key observation that is
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00:07:54,000 --> 00:07:58,000
valid on this picture,
and that actually remain true
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00:07:58,000 --> 00:08:03,000
in the completely general case,
is that when we have a minimum
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00:08:03,000 --> 00:08:09,000
the level curve of f is actually
tangent to our hyperbola.
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00:08:09,000 --> 00:08:15,000
It is tangent to the set of
points where x,
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00:08:15,000 --> 00:08:20,000
y equals three,
to the hyperbola.
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00:08:20,000 --> 00:08:32,000
Let's write that down.
We observe that at the minimum
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00:08:32,000 --> 00:08:49,000
the level curve of f is tangent
to the hyperbola.
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00:08:49,000 --> 00:08:53,000
Remember, the hyperbola is
given by the equal g equals
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00:08:53,000 --> 00:08:56,000
three, so it is a level curve of
g.
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00:08:56,000 --> 00:08:59,000
We have a level curve of f and
a level curve of g that are
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00:08:59,000 --> 00:09:03,000
tangent to each other.
And I claim that is going to be
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00:09:03,000 --> 00:09:07,000
the general situation that we
are interested in.
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00:09:07,000 --> 00:09:12,000
How do we try to solve for
points where this happens?
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00:09:28,000 --> 00:09:36,000
How do we find x,
y where the level curves of f
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00:09:36,000 --> 00:09:47,000
and g are tangent to each other?
Let's think for a second.
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00:09:47,000 --> 00:09:51,000
If the two level curves are
tangent to each other that means
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00:09:51,000 --> 00:09:57,000
they have the same tangent line.
That means that the normal
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00:09:57,000 --> 00:10:03,000
vectors should be parallel.
Let me maybe draw a picture
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00:10:03,000 --> 00:10:06,000
here.
This is the level curve maybe f
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00:10:06,000 --> 00:10:11,000
equals something.
And this is the level curve g
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00:10:11,000 --> 00:10:16,000
equals constant.
Here my constant is three.
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00:10:16,000 --> 00:10:20,000
Well, if I look for gradient
vectors, the gradient of f will
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00:10:20,000 --> 00:10:23,000
be perpendicular to the level
curve of f.
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00:10:23,000 --> 00:10:27,000
The gradient of g will be
perpendicular to the level curve
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00:10:27,000 --> 00:10:29,000
of g.
They don't have any reason to
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00:10:29,000 --> 00:10:32,000
be of the same size,
but they have to be parallel to
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00:10:32,000 --> 00:10:35,000
each other.
Of course, they could also be
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00:10:35,000 --> 00:10:38,000
parallel pointing in opposite
directions.
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00:10:38,000 --> 00:10:48,000
But the key point is that when
this happens the gradient of f
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00:10:48,000 --> 00:10:54,000
is parallel to the gradient of
g.
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00:10:54,000 --> 00:11:03,000
Well, let's check that.
Here is a point.
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00:11:03,000 --> 00:11:05,000
And I can plot the gradient of
f in blue.
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00:11:05,000 --> 00:11:08,000
The gradient of g in yellow.
And you see,
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00:11:08,000 --> 00:11:12,000
in most of these places,
somehow the two gradients are
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00:11:12,000 --> 00:11:14,000
not really parallel.
Actually, I should not be
145
00:11:14,000 --> 00:11:17,000
looking at random points.
I should be looking only on the
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00:11:17,000 --> 00:11:19,000
hyperbola.
I want points on the hyperbola
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00:11:19,000 --> 00:11:22,000
where the two gradients are
parallel.
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00:11:22,000 --> 00:11:28,000
Well, when does that happen?
Well, it looks like it will
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00:11:28,000 --> 00:11:31,000
happen here.
When I am at a minimum,
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00:11:31,000 --> 00:11:34,000
the two gradient vectors are
parallel.
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00:11:34,000 --> 00:11:37,000
It is not really proof.
It is an example that seems to
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00:11:37,000 --> 00:11:43,000
be convincing.
So far things work pretty well.
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00:11:43,000 --> 00:11:46,000
How do we decide if two vectors
are parallel?
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00:11:46,000 --> 00:11:50,000
Well, they are parallel when
they are proportional to each
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00:11:50,000 --> 00:11:54,000
other.
You can write one of them as a
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00:11:54,000 --> 00:12:02,000
constant times the other one,
and that constant usually one
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00:12:02,000 --> 00:12:07,000
uses the Greek letter lambda.
I don't know if you have seen
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00:12:07,000 --> 00:12:10,000
it before.
It is the Greek letter for L.
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00:12:10,000 --> 00:12:15,000
And probably,
I am sure, it is somebody's
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00:12:15,000 --> 00:12:22,000
idea of paying tribute to
Lagrange by putting an L in
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00:12:22,000 --> 00:12:25,000
there.
Lambda is just a constant.
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00:12:25,000 --> 00:12:31,000
And we are looking for a scalar
lambda and points x and y where
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00:12:31,000 --> 00:12:33,000
this holds.
In fact,
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00:12:33,000 --> 00:12:37,000
what we are doing is replacing
min/max problems in two
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00:12:37,000 --> 00:12:41,000
variables with a constraint
between them by a set of
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00:12:41,000 --> 00:12:47,000
equations involving,
you will see, three variables.
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00:12:47,000 --> 00:12:54,000
We had min/max with two
variables x, y,
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00:12:54,000 --> 00:13:00,000
but no independent.
We had a constraint g of x,
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00:13:00,000 --> 00:13:06,000
y equals constant.
And that becomes something new.
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00:13:06,000 --> 00:13:12,000
That becomes a system of
equations where we have to
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00:13:12,000 --> 00:13:19,000
solve, well, let's write down
what it means for gradient f to
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00:13:19,000 --> 00:13:26,000
be proportional to gradient g.
That means that f sub x should
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00:13:26,000 --> 00:13:32,000
be lambda times g sub x,
and f sub y should be lambda
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00:13:32,000 --> 00:13:36,000
times g sub y.
Because the gradient vectors
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00:13:36,000 --> 00:13:39,000
here are f sub x,
f sub y and g sub x,
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00:13:39,000 --> 00:13:43,000
g sub y.
If you have a third variable z
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00:13:43,000 --> 00:13:49,000
then you have also an equation f
sub z equals lambda g sub z.
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00:13:49,000 --> 00:13:53,000
Now, let's see.
How many unknowns do we have in
179
00:13:53,000 --> 00:13:55,000
these equations?
Well, there is x,
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00:13:55,000 --> 00:14:01,000
there is y and there is lambda.
We have three unknowns and have
181
00:14:01,000 --> 00:14:06,000
only two equations.
Something is missing.
182
00:14:06,000 --> 00:14:10,000
Well, I mean x and y are not
actually independent.
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00:14:10,000 --> 00:14:14,000
They are related by the
equation g of x,
184
00:14:14,000 --> 00:14:21,000
y equals c, so we need to add
the constraint g equals c.
185
00:14:21,000 --> 00:14:26,000
And now we have three equations
involving three variables.
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00:14:26,000 --> 00:14:39,000
Let's see how that works.
Here remember we have f equals
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00:14:39,000 --> 00:14:45,000
x squared y squared and g = xy.
What is f sub x?
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00:14:45,000 --> 00:14:52,000
It is going to be 2x equals
lambda times,
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00:14:52,000 --> 00:14:55,000
what is g sub x,
y.
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00:14:55,000 --> 00:14:59,000
Maybe I should write here f sub
x equals lambda g sub x just to
191
00:14:59,000 --> 00:15:03,000
remind you.
Then we have f sub y equals
192
00:15:03,000 --> 00:15:10,000
lambda g sub y.
F sub y is 2y equals lambda
193
00:15:10,000 --> 00:15:18,000
times g sub y is x.
And then our third equation g
194
00:15:18,000 --> 00:15:22,000
equals c becomes xy equals
three.
195
00:15:22,000 --> 00:15:26,000
So, that is what you would have
to solve.
196
00:15:26,000 --> 00:15:33,000
Any questions at this point?
No.
197
00:15:33,000 --> 00:15:44,000
Yes?
How do I know the direction of
198
00:15:44,000 --> 00:15:47,000
a gradient?
Do you mean how do I know that
199
00:15:47,000 --> 00:15:50,000
it is perpendicular to a level
curve?
200
00:15:50,000 --> 00:15:54,000
Oh, how do I know if it points
in that direction on the
201
00:15:54,000 --> 00:15:56,000
opposite one?
Well, that depends.
202
00:15:56,000 --> 00:15:59,000
I mean we'd seen in last time,
but the gradient is
203
00:15:59,000 --> 00:16:02,000
perpendicular to the level and
points towards higher values of
204
00:16:02,000 --> 00:16:05,000
a function.
So it could be -- Wait.
205
00:16:05,000 --> 00:16:08,000
What did I have?
It could be that my gradient
206
00:16:08,000 --> 00:16:11,000
vectors up there actually point
in opposite directions.
207
00:16:11,000 --> 00:16:15,000
It doesn't matter to me because
it will still look the same in
208
00:16:15,000 --> 00:16:18,000
terms of the equation,
just lambda will be positive or
209
00:16:18,000 --> 00:16:22,000
negative, depending on the case.
I can handle both situations.
210
00:16:22,000 --> 00:16:30,000
It's not a problem.
I can allow lambda to be
211
00:16:30,000 --> 00:16:34,000
positive or negative.
Well, in this example,
212
00:16:34,000 --> 00:16:35,000
it looks like lambda will be
positive.
213
00:16:35,000 --> 00:16:38,000
If you look at the picture on
the plot.
214
00:16:38,000 --> 00:16:48,000
Yes?
Well, because actually they are
215
00:16:48,000 --> 00:16:51,000
not equal to each other.
If you look at this point where
216
00:16:51,000 --> 00:16:55,000
the hyperbola and the circle
touch each other,
217
00:16:55,000 --> 00:16:58,000
first of all,
I don't know which circle I am
218
00:16:58,000 --> 00:17:01,000
going to look at.
I am trying to solve,
219
00:17:01,000 --> 00:17:04,000
actually, for the radius of the
circle.
220
00:17:04,000 --> 00:17:07,000
I am trying to find what the
minimum value of f is.
221
00:17:07,000 --> 00:17:10,000
And, second,
at that point,
222
00:17:10,000 --> 00:17:14,000
the value of f and the value of
g are not equal.
223
00:17:14,000 --> 00:17:17,000
g is equal to three because I
want the hyperbola x equals
224
00:17:17,000 --> 00:17:19,000
three.
The value of f will be the
225
00:17:19,000 --> 00:17:22,000
square of a distance,
whatever that is.
226
00:17:22,000 --> 00:17:27,000
I think it will end up being 6,
but we will see.
227
00:17:27,000 --> 00:17:29,000
So, you cannot really set them
equal because you don't know
228
00:17:29,000 --> 00:17:45,000
what f is equal to in advance.
Yes?
229
00:17:45,000 --> 00:17:49,000
Not quite.
Actually, here I am just using
230
00:17:49,000 --> 00:17:52,000
this idea of finding a point
closest to the origin to
231
00:17:52,000 --> 00:17:55,000
illustrate an example of a
min/max problem.
232
00:17:55,000 --> 00:17:59,000
The general problem we are
trying to solve is minimize f
233
00:17:59,000 --> 00:18:03,000
subject to g equals constant.
And what we are going to do for
234
00:18:03,000 --> 00:18:07,000
that is we are really going to
say instead let's look at places
235
00:18:07,000 --> 00:18:10,000
where gradient f and gradient g
are parallel to each other and
236
00:18:10,000 --> 00:18:14,000
solve for equations of that.
I think we completely lose the
237
00:18:14,000 --> 00:18:19,000
notion of closest point if we
just look at these equations.
238
00:18:19,000 --> 00:18:21,000
We don't really say anything
about closest points anymore.
239
00:18:21,000 --> 00:18:24,000
Of course, that is what they
mean in the end.
240
00:18:24,000 --> 00:18:28,000
But, in the general setting,
there is no closest point
241
00:18:28,000 --> 00:18:31,000
involved anymore.
OK.
242
00:18:31,000 --> 00:18:40,000
Yes?
Yes.
243
00:18:40,000 --> 00:18:43,000
It is always going to be the
case that,
244
00:18:43,000 --> 00:18:46,000
at the minimum,
or at the maximum of a function
245
00:18:46,000 --> 00:18:49,000
subject to a constraint,
the level curves of f and the
246
00:18:49,000 --> 00:18:52,000
level curves of g will be
tangent to each other.
247
00:18:52,000 --> 00:18:54,000
That is the basis for this
method.
248
00:18:54,000 --> 00:19:00,000
I am going to justify that soon.
It could be minimum or maximum.
249
00:19:00,000 --> 00:19:02,000
In three-dimensions it could
even be a saddle point.
250
00:19:02,000 --> 00:19:03,000
And, in fact,
I should say in advance,
251
00:19:03,000 --> 00:19:06,000
this method will not tell us
whether it is a minimum or a
252
00:19:06,000 --> 00:19:08,000
maximum.
We do not have any way of
253
00:19:08,000 --> 00:19:10,000
knowing, except for testing
values.
254
00:19:10,000 --> 00:19:13,000
We cannot use second derivative
tests or anything like that.
255
00:19:13,000 --> 00:19:21,000
I will get back to that.
Yes?
256
00:19:21,000 --> 00:19:23,000
Yes.
Here you can set y equals to
257
00:19:23,000 --> 00:19:26,000
favor x.
Then you can minimize x squared
258
00:19:26,000 --> 00:19:30,000
plus nine over x squared.
In general, if I am trying to
259
00:19:30,000 --> 00:19:33,000
solve a more complicated
problem, I might not be able to
260
00:19:33,000 --> 00:19:35,000
solve.
I am doing an example where,
261
00:19:35,000 --> 00:19:38,000
indeed, here you could solve
and remove one variable,
262
00:19:38,000 --> 00:19:41,000
but you cannot always do that.
And this method will still work.
263
00:19:41,000 --> 00:19:47,000
The other one won't.
OK.
264
00:19:47,000 --> 00:19:53,000
I don't see any other questions.
Are there any other questions?
265
00:19:53,000 --> 00:19:56,000
No.
OK.
266
00:19:56,000 --> 00:20:02,000
I see a lot of students
stretching and so on,
267
00:20:02,000 --> 00:20:08,000
so it is very confusing for me.
How do we solve these equations?
268
00:20:08,000 --> 00:20:14,000
Well, the answer is in general
we might be in deep trouble.
269
00:20:14,000 --> 00:20:18,000
There is no general method for
solving the equations that you
270
00:20:18,000 --> 00:20:21,000
get from this method.
You just have to think about
271
00:20:21,000 --> 00:20:25,000
them.
Sometimes it will be very easy.
272
00:20:25,000 --> 00:20:28,000
Sometimes it will be so hard
that you cannot actually do it
273
00:20:28,000 --> 00:20:31,000
without the computer.
Sometimes it will be just hard
274
00:20:31,000 --> 00:20:33,000
enough to be on Part B of this
week's problem set.
275
00:20:50,000 --> 00:20:56,000
I claim in this case we can
actually do it without so much
276
00:20:56,000 --> 00:21:03,000
trouble, because actually we can
think of this as a two by two
277
00:21:03,000 --> 00:21:10,000
linear system in x and y.
Well, let me do something.
278
00:21:10,000 --> 00:21:18,000
Let me rewrite the first two
equations as 2x - lambda y = 0.
279
00:21:18,000 --> 00:21:30,000
And lambda x - 2y = 0.
And xy = 3.
280
00:21:30,000 --> 00:21:36,000
That is what we want to solve.
Well, I can put this into
281
00:21:36,000 --> 00:21:41,000
matrix form.
Two minus lambda,
282
00:21:41,000 --> 00:21:48,000
lambda minus two times x,
y equals 0,0.
283
00:21:48,000 --> 00:21:52,000
Now, how do I solve a linear
system matrix times x,
284
00:21:52,000 --> 00:21:54,000
y equals zero?
Well, I always have an obvious
285
00:21:54,000 --> 00:21:56,000
solution.
X and y both equal to zero.
286
00:21:56,000 --> 00:22:02,000
Is that a good solution?
No, because zero times zero is
287
00:22:02,000 --> 00:22:07,000
not three.
We want another solution,
288
00:22:07,000 --> 00:22:14,000
the trivial solution.
0,0 does not solve the
289
00:22:14,000 --> 00:22:20,000
constraint equation xy equals
three, so we want another
290
00:22:20,000 --> 00:22:24,000
solution.
When do we have another
291
00:22:24,000 --> 00:22:29,000
solution?
Well, when the determinant of a
292
00:22:29,000 --> 00:22:37,000
matrix is zero.
We have other solutions that
293
00:22:37,000 --> 00:22:46,000
exist only if determinant of a
matrix is zero.
294
00:22:46,000 --> 00:23:01,000
M is this guy.
Let's compute the determinant.
295
00:23:01,000 --> 00:23:08,000
Well, that seems to be negative
four plus lambda squared.
296
00:23:08,000 --> 00:23:15,000
That is zero exactly when
lambda squared equals four,
297
00:23:15,000 --> 00:23:20,000
which is lambda is plus or
minus two.
298
00:23:20,000 --> 00:23:25,000
Already you see here it is a
the level of difficulty that is
299
00:23:25,000 --> 00:23:30,000
a little bit much for an exam
but perfectly fine for a problem
300
00:23:30,000 --> 00:23:33,000
set or for a beautiful lecture
like this one.
301
00:23:33,000 --> 00:23:37,000
How do we deal with -- Well,
we have two cases to look at.
302
00:23:37,000 --> 00:23:40,000
Lambda equals two or lambda
equals minus two.
303
00:23:40,000 --> 00:23:43,000
Let's start with lambda equals
two.
304
00:23:43,000 --> 00:23:47,000
If I set lambda equals two,
what does this equation become?
305
00:23:47,000 --> 00:23:53,000
Well, it becomes x equals y.
This one becomes y equals x.
306
00:23:53,000 --> 00:23:57,000
Well, they seem to be the same.
x equals y.
307
00:23:57,000 --> 00:24:01,000
And then the equation xy equals
three becomes,
308
00:24:01,000 --> 00:24:06,000
well, x squared equals three.
I have two solutions.
309
00:24:06,000 --> 00:24:15,000
One is x equals root three and,
therefore, y equals root three
310
00:24:15,000 --> 00:24:23,000
as well, or negative root three
and negative root three.
311
00:24:23,000 --> 00:24:26,000
Let's look at the other case.
If I set lambda equal to
312
00:24:26,000 --> 00:24:30,000
negative two then I get 2x
equals negative 2y.
313
00:24:30,000 --> 00:24:37,000
That means x equals negative y.
The second one,
314
00:24:37,000 --> 00:24:40,000
2y equals negative 2x.
That is y equals negative x.
315
00:24:40,000 --> 00:24:45,000
Well, that is the same thing.
And xy equals three becomes
316
00:24:45,000 --> 00:24:51,000
negative x squared equals three.
Can we solve that?
317
00:24:51,000 --> 00:24:58,000
No.
There are no solutions here.
318
00:24:58,000 --> 00:25:03,000
Now we have two candidate
points which are these two
319
00:25:03,000 --> 00:25:07,000
points, root three,
root three or negative root
320
00:25:07,000 --> 00:25:13,000
three, negative root three.
OK.
321
00:25:13,000 --> 00:25:16,000
Let's actually look at what we
have here.
322
00:25:16,000 --> 00:25:20,000
Maybe you cannot read the
coordinates, but the point that
323
00:25:20,000 --> 00:25:23,000
I have here is indeed root
three, root three.
324
00:25:23,000 --> 00:25:26,000
How do we see that lambda
equals two?
325
00:25:26,000 --> 00:25:29,000
Well, if you look at this
picture, the gradient of f,
326
00:25:29,000 --> 00:25:32,000
that is the blue vector,
is indeed twice the yellow
327
00:25:32,000 --> 00:25:36,000
vector, gradient g.
That is where you read the
328
00:25:36,000 --> 00:25:41,000
value of lambda.
And we have the other solution
329
00:25:41,000 --> 00:25:45,000
which is somewhere here.
Negative root three,
330
00:25:45,000 --> 00:25:48,000
negative root there.
And there, again,
331
00:25:48,000 --> 00:25:51,000
lambda equals two.
The two vectors are
332
00:25:51,000 --> 00:25:59,000
proportional by a factor of two.
Yes?
333
00:25:59,000 --> 00:26:01,000
No, solutions are not quite
guaranteed to be absolute minima
334
00:26:01,000 --> 00:26:03,000
or maxima.
They are guaranteed to be
335
00:26:03,000 --> 00:26:06,000
somehow critical points end of a
constraint.
336
00:26:06,000 --> 00:26:09,000
That means if you were able to
solve and eliminate the variable
337
00:26:09,000 --> 00:26:12,000
that would be a critical point.
When you have the same problem,
338
00:26:12,000 --> 00:26:14,000
as we have critical points,
are they maxima or minima?
339
00:26:14,000 --> 00:26:22,000
And the answer is,
well, we won't know until we
340
00:26:22,000 --> 00:26:28,000
check.
More questions?
341
00:26:28,000 --> 00:26:32,000
No.
Yes?
342
00:26:32,000 --> 00:26:36,000
What is a Lagrange multiplier?
Well, it is this number lambda
343
00:26:36,000 --> 00:26:39,000
that is called the multiplier
here.
344
00:26:39,000 --> 00:26:44,000
It is a multiplier because it
is what you have to multiply
345
00:26:44,000 --> 00:26:48,000
gradient of g by to get gradient
of f.
346
00:26:48,000 --> 00:26:49,000
It multiplies.
347
00:27:04,000 --> 00:27:11,000
Let's try to see why is this
method valid?
348
00:27:11,000 --> 00:27:18,000
Because so far I have shown you
pictures and have said see they
349
00:27:18,000 --> 00:27:23,000
are tangent.
But why is it that they have to
350
00:27:23,000 --> 00:27:28,000
be tangent in general?
Let's think about it.
351
00:27:28,000 --> 00:27:37,000
Let's say that we are at
constrained min or max.
352
00:27:37,000 --> 00:27:42,000
What that means is that if I
move on the level g equals
353
00:27:42,000 --> 00:27:46,000
constant then the value of f
should only increase or only
354
00:27:46,000 --> 00:27:49,000
decrease.
But it means,
355
00:27:49,000 --> 00:27:53,000
in particular,
to first order it will not
356
00:27:53,000 --> 00:27:56,000
change.
At an unconstrained min or max,
357
00:27:56,000 --> 00:27:59,000
partial derivatives are zero.
In this case,
358
00:27:59,000 --> 00:28:02,000
derivatives are zero only in
the allowed directions.
359
00:28:02,000 --> 00:28:09,000
And the allowed directions are
those that stay on the levels of
360
00:28:09,000 --> 00:28:21,000
this g equals constant.
In any direction along the
361
00:28:21,000 --> 00:28:40,000
level set g = c the rate of
change of f must be zero.
362
00:28:40,000 --> 00:28:44,000
That is what happens at minima
or maxima.
363
00:28:44,000 --> 00:28:49,000
Except here,
of course, we look only at the
364
00:28:49,000 --> 00:28:54,000
allowed directions.
Let's say the same thing in
365
00:28:54,000 --> 00:28:57,000
terms of directional
derivatives.
366
00:29:23,000 --> 00:29:35,000
That means for any direction
that is tangent to the
367
00:29:35,000 --> 00:29:49,000
constraint level g equal c,
we must have df over ds in the
368
00:29:49,000 --> 00:30:00,000
direction of u equals zero.
I will draw a picture.
369
00:30:00,000 --> 00:30:05,000
Let's say now I am in three
variables just to give you
370
00:30:05,000 --> 00:30:09,000
different examples.
Here I have a level surface g
371
00:30:09,000 --> 00:30:11,000
equals c.
I am at my point.
372
00:30:11,000 --> 00:30:18,000
And if I move in any direction
that is on the level surface,
373
00:30:18,000 --> 00:30:24,000
so I move in the direction u
tangent to the level surface,
374
00:30:24,000 --> 00:30:32,000
then the rate of change of f in
that direction should be zero.
375
00:30:32,000 --> 00:30:34,000
Now, remember what the formula
is for this guy.
376
00:30:34,000 --> 00:30:44,000
Well, we have seen that this
guy is actually radiant f dot u.
377
00:30:44,000 --> 00:30:58,000
That means any such vector u
must be perpendicular to the
378
00:30:58,000 --> 00:31:05,000
gradient of f.
That means that the gradient of
379
00:31:05,000 --> 00:31:10,000
f should be perpendicular to
anything that is tangent to this
380
00:31:10,000 --> 00:31:12,000
level.
That means the gradient of f
381
00:31:12,000 --> 00:31:16,000
should be perpendicular to the
level set.
382
00:31:16,000 --> 00:31:17,000
That is what we have shown.
383
00:31:37,000 --> 00:31:40,000
But we know another vector that
is also perpendicular to the
384
00:31:40,000 --> 00:31:57,000
level set of g.
That is the gradient of g.
385
00:31:57,000 --> 00:32:02,000
We conclude that the gradient
of f must be parallel to the
386
00:32:02,000 --> 00:32:07,000
gradient of g because both are
perpendicular to the level set
387
00:32:07,000 --> 00:32:09,000
of g.
I see confused faces,
388
00:32:09,000 --> 00:32:13,000
so let me try to tell you again
where that comes from.
389
00:32:13,000 --> 00:32:16,000
We said if we had a constrained
minimum or maximum,
390
00:32:16,000 --> 00:32:19,000
if we move in the level set of
g, f doesn't change.
391
00:32:19,000 --> 00:32:20,000
Well, it doesn't change to
first order.
392
00:32:20,000 --> 00:32:24,000
It is the same idea as when you
are looking for a minimum you
393
00:32:24,000 --> 00:32:26,000
set the derivative equal to
zero.
394
00:32:26,000 --> 00:32:31,000
So the derivative in any
direction, tangent to g equals
395
00:32:31,000 --> 00:32:34,000
c, should be the directional
derivative of f,
396
00:32:34,000 --> 00:32:38,000
in any such direction,
should be zero.
397
00:32:38,000 --> 00:32:43,000
That is what we mean by
critical point of f.
398
00:32:43,000 --> 00:32:48,000
And so that means that any
vector u, any unit vector
399
00:32:48,000 --> 00:32:55,000
tangent to the level set of g is
going to be perpendicular to the
400
00:32:55,000 --> 00:33:00,000
gradient of f.
That means that the gradient of
401
00:33:00,000 --> 00:33:04,000
f is perpendicular to the level
set of g.
402
00:33:04,000 --> 00:33:06,000
If you want,
that means the level sets of f
403
00:33:06,000 --> 00:33:10,000
and g are tangent to each other.
That is justifying what we have
404
00:33:10,000 --> 00:33:15,000
observed in the picture that the
two level sets have to be
405
00:33:15,000 --> 00:33:20,000
tangent to each other at the
prime minimum or maximum.
406
00:33:20,000 --> 00:33:23,000
Does that make a little bit of
sense?
407
00:33:23,000 --> 00:33:28,000
Kind of.
I see at least a few faces
408
00:33:28,000 --> 00:33:35,000
nodding so I take that to be a
positive answer.
409
00:33:35,000 --> 00:33:39,000
Since I have been asked by
several of you,
410
00:33:39,000 --> 00:33:43,000
how do I know if it is a
maximum or a minimum?
411
00:33:43,000 --> 00:33:57,000
Well, warning,
the method doesn't tell whether
412
00:33:57,000 --> 00:34:09,000
a solution is a minimum or a
maximum.
413
00:34:09,000 --> 00:34:13,000
How do we do it?
Well, more bad news.
414
00:34:13,000 --> 00:34:26,000
We cannot use the second
derivative test.
415
00:34:26,000 --> 00:34:30,000
And the reason for that is that
we care actually only about
416
00:34:30,000 --> 00:34:34,000
these specific directions that
are tangent to variable of g.
417
00:34:34,000 --> 00:34:39,000
And we don't want to bother to
try to define directional second
418
00:34:39,000 --> 00:34:42,000
derivatives.
Not to mention that actually it
419
00:34:42,000 --> 00:34:45,000
wouldn't work.
There is a criterion but it is
420
00:34:45,000 --> 00:34:49,000
much more complicated than that.
Basically, the answer for us is
421
00:34:49,000 --> 00:34:52,000
that we don't have a second
derivative test in this
422
00:34:52,000 --> 00:34:54,000
situation.
What are we left with?
423
00:34:54,000 --> 00:34:57,000
Well, we are just left with
comparing values.
424
00:34:57,000 --> 00:35:00,000
Say that in this problem you
found a point where f equals
425
00:35:00,000 --> 00:35:04,000
three, a point where f equals
nine, a point where f equals 15.
426
00:35:04,000 --> 00:35:08,000
Well, then probably the minimum
is the point where f equals
427
00:35:08,000 --> 00:35:12,000
three and the maximum is 15.
Actually, in this case,
428
00:35:12,000 --> 00:35:17,000
where we found minima,
these two points are tied for
429
00:35:17,000 --> 00:35:19,000
minimum.
What about the maximum?
430
00:35:19,000 --> 00:35:22,000
What is the maximum of f on the
hyperbola?
431
00:35:22,000 --> 00:35:25,000
Well, it is infinity because
the point can go as far as you
432
00:35:25,000 --> 00:35:29,000
want from the origin.
But the general idea is if we
433
00:35:29,000 --> 00:35:35,000
have a good reason to believe
that there should be a minimum,
434
00:35:35,000 --> 00:35:38,000
and it's not like at infinity
or something weird like that,
435
00:35:38,000 --> 00:35:42,000
then the minimum will be a
solution of the Lagrange
436
00:35:42,000 --> 00:35:46,000
multiplier equations.
We just look for all the
437
00:35:46,000 --> 00:35:51,000
solutions and then we choose the
one that gives us the lowest
438
00:35:51,000 --> 00:35:55,000
value.
Is that good enough?
439
00:35:55,000 --> 00:35:57,000
Let me actually write that down.
440
00:36:23,000 --> 00:36:35,000
To find the minimum or the
maximum, we compare values of f
441
00:36:35,000 --> 00:36:46,000
at the various solutions -- --
to Lagrange multiplier
442
00:36:46,000 --> 00:36:49,000
equations.
443
00:37:08,000 --> 00:37:11,000
I should say also that
sometimes you can just conclude
444
00:37:11,000 --> 00:37:14,000
by thinking geometrically.
In this case,
445
00:37:14,000 --> 00:37:18,000
when it is asking you which
point is closest to the origin
446
00:37:18,000 --> 00:37:23,000
you can just see that your
answer is the correct one.
447
00:37:23,000 --> 00:37:32,000
Let's do an advanced example.
Advanced means that -- Well,
448
00:37:32,000 --> 00:37:37,000
this one I didn't actually dare
to put on top of the other
449
00:37:37,000 --> 00:37:48,000
problem sets.
Instead, I am going to do it.
450
00:37:48,000 --> 00:37:51,000
What is this going to be about?
We are going to look for a
451
00:37:51,000 --> 00:38:03,000
surface minimizing pyramid.
Let's say that we want to build
452
00:38:03,000 --> 00:38:19,000
a pyramid with a given
triangular base -- -- and a
453
00:38:19,000 --> 00:38:28,000
given volume.
Say that I have maybe in the x,
454
00:38:28,000 --> 00:38:33,000
y plane I am giving you some
triangle.
455
00:38:33,000 --> 00:38:40,000
And I am going to try to build
a pyramid.
456
00:38:40,000 --> 00:38:48,000
Of course, I can choose where
to put the top of a pyramid.
457
00:38:48,000 --> 00:38:53,000
This guy will end up being
behind now.
458
00:38:53,000 --> 00:39:09,000
And the constraint and the goal
is to minimize the total surface
459
00:39:09,000 --> 00:39:13,000
area.
The first time I taught this
460
00:39:13,000 --> 00:39:15,000
class, it was a few years ago,
was just before they built the
461
00:39:15,000 --> 00:39:17,000
Stata Center.
And then I used to motivate
462
00:39:17,000 --> 00:39:20,000
this problem by saying Frank
Gehry has gone crazy and has
463
00:39:20,000 --> 00:39:23,000
been given a triangular plot of
land he wants to put a pyramid.
464
00:39:23,000 --> 00:39:26,000
There needs to be the right
amount of volume so that you can
465
00:39:26,000 --> 00:39:28,000
put all the offices in there.
And he wants it to be,
466
00:39:28,000 --> 00:39:31,000
actually, covered in solid
gold.
467
00:39:31,000 --> 00:39:34,000
And because that is expensive,
the administration wants him to
468
00:39:34,000 --> 00:39:38,000
cut the costs a bit.
And so you have to minimize the
469
00:39:38,000 --> 00:39:42,000
total size so that it doesn't
cost too much.
470
00:39:42,000 --> 00:39:45,000
We will see if MIT comes up
with a triangular pyramid
471
00:39:45,000 --> 00:39:48,000
building.
Hopefully not.
472
00:39:48,000 --> 00:39:58,000
It could be our next dorm,
you never know.
473
00:39:58,000 --> 00:40:01,000
Anyway, it is a fine geometry
problem.
474
00:40:01,000 --> 00:40:07,000
Let's try to think about how we
can do this.
475
00:40:07,000 --> 00:40:10,000
The natural way to think about
it would be -- Well,
476
00:40:10,000 --> 00:40:11,000
what do we have to look for
first?
477
00:40:11,000 --> 00:40:18,000
We have to look for the
position of that top point.
478
00:40:18,000 --> 00:40:29,000
Remember we know that the
volume of a pyramid is one-third
479
00:40:29,000 --> 00:40:37,000
the area of base times height.
In fact, fixing the volume,
480
00:40:37,000 --> 00:40:39,000
knowing that we have fixed the
area of a base,
481
00:40:39,000 --> 00:40:43,000
means that we are fixing the
height of the pyramid.
482
00:40:43,000 --> 00:40:47,000
The height is completely fixed.
What we have to choose just is
483
00:40:47,000 --> 00:40:52,000
where do we put that top point?
Do we put it smack in the
484
00:40:52,000 --> 00:40:58,000
middle of a triangle or to a
side or even anywhere we want?
485
00:40:58,000 --> 00:41:15,000
Its z coordinate is fixed.
Let's call h the height.
486
00:41:15,000 --> 00:41:20,000
What we could do is something
like this.
487
00:41:20,000 --> 00:41:24,000
We say we have three points of
a base.
488
00:41:24,000 --> 00:41:32,000
Let's call them p1 at (x1,
y1,0); p2 at (x2,
489
00:41:32,000 --> 00:41:36,000
y2,0); p3 at (x3,
y3,0).
490
00:41:36,000 --> 00:41:40,000
This point p is the unknown
point at (x, y,
491
00:41:40,000 --> 00:41:42,000
h).
We know the height.
492
00:41:42,000 --> 00:41:46,000
And then we want to minimize
the sum of the areas of these
493
00:41:46,000 --> 00:41:50,000
three triangles.
One here, one here and one at
494
00:41:50,000 --> 00:41:53,000
the back.
And areas of triangles we know
495
00:41:53,000 --> 00:41:57,000
how to express by using length
of cross-product.
496
00:41:57,000 --> 00:42:00,000
It becomes a function of x and
y.
497
00:42:00,000 --> 00:42:04,000
And you can try to minimize it.
Actually, it doesn't quite work.
498
00:42:04,000 --> 00:42:05,000
The formulas are just too
complicated.
499
00:42:05,000 --> 00:42:14,000
You will never get there.
What happens is actually maybe
500
00:42:14,000 --> 00:42:18,000
we need better coordinates.
Why do we need better
501
00:42:18,000 --> 00:42:21,000
coordinates?
That is because the geometry is
502
00:42:21,000 --> 00:42:24,000
kind of difficult to do if you
use x, y coordinates.
503
00:42:24,000 --> 00:42:28,000
I mean formula for
cross-product is fine,
504
00:42:28,000 --> 00:42:33,000
but then the length of the
vector will be annoying and just
505
00:42:33,000 --> 00:42:37,000
doesn't look good.
Instead, let's think about it
506
00:42:37,000 --> 00:42:38,000
differently.
507
00:42:54,000 --> 00:43:01,000
I claim if we do it this way
and we express the area as a
508
00:43:01,000 --> 00:43:06,000
function of x,
y, well, actually we can't
509
00:43:06,000 --> 00:43:13,000
solve for a minimum.
Here is another way to do it.
510
00:43:13,000 --> 00:43:17,000
Well, what has worked pretty
well for us so far is this
511
00:43:17,000 --> 00:43:19,000
geometric idea of base times
height.
512
00:43:19,000 --> 00:43:29,000
So let's think in terms of the
heights of side triangles.
513
00:43:29,000 --> 00:43:37,000
I am going to use the height of
these things.
514
00:43:37,000 --> 00:43:43,000
And I am going to say that the
area will be the sum of three
515
00:43:43,000 --> 00:43:48,000
terms, which are three bases
times three heights.
516
00:43:48,000 --> 00:43:53,000
Let's give names to these
quantities.
517
00:43:53,000 --> 00:43:58,000
Actually, for that it is going
to be good to have the point in
518
00:43:58,000 --> 00:44:01,000
the xy plane that lives directly
below p.
519
00:44:01,000 --> 00:44:08,000
Let's call it q.
P is the point that coordinates
520
00:44:08,000 --> 00:44:13,000
x, y, h.
And let's call q the point that
521
00:44:13,000 --> 00:44:19,000
is just below it and so it'
coordinates are x,
522
00:44:19,000 --> 00:44:22,000
y, 0.
Let's see.
523
00:44:22,000 --> 00:44:34,000
Let me draw a map of this thing.
p1, p2, p3 and I have my point
524
00:44:34,000 --> 00:44:37,000
q in the middle.
Let's see.
525
00:44:37,000 --> 00:44:40,000
To know these areas,
I need to know the base.
526
00:44:40,000 --> 00:44:44,000
Well, the base I can decide
that I know it because it is
527
00:44:44,000 --> 00:44:48,000
part of my given data.
I know the sides of this
528
00:44:48,000 --> 00:44:53,000
triangle.
Let me call the lengths a1,
529
00:44:53,000 --> 00:44:56,000
a2, a3.
I also need to know the height,
530
00:44:56,000 --> 00:44:58,000
so I need to know these
lengths.
531
00:44:58,000 --> 00:45:01,000
How do I know these lengths?
Well, its distance in space,
532
00:45:01,000 --> 00:45:03,000
but it is a little bit
annoying.
533
00:45:03,000 --> 00:45:10,000
But maybe I can reduce it to a
distance in the plane by looking
534
00:45:10,000 --> 00:45:17,000
instead at this distance here.
Let me give names to the
535
00:45:17,000 --> 00:45:24,000
distances from q to the sides.
Let's call u1,
536
00:45:24,000 --> 00:45:35,000
u2, u3 the distances from q to
the sides.
537
00:45:47,000 --> 00:45:49,000
Well, now I can claim I can
find, actually,
538
00:45:49,000 --> 00:45:53,000
sorry.
I need to draw one more thing.
539
00:45:53,000 --> 00:45:57,000
I claim I have a nice formula
for the area,
540
00:45:57,000 --> 00:46:01,000
because this is vertical and
this is horizontal so this
541
00:46:01,000 --> 00:46:05,000
length here is u3,
this length here is h.
542
00:46:05,000 --> 00:46:13,000
So what is this length here?
It is the square root of u3
543
00:46:13,000 --> 00:46:17,000
squared plus h squared.
And similarly for these other
544
00:46:17,000 --> 00:46:23,000
guys.
They are square roots of a u
545
00:46:23,000 --> 00:46:31,000
squared plus h squared.
The heights of the faces are
546
00:46:31,000 --> 00:46:36,000
square root of u1 squared times
h squared.
547
00:46:36,000 --> 00:46:43,000
And similarly with u2 and u3.
So the total side area is going
548
00:46:43,000 --> 00:46:47,000
to be the area of the first
faces,
549
00:46:47,000 --> 00:46:58,000
one-half of base times height,
plus one-half of a base times a
550
00:46:58,000 --> 00:47:06,000
height plus one-half of the
third one.
551
00:47:06,000 --> 00:47:09,000
It doesn't look so much better.
But, trust me,
552
00:47:09,000 --> 00:47:15,000
it will get better.
Now, that is a function of
553
00:47:15,000 --> 00:47:19,000
three variables,
u1, u2, u3.
554
00:47:19,000 --> 00:47:22,000
And how do we relate u1,
u2, u3 to each other?
555
00:47:22,000 --> 00:47:25,000
They are probably not
independent.
556
00:47:25,000 --> 00:47:32,000
Well, let's cut this triangle
here into three pieces like
557
00:47:32,000 --> 00:47:35,000
that.
Then each piece has side --
558
00:47:35,000 --> 00:47:40,000
Well, let's look at it the piece
of the bottom.
559
00:47:40,000 --> 00:47:50,000
It has base a3, height u3.
Cutting base into three tells
560
00:47:50,000 --> 00:47:57,000
you that the area of a base is
one-half of a1,
561
00:47:57,000 --> 00:48:04,000
u1 plus one-half of a2,
u2 plus one-half of a3,
562
00:48:04,000 --> 00:48:09,000
u3.
And that is our constraint.
563
00:48:09,000 --> 00:48:12,000
My three variables,
u1, u2, u3, are constrained in
564
00:48:12,000 --> 00:48:14,000
this way.
The sum of this figure must be
565
00:48:14,000 --> 00:48:17,000
the area of a base.
And I want to minimize that guy.
566
00:48:17,000 --> 00:48:23,000
So that is my g and that guy
here is my f.
567
00:48:23,000 --> 00:48:28,000
Now we try to apply our
Lagrange multiplier equations.
568
00:48:28,000 --> 00:48:33,000
Well, partial f of a partial u1
is -- Well,
569
00:48:33,000 --> 00:48:36,000
if you do the calculation,
you will see it is one-half a1,
570
00:48:36,000 --> 00:48:43,000
u1 over square root of u1^2
plus h^2 equals lambda,
571
00:48:43,000 --> 00:48:46,000
what is partial g,
partial a1?
572
00:48:46,000 --> 00:48:50,000
That one you can do, I am sure.
It is one-half a1.
573
00:48:50,000 --> 00:49:00,000
Oh, these guys simplify.
If you do the same with the
574
00:49:00,000 --> 00:49:09,000
second one -- -- things simplify
again.
575
00:49:09,000 --> 00:49:17,000
And the same with the third one.
Well, you will get,
576
00:49:17,000 --> 00:49:21,000
after simplifying,
u3 over square root of u3
577
00:49:21,000 --> 00:49:24,000
squared plus h squared equals
lambda.
578
00:49:24,000 --> 00:49:27,000
Now, that means this guy equals
this guy equals this guy.
579
00:49:27,000 --> 00:49:33,000
They are all equal to lambda.
And, if you think about it,
580
00:49:33,000 --> 00:49:39,000
that means that u1 = u2 = u3.
See, it looked like scary
581
00:49:39,000 --> 00:49:42,000
equations but the solution is
very simple.
582
00:49:42,000 --> 00:49:45,000
What does it mean?
It means that our point q
583
00:49:45,000 --> 00:49:47,000
should be equidistant from all
three sides.
584
00:49:47,000 --> 00:49:52,000
That is called the incenter.
Q should be in the incenter.
585
00:49:52,000 --> 00:49:56,000
The next time you have to build
a golden pyramid and don't want
586
00:49:56,000 --> 00:49:59,000
to go broke, well,
you know where to put the top.
587
00:49:59,000 --> 00:50:03,000
If that was a bit fast, sorry.
Anyway, it is not completely
588
00:50:03,000 --> 00:50:06,000
crucial.
But go over it and you will see
589
00:50:06,000 --> 00:50:08,000
it works.
Have a nice weekend.