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Let me start by basically
listing the main things we have

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learned over the past three
weeks or so.

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And I will add a few
complements of information about

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that because there are a few
small details that I didn't

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quite clarify and that I should
probably make a bit clearer,

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especially what happened at the
very end of yesterday's class.

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Here is a list of things that
should be on your review sheet

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for the exam.
The first thing we learned

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00:01:01,000 --> 00:01:08,000
about, the main topic of this
unit is about functions of

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several variables.
We have learned how to think of

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functions of two or three
variables in terms of plotting

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00:01:16,000 --> 00:01:17,000
them.
In particular,

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well, not only the graph but
also the contour plot and how to

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read a contour plot.
And we have learned how to

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study variations of these
functions using partial

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derivatives.
Remember, we have defined the

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00:01:44,000 --> 00:01:47,000
partial of f with respect to
some variable,

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say, x to be the rate of change
with respect to x when we hold

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all the other variables
constant.

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00:01:55,000 --> 00:02:01,000
If you have a function of x and
y, this symbol means you

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00:02:01,000 --> 00:02:07,000
differentiate with respect to x
treating y as a constant.

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00:02:07,000 --> 00:02:15,000
And we have learned how to
package partial derivatives into

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a vector,the gradient vector.
For example,

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if we have a function of three
variables, the vector whose

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components are the partial
derivatives.

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00:02:26,000 --> 00:02:33,000
And we have seen how to use the
gradient vector or the partial

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00:02:33,000 --> 00:02:39,000
derivatives to derive various
things such as approximation

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formulas.
The change in f,

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when we change x,
y, z slightly,

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is approximately equal to,
well, there are several terms.

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00:02:57,000 --> 00:03:03,000
And I can rewrite this in
vector form as the gradient dot

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product the amount by which the
position vector has changed.

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00:03:08,000 --> 00:03:11,000
Basically, what causes f to
change is that I am changing x,

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00:03:11,000 --> 00:03:16,000
y and z by small amounts and
how sensitive f is to each

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variable is precisely what the
partial derivatives measure.

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00:03:22,000 --> 00:03:26,000
And, in particular,
this approximation is called

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the tangent plane approximation
because it tells us,

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in fact,
it amounts to identifying the

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graph of the function with its
tangent plane.

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00:03:38,000 --> 00:03:43,000
It means that we assume that
the function depends more or

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00:03:43,000 --> 00:03:45,000
less linearly on x,
y and z.

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00:03:45,000 --> 00:03:48,000
And, if we set these things
equal, what we get is actually,

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we are replacing the function
by its linear approximation.

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We are replacing the graph by
its tangent plane.

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Except, of course,
we haven't see the graph of a

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function of three variables
because that would live in

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4-dimensional space.
So, when we think of a graph,

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00:04:04,000 --> 00:04:08,000
really, it is a function of two
variables.

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That also tells us how to find
tangent planes to level

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surfaces.

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Recall that the tangent plane
to a surface,

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given by the equation f of x,
y, z equals z,

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at a given point can be found
by looking first for its normal

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vector.
And we know that the normal

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vector is actually,
well,

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one normal vector is given by
the gradient of a function

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because we know that the
gradient is actually pointing

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perpendicularly to the level
sets towards higher values of a

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00:05:01,000 --> 00:05:05,000
function.
And it gives us the direction

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of fastest increase of a
function.

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OK.
Any questions about these

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topics?
No.

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OK.
Let me add, actually,

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a cultural note to what we have
seen so far about partial

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derivatives and how to use them,
which is maybe something I

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should have mentioned a couple
of weeks ago.

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Why do we like partial
derivatives?

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Well, one obvious reason is we
can do all these things.

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But another reason is that,
really,

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you need partial derivatives to
do physics and to understand

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much of the world that is around
you because a lot of things

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actually are governed by what is
called partial differentiation

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equations.

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So if you want a cultural
remark about what this is good

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00:06:07,000 --> 00:06:09,000
for.
A partial differential equation

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is an equation that involves the
partial derivatives of a

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function.
So you have some function that

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is unknown that depends on a
bunch of variables.

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And a partial differential
equation is some relation

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between its partial derivatives.
Let me see.

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These are equations involving
the partial derivatives -- -- of

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an unknown function.
Let me give you an example to

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see how that works.
For example,

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the heat equation is one
example of a partial

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00:07:02,000 --> 00:07:09,000
differential equation.
It is the equation -- Well,

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let me write for you the space
version of it.

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It is the equation partial f
over partial t equals some

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constant times the sum of the
second partials with respect to

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x, y and z.
So this is an equation where we

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are trying to solve for a
function f that depends,

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actually, on four variables,
x, y, z, t.

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And what should you have in
mind?

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Well, this equation governs
temperature.

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If you think that f of x, y, z,
t will be the temperature at a

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point in space at position x,
y, z and at time t,

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00:07:59,000 --> 00:08:04,000
then this tells you how
temperature changes over time.

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00:08:04,000 --> 00:08:07,000
It tells you that at any given
point,

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00:08:07,000 --> 00:08:10,000
the rate of change of
temperature over time is given

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00:08:10,000 --> 00:08:15,000
by this complicated expression
in the partial derivatives in

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terms of the space coordinates
x, y, z.

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00:08:18,000 --> 00:08:21,000
If you know, for example,
the initial distribution of

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temperature in this room,
and if you assume that nothing

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is generating heat or taking
heat away,

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00:08:26,000 --> 00:08:29,000
so if you don't have any air
conditioning or heating going

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on,
then it will tell you how the

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temperature will change over
time and eventually stabilize to

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some final value.
Yes?

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Why do we take the partial
derivative twice?

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Well, that is a question,
I would say,

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for a physics person.
But in a few weeks we will

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actually see a derivation of
where this equation comes from

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and try to justify it.
But, really,

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that is something you will see
in a physics class.

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The reason for that is
basically physics of how heat is

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00:09:02,000 --> 00:09:09,000
transported between particles in
fluid, or actually any medium.

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00:09:09,000 --> 00:09:12,000
This constant k actually is
called the heat conductivity.

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00:09:12,000 --> 00:09:17,000
It tells you how well the heat
flows through the material that

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you are looking at.
Anyway, I am giving it to you

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00:09:20,000 --> 00:09:23,000
just to show you an example of a
real life problem where,

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in fact, you have to solve one
of these things.

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00:09:26,000 --> 00:09:29,000
Now, how to solve partial
differential equations is not a

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topic for this class.
It is not even a topic for

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18.03 which is called
Differential Equations,

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00:09:34,000 --> 00:09:38,000
without partial,
which means there actually you

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00:09:38,000 --> 00:09:41,000
will learn tools to study and
solve these equations but when

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00:09:41,000 --> 00:09:43,000
there is only one variable
involved.

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00:09:43,000 --> 00:09:47,000
And you will see it is already
quite hard.

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00:09:47,000 --> 00:09:50,000
And, if you want more on that
one, we have many fine classes

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about partial differential
equations.

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00:09:52,000 --> 00:09:58,000
But one thing at a time.
I wanted to point out to you

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that very often functions that
you see in real life satisfy

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00:10:03,000 --> 00:10:08,000
many nice relations between the
partial derivatives.

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00:10:08,000 --> 00:10:10,000
That was in case you were
wondering why on the syllabus

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00:10:10,000 --> 00:10:13,000
for today it said partial
differential equations.

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00:10:13,000 --> 00:10:15,000
Now we have officially covered
the topic.

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00:10:15,000 --> 00:10:20,000
That is basically all we need
to know about it.

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00:10:20,000 --> 00:10:22,000
But we will come back to that a
bit later.

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00:10:22,000 --> 00:10:27,000
You will see.
OK.

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00:10:27,000 --> 00:10:30,000
If there are no further
questions, let me continue and

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00:10:30,000 --> 00:10:33,000
go back to my list of topics.
Oh, sorry.

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I should have written down that
this equation is solved by

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00:10:42,000 --> 00:10:48,000
temperature for point x,
y, z at time t.

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00:10:48,000 --> 00:10:52,000
OK.
And there are, actually,

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00:10:52,000 --> 00:10:56,000
many other interesting partial
differential equations you will

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maybe sometimes learn about the
wave equation that governs how

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waves propagate in space,
about the diffusion equation,

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00:11:02,000 --> 00:11:07,000
when you have maybe a mixture
of two fluids,

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00:11:07,000 --> 00:11:11,000
how they somehow mix over time
and so on.

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00:11:11,000 --> 00:11:16,000
Basically, to every problem you
might want to consider there is

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00:11:16,000 --> 00:11:19,000
a partial differential equation
to solve.

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00:11:19,000 --> 00:11:23,000
OK. Anyway. Sorry.
Back to my list of topics.

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One important application we
have seen of partial derivatives

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is to try to optimize things,
try to solve minimum/maximum

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problems.

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Remember that we have
introduced the notion of

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critical points of a function.
A critical point is when all

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the partial derivatives are
zero.

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00:12:03,000 --> 00:12:05,000
And then there are various
kinds of critical points.

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00:12:05,000 --> 00:12:09,000
There is maxima and there is
minimum, but there is also

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00:12:09,000 --> 00:12:15,000
saddle points.
And we have seen a method using

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second derivatives -- -- to
decide which kind of critical

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00:12:24,000 --> 00:12:29,000
point we have.
I should say that is for a

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00:12:29,000 --> 00:12:35,000
function of two variables to try
to decide whether a given

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00:12:35,000 --> 00:12:41,000
critical point is a minimum,
a maximum or a saddle point.

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00:12:41,000 --> 00:12:44,000
And we have also seen that
actually that is not enough to

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find the minimum of a maximum of
a function because the minimum

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00:12:48,000 --> 00:12:50,000
of a maximum could occur on the
boundary.

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00:12:50,000 --> 00:12:53,000
Just to give you a small
reminder,

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00:12:53,000 --> 00:12:55,000
when you have a function of one
variables,

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00:12:55,000 --> 00:13:00,000
if you are trying to find the
minimum and the maximum of a

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00:13:00,000 --> 00:13:03,000
function whose graph looks like
this,

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00:13:03,000 --> 00:13:05,000
well, you are going to tell me,
quite obviously,

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00:13:05,000 --> 00:13:07,000
that the maximum is this point
up here.

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00:13:07,000 --> 00:13:11,000
And that is a point where the
first derivative is zero.

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00:13:11,000 --> 00:13:14,000
That is a critical point.
And we used the second

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00:13:14,000 --> 00:13:18,000
derivative to see that this
critical point is a local

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00:13:18,000 --> 00:13:20,000
maximum.
But then, when we are looking

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for the minimum of a function,
well, it is not at a critical

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00:13:23,000 --> 00:13:26,000
point.
It is actually here at the

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00:13:26,000 --> 00:13:30,000
boundary of the domain,
you know, the range of values

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00:13:30,000 --> 00:13:38,000
that we are going to consider.
Here the minimum is at the

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boundary.
And the maximum is at a

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critical point.
Similarly, when you have a

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00:13:50,000 --> 00:13:53,000
function of several variables,
say of two variables,

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for example,
then the minimum and the

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00:13:55,000 --> 00:13:58,000
maximum will be achieved either
at a critical point.

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00:13:58,000 --> 00:14:01,000
And then we can use these
methods to find where they are.

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00:14:01,000 --> 00:14:06,000
Or, somewhere on the boundary
of a set of values that are

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00:14:06,000 --> 00:14:09,000
allowed.
It could be that we actually

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00:14:09,000 --> 00:14:13,000
achieve a minimum by making x
and y as small as possible.

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00:14:13,000 --> 00:14:16,000
Maybe letting them go to zero
if they had to be positive or

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00:14:16,000 --> 00:14:19,000
maybe by making them go to
infinity.

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00:14:19,000 --> 00:14:23,000
So, we have to keep our minds
open and look at various

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00:14:23,000 --> 00:14:26,000
possibilities.
We are going to do a problem

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00:14:26,000 --> 00:14:29,000
like that.
We are going to go over a

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00:14:29,000 --> 00:14:34,000
practice problem from the
practice test to clarify this.

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00:14:34,000 --> 00:14:38,000
Another important cultural
application of minimum/maximum

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00:14:38,000 --> 00:14:42,000
problems in two variables that
we have seen in class is the

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00:14:42,000 --> 00:14:45,000
least squared method to find the
best fit line,

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00:14:45,000 --> 00:14:49,000
or the best fit anything,
really,

207
00:14:49,000 --> 00:14:56,000
to find when you have a set of
data points what is the best

208
00:14:56,000 --> 00:15:01,000
linear approximately for these
data points.

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00:15:01,000 --> 00:15:03,000
And here I have some good news
for you.

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00:15:03,000 --> 00:15:07,000
While you should definitely
know what this is about,

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00:15:07,000 --> 00:15:09,000
it will not be on the test.

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00:15:30,000 --> 00:15:35,000
[APPLAUSE]
That doesn't mean that you

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00:15:35,000 --> 00:15:41,000
should forget everything we have
seen about it,

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OK?
Now what is next on my list of

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00:15:51,000 --> 00:15:58,000
topics?
We have seen differentials.

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00:15:58,000 --> 00:16:03,000
Remember the differential of f,
by definition,

217
00:16:03,000 --> 00:16:09,000
would be this kind of quantity.
At first it looks just like a

218
00:16:09,000 --> 00:16:12,000
new way to package partial
derivatives together into some

219
00:16:12,000 --> 00:16:15,000
new kind of object.
Now, what is this good for?

220
00:16:15,000 --> 00:16:18,000
Well, it is a good way to
remember approximation formulas.

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00:16:18,000 --> 00:16:22,000
It is a good way to also study
how variations in x,

222
00:16:22,000 --> 00:16:26,000
y, z relate to variations in f.
In particular,

223
00:16:26,000 --> 00:16:30,000
we can divide this by
variations,

224
00:16:30,000 --> 00:16:34,000
actually, by dx or by dy or by
dz in any situation that we

225
00:16:34,000 --> 00:16:40,000
want,
or by d of some other variable

226
00:16:40,000 --> 00:16:46,000
to get chain rules.
The chain rule says,

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00:16:46,000 --> 00:16:50,000
for example,
there are many situations.

228
00:16:50,000 --> 00:16:56,000
But, for example,
if x, y and z depend on some

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00:16:56,000 --> 00:17:04,000
other variable,
say of variables maybe even u

230
00:17:04,000 --> 00:17:08,000
and v,
then that means that f becomes

231
00:17:08,000 --> 00:17:13,000
a function of u and v.
And then we can ask ourselves,

232
00:17:13,000 --> 00:17:18,000
how sensitive is f to a value
of u?

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00:17:18,000 --> 00:17:25,000
Well, we can answer that.
The chain rule is something

234
00:17:25,000 --> 00:17:33,000
like this.
And let me explain to you again

235
00:17:33,000 --> 00:17:41,000
where this comes from.
Basically, what this quantity

236
00:17:41,000 --> 00:17:46,000
means is if we change u and keep
v constant, what happens to the

237
00:17:46,000 --> 00:17:48,000
value of f?
Well, why would the value of f

238
00:17:48,000 --> 00:17:51,000
change in the first place when f
is just a function of x,

239
00:17:51,000 --> 00:17:55,000
y, z and not directly of you?
Well, it changes because x,

240
00:17:55,000 --> 00:17:59,000
y and z depend on u.
First we have to figure out how

241
00:17:59,000 --> 00:18:02,000
quickly x, y and z change when
we change u.

242
00:18:02,000 --> 00:18:05,000
Well, how quickly they do that
is precisely partial x over

243
00:18:05,000 --> 00:18:08,000
partial u, partial y over
partial u, partial z over

244
00:18:08,000 --> 00:18:10,000
partial u.
These are the rates of change

245
00:18:10,000 --> 00:18:14,000
of x, y, z when we change u.
And now, when we change x,

246
00:18:14,000 --> 00:18:17,000
y and z, that causes f to
change.

247
00:18:17,000 --> 00:18:21,000
How much does f change?
Well, partial f over partial x

248
00:18:21,000 --> 00:18:25,000
tells us how quickly f changes
if I just change x.

249
00:18:25,000 --> 00:18:29,000
I get this.
That is the change in f caused

250
00:18:29,000 --> 00:18:33,000
just by the fact that x changes
when u changes.

251
00:18:33,000 --> 00:18:36,000
But then y also changes.
y changes at this rate.

252
00:18:36,000 --> 00:18:39,000
And that causes f to change at
that rate.

253
00:18:39,000 --> 00:18:42,000
And z changes as well,
and that causes f to change at

254
00:18:42,000 --> 00:18:45,000
that rate.
And the effects add up together.

255
00:18:45,000 --> 00:18:57,000
Does that make sense?
OK.

256
00:18:57,000 --> 00:19:00,000
And so, in particular,
we can use the chain rule to do

257
00:19:00,000 --> 00:19:03,000
changes of variables.
If we have, say,

258
00:19:03,000 --> 00:19:08,000
a function in terms of polar
coordinates on theta and we like

259
00:19:08,000 --> 00:19:14,000
to switch it to rectangular
coordinates x and y then we can

260
00:19:14,000 --> 00:19:19,000
use chain rules to relate the
partial derivatives.

261
00:19:19,000 --> 00:19:23,000
And finally,
last but not least,

262
00:19:23,000 --> 00:19:31,000
we have seen how to deal with
non-independent variables.

263
00:19:31,000 --> 00:19:37,000
When our variables say x,
y, z related by some equation.

264
00:19:37,000 --> 00:19:41,000
One way we can deal with this
is to solve for one of the

265
00:19:41,000 --> 00:19:44,000
variables and go back to two
independent variables,

266
00:19:44,000 --> 00:19:47,000
but we cannot always do that.
Of course, on the exam,

267
00:19:47,000 --> 00:19:50,000
you can be sure that I will
make sure that you cannot solve

268
00:19:50,000 --> 00:19:53,000
for a variable you want to
remove because that would be too

269
00:19:53,000 --> 00:19:56,000
easy.
Then when we have to look at

270
00:19:56,000 --> 00:20:02,000
all of them, we will have to
take into account this relation,

271
00:20:02,000 --> 00:20:05,000
we have seen two useful
methods.

272
00:20:05,000 --> 00:20:09,000
One of them is to find the
minimum of a maximum of a

273
00:20:09,000 --> 00:20:13,000
function when the variables are
not independent,

274
00:20:13,000 --> 00:20:17,000
and that is the method of
Lagrange multipliers.

275
00:20:33,000 --> 00:20:39,000
Remember, to find the minimum
or the maximum of the function

276
00:20:39,000 --> 00:20:45,000
f,
subject to the constraint g

277
00:20:45,000 --> 00:20:52,000
equals constant,
well, we write down equations

278
00:20:52,000 --> 00:20:59,000
that say that the gradient of f
is actually proportional to the

279
00:20:59,000 --> 00:21:04,000
gradient of g.
There is a new variable here,

280
00:21:04,000 --> 00:21:08,000
lambda, the multiplier.
And so, for example,

281
00:21:08,000 --> 00:21:12,000
well, I guess here I had
functions of three variables,

282
00:21:12,000 --> 00:21:14,000
so this becomes three
equations.

283
00:21:14,000 --> 00:21:21,000
f sub x equals lambda g sub x,
f sub y equals lambda g sub y,

284
00:21:21,000 --> 00:21:25,000
and f sub z equals lambda g sub
z.

285
00:21:25,000 --> 00:21:27,000
And, when we plug in the
formulas for f and g,

286
00:21:27,000 --> 00:21:31,000
well, we are left with three
equations involving the four

287
00:21:31,000 --> 00:21:33,000
variables, x,
y, z and lambda.

288
00:21:33,000 --> 00:21:36,000
What is wrong?
Well, we don't have actually

289
00:21:36,000 --> 00:21:41,000
four independent variables.
We also have this relation,

290
00:21:41,000 --> 00:21:48,000
whatever the constraint was
relating x, y and z together.

291
00:21:48,000 --> 00:21:51,000
Then we can try to solve this.
And, depending on the

292
00:21:51,000 --> 00:21:56,000
situation, it is sometimes easy.
And it sometimes it is very

293
00:21:56,000 --> 00:22:01,000
hard or even impossible.
But on the test,

294
00:22:01,000 --> 00:22:03,000
I haven't decided yet,
but it could well be that the

295
00:22:03,000 --> 00:22:06,000
problem about Lagrange
multipliers just asks you to

296
00:22:06,000 --> 00:22:08,000
write the equations and not to
solve them.

297
00:22:08,000 --> 00:22:14,000
[APPLAUSE]
Well, I don't know yet.

298
00:22:14,000 --> 00:22:18,000
I am not promising anything.
But, before you start solving,

299
00:22:18,000 --> 00:22:23,000
check whether the problem asks
you to solve them or not.

300
00:22:23,000 --> 00:22:26,000
If it doesn't then probably you
shouldn't.

301
00:23:02,000 --> 00:23:09,000
Another topic that we solved
just yesterday is constrained

302
00:23:09,000 --> 00:23:13,000
partial derivatives.
And I guess I have to

303
00:23:13,000 --> 00:23:19,000
re-explain a little bit because
my guess is that things were not

304
00:23:19,000 --> 00:23:23,000
extremely clear at the end of
class yesterday.

305
00:23:23,000 --> 00:23:25,000
Now we are in the same
situation.

306
00:23:25,000 --> 00:23:29,000
We have a function,
let's say, f of x,

307
00:23:29,000 --> 00:23:34,000
y, z where variables x,
y and z are not independent but

308
00:23:34,000 --> 00:23:39,000
are constrained by some relation
of this form.

309
00:23:39,000 --> 00:23:43,000
Some quantity involving x,
y and z is equal to maybe zero

310
00:23:43,000 --> 00:23:47,000
or some other constant.
And then, what we want to know,

311
00:23:47,000 --> 00:23:51,000
is what is the rate of change
of f with respect to one of the

312
00:23:51,000 --> 00:23:57,000
variables,
say, x, y or z when I keep the

313
00:23:57,000 --> 00:24:02,000
others constant?
Well, I cannot keep all the

314
00:24:02,000 --> 00:24:07,000
other constant because that
would not be compatible with

315
00:24:07,000 --> 00:24:11,000
this condition.
I mean that would be the usual

316
00:24:11,000 --> 00:24:15,000
or so-called formal partial
derivative of f ignoring the

317
00:24:15,000 --> 00:24:18,000
constraint.
To take this into account means

318
00:24:18,000 --> 00:24:23,000
that if we vary one variable
while keeping another one fixed

319
00:24:23,000 --> 00:24:26,000
then the third one,
since it depends on them,

320
00:24:26,000 --> 00:24:31,000
must also change somehow.
And we must take that into

321
00:24:31,000 --> 00:24:34,000
account.
Let's say, for example,

322
00:24:34,000 --> 00:24:39,000
we want to find -- I am going
to do a different example from

323
00:24:39,000 --> 00:24:42,000
yesterday.
So, if you really didn't like

324
00:24:42,000 --> 00:24:46,000
that one, you don't have to see
it again.

325
00:24:46,000 --> 00:24:51,000
Let's say that we want to find
the partial derivative of f with

326
00:24:51,000 --> 00:24:56,000
respect to z keeping y constant.
What does that mean?

327
00:24:56,000 --> 00:25:03,000
That means y is constant,
z varies and x somehow is

328
00:25:03,000 --> 00:25:11,000
mysteriously a function of y and
z for this equation.

329
00:25:11,000 --> 00:25:14,000
And then, of course because it
depends on y,

330
00:25:14,000 --> 00:25:19,000
that means x will vary.
Sorry, depends on y and z and z

331
00:25:19,000 --> 00:25:21,000
varies.
Now we are asking ourselves

332
00:25:21,000 --> 00:25:25,000
what is the rate of change of f
with respect to z in this

333
00:25:25,000 --> 00:25:26,000
situation?

334
00:25:42,000 --> 00:25:47,000
And so we have two methods to
do that.

335
00:25:47,000 --> 00:25:55,000
Let me start with the one with
differentials that hopefully you

336
00:25:55,000 --> 00:26:02,000
kind of understood yesterday,
but if not here is a second

337
00:26:02,000 --> 00:26:06,000
chance.
Using differentials means that

338
00:26:06,000 --> 00:26:10,000
we will try to express df in
terms of dz in this particular

339
00:26:10,000 --> 00:26:14,000
situation.
What do we know about df in

340
00:26:14,000 --> 00:26:19,000
general?
Well, we know that df is f sub

341
00:26:19,000 --> 00:26:25,000
x dx plus f sub y dy plus f sub
z dz.

342
00:26:25,000 --> 00:26:28,000
That is the general statement.
But, of course,

343
00:26:28,000 --> 00:26:31,000
we are in a special case.
We are in a special case where

344
00:26:31,000 --> 00:26:41,000
first y is constant.
y is constant means that we can

345
00:26:41,000 --> 00:26:50,000
set dy to be zero.
This goes away and becomes zero.

346
00:26:50,000 --> 00:26:53,000
The second thing is actually we
don't care about x.

347
00:26:53,000 --> 00:26:57,000
We would like to get rid of x
because it is this dependent

348
00:26:57,000 --> 00:27:00,000
variable.
What we really want to do is

349
00:27:00,000 --> 00:27:12,000
express df only in terms of dz.
What we need is to relate dx

350
00:27:12,000 --> 00:27:16,000
with dz.
Well, to do that,

351
00:27:16,000 --> 00:27:20,000
we need to look at how the
variables are related so we need

352
00:27:20,000 --> 00:27:24,000
to look at the constraint g.
Well, how do we do that?

353
00:27:24,000 --> 00:27:31,000
We look at the differential g.
So dg is g sub x dx plus g sub

354
00:27:31,000 --> 00:27:37,000
y dy plus g sub z dz.
And that is zero because we are

355
00:27:37,000 --> 00:27:40,000
setting g to always stay
constant.

356
00:27:40,000 --> 00:27:44,000
So, g doesn't change.
If g doesn't change then we

357
00:27:44,000 --> 00:27:48,000
have a relation between dx,
dy and dz.

358
00:27:48,000 --> 00:27:50,000
Well, in fact,
we say we are going to look

359
00:27:50,000 --> 00:27:52,000
only at the case where y is
constant.

360
00:27:52,000 --> 00:27:56,000
y doesn't change and this
becomes zero.

361
00:27:56,000 --> 00:27:59,000
Well, now we have a relation
between dx and dz.

362
00:27:59,000 --> 00:28:05,000
We know how x depends on z.
And when we know how x depends

363
00:28:05,000 --> 00:28:10,000
on z, we can plug that into here
and get how f depends on z.

364
00:28:10,000 --> 00:28:11,000
Let's do that.

365
00:28:28,000 --> 00:28:33,000
Again, saying that g cannot
change and keeping y constant

366
00:28:33,000 --> 00:28:39,000
tells us g sub x dx plus g sub z
dz is zero and we would like to

367
00:28:39,000 --> 00:28:46,000
solve for dx in terms of dz.
That tells us dx should be

368
00:28:46,000 --> 00:28:53,000
minus g sub z dz divided by g
sub x.

369
00:28:53,000 --> 00:28:57,000
If you want,
this is the rate of change of x

370
00:28:57,000 --> 00:29:00,000
with respect to z when we keep y
constant.

371
00:29:00,000 --> 00:29:13,000
In our new terminology this is
partial x over partial z with y

372
00:29:13,000 --> 00:29:18,000
held constant.
This is the rate of change of x

373
00:29:18,000 --> 00:29:23,000
with respect to z.
Now, when we know that,

374
00:29:23,000 --> 00:29:30,000
we are going to plug that into
this equation.

375
00:29:30,000 --> 00:29:37,000
And that will tell us that df
is f sub x times dx.

376
00:29:37,000 --> 00:29:43,000
Well, what is dx?
dx is now minus g sub z over g

377
00:29:43,000 --> 00:29:51,000
sub x dz plus f sub z dz.
So that will be minus fx g sub

378
00:29:51,000 --> 00:29:56,000
z over g sub x plus f sub z
times dz.

379
00:29:56,000 --> 00:30:02,000
And so this coefficient here is
the rate of change of f with

380
00:30:02,000 --> 00:30:06,000
respect to z in the situation we
are considering.

381
00:30:06,000 --> 00:30:13,000
This quantity is what we call
partial f over partial z with y

382
00:30:13,000 --> 00:30:21,000
held constant.
That is what we wanted to find.

383
00:30:21,000 --> 00:30:25,000
Now, let's see another way to
do the same calculation and then

384
00:30:25,000 --> 00:30:28,000
you can choose which one you
prefer.

385
00:30:57,000 --> 00:31:09,000
The other method is using the
chain rule.

386
00:31:09,000 --> 00:31:14,000
We use the chain rule to
understand how f depends on z

387
00:31:14,000 --> 00:31:19,000
when y is held constant.
Let me first try the chain rule

388
00:31:19,000 --> 00:31:24,000
brutally and then we will try to
analyze what is going on.

389
00:31:24,000 --> 00:31:29,000
You can just use the version
that I have up there as a

390
00:31:29,000 --> 00:31:35,000
template to see what is going
on, but I am going to explain it

391
00:31:35,000 --> 00:31:37,000
more carefully again.

392
00:31:50,000 --> 00:31:57,000
That is the most mechanical and
mindless way of writing down the

393
00:31:57,000 --> 00:32:01,000
chain rule.
I am just saying here that I am

394
00:32:01,000 --> 00:32:04,000
varying z, keeping y constant,
and I want to know how f

395
00:32:04,000 --> 00:32:07,000
changes.
Well, f might change because x

396
00:32:07,000 --> 00:32:10,000
might change,
y might change and z might

397
00:32:10,000 --> 00:32:14,000
change.
Now, how quickly does x change?

398
00:32:14,000 --> 00:32:18,000
Well, the rate of change of x
in this situation is partial x,

399
00:32:18,000 --> 00:32:24,000
partial z with y held constant.
If I change x at this rate then

400
00:32:24,000 --> 00:32:29,000
f will change at that rate.
Now, y might change,

401
00:32:29,000 --> 00:32:32,000
so the rate of change of y
would be the rate of change of y

402
00:32:32,000 --> 00:32:35,000
with respect to z holding y
constant.

403
00:32:35,000 --> 00:32:38,000
Wait a second.
If y is held constant then y

404
00:32:38,000 --> 00:32:40,000
doesn't change.
So, actually,

405
00:32:40,000 --> 00:32:43,000
this guy is zero and you didn't
really have to write that term.

406
00:32:43,000 --> 00:32:47,000
But I wrote it just to be
systematic.

407
00:32:47,000 --> 00:32:51,000
If y had been somehow able to
change at a certain rate then

408
00:32:51,000 --> 00:32:54,000
that would have caused f to
change at that rate.

409
00:32:54,000 --> 00:32:57,000
And, of course,
if y is held constant then

410
00:32:57,000 --> 00:33:01,000
nothing happens here.
Finally, while z is changing at

411
00:33:01,000 --> 00:33:05,000
a certain rate,
this rate is this one and that

412
00:33:05,000 --> 00:33:10,000
causes f to change at that rate.
And then we add the effects

413
00:33:10,000 --> 00:33:12,000
together.
See, it is nothing but the

414
00:33:12,000 --> 00:33:16,000
good-old chain rule.
Just I have put these extra

415
00:33:16,000 --> 00:33:22,000
subscripts to tell us what is
held constant and what isn't.

416
00:33:22,000 --> 00:33:23,000
Now, of course we can simplify
it a little bit more.

417
00:33:23,000 --> 00:33:27,000
Because, here,
how quickly does z change if I

418
00:33:27,000 --> 00:33:32,000
am changing z?
Well, the rate of change of z,

419
00:33:32,000 --> 00:33:37,000
with respect to itself,
is just one.

420
00:33:37,000 --> 00:33:41,000
In fact, the really mysterious
part of this is the one here,

421
00:33:41,000 --> 00:33:45,000
which is the rate of change of
x with respect to z.

422
00:33:45,000 --> 00:33:49,000
And, to find that,
we have to understand the

423
00:33:49,000 --> 00:33:52,000
constraint.
How can we find the rate of

424
00:33:52,000 --> 00:33:54,000
change of x with respect to z?
Well, we could use

425
00:33:54,000 --> 00:33:56,000
differentials,
like we did here,

426
00:33:56,000 --> 00:33:58,000
but we can also keep using the
chain rule.

427
00:34:17,000 --> 00:34:20,000
How can I do that?
Well, I can just look at how g

428
00:34:20,000 --> 00:34:24,000
would change with respect to z
when y is held constant.

429
00:34:24,000 --> 00:34:33,000
I just do the same calculation
with g instead of f.

430
00:34:33,000 --> 00:34:37,000
But, before I do it,
let's ask ourselves first what

431
00:34:37,000 --> 00:34:40,000
is this equal to.
Well, if g is held constant

432
00:34:40,000 --> 00:34:44,000
then, when we vary z keeping y
constant and changing x,

433
00:34:44,000 --> 00:34:53,000
well, g still doesn't change.
It is held constant.

434
00:34:53,000 --> 00:34:58,000
In fact, that should be zero.
But, if we just say that,

435
00:34:58,000 --> 00:35:01,000
we are not going to get to
that.

436
00:35:01,000 --> 00:35:04,000
Let's see how we can compute
that using the chain rule.

437
00:35:04,000 --> 00:35:09,000
Well, the chain rule tells us g
changes because x,

438
00:35:09,000 --> 00:35:12,000
y and z change.
How does it change because of x?

439
00:35:12,000 --> 00:35:18,000
Well, partial g over partial x
times the rate of change of x.

440
00:35:18,000 --> 00:35:21,000
How does it change because of y?
Well, partial g over partial y

441
00:35:21,000 --> 00:35:24,000
times the rate of change of y.
But, of course,

442
00:35:24,000 --> 00:35:28,000
if you are smarter than me then
you don't need to actually write

443
00:35:28,000 --> 00:35:31,000
this one because y is held
constant.

444
00:35:31,000 --> 00:35:38,000
And then there is the rate of
change because z changes.

445
00:35:38,000 --> 00:35:45,000
And how quickly z changes here,
of course, is one.

446
00:35:45,000 --> 00:35:50,000
Out of this you get,
well, I am tired of writing

447
00:35:50,000 --> 00:35:58,000
partial g over partial x.
We can just write g sub x times

448
00:35:58,000 --> 00:36:05,000
partial x over partial z y
constant plus g sub z.

449
00:36:05,000 --> 00:36:11,000
And now we found how x depends
on z.

450
00:36:11,000 --> 00:36:17,000
Partial x over partial z with y
held constant is negative g sub

451
00:36:17,000 --> 00:36:24,000
z over g sub x.
Now we plug that into that and

452
00:36:24,000 --> 00:36:32,000
we get our answer.
It goes all the way up here.

453
00:36:32,000 --> 00:36:34,000
And then we get the answer.
I am not going to,

454
00:36:34,000 --> 00:36:35,000
well, I guess I can write it
again.

455
00:36:47,000 --> 00:36:52,000
There was partial f over
partial x times this guy,

456
00:36:52,000 --> 00:36:59,000
minus g sub z over g sub x,
plus partial f over partial z.

457
00:36:59,000 --> 00:37:03,000
And you can observe that this
is exactly the same formula that

458
00:37:03,000 --> 00:37:07,000
we had over here.
In fact, let's compare this to

459
00:37:07,000 --> 00:37:10,000
make it side by side.
I claim we did exactly the same

460
00:37:10,000 --> 00:37:13,000
thing, just with different
notations.

461
00:37:13,000 --> 00:37:17,000
If you take the differential of
f and you divide it by dz in

462
00:37:17,000 --> 00:37:20,000
this situation where y is held
constant and so on,

463
00:37:20,000 --> 00:37:23,000
you get exactly this chain rule
up there.

464
00:37:23,000 --> 00:37:28,000
That chain rule up there is
this guy, df,

465
00:37:28,000 --> 00:37:33,000
divided by dz with y held
constant.

466
00:37:33,000 --> 00:37:38,000
And the term involving dy was
replaced by zero on both sides

467
00:37:38,000 --> 00:37:41,000
because we knew,
actually, that y is held

468
00:37:41,000 --> 00:37:44,000
constant.
Now, the real difficulty in

469
00:37:44,000 --> 00:37:48,000
both cases comes from dx.
And what we do about dx is we

470
00:37:48,000 --> 00:37:52,000
use the constant.
Here we use it by writing dg

471
00:37:52,000 --> 00:37:55,000
equals zero.
Here we write the chain rule

472
00:37:55,000 --> 00:38:00,000
for g, which is the same thing,
just divided by dz with y held

473
00:38:00,000 --> 00:38:03,000
constant.
This formula or that formula

474
00:38:03,000 --> 00:38:07,000
are the same,
just divided by dz with y held

475
00:38:07,000 --> 00:38:11,000
constant.
And then, in both cases,

476
00:38:11,000 --> 00:38:16,000
we used that to solve for dx.
And then we plugged into the

477
00:38:16,000 --> 00:38:21,000
formula of df to express df over
dz, or partial f,

478
00:38:21,000 --> 00:38:26,000
partial z with y held constant.
So, the two methods are pretty

479
00:38:26,000 --> 00:38:27,000
much the same.
Quick poll.

480
00:38:27,000 --> 00:38:33,000
Who prefers this one?
Who prefers that one?

481
00:38:33,000 --> 00:38:34,000
OK.
Majority vote seems to be for

482
00:38:34,000 --> 00:38:36,000
differentials,
but it doesn't mean that it is

483
00:38:36,000 --> 00:38:39,000
better.
Both are fine.

484
00:38:39,000 --> 00:38:42,000
You can use whichever one you
want.

485
00:38:42,000 --> 00:38:50,000
But you should give both a try.
OK. Any questions?

486
00:38:50,000 --> 00:38:58,000
Yes?
Yes. Thank you.

487
00:38:58,000 --> 00:39:02,000
I forgot to mention it.
Where did that go?

488
00:39:02,000 --> 00:39:11,000
I think I erased that part.
We need to know -- --

489
00:39:11,000 --> 00:39:20,000
directional derivatives.
Pretty much the only thing to

490
00:39:20,000 --> 00:39:23,000
remember about them is that df
over ds,

491
00:39:23,000 --> 00:39:25,000
in the direction of some unit
vector u,

492
00:39:25,000 --> 00:39:30,000
is just the gradient f dot
product with u.

493
00:39:30,000 --> 00:39:35,000
That is pretty much all we know
about them.

494
00:39:35,000 --> 00:39:39,000
Any other topics that I forgot
to list?

495
00:39:39,000 --> 00:39:45,000
No.
Yes?

496
00:39:45,000 --> 00:39:46,000
Can I erase three boards at a
time?

497
00:39:46,000 --> 00:39:47,000
No, I would need three hands to
do that.

498
00:40:03,000 --> 00:40:07,000
I think what we should do now
is look quickly at the practice

499
00:40:07,000 --> 00:40:10,000
test.
I mean, given the time,

500
00:40:10,000 --> 00:40:15,000
you will mostly have to think
about it yourselves.

501
00:40:15,000 --> 00:40:23,000
Hopefully you have a copy of
the practice exam.

502
00:40:23,000 --> 00:40:26,000
The first problem is a simple
problem.

503
00:40:26,000 --> 00:40:28,000
Find the gradient.
Find an approximation formula.

504
00:40:28,000 --> 00:40:30,000
Hopefully you know how to do
that.

505
00:40:30,000 --> 00:40:33,000
The second problem is one about
writing a contour plot.

506
00:40:33,000 --> 00:40:41,000
And so, before I let you go for
the weekend, I want to make sure

507
00:40:41,000 --> 00:40:47,000
that you actually know how to
read a contour plot.

508
00:40:47,000 --> 00:40:51,000
One thing I should mention is
this problem asks you to

509
00:40:51,000 --> 00:40:55,000
estimate partial derivatives by
writing a contour plot.

510
00:40:55,000 --> 00:40:57,000
We have not done that,
so that will not actually be on

511
00:40:57,000 --> 00:40:59,000
the test.
We will be doing qualitative

512
00:40:59,000 --> 00:41:01,000
questions like what is the sine
of a partial derivative.

513
00:41:01,000 --> 00:41:04,000
Is it zero, less than zero or
more than zero?

514
00:41:04,000 --> 00:41:07,000
You don't need to bring a ruler
to estimate partial derivatives

515
00:41:07,000 --> 00:41:09,000
the way that this problem asks
you to.

516
00:41:35,000 --> 00:41:38,000
[APPLAUSE]
Let's look at problem 2B.

517
00:41:38,000 --> 00:41:43,000
Problem 2B is asking you to
find the point at which h equals

518
00:41:43,000 --> 00:41:46,000
2200,
partial h over partial x equals

519
00:41:46,000 --> 00:41:49,000
zero and partial h over partial
y is less than zero.

520
00:41:49,000 --> 00:41:53,000
Let's try and see what is going
on here.

521
00:41:53,000 --> 00:41:57,000
A point where f equals 2200,
well, that should be probably

522
00:41:57,000 --> 00:41:59,000
on the level curve that says
2200.

523
00:41:59,000 --> 00:42:09,000
We can actually zoom in.
Here is the level 2200.

524
00:42:09,000 --> 00:42:12,000
Now I want partial h over
partial x to be zero.

525
00:42:12,000 --> 00:42:17,000
That means if I change x,
keeping y constant,

526
00:42:17,000 --> 00:42:24,000
the value of h doesn't change.
Which points on the level curve

527
00:42:24,000 --> 00:42:30,000
satisfy that property?
It is the top and the bottom.

528
00:42:30,000 --> 00:42:34,000
If you are here, for example,
and you move in the x

529
00:42:34,000 --> 00:42:36,000
direction,
well, you see,

530
00:42:36,000 --> 00:42:38,000
as you get to there from the
left,

531
00:42:38,000 --> 00:42:41,000
the height first increases and
then decreases.

532
00:42:41,000 --> 00:42:44,000
It goes for a maximum at that
point.

533
00:42:44,000 --> 00:42:47,000
So, at that point,
the partial derivative is zero

534
00:42:47,000 --> 00:42:53,000
with respect to x.
And the same here.

535
00:42:53,000 --> 00:42:59,000
Now, let's find partial h over
partial y less than zero.

536
00:42:59,000 --> 00:43:03,000
That means if we go north we
should go down.

537
00:43:03,000 --> 00:43:07,000
Well, which one is it,
top or bottom?

538
00:43:07,000 --> 00:43:11,000
Top. Yes.
Here, if you go north,

539
00:43:11,000 --> 00:43:16,000
then you go from 2200 down to
2100.

540
00:43:16,000 --> 00:43:23,000
This is where the point is.
Now, the problem here was also

541
00:43:23,000 --> 00:43:25,000
asking you to estimate partial h
over partial y.

542
00:43:25,000 --> 00:43:28,000
And if you were curious how you
would do that,

543
00:43:28,000 --> 00:43:33,000
well, you would try to figure
out how long it takes before you

544
00:43:33,000 --> 00:43:42,000
reach the next level curve.
To go from here to here,

545
00:43:42,000 --> 00:43:47,000
to go from Q to this new point,
say Q prime,

546
00:43:47,000 --> 00:43:49,000
the change in y,
well, you would have to read

547
00:43:49,000 --> 00:43:56,000
the scale,
which was down here,

548
00:43:56,000 --> 00:44:00,000
would be about something like
300.

549
00:44:00,000 --> 00:44:04,000
What is the change in height
when you go from Q to Q prime?

550
00:44:04,000 --> 00:44:07,000
Well, you go down from 2200 to
2100.

551
00:44:07,000 --> 00:44:14,000
That is actually minus 100
exactly.

552
00:44:14,000 --> 00:44:19,000
OK?
And so delta h over delta y is

553
00:44:19,000 --> 00:44:27,000
about minus one-third,
well, minus 100 over 300 which

554
00:44:27,000 --> 00:44:35,000
is minus one-third.
And that is an approximation

555
00:44:35,000 --> 00:44:43,000
for partial derivative.
So, that is how you would do it.

556
00:44:43,000 --> 00:44:48,000
Now, let me go back to other
things.

557
00:44:48,000 --> 00:44:52,000
If you look at this practice
exam, basically there is a bit

558
00:44:52,000 --> 00:44:56,000
of everything and it is kind of
fairly representative of what

559
00:44:56,000 --> 00:45:00,000
might happen on Tuesday.
There will be a mix of easy

560
00:45:00,000 --> 00:45:03,000
problems and of harder problems.
Expect something about

561
00:45:03,000 --> 00:45:05,000
computing gradients,
approximations,

562
00:45:05,000 --> 00:45:08,000
rate of change.
Expect a problem about reading

563
00:45:08,000 --> 00:45:13,000
a contour plot.
Expect one about a min/max

564
00:45:13,000 --> 00:45:15,000
problem,
something about Lagrange

565
00:45:15,000 --> 00:45:17,000
multipliers,
something about the chain rule

566
00:45:17,000 --> 00:45:20,000
and something about constrained
partial derivatives.

567
00:45:20,000 --> 00:45:22,000
I mean pretty much all the
topics are going to be there.