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Let me start by basically
listing the main things we have

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00:00:25 --> 00:00:28
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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00:00:31 --> 00:00:34
that because there are a few
small details that I didn't

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00:00:34 --> 00:00:38
quite clarify and that I should
probably make a bit clearer,

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00:00:38 --> 00:00:48
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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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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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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partial of f with respect to
some variable,

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00:01:47 --> 00:01:52
say, x to be the rate of change
with respect to x when we hold

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00:01:52 --> 00:01:55
all the other variables
constant.

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

27
00:02:01 --> 00:02:07
differentiate with respect to x
treating y as a constant.

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

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00:02:15 --> 00:02:20
a vector,the gradient vector.
For example,

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00:02:20 --> 00:02:24
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 --> 00:02:33
And we have seen how to use the
gradient vector or the partial

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derivatives to derive various
things such as approximation

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00:02:39 --> 00:02:43
formulas.
The change in f,

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00:02:43 --> 00:02:48
when we change x,
y, z slightly,

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00:02:48 --> 00:02:57
is approximately equal to,
well, there are several terms.

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00:02:57 --> 00:03:03
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 --> 00:03:11
Basically, what causes f to
change is that I am changing x,

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

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00:03:16 --> 00:03:22
variable is precisely what the
partial derivatives measure.

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

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00:03:26 --> 00:03:30
the tangent plane approximation
because it tells us,

44
00:03:30 --> 00:03:35
in fact, 
it amounts to identifying the

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00:03:35 --> 00:03:38
graph of the function with its
tangent plane.

46
00:03:38 --> 00:03:43
It means that we assume that
the function depends more or

47
00:03:43 --> 00:03:45
less linearly on x,
y and z.

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

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00:03:48 --> 00:03:52
we are replacing the function
by its linear approximation.

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00:03:52 --> 00:03:56
We are replacing the graph by
its tangent plane.

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00:03:56 --> 00:03:58
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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00:04:00 --> 00:04:04
4-dimensional space.
So, when we think of a graph,

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

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00:04:08 --> 00:04:12
That also tells us how to find
tangent planes to level

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surfaces.

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00:04:12 --> 00:04:22

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00:04:22 --> 00:04:30
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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00:04:37 --> 00:04:43
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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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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00:05:50 --> 00:05:51
equations.

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00:05:51 --> 00:05:59

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

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

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00:06:09 --> 00:06:13
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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00:06:15 --> 00:06:18
is unknown that depends on a
bunch of variables.

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00:06:18 --> 00:06:23
And a partial differential
equation is some relation

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00:06:23 --> 00:06:28
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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00:06:57 --> 00:07:02
the heat equation is one
example of a partial

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

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00:07:09 --> 00:07:15
let me write for you the space
version of it.

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00:07:15 --> 00:07:21
It is the equation partial f
over partial t equals some

96
00:07:21 --> 00:07:27
constant times the sum of the
second partials with respect to

97
00:07:27 --> 00:07:32
x, y and z.
So this is an equation where we

98
00:07:32 --> 00:07:38
are trying to solve for a
function f that depends,

99
00:07:38 --> 00:07:42
actually, on four variables,
x, y, z, t.

100
00:07:42 --> 00:07:47
And what should you have in
mind?

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00:07:47 --> 00:07:50
Well, this equation governs
temperature.

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00:07:50 --> 00:07:55
If you think that f of x, y, z, 
t will be the temperature at a

103
00:07:55 --> 00:07:59
point in space at position x,
y, z and at time t,

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

105
00:08:04 --> 00:08:07
It tells you that at any given
point,

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

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

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00:08:15 --> 00:08:18
terms of the space coordinates
x, y, z.

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

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00:08:21 --> 00:08:24
temperature in this room,
and if you assume that nothing

111
00:08:24 --> 00:08:26
is generating heat or taking
heat away,

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

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00:08:29 --> 00:08:31
on,
then it will tell you how the

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00:08:31 --> 00:08:35
temperature will change over
time and eventually stabilize to

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00:08:35 --> 00:08:41
some final value.
Yes?

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00:08:41 --> 00:08:43
Why do we take the partial
derivative twice?

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00:08:43 --> 00:08:45
Well, that is a question,
I would say,

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00:08:45 --> 00:08:48
for a physics person.
But in a few weeks we will

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00:08:48 --> 00:08:52
actually see a derivation of
where this equation comes from

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00:08:52 --> 00:08:55
and try to justify it.
But, really,

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00:08:55 --> 00:08:57
that is something you will see
in a physics class.

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00:08:57 --> 00:09:02
The reason for that is
basically physics of how heat is

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

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

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

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00:09:17 --> 00:09:20
you are looking at.
Anyway, I am giving it to you

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

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00:09:23 --> 00:09:26
in fact, you have to solve one
of these things.

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

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00:09:29 --> 00:09:32
topic for this class.
It is not even a topic for

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00:09:32 --> 00:09:34
18.03 which is called
Differential Equations,

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

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

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

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

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

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00:09:50 --> 00:09:52
about partial differential
equations.

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

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00:09:58 --> 00:10:03
that very often functions that
you see in real life satisfy

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

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

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

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

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

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

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

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

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

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00:10:33 --> 00:10:42
I should have written down that
this equation is solved by

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

151
00:10:48 --> 00:10:52
OK.
And there are, actually, 

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

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00:10:56 --> 00:10:59
maybe sometimes learn about the
wave equation that governs how

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00:10:59 --> 00:11:02
waves propagate in space,
about the diffusion equation, 

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

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

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

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

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

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00:11:23 --> 00:11:27
One important application we
have seen of partial derivatives

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00:11:27 --> 00:11:30
is to try to optimize things,
try to solve minimum/maximum

162
00:11:30 --> 00:11:31
problems.

163
00:11:31 --> 00:11:42

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00:11:42 --> 00:11:47
Remember that we have
introduced the notion of

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00:11:47 --> 00:11:56
critical points of a function.
A critical point is when all

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00:11:56 --> 00:12:03
the partial derivatives are
zero.

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

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

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

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00:12:15 --> 00:12:24
second derivatives -- -- to
decide which kind of critical

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

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

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

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

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00:12:44 --> 00:12:48
find the minimum of a maximum of
a function because the minimum

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

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

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

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

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

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

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

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

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

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

186
00:13:18 --> 00:13:20
maximum.
But then, when we are looking

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00:13:20 --> 00:13:23
for the minimum of a function,
well, it is not at a critical

188
00:13:23 --> 00:13:26
point.
It is actually here at the

189
00:13:26 --> 00:13:30
boundary of the domain,
you know, the range of values

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

191
00:13:38 --> 00:13:44
boundary.
And the maximum is at a

192
00:13:44 --> 00:13:50
critical point.
Similarly, when you have a

193
00:13:50 --> 00:13:53
function of several variables,
say of two variables,

194
00:13:53 --> 00:13:55
for example,
then the minimum and the

195
00:13:55 --> 00:13:58
maximum will be achieved either
at a critical point.

196
00:13:58 --> 00:14:01
And then we can use these
methods to find where they are.

197
00:14:01 --> 00:14:06
Or, somewhere on the boundary
of a set of values that are

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

199
00:14:09 --> 00:14:13
achieve a minimum by making x
and y as small as possible.

200
00:14:13 --> 00:14:16
Maybe letting them go to zero
if they had to be positive or

201
00:14:16 --> 00:14:19
maybe by making them go to
infinity.

202
00:14:19 --> 00:14:23
So, we have to keep our minds
open and look at various

203
00:14:23 --> 00:14:26
possibilities.
We are going to do a problem

204
00:14:26 --> 00:14:29
like that.
We are going to go over a

205
00:14:29 --> 00:14:34
practice problem from the
practice test to clarify this.

206
00:14:34 --> 00:14:38
Another important cultural
application of minimum/maximum

207
00:14:38 --> 00:14:42
problems in two variables that
we have seen in class is the

208
00:14:42 --> 00:14:45
least squared method to find the
best fit line,

209
00:14:45 --> 00:14:49
or the best fit anything,
really,

210
00:14:49 --> 00:14:56
to find when you have a set of
data points what is the best

211
00:14:56 --> 00:15:01
linear approximately for these
data points.

212
00:15:01 --> 00:15:03
And here I have some good news
for you.

213
00:15:03 --> 00:15:07
While you should definitely
know what this is about,

214
00:15:07 --> 00:15:09
it will not be on the test.

215
00:15:09 --> 00:15:30

216
00:15:30 --> 00:15:35
[APPLAUSE]
That doesn't mean that you

217
00:15:35 --> 00:15:41
should forget everything we have
seen about it,

218
00:15:41 --> 00:15:51
OK?
Now what is next on my list of

219
00:15:51 --> 00:15:58
topics?
We have seen differentials.

220
00:15:58 --> 00:16:03
Remember the differential of f,
by definition,

221
00:16:03 --> 00:16:09
would be this kind of quantity.
At first it looks just like a

222
00:16:09 --> 00:16:12
new way to package partial
derivatives together into some

223
00:16:12 --> 00:16:15
new kind of object.
Now, what is this good for?

224
00:16:15 --> 00:16:18
Well, it is a good way to
remember approximation formulas.

225
00:16:18 --> 00:16:22
It is a good way to also study
how variations in x,

226
00:16:22 --> 00:16:26
y, z relate to variations in f.
In particular, 

227
00:16:26 --> 00:16:30
we can divide this by
variations,

228
00:16:30 --> 00:16:34
actually, by dx or by dy or by
dz in any situation that we

229
00:16:34 --> 00:16:40
want,
or by d of some other variable

230
00:16:40 --> 00:16:46
to get chain rules.
The chain rule says,

231
00:16:46 --> 00:16:50
for example,
there are many situations.

232
00:16:50 --> 00:16:56
But, for example, 
if x, y and z depend on some

233
00:16:56 --> 00:17:04
other variable,
say of variables maybe even u

234
00:17:04 --> 00:17:08
and v,
then that means that f becomes

235
00:17:08 --> 00:17:13
a function of u and v.
And then we can ask ourselves,

236
00:17:13 --> 00:17:18
how sensitive is f to a value
of u?

237
00:17:18 --> 00:17:25
Well, we can answer that.
The chain rule is something

238
00:17:25 --> 00:17:33
like this.
And let me explain to you again

239
00:17:33 --> 00:17:41
where this comes from.
Basically, what this quantity

240
00:17:41 --> 00:17:46
means is if we change u and keep
v constant, what happens to the

241
00:17:46 --> 00:17:48
value of f?
Well, why would the value of f

242
00:17:48 --> 00:17:51
change in the first place when f
is just a function of x,

243
00:17:51 --> 00:17:55
y, z and not directly of you?
Well, it changes because x,

244
00:17:55 --> 00:17:59
y and z depend on u.
First we have to figure out how

245
00:17:59 --> 00:18:02
quickly x, y and z change when
we change u.

246
00:18:02 --> 00:18:05
Well, how quickly they do that
is precisely partial x over

247
00:18:05 --> 00:18:08
partial u, partial y over
partial u, partial z over

248
00:18:08 --> 00:18:10
partial u.
These are the rates of change

249
00:18:10 --> 00:18:14
of x, y, z when we change u.
And now, when we change x,

250
00:18:14 --> 00:18:17
y and z, that causes f to
change.

251
00:18:17 --> 00:18:21
How much does f change?
Well, partial f over partial x

252
00:18:21 --> 00:18:25
tells us how quickly f changes
if I just change x.

253
00:18:25 --> 00:18:29
I get this.
That is the change in f caused

254
00:18:29 --> 00:18:33
just by the fact that x changes
when u changes.

255
00:18:33 --> 00:18:36
But then y also changes.
y changes at this rate.

256
00:18:36 --> 00:18:39
And that causes f to change at
that rate.

257
00:18:39 --> 00:18:42
And z changes as well,
and that causes f to change at

258
00:18:42 --> 00:18:45
that rate.
And the effects add up together.

259
00:18:45 --> 00:18:57
Does that make sense?
OK.

260
00:18:57 --> 00:19:00
And so, in particular,
we can use the chain rule to do

261
00:19:00 --> 00:19:03
changes of variables.
If we have, say, 

262
00:19:03 --> 00:19:08
a function in terms of polar
coordinates on theta and we like

263
00:19:08 --> 00:19:14
to switch it to rectangular
coordinates x and y then we can

264
00:19:14 --> 00:19:19
use chain rules to relate the
partial derivatives.

265
00:19:19 --> 00:19:23
And finally,
last but not least,

266
00:19:23 --> 00:19:31
we have seen how to deal with
non-independent variables.

267
00:19:31 --> 00:19:37
When our variables say x,
y, z related by some equation.

268
00:19:37 --> 00:19:41
One way we can deal with this
is to solve for one of the

269
00:19:41 --> 00:19:44
variables and go back to two
independent variables,

270
00:19:44 --> 00:19:47
but we cannot always do that.
Of course, on the exam,

271
00:19:47 --> 00:19:50
you can be sure that I will
make sure that you cannot solve

272
00:19:50 --> 00:19:53
for a variable you want to
remove because that would be too

273
00:19:53 --> 00:19:56
easy.
Then when we have to look at

274
00:19:56 --> 00:20:02
all of them, we will have to
take into account this relation,

275
00:20:02 --> 00:20:05
we have seen two useful
methods.

276
00:20:05 --> 00:20:09
One of them is to find the
minimum of a maximum of a

277
00:20:09 --> 00:20:13
function when the variables are
not independent,

278
00:20:13 --> 00:20:17
and that is the method of
Lagrange multipliers.

279
00:20:17 --> 00:20:33
 
 

280
00:20:33 --> 00:20:39
Remember, to find the minimum
or the maximum of the function

281
00:20:39 --> 00:20:45
f,
subject to the constraint g

282
00:20:45 --> 00:20:52
equals constant,
well, we write down equations

283
00:20:52 --> 00:20:59
that say that the gradient of f
is actually proportional to the

284
00:20:59 --> 00:21:04
gradient of g.
There is a new variable here,

285
00:21:04 --> 00:21:08
lambda, the multiplier.
And so, for example,

286
00:21:08 --> 00:21:12
well, I guess here I had
functions of three variables,

287
00:21:12 --> 00:21:14
so this becomes three
equations.

288
00:21:14 --> 00:21:21
f sub x equals lambda g sub x,
f sub y equals lambda g sub y,

289
00:21:21 --> 00:21:25
and f sub z equals lambda g sub
z.

290
00:21:25 --> 00:21:27
And, when we plug in the
formulas for f and g,

291
00:21:27 --> 00:21:31
well, we are left with three
equations involving the four

292
00:21:31 --> 00:21:33
variables, x,
y, z and lambda.

293
00:21:33 --> 00:21:36
What is wrong?
Well, we don't have actually

294
00:21:36 --> 00:21:41
four independent variables.
We also have this relation,

295
00:21:41 --> 00:21:48
whatever the constraint was
relating x, y and z together.

296
00:21:48 --> 00:21:51
Then we can try to solve this.
And, depending on the

297
00:21:51 --> 00:21:56
situation, it is sometimes easy.
And it sometimes it is very

298
00:21:56 --> 00:22:01
hard or even impossible.
But on the test, 

299
00:22:01 --> 00:22:03
I haven't decided yet, 
but it could well be that the

300
00:22:03 --> 00:22:06
problem about Lagrange
multipliers just asks you to

301
00:22:06 --> 00:22:08
write the equations and not to
solve them.

302
00:22:08 --> 00:22:14
[APPLAUSE]
Well, I don't know yet.

303
00:22:14 --> 00:22:18
I am not promising anything.
But, before you start solving,

304
00:22:18 --> 00:22:23
check whether the problem asks
you to solve them or not.

305
00:22:23 --> 00:22:26
If it doesn't then probably you
shouldn't.

306
00:22:26 --> 00:23:02
 
 

307
00:23:02 --> 00:23:09
Another topic that we solved
just yesterday is constrained

308
00:23:09 --> 00:23:13
partial derivatives.
And I guess I have to

309
00:23:13 --> 00:23:19
re-explain a little bit because
my guess is that things were not

310
00:23:19 --> 00:23:23
extremely clear at the end of
class yesterday.

311
00:23:23 --> 00:23:25
Now we are in the same
situation.

312
00:23:25 --> 00:23:29
We have a function,
let's say, f of x,

313
00:23:29 --> 00:23:34
y, z where variables x,
y and z are not independent but

314
00:23:34 --> 00:23:39
are constrained by some relation
of this form.

315
00:23:39 --> 00:23:43
Some quantity involving x,
y and z is equal to maybe zero

316
00:23:43 --> 00:23:47
or some other constant.
And then, what we want to know, 

317
00:23:47 --> 00:23:51
is what is the rate of change
of f with respect to one of the

318
00:23:51 --> 00:23:57
variables,
say, x, y or z when I keep the

319
00:23:57 --> 00:24:02
others constant?
Well, I cannot keep all the

320
00:24:02 --> 00:24:07
other constant because that
would not be compatible with

321
00:24:07 --> 00:24:11
this condition.
I mean that would be the usual

322
00:24:11 --> 00:24:15
or so-called formal partial
derivative of f ignoring the

323
00:24:15 --> 00:24:18
constraint.
To take this into account means

324
00:24:18 --> 00:24:23
that if we vary one variable
while keeping another one fixed

325
00:24:23 --> 00:24:26
then the third one,
since it depends on them, 

326
00:24:26 --> 00:24:31
must also change somehow.
And we must take that into

327
00:24:31 --> 00:24:34
account.
Let's say, for example,

328
00:24:34 --> 00:24:39
we want to find -- I am going
to do a different example from

329
00:24:39 --> 00:24:42
yesterday.
So, if you really didn't like

330
00:24:42 --> 00:24:46
that one, you don't have to see
it again.

331
00:24:46 --> 00:24:51
Let's say that we want to find
the partial derivative of f with

332
00:24:51 --> 00:24:56
respect to z keeping y constant.
What does that mean?

333
00:24:56 --> 00:25:03
That means y is constant,
z varies and x somehow is

334
00:25:03 --> 00:25:11
mysteriously a function of y and
z for this equation.

335
00:25:11 --> 00:25:14
And then, of course because it
depends on y,

336
00:25:14 --> 00:25:19
that means x will vary.
Sorry, depends on y and z and z

337
00:25:19 --> 00:25:21
varies.
Now we are asking ourselves

338
00:25:21 --> 00:25:25
what is the rate of change of f
with respect to z in this

339
00:25:25 --> 00:25:26
situation?

340
00:25:26 --> 00:25:42

341
00:25:42 --> 00:25:47
And so we have two methods to
do that.

342
00:25:47 --> 00:25:55
Let me start with the one with
differentials that hopefully you

343
00:25:55 --> 00:26:02
kind of understood yesterday,
but if not here is a second

344
00:26:02 --> 00:26:06
chance.
Using differentials means that

345
00:26:06 --> 00:26:10
we will try to express df in
terms of dz in this particular

346
00:26:10 --> 00:26:14
situation.
What do we know about df in

347
00:26:14 --> 00:26:19
general?
Well, we know that df is f sub

348
00:26:19 --> 00:26:25
x dx plus f sub y dy plus f sub
z dz.

349
00:26:25 --> 00:26:28
That is the general statement.
But, of course,

350
00:26:28 --> 00:26:31
we are in a special case.
We are in a special case where

351
00:26:31 --> 00:26:41
first y is constant.
y is constant means that we can

352
00:26:41 --> 00:26:50
set dy to be zero.
This goes away and becomes zero.

353
00:26:50 --> 00:26:53
The second thing is actually we
don't care about x.

354
00:26:53 --> 00:26:57
We would like to get rid of x
because it is this dependent

355
00:26:57 --> 00:27:00
variable.
What we really want to do is

356
00:27:00 --> 00:27:12
express df only in terms of dz.
What we need is to relate dx

357
00:27:12 --> 00:27:16
with dz.
Well, to do that,

358
00:27:16 --> 00:27:20
we need to look at how the
variables are related so we need

359
00:27:20 --> 00:27:24
to look at the constraint g.
Well, how do we do that?

360
00:27:24 --> 00:27:31
We look at the differential g.
So dg is g sub x dx plus g sub

361
00:27:31 --> 00:27:37
y dy plus g sub z dz.
And that is zero because we are

362
00:27:37 --> 00:27:40
setting g to always stay
constant.

363
00:27:40 --> 00:27:44
So, g doesn't change.
If g doesn't change then we

364
00:27:44 --> 00:27:48
have a relation between dx,
dy and dz.

365
00:27:48 --> 00:27:50
Well, in fact,
we say we are going to look

366
00:27:50 --> 00:27:52
only at the case where y is
constant.

367
00:27:52 --> 00:27:56
y doesn't change and this
becomes zero.

368
00:27:56 --> 00:27:59
Well, now we have a relation
between dx and dz.

369
00:27:59 --> 00:28:05
We know how x depends on z.
And when we know how x depends

370
00:28:05 --> 00:28:10
on z, we can plug that into here
and get how f depends on z.

371
00:28:10 --> 00:28:11
Let's do that.

372
00:28:11 --> 00:28:28

373
00:28:28 --> 00:28:33
Again, saying that g cannot
change and keeping y constant

374
00:28:33 --> 00:28:39
tells us g sub x dx plus g sub z
dz is zero and we would like to

375
00:28:39 --> 00:28:46
solve for dx in terms of dz.
That tells us dx should be

376
00:28:46 --> 00:28:53
minus g sub z dz divided by g
sub x.

377
00:28:53 --> 00:28:57
If you want,
this is the rate of change of x

378
00:28:57 --> 00:29:00
with respect to z when we keep y
constant.

379
00:29:00 --> 00:29:13
In our new terminology this is
partial x over partial z with y

380
00:29:13 --> 00:29:18
held constant.
This is the rate of change of x

381
00:29:18 --> 00:29:23
with respect to z.
Now, when we know that,

382
00:29:23 --> 00:29:30
we are going to plug that into
this equation.

383
00:29:30 --> 00:29:37
And that will tell us that df
is f sub x times dx.

384
00:29:37 --> 00:29:43
Well, what is dx?
dx is now minus g sub z over g

385
00:29:43 --> 00:29:51
sub x dz plus f sub z dz.
So that will be minus fx g sub

386
00:29:51 --> 00:29:56
z over g sub x plus f sub z
times dz.

387
00:29:56 --> 00:30:02
And so this coefficient here is
the rate of change of f with

388
00:30:02 --> 00:30:06
respect to z in the situation we
are considering.

389
00:30:06 --> 00:30:13
This quantity is what we call
partial f over partial z with y

390
00:30:13 --> 00:30:21
held constant.
That is what we wanted to find.

391
00:30:21 --> 00:30:25
Now, let's see another way to
do the same calculation and then

392
00:30:25 --> 00:30:28
you can choose which one you
prefer.

393
00:30:28 --> 00:30:57
 
 

394
00:30:57 --> 00:31:09
The other method is using the
chain rule.

395
00:31:09 --> 00:31:14
We use the chain rule to
understand how f depends on z

396
00:31:14 --> 00:31:19
when y is held constant.
Let me first try the chain rule

397
00:31:19 --> 00:31:24
brutally and then we will try to
analyze what is going on.

398
00:31:24 --> 00:31:29
You can just use the version
that I have up there as a

399
00:31:29 --> 00:31:35
template to see what is going
on, but I am going to explain it

400
00:31:35 --> 00:31:37
more carefully again.

401
00:31:37 --> 00:31:50

402
00:31:50 --> 00:31:57
That is the most mechanical and
mindless way of writing down the

403
00:31:57 --> 00:32:01
chain rule.
I am just saying here that I am

404
00:32:01 --> 00:32:04
varying z, keeping y constant,
and I want to know how f

405
00:32:04 --> 00:32:07
changes.
Well, f might change because x

406
00:32:07 --> 00:32:10
might change,
y might change and z might

407
00:32:10 --> 00:32:14
change.
Now, how quickly does x change?

408
00:32:14 --> 00:32:18
Well, the rate of change of x
in this situation is partial x,

409
00:32:18 --> 00:32:24
partial z with y held constant.
If I change x at this rate then

410
00:32:24 --> 00:32:29
f will change at that rate.
Now, y might change,

411
00:32:29 --> 00:32:32
so the rate of change of y
would be the rate of change of y

412
00:32:32 --> 00:32:35
with respect to z holding y
constant.

413
00:32:35 --> 00:32:38
Wait a second.
If y is held constant then y

414
00:32:38 --> 00:32:40
doesn't change.
So, actually,

415
00:32:40 --> 00:32:43
this guy is zero and you didn't
really have to write that term.

416
00:32:43 --> 00:32:47
But I wrote it just to be
systematic.

417
00:32:47 --> 00:32:51
If y had been somehow able to
change at a certain rate then

418
00:32:51 --> 00:32:54
that would have caused f to
change at that rate.

419
00:32:54 --> 00:32:57
And, of course,
if y is held constant then

420
00:32:57 --> 00:33:01
nothing happens here.
Finally, while z is changing at

421
00:33:01 --> 00:33:05
a certain rate,
this rate is this one and that

422
00:33:05 --> 00:33:10
causes f to change at that rate.
And then we add the effects

423
00:33:10 --> 00:33:12
together.
See, it is nothing but the

424
00:33:12 --> 00:33:16
good-old chain rule.
Just I have put these extra

425
00:33:16 --> 00:33:22
subscripts to tell us what is
held constant and what isn't.

426
00:33:22 --> 00:33:23
Now, of course we can simplify
it a little bit more.

427
00:33:23 --> 00:33:27
Because, here,
how quickly does z change if I

428
00:33:27 --> 00:33:32
am changing z?
Well, the rate of change of z,

429
00:33:32 --> 00:33:37
with respect to itself,
is just one.

430
00:33:37 --> 00:33:41
In fact, the really mysterious
part of this is the one here,

431
00:33:41 --> 00:33:45
which is the rate of change of
x with respect to z.

432
00:33:45 --> 00:33:49
And, to find that,
we have to understand the

433
00:33:49 --> 00:33:52
constraint.
How can we find the rate of

434
00:33:52 --> 00:33:54
change of x with respect to z?
Well, we could use

435
00:33:54 --> 00:33:56
differentials,
like we did here,

436
00:33:56 --> 00:33:58
but we can also keep using the
chain rule.

437
00:33:58 --> 00:34:17
 
 

438
00:34:17 --> 00:34:20
How can I do that?
Well, I can just look at how g

439
00:34:20 --> 00:34:24
would change with respect to z
when y is held constant.

440
00:34:24 --> 00:34:33
I just do the same calculation
with g instead of f.

441
00:34:33 --> 00:34:37
But, before I do it,
let's ask ourselves first what

442
00:34:37 --> 00:34:40
is this equal to.
Well, if g is held constant

443
00:34:40 --> 00:34:44
then, when we vary z keeping y
constant and changing x,

444
00:34:44 --> 00:34:53
well, g still doesn't change.
It is held constant.

445
00:34:53 --> 00:34:58
In fact, that should be zero.
But, if we just say that,

446
00:34:58 --> 00:35:01
we are not going to get to
that.

447
00:35:01 --> 00:35:04
Let's see how we can compute
that using the chain rule.

448
00:35:04 --> 00:35:09
Well, the chain rule tells us g
changes because x,

449
00:35:09 --> 00:35:12
y and z change.
How does it change because of x?

450
00:35:12 --> 00:35:18
Well, partial g over partial x
times the rate of change of x.

451
00:35:18 --> 00:35:21
How does it change because of y?
Well, partial g over partial y

452
00:35:21 --> 00:35:24
times the rate of change of y.
But, of course,

453
00:35:24 --> 00:35:28
if you are smarter than me then
you don't need to actually write

454
00:35:28 --> 00:35:31
this one because y is held
constant.

455
00:35:31 --> 00:35:38
And then there is the rate of
change because z changes.

456
00:35:38 --> 00:35:45
And how quickly z changes here,
of course, is one.

457
00:35:45 --> 00:35:50
Out of this you get,
well, I am tired of writing

458
00:35:50 --> 00:35:58
partial g over partial x.
We can just write g sub x times

459
00:35:58 --> 00:36:05
partial x over partial z y
constant plus g sub z.

460
00:36:05 --> 00:36:11
And now we found how x depends
on z.

461
00:36:11 --> 00:36:17
Partial x over partial z with y
held constant is negative g sub

462
00:36:17 --> 00:36:24
z over g sub x.
Now we plug that into that and

463
00:36:24 --> 00:36:32
we get our answer.
It goes all the way up here.

464
00:36:32 --> 00:36:34
And then we get the answer.
I am not going to,

465
00:36:34 --> 00:36:35
well, I guess I can write it
again.

466
00:36:35 --> 00:36:47
 
 

467
00:36:47 --> 00:36:52
There was partial f over
partial x times this guy,

468
00:36:52 --> 00:36:59
minus g sub z over g sub x,
plus partial f over partial z.

469
00:36:59 --> 00:37:03
And you can observe that this
is exactly the same formula that

470
00:37:03 --> 00:37:07
we had over here.
In fact, let's compare this to

471
00:37:07 --> 00:37:10
make it side by side.
I claim we did exactly the same

472
00:37:10 --> 00:37:13
thing, just with different
notations.

473
00:37:13 --> 00:37:17
If you take the differential of
f and you divide it by dz in

474
00:37:17 --> 00:37:20
this situation where y is held
constant and so on,

475
00:37:20 --> 00:37:23
you get exactly this chain rule
up there.

476
00:37:23 --> 00:37:28
That chain rule up there is
this guy, df,

477
00:37:28 --> 00:37:33
divided by dz with y held
constant.

478
00:37:33 --> 00:37:38
And the term involving dy was
replaced by zero on both sides

479
00:37:38 --> 00:37:41
because we knew,
actually, that y is held

480
00:37:41 --> 00:37:44
constant.
Now, the real difficulty in

481
00:37:44 --> 00:37:48
both cases comes from dx.
And what we do about dx is we

482
00:37:48 --> 00:37:52
use the constant.
Here we use it by writing dg

483
00:37:52 --> 00:37:55
equals zero.
Here we write the chain rule

484
00:37:55 --> 00:38:00
for g, which is the same thing,
just divided by dz with y held

485
00:38:00 --> 00:38:03
constant.
This formula or that formula

486
00:38:03 --> 00:38:07
are the same,
just divided by dz with y held

487
00:38:07 --> 00:38:11
constant.
And then, in both cases,

488
00:38:11 --> 00:38:16
we used that to solve for dx.
And then we plugged into the

489
00:38:16 --> 00:38:21
formula of df to express df over
dz, or partial f,

490
00:38:21 --> 00:38:26
partial z with y held constant.
So, the two methods are pretty

491
00:38:26 --> 00:38:27
much the same.
Quick poll.

492
00:38:27 --> 00:38:33
Who prefers this one?
Who prefers that one?

493
00:38:33 --> 00:38:34
OK.
Majority vote seems to be for

494
00:38:34 --> 00:38:36
differentials,
but it doesn't mean that it is

495
00:38:36 --> 00:38:39
better.
Both are fine.

496
00:38:39 --> 00:38:42
You can use whichever one you
want.

497
00:38:42 --> 00:38:50
But you should give both a try.
OK. Any questions?

498
00:38:50 --> 00:38:58
Yes?
Yes. Thank you.

499
00:38:58 --> 00:39:02
I forgot to mention it.
Where did that go?

500
00:39:02 --> 00:39:11
I think I erased that part.
We need to know -- --

501
00:39:11 --> 00:39:20
directional derivatives.
Pretty much the only thing to

502
00:39:20 --> 00:39:23
remember about them is that df
over ds,

503
00:39:23 --> 00:39:25
in the direction of some unit
vector u,

504
00:39:25 --> 00:39:30
is just the gradient f dot
product with u.

505
00:39:30 --> 00:39:35
That is pretty much all we know
about them.

506
00:39:35 --> 00:39:39
Any other topics that I forgot
to list?

507
00:39:39 --> 00:39:45
No.
Yes?

508
00:39:45 --> 00:39:46
Can I erase three boards at a
time?

509
00:39:46 --> 00:39:47
No, I would need three hands to
do that.

510
00:39:47 --> 00:40:03
 
 

511
00:40:03 --> 00:40:07
I think what we should do now
is look quickly at the practice

512
00:40:07 --> 00:40:10
test.
I mean, given the time,

513
00:40:10 --> 00:40:15
you will mostly have to think
about it yourselves.

514
00:40:15 --> 00:40:23
Hopefully you have a copy of
the practice exam.

515
00:40:23 --> 00:40:26
The first problem is a simple
problem.

516
00:40:26 --> 00:40:28
Find the gradient.
Find an approximation formula.

517
00:40:28 --> 00:40:30
Hopefully you know how to do
that.

518
00:40:30 --> 00:40:33
The second problem is one about
writing a contour plot.

519
00:40:33 --> 00:40:41
And so, before I let you go for
the weekend, I want to make sure

520
00:40:41 --> 00:40:47
that you actually know how to
read a contour plot.

521
00:40:47 --> 00:40:51
One thing I should mention is
this problem asks you to

522
00:40:51 --> 00:40:55
estimate partial derivatives by
writing a contour plot.

523
00:40:55 --> 00:40:57
We have not done that,
so that will not actually be on

524
00:40:57 --> 00:40:59
the test.
We will be doing qualitative

525
00:40:59 --> 00:41:01
questions like what is the sine
of a partial derivative.

526
00:41:01 --> 00:41:04
Is it zero, less than zero or
more than zero?

527
00:41:04 --> 00:41:07
You don't need to bring a ruler
to estimate partial derivatives

528
00:41:07 --> 00:41:09
the way that this problem asks
you to.

529
00:41:09 --> 00:41:35
 
 

530
00:41:35 --> 00:41:38
[APPLAUSE]
Let's look at problem 2B.

531
00:41:38 --> 00:41:43
Problem 2B is asking you to
find the point at which h equals

532
00:41:43 --> 00:41:46
2200,
partial h over partial x equals

533
00:41:46 --> 00:41:49
zero and partial h over partial
y is less than zero.

534
00:41:49 --> 00:41:53
Let's try and see what is going
on here.

535
00:41:53 --> 00:41:57
A point where f equals 2200,
well, that should be probably

536
00:41:57 --> 00:41:59
on the level curve that says
2200.

537
00:41:59 --> 00:42:09
We can actually zoom in.
Here is the level 2200.

538
00:42:09 --> 00:42:12
Now I want partial h over
partial x to be zero.

539
00:42:12 --> 00:42:17
That means if I change x,
keeping y constant,

540
00:42:17 --> 00:42:24
the value of h doesn't change.
Which points on the level curve

541
00:42:24 --> 00:42:30
satisfy that property?
It is the top and the bottom.

542
00:42:30 --> 00:42:34
If you are here, for example, 
and you move in the x

543
00:42:34 --> 00:42:36
direction,
well, you see,

544
00:42:36 --> 00:42:38
as you get to there from the
left,

545
00:42:38 --> 00:42:41
the height first increases and
then decreases.

546
00:42:41 --> 00:42:44
It goes for a maximum at that
point.

547
00:42:44 --> 00:42:47
So, at that point,
the partial derivative is zero

548
00:42:47 --> 00:42:53
with respect to x.
And the same here.

549
00:42:53 --> 00:42:59
Now, let's find partial h over
partial y less than zero.

550
00:42:59 --> 00:43:03
That means if we go north we
should go down.

551
00:43:03 --> 00:43:07
Well, which one is it,
top or bottom?

552
00:43:07 --> 00:43:11
Top. Yes.
Here, if you go north,

553
00:43:11 --> 00:43:16
then you go from 2200 down to
2100.

554
00:43:16 --> 00:43:23
This is where the point is.
Now, the problem here was also

555
00:43:23 --> 00:43:25
asking you to estimate partial h
over partial y.

556
00:43:25 --> 00:43:28
And if you were curious how you
would do that,

557
00:43:28 --> 00:43:33
well, you would try to figure
out how long it takes before you

558
00:43:33 --> 00:43:42
reach the next level curve.
To go from here to here, 

559
00:43:42 --> 00:43:47
to go from Q to this new point, 
say Q prime, 

560
00:43:47 --> 00:43:49
the change in y,
well, you would have to read

561
00:43:49 --> 00:43:56
the scale,
which was down here, 

562
00:43:56 --> 00:44:00
would be about something like
300.

563
00:44:00 --> 00:44:04
What is the change in height
when you go from Q to Q prime?

564
00:44:04 --> 00:44:07
Well, you go down from 2200 to
2100.

565
00:44:07 --> 00:44:14
That is actually minus 100
exactly.

566
00:44:14 --> 00:44:19
OK?
And so delta h over delta y is

567
00:44:19 --> 00:44:27
about minus one-third,
well, minus 100 over 300 which

568
00:44:27 --> 00:44:35
is minus one-third.
And that is an approximation

569
00:44:35 --> 00:44:43
for partial derivative.
So, that is how you would do it.

570
00:44:43 --> 00:44:48
Now, let me go back to other
things.

571
00:44:48 --> 00:44:52
If you look at this practice
exam, basically there is a bit

572
00:44:52 --> 00:44:56
of everything and it is kind of
fairly representative of what

573
00:44:56 --> 00:45:00
might happen on Tuesday.
There will be a mix of easy

574
00:45:00 --> 00:45:03
problems and of harder problems.
Expect something about

575
00:45:03 --> 00:45:05
computing gradients,
approximations,

576
00:45:05 --> 00:45:08
rate of change.
Expect a problem about reading

577
00:45:08 --> 00:45:13
a contour plot.
Expect one about a min/max

578
00:45:13 --> 00:45:15
problem,
something about Lagrange

579
00:45:15 --> 00:45:17
multipliers,
something about the chain rule

580
00:45:17 --> 00:45:20
and something about constrained
partial derivatives.

581
00:45:20 --> 00:45:22
I mean pretty much all the
topics are going to be there.

582
00:45:22 --> 00:45:23