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Let me start by basically
listing the main things we have
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00:00:25,000 --> 00:00:28,000
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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00:00:31,000 --> 00:00:34,000
that because there are a few
small details that I didn't
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00:00:34,000 --> 00:00:38,000
quite clarify and that I should
probably make a bit clearer,
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00:00:38,000 --> 00:00:48,000
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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00:01:08,000 --> 00:01:12,000
several variables.
We have learned how to think of
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00:01:12,000 --> 00:01:16,000
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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00:01:17,000 --> 00:01:19,000
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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00:01:27,000 --> 00:01:38,000
study variations of these
functions using partial
22
00:01:38,000 --> 00:01:44,000
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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00:01:47,000 --> 00:01:52,000
say, x to be the rate of change
with respect to x when we hold
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00:01:52,000 --> 00:01:55,000
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
27
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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00:02:15,000 --> 00:02:20,000
a vector,the gradient vector.
For example,
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00:02:20,000 --> 00:02:24,000
if we have a function of three
variables, the vector whose
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00:02:24,000 --> 00:02:26,000
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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00:02:39,000 --> 00:02:43,000
formulas.
The change in f,
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00:02:43,000 --> 00:02:48,000
when we change x,
y, z slightly,
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00:02:48,000 --> 00:02:57,000
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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00:03:03,000 --> 00:03:08,000
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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00:03:16,000 --> 00:03:22,000
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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00:03:26,000 --> 00:03:30,000
the tangent plane approximation
because it tells us,
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00:03:30,000 --> 00:03:35,000
in fact,
it amounts to identifying the
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00:03:35,000 --> 00:03:38,000
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
47
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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00:03:48,000 --> 00:03:52,000
we are replacing the function
by its linear approximation.
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00:03:52,000 --> 00:03:56,000
We are replacing the graph by
its tangent plane.
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00:03:56,000 --> 00:03:58,000
Except, of course,
we haven't see the graph of a
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00:03:58,000 --> 00:04:00,000
function of three variables
because that would live in
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00:04:00,000 --> 00:04:04,000
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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00:04:08,000 --> 00:04:12,000
That also tells us how to find
tangent planes to level
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00:04:12,000 --> 00:04:12,000
surfaces.
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00:04:22,000 --> 00:04:30,000
Recall that the tangent plane
to a surface,
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00:04:30,000 --> 00:04:37,000
given by the equation f of x,
y, z equals z,
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00:04:37,000 --> 00:04:43,000
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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00:04:47,000 --> 00:04:49,000
vector is actually,
well,
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00:04:49,000 --> 00:04:53,000
one normal vector is given by
the gradient of a function
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00:04:53,000 --> 00:04:56,000
because we know that the
gradient is actually pointing
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00:04:56,000 --> 00:05:01,000
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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00:05:05,000 --> 00:05:08,000
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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00:05:18,000 --> 00:05:20,000
OK.
Let me add, actually,
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00:05:20,000 --> 00:05:23,000
a cultural note to what we have
seen so far about partial
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00:05:23,000 --> 00:05:28,000
derivatives and how to use them,
which is maybe something I
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00:05:28,000 --> 00:05:32,000
should have mentioned a couple
of weeks ago.
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00:05:32,000 --> 00:05:33,000
Why do we like partial
derivatives?
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00:05:33,000 --> 00:05:37,000
Well, one obvious reason is we
can do all these things.
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00:05:37,000 --> 00:05:39,000
But another reason is that,
really,
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00:05:39,000 --> 00:05:42,000
you need partial derivatives to
do physics and to understand
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00:05:42,000 --> 00:05:46,000
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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00:05:50,000 --> 00:05:51,000
equations.
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00:05:59,000 --> 00:06:07,000
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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00:06:09,000 --> 00:06:13,000
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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00:06:15,000 --> 00:06:18,000
is unknown that depends on a
bunch of variables.
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00:06:18,000 --> 00:06:23,000
And a partial differential
equation is some relation
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00:06:23,000 --> 00:06:28,000
between its partial derivatives.
Let me see.
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00:06:28,000 --> 00:06:45,000
These are equations involving
the partial derivatives -- -- of
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00:06:45,000 --> 00:06:54,000
an unknown function.
Let me give you an example to
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00:06:54,000 --> 00:06:57,000
see how that works.
For example,
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00:06:57,000 --> 00:07:02,000
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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00:07:09,000 --> 00:07:15,000
let me write for you the space
version of it.
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00:07:15,000 --> 00:07:21,000
It is the equation partial f
over partial t equals some
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00:07:21,000 --> 00:07:27,000
constant times the sum of the
second partials with respect to
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00:07:27,000 --> 00:07:32,000
x, y and z.
So this is an equation where we
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00:07:32,000 --> 00:07:38,000
are trying to solve for a
function f that depends,
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00:07:38,000 --> 00:07:42,000
actually, on four variables,
x, y, z, t.
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00:07:42,000 --> 00:07:47,000
And what should you have in
mind?
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00:07:47,000 --> 00:07:50,000
Well, this equation governs
temperature.
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00:07:50,000 --> 00:07:55,000
If you think that f of x, y, z,
t will be the temperature at a
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00:07:55,000 --> 00:07:59,000
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
106
00:08:15,000 --> 00:08:18,000
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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00:08:21,000 --> 00:08:24,000
temperature in this room,
and if you assume that nothing
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00:08:24,000 --> 00:08:26,000
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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00:08:29,000 --> 00:08:31,000
on,
then it will tell you how the
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00:08:31,000 --> 00:08:35,000
temperature will change over
time and eventually stabilize to
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00:08:35,000 --> 00:08:41,000
some final value.
Yes?
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00:08:41,000 --> 00:08:43,000
Why do we take the partial
derivative twice?
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00:08:43,000 --> 00:08:45,000
Well, that is a question,
I would say,
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00:08:45,000 --> 00:08:48,000
for a physics person.
But in a few weeks we will
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00:08:48,000 --> 00:08:52,000
actually see a derivation of
where this equation comes from
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00:08:52,000 --> 00:08:55,000
and try to justify it.
But, really,
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00:08:55,000 --> 00:08:57,000
that is something you will see
in a physics class.
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00:08:57,000 --> 00:09:02,000
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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00:09:17,000 --> 00:09:20,000
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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00:09:23,000 --> 00:09:26,000
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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00:09:29,000 --> 00:09:32,000
topic for this class.
It is not even a topic for
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00:09:32,000 --> 00:09:34,000
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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00:09:50,000 --> 00:09:52,000
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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00:09:58,000 --> 00:10:03,000
that very often functions that
you see in real life satisfy
138
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.
142
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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00:10:33,000 --> 00:10:42,000
I should have written down that
this equation is solved by
148
00:10:42,000 --> 00:10:48,000
temperature for point x,
y, z at time t.
149
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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00:10:56,000 --> 00:10:59,000
maybe sometimes learn about the
wave equation that governs how
152
00:10:59,000 --> 00:11:02,000
waves propagate in space,
about the diffusion equation,
153
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
156
00:11:16,000 --> 00:11:19,000
a partial differential equation
to solve.
157
00:11:19,000 --> 00:11:23,000
OK. Anyway. Sorry.
Back to my list of topics.
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00:11:23,000 --> 00:11:27,000
One important application we
have seen of partial derivatives
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00:11:27,000 --> 00:11:30,000
is to try to optimize things,
try to solve minimum/maximum
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00:11:30,000 --> 00:11:31,000
problems.
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00:11:42,000 --> 00:11:47,000
Remember that we have
introduced the notion of
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00:11:47,000 --> 00:11:56,000
critical points of a function.
A critical point is when all
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00:11:56,000 --> 00:12:03,000
the partial derivatives are
zero.
164
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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00:12:15,000 --> 00:12:24,000
second derivatives -- -- to
decide which kind of critical
168
00:12:24,000 --> 00:12:29,000
point we have.
I should say that is for a
169
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.
171
00:12:41,000 --> 00:12:44,000
And we have also seen that
actually that is not enough to
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00:12:44,000 --> 00:12:48,000
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.
174
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,
178
00:13:03,000 --> 00:13:05,000
well, you are going to tell me,
quite obviously,
179
00:13:05,000 --> 00:13:07,000
that the maximum is this point
up here.
180
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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00:13:20,000 --> 00:13:23,000
for the minimum of a function,
well, it is not at a critical
185
00:13:23,000 --> 00:13:26,000
point.
It is actually here at the
186
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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00:13:38,000 --> 00:13:44,000
boundary.
And the maximum is at a
189
00:13:44,000 --> 00:13:50,000
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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00:13:53,000 --> 00:13:55,000
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.
193
00:13:58,000 --> 00:14:01,000
And then we can use these
methods to find where they are.
194
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
196
00:14:09,000 --> 00:14:13,000
achieve a minimum by making x
and y as small as possible.
197
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
200
00:14:23,000 --> 00:14:26,000
possibilities.
We are going to do a problem
201
00:14:26,000 --> 00:14:29,000
like that.
We are going to go over a
202
00:14:29,000 --> 00:14:34,000
practice problem from the
practice test to clarify this.
203
00:14:34,000 --> 00:14:38,000
Another important cultural
application of minimum/maximum
204
00:14:38,000 --> 00:14:42,000
problems in two variables that
we have seen in class is the
205
00:14:42,000 --> 00:14:45,000
least squared method to find the
best fit line,
206
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.
209
00:15:01,000 --> 00:15:03,000
And here I have some good news
for you.
210
00:15:03,000 --> 00:15:07,000
While you should definitely
know what this is about,
211
00:15:07,000 --> 00:15:09,000
it will not be on the test.
212
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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00:15:41,000 --> 00:15:51,000
OK?
Now what is next on my list of
215
00:15:51,000 --> 00:15:58,000
topics?
We have seen differentials.
216
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.
221
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,
227
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
229
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?
233
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.