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So far we have learned about
partial derivatives and how to

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use them to find minima and
maxima of functions of two

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variables or several variables.
And now we are going to try to

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00:00:35,000 --> 00:00:38,000
study, in more detail,
how functions of several

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variables behave,
how to compete their

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variations.
How to estimate the variation

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in arbitrary directions.
And so for that we are going to

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00:00:50,000 --> 00:00:56,000
need some more tools actually to
study this things.

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00:00:56,000 --> 00:01:00,000
More tools to study functions.

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00:01:15,000 --> 00:01:26,000
Today's topic is going to be
differentials.

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00:01:26,000 --> 00:01:34,000
And, just to motivate that,
let me remind you about one

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00:01:34,000 --> 00:01:43,000
trick that you probably know
from single variable calculus,

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namely implicit
differentiation.

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00:01:48,000 --> 00:01:56,000
Let's say that you have a
function y equals f of x then

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00:01:56,000 --> 00:02:05,000
you would sometimes write dy
equals f prime of x times dx.

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00:02:05,000 --> 00:02:17,000
And then maybe you would -- We
use implicit differentiation to

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00:02:17,000 --> 00:02:29,000
actually relate infinitesimal
changes in y with infinitesimal

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00:02:29,000 --> 00:02:35,000
changes in x.
And one thing we can do with

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00:02:35,000 --> 00:02:39,000
that, for example,
is actually figure out the rate

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00:02:39,000 --> 00:02:43,000
of change dy by dx,
but also the reciprocal dx by

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00:02:43,000 --> 00:02:48,000
dy.
And so, for example,

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00:02:48,000 --> 00:02:58,000
let's say that we have y equals
inverse sin(x).

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00:02:58,000 --> 00:03:03,000
Then we can write x equals
sin(y).

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00:03:03,000 --> 00:03:08,000
And, from there,
we can actually find out what

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00:03:08,000 --> 00:03:13,000
is the derivative of this
function if we didn't know the

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00:03:13,000 --> 00:03:18,000
answer already by writing dx
equals cosine y dy.

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00:03:18,000 --> 00:03:28,000
That tells us that dy over dx
is going to be one over cosine

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00:03:28,000 --> 00:03:32,000
y.
And now cosine for relation to

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00:03:32,000 --> 00:03:40,000
sine is basically one over
square root of one minus x^2.

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00:03:40,000 --> 00:03:44,000
And that is how you find the
formula for the derivative of

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00:03:44,000 --> 00:03:50,000
the inverse sine function.
A formula that you probably

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00:03:50,000 --> 00:03:54,000
already knew,
but that is one way to derive

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00:03:54,000 --> 00:03:57,000
it.
Now we are going to use also

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00:03:57,000 --> 00:03:59,000
these kinds of notations,
dx, dy and so on,

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00:03:59,000 --> 00:04:03,000
but use them for functions of
several variables.

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00:04:03,000 --> 00:04:05,000
And, of course,
we will have to learn what the

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00:04:05,000 --> 00:04:08,000
rules of manipulation are and
what we can do with them.

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00:04:17,000 --> 00:04:20,000
The actual name of that is the
total differential,

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00:04:20,000 --> 00:04:23,000
as opposed to the partial
derivatives.

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00:04:23,000 --> 00:04:28,000
The total differential includes
all of the various causes that

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00:04:28,000 --> 00:04:33,000
can change -- Sorry.
All the contributions that can

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00:04:33,000 --> 00:04:38,000
cause the value of your function
f to change.

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00:04:38,000 --> 00:04:43,000
Namely, let's say that you have
a function maybe of three

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00:04:43,000 --> 00:04:44,000
variables, x,
y, z,

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00:04:44,000 --> 00:04:56,000
then you would write df equals
f sub x dx plus f sub y dy plus

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00:04:56,000 --> 00:05:02,000
f sub z dz.
Maybe, just to remind you of

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00:05:02,000 --> 00:05:07,000
the other notation,
partial f over partial x dx

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00:05:07,000 --> 00:05:14,000
plus partial f over partial y dy
plus partial f over partial z

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00:05:14,000 --> 00:05:18,000
dz.
Now, what is this object?

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00:05:18,000 --> 00:05:22,000
What are the things on either
side of this equality?

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Well, they are called
differentials.

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00:05:24,000 --> 00:05:26,000
And they are not numbers,
they are not vectors,

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00:05:26,000 --> 00:05:29,000
they are not matrices,
they are a different kind of

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object.
These things have their own

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00:05:32,000 --> 00:05:36,000
rules of manipulations,
and we have to learn what we

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00:05:36,000 --> 00:05:40,000
can do with them.
So how do we think about them?

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00:05:40,000 --> 00:05:51,000
First of all,
how do we not think about them?

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00:05:51,000 --> 00:05:55,000
Here is an important thing to
know.

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00:05:55,000 --> 00:06:07,000
Important.
df is not the same thing as

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00:06:07,000 --> 00:06:12,000
delta f.
That is meant to be a number.

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00:06:12,000 --> 00:06:16,000
It is going to be a number once
you have a small variation of x,

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00:06:16,000 --> 00:06:19,000
a small variation of y,
a small variation of z.

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00:06:19,000 --> 00:06:21,000
These are numbers.
Delta x, delta y and delta z

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00:06:21,000 --> 00:06:24,000
are actual numbers,
and this becomes a number.

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00:06:24,000 --> 00:06:26,000
This guy actually is not a
number.

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00:06:26,000 --> 00:06:30,000
You cannot give it a particular
value.

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00:06:30,000 --> 00:06:33,000
All you can do with a
differential is express it in

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00:06:33,000 --> 00:06:36,000
terms of other differentials.
In fact, this dx,

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00:06:36,000 --> 00:06:38,000
dy and dz, well,
they are mostly symbols out

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00:06:38,000 --> 00:06:42,000
there.
But if you want to think about

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them, they are the differentials
of x, y and z.

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00:06:46,000 --> 00:06:52,000
In fact, you can think of these
differentials as placeholders

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00:06:52,000 --> 00:06:57,000
where you will put other things.
Of course, they represent,

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00:06:57,000 --> 00:07:02,000
you know, there is this idea of
changes in x,

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00:07:02,000 --> 00:07:05,000
y, z and f.
One way that one could explain

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00:07:05,000 --> 00:07:09,000
it, and I don't really like it,
is to say they represent

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00:07:09,000 --> 00:07:12,000
infinitesimal changes.
Another way to say it,

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00:07:12,000 --> 00:07:14,000
and I think that is probably
closer to the truth,

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00:07:14,000 --> 00:07:19,000
is that these things are
somehow placeholders to put

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00:07:19,000 --> 00:07:22,000
values and get a tangent
approximation.

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00:07:22,000 --> 00:07:25,000
For example,
if I do replace these symbols

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00:07:25,000 --> 00:07:30,000
by delta x, delta y and delta z
numbers then I will actually get

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00:07:30,000 --> 00:07:33,000
a numerical quantity.
And that will be an

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00:07:33,000 --> 00:07:39,000
approximation formula for delta.
It will be the linear

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00:07:39,000 --> 00:07:44,000
approximation,
a tangent plane approximation.

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00:07:44,000 --> 00:07:52,000
What we can do -- Well,
let me start first with maybe

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00:07:52,000 --> 00:08:00,000
something even before that.
The first thing that it does is

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00:08:00,000 --> 00:08:10,000
it can encode how changes in x,
y, z affect the value of f.

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00:08:10,000 --> 00:08:15,000
I would say that is the most
general answer to what is this

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00:08:15,000 --> 00:08:18,000
formula, what are these
differentials.

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00:08:18,000 --> 00:08:24,000
It is a relation between x,
y, z and f.

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00:08:24,000 --> 00:08:36,000
And this is a placeholder for
small variations,

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00:08:36,000 --> 00:08:53,000
delta x, delta y and delta z to
get an approximation formula.

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00:08:53,000 --> 00:09:00,000
Which is delta f is
approximately equal to fx delta

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00:09:00,000 --> 00:09:06,000
x fy delta y fz delta z.
It is getting cramped,

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00:09:06,000 --> 00:09:11,000
but I am sure you know what is
going on here.

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00:09:11,000 --> 00:09:15,000
And observe how this one is
actually equal while that one is

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00:09:15,000 --> 00:09:19,000
approximately equal.
So they are really not the same.

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00:09:19,000 --> 00:09:22,000
Another thing that the notation
suggests we can do,

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00:09:22,000 --> 00:09:26,000
and they claim we can do,
is divide everything by some

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00:09:26,000 --> 00:09:29,000
variable that everybody depends
on.

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00:09:29,000 --> 00:09:33,000
Say, for example,
that x, y and z actually depend

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00:09:33,000 --> 00:09:39,000
on some parameter t then they
will vary, at a certain rate,

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00:09:39,000 --> 00:09:42,000
dx over dt, dy over dt,
dz over dt.

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00:09:42,000 --> 00:09:46,000
And what the differential will
tell us then is the rate of

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00:09:46,000 --> 00:09:51,000
change of f as a function of t,
when you plug in these values

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00:09:51,000 --> 00:09:57,000
of x, y, z,
you will get df over dt by

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00:09:57,000 --> 00:10:05,000
dividing everything by dt in
here.

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00:10:05,000 --> 00:10:21,000
The first thing we can do is
divide by something like dt to

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00:10:21,000 --> 00:10:30,000
get infinitesimal rate of
change.

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00:10:30,000 --> 00:10:43,000
Well, let me just say rate of
change.

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00:10:43,000 --> 00:10:52,000
df over dt equals f sub x dx
over dt plus f sub y dy over dt

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00:10:52,000 --> 00:11:00,000
plus f sub z dz over dt.
And that corresponds to the

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00:11:00,000 --> 00:11:09,000
situation where x is a function
of t, y is a function of t and z

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00:11:09,000 --> 00:11:14,000
is a function of t.
That means you can plug in

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00:11:14,000 --> 00:11:18,000
these values into f to get,
well, the value of f will

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00:11:18,000 --> 00:11:23,000
depend on t,
and then you can find the rate

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00:11:23,000 --> 00:11:27,000
of change with t of a value of
f.

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00:11:27,000 --> 00:11:35,000
These are the basic rules.
And this is known as the chain

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00:11:35,000 --> 00:11:38,000
rule.
It is one instance of a chain

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00:11:38,000 --> 00:11:40,000
rule,
which tells you when you have a

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00:11:40,000 --> 00:11:42,000
function that depends on
something,

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00:11:42,000 --> 00:11:45,000
and that something in turn
depends on something else,

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00:11:45,000 --> 00:11:51,000
how to find the rate of change
of a function on the new

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00:11:51,000 --> 00:11:56,000
variable in terms of the
derivatives of a function and

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00:11:56,000 --> 00:12:01,000
also the dependence between the
various variables.

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00:12:01,000 --> 00:12:08,000
Any questions so far?
No.

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00:12:08,000 --> 00:12:11,000
OK.
A word of warming,

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00:12:11,000 --> 00:12:15,000
in particular,
about what I said up here.

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00:12:15,000 --> 00:12:19,000
It is kind of unfortunate,
but the textbook actually has a

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00:12:19,000 --> 00:12:23,000
serious mistake on that.
I mean they do have a couple of

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00:12:23,000 --> 00:12:29,000
formulas where they mix a d with
a delta, and I warn you not to

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00:12:29,000 --> 00:12:32,000
do that, please.
I mean there are d's and there

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00:12:32,000 --> 00:12:34,000
are delta's, and basically they
don't live in the same world.

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00:12:34,000 --> 00:12:53,000
They don't see each other.
The textbook is lying to you.

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00:12:53,000 --> 00:12:59,000
Let's see.
The first and the second

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00:12:59,000 --> 00:13:01,000
claims,
I don't really need to justify

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00:13:01,000 --> 00:13:05,000
because the first one is just
stating some general principle,

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00:13:05,000 --> 00:13:08,000
but I am not making a precise
mathematical claim.

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00:13:08,000 --> 00:13:11,000
The second one,
well, we know the approximation

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00:13:11,000 --> 00:13:14,000
formula already,
so I don't need to justify it

148
00:13:14,000 --> 00:13:16,000
for you.
But, on the other hand,

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00:13:16,000 --> 00:13:20,000
this formula here,
I mean, you probably have a

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00:13:20,000 --> 00:13:24,000
right to expect some reason for
why this works.

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00:13:24,000 --> 00:13:27,000
Why is this valid?
After all, I first told you we

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00:13:27,000 --> 00:13:29,000
have these new mysterious
objects.

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00:13:29,000 --> 00:13:32,000
And then I am telling you we
can do that, but I kind of

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00:13:32,000 --> 00:13:44,000
pulled it out of my hat.
I mean I don't have a hat.

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00:13:44,000 --> 00:13:53,000
Why is this valid?
How can I get to this?

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00:13:53,000 --> 00:14:06,000
Here is a first attempt of
justifying how to get there.

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00:14:06,000 --> 00:14:13,000
Let's see.
Well, we said df is f sub x dx

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00:14:13,000 --> 00:14:25,000
plus f sub y dy plus f sub z dz.
But we know if x is a function

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00:14:25,000 --> 00:14:37,000
of t then dx is x prime of t dt,
dy is y prime of t dt,

160
00:14:37,000 --> 00:14:47,000
dz is z prime of t dt.
If we plug these into that

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00:14:47,000 --> 00:14:58,000
formula, we will get that df is
f sub x times x prime t dt plus

162
00:14:58,000 --> 00:15:08,000
f sub y y prime of t dt plus f
sub z z prime of t dt.

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00:15:08,000 --> 00:15:14,000
And now I have a relation
between df and dt.

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00:15:14,000 --> 00:15:17,000
See, I got df equals sometimes
times dt.

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00:15:17,000 --> 00:15:23,000
That means the rate of change
of f with respect to t should be

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00:15:23,000 --> 00:15:38,000
that coefficient.
If I divide by dt then I get

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00:15:38,000 --> 00:15:46,000
the chain rule.
That kind of works,

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00:15:46,000 --> 00:15:49,000
but that shouldn't be
completely satisfactory.

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00:15:49,000 --> 00:15:53,000
Let's say that you are a true
skeptic and you don't believe in

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00:15:53,000 --> 00:15:57,000
differentials yet then it is
maybe not very good that I

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00:15:57,000 --> 00:16:01,000
actually used more of these
differential notations in

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00:16:01,000 --> 00:16:05,000
deriving the answer.
That is actually not how it is

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00:16:05,000 --> 00:16:08,000
proved.
The way in which you prove the

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00:16:08,000 --> 00:16:13,000
chain rule is not this way
because we shouldn't have too

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00:16:13,000 --> 00:16:16,000
much trust in differentials just
yet.

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00:16:16,000 --> 00:16:18,000
I mean at the end of today's
lecture, yes,

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00:16:18,000 --> 00:16:20,000
probably we should believe in
them,

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00:16:20,000 --> 00:16:26,000
but so far we should be a
little bit reluctant to believe

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00:16:26,000 --> 00:16:32,000
these kind of strange objects
telling us weird things.

180
00:16:32,000 --> 00:16:39,000
Here is a better way to think
about it.

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00:16:39,000 --> 00:16:43,000
One thing that we have trust in
so far are approximation

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00:16:43,000 --> 00:16:48,000
formulas.
We should have trust in them.

183
00:16:48,000 --> 00:16:54,000
We should believe that if we
change x a little bit,

184
00:16:54,000 --> 00:17:02,000
if we change y a little bit
then we are actually going to

185
00:17:02,000 --> 00:17:11,000
get a change in f that is
approximately given by these

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00:17:11,000 --> 00:17:13,000
guys.
And this is true for any

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00:17:13,000 --> 00:17:14,000
changes in x,
y, z,

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00:17:14,000 --> 00:17:20,000
but in particular let's look at
the changes that we get if we

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00:17:20,000 --> 00:17:26,000
just take these formulas as
function of time and change time

190
00:17:26,000 --> 00:17:32,000
a little bit by delta t.
We will actually use the

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00:17:32,000 --> 00:17:39,000
changes in x,
y, z in a small time delta t.

192
00:17:39,000 --> 00:17:47,000
Let's divide everybody by delta
t.

193
00:17:47,000 --> 00:17:52,000
Here I am just dividing numbers
so I am not actually playing any

194
00:17:52,000 --> 00:17:54,000
tricks on you.
I mean we don't really know

195
00:17:54,000 --> 00:17:57,000
what it means to divide
differentials,

196
00:17:57,000 --> 00:17:59,000
but dividing numbers is
something we know.

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00:17:59,000 --> 00:18:11,000
And now, if I take delta t very
small, this guy tends to the

198
00:18:11,000 --> 00:18:19,000
derivative, df over dt.
Remember, the definition of df

199
00:18:19,000 --> 00:18:23,000
over dt is the limit of this
ratio when the time interval

200
00:18:23,000 --> 00:18:28,000
delta t tends to zero.
That means if I choose smaller

201
00:18:28,000 --> 00:18:32,000
and smaller values of delta t
then these ratios of numbers

202
00:18:32,000 --> 00:18:35,000
will actually tend to some
value,

203
00:18:35,000 --> 00:18:41,000
and that value is the
derivative.

204
00:18:41,000 --> 00:18:51,000
Similarly, here delta x over
delta t, when delta t is really

205
00:18:51,000 --> 00:18:59,000
small, will tend to the
derivative dx/dt.

206
00:18:59,000 --> 00:19:00,000
And similarly for the others.

207
00:19:18,000 --> 00:19:28,000
That means, in particular,
we take the limit as delta t

208
00:19:28,000 --> 00:19:35,000
tends to zero and we get df over
dt on one side and on the other

209
00:19:35,000 --> 00:19:42,000
side we get f sub x dx over dt
plus f sub y dy over dt plus f

210
00:19:42,000 --> 00:19:46,000
sub z dz over dt.
And the approximation becomes

211
00:19:46,000 --> 00:19:49,000
better and better.
Remember when we write

212
00:19:49,000 --> 00:19:53,000
approximately equal that means
it is not quite the same,

213
00:19:53,000 --> 00:19:57,000
but if we take smaller
variations then actually we will

214
00:19:57,000 --> 00:20:01,000
end up with values that are
closer and closer.

215
00:20:01,000 --> 00:20:04,000
When we take the limit,
as delta t tends to zero,

216
00:20:04,000 --> 00:20:06,000
eventually we get an equality.

217
00:20:21,000 --> 00:20:24,000
I mean mathematicians have more
complicated words to justify

218
00:20:24,000 --> 00:20:28,000
this statement.
I will spare them for now,

219
00:20:28,000 --> 00:20:36,000
and you will see them when you
take analysis if you go in that

220
00:20:36,000 --> 00:20:42,000
direction.
Any questions so far?

221
00:20:42,000 --> 00:20:46,000
No.
OK.

222
00:20:46,000 --> 00:20:47,000
Let's check this with an
example.

223
00:20:47,000 --> 00:20:58,000
Let's say that we really don't
have any faith in these things

224
00:20:58,000 --> 00:21:06,000
so let's try to do it.
Let's say I give you a function

225
00:21:06,000 --> 00:21:14,000
that is x ^2 y z.
And let's say that maybe x will

226
00:21:14,000 --> 00:21:20,000
be t, y will be e^t and z will
be sin(t).

227
00:21:34,000 --> 00:21:40,000
What does the chain rule say?
Well, the chain rule tells us

228
00:21:40,000 --> 00:21:46,000
that dw/dt is,
we start with partial w over

229
00:21:46,000 --> 00:21:51,000
partial x, well,
what is that?

230
00:21:51,000 --> 00:21:58,000
That is 2xy,
and maybe I should point out

231
00:21:58,000 --> 00:22:08,000
that this is w sub x,
times dx over dt plus -- Well,

232
00:22:08,000 --> 00:22:21,000
w sub y is x squared times dy
over dt plus w sub z,

233
00:22:21,000 --> 00:22:28,000
which is going to be just one,
dz over dt.

234
00:22:28,000 --> 00:22:33,000
And so now let's plug in the
actual values of these things.

235
00:22:33,000 --> 00:22:38,000
x is t and y is e^t,
so that will be 2t e to the t,

236
00:22:38,000 --> 00:22:47,000
dx over dt is one plus x
squared is t squared,

237
00:22:47,000 --> 00:23:00,000
dy over dt is e over t,
plus dz over dt is cosine t.

238
00:23:00,000 --> 00:23:06,000
At the end of calculation we
get 2t e to the t plus t squared

239
00:23:06,000 --> 00:23:11,000
e to the t plus cosine t.
That is what the chain rule

240
00:23:11,000 --> 00:23:16,000
tells us.
How else could we find that?

241
00:23:16,000 --> 00:23:20,000
Well, we could just plug in
values of x, y and z,

242
00:23:20,000 --> 00:23:23,000
x plus w is a function of t,
and take its derivative.

243
00:23:23,000 --> 00:23:26,000
Let's do that just for
verification.

244
00:23:26,000 --> 00:23:30,000
It should be exactly the same
answer.

245
00:23:30,000 --> 00:23:32,000
And, in fact,
in this case,

246
00:23:32,000 --> 00:23:35,000
the two calculations are
roughly equal in complication.

247
00:23:35,000 --> 00:23:39,000
But say that your function of
x, y, z was much more

248
00:23:39,000 --> 00:23:43,000
complicated than that,
or maybe you actually didn't

249
00:23:43,000 --> 00:23:45,000
know a formula for it,
you only knew its partial

250
00:23:45,000 --> 00:23:48,000
derivatives,
then you would need to use the

251
00:23:48,000 --> 00:23:51,000
chain rule.
So, sometimes plugging in

252
00:23:51,000 --> 00:23:54,000
values is easier but not always.

253
00:24:13,000 --> 00:24:18,000
Let's just check quickly.
The other method would be to

254
00:24:18,000 --> 00:24:23,000
substitute.
W as a function of t.

255
00:24:23,000 --> 00:24:36,000
Remember w was x^2y z.
x was t, so you get t squared,

256
00:24:36,000 --> 00:24:41,000
y is e to the t,
plus z was sine t.

257
00:24:41,000 --> 00:24:47,000
dw over dt, we know how to take
the derivative using single

258
00:24:47,000 --> 00:24:50,000
variable calculus.
Well, we should know.

259
00:24:50,000 --> 00:24:55,000
If we don't know then we should
take a look at 18.01 again.

260
00:24:55,000 --> 00:25:02,000
The product rule that will be
derivative of t squared is 2t

261
00:25:02,000 --> 00:25:08,000
times e to the t plus t squared
time the derivative of e to the

262
00:25:08,000 --> 00:25:16,000
t is e to the t plus cosine t.
And that is the same answer as

263
00:25:16,000 --> 00:25:19,000
over there.
I ended up writing,

264
00:25:19,000 --> 00:25:23,000
you know, maybe I wrote
slightly more here,

265
00:25:23,000 --> 00:25:28,000
but actually the amount of
calculations really was pretty

266
00:25:28,000 --> 00:25:32,000
much the same.
Any questions about that?

267
00:25:32,000 --> 00:25:39,000
Yes?
What kind of object is w?

268
00:25:39,000 --> 00:25:43,000
Well, you can think of w as
just another variable that is

269
00:25:43,000 --> 00:25:47,000
given as a function of x,
y and z, for example.

270
00:25:47,000 --> 00:25:51,000
You would have a function of x,
y, z defined by this formula,

271
00:25:51,000 --> 00:25:57,000
and I call it w.
I call its value w so that I

272
00:25:57,000 --> 00:26:04,000
can substitute t instead of x,
y, z.

273
00:26:04,000 --> 00:26:07,000
Well, let's think of w as a
function of three variables.

274
00:26:07,000 --> 00:26:12,000
And then, when I plug in the
dependents of these three

275
00:26:12,000 --> 00:26:17,000
variables on t,
then it becomes just a function

276
00:26:17,000 --> 00:26:19,000
of t.
I mean, really,

277
00:26:19,000 --> 00:26:23,000
my w here is pretty much what I
called f before.

278
00:26:23,000 --> 00:26:31,000
There is no major difference
between the two.

279
00:26:31,000 --> 00:26:38,000
Any other questions?
No.

280
00:26:38,000 --> 00:26:45,000
OK.
Let's see.

281
00:26:45,000 --> 00:26:49,000
Here is an application of what
we have seen.

282
00:26:49,000 --> 00:26:53,000
Let's say that you want to
understand actually all these

283
00:26:53,000 --> 00:26:57,000
rules about taking derivatives
in single variable calculus.

284
00:26:57,000 --> 00:27:00,000
What I showed you at the
beginning, and then erased,

285
00:27:00,000 --> 00:27:04,000
basically justifies how to take
the derivative of a reciprocal

286
00:27:04,000 --> 00:27:06,000
function.
And for that you didn't need

287
00:27:06,000 --> 00:27:10,000
multivariable calculus.
But let's try to justify the

288
00:27:10,000 --> 00:27:12,000
product rule,
for example,

289
00:27:12,000 --> 00:27:21,000
for the derivative.
An application of this actually

290
00:27:21,000 --> 00:27:31,000
is to justify the product and
quotient rules.

291
00:27:31,000 --> 00:27:33,000
Let's think,
for example,

292
00:27:33,000 --> 00:27:39,000
of a function of two variables,
u and v, that is just the

293
00:27:39,000 --> 00:27:44,000
product uv.
And let's say that u and v are

294
00:27:44,000 --> 00:27:48,000
actually functions of one
variable t.

295
00:27:48,000 --> 00:28:00,000
Then, well, d of uv over dt is
given by the chain rule applied

296
00:28:00,000 --> 00:28:04,000
to f.
This is df over dt.

297
00:28:04,000 --> 00:28:15,000
So df over dt should be f sub q
du over dt plus f sub v plus dv

298
00:28:15,000 --> 00:28:19,000
over dt.
But now what is the partial of

299
00:28:19,000 --> 00:28:23,000
f with respect to u?
It is v.

300
00:28:23,000 --> 00:28:31,000
That is v du over dt.
And partial of f with respect

301
00:28:31,000 --> 00:28:38,000
to v is going to be just u,
dv over dt.

302
00:28:38,000 --> 00:28:42,000
So you get back the usual
product rule.

303
00:28:42,000 --> 00:28:46,000
That is a slightly complicated
way of deriving it,

304
00:28:46,000 --> 00:28:50,000
but that is a valid way of
understanding how to take the

305
00:28:50,000 --> 00:28:54,000
derivative of a product by
thinking of the product first as

306
00:28:54,000 --> 00:28:57,000
a function of variables,
which are u and v.

307
00:28:57,000 --> 00:29:00,000
And then say,
oh, but u and v were actually

308
00:29:00,000 --> 00:29:03,000
functions of a variable t.
And then you do the

309
00:29:03,000 --> 00:29:08,000
differentiation in two stages
using the chain rule.

310
00:29:08,000 --> 00:29:16,000
Similarly, you can do the
quotient rule just for practice.

311
00:29:16,000 --> 00:29:21,000
If I give you the function g
equals u of v.

312
00:29:21,000 --> 00:29:25,000
Right now I am thinking of it
as a function of two variables,

313
00:29:25,000 --> 00:29:29,000
u and v.
U and v themselves are actually

314
00:29:29,000 --> 00:29:39,000
going to be functions of t.
Then, well, dg over dt is going

315
00:29:39,000 --> 00:29:44,000
to be partial g,
partial u.

316
00:29:44,000 --> 00:29:48,000
How much is that?
How much is partial g,

317
00:29:48,000 --> 00:29:53,000
partial u?
One over v times du over dt

318
00:29:53,000 --> 00:29:58,000
plus -- Well,
next we need to have partial g

319
00:29:58,000 --> 00:30:01,000
over partial v.
Well, what is the derivative of

320
00:30:01,000 --> 00:30:04,000
this with respect to v?
Here we need to know how to

321
00:30:04,000 --> 00:30:11,000
differentiate the inverse.
It is minus u over v squared

322
00:30:11,000 --> 00:30:20,000
times dv over dt.
And that is actually the usual

323
00:30:20,000 --> 00:30:28,000
quotient rule just written in a
slightly different way.

324
00:30:28,000 --> 00:30:30,000
I mean, just in case you really
want to see it,

325
00:30:30,000 --> 00:30:36,000
if you clear denominators for v
squared then you will see

326
00:30:36,000 --> 00:30:41,000
basically u prime times v minus
v prime times u.

327
00:31:25,000 --> 00:31:32,000
Now let's go to something even
more crazy.

328
00:31:32,000 --> 00:31:45,000
I claim we can do chain rules
with more variables.

329
00:31:45,000 --> 00:31:50,000
Let's say that I have a
quantity.

330
00:31:50,000 --> 00:31:55,000
Let's call it w for now.
Let's say I have quantity w as

331
00:31:55,000 --> 00:31:58,000
a function of say variables x
and y.

332
00:31:58,000 --> 00:32:02,000
And so in the previous setup x
and y depended on some

333
00:32:02,000 --> 00:32:04,000
parameters t.
But, actually,

334
00:32:04,000 --> 00:32:07,000
let's now look at the case
where x and y themselves are

335
00:32:07,000 --> 00:32:10,000
functions of several variables.
Let's say of two more variables.

336
00:32:10,000 --> 00:32:25,000
Let's call them u and v.
I am going to stay with these

337
00:32:25,000 --> 00:32:27,000
abstract letters,
but if it bothers you,

338
00:32:27,000 --> 00:32:31,000
if it sounds completely
unmotivated think about it maybe

339
00:32:31,000 --> 00:32:33,000
in terms of something you might
now.

340
00:32:33,000 --> 00:32:36,000
Say, polar coordinates.
Let's say that I have a

341
00:32:36,000 --> 00:32:40,000
function but is defined in terms
of the polar coordinate

342
00:32:40,000 --> 00:32:43,000
variables on theta.
And then I know I want to

343
00:32:43,000 --> 00:32:45,000
switch to usual coordinates x
and y.

344
00:32:45,000 --> 00:32:49,000
Or, the other way around,
I have a function of x and y

345
00:32:49,000 --> 00:32:53,000
and I want to express it in
terms of the polar coordinates r

346
00:32:53,000 --> 00:32:57,000
and theta.
Then I would want to know maybe

347
00:32:57,000 --> 00:33:02,000
how the derivatives,
with respect to the various

348
00:33:02,000 --> 00:33:07,000
sets of variables,
related to each other.

349
00:33:07,000 --> 00:33:10,000
One way I could do it is,
of course,

350
00:33:10,000 --> 00:33:16,000
to say now if I plug the
formula for x and the formula

351
00:33:16,000 --> 00:33:23,000
for y into the formula for f
then w becomes a function of u

352
00:33:23,000 --> 00:33:27,000
and v,
and it can try to take partial

353
00:33:27,000 --> 00:33:29,000
derivatives.
If I have explicit formulas,

354
00:33:29,000 --> 00:33:32,000
well, that could work.
But maybe the formulas are

355
00:33:32,000 --> 00:33:35,000
complicated.
Typically, if I switch between

356
00:33:35,000 --> 00:33:37,000
rectangular and polar
coordinates,

357
00:33:37,000 --> 00:33:41,000
there might be inverse trig,
there might be maybe arctangent

358
00:33:41,000 --> 00:33:45,000
to express the polar angle in
terms of x and y.

359
00:33:45,000 --> 00:33:51,000
And when I don't really want to
actually substitute arctangents

360
00:33:51,000 --> 00:33:56,000
everywhere, maybe I would rather
deal with the derivatives.

361
00:33:56,000 --> 00:34:03,000
How do I do that?
The question is what are

362
00:34:03,000 --> 00:34:11,000
partial w over partial u and
partial w over partial v in

363
00:34:11,000 --> 00:34:17,000
terms of, let's see,
what do we need to know to

364
00:34:17,000 --> 00:34:22,000
understand that?
Well, probably we should know

365
00:34:22,000 --> 00:34:28,000
how w depends on x and y.
If we don't know that then we

366
00:34:28,000 --> 00:34:32,000
are probably toast.
Partial w over partial x,

367
00:34:32,000 --> 00:34:36,000
partial w over partial y should
be required.

368
00:34:36,000 --> 00:34:39,000
What else should we know?
Well, it would probably help to

369
00:34:39,000 --> 00:34:42,000
know how x and y depend on u and
v.

370
00:34:42,000 --> 00:34:46,000
If we don't know that then we
don't really know how to do it.

371
00:34:46,000 --> 00:34:55,000
We need also x sub u,
x sub v, y sub u,

372
00:34:55,000 --> 00:35:00,000
y sub v.
We have a lot of partials in

373
00:35:00,000 --> 00:35:07,000
there.
Well, let's see how we can do

374
00:35:07,000 --> 00:35:13,000
that.
Let's start by writing dw.

375
00:35:13,000 --> 00:35:19,000
We know that dw is partial f,
well, I don't know why I have

376
00:35:19,000 --> 00:35:25,000
two names, w and f.
I mean w and f are really the

377
00:35:25,000 --> 00:35:30,000
same thing here,
but let's say f sub x dx plus f

378
00:35:30,000 --> 00:35:35,000
sub y dy.
So far that is our new friend,

379
00:35:35,000 --> 00:35:39,000
the differential.
Now what do we want to do with

380
00:35:39,000 --> 00:35:42,000
it?
Well, we would like to get rid

381
00:35:42,000 --> 00:35:47,000
of dx and dy because we like to
express things in terms of,

382
00:35:47,000 --> 00:35:50,000
you know, the question we are
asking ourselves is let's say

383
00:35:50,000 --> 00:35:55,000
that I change u a little bit,
how does w change?

384
00:35:55,000 --> 00:35:58,000
Of course, what happens,
if I change u a little bit,

385
00:35:58,000 --> 00:36:01,000
is y and y will change.
How do they change?

386
00:36:01,000 --> 00:36:05,000
Well, that is given to me by
the differential.

387
00:36:05,000 --> 00:36:13,000
dx is going to be,
well, I can use the

388
00:36:13,000 --> 00:36:19,000
differential again.
Well, x is a function of u and

389
00:36:19,000 --> 00:36:24,000
v.
That will be x sub u times du

390
00:36:24,000 --> 00:36:28,000
plus x sub v times dv.
That is, again,

391
00:36:28,000 --> 00:36:31,000
taking the differential of a
function of two variables.

392
00:36:31,000 --> 00:36:37,000
Does that make sense?
And then we have the other guy,

393
00:36:37,000 --> 00:36:39,000
f sub y times,
what is dy?

394
00:36:39,000 --> 00:36:49,000
Well, similarly dy is y sub u
du plus y sub v dv.

395
00:36:49,000 --> 00:36:54,000
And now we have a relation
between dw and du and dv.

396
00:36:54,000 --> 00:37:00,000
We are expressing how w reacts
to changes in u and v,

397
00:37:00,000 --> 00:37:04,000
which was our goal.
Now, let's actually collect

398
00:37:04,000 --> 00:37:08,000
terms so that we see it a bit
better.

399
00:37:08,000 --> 00:37:19,000
It is going to be f sub x times
x sub u times f sub y times y

400
00:37:19,000 --> 00:37:28,000
sub u du plus f sub x,
x sub v plus f sub y y sub v

401
00:37:28,000 --> 00:37:32,000
dv.
Now we have dw equals something

402
00:37:32,000 --> 00:37:38,000
du plus something dv.
Well, the coefficient here has

403
00:37:38,000 --> 00:37:44,000
to be partial f over partial u.
What else could it be?

404
00:37:44,000 --> 00:37:49,000
That's the rate of change of w
with respect to u if I forget

405
00:37:49,000 --> 00:37:54,000
what happens when I change v.
That is the definition of a

406
00:37:54,000 --> 00:37:58,000
partial.
Similarly, this one has to be

407
00:37:58,000 --> 00:38:04,000
partial f over partial v.
That is because it is the rate

408
00:38:04,000 --> 00:38:09,000
of change with respect to v,
if I keep u constant,

409
00:38:09,000 --> 00:38:13,000
so that these guys are
completely ignored.

410
00:38:13,000 --> 00:38:16,000
Now you see how the total
differential accounts for,

411
00:38:16,000 --> 00:38:21,000
somehow, all the partial
derivatives that come as

412
00:38:21,000 --> 00:38:27,000
coefficients of the individual
variables in these expressions.

413
00:38:27,000 --> 00:38:33,000
Let me maybe rewrite these
formulas in a more visible way

414
00:38:33,000 --> 00:38:40,000
and then re-explain them to you.
Here is the chain rule for this

415
00:38:40,000 --> 00:38:46,000
situation, with two intermediate
variables and two variables that

416
00:38:46,000 --> 00:38:50,000
you express these in terms of.
In our setting,

417
00:38:50,000 --> 00:38:56,000
we get partial f over partial u
equals partial f over partial x

418
00:38:56,000 --> 00:39:02,000
time partial x over partial u
plus partial f over partial y

419
00:39:02,000 --> 00:39:08,000
times partial y over partial u.
And the other one,

420
00:39:08,000 --> 00:39:15,000
the same thing with v instead
of u,

421
00:39:15,000 --> 00:39:22,000
partial f over partial x times
partial x over partial v plus

422
00:39:22,000 --> 00:39:28,000
partial f over partial u partial
y over partial v.

423
00:39:28,000 --> 00:39:31,000
I have to explain various
things about these formulas

424
00:39:31,000 --> 00:39:34,000
because they look complicated.
And, actually,

425
00:39:34,000 --> 00:39:39,000
they are not that complicated.
A couple of things to know.

426
00:39:39,000 --> 00:39:42,000
The first thing,
how do we remember a formula

427
00:39:42,000 --> 00:39:44,000
like that?
Well, that is easy.

428
00:39:44,000 --> 00:39:47,000
We want to know how f depends
on u.

429
00:39:47,000 --> 00:39:51,000
Well, what does f depend on?
It depends on x and y.

430
00:39:51,000 --> 00:39:55,000
So we will put partial f over
partial x and partial f over

431
00:39:55,000 --> 00:39:59,000
partial y.
Now, x and y, why are they here?

432
00:39:59,000 --> 00:40:01,000
Well, they are here because
they actually depend on u as

433
00:40:01,000 --> 00:40:04,000
well.
How does x depend on u?

434
00:40:04,000 --> 00:40:06,000
Well, the answer is partial x
over partial u.

435
00:40:06,000 --> 00:40:10,000
How does y depend on u?
The answer is partial y over

436
00:40:10,000 --> 00:40:12,000
partial u.
See, the structure of this

437
00:40:12,000 --> 00:40:16,000
formula is simple.
To find the partial of f with

438
00:40:16,000 --> 00:40:20,000
respect to some new variable you
use the partials with respect to

439
00:40:20,000 --> 00:40:24,000
the variables that f was
initially defined in terms of x

440
00:40:24,000 --> 00:40:28,000
and y.
And you multiply them by the

441
00:40:28,000 --> 00:40:33,000
partials of x and y in terms of
the new variable that you want

442
00:40:33,000 --> 00:40:37,000
to look at, v here,
and you sum these things

443
00:40:37,000 --> 00:40:40,000
together.
That is the structure of the

444
00:40:40,000 --> 00:40:42,000
formula.
Why does it work?

445
00:40:42,000 --> 00:40:45,000
Well, let me explain it to you
in a slightly different

446
00:40:45,000 --> 00:40:49,000
language.
This asks us how does f change

447
00:40:49,000 --> 00:40:54,000
if I change u a little bit?
Well, why would f change if u

448
00:40:54,000 --> 00:40:57,000
changes a little bit?
Well, it would change because f

449
00:40:57,000 --> 00:41:00,000
actually depends on x and y and
x and y depend on u.

450
00:41:00,000 --> 00:41:03,000
If I change u,
how quickly does x change?

451
00:41:03,000 --> 00:41:06,000
Well, the answer is partial x
over partial u.

452
00:41:06,000 --> 00:41:09,000
And now, if I change x at this
rate, how does that have to

453
00:41:09,000 --> 00:41:13,000
change?
Well, the answer is partial f

454
00:41:13,000 --> 00:41:17,000
over partial x times this guy.
Well, at the same time,

455
00:41:17,000 --> 00:41:21,000
y is also changing.
How fast is y changing if I

456
00:41:21,000 --> 00:41:24,000
change u?
Well, at the rate of partial y

457
00:41:24,000 --> 00:41:27,000
over partial u.
But now if I change this how

458
00:41:27,000 --> 00:41:30,000
does f change?
Well, the rate of change is

459
00:41:30,000 --> 00:41:34,000
partial f over partial y.
The product is the effect of

460
00:41:34,000 --> 00:41:37,000
how you change it,
changing u, and therefore

461
00:41:37,000 --> 00:41:40,000
changing f.
Now, what happens in real life,

462
00:41:40,000 --> 00:41:43,000
if I change u a little bit?
Well, both x and y change at

463
00:41:43,000 --> 00:41:46,000
the same time.
So how does f change?

464
00:41:46,000 --> 00:41:50,000
Well, it is the sum of the two
effects.

465
00:41:50,000 --> 00:41:54,000
Does that make sense?
Good.

466
00:41:54,000 --> 00:42:00,000
Of course, if f depends on more
variables then you just have

467
00:42:00,000 --> 00:42:02,000
more terms in here.
OK.

468
00:42:02,000 --> 00:42:05,000
Here is another thing that may
be a little bit confusing.

469
00:42:05,000 --> 00:42:09,000
What is tempting?
Well, what is tempting here

470
00:42:09,000 --> 00:42:12,000
would be to simplify these
formulas by removing these

471
00:42:12,000 --> 00:42:15,000
partial x's.
Let's simplify by partial x.

472
00:42:15,000 --> 00:42:18,000
Let's simplify by partial y.
We get partial f over partial u

473
00:42:18,000 --> 00:42:21,000
equals partial f over partial u
plus partial f over partial u.

474
00:42:21,000 --> 00:42:25,000
Something is not working
properly.

475
00:42:25,000 --> 00:42:28,000
Why doesn't it work?
The answer is precisely because

476
00:42:28,000 --> 00:42:32,000
these are partial derivatives.
These are not total derivatives.

477
00:42:32,000 --> 00:42:36,000
And so you cannot simplify them
in that way.

478
00:42:36,000 --> 00:42:39,000
And that is actually the reason
why we use this curly d rather

479
00:42:39,000 --> 00:42:41,000
than a straight d.
It is to remind us,

480
00:42:41,000 --> 00:42:44,000
beware, there are these
simplifications that we can do

481
00:42:44,000 --> 00:42:47,000
with straight d's that are not
legal here.

482
00:42:47,000 --> 00:42:52,000
Somehow, when you have a
partial derivative,

483
00:42:52,000 --> 00:42:57,000
you must resist the urge of
simplifying things.

484
00:42:57,000 --> 00:43:02,000
No simplifications in here.
That is the simplest formula

485
00:43:02,000 --> 00:43:10,000
you can get.
Any questions at this point?

486
00:43:10,000 --> 00:43:21,000
No.
Yes?

487
00:43:21,000 --> 00:43:23,000
When would you use this and
what does it describe?

488
00:43:23,000 --> 00:43:26,000
Well, it is basically when you
have a function given in terms

489
00:43:26,000 --> 00:43:29,000
of a certain set of variables
because maybe there is a simply

490
00:43:29,000 --> 00:43:31,000
expression in terms of those
variables.

491
00:43:31,000 --> 00:43:35,000
But ultimately what you care
about is not those variables,

492
00:43:35,000 --> 00:43:39,000
z and y, but another set of
variables, here u and v.

493
00:43:39,000 --> 00:43:42,000
So x and y are giving you a
nice formula for f,

494
00:43:42,000 --> 00:43:46,000
but actually the relevant
variables for your problem are u

495
00:43:46,000 --> 00:43:48,000
and v.
And you know x and y are

496
00:43:48,000 --> 00:43:50,000
related to u and v.
So, of course,

497
00:43:50,000 --> 00:43:53,000
what you could do is plug the
formulas the way that we did

498
00:43:53,000 --> 00:43:55,000
substituting.
But maybe that will give you

499
00:43:55,000 --> 00:43:59,000
very complicated expressions.
And maybe it is actually easier

500
00:43:59,000 --> 00:44:02,000
to just work with the derivates.
The important claim here is

501
00:44:02,000 --> 00:44:05,000
basically we don't need to know
the actual formulas.

502
00:44:05,000 --> 00:44:07,000
All we need to know are the
rate of changes.

503
00:44:07,000 --> 00:44:11,000
If we know all these rates of
change then we know how to take

504
00:44:11,000 --> 00:44:14,000
these derivatives without
actually having to plug in

505
00:44:14,000 --> 00:44:22,000
values.
Yes?

506
00:44:22,000 --> 00:44:25,000
Yes, you could certain do the
same things in terms of t.

507
00:44:25,000 --> 00:44:29,000
If x and y were functions of t
instead of being functions of u

508
00:44:29,000 --> 00:44:31,000
and v then it would be the same
thing.

509
00:44:31,000 --> 00:44:34,000
And you would have the same
formulas that I had,

510
00:44:34,000 --> 00:44:37,000
well, over there I still have
it.

511
00:44:37,000 --> 00:44:39,000
Why does that one have straight
d's?

512
00:44:39,000 --> 00:44:42,000
Well, the answer is I could put
curly d's if I wanted,

513
00:44:42,000 --> 00:44:45,000
but I end up with a function of
a single variable.

514
00:44:45,000 --> 00:44:48,000
If you have a single variable
then the partial,

515
00:44:48,000 --> 00:44:50,000
with respect to that variable,
is the same thing as the usual

516
00:44:50,000 --> 00:44:53,000
derivative.
We don't actually need to worry

517
00:44:53,000 --> 00:44:57,000
about curly in that case.
But that one is indeed special

518
00:44:57,000 --> 00:45:00,000
case of this one where instead
of x and y depending on two

519
00:45:00,000 --> 00:45:03,000
variables, u and v,
they depend on a single

520
00:45:03,000 --> 00:45:04,000
variable t.
Now, of course,

521
00:45:04,000 --> 00:45:06,000
you can call variables any name
you want.

522
00:45:06,000 --> 00:45:12,000
It doesn't matter.
This is just a slight

523
00:45:12,000 --> 00:45:16,000
generalization of that.
Well, not quite because here I

524
00:45:16,000 --> 00:45:18,000
also had a z.
See, I am trying to just

525
00:45:18,000 --> 00:45:21,000
confuse you by giving you
functions that depend on various

526
00:45:21,000 --> 00:45:25,000
numbers of variables.
If you have a function of 30

527
00:45:25,000 --> 00:45:28,000
variables, things work the same
way, just longer,

528
00:45:28,000 --> 00:45:33,000
and you are going to run out of
letters in the alphabet before

529
00:45:33,000 --> 00:45:38,000
the end.
Any other questions?

530
00:45:38,000 --> 00:45:43,000
No.
What?

531
00:45:43,000 --> 00:45:51,000
Yes?
If u and v themselves depended

532
00:45:51,000 --> 00:45:55,000
on another variable then you
would continue with your chain

533
00:45:55,000 --> 00:45:58,000
rules.
Maybe you would know to express

534
00:45:58,000 --> 00:46:02,000
partial x over partial u in
terms using that chain rule.

535
00:46:02,000 --> 00:46:05,000
Sorry.
If u and v are dependent on yet

536
00:46:05,000 --> 00:46:08,000
another variable then you could
get the derivative with respect

537
00:46:08,000 --> 00:46:11,000
to that using first the chain
rule to pass from u v to that

538
00:46:11,000 --> 00:46:14,000
new variable,
and then you would plug in

539
00:46:14,000 --> 00:46:17,000
these formulas for partials of f
with respect to u and v.

540
00:46:17,000 --> 00:46:19,000
In fact, if you have several
substitutions to do,

541
00:46:19,000 --> 00:46:21,000
you can always arrange to use
one chain rule at a time.

542
00:46:21,000 --> 00:46:25,000
You just have to do them in
sequence.

543
00:46:25,000 --> 00:46:28,000
That's why we don't actually
learn that, but you can just do

544
00:46:28,000 --> 00:46:32,000
it be repeating the process.
I mean, probably at that stage,

545
00:46:32,000 --> 00:46:35,000
the easiest to not get confused
actually is to manipulate

546
00:46:35,000 --> 00:46:38,000
differentials because that is
probably easier.

547
00:46:38,000 --> 00:46:47,000
Yes?
Curly f does not exist.

548
00:46:47,000 --> 00:46:50,000
That's easy.
Curly f makes no sense by

549
00:46:50,000 --> 00:46:52,000
itself.
It doesn't exist alone.

550
00:46:52,000 --> 00:46:58,000
What exists is only curly df
over curly d some variable.

551
00:46:58,000 --> 00:47:02,000
And then that accounts only for
the rate of change with respect

552
00:47:02,000 --> 00:47:05,000
to that variable leaving the
others fixed,

553
00:47:05,000 --> 00:47:11,000
while straight df is somehow a
total variation of f.

554
00:47:11,000 --> 00:47:16,000
It accounts for all of the
partial derivatives and their

555
00:47:16,000 --> 00:47:25,000
combined effects.
OK. Any more questions? No.

556
00:47:25,000 --> 00:47:29,000
Let me just finish up very
quickly by telling you again one

557
00:47:29,000 --> 00:47:33,000
example where completely you
might want to do this.

558
00:47:33,000 --> 00:47:40,000
You have a function that you
want to switch between

559
00:47:40,000 --> 00:47:45,000
rectangular and polar
coordinates.

560
00:47:45,000 --> 00:47:48,000
To make things a little bit
concrete.

561
00:47:48,000 --> 00:47:55,000
If you have polar coordinates
that means in the plane,

562
00:47:55,000 --> 00:48:00,000
instead of using x and y,
you will use coordinates r,

563
00:48:00,000 --> 00:48:05,000
distance to the origin,
and theta, the angles from the

564
00:48:05,000 --> 00:48:08,000
x-axis.
The change of variables for

565
00:48:08,000 --> 00:48:14,000
that is x equals r cosine theta
and y equals r sine theta.

566
00:48:14,000 --> 00:48:21,000
And so that means if you have a
function f that depends on x and

567
00:48:21,000 --> 00:48:29,000
y, in fact, you can plug these
in as a function of r and theta.

568
00:48:29,000 --> 00:48:34,000
Then you can ask yourself,
well, what is partial f over

569
00:48:34,000 --> 00:48:37,000
partial r?
And that is going to be,

570
00:48:37,000 --> 00:48:42,000
well, you want to take partial
f over partial x times partial x

571
00:48:42,000 --> 00:48:48,000
partial r plus partial f over
partial y times partial y over

572
00:48:48,000 --> 00:48:53,000
partial r.
That will end up being actually

573
00:48:53,000 --> 00:48:59,000
f sub x times cosine theta plus
f sub y times sine theta.

574
00:48:59,000 --> 00:49:02,000
And you can do the same thing
to find partial f,

575
00:49:02,000 --> 00:49:05,000
partial theta.
And so you can express

576
00:49:05,000 --> 00:49:10,000
derivatives either in terms of
x, y or in terms of r and theta

577
00:49:10,000 --> 00:49:13,000
with simple relations between
them.

578
00:49:13,000 --> 00:49:20,000
And the one last thing I should
say.

579
00:49:20,000 --> 00:49:23,000
On Thursday we will learn about
more tricks we can play with

580
00:49:23,000 --> 00:49:27,000
variations of functions.
And one that is important,

581
00:49:27,000 --> 00:49:29,000
because you need to know it
actually to do the p-set,

582
00:49:29,000 --> 00:49:38,000
is the gradient vector.
The gradient vector is simply a

583
00:49:38,000 --> 00:49:41,000
vector.
You use this downward pointing

584
00:49:41,000 --> 00:49:44,000
triangle as the notation for the
gradient.

585
00:49:44,000 --> 00:49:49,000
It is simply is a vector whose
components are the partial

586
00:49:49,000 --> 00:49:53,000
derivatives of a function.
I mean, in a way,

587
00:49:53,000 --> 00:49:56,000
you can think of a differential
as a way to package partial

588
00:49:56,000 --> 00:49:59,000
derivatives together into some
weird object.

589
00:49:59,000 --> 00:50:01,000
Well, the gradient is also a
way to package partials

590
00:50:01,000 --> 00:50:04,000
together.
We will see on Thursday what it

591
00:50:04,000 --> 00:50:07,000
is good for, but some of the
problems on the p-set use it.