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So far we have learned about
partial derivatives and how to

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00:00:27 --> 00:00:31
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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study, in more detail,
how functions of several

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variables behave,
how to compete their

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00:00:41 --> 00:00:44
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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need some more tools actually to
study this things.

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More tools to study functions.

16
00:01:00 --> 00:01:15

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

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

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

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00:01:43 --> 00:01:48
namely implicit
differentiation.

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

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you would sometimes write dy
equals f prime of x times dx.

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

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actually relate infinitesimal
changes in y with infinitesimal

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

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

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

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

29
00:02:48 --> 00:02:58
let's say that we have y equals
inverse sin(x).

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

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

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

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

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

35
00:03:28 --> 00:03:32
y.
And now cosine for relation to

36
00:03:32 --> 00:03:40
sine is basically one over
square root of one minus x^2.

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

38
00:03:44 --> 00:03:50
the inverse sine function.
A formula that you probably

39
00:03:50 --> 00:03:54
already knew,
but that is one way to derive

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

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

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

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

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

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00:04:08 --> 00:04:17
 
 

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

47
00:04:20 --> 00:04:23
as opposed to the partial
derivatives.

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

49
00:04:28 --> 00:04:33
can change -- Sorry.
All the contributions that can

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

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

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

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

54
00:04:56 --> 00:05:02
f sub z dz.
Maybe, just to remind you of

55
00:05:02 --> 00:05:07
the other notation,
partial f over partial x dx

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

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

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

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00:05:22 --> 00:05:24
Well, they are called
differentials.

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

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

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object.
These things have their own

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rules of manipulations,
and we have to learn what we

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can do with them.
So how do we think about them?

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

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Here is an important thing to
know.

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Important.
df is not the same thing as

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delta f.
That is meant to be a number.

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

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

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These are numbers.
Delta x, delta y and delta z

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are actual numbers,
and this becomes a number.

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This guy actually is not a
number.

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You cannot give it a particular
value.

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All you can do with a
differential is express it in

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terms of other differentials.
In fact, this dx,

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

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

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00:06:42 --> 00:06:46
them, they are the differentials
of x, y and z.

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

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where you will put other things.
Of course, they represent,

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

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

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

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infinitesimal changes.
Another way to say it,

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

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

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

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

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

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

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

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

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

95
00:07:52 --> 00:08:00
something even before that.
The first thing that it does is

96
00:08:00 --> 00:08:10
it can encode how changes in x,
y, z affect the value of f.

97
00:08:10 --> 00:08:15
I would say that is the most
general answer to what is this

98
00:08:15 --> 00:08:18
formula, what are these
differentials.

99
00:08:18 --> 00:08:24
It is a relation between x,
y, z and f.

100
00:08:24 --> 00:08:36
And this is a placeholder for
small variations,

101
00:08:36 --> 00:08:53
delta x, delta y and delta z to
get an approximation formula.

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

103
00:09:00 --> 00:09:06
x fy delta y fz delta z.
It is getting cramped,

104
00:09:06 --> 00:09:11
but I am sure you know what is
going on here.

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

106
00:09:15 --> 00:09:19
approximately equal.
So they are really not the same.

107
00:09:19 --> 00:09:22
Another thing that the notation
suggests we can do,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

122
00:11:00 --> 00:11:09
situation where x is a function
of t, y is a function of t and z

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

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

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

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

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

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

129
00:11:38 --> 00:11:40
rule,
which tells you when you have a

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

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

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

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

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

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

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

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

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

139
00:12:19 --> 00:12:23
serious mistake on that.
I mean they do have a couple of

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

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

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

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

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

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

146
00:13:01 --> 00:13:05
because the first one is just
stating some general principle,

147
00:13:05 --> 00:13:08
but I am not making a precise
mathematical claim.

148
00:13:08 --> 00:13:11
The second one,
well, we know the approximation

149
00:13:11 --> 00:13:14
formula already,
so I don't need to justify it

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

151
00:13:16 --> 00:13:20
this formula here,
I mean, you probably have a

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

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

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

155
00:13:29 --> 00:13:32
And then I am telling you we
can do that, but I kind of

156
00:13:32 --> 00:13:44
pulled it out of my hat.
I mean I don't have a hat.

157
00:13:44 --> 00:13:53
Why is this valid?
How can I get to this?

158
00:13:53 --> 00:14:06
Here is a first attempt of
justifying how to get there.

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

160
00:14:13 --> 00:14:25
plus f sub y dy plus f sub z dz.
But we know if x is a function

161
00:14:25 --> 00:14:37
of t then dx is x prime of t dt,
dy is y prime of t dt,

162
00:14:37 --> 00:14:47
dz is z prime of t dt.
If we plug these into that

163
00:14:47 --> 00:14:58
formula, we will get that df is
f sub x times x prime t dt plus

164
00:14:58 --> 00:15:08
f sub y y prime of t dt plus f
sub z z prime of t dt.

165
00:15:08 --> 00:15:14
And now I have a relation
between df and dt.

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

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

168
00:15:23 --> 00:15:38
that coefficient.
If I divide by dt then I get

169
00:15:38 --> 00:15:46
the chain rule.
That kind of works,

170
00:15:46 --> 00:15:49
but that shouldn't be
completely satisfactory.

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

172
00:15:53 --> 00:15:57
differentials yet then it is
maybe not very good that I

173
00:15:57 --> 00:16:01
actually used more of these
differential notations in

174
00:16:01 --> 00:16:05
deriving the answer.
That is actually not how it is

175
00:16:05 --> 00:16:08
proved.
The way in which you prove the

176
00:16:08 --> 00:16:13
chain rule is not this way
because we shouldn't have too

177
00:16:13 --> 00:16:16
much trust in differentials just
yet.

178
00:16:16 --> 00:16:18
I mean at the end of today's
lecture, yes,

179
00:16:18 --> 00:16:20
probably we should believe in
them,

180
00:16:20 --> 00:16:26
but so far we should be a
little bit reluctant to believe

181
00:16:26 --> 00:16:32
these kind of strange objects
telling us weird things.

182
00:16:32 --> 00:16:39
Here is a better way to think
about it.

183
00:16:39 --> 00:16:43
One thing that we have trust in
so far are approximation

184
00:16:43 --> 00:16:48
formulas.
We should have trust in them.

185
00:16:48 --> 00:16:54
We should believe that if we
change x a little bit,

186
00:16:54 --> 00:17:02
if we change y a little bit
then we are actually going to

187
00:17:02 --> 00:17:11
get a change in f that is
approximately given by these

188
00:17:11 --> 00:17:13
guys.
And this is true for any

189
00:17:13 --> 00:17:14
changes in x,
y, z,

190
00:17:14 --> 00:17:20
but in particular let's look at
the changes that we get if we

191
00:17:20 --> 00:17:26
just take these formulas as
function of time and change time

192
00:17:26 --> 00:17:32
a little bit by delta t.
We will actually use the

193
00:17:32 --> 00:17:39
changes in x,
y, z in a small time delta t.

194
00:17:39 --> 00:17:47
Let's divide everybody by delta
t.

195
00:17:47 --> 00:17:52
Here I am just dividing numbers
so I am not actually playing any

196
00:17:52 --> 00:17:54
tricks on you.
I mean we don't really know

197
00:17:54 --> 00:17:57
what it means to divide
differentials,

198
00:17:57 --> 00:17:59
but dividing numbers is
something we know.

199
00:17:59 --> 00:18:11
And now, if I take delta t very
small, this guy tends to the

200
00:18:11 --> 00:18:19
derivative, df over dt.
Remember, the definition of df

201
00:18:19 --> 00:18:23
over dt is the limit of this
ratio when the time interval

202
00:18:23 --> 00:18:28
delta t tends to zero.
That means if I choose smaller

203
00:18:28 --> 00:18:32
and smaller values of delta t
then these ratios of numbers

204
00:18:32 --> 00:18:35
will actually tend to some
value,

205
00:18:35 --> 00:18:41
and that value is the
derivative.

206
00:18:41 --> 00:18:51
Similarly, here delta x over
delta t, when delta t is really

207
00:18:51 --> 00:18:59
small, will tend to the
derivative dx/dt.

208
00:18:59 --> 00:19:00
And similarly for the others.

209
00:19:00 --> 00:19:18

210
00:19:18 --> 00:19:28
That means, in particular, 
we take the limit as delta t

211
00:19:28 --> 00:19:35
tends to zero and we get df over
dt on one side and on the other

212
00:19:35 --> 00:19:42
side we get f sub x dx over dt
plus f sub y dy over dt plus f

213
00:19:42 --> 00:19:46
sub z dz over dt.
And the approximation becomes

214
00:19:46 --> 00:19:49
better and better.
Remember when we write

215
00:19:49 --> 00:19:53
approximately equal that means
it is not quite the same,

216
00:19:53 --> 00:19:57
but if we take smaller
variations then actually we will

217
00:19:57 --> 00:20:01
end up with values that are
closer and closer.

218
00:20:01 --> 00:20:04
When we take the limit,
as delta t tends to zero,

219
00:20:04 --> 00:20:06
eventually we get an equality.

220
00:20:06 --> 00:20:21

221
00:20:21 --> 00:20:24
I mean mathematicians have more
complicated words to justify

222
00:20:24 --> 00:20:28
this statement.
I will spare them for now,

223
00:20:28 --> 00:20:36
and you will see them when you
take analysis if you go in that

224
00:20:36 --> 00:20:42
direction.
Any questions so far?

225
00:20:42 --> 00:20:46
No.
OK.

226
00:20:46 --> 00:20:47
Let's check this with an
example.

227
00:20:47 --> 00:20:58
Let's say that we really don't
have any faith in these things

228
00:20:58 --> 00:21:06
so let's try to do it.
Let's say I give you a function

229
00:21:06 --> 00:21:14
that is x ^2 y z.
And let's say that maybe x will

230
00:21:14 --> 00:21:20
be t, y will be e^t and z will
be sin(t).

231
00:21:20 --> 00:21:34
 
 

232
00:21:34 --> 00:21:40
What does the chain rule say?
Well, the chain rule tells us

233
00:21:40 --> 00:21:46
that dw/dt is,
we start with partial w over

234
00:21:46 --> 00:21:51
partial x, well,
what is that?

235
00:21:51 --> 00:21:58
That is 2xy, 
and maybe I should point out

236
00:21:58 --> 00:22:08
that this is w sub x,
times dx over dt plus -- Well, 

237
00:22:08 --> 00:22:21
w sub y is x squared times dy
over dt plus w sub z,

238
00:22:21 --> 00:22:28
which is going to be just one, 
dz over dt.

239
00:22:28 --> 00:22:33
And so now let's plug in the
actual values of these things.

240
00:22:33 --> 00:22:38
x is t and y is e^t, 
so that will be 2t e to the t, 

241
00:22:38 --> 00:22:47
dx over dt is one plus x
squared is t squared,

242
00:22:47 --> 00:23:00
dy over dt is e over t, 
plus dz over dt is cosine t.

243
00:23:00 --> 00:23:06
At the end of calculation we
get 2t e to the t plus t squared

244
00:23:06 --> 00:23:11
e to the t plus cosine t.
That is what the chain rule

245
00:23:11 --> 00:23:16
tells us.
How else could we find that?

246
00:23:16 --> 00:23:20
Well, we could just plug in
values of x, y and z,

247
00:23:20 --> 00:23:23
x plus w is a function of t,
and take its derivative.

248
00:23:23 --> 00:23:26
Let's do that just for
verification.

249
00:23:26 --> 00:23:30
It should be exactly the same
answer.

250
00:23:30 --> 00:23:32
And, in fact,
in this case,

251
00:23:32 --> 00:23:35
the two calculations are
roughly equal in complication.

252
00:23:35 --> 00:23:39
But say that your function of
x, y, z was much more

253
00:23:39 --> 00:23:43
complicated than that,
or maybe you actually didn't

254
00:23:43 --> 00:23:45
know a formula for it,
you only knew its partial

255
00:23:45 --> 00:23:48
derivatives,
then you would need to use the

256
00:23:48 --> 00:23:51
chain rule.
So, sometimes plugging in

257
00:23:51 --> 00:23:54
values is easier but not always.

258
00:23:54 --> 00:24:13

259
00:24:13 --> 00:24:18
Let's just check quickly.
The other method would be to

260
00:24:18 --> 00:24:23
substitute.
W as a function of t.

261
00:24:23 --> 00:24:36
Remember w was x^2y z.
x was t, so you get t squared,

262
00:24:36 --> 00:24:41
y is e to the t,
plus z was sine t.

263
00:24:41 --> 00:24:47
dw over dt, we know how to take
the derivative using single

264
00:24:47 --> 00:24:50
variable calculus.
Well, we should know.

265
00:24:50 --> 00:24:55
If we don't know then we should
take a look at 18.01 again.

266
00:24:55 --> 00:25:02
The product rule that will be
derivative of t squared is 2t

267
00:25:02 --> 00:25:08
times e to the t plus t squared
time the derivative of e to the

268
00:25:08 --> 00:25:16
t is e to the t plus cosine t.
And that is the same answer as

269
00:25:16 --> 00:25:19
over there.
I ended up writing,

270
00:25:19 --> 00:25:23
you know, maybe I wrote
slightly more here,

271
00:25:23 --> 00:25:28
but actually the amount of
calculations really was pretty

272
00:25:28 --> 00:25:32
much the same.
Any questions about that?

273
00:25:32 --> 00:25:39
Yes?
What kind of object is w?

274
00:25:39 --> 00:25:43
Well, you can think of w as
just another variable that is

275
00:25:43 --> 00:25:47
given as a function of x,
y and z, for example.

276
00:25:47 --> 00:25:51
You would have a function of x,
y, z defined by this formula,

277
00:25:51 --> 00:25:57
and I call it w.
I call its value w so that I

278
00:25:57 --> 00:26:04
can substitute t instead of x,
y, z.

279
00:26:04 --> 00:26:07
Well, let's think of w as a
function of three variables.

280
00:26:07 --> 00:26:12
And then, when I plug in the
dependents of these three

281
00:26:12 --> 00:26:17
variables on t,
then it becomes just a function

282
00:26:17 --> 00:26:19
of t.
I mean, really,

283
00:26:19 --> 00:26:23
my w here is pretty much what I
called f before.

284
00:26:23 --> 00:26:31
There is no major difference
between the two.

285
00:26:31 --> 00:26:38
Any other questions?
No.

286
00:26:38 --> 00:26:45
OK.
Let's see.

287
00:26:45 --> 00:26:49
Here is an application of what
we have seen.

288
00:26:49 --> 00:26:53
Let's say that you want to
understand actually all these

289
00:26:53 --> 00:26:57
rules about taking derivatives
in single variable calculus.

290
00:26:57 --> 00:27:00
What I showed you at the
beginning, and then erased,

291
00:27:00 --> 00:27:04
basically justifies how to take
the derivative of a reciprocal

292
00:27:04 --> 00:27:06
function.
And for that you didn't need

293
00:27:06 --> 00:27:10
multivariable calculus.
But let's try to justify the

294
00:27:10 --> 00:27:12
product rule,
for example,

295
00:27:12 --> 00:27:21
for the derivative.
An application of this actually

296
00:27:21 --> 00:27:31
is to justify the product and
quotient rules.

297
00:27:31 --> 00:27:33
Let's think,
for example,

298
00:27:33 --> 00:27:39
of a function of two variables,
u and v, that is just the

299
00:27:39 --> 00:27:44
product uv.
And let's say that u and v are

300
00:27:44 --> 00:27:48
actually functions of one
variable t.

301
00:27:48 --> 00:28:00
Then, well, d of uv over dt is
given by the chain rule applied

302
00:28:00 --> 00:28:04
to f.
This is df over dt.

303
00:28:04 --> 00:28:15
So df over dt should be f sub q
du over dt plus f sub v plus dv

304
00:28:15 --> 00:28:19
over dt.
But now what is the partial of

305
00:28:19 --> 00:28:23
f with respect to u?
It is v.

306
00:28:23 --> 00:28:31
That is v du over dt.
And partial of f with respect

307
00:28:31 --> 00:28:38
to v is going to be just u,
dv over dt.

308
00:28:38 --> 00:28:42
So you get back the usual
product rule.

309
00:28:42 --> 00:28:46
That is a slightly complicated
way of deriving it,

310
00:28:46 --> 00:28:50
but that is a valid way of
understanding how to take the

311
00:28:50 --> 00:28:54
derivative of a product by
thinking of the product first as

312
00:28:54 --> 00:28:57
a function of variables,
which are u and v.

313
00:28:57 --> 00:29:00
And then say,
oh, but u and v were actually

314
00:29:00 --> 00:29:03
functions of a variable t.
And then you do the

315
00:29:03 --> 00:29:08
differentiation in two stages
using the chain rule.

316
00:29:08 --> 00:29:16
Similarly, you can do the
quotient rule just for practice.

317
00:29:16 --> 00:29:21
If I give you the function g
equals u of v.

318
00:29:21 --> 00:29:25
Right now I am thinking of it
as a function of two variables,

319
00:29:25 --> 00:29:29
u and v.
U and v themselves are actually

320
00:29:29 --> 00:29:39
going to be functions of t.
Then, well, dg over dt is going

321
00:29:39 --> 00:29:44
to be partial g,
partial u.

322
00:29:44 --> 00:29:48
How much is that?
How much is partial g,

323
00:29:48 --> 00:29:53
partial u?
One over v times du over dt

324
00:29:53 --> 00:29:58
plus -- Well,
next we need to have partial g

325
00:29:58 --> 00:30:01
over partial v.
Well, what is the derivative of

326
00:30:01 --> 00:30:04
this with respect to v?
Here we need to know how to

327
00:30:04 --> 00:30:11
differentiate the inverse.
It is minus u over v squared

328
00:30:11 --> 00:30:20
times dv over dt.
And that is actually the usual

329
00:30:20 --> 00:30:28
quotient rule just written in a
slightly different way.

330
00:30:28 --> 00:30:30
I mean, just in case you really
want to see it,

331
00:30:30 --> 00:30:36
if you clear denominators for v
squared then you will see

332
00:30:36 --> 00:30:41
basically u prime times v minus
v prime times u.

333
00:30:41 --> 00:31:25
 
 

334
00:31:25 --> 00:31:32
Now let's go to something even
more crazy.

335
00:31:32 --> 00:31:45
I claim we can do chain rules
with more variables.

336
00:31:45 --> 00:31:50
Let's say that I have a
quantity.

337
00:31:50 --> 00:31:55
Let's call it w for now.
Let's say I have quantity w as

338
00:31:55 --> 00:31:58
a function of say variables x
and y.

339
00:31:58 --> 00:32:02
And so in the previous setup x
and y depended on some

340
00:32:02 --> 00:32:04
parameters t.
But, actually,

341
00:32:04 --> 00:32:07
let's now look at the case
where x and y themselves are

342
00:32:07 --> 00:32:10
functions of several variables.
Let's say of two more variables.

343
00:32:10 --> 00:32:25
Let's call them u and v.
I am going to stay with these

344
00:32:25 --> 00:32:27
abstract letters,
but if it bothers you,

345
00:32:27 --> 00:32:31
if it sounds completely
unmotivated think about it maybe

346
00:32:31 --> 00:32:33
in terms of something you might
now.

347
00:32:33 --> 00:32:36
Say, polar coordinates.
Let's say that I have a

348
00:32:36 --> 00:32:40
function but is defined in terms
of the polar coordinate

349
00:32:40 --> 00:32:43
variables on theta.
And then I know I want to

350
00:32:43 --> 00:32:45
switch to usual coordinates x
and y.

351
00:32:45 --> 00:32:49
Or, the other way around,
I have a function of x and y

352
00:32:49 --> 00:32:53
and I want to express it in
terms of the polar coordinates r

353
00:32:53 --> 00:32:57
and theta.
Then I would want to know maybe

354
00:32:57 --> 00:33:02
how the derivatives,
with respect to the various

355
00:33:02 --> 00:33:07
sets of variables,
related to each other.

356
00:33:07 --> 00:33:10
One way I could do it is,
of course,

357
00:33:10 --> 00:33:16
to say now if I plug the
formula for x and the formula

358
00:33:16 --> 00:33:23
for y into the formula for f
then w becomes a function of u

359
00:33:23 --> 00:33:27
and v,
and it can try to take partial

360
00:33:27 --> 00:33:29
derivatives.
If I have explicit formulas,

361
00:33:29 --> 00:33:32
well, that could work.
But maybe the formulas are

362
00:33:32 --> 00:33:35
complicated.
Typically, if I switch between

363
00:33:35 --> 00:33:37
rectangular and polar
coordinates,

364
00:33:37 --> 00:33:41
there might be inverse trig, 
there might be maybe arctangent

365
00:33:41 --> 00:33:45
to express the polar angle in
terms of x and y.

366
00:33:45 --> 00:33:51
And when I don't really want to
actually substitute arctangents

367
00:33:51 --> 00:33:56
everywhere, maybe I would rather
deal with the derivatives.

368
00:33:56 --> 00:34:03
How do I do that?
The question is what are

369
00:34:03 --> 00:34:11
partial w over partial u and
partial w over partial v in

370
00:34:11 --> 00:34:17
terms of, let's see,
what do we need to know to

371
00:34:17 --> 00:34:22
understand that?
Well, probably we should know

372
00:34:22 --> 00:34:28
how w depends on x and y.
If we don't know that then we

373
00:34:28 --> 00:34:32
are probably toast.
Partial w over partial x,

374
00:34:32 --> 00:34:36
partial w over partial y should
be required.

375
00:34:36 --> 00:34:39
What else should we know?
Well, it would probably help to

376
00:34:39 --> 00:34:42
know how x and y depend on u and
v.

377
00:34:42 --> 00:34:46
If we don't know that then we
don't really know how to do it.

378
00:34:46 --> 00:34:55
We need also x sub u,
x sub v, y sub u,

379
00:34:55 --> 00:35:00
y sub v.
We have a lot of partials in

380
00:35:00 --> 00:35:07
there.
Well, let's see how we can do

381
00:35:07 --> 00:35:13
that.
Let's start by writing dw.

382
00:35:13 --> 00:35:19
We know that dw is partial f,
well, I don't know why I have

383
00:35:19 --> 00:35:25
two names, w and f.
I mean w and f are really the

384
00:35:25 --> 00:35:30
same thing here,
but let's say f sub x dx plus f

385
00:35:30 --> 00:35:35
sub y dy.
So far that is our new friend,

386
00:35:35 --> 00:35:39
the differential.
Now what do we want to do with

387
00:35:39 --> 00:35:42
it?
Well, we would like to get rid

388
00:35:42 --> 00:35:47
of dx and dy because we like to
express things in terms of,

389
00:35:47 --> 00:35:50
you know, the question we are
asking ourselves is let's say

390
00:35:50 --> 00:35:55
that I change u a little bit,
how does w change?

391
00:35:55 --> 00:35:58
Of course, what happens,
if I change u a little bit,

392
00:35:58 --> 00:36:01
is y and y will change.
How do they change?

393
00:36:01 --> 00:36:05
Well, that is given to me by
the differential.

394
00:36:05 --> 00:36:13
dx is going to be,
well, I can use the

395
00:36:13 --> 00:36:19
differential again.
Well, x is a function of u and

396
00:36:19 --> 00:36:24
v.
That will be x sub u times du

397
00:36:24 --> 00:36:28
plus x sub v times dv.
That is, again,

398
00:36:28 --> 00:36:31
taking the differential of a
function of two variables.

399
00:36:31 --> 00:36:37
Does that make sense?
And then we have the other guy,

400
00:36:37 --> 00:36:39
f sub y times,
what is dy?

401
00:36:39 --> 00:36:49
Well, similarly dy is y sub u
du plus y sub v dv.

402
00:36:49 --> 00:36:54
And now we have a relation
between dw and du and dv.

403
00:36:54 --> 00:37:00
We are expressing how w reacts
to changes in u and v,

404
00:37:00 --> 00:37:04
which was our goal.
Now, let's actually collect

405
00:37:04 --> 00:37:08
terms so that we see it a bit
better.

406
00:37:08 --> 00:37:19
It is going to be f sub x times
x sub u times f sub y times y

407
00:37:19 --> 00:37:28
sub u du plus f sub x,
x sub v plus f sub y y sub v

408
00:37:28 --> 00:37:32
dv.
Now we have dw equals something

409
00:37:32 --> 00:37:38
du plus something dv.
Well, the coefficient here has

410
00:37:38 --> 00:37:44
to be partial f over partial u.
What else could it be?

411
00:37:44 --> 00:37:49
That's the rate of change of w
with respect to u if I forget

412
00:37:49 --> 00:37:54
what happens when I change v.
That is the definition of a

413
00:37:54 --> 00:37:58
partial.
Similarly, this one has to be

414
00:37:58 --> 00:38:04
partial f over partial v.
That is because it is the rate

415
00:38:04 --> 00:38:09
of change with respect to v,
if I keep u constant,

416
00:38:09 --> 00:38:13
so that these guys are
completely ignored.

417
00:38:13 --> 00:38:16
Now you see how the total
differential accounts for,

418
00:38:16 --> 00:38:21
somehow, all the partial
derivatives that come as

419
00:38:21 --> 00:38:27
coefficients of the individual
variables in these expressions.

420
00:38:27 --> 00:38:33
Let me maybe rewrite these
formulas in a more visible way

421
00:38:33 --> 00:38:40
and then re-explain them to you.
Here is the chain rule for this

422
00:38:40 --> 00:38:46
situation, with two intermediate
variables and two variables that

423
00:38:46 --> 00:38:50
you express these in terms of.
In our setting, 

424
00:38:50 --> 00:38:56
we get partial f over partial u
equals partial f over partial x

425
00:38:56 --> 00:39:02
time partial x over partial u
plus partial f over partial y

426
00:39:02 --> 00:39:08
times partial y over partial u.
And the other one, 

427
00:39:08 --> 00:39:15
the same thing with v instead
of u,

428
00:39:15 --> 00:39:22
partial f over partial x times
partial x over partial v plus

429
00:39:22 --> 00:39:28
partial f over partial u partial
y over partial v.

430
00:39:28 --> 00:39:31
I have to explain various
things about these formulas

431
00:39:31 --> 00:39:34
because they look complicated.
And, actually,

432
00:39:34 --> 00:39:39
they are not that complicated.
A couple of things to know.

433
00:39:39 --> 00:39:42
The first thing,
how do we remember a formula

434
00:39:42 --> 00:39:44
like that?
Well, that is easy.

435
00:39:44 --> 00:39:47
We want to know how f depends
on u.

436
00:39:47 --> 00:39:51
Well, what does f depend on?
It depends on x and y.

437
00:39:51 --> 00:39:55
So we will put partial f over
partial x and partial f over

438
00:39:55 --> 00:39:59
partial y.
Now, x and y, why are they here?

439
00:39:59 --> 00:40:01
Well, they are here because
they actually depend on u as

440
00:40:01 --> 00:40:04
well.
How does x depend on u?

441
00:40:04 --> 00:40:06
Well, the answer is partial x
over partial u.

442
00:40:06 --> 00:40:10
How does y depend on u?
The answer is partial y over

443
00:40:10 --> 00:40:12
partial u.
See, the structure of this

444
00:40:12 --> 00:40:16
formula is simple.
To find the partial of f with

445
00:40:16 --> 00:40:20
respect to some new variable you
use the partials with respect to

446
00:40:20 --> 00:40:24
the variables that f was
initially defined in terms of x

447
00:40:24 --> 00:40:28
and y.
And you multiply them by the

448
00:40:28 --> 00:40:33
partials of x and y in terms of
the new variable that you want

449
00:40:33 --> 00:40:37
to look at, v here,
and you sum these things

450
00:40:37 --> 00:40:40
together.
That is the structure of the

451
00:40:40 --> 00:40:42
formula.
Why does it work?

452
00:40:42 --> 00:40:45
Well, let me explain it to you
in a slightly different

453
00:40:45 --> 00:40:49
language.
This asks us how does f change

454
00:40:49 --> 00:40:54
if I change u a little bit?
Well, why would f change if u

455
00:40:54 --> 00:40:57
changes a little bit?
Well, it would change because f

456
00:40:57 --> 00:41:00
actually depends on x and y and
x and y depend on u.

457
00:41:00 --> 00:41:03
If I change u,
how quickly does x change?

458
00:41:03 --> 00:41:06
Well, the answer is partial x
over partial u.

459
00:41:06 --> 00:41:09
And now, if I change x at this
rate, how does that have to

460
00:41:09 --> 00:41:13
change?
Well, the answer is partial f

461
00:41:13 --> 00:41:17
over partial x times this guy.
Well, at the same time,

462
00:41:17 --> 00:41:21
y is also changing.
How fast is y changing if I

463
00:41:21 --> 00:41:24
change u?
Well, at the rate of partial y

464
00:41:24 --> 00:41:27
over partial u.
But now if I change this how

465
00:41:27 --> 00:41:30
does f change?
Well, the rate of change is

466
00:41:30 --> 00:41:34
partial f over partial y.
The product is the effect of

467
00:41:34 --> 00:41:37
how you change it,
changing u, and therefore

468
00:41:37 --> 00:41:40
changing f.
Now, what happens in real life,

469
00:41:40 --> 00:41:43
if I change u a little bit?
Well, both x and y change at

470
00:41:43 --> 00:41:46
the same time.
So how does f change?

471
00:41:46 --> 00:41:50
Well, it is the sum of the two
effects.

472
00:41:50 --> 00:41:54
Does that make sense?
Good.

473
00:41:54 --> 00:42:00
Of course, if f depends on more
variables then you just have

474
00:42:00 --> 00:42:02
more terms in here.
OK.

475
00:42:02 --> 00:42:05
Here is another thing that may
be a little bit confusing.

476
00:42:05 --> 00:42:09
What is tempting?
Well, what is tempting here

477
00:42:09 --> 00:42:12
would be to simplify these
formulas by removing these

478
00:42:12 --> 00:42:15
partial x's.
Let's simplify by partial x.

479
00:42:15 --> 00:42:18
Let's simplify by partial y.
We get partial f over partial u

480
00:42:18 --> 00:42:21
equals partial f over partial u
plus partial f over partial u.

481
00:42:21 --> 00:42:25
Something is not working
properly.

482
00:42:25 --> 00:42:28
Why doesn't it work?
The answer is precisely because

483
00:42:28 --> 00:42:32
these are partial derivatives.
These are not total derivatives.

484
00:42:32 --> 00:42:36
And so you cannot simplify them
in that way.

485
00:42:36 --> 00:42:39
And that is actually the reason
why we use this curly d rather

486
00:42:39 --> 00:42:41
than a straight d.
It is to remind us,

487
00:42:41 --> 00:42:44
beware, there are these
simplifications that we can do

488
00:42:44 --> 00:42:47
with straight d's that are not
legal here.

489
00:42:47 --> 00:42:52
Somehow, when you have a
partial derivative,

490
00:42:52 --> 00:42:57
you must resist the urge of
simplifying things.

491
00:42:57 --> 00:43:02
No simplifications in here.
That is the simplest formula

492
00:43:02 --> 00:43:10
you can get.
Any questions at this point?

493
00:43:10 --> 00:43:21
No.
Yes?

494
00:43:21 --> 00:43:23
When would you use this and
what does it describe?

495
00:43:23 --> 00:43:26
Well, it is basically when you
have a function given in terms

496
00:43:26 --> 00:43:29
of a certain set of variables
because maybe there is a simply

497
00:43:29 --> 00:43:31
expression in terms of those
variables.

498
00:43:31 --> 00:43:35
But ultimately what you care
about is not those variables,

499
00:43:35 --> 00:43:39
z and y, but another set of
variables, here u and v.

500
00:43:39 --> 00:43:42
So x and y are giving you a
nice formula for f,

501
00:43:42 --> 00:43:46
but actually the relevant
variables for your problem are u

502
00:43:46 --> 00:43:48
and v.
And you know x and y are

503
00:43:48 --> 00:43:50
related to u and v.
So, of course,

504
00:43:50 --> 00:43:53
what you could do is plug the
formulas the way that we did

505
00:43:53 --> 00:43:55
substituting.
But maybe that will give you

506
00:43:55 --> 00:43:59
very complicated expressions.
And maybe it is actually easier

507
00:43:59 --> 00:44:02
to just work with the derivates.
The important claim here is

508
00:44:02 --> 00:44:05
basically we don't need to know
the actual formulas.

509
00:44:05 --> 00:44:07
All we need to know are the
rate of changes.

510
00:44:07 --> 00:44:11
If we know all these rates of
change then we know how to take

511
00:44:11 --> 00:44:14
these derivatives without
actually having to plug in

512
00:44:14 --> 00:44:22
values.
Yes?

513
00:44:22 --> 00:44:25
Yes, you could certain do the
same things in terms of t.

514
00:44:25 --> 00:44:29
If x and y were functions of t
instead of being functions of u

515
00:44:29 --> 00:44:31
and v then it would be the same
thing.

516
00:44:31 --> 00:44:34
And you would have the same
formulas that I had,

517
00:44:34 --> 00:44:37
well, over there I still have
it.

518
00:44:37 --> 00:44:39
Why does that one have straight
d's?

519
00:44:39 --> 00:44:42
Well, the answer is I could put
curly d's if I wanted,

520
00:44:42 --> 00:44:45
but I end up with a function of
a single variable.

521
00:44:45 --> 00:44:48
If you have a single variable
then the partial,

522
00:44:48 --> 00:44:50
with respect to that variable,
is the same thing as the usual

523
00:44:50 --> 00:44:53
derivative.
We don't actually need to worry

524
00:44:53 --> 00:44:57
about curly in that case.
But that one is indeed special

525
00:44:57 --> 00:45:00
case of this one where instead
of x and y depending on two

526
00:45:00 --> 00:45:03
variables, u and v,
they depend on a single

527
00:45:03 --> 00:45:04
variable t.
Now, of course,

528
00:45:04 --> 00:45:06
you can call variables any name
you want.

529
00:45:06 --> 00:45:12
It doesn't matter.
This is just a slight

530
00:45:12 --> 00:45:16
generalization of that.
Well, not quite because here I

531
00:45:16 --> 00:45:18
also had a z.
See, I am trying to just

532
00:45:18 --> 00:45:21
confuse you by giving you
functions that depend on various

533
00:45:21 --> 00:45:25
numbers of variables.
If you have a function of 30

534
00:45:25 --> 00:45:28
variables, things work the same
way, just longer,

535
00:45:28 --> 00:45:33
and you are going to run out of
letters in the alphabet before

536
00:45:33 --> 00:45:38
the end.
Any other questions?

537
00:45:38 --> 00:45:43
No.
What?

538
00:45:43 --> 00:45:51
Yes?
If u and v themselves depended

539
00:45:51 --> 00:45:55
on another variable then you
would continue with your chain

540
00:45:55 --> 00:45:58
rules.
Maybe you would know to express

541
00:45:58 --> 00:46:02
partial x over partial u in
terms using that chain rule.

542
00:46:02 --> 00:46:05
Sorry.
If u and v are dependent on yet

543
00:46:05 --> 00:46:08
another variable then you could
get the derivative with respect

544
00:46:08 --> 00:46:11
to that using first the chain
rule to pass from u v to that

545
00:46:11 --> 00:46:14
new variable,
and then you would plug in

546
00:46:14 --> 00:46:17
these formulas for partials of f
with respect to u and v.

547
00:46:17 --> 00:46:19
In fact, if you have several
substitutions to do,

548
00:46:19 --> 00:46:21
you can always arrange to use
one chain rule at a time.

549
00:46:21 --> 00:46:25
You just have to do them in
sequence.

550
00:46:25 --> 00:46:28
That's why we don't actually
learn that, but you can just do

551
00:46:28 --> 00:46:32
it be repeating the process.
I mean, probably at that stage,

552
00:46:32 --> 00:46:35
the easiest to not get confused
actually is to manipulate

553
00:46:35 --> 00:46:38
differentials because that is
probably easier.

554
00:46:38 --> 00:46:47
Yes?
Curly f does not exist.

555
00:46:47 --> 00:46:50
That's easy.
Curly f makes no sense by

556
00:46:50 --> 00:46:52
itself.
It doesn't exist alone.

557
00:46:52 --> 00:46:58
What exists is only curly df
over curly d some variable.

558
00:46:58 --> 00:47:02
And then that accounts only for
the rate of change with respect

559
00:47:02 --> 00:47:05
to that variable leaving the
others fixed,

560
00:47:05 --> 00:47:11
while straight df is somehow a
total variation of f.

561
00:47:11 --> 00:47:16
It accounts for all of the
partial derivatives and their

562
00:47:16 --> 00:47:25
combined effects.
OK. Any more questions? No.

563
00:47:25 --> 00:47:29
Let me just finish up very
quickly by telling you again one

564
00:47:29 --> 00:47:33
example where completely you
might want to do this.

565
00:47:33 --> 00:47:40
You have a function that you
want to switch between

566
00:47:40 --> 00:47:45
rectangular and polar
coordinates.

567
00:47:45 --> 00:47:48
To make things a little bit
concrete.

568
00:47:48 --> 00:47:55
If you have polar coordinates
that means in the plane,

569
00:47:55 --> 00:48:00
instead of using x and y, 
you will use coordinates r,

570
00:48:00 --> 00:48:05
distance to the origin,
and theta, the angles from the

571
00:48:05 --> 00:48:08
x-axis.
The change of variables for

572
00:48:08 --> 00:48:14
that is x equals r cosine theta
and y equals r sine theta.

573
00:48:14 --> 00:48:21
And so that means if you have a
function f that depends on x and

574
00:48:21 --> 00:48:29
y, in fact, you can plug these
in as a function of r and theta.

575
00:48:29 --> 00:48:34
Then you can ask yourself,
well, what is partial f over

576
00:48:34 --> 00:48:37
partial r?
And that is going to be, 

577
00:48:37 --> 00:48:42
well, you want to take partial
f over partial x times partial x

578
00:48:42 --> 00:48:48
partial r plus partial f over
partial y times partial y over

579
00:48:48 --> 00:48:53
partial r.
That will end up being actually

580
00:48:53 --> 00:48:59
f sub x times cosine theta plus
f sub y times sine theta.

581
00:48:59 --> 00:49:02
And you can do the same thing
to find partial f,

582
00:49:02 --> 00:49:05
partial theta.
And so you can express

583
00:49:05 --> 00:49:10
derivatives either in terms of
x, y or in terms of r and theta

584
00:49:10 --> 00:49:13
with simple relations between
them.

585
00:49:13 --> 00:49:20
And the one last thing I should
say.

586
00:49:20 --> 00:49:23
On Thursday we will learn about
more tricks we can play with

587
00:49:23 --> 00:49:27
variations of functions.
And one that is important,

588
00:49:27 --> 00:49:29
because you need to know it
actually to do the p-set,

589
00:49:29 --> 00:49:38
is the gradient vector.
The gradient vector is simply a

590
00:49:38 --> 00:49:41
vector.
You use this downward pointing

591
00:49:41 --> 00:49:44
triangle as the notation for the
gradient.

592
00:49:44 --> 00:49:49
It is simply is a vector whose
components are the partial

593
00:49:49 --> 00:49:53
derivatives of a function.
I mean, in a way,

594
00:49:53 --> 00:49:56
you can think of a differential
as a way to package partial

595
00:49:56 --> 00:49:59
derivatives together into some
weird object.

596
00:49:59 --> 00:50:01
Well, the gradient is also a
way to package partials

597
00:50:01 --> 00:50:04
together.
We will see on Thursday what it

598
00:50:04 --> 00:50:07
is good for, but some of the
problems on the p-set use it.

599
00:50:07 --> 00:50:09