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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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study, in more detail,
how functions of several
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variables behave,
how to compete their
00:00:41.000 --> 00:00:44.000
variations.
How to estimate the variation
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in arbitrary directions.
And so for that we are going to
00:00:50.000 --> 00:00:56.000
need some more tools actually to
study this things.
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More tools to study functions.
00:01:15.000 --> 00:01:26.000
Today's topic is going to be
differentials.
00:01:26.000 --> 00:01:34.000
And, just to motivate that,
let me remind you about one
00:01:34.000 --> 00:01:43.000
trick that you probably know
from single variable calculus,
00:01:43.000 --> 00:01:48.000
namely implicit
differentiation.
00:01:48.000 --> 00:01:56.000
Let's say that you have a
function y equals f of x then
00:01:56.000 --> 00:02:05.000
you would sometimes write dy
equals f prime of x times dx.
00:02:05.000 --> 00:02:17.000
And then maybe you would -- We
use implicit differentiation to
00:02:17.000 --> 00:02:29.000
actually relate infinitesimal
changes in y with infinitesimal
00:02:29.000 --> 00:02:35.000
changes in x.
And one thing we can do with
00:02:35.000 --> 00:02:39.000
that, for example,
is actually figure out the rate
00:02:39.000 --> 00:02:43.000
of change dy by dx,
but also the reciprocal dx by
00:02:43.000 --> 00:02:48.000
dy.
And so, for example,
00:02:48.000 --> 00:02:58.000
let's say that we have y equals
inverse sin(x).
00:02:58.000 --> 00:03:03.000
Then we can write x equals
sin(y).
00:03:03.000 --> 00:03:08.000
And, from there,
we can actually find out what
00:03:08.000 --> 00:03:13.000
is the derivative of this
function if we didn't know the
00:03:13.000 --> 00:03:18.000
answer already by writing dx
equals cosine y dy.
00:03:18.000 --> 00:03:28.000
That tells us that dy over dx
is going to be one over cosine
00:03:28.000 --> 00:03:32.000
y.
And now cosine for relation to
00:03:32.000 --> 00:03:40.000
sine is basically one over
square root of one minus x^2.
00:03:40.000 --> 00:03:44.000
And that is how you find the
formula for the derivative of
00:03:44.000 --> 00:03:50.000
the inverse sine function.
A formula that you probably
00:03:50.000 --> 00:03:54.000
already knew,
but that is one way to derive
00:03:54.000 --> 00:03:57.000
it.
Now we are going to use also
00:03:57.000 --> 00:03:59.000
these kinds of notations,
dx, dy and so on,
00:03:59.000 --> 00:04:03.000
but use them for functions of
several variables.
00:04:03.000 --> 00:04:05.000
And, of course,
we will have to learn what the
00:04:05.000 --> 00:04:08.000
rules of manipulation are and
what we can do with them.
00:04:17.000 --> 00:04:20.000
The actual name of that is the
total differential,
00:04:20.000 --> 00:04:23.000
as opposed to the partial
derivatives.
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The total differential includes
all of the various causes that
00:04:28.000 --> 00:04:33.000
can change -- Sorry.
All the contributions that can
00:04:33.000 --> 00:04:38.000
cause the value of your function
f to change.
00:04:38.000 --> 00:04:43.000
Namely, let's say that you have
a function maybe of three
00:04:43.000 --> 00:04:44.000
variables, x,
y, z,
00:04:44.000 --> 00:04:56.000
then you would write df equals
f sub x dx plus f sub y dy plus
00:04:56.000 --> 00:05:02.000
f sub z dz.
Maybe, just to remind you of
00:05:02.000 --> 00:05:07.000
the other notation,
partial f over partial x dx
00:05:07.000 --> 00:05:14.000
plus partial f over partial y dy
plus partial f over partial z
00:05:14.000 --> 00:05:18.000
dz.
Now, what is this object?
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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And they are not numbers,
they are not vectors,
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
00:05:32.000 --> 00:05:36.000
rules of manipulations,
and we have to learn what we
00:05:36.000 --> 00:05:40.000
can do with them.
So how do we think about them?
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First of all,
how do we not think about them?
00:05:51.000 --> 00:05:55.000
Here is an important thing to
know.
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Important.
df is not the same thing as
00:06:07.000 --> 00:06:12.000
delta f.
That is meant to be a number.
00:06:12.000 --> 00:06:16.000
It is going to be a number once
you have a small variation of x,
00:06:16.000 --> 00:06:19.000
a small variation of y,
a small variation of z.
00:06:19.000 --> 00:06:21.000
These are numbers.
Delta x, delta y and delta z
00:06:21.000 --> 00:06:24.000
are actual numbers,
and this becomes a number.
00:06:24.000 --> 00:06:26.000
This guy actually is not a
number.
00:06:26.000 --> 00:06:30.000
You cannot give it a particular
value.
00:06:30.000 --> 00:06:33.000
All you can do with a
differential is express it in
00:06:33.000 --> 00:06:36.000
terms of other differentials.
In fact, this dx,
00:06:36.000 --> 00:06:38.000
dy and dz, well,
they are mostly symbols out
00:06:38.000 --> 00:06:42.000
there.
But if you want to think about
00:06:42.000 --> 00:06:46.000
them, they are the differentials
of x, y and z.
00:06:46.000 --> 00:06:52.000
In fact, you can think of these
differentials as placeholders
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where you will put other things.
Of course, they represent,
00:06:57.000 --> 00:07:02.000
you know, there is this idea of
changes in x,
00:07:02.000 --> 00:07:05.000
y, z and f.
One way that one could explain
00:07:05.000 --> 00:07:09.000
it, and I don't really like it,
is to say they represent
00:07:09.000 --> 00:07:12.000
infinitesimal changes.
Another way to say it,
00:07:12.000 --> 00:07:14.000
and I think that is probably
closer to the truth,
00:07:14.000 --> 00:07:19.000
is that these things are
somehow placeholders to put
00:07:19.000 --> 00:07:22.000
values and get a tangent
approximation.
00:07:22.000 --> 00:07:25.000
For example,
if I do replace these symbols
00:07:25.000 --> 00:07:30.000
by delta x, delta y and delta z
numbers then I will actually get
00:07:30.000 --> 00:07:33.000
a numerical quantity.
And that will be an
00:07:33.000 --> 00:07:39.000
approximation formula for delta.
It will be the linear
00:07:39.000 --> 00:07:44.000
approximation,
a tangent plane approximation.
00:07:44.000 --> 00:07:52.000
What we can do -- Well,
let me start first with maybe
00:07:52.000 --> 00:08:00.000
something even before that.
The first thing that it does is
00:08:00.000 --> 00:08:10.000
it can encode how changes in x,
y, z affect the value of f.
00:08:10.000 --> 00:08:15.000
I would say that is the most
general answer to what is this
00:08:15.000 --> 00:08:18.000
formula, what are these
differentials.
00:08:18.000 --> 00:08:24.000
It is a relation between x,
y, z and f.
00:08:24.000 --> 00:08:36.000
And this is a placeholder for
small variations,
00:08:36.000 --> 00:08:53.000
delta x, delta y and delta z to
get an approximation formula.
00:08:53.000 --> 00:09:00.000
Which is delta f is
approximately equal to fx delta
00:09:00.000 --> 00:09:06.000
x fy delta y fz delta z.
It is getting cramped,
00:09:06.000 --> 00:09:11.000
but I am sure you know what is
going on here.
00:09:11.000 --> 00:09:15.000
And observe how this one is
actually equal while that one is
00:09:15.000 --> 00:09:19.000
approximately equal.
So they are really not the same.
00:09:19.000 --> 00:09:22.000
Another thing that the notation
suggests we can do,
00:09:22.000 --> 00:09:26.000
and they claim we can do,
is divide everything by some
00:09:26.000 --> 00:09:29.000
variable that everybody depends
on.
00:09:29.000 --> 00:09:33.000
Say, for example,
that x, y and z actually depend
00:09:33.000 --> 00:09:39.000
on some parameter t then they
will vary, at a certain rate,
00:09:39.000 --> 00:09:42.000
dx over dt, dy over dt,
dz over dt.
00:09:42.000 --> 00:09:46.000
And what the differential will
tell us then is the rate of
00:09:46.000 --> 00:09:51.000
change of f as a function of t,
when you plug in these values
00:09:51.000 --> 00:09:57.000
of x, y, z,
you will get df over dt by
00:09:57.000 --> 00:10:05.000
dividing everything by dt in
here.
00:10:05.000 --> 00:10:21.000
The first thing we can do is
divide by something like dt to
00:10:21.000 --> 00:10:30.000
get infinitesimal rate of
change.
00:10:30.000 --> 00:10:43.000
Well, let me just say rate of
change.
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
00:10:52.000 --> 00:11:00.000
plus f sub z dz over dt.
And that corresponds to the
00:11:00.000 --> 00:11:09.000
situation where x is a function
of t, y is a function of t and z
00:11:09.000 --> 00:11:14.000
is a function of t.
That means you can plug in
00:11:14.000 --> 00:11:18.000
these values into f to get,
well, the value of f will
00:11:18.000 --> 00:11:23.000
depend on t,
and then you can find the rate
00:11:23.000 --> 00:11:27.000
of change with t of a value of
f.
00:11:27.000 --> 00:11:35.000
These are the basic rules.
And this is known as the chain
00:11:35.000 --> 00:11:38.000
rule.
It is one instance of a chain
00:11:38.000 --> 00:11:40.000
rule,
which tells you when you have a
00:11:40.000 --> 00:11:42.000
function that depends on
something,
00:11:42.000 --> 00:11:45.000
and that something in turn
depends on something else,
00:11:45.000 --> 00:11:51.000
how to find the rate of change
of a function on the new
00:11:51.000 --> 00:11:56.000
variable in terms of the
derivatives of a function and
00:11:56.000 --> 00:12:01.000
also the dependence between the
various variables.
00:12:01.000 --> 00:12:08.000
Any questions so far?
No.
00:12:08.000 --> 00:12:11.000
OK.
A word of warming,
00:12:11.000 --> 00:12:15.000
in particular,
about what I said up here.
00:12:15.000 --> 00:12:19.000
It is kind of unfortunate,
but the textbook actually has a
00:12:19.000 --> 00:12:23.000
serious mistake on that.
I mean they do have a couple of
00:12:23.000 --> 00:12:29.000
formulas where they mix a d with
a delta, and I warn you not to
00:12:29.000 --> 00:12:32.000
do that, please.
I mean there are d's and there
00:12:32.000 --> 00:12:34.000
are delta's, and basically they
don't live in the same world.
00:12:34.000 --> 00:12:53.000
They don't see each other.
The textbook is lying to you.
00:12:53.000 --> 00:12:59.000
Let's see.
The first and the second
00:12:59.000 --> 00:13:01.000
claims,
I don't really need to justify
00:13:01.000 --> 00:13:05.000
because the first one is just
stating some general principle,
00:13:05.000 --> 00:13:08.000
but I am not making a precise
mathematical claim.
00:13:08.000 --> 00:13:11.000
The second one,
well, we know the approximation
00:13:11.000 --> 00:13:14.000
formula already,
so I don't need to justify it
00:13:14.000 --> 00:13:16.000
for you.
But, on the other hand,
00:13:16.000 --> 00:13:20.000
this formula here,
I mean, you probably have a
00:13:20.000 --> 00:13:24.000
right to expect some reason for
why this works.
00:13:24.000 --> 00:13:27.000
Why is this valid?
After all, I first told you we
00:13:27.000 --> 00:13:29.000
have these new mysterious
objects.
00:13:29.000 --> 00:13:32.000
And then I am telling you we
can do that, but I kind of
00:13:32.000 --> 00:13:44.000
pulled it out of my hat.
I mean I don't have a hat.
00:13:44.000 --> 00:13:53.000
Why is this valid?
How can I get to this?
00:13:53.000 --> 00:14:06.000
Here is a first attempt of
justifying how to get there.
00:14:06.000 --> 00:14:13.000
Let's see.
Well, we said df is f sub x dx
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
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,
00:14:37.000 --> 00:14:47.000
dz is z prime of t dt.
If we plug these into that
00:14:47.000 --> 00:14:58.000
formula, we will get that df is
f sub x times x prime t dt plus
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.
00:15:08.000 --> 00:15:14.000
And now I have a relation
between df and dt.
00:15:14.000 --> 00:15:17.000
See, I got df equals sometimes
times dt.
00:15:17.000 --> 00:15:23.000
That means the rate of change
of f with respect to t should be
00:15:23.000 --> 00:15:38.000
that coefficient.
If I divide by dt then I get
00:15:38.000 --> 00:15:46.000
the chain rule.
That kind of works,
00:15:46.000 --> 00:15:49.000
but that shouldn't be
completely satisfactory.
00:15:49.000 --> 00:15:53.000
Let's say that you are a true
skeptic and you don't believe in
00:15:53.000 --> 00:15:57.000
differentials yet then it is
maybe not very good that I
00:15:57.000 --> 00:16:01.000
actually used more of these
differential notations in
00:16:01.000 --> 00:16:05.000
deriving the answer.
That is actually not how it is
00:16:05.000 --> 00:16:08.000
proved.
The way in which you prove the
00:16:08.000 --> 00:16:13.000
chain rule is not this way
because we shouldn't have too
00:16:13.000 --> 00:16:16.000
much trust in differentials just
yet.
00:16:16.000 --> 00:16:18.000
I mean at the end of today's
lecture, yes,
00:16:18.000 --> 00:16:20.000
probably we should believe in
them,
00:16:20.000 --> 00:16:26.000
but so far we should be a
little bit reluctant to believe
00:16:26.000 --> 00:16:32.000
these kind of strange objects
telling us weird things.
00:16:32.000 --> 00:16:39.000
Here is a better way to think
about it.
00:16:39.000 --> 00:16:43.000
One thing that we have trust in
so far are approximation
00:16:43.000 --> 00:16:48.000
formulas.
We should have trust in them.
00:16:48.000 --> 00:16:54.000
We should believe that if we
change x a little bit,
00:16:54.000 --> 00:17:02.000
if we change y a little bit
then we are actually going to
00:17:02.000 --> 00:17:11.000
get a change in f that is
approximately given by these
00:17:11.000 --> 00:17:13.000
guys.
And this is true for any
00:17:13.000 --> 00:17:14.000
changes in x,
y, z,
00:17:14.000 --> 00:17:20.000
but in particular let's look at
the changes that we get if we
00:17:20.000 --> 00:17:26.000
just take these formulas as
function of time and change time
00:17:26.000 --> 00:17:32.000
a little bit by delta t.
We will actually use the
00:17:32.000 --> 00:17:39.000
changes in x,
y, z in a small time delta t.
00:17:39.000 --> 00:17:47.000
Let's divide everybody by delta
t.
00:17:47.000 --> 00:17:52.000
Here I am just dividing numbers
so I am not actually playing any
00:17:52.000 --> 00:17:54.000
tricks on you.
I mean we don't really know
00:17:54.000 --> 00:17:57.000
what it means to divide
differentials,
00:17:57.000 --> 00:17:59.000
but dividing numbers is
something we know.
00:17:59.000 --> 00:18:11.000
And now, if I take delta t very
small, this guy tends to the
00:18:11.000 --> 00:18:19.000
derivative, df over dt.
Remember, the definition of df
00:18:19.000 --> 00:18:23.000
over dt is the limit of this
ratio when the time interval
00:18:23.000 --> 00:18:28.000
delta t tends to zero.
That means if I choose smaller
00:18:28.000 --> 00:18:32.000
and smaller values of delta t
then these ratios of numbers
00:18:32.000 --> 00:18:35.000
will actually tend to some
value,
00:18:35.000 --> 00:18:41.000
and that value is the
derivative.
00:18:41.000 --> 00:18:51.000
Similarly, here delta x over
delta t, when delta t is really
00:18:51.000 --> 00:18:59.000
small, will tend to the
derivative dx/dt.
00:18:59.000 --> 00:19:00.000
And similarly for the others.
00:19:18.000 --> 00:19:28.000
That means, in particular,
we take the limit as delta t
00:19:28.000 --> 00:19:35.000
tends to zero and we get df over
dt on one side and on the other
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
00:19:42.000 --> 00:19:46.000
sub z dz over dt.
And the approximation becomes
00:19:46.000 --> 00:19:49.000
better and better.
Remember when we write
00:19:49.000 --> 00:19:53.000
approximately equal that means
it is not quite the same,
00:19:53.000 --> 00:19:57.000
but if we take smaller
variations then actually we will
00:19:57.000 --> 00:20:01.000
end up with values that are
closer and closer.
00:20:01.000 --> 00:20:04.000
When we take the limit,
as delta t tends to zero,
00:20:04.000 --> 00:20:06.000
eventually we get an equality.
00:20:21.000 --> 00:20:24.000
I mean mathematicians have more
complicated words to justify
00:20:24.000 --> 00:20:28.000
this statement.
I will spare them for now,
00:20:28.000 --> 00:20:36.000
and you will see them when you
take analysis if you go in that
00:20:36.000 --> 00:20:42.000
direction.
Any questions so far?
00:20:42.000 --> 00:20:46.000
No.
OK.
00:20:46.000 --> 00:20:47.000
Let's check this with an
example.
00:20:47.000 --> 00:20:58.000
Let's say that we really don't
have any faith in these things
00:20:58.000 --> 00:21:06.000
so let's try to do it.
Let's say I give you a function
00:21:06.000 --> 00:21:14.000
that is x ^2 y z.
And let's say that maybe x will
00:21:14.000 --> 00:21:20.000
be t, y will be e^t and z will
be sin(t).
00:21:34.000 --> 00:21:40.000
What does the chain rule say?
Well, the chain rule tells us
00:21:40.000 --> 00:21:46.000
that dw/dt is,
we start with partial w over
00:21:46.000 --> 00:21:51.000
partial x, well,
what is that?
00:21:51.000 --> 00:21:58.000
That is 2xy,
and maybe I should point out
00:21:58.000 --> 00:22:08.000
that this is w sub x,
times dx over dt plus -- Well,
00:22:08.000 --> 00:22:21.000
w sub y is x squared times dy
over dt plus w sub z,
00:22:21.000 --> 00:22:28.000
which is going to be just one,
dz over dt.
00:22:28.000 --> 00:22:33.000
And so now let's plug in the
actual values of these things.
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,
00:22:38.000 --> 00:22:47.000
dx over dt is one plus x
squared is t squared,
00:22:47.000 --> 00:23:00.000
dy over dt is e over t,
plus dz over dt is cosine t.
00:23:00.000 --> 00:23:06.000
At the end of calculation we
get 2t e to the t plus t squared
00:23:06.000 --> 00:23:11.000
e to the t plus cosine t.
That is what the chain rule
00:23:11.000 --> 00:23:16.000
tells us.
How else could we find that?
00:23:16.000 --> 00:23:20.000
Well, we could just plug in
values of x, y and z,
00:23:20.000 --> 00:23:23.000
x plus w is a function of t,
and take its derivative.
00:23:23.000 --> 00:23:26.000
Let's do that just for
verification.
00:23:26.000 --> 00:23:30.000
It should be exactly the same
answer.
00:23:30.000 --> 00:23:32.000
And, in fact,
in this case,
00:23:32.000 --> 00:23:35.000
the two calculations are
roughly equal in complication.
00:23:35.000 --> 00:23:39.000
But say that your function of
x, y, z was much more
00:23:39.000 --> 00:23:43.000
complicated than that,
or maybe you actually didn't
00:23:43.000 --> 00:23:45.000
know a formula for it,
you only knew its partial
00:23:45.000 --> 00:23:48.000
derivatives,
then you would need to use the
00:23:48.000 --> 00:23:51.000
chain rule.
So, sometimes plugging in
00:23:51.000 --> 00:23:54.000
values is easier but not always.
00:24:13.000 --> 00:24:18.000
Let's just check quickly.
The other method would be to
00:24:18.000 --> 00:24:23.000
substitute.
W as a function of t.
00:24:23.000 --> 00:24:36.000
Remember w was x^2y z.
x was t, so you get t squared,
00:24:36.000 --> 00:24:41.000
y is e to the t,
plus z was sine t.
00:24:41.000 --> 00:24:47.000
dw over dt, we know how to take
the derivative using single
00:24:47.000 --> 00:24:50.000
variable calculus.
Well, we should know.
00:24:50.000 --> 00:24:55.000
If we don't know then we should
take a look at 18.01 again.
00:24:55.000 --> 00:25:02.000
The product rule that will be
derivative of t squared is 2t
00:25:02.000 --> 00:25:08.000
times e to the t plus t squared
time the derivative of e to the
00:25:08.000 --> 00:25:16.000
t is e to the t plus cosine t.
And that is the same answer as
00:25:16.000 --> 00:25:19.000
over there.
I ended up writing,
00:25:19.000 --> 00:25:23.000
you know, maybe I wrote
slightly more here,
00:25:23.000 --> 00:25:28.000
but actually the amount of
calculations really was pretty
00:25:28.000 --> 00:25:32.000
much the same.
Any questions about that?
00:25:32.000 --> 00:25:39.000
Yes?
What kind of object is w?
00:25:39.000 --> 00:25:43.000
Well, you can think of w as
just another variable that is
00:25:43.000 --> 00:25:47.000
given as a function of x,
y and z, for example.
00:25:47.000 --> 00:25:51.000
You would have a function of x,
y, z defined by this formula,
00:25:51.000 --> 00:25:57.000
and I call it w.
I call its value w so that I
00:25:57.000 --> 00:26:04.000
can substitute t instead of x,
y, z.
00:26:04.000 --> 00:26:07.000
Well, let's think of w as a
function of three variables.
00:26:07.000 --> 00:26:12.000
And then, when I plug in the
dependents of these three
00:26:12.000 --> 00:26:17.000
variables on t,
then it becomes just a function
00:26:17.000 --> 00:26:19.000
of t.
I mean, really,
00:26:19.000 --> 00:26:23.000
my w here is pretty much what I
called f before.
00:26:23.000 --> 00:26:31.000
There is no major difference
between the two.
00:26:31.000 --> 00:26:38.000
Any other questions?
No.
00:26:38.000 --> 00:26:45.000
OK.
Let's see.
00:26:45.000 --> 00:26:49.000
Here is an application of what
we have seen.
00:26:49.000 --> 00:26:53.000
Let's say that you want to
understand actually all these
00:26:53.000 --> 00:26:57.000
rules about taking derivatives
in single variable calculus.
00:26:57.000 --> 00:27:00.000
What I showed you at the
beginning, and then erased,
00:27:00.000 --> 00:27:04.000
basically justifies how to take
the derivative of a reciprocal
00:27:04.000 --> 00:27:06.000
function.
And for that you didn't need
00:27:06.000 --> 00:27:10.000
multivariable calculus.
But let's try to justify the
00:27:10.000 --> 00:27:12.000
product rule,
for example,
00:27:12.000 --> 00:27:21.000
for the derivative.
An application of this actually
00:27:21.000 --> 00:27:31.000
is to justify the product and
quotient rules.
00:27:31.000 --> 00:27:33.000
Let's think,
for example,
00:27:33.000 --> 00:27:39.000
of a function of two variables,
u and v, that is just the
00:27:39.000 --> 00:27:44.000
product uv.
And let's say that u and v are
00:27:44.000 --> 00:27:48.000
actually functions of one
variable t.
00:27:48.000 --> 00:28:00.000
Then, well, d of uv over dt is
given by the chain rule applied
00:28:00.000 --> 00:28:04.000
to f.
This is df over dt.
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
00:28:15.000 --> 00:28:19.000
over dt.
But now what is the partial of
00:28:19.000 --> 00:28:23.000
f with respect to u?
It is v.
00:28:23.000 --> 00:28:31.000
That is v du over dt.
And partial of f with respect
00:28:31.000 --> 00:28:38.000
to v is going to be just u,
dv over dt.
00:28:38.000 --> 00:28:42.000
So you get back the usual
product rule.
00:28:42.000 --> 00:28:46.000
That is a slightly complicated
way of deriving it,
00:28:46.000 --> 00:28:50.000
but that is a valid way of
understanding how to take the
00:28:50.000 --> 00:28:54.000
derivative of a product by
thinking of the product first as
00:28:54.000 --> 00:28:57.000
a function of variables,
which are u and v.
00:28:57.000 --> 00:29:00.000
And then say,
oh, but u and v were actually
00:29:00.000 --> 00:29:03.000
functions of a variable t.
And then you do the
00:29:03.000 --> 00:29:08.000
differentiation in two stages
using the chain rule.
00:29:08.000 --> 00:29:16.000
Similarly, you can do the
quotient rule just for practice.
00:29:16.000 --> 00:29:21.000
If I give you the function g
equals u of v.
00:29:21.000 --> 00:29:25.000
Right now I am thinking of it
as a function of two variables,
00:29:25.000 --> 00:29:29.000
u and v.
U and v themselves are actually
00:29:29.000 --> 00:29:39.000
going to be functions of t.
Then, well, dg over dt is going
00:29:39.000 --> 00:29:44.000
to be partial g,
partial u.
00:29:44.000 --> 00:29:48.000
How much is that?
How much is partial g,
00:29:48.000 --> 00:29:53.000
partial u?
One over v times du over dt
00:29:53.000 --> 00:29:58.000
plus -- Well,
next we need to have partial g
00:29:58.000 --> 00:30:01.000
over partial v.
Well, what is the derivative of
00:30:01.000 --> 00:30:04.000
this with respect to v?
Here we need to know how to
00:30:04.000 --> 00:30:11.000
differentiate the inverse.
It is minus u over v squared
00:30:11.000 --> 00:30:20.000
times dv over dt.
And that is actually the usual
00:30:20.000 --> 00:30:28.000
quotient rule just written in a
slightly different way.
00:30:28.000 --> 00:30:30.000
I mean, just in case you really
want to see it,
00:30:30.000 --> 00:30:36.000
if you clear denominators for v
squared then you will see
00:30:36.000 --> 00:30:41.000
basically u prime times v minus
v prime times u.
00:31:25.000 --> 00:31:32.000
Now let's go to something even
more crazy.
00:31:32.000 --> 00:31:45.000
I claim we can do chain rules
with more variables.
00:31:45.000 --> 00:31:50.000
Let's say that I have a
quantity.
00:31:50.000 --> 00:31:55.000
Let's call it w for now.
Let's say I have quantity w as
00:31:55.000 --> 00:31:58.000
a function of say variables x
and y.
00:31:58.000 --> 00:32:02.000
And so in the previous setup x
and y depended on some
00:32:02.000 --> 00:32:04.000
parameters t.
But, actually,
00:32:04.000 --> 00:32:07.000
let's now look at the case
where x and y themselves are
00:32:07.000 --> 00:32:10.000
functions of several variables.
Let's say of two more variables.
00:32:10.000 --> 00:32:25.000
Let's call them u and v.
I am going to stay with these
00:32:25.000 --> 00:32:27.000
abstract letters,
but if it bothers you,
00:32:27.000 --> 00:32:31.000
if it sounds completely
unmotivated think about it maybe
00:32:31.000 --> 00:32:33.000
in terms of something you might
now.
00:32:33.000 --> 00:32:36.000
Say, polar coordinates.
Let's say that I have a
00:32:36.000 --> 00:32:40.000
function but is defined in terms
of the polar coordinate
00:32:40.000 --> 00:32:43.000
variables on theta.
And then I know I want to
00:32:43.000 --> 00:32:45.000
switch to usual coordinates x
and y.
00:32:45.000 --> 00:32:49.000
Or, the other way around,
I have a function of x and y
00:32:49.000 --> 00:32:53.000
and I want to express it in
terms of the polar coordinates r
00:32:53.000 --> 00:32:57.000
and theta.
Then I would want to know maybe
00:32:57.000 --> 00:33:02.000
how the derivatives,
with respect to the various
00:33:02.000 --> 00:33:07.000
sets of variables,
related to each other.
00:33:07.000 --> 00:33:10.000
One way I could do it is,
of course,
00:33:10.000 --> 00:33:16.000
to say now if I plug the
formula for x and the formula
00:33:16.000 --> 00:33:23.000
for y into the formula for f
then w becomes a function of u
00:33:23.000 --> 00:33:27.000
and v,
and it can try to take partial
00:33:27.000 --> 00:33:29.000
derivatives.
If I have explicit formulas,
00:33:29.000 --> 00:33:32.000
well, that could work.
But maybe the formulas are
00:33:32.000 --> 00:33:35.000
complicated.
Typically, if I switch between
00:33:35.000 --> 00:33:37.000
rectangular and polar
coordinates,
00:33:37.000 --> 00:33:41.000
there might be inverse trig,
there might be maybe arctangent
00:33:41.000 --> 00:33:45.000
to express the polar angle in
terms of x and y.
00:33:45.000 --> 00:33:51.000
And when I don't really want to
actually substitute arctangents
00:33:51.000 --> 00:33:56.000
everywhere, maybe I would rather
deal with the derivatives.
00:33:56.000 --> 00:34:03.000
How do I do that?
The question is what are
00:34:03.000 --> 00:34:11.000
partial w over partial u and
partial w over partial v in
00:34:11.000 --> 00:34:17.000
terms of, let's see,
what do we need to know to
00:34:17.000 --> 00:34:22.000
understand that?
Well, probably we should know
00:34:22.000 --> 00:34:28.000
how w depends on x and y.
If we don't know that then we
00:34:28.000 --> 00:34:32.000
are probably toast.
Partial w over partial x,
00:34:32.000 --> 00:34:36.000
partial w over partial y should
be required.
00:34:36.000 --> 00:34:39.000
What else should we know?
Well, it would probably help to
00:34:39.000 --> 00:34:42.000
know how x and y depend on u and
v.
00:34:42.000 --> 00:34:46.000
If we don't know that then we
don't really know how to do it.
00:34:46.000 --> 00:34:55.000
We need also x sub u,
x sub v, y sub u,
00:34:55.000 --> 00:35:00.000
y sub v.
We have a lot of partials in
00:35:00.000 --> 00:35:07.000
there.
Well, let's see how we can do
00:35:07.000 --> 00:35:13.000
that.
Let's start by writing dw.
00:35:13.000 --> 00:35:19.000
We know that dw is partial f,
well, I don't know why I have
00:35:19.000 --> 00:35:25.000
two names, w and f.
I mean w and f are really the
00:35:25.000 --> 00:35:30.000
same thing here,
but let's say f sub x dx plus f
00:35:30.000 --> 00:35:35.000
sub y dy.
So far that is our new friend,
00:35:35.000 --> 00:35:39.000
the differential.
Now what do we want to do with
00:35:39.000 --> 00:35:42.000
it?
Well, we would like to get rid
00:35:42.000 --> 00:35:47.000
of dx and dy because we like to
express things in terms of,
00:35:47.000 --> 00:35:50.000
you know, the question we are
asking ourselves is let's say
00:35:50.000 --> 00:35:55.000
that I change u a little bit,
how does w change?
00:35:55.000 --> 00:35:58.000
Of course, what happens,
if I change u a little bit,
00:35:58.000 --> 00:36:01.000
is y and y will change.
How do they change?
00:36:01.000 --> 00:36:05.000
Well, that is given to me by
the differential.
00:36:05.000 --> 00:36:13.000
dx is going to be,
well, I can use the
00:36:13.000 --> 00:36:19.000
differential again.
Well, x is a function of u and
00:36:19.000 --> 00:36:24.000
v.
That will be x sub u times du
00:36:24.000 --> 00:36:28.000
plus x sub v times dv.
That is, again,
00:36:28.000 --> 00:36:31.000
taking the differential of a
function of two variables.
00:36:31.000 --> 00:36:37.000
Does that make sense?
And then we have the other guy,
00:36:37.000 --> 00:36:39.000
f sub y times,
what is dy?
00:36:39.000 --> 00:36:49.000
Well, similarly dy is y sub u
du plus y sub v dv.
00:36:49.000 --> 00:36:54.000
And now we have a relation
between dw and du and dv.
00:36:54.000 --> 00:37:00.000
We are expressing how w reacts
to changes in u and v,
00:37:00.000 --> 00:37:04.000
which was our goal.
Now, let's actually collect
00:37:04.000 --> 00:37:08.000
terms so that we see it a bit
better.
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
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
00:37:28.000 --> 00:37:32.000
dv.
Now we have dw equals something
00:37:32.000 --> 00:37:38.000
du plus something dv.
Well, the coefficient here has
00:37:38.000 --> 00:37:44.000
to be partial f over partial u.
What else could it be?
00:37:44.000 --> 00:37:49.000
That's the rate of change of w
with respect to u if I forget
00:37:49.000 --> 00:37:54.000
what happens when I change v.
That is the definition of a
00:37:54.000 --> 00:37:58.000
partial.
Similarly, this one has to be
00:37:58.000 --> 00:38:04.000
partial f over partial v.
That is because it is the rate
00:38:04.000 --> 00:38:09.000
of change with respect to v,
if I keep u constant,
00:38:09.000 --> 00:38:13.000
so that these guys are
completely ignored.
00:38:13.000 --> 00:38:16.000
Now you see how the total
differential accounts for,
00:38:16.000 --> 00:38:21.000
somehow, all the partial
derivatives that come as
00:38:21.000 --> 00:38:27.000
coefficients of the individual
variables in these expressions.
00:38:27.000 --> 00:38:33.000
Let me maybe rewrite these
formulas in a more visible way
00:38:33.000 --> 00:38:40.000
and then re-explain them to you.
Here is the chain rule for this
00:38:40.000 --> 00:38:46.000
situation, with two intermediate
variables and two variables that
00:38:46.000 --> 00:38:50.000
you express these in terms of.
In our setting,
00:38:50.000 --> 00:38:56.000
we get partial f over partial u
equals partial f over partial x
00:38:56.000 --> 00:39:02.000
time partial x over partial u
plus partial f over partial y
00:39:02.000 --> 00:39:08.000
times partial y over partial u.
And the other one,
00:39:08.000 --> 00:39:15.000
the same thing with v instead
of u,
00:39:15.000 --> 00:39:22.000
partial f over partial x times
partial x over partial v plus
00:39:22.000 --> 00:39:28.000
partial f over partial u partial
y over partial v.
00:39:28.000 --> 00:39:31.000
I have to explain various
things about these formulas
00:39:31.000 --> 00:39:34.000
because they look complicated.
And, actually,
00:39:34.000 --> 00:39:39.000
they are not that complicated.
A couple of things to know.
00:39:39.000 --> 00:39:42.000
The first thing,
how do we remember a formula
00:39:42.000 --> 00:39:44.000
like that?
Well, that is easy.
00:39:44.000 --> 00:39:47.000
We want to know how f depends
on u.
00:39:47.000 --> 00:39:51.000
Well, what does f depend on?
It depends on x and y.
00:39:51.000 --> 00:39:55.000
So we will put partial f over
partial x and partial f over
00:39:55.000 --> 00:39:59.000
partial y.
Now, x and y, why are they here?
00:39:59.000 --> 00:40:01.000
Well, they are here because
they actually depend on u as
00:40:01.000 --> 00:40:04.000
well.
How does x depend on u?
00:40:04.000 --> 00:40:06.000
Well, the answer is partial x
over partial u.
00:40:06.000 --> 00:40:10.000
How does y depend on u?
The answer is partial y over
00:40:10.000 --> 00:40:12.000
partial u.
See, the structure of this
00:40:12.000 --> 00:40:16.000
formula is simple.
To find the partial of f with
00:40:16.000 --> 00:40:20.000
respect to some new variable you
use the partials with respect to
00:40:20.000 --> 00:40:24.000
the variables that f was
initially defined in terms of x
00:40:24.000 --> 00:40:28.000
and y.
And you multiply them by the
00:40:28.000 --> 00:40:33.000
partials of x and y in terms of
the new variable that you want
00:40:33.000 --> 00:40:37.000
to look at, v here,
and you sum these things
00:40:37.000 --> 00:40:40.000
together.
That is the structure of the
00:40:40.000 --> 00:40:42.000
formula.
Why does it work?
00:40:42.000 --> 00:40:45.000
Well, let me explain it to you
in a slightly different
00:40:45.000 --> 00:40:49.000
language.
This asks us how does f change
00:40:49.000 --> 00:40:54.000
if I change u a little bit?
Well, why would f change if u
00:40:54.000 --> 00:40:57.000
changes a little bit?
Well, it would change because f
00:40:57.000 --> 00:41:00.000
actually depends on x and y and
x and y depend on u.
00:41:00.000 --> 00:41:03.000
If I change u,
how quickly does x change?
00:41:03.000 --> 00:41:06.000
Well, the answer is partial x
over partial u.
00:41:06.000 --> 00:41:09.000
And now, if I change x at this
rate, how does that have to
00:41:09.000 --> 00:41:13.000
change?
Well, the answer is partial f
00:41:13.000 --> 00:41:17.000
over partial x times this guy.
Well, at the same time,
00:41:17.000 --> 00:41:21.000
y is also changing.
How fast is y changing if I
00:41:21.000 --> 00:41:24.000
change u?
Well, at the rate of partial y
00:41:24.000 --> 00:41:27.000
over partial u.
But now if I change this how
00:41:27.000 --> 00:41:30.000
does f change?
Well, the rate of change is
00:41:30.000 --> 00:41:34.000
partial f over partial y.
The product is the effect of
00:41:34.000 --> 00:41:37.000
how you change it,
changing u, and therefore
00:41:37.000 --> 00:41:40.000
changing f.
Now, what happens in real life,
00:41:40.000 --> 00:41:43.000
if I change u a little bit?
Well, both x and y change at
00:41:43.000 --> 00:41:46.000
the same time.
So how does f change?
00:41:46.000 --> 00:41:50.000
Well, it is the sum of the two
effects.
00:41:50.000 --> 00:41:54.000
Does that make sense?
Good.
00:41:54.000 --> 00:42:00.000
Of course, if f depends on more
variables then you just have
00:42:00.000 --> 00:42:02.000
more terms in here.
OK.
00:42:02.000 --> 00:42:05.000
Here is another thing that may
be a little bit confusing.
00:42:05.000 --> 00:42:09.000
What is tempting?
Well, what is tempting here
00:42:09.000 --> 00:42:12.000
would be to simplify these
formulas by removing these
00:42:12.000 --> 00:42:15.000
partial x's.
Let's simplify by partial x.
00:42:15.000 --> 00:42:18.000
Let's simplify by partial y.
We get partial f over partial u
00:42:18.000 --> 00:42:21.000
equals partial f over partial u
plus partial f over partial u.
00:42:21.000 --> 00:42:25.000
Something is not working
properly.
00:42:25.000 --> 00:42:28.000
Why doesn't it work?
The answer is precisely because
00:42:28.000 --> 00:42:32.000
these are partial derivatives.
These are not total derivatives.
00:42:32.000 --> 00:42:36.000
And so you cannot simplify them
in that way.
00:42:36.000 --> 00:42:39.000
And that is actually the reason
why we use this curly d rather
00:42:39.000 --> 00:42:41.000
than a straight d.
It is to remind us,
00:42:41.000 --> 00:42:44.000
beware, there are these
simplifications that we can do
00:42:44.000 --> 00:42:47.000
with straight d's that are not
legal here.
00:42:47.000 --> 00:42:52.000
Somehow, when you have a
partial derivative,
00:42:52.000 --> 00:42:57.000
you must resist the urge of
simplifying things.
00:42:57.000 --> 00:43:02.000
No simplifications in here.
That is the simplest formula
00:43:02.000 --> 00:43:10.000
you can get.
Any questions at this point?
00:43:10.000 --> 00:43:21.000
No.
Yes?
00:43:21.000 --> 00:43:23.000
When would you use this and
what does it describe?
00:43:23.000 --> 00:43:26.000
Well, it is basically when you
have a function given in terms
00:43:26.000 --> 00:43:29.000
of a certain set of variables
because maybe there is a simply
00:43:29.000 --> 00:43:31.000
expression in terms of those
variables.
00:43:31.000 --> 00:43:35.000
But ultimately what you care
about is not those variables,
00:43:35.000 --> 00:43:39.000
z and y, but another set of
variables, here u and v.
00:43:39.000 --> 00:43:42.000
So x and y are giving you a
nice formula for f,
00:43:42.000 --> 00:43:46.000
but actually the relevant
variables for your problem are u
00:43:46.000 --> 00:43:48.000
and v.
And you know x and y are
00:43:48.000 --> 00:43:50.000
related to u and v.
So, of course,
00:43:50.000 --> 00:43:53.000
what you could do is plug the
formulas the way that we did
00:43:53.000 --> 00:43:55.000
substituting.
But maybe that will give you
00:43:55.000 --> 00:43:59.000
very complicated expressions.
And maybe it is actually easier
00:43:59.000 --> 00:44:02.000
to just work with the derivates.
The important claim here is
00:44:02.000 --> 00:44:05.000
basically we don't need to know
the actual formulas.
00:44:05.000 --> 00:44:07.000
All we need to know are the
rate of changes.
00:44:07.000 --> 00:44:11.000
If we know all these rates of
change then we know how to take
00:44:11.000 --> 00:44:14.000
these derivatives without
actually having to plug in
00:44:14.000 --> 00:44:22.000
values.
Yes?
00:44:22.000 --> 00:44:25.000
Yes, you could certain do the
same things in terms of t.
00:44:25.000 --> 00:44:29.000
If x and y were functions of t
instead of being functions of u
00:44:29.000 --> 00:44:31.000
and v then it would be the same
thing.
00:44:31.000 --> 00:44:34.000
And you would have the same
formulas that I had,
00:44:34.000 --> 00:44:37.000
well, over there I still have
it.
00:44:37.000 --> 00:44:39.000
Why does that one have straight
d's?
00:44:39.000 --> 00:44:42.000
Well, the answer is I could put
curly d's if I wanted,
00:44:42.000 --> 00:44:45.000
but I end up with a function of
a single variable.
00:44:45.000 --> 00:44:48.000
If you have a single variable
then the partial,
00:44:48.000 --> 00:44:50.000
with respect to that variable,
is the same thing as the usual
00:44:50.000 --> 00:44:53.000
derivative.
We don't actually need to worry
00:44:53.000 --> 00:44:57.000
about curly in that case.
But that one is indeed special
00:44:57.000 --> 00:45:00.000
case of this one where instead
of x and y depending on two
00:45:00.000 --> 00:45:03.000
variables, u and v,
they depend on a single
00:45:03.000 --> 00:45:04.000
variable t.
Now, of course,
00:45:04.000 --> 00:45:06.000
you can call variables any name
you want.
00:45:06.000 --> 00:45:12.000
It doesn't matter.
This is just a slight
00:45:12.000 --> 00:45:16.000
generalization of that.
Well, not quite because here I
00:45:16.000 --> 00:45:18.000
also had a z.
See, I am trying to just
00:45:18.000 --> 00:45:21.000
confuse you by giving you
functions that depend on various
00:45:21.000 --> 00:45:25.000
numbers of variables.
If you have a function of 30
00:45:25.000 --> 00:45:28.000
variables, things work the same
way, just longer,
00:45:28.000 --> 00:45:33.000
and you are going to run out of
letters in the alphabet before
00:45:33.000 --> 00:45:38.000
the end.
Any other questions?
00:45:38.000 --> 00:45:43.000
No.
What?
00:45:43.000 --> 00:45:51.000
Yes?
If u and v themselves depended
00:45:51.000 --> 00:45:55.000
on another variable then you
would continue with your chain
00:45:55.000 --> 00:45:58.000
rules.
Maybe you would know to express
00:45:58.000 --> 00:46:02.000
partial x over partial u in
terms using that chain rule.
00:46:02.000 --> 00:46:05.000
Sorry.
If u and v are dependent on yet
00:46:05.000 --> 00:46:08.000
another variable then you could
get the derivative with respect
00:46:08.000 --> 00:46:11.000
to that using first the chain
rule to pass from u v to that
00:46:11.000 --> 00:46:14.000
new variable,
and then you would plug in
00:46:14.000 --> 00:46:17.000
these formulas for partials of f
with respect to u and v.
00:46:17.000 --> 00:46:19.000
In fact, if you have several
substitutions to do,
00:46:19.000 --> 00:46:21.000
you can always arrange to use
one chain rule at a time.
00:46:21.000 --> 00:46:25.000
You just have to do them in
sequence.
00:46:25.000 --> 00:46:28.000
That's why we don't actually
learn that, but you can just do
00:46:28.000 --> 00:46:32.000
it be repeating the process.
I mean, probably at that stage,
00:46:32.000 --> 00:46:35.000
the easiest to not get confused
actually is to manipulate
00:46:35.000 --> 00:46:38.000
differentials because that is
probably easier.
00:46:38.000 --> 00:46:47.000
Yes?
Curly f does not exist.
00:46:47.000 --> 00:46:50.000
That's easy.
Curly f makes no sense by
00:46:50.000 --> 00:46:52.000
itself.
It doesn't exist alone.
00:46:52.000 --> 00:46:58.000
What exists is only curly df
over curly d some variable.
00:46:58.000 --> 00:47:02.000
And then that accounts only for
the rate of change with respect
00:47:02.000 --> 00:47:05.000
to that variable leaving the
others fixed,
00:47:05.000 --> 00:47:11.000
while straight df is somehow a
total variation of f.
00:47:11.000 --> 00:47:16.000
It accounts for all of the
partial derivatives and their
00:47:16.000 --> 00:47:25.000
combined effects.
OK. Any more questions? No.
00:47:25.000 --> 00:47:29.000
Let me just finish up very
quickly by telling you again one
00:47:29.000 --> 00:47:33.000
example where completely you
might want to do this.
00:47:33.000 --> 00:47:40.000
You have a function that you
want to switch between
00:47:40.000 --> 00:47:45.000
rectangular and polar
coordinates.
00:47:45.000 --> 00:47:48.000
To make things a little bit
concrete.
00:47:48.000 --> 00:47:55.000
If you have polar coordinates
that means in the plane,
00:47:55.000 --> 00:48:00.000
instead of using x and y,
you will use coordinates r,
00:48:00.000 --> 00:48:05.000
distance to the origin,
and theta, the angles from the
00:48:05.000 --> 00:48:08.000
x-axis.
The change of variables for
00:48:08.000 --> 00:48:14.000
that is x equals r cosine theta
and y equals r sine theta.
00:48:14.000 --> 00:48:21.000
And so that means if you have a
function f that depends on x and
00:48:21.000 --> 00:48:29.000
y, in fact, you can plug these
in as a function of r and theta.
00:48:29.000 --> 00:48:34.000
Then you can ask yourself,
well, what is partial f over
00:48:34.000 --> 00:48:37.000
partial r?
And that is going to be,
00:48:37.000 --> 00:48:42.000
well, you want to take partial
f over partial x times partial x
00:48:42.000 --> 00:48:48.000
partial r plus partial f over
partial y times partial y over
00:48:48.000 --> 00:48:53.000
partial r.
That will end up being actually
00:48:53.000 --> 00:48:59.000
f sub x times cosine theta plus
f sub y times sine theta.
00:48:59.000 --> 00:49:02.000
And you can do the same thing
to find partial f,
00:49:02.000 --> 00:49:05.000
partial theta.
And so you can express
00:49:05.000 --> 00:49:10.000
derivatives either in terms of
x, y or in terms of r and theta
00:49:10.000 --> 00:49:13.000
with simple relations between
them.
00:49:13.000 --> 00:49:20.000
And the one last thing I should
say.
00:49:20.000 --> 00:49:23.000
On Thursday we will learn about
more tricks we can play with
00:49:23.000 --> 00:49:27.000
variations of functions.
And one that is important,
00:49:27.000 --> 00:49:29.000
because you need to know it
actually to do the p-set,
00:49:29.000 --> 00:49:38.000
is the gradient vector.
The gradient vector is simply a
00:49:38.000 --> 00:49:41.000
vector.
You use this downward pointing
00:49:41.000 --> 00:49:44.000
triangle as the notation for the
gradient.
00:49:44.000 --> 00:49:49.000
It is simply is a vector whose
components are the partial
00:49:49.000 --> 00:49:53.000
derivatives of a function.
I mean, in a way,
00:49:53.000 --> 00:49:56.000
you can think of a differential
as a way to package partial
00:49:56.000 --> 00:49:59.000
derivatives together into some
weird object.
00:49:59.000 --> 00:50:01.000
Well, the gradient is also a
way to package partials
00:50:01.000 --> 00:50:04.000
together.
We will see on Thursday what it
00:50:04.000 --> 00:50:07.000
is good for, but some of the
problems on the p-set use it.