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OK, so we're going to continue
looking at what happens when we
00:00:35.000 --> 00:00:45.000
have non-independent variables.
So, I'm afraid we don't take
00:00:45.000 --> 00:00:50.000
deliveries during class time,
sorry.
00:00:50.000 --> 00:01:00.000
Please take a seat, thanks.
[LAUGHTER]
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[APPLAUSE]
OK, so Jason,
00:01:05.000 --> 00:01:17.000
you please claim your package
at the end of lecture.
00:01:17.000 --> 00:01:19.000
OK,
so last time we saw how to use
00:01:19.000 --> 00:01:23.000
Lagrange multipliers to find the
minimum or maximum of a function
00:01:23.000 --> 00:01:27.000
of several variables when the
variables are not independent.
00:01:27.000 --> 00:01:29.000
And, today we're going to try
to figure out more about
00:01:29.000 --> 00:01:33.000
relations between the variables,
and how to handle functions
00:01:33.000 --> 00:01:36.000
that depend on several variables
when they're related.
00:01:36.000 --> 00:01:40.000
So, just to give you an
example,
00:01:40.000 --> 00:01:44.000
in physics, very often,
you have functions that depend
00:01:44.000 --> 00:01:49.000
on pressure, volume,
and temperature where pressure,
00:01:49.000 --> 00:01:52.000
volume,
and temperature are actually
00:01:52.000 --> 00:01:55.000
not independent.
But they are related,
00:01:55.000 --> 00:01:58.000
say, by PV=nRT.
So, of course,
00:01:58.000 --> 00:02:01.000
then you can substitute and
expressed a function in terms of
00:02:01.000 --> 00:02:04.000
two of them only,
but very often it's convenient
00:02:04.000 --> 00:02:06.000
to keep all three.
But then we have to figure out,
00:02:06.000 --> 00:02:11.000
what are the rates of change
with respect to t,
00:02:11.000 --> 00:02:14.000
with respect to each other,
the rate of change of f with
00:02:14.000 --> 00:02:16.000
respect to these variables,
and so on.
00:02:16.000 --> 00:02:21.000
So, we have to figure out what
we mean by partial derivatives
00:02:21.000 --> 00:02:24.000
again.
So,
00:02:24.000 --> 00:02:31.000
OK, more generally,
let's say just for the sake of
00:02:31.000 --> 00:02:33.000
notation,
I'm going to think of a
00:02:33.000 --> 00:02:35.000
function of three variables,
x, y, z,
00:02:35.000 --> 00:02:39.000
where the variables are related
by some equation,
00:02:39.000 --> 00:02:44.000
but I will put in the form g of
x, y, z equals some constant.
00:02:44.000 --> 00:02:48.000
OK, so that's the same kind of
setup as we had last time,
00:02:48.000 --> 00:02:52.000
except now we are not just
looking for minima and maxima.
00:02:52.000 --> 00:02:59.000
We are trying to understand
partial derivatives.
00:02:59.000 --> 00:03:06.000
So, the first observation is
that if x, y,
00:03:06.000 --> 00:03:09.000
and z are related,
then that means,
00:03:09.000 --> 00:03:11.000
in principle,
we could solve for one of them,
00:03:11.000 --> 00:03:15.000
and express it as a function of
the two others.
00:03:15.000 --> 00:03:19.000
So, in particular,
can we understand even without
00:03:19.000 --> 00:03:21.000
solving?
Maybe we can not solve.
00:03:21.000 --> 00:03:27.000
Can we understand how the
variables are related to each
00:03:27.000 --> 00:03:29.000
other?
So, for example,
00:03:29.000 --> 00:03:33.000
z, you can think of z as a
function of x and y.
00:03:33.000 --> 00:03:40.000
So, we can ask ourselves,
what are the rates of change of
00:03:40.000 --> 00:03:44.000
z with respect to x,
keeping y constant,
00:03:44.000 --> 00:03:49.000
or with respect to y keeping x
constant?
00:03:49.000 --> 00:03:51.000
And, of course,
if we can solve,
00:03:51.000 --> 00:03:53.000
that we know the formula for
this.
00:03:53.000 --> 00:03:55.000
And then we can compute these
guys.
00:03:55.000 --> 00:04:03.000
But, what if we can't solve?
So, how do we find these things
00:04:03.000 --> 00:04:11.000
without solving?
Well, so let's do an example.
00:04:11.000 --> 00:04:19.000
Let's say that my relation is
x^2 yz z^3=8.
00:04:19.000 --> 00:04:24.000
And, let's say that I'm looking
near the point (x,
00:04:24.000 --> 00:04:27.000
y, z) equals (2,3,
1).
00:04:27.000 --> 00:04:33.000
So, let me check 2^2 plus three
times one plus 1^3 is indeed
00:04:33.000 --> 00:04:34.000
eight.
OK, but now,
00:04:34.000 --> 00:04:38.000
if I change x and y a little
bit, how does z change?
00:04:38.000 --> 00:04:41.000
Well, of course I could solve
for z in here.
00:04:41.000 --> 00:04:43.000
It's a cubic equation.
There is actually a formula.
00:04:43.000 --> 00:04:45.000
But that formula is quite
complicated.
00:04:45.000 --> 00:04:47.000
We actually don't want to do
that.
00:04:47.000 --> 00:04:58.000
There's an easier way.
So, how can we do it?
00:04:58.000 --> 00:05:07.000
Well, let's look at the
differential -- -- of this
00:05:07.000 --> 00:05:15.000
constraint quantity.
OK, so if we called this g,
00:05:15.000 --> 00:05:21.000
let's look at dg.
So, what's the differential of
00:05:21.000 --> 00:05:26.000
this?
So, the differential of x^2 is
00:05:26.000 --> 00:05:32.000
2x dx plus, I think there's a
zdy.
00:05:32.000 --> 00:05:38.000
There's a ydz,
and there's also a 3z^2 dz.
00:05:38.000 --> 00:05:42.000
OK, you can get this either by
implicit differentiation and the
00:05:42.000 --> 00:05:45.000
product rule,
or you could get this just by
00:05:45.000 --> 00:05:46.000
putting here,
here,
00:05:46.000 --> 00:05:51.000
and here the partial
derivatives of this with respect
00:05:51.000 --> 00:05:56.000
to x, y, and z.
OK, any questions about how I
00:05:56.000 --> 00:05:58.000
got this?
No?
00:05:58.000 --> 00:06:03.000
OK.
So, now, what do I do with this?
00:06:03.000 --> 00:06:07.000
Well, this represents,
somehow, variations of g.
00:06:07.000 --> 00:06:12.000
But, well, I've set this thing
equal to eight.
00:06:12.000 --> 00:06:16.000
And, eight is a constant.
So, it doesn't change.
00:06:16.000 --> 00:06:26.000
So, in fact,
well, we can set this to zero
00:06:26.000 --> 00:06:35.000
because, well,
they call this g.
00:06:35.000 --> 00:06:39.000
Then, g equals eight is
constant.
00:06:39.000 --> 00:06:43.000
That means we set dg equal to
zero.
00:06:43.000 --> 00:06:53.000
OK, so, now let's just plug in
some values at this point.
00:06:53.000 --> 00:06:58.000
That tells us,
well, so if x equals two,
00:06:58.000 --> 00:07:07.000
that's 4dx plus z is one.
So, dy plus y 3z^2 should be
00:07:07.000 --> 00:07:13.000
6dz equals zero.
And now, this equation,
00:07:13.000 --> 00:07:17.000
here, tells us about a relation
between the changes in x,
00:07:17.000 --> 00:07:21.000
y, and z near that point.
It tells us how you change x
00:07:21.000 --> 00:07:25.000
and y, well, how z will change.
Or, it tells you actually
00:07:25.000 --> 00:07:28.000
anything you might want to know
about the relations between
00:07:28.000 --> 00:07:30.000
these variables so,
for example,
00:07:30.000 --> 00:07:34.000
you can move dz to that side,
and then express dz in terms of
00:07:34.000 --> 00:07:37.000
dx and dy.
Or, you can move dy to that
00:07:37.000 --> 00:07:41.000
side and express dy in terms of
dx and dz, and so on.
00:07:41.000 --> 00:07:46.000
It tells you at the level of
the derivatives how each of the
00:07:46.000 --> 00:07:49.000
variables depends on the two
others.
00:07:49.000 --> 00:07:57.000
OK, so, just to clarify this:
if we want to view z as a
00:07:57.000 --> 00:08:03.000
function of x and y,
then what we will do is we will
00:08:03.000 --> 00:08:06.000
just move the dz's to the other
side,
00:08:06.000 --> 00:08:15.000
and it will tell us dz equals
minus one over six times 4dx
00:08:15.000 --> 00:08:20.000
plus dy.
And, so that should tell you
00:08:20.000 --> 00:08:26.000
that partial z over partial x is
minus four over six.
00:08:26.000 --> 00:08:35.000
Well, that's minus two thirds,
and partial z over partial y is
00:08:35.000 --> 00:08:42.000
going to be minus one sixth.
OK, another way to think about
00:08:42.000 --> 00:08:47.000
this: when we compute partial z
over partial x,
00:08:47.000 --> 00:08:51.000
that means that actually we
keep y constant.
00:08:51.000 --> 00:08:55.000
OK, let me actually add some
subtitles here.
00:08:55.000 --> 00:09:00.000
So, here that means we keep y
constant.
00:09:00.000 --> 00:09:05.000
And so, if we keep y constant,
another way to think about it
00:09:05.000 --> 00:09:10.000
is we set dy to zero.
We set dy equals zero.
00:09:10.000 --> 00:09:14.000
So if we do that,
we get dx equals negative four
00:09:14.000 --> 00:09:17.000
sixths dx.
That tells us the rate of
00:09:17.000 --> 00:09:24.000
change of z with respect to x.
Here, we set x constant.
00:09:24.000 --> 00:09:29.000
So, that means we set dx equal
to zero.
00:09:29.000 --> 00:09:32.000
And, if we set dx equal to
zero, then we have dz equals
00:09:32.000 --> 00:09:36.000
negative one sixth of dy.
That tells us the rate of
00:09:36.000 --> 00:09:47.000
change of z with respect to y.
OK, any questions about that?
00:09:47.000 --> 00:09:57.000
No?
What, yes?
00:09:57.000 --> 00:09:59.000
Yes, OK, let me explain that
again.
00:09:59.000 --> 00:10:03.000
So we found an expression for
dz in terms of dx and dy.
00:10:03.000 --> 00:10:07.000
That means that this thing,
the differential,
00:10:07.000 --> 00:10:11.000
is the total differential of z
viewed as a function of x and y.
00:10:11.000 --> 00:10:16.000
OK, and so the coefficients of
dx and dy are the partials.
00:10:16.000 --> 00:10:20.000
Or, another way to think about
it, if you want to know partial
00:10:20.000 --> 00:10:22.000
z partial x, it means you set y
to be constant.
00:10:22.000 --> 00:10:27.000
Setting y to be constant means
that you will put zero in the
00:10:27.000 --> 00:10:30.000
place of dy.
So, you will be left with dz
00:10:30.000 --> 00:10:35.000
equals minus four sixths dx.
And, that will give you the
00:10:35.000 --> 00:10:41.000
rate of change of z with respect
to x when you keep y constant,
00:10:41.000 --> 00:10:46.000
OK?
So, there are various ways to
00:10:46.000 --> 00:10:53.000
think about this,
but hopefully it makes sense.
00:10:53.000 --> 00:11:03.000
OK, so how do we think about
this in general?
00:11:03.000 --> 00:11:15.000
Well, if we know that g of x,
y, z equals a constant,
00:11:15.000 --> 00:11:27.000
then dg, which is gxdx gydy
gzdz should be set equal to
00:11:27.000 --> 00:11:32.000
zero.
OK, and now we can solve for
00:11:32.000 --> 00:11:37.000
whichever variable we want to
express in terms of the others.
00:11:37.000 --> 00:11:47.000
So, for example,
if we care about z as a
00:11:47.000 --> 00:12:02.000
function of x and y -- -- we'll
get that dz is negative gx over
00:12:02.000 --> 00:12:17.000
gz dx minus gy over gz dy.
And, so if we want partial z
00:12:17.000 --> 00:12:23.000
over partial x,
well, so one way is just to say
00:12:23.000 --> 00:12:26.000
that's going to be the
coefficient of dx in here,
00:12:26.000 --> 00:12:29.000
or just to write down the other
way.
00:12:29.000 --> 00:12:34.000
We are setting y equals
constant.
00:12:34.000 --> 00:12:39.000
So, that means we set dy equal
to zero.
00:12:39.000 --> 00:12:48.000
And then, we get dz equals
negative gx over gz dx.
00:12:48.000 --> 00:12:56.000
So, that means partial z over
partial x is minus gx over gz.
00:12:56.000 --> 00:12:59.000
And, see,
that's a very counterintuitive
00:12:59.000 --> 00:13:02.000
formula because you have this
minus sign that you somehow
00:13:02.000 --> 00:13:06.000
probably couldn't have seen come
if you hadn't actually derived
00:13:06.000 --> 00:13:11.000
things this way.
I mean, it's pretty surprising
00:13:11.000 --> 00:13:17.000
to see that minus sign come out
of nowhere the first time you
00:13:17.000 --> 00:13:22.000
see it.
OK, so now we know how to find
00:13:22.000 --> 00:13:26.000
the rate of change of
constrained variables with
00:13:26.000 --> 00:13:31.000
respect to each other.
You can apply the same to find,
00:13:31.000 --> 00:13:35.000
if you want partial x,
partial y, or any of them,
00:13:35.000 --> 00:13:41.000
you can do it.
Any questions so far?
00:13:41.000 --> 00:13:48.000
No?
OK, so, before we proceed
00:13:48.000 --> 00:13:55.000
further, I should probably
expose some problem with the
00:13:55.000 --> 00:14:03.000
notations that we have so far.
So, let me try to get you a bit
00:14:03.000 --> 00:14:07.000
confused, OK?
So, let's take a very simple
00:14:07.000 --> 00:14:10.000
example.
Let's say I have a function,
00:14:10.000 --> 00:14:15.000
f of x, y equals x y.
OK, so far it doesn't sound
00:14:15.000 --> 00:14:20.000
very confusing.
And then, I can write partial f
00:14:20.000 --> 00:14:24.000
over partial x.
And, I think you all know how
00:14:24.000 --> 00:14:28.000
to compute it.
It's going to be just one.
00:14:28.000 --> 00:14:34.000
OK, so far we are pretty happy.
Now let's do a change of
00:14:34.000 --> 00:14:44.000
variables.
Let's set x=u and y=u v.
00:14:44.000 --> 00:14:46.000
It's not very complicated
change of variables.
00:14:46.000 --> 00:14:54.000
But let's do it.
Then, f in terms of u and v,
00:14:54.000 --> 00:15:02.000
well, so f, remember f was x y
becomes u plus u plus v.
00:15:02.000 --> 00:15:13.000
That's twice u plus v.
What's partial f over partial u?
00:15:13.000 --> 00:15:18.000
It's two.
So, x and u are the same thing.
00:15:18.000 --> 00:15:21.000
Partial f over partial x,
and partial f over partial u,
00:15:21.000 --> 00:15:24.000
well, unless you believe that
one equals two,
00:15:24.000 --> 00:15:26.000
they are really not the same
thing, OK?
00:15:26.000 --> 00:15:36.000
So, that's an interesting,
slightly strange phenomenon.
00:15:36.000 --> 00:15:46.000
x equals u, but partial f
partial x is not the same as
00:15:46.000 --> 00:15:52.000
partial f partial u.
So, how do we get rid of this
00:15:52.000 --> 00:15:55.000
contradiction?
Well, we have to think a bit
00:15:55.000 --> 00:15:59.000
more about what these notations
mean, OK?
00:15:59.000 --> 00:16:03.000
So, when we write partial f
over partial x,
00:16:03.000 --> 00:16:08.000
it means that we are varying x,
keeping y constant.
00:16:08.000 --> 00:16:11.000
When we write partial f over
partial u, it means we are
00:16:11.000 --> 00:16:15.000
varying u, keeping v constant.
So, varying u or varying x is
00:16:15.000 --> 00:16:17.000
the same thing.
But, keeping v constant,
00:16:17.000 --> 00:16:20.000
or keeping y constant are not
the same thing.
00:16:20.000 --> 00:16:23.000
If I keep y constant,
then when I change x,
00:16:23.000 --> 00:16:27.000
so when I change u,
then v will also have to change
00:16:27.000 --> 00:16:29.000
so that their sum stays the
same.
00:16:29.000 --> 00:16:32.000
Or, if you prefer the other way
around, when I do this one I
00:16:32.000 --> 00:16:35.000
keep v constant.
If I keep v constant and I
00:16:35.000 --> 00:16:39.000
change u, then y will change.
It won't be constant.
00:16:39.000 --> 00:16:43.000
So, that means,
well, life looked quite nice
00:16:43.000 --> 00:16:49.000
and easy with these notations.
But, what's dangerous about
00:16:49.000 --> 00:16:55.000
them is they are not making
explicit what it is exactly that
00:16:55.000 --> 00:17:01.000
we are keeping constant.
OK, so just to write things,
00:17:01.000 --> 00:17:08.000
so here we change u and x that
are the same thing.
00:17:08.000 --> 00:17:14.000
But we keep y constant,
while here we change u,
00:17:14.000 --> 00:17:19.000
which is still the same thing
as x.
00:17:19.000 --> 00:17:26.000
But, what we keep constant is
v, or in terms of x and y,
00:17:26.000 --> 00:17:33.000
that's y minus x constant.
And, that's why they are not
00:17:33.000 --> 00:17:36.000
the same.
So, whenever there's any risk
00:17:36.000 --> 00:17:39.000
of confusion,
OK, so not in the cases that we
00:17:39.000 --> 00:17:42.000
had before because what we've
done until now,
00:17:42.000 --> 00:17:46.000
we didn't really have a problem.
But, in a situation like this,
00:17:46.000 --> 00:17:50.000
to clarify things,
we'll actually say explicitly
00:17:50.000 --> 00:17:53.000
what it is that we want to keep
constant.
00:18:04.000 --> 00:18:07.000
OK, so what's going to be our
new notation?
00:18:07.000 --> 00:18:14.000
Well, so it's not particularly
pleasant because it uses,
00:18:14.000 --> 00:18:16.000
now, a subscript not to
indicate what you are
00:18:16.000 --> 00:18:18.000
differentiating,
but rather what you were
00:18:18.000 --> 00:18:22.000
holding constant.
So, that's quite a conflict of
00:18:22.000 --> 00:18:25.000
notation with what we had
before.
00:18:25.000 --> 00:18:32.000
I think I can safely blame it
on physicists or chemists.
00:18:32.000 --> 00:18:43.000
OK, so this one means we keep y
constant, and partial f over
00:18:43.000 --> 00:18:51.000
partial u with v held constant,
similarly.
00:18:51.000 --> 00:18:54.000
OK, so now what happens is we
no longer have any
00:18:54.000 --> 00:18:59.000
contradiction.
We have partial f over partial
00:18:59.000 --> 00:19:06.000
x with y constant is different
from partial f over partial x
00:19:06.000 --> 00:19:12.000
with v constant,
which is the same as partial f
00:19:12.000 --> 00:19:18.000
over partial u with v constant.
OK, so this guy is one.
00:19:18.000 --> 00:19:28.000
And these guys are two.
So, now we can safely use the
00:19:28.000 --> 00:19:33.000
fact that x equals u if we are
keeping track of what is
00:19:33.000 --> 00:19:36.000
actually held constant,
OK?
00:19:36.000 --> 00:19:39.000
So now, that's going to be
particularly important when we
00:19:39.000 --> 00:19:41.000
have variables that are related
because,
00:19:41.000 --> 00:19:45.000
let's say now that I have a
function that depends on x,
00:19:45.000 --> 00:19:48.000
y, and z.
But, x, y, and z are related.
00:19:48.000 --> 00:19:54.000
Then, it means that I look at,
say, x and y as my independent
00:19:54.000 --> 00:19:59.000
variables, and z as a function
of x and y.
00:19:59.000 --> 00:20:01.000
Then, it means that when I do
partials, say,
00:20:01.000 --> 00:20:04.000
with respect to x,
I will hold y constant.
00:20:04.000 --> 00:20:08.000
But, I will let z vary as a
function of x and y.
00:20:08.000 --> 00:20:10.000
Or, I could do it the other way
around.
00:20:10.000 --> 00:20:12.000
I could vary x,
keep z constant,
00:20:12.000 --> 00:20:15.000
and let y be a function of x
and z.
00:20:15.000 --> 00:20:24.000
And so, I will need to use this
kind of notation to indicate
00:20:24.000 --> 00:20:34.000
which one I mean.
OK, any questions?
00:20:34.000 --> 00:20:39.000
No?
All right, so let's try to do
00:20:39.000 --> 00:20:42.000
an example where we have a
function that depends on
00:20:42.000 --> 00:20:46.000
variables that are related.
OK, so I don't want to do one
00:20:46.000 --> 00:20:50.000
with PV=nRT because probably,
I mean, if you've seen it,
00:20:50.000 --> 00:20:53.000
then you've seen too much of
it.
00:20:53.000 --> 00:20:58.000
And, if you haven't seen it,
then maybe it's not the best
00:20:58.000 --> 00:21:02.000
example.
So, let's do a geometric
00:21:02.000 --> 00:21:08.000
example.
So, let's look at the area of
00:21:08.000 --> 00:21:14.000
the triangle.
So, let's say I have a
00:21:14.000 --> 00:21:21.000
triangle, and my variables will
be the sides a and b.
00:21:21.000 --> 00:21:26.000
And the angle here, theta.
OK, so what's the area of this
00:21:26.000 --> 00:21:29.000
triangle?
Well, its base times height
00:21:29.000 --> 00:21:34.000
over two.
So, it's one half of the base
00:21:34.000 --> 00:21:39.000
is a, and the height is b sine
theta.
00:21:39.000 --> 00:21:45.000
OK, so that's a function of a,
b, and theta.
00:21:45.000 --> 00:21:47.000
Now, let's say,
actually, there is a relation
00:21:47.000 --> 00:21:49.000
between a, b,
and theta that I didn't tell
00:21:49.000 --> 00:21:52.000
you about,
namely, actually,
00:21:52.000 --> 00:21:58.000
I want to assume that it's a
right triangle,
00:21:58.000 --> 00:22:05.000
OK?
So, let's now assume it's a
00:22:05.000 --> 00:22:16.000
right triangle with,
let's say, the hypotenuse is b.
00:22:16.000 --> 00:22:19.000
So, we have the right angle
here, actually.
00:22:19.000 --> 00:22:23.000
So, a is here. b is here.
Theta is here.
00:22:23.000 --> 00:22:28.000
So, saying it's a right
triangle is the same thing as
00:22:28.000 --> 00:22:31.000
saying that b equals sine theta,
OK?
00:22:31.000 --> 00:22:37.000
So that's our constraint.
That's the relation between a,
00:22:37.000 --> 00:22:46.000
b, and theta.
And, this is a function of a,
00:22:46.000 --> 00:22:53.000
b, and theta.
And, let's say that we want to
00:22:53.000 --> 00:22:57.000
understand how the area depends
on theta.
00:22:57.000 --> 00:23:00.000
OK, what's the rate of change
of the area of this triangle
00:23:00.000 --> 00:23:06.000
with respect to theta?
So, I claim there's various
00:23:06.000 --> 00:23:09.000
answers.
I can think of at least three
00:23:09.000 --> 00:23:10.000
possible answers.
00:23:44.000 --> 00:23:52.000
So, what can we possibly mean
by the rate of change of A with
00:23:52.000 --> 00:23:57.000
respect to theta?
So, these are all things that
00:23:57.000 --> 00:23:59.000
we might want to call partial A
partial theta.
00:23:59.000 --> 00:24:03.000
But of course,
we'll have to actually use
00:24:03.000 --> 00:24:06.000
different notations to
distinguish them.
00:24:06.000 --> 00:24:11.000
So, the first way that we
actually already know about is
00:24:11.000 --> 00:24:17.000
if we just forget about the fact
that the variables are related,
00:24:17.000 --> 00:24:20.000
OK?
So, if we just think of little
00:24:20.000 --> 00:24:23.000
a, b,
and theta as independent
00:24:23.000 --> 00:24:25.000
variables,
and we just change theta,
00:24:25.000 --> 00:24:48.000
keeping a and b constant -- So,
that's exactly what we meant by
00:24:48.000 --> 00:24:51.000
partial A, partial theta,
right?
00:24:51.000 --> 00:24:59.000
I'm not putting any constraints.
So, just to use some new
00:24:59.000 --> 00:25:03.000
notation, that would be the rate
of change of A with respect to
00:25:03.000 --> 00:25:07.000
theta, keeping a and b fixed at
the same time.
00:25:07.000 --> 00:25:11.000
Of course, if we are keeping a
and b fixed, and we are changing
00:25:11.000 --> 00:25:14.000
theta, it means we completely
ignore this property of being a
00:25:14.000 --> 00:25:16.000
right triangle.
So, in fact,
00:25:16.000 --> 00:25:20.000
it corresponds to changing the
area by changing the angle,
00:25:20.000 --> 00:25:23.000
keeping these lengths fixed.
And, of course,
00:25:23.000 --> 00:25:27.000
we lose the right angle.
When we rotate this side here,
00:25:27.000 --> 00:25:32.000
but the angle doesn't stay at a
right angle.
00:25:32.000 --> 00:25:35.000
And that one,
we know how to compute,
00:25:35.000 --> 00:25:40.000
right, because it's the one
we've been computing all along.
00:25:40.000 --> 00:25:44.000
So, that means we keep a and b
fixed.
00:25:44.000 --> 00:25:51.000
And then, so let's see,
what's the derivatives of A
00:25:51.000 --> 00:26:02.000
with respect to theta?
It's one half ab cosine theta.
00:26:02.000 --> 00:26:11.000
OK, now that one we know.
Any questions?
00:26:11.000 --> 00:26:14.000
No?
OK, the two other guys will be
00:26:14.000 --> 00:26:18.000
more interesting.
So far, I'm not really doing
00:26:18.000 --> 00:26:23.000
anything with my constraint.
Let's say that actually I do
00:26:23.000 --> 00:26:27.000
want to keep the right angle.
Then, when I change theta,
00:26:27.000 --> 00:26:31.000
there's two options.
One is I keep a constant,
00:26:31.000 --> 00:26:35.000
and then of course b will have
to change because if this width
00:26:35.000 --> 00:26:38.000
stays the same,
then when I change theta,
00:26:38.000 --> 00:26:41.000
the height increases,
and then this side length
00:26:41.000 --> 00:26:45.000
increases.
The other option is to change
00:26:45.000 --> 00:26:47.000
the angle, keeping b constant.
So, actually,
00:26:47.000 --> 00:26:49.000
this side stays the same
length.
00:26:49.000 --> 00:26:53.000
But then, a has to become a bit
shorter.
00:26:53.000 --> 00:26:56.000
And, of course,
the area will change in
00:26:56.000 --> 00:26:59.000
different ways depending on what
I do.
00:26:59.000 --> 00:27:05.000
So, that's why I said we have
three different answers.
00:27:05.000 --> 00:27:10.000
So, the next one is keep,
I forgot which one I said
00:27:10.000 --> 00:27:17.000
first.
Let's say keep a constant.
00:27:17.000 --> 00:27:26.000
And, that means that b will
change.
00:27:26.000 --> 00:27:30.000
b is going to be some function
of a and theta.
00:27:30.000 --> 00:27:34.000
Well, in fact here,
we know what the function is
00:27:34.000 --> 00:27:37.000
because we can solve the
constraint, namely,
00:27:37.000 --> 00:27:45.000
b is a over cosine theta.
But we don't actually need to
00:27:45.000 --> 00:27:55.000
know that so that the triangle,
so that the right angle,
00:27:55.000 --> 00:28:05.000
so that we keep a right angle.
And, so the name we will have
00:28:05.000 --> 00:28:11.000
for this is partial a over
partial theta with a held
00:28:11.000 --> 00:28:14.000
constant, OK?
And, the fact that I'm not
00:28:14.000 --> 00:28:17.000
putting b in my subscript there
means that actually b will be a
00:28:17.000 --> 00:28:20.000
dependent variable.
It changes in whatever way it
00:28:20.000 --> 00:28:26.000
has to change so that when theta
changes, a stays the same while
00:28:26.000 --> 00:28:29.000
b changes so that we keep a
right triangle.
00:28:38.000 --> 00:28:46.000
And, the third guy is the one
where we actually keep b
00:28:46.000 --> 00:28:51.000
constant,
and now a,
00:28:51.000 --> 00:28:54.000
we think a as a function of b
and theta,
00:28:54.000 --> 00:28:58.000
and it changes so that we keep
the right angle.
00:28:58.000 --> 00:29:01.000
So actually as a function of b
and theta, it's given over
00:29:01.000 --> 00:29:06.000
there.
A equals b cosine theta.
00:29:06.000 --> 00:29:13.000
And so, this guy is called
partial a over partial theta
00:29:13.000 --> 00:29:19.000
with b held constant.
OK, so we've just defined them.
00:29:19.000 --> 00:29:21.000
We don't know yet how to
compute these things.
00:29:21.000 --> 00:29:22.000
That's what we're going to do
now.
00:29:22.000 --> 00:29:25.000
That is the definition,
and what these things mean.
00:29:25.000 --> 00:29:33.000
Is that clear to everyone?
Yes, OK.
00:29:33.000 --> 00:29:41.000
Yes?
OK, so the second answer,
00:29:41.000 --> 00:29:46.000
again, so one way to ask
ourselves,
00:29:46.000 --> 00:29:48.000
how does the area depend on
theta,
00:29:48.000 --> 00:29:53.000
is to say, well,
actually look at the area of
00:29:53.000 --> 00:29:59.000
the right triangle as a function
of a and theta only by solving
00:29:59.000 --> 00:30:03.000
for b.
And then, we'll change theta,
00:30:03.000 --> 00:30:06.000
keep a constant,
and ask, how does the area
00:30:06.000 --> 00:30:08.000
change?
So, when we do that,
00:30:08.000 --> 00:30:11.000
when we change theta and keep a
the same,
00:30:11.000 --> 00:30:14.000
then b has to change so that it
stays a right triangle,
00:30:14.000 --> 00:30:18.000
right, so that this relation
still holds.
00:30:18.000 --> 00:30:22.000
That requires us to change b.
So, when we write partial a
00:30:22.000 --> 00:30:26.000
over partial theta with a
constant, it means that,
00:30:26.000 --> 00:30:30.000
actually, b will be the
dependent variable.
00:30:30.000 --> 00:30:35.000
It depends on a and theta.
And so, the area depends on
00:30:35.000 --> 00:30:40.000
theta, not only because theta is
in the formula,
00:30:40.000 --> 00:30:46.000
but also because b changes,
and b is in the formula.
00:30:46.000 --> 00:30:53.000
Yes?
No, no, we don't keep theta
00:30:53.000 --> 00:30:54.000
constant.
We vary theta, right?
00:30:54.000 --> 00:30:58.000
The goal is to see how things
change when I change theta by a
00:30:58.000 --> 00:31:01.000
little bit.
OK, so if I change theta a
00:31:01.000 --> 00:31:04.000
little bit in this one,
if I change theta a little bit
00:31:04.000 --> 00:31:07.000
and I keep a the same,
then b has to change also in
00:31:07.000 --> 00:31:09.000
some way.
There's a right triangle.
00:31:09.000 --> 00:31:16.000
And then, because theta and b
change, that causes the area to
00:31:16.000 --> 00:31:18.000
change.
OK, so maybe I should
00:31:18.000 --> 00:31:23.000
re-explain that again.
So, theta changes.
00:31:23.000 --> 00:31:30.000
A is constant.
But, we have the constraint,
00:31:30.000 --> 00:31:37.000
a equals be plus sine theta.
That means that b changes.
00:31:37.000 --> 00:31:43.000
And then, the question is,
how does A change?
00:31:43.000 --> 00:31:46.000
Well, it will change in part
because theta changes,
00:31:46.000 --> 00:31:50.000
and in part because b changes.
But, we want to know how it
00:31:50.000 --> 00:31:54.000
depends on theta in this
situation.
00:31:54.000 --> 00:32:04.000
Yes?
Ah, that's a very good question.
00:32:04.000 --> 00:32:08.000
So, what about,
I don't keep a and b constant?
00:32:08.000 --> 00:32:10.000
Well, then there's too many
choices.
00:32:10.000 --> 00:32:13.000
So I have to decide actually
how I'm going to change things.
00:32:13.000 --> 00:32:17.000
See, if I just say I have this
relation, that means I have two
00:32:17.000 --> 00:32:20.000
independent variables left,
whichever two of the three I
00:32:20.000 --> 00:32:23.000
want.
But, I still have to specify
00:32:23.000 --> 00:32:27.000
two of them to say exactly which
triangle I mean.
00:32:27.000 --> 00:32:31.000
So, I cannot ask myself just
how will it change if I change
00:32:31.000 --> 00:32:34.000
theta and do random things with
a and b?
00:32:34.000 --> 00:32:36.000
It depends what I do with a and
b.
00:32:36.000 --> 00:32:40.000
Of course, I could choose to
change them simultaneously,
00:32:40.000 --> 00:32:45.000
but then I have to specify how
exactly I'm going to do that.
00:32:45.000 --> 00:32:49.000
Ah, yes, if you wanted to,
indeed, we could also change
00:32:49.000 --> 00:32:53.000
things in such a way that the
third side remains constant.
00:32:53.000 --> 00:32:55.000
And that would be,
yet, a different way to attack
00:32:55.000 --> 00:32:57.000
the problem.
I mean, we don't have good
00:32:57.000 --> 00:33:00.000
notation for this,
here, because we didn't give it
00:33:00.000 --> 00:33:01.000
a name.
But, yeah, I mean, we could.
00:33:01.000 --> 00:33:07.000
We could call this guy c,
and then we'd have a different
00:33:07.000 --> 00:33:11.000
formula, and so on.
So, I mean, I'm not looking at
00:33:11.000 --> 00:33:17.000
it for simplicity.
But, you could have many more.
00:33:17.000 --> 00:33:19.000
I mean, in general,
you will want,
00:33:19.000 --> 00:33:22.000
once you have a set of nice,
natural variables,
00:33:22.000 --> 00:33:25.000
you will want to look mostly at
situations where one of the
00:33:25.000 --> 00:33:29.000
variables changes.
Some of them are held fixed,
00:33:29.000 --> 00:33:33.000
and then some dependent
variable does whatever it must
00:33:33.000 --> 00:33:36.000
so that the constraint keeps
holding.
00:33:36.000 --> 00:33:39.000
OK, so let's try to compute one
of them.
00:33:39.000 --> 00:33:44.000
Let's say I decide that we will
compute this one.
00:33:44.000 --> 00:33:46.000
OK, let's see how we can
compute partial a,
00:33:46.000 --> 00:33:49.000
partial theta with a held
fixed.
00:34:21.000 --> 00:34:27.000
[APPLAUSE]
OK, so let's try to compute
00:34:27.000 --> 00:34:34.000
partial A, partial theta with a
held constant.
00:34:34.000 --> 00:34:40.000
So, let's see three different
ways of doing that.
00:34:40.000 --> 00:34:45.000
So, let me start with method
zero.
00:34:45.000 --> 00:34:50.000
OK, it's not a real method.
That's why I'm not getting a
00:34:50.000 --> 00:34:54.000
positive number.
So, that one is just,
00:34:54.000 --> 00:34:58.000
we solve for b,
and we remove b from the
00:34:58.000 --> 00:35:01.000
formulas.
OK, so here it works well
00:35:01.000 --> 00:35:04.000
because we know how to solve for
b.
00:35:04.000 --> 00:35:07.000
But I'm not considering this to
be a real method because in
00:35:07.000 --> 00:35:08.000
general we don't know how to do
that.
00:35:08.000 --> 00:35:12.000
I mean, in the beginning I had
this relation that was an
00:35:12.000 --> 00:35:16.000
equation of degree three.
You don't really want to solve
00:35:16.000 --> 00:35:19.000
your equation for the dependent
variable usually.
00:35:19.000 --> 00:35:33.000
Here, we can.
So, solve for b and substitute.
00:35:33.000 --> 00:35:38.000
So, how do we do that?
Well, the constraint is a=b
00:35:38.000 --> 00:35:45.000
cosine theta.
That means b is a over cosine
00:35:45.000 --> 00:35:48.000
theta.
Some of you know that as a
00:35:48.000 --> 00:35:56.000
secan theta.
That's the same.
00:35:56.000 --> 00:36:04.000
And now, if we express the area
in terms of a and theta only,
00:36:04.000 --> 00:36:13.000
A is one half of ab cosine,
sorry, ab sine theta is now one
00:36:13.000 --> 00:36:20.000
half of a^2 sine theta over
cosine theta.
00:36:20.000 --> 00:36:29.000
Or, if you prefer,
one half of a^2 tangent theta.
00:36:29.000 --> 00:36:32.000
Well, now that it's only a
function of a and theta,
00:36:32.000 --> 00:36:35.000
I know what it means to take
the partial derivative with
00:36:35.000 --> 00:36:38.000
respect to theta,
keeping a constant.
00:36:38.000 --> 00:36:51.000
I know how to do it.
So, partial A over partial
00:36:51.000 --> 00:36:55.000
theta,
a held constant,
00:36:55.000 --> 00:36:59.000
well, if a is a constant,
then I get this one half a^2
00:36:59.000 --> 00:37:03.000
coming out times,
what's the derivative of
00:37:03.000 --> 00:37:09.000
tangent?
Secan squared, very good.
00:37:09.000 --> 00:37:12.000
If you're European and you've
never heard of secan,
00:37:12.000 --> 00:37:15.000
that's one over cosine.
And, if you know the derivative
00:37:15.000 --> 00:37:18.000
as one plus tangent squared,
that's the same thing.
00:37:18.000 --> 00:37:24.000
And, it's also correct.
OK, so, that's one way of doing
00:37:24.000 --> 00:37:26.000
it.
But, as I've already said,
00:37:26.000 --> 00:37:30.000
it doesn't get us very far if
we don't know how to solve for
00:37:30.000 --> 00:37:33.000
b.
We really used the fact that we
00:37:33.000 --> 00:37:36.000
could solve for b and get rid of
it.
00:37:36.000 --> 00:37:45.000
So, there's two systematic
methods, and let's say the basic
00:37:45.000 --> 00:37:53.000
rule is that you should give
both of them a chance.
00:37:53.000 --> 00:37:56.000
You should see which one you
prefer, and you should be able
00:37:56.000 --> 00:37:59.000
to use one or the other on the
exam.
00:37:59.000 --> 00:38:04.000
OK, most likely you'll actually
have a choice between one or the
00:38:04.000 --> 00:38:06.000
other.
It will be up to you to decide
00:38:06.000 --> 00:38:10.000
which one you want to use.
But, you cannot use solving in
00:38:10.000 --> 00:38:14.000
substitution.
That's not fair.
00:38:14.000 --> 00:38:25.000
OK, so the first one is to use
differentials.
00:38:25.000 --> 00:38:29.000
By the way, in the notes they
are called also method one and
00:38:29.000 --> 00:38:32.000
method two.
I'm not promising that I have
00:38:32.000 --> 00:38:32.000
the same one,
am I?
I mean, I might have one and
two switched.
00:38:35.000 --> 00:38:39.000
It doesn't really matter.
So, how do we do things using
00:38:39.000 --> 00:38:43.000
differentials?
Well, first,
00:38:43.000 --> 00:38:52.000
we know that we want to keep a
fixed, and that means that we'll
00:38:52.000 --> 00:38:56.000
set da equal to zero,
OK?
00:38:56.000 --> 00:39:00.000
The second thing that we want
to do is we want to look at the
00:39:00.000 --> 00:39:04.000
constraint.
The constraint is a equals b
00:39:04.000 --> 00:39:08.000
cosine theta.
And, we want to differentiate
00:39:08.000 --> 00:39:10.000
that.
Well, differentiate the
00:39:10.000 --> 00:39:15.000
left-hand side.
You get da.
00:39:15.000 --> 00:39:18.000
And, differentiate the
right-hand side as a function of
00:39:18.000 --> 00:39:20.000
b and theta.
You should get,
00:39:20.000 --> 00:39:23.000
well, how many db's?
Well, that's the rate of change
00:39:23.000 --> 00:39:28.000
with respect to b.
That's cosine theta db minus b
00:39:28.000 --> 00:39:35.000
sine theta d theta.
That's a product rule applied
00:39:35.000 --> 00:39:47.000
to b times cosine theta.
So -- Well, now,
00:39:47.000 --> 00:39:51.000
if we have a constraint that's
relating da, db,
00:39:51.000 --> 00:39:54.000
and d theta,
OK, so that's actually what we
00:39:54.000 --> 00:39:56.000
did, right,
that's the same sort of thing
00:39:56.000 --> 00:39:59.000
as what we did at the beginning
when we related dx,
00:39:59.000 --> 00:40:02.000
dy, and dz.
That's really the same thing,
00:40:02.000 --> 00:40:05.000
except now are variables are a,
b, and theta.
00:40:05.000 --> 00:40:07.000
Now, we know that also we are
keeping a fixed.
00:40:07.000 --> 00:40:10.000
So actually,
we set this equal to zero.
00:40:10.000 --> 00:40:18.000
So, we have zero equals da
equals cosine theta db minus b
00:40:18.000 --> 00:40:23.000
sine theta d theta.
That means that actually we
00:40:23.000 --> 00:40:33.000
know how to solve for db.
OK, so cosine theta db equals b
00:40:33.000 --> 00:40:45.000
sine theta d theta or db is b
tangent theta d theta.
00:40:45.000 --> 00:40:47.000
OK, so in fact,
what we found,
00:40:47.000 --> 00:40:50.000
if you want,
is the rate of change of b with
00:40:50.000 --> 00:40:53.000
respect to theta.
Why do we care?
00:40:53.000 --> 00:40:59.000
Well, we care because let's
look, now, at dA,
00:40:59.000 --> 00:41:03.000
the function that we want to
look at.
00:41:03.000 --> 00:41:12.000
OK, so the function is A equals
one half ab sine theta.
00:41:12.000 --> 00:41:15.000
Well, then, dA,
so we had to use the product
00:41:15.000 --> 00:41:18.000
rule carefully,
or we use the partials.
00:41:18.000 --> 00:41:21.000
So, the coefficient of d little
a will be partial with respect
00:41:21.000 --> 00:41:26.000
to little a.
That's one half b sine theta da
00:41:26.000 --> 00:41:36.000
plus coefficient of db will be
one half a sine theta db plus
00:41:36.000 --> 00:41:45.000
coefficient of d theta will be
one half ab cosine theta d
00:41:45.000 --> 00:41:48.000
theta.
But now, what do I do with that?
00:41:48.000 --> 00:41:52.000
Well, first I said a is
constant.
00:41:52.000 --> 00:41:56.000
So, da is zero.
Second, well,
00:41:56.000 --> 00:41:59.000
actually we don't like b at
all, right?
00:41:59.000 --> 00:42:03.000
We want to view a as a function
of theta.
00:42:03.000 --> 00:42:13.000
So, well, maybe we actually
want to use this formula for db
00:42:13.000 --> 00:42:18.000
that we found in here.
OK, and then we'll be left only
00:42:18.000 --> 00:42:20.000
with d thetas,
which is what we want.
00:42:56.000 --> 00:43:06.000
So, if we plug this one into
that one, we get da equals one
00:43:06.000 --> 00:43:16.000
half a sine theta times b
tangent theta d theta plus one
00:43:16.000 --> 00:43:26.000
half ab cosine theta d theta.
And, if we collect these things
00:43:26.000 --> 00:43:35.000
together, we get one half of ab
times sine theta times tangent
00:43:35.000 --> 00:43:41.000
theta plus cosine theta d theta.
And, if you know your trig,
00:43:41.000 --> 00:43:44.000
but you'll see that this is
sine squared over cosine plus
00:43:44.000 --> 00:43:49.000
cosine squared over cosine.
That's the same as secan theta.
00:43:49.000 --> 00:43:54.000
So, now you have expressed da
as something times d theta.
00:43:54.000 --> 00:43:59.000
Well, that coefficient is the
rate of change of A with respect
00:43:59.000 --> 00:44:04.000
to theta with the understanding
that we are keeping a fixed,
00:44:04.000 --> 00:44:10.000
and letting b vary as a
dependent variable.
00:44:10.000 --> 00:44:11.000
Not enough space: sorry.
00:44:26.000 --> 00:44:29.000
OK, in case it's clearer for
you, let's think about it
00:44:29.000 --> 00:44:32.000
backwards.
So, we wanted to find how A
00:44:32.000 --> 00:44:35.000
changes.
To find how A changes,
00:44:35.000 --> 00:44:38.000
we write da.
But now, this tells us how A
00:44:38.000 --> 00:44:41.000
depends on little a,
little b, and theta.
00:44:41.000 --> 00:44:45.000
Well, we know actually we want
to keep little a constant.
00:44:45.000 --> 00:44:49.000
So, we set this to be zero.
Theta, well,
00:44:49.000 --> 00:44:52.000
we are very happy because we
want to express things in terms
00:44:52.000 --> 00:44:55.000
of theta.
Db we want to get rid of.
00:44:55.000 --> 00:45:00.000
How do we get rid of db?
Well, we do that by figuring
00:45:00.000 --> 00:45:05.000
out how b depends on theta when
a is fixed.
00:45:05.000 --> 00:45:08.000
And, we do that by
differentiating the constraint
00:45:08.000 --> 00:45:12.000
equation, and setting da equal
to zero.
00:45:12.000 --> 00:45:31.000
OK, so -- I guess to summarize
the method, we wrote dA in terms
00:45:31.000 --> 00:45:41.000
of da, db, d theta.
Then, we say that a is constant
00:45:41.000 --> 00:45:50.000
means we set da equals zero.
And, the third thing is that
00:45:50.000 --> 00:45:57.000
because, well,
we differentiate the
00:45:57.000 --> 00:46:06.000
constraint.
And, we can solve for db in
00:46:06.000 --> 00:46:19.000
terms of d theta.
And then, we plug into dA,
00:46:19.000 --> 00:46:32.000
and we get the answer.
OK, oops.
00:46:32.000 --> 00:46:38.000
So, here's another method to do
the same thing differently is to
00:46:38.000 --> 00:46:43.000
use the chain rule.
So, we can use the chain rule
00:46:43.000 --> 00:46:45.000
with dependent variables,
OK?
00:46:45.000 --> 00:46:48.000
So, what does the chain rule
tell us?
00:46:48.000 --> 00:46:54.000
The chain rule tells us,
so we will want to
00:46:54.000 --> 00:47:02.000
differentiate -- -- the formula
for a with respect to theta
00:47:02.000 --> 00:47:06.000
holding a constant.
So, I claim,
00:47:06.000 --> 00:47:10.000
well, what does the chain rule
tell us?
00:47:10.000 --> 00:47:14.000
It tells us that,
well, when we change things,
00:47:14.000 --> 00:47:19.000
a changes because of the
changes in the variables.
00:47:19.000 --> 00:47:24.000
So, part of it is that A
depends on theta and theta
00:47:24.000 --> 00:47:28.000
changes.
How fast does theta change?
00:47:28.000 --> 00:47:31.000
Well, you could call that the
rate of change of theta with
00:47:31.000 --> 00:47:33.000
respect to theta with a
constant.
00:47:33.000 --> 00:47:35.000
But of course,
how fast does theta depend to
00:47:35.000 --> 00:47:38.000
itself?
The answer is one.
00:47:38.000 --> 00:47:44.000
So, that's pretty easy.
Plus, then we have the partial
00:47:44.000 --> 00:47:49.000
derivative, formal partial
derivative, of A with respect to
00:47:49.000 --> 00:47:55.000
little a times the rate of
change of a in our situation.
00:47:55.000 --> 00:47:58.000
Well, how does little a change
if a is constant?
00:47:58.000 --> 00:48:08.000
Well, it doesn't change.
And then, there is Ab,
00:48:08.000 --> 00:48:14.000
the formal partial derivative
times, sorry,
00:48:14.000 --> 00:48:20.000
the rate of change of b.
OK, and how do we find this one?
00:48:20.000 --> 00:48:27.000
Well, here we have to use the
constraint.
00:48:27.000 --> 00:48:30.000
OK, and we can find this one
from the constraint as we've
00:48:30.000 --> 00:48:32.000
seen at the beginning either by
differentiating the constraint,
00:48:32.000 --> 00:48:36.000
or by using the chain rule on
the constraint.
00:48:36.000 --> 00:48:39.000
So, of course the calculations
are exactly the same.
00:48:39.000 --> 00:48:44.000
See, this is the same formula
as the one over there,
00:48:44.000 --> 00:48:48.000
just dividing everything by
partial theta and with
00:48:48.000 --> 00:48:54.000
subscripts little a.
But, if it's easier to think
00:48:54.000 --> 00:48:59.000
about it this way,
then that's also valid.
00:48:59.000 --> 00:49:03.000
OK, so tomorrow we are going to
review for the test,
00:49:03.000 --> 00:49:06.000
so I'm going to tell you a bit
more about this also as we go
00:49:06.000 --> 00:49:09.000
over one practice problem on
that.