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OK, so we're going to continue
looking at what happens when we

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have non-independent variables.
So, I'm afraid we don't take

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deliveries during class time,
sorry.

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00:00:50,000 --> 00:01:00,000
Please take a seat, thanks.
[LAUGHTER]

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00:01:00,000 --> 00:01:05,000
[APPLAUSE]
OK, so Jason,

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00:01:05,000 --> 00:01:17,000
you please claim your package
at the end of lecture.

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00:01:17,000 --> 00:01:19,000
OK,
so last time we saw how to use

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00:01:19,000 --> 00:01:23,000
Lagrange multipliers to find the
minimum or maximum of a function

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00:01:23,000 --> 00:01:27,000
of several variables when the
variables are not independent.

16
00:01:27,000 --> 00:01:29,000
And, today we're going to try
to figure out more about

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00:01:29,000 --> 00:01:33,000
relations between the variables,
and how to handle functions

18
00:01:33,000 --> 00:01:36,000
that depend on several variables
when they're related.

19
00:01:36,000 --> 00:01:40,000
So, just to give you an
example,

20
00:01:40,000 --> 00:01:44,000
in physics, very often,
you have functions that depend

21
00:01:44,000 --> 00:01:49,000
on pressure, volume,
and temperature where pressure,

22
00:01:49,000 --> 00:01:52,000
volume,
and temperature are actually

23
00:01:52,000 --> 00:01:55,000
not independent.
But they are related,

24
00:01:55,000 --> 00:01:58,000
say, by PV=nRT.
So, of course,

25
00:01:58,000 --> 00:02:01,000
then you can substitute and
expressed a function in terms of

26
00:02:01,000 --> 00:02:04,000
two of them only,
but very often it's convenient

27
00:02:04,000 --> 00:02:06,000
to keep all three.
But then we have to figure out,

28
00:02:06,000 --> 00:02:11,000
what are the rates of change
with respect to t,

29
00:02:11,000 --> 00:02:14,000
with respect to each other,
the rate of change of f with

30
00:02:14,000 --> 00:02:16,000
respect to these variables,
and so on.

31
00:02:16,000 --> 00:02:21,000
So, we have to figure out what
we mean by partial derivatives

32
00:02:21,000 --> 00:02:24,000
again.
So,

33
00:02:24,000 --> 00:02:31,000
OK, more generally,
let's say just for the sake of

34
00:02:31,000 --> 00:02:33,000
notation,
I'm going to think of a

35
00:02:33,000 --> 00:02:35,000
function of three variables,
x, y, z,

36
00:02:35,000 --> 00:02:39,000
where the variables are related
by some equation,

37
00:02:39,000 --> 00:02:44,000
but I will put in the form g of
x, y, z equals some constant.

38
00:02:44,000 --> 00:02:48,000
OK, so that's the same kind of
setup as we had last time,

39
00:02:48,000 --> 00:02:52,000
except now we are not just
looking for minima and maxima.

40
00:02:52,000 --> 00:02:59,000
We are trying to understand
partial derivatives.

41
00:02:59,000 --> 00:03:06,000
So, the first observation is
that if x, y,

42
00:03:06,000 --> 00:03:09,000
and z are related,
then that means,

43
00:03:09,000 --> 00:03:11,000
in principle,
we could solve for one of them,

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00:03:11,000 --> 00:03:15,000
and express it as a function of
the two others.

45
00:03:15,000 --> 00:03:19,000
So, in particular,
can we understand even without

46
00:03:19,000 --> 00:03:21,000
solving?
Maybe we can not solve.

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00:03:21,000 --> 00:03:27,000
Can we understand how the
variables are related to each

48
00:03:27,000 --> 00:03:29,000
other?
So, for example,

49
00:03:29,000 --> 00:03:33,000
z, you can think of z as a
function of x and y.

50
00:03:33,000 --> 00:03:40,000
So, we can ask ourselves,
what are the rates of change of

51
00:03:40,000 --> 00:03:44,000
z with respect to x,
keeping y constant,

52
00:03:44,000 --> 00:03:49,000
or with respect to y keeping x
constant?

53
00:03:49,000 --> 00:03:51,000
And, of course,
if we can solve,

54
00:03:51,000 --> 00:03:53,000
that we know the formula for
this.

55
00:03:53,000 --> 00:03:55,000
And then we can compute these
guys.

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00:03:55,000 --> 00:04:03,000
But, what if we can't solve?
So, how do we find these things

57
00:04:03,000 --> 00:04:11,000
without solving?
Well, so let's do an example.

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00:04:11,000 --> 00:04:19,000
Let's say that my relation is
x^2 yz z^3=8.

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00:04:19,000 --> 00:04:24,000
And, let's say that I'm looking
near the point (x,

60
00:04:24,000 --> 00:04:27,000
y, z) equals (2,3,
1).

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00:04:27,000 --> 00:04:33,000
So, let me check 2^2 plus three
times one plus 1^3 is indeed

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00:04:33,000 --> 00:04:34,000
eight.
OK, but now,

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00:04:34,000 --> 00:04:38,000
if I change x and y a little
bit, how does z change?

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00:04:38,000 --> 00:04:41,000
Well, of course I could solve
for z in here.

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00:04:41,000 --> 00:04:43,000
It's a cubic equation.
There is actually a formula.

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00:04:43,000 --> 00:04:45,000
But that formula is quite
complicated.

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00:04:45,000 --> 00:04:47,000
We actually don't want to do
that.

68
00:04:47,000 --> 00:04:58,000
There's an easier way.
So, how can we do it?

69
00:04:58,000 --> 00:05:07,000
Well, let's look at the
differential -- -- of this

70
00:05:07,000 --> 00:05:15,000
constraint quantity.
OK, so if we called this g,

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00:05:15,000 --> 00:05:21,000
let's look at dg.
So, what's the differential of

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00:05:21,000 --> 00:05:26,000
this?
So, the differential of x^2 is

73
00:05:26,000 --> 00:05:32,000
2x dx plus, I think there's a
zdy.

74
00:05:32,000 --> 00:05:38,000
There's a ydz,
and there's also a 3z^2 dz.

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00:05:38,000 --> 00:05:42,000
OK, you can get this either by
implicit differentiation and the

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00:05:42,000 --> 00:05:45,000
product rule,
or you could get this just by

77
00:05:45,000 --> 00:05:46,000
putting here,
here,

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00:05:46,000 --> 00:05:51,000
and here the partial
derivatives of this with respect

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00:05:51,000 --> 00:05:56,000
to x, y, and z.
OK, any questions about how I

80
00:05:56,000 --> 00:05:58,000
got this?
No?

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00:05:58,000 --> 00:06:03,000
OK.
So, now, what do I do with this?

82
00:06:03,000 --> 00:06:07,000
Well, this represents,
somehow, variations of g.

83
00:06:07,000 --> 00:06:12,000
But, well, I've set this thing
equal to eight.

84
00:06:12,000 --> 00:06:16,000
And, eight is a constant.
So, it doesn't change.

85
00:06:16,000 --> 00:06:26,000
So, in fact,
well, we can set this to zero

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00:06:26,000 --> 00:06:35,000
because, well,
they call this g.

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00:06:35,000 --> 00:06:39,000
Then, g equals eight is
constant.

88
00:06:39,000 --> 00:06:43,000
That means we set dg equal to
zero.

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00:06:43,000 --> 00:06:53,000
OK, so, now let's just plug in
some values at this point.

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00:06:53,000 --> 00:06:58,000
That tells us,
well, so if x equals two,

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00:06:58,000 --> 00:07:07,000
that's 4dx plus z is one.
So, dy plus y 3z^2 should be

92
00:07:07,000 --> 00:07:13,000
6dz equals zero.
And now, this equation,

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00:07:13,000 --> 00:07:17,000
here, tells us about a relation
between the changes in x,

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00:07:17,000 --> 00:07:21,000
y, and z near that point.
It tells us how you change x

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00:07:21,000 --> 00:07:25,000
and y, well, how z will change.
Or, it tells you actually

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00:07:25,000 --> 00:07:28,000
anything you might want to know
about the relations between

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00:07:28,000 --> 00:07:30,000
these variables so,
for example,

98
00:07:30,000 --> 00:07:34,000
you can move dz to that side,
and then express dz in terms of

99
00:07:34,000 --> 00:07:37,000
dx and dy.
Or, you can move dy to that

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00:07:37,000 --> 00:07:41,000
side and express dy in terms of
dx and dz, and so on.

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00:07:41,000 --> 00:07:46,000
It tells you at the level of
the derivatives how each of the

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00:07:46,000 --> 00:07:49,000
variables depends on the two
others.

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00:07:49,000 --> 00:07:57,000
OK, so, just to clarify this:
if we want to view z as a

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00:07:57,000 --> 00:08:03,000
function of x and y,
then what we will do is we will

105
00:08:03,000 --> 00:08:06,000
just move the dz's to the other
side,

106
00:08:06,000 --> 00:08:15,000
and it will tell us dz equals
minus one over six times 4dx

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00:08:15,000 --> 00:08:20,000
plus dy.
And, so that should tell you

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00:08:20,000 --> 00:08:26,000
that partial z over partial x is
minus four over six.

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00:08:26,000 --> 00:08:35,000
Well, that's minus two thirds,
and partial z over partial y is

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00:08:35,000 --> 00:08:42,000
going to be minus one sixth.
OK, another way to think about

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00:08:42,000 --> 00:08:47,000
this: when we compute partial z
over partial x,

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00:08:47,000 --> 00:08:51,000
that means that actually we
keep y constant.

113
00:08:51,000 --> 00:08:55,000
OK, let me actually add some
subtitles here.

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00:08:55,000 --> 00:09:00,000
So, here that means we keep y
constant.

115
00:09:00,000 --> 00:09:05,000
And so, if we keep y constant,
another way to think about it

116
00:09:05,000 --> 00:09:10,000
is we set dy to zero.
We set dy equals zero.

117
00:09:10,000 --> 00:09:14,000
So if we do that,
we get dx equals negative four

118
00:09:14,000 --> 00:09:17,000
sixths dx.
That tells us the rate of

119
00:09:17,000 --> 00:09:24,000
change of z with respect to x.
Here, we set x constant.

120
00:09:24,000 --> 00:09:29,000
So, that means we set dx equal
to zero.

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00:09:29,000 --> 00:09:32,000
And, if we set dx equal to
zero, then we have dz equals

122
00:09:32,000 --> 00:09:36,000
negative one sixth of dy.
That tells us the rate of

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00:09:36,000 --> 00:09:47,000
change of z with respect to y.
OK, any questions about that?

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00:09:47,000 --> 00:09:57,000
No?
What, yes?

125
00:09:57,000 --> 00:09:59,000
Yes, OK, let me explain that
again.

126
00:09:59,000 --> 00:10:03,000
So we found an expression for
dz in terms of dx and dy.

127
00:10:03,000 --> 00:10:07,000
That means that this thing,
the differential,

128
00:10:07,000 --> 00:10:11,000
is the total differential of z
viewed as a function of x and y.

129
00:10:11,000 --> 00:10:16,000
OK, and so the coefficients of
dx and dy are the partials.

130
00:10:16,000 --> 00:10:20,000
Or, another way to think about
it, if you want to know partial

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00:10:20,000 --> 00:10:22,000
z partial x, it means you set y
to be constant.

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00:10:22,000 --> 00:10:27,000
Setting y to be constant means
that you will put zero in the

133
00:10:27,000 --> 00:10:30,000
place of dy.
So, you will be left with dz

134
00:10:30,000 --> 00:10:35,000
equals minus four sixths dx.
And, that will give you the

135
00:10:35,000 --> 00:10:41,000
rate of change of z with respect
to x when you keep y constant,

136
00:10:41,000 --> 00:10:46,000
OK?
So, there are various ways to

137
00:10:46,000 --> 00:10:53,000
think about this,
but hopefully it makes sense.

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00:10:53,000 --> 00:11:03,000
OK, so how do we think about
this in general?

139
00:11:03,000 --> 00:11:15,000
Well, if we know that g of x,
y, z equals a constant,

140
00:11:15,000 --> 00:11:27,000
then dg, which is gxdx gydy
gzdz should be set equal to

141
00:11:27,000 --> 00:11:32,000
zero.
OK, and now we can solve for

142
00:11:32,000 --> 00:11:37,000
whichever variable we want to
express in terms of the others.

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00:11:37,000 --> 00:11:47,000
So, for example,
if we care about z as a

144
00:11:47,000 --> 00:12:02,000
function of x and y -- -- we'll
get that dz is negative gx over

145
00:12:02,000 --> 00:12:17,000
gz dx minus gy over gz dy.
And, so if we want partial z

146
00:12:17,000 --> 00:12:23,000
over partial x,
well, so one way is just to say

147
00:12:23,000 --> 00:12:26,000
that's going to be the
coefficient of dx in here,

148
00:12:26,000 --> 00:12:29,000
or just to write down the other
way.

149
00:12:29,000 --> 00:12:34,000
We are setting y equals
constant.

150
00:12:34,000 --> 00:12:39,000
So, that means we set dy equal
to zero.

151
00:12:39,000 --> 00:12:48,000
And then, we get dz equals
negative gx over gz dx.

152
00:12:48,000 --> 00:12:56,000
So, that means partial z over
partial x is minus gx over gz.

153
00:12:56,000 --> 00:12:59,000
And, see,
that's a very counterintuitive

154
00:12:59,000 --> 00:13:02,000
formula because you have this
minus sign that you somehow

155
00:13:02,000 --> 00:13:06,000
probably couldn't have seen come
if you hadn't actually derived

156
00:13:06,000 --> 00:13:11,000
things this way.
I mean, it's pretty surprising

157
00:13:11,000 --> 00:13:17,000
to see that minus sign come out
of nowhere the first time you

158
00:13:17,000 --> 00:13:22,000
see it.
OK, so now we know how to find

159
00:13:22,000 --> 00:13:26,000
the rate of change of
constrained variables with

160
00:13:26,000 --> 00:13:31,000
respect to each other.
You can apply the same to find,

161
00:13:31,000 --> 00:13:35,000
if you want partial x,
partial y, or any of them,

162
00:13:35,000 --> 00:13:41,000
you can do it.
Any questions so far?

163
00:13:41,000 --> 00:13:48,000
No?
OK, so, before we proceed

164
00:13:48,000 --> 00:13:55,000
further, I should probably
expose some problem with the

165
00:13:55,000 --> 00:14:03,000
notations that we have so far.
So, let me try to get you a bit

166
00:14:03,000 --> 00:14:07,000
confused, OK?
So, let's take a very simple

167
00:14:07,000 --> 00:14:10,000
example.
Let's say I have a function,

168
00:14:10,000 --> 00:14:15,000
f of x, y equals x y.
OK, so far it doesn't sound

169
00:14:15,000 --> 00:14:20,000
very confusing.
And then, I can write partial f

170
00:14:20,000 --> 00:14:24,000
over partial x.
And, I think you all know how

171
00:14:24,000 --> 00:14:28,000
to compute it.
It's going to be just one.

172
00:14:28,000 --> 00:14:34,000
OK, so far we are pretty happy.
Now let's do a change of

173
00:14:34,000 --> 00:14:44,000
variables.
Let's set x=u and y=u v.

174
00:14:44,000 --> 00:14:46,000
It's not very complicated
change of variables.

175
00:14:46,000 --> 00:14:54,000
But let's do it.
Then, f in terms of u and v,

176
00:14:54,000 --> 00:15:02,000
well, so f, remember f was x y
becomes u plus u plus v.

177
00:15:02,000 --> 00:15:13,000
That's twice u plus v.
What's partial f over partial u?

178
00:15:13,000 --> 00:15:18,000
It's two.
So, x and u are the same thing.

179
00:15:18,000 --> 00:15:21,000
Partial f over partial x,
and partial f over partial u,

180
00:15:21,000 --> 00:15:24,000
well, unless you believe that
one equals two,

181
00:15:24,000 --> 00:15:26,000
they are really not the same
thing, OK?

182
00:15:26,000 --> 00:15:36,000
So, that's an interesting,
slightly strange phenomenon.

183
00:15:36,000 --> 00:15:46,000
x equals u, but partial f
partial x is not the same as

184
00:15:46,000 --> 00:15:52,000
partial f partial u.
So, how do we get rid of this

185
00:15:52,000 --> 00:15:55,000
contradiction?
Well, we have to think a bit

186
00:15:55,000 --> 00:15:59,000
more about what these notations
mean, OK?

187
00:15:59,000 --> 00:16:03,000
So, when we write partial f
over partial x,

188
00:16:03,000 --> 00:16:08,000
it means that we are varying x,
keeping y constant.

189
00:16:08,000 --> 00:16:11,000
When we write partial f over
partial u, it means we are

190
00:16:11,000 --> 00:16:15,000
varying u, keeping v constant.
So, varying u or varying x is

191
00:16:15,000 --> 00:16:17,000
the same thing.
But, keeping v constant,

192
00:16:17,000 --> 00:16:20,000
or keeping y constant are not
the same thing.

193
00:16:20,000 --> 00:16:23,000
If I keep y constant,
then when I change x,

194
00:16:23,000 --> 00:16:27,000
so when I change u,
then v will also have to change

195
00:16:27,000 --> 00:16:29,000
so that their sum stays the
same.

196
00:16:29,000 --> 00:16:32,000
Or, if you prefer the other way
around, when I do this one I

197
00:16:32,000 --> 00:16:35,000
keep v constant.
If I keep v constant and I

198
00:16:35,000 --> 00:16:39,000
change u, then y will change.
It won't be constant.

199
00:16:39,000 --> 00:16:43,000
So, that means,
well, life looked quite nice

200
00:16:43,000 --> 00:16:49,000
and easy with these notations.
But, what's dangerous about

201
00:16:49,000 --> 00:16:55,000
them is they are not making
explicit what it is exactly that

202
00:16:55,000 --> 00:17:01,000
we are keeping constant.
OK, so just to write things,

203
00:17:01,000 --> 00:17:08,000
so here we change u and x that
are the same thing.

204
00:17:08,000 --> 00:17:14,000
But we keep y constant,
while here we change u,

205
00:17:14,000 --> 00:17:19,000
which is still the same thing
as x.

206
00:17:19,000 --> 00:17:26,000
But, what we keep constant is
v, or in terms of x and y,

207
00:17:26,000 --> 00:17:33,000
that's y minus x constant.
And, that's why they are not

208
00:17:33,000 --> 00:17:36,000
the same.
So, whenever there's any risk

209
00:17:36,000 --> 00:17:39,000
of confusion,
OK, so not in the cases that we

210
00:17:39,000 --> 00:17:42,000
had before because what we've
done until now,

211
00:17:42,000 --> 00:17:46,000
we didn't really have a problem.
But, in a situation like this,

212
00:17:46,000 --> 00:17:50,000
to clarify things,
we'll actually say explicitly

213
00:17:50,000 --> 00:17:53,000
what it is that we want to keep
constant.

214
00:18:04,000 --> 00:18:07,000
OK, so what's going to be our
new notation?

215
00:18:07,000 --> 00:18:14,000
Well, so it's not particularly
pleasant because it uses,

216
00:18:14,000 --> 00:18:16,000
now, a subscript not to
indicate what you are

217
00:18:16,000 --> 00:18:18,000
differentiating,
but rather what you were

218
00:18:18,000 --> 00:18:22,000
holding constant.
So, that's quite a conflict of

219
00:18:22,000 --> 00:18:25,000
notation with what we had
before.

220
00:18:25,000 --> 00:18:32,000
I think I can safely blame it
on physicists or chemists.

221
00:18:32,000 --> 00:18:43,000
OK, so this one means we keep y
constant, and partial f over

222
00:18:43,000 --> 00:18:51,000
partial u with v held constant,
similarly.

223
00:18:51,000 --> 00:18:54,000
OK, so now what happens is we
no longer have any

224
00:18:54,000 --> 00:18:59,000
contradiction.
We have partial f over partial

225
00:18:59,000 --> 00:19:06,000
x with y constant is different
from partial f over partial x

226
00:19:06,000 --> 00:19:12,000
with v constant,
which is the same as partial f

227
00:19:12,000 --> 00:19:18,000
over partial u with v constant.
OK, so this guy is one.

228
00:19:18,000 --> 00:19:28,000
And these guys are two.
So, now we can safely use the

229
00:19:28,000 --> 00:19:33,000
fact that x equals u if we are
keeping track of what is

230
00:19:33,000 --> 00:19:36,000
actually held constant,
OK?

231
00:19:36,000 --> 00:19:39,000
So now, that's going to be
particularly important when we

232
00:19:39,000 --> 00:19:41,000
have variables that are related
because,

233
00:19:41,000 --> 00:19:45,000
let's say now that I have a
function that depends on x,

234
00:19:45,000 --> 00:19:48,000
y, and z.
But, x, y, and z are related.

235
00:19:48,000 --> 00:19:54,000
Then, it means that I look at,
say, x and y as my independent

236
00:19:54,000 --> 00:19:59,000
variables, and z as a function
of x and y.

237
00:19:59,000 --> 00:20:01,000
Then, it means that when I do
partials, say,

238
00:20:01,000 --> 00:20:04,000
with respect to x,
I will hold y constant.

239
00:20:04,000 --> 00:20:08,000
But, I will let z vary as a
function of x and y.

240
00:20:08,000 --> 00:20:10,000
Or, I could do it the other way
around.

241
00:20:10,000 --> 00:20:12,000
I could vary x,
keep z constant,

242
00:20:12,000 --> 00:20:15,000
and let y be a function of x
and z.

243
00:20:15,000 --> 00:20:24,000
And so, I will need to use this
kind of notation to indicate

244
00:20:24,000 --> 00:20:34,000
which one I mean.
OK, any questions?

245
00:20:34,000 --> 00:20:39,000
No?
All right, so let's try to do

246
00:20:39,000 --> 00:20:42,000
an example where we have a
function that depends on

247
00:20:42,000 --> 00:20:46,000
variables that are related.
OK, so I don't want to do one

248
00:20:46,000 --> 00:20:50,000
with PV=nRT because probably,
I mean, if you've seen it,

249
00:20:50,000 --> 00:20:53,000
then you've seen too much of
it.

250
00:20:53,000 --> 00:20:58,000
And, if you haven't seen it,
then maybe it's not the best

251
00:20:58,000 --> 00:21:02,000
example.
So, let's do a geometric

252
00:21:02,000 --> 00:21:08,000
example.
So, let's look at the area of

253
00:21:08,000 --> 00:21:14,000
the triangle.
So, let's say I have a

254
00:21:14,000 --> 00:21:21,000
triangle, and my variables will
be the sides a and b.

255
00:21:21,000 --> 00:21:26,000
And the angle here, theta.
OK, so what's the area of this

256
00:21:26,000 --> 00:21:29,000
triangle?
Well, its base times height

257
00:21:29,000 --> 00:21:34,000
over two.
So, it's one half of the base

258
00:21:34,000 --> 00:21:39,000
is a, and the height is b sine
theta.

259
00:21:39,000 --> 00:21:45,000
OK, so that's a function of a,
b, and theta.

260
00:21:45,000 --> 00:21:47,000
Now, let's say,
actually, there is a relation

261
00:21:47,000 --> 00:21:49,000
between a, b,
and theta that I didn't tell

262
00:21:49,000 --> 00:21:52,000
you about,
namely, actually,

263
00:21:52,000 --> 00:21:58,000
I want to assume that it's a
right triangle,

264
00:21:58,000 --> 00:22:05,000
OK?
So, let's now assume it's a

265
00:22:05,000 --> 00:22:16,000
right triangle with,
let's say, the hypotenuse is b.

266
00:22:16,000 --> 00:22:19,000
So, we have the right angle
here, actually.

267
00:22:19,000 --> 00:22:23,000
So, a is here. b is here.
Theta is here.

268
00:22:23,000 --> 00:22:28,000
So, saying it's a right
triangle is the same thing as

269
00:22:28,000 --> 00:22:31,000
saying that b equals sine theta,
OK?

270
00:22:31,000 --> 00:22:37,000
So that's our constraint.
That's the relation between a,

271
00:22:37,000 --> 00:22:46,000
b, and theta.
And, this is a function of a,

272
00:22:46,000 --> 00:22:53,000
b, and theta.
And, let's say that we want to

273
00:22:53,000 --> 00:22:57,000
understand how the area depends
on theta.

274
00:22:57,000 --> 00:23:00,000
OK, what's the rate of change
of the area of this triangle

275
00:23:00,000 --> 00:23:06,000
with respect to theta?
So, I claim there's various

276
00:23:06,000 --> 00:23:09,000
answers.
I can think of at least three

277
00:23:09,000 --> 00:23:10,000
possible answers.

278
00:23:44,000 --> 00:23:52,000
So, what can we possibly mean
by the rate of change of A with

279
00:23:52,000 --> 00:23:57,000
respect to theta?
So, these are all things that

280
00:23:57,000 --> 00:23:59,000
we might want to call partial A
partial theta.

281
00:23:59,000 --> 00:24:03,000
But of course,
we'll have to actually use

282
00:24:03,000 --> 00:24:06,000
different notations to
distinguish them.

283
00:24:06,000 --> 00:24:11,000
So, the first way that we
actually already know about is

284
00:24:11,000 --> 00:24:17,000
if we just forget about the fact
that the variables are related,

285
00:24:17,000 --> 00:24:20,000
OK?
So, if we just think of little

286
00:24:20,000 --> 00:24:23,000
a, b,
and theta as independent

287
00:24:23,000 --> 00:24:25,000
variables,
and we just change theta,

288
00:24:25,000 --> 00:24:48,000
keeping a and b constant -- So,
that's exactly what we meant by

289
00:24:48,000 --> 00:24:51,000
partial A, partial theta,
right?

290
00:24:51,000 --> 00:24:59,000
I'm not putting any constraints.
So, just to use some new

291
00:24:59,000 --> 00:25:03,000
notation, that would be the rate
of change of A with respect to

292
00:25:03,000 --> 00:25:07,000
theta, keeping a and b fixed at
the same time.

293
00:25:07,000 --> 00:25:11,000
Of course, if we are keeping a
and b fixed, and we are changing

294
00:25:11,000 --> 00:25:14,000
theta, it means we completely
ignore this property of being a

295
00:25:14,000 --> 00:25:16,000
right triangle.
So, in fact,

296
00:25:16,000 --> 00:25:20,000
it corresponds to changing the
area by changing the angle,

297
00:25:20,000 --> 00:25:23,000
keeping these lengths fixed.
And, of course,

298
00:25:23,000 --> 00:25:27,000
we lose the right angle.
When we rotate this side here,

299
00:25:27,000 --> 00:25:32,000
but the angle doesn't stay at a
right angle.

300
00:25:32,000 --> 00:25:35,000
And that one,
we know how to compute,

301
00:25:35,000 --> 00:25:40,000
right, because it's the one
we've been computing all along.

302
00:25:40,000 --> 00:25:44,000
So, that means we keep a and b
fixed.

303
00:25:44,000 --> 00:25:51,000
And then, so let's see,
what's the derivatives of A

304
00:25:51,000 --> 00:26:02,000
with respect to theta?
It's one half ab cosine theta.

305
00:26:02,000 --> 00:26:11,000
OK, now that one we know.
Any questions?

306
00:26:11,000 --> 00:26:14,000
No?
OK, the two other guys will be

307
00:26:14,000 --> 00:26:18,000
more interesting.
So far, I'm not really doing

308
00:26:18,000 --> 00:26:23,000
anything with my constraint.
Let's say that actually I do

309
00:26:23,000 --> 00:26:27,000
want to keep the right angle.
Then, when I change theta,

310
00:26:27,000 --> 00:26:31,000
there's two options.
One is I keep a constant,

311
00:26:31,000 --> 00:26:35,000
and then of course b will have
to change because if this width

312
00:26:35,000 --> 00:26:38,000
stays the same,
then when I change theta,

313
00:26:38,000 --> 00:26:41,000
the height increases,
and then this side length

314
00:26:41,000 --> 00:26:45,000
increases.
The other option is to change

315
00:26:45,000 --> 00:26:47,000
the angle, keeping b constant.
So, actually,

316
00:26:47,000 --> 00:26:49,000
this side stays the same
length.

317
00:26:49,000 --> 00:26:53,000
But then, a has to become a bit
shorter.

318
00:26:53,000 --> 00:26:56,000
And, of course,
the area will change in

319
00:26:56,000 --> 00:26:59,000
different ways depending on what
I do.

320
00:26:59,000 --> 00:27:05,000
So, that's why I said we have
three different answers.

321
00:27:05,000 --> 00:27:10,000
So, the next one is keep,
I forgot which one I said

322
00:27:10,000 --> 00:27:17,000
first.
Let's say keep a constant.

323
00:27:17,000 --> 00:27:26,000
And, that means that b will
change.

324
00:27:26,000 --> 00:27:30,000
b is going to be some function
of a and theta.

325
00:27:30,000 --> 00:27:34,000
Well, in fact here,
we know what the function is

326
00:27:34,000 --> 00:27:37,000
because we can solve the
constraint, namely,

327
00:27:37,000 --> 00:27:45,000
b is a over cosine theta.
But we don't actually need to

328
00:27:45,000 --> 00:27:55,000
know that so that the triangle,
so that the right angle,

329
00:27:55,000 --> 00:28:05,000
so that we keep a right angle.
And, so the name we will have

330
00:28:05,000 --> 00:28:11,000
for this is partial a over
partial theta with a held

331
00:28:11,000 --> 00:28:14,000
constant, OK?
And, the fact that I'm not

332
00:28:14,000 --> 00:28:17,000
putting b in my subscript there
means that actually b will be a

333
00:28:17,000 --> 00:28:20,000
dependent variable.
It changes in whatever way it

334
00:28:20,000 --> 00:28:26,000
has to change so that when theta
changes, a stays the same while

335
00:28:26,000 --> 00:28:29,000
b changes so that we keep a
right triangle.

336
00:28:38,000 --> 00:28:46,000
And, the third guy is the one
where we actually keep b

337
00:28:46,000 --> 00:28:51,000
constant,
and now a,

338
00:28:51,000 --> 00:28:54,000
we think a as a function of b
and theta,

339
00:28:54,000 --> 00:28:58,000
and it changes so that we keep
the right angle.

340
00:28:58,000 --> 00:29:01,000
So actually as a function of b
and theta, it's given over

341
00:29:01,000 --> 00:29:06,000
there.
A equals b cosine theta.

342
00:29:06,000 --> 00:29:13,000
And so, this guy is called
partial a over partial theta

343
00:29:13,000 --> 00:29:19,000
with b held constant.
OK, so we've just defined them.

344
00:29:19,000 --> 00:29:21,000
We don't know yet how to
compute these things.

345
00:29:21,000 --> 00:29:22,000
That's what we're going to do
now.

346
00:29:22,000 --> 00:29:25,000
That is the definition,
and what these things mean.

347
00:29:25,000 --> 00:29:33,000
Is that clear to everyone?
Yes, OK.

348
00:29:33,000 --> 00:29:41,000
Yes?
OK, so the second answer,

349
00:29:41,000 --> 00:29:46,000
again, so one way to ask
ourselves,

350
00:29:46,000 --> 00:29:48,000
how does the area depend on
theta,

351
00:29:48,000 --> 00:29:53,000
is to say, well,
actually look at the area of

352
00:29:53,000 --> 00:29:59,000
the right triangle as a function
of a and theta only by solving

353
00:29:59,000 --> 00:30:03,000
for b.
And then, we'll change theta,

354
00:30:03,000 --> 00:30:06,000
keep a constant,
and ask, how does the area

355
00:30:06,000 --> 00:30:08,000
change?
So, when we do that,

356
00:30:08,000 --> 00:30:11,000
when we change theta and keep a
the same,

357
00:30:11,000 --> 00:30:14,000
then b has to change so that it
stays a right triangle,

358
00:30:14,000 --> 00:30:18,000
right, so that this relation
still holds.

359
00:30:18,000 --> 00:30:22,000
That requires us to change b.
So, when we write partial a

360
00:30:22,000 --> 00:30:26,000
over partial theta with a
constant, it means that,

361
00:30:26,000 --> 00:30:30,000
actually, b will be the
dependent variable.

362
00:30:30,000 --> 00:30:35,000
It depends on a and theta.
And so, the area depends on

363
00:30:35,000 --> 00:30:40,000
theta, not only because theta is
in the formula,

364
00:30:40,000 --> 00:30:46,000
but also because b changes,
and b is in the formula.

365
00:30:46,000 --> 00:30:53,000
Yes?
No, no, we don't keep theta

366
00:30:53,000 --> 00:30:54,000
constant.
We vary theta, right?

367
00:30:54,000 --> 00:30:58,000
The goal is to see how things
change when I change theta by a

368
00:30:58,000 --> 00:31:01,000
little bit.
OK, so if I change theta a

369
00:31:01,000 --> 00:31:04,000
little bit in this one,
if I change theta a little bit

370
00:31:04,000 --> 00:31:07,000
and I keep a the same,
then b has to change also in

371
00:31:07,000 --> 00:31:09,000
some way.
There's a right triangle.

372
00:31:09,000 --> 00:31:16,000
And then, because theta and b
change, that causes the area to

373
00:31:16,000 --> 00:31:18,000
change.
OK, so maybe I should

374
00:31:18,000 --> 00:31:23,000
re-explain that again.
So, theta changes.

375
00:31:23,000 --> 00:31:30,000
A is constant.
But, we have the constraint,

376
00:31:30,000 --> 00:31:37,000
a equals be plus sine theta.
That means that b changes.

377
00:31:37,000 --> 00:31:43,000
And then, the question is,
how does A change?

378
00:31:43,000 --> 00:31:46,000
Well, it will change in part
because theta changes,

379
00:31:46,000 --> 00:31:50,000
and in part because b changes.
But, we want to know how it

380
00:31:50,000 --> 00:31:54,000
depends on theta in this
situation.

381
00:31:54,000 --> 00:32:04,000
Yes?
Ah, that's a very good question.

382
00:32:04,000 --> 00:32:08,000
So, what about,
I don't keep a and b constant?

383
00:32:08,000 --> 00:32:10,000
Well, then there's too many
choices.

384
00:32:10,000 --> 00:32:13,000
So I have to decide actually
how I'm going to change things.

385
00:32:13,000 --> 00:32:17,000
See, if I just say I have this
relation, that means I have two

386
00:32:17,000 --> 00:32:20,000
independent variables left,
whichever two of the three I

387
00:32:20,000 --> 00:32:23,000
want.
But, I still have to specify

388
00:32:23,000 --> 00:32:27,000
two of them to say exactly which
triangle I mean.

389
00:32:27,000 --> 00:32:31,000
So, I cannot ask myself just
how will it change if I change

390
00:32:31,000 --> 00:32:34,000
theta and do random things with
a and b?

391
00:32:34,000 --> 00:32:36,000
It depends what I do with a and
b.

392
00:32:36,000 --> 00:32:40,000
Of course, I could choose to
change them simultaneously,

393
00:32:40,000 --> 00:32:45,000
but then I have to specify how
exactly I'm going to do that.

394
00:32:45,000 --> 00:32:49,000
Ah, yes, if you wanted to,
indeed, we could also change

395
00:32:49,000 --> 00:32:53,000
things in such a way that the
third side remains constant.

396
00:32:53,000 --> 00:32:55,000
And that would be,
yet, a different way to attack

397
00:32:55,000 --> 00:32:57,000
the problem.
I mean, we don't have good

398
00:32:57,000 --> 00:33:00,000
notation for this,
here, because we didn't give it

399
00:33:00,000 --> 00:33:01,000
a name.
But, yeah, I mean, we could.

400
00:33:01,000 --> 00:33:07,000
We could call this guy c,
and then we'd have a different

401
00:33:07,000 --> 00:33:11,000
formula, and so on.
So, I mean, I'm not looking at

402
00:33:11,000 --> 00:33:17,000
it for simplicity.
But, you could have many more.

403
00:33:17,000 --> 00:33:19,000
I mean, in general,
you will want,

404
00:33:19,000 --> 00:33:22,000
once you have a set of nice,
natural variables,

405
00:33:22,000 --> 00:33:25,000
you will want to look mostly at
situations where one of the

406
00:33:25,000 --> 00:33:29,000
variables changes.
Some of them are held fixed,

407
00:33:29,000 --> 00:33:33,000
and then some dependent
variable does whatever it must

408
00:33:33,000 --> 00:33:36,000
so that the constraint keeps
holding.

409
00:33:36,000 --> 00:33:39,000
OK, so let's try to compute one
of them.

410
00:33:39,000 --> 00:33:44,000
Let's say I decide that we will
compute this one.

411
00:33:44,000 --> 00:33:46,000
OK, let's see how we can
compute partial a,

412
00:33:46,000 --> 00:33:49,000
partial theta with a held
fixed.

413
00:34:21,000 --> 00:34:27,000
[APPLAUSE]
OK, so let's try to compute

414
00:34:27,000 --> 00:34:34,000
partial A, partial theta with a
held constant.

415
00:34:34,000 --> 00:34:40,000
So, let's see three different
ways of doing that.

416
00:34:40,000 --> 00:34:45,000
So, let me start with method
zero.

417
00:34:45,000 --> 00:34:50,000
OK, it's not a real method.
That's why I'm not getting a

418
00:34:50,000 --> 00:34:54,000
positive number.
So, that one is just,

419
00:34:54,000 --> 00:34:58,000
we solve for b,
and we remove b from the

420
00:34:58,000 --> 00:35:01,000
formulas.
OK, so here it works well

421
00:35:01,000 --> 00:35:04,000
because we know how to solve for
b.

422
00:35:04,000 --> 00:35:07,000
But I'm not considering this to
be a real method because in

423
00:35:07,000 --> 00:35:08,000
general we don't know how to do
that.

424
00:35:08,000 --> 00:35:12,000
I mean, in the beginning I had
this relation that was an

425
00:35:12,000 --> 00:35:16,000
equation of degree three.
You don't really want to solve

426
00:35:16,000 --> 00:35:19,000
your equation for the dependent
variable usually.

427
00:35:19,000 --> 00:35:33,000
Here, we can.
So, solve for b and substitute.

428
00:35:33,000 --> 00:35:38,000
So, how do we do that?
Well, the constraint is a=b

429
00:35:38,000 --> 00:35:45,000
cosine theta.
That means b is a over cosine

430
00:35:45,000 --> 00:35:48,000
theta.
Some of you know that as a

431
00:35:48,000 --> 00:35:56,000
secan theta.
That's the same.

432
00:35:56,000 --> 00:36:04,000
And now, if we express the area
in terms of a and theta only,

433
00:36:04,000 --> 00:36:13,000
A is one half of ab cosine,
sorry, ab sine theta is now one

434
00:36:13,000 --> 00:36:20,000
half of a^2 sine theta over
cosine theta.

435
00:36:20,000 --> 00:36:29,000
Or, if you prefer,
one half of a^2 tangent theta.

436
00:36:29,000 --> 00:36:32,000
Well, now that it's only a
function of a and theta,

437
00:36:32,000 --> 00:36:35,000
I know what it means to take
the partial derivative with

438
00:36:35,000 --> 00:36:38,000
respect to theta,
keeping a constant.

439
00:36:38,000 --> 00:36:51,000
I know how to do it.
So, partial A over partial

440
00:36:51,000 --> 00:36:55,000
theta,
a held constant,

441
00:36:55,000 --> 00:36:59,000
well, if a is a constant,
then I get this one half a^2

442
00:36:59,000 --> 00:37:03,000
coming out times,
what's the derivative of

443
00:37:03,000 --> 00:37:09,000
tangent?
Secan squared, very good.

444
00:37:09,000 --> 00:37:12,000
If you're European and you've
never heard of secan,

445
00:37:12,000 --> 00:37:15,000
that's one over cosine.
And, if you know the derivative

446
00:37:15,000 --> 00:37:18,000
as one plus tangent squared,
that's the same thing.

447
00:37:18,000 --> 00:37:24,000
And, it's also correct.
OK, so, that's one way of doing

448
00:37:24,000 --> 00:37:26,000
it.
But, as I've already said,

449
00:37:26,000 --> 00:37:30,000
it doesn't get us very far if
we don't know how to solve for

450
00:37:30,000 --> 00:37:33,000
b.
We really used the fact that we

451
00:37:33,000 --> 00:37:36,000
could solve for b and get rid of
it.

452
00:37:36,000 --> 00:37:45,000
So, there's two systematic
methods, and let's say the basic

453
00:37:45,000 --> 00:37:53,000
rule is that you should give
both of them a chance.

454
00:37:53,000 --> 00:37:56,000
You should see which one you
prefer, and you should be able

455
00:37:56,000 --> 00:37:59,000
to use one or the other on the
exam.

456
00:37:59,000 --> 00:38:04,000
OK, most likely you'll actually
have a choice between one or the

457
00:38:04,000 --> 00:38:06,000
other.
It will be up to you to decide

458
00:38:06,000 --> 00:38:10,000
which one you want to use.
But, you cannot use solving in

459
00:38:10,000 --> 00:38:14,000
substitution.
That's not fair.

460
00:38:14,000 --> 00:38:25,000
OK, so the first one is to use
differentials.

461
00:38:25,000 --> 00:38:29,000
By the way, in the notes they
are called also method one and

462
00:38:29,000 --> 00:38:32,000
method two.
I'm not promising that I have

463
00:38:32,000 --> 00:38:32,000
the same one,
am I?
I mean, I might have one and
two switched.

464
00:38:35,000 --> 00:38:39,000
It doesn't really matter.
So, how do we do things using

465
00:38:39,000 --> 00:38:43,000
differentials?
Well, first,

466
00:38:43,000 --> 00:38:52,000
we know that we want to keep a
fixed, and that means that we'll

467
00:38:52,000 --> 00:38:56,000
set da equal to zero,
OK?

468
00:38:56,000 --> 00:39:00,000
The second thing that we want
to do is we want to look at the

469
00:39:00,000 --> 00:39:04,000
constraint.
The constraint is a equals b

470
00:39:04,000 --> 00:39:08,000
cosine theta.
And, we want to differentiate

471
00:39:08,000 --> 00:39:10,000
that.
Well, differentiate the

472
00:39:10,000 --> 00:39:15,000
left-hand side.
You get da.

473
00:39:15,000 --> 00:39:18,000
And, differentiate the
right-hand side as a function of

474
00:39:18,000 --> 00:39:20,000
b and theta.
You should get,

475
00:39:20,000 --> 00:39:23,000
well, how many db's?
Well, that's the rate of change

476
00:39:23,000 --> 00:39:28,000
with respect to b.
That's cosine theta db minus b

477
00:39:28,000 --> 00:39:35,000
sine theta d theta.
That's a product rule applied

478
00:39:35,000 --> 00:39:47,000
to b times cosine theta.
So -- Well, now,

479
00:39:47,000 --> 00:39:51,000
if we have a constraint that's
relating da, db,

480
00:39:51,000 --> 00:39:54,000
and d theta,
OK, so that's actually what we

481
00:39:54,000 --> 00:39:56,000
did, right,
that's the same sort of thing

482
00:39:56,000 --> 00:39:59,000
as what we did at the beginning
when we related dx,

483
00:39:59,000 --> 00:40:02,000
dy, and dz.
That's really the same thing,

484
00:40:02,000 --> 00:40:05,000
except now are variables are a,
b, and theta.

485
00:40:05,000 --> 00:40:07,000
Now, we know that also we are
keeping a fixed.

486
00:40:07,000 --> 00:40:10,000
So actually,
we set this equal to zero.

487
00:40:10,000 --> 00:40:18,000
So, we have zero equals da
equals cosine theta db minus b

488
00:40:18,000 --> 00:40:23,000
sine theta d theta.
That means that actually we

489
00:40:23,000 --> 00:40:33,000
know how to solve for db.
OK, so cosine theta db equals b

490
00:40:33,000 --> 00:40:45,000
sine theta d theta or db is b
tangent theta d theta.

491
00:40:45,000 --> 00:40:47,000
OK, so in fact,
what we found,

492
00:40:47,000 --> 00:40:50,000
if you want,
is the rate of change of b with

493
00:40:50,000 --> 00:40:53,000
respect to theta.
Why do we care?

494
00:40:53,000 --> 00:40:59,000
Well, we care because let's
look, now, at dA,

495
00:40:59,000 --> 00:41:03,000
the function that we want to
look at.

496
00:41:03,000 --> 00:41:12,000
OK, so the function is A equals
one half ab sine theta.

497
00:41:12,000 --> 00:41:15,000
Well, then, dA,
so we had to use the product

498
00:41:15,000 --> 00:41:18,000
rule carefully,
or we use the partials.

499
00:41:18,000 --> 00:41:21,000
So, the coefficient of d little
a will be partial with respect

500
00:41:21,000 --> 00:41:26,000
to little a.
That's one half b sine theta da

501
00:41:26,000 --> 00:41:36,000
plus coefficient of db will be
one half a sine theta db plus

502
00:41:36,000 --> 00:41:45,000
coefficient of d theta will be
one half ab cosine theta d

503
00:41:45,000 --> 00:41:48,000
theta.
But now, what do I do with that?

504
00:41:48,000 --> 00:41:52,000
Well, first I said a is
constant.

505
00:41:52,000 --> 00:41:56,000
So, da is zero.
Second, well,

506
00:41:56,000 --> 00:41:59,000
actually we don't like b at
all, right?

507
00:41:59,000 --> 00:42:03,000
We want to view a as a function
of theta.

508
00:42:03,000 --> 00:42:13,000
So, well, maybe we actually
want to use this formula for db

509
00:42:13,000 --> 00:42:18,000
that we found in here.
OK, and then we'll be left only

510
00:42:18,000 --> 00:42:20,000
with d thetas,
which is what we want.

511
00:42:56,000 --> 00:43:06,000
So, if we plug this one into
that one, we get da equals one

512
00:43:06,000 --> 00:43:16,000
half a sine theta times b
tangent theta d theta plus one

513
00:43:16,000 --> 00:43:26,000
half ab cosine theta d theta.
And, if we collect these things

514
00:43:26,000 --> 00:43:35,000
together, we get one half of ab
times sine theta times tangent

515
00:43:35,000 --> 00:43:41,000
theta plus cosine theta d theta.
And, if you know your trig,

516
00:43:41,000 --> 00:43:44,000
but you'll see that this is
sine squared over cosine plus

517
00:43:44,000 --> 00:43:49,000
cosine squared over cosine.
That's the same as secan theta.

518
00:43:49,000 --> 00:43:54,000
So, now you have expressed da
as something times d theta.

519
00:43:54,000 --> 00:43:59,000
Well, that coefficient is the
rate of change of A with respect

520
00:43:59,000 --> 00:44:04,000
to theta with the understanding
that we are keeping a fixed,

521
00:44:04,000 --> 00:44:10,000
and letting b vary as a
dependent variable.

522
00:44:10,000 --> 00:44:11,000
Not enough space: sorry.

523
00:44:26,000 --> 00:44:29,000
OK, in case it's clearer for
you, let's think about it

524
00:44:29,000 --> 00:44:32,000
backwards.
So, we wanted to find how A

525
00:44:32,000 --> 00:44:35,000
changes.
To find how A changes,

526
00:44:35,000 --> 00:44:38,000
we write da.
But now, this tells us how A

527
00:44:38,000 --> 00:44:41,000
depends on little a,
little b, and theta.

528
00:44:41,000 --> 00:44:45,000
Well, we know actually we want
to keep little a constant.

529
00:44:45,000 --> 00:44:49,000
So, we set this to be zero.
Theta, well,

530
00:44:49,000 --> 00:44:52,000
we are very happy because we
want to express things in terms

531
00:44:52,000 --> 00:44:55,000
of theta.
Db we want to get rid of.

532
00:44:55,000 --> 00:45:00,000
How do we get rid of db?
Well, we do that by figuring

533
00:45:00,000 --> 00:45:05,000
out how b depends on theta when
a is fixed.

534
00:45:05,000 --> 00:45:08,000
And, we do that by
differentiating the constraint

535
00:45:08,000 --> 00:45:12,000
equation, and setting da equal
to zero.

536
00:45:12,000 --> 00:45:31,000
OK, so -- I guess to summarize
the method, we wrote dA in terms

537
00:45:31,000 --> 00:45:41,000
of da, db, d theta.
Then, we say that a is constant

538
00:45:41,000 --> 00:45:50,000
means we set da equals zero.
And, the third thing is that

539
00:45:50,000 --> 00:45:57,000
because, well,
we differentiate the

540
00:45:57,000 --> 00:46:06,000
constraint.
And, we can solve for db in

541
00:46:06,000 --> 00:46:19,000
terms of d theta.
And then, we plug into dA,

542
00:46:19,000 --> 00:46:32,000
and we get the answer.
OK, oops.

543
00:46:32,000 --> 00:46:38,000
So, here's another method to do
the same thing differently is to

544
00:46:38,000 --> 00:46:43,000
use the chain rule.
So, we can use the chain rule

545
00:46:43,000 --> 00:46:45,000
with dependent variables,
OK?

546
00:46:45,000 --> 00:46:48,000
So, what does the chain rule
tell us?

547
00:46:48,000 --> 00:46:54,000
The chain rule tells us,
so we will want to

548
00:46:54,000 --> 00:47:02,000
differentiate -- -- the formula
for a with respect to theta

549
00:47:02,000 --> 00:47:06,000
holding a constant.
So, I claim,

550
00:47:06,000 --> 00:47:10,000
well, what does the chain rule
tell us?

551
00:47:10,000 --> 00:47:14,000
It tells us that,
well, when we change things,

552
00:47:14,000 --> 00:47:19,000
a changes because of the
changes in the variables.

553
00:47:19,000 --> 00:47:24,000
So, part of it is that A
depends on theta and theta

554
00:47:24,000 --> 00:47:28,000
changes.
How fast does theta change?

555
00:47:28,000 --> 00:47:31,000
Well, you could call that the
rate of change of theta with

556
00:47:31,000 --> 00:47:33,000
respect to theta with a
constant.

557
00:47:33,000 --> 00:47:35,000
But of course,
how fast does theta depend to

558
00:47:35,000 --> 00:47:38,000
itself?
The answer is one.

559
00:47:38,000 --> 00:47:44,000
So, that's pretty easy.
Plus, then we have the partial

560
00:47:44,000 --> 00:47:49,000
derivative, formal partial
derivative, of A with respect to

561
00:47:49,000 --> 00:47:55,000
little a times the rate of
change of a in our situation.

562
00:47:55,000 --> 00:47:58,000
Well, how does little a change
if a is constant?

563
00:47:58,000 --> 00:48:08,000
Well, it doesn't change.
And then, there is Ab,

564
00:48:08,000 --> 00:48:14,000
the formal partial derivative
times, sorry,

565
00:48:14,000 --> 00:48:20,000
the rate of change of b.
OK, and how do we find this one?

566
00:48:20,000 --> 00:48:27,000
Well, here we have to use the
constraint.

567
00:48:27,000 --> 00:48:30,000
OK, and we can find this one
from the constraint as we've

568
00:48:30,000 --> 00:48:32,000
seen at the beginning either by
differentiating the constraint,

569
00:48:32,000 --> 00:48:36,000
or by using the chain rule on
the constraint.

570
00:48:36,000 --> 00:48:39,000
So, of course the calculations
are exactly the same.

571
00:48:39,000 --> 00:48:44,000
See, this is the same formula
as the one over there,

572
00:48:44,000 --> 00:48:48,000
just dividing everything by
partial theta and with

573
00:48:48,000 --> 00:48:54,000
subscripts little a.
But, if it's easier to think

574
00:48:54,000 --> 00:48:59,000
about it this way,
then that's also valid.

575
00:48:59,000 --> 00:49:03,000
OK, so tomorrow we are going to
review for the test,

576
00:49:03,000 --> 00:49:06,000
so I'm going to tell you a bit
more about this also as we go

577
00:49:06,000 --> 00:49:09,000
over one practice problem on
that.