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OK, so we're going to continue
looking at what happens when we
8
00:00:35 --> 00:00:45
have non-independent variables.
So, I'm afraid we don't take
9
00:00:45 --> 00:00:50
deliveries during class time,
sorry.
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00:00:50 --> 00:01:00
Please take a seat, thanks.
[LAUGHTER]
11
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[APPLAUSE]
OK, so Jason,
12
00:01:05 --> 00:01:17
you please claim your package
at the end of lecture.
13
00:01:17 --> 00:01:19
OK,
so last time we saw how to use
14
00:01:19 --> 00:01:23
Lagrange multipliers to find the
minimum or maximum of a function
15
00:01:23 --> 00:01:27
of several variables when the
variables are not independent.
16
00:01:27 --> 00:01:29
And, today we're going to try
to figure out more about
17
00:01:29 --> 00:01:33
relations between the variables,
and how to handle functions
18
00:01:33 --> 00:01:36
that depend on several variables
when they're related.
19
00:01:36 --> 00:01:40
So, just to give you an
example,
20
00:01:40 --> 00:01:44
in physics, very often,
you have functions that depend
21
00:01:44 --> 00:01:49
on pressure, volume,
and temperature where pressure,
22
00:01:49 --> 00:01:52
volume,
and temperature are actually
23
00:01:52 --> 00:01:55
not independent.
But they are related,
24
00:01:55 --> 00:01:58
say, by PV=nRT.
So, of course,
25
00:01:58 --> 00:02:01
then you can substitute and
expressed a function in terms of
26
00:02:01 --> 00:02:04
two of them only,
but very often it's convenient
27
00:02:04 --> 00:02:06
to keep all three.
But then we have to figure out,
28
00:02:06 --> 00:02:11
what are the rates of change
with respect to t,
29
00:02:11 --> 00:02:14
with respect to each other,
the rate of change of f with
30
00:02:14 --> 00:02:16
respect to these variables,
and so on.
31
00:02:16 --> 00:02:21
So, we have to figure out what
we mean by partial derivatives
32
00:02:21 --> 00:02:24
again.
So,
33
00:02:24 --> 00:02:31
OK, more generally,
let's say just for the sake of
34
00:02:31 --> 00:02:33
notation,
I'm going to think of a
35
00:02:33 --> 00:02:35
function of three variables,
x, y, z,
36
00:02:35 --> 00:02:39
where the variables are related
by some equation,
37
00:02:39 --> 00:02:44
but I will put in the form g of
x, y, z equals some constant.
38
00:02:44 --> 00:02:48
OK, so that's the same kind of
setup as we had last time,
39
00:02:48 --> 00:02:52
except now we are not just
looking for minima and maxima.
40
00:02:52 --> 00:02:59
We are trying to understand
partial derivatives.
41
00:02:59 --> 00:03:06
So, the first observation is
that if x, y,
42
00:03:06 --> 00:03:09
and z are related,
then that means,
43
00:03:09 --> 00:03:11
in principle,
we could solve for one of them,
44
00:03:11 --> 00:03:15
and express it as a function of
the two others.
45
00:03:15 --> 00:03:19
So, in particular,
can we understand even without
46
00:03:19 --> 00:03:21
solving?
Maybe we can not solve.
47
00:03:21 --> 00:03:27
Can we understand how the
variables are related to each
48
00:03:27 --> 00:03:29
other?
So, for example,
49
00:03:29 --> 00:03:33
z, you can think of z as a
function of x and y.
50
00:03:33 --> 00:03:40
So, we can ask ourselves,
what are the rates of change of
51
00:03:40 --> 00:03:44
z with respect to x,
keeping y constant,
52
00:03:44 --> 00:03:49
or with respect to y keeping x
constant?
53
00:03:49 --> 00:03:51
And, of course,
if we can solve,
54
00:03:51 --> 00:03:53
that we know the formula for
this.
55
00:03:53 --> 00:03:55
And then we can compute these
guys.
56
00:03:55 --> 00:04:03
But, what if we can't solve?
So, how do we find these things
57
00:04:03 --> 00:04:11
without solving?
Well, so let's do an example.
58
00:04:11 --> 00:04:19
Let's say that my relation is
x^2 yz z^3=8.
59
00:04:19 --> 00:04:24
And, let's say that I'm looking
near the point (x,
60
00:04:24 --> 00:04:27
y, z) equals (2,3,
1).
61
00:04:27 --> 00:04:33
So, let me check 2^2 plus three
times one plus 1^3 is indeed
62
00:04:33 --> 00:04:34
eight.
OK, but now,
63
00:04:34 --> 00:04:38
if I change x and y a little
bit, how does z change?
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00:04:38 --> 00:04:41
Well, of course I could solve
for z in here.
65
00:04:41 --> 00:04:43
It's a cubic equation.
There is actually a formula.
66
00:04:43 --> 00:04:45
But that formula is quite
complicated.
67
00:04:45 --> 00:04:47
We actually don't want to do
that.
68
00:04:47 --> 00:04:58
There's an easier way.
So, how can we do it?
69
00:04:58 --> 00:05:07
Well, let's look at the
differential -- -- of this
70
00:05:07 --> 00:05:15
constraint quantity.
OK, so if we called this g,
71
00:05:15 --> 00:05:21
let's look at dg.
So, what's the differential of
72
00:05:21 --> 00:05:26
this?
So, the differential of x^2 is
73
00:05:26 --> 00:05:32
2x dx plus, I think there's a
zdy.
74
00:05:32 --> 00:05:38
There's a ydz,
and there's also a 3z^2 dz.
75
00:05:38 --> 00:05:42
OK, you can get this either by
implicit differentiation and the
76
00:05:42 --> 00:05:45
product rule,
or you could get this just by
77
00:05:45 --> 00:05:46
putting here,
here,
78
00:05:46 --> 00:05:51
and here the partial
derivatives of this with respect
79
00:05:51 --> 00:05:56
to x, y, and z.
OK, any questions about how I
80
00:05:56 --> 00:05:58
got this?
No?
81
00:05:58 --> 00:06:03
OK.
So, now, what do I do with this?
82
00:06:03 --> 00:06:07
Well, this represents,
somehow, variations of g.
83
00:06:07 --> 00:06:12
But, well, I've set this thing
equal to eight.
84
00:06:12 --> 00:06:16
And, eight is a constant.
So, it doesn't change.
85
00:06:16 --> 00:06:26
So, in fact,
well, we can set this to zero
86
00:06:26 --> 00:06:35
because, well,
they call this g.
87
00:06:35 --> 00:06:39
Then, g equals eight is
constant.
88
00:06:39 --> 00:06:43
That means we set dg equal to
zero.
89
00:06:43 --> 00:06:53
OK, so, now let's just plug in
some values at this point.
90
00:06:53 --> 00:06:58
That tells us,
well, so if x equals two,
91
00:06:58 --> 00:07:07
that's 4dx plus z is one.
So, dy plus y 3z^2 should be
92
00:07:07 --> 00:07:13
6dz equals zero.
And now, this equation,
93
00:07:13 --> 00:07:17
here, tells us about a relation
between the changes in x,
94
00:07:17 --> 00:07:21
y, and z near that point.
It tells us how you change x
95
00:07:21 --> 00:07:25
and y, well, how z will change.
Or, it tells you actually
96
00:07:25 --> 00:07:28
anything you might want to know
about the relations between
97
00:07:28 --> 00:07:30
these variables so,
for example,
98
00:07:30 --> 00:07:34
you can move dz to that side,
and then express dz in terms of
99
00:07:34 --> 00:07:37
dx and dy.
Or, you can move dy to that
100
00:07:37 --> 00:07:41
side and express dy in terms of
dx and dz, and so on.
101
00:07:41 --> 00:07:46
It tells you at the level of
the derivatives how each of the
102
00:07:46 --> 00:07:49
variables depends on the two
others.
103
00:07:49 --> 00:07:57
OK, so, just to clarify this:
if we want to view z as a
104
00:07:57 --> 00:08:03
function of x and y,
then what we will do is we will
105
00:08:03 --> 00:08:06
just move the dz's to the other
side,
106
00:08:06 --> 00:08:15
and it will tell us dz equals
minus one over six times 4dx
107
00:08:15 --> 00:08:20
plus dy.
And, so that should tell you
108
00:08:20 --> 00:08:26
that partial z over partial x is
minus four over six.
109
00:08:26 --> 00:08:35
Well, that's minus two thirds,
and partial z over partial y is
110
00:08:35 --> 00:08:42
going to be minus one sixth.
OK, another way to think about
111
00:08:42 --> 00:08:47
this: when we compute partial z
over partial x,
112
00:08:47 --> 00:08:51
that means that actually we
keep y constant.
113
00:08:51 --> 00:08:55
OK, let me actually add some
subtitles here.
114
00:08:55 --> 00:09:00
So, here that means we keep y
constant.
115
00:09:00 --> 00:09:05
And so, if we keep y constant,
another way to think about it
116
00:09:05 --> 00:09:10
is we set dy to zero.
We set dy equals zero.
117
00:09:10 --> 00:09:14
So if we do that,
we get dx equals negative four
118
00:09:14 --> 00:09:17
sixths dx.
That tells us the rate of
119
00:09:17 --> 00:09:24
change of z with respect to x.
Here, we set x constant.
120
00:09:24 --> 00:09:29
So, that means we set dx equal
to zero.
121
00:09:29 --> 00:09:32
And, if we set dx equal to
zero, then we have dz equals
122
00:09:32 --> 00:09:36
negative one sixth of dy.
That tells us the rate of
123
00:09:36 --> 00:09:47
change of z with respect to y.
OK, any questions about that?
124
00:09:47 --> 00:09:57
No?
What, yes?
125
00:09:57 --> 00:09:59
Yes, OK, let me explain that
again.
126
00:09:59 --> 00:10:03
So we found an expression for
dz in terms of dx and dy.
127
00:10:03 --> 00:10:07
That means that this thing,
the differential,
128
00:10:07 --> 00:10:11
is the total differential of z
viewed as a function of x and y.
129
00:10:11 --> 00:10:16
OK, and so the coefficients of
dx and dy are the partials.
130
00:10:16 --> 00:10:20
Or, another way to think about
it, if you want to know partial
131
00:10:20 --> 00:10:22
z partial x, it means you set y
to be constant.
132
00:10:22 --> 00:10:27
Setting y to be constant means
that you will put zero in the
133
00:10:27 --> 00:10:30
place of dy.
So, you will be left with dz
134
00:10:30 --> 00:10:35
equals minus four sixths dx.
And, that will give you the
135
00:10:35 --> 00:10:41
rate of change of z with respect
to x when you keep y constant,
136
00:10:41 --> 00:10:46
OK?
So, there are various ways to
137
00:10:46 --> 00:10:53
think about this,
but hopefully it makes sense.
138
00:10:53 --> 00:11:03
OK, so how do we think about
this in general?
139
00:11:03 --> 00:11:15
Well, if we know that g of x,
y, z equals a constant,
140
00:11:15 --> 00:11:27
then dg, which is gxdx gydy
gzdz should be set equal to
141
00:11:27 --> 00:11:32
zero.
OK, and now we can solve for
142
00:11:32 --> 00:11:37
whichever variable we want to
express in terms of the others.
143
00:11:37 --> 00:11:47
So, for example,
if we care about z as a
144
00:11:47 --> 00:12:02
function of x and y -- -- we'll
get that dz is negative gx over
145
00:12:02 --> 00:12:17
gz dx minus gy over gz dy.
And, so if we want partial z
146
00:12:17 --> 00:12:23
over partial x,
well, so one way is just to say
147
00:12:23 --> 00:12:26
that's going to be the
coefficient of dx in here,
148
00:12:26 --> 00:12:29
or just to write down the other
way.
149
00:12:29 --> 00:12:34
We are setting y equals
constant.
150
00:12:34 --> 00:12:39
So, that means we set dy equal
to zero.
151
00:12:39 --> 00:12:48
And then, we get dz equals
negative gx over gz dx.
152
00:12:48 --> 00:12:56
So, that means partial z over
partial x is minus gx over gz.
153
00:12:56 --> 00:12:59
And, see,
that's a very counterintuitive
154
00:12:59 --> 00:13:02
formula because you have this
minus sign that you somehow
155
00:13:02 --> 00:13:06
probably couldn't have seen come
if you hadn't actually derived
156
00:13:06 --> 00:13:11
things this way.
I mean, it's pretty surprising
157
00:13:11 --> 00:13:17
to see that minus sign come out
of nowhere the first time you
158
00:13:17 --> 00:13:22
see it.
OK, so now we know how to find
159
00:13:22 --> 00:13:26
the rate of change of
constrained variables with
160
00:13:26 --> 00:13:31
respect to each other.
You can apply the same to find,
161
00:13:31 --> 00:13:35
if you want partial x,
partial y, or any of them,
162
00:13:35 --> 00:13:41
you can do it.
Any questions so far?
163
00:13:41 --> 00:13:48
No?
OK, so, before we proceed
164
00:13:48 --> 00:13:55
further, I should probably
expose some problem with the
165
00:13:55 --> 00:14:03
notations that we have so far.
So, let me try to get you a bit
166
00:14:03 --> 00:14:07
confused, OK?
So, let's take a very simple
167
00:14:07 --> 00:14:10
example.
Let's say I have a function,
168
00:14:10 --> 00:14:15
f of x, y equals x y.
OK, so far it doesn't sound
169
00:14:15 --> 00:14:20
very confusing.
And then, I can write partial f
170
00:14:20 --> 00:14:24
over partial x.
And, I think you all know how
171
00:14:24 --> 00:14:28
to compute it.
It's going to be just one.
172
00:14:28 --> 00:14:34
OK, so far we are pretty happy.
Now let's do a change of
173
00:14:34 --> 00:14:44
variables.
Let's set x=u and y=u v.
174
00:14:44 --> 00:14:46
It's not very complicated
change of variables.
175
00:14:46 --> 00:14:54
But let's do it.
Then, f in terms of u and v,
176
00:14:54 --> 00:15:02
well, so f, remember f was x y
becomes u plus u plus v.
177
00:15:02 --> 00:15:13
That's twice u plus v.
What's partial f over partial u?
178
00:15:13 --> 00:15:18
It's two.
So, x and u are the same thing.
179
00:15:18 --> 00:15:21
Partial f over partial x,
and partial f over partial u,
180
00:15:21 --> 00:15:24
well, unless you believe that
one equals two,
181
00:15:24 --> 00:15:26
they are really not the same
thing, OK?
182
00:15:26 --> 00:15:36
So, that's an interesting,
slightly strange phenomenon.
183
00:15:36 --> 00:15:46
x equals u, but partial f
partial x is not the same as
184
00:15:46 --> 00:15:52
partial f partial u.
So, how do we get rid of this
185
00:15:52 --> 00:15:55
contradiction?
Well, we have to think a bit
186
00:15:55 --> 00:15:59
more about what these notations
mean, OK?
187
00:15:59 --> 00:16:03
So, when we write partial f
over partial x,
188
00:16:03 --> 00:16:08
it means that we are varying x,
keeping y constant.
189
00:16:08 --> 00:16:11
When we write partial f over
partial u, it means we are
190
00:16:11 --> 00:16:15
varying u, keeping v constant.
So, varying u or varying x is
191
00:16:15 --> 00:16:17
the same thing.
But, keeping v constant,
192
00:16:17 --> 00:16:20
or keeping y constant are not
the same thing.
193
00:16:20 --> 00:16:23
If I keep y constant,
then when I change x,
194
00:16:23 --> 00:16:27
so when I change u,
then v will also have to change
195
00:16:27 --> 00:16:29
so that their sum stays the
same.
196
00:16:29 --> 00:16:32
Or, if you prefer the other way
around, when I do this one I
197
00:16:32 --> 00:16:35
keep v constant.
If I keep v constant and I
198
00:16:35 --> 00:16:39
change u, then y will change.
It won't be constant.
199
00:16:39 --> 00:16:43
So, that means,
well, life looked quite nice
200
00:16:43 --> 00:16:49
and easy with these notations.
But, what's dangerous about
201
00:16:49 --> 00:16:55
them is they are not making
explicit what it is exactly that
202
00:16:55 --> 00:17:01
we are keeping constant.
OK, so just to write things,
203
00:17:01 --> 00:17:08
so here we change u and x that
are the same thing.
204
00:17:08 --> 00:17:14
But we keep y constant,
while here we change u,
205
00:17:14 --> 00:17:19
which is still the same thing
as x.
206
00:17:19 --> 00:17:26
But, what we keep constant is
v, or in terms of x and y,
207
00:17:26 --> 00:17:33
that's y minus x constant.
And, that's why they are not
208
00:17:33 --> 00:17:36
the same.
So, whenever there's any risk
209
00:17:36 --> 00:17:39
of confusion,
OK, so not in the cases that we
210
00:17:39 --> 00:17:42
had before because what we've
done until now,
211
00:17:42 --> 00:17:46
we didn't really have a problem.
But, in a situation like this,
212
00:17:46 --> 00:17:50
to clarify things,
we'll actually say explicitly
213
00:17:50 --> 00:17:53
what it is that we want to keep
constant.
214
00:17:53 --> 00:18:04
215
00:18:04 --> 00:18:07
OK, so what's going to be our
new notation?
216
00:18:07 --> 00:18:14
Well, so it's not particularly
pleasant because it uses,
217
00:18:14 --> 00:18:16
now, a subscript not to
indicate what you are
218
00:18:16 --> 00:18:18
differentiating,
but rather what you were
219
00:18:18 --> 00:18:22
holding constant.
So, that's quite a conflict of
220
00:18:22 --> 00:18:25
notation with what we had
before.
221
00:18:25 --> 00:18:32
I think I can safely blame it
on physicists or chemists.
222
00:18:32 --> 00:18:43
OK, so this one means we keep y
constant, and partial f over
223
00:18:43 --> 00:18:51
partial u with v held constant,
similarly.
224
00:18:51 --> 00:18:54
OK, so now what happens is we
no longer have any
225
00:18:54 --> 00:18:59
contradiction.
We have partial f over partial
226
00:18:59 --> 00:19:06
x with y constant is different
from partial f over partial x
227
00:19:06 --> 00:19:12
with v constant,
which is the same as partial f
228
00:19:12 --> 00:19:18
over partial u with v constant.
OK, so this guy is one.
229
00:19:18 --> 00:19:28
And these guys are two.
So, now we can safely use the
230
00:19:28 --> 00:19:33
fact that x equals u if we are
keeping track of what is
231
00:19:33 --> 00:19:36
actually held constant,
OK?
232
00:19:36 --> 00:19:39
So now, that's going to be
particularly important when we
233
00:19:39 --> 00:19:41
have variables that are related
because,
234
00:19:41 --> 00:19:45
let's say now that I have a
function that depends on x,
235
00:19:45 --> 00:19:48
y, and z.
But, x, y, and z are related.
236
00:19:48 --> 00:19:54
Then, it means that I look at,
say, x and y as my independent
237
00:19:54 --> 00:19:59
variables, and z as a function
of x and y.
238
00:19:59 --> 00:20:01
Then, it means that when I do
partials, say,
239
00:20:01 --> 00:20:04
with respect to x,
I will hold y constant.
240
00:20:04 --> 00:20:08
But, I will let z vary as a
function of x and y.
241
00:20:08 --> 00:20:10
Or, I could do it the other way
around.
242
00:20:10 --> 00:20:12
I could vary x,
keep z constant,
243
00:20:12 --> 00:20:15
and let y be a function of x
and z.
244
00:20:15 --> 00:20:24
And so, I will need to use this
kind of notation to indicate
245
00:20:24 --> 00:20:34
which one I mean.
OK, any questions?
246
00:20:34 --> 00:20:39
No?
All right, so let's try to do
247
00:20:39 --> 00:20:42
an example where we have a
function that depends on
248
00:20:42 --> 00:20:46
variables that are related.
OK, so I don't want to do one
249
00:20:46 --> 00:20:50
with PV=nRT because probably,
I mean, if you've seen it,
250
00:20:50 --> 00:20:53
then you've seen too much of
it.
251
00:20:53 --> 00:20:58
And, if you haven't seen it,
then maybe it's not the best
252
00:20:58 --> 00:21:02
example.
So, let's do a geometric
253
00:21:02 --> 00:21:08
example.
So, let's look at the area of
254
00:21:08 --> 00:21:14
the triangle.
So, let's say I have a
255
00:21:14 --> 00:21:21
triangle, and my variables will
be the sides a and b.
256
00:21:21 --> 00:21:26
And the angle here, theta.
OK, so what's the area of this
257
00:21:26 --> 00:21:29
triangle?
Well, its base times height
258
00:21:29 --> 00:21:34
over two.
So, it's one half of the base
259
00:21:34 --> 00:21:39
is a, and the height is b sine
theta.
260
00:21:39 --> 00:21:45
OK, so that's a function of a,
b, and theta.
261
00:21:45 --> 00:21:47
Now, let's say,
actually, there is a relation
262
00:21:47 --> 00:21:49
between a, b,
and theta that I didn't tell
263
00:21:49 --> 00:21:52
you about,
namely, actually,
264
00:21:52 --> 00:21:58
I want to assume that it's a
right triangle,
265
00:21:58 --> 00:22:05
OK?
So, let's now assume it's a
266
00:22:05 --> 00:22:16
right triangle with,
let's say, the hypotenuse is b.
267
00:22:16 --> 00:22:19
So, we have the right angle
here, actually.
268
00:22:19 --> 00:22:23
So, a is here. b is here.
Theta is here.
269
00:22:23 --> 00:22:28
So, saying it's a right
triangle is the same thing as
270
00:22:28 --> 00:22:31
saying that b equals sine theta,
OK?
271
00:22:31 --> 00:22:37
So that's our constraint.
That's the relation between a,
272
00:22:37 --> 00:22:46
b, and theta.
And, this is a function of a,
273
00:22:46 --> 00:22:53
b, and theta.
And, let's say that we want to
274
00:22:53 --> 00:22:57
understand how the area depends
on theta.
275
00:22:57 --> 00:23:00
OK, what's the rate of change
of the area of this triangle
276
00:23:00 --> 00:23:06
with respect to theta?
So, I claim there's various
277
00:23:06 --> 00:23:09
answers.
I can think of at least three
278
00:23:09 --> 00:23:10
possible answers.
279
00:23:10 --> 00:23:44
280
00:23:44 --> 00:23:52
So, what can we possibly mean
by the rate of change of A with
281
00:23:52 --> 00:23:57
respect to theta?
So, these are all things that
282
00:23:57 --> 00:23:59
we might want to call partial A
partial theta.
283
00:23:59 --> 00:24:03
But of course,
we'll have to actually use
284
00:24:03 --> 00:24:06
different notations to
distinguish them.
285
00:24:06 --> 00:24:11
So, the first way that we
actually already know about is
286
00:24:11 --> 00:24:17
if we just forget about the fact
that the variables are related,
287
00:24:17 --> 00:24:20
OK?
So, if we just think of little
288
00:24:20 --> 00:24:23
a, b,
and theta as independent
289
00:24:23 --> 00:24:25
variables,
and we just change theta,
290
00:24:25 --> 00:24:48
keeping a and b constant -- So,
that's exactly what we meant by
291
00:24:48 --> 00:24:51
partial A, partial theta,
right?
292
00:24:51 --> 00:24:59
I'm not putting any constraints.
So, just to use some new
293
00:24:59 --> 00:25:03
notation, that would be the rate
of change of A with respect to
294
00:25:03 --> 00:25:07
theta, keeping a and b fixed at
the same time.
295
00:25:07 --> 00:25:11
Of course, if we are keeping a
and b fixed, and we are changing
296
00:25:11 --> 00:25:14
theta, it means we completely
ignore this property of being a
297
00:25:14 --> 00:25:16
right triangle.
So, in fact,
298
00:25:16 --> 00:25:20
it corresponds to changing the
area by changing the angle,
299
00:25:20 --> 00:25:23
keeping these lengths fixed.
And, of course,
300
00:25:23 --> 00:25:27
we lose the right angle.
When we rotate this side here,
301
00:25:27 --> 00:25:32
but the angle doesn't stay at a
right angle.
302
00:25:32 --> 00:25:35
And that one,
we know how to compute,
303
00:25:35 --> 00:25:40
right, because it's the one
we've been computing all along.
304
00:25:40 --> 00:25:44
So, that means we keep a and b
fixed.
305
00:25:44 --> 00:25:51
And then, so let's see,
what's the derivatives of A
306
00:25:51 --> 00:26:02
with respect to theta?
It's one half ab cosine theta.
307
00:26:02 --> 00:26:11
OK, now that one we know.
Any questions?
308
00:26:11 --> 00:26:14
No?
OK, the two other guys will be
309
00:26:14 --> 00:26:18
more interesting.
So far, I'm not really doing
310
00:26:18 --> 00:26:23
anything with my constraint.
Let's say that actually I do
311
00:26:23 --> 00:26:27
want to keep the right angle.
Then, when I change theta,
312
00:26:27 --> 00:26:31
there's two options.
One is I keep a constant,
313
00:26:31 --> 00:26:35
and then of course b will have
to change because if this width
314
00:26:35 --> 00:26:38
stays the same,
then when I change theta,
315
00:26:38 --> 00:26:41
the height increases,
and then this side length
316
00:26:41 --> 00:26:45
increases.
The other option is to change
317
00:26:45 --> 00:26:47
the angle, keeping b constant.
So, actually,
318
00:26:47 --> 00:26:49
this side stays the same
length.
319
00:26:49 --> 00:26:53
But then, a has to become a bit
shorter.
320
00:26:53 --> 00:26:56
And, of course,
the area will change in
321
00:26:56 --> 00:26:59
different ways depending on what
I do.
322
00:26:59 --> 00:27:05
So, that's why I said we have
three different answers.
323
00:27:05 --> 00:27:10
So, the next one is keep,
I forgot which one I said
324
00:27:10 --> 00:27:17
first.
Let's say keep a constant.
325
00:27:17 --> 00:27:26
And, that means that b will
change.
326
00:27:26 --> 00:27:30
b is going to be some function
of a and theta.
327
00:27:30 --> 00:27:34
Well, in fact here,
we know what the function is
328
00:27:34 --> 00:27:37
because we can solve the
constraint, namely,
329
00:27:37 --> 00:27:45
b is a over cosine theta.
But we don't actually need to
330
00:27:45 --> 00:27:55
know that so that the triangle,
so that the right angle,
331
00:27:55 --> 00:28:05
so that we keep a right angle.
And, so the name we will have
332
00:28:05 --> 00:28:11
for this is partial a over
partial theta with a held
333
00:28:11 --> 00:28:14
constant, OK?
And, the fact that I'm not
334
00:28:14 --> 00:28:17
putting b in my subscript there
means that actually b will be a
335
00:28:17 --> 00:28:20
dependent variable.
It changes in whatever way it
336
00:28:20 --> 00:28:26
has to change so that when theta
changes, a stays the same while
337
00:28:26 --> 00:28:29
b changes so that we keep a
right triangle.
338
00:28:29 --> 00:28:38
339
00:28:38 --> 00:28:46
And, the third guy is the one
where we actually keep b
340
00:28:46 --> 00:28:51
constant,
and now a,
341
00:28:51 --> 00:28:54
we think a as a function of b
and theta,
342
00:28:54 --> 00:28:58
and it changes so that we keep
the right angle.
343
00:28:58 --> 00:29:01
So actually as a function of b
and theta, it's given over
344
00:29:01 --> 00:29:06
there.
A equals b cosine theta.
345
00:29:06 --> 00:29:13
And so, this guy is called
partial a over partial theta
346
00:29:13 --> 00:29:19
with b held constant.
OK, so we've just defined them.
347
00:29:19 --> 00:29:21
We don't know yet how to
compute these things.
348
00:29:21 --> 00:29:22
That's what we're going to do
now.
349
00:29:22 --> 00:29:25
That is the definition,
and what these things mean.
350
00:29:25 --> 00:29:33
Is that clear to everyone?
Yes, OK.
351
00:29:33 --> 00:29:41
Yes?
OK, so the second answer,
352
00:29:41 --> 00:29:46
again, so one way to ask
ourselves,
353
00:29:46 --> 00:29:48
how does the area depend on
theta,
354
00:29:48 --> 00:29:53
is to say, well,
actually look at the area of
355
00:29:53 --> 00:29:59
the right triangle as a function
of a and theta only by solving
356
00:29:59 --> 00:30:03
for b.
And then, we'll change theta,
357
00:30:03 --> 00:30:06
keep a constant,
and ask, how does the area
358
00:30:06 --> 00:30:08
change?
So, when we do that,
359
00:30:08 --> 00:30:11
when we change theta and keep a
the same,
360
00:30:11 --> 00:30:14
then b has to change so that it
stays a right triangle,
361
00:30:14 --> 00:30:18
right, so that this relation
still holds.
362
00:30:18 --> 00:30:22
That requires us to change b.
So, when we write partial a
363
00:30:22 --> 00:30:26
over partial theta with a
constant, it means that,
364
00:30:26 --> 00:30:30
actually, b will be the
dependent variable.
365
00:30:30 --> 00:30:35
It depends on a and theta.
And so, the area depends on
366
00:30:35 --> 00:30:40
theta, not only because theta is
in the formula,
367
00:30:40 --> 00:30:46
but also because b changes,
and b is in the formula.
368
00:30:46 --> 00:30:53
Yes?
No, no, we don't keep theta
369
00:30:53 --> 00:30:54
constant.
We vary theta, right?
370
00:30:54 --> 00:30:58
The goal is to see how things
change when I change theta by a
371
00:30:58 --> 00:31:01
little bit.
OK, so if I change theta a
372
00:31:01 --> 00:31:04
little bit in this one,
if I change theta a little bit
373
00:31:04 --> 00:31:07
and I keep a the same,
then b has to change also in
374
00:31:07 --> 00:31:09
some way.
There's a right triangle.
375
00:31:09 --> 00:31:16
And then, because theta and b
change, that causes the area to
376
00:31:16 --> 00:31:18
change.
OK, so maybe I should
377
00:31:18 --> 00:31:23
re-explain that again.
So, theta changes.
378
00:31:23 --> 00:31:30
A is constant.
But, we have the constraint,
379
00:31:30 --> 00:31:37
a equals be plus sine theta.
That means that b changes.
380
00:31:37 --> 00:31:43
And then, the question is,
how does A change?
381
00:31:43 --> 00:31:46
Well, it will change in part
because theta changes,
382
00:31:46 --> 00:31:50
and in part because b changes.
But, we want to know how it
383
00:31:50 --> 00:31:54
depends on theta in this
situation.
384
00:31:54 --> 00:32:04
Yes?
Ah, that's a very good question.
385
00:32:04 --> 00:32:08
So, what about,
I don't keep a and b constant?
386
00:32:08 --> 00:32:10
Well, then there's too many
choices.
387
00:32:10 --> 00:32:13
So I have to decide actually
how I'm going to change things.
388
00:32:13 --> 00:32:17
See, if I just say I have this
relation, that means I have two
389
00:32:17 --> 00:32:20
independent variables left,
whichever two of the three I
390
00:32:20 --> 00:32:23
want.
But, I still have to specify
391
00:32:23 --> 00:32:27
two of them to say exactly which
triangle I mean.
392
00:32:27 --> 00:32:31
So, I cannot ask myself just
how will it change if I change
393
00:32:31 --> 00:32:34
theta and do random things with
a and b?
394
00:32:34 --> 00:32:36
It depends what I do with a and
b.
395
00:32:36 --> 00:32:40
Of course, I could choose to
change them simultaneously,
396
00:32:40 --> 00:32:45
but then I have to specify how
exactly I'm going to do that.
397
00:32:45 --> 00:32:49
Ah, yes, if you wanted to,
indeed, we could also change
398
00:32:49 --> 00:32:53
things in such a way that the
third side remains constant.
399
00:32:53 --> 00:32:55
And that would be,
yet, a different way to attack
400
00:32:55 --> 00:32:57
the problem.
I mean, we don't have good
401
00:32:57 --> 00:33:00
notation for this,
here, because we didn't give it
402
00:33:00 --> 00:33:01
a name.
But, yeah, I mean, we could.
403
00:33:01 --> 00:33:07
We could call this guy c,
and then we'd have a different
404
00:33:07 --> 00:33:11
formula, and so on.
So, I mean, I'm not looking at
405
00:33:11 --> 00:33:17
it for simplicity.
But, you could have many more.
406
00:33:17 --> 00:33:19
I mean, in general,
you will want,
407
00:33:19 --> 00:33:22
once you have a set of nice,
natural variables,
408
00:33:22 --> 00:33:25
you will want to look mostly at
situations where one of the
409
00:33:25 --> 00:33:29
variables changes.
Some of them are held fixed,
410
00:33:29 --> 00:33:33
and then some dependent
variable does whatever it must
411
00:33:33 --> 00:33:36
so that the constraint keeps
holding.
412
00:33:36 --> 00:33:39
OK, so let's try to compute one
of them.
413
00:33:39 --> 00:33:44
Let's say I decide that we will
compute this one.
414
00:33:44 --> 00:33:46
OK, let's see how we can
compute partial a,
415
00:33:46 --> 00:33:49
partial theta with a held
fixed.
416
00:33:49 --> 00:34:21
417
00:34:21 --> 00:34:27
[APPLAUSE]
OK, so let's try to compute
418
00:34:27 --> 00:34:34
partial A, partial theta with a
held constant.
419
00:34:34 --> 00:34:40
So, let's see three different
ways of doing that.
420
00:34:40 --> 00:34:45
So, let me start with method
zero.
421
00:34:45 --> 00:34:50
OK, it's not a real method.
That's why I'm not getting a
422
00:34:50 --> 00:34:54
positive number.
So, that one is just,
423
00:34:54 --> 00:34:58
we solve for b,
and we remove b from the
424
00:34:58 --> 00:35:01
formulas.
OK, so here it works well
425
00:35:01 --> 00:35:04
because we know how to solve for
b.
426
00:35:04 --> 00:35:07
But I'm not considering this to
be a real method because in
427
00:35:07 --> 00:35:08
general we don't know how to do
that.
428
00:35:08 --> 00:35:12
I mean, in the beginning I had
this relation that was an
429
00:35:12 --> 00:35:16
equation of degree three.
You don't really want to solve
430
00:35:16 --> 00:35:19
your equation for the dependent
variable usually.
431
00:35:19 --> 00:35:33
Here, we can.
So, solve for b and substitute.
432
00:35:33 --> 00:35:38
So, how do we do that?
Well, the constraint is a=b
433
00:35:38 --> 00:35:45
cosine theta.
That means b is a over cosine
434
00:35:45 --> 00:35:48
theta.
Some of you know that as a
435
00:35:48 --> 00:35:56
secan theta.
That's the same.
436
00:35:56 --> 00:36:04
And now, if we express the area
in terms of a and theta only,
437
00:36:04 --> 00:36:13
A is one half of ab cosine,
sorry, ab sine theta is now one
438
00:36:13 --> 00:36:20
half of a^2 sine theta over
cosine theta.
439
00:36:20 --> 00:36:29
Or, if you prefer,
one half of a^2 tangent theta.
440
00:36:29 --> 00:36:32
Well, now that it's only a
function of a and theta,
441
00:36:32 --> 00:36:35
I know what it means to take
the partial derivative with
442
00:36:35 --> 00:36:38
respect to theta,
keeping a constant.
443
00:36:38 --> 00:36:51
I know how to do it.
So, partial A over partial
444
00:36:51 --> 00:36:55
theta,
a held constant,
445
00:36:55 --> 00:36:59
well, if a is a constant,
then I get this one half a^2
446
00:36:59 --> 00:37:03
coming out times,
what's the derivative of
447
00:37:03 --> 00:37:09
tangent?
Secan squared, very good.
448
00:37:09 --> 00:37:12
If you're European and you've
never heard of secan,
449
00:37:12 --> 00:37:15
that's one over cosine.
And, if you know the derivative
450
00:37:15 --> 00:37:18
as one plus tangent squared,
that's the same thing.
451
00:37:18 --> 00:37:24
And, it's also correct.
OK, so, that's one way of doing
452
00:37:24 --> 00:37:26
it.
But, as I've already said,
453
00:37:26 --> 00:37:30
it doesn't get us very far if
we don't know how to solve for
454
00:37:30 --> 00:37:33
b.
We really used the fact that we
455
00:37:33 --> 00:37:36
could solve for b and get rid of
it.
456
00:37:36 --> 00:37:45
So, there's two systematic
methods, and let's say the basic
457
00:37:45 --> 00:37:53
rule is that you should give
both of them a chance.
458
00:37:53 --> 00:37:56
You should see which one you
prefer, and you should be able
459
00:37:56 --> 00:37:59
to use one or the other on the
exam.
460
00:37:59 --> 00:38:04
OK, most likely you'll actually
have a choice between one or the
461
00:38:04 --> 00:38:06
other.
It will be up to you to decide
462
00:38:06 --> 00:38:10
which one you want to use.
But, you cannot use solving in
463
00:38:10 --> 00:38:14
substitution.
That's not fair.
464
00:38:14 --> 00:38:25
OK, so the first one is to use
differentials.
465
00:38:25 --> 00:38:29
By the way, in the notes they
are called also method one and
466
00:38:29 --> 00:38:32
method two.
I'm not promising that I have
467
00:38:32 --> 00:38:32
the same one,
am I?
468
00:38:32 --> 00:38:35
I mean, I might have one and
two switched.
469
00:38:35 --> 00:38:39
It doesn't really matter.
So, how do we do things using
470
00:38:39 --> 00:38:43
differentials?
Well, first,
471
00:38:43 --> 00:38:52
we know that we want to keep a
fixed, and that means that we'll
472
00:38:52 --> 00:38:56
set da equal to zero,
OK?
473
00:38:56 --> 00:39:00
The second thing that we want
to do is we want to look at the
474
00:39:00 --> 00:39:04
constraint.
The constraint is a equals b
475
00:39:04 --> 00:39:08
cosine theta.
And, we want to differentiate
476
00:39:08 --> 00:39:10
that.
Well, differentiate the
477
00:39:10 --> 00:39:15
left-hand side.
You get da.
478
00:39:15 --> 00:39:18
And, differentiate the
right-hand side as a function of
479
00:39:18 --> 00:39:20
b and theta.
You should get,
480
00:39:20 --> 00:39:23
well, how many db's?
Well, that's the rate of change
481
00:39:23 --> 00:39:28
with respect to b.
That's cosine theta db minus b
482
00:39:28 --> 00:39:35
sine theta d theta.
That's a product rule applied
483
00:39:35 --> 00:39:47
to b times cosine theta.
So -- Well, now,
484
00:39:47 --> 00:39:51
if we have a constraint that's
relating da, db,
485
00:39:51 --> 00:39:54
and d theta,
OK, so that's actually what we
486
00:39:54 --> 00:39:56
did, right,
that's the same sort of thing
487
00:39:56 --> 00:39:59
as what we did at the beginning
when we related dx,
488
00:39:59 --> 00:40:02
dy, and dz.
That's really the same thing,
489
00:40:02 --> 00:40:05
except now are variables are a,
b, and theta.
490
00:40:05 --> 00:40:07
Now, we know that also we are
keeping a fixed.
491
00:40:07 --> 00:40:10
So actually,
we set this equal to zero.
492
00:40:10 --> 00:40:18
So, we have zero equals da
equals cosine theta db minus b
493
00:40:18 --> 00:40:23
sine theta d theta.
That means that actually we
494
00:40:23 --> 00:40:33
know how to solve for db.
OK, so cosine theta db equals b
495
00:40:33 --> 00:40:45
sine theta d theta or db is b
tangent theta d theta.
496
00:40:45 --> 00:40:47
OK, so in fact,
what we found,
497
00:40:47 --> 00:40:50
if you want,
is the rate of change of b with
498
00:40:50 --> 00:40:53
respect to theta.
Why do we care?
499
00:40:53 --> 00:40:59
Well, we care because let's
look, now, at dA,
500
00:40:59 --> 00:41:03
the function that we want to
look at.
501
00:41:03 --> 00:41:12
OK, so the function is A equals
one half ab sine theta.
502
00:41:12 --> 00:41:15
Well, then, dA,
so we had to use the product
503
00:41:15 --> 00:41:18
rule carefully,
or we use the partials.
504
00:41:18 --> 00:41:21
So, the coefficient of d little
a will be partial with respect
505
00:41:21 --> 00:41:26
to little a.
That's one half b sine theta da
506
00:41:26 --> 00:41:36
plus coefficient of db will be
one half a sine theta db plus
507
00:41:36 --> 00:41:45
coefficient of d theta will be
one half ab cosine theta d
508
00:41:45 --> 00:41:48
theta.
But now, what do I do with that?
509
00:41:48 --> 00:41:52
Well, first I said a is
constant.
510
00:41:52 --> 00:41:56
So, da is zero.
Second, well,
511
00:41:56 --> 00:41:59
actually we don't like b at
all, right?
512
00:41:59 --> 00:42:03
We want to view a as a function
of theta.
513
00:42:03 --> 00:42:13
So, well, maybe we actually
want to use this formula for db
514
00:42:13 --> 00:42:18
that we found in here.
OK, and then we'll be left only
515
00:42:18 --> 00:42:20
with d thetas,
which is what we want.
516
00:42:20 --> 00:42:56
517
00:42:56 --> 00:43:06
So, if we plug this one into
that one, we get da equals one
518
00:43:06 --> 00:43:16
half a sine theta times b
tangent theta d theta plus one
519
00:43:16 --> 00:43:26
half ab cosine theta d theta.
And, if we collect these things
520
00:43:26 --> 00:43:35
together, we get one half of ab
times sine theta times tangent
521
00:43:35 --> 00:43:41
theta plus cosine theta d theta.
And, if you know your trig,
522
00:43:41 --> 00:43:44
but you'll see that this is
sine squared over cosine plus
523
00:43:44 --> 00:43:49
cosine squared over cosine.
That's the same as secan theta.
524
00:43:49 --> 00:43:54
So, now you have expressed da
as something times d theta.
525
00:43:54 --> 00:43:59
Well, that coefficient is the
rate of change of A with respect
526
00:43:59 --> 00:44:04
to theta with the understanding
that we are keeping a fixed,
527
00:44:04 --> 00:44:10
and letting b vary as a
dependent variable.
528
00:44:10 --> 00:44:11
Not enough space: sorry.
529
00:44:11 --> 00:44:26
530
00:44:26 --> 00:44:29
OK, in case it's clearer for
you, let's think about it
531
00:44:29 --> 00:44:32
backwards.
So, we wanted to find how A
532
00:44:32 --> 00:44:35
changes.
To find how A changes,
533
00:44:35 --> 00:44:38
we write da.
But now, this tells us how A
534
00:44:38 --> 00:44:41
depends on little a,
little b, and theta.
535
00:44:41 --> 00:44:45
Well, we know actually we want
to keep little a constant.
536
00:44:45 --> 00:44:49
So, we set this to be zero.
Theta, well,
537
00:44:49 --> 00:44:52
we are very happy because we
want to express things in terms
538
00:44:52 --> 00:44:55
of theta.
Db we want to get rid of.
539
00:44:55 --> 00:45:00
How do we get rid of db?
Well, we do that by figuring
540
00:45:00 --> 00:45:05
out how b depends on theta when
a is fixed.
541
00:45:05 --> 00:45:08
And, we do that by
differentiating the constraint
542
00:45:08 --> 00:45:12
equation, and setting da equal
to zero.
543
00:45:12 --> 00:45:31
OK, so -- I guess to summarize
the method, we wrote dA in terms
544
00:45:31 --> 00:45:41
of da, db, d theta.
Then, we say that a is constant
545
00:45:41 --> 00:45:50
means we set da equals zero.
And, the third thing is that
546
00:45:50 --> 00:45:57
because, well,
we differentiate the
547
00:45:57 --> 00:46:06
constraint.
And, we can solve for db in
548
00:46:06 --> 00:46:19
terms of d theta.
And then, we plug into dA,
549
00:46:19 --> 00:46:32
and we get the answer.
OK, oops.
550
00:46:32 --> 00:46:38
So, here's another method to do
the same thing differently is to
551
00:46:38 --> 00:46:43
use the chain rule.
So, we can use the chain rule
552
00:46:43 --> 00:46:45
with dependent variables,
OK?
553
00:46:45 --> 00:46:48
So, what does the chain rule
tell us?
554
00:46:48 --> 00:46:54
The chain rule tells us,
so we will want to
555
00:46:54 --> 00:47:02
differentiate -- -- the formula
for a with respect to theta
556
00:47:02 --> 00:47:06
holding a constant.
So, I claim,
557
00:47:06 --> 00:47:10
well, what does the chain rule
tell us?
558
00:47:10 --> 00:47:14
It tells us that,
well, when we change things,
559
00:47:14 --> 00:47:19
a changes because of the
changes in the variables.
560
00:47:19 --> 00:47:24
So, part of it is that A
depends on theta and theta
561
00:47:24 --> 00:47:28
changes.
How fast does theta change?
562
00:47:28 --> 00:47:31
Well, you could call that the
rate of change of theta with
563
00:47:31 --> 00:47:33
respect to theta with a
constant.
564
00:47:33 --> 00:47:35
But of course,
how fast does theta depend to
565
00:47:35 --> 00:47:38
itself?
The answer is one.
566
00:47:38 --> 00:47:44
So, that's pretty easy.
Plus, then we have the partial
567
00:47:44 --> 00:47:49
derivative, formal partial
derivative, of A with respect to
568
00:47:49 --> 00:47:55
little a times the rate of
change of a in our situation.
569
00:47:55 --> 00:47:58
Well, how does little a change
if a is constant?
570
00:47:58 --> 00:48:08
Well, it doesn't change.
And then, there is Ab,
571
00:48:08 --> 00:48:14
the formal partial derivative
times, sorry,
572
00:48:14 --> 00:48:20
the rate of change of b.
OK, and how do we find this one?
573
00:48:20 --> 00:48:27
Well, here we have to use the
constraint.
574
00:48:27 --> 00:48:30
OK, and we can find this one
from the constraint as we've
575
00:48:30 --> 00:48:32
seen at the beginning either by
differentiating the constraint,
576
00:48:32 --> 00:48:36
or by using the chain rule on
the constraint.
577
00:48:36 --> 00:48:39
So, of course the calculations
are exactly the same.
578
00:48:39 --> 00:48:44
See, this is the same formula
as the one over there,
579
00:48:44 --> 00:48:48
just dividing everything by
partial theta and with
580
00:48:48 --> 00:48:54
subscripts little a.
But, if it's easier to think
581
00:48:54 --> 00:48:59
about it this way,
then that's also valid.
582
00:48:59 --> 00:49:03
OK, so tomorrow we are going to
review for the test,
583
00:49:03 --> 00:49:06
so I'm going to tell you a bit
more about this also as we go
584
00:49:06 --> 00:49:09
over one practice problem on
that.
585
00:49:09 --> 00:49:10