1
00:00:00 --> 00:00:06
2
00:00:01 --> 00:00:07
I usually like to start a
lecture with something new,
3
00:00:04 --> 00:00:10
but this time I'm going to make
an exception,
4
00:00:07 --> 00:00:13
and start with the finishing up
on Friday because it involves a
5
00:00:11 --> 00:00:17
little more practice with
complex numbers.
6
00:00:14 --> 00:00:20
I think that's what a large
number of you are still fairly
7
00:00:18 --> 00:00:24
weak in.
So, to briefly remind you,
8
00:00:20 --> 00:00:26
it will be sort of
self-contained,
9
00:00:23 --> 00:00:29
but still, it will use complex
numbers.
10
00:00:25 --> 00:00:31
And, I think it's a good way to
start today.
11
00:00:30 --> 00:00:36
So, remember,
the basic problem was to solve
12
00:00:33 --> 00:00:39
something with,
where the input was sinusoidal
13
00:00:38 --> 00:00:44
in particular.
The k was on both sides,
14
00:00:41 --> 00:00:47
and the input looked like
cosine omega t.
15
00:00:46 --> 00:00:52
And, the plan of the solution
consisted of transporting the
16
00:00:52 --> 00:00:58
problem to the complex domain.
So, you look for a complex
17
00:00:57 --> 00:01:03
solution, and you complexify the
right hand side of the equation,
18
00:01:02 --> 00:01:08
as well.
So, cosine omega t
19
00:01:07 --> 00:01:13
becomes the real part of
this complex function.
20
00:01:11 --> 00:01:17
The reason for doing that,
remember, was because it's
21
00:01:15 --> 00:01:21
easier to handle when you solve
linear equations.
22
00:01:19 --> 00:01:25
It's much easier to handle
exponentials on the right-hand
23
00:01:23 --> 00:01:29
side than it is to handle sines
and cosines because exponentials
24
00:01:28 --> 00:01:34
are so easy to integrate when
you multiply them by other
25
00:01:33 --> 00:01:39
exponentials.
So, the result was,
26
00:01:35 --> 00:01:41
after doing that,
y tilda turned out to be one,
27
00:01:39 --> 00:01:45
after I scale the coefficient,
one over one plus omega over k
28
00:01:44 --> 00:01:50
And then, the rest was e to the
29
00:01:51 --> 00:01:57
i times (omega t minus phi),
30
00:01:56 --> 00:02:02
where phi had a certain
meaning.
31
00:02:00 --> 00:02:06
It was the arc tangent of a,
it was a phase lag.
32
00:02:06 --> 00:02:12
And, this was then,
I had to take the real part of
33
00:02:09 --> 00:02:15
this to get the final answer,
which came out to be something
34
00:02:13 --> 00:02:19
like one over the square root of
one plus the amplitude one omega
35
00:02:18 --> 00:02:24
/ k squared, and then the rest
was cosine omega t plus minus
36
00:02:22 --> 00:02:28
phi.
37
00:02:26 --> 00:02:32
It's easier to see that that
part is the real part of this;
38
00:02:30 --> 00:02:36
the problem is,
of course you have to convert
39
00:02:33 --> 00:02:39
this.
Sorry, this should be i omega
40
00:02:37 --> 00:02:43
t, in which case you don't need
the parentheses,
41
00:02:41 --> 00:02:47
either.
So, the problem was to use the
42
00:02:45 --> 00:02:51
polar representation of this
complex number to convert it
43
00:02:50 --> 00:02:56
into something whose amplitude
was this, and whose angle was
44
00:02:55 --> 00:03:01
minus phi.
Now, that's what we call the
45
00:02:58 --> 00:03:04
polar method,
going polar.
46
00:03:02 --> 00:03:08
I'd like, now,
for the first few minutes of
47
00:03:04 --> 00:03:10
the period, to talk about the
other method,
48
00:03:07 --> 00:03:13
the Cartesian method.
I think for a long while,
49
00:03:10 --> 00:03:16
many of you will be more
comfortable with that anyway.
50
00:03:14 --> 00:03:20
Although, one of the objects of
the course should be to get you
51
00:03:18 --> 00:03:24
equally comfortable with the
polar representation of complex
52
00:03:22 --> 00:03:28
numbers.
So, if we try to do the same
53
00:03:25 --> 00:03:31
thing going Cartesian,
what's going to happen?
54
00:03:28 --> 00:03:34
Well, I guess the same point
here.
55
00:03:32 --> 00:03:38
So, the starting point is still
y tilde equals one over,
56
00:03:37 --> 00:03:43
sorry, this should have an i
here, one plus i times omega
57
00:03:42 --> 00:03:48
over k, e to the i omega t.
58
00:03:48 --> 00:03:54
But now, what you're going to
59
00:03:52 --> 00:03:58
do is turn this into its
Cartesian, turn both of these
60
00:03:57 --> 00:04:03
into their Cartesian
representations as a plus ib.
61
00:04:02 --> 00:04:08
So, if you do that Cartesianly,
62
00:04:06 --> 00:04:12
of course, what you have to do
is the standard thing about
63
00:04:09 --> 00:04:15
dividing complex numbers or
taking the reciprocals that I
64
00:04:13 --> 00:04:19
told you at the very beginning
of complex numbers.
65
00:04:16 --> 00:04:22
You multiply the top and bottom
by the complex conjugate of this
66
00:04:21 --> 00:04:27
in order to make the bottom
real.
67
00:04:23 --> 00:04:29
So, what does this become?
This becomes one minus i times
68
00:04:27 --> 00:04:33
omega over k divided by the
product of this in its complex
69
00:04:30 --> 00:04:36
conjugate, which is the real
number, one plus omega over k
70
00:04:34 --> 00:04:40
squared
71
00:04:39 --> 00:04:45
So, I've now converted this to
the a plus bi form.
72
00:04:43 --> 00:04:49
I have also to convert the
right-hand side to the a plus bi
73
00:04:47 --> 00:04:53
form.
So, it will look like cosine
74
00:04:49 --> 00:04:55
omega t plus i sine omega t.
75
00:04:53 --> 00:04:59
Having done that,
76
00:04:55 --> 00:05:01
I take the last step,
which is to take the real part
77
00:04:59 --> 00:05:05
of that.
Remember, the reason I want the
78
00:05:01 --> 00:05:07
real part is because this input
was the real part of the complex
79
00:05:06 --> 00:05:12
input.
So, once you've got the complex
80
00:05:10 --> 00:05:16
solution, you have to take its
real part to go back into the
81
00:05:14 --> 00:05:20
domain you started with,
of real numbers,
82
00:05:17 --> 00:05:23
from the domain of complex
numbers.
83
00:05:19 --> 00:05:25
So, I want the real part is
going to be, the real part of
84
00:05:23 --> 00:05:29
that is, first of all,
there's a factor out in front.
85
00:05:27 --> 00:05:33
That's entirely real.
Let's put that out in front,
86
00:05:32 --> 00:05:38
so doesn't bother us
particularly.
87
00:05:34 --> 00:05:40
And now, I need the product of
this complex number and that
88
00:05:39 --> 00:05:45
complex number.
But, I only want the real part
89
00:05:42 --> 00:05:48
of it.
So, I'm not going to multiply
90
00:05:45 --> 00:05:51
it out and get four terms.
I'm just going to look at the
91
00:05:49 --> 00:05:55
two terms that I do want.
I don't want the others.
92
00:05:53 --> 00:05:59
All right, the real part is
cosine omega t,
93
00:05:58 --> 00:06:04
from the product of this and
that.
94
00:06:02 --> 00:06:08
And, the rest of the real part
will be the product of the two i
95
00:06:07 --> 00:06:13
terms.
But, it's i times negative i,
96
00:06:11 --> 00:06:17
which makes one.
And therefore,
97
00:06:13 --> 00:06:19
it's omega over k times sine
omega t.
98
00:06:18 --> 00:06:24
Now, that's the answer.
99
00:06:22 --> 00:06:28
And that's the answer,
too; they must be equal,
100
00:06:26 --> 00:06:32
unless there's a contradiction
in mathematics.
101
00:06:32 --> 00:06:38
But, it's extremely important.
And that's the other reason why
102
00:06:36 --> 00:06:42
I'm giving you this,
that you learn in this course
103
00:06:40 --> 00:06:46
to be able to convert quickly
and automatically things that
104
00:06:44 --> 00:06:50
look like this into things that
look like that.
105
00:06:47 --> 00:06:53
And, that's done by means of a
basic formula,
106
00:06:50 --> 00:06:56
which occurs at the end of the
notes for reference,
107
00:06:54 --> 00:07:00
as I optimistically say,
although I think for a lot of
108
00:06:58 --> 00:07:04
you will not be referenced,
stuff in the category of,
109
00:07:02 --> 00:07:08
yeah, I think I've vaguely seen
that somewhere.
110
00:07:07 --> 00:07:13
But, well, we never used it for
anything.
111
00:07:10 --> 00:07:16
Okay, you're going to use it
all term.
112
00:07:12 --> 00:07:18
So, the formula is,
the famous trigonometric
113
00:07:16 --> 00:07:22
identity, which is,
so, the problem is to convert
114
00:07:19 --> 00:07:25
this into the other guy.
And, the thing which is going
115
00:07:24 --> 00:07:30
to do that, enable one to
combine the sine and the cosine
116
00:07:28 --> 00:07:34
terms, is the famous formula
that a times the cosine,
117
00:07:32 --> 00:07:38
I'm going to use theta to make
it as neutral as possible,
118
00:07:36 --> 00:07:42
--
-- so, theta you can think of
119
00:07:40 --> 00:07:46
as being replaced by omega t in
this particular application of
120
00:07:44 --> 00:07:50
the formula.
But, I'll just use a general
121
00:07:47 --> 00:07:53
angle theta, which doesn't
suggest anything in particular.
122
00:07:51 --> 00:07:57
So, the problem is,
you have something which is a
123
00:07:54 --> 00:08:00
combination with real
coefficients of cosine and sine,
124
00:07:58 --> 00:08:04
and the important thing is that
these numbers be the same.
125
00:08:03 --> 00:08:09
In practice,
that means that the omega t,
126
00:08:06 --> 00:08:12
you're not allowed to have
omega one t here,
127
00:08:10 --> 00:08:16
and some other frequency,
omega two t here.
128
00:08:14 --> 00:08:20
That would correspond to using
theta one here,
129
00:08:17 --> 00:08:23
and theta two here.
And, though there is a formula
130
00:08:21 --> 00:08:27
for combining that,
nobody remembers it,
131
00:08:24 --> 00:08:30
and it's, in general,
less universally useful than
132
00:08:28 --> 00:08:34
the first.
If you're going to memorize a
133
00:08:32 --> 00:08:38
formula, and learn this one,
it's best to start with the
134
00:08:38 --> 00:08:44
ones where the two are equal.
That's the basic formula.
135
00:08:44 --> 00:08:50
The others are variations of
it, but there is a sizable
136
00:08:49 --> 00:08:55
variations.
All right, so the answer is
137
00:08:53 --> 00:08:59
that this is equal to some other
constant, real constant,
138
00:08:59 --> 00:09:05
times the cosine of theta minus
phi.
139
00:09:06 --> 00:09:12
Of course, most people remember
this vaguely.
140
00:09:09 --> 00:09:15
What they don't remember is
what the c and the phi are,
141
00:09:14 --> 00:09:20
how to calculate them.
I don't suggest you memorize
142
00:09:18 --> 00:09:24
the formulas for them.
You can if you wish.
143
00:09:22 --> 00:09:28
Instead, memorize the picture,
which is much easier.
144
00:09:26 --> 00:09:32
Memorize that a and b are the
two sides of a right triangle.
145
00:09:31 --> 00:09:37
Phi is the angle opposite the b
side, and c is the length of the
146
00:09:36 --> 00:09:42
hypotenuse.
Okay, that's worth putting up.
147
00:09:42 --> 00:09:48
I think that's a pink formula.
It's even worth two of those,
148
00:09:48 --> 00:09:54
but I will thrift.
Now, let's apply it to this
149
00:09:52 --> 00:09:58
case to see that it gives the
right answer.
150
00:09:57 --> 00:10:03
So, to use this formula,
how I use it?
151
00:10:02 --> 00:10:08
Well, I should take,
I will reproduce the left-hand
152
00:10:07 --> 00:10:13
side.
So that part,
153
00:10:09 --> 00:10:15
I just copy.
And, how about the right?
154
00:10:13 --> 00:10:19
Well, the amplitude,
it's combined into a single
155
00:10:19 --> 00:10:25
cosine term whose amplitude is,
well, the two sides of the
156
00:10:25 --> 00:10:31
right triangle are one,
and omega over k.
157
00:10:33 --> 00:10:39
The hypotenuse in that case is
going to be, well,
158
00:10:36 --> 00:10:42
why don't we write it here?
So, we have one,
159
00:10:39 --> 00:10:45
and omega over k.
And, here's phi.
160
00:10:43 --> 00:10:49
So, the hypotenuse is going to
be the square root of one plus
161
00:10:47 --> 00:10:53
omega over k squared.
162
00:10:51 --> 00:10:57
And, that's going to be
multiplied by the cosine of
163
00:10:55 --> 00:11:01
omega t minus this phase lag
angle phi.
164
00:10:59 --> 00:11:05
You can write,
165
00:11:02 --> 00:11:08
if you wish,
phi equals the arc tangent,
166
00:11:05 --> 00:11:11
but you are not learning a lot
by that.
167
00:11:09 --> 00:11:15
Phi is the arc tangent of omega
over k.
168
00:11:15 --> 00:11:21
That's okay,
169
00:11:16 --> 00:11:22
but it's true.
But, notice there's
170
00:11:19 --> 00:11:25
cancellation now.
This over that is equal to
171
00:11:23 --> 00:11:29
what?
Well, it's equal to this.
172
00:11:28 --> 00:11:34
And, so when we get in this
way, by combining these two
173
00:11:31 --> 00:11:37
factors, one gets exactly the
same formula that we got before.
174
00:11:36 --> 00:11:42
So, as you can see,
in some sense,
175
00:11:38 --> 00:11:44
there's not,
if you can remember this
176
00:11:41 --> 00:11:47
trigonometric identity,
there's not a lot of difference
177
00:11:45 --> 00:11:51
between the two methods except
that this one requires this
178
00:11:49 --> 00:11:55
extra step.
The answer will come out in
179
00:11:51 --> 00:11:57
this form, and you then,
to see what it really looks
180
00:11:55 --> 00:12:01
like, really have to convert it
to this form,
181
00:11:58 --> 00:12:04
the form in which you can see
what the phase lag and the
182
00:12:02 --> 00:12:08
amplitude is.
It's amazing how many people
183
00:12:07 --> 00:12:13
who should know,
this includes working
184
00:12:10 --> 00:12:16
mathematicians,
theoretical mathematicians,
185
00:12:13 --> 00:12:19
includes even possibly the
authors of your textbooks.
186
00:12:17 --> 00:12:23
I'm not sure,
but I've caught them in this,
187
00:12:20 --> 00:12:26
too, who in this form,
everybody remembers that it's
188
00:12:24 --> 00:12:30
something like that.
Unfortunately,
189
00:12:27 --> 00:12:33
when it occurs as the answer in
an answer book,
190
00:12:31 --> 00:12:37
the numbers are some colossal
mess here plus some colossal
191
00:12:35 --> 00:12:41
mess here.
And theta is,
192
00:12:39 --> 00:12:45
again, a real mess,
involving roots and some cube
193
00:12:44 --> 00:12:50
roots, and whatnot.
The only thing is,
194
00:12:48 --> 00:12:54
these two are the same real
mess.
195
00:12:52 --> 00:12:58
That amounts to just another
pure oscillation with the same
196
00:12:58 --> 00:13:04
frequency as the old guy,
and with the amplitude changed,
197
00:13:05 --> 00:13:11
and with a phase shift,
move to the right or left.
198
00:13:12 --> 00:13:18
So, this is no more general
than that.
199
00:13:14 --> 00:13:20
Notice they both have two
parameters in them,
200
00:13:18 --> 00:13:24
these two coefficients.
This one has the two parameters
201
00:13:22 --> 00:13:28
in an altered form.
Okay, well, I wanted,
202
00:13:26 --> 00:13:32
because of the importance of
this formula,
203
00:13:29 --> 00:13:35
I wanted to take a couple of
minutes out for a proof of the
204
00:13:34 --> 00:13:40
formula, --
205
00:13:36 --> 00:13:42
206
00:13:42 --> 00:13:48
-- just to give you chance to
stare at it a little more now.
207
00:13:46 --> 00:13:52
There are three proofs I know.
I'm sure there are 27.
208
00:13:50 --> 00:13:56
The Pythagorean theorem now has
several hundred.
209
00:13:53 --> 00:13:59
But, there are three basic
proofs.
210
00:13:56 --> 00:14:02
There is the one I will not
give you, I'll call the high
211
00:14:00 --> 00:14:06
school proof,
which is the only one one
212
00:14:03 --> 00:14:09
normally finds in books,
physics textbooks or other
213
00:14:07 --> 00:14:13
textbooks.
The high school proof takes the
214
00:14:11 --> 00:14:17
right-hand side,
applies the formula for the
215
00:14:14 --> 00:14:20
cosine of the difference of two
angles, which it assumes you had
216
00:14:19 --> 00:14:25
in trigonometry,
and then converts it into this.
217
00:14:22 --> 00:14:28
It shows you that once you've
done that, that a turns out to
218
00:14:27 --> 00:14:33
be c cosine phi
and b, the number b is c sine
219
00:14:31 --> 00:14:37
phi,
and therefore it identifies the
220
00:14:35 --> 00:14:41
two sides.
Now, the thing that's of course
221
00:14:39 --> 00:14:45
correct and it's the simplest
possible argument,
222
00:14:42 --> 00:14:48
the thing that's no good about
it is that the direction at
223
00:14:47 --> 00:14:53
which it goes is from here to
here.
224
00:14:49 --> 00:14:55
Well, everybody knew that.
If I gave you this and told
225
00:14:53 --> 00:14:59
you, write it out in terms of
cosine and sine,
226
00:14:56 --> 00:15:02
I would assume it dearly hope
that practically all of you can
227
00:15:01 --> 00:15:07
do that.
Unfortunately,
228
00:15:03 --> 00:15:09
when you want to use the
formula, it's this way you want
229
00:15:07 --> 00:15:13
to use it in the opposite
direction.
230
00:15:09 --> 00:15:15
You are starting with this,
and want to convert it to that.
231
00:15:13 --> 00:15:19
Now, the proof,
therefore, will not be of much
232
00:15:15 --> 00:15:21
help.
It requires you to go in the
233
00:15:17 --> 00:15:23
backwards direction,
and match up coefficients.
234
00:15:20 --> 00:15:26
It's much better to go
forwards.
235
00:15:22 --> 00:15:28
Now, there are two proofs that
go forwards.
236
00:15:25 --> 00:15:31
There's the 18.02 proof.
Since I didn't teach most of
237
00:15:28 --> 00:15:34
you 18.02, I can't be sure you
had it.
238
00:15:32 --> 00:15:38
So, I'll spend one minute
giving it to you.
239
00:15:36 --> 00:15:42
What is the 18.02 proof?
It is the following picture.
240
00:15:42 --> 00:15:48
I think this requires deep
colored chalk.
241
00:15:46 --> 00:15:52
This is going to be pretty
heavy.
242
00:15:50 --> 00:15:56
All right, first of all,
the a and the b are the given.
243
00:15:55 --> 00:16:01
So, I'm going to put in that
vector.
244
00:16:01 --> 00:16:07
So, there is the vector whose
sides are, whose components are
245
00:16:05 --> 00:16:11
a and b.
I'll write it without the i and
246
00:16:08 --> 00:16:14
j.
I hope you had from Jerison
247
00:16:11 --> 00:16:17
that form for the vector,
if you don't like that,
248
00:16:14 --> 00:16:20
write ai plus bj,
okay?
249
00:16:17 --> 00:16:23
Now, there's another vector
lurking around.
250
00:16:20 --> 00:16:26
It's the unit vector whose,
I'll write it this way,
251
00:16:24 --> 00:16:30
u because it's a unit vector,
and theta to indicate that it's
252
00:16:29 --> 00:16:35
angle is theta.
Now, the reason for doing that
253
00:16:34 --> 00:16:40
is because you see that the
left-hand side is a dot product
254
00:16:38 --> 00:16:44
of two vectors.
The left-hand side of the
255
00:16:41 --> 00:16:47
identity is the dot product of
the vector 00:16:51
b> with the vector whose
components are cosine theta and
257
00:16:49 --> 00:16:55
sine theta.
258
00:16:52 --> 00:16:58
That's what I'm calling this
unit vector.
259
00:16:55 --> 00:17:01
It's a unit vector because
cosine squared plus sine squared
260
00:16:59 --> 00:17:05
is one.
261
00:17:04 --> 00:17:10
Now, all this formula is,
is saying that scalar product,
262
00:17:08 --> 00:17:14
the dot product of those two
vectors, can be evaluated if you
263
00:17:13 --> 00:17:19
know their components by the
left-hand side of the formula.
264
00:17:18 --> 00:17:24
And, if you don't know their
components, it can be evaluated
265
00:17:24 --> 00:17:30
in another way,
the geometric evaluation,
266
00:17:27 --> 00:17:33
which goes, what is it?
It's a magnitude of one,
267
00:17:31 --> 00:17:37
times the magnitude of the
other, times the cosine of the
268
00:17:36 --> 00:17:42
included angle.
Now, what's the included angle?
269
00:17:42 --> 00:17:48
Well, theta is this angle from
the horizontal to that unit
270
00:17:49 --> 00:17:55
vector.
The angle phi is this angle,
271
00:17:54 --> 00:18:00
from this picture here.
And therefore,
272
00:17:58 --> 00:18:04
the included angle between
(u)theta and my pink vector is
273
00:18:05 --> 00:18:11
theta minus phi.
That's the formula.
274
00:18:12 --> 00:18:18
It comes from two ways of
calculating the scalar product
275
00:18:16 --> 00:18:22
of the vector whose coefficients
are, and the unit vector
276
00:18:21 --> 00:18:27
whose components are cosine
theta and sine theta.
277
00:18:25 --> 00:18:31
All right, well,
278
00:18:28 --> 00:18:34
you should, that was 18.02.
279
00:18:35 --> 00:18:41
There must be an 18.03 proof
also. Yes.
280
00:18:36 --> 00:18:42
What's the 18.03 proof?
The 18.03 proof uses complex
281
00:18:43 --> 00:18:49
numbers.
It says, look,
282
00:18:46 --> 00:18:52
take the left side.
Instead of viewing it as the
283
00:18:53 --> 00:18:59
dot product of two vectors,
there's another way.
284
00:19:02 --> 00:19:08
You can think of it as the part
of the products of two complex
285
00:19:06 --> 00:19:12
numbers.
So, the 18.03 argument,
286
00:19:09 --> 00:19:15
really, the complex number
argument says,
287
00:19:12 --> 00:19:18
look, multiply together a minus
bi and the complex
288
00:19:17 --> 00:19:23
number cosine theta plus i sine
theta.
289
00:19:21 --> 00:19:27
There are different ways of
290
00:19:24 --> 00:19:30
explaining why I want to put the
minus i there instead of i.
291
00:19:28 --> 00:19:34
But, the simplest is because I
want, when I take the real part,
292
00:19:33 --> 00:19:39
to get the left-hand side.
I will.
293
00:19:37 --> 00:19:43
If I take the real part of
this, I'm going to get a cosine
294
00:19:42 --> 00:19:48
theta plus b sine theta
295
00:19:46 --> 00:19:52
because of negative i and i make
one,
296
00:19:51 --> 00:19:57
multiplied together.
All right, that's the left-hand
297
00:19:55 --> 00:20:01
side.
And now, the right-hand side,
298
00:19:58 --> 00:20:04
I'm going to use polar
representation instead.
299
00:20:03 --> 00:20:09
What's the polar representation
of this guy?
300
00:20:06 --> 00:20:12
Well, if has the angle theta,
301
00:20:09 --> 00:20:15
then a negative b,
a minus bi goes down
302
00:20:13 --> 00:20:19
below.
It has the angle minus phi.
303
00:20:16 --> 00:20:22
So, this is,
has magnitude.
304
00:20:18 --> 00:20:24
It is polar representation.
Its magnitude is a squared plus
305
00:20:23 --> 00:20:29
b squared,
and its angle is negative phi,
306
00:20:27 --> 00:20:33
not positive phi because this a
minus bi goes below
307
00:20:32 --> 00:20:38
the axis if a and b are
positive.
308
00:20:36 --> 00:20:42
So, it's e to the minus i phi.
309
00:20:39 --> 00:20:45
That's the first guy.
And, how about the second guy?
310
00:20:43 --> 00:20:49
Well, the second guy is e to
the i theta.
311
00:20:47 --> 00:20:53
So, what's the product?
It is a squared plus b squared,
312
00:20:51 --> 00:20:57
the square root,
times e to the i times (theta
313
00:20:54 --> 00:21:00
minus phi).
314
00:20:58 --> 00:21:04
And now, what do I want?
The real part of this,
315
00:21:01 --> 00:21:07
and I want the real part of
this.
316
00:21:05 --> 00:21:11
So, let's just say take the
real parts of both sides.
317
00:21:08 --> 00:21:14
If I take the real part of the
left-hand side,
318
00:21:11 --> 00:21:17
I get a cosine theta plus b
sine theta.
319
00:21:14 --> 00:21:20
If I take a real part of this
320
00:21:17 --> 00:21:23
side, I get square root of a
squared plus b squared times e,
321
00:21:21 --> 00:21:27
times the cosine,
that's the real part,
322
00:21:23 --> 00:21:29
right, of theta minus phi,
which is just what it's
323
00:21:26 --> 00:21:32
supposed to be.
324
00:21:31 --> 00:21:37
Well, with three different
arguments, I'm really pounding
325
00:21:35 --> 00:21:41
the table on this formula.
But, I think there's something
326
00:21:40 --> 00:21:46
to be learned from at least two
of them.
327
00:21:44 --> 00:21:50
And, you know,
I'm still, for awhile,
328
00:21:47 --> 00:21:53
I will never miss an
opportunity to bang complex
329
00:21:51 --> 00:21:57
numbers into your head because,
in some sense,
330
00:21:55 --> 00:22:01
you have to reproduce the
experience of the race.
331
00:22:01 --> 00:22:07
As I mentioned in the notes,
it took mathematicians 300 or
332
00:22:04 --> 00:22:10
400 years to get used to complex
numbers.
333
00:22:07 --> 00:22:13
So, if it takes you three or
four weeks, that's not too bad.
334
00:22:11 --> 00:22:17
335
00:22:32 --> 00:22:38
Now, for the rest of the period
I'd like to go back to the
336
00:22:36 --> 00:22:42
linear equations,
and try to put into perspective
337
00:22:40 --> 00:22:46
and summarize,
and tell you a couple of things
338
00:22:43 --> 00:22:49
which I had to leave out,
but which are,
339
00:22:46 --> 00:22:52
in my view, extremely
important.
340
00:22:48 --> 00:22:54
And, up to now,
I don't want to leave you with
341
00:22:52 --> 00:22:58
any misapprehensions.
So, I'm going to summarize this
342
00:22:56 --> 00:23:02
way, whereas last lecture I went
from the most general equation
343
00:23:01 --> 00:23:07
to the most special.
I'd like to just write them
344
00:23:06 --> 00:23:12
down in the reverse order,
now.
345
00:23:09 --> 00:23:15
So, we are talking about basic
linear equations.
346
00:23:13 --> 00:23:19
First order,
of course, we haven't moved as
347
00:23:16 --> 00:23:22
a second order yet.
So, the most special one,
348
00:23:20 --> 00:23:26
and the one we talked about
essentially all of the previous
349
00:23:25 --> 00:23:31
two times, or last Friday,
anyway, was the equation where
350
00:23:30 --> 00:23:36
the k, the coefficient of y,
is constant,
351
00:23:34 --> 00:23:40
and where you also get it on
the right-hand side quite
352
00:23:39 --> 00:23:45
providentially.
So, this is the most special
353
00:23:44 --> 00:23:50
form, and it's the one which
governed what I will call the
354
00:23:48 --> 00:23:54
temperature-concentration model,
or if you want to be grown up,
355
00:23:53 --> 00:23:59
the conduction-diffusion model,
conduction-diffusion which
356
00:23:57 --> 00:24:03
describes the processes,
which the equation is modeling,
357
00:24:01 --> 00:24:07
whereas these simply described
the variables of things,
358
00:24:05 --> 00:24:11
which you usually are trying to
calculate when you use the
359
00:24:10 --> 00:24:16
equation.
Now, there are a class of
360
00:24:13 --> 00:24:19
things where the thing is
constant, but where the k does
361
00:24:17 --> 00:24:23
not appear naturally on the
right hand side.
362
00:24:20 --> 00:24:26
And, you're going to encounter
them pretty quickly in physics,
363
00:24:24 --> 00:24:30
for one place.
So, I better not try to sweep
364
00:24:27 --> 00:24:33
those under the rug.
Let's just call that q of t.
365
00:24:32 --> 00:24:38
And finally,
there is the most general case,
366
00:24:36 --> 00:24:42
where you allow k to be
non-constant.
367
00:24:40 --> 00:24:46
That's the one we began,
when we talked about the linear
368
00:24:45 --> 00:24:51
equation.
And you know how to solve it in
369
00:24:49 --> 00:24:55
general by a definite or an
indefinite integral.
370
00:24:54 --> 00:25:00
Now, there's one other thing,
which I want to talk about.
371
00:25:01 --> 00:25:07
I will do all these in a
certain order.
372
00:25:03 --> 00:25:09
But, from the beginning,
you should keep in mind that
373
00:25:07 --> 00:25:13
there's another between the
first two cases.
374
00:25:10 --> 00:25:16
Between the first two cases,
there's another extremely
375
00:25:14 --> 00:25:20
important distinction,
and that is as to whether k is
376
00:25:18 --> 00:25:24
positive or not.
Up to now, we've always had k
377
00:25:21 --> 00:25:27
positive.
So, I'm going to put that here.
378
00:25:24 --> 00:25:30
So, it's understood when I
write these, that k is positive.
379
00:25:30 --> 00:25:36
I want to talk about that,
too.
380
00:25:32 --> 00:25:38
But, first things first.
The first thing I wanted to do
381
00:25:36 --> 00:25:42
was to show you that this,
the first case,
382
00:25:39 --> 00:25:45
the most special case,
does not just apply to this.
383
00:25:43 --> 00:25:49
It applies to other things,
too.
384
00:25:45 --> 00:25:51
Let me give you a mixing
problem.
385
00:25:48 --> 00:25:54
The typical mixing problem
gives another example.
386
00:25:51 --> 00:25:57
You've already done in
recitation, and you did one for
387
00:25:55 --> 00:26:01
the problem set,
the problem of the two rooms
388
00:25:59 --> 00:26:05
filled with smoke.
But, let me do it just using
389
00:26:04 --> 00:26:10
letters, so that the ideas stand
out a little more clearly,
390
00:26:08 --> 00:26:14
and you are not preoccupied
with the numbers,
391
00:26:11 --> 00:26:17
and calculating with the
numbers, and trying to get
392
00:26:15 --> 00:26:21
numerical examples.
So, it's as simple as k sub
393
00:26:19 --> 00:26:25
mixing.
It looks like this.
394
00:26:21 --> 00:26:27
You have a tank,
a room, I don't know,
395
00:26:23 --> 00:26:29
where everything's getting
mixed in.
396
00:26:26 --> 00:26:32
It has a certain volume,
which I will call v.
397
00:26:31 --> 00:26:37
Something is flowing in,
a gas or a liquid.
398
00:26:34 --> 00:26:40
And, r will be the flow rate,
in some units.
399
00:26:38 --> 00:26:44
Now, since it can't pile up
inside this sealed container,
400
00:26:44 --> 00:26:50
which I'm sure is full,
the flow rate out must also be
401
00:26:49 --> 00:26:55
r.
And, what we're interested in
402
00:26:51 --> 00:26:57
is the amount of salt.
So, x, let's suppose these are
403
00:26:56 --> 00:27:02
fluid flows, and the dissolved
substance that I'm talking about
404
00:27:02 --> 00:27:08
is not carbon monoxide,
it's salt, any dissolved
405
00:27:06 --> 00:27:12
substance, some pollutant or
whatever the problem calls for.
406
00:27:14 --> 00:27:20
Let's use salt.
So, it's the amount of salt in
407
00:27:20 --> 00:27:26
the tank at time t.
I'm interested in knowing how
408
00:27:28 --> 00:27:34
that varies with time.
Now, there's nothing to be said
409
00:27:34 --> 00:27:40
about how it flows out.
What flows out,
410
00:27:36 --> 00:27:42
of course, is what happens to
be in the tank.
411
00:27:39 --> 00:27:45
But, I do have to say what
flows in.
412
00:27:41 --> 00:27:47
Now, the only convenient way to
describe the in-flow is in terms
413
00:27:46 --> 00:27:52
of its concentration.
The salt will be dissolved in
414
00:27:49 --> 00:27:55
the in-flowing water,
and so there will be a certain
415
00:27:52 --> 00:27:58
concentration.
And, as you will see,
416
00:27:55 --> 00:28:01
for a secret reason,
I'm going to give that the
417
00:27:58 --> 00:28:04
subscript e.
So, e is the concentration of
418
00:28:02 --> 00:28:08
the incoming salt,
in other words,
419
00:28:05 --> 00:28:11
in the fluid,
how many grams are there per
420
00:28:09 --> 00:28:15
liter in the incoming fluid.
That's the data.
421
00:28:13 --> 00:28:19
So, this is the data.
r is part of the data.
422
00:28:17 --> 00:28:23
r is the flow rate.
v is the volume.
423
00:28:20 --> 00:28:26
I think I won't bother writing
that down, and the problem is to
424
00:28:25 --> 00:28:31
determine what happens to x of
t.
425
00:28:30 --> 00:28:36
Now, I strongly recommend you
not attempt to work directly
426
00:28:34 --> 00:28:40
with the concentrations unless
you feel you have a really good
427
00:28:39 --> 00:28:45
physical feeling for
concentrations.
428
00:28:42 --> 00:28:48
I strongly recommend you work
with a variable that you are
429
00:28:46 --> 00:28:52
given, namely,
the dependent variable,
430
00:28:49 --> 00:28:55
which is the amount of salt,
grams.
431
00:28:52 --> 00:28:58
Well, because it's something
you can physically think about.
432
00:28:57 --> 00:29:03
It's coming in,
it's getting mixed up,
433
00:29:00 --> 00:29:06
and some of it is going out.
So, the basic equation is going
434
00:29:08 --> 00:29:14
to be that the rate of change of
salt in the tank is the rate of
435
00:29:18 --> 00:29:24
salt inflow, let me write salt
inflow, minus the rate of salt
436
00:29:27 --> 00:29:33
outflow.
Okay, at what rate is salt
437
00:29:33 --> 00:29:39
flowing in?
Well, the flow rate is the flow
438
00:29:39 --> 00:29:45
rate of the liquid.
I multiply the flow rate,
439
00:29:43 --> 00:29:49
1 L per minute times the
concentration,
440
00:29:47 --> 00:29:53
3 g per liter.
That means 3 g per minute.
441
00:29:51 --> 00:29:57
It's going to be,
therefore, the product of the
442
00:29:56 --> 00:30:02
flow rate and the concentration,
incoming concentration.
443
00:30:03 --> 00:30:09
How about the rate of the salt
outflow?
444
00:30:05 --> 00:30:11
Well, again,
the rate of the liquid outflow
445
00:30:08 --> 00:30:14
is r.
And, what is the concentration
446
00:30:11 --> 00:30:17
of salt in the outflow?
I must use, when you talk flow
447
00:30:15 --> 00:30:21
rates, the other factor must be
the concentration,
448
00:30:18 --> 00:30:24
not the amount.
So, what is the concentration
449
00:30:22 --> 00:30:28
in the outflow?
Well, it's the amount of salt
450
00:30:25 --> 00:30:31
in the tank divided by its
volume.
451
00:30:29 --> 00:30:35
So, the analog,
the concentration here is x
452
00:30:32 --> 00:30:38
divided by v.
Now, here's a typical messy
453
00:30:36 --> 00:30:42
equation, dx / dt,
let's write it in the standard,
454
00:30:40 --> 00:30:46
linear form,
plus r times x over v equals r
455
00:30:44 --> 00:30:50
times the given concentration,
which is a function of time.
456
00:30:50 --> 00:30:56
Now, this is going to be some
given function,
457
00:30:54 --> 00:31:00
and there will be no reason
whatsoever why you can't solve
458
00:30:59 --> 00:31:05
it in that form.
And, that's normally what you
459
00:31:04 --> 00:31:10
will do.
Nonetheless,
460
00:31:05 --> 00:31:11
in trying to understand how it
fits into this paradigm,
461
00:31:09 --> 00:31:15
which kind of equation is it?
Well, clearly there's an
462
00:31:12 --> 00:31:18
awkwardness in that on the
right-hand side,
463
00:31:15 --> 00:31:21
we have concentration,
and on the left-hand side,
464
00:31:19 --> 00:31:25
we seem to have amounts.
Now, the way to understand the
465
00:31:22 --> 00:31:28
equation as opposed to the way
to solve it, well,
466
00:31:26 --> 00:31:32
it's a step on the way to
solving it.
467
00:31:28 --> 00:31:34
But, I emphasize,
you can and normally will solve
468
00:31:32 --> 00:31:38
it in exactly that form.
But, to understand what's
469
00:31:36 --> 00:31:42
happening, it's better to
express it in terms of
470
00:31:40 --> 00:31:46
concentration entirely,
and that's why it's called the
471
00:31:43 --> 00:31:49
concentration,
or the diffusion,
472
00:31:45 --> 00:31:51
concentration-diffusion
equation.
473
00:31:48 --> 00:31:54
So, I'm going to convert this
to concentrations.
474
00:31:51 --> 00:31:57
Now, there's no problem here.
x over v is the concentration
475
00:31:55 --> 00:32:01
in the tank.
And now, immediately,
476
00:31:57 --> 00:32:03
you see, hey,
it looks like it's going to
477
00:32:00 --> 00:32:06
come out just in the first form.
But, wait a minute.
478
00:32:06 --> 00:32:12
How about the x?
How do I convert that?
479
00:32:10 --> 00:32:16
Well, what's the relation
between x?
480
00:32:13 --> 00:32:19
So, if the concentration in the
tank is equal to x over v,
481
00:32:19 --> 00:32:25
so the tank concentration,
then x is equal to C times the
482
00:32:25 --> 00:32:31
constant, V, and dx / dt,
therefore, will be c times dC /
483
00:32:31 --> 00:32:37
dt.
You see that?
484
00:32:34 --> 00:32:40
Now, that's not in standard
form.
485
00:32:38 --> 00:32:44
Let's put it in standard form.
To put it in standard form,
486
00:32:44 --> 00:32:50
I see, now, that it's not r
that's the critical quantity.
487
00:32:51 --> 00:32:57
It's r divided by v.
So, it's dC / dt,
488
00:32:55 --> 00:33:01
C prime, plus r divided by v,
--
489
00:33:00 --> 00:33:06
-- I'm going to call that k,
k C, no let's not,
490
00:33:04 --> 00:33:10
r divided by v is equal to r
divided by v times Ce.
491
00:33:09 --> 00:33:15
That's the equation expressed
492
00:33:14 --> 00:33:20
in a form where the
concentration is the dependent
493
00:33:18 --> 00:33:24
variable, rather than the amount
of salt itself.
494
00:33:23 --> 00:33:29
And, you can see it falls
exactly in this category.
495
00:33:27 --> 00:33:33
That means that I can talk
about it.
496
00:33:31 --> 00:33:37
The natural way to talk about
this equation is in terms of,
497
00:33:36 --> 00:33:42
the same way we talked about
the temperature equation.
498
00:33:43 --> 00:33:49
I said concentration.
I mean, that concentration has
499
00:33:47 --> 00:33:53
nothing to do with this
concentration.
500
00:33:50 --> 00:33:56
This is the diffusion model,
where salt solution outside,
501
00:33:54 --> 00:34:00
cell in the middle,
salt diffusing through a
502
00:33:58 --> 00:34:04
semi-permeable membrane into
that, uses Newton's law of
503
00:34:03 --> 00:34:09
diffusion, except he didn't do a
law of diffusion.
504
00:34:08 --> 00:34:14
But, he is sticky.
His name is attached to
505
00:34:10 --> 00:34:16
everything.
So, that's this concentration
506
00:34:13 --> 00:34:19
model.
It's the one entirely analogous
507
00:34:15 --> 00:34:21
to the temperature.
And the physical setup is the
508
00:34:18 --> 00:34:24
same.
This one is entirely different.
509
00:34:21 --> 00:34:27
Mixing in this form of this
problem has really nothing to do
510
00:34:24 --> 00:34:30
with this model whatsoever.
But, nor does that
511
00:34:27 --> 00:34:33
concentration had anything to do
with this concentration,
512
00:34:31 --> 00:34:37
which refers to the result of
the mixing in the tank.
513
00:34:36 --> 00:34:42
But, what happens is the
differential equation is the
514
00:34:40 --> 00:34:46
same.
The language of input and
515
00:34:43 --> 00:34:49
response that we talked about is
also available here.
516
00:34:47 --> 00:34:53
So, everything is the same.
And, the most interesting thing
517
00:34:52 --> 00:34:58
is that it shows that the analog
of the conductivity,
518
00:34:57 --> 00:35:03
the k, the analog of
conductivity and diffusivity is
519
00:35:01 --> 00:35:07
this quantity.
I should not be considering r
520
00:35:06 --> 00:35:12
and v by themselves.
I should be considering as the
521
00:35:11 --> 00:35:17
basic quantity,
the ratio of those two.
522
00:35:15 --> 00:35:21
Now, why is that,
is the basic parameter.
523
00:35:19 --> 00:35:25
What is this?
Well, r is the rate of outflow,
524
00:35:23 --> 00:35:29
and the rate of inflow,
what's r over v?
525
00:35:27 --> 00:35:33
r over v is the fractional rate
of outflow.
526
00:35:31 --> 00:35:37
In other words,
if r over v is one tenth,
527
00:35:35 --> 00:35:41
it means that 1/10 of the tank
will be emptied in a minute,
528
00:35:40 --> 00:35:46
say.
In other words,
529
00:35:44 --> 00:35:50
we lumped these two constants
into a single k,
530
00:35:49 --> 00:35:55
and at the same time have
simplified the units.
531
00:35:53 --> 00:35:59
What are the units?
This is volume per minute.
532
00:35:58 --> 00:36:04
This is volume.
So, it's simply reciprocal
533
00:36:02 --> 00:36:08
minutes, reciprocal time,
which was the same units of
534
00:36:07 --> 00:36:13
that diffusivity and
conductivity had,
535
00:36:11 --> 00:36:17
reciprocal time.
The space variables have
536
00:36:16 --> 00:36:22
entirely disappeared.
So, it that way,
537
00:36:19 --> 00:36:25
it's simplified.
It's simplified conceptually,
538
00:36:22 --> 00:36:28
and now, you can answer the
same type of questions we asked
539
00:36:27 --> 00:36:33
before about this.
I think it would be better for
540
00:36:32 --> 00:36:38
us to move on,
though.
541
00:36:33 --> 00:36:39
Well, just an example,
one really simple thing,
542
00:36:37 --> 00:36:43
so, suppose since we spent so
much time worrying about what
543
00:36:41 --> 00:36:47
was happening with sinusoid
inputs, I mean,
544
00:36:45 --> 00:36:51
when could Ce be sinusoidal,
for example?
545
00:36:48 --> 00:36:54
Well, roughly sinusoidal if,
for example,
546
00:36:51 --> 00:36:57
some factory were polluting.
If this were a lake,
547
00:36:55 --> 00:37:01
and some factory were polluting
it, but in the beginning,
548
00:36:59 --> 00:37:05
at the beginning of the day,
they produced a lot of the
549
00:37:03 --> 00:37:09
pollutant, and by the end of the
day when it wound down,
550
00:37:07 --> 00:37:13
it might well happen that the
concentration of pollutants in
551
00:37:12 --> 00:37:18
the incoming stream would vary
sinusoidally with a 24 hour
552
00:37:16 --> 00:37:22
cycle.
And then, we would be asking,
553
00:37:22 --> 00:37:28
so, suppose this varies
sinusoidally.
554
00:37:26 --> 00:37:32
In other words,
it's like cosine omega t.
555
00:37:31 --> 00:37:37
I'm asking, how closely does
556
00:37:36 --> 00:37:42
the concentration in the tank
follow C sub e?
557
00:37:44 --> 00:37:50
Now, what would that depend
upon?
558
00:37:48 --> 00:37:54
Think about it.
Well, the answer,
559
00:37:52 --> 00:37:58
suppose k is large.
Closely, let's just analyze one
560
00:37:58 --> 00:38:04
case, if k is big.
Now, what would make k big?
561
00:38:03 --> 00:38:09
We know that from the
temperature thing.
562
00:38:06 --> 00:38:12
If the conductivity is high,
then the inner temperature will
563
00:38:10 --> 00:38:16
follow the outer temperature
closely, and the same thing with
564
00:38:15 --> 00:38:21
the diffusion model.
But we, of course,
565
00:38:17 --> 00:38:23
therefore, since the equation
is the same, we must get the
566
00:38:21 --> 00:38:27
same result here.
Now, what would make k big?
567
00:38:25 --> 00:38:31
If r is big,
if the flow rate is very fast,
568
00:38:28 --> 00:38:34
we will expect the
concentration of the inside of
569
00:38:31 --> 00:38:37
that tank to match fairly
closely the concentration of the
570
00:38:35 --> 00:38:41
pollutant, of the incoming salt
solution, or,
571
00:38:38 --> 00:38:44
if the tank is very small.
For fixed flow rates,
572
00:38:43 --> 00:38:49
if the tank is very small,
well, then it gets emptied
573
00:38:47 --> 00:38:53
quickly.
So, both of these are,
574
00:38:49 --> 00:38:55
I think, intuitive results.
And, of course,
575
00:38:52 --> 00:38:58
as before, we got them from
that, by trying to analyze the
576
00:38:56 --> 00:39:02
final form of the solution.
In other words,
577
00:38:59 --> 00:39:05
we got them by looking at that
form of the solution up there,
578
00:39:03 --> 00:39:09
and seeing if k is big.
As k increases,
579
00:39:07 --> 00:39:13
what happens to the amplitude,
and what happens to the phase
580
00:39:12 --> 00:39:18
lag?
But, that summarizes the two.
581
00:39:14 --> 00:39:20
So, this means,
closely means,
582
00:39:17 --> 00:39:23
that the phase lag is,
big or small?
583
00:39:20 --> 00:39:26
The lag is small.
And, the amplitude is,
584
00:39:23 --> 00:39:29
well, the amplitude,
the biggest the amplitude could
585
00:39:27 --> 00:39:33
ever be is one because that's
the amplitude of this.
586
00:39:33 --> 00:39:39
So, the amplitude is near one,
one because that's the
587
00:39:38 --> 00:39:44
amplitude of the incoming
signal, input,
588
00:39:41 --> 00:39:47
whatever you want to call it.
Okay, now, I'd like to spend
589
00:39:47 --> 00:39:53
the rest of the time talking
about the failures of number
590
00:39:52 --> 00:39:58
one, and when you have to use
number two, and when even number
591
00:39:58 --> 00:40:04
two is no good.
So, let me end first-order
592
00:40:03 --> 00:40:09
equations by putting my worst
foot forward.
593
00:40:08 --> 00:40:14
Well, I'm just trying to avoid
disappointment at
594
00:40:13 --> 00:40:19
misapprehensions from you.
I'll watch you leave this room
595
00:40:19 --> 00:40:25
and say, well,
he said that,
596
00:40:22 --> 00:40:28
okay.
So, the first one you're going
597
00:40:26 --> 00:40:32
to encounter very shortly where
one is not satisfied,
598
00:40:32 --> 00:40:38
but two is, so on some examples
where you need two,
599
00:40:38 --> 00:40:44
well, it's going to happen
right here.
600
00:40:44 --> 00:40:50
Somebody, sooner or later,
it's going to draw on that
601
00:40:47 --> 00:40:53
loathsome orange chalk,
which is unerasable,
602
00:40:50 --> 00:40:56
something which looks like
that.
603
00:40:52 --> 00:40:58
Remember, you saw it here
first.
604
00:40:54 --> 00:41:00
r, yeah, we had that.
Okay, see, I had it in high
605
00:40:58 --> 00:41:04
school too.
That's the capacitance.
606
00:41:00 --> 00:41:06
This is the resistance.
That's the electromotive force:
607
00:41:04 --> 00:41:10
battery, or a source of
alternating current,
608
00:41:07 --> 00:41:13
something like that.
Now, of course,
609
00:41:11 --> 00:41:17
what you're interested in is
how the current flows in the
610
00:41:15 --> 00:41:21
circuit.
Since current across the
611
00:41:18 --> 00:41:24
capacitance doesn't make sense,
you have to talk about the
612
00:41:23 --> 00:41:29
charge on the capacitance.
So, q, it's customary in a
613
00:41:27 --> 00:41:33
circle this simple to use as the
variable not current,
614
00:41:32 --> 00:41:38
but the charge on the
capacitance.
615
00:41:36 --> 00:41:42
And then Kirchhoff's,
you are also supposed to know
616
00:41:40 --> 00:41:46
that the derivative,
that the time derivative of q
617
00:41:45 --> 00:41:51
is what's called the current in
the circuit.
618
00:41:49 --> 00:41:55
That sort of intuitive.
But, i in a physics class,
619
00:41:53 --> 00:41:59
j in an electrical engineering
class, and why,
620
00:41:57 --> 00:42:03
not the letter Y,
but why is that?
621
00:42:02 --> 00:42:08
That's because of electrical
engineers use lots of lots of
622
00:42:06 --> 00:42:12
complex numbers and therefore,
you have to call current j,
623
00:42:11 --> 00:42:17
I guess.
I think they do j in physics,
624
00:42:15 --> 00:42:21
too, now.
No, no they don't.
625
00:42:17 --> 00:42:23
I don't know.
So, i is ambiguous if you are
626
00:42:21 --> 00:42:27
in that particular subject.
And it's customary to use,
627
00:42:25 --> 00:42:31
I don't know.
Now it's completely safe.
628
00:42:29 --> 00:42:35
Okay, where are we?
Well, the law is,
629
00:42:32 --> 00:42:38
the basic differential equation
is Kirchhoff's voltage law,
630
00:42:36 --> 00:42:42
but the sum of the voltage
drops across these three has to
631
00:42:40 --> 00:42:46
be zero.
So, it's R times i,
632
00:42:42 --> 00:42:48
which is dq / dt.
That's Ohm's law.
633
00:42:44 --> 00:42:50
That's the voltage drop across
resistance.
634
00:42:47 --> 00:42:53
The voltage drop across the
capacitance is Coulomb's law,
635
00:42:51 --> 00:42:57
one form of Coulomb's law.
It's q divided by C.
636
00:42:55 --> 00:43:01
And, that has to be the voltage
drop.
637
00:42:57 --> 00:43:03
And then, there is some sign
convention.
638
00:43:02 --> 00:43:08
So, this is either plus or
minus, depending on your sign
639
00:43:06 --> 00:43:12
conventions, but it's E of t.
Now, if I put that in standard
640
00:43:11 --> 00:43:17
form, in standard form I
probably should say q prime plus
641
00:43:15 --> 00:43:21
q over RC equals,
well, I suppose,
642
00:43:18 --> 00:43:24
E over R.
And, this is what would appear
643
00:43:23 --> 00:43:29
in the equation.
But, it's not the natural
644
00:43:26 --> 00:43:32
thing.
The k is one over RC.
645
00:43:30 --> 00:43:36
And, that's the reciprocal.
646
00:43:32 --> 00:43:38
The RC constant is what
everybody knows is important
647
00:43:35 --> 00:43:41
when you talk about a little
circuit of that form.
648
00:43:39 --> 00:43:45
On the other hand,
the right-hand side,
649
00:43:41 --> 00:43:47
it's quite unnatural to try to
stick in the right-hand side
650
00:43:46 --> 00:43:52
that same RC.
Call this EC over RC.
651
00:43:48 --> 00:43:54
People don't do that,
and therefore,
652
00:43:50 --> 00:43:56
it doesn't really fall into the
paradigm of that first equation.
653
00:43:55 --> 00:44:01
It's the second equation that
really falls into the category.
654
00:43:59 --> 00:44:05
Another simple example of this
is chained to k,
655
00:44:02 --> 00:44:08
radioactively changed to k.
Well, let's say the radioactive
656
00:44:08 --> 00:44:14
substance, A,
decays into,
657
00:44:10 --> 00:44:16
let's say, one atom of this
produces one atom of that for
658
00:44:14 --> 00:44:20
simplicity.
So, it decays into B,
659
00:44:17 --> 00:44:23
which then still is radioactive
and decays further.
660
00:44:21 --> 00:44:27
Okay, what's the differential
equation, which is going to be,
661
00:44:26 --> 00:44:32
it's going to govern this
situation?
662
00:44:29 --> 00:44:35
What I want to know is how much
B there is at any given time.
663
00:44:35 --> 00:44:41
So, I want a differential
equation for the quantity of the
664
00:44:39 --> 00:44:45
radioactive product at any given
time.
665
00:44:41 --> 00:44:47
Well, what's it going to look
like?
666
00:44:44 --> 00:44:50
Well, it's the amount coming in
minus the amount going out,
667
00:44:48 --> 00:44:54
so to speak.
The rate of inflow minus the
668
00:44:51 --> 00:44:57
rate of outflow,
except it's not the same type
669
00:44:55 --> 00:45:01
of physical flow we had before.
How fast is it coming in?
670
00:44:59 --> 00:45:05
Well, A is decaying at a
certain rate,
671
00:45:01 --> 00:45:07
and so the rate at which A
decays is by the basic
672
00:45:05 --> 00:45:11
radioactive law.
It's k1, it's constant,
673
00:45:09 --> 00:45:15
decay constant,
times the amount of A present.
674
00:45:12 --> 00:45:18
If I used the differential
675
00:45:15 --> 00:45:21
equation with A here,
I'd have to put a negative sign
676
00:45:18 --> 00:45:24
because it's the rate at which
that stuff is leaving A.
677
00:45:22 --> 00:45:28
But, I'm interested in the rate
at which it's coming in to B.
678
00:45:26 --> 00:45:32
So, it has a positive sign.
And then, the rate at which B
679
00:45:30 --> 00:45:36
is decaying, and therefore the
quantity of good B is gone,
680
00:45:34 --> 00:45:40
--
-- that will have some other
681
00:45:38 --> 00:45:44
constant, B.
So, that will be the equation,
682
00:45:41 --> 00:45:47
and to avoid having two
dependent variables in there,
683
00:45:46 --> 00:45:52
we know how A is decaying.
So, it's k1,
684
00:45:49 --> 00:45:55
some constant times A,
sorry, A will be e to the
685
00:45:53 --> 00:45:59
negative, you know,
the decay law,
686
00:45:56 --> 00:46:02
so, times the initial amount
that was there times e to the
687
00:46:01 --> 00:46:07
negative k1 t.
That's how much A there is at
688
00:46:07 --> 00:46:13
any given time.
It's decaying by the
689
00:46:10 --> 00:46:16
radioactive decay law,
minus k2 B.
690
00:46:14 --> 00:46:20
Okay, so how does the
differential equation look like?
691
00:46:18 --> 00:46:24
It looks like B prime plus k2 B
equals an exponential,
692
00:46:23 --> 00:46:29
k1 A zero e to the negative k1
t.
693
00:46:28 --> 00:46:34
But, there's no reason to
694
00:46:31 --> 00:46:37
expect that that constant really
has anything to do with k2.
695
00:46:36 --> 00:46:42
It's unnatural to put it in
that form, which is the correct
696
00:46:41 --> 00:46:47
one.
Now, in the last two minutes,
697
00:46:49 --> 00:46:55
I wish to alienate half the
class by pointing out that if k
698
00:47:02 --> 00:47:08
is less than zero,
none of the terminology of
699
00:47:13 --> 00:47:19
transient, steady-state input
response applies.
700
00:47:25 --> 00:47:31
The technique of solving the
equation is identical.
701
00:47:28 --> 00:47:34
But, you cannot interpret.
So, the technique is the same,
702
00:47:33 --> 00:47:39
and therefore it's worth
learning.
703
00:47:37 --> 00:47:43
The technique is the same.
In other words,
704
00:47:41 --> 00:47:47
the solution will be still e to
the negative kt integral q of t
705
00:47:48 --> 00:47:54
e to the kt dt plus
706
00:47:54 --> 00:48:00
a constant times e to the k,
oh, this is terrible,
707
00:48:00 --> 00:48:06
no.
dy / dt, let's give an example.
708
00:48:04 --> 00:48:10
The equation I'm going to look
at is something that looks like
709
00:48:10 --> 00:48:16
this: y equals q of t,
let's say, okay,
710
00:48:13 --> 00:48:19
a constant, but the constant a
is positive.
711
00:48:17 --> 00:48:23
So, the constant here is
negative.
712
00:48:19 --> 00:48:25
Then, when I solve,
my k, in other words,
713
00:48:23 --> 00:48:29
is now properly written as
negative a.
714
00:48:26 --> 00:48:32
And therefore,
this formula should now become
715
00:48:30 --> 00:48:36
not this, but the negative k is
a t.
716
00:48:36 --> 00:48:42
And, here it's negative a t.
And, here it is positive a t.
717
00:48:41 --> 00:48:47
Now, why is it,
if this is going to be the
718
00:48:44 --> 00:48:50
solution, why are all those
things totally irrelevant?
719
00:48:49 --> 00:48:55
This is not a transient any
longer because if a is positive,
720
00:48:54 --> 00:49:00
this goes to infinity.
Or, if I go to minus infinity,
721
00:48:59 --> 00:49:05
then C is negative.
So, it's not transient.
722
00:49:03 --> 00:49:09
It's not going to zero,
and it depends heavily on the
723
00:49:07 --> 00:49:13
initial conditions.
That means that of these two
724
00:49:10 --> 00:49:16
functions, this is the important
guy.
725
00:49:12 --> 00:49:18
This is just fixed,
some fixed function.
726
00:49:15 --> 00:49:21
Everything, in other words,
is going to depend upon the
727
00:49:18 --> 00:49:24
initial conditions,
whereas in the other cases we
728
00:49:21 --> 00:49:27
have been studying,
the initial conditions after a
729
00:49:25 --> 00:49:31
while don't matter anymore.
Now, why did I say I would
730
00:49:30 --> 00:49:36
alienate half of you?
Well, because in what subjects
731
00:49:35 --> 00:49:41
will a be positive?
In what subjects will k be
732
00:49:39 --> 00:49:45
negative?
k is typically negative in
733
00:49:43 --> 00:49:49
biology, economics,
Sloan.
734
00:49:45 --> 00:49:51
In other words,
the simple thing is think of it
735
00:49:50 --> 00:49:56
in biology.
What's the simplest equation
736
00:49:53 --> 00:49:59
for population growth?
Well, it is dP / dt equals
737
00:49:58 --> 00:50:04
some, if the population is
growing, a times P,
738
00:50:02 --> 00:50:08
and a is a positive number.
That means P prime minus a P is
739
00:50:09 --> 00:50:15
zero.
So, the thing I want to leave
740
00:50:13 --> 00:50:19
you with is this.
If life is involved,
741
00:50:16 --> 00:50:22
k is likely to be negative.
k is positive when inanimate
742
00:50:22 --> 00:50:28
things are involved;
I won't say dead,
743
00:50:25 --> 00:50:31
inanimate.