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PROFESSOR STRANG: OK,
let's start with a
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review and preview.
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I put a P up there because
we're really looking into
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the Fourier part that just
started this morning.
13
00:00:36 --> 00:00:41
And there'll be some homework
from these early sections about
14
00:00:41 --> 00:00:44
the Fourier stuff, so we maybe
we should just do a few
15
00:00:44 --> 00:00:51
of those problems or
discuss here today.
16
00:00:51 --> 00:00:54
Just in advance.
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00:00:54 --> 00:00:57
Can I say one thing about
MATLAB and the MATLAB
18
00:00:57 --> 00:00:58
homework first?
19
00:00:58 --> 00:01:03
And maybe open a
conversation about it?
20
00:01:03 --> 00:01:10
So there's really two different
problems that I'm personally
21
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quite interested in.
22
00:01:13 --> 00:01:16
Two model, I'll say model
problems because they're
23
00:01:16 --> 00:01:20
there for regular
polyons in a circle.
24
00:01:20 --> 00:01:31
And I'll draw an octagon
again, so M sides.
25
00:01:31 --> 00:01:34
And I'm interested in
if M goes to infinity.
26
00:01:34 --> 00:01:37
And I'm interested in
two different problems.
27
00:01:37 --> 00:01:41
So one of them is our
MATLAB problem, minus
28
00:01:41 --> 00:01:46
is Laplace's equation.
29
00:01:46 --> 00:01:50
What was it, four?
30
00:01:50 --> 00:01:58
With u=0 on the circle.
31
00:01:58 --> 00:02:02
OK, so that's our problem,
totally open for discussion.
32
00:02:02 --> 00:02:05
How many have started on that?
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00:02:05 --> 00:02:06
Oh, good.
34
00:02:06 --> 00:02:06
OK.
35
00:02:06 --> 00:02:08
Well, then you all know
more about it than I.
36
00:02:08 --> 00:02:10
And that's great.
37
00:02:10 --> 00:02:14
I'd be happy to learn.
38
00:02:14 --> 00:02:15
So have I said
everything there?
39
00:02:15 --> 00:02:18
Yeah, we've got Poisson's
equation inside.
40
00:02:18 --> 00:02:23
We've got u=0 on the circle, so
the problem's well defined and
41
00:02:23 --> 00:02:28
the solution should be
one minus x squared
42
00:02:28 --> 00:02:33
minus y squared.
43
00:02:33 --> 00:02:35
So that's the correct solution.
44
00:02:35 --> 00:02:38
Maybe I can also tell you
about the second problem
45
00:02:38 --> 00:02:40
that I'm interested in.
46
00:02:40 --> 00:02:43
Because it hasn't come
up in class but it's
47
00:02:43 --> 00:02:45
very important, too.
48
00:02:45 --> 00:02:47
It would be the
eigenvalue problem.
49
00:02:47 --> 00:02:52
So this is problem number one,
the steady state problem
50
00:02:52 --> 00:02:57
when you've got a source and
you want to find out the
51
00:02:57 --> 00:02:59
temperature distribution.
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00:02:59 --> 00:03:01
The problem number two
would be the eigenvalue
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problem, -u_xx-u_yy.
54
00:03:03 --> 00:03:06
55
00:03:06 --> 00:03:10
I take those minuses so
that the eigenvalue
56
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will be positive.
57
00:03:12 --> 00:03:15
So that's what the eigenvalue
problem might look like.
58
00:03:15 --> 00:03:20
And again let me say, with
u=0 on the boundary.
59
00:03:20 --> 00:03:25
On the third one.
60
00:03:25 --> 00:03:33
OK, so a person would say
this is Laplace's eigenvalue
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problem because we have
Laplace's equation.
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We've got eigenvalue.
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As always, it's not linear
because we have two unknowns,
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lambda's multiplying u.
65
00:03:45 --> 00:03:48
And we have boundary
conditions, and this would
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00:03:48 --> 00:03:56
describe the normal modes, for
example, of a circular drum.
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00:03:56 --> 00:04:00
If I had a drum, or
a polygon drum.
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So maybe I connect to actually
build the drum, I might fold in
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the sides there and
have a polygon.
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And again, I hope that the
eigenvalues of the polygon,
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this equation in the
polygon, which are not
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known, by the way.
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To the best to my knowledge, we
know it only for M=3, which
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would be an equilateral
triangle, and M=4, which
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would be a square.
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And those eigenvalues, because
of Fourier or something
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are humanly doable.
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But I think five on up is, I
may be wrong about six I'm
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not sure about M=6,
a hexagon sometimes
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gives you enough help.
81
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But beyond that
you're on your own.
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With finite elements
to help you.
83
00:05:03 --> 00:05:09
So there's a whole sequence of
eigenfunctions, u, eigenvalues,
84
00:05:09 --> 00:05:15
lambda, just the way there
were in one dimension.
85
00:05:15 --> 00:05:18
And on the circle
they involve Bessel.
86
00:05:18 --> 00:05:21
That's where Bessel showed up.
87
00:05:21 --> 00:05:25
He figured out the functions
and they're not especially
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nice functions.
89
00:05:29 --> 00:05:32
But they're studied
for centuries.
90
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Bessel functions
come into that.
91
00:05:36 --> 00:05:40
But, here I have
the same question.
92
00:05:40 --> 00:05:45
I mean, let me just say, for me
this could be a UROP project if
93
00:05:45 --> 00:05:48
anybody was an undergraduate,
or it could be a project
94
00:05:48 --> 00:05:51
over January or something.
95
00:05:51 --> 00:05:57
I'd like to know something
about what happens as M goes
96
00:05:57 --> 00:06:03
to infinity as the polygon
approaches the circle.
97
00:06:03 --> 00:06:11
So I'm hoping maybe on the
homework that comes, if it's
98
00:06:11 --> 00:06:18
not too difficult, and maybe
it's not, to let m go up a bit.
99
00:06:18 --> 00:06:20
There is one thing.
100
00:06:20 --> 00:06:25
That the code we're working
with is linear elements, right?
101
00:06:25 --> 00:06:27
We're using linear
finite elements.
102
00:06:27 --> 00:06:32
So we're not getting
high accuracy.
103
00:06:32 --> 00:06:38
So I would really like to move
up to quadratic elements, at
104
00:06:38 --> 00:06:42
least, you remember quadratic
elements would be ones where -
105
00:06:42 --> 00:06:52
well, let me draw the one that
we've drawn in class before.
106
00:06:52 --> 00:06:56
We only have to look at one
triangle and then we cut it up
107
00:06:56 --> 00:07:02
into triangular elements by
taking some pieces here, taking
108
00:07:02 --> 00:07:07
the points above, which I hope
are now correct on the website.
109
00:07:07 --> 00:07:11
Connecting those edges, and
then connecting these.
110
00:07:11 --> 00:07:12
Is that right?
111
00:07:12 --> 00:07:18
Is that our mesh?
112
00:07:18 --> 00:07:20
So that mesh is
controlled by N.
113
00:07:20 --> 00:07:23
One, two, N points.
114
00:07:23 --> 00:07:32
Also, N is going to have to get
large too, to give me accuracy.
115
00:07:32 --> 00:07:38
And another way toward more
accuracy is, instead of linear
116
00:07:38 --> 00:07:42
elements, second degree.
117
00:07:42 --> 00:07:44
So do you remember I
wrote those down?
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Let me take that little
triangle out here as
119
00:07:48 --> 00:07:51
a bigger triangle.
120
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It would look something
like that, I guess.
121
00:07:53 --> 00:07:58
The second degree elements
have those six mesh points.
122
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You remember I drew those but
we didn't really have time
123
00:08:01 --> 00:08:07
to get further with them.
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The trial functions phi, which
are one at a typical mesh point
125
00:08:12 --> 00:08:18
and zero at all the others,
they are computable.
126
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We're up to second degree, so
it's a little, second degree
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00:08:22 --> 00:08:26
things, then the first
derivatives, which come into
128
00:08:26 --> 00:08:28
the integrations, are linear.
129
00:08:28 --> 00:08:29
And not constant.
130
00:08:29 --> 00:08:31
So a little bit harder.
131
00:08:31 --> 00:08:37
But finite elements,
linear or quadratic.
132
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Or higher.
133
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Could be used for this problem.
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As we know, and
for this problem.
135
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What I wanted to add, that I've
not mentioned in class, and I
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00:08:48 --> 00:08:55
think we may just not get a
chance to do it, is what does
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00:08:55 --> 00:08:58
the finite element method look
like for an eigenvalue problem?
138
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Because eigenvalues
are highly important.
139
00:09:02 --> 00:09:06
That's the different
way to understand.
140
00:09:06 --> 00:09:08
There's the matrix
K and its entries.
141
00:09:08 --> 00:09:11
But then there eigenvalues.
142
00:09:11 --> 00:09:14
And you might think that, what
do you think is the discrete
143
00:09:14 --> 00:09:19
eigenvalue problem
copying this one?
144
00:09:19 --> 00:09:21
Here's my point.
145
00:09:21 --> 00:09:26
Your first guess would be,
well this is like K, right?
146
00:09:26 --> 00:09:34
This is like KU, right? (K2D)U,
I should call it, maybe.
147
00:09:34 --> 00:09:39
Well, I'll call it K, because
K2D I have specifically
148
00:09:39 --> 00:09:47
reserved for the Laplace
stiffness matrix on a square
149
00:09:47 --> 00:09:50
mesh, square mesh with
triangles, the K2D.
150
00:09:51 --> 00:09:57
That was one specific matrix
for one specific mesh, and here
151
00:09:57 --> 00:09:59
we have a different mesh.
152
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So I should just call it K.
153
00:10:01 --> 00:10:06
Ok, I think if anybody was
going to make a guess, they
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00:10:06 --> 00:10:07
would say OK, KU=LAMBDA*U.
155
00:10:09 --> 00:10:13
Maybe I'll use capital LAMBDA,
because I'm using capital U.
156
00:10:13 --> 00:10:28
Is this the finite element
method eigenvalue problem.
157
00:10:28 --> 00:10:33
And if you answered yes, I
would have to say, well
158
00:10:33 --> 00:10:35
that's a reasonable answer.
159
00:10:35 --> 00:10:38
But it's wrong.
160
00:10:38 --> 00:10:43
The eigenvalue problem, when I
take the differential equation
161
00:10:43 --> 00:10:47
for the Laplace, Laplace's
equation, lambda u on the right
162
00:10:47 --> 00:10:55
side, and I go to do finite
elements, it produces K.
163
00:10:55 --> 00:11:00
Out of this stuff, out of the
weak form, all that stuff.
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00:11:00 --> 00:11:04
But it produces another matrix
on the right-hand side from the
165
00:11:04 --> 00:11:07
constant term, and we have
not really mentioned it,
166
00:11:07 --> 00:11:09
it's the mass matrix.
167
00:11:09 --> 00:11:13
So this, instead of just
the identity here,
168
00:11:13 --> 00:11:16
there's a mass matrix.
169
00:11:16 --> 00:11:21
So that is the problem
that you could do.
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00:11:21 --> 00:11:28
I could've made a
MATLAB project.
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00:11:28 --> 00:11:32
I bet I'd do it
next fall. right?
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00:11:32 --> 00:11:39
You guys did the
first one, this one.
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00:11:39 --> 00:11:40
Or you are doing it now.
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00:11:40 --> 00:11:43
And I'm going to pause in a
minute for questions about
175
00:11:43 --> 00:11:46
it, or discussion of it.
176
00:11:46 --> 00:11:49
But this one brings
in something called
177
00:11:49 --> 00:11:50
the mass matrix.
178
00:11:50 --> 00:11:58
So let me just say
what those are.
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00:11:58 --> 00:12:02
If I write down the entries in
the mass matrix, you'll sort of
180
00:12:02 --> 00:12:04
get an idea of why they are.
181
00:12:04 --> 00:12:07
So what are the entries
in the stiffness matrix?
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00:12:07 --> 00:12:15
K_ij, you remember, is
the integral of the d
183
00:12:15 --> 00:12:18
phi_i/dx, d phi_j/dx.
184
00:12:20 --> 00:12:33
Plus d phi_i/dy, d phi_j/dy,
dxdy, and that's what's
185
00:12:33 --> 00:12:34
you're computing.
186
00:12:34 --> 00:12:36
And that's what that
code is computing.
187
00:12:36 --> 00:12:42
And when phi is linear,
phi linear, then
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00:12:42 --> 00:12:47
slopes are constant.
189
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So all you have to do, and what
that code in the book is doing,
190
00:12:52 --> 00:12:55
is figuring out what
are the slopes.
191
00:12:55 --> 00:13:01
These things are constant, so
we just need to know the area
192
00:13:01 --> 00:13:06
of the integration where
we're integrating.
193
00:13:06 --> 00:13:09
The area, triangle by triangle.
194
00:13:09 --> 00:13:11
Fine.
195
00:13:11 --> 00:13:12
That's what we're doing.
196
00:13:12 --> 00:13:16
That's what that code
is just set up to do.
197
00:13:16 --> 00:13:20
Now, I have to tell you what
is M_ij, the mass matrix.
198
00:13:20 --> 00:13:25
I just think you don't want to
have - we haven't done too
199
00:13:25 --> 00:13:27
badly with finite that
elements in here.
200
00:13:27 --> 00:13:31
We did it in 1-D, where we
got it kind of straight.
201
00:13:31 --> 00:13:34
And now we're seeing
what it looks like 2-D.
202
00:13:34 --> 00:13:38
But I had not really
mentioned a mass matrix.
203
00:13:38 --> 00:13:42
So here it comes.
204
00:13:42 --> 00:13:45
The mass matrix will
be the integral of
205
00:13:45 --> 00:13:48
phi_i i times phi_j.
206
00:13:49 --> 00:13:50
dxdy.
207
00:13:50 --> 00:13:57
It's the zero order, no
derivatives, just plain zero
208
00:13:57 --> 00:14:07
order, as you'd expect from the
fact that the term in the
209
00:14:07 --> 00:14:09
continuous part is zero order.
210
00:14:09 --> 00:14:12
So it's this mass
matrix that comes in.
211
00:14:12 --> 00:14:20
And maybe we could just look
to see which entries will
212
00:14:20 --> 00:14:23
be zero and which will not.
213
00:14:23 --> 00:14:25
How sparse is it?
214
00:14:25 --> 00:14:28
What does the mass
matrix look like?
215
00:14:28 --> 00:14:33
And can we just, let
me do 1-D first.
216
00:14:33 --> 00:14:36
So there's a phi, right?
217
00:14:36 --> 00:14:38
There's another one.
218
00:14:38 --> 00:14:40
There's another one.
219
00:14:40 --> 00:14:45
So, what do you think about the
mass matrix, one phi multiplied
220
00:14:45 --> 00:14:48
by another phi and integrated?
221
00:14:48 --> 00:14:51
Is it diagonal?
222
00:14:51 --> 00:15:00
No, because each phi
overlaps its two neighbors.
223
00:15:00 --> 00:15:02
So tell me what kind of a
matrix m is going to be?
224
00:15:02 --> 00:15:04
In 1-D.
225
00:15:04 --> 00:15:07
Tri-diagonal.
226
00:15:07 --> 00:15:08
It'll be tri-diagonal.
227
00:15:08 --> 00:15:11
Now, so was K.
228
00:15:11 --> 00:15:15
So K and M actually have
non-zeroes in the same places.
229
00:15:15 --> 00:15:17
the same sparsity pattern.
230
00:15:17 --> 00:15:20
But, of course, not the
same numbers in there.
231
00:15:20 --> 00:15:31
K had minus ones and twos
and fours and minus ones.
232
00:15:31 --> 00:15:36
What can you tell me about
this tri-diagonal matrix?
233
00:15:36 --> 00:15:42
When I integrate that against
this, well, again I would do it
234
00:15:42 --> 00:15:45
element by element because this
against this, they
235
00:15:45 --> 00:15:48
only overlap here.
236
00:15:48 --> 00:15:48
Right?
237
00:15:48 --> 00:15:51
I'll just draw the one
place that they overlap.
238
00:15:51 --> 00:15:54
And what's the point?
239
00:15:54 --> 00:15:56
They're both positive.
240
00:15:56 --> 00:16:02
So the mass matrix is, its
rows don't add to zero.
241
00:16:02 --> 00:16:04
Its rows tend to add to one.
242
00:16:04 --> 00:16:08
But it's not diagonal,
that's the difference.
243
00:16:08 --> 00:16:16
OK, so I just felt I couldn't
feel as though I'd done a
244
00:16:16 --> 00:16:21
decent job in describing
finite elements if I
245
00:16:21 --> 00:16:23
didn't describe this.
246
00:16:23 --> 00:16:26
Didn't mention
this mass matrix.
247
00:16:26 --> 00:16:30
And maybe I'd better say
where it comes from.
248
00:16:30 --> 00:16:35
Because eigenvalue problems, it
may come number two, but that's
249
00:16:35 --> 00:16:37
pretty high up the list.
250
00:16:37 --> 00:16:50
So let me tell you where does
this mass matrix come from.
251
00:16:50 --> 00:16:52
First, let me tell you
about eigenvalues of
252
00:16:52 --> 00:16:56
a, matrix eigenvalues.
253
00:16:56 --> 00:17:00
So the answer was, is
this the finite element
254
00:17:00 --> 00:17:01
eigenvalue problem?
255
00:17:01 --> 00:17:03
Only if there's an M there.
256
00:17:03 --> 00:17:11
And now I want to, OK, first
of all, what MATLAB command
257
00:17:11 --> 00:17:13
solves that problem?
258
00:17:13 --> 00:17:16
Let's just be a little
practical for a moment.
259
00:17:16 --> 00:17:23
What MATLAB command gives me
the matrix of eigenvectors, the
260
00:17:23 --> 00:17:32
matrix of eigenvalues would
come from eig of what?
261
00:17:32 --> 00:17:35
I'd call this the generalized
eigenvalue problem.
262
00:17:35 --> 00:17:38
Generalized because it's
got somebody over here.
263
00:17:38 --> 00:17:39
And it's just K,M.
264
00:17:42 --> 00:17:46
Or of course you get the same
answer, well you get the same
265
00:17:46 --> 00:17:50
eigenvalues, I guess the same
eigenvectors, yeah, if you
266
00:17:50 --> 00:17:57
or, eig(M^-1,K), of course.
267
00:17:57 --> 00:18:00
If you want to do it with
just one matrix then bring
268
00:18:00 --> 00:18:01
M inverse over here.
269
00:18:01 --> 00:18:06
But and M inverse, the
inverse of this tridiagonal
270
00:18:06 --> 00:18:08
matrix, is full.
271
00:18:08 --> 00:18:11
No zeroes in the inverse.
272
00:18:11 --> 00:18:14
So everybody would much
prefer this tridiagonal
273
00:18:14 --> 00:18:16
tridiagonal one.
274
00:18:16 --> 00:18:19
So that's how MATLAB
would do it.
275
00:18:19 --> 00:18:27
And what I want to know is,
back in this problem, how close
276
00:18:27 --> 00:18:39
do the finite element guys
come, on polygons, come to the
277
00:18:39 --> 00:18:41
correct solution on circles.
278
00:18:41 --> 00:18:47
I'm hoping that for problem
one you can maybe keep M
279
00:18:47 --> 00:18:53
and N equal, or maybe four
times M or something.
280
00:18:53 --> 00:18:58
And let them grow and see.
281
00:18:58 --> 00:19:02
Well, for example, at the
center of the circle, or how
282
00:19:02 --> 00:19:05
quickly do you approach the
correct answer one at the
283
00:19:05 --> 00:19:07
center of the circle?
284
00:19:07 --> 00:19:09
I think it's going to
be a good problem.
285
00:19:09 --> 00:19:13
Let me open to, so I started
out just talking there.
286
00:19:13 --> 00:19:17
What about the MATLAB problem.
287
00:19:17 --> 00:19:24
You made a start on
it, is it going?
288
00:19:24 --> 00:19:30
Have you got a graph, maybe, or
what's reasonable to graph,
289
00:19:30 --> 00:19:35
to give Peter to look at?
290
00:19:35 --> 00:19:39
Who's done something on
that MATLAB problem?
291
00:19:39 --> 00:19:43
Yeah, go ahead tell
us all what to do.
292
00:19:43 --> 00:19:45
AUDIENCE: I made the
triangle pi section
293
00:19:45 --> 00:19:48
PROFESSOR STRANG: OK, right
294
00:19:48 --> 00:19:49
AUDIENCE: [INAUDIBLE]
295
00:19:49 --> 00:19:55
and I found that
the [INAUDIBLE]
296
00:19:55 --> 00:19:57
changes to M.
297
00:19:57 --> 00:19:59
PROFESSOR STRANG:
With M more, I see.
298
00:19:59 --> 00:20:07
So if you just fixed M like
eight, and let n get, it
299
00:20:07 --> 00:20:09
didn't change significantly.
300
00:20:09 --> 00:20:12
It wouldn't, of course,
converge to the right answer.
301
00:20:12 --> 00:20:15
It'll converge, if it
does, to some kind of an
302
00:20:15 --> 00:20:18
answer, for the polygon.
303
00:20:18 --> 00:20:19
Right.
304
00:20:19 --> 00:20:19
That's right.
305
00:20:19 --> 00:20:26
So you know, as I wrote the
problem I didn't know whether I
306
00:20:26 --> 00:20:31
dared say let M get increased
too, but of course that's
307
00:20:31 --> 00:20:32
the real question.
308
00:20:32 --> 00:20:34
And what happened then?
309
00:20:34 --> 00:20:37
Did error shrink?
310
00:20:37 --> 00:20:41
OK, and now maybe it's
possible to see how fast or
311
00:20:41 --> 00:20:42
something that's always--
312
00:20:42 --> 00:20:46
AUDIENCE: [INAUDIBLE]
313
00:20:46 --> 00:20:47
PROFESSOR STRANG: Ah.
314
00:20:47 --> 00:20:49
OK, at the center.
315
00:20:49 --> 00:20:53
OK, then I hope
for more comment.
316
00:20:53 --> 00:20:55
Let me say one more thing.
317
00:20:55 --> 00:21:01
My theory is that the error
at the center is quite a
318
00:21:01 --> 00:21:06
bit smaller than the error
closer to the boundary.
319
00:21:06 --> 00:21:13
I would be interested in an
error, is it fairly even?
320
00:21:13 --> 00:21:16
Oh, my theory's wrong.
321
00:21:16 --> 00:21:18
It wouldn't be the first time.
322
00:21:18 --> 00:21:21
And maybe because it's linear.
323
00:21:21 --> 00:21:28
Yeah, my theory is more for
better elements, like these.
324
00:21:28 --> 00:21:30
I'd be interested to know.
325
00:21:30 --> 00:21:37
Why do I think, why do I have
this theory, which you guys
326
00:21:37 --> 00:21:41
are going to prove wrong
anyway, but still.
327
00:21:41 --> 00:21:43
After you've proved it wrong,
you won't listen to me
328
00:21:43 --> 00:21:44
if I tell it to you.
329
00:21:44 --> 00:21:45
So now I'll tell it.
330
00:21:45 --> 00:21:54
My theory is that the error
around the boundary is, there's
331
00:21:54 --> 00:21:57
no error at these vertices, and
then there's sort of a going to
332
00:21:57 --> 00:22:01
be an error because the real
answer is not zero along here.
333
00:22:01 --> 00:22:04
It's sort of near
zero, but not quite.
334
00:22:04 --> 00:22:07
You know, there's an error.
335
00:22:07 --> 00:22:10
So there's errors around
here, from getting
336
00:22:10 --> 00:22:13
the boundary wrong.
337
00:22:13 --> 00:22:16
Squaring it off.
338
00:22:16 --> 00:22:20
But my theory is that errors,
the boundary stuff, drops off
339
00:22:20 --> 00:22:22
quickly as you go inside.
340
00:22:22 --> 00:22:26
That's why I think, from those,
you remember those - well,
341
00:22:26 --> 00:22:31
we'll see them again either
today or Friday, those
342
00:22:31 --> 00:22:36
r^n*cos(nx) type things?
343
00:22:36 --> 00:22:37
That cos(n*theta)?
344
00:22:39 --> 00:22:42
Yeah, you remember those
are the typical solutions
345
00:22:42 --> 00:22:44
to Laplace's equation.
346
00:22:44 --> 00:22:48
And then so that if, and
it has some coefficient,
347
00:22:48 --> 00:22:50
of course, a n.
348
00:22:50 --> 00:22:56
And I look at that, that
might be a piece of error.
349
00:22:56 --> 00:22:59
And it's way bigger
when r is one and way
350
00:22:59 --> 00:23:01
smaller when r is zero.
351
00:23:01 --> 00:23:05
So anyway, that's
sort of my theory.
352
00:23:05 --> 00:23:10
That if you have,
like physically.
353
00:23:10 --> 00:23:18
You have a circular plate
and you're maintaining the
354
00:23:18 --> 00:23:22
boundary temperature at
some sort of oscillation.
355
00:23:22 --> 00:23:27
Like, near one but up
and down from one.
356
00:23:27 --> 00:23:33
Then I think further
inside, it doesn't know.
357
00:23:33 --> 00:23:36
It hardly knows about
that oscillation.
358
00:23:36 --> 00:23:38
This is my theory.
359
00:23:38 --> 00:23:43
That toward the center of the
circle it only sees kind of an
360
00:23:43 --> 00:23:47
average boundary temperature
and not your little
361
00:23:47 --> 00:23:49
ups and downs.
362
00:23:49 --> 00:23:56
So when M is big, I expect that
part of the up and down part to
363
00:23:56 --> 00:23:59
be not so significant
in the center.
364
00:23:59 --> 00:24:01
Anyway, now that's my theory.
365
00:24:01 --> 00:24:04
AUDIENCE: [INAUDIBLE]
366
00:24:04 --> 00:24:11
PROFESSOR STRANG:
Ah, good question.
367
00:24:11 --> 00:24:15
So if we only looked
at the center, would
368
00:24:15 --> 00:24:17
it all be the same?
369
00:24:17 --> 00:24:20
I mean, if we're only looking
at that one point where it
370
00:24:20 --> 00:24:32
should be 1 at the center, but
along the thing, I don't know.
371
00:24:32 --> 00:24:38
If you look at both, and see a
significant difference in the
372
00:24:38 --> 00:24:40
behavior I'd be interested.
373
00:24:40 --> 00:24:41
Yeah, yeah.
374
00:24:41 --> 00:24:43
You know, all these problems
are things that there's
375
00:24:43 --> 00:24:47
no single solution to.
376
00:24:47 --> 00:24:53
AUDIENCE: [INAUDIBLE]
377
00:24:53 --> 00:24:55
PROFESSOR STRANG: The error
between one minus r squared
378
00:24:55 --> 00:25:04
AUDIENCE: [INAUDIBLE]
379
00:25:04 --> 00:25:06
PROFESSOR STRANG: Oh, right,
we've got slope error, too.
380
00:25:06 --> 00:25:10
That's a very
significant point.
381
00:25:10 --> 00:25:13
I see, right.
382
00:25:13 --> 00:25:14
So the slope error's in there.
383
00:25:14 --> 00:25:19
Everybody knows, then,
everybody in working the
384
00:25:19 --> 00:25:26
problem, I mentioned that the
boundary conditions in this
385
00:25:26 --> 00:25:32
piece of pie were zero along
here and normal derivative,
386
00:25:32 --> 00:25:36
somehow it got printed du/dh,
but that was an accident.
387
00:25:36 --> 00:25:42
It should've been
du/dn, dn is zero.
388
00:25:42 --> 00:25:46
So Neumann conditions on this
thing and then I was a little
389
00:25:46 --> 00:25:50
scared about that point,
but I think phooey on it.
390
00:25:50 --> 00:25:56
It's just, don't
worry about it.
391
00:25:56 --> 00:25:59
But what I was going to say.
392
00:25:59 --> 00:26:04
How do you, what do you do to
take into account this du/dn=0?
393
00:26:04 --> 00:26:07
394
00:26:07 --> 00:26:13
This slope condition on
these long boundaries?
395
00:26:13 --> 00:26:15
What should you do in
finite elements to
396
00:26:15 --> 00:26:17
take account for that?
397
00:26:17 --> 00:26:21
And the answer is,
in one nice word?
398
00:26:21 --> 00:26:22
Nothing.
399
00:26:22 --> 00:26:24
Right, nothing.
400
00:26:24 --> 00:26:27
Your finite element method
should not, you don't
401
00:26:27 --> 00:26:30
impose any condition
along these boundaries.
402
00:26:30 --> 00:26:35
Just use the code as it is
with zeroes on this boundary.
403
00:26:35 --> 00:26:39
And it should work, yeah.
404
00:26:39 --> 00:26:39
It should work.
405
00:26:39 --> 00:26:43
Any comments on other people.
406
00:26:43 --> 00:26:48
Did you get reasonable
results, or?
407
00:26:48 --> 00:26:49
Tell me something.
408
00:26:49 --> 00:26:55
Because you guys looked at
those graphs and I have not.
409
00:26:55 --> 00:26:57
Any feedback yet?
410
00:26:57 --> 00:26:58
On those?
411
00:26:58 --> 00:27:01
I'm happy to get
email, too, about.
412
00:27:01 --> 00:27:04
So all the email, first of
all they've corrected the
413
00:27:04 --> 00:27:09
typos in the original
coordinate positions.
414
00:27:09 --> 00:27:14
And now they've pointed
out I'd better look at M
415
00:27:14 --> 00:27:19
is very, very welcome.
416
00:27:19 --> 00:27:22
It doesn't mean that everybody
has to do this, if you've
417
00:27:22 --> 00:27:25
completed that MATLAB
assignment, you never want to
418
00:27:25 --> 00:27:30
see it again, and you've
kept M=8, it's ok.
419
00:27:30 --> 00:27:36
But if you're interested to see
what happens if M goes to 16
420
00:27:36 --> 00:27:39
or 32, I'm interested also.
421
00:27:39 --> 00:27:41
Right, yeah.
422
00:27:41 --> 00:27:45
OK, so anyway that's the
problem we're really
423
00:27:45 --> 00:27:45
thinking about.
424
00:27:45 --> 00:27:51
And that's the problem that is
equally important, but it
425
00:27:51 --> 00:27:55
seemed reasonable just
to do one of the two.
426
00:27:55 --> 00:27:59
And we were set up to do, we
have the code for the stiffness
427
00:27:59 --> 00:28:07
matrix, we would need a new
code to do these integrals.
428
00:28:07 --> 00:28:13
Because this will be linear
times linear, right?
429
00:28:13 --> 00:28:19
I'll have to compute that
one times this one and I
430
00:28:19 --> 00:28:23
would need new formulas
that are not there.
431
00:28:23 --> 00:28:26
I'd need formulas for, this
will be linear times linear
432
00:28:26 --> 00:28:31
so I'll be integrating
x squared type stuff.
433
00:28:31 --> 00:28:36
And xy's, because I'm
2-D, and y squareds.
434
00:28:36 --> 00:28:43
So it would take a little
more code, but not much.
435
00:28:43 --> 00:28:47
I think the math, oh here's
a question for you.
436
00:28:47 --> 00:28:49
Here's a question for you.
437
00:28:49 --> 00:28:52
Suppose I have my trial
functions, phi_i(x).
438
00:28:52 --> 00:28:56
439
00:28:56 --> 00:29:00
What do they add up to?
440
00:29:00 --> 00:29:05
Let me again draw a mesh,
so I've got a mesh.
441
00:29:05 --> 00:29:10
These are you know, I'm
sorry, I want to put in
442
00:29:10 --> 00:29:14
some more triangles here.
443
00:29:14 --> 00:29:18
Lots of triangles, whatever.
444
00:29:18 --> 00:29:23
Let me get some more
vertices, too.
445
00:29:23 --> 00:29:25
I'm getting in trouble.
446
00:29:25 --> 00:29:28
OK, whatever.
447
00:29:28 --> 00:29:33
So phi_i, is the piecewise
linear guy that
448
00:29:33 --> 00:29:35
is one at node i.
449
00:29:35 --> 00:29:38
So I've got all these
different nodes.
450
00:29:38 --> 00:29:41
I need a node there, so I've
got one, two, three, there's
451
00:29:41 --> 00:29:44
a node, there's more nodes.
452
00:29:44 --> 00:29:48
If I add them all up,
this is just like in
453
00:29:48 --> 00:29:51
an insights question.
454
00:29:51 --> 00:29:57
I've got all these, you could
add up these hats in 1-D.
455
00:29:57 --> 00:30:00
What's the sum of the
hats in one dimension?
456
00:30:00 --> 00:30:01
One.
457
00:30:01 --> 00:30:03
Good.
458
00:30:03 --> 00:30:05
The sum is one.
459
00:30:05 --> 00:30:09
It's a nice fact that
these guys add up to one.
460
00:30:09 --> 00:30:14
And now why is it still true
here in 2-D, that these little
461
00:30:14 --> 00:30:18
pyramids will add to one?
462
00:30:18 --> 00:30:22
That's an inside question, but
it's worth thinking about.
463
00:30:22 --> 00:30:25
Why do those pyramids
add to one?
464
00:30:25 --> 00:30:29
Let me leave that question.
465
00:30:29 --> 00:30:32
I'm thinking about, we
haven't imposed any
466
00:30:32 --> 00:30:34
boundary conditions yet.
467
00:30:34 --> 00:30:39
We've got them all. and I claim
that if we add up all the
468
00:30:39 --> 00:30:44
pyramids including the boundary
chopped off pyramids from the
469
00:30:44 --> 00:30:48
boundary, that we'll get
one throughout the whole,
470
00:30:48 --> 00:30:49
now it'll be phi(x,y).
471
00:30:49 --> 00:30:52
472
00:30:52 --> 00:30:56
Because now I'm moving
to 2-D, with pyramids.
473
00:30:56 --> 00:31:00
I think we'll still have one.
474
00:31:00 --> 00:31:02
Let me give you a minute
to think about that one.
475
00:31:02 --> 00:31:08
And then we could turn to
Fourier questions if you
476
00:31:08 --> 00:31:15
would like, we could do some
problems from the text.
477
00:31:15 --> 00:31:17
Any thoughts about this guy?
478
00:31:17 --> 00:31:24
Why should all those
individual pyramids add
479
00:31:24 --> 00:31:30
up to a flat group?
480
00:31:30 --> 00:31:31
Why did it work here?
481
00:31:31 --> 00:31:41
Well, it worked because you
could see it, right, somehow?
482
00:31:41 --> 00:31:47
Does it still work if the
nodes are not equally spaced?
483
00:31:47 --> 00:31:51
So we've got a hat function for
that guy, and a hat function
484
00:31:51 --> 00:31:54
for this guy, and a hat
function for this guy.
485
00:31:54 --> 00:31:57
And these guys are
in there, too.
486
00:31:57 --> 00:32:00
We haven't imposed anything.
487
00:32:00 --> 00:32:08
So those one, two, three, four,
five functions, five phis,
488
00:32:08 --> 00:32:14
they add up to one and y.
489
00:32:14 --> 00:32:18
Well, you're going to say it's
obvious, but that's what
490
00:32:18 --> 00:32:20
professors are allowed to say.
491
00:32:20 --> 00:32:24
Things are obvious, you
have to actually say why.
492
00:32:24 --> 00:32:26
Which is not as easy.
493
00:32:26 --> 00:32:36
So, why do they add to one?
494
00:32:36 --> 00:32:41
Let me look inside one element.
495
00:32:41 --> 00:32:47
Why does the sum of these
two guys add to a flat
496
00:32:47 --> 00:32:51
top inside that interval?
497
00:32:51 --> 00:32:57
AUDIENCE: [INAUDIBLE]
498
00:32:57 --> 00:33:00
PROFESSOR STRANG: At the
end points, you've got it.
499
00:33:00 --> 00:33:04
Because what's happening
at the end points?
500
00:33:04 --> 00:33:10
This guy, one of the guys,
the right guy is one.
501
00:33:10 --> 00:33:14
And all other guys
are zero, right.
502
00:33:14 --> 00:33:17
And this guy is also at one.
503
00:33:17 --> 00:33:20
Because it's the right guy.
504
00:33:20 --> 00:33:23
It has height one and
all others zero.
505
00:33:23 --> 00:33:27
So at the nodes we are at
one, because of one person,
506
00:33:27 --> 00:33:29
really, one element.
507
00:33:29 --> 00:33:30
And then?
508
00:33:30 --> 00:33:32
AUDIENCE: [INAUDIBLE]
509
00:33:32 --> 00:33:38
PROFESSOR STRANG: Right.
510
00:33:38 --> 00:33:42
But the sum of them is, why
is the sum of them always
511
00:33:42 --> 00:33:43
one, why is slope zero?
512
00:33:44 --> 00:33:48
Yeah.
513
00:33:48 --> 00:33:51
The slopes cancel, right.
514
00:33:51 --> 00:33:54
We know that in between it
will be a linear function.
515
00:33:54 --> 00:33:56
That would be one
way to look at it.
516
00:33:56 --> 00:33:59
If I add up a linear function
and a linear function the
517
00:33:59 --> 00:34:01
sum is a linear function.
518
00:34:01 --> 00:34:04
So I'm getting a linear
function, which is one at
519
00:34:04 --> 00:34:08
those points, so what
is that function?
520
00:34:08 --> 00:34:09
One.
521
00:34:09 --> 00:34:11
Right, you know that's
the straight line.
522
00:34:11 --> 00:34:16
So, that idea will
work here too.
523
00:34:16 --> 00:34:20
Look inside some
little triangle here.
524
00:34:20 --> 00:34:25
OK, that's got one, two,
three corners, OK.
525
00:34:25 --> 00:34:31
And if I look at this sum,
what is it at this point?
526
00:34:31 --> 00:34:36
If I look at that sum at this
corner, one guy is one,
527
00:34:36 --> 00:34:37
the one for that pyramid.
528
00:34:37 --> 00:34:40
And all others are?
529
00:34:40 --> 00:34:41
Zero.
530
00:34:41 --> 00:34:44
So the sum is one there, the
sum is one there, the sum is
531
00:34:44 --> 00:34:50
one there, so that blowing up
this little triangle, this is
532
00:34:50 --> 00:34:53
at height one, this is at
height one, this is at height
533
00:34:53 --> 00:34:56
one, so what's the roof?
534
00:34:56 --> 00:34:59
Flat.
535
00:34:59 --> 00:35:04
It's just a nice way to see the
nice property of these phis.
536
00:35:04 --> 00:35:14
That there's a phi for every
node, and they add to one.
537
00:35:14 --> 00:35:17
To that's it.
538
00:35:17 --> 00:35:24
OK, well I was going to say one
more thing and I am, about this
539
00:35:24 --> 00:35:27
eigenvalue problem, just
because I'll never
540
00:35:27 --> 00:35:29
have a chance again.
541
00:35:29 --> 00:35:33
So this is the moment
to say something about
542
00:35:33 --> 00:35:34
the eigenvalues.
543
00:35:34 --> 00:35:35
Lambda.
544
00:35:35 --> 00:35:41
Eigenvalue.
545
00:35:41 --> 00:35:44
I'm answering the question
where does K come from,
546
00:35:44 --> 00:35:45
where does M come from?
547
00:35:45 --> 00:35:56
Well, eigenvalue is, boy we
really got dramatic music here.
548
00:35:56 --> 00:36:00
That's a great Gates of
Kiev, I think might be.
549
00:36:00 --> 00:36:01
Mussorgski.
550
00:36:01 --> 00:36:05
If you like drums and big
noise, it's not music
551
00:36:05 --> 00:36:11
actually, but you got a
lot of noise out of it.
552
00:36:11 --> 00:36:16
Well, of course, he'd know
more than we do, but still.
553
00:36:16 --> 00:36:26
OK, so the eigenvalues in the
matrix case for Kx=lambda*M*x,
554
00:36:26 --> 00:36:31
the eigenvalue problem, lambda,
the lowest eigenvalue, lambda
555
00:36:31 --> 00:36:34
lowest, has a nice property.
556
00:36:34 --> 00:36:46
It's the minimum of sort of our
energy over our other energy.
557
00:36:46 --> 00:36:51
I just think, well this is
something you should see.
558
00:36:51 --> 00:36:54
This is a quotient here.
559
00:36:54 --> 00:36:56
It's got a name called
the Rayleigh quotient.
560
00:36:56 --> 00:36:59
And it would appear
in the book.
561
00:36:59 --> 00:37:02
So really, I guess what
I'm doing is calling your
562
00:37:02 --> 00:37:06
attention to something
that's in the book.
563
00:37:06 --> 00:37:10
That this a ratio of x
transpose K x to x transpose M
564
00:37:10 --> 00:37:19
x, if I look over all vectors
x, the lowest one is
565
00:37:19 --> 00:37:20
the eigenvector.
566
00:37:20 --> 00:37:23
The best x is the eigenvector
and the ratio is
567
00:37:23 --> 00:37:26
the eigenvalue.
568
00:37:26 --> 00:37:29
This is like my point
that I wanted to mention
569
00:37:29 --> 00:37:31
the Rayleigh quotient.
570
00:37:31 --> 00:37:34
Here it is in the matrix case,
and there would be similar
571
00:37:34 --> 00:37:38
Rayleigh quotient in
the continuous case.
572
00:37:38 --> 00:37:41
I'll just leave it at that.
573
00:37:41 --> 00:37:45
That in describing eigenvalues,
we can talk about
574
00:37:45 --> 00:37:49
Kx=lambda*M*x, like this.
575
00:37:49 --> 00:37:51
Or we can get energy into it.
576
00:37:51 --> 00:37:54
And you remember the whole
point about finite elements
577
00:37:54 --> 00:37:57
is, look at the energy.
578
00:37:57 --> 00:37:59
Look at that the quadratics.
579
00:37:59 --> 00:38:04
Multiply things by things.
580
00:38:04 --> 00:38:07
It came from the weak
form, it didn't come
581
00:38:07 --> 00:38:10
from the strong form.
582
00:38:10 --> 00:38:13
In the differential equation
here, we just have
583
00:38:13 --> 00:38:15
single terms.
584
00:38:15 --> 00:38:20
We got to these things through
that process of multiplying
585
00:38:20 --> 00:38:23
by u's and integrating.
586
00:38:23 --> 00:38:25
That's what gave us these
products and it works
587
00:38:25 --> 00:38:29
also in the matrix case.
588
00:38:29 --> 00:38:36
OK, that was a lot of
speechmaking about topics
589
00:38:36 --> 00:38:41
that we simply didn't
have time for in class.
590
00:38:41 --> 00:38:45
I'm ready for any question, or
I'm ready to maybe do a Fourier
591
00:38:45 --> 00:38:48
example, would you like that?
592
00:38:48 --> 00:38:51
Because this is where
we really are.
593
00:38:51 --> 00:38:55
I'll even take one that
will be on the homework.
594
00:38:55 --> 00:39:01
Just so you'll have a start.
595
00:39:01 --> 00:39:09
OK, let me take a square
pulse, yeah this is
596
00:39:09 --> 00:39:16
a good one, I think.
597
00:39:16 --> 00:39:19
In Section 4.1, there's a
question for the Fourier
598
00:39:19 --> 00:39:21
series of a square pulse.
599
00:39:21 --> 00:39:24
OK, so what does the
square pulse look like?
600
00:39:24 --> 00:39:29
Here's minus pi to pi.
601
00:39:29 --> 00:39:30
Here's zero.
602
00:39:30 --> 00:39:35
The square pulse goes along
here, up square pulse and down.
603
00:39:35 --> 00:39:48
Actually, let me go to L/2,
oh I'll just call it h.
604
00:39:48 --> 00:39:56
Let me find the Fourier
series for this function.
605
00:39:56 --> 00:40:02
It goes along at 0, it jumps up
to 1 over a interval of length
606
00:40:02 --> 00:40:06
2 h, going from minus h to h,
and then back down to
607
00:40:06 --> 00:40:08
0 and then repeat.
608
00:40:08 --> 00:40:11
So bip bip bip, square pulse.
609
00:40:11 --> 00:40:14
So that's my function.
610
00:40:14 --> 00:40:18
Is that function odd, or
even, or neither one?
611
00:40:18 --> 00:40:21
It's even, so I can
call that C(x).
612
00:40:22 --> 00:40:25
And figure that I'm going
to use cosines for
613
00:40:25 --> 00:40:27
that one, right?
614
00:40:27 --> 00:40:31
So tell me a formula for the
coefficients, what's the
615
00:40:31 --> 00:40:33
integral that I have to do?
616
00:40:33 --> 00:40:40
So my C(x) o is going to be
some a_0, we have to think
617
00:40:40 --> 00:40:46
what's a_0, then a_1*cos(x),
a_2*cos, and so on.
618
00:40:46 --> 00:40:48
So on. a_k*cos(kx).
619
00:40:48 --> 00:40:52
620
00:40:52 --> 00:41:01
OK, what's the formula for a_k?
621
00:41:01 --> 00:41:03
Before I plug in that
function I would like
622
00:41:03 --> 00:41:04
to get the formula.
623
00:41:04 --> 00:41:07
So I'm looking for the formula.
624
00:41:07 --> 00:41:10
It's a formula to remember.
625
00:41:10 --> 00:41:12
So I'm not wasting your time.
626
00:41:12 --> 00:41:14
Because you're going to see it
on the board and it'll just
627
00:41:14 --> 00:41:16
take a mental photograph of it.
628
00:41:16 --> 00:41:18
What do you think
it's going to be?
629
00:41:18 --> 00:41:20
How am I going to get it?
630
00:41:20 --> 00:41:26
I'll multiply both sides of the
equation by cos(kx), right?
631
00:41:26 --> 00:41:28
And I'll integrate.
632
00:41:28 --> 00:41:32
So and then when I integrate,
the cosines are orthogonal.
633
00:41:32 --> 00:41:34
Just like the sines
this morning.
634
00:41:34 --> 00:41:38
All those terms will go,
except for this term.
635
00:41:38 --> 00:41:40
When I multiply this
by cos(kx), I'll have
636
00:41:40 --> 00:41:42
cos(kx) squared.
637
00:41:42 --> 00:41:46
Here I'll have a cos(kx), and
here I'll have a whole lot of
638
00:41:46 --> 00:41:51
cos(kx)'s but when I
integrate, all this stuff is
639
00:41:51 --> 00:41:55
going to disappear.
640
00:41:55 --> 00:41:57
And this will all disappear.
641
00:41:57 --> 00:41:58
This is it.
642
00:41:58 --> 00:42:04
So a_k is going to be the
integral of my function,
643
00:42:04 --> 00:42:04
times cos(kx)dx.
644
00:42:04 --> 00:42:07
645
00:42:07 --> 00:42:09
Divided by what?
646
00:42:09 --> 00:42:14
Divided by the integral
of cos(kx) squared.
647
00:42:14 --> 00:42:18
Because I haven't
normalized things.
648
00:42:18 --> 00:42:21
So I don't know that that's
one, and in fact it isn't one.
649
00:42:21 --> 00:42:25
So I have to remember
to put that number in.
650
00:42:25 --> 00:42:28
OK, so that's the formula
and that number turns
651
00:42:28 --> 00:42:32
out to be pi, again.
652
00:42:32 --> 00:42:37
If I'm integrating from minus
pi to pi, then the average
653
00:42:37 --> 00:42:41
value of the cosine squared is
a 1/2, it's sort of as much
654
00:42:41 --> 00:42:47
above 1/2 as it is below 1/2,
and so the average of the half,
655
00:42:47 --> 00:42:51
the interval is 2pi, so pi.
656
00:42:51 --> 00:42:54
OK, that's the formula.
657
00:42:54 --> 00:42:59
Please just take a
mental photograph.
658
00:42:59 --> 00:43:00
Catch that one.
659
00:43:00 --> 00:43:07
Alright, now I've got my
particular C(x), my square
660
00:43:07 --> 00:43:10
wave, square pulse.
661
00:43:10 --> 00:43:11
Very, very important.
662
00:43:11 --> 00:43:16
Very important
Fourier series here.
663
00:43:16 --> 00:43:18
Famous one.
664
00:43:18 --> 00:43:20
OK, so what do I have?
665
00:43:20 --> 00:43:24
From minus pi to pi, so
what's my integral?
666
00:43:24 --> 00:43:27
Well, my integral really
doesn't go from minus
667
00:43:27 --> 00:43:31
pi to pi because my
function is mostly zero.
668
00:43:31 --> 00:43:34
Where does my integral go?
669
00:43:34 --> 00:43:36
Negative h to h, right?
670
00:43:36 --> 00:43:39
And in that region,
what is C(x)?
671
00:43:40 --> 00:43:44
One.
672
00:43:44 --> 00:43:47
So you see it's
going to be nice.
673
00:43:47 --> 00:43:53
From negative h to h, where
this is one, I just have to
674
00:43:53 --> 00:44:00
integrate cos(kx), so what do I
get? sin(kx), over k, and the
675
00:44:00 --> 00:44:05
pi so you see again that that k
is showing up in the
676
00:44:05 --> 00:44:09
denominator, and that's going
to give me the typical
677
00:44:09 --> 00:44:20
decay rate of 1/k for
functions with steps.
678
00:44:20 --> 00:44:21
For step functions.
679
00:44:21 --> 00:44:28
And now I have to evaluate
this between minus pi and pi.
680
00:44:28 --> 00:44:30
And no, h.
681
00:44:30 --> 00:44:32
Better be h.
682
00:44:32 --> 00:44:35
I mean, minus h and h.
683
00:44:35 --> 00:44:37
So what do I get for that?
684
00:44:37 --> 00:44:42
I get sine(kh), right?
685
00:44:42 --> 00:44:48
At the top, and what
do I get at minus?
686
00:44:48 --> 00:44:51
So I now I want to subtract,
what is the sin(-kh)?
687
00:44:51 --> 00:44:54
688
00:44:54 --> 00:44:57
It's a negative, right?
689
00:44:57 --> 00:45:04
So as I expect with an even
function like cosine, am
690
00:45:04 --> 00:45:08
I just getting twice?
691
00:45:08 --> 00:45:13
I could take it from 0 to h,
and it would give me one
692
00:45:13 --> 00:45:14
of them and the other one.
693
00:45:14 --> 00:45:21
Yep, I think so, and
divide by k pi.
694
00:45:21 --> 00:45:24
So those are the
Fourier coefficients.
695
00:45:24 --> 00:45:25
Except for a_0.
696
00:45:27 --> 00:45:33
a_0 has a slightly different
formula, because for a_0,
697
00:45:33 --> 00:45:34
why is a_0 different?
698
00:45:34 --> 00:45:39
How do you come up with a_0,
and what's its meaning? a_0
699
00:45:39 --> 00:45:45
has a nice meeting, so this
is worth having come
700
00:45:45 --> 00:45:46
this afternoon for.
701
00:45:46 --> 00:45:50
a_0 will be what?
702
00:45:50 --> 00:45:52
Well, I could get
it the same way.
703
00:45:52 --> 00:45:57
What will I multiply
both sides by?
704
00:45:57 --> 00:45:58
If I want to pick off a_0?
705
00:46:00 --> 00:46:01
Just one.
706
00:46:01 --> 00:46:06
It's not a cosine, it's the
cos(0x), it's the one.
707
00:46:06 --> 00:46:08
And then I integrate.
708
00:46:08 --> 00:46:13
I'm just going to get the
integral from minus pi to pi
709
00:46:13 --> 00:46:19
of C(x) times one, divided
by the integral from minus
710
00:46:19 --> 00:46:30
pi of one times one.
711
00:46:30 --> 00:46:37
Same method. multiply both
sides by one, which was
712
00:46:37 --> 00:46:41
the very first of my
orthogonal functions.
713
00:46:41 --> 00:46:45
Integrate it, all the other
integrals went away, right?
714
00:46:45 --> 00:46:48
The integral of cosine
over a whole interval.
715
00:46:48 --> 00:46:51
Its periodic.
716
00:46:51 --> 00:46:53
You get the same at
the two ends, so the
717
00:46:53 --> 00:46:56
difference is zero.
718
00:46:56 --> 00:47:00
So we just, the only term
left was a constant.
719
00:47:00 --> 00:47:03
And now what is the integral,
row what's the denominator now?
720
00:47:03 --> 00:47:06
That was the little,
slight twist.
721
00:47:06 --> 00:47:06
2pi.
722
00:47:07 --> 00:47:08
The denominator's 2pi.
723
00:47:09 --> 00:47:10
Yeah.
724
00:47:10 --> 00:47:15
That's that's why it's not,
yeah, it's slightly irregular,
725
00:47:15 --> 00:47:16
I have to divide by 2pi.
726
00:47:18 --> 00:47:22
And now, what word would you
use to describe, if I have a
727
00:47:22 --> 00:47:25
function, and integrate
it, and I divide by the
728
00:47:25 --> 00:47:30
length, what am I getting?
729
00:47:30 --> 00:47:35
There's an English word that
would describe what this is.
730
00:47:35 --> 00:47:37
Average.
731
00:47:37 --> 00:47:44
This is the average.
732
00:47:44 --> 00:47:46
And it has to be.
733
00:47:46 --> 00:47:49
This constant term is
always the average.
734
00:47:49 --> 00:47:52
And what will it be for this?
735
00:47:52 --> 00:47:59
So this was a_k, and
what is a_0, then?
736
00:47:59 --> 00:48:03
So you can now tell me, so
everybody's remembering this
737
00:48:03 --> 00:48:06
formula, you integrate the
function and divide by the 2pi.
738
00:48:07 --> 00:48:10
Now we've got a particular
function, so what is the
739
00:48:10 --> 00:48:12
integral of that function?
740
00:48:12 --> 00:48:16
So what does this equal?
741
00:48:16 --> 00:48:17
For this particular C(x)?
742
00:48:18 --> 00:48:20
What's the area under
that function C(x)?
743
00:48:22 --> 00:48:24
2h.
744
00:48:24 --> 00:48:26
Right?
745
00:48:26 --> 00:48:29
So 2h/2pi cancel twos.
746
00:48:29 --> 00:48:40
So there's a constant term, a_0
is h/pi and the and the cosine
747
00:48:40 --> 00:48:43
terms are, yeah, actually we're
going to get something nice.
748
00:48:43 --> 00:48:50
A really nice way to complete
this will be if I put this
749
00:48:50 --> 00:48:55
together, put this
series together.
750
00:48:55 --> 00:49:05
So now I'm saying that this
square pulse is that constant
751
00:49:05 --> 00:49:13
term h/pi plus the next
term a_1, you can see all
752
00:49:13 --> 00:49:14
these terms have 2/pi's.
753
00:49:14 --> 00:49:17
754
00:49:17 --> 00:49:21
I'm a little surprised that h
over, yeah, I guess it's right.
755
00:49:21 --> 00:49:22
2/pi.
756
00:49:22 --> 00:49:29
757
00:49:29 --> 00:49:33
So I've got sin(h), I think.
758
00:49:33 --> 00:49:35
And now I'm just copying this.
759
00:49:35 --> 00:49:47
2/pi*sin(h), sin(h),
is that what I want?
760
00:49:47 --> 00:49:51
Over one.
761
00:49:51 --> 00:49:54
That's the coefficient
version of sine, of cos(x).
762
00:49:54 --> 00:49:58
763
00:49:58 --> 00:50:00
a_1 was the coefficient
of cos(1x).
764
00:50:01 --> 00:50:04
And then a_2 is the
coefficient of cos(2x).
765
00:50:06 --> 00:50:07
So that will be sin(2h).
766
00:50:07 --> 00:50:09
767
00:50:09 --> 00:50:13
k is two, so I have a
two down here, cos(2x).
768
00:50:14 --> 00:50:20
And so on.
769
00:50:20 --> 00:50:23
Yeah, I think that's
the Fourier series.
770
00:50:23 --> 00:50:33
That would be the Fourier
series for the square pulse.
771
00:50:33 --> 00:50:34
Yeah.
772
00:50:34 --> 00:50:38
That would be the Fourier
series for the square pulse.
773
00:50:38 --> 00:50:40
Could I test any
interesting cases?
774
00:50:40 --> 00:50:45
Suppose h is all
the way out to pi.
775
00:50:45 --> 00:50:46
Suppose I take that case.
776
00:50:46 --> 00:50:55
Let h go all the way out to
pi, then what's my function?
777
00:50:55 --> 00:51:01
If h=pi, then what have
I got a graph of?
778
00:51:01 --> 00:51:02
Just one.
779
00:51:02 --> 00:51:03
It's just a one.
780
00:51:03 --> 00:51:07
If h is pi, what happens?
781
00:51:07 --> 00:51:11
That becomes a one, and what
about these other things?
782
00:51:11 --> 00:51:15
What is this thing
when h is pi?
783
00:51:15 --> 00:51:15
Zero.
784
00:51:15 --> 00:51:18
All the other terms go away.
785
00:51:18 --> 00:51:22
It's just a sin(2pi)
that would go away.
786
00:51:22 --> 00:51:27
Yeah, so if h is pi, if I go
out to the place where I don't
787
00:51:27 --> 00:51:33
have any jumps at all because
it's now all the way out there,
788
00:51:33 --> 00:51:37
then these terms all disappear
and I just have this.
789
00:51:37 --> 00:51:40
And I would like to ask you
and it's going to come up on
790
00:51:40 --> 00:51:47
Friday, too, what happens
if h goes to zero?
791
00:51:47 --> 00:51:50
Well, let me just take
h going to zero.
792
00:51:50 --> 00:51:52
What happens to
this whole thing?
793
00:51:52 --> 00:51:55
What happens to my function
if h goes to zero?
794
00:51:55 --> 00:51:58
Goes to zero, right, then
squeezed it to nothing.
795
00:51:58 --> 00:52:05
And if h is zero then sin(h) is
zero, I get 0=0, that's not
796
00:52:05 --> 00:52:10
interesting enough to mention
on Friday But there is one
797
00:52:10 --> 00:52:13
case that is important.
798
00:52:13 --> 00:52:16
Suppose I make the
height, yeah.
799
00:52:16 --> 00:52:17
Make a guess.
800
00:52:17 --> 00:52:24
Suppose I make the
height higher as I
801
00:52:24 --> 00:52:27
make the base smaller.
802
00:52:27 --> 00:52:31
I'm going to keep the area as
one, so if this has a base of
803
00:52:31 --> 00:52:34
2h, I'm going to have
a height of 1/2h.
804
00:52:35 --> 00:52:41
So if I keep the area at one,
so the height now is 1/2h,
805
00:52:41 --> 00:52:45
so now my square pulse
I've divided it by 2h.
806
00:52:46 --> 00:52:50
I have a 1/2h
multiplying everything.
807
00:52:50 --> 00:52:56
And now if I let h go to
zero, something more
808
00:52:56 --> 00:52:58
interesting will happen.
809
00:52:58 --> 00:52:59
And what?
810
00:52:59 --> 00:53:05
Just tell me first, what
would you expect to happen?
811
00:53:05 --> 00:53:05
Delta.
812
00:53:05 --> 00:53:08
Right, delta.
813
00:53:08 --> 00:53:12
So what I'll see show up
will be the Fourier series
814
00:53:12 --> 00:53:15
for the delta function.
815
00:53:15 --> 00:53:23
When I divide by 2h, so I have
sin(h)'s over h's, and of
816
00:53:23 --> 00:53:25
course what's the great
fact about sin(h)/h?
817
00:53:27 --> 00:53:34
As h goes to zero, it
goes to, everybody know,
818
00:53:34 --> 00:53:36
that's the big deal.
819
00:53:36 --> 00:53:36
Yeah.
820
00:53:36 --> 00:53:42
One. sin(h) is the same
size as h for a very small
821
00:53:42 --> 00:53:44
h, and approaches one.
822
00:53:44 --> 00:53:49
Yeah so we'll see the
delta function Friday.
823
00:53:49 --> 00:53:53
OK, so you've got a sort of
mini-lecture instead of a real
824
00:53:53 --> 00:53:56
chance to ask about homework.
825
00:53:56 --> 00:53:59
Next Wednesday should be
different because there will be
826
00:53:59 --> 00:54:04
Fourier series homework, and
I'll be ready to answer
827
00:54:04 --> 00:54:05
questions about it.
828
00:54:05 --> 00:54:07
OK, thanks.