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PROFESSOR STRANG:
OK, let's start

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with a review and preview.

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I put a P up there
because we're really

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00:00:29,350 --> 00:00:36,080
looking into the Fourier part
that just started this morning.

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And there'll be some homework
from these early sections

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about the Fourier
stuff, so we maybe we

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should just do a few of those
problems or discuss here today.

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00:00:51,400 --> 00:00:54,880
Just in advance.

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00:00:54,880 --> 00:00:58,810
Can I say one thing about MATLAB
and the MATLAB homework first?

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And maybe open a
conversation about it?

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00:01:03,990 --> 00:01:09,340
So there's really two
different problems

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that I'm personally
quite interested in.

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Two model, I'll
say model problems

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because they're for regular
polygons in a circle.

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And I'll draw an octagon again.

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So M sides.

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00:01:31,170 --> 00:01:34,800
And I'm interested in
as M goes to infinity.

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00:01:34,800 --> 00:01:37,230
And I'm interested in
two different problems.

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00:01:37,230 --> 00:01:46,690
So one of them is our MATLAB
problem, is Laplace's equation.

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What was it, four?

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00:01:50,920 --> 00:01:58,500
With u=0 on the circle.

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OK, so that's our problem,
totally open for discussion.

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How many have started on that?

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00:02:05,230 --> 00:02:06,051
Oh, good.

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00:02:06,051 --> 00:02:06,550
OK.

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00:02:06,550 --> 00:02:08,900
Well, then you all know
more about it than I.

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00:02:08,900 --> 00:02:10,580
And that's great.

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00:02:10,580 --> 00:02:14,050
I'd be happy to learn.

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00:02:14,050 --> 00:02:15,700
So have I said everything there?

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00:02:15,700 --> 00:02:18,500
Yeah, we've got Poisson's
equation inside.

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00:02:18,500 --> 00:02:22,940
We've got u=0 on the circle,
so the problem's well defined

40
00:02:22,940 --> 00:02:29,490
and the solution should be
one minus x squared minus y

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squared.

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So that's the correct solution.

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00:02:35,490 --> 00:02:38,360
Maybe I can also tell you
about the second problem

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that I'm interested in.

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00:02:40,270 --> 00:02:43,450
Because it hasn't come
up in class but it's

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very important too.

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It would be the
eigenvalue problem.

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00:02:47,370 --> 00:02:52,140
So this is problem number
one, the steady state problem

49
00:02:52,140 --> 00:02:57,330
when you've got a source
and you want to find out

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00:02:57,330 --> 00:02:59,590
the temperature distribution.

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00:02:59,590 --> 00:03:04,220
The problem number two would
be the eigenvalue problem,

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00:03:04,220 --> 00:03:06,050
-u_xx-u_yy.

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00:03:06,050 --> 00:03:12,590
I take those minuses so that
the eigenvalue will be positive.

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00:03:12,590 --> 00:03:15,540
So that's what the eigenvalue
problem might look like.

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00:03:15,540 --> 00:03:20,240
And again let me say,
with u=0 on the boundary.

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On the circle.

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00:03:25,090 --> 00:03:33,500
OK, so a person would say this
is Laplace's eigenvalue problem

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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.

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And we have boundary
conditions, and this

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would describe the normal
modes, for example,

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of a circular drum.

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If I had a drum--
Or a polygon drum.

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00:04:00,520 --> 00:04:06,780
So maybe I connect, to
actually build the drum,

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I might fold in the sides
there and have a polygon.

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00:04:11,140 --> 00:04:15,090
And again, I hope
that the eigenvalues

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of the polygon, this
equation in the polygon,

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which are not known, by the way.

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00:04:22,660 --> 00:04:27,700
To the best to my knowledge,
we know it only for M=3,

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which would be an equilateral
triangle, and M=4,

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00:04:32,290 --> 00:04:34,100
which would be a square.

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00:04:34,100 --> 00:04:39,060
And those eigenvalues, because
of Fourier or something

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00:04:39,060 --> 00:04:42,130
are humanly doable.

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00:04:42,130 --> 00:04:48,980
But I think five on up is,
I may be wrong about six,

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I'm not sure about M=6, a
hexagon sometimes gives you

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enough help.

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00:04:53,210 --> 00:04:59,120
But beyond that
you're on your own.

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With finite elements
to help you.

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00:05:03,720 --> 00:05:09,080
So there's a whole sequence
of eigenfunctions u,

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00:05:09,080 --> 00:05:15,450
eigenvalues lambda, just the
way there were in one dimension.

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00:05:15,450 --> 00:05:18,090
And on the circle
they involve Bessel.

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00:05:18,090 --> 00:05:21,970
That's where Bessel showed up.

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He figured out the
functions and they're not

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00:05:24,640 --> 00:05:29,270
especially nice functions.

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00:05:29,270 --> 00:05:32,630
But they're studied
for centuries.

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00:05:32,630 --> 00:05:36,270
Bessel functions come into that.

89
00:05:36,270 --> 00:05:40,420
But, here I have
the same question.

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00:05:40,420 --> 00:05:44,950
I mean, let me just say, for
me this could be a UROP project

91
00:05:44,950 --> 00:05:47,120
if anybody was an
undergraduate, or it

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00:05:47,120 --> 00:05:51,000
could be a project over
January or something.

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00:05:51,000 --> 00:05:55,900
I'd like to know
something about what

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00:05:55,900 --> 00:06:02,500
happens as M goes to infinity,
as the polygon approaches

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00:06:02,500 --> 00:06:03,520
the circle.

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00:06:03,520 --> 00:06:11,280
So I'm hoping maybe on
the homework that come in,

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00:06:11,280 --> 00:06:14,720
if it's not too difficult,
and maybe it's not,

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00:06:14,720 --> 00:06:18,750
to let M go up a bit.

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00:06:18,750 --> 00:06:20,910
There is one thing.

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00:06:20,910 --> 00:06:25,680
That the code we're working
with is linear elements, right?

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00:06:25,680 --> 00:06:27,820
We're using linear
finite elements.

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00:06:27,820 --> 00:06:32,050
So we're not getting
high accuracy.

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00:06:32,050 --> 00:06:34,540
So I would really
like to move up

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00:06:34,540 --> 00:06:39,810
to quadratic elements,
at least, you

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00:06:39,810 --> 00:06:41,420
remember quadratic
elements would

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00:06:41,420 --> 00:06:48,260
be ones where-- Well, let
me draw the one that we've

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00:06:48,260 --> 00:06:52,160
drawn in class before.

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00:06:52,160 --> 00:06:54,630
We only have to
look at one triangle

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00:06:54,630 --> 00:06:57,790
and then we cut it up
into triangular elements

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00:06:57,790 --> 00:07:03,470
by taking some pieces here,
taking the points above,

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00:07:03,470 --> 00:07:07,650
which I hope are now
correct on the website.

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00:07:07,650 --> 00:07:11,150
Connecting those edges,
and then connecting these.

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Is that right?

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00:07:12,170 --> 00:07:18,210
Is that our mesh?

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00:07:18,210 --> 00:07:23,640
So that mesh is controlled
by N. One, two, N points.

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00:07:23,640 --> 00:07:32,810
Also, N is going to have to get
large too, to give me accuracy.

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00:07:32,810 --> 00:07:36,210
And another way
toward more accuracy

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00:07:36,210 --> 00:07:42,320
is, instead of linear
elements, second degree.

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00:07:42,320 --> 00:07:44,710
So do you remember
I wrote those down?

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00:07:44,710 --> 00:07:51,230
Let me take that little triangle
out here as a bigger triangle.

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00:07:51,230 --> 00:07:53,920
It would look something
like that, I guess.

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00:07:53,920 --> 00:07:58,170
The second degree elements
have those six mesh points.

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00:07:58,170 --> 00:08:01,590
You remember I drew those but
we didn't really have time

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00:08:01,590 --> 00:08:07,330
to get further with them.

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00:08:07,330 --> 00:08:12,640
The trial functions phi, which
are one at a typical mesh

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00:08:12,640 --> 00:08:18,800
point and zero at all the
others, they are computable.

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00:08:18,800 --> 00:08:22,640
We're up to second degree, so
it's a little-- Second degree

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00:08:22,640 --> 00:08:25,400
things, then the
first derivatives,

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00:08:25,400 --> 00:08:28,420
which come into the
integrations, are linear.

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00:08:28,420 --> 00:08:29,370
And not constant.

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00:08:29,370 --> 00:08:31,960
So a little bit harder.

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00:08:31,960 --> 00:08:38,290
But finite elements, linear
or quadratic, or higher,

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00:08:38,290 --> 00:08:40,510
could be used for this problem.

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00:08:40,510 --> 00:08:45,280
As we know, and
for this problem.

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00:08:45,280 --> 00:08:48,670
What I wanted to add, that
I've not mentioned in class,

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00:08:48,670 --> 00:08:53,360
and I think we may just
not get a chance to do it,

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00:08:53,360 --> 00:08:56,710
is what does the finite
element method look

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00:08:56,710 --> 00:08:58,700
like for an eigenvalue problem?

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00:08:58,700 --> 00:09:02,120
Because eigenvalues
are highly important.

140
00:09:02,120 --> 00:09:06,120
That's the different
way to understand.

141
00:09:06,120 --> 00:09:08,890
There's the matrix
K and its entries.

142
00:09:08,890 --> 00:09:11,420
But then there are
the eigenvalues.

143
00:09:11,420 --> 00:09:13,140
And you might
think that, what do

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00:09:13,140 --> 00:09:17,710
you think is the discrete
eigenvalue problem copying

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00:09:17,710 --> 00:09:19,390
this one?

146
00:09:19,390 --> 00:09:21,510
Here's my point.

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00:09:21,510 --> 00:09:26,370
Your first guess would be,
well this is like K, right?

148
00:09:26,370 --> 00:09:34,110
This is like KU, right? (K2D)U,
I should call it, maybe.

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00:09:34,110 --> 00:09:39,420
Well, I'll call it K, because
K2D I have specifically

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00:09:39,420 --> 00:09:46,170
reserved for the
Laplace stiffness

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00:09:46,170 --> 00:09:51,600
matrix on a square mesh, square
mesh with triangles, the K2D.

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00:09:51,600 --> 00:09:56,970
That was one specific matrix
for one specific mesh,

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00:09:56,970 --> 00:09:59,140
and here we have
a different mesh.

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00:09:59,140 --> 00:10:05,060
So I should just call it K. Ok,
I think if anybody was going

155
00:10:05,060 --> 00:10:09,290
to make a guess, they
would say OK, KU=Lambda*U.

156
00:10:09,290 --> 00:10:13,760
Maybe I'll use capital Lambda,
because I'm using capital U.

157
00:10:13,760 --> 00:10:28,330
Is this the finite element
method eigenvalue problem?

158
00:10:28,330 --> 00:10:33,270
And if you answered yes,
I would have to say, well

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that's a reasonable answer.

160
00:10:35,930 --> 00:10:38,120
But it's wrong.

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00:10:38,120 --> 00:10:43,990
The eigenvalue problem, when I
take the differential equation

162
00:10:43,990 --> 00:10:46,760
for the Laplace,
Laplace's equation,

163
00:10:46,760 --> 00:10:53,240
lambda u on the right side,
and I go to do finite elements,

164
00:10:53,240 --> 00:10:57,110
it produces K.
Out of this stuff,

165
00:10:57,110 --> 00:11:00,160
out of the weak
form, all that stuff.

166
00:11:00,160 --> 00:11:04,110
But it produces another
matrix on the right-hand side

167
00:11:04,110 --> 00:11:07,840
from the constant term, and we
have not really mentioned it,

168
00:11:07,840 --> 00:11:09,910
it's the mass matrix.

169
00:11:09,910 --> 00:11:13,580
So this, instead of
just the identity here,

170
00:11:13,580 --> 00:11:16,170
there's a mass matrix.

171
00:11:16,170 --> 00:11:21,570
So that is the problem
that you could do.

172
00:11:21,570 --> 00:11:28,080
I could've made
a MATLAB project.

173
00:11:28,080 --> 00:11:31,730
I bet I'd do it next fall.

174
00:11:31,730 --> 00:11:32,680
Right?

175
00:11:32,680 --> 00:11:39,250
You guys did the
first one, this one.

176
00:11:39,250 --> 00:11:40,850
Or you are doing it now.

177
00:11:40,850 --> 00:11:44,460
And I'm going to pause in a
minute for questions about it,

178
00:11:44,460 --> 00:11:46,000
or discussion of it.

179
00:11:46,000 --> 00:11:50,850
But this one brings in something
called the mass matrix.

180
00:11:50,850 --> 00:11:58,620
So let me just say
what those are.

181
00:11:58,620 --> 00:12:02,000
If I write down the
entries in the mass matrix,

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00:12:02,000 --> 00:12:04,930
you'll sort of get an
idea of why they are.

183
00:12:04,930 --> 00:12:07,680
So what are the entries
in the stiffness matrix?

184
00:12:07,680 --> 00:12:18,500
K_ij, you remember, is the
integral of the d phi_i/dx,

185
00:12:18,500 --> 00:12:20,940
d phi_j/dx.

186
00:12:20,940 --> 00:12:31,810
Plus d phi_i/dy,
d phi_j/dy, dxdy,

187
00:12:31,810 --> 00:12:34,190
and that's what's
you're computing.

188
00:12:34,190 --> 00:12:36,700
And that's what that
code is computing.

189
00:12:36,700 --> 00:12:41,270
And when phi is
linear, phi linear,

190
00:12:41,270 --> 00:12:47,500
then slopes are constant.

191
00:12:47,500 --> 00:12:52,300
So all you have to do, and what
that code in the book is doing,

192
00:12:52,300 --> 00:12:55,180
is figuring out
what are the slopes.

193
00:12:55,180 --> 00:12:58,190
These things are
constant, so we just

194
00:12:58,190 --> 00:13:04,420
need to know the area
of the integration

195
00:13:04,420 --> 00:13:06,180
where we're integrating.

196
00:13:06,180 --> 00:13:09,300
The area, triangle by triangle.

197
00:13:09,300 --> 00:13:11,310
Fine.

198
00:13:11,310 --> 00:13:12,970
That's what we're doing.

199
00:13:12,970 --> 00:13:16,860
That's what that code
is just set up to do.

200
00:13:16,860 --> 00:13:20,550
Now, I have to tell you what
is M_ij, the mass matrix.

201
00:13:20,550 --> 00:13:24,630
I just think you
don't want to have--

202
00:13:24,630 --> 00:13:27,820
we haven't done too badly
with finite elements in here.

203
00:13:27,820 --> 00:13:31,180
We did it in 1-D, where we
got it kind of straight.

204
00:13:31,180 --> 00:13:34,520
And now we're seeing what
it looks like in 2-D.

205
00:13:34,520 --> 00:13:38,170
But I had not really
mentioned a mass matrix.

206
00:13:38,170 --> 00:13:42,400
So here it comes.

207
00:13:42,400 --> 00:13:45,370
The mass matrix
will be the integral

208
00:13:45,370 --> 00:13:50,870
of phi_i times phi_j dxdy.

209
00:13:50,870 --> 00:13:57,680
It's the zero order, no
derivatives, just plain zero

210
00:13:57,680 --> 00:14:03,340
order, as you'd
expect from the fact

211
00:14:03,340 --> 00:14:09,280
that the term in the
continuous part is zero order.

212
00:14:09,280 --> 00:14:12,100
So it's this mass
matrix that comes in.

213
00:14:12,100 --> 00:14:20,690
And maybe we could just look to
see which entries will be zero

214
00:14:20,690 --> 00:14:23,730
and which will not.

215
00:14:23,730 --> 00:14:25,330
How sparse is it?

216
00:14:25,330 --> 00:14:28,710
What does the mass
matrix look like?

217
00:14:28,710 --> 00:14:33,040
And can we just,
let me do 1-D first.

218
00:14:33,040 --> 00:14:36,760
So there's a phi, right?

219
00:14:36,760 --> 00:14:38,470
There's another one.

220
00:14:38,470 --> 00:14:40,070
There's another one.

221
00:14:40,070 --> 00:14:45,960
So, what do you think about the
mass matrix, one phi multiplied

222
00:14:45,960 --> 00:14:48,770
by another phi and integrated?

223
00:14:48,770 --> 00:14:51,570
Is it diagonal?

224
00:14:51,570 --> 00:15:00,170
No, because each phi
overlaps its two neighbors.

225
00:15:00,170 --> 00:15:02,850
So tell me what kind of a
matrix M is going to be?

226
00:15:02,850 --> 00:15:07,080
In 1-D. Tridiagonal.

227
00:15:07,080 --> 00:15:08,560
It'll be tridiagonal.

228
00:15:08,560 --> 00:15:12,550
Now, so was K. So
K and M actually

229
00:15:12,550 --> 00:15:15,990
have non-zeroes in
the same places.

230
00:15:15,990 --> 00:15:17,860
the same sparsity pattern.

231
00:15:17,860 --> 00:15:20,830
But, of course, not the
same numbers in there.

232
00:15:20,830 --> 00:15:31,690
K had minus ones and twos
and fours and minus ones.

233
00:15:31,690 --> 00:15:36,190
What can you tell me about
this tridiagonal matrix?

234
00:15:36,190 --> 00:15:41,150
When I integrate that
against this, well,

235
00:15:41,150 --> 00:15:43,130
again I would do it
element by element

236
00:15:43,130 --> 00:15:48,330
because this against this,
they only overlap here.

237
00:15:48,330 --> 00:15:48,830
Right?

238
00:15:48,830 --> 00:15:51,920
I'll just draw the one
place that they overlap.

239
00:15:51,920 --> 00:15:54,160
And what's the point?

240
00:15:54,160 --> 00:15:56,510
They're both positive.

241
00:15:56,510 --> 00:16:02,080
So the mass matrix is, its
rows don't add to zero.

242
00:16:02,080 --> 00:16:04,840
Its rows tend to add to one.

243
00:16:04,840 --> 00:16:08,370
But it's not diagonal,
that's the difference.

244
00:16:08,370 --> 00:16:14,230
OK, so I just felt
I couldn't feel--

245
00:16:14,230 --> 00:16:20,100
I wouldn't have done a
decent job in describing

246
00:16:20,100 --> 00:16:23,780
finite elements if I
didn't describe this.

247
00:16:23,780 --> 00:16:26,480
Didn't mention this mass matrix.

248
00:16:26,480 --> 00:16:30,880
And maybe I'd better
say where it comes from.

249
00:16:30,880 --> 00:16:35,190
Because eigenvalue problems, it
may come number two, but that's

250
00:16:35,190 --> 00:16:37,440
pretty high up the list.

251
00:16:37,440 --> 00:16:50,380
So let me tell you where does
this mass matrix come from.

252
00:16:50,380 --> 00:16:52,470
First, let me tell
you about eigenvalues

253
00:16:52,470 --> 00:16:56,750
of a-- matrix eigenvalues.

254
00:16:56,750 --> 00:17:01,090
So the answer was, is this
the finite element eigenvalue

255
00:17:01,090 --> 00:17:01,770
problem?

256
00:17:01,770 --> 00:17:03,690
Only if there's an M there.

257
00:17:03,690 --> 00:17:12,260
And now I want to, OK, first of
all, what MATLAB command solves

258
00:17:12,260 --> 00:17:13,960
that problem?

259
00:17:13,960 --> 00:17:16,770
Let's just be a little
practical for a moment.

260
00:17:16,770 --> 00:17:21,560
What MATLAB command
gives me the matrix

261
00:17:21,560 --> 00:17:25,230
of eigenvectors, the
matrix of eigenvalues

262
00:17:25,230 --> 00:17:32,450
would come from eig of what?

263
00:17:32,450 --> 00:17:35,320
I'd call this the generalized
eigenvalue problem.

264
00:17:35,320 --> 00:17:38,690
Generalized because it's
got somebody over here.

265
00:17:38,690 --> 00:17:45,270
And it's just K,M. Or of
course you get the same answer,

266
00:17:45,270 --> 00:17:47,970
well you get the
same eigenvalues,

267
00:17:47,970 --> 00:17:50,770
I guess the same
eigenvectors, yeah, if you--

268
00:17:50,770 --> 00:17:57,660
Or eig of M inverse
K, of course.

269
00:17:57,660 --> 00:17:59,660
If you want to do it
with just one matrix

270
00:17:59,660 --> 00:18:01,690
then bring M inverse over here.

271
00:18:01,690 --> 00:18:06,700
But, M inverse, the inverse
of this tridiagonal matrix,

272
00:18:06,700 --> 00:18:08,350
is full.

273
00:18:08,350 --> 00:18:11,190
No zeroes in the inverse.

274
00:18:11,190 --> 00:18:15,630
So everybody would much prefer
this tridiagonal-tridiagonal

275
00:18:15,630 --> 00:18:16,290
one.

276
00:18:16,290 --> 00:18:19,860
So that's how
MATLAB would do it.

277
00:18:19,860 --> 00:18:25,590
And what I want to know
is, back in this problem,

278
00:18:25,590 --> 00:18:29,910
how close do the
finite element guys

279
00:18:29,910 --> 00:18:41,500
come, on polygons, come to the
correct solution on circles.

280
00:18:41,500 --> 00:18:46,480
I'm hoping that for
problem one you can maybe

281
00:18:46,480 --> 00:18:49,720
keep M and N equal,
or maybe N may

282
00:18:49,720 --> 00:18:53,050
be four times M or something.

283
00:18:53,050 --> 00:18:58,270
And let them grow and see.

284
00:18:58,270 --> 00:19:01,230
Well, for example, at
the center of the circle,

285
00:19:01,230 --> 00:19:05,360
or how quickly do you approach
the correct answer, one,

286
00:19:05,360 --> 00:19:07,700
at the center of the circle?

287
00:19:07,700 --> 00:19:09,920
I think it's going
to be a good problem.

288
00:19:09,920 --> 00:19:13,950
Let me open to, so I started
out just talking there.

289
00:19:13,950 --> 00:19:17,390
What about the MATLAB problem.

290
00:19:17,390 --> 00:19:24,240
You made a start
on it, is it going?

291
00:19:24,240 --> 00:19:30,320
Have you got a graph, maybe,
or what's reasonable to graph,

292
00:19:30,320 --> 00:19:35,740
to give Peter to look at?

293
00:19:35,740 --> 00:19:39,420
Who's done something
on that MATLAB problem?

294
00:19:39,420 --> 00:19:43,240
Yeah, go ahead tell
us all what to do.

295
00:19:43,240 --> 00:19:45,840
AUDIENCE: I made the
triangle bisection and--

296
00:19:45,840 --> 00:19:48,270
PROFESSOR STRANG: OK, right.

297
00:19:48,270 --> 00:19:55,580
AUDIENCE: [INAUDIBLE] and I
found that the [INAUDIBLE]

298
00:19:55,580 --> 00:19:57,070
changes to M.

299
00:19:57,070 --> 00:19:59,040
PROFESSOR STRANG:
With M more, I see.

300
00:19:59,040 --> 00:20:07,330
So if you just fixed M
like eight, and let N get,

301
00:20:07,330 --> 00:20:09,080
it didn't change significantly.

302
00:20:09,080 --> 00:20:12,070
It wouldn't, of course,
converge to the right answer.

303
00:20:12,070 --> 00:20:16,840
It'll converge, if it does,
to some kind of an answer,

304
00:20:16,840 --> 00:20:18,650
for the polygon.

305
00:20:18,650 --> 00:20:19,150
Right.

306
00:20:19,150 --> 00:20:19,730
That's right.

307
00:20:19,730 --> 00:20:23,490
So you know, as I
wrote the problem

308
00:20:23,490 --> 00:20:27,980
I didn't know
whether I dared say

309
00:20:27,980 --> 00:20:31,030
let M get increased
too, but of course

310
00:20:31,030 --> 00:20:32,770
that's the real question.

311
00:20:32,770 --> 00:20:34,190
And what happened then?

312
00:20:34,190 --> 00:20:37,400
Did error shrink?

313
00:20:37,400 --> 00:20:41,324
OK, and now maybe it's
possible to see how fast

314
00:20:41,324 --> 00:20:42,490
or something that's always--

315
00:20:42,490 --> 00:20:46,206
AUDIENCE: [INAUDIBLE]

316
00:20:46,206 --> 00:20:47,080
PROFESSOR STRANG: Ah.

317
00:20:47,080 --> 00:20:49,800
OK, at the center.

318
00:20:49,800 --> 00:20:53,930
OK, then I hope
for more comment.

319
00:20:53,930 --> 00:20:55,110
Let me say one more thing.

320
00:20:55,110 --> 00:21:00,190
My theory is that the
error at the center

321
00:21:00,190 --> 00:21:06,740
is quite a bit smaller than the
error closer to the boundary.

322
00:21:06,740 --> 00:21:13,340
I would be interested in an
error-- Is it fairly even?

323
00:21:13,340 --> 00:21:16,200
Oh, my theory's wrong.

324
00:21:16,200 --> 00:21:18,860
It wouldn't be the first time.

325
00:21:18,860 --> 00:21:21,390
And maybe because it's linear.

326
00:21:21,390 --> 00:21:28,040
Yeah, my theory is more for
better elements, like these.

327
00:21:28,040 --> 00:21:30,920
I'd be interested to know.

328
00:21:30,920 --> 00:21:37,070
Why do I think, why
do I have this theory,

329
00:21:37,070 --> 00:21:41,036
which you guys are going to
prove wrong anyway, but still.

330
00:21:41,036 --> 00:21:42,410
After you've proved
it wrong, you

331
00:21:42,410 --> 00:21:44,190
won't listen to me
if I tell it to you.

332
00:21:44,190 --> 00:21:45,970
So now I'll tell it.

333
00:21:45,970 --> 00:21:51,730
My theory is that the
error around the boundary

334
00:21:51,730 --> 00:21:56,217
is, there's no error
at these vertices,

335
00:21:56,217 --> 00:21:57,800
and then there's
sort of a going to be

336
00:21:57,800 --> 00:22:01,540
an error because the real
answer is not zero along here.

337
00:22:01,540 --> 00:22:04,670
It's sort of near
zero, but not quite.

338
00:22:04,670 --> 00:22:07,060
You know, there's an error.

339
00:22:07,060 --> 00:22:10,900
So there's errors around
here, from getting

340
00:22:10,900 --> 00:22:13,620
the boundary wrong.

341
00:22:13,620 --> 00:22:16,190
Squaring it off.

342
00:22:16,190 --> 00:22:19,910
But my theory is that
errors, the boundary stuff,

343
00:22:19,910 --> 00:22:22,360
drops off quickly
as you go inside.

344
00:22:22,360 --> 00:22:26,480
That's why I think, from those,
you remember those-- Well,

345
00:22:26,480 --> 00:22:31,190
we'll see them again
either today or Friday,

346
00:22:31,190 --> 00:22:36,320
those r^n*cos(nx) type things?

347
00:22:36,320 --> 00:22:39,050
That cos(n*theta)?

348
00:22:39,050 --> 00:22:42,580
Yeah, you remember those
are the typical solutions

349
00:22:42,580 --> 00:22:44,780
to Laplace's equation.

350
00:22:44,780 --> 00:22:49,070
And then so that if-- And it
has some coefficient, of course,

351
00:22:49,070 --> 00:22:50,390
a_n.

352
00:22:50,390 --> 00:22:56,080
And I look at that, that
might be a piece of error.

353
00:22:56,080 --> 00:23:00,140
And it's way bigger when
r is one and way smaller

354
00:23:00,140 --> 00:23:01,120
when r is zero.

355
00:23:01,120 --> 00:23:05,660
So anyway, that's
sort of my theory.

356
00:23:05,660 --> 00:23:10,470
That if you have--
Like, physically.

357
00:23:10,470 --> 00:23:17,890
You have a circular
plate and you're

358
00:23:17,890 --> 00:23:20,850
maintaining the
boundary temperature

359
00:23:20,850 --> 00:23:22,720
at some sort of oscillation.

360
00:23:22,720 --> 00:23:27,050
Like, near one but
up and down from one.

361
00:23:27,050 --> 00:23:33,080
Then I think further
inside, it doesn't know.

362
00:23:33,080 --> 00:23:36,290
It hardly knows about
that oscillation.

363
00:23:36,290 --> 00:23:38,020
This is my theory.

364
00:23:38,020 --> 00:23:42,280
That toward the center
of the circle it

365
00:23:42,280 --> 00:23:46,260
only sees kind of an
average boundary temperature

366
00:23:46,260 --> 00:23:49,390
and not your little
ups and downs.

367
00:23:49,390 --> 00:23:53,970
So when M is big, I
expect that part of error,

368
00:23:53,970 --> 00:23:59,250
the up and down part, to be not
so significant in the center.

369
00:23:59,250 --> 00:24:01,660
Anyway, now that's my theory.

370
00:24:01,660 --> 00:24:04,790
AUDIENCE: [INAUDIBLE]

371
00:24:04,790 --> 00:24:11,320
PROFESSOR STRANG:
Ah, good question.

372
00:24:11,320 --> 00:24:14,580
So if we only looked
at the center,

373
00:24:14,580 --> 00:24:17,060
would it all be the same?

374
00:24:17,060 --> 00:24:19,900
I mean, if we're only
looking at that one point

375
00:24:19,900 --> 00:24:28,900
where it should be one at the
center, but along the thing,

376
00:24:28,900 --> 00:24:32,910
I don't know.

377
00:24:32,910 --> 00:24:36,770
If you look at both, and
see a significant difference

378
00:24:36,770 --> 00:24:40,110
in the behavior
I'd be interested.

379
00:24:40,110 --> 00:24:41,560
Yeah, yeah.

380
00:24:41,560 --> 00:24:43,960
You know, all these problems
are things that there's

381
00:24:43,960 --> 00:24:47,600
no single solution to.

382
00:24:47,600 --> 00:24:53,416
AUDIENCE: [INAUDIBLE]

383
00:24:53,416 --> 00:24:55,790
PROFESSOR STRANG: The error
between one minus r squared--

384
00:24:55,790 --> 00:25:04,010
AUDIENCE: [INAUDIBLE]

385
00:25:04,010 --> 00:25:06,770
PROFESSOR STRANG: Oh, right,
we've got slope error, too.

386
00:25:06,770 --> 00:25:10,100
That's a very significant point.

387
00:25:10,100 --> 00:25:13,130
I see, right.

388
00:25:13,130 --> 00:25:14,930
So the slope error's in there.

389
00:25:14,930 --> 00:25:18,440
Everybody knows,
then, everybody--

390
00:25:18,440 --> 00:25:22,810
In working the
problem, I mentioned

391
00:25:22,810 --> 00:25:29,420
that the boundary conditions
in this piece of pie

392
00:25:29,420 --> 00:25:32,460
were zero along here
and normal derivative,

393
00:25:32,460 --> 00:25:36,300
somehow it got printed du/dh,
but that was an accident.

394
00:25:36,300 --> 00:25:42,170
It should've been
du/dn, dn is zero.

395
00:25:42,170 --> 00:25:45,610
So Neumann conditions
on the thing

396
00:25:45,610 --> 00:25:48,020
and then I was a little
scared about that point,

397
00:25:48,020 --> 00:25:50,560
but I think phooey on it.

398
00:25:50,560 --> 00:25:56,640
It's just, don't worry about it.

399
00:25:56,640 --> 00:25:59,040
But what I was going to say.

400
00:25:59,040 --> 00:26:07,430
How do you, what do you do to
take into account this du/dn=0,

401
00:26:07,430 --> 00:26:13,520
this slope condition on
these long boundaries?

402
00:26:13,520 --> 00:26:15,790
What should you do
in finite elements

403
00:26:15,790 --> 00:26:17,790
to take account for that?

404
00:26:17,790 --> 00:26:21,490
And the answer is,
in one nice word?

405
00:26:21,490 --> 00:26:22,510
Nothing.

406
00:26:22,510 --> 00:26:24,180
Right, nothing.

407
00:26:24,180 --> 00:26:26,840
Your finite element
method should not,

408
00:26:26,840 --> 00:26:30,010
you don't impose any condition
along these boundaries.

409
00:26:30,010 --> 00:26:35,260
Just use the code as it is
with zeroes on this boundary.

410
00:26:35,260 --> 00:26:39,060
And it should work, yeah.

411
00:26:39,060 --> 00:26:39,910
It should work.

412
00:26:39,910 --> 00:26:43,910
Any comments on--
Other people, did you

413
00:26:43,910 --> 00:26:48,390
get reasonable results, or?

414
00:26:48,390 --> 00:26:49,150
Tell me something.

415
00:26:49,150 --> 00:26:55,310
Because you guys looked at
those graphs and I have not.

416
00:26:55,310 --> 00:26:57,930
Any feedback yet?

417
00:26:57,930 --> 00:26:58,870
On those?

418
00:26:58,870 --> 00:27:01,930
I'm happy to get
email, too, about.

419
00:27:01,930 --> 00:27:03,780
So all the email,
first of all they've

420
00:27:03,780 --> 00:27:09,310
corrected the typos in the
original coordinate positions.

421
00:27:09,310 --> 00:27:12,650
And now they've
pointed out I'd better

422
00:27:12,650 --> 00:27:19,000
look at M is very, very welcome.

423
00:27:19,000 --> 00:27:21,760
It doesn't mean that
everybody has to do this,

424
00:27:21,760 --> 00:27:24,450
if you've completed
that MATLAB assignment,

425
00:27:24,450 --> 00:27:30,970
you never want to see it again,
and you've kept M=8, it's OK.

426
00:27:30,970 --> 00:27:34,290
But if you're
interested to see what

427
00:27:34,290 --> 00:27:39,540
happens if M goes to 16 or
32, I'm interested also.

428
00:27:39,540 --> 00:27:41,500
Right, yeah.

429
00:27:41,500 --> 00:27:45,730
OK, so anyway that's the problem
we're really thinking about.

430
00:27:45,730 --> 00:27:50,100
And that's the problem
that is equally important,

431
00:27:50,100 --> 00:27:55,540
but it seemed reasonable
just to do one of the two.

432
00:27:55,540 --> 00:27:58,690
And we were set
up to do, we have

433
00:27:58,690 --> 00:28:00,490
the code for the
stiffness matrix,

434
00:28:00,490 --> 00:28:07,140
we would need a new code
to do these integrals.

435
00:28:07,140 --> 00:28:13,860
Because this will be
linear times linear, right?

436
00:28:13,860 --> 00:28:17,430
I'll have to compute
that one times this one

437
00:28:17,430 --> 00:28:23,480
and I would need new
formulas that are not there.

438
00:28:23,480 --> 00:28:26,830
I'd need formulas for, this
will be linear times linear

439
00:28:26,830 --> 00:28:31,710
so I'll be integrating
x squared type stuff.

440
00:28:31,710 --> 00:28:36,060
And xy's, because I'm
2-D, and y squareds.

441
00:28:36,060 --> 00:28:43,240
So it would take a little
more code, but not much.

442
00:28:43,240 --> 00:28:47,720
I think the math-- Oh,
here's a question for you.

443
00:28:47,720 --> 00:28:49,190
Here's a question for you.

444
00:28:49,190 --> 00:28:53,100
Suppose I have my trial
functions, phi_i(x).

445
00:28:53,100 --> 00:28:56,110


446
00:28:56,110 --> 00:29:00,920
What do they add up to?

447
00:29:00,920 --> 00:29:05,420
Let me again draw a
mesh, so I've got a mesh.

448
00:29:05,420 --> 00:29:09,500
These are, you
know-- I'm sorry, I

449
00:29:09,500 --> 00:29:14,890
want to put in some
more triangles here.

450
00:29:14,890 --> 00:29:18,930
Lots of triangles, whatever.

451
00:29:18,930 --> 00:29:23,640
Let me get some
more vertices, too.

452
00:29:23,640 --> 00:29:25,800
I'm getting in trouble.

453
00:29:25,800 --> 00:29:28,190
OK, whatever.

454
00:29:28,190 --> 00:29:35,030
So phi_i is the piecewise linear
guy that is one at node i.

455
00:29:35,030 --> 00:29:38,270
So I've got all these
different nodes.

456
00:29:38,270 --> 00:29:41,800
I need a node there, so I've
got one, two, three, there's

457
00:29:41,800 --> 00:29:44,670
a node, there's more nodes.

458
00:29:44,670 --> 00:29:47,610
If I add them all
up, this is just

459
00:29:47,610 --> 00:29:51,750
like in an insight question.

460
00:29:51,750 --> 00:29:55,260
I've got all these, you
could add up these hats

461
00:29:55,260 --> 00:30:00,620
in 1-D. What's the sum of
the hats in one dimension?

462
00:30:00,620 --> 00:30:01,670
One.

463
00:30:01,670 --> 00:30:03,470
Good.

464
00:30:03,470 --> 00:30:05,820
The sum is one.

465
00:30:05,820 --> 00:30:09,600
It's a nice fact that
these guys add up to one.

466
00:30:09,600 --> 00:30:12,880
And now why is it
still true here in 2-D,

467
00:30:12,880 --> 00:30:18,080
that these little
pyramids will add to one?

468
00:30:18,080 --> 00:30:22,720
That's an insight question,
but it's worth thinking about.

469
00:30:22,720 --> 00:30:25,570
Why do those
pyramids add to one?

470
00:30:25,570 --> 00:30:29,960
Let me leave that question.

471
00:30:29,960 --> 00:30:33,620
I'm thinking about, we haven't
imposed any boundary conditions

472
00:30:33,620 --> 00:30:34,160
yet.

473
00:30:34,160 --> 00:30:38,680
We've got them all. and I
claim that if we add up all

474
00:30:38,680 --> 00:30:43,740
the pyramids including the
boundary chopped off pyramids

475
00:30:43,740 --> 00:30:48,020
from the boundary, that we'll
get one throughout the whole,

476
00:30:48,020 --> 00:30:52,130
now it'll be phi(x,y).

477
00:30:52,130 --> 00:30:56,440
Because now I'm moving
to 2-D, with pyramids.

478
00:30:56,440 --> 00:31:00,110
I think we'll still have one.

479
00:31:00,110 --> 00:31:02,890
Let me give you a minute
to think about that one.

480
00:31:02,890 --> 00:31:08,170
And then we could turn
to Fourier questions

481
00:31:08,170 --> 00:31:15,340
if you would like, we could do
some problems from the text.

482
00:31:15,340 --> 00:31:17,450
Any thoughts about this guy?

483
00:31:17,450 --> 00:31:30,160
Why should all those individual
pyramids add up to a flat roof?

484
00:31:30,160 --> 00:31:31,540
Why did it work here?

485
00:31:31,540 --> 00:31:41,580
Well, it worked because you
could see it, right, somehow?

486
00:31:41,580 --> 00:31:47,110
Does it still work if the
nodes are not equally spaced?

487
00:31:47,110 --> 00:31:50,070
So we've got a hat
function for that guy,

488
00:31:50,070 --> 00:31:52,410
and a hat function for
this guy, and a hat

489
00:31:52,410 --> 00:31:54,010
function for this guy.

490
00:31:54,010 --> 00:31:57,690
And these guys
are in there, too.

491
00:31:57,690 --> 00:32:00,080
We haven't imposed anything.

492
00:32:00,080 --> 00:32:07,670
So those one, two, three,
four, five functions,

493
00:32:07,670 --> 00:32:14,296
five phis, they add
up to one, and why?

494
00:32:14,296 --> 00:32:15,920
Well, you're going
to say it's obvious,

495
00:32:15,920 --> 00:32:20,110
but that's what professors
are allowed to say.

496
00:32:20,110 --> 00:32:24,100
Things are obvious, you
have to actually say why.

497
00:32:24,100 --> 00:32:26,500
Which is not as easy.

498
00:32:26,500 --> 00:32:36,880
So, why do they add to one?

499
00:32:36,880 --> 00:32:41,700
Let me look inside one element.

500
00:32:41,700 --> 00:32:45,760
Why does the sum
of these two guys

501
00:32:45,760 --> 00:32:51,310
add to a flat top
inside that interval?

502
00:32:51,310 --> 00:32:57,700
AUDIENCE: [INAUDIBLE]

503
00:32:57,700 --> 00:33:00,170
PROFESSOR STRANG: At the
end points, you've got it.

504
00:33:00,170 --> 00:33:04,430
Because what's happening
at the end points?

505
00:33:04,430 --> 00:33:10,480
This guy, one of the guys,
the right guy is one.

506
00:33:10,480 --> 00:33:14,160
And all other guys
are zero, right.

507
00:33:14,160 --> 00:33:17,010
And this guy is also at one.

508
00:33:17,010 --> 00:33:20,530
Because it's the right guy.

509
00:33:20,530 --> 00:33:23,280
It has height one
and all others zero.

510
00:33:23,280 --> 00:33:26,960
So at the nodes we
are at one, because

511
00:33:26,960 --> 00:33:29,570
of one person,
really, one element.

512
00:33:29,570 --> 00:33:30,110
And then?

513
00:33:30,110 --> 00:33:32,870
AUDIENCE: [INAUDIBLE]

514
00:33:32,870 --> 00:33:38,370
PROFESSOR STRANG: Right.

515
00:33:38,370 --> 00:33:42,965
But the sum of them is, why
is the sum of them always one,

516
00:33:42,965 --> 00:33:44,510
why is slope zero?

517
00:33:44,510 --> 00:33:48,010
Yeah.

518
00:33:48,010 --> 00:33:51,270
The slopes cancel, right.

519
00:33:51,270 --> 00:33:54,340
We know that in between it
will be a linear function.

520
00:33:54,340 --> 00:33:56,390
That would be one
way to look at it.

521
00:33:56,390 --> 00:33:59,770
If I add up a linear function
and a linear function

522
00:33:59,770 --> 00:34:01,200
the sum is a linear function.

523
00:34:01,200 --> 00:34:03,090
So I'm getting a
linear function,

524
00:34:03,090 --> 00:34:08,800
which is one at those points,
so what is that function?

525
00:34:08,800 --> 00:34:09,570
One.

526
00:34:09,570 --> 00:34:11,790
Right, you know that's
the straight line.

527
00:34:11,790 --> 00:34:16,010
So, that idea will
work here too.

528
00:34:16,010 --> 00:34:20,900
Look inside some
little triangle here.

529
00:34:20,900 --> 00:34:25,540
OK, that's got one,
two, three corners, OK.

530
00:34:25,540 --> 00:34:31,900
And if I look at this sum,
what is it at this point?

531
00:34:31,900 --> 00:34:35,010
If I look at that
sum at this corner,

532
00:34:35,010 --> 00:34:37,990
one guy is one, the
one for that pyramid.

533
00:34:37,990 --> 00:34:40,560
And all others are?

534
00:34:40,560 --> 00:34:41,430
Zero.

535
00:34:41,430 --> 00:34:44,410
So the sum is one there,
the sum is one there,

536
00:34:44,410 --> 00:34:48,070
the sum is one there,
so that blowing up

537
00:34:48,070 --> 00:34:51,140
this little triangle,
this is at height one,

538
00:34:51,140 --> 00:34:53,860
this is at height one,
this is at height one,

539
00:34:53,860 --> 00:34:56,440
so what's the roof?

540
00:34:56,440 --> 00:34:59,530
Flat.

541
00:34:59,530 --> 00:35:04,270
It's just a nice way to see the
nice property of these phis.

542
00:35:04,270 --> 00:35:14,580
That there's a phi for every
node, and they add to one.

543
00:35:14,580 --> 00:35:17,510
To that's it.

544
00:35:17,510 --> 00:35:22,870
OK, well I was going
to say one more thing

545
00:35:22,870 --> 00:35:25,600
and I am, about this
eigenvalue problem, just

546
00:35:25,600 --> 00:35:29,680
because I'll never
have a chance again.

547
00:35:29,680 --> 00:35:33,540
So this is the moment
to say something

548
00:35:33,540 --> 00:35:34,630
about the eigenvalues.

549
00:35:34,630 --> 00:35:35,940
Lambda.

550
00:35:35,940 --> 00:35:41,290
Eigenvalue.

551
00:35:41,290 --> 00:35:44,000
I'm answering the question
where does K come from,

552
00:35:44,000 --> 00:35:45,680
where does M come from?

553
00:35:45,680 --> 00:35:56,610
Well, the eigenvalue is-- Voy we
really got dramatic music here.

554
00:35:56,610 --> 00:36:00,410
Is that the Great Gate of
Kiev, I think might be.

555
00:36:00,410 --> 00:36:01,940
Mussorgsky.

556
00:36:01,940 --> 00:36:06,230
If you like drums and big
noise, it's not music actually,

557
00:36:06,230 --> 00:36:11,190
but you got a lot
of noise out of it.

558
00:36:11,190 --> 00:36:16,710
Well, of course, he'd know
more than we do, but still.

559
00:36:16,710 --> 00:36:26,650
OK, so the eigenvalues in the
matrix case, for Kx=lambda*M*x,

560
00:36:26,650 --> 00:36:31,000
the eigenvalue problem,
lambda, the lowest eigenvalue,

561
00:36:31,000 --> 00:36:34,980
lambda_lowest, has
a nice property.

562
00:36:34,980 --> 00:36:46,020
It's the minimum of sort of our
energy over our other energy.

563
00:36:46,020 --> 00:36:51,470
I just think, well this is
something you should see.

564
00:36:51,470 --> 00:36:54,020
This is a quotient here.

565
00:36:54,020 --> 00:36:56,880
It's got a name called
the Rayleigh quotient.

566
00:36:56,880 --> 00:36:59,230
And it would appear in the book.

567
00:36:59,230 --> 00:37:01,350
So really, I guess
what I'm doing

568
00:37:01,350 --> 00:37:06,170
is calling your attention to
something that's in the book.

569
00:37:06,170 --> 00:37:10,340
That this a ratio of x
transpose K x to x transpose M

570
00:37:10,340 --> 00:37:19,560
x, if I look over all
vectors x, the lowest one

571
00:37:19,560 --> 00:37:20,870
is the eigenvector.

572
00:37:20,870 --> 00:37:23,590
The best x is the
eigenvector and the ratio

573
00:37:23,590 --> 00:37:26,410
is the eigenvalue.

574
00:37:26,410 --> 00:37:30,350
This is like my point that I
wanted to mention the Rayleigh

575
00:37:30,350 --> 00:37:31,150
quotient.

576
00:37:31,150 --> 00:37:33,340
Here it is in the
matrix case, and there

577
00:37:33,340 --> 00:37:35,080
would be similar
Rayleigh quotient

578
00:37:35,080 --> 00:37:38,490
in the continuous case.

579
00:37:38,490 --> 00:37:41,200
I'll just leave it at that.

580
00:37:41,200 --> 00:37:46,070
That in describing
eigenvalues, we can talk about

581
00:37:46,070 --> 00:37:49,070
Kx=lambda*M*x, like this.

582
00:37:49,070 --> 00:37:51,980
Or we can get energy into it.

583
00:37:51,980 --> 00:37:54,870
And you remember the whole
point about finite elements

584
00:37:54,870 --> 00:37:57,500
is, look at the energy.

585
00:37:57,500 --> 00:37:59,470
Look at that the quadratics.

586
00:37:59,470 --> 00:38:04,980
Multiply things by things.

587
00:38:04,980 --> 00:38:07,150
It came from the
weak form, it didn't

588
00:38:07,150 --> 00:38:10,220
come from the strong form.

589
00:38:10,220 --> 00:38:12,370
In the differential
equation here, we just

590
00:38:12,370 --> 00:38:15,280
have single terms.

591
00:38:15,280 --> 00:38:18,870
We got to these things
through that process

592
00:38:18,870 --> 00:38:23,460
of multiplying by
u's and integrating.

593
00:38:23,460 --> 00:38:25,090
That's what gave
us these products

594
00:38:25,090 --> 00:38:29,790
and it works also
in the matrix case.

595
00:38:29,790 --> 00:38:32,420
OK, that was a lot
of speech-making

596
00:38:32,420 --> 00:38:41,970
about topics that we simply
didn't have time for in class.

597
00:38:41,970 --> 00:38:44,050
I'm ready for any
question, or I'm

598
00:38:44,050 --> 00:38:48,210
ready to maybe do a Fourier
example, would you like that?

599
00:38:48,210 --> 00:38:51,890
Because this is
where we really are.

600
00:38:51,890 --> 00:38:55,660
I'll even take one that
will be on the homework.

601
00:38:55,660 --> 00:39:01,510
Just so you'll have a start.

602
00:39:01,510 --> 00:39:10,780
OK, let me take a square
pulse, yeah this is a good one,

603
00:39:10,780 --> 00:39:16,330
I think.

604
00:39:16,330 --> 00:39:20,320
In Section 4.1, there's a
question for the Fourier series

605
00:39:20,320 --> 00:39:21,970
of a square pulse.

606
00:39:21,970 --> 00:39:24,900
OK, so what does the
square pulse look like?

607
00:39:24,900 --> 00:39:29,090
Here's minus pi to pi.

608
00:39:29,090 --> 00:39:30,170
Here's zero.

609
00:39:30,170 --> 00:39:33,060
The square pulse
goes along here, up,

610
00:39:33,060 --> 00:39:35,360
square pulse, and down.

611
00:39:35,360 --> 00:39:48,120
Actually, let me go to L/2,
oh I'll just call it h.

612
00:39:48,120 --> 00:39:56,450
Let me find the Fourier
series for this function.

613
00:39:56,450 --> 00:40:00,160
It goes along at
0, it jumps up to 1

614
00:40:00,160 --> 00:40:05,010
over a interval of length
2h, going from minus h to h,

615
00:40:05,010 --> 00:40:08,410
and then back down to
0 and then repeats.

616
00:40:08,410 --> 00:40:11,810
So bip bip bip, square pulse.

617
00:40:11,810 --> 00:40:14,200
So that's my function.

618
00:40:14,200 --> 00:40:18,860
Is that function odd,
or even, or neither one?

619
00:40:18,860 --> 00:40:22,620
It's even, so I
can call that C(x).

620
00:40:22,620 --> 00:40:27,480
And figure that I'm going to
use cosines for that one, right?

621
00:40:27,480 --> 00:40:31,590
So tell me a formula for
the coefficients, what's

622
00:40:31,590 --> 00:40:33,890
the integral that I have to do?

623
00:40:33,890 --> 00:40:39,210
So my C(x) is going
to be some a_0,

624
00:40:39,210 --> 00:40:45,170
we have to think what's a_0,
then a_1*cos(x), a_2*cos,

625
00:40:45,170 --> 00:40:46,540
and so on.

626
00:40:46,540 --> 00:40:48,700
So on.

627
00:40:48,700 --> 00:40:49,220
a_k*cos(kx).

628
00:40:49,220 --> 00:40:52,950


629
00:40:52,950 --> 00:41:01,090
OK, what's the formula for a_k?

630
00:41:01,090 --> 00:41:04,380
Before I plug in that function
I would like to get the formula.

631
00:41:04,380 --> 00:41:07,350
So I'm looking for the formula.

632
00:41:07,350 --> 00:41:10,110
It's a formula to remember.

633
00:41:10,110 --> 00:41:11,912
So I'm not wasting your time.

634
00:41:11,912 --> 00:41:14,120
Because you're going to see
it on the board and it'll

635
00:41:14,120 --> 00:41:16,580
just take a mental
photograph of it.

636
00:41:16,580 --> 00:41:18,420
What do you think
it's going to be?

637
00:41:18,420 --> 00:41:20,170
How am I going to get it?

638
00:41:20,170 --> 00:41:26,670
I'll multiply both sides of
the equation by cos(kx), right?

639
00:41:26,670 --> 00:41:28,600
And I'll integrate.

640
00:41:28,600 --> 00:41:32,440
So and then when I integrate,
the cosines are orthogonal.

641
00:41:32,440 --> 00:41:34,970
Just like the
sines this morning.

642
00:41:34,970 --> 00:41:38,030
All those terms will go,
except for this term.

643
00:41:38,030 --> 00:41:42,750
When I multiply this by cos(kx),
I'll have cos(kx) squared.

644
00:41:42,750 --> 00:41:46,250
Here I'll have a
cos(kx), and here I'll

645
00:41:46,250 --> 00:41:50,800
have a whole lot of cos(kx)'s
but when I integrate,

646
00:41:50,800 --> 00:41:55,150
all this stuff is
going to disappear.

647
00:41:55,150 --> 00:41:57,870
And this will all disappear.

648
00:41:57,870 --> 00:41:58,970
This is it.

649
00:41:58,970 --> 00:42:04,070
So a_k is going to be the
integral of my function,

650
00:42:04,070 --> 00:42:07,740
times cos(kx) dx.

651
00:42:07,740 --> 00:42:09,010
Divided by what?

652
00:42:09,010 --> 00:42:14,170
Divided by the integral
of cos(kx) squared.

653
00:42:14,170 --> 00:42:18,130
Because I haven't
normalized things.

654
00:42:18,130 --> 00:42:21,620
So I don't know that that's
one, and in fact it isn't one.

655
00:42:21,620 --> 00:42:25,040
So I have to remember
to put that number in.

656
00:42:25,040 --> 00:42:28,030
OK, so that's the
formula and that number

657
00:42:28,030 --> 00:42:32,310
turns out to be pi, again.

658
00:42:32,310 --> 00:42:36,830
If I'm integrating
from minus pi to pi,

659
00:42:36,830 --> 00:42:40,150
then the average value of
the cosine squared is a 1/2,

660
00:42:40,150 --> 00:42:44,450
it's sort of as much above
1/2 as it is below 1/2,

661
00:42:44,450 --> 00:42:51,500
and so the average is 1/2,
the interval is 2pi, so pi.

662
00:42:51,500 --> 00:42:54,940
OK, that's the formula.

663
00:42:54,940 --> 00:42:59,600
Please just take a
mental photograph.

664
00:42:59,600 --> 00:43:00,740
Catch that one.

665
00:43:00,740 --> 00:43:05,990
Alright, now I've got
my particular C(x),

666
00:43:05,990 --> 00:43:10,340
my square wave, square pulse.

667
00:43:10,340 --> 00:43:11,760
Very, very important.

668
00:43:11,760 --> 00:43:16,160
Very important
Fourier series here.

669
00:43:16,160 --> 00:43:18,210
Famous one.

670
00:43:18,210 --> 00:43:20,250
OK, so what do I have?

671
00:43:20,250 --> 00:43:24,560
From minus pi to pi,
so what's my integral?

672
00:43:24,560 --> 00:43:28,370
Well, my integral really
doesn't go from minus pi to pi

673
00:43:28,370 --> 00:43:31,430
because my function
is mostly zero.

674
00:43:31,430 --> 00:43:34,330
Where does my integral go?

675
00:43:34,330 --> 00:43:36,950
Negative h to h, right?

676
00:43:36,950 --> 00:43:40,840
And in that region,
what is C(x)?

677
00:43:40,840 --> 00:43:44,320
One.

678
00:43:44,320 --> 00:43:47,960
So you see it's
going to be nice.

679
00:43:47,960 --> 00:43:52,180
From negative h to
h, where this is one,

680
00:43:52,180 --> 00:43:55,630
I just have to integrate
cos(kx), so what do I get?

681
00:43:55,630 --> 00:44:01,930
sin(kx), over k, and a pi.

682
00:44:01,930 --> 00:44:06,490
So you see again that that k is
showing up in the denominator,

683
00:44:06,490 --> 00:44:13,650
and that's going to give me
the typical decay rate of 1/k

684
00:44:13,650 --> 00:44:20,110
for functions with steps.

685
00:44:20,110 --> 00:44:21,480
For step functions.

686
00:44:21,480 --> 00:44:28,960
And now I have to evaluate
this between minus pi and pi.

687
00:44:28,960 --> 00:44:30,200
And-- No, h.

688
00:44:30,200 --> 00:44:32,090
Better be h.

689
00:44:32,090 --> 00:44:35,090
I mean, minus h and h.

690
00:44:35,090 --> 00:44:37,830
So what do I get for that?

691
00:44:37,830 --> 00:44:42,140
I get sin(kh), right?

692
00:44:42,140 --> 00:44:48,520
At the top, and what
do I get at minus?

693
00:44:48,520 --> 00:44:54,930
So I now I want to subtract,
what is the sine of minus kh?

694
00:44:54,930 --> 00:44:57,360
It's negative, right?

695
00:44:57,360 --> 00:45:04,070
So as I expect with an
even function like cosine,

696
00:45:04,070 --> 00:45:08,700
am I just getting twice?

697
00:45:08,700 --> 00:45:12,930
I could take it from 0 to
h, and it would give me

698
00:45:12,930 --> 00:45:14,490
one of them and the other one.

699
00:45:14,490 --> 00:45:16,500
Yep, I think so,
and divide by k*pi.

700
00:45:16,500 --> 00:45:21,390


701
00:45:21,390 --> 00:45:24,480
So those are the
Fourier coefficients.

702
00:45:24,480 --> 00:45:27,060
Except for a_0.

703
00:45:27,060 --> 00:45:31,090
a_0 has a slightly
different formula,

704
00:45:31,090 --> 00:45:34,850
because for a_0, why
is a_0 different?

705
00:45:34,850 --> 00:45:39,170
How do you come up with
a_0, and what's its meaning?

706
00:45:39,170 --> 00:45:44,940
a_0 has a nice meaning,
so this is worth

707
00:45:44,940 --> 00:45:46,900
having come this afternoon for.

708
00:45:46,900 --> 00:45:50,900
a_0 will be what?

709
00:45:50,900 --> 00:45:52,610
Well, I could get
it the same way.

710
00:45:52,610 --> 00:45:57,110
What will I multiply
both sides by?

711
00:45:57,110 --> 00:46:00,200
If I want to pick off a_0?

712
00:46:00,200 --> 00:46:01,810
Just one.

713
00:46:01,810 --> 00:46:06,530
It's not a cosine, it's
the cos(0x), it's the one.

714
00:46:06,530 --> 00:46:08,460
And then I integrate.

715
00:46:08,460 --> 00:46:11,350
I'm just going to
get the integral

716
00:46:11,350 --> 00:46:15,970
from minus pi to
pi of C(x) times

717
00:46:15,970 --> 00:46:19,796
one, divided by the
integral from minus pi

718
00:46:19,796 --> 00:46:31,190
to pi of one times one. dx.

719
00:46:31,190 --> 00:46:33,190
Same method.

720
00:46:33,190 --> 00:46:39,170
Multiply both sides by one,
which was the very first

721
00:46:39,170 --> 00:46:41,660
of my orthogonal functions.

722
00:46:41,660 --> 00:46:45,250
Integrate it, all the other
integrals went away, right?

723
00:46:45,250 --> 00:46:48,900
The integral of cosine
over a whole interval.

724
00:46:48,900 --> 00:46:51,000
It's periodic.

725
00:46:51,000 --> 00:46:52,630
You get the same
at the two ends,

726
00:46:52,630 --> 00:46:56,620
so the difference is zero.

727
00:46:56,620 --> 00:47:00,230
So we just, the only term
left was the constant.

728
00:47:00,230 --> 00:47:03,060
And now what is the integral,
what's the denominator now?

729
00:47:03,060 --> 00:47:06,490
That was the little,
slight twist.

730
00:47:06,490 --> 00:47:07,780
2pi.

731
00:47:07,780 --> 00:47:09,910
The denominator's 2pi.

732
00:47:09,910 --> 00:47:10,630
Yeah.

733
00:47:10,630 --> 00:47:15,490
That's that's why it's not,
yeah, it's slightly irregular,

734
00:47:15,490 --> 00:47:18,010
I have to divide by 2pi.

735
00:47:18,010 --> 00:47:21,540
And now, what word would
you use to describe,

736
00:47:21,540 --> 00:47:24,180
if I have a function,
and integrate it,

737
00:47:24,180 --> 00:47:30,050
and I divide by the
length, what am I getting?

738
00:47:30,050 --> 00:47:35,710
There's an English word that
would describe what this is.

739
00:47:35,710 --> 00:47:37,780
Average.

740
00:47:37,780 --> 00:47:44,880
This is the average.

741
00:47:44,880 --> 00:47:46,530
And it has to be.

742
00:47:46,530 --> 00:47:49,490
This constant term is
always the average.

743
00:47:49,490 --> 00:47:52,060
And what will it be for this?

744
00:47:52,060 --> 00:47:59,130
So this was a_k, and
what is a_0, then?

745
00:47:59,130 --> 00:48:02,430
So you can now tell
me, so everybody's

746
00:48:02,430 --> 00:48:05,090
remembering this formula,
you integrate the function

747
00:48:05,090 --> 00:48:07,980
and divide by the 2pi.

748
00:48:07,980 --> 00:48:10,070
Now we've got a
particular function,

749
00:48:10,070 --> 00:48:12,220
so what is the integral
of that function?

750
00:48:12,220 --> 00:48:16,210
So what does this equal?

751
00:48:16,210 --> 00:48:18,180
For this particular C(x)?

752
00:48:18,180 --> 00:48:22,210
What's the area under
that function C(x)?

753
00:48:22,210 --> 00:48:24,780
2h.

754
00:48:24,780 --> 00:48:26,040
Right?

755
00:48:26,040 --> 00:48:29,070
So 2h/(2pi), cancel the twos.

756
00:48:29,070 --> 00:48:34,060
So there's a constant
term, a_0 is h/pi

757
00:48:34,060 --> 00:48:42,130
and the cosine terms
are-- yeah, actually

758
00:48:42,130 --> 00:48:43,680
we're going to get
something nice.

759
00:48:43,680 --> 00:48:47,610
A really nice way
to complete this

760
00:48:47,610 --> 00:48:55,220
will be if I put this together,
put this series together.

761
00:48:55,220 --> 00:49:05,960
So now I'm saying that this
square pulse is that constant

762
00:49:05,960 --> 00:49:12,820
term h/pi plus
the next term a_1,

763
00:49:12,820 --> 00:49:17,990
you can see all these
terms have 2/pi's.

764
00:49:17,990 --> 00:49:20,660
I'm a little surprised
that h over-- Yeah, no, I

765
00:49:20,660 --> 00:49:21,710
guess it's right.

766
00:49:21,710 --> 00:49:29,170
2/pi, yeah, right?

767
00:49:29,170 --> 00:49:33,700
So I've got sin(h), I think.

768
00:49:33,700 --> 00:49:35,380
And now I'm just copying this.

769
00:49:35,380 --> 00:49:47,030
2/pi*sin(h), sin(h),
is that what I want?

770
00:49:47,030 --> 00:49:51,500
Over one.

771
00:49:51,500 --> 00:49:54,810
That's the coefficient
of sine, of cos(x).

772
00:49:54,810 --> 00:49:58,060


773
00:49:58,060 --> 00:50:01,550
a_1 was the
coefficient of cos(1x).

774
00:50:01,550 --> 00:50:06,160
And then a_2 is the
coefficient of cos(2x).

775
00:50:06,160 --> 00:50:10,870
So that will be
sin(2h). k is two,

776
00:50:10,870 --> 00:50:14,770
so I have a two
down here, cos(2x).

777
00:50:14,770 --> 00:50:20,470
And so on.

778
00:50:20,470 --> 00:50:23,840
Yeah, I think that's
the Fourier series.

779
00:50:23,840 --> 00:50:33,240
That would be the Fourier
series for the square pulse.

780
00:50:33,240 --> 00:50:34,350
Yeah.

781
00:50:34,350 --> 00:50:38,200
That would be the Fourier
series for the square pulse.

782
00:50:38,200 --> 00:50:40,470
Could I test any
interesting cases?

783
00:50:40,470 --> 00:50:45,350
Suppose h is all
the way out to pi.

784
00:50:45,350 --> 00:50:46,990
Suppose I take that case.

785
00:50:46,990 --> 00:50:55,190
Let h go all the way out to
pi, then what's my function?

786
00:50:55,190 --> 00:51:01,080
If h=pi, then what
have I got a graph of?

787
00:51:01,080 --> 00:51:02,160
Just one.

788
00:51:02,160 --> 00:51:03,530
It's just a one.

789
00:51:03,530 --> 00:51:07,110
If h is pi, what happens?

790
00:51:07,110 --> 00:51:11,760
That becomes a one, and what
about these other things?

791
00:51:11,760 --> 00:51:15,030
What is this thing when h is pi?

792
00:51:15,030 --> 00:51:15,620
Zero.

793
00:51:15,620 --> 00:51:18,470
All the other terms go away.

794
00:51:18,470 --> 00:51:22,230
It's just a sin(2pi)
that would go away.

795
00:51:22,230 --> 00:51:27,470
Yeah, so if h is pi, if I go out
to the place where I don't have

796
00:51:27,470 --> 00:51:33,200
any jumps at all because it's
now all the way out there,

797
00:51:33,200 --> 00:51:37,410
then these terms all disappear
and I just have this.

798
00:51:37,410 --> 00:51:39,960
And I would like
to ask you and it's

799
00:51:39,960 --> 00:51:47,850
going to come up on Friday too,
what happens if h goes to zero?

800
00:51:47,850 --> 00:51:50,570
Well, let me just
take h going to zero.

801
00:51:50,570 --> 00:51:52,140
What happens to
this whole thing?

802
00:51:52,140 --> 00:51:55,920
What happens to my
function as h goes to zero?

803
00:51:55,920 --> 00:51:58,890
Goes to zero, right, we've
squeezed it to nothing.

804
00:51:58,890 --> 00:52:04,780
And if h is zero then
sin(h) is zero, I get 0=0,

805
00:52:04,780 --> 00:52:09,780
that's not interesting
enough to mention on Friday.

806
00:52:09,780 --> 00:52:13,960
But there is one case
that is important.

807
00:52:13,960 --> 00:52:16,600
Suppose I make the height, yeah.

808
00:52:16,600 --> 00:52:17,600
Make a guess.

809
00:52:17,600 --> 00:52:27,680
Suppose I make the height higher
as I make the base smaller.

810
00:52:27,680 --> 00:52:29,410
I'm going to keep
the area as one,

811
00:52:29,410 --> 00:52:33,640
so if this has a base of 2h,
I'm going to have a height

812
00:52:33,640 --> 00:52:39,070
of 1/(2h), So if I
keep the area at one,

813
00:52:39,070 --> 00:52:43,910
so the height now is 1/(2h),
so now my square pulse,

814
00:52:43,910 --> 00:52:46,430
I've divided it by 2h.

815
00:52:46,430 --> 00:52:50,980
I have a 1/(2h)
multiplying everything.

816
00:52:50,980 --> 00:52:56,820
And now if I let h go to zero,
something more interesting

817
00:52:56,820 --> 00:52:58,320
will happen.

818
00:52:58,320 --> 00:52:59,570
And what?

819
00:52:59,570 --> 00:53:05,370
Just tell me first, what
would you expect to happen?

820
00:53:05,370 --> 00:53:05,980
Delta.

821
00:53:05,980 --> 00:53:08,310
Right, delta.

822
00:53:08,310 --> 00:53:12,500
So what I'll see show up
will be the Fourier series

823
00:53:12,500 --> 00:53:15,180
for the delta function.

824
00:53:15,180 --> 00:53:22,300
When I divide by 2h,
so I have sin(h)'s over

825
00:53:22,300 --> 00:53:27,840
h's, and of course what's the
great fact about sin(h)/h?

826
00:53:27,840 --> 00:53:33,440
As h goes to zero,
it goes to, everybody

827
00:53:33,440 --> 00:53:36,170
know what, that's the big deal.

828
00:53:36,170 --> 00:53:36,880
Yeah.

829
00:53:36,880 --> 00:53:43,030
One. sin(h) is the same size
as h for a very small h,

830
00:53:43,030 --> 00:53:44,780
and approaches one.

831
00:53:44,780 --> 00:53:49,380
Yeah so we'll see the
delta function Friday.

832
00:53:49,380 --> 00:53:52,470
OK, so you've got a
sort of mini-lecture

833
00:53:52,470 --> 00:53:56,410
instead of a real chance
to ask about homework.

834
00:53:56,410 --> 00:53:58,090
Next Wednesday
should be different

835
00:53:58,090 --> 00:54:01,010
because there will be
Fourier series homework,

836
00:54:01,010 --> 00:54:05,570
and I'll be ready to
answer questions about it.

837
00:54:05,570 --> 00:54:07,340
OK, thanks.