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PROFESSOR STRANG: So,
shall we start, as always,

00:00:24.750 --> 00:00:26.570
just open for questions.

00:00:26.570 --> 00:00:29.150
About any topic.

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I listed again trusses,
1-D finite elements,

00:00:34.080 --> 00:00:35.540
grad, div, curl.

00:00:35.540 --> 00:00:39.970
And I should have
squeezed in x+iy too,

00:00:39.970 --> 00:00:46.000
as the magic trick for finding
solutions to Laplace's equation

00:00:46.000 --> 00:00:50.400
in 2-D. So those are all
certainly topics that are

00:00:50.400 --> 00:00:52.210
in this part of the course.

00:00:52.210 --> 00:00:57.330
We didn't really get to
3-D, I'm sorry about that,

00:00:57.330 --> 00:00:59.160
where the curl comes in.

00:00:59.160 --> 00:01:01.900
Maybe I can say a few
words about curl today.

00:01:01.900 --> 00:01:03.050
Anyway, questions.

00:01:03.050 --> 00:01:03.830
Discussion.

00:01:03.830 --> 00:01:04.530
Yes, thanks.

00:01:04.530 --> 00:01:10.600
AUDIENCE: [INAUDIBLE]

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PROFESSOR STRANG:
Ooh, let's see.

00:01:12.520 --> 00:01:14.070
So A^T A for a truss.

00:01:14.070 --> 00:01:16.620
That's a good question.

00:01:16.620 --> 00:01:27.490
Trusses, A^T A. I guess I don't
know any magic tricks either,

00:01:27.490 --> 00:01:32.420
so one way is to construct
A, or A transpose,

00:01:32.420 --> 00:01:35.420
and then just multiply.

00:01:35.420 --> 00:01:43.290
A second way to do it would be
by the four by four bar element

00:01:43.290 --> 00:01:47.700
matrices, so go bar by bar.

00:01:47.700 --> 00:01:54.650
So four by four bar
matrices, four by four.

00:01:54.650 --> 00:02:00.270
So those are already in
the A transpose A form.

00:02:00.270 --> 00:02:08.480
They're little A element,
A bar transpose A, jeez,

00:02:08.480 --> 00:02:13.380
this isn't a great
as it should be.

00:02:13.380 --> 00:02:20.010
A element, but I don't
know if that would be,

00:02:20.010 --> 00:02:24.150
so and then you pop those
into their correct places.

00:02:24.150 --> 00:02:28.880
I don't think I know any
great idea beyond that.

00:02:28.880 --> 00:02:34.450
I think you should really be
ready to construct a matrix A,

00:02:34.450 --> 00:02:35.890
yeah.

00:02:35.890 --> 00:02:44.250
For a reasonably small
truss, of course.

00:02:44.250 --> 00:02:47.560
And of course the
other part of trusses,

00:02:47.560 --> 00:02:56.070
the fun part is to be able to
recognize solutions to Au=0.

00:02:56.070 --> 00:03:03.190
Possibly by looking at the
truss more than by solving Au=0.

00:03:03.190 --> 00:03:05.700
Yeah, any particular example?

00:03:05.700 --> 00:03:10.790
Of a truss that I should look
at just to pin this down?

00:03:10.790 --> 00:03:14.530
Any favorite trusses?

00:03:14.530 --> 00:03:19.150
There was an exam question, what
was it, a complicated truss?

00:03:19.150 --> 00:03:22.210
Let's just create a truss.

00:03:22.210 --> 00:03:25.440
And just think about it.

00:03:25.440 --> 00:03:35.160
Maybe I won't create the whole
matrix A. Here's a truss.

00:03:35.160 --> 00:03:37.650
How's that for a truss?

00:03:37.650 --> 00:03:43.430
So it's got-- And let me
put no supports on it.

00:03:43.430 --> 00:03:48.260
Just, there's a
truss to think about.

00:03:48.260 --> 00:03:52.290
Probably we won't get
to all the gory details

00:03:52.290 --> 00:03:57.280
but if you look at that
truss, what's the shape of A?

00:03:57.280 --> 00:04:06.680
Shape of the matrix A. I
have a row for every bar.

00:04:06.680 --> 00:04:11.340
So one, two, three, four, five.

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And how many columns
have I got, how many

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unknown u's, unknown
displacements have I got?

00:04:18.620 --> 00:04:19.740
Eight.

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Two four, six, eight.

00:04:21.950 --> 00:04:30.980
So I would expect Au=0
would probably have how many

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independent solutions?

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Three.

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You don't know the exact
rank, that's exactly true.

00:04:41.690 --> 00:04:43.880
There could be more
than three, right.

00:04:43.880 --> 00:04:46.520
So to really pin
it down you'd have

00:04:46.520 --> 00:04:51.370
to be sure you were
right about that.

00:04:51.370 --> 00:04:54.050
So three, at least.

00:04:54.050 --> 00:05:01.800
And I guess here you could tell
me the three solutions to Au=0.

00:05:01.800 --> 00:05:03.640
Three rigid motions.

00:05:03.640 --> 00:05:07.830
I could translate it to the
right, I could translate it up,

00:05:07.830 --> 00:05:09.790
and you would know
what the u is,

00:05:09.790 --> 00:05:16.910
so u translating to the right
would be [1, 0, 1, 0, 1, 0, 1,

00:05:16.910 --> 00:05:19.360
0], right?

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That's horizontal motion
all the same, rigid motion.

00:05:25.370 --> 00:05:29.200
And we should certainly
discover that if we created A

00:05:29.200 --> 00:05:31.830
for this truss,
that Au was zero.

00:05:31.830 --> 00:05:36.010
And similarly vertical motion
and the third one would be?

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Rotation, rotation.

00:05:37.640 --> 00:05:39.690
Yeah.

00:05:39.690 --> 00:05:43.740
So if I did the rotation
around there, for example,

00:05:43.740 --> 00:05:49.250
this guy would-- this u
would also be a [1, 0] here.

00:05:49.250 --> 00:05:50.900
This wouldn't move.

00:05:50.900 --> 00:05:55.950
So I'm putting in the four
pieces that would go into u.

00:05:55.950 --> 00:06:00.550
This one, what would be the
u for this, the displacement

00:06:00.550 --> 00:06:05.600
of that corner of the truss?

00:06:05.600 --> 00:06:06.530
In a rotation?

00:06:06.530 --> 00:06:12.470
So my rotation is just swing
this whole thing around.

00:06:12.470 --> 00:06:15.660
Zero? [0, -1], I think.

00:06:15.660 --> 00:06:16.470
Right.

00:06:16.470 --> 00:06:18.380
Because it's not
going to go out,

00:06:18.380 --> 00:06:21.430
it's going to go
straight down, [0, -1].

00:06:21.430 --> 00:06:26.520
And what do you think this
guy is? [1, -1], let's see.

00:06:26.520 --> 00:06:29.100
It's going to go
this way, so it's

00:06:29.100 --> 00:06:31.020
going to go forward and down.

00:06:31.020 --> 00:06:32.570
And I think you're right.

00:06:32.570 --> 00:06:35.370
One and negative one, yeah.

00:06:35.370 --> 00:06:37.190
I think that would be right.

00:06:37.190 --> 00:06:46.310
Yeah, then the truss,
we could check each bar.

00:06:46.310 --> 00:06:49.010
That bar, for example,
should not change length

00:06:49.010 --> 00:06:53.630
because the movement is
perpendicular to the bar

00:06:53.630 --> 00:06:55.830
and I'm writing
ones, but I really

00:06:55.830 --> 00:07:00.550
should write some much smaller
number like .1 everywhere,

00:07:00.550 --> 00:07:05.350
or something just so that
this isn't a very big angle.

00:07:05.350 --> 00:07:09.910
And it wouldn't change
length to first order.

00:07:09.910 --> 00:07:16.980
So that that's maybe an example
where we see the motions,

00:07:16.980 --> 00:07:19.380
but we didn't actually
create A, and we should

00:07:19.380 --> 00:07:23.420
be able to create A,
don't let me prevent you

00:07:23.420 --> 00:07:28.380
from thinking about A. Yeah.

00:07:28.380 --> 00:07:30.929
AUDIENCE: [INAUDIBLE]
Should those be ones,

00:07:30.929 --> 00:07:32.470
or should they be
root two over twos?

00:07:32.470 --> 00:07:34.386
PROFESSOR STRANG: Well,
that's a good question

00:07:34.386 --> 00:07:39.590
and after many years I've
figured out that they're ones.

00:07:39.590 --> 00:07:41.730
But it's a very good question.

00:07:41.730 --> 00:07:45.000
Let's just see why.

00:07:45.000 --> 00:07:49.970
Let's look at this bar to
be sure it's not stretched.

00:07:49.970 --> 00:07:50.570
Right?

00:07:50.570 --> 00:07:54.910
So this guy is moving over
by one, and this also by one,

00:07:54.910 --> 00:07:55.980
is my claim.

00:07:55.980 --> 00:07:59.180
And then this movement
down doesn't stretch it

00:07:59.180 --> 00:08:00.890
to first order, yeah.

00:08:00.890 --> 00:08:04.450
So I needed to make
those guys the same.

00:08:04.450 --> 00:08:07.320
I guess what I figured out is
that if you're rotating around

00:08:07.320 --> 00:08:14.340
here, then somehow it's the
x and y, is that-- Anyway.

00:08:14.340 --> 00:08:19.330
Whatever.

00:08:19.330 --> 00:08:26.650
So that gives us a chance
to do a specific example.

00:08:26.650 --> 00:08:31.070
OK, but I've dodged the
creation of A. Yep, thanks.

00:08:31.070 --> 00:08:37.600
AUDIENCE: [INAUDIBLE]

00:08:37.600 --> 00:08:41.040
PROFESSOR STRANG: Sorry, the
solution to A transpose A?

00:08:41.040 --> 00:08:46.424
AUDIENCE: [INAUDIBLE]

00:08:46.424 --> 00:08:47.840
PROFESSOR STRANG:
Yeah, I see, OK.

00:08:47.840 --> 00:08:52.180
So the reason I stopped
here was that A transpose A

00:08:52.180 --> 00:08:53.790
will be singular.

00:08:53.790 --> 00:09:00.440
So I wouldn't, like, go ahead,
go forward to A transpose Au=f.

00:09:00.440 --> 00:09:08.430
But if I put on some supports,
then of course now all good.

00:09:08.430 --> 00:09:14.840
So now I have, what's
now the shape of A?

00:09:14.840 --> 00:09:17.210
For this one.

00:09:17.210 --> 00:09:22.270
I now have this bar is
now, forget it, right?

00:09:22.270 --> 00:09:24.840
This bar is just
between two supports.

00:09:24.840 --> 00:09:31.100
So if we put it in the matrix
it'll just be a row of zeroes.

00:09:31.100 --> 00:09:33.520
Nothing will happen,
and we're better off

00:09:33.520 --> 00:09:34.990
to just knock it out.

00:09:34.990 --> 00:09:39.470
So I think, now I have,
I now have four bars.

00:09:39.470 --> 00:09:41.720
And how many unknowns?

00:09:41.720 --> 00:09:44.420
Four: two there and two there.

00:09:44.420 --> 00:09:48.340
And, do you guess
that it's stable?

00:09:48.340 --> 00:09:49.570
That truss?

00:09:49.570 --> 00:09:51.780
Yeah, that looks stable to me.

00:09:51.780 --> 00:09:55.550
So the four by four
matrix would be invertible

00:09:55.550 --> 00:09:58.050
and then I could solve.

00:09:58.050 --> 00:09:59.750
Good point.

00:09:59.750 --> 00:10:03.800
Then, A would be four
by four, A transpose

00:10:03.800 --> 00:10:07.560
would be four by four, C would
be four by four in between.

00:10:07.560 --> 00:10:11.450
This is the case that-- I
gave a name for this case.

00:10:11.450 --> 00:10:14.770
When I have a square matrix,
do you remember the name just

00:10:14.770 --> 00:10:15.900
for the hell of it?

00:10:15.900 --> 00:10:17.690
Statically determinate.

00:10:17.690 --> 00:10:21.520
It's determinate because each
step determines everything

00:10:21.520 --> 00:10:22.580
completely.

00:10:22.580 --> 00:10:26.360
Normally, if I have
another bar there,

00:10:26.360 --> 00:10:31.090
now it would be five
by four, and now

00:10:31.090 --> 00:10:36.020
I really have to do the A
transpose C A to get to a four

00:10:36.020 --> 00:10:38.590
by four invertible.

00:10:38.590 --> 00:10:43.350
By itself, A would
not be invertible.

00:10:43.350 --> 00:10:47.230
This is the more typical
case, where you really

00:10:47.230 --> 00:10:49.190
have to put all three together.

00:10:49.190 --> 00:10:51.410
Right.

00:10:51.410 --> 00:10:55.920
I hope you enjoyed the trusses
part, though, and continue

00:10:55.920 --> 00:11:01.050
to enjoy them this evening.

00:11:01.050 --> 00:11:04.410
OK, I'll just keep
moving to be sure

00:11:04.410 --> 00:11:07.220
that we cover any other topics.

00:11:07.220 --> 00:11:07.840
Yeah, thanks.

00:11:07.840 --> 00:11:09.834
AUDIENCE: [INAUDIBLE]

00:11:09.834 --> 00:11:11.250
PROFESSOR STRANG:
For this matrix?

00:11:11.250 --> 00:11:13.036
AUDIENCE: [INAUDIBLE]

00:11:13.036 --> 00:11:14.410
PROFESSOR STRANG:
For that truss?

00:11:14.410 --> 00:11:15.960
Oh my God.

00:11:15.960 --> 00:11:20.260
OK, let me see.

00:11:20.260 --> 00:11:23.680
Then can I do one row?

00:11:23.680 --> 00:11:27.990
OK, of course, you guys are
responsible for much more.

00:11:27.990 --> 00:11:29.310
Alright, which row shall I do?

00:11:29.310 --> 00:11:32.310
Which bar?

00:11:32.310 --> 00:11:33.430
A diagonal bar?

00:11:33.430 --> 00:11:38.210
I knew you'd make it like,
you could make up quizzes

00:11:38.210 --> 00:11:40.180
and I could just sit back here.

00:11:40.180 --> 00:11:43.860
OK, so let's take this
diagonal bar, alright.

00:11:43.860 --> 00:11:47.470
And are we going to keep
that supported, or not?

00:11:47.470 --> 00:11:49.590
Do you want to
keep it supported?

00:11:49.590 --> 00:11:53.530
OK, so in this case then
that end is not moving.

00:11:53.530 --> 00:11:55.960
So this will be in
this, and this bar

00:11:55.960 --> 00:12:01.470
corresponds to a row of A.
And how many non-zeroes will

00:12:01.470 --> 00:12:03.540
I expect in that row?

00:12:03.540 --> 00:12:04.750
Just two.

00:12:04.750 --> 00:12:08.410
Normally four, but I'm not
getting any motion down here.

00:12:08.410 --> 00:12:11.500
So it'll just be two and
if that's 45 degrees,

00:12:11.500 --> 00:12:16.370
shall we say, then that
row, I think, would be what?

00:12:16.370 --> 00:12:19.320
Well, OK.

00:12:19.320 --> 00:12:21.680
Where do my non-zeroes appear?

00:12:21.680 --> 00:12:26.080
This is node number
one with an H and a V.

00:12:26.080 --> 00:12:30.910
So I think we have zeroes.

00:12:30.910 --> 00:12:37.400
If that bar stretches-- The
connection between displacement

00:12:37.400 --> 00:12:40.840
and stretching of this bar
does not involve this guy.

00:12:40.840 --> 00:12:42.800
So I think it's
zero and zero there.

00:12:42.800 --> 00:12:45.350
And now, what else is it?

00:12:45.350 --> 00:12:47.200
So now come the
real numbers, which

00:12:47.200 --> 00:12:51.320
I believe to be cosine
and sine of that angle.

00:12:51.320 --> 00:12:56.400
Because if I, how much
does that bar stretch?

00:12:56.400 --> 00:12:59.865
I know that I'm looking
for a cosine and a sine,

00:12:59.865 --> 00:13:04.840
and if this goes positively,
then the bar does stretch.

00:13:04.840 --> 00:13:08.060
If this goes positively,
that does stretch the bar

00:13:08.060 --> 00:13:10.810
so I'm expecting
positive numbers there,

00:13:10.810 --> 00:13:14.270
like the cosine, square
root of two over two,

00:13:14.270 --> 00:13:20.360
and the sine, square
root of two over two.

00:13:20.360 --> 00:13:25.670
Well, I dodged the bullet
of getting the whole matrix,

00:13:25.670 --> 00:13:27.680
but maybe that would do it.

00:13:27.680 --> 00:13:30.280
Why don't we do
this the top one?

00:13:30.280 --> 00:13:31.790
Yeah.

00:13:31.790 --> 00:13:36.370
Tell me the first, if
that's bar one, what would

00:13:36.370 --> 00:13:38.950
be the first row of the matrix?

00:13:38.950 --> 00:13:42.410
OK, it involves
both of these nodes.

00:13:42.410 --> 00:13:44.080
But the angle is zero.

00:13:44.080 --> 00:13:48.620
So that's going to
be a little special.

00:13:48.620 --> 00:13:55.550
So if this goes
out horizontally,

00:13:55.550 --> 00:13:57.350
I should really start
with the first one.

00:13:57.350 --> 00:14:00.160
Is this goes horizontally
it compresses the bar,

00:14:00.160 --> 00:14:02.500
I think we get a
minus one there.

00:14:02.500 --> 00:14:05.670
If it goes vertically,
that doesn't do anything.

00:14:05.670 --> 00:14:08.430
If this goes horizontally
it does do something.

00:14:08.430 --> 00:14:11.090
If it goes vertically
it doesn't.

00:14:11.090 --> 00:14:22.370
I'd say that would be the row of
the matrix coming from the top.

00:14:22.370 --> 00:14:24.100
That would give
me the stretching.

00:14:24.100 --> 00:14:25.840
You remember, I'm
always going to,

00:14:25.840 --> 00:14:28.490
I think of
multiplying this by u.

00:14:28.490 --> 00:14:37.200
I think of multiplying that by
[u 1 H, u 1 V, u 2 H, u 2 V],

00:14:37.200 --> 00:14:41.960
and this top row
should give me you

00:14:41.960 --> 00:14:57.930
u 2 H minus u 1 H, which
is the stretch in bar one.

00:14:57.930 --> 00:15:00.470
So that would be a
typical one, this

00:15:00.470 --> 00:15:03.870
would be at least
typical of one where

00:15:03.870 --> 00:15:06.500
I do see a cosine and a sine.

00:15:06.500 --> 00:15:15.270
And let me just finally add,
suppose this was not supported.

00:15:15.270 --> 00:15:17.280
OK, suppose that's
not supported,

00:15:17.280 --> 00:15:20.430
now I've got a couple more
columns to squeeze in.

00:15:20.430 --> 00:15:23.220
Maybe I can somehow do it here.

00:15:23.220 --> 00:15:25.820
Can I squeeze in the
two more columns?

00:15:25.820 --> 00:15:31.240
So can you complete the
top row of the matrix?

00:15:31.240 --> 00:15:33.310
Now I've got six columns.

00:15:33.310 --> 00:15:37.550
Because here's two, here's
two, here's two more.

00:15:37.550 --> 00:15:40.850
What goes on the top
row of a matrix now?

00:15:40.850 --> 00:15:44.580
Zeroes, because this is
not affected by bar one.

00:15:44.580 --> 00:15:47.920
But it is affected by this bar.

00:15:47.920 --> 00:15:52.070
So it's going to show up in this
row, and how will it show up?

00:15:52.070 --> 00:15:53.290
Two negatives, right.

00:15:53.290 --> 00:15:59.450
A negative cosine and a negative
sine. and at 45 degrees I

00:15:59.450 --> 00:16:02.220
can't tell the difference.

00:16:02.220 --> 00:16:05.940
Because if these move forward,
that compresses the bar.

00:16:05.940 --> 00:16:07.210
So the minus sign.

00:16:07.210 --> 00:16:11.430
So again, the rows
add up to zero,

00:16:11.430 --> 00:16:18.220
as we expect when the bar
is not touching a support.

00:16:18.220 --> 00:16:26.240
This is not touching a
support, so it adds up to zero.

00:16:26.240 --> 00:16:34.840
OK, we'll have a truss
problem this evening,

00:16:34.840 --> 00:16:39.500
but not a big messy one.

00:16:39.500 --> 00:16:42.320
How about finite elements?

00:16:42.320 --> 00:16:45.000
You guys, do you
like finite elements?

00:16:45.000 --> 00:16:49.180
I'm sort of hoping to
make them attractive.

00:16:49.180 --> 00:16:52.960
I noticed a problem, just
to give us some specific one

00:16:52.960 --> 00:16:56.850
to work on, and I don't remember
that it was a homework problem.

00:16:56.850 --> 00:17:07.590
This is Section 3.1, number 18,
asks about the equation u''=0.

00:17:07.590 --> 00:17:11.260
Well, we've talked
about it in class.

00:17:11.260 --> 00:17:19.220
With u(0)=0 but u' of-- the
slope equal zero at the other

00:17:19.220 --> 00:17:19.860
end.

00:17:19.860 --> 00:17:23.130
So what's the picture if
I use linear elements?

00:17:23.130 --> 00:17:26.300
I don't remember how many
I used in the problem.

00:17:26.300 --> 00:17:34.500
Well, it allows you to
use N interior guys, one,

00:17:34.500 --> 00:17:41.060
two, up to N, and then
another-- This will come in.

00:17:41.060 --> 00:17:45.710
Or that's the point.

00:17:45.710 --> 00:17:49.340
OK, so what's the
finite element method,

00:17:49.340 --> 00:17:52.030
finite element
matrix K for this--

00:17:52.030 --> 00:18:01.360
So I want to do linear elements
and I want to construct K.

00:18:01.360 --> 00:18:03.880
And, yeah, I guess.

00:18:03.880 --> 00:18:07.580
Oh, I haven't
actually made anything

00:18:07.580 --> 00:18:09.370
happen to this problem.

00:18:09.370 --> 00:18:12.970
All zeroes is kind
of slow going.

00:18:12.970 --> 00:18:16.840
u will be, the solution
will certainly be zero.

00:18:16.840 --> 00:18:24.800
So maybe I'd better put in a
load here to get some action.

00:18:24.800 --> 00:18:32.350
OK, well, yeah.

00:18:32.350 --> 00:18:36.430
So I proposed this question
but now, is this a question

00:18:36.430 --> 00:18:38.610
to think about?

00:18:38.610 --> 00:18:41.490
I think that's a
reasonable example to do.

00:18:41.490 --> 00:18:43.990
It's got the two types
of boundary conditions.

00:18:43.990 --> 00:18:46.100
It's got the
right-hand side f, it's

00:18:46.100 --> 00:18:51.440
got linear elements which means
it's kind of doable by hand.

00:18:51.440 --> 00:19:00.760
And we kind of know what
matrix to expect out of it.

00:19:00.760 --> 00:19:05.280
What matrix do we expect?

00:19:05.280 --> 00:19:08.370
What do I expect out
of linear elements,

00:19:08.370 --> 00:19:11.690
do you remember the point about
linear elements on equally

00:19:11.690 --> 00:19:13.300
spaced meshes?

00:19:13.300 --> 00:19:17.510
That just brought back our
regular difference matrices.

00:19:17.510 --> 00:19:21.300
So I'm expecting
this stiffness matrix

00:19:21.300 --> 00:19:22.620
to be a difference matrix.

00:19:22.620 --> 00:19:24.890
Anyway, the point
of this question

00:19:24.890 --> 00:19:28.290
is, OK, I have a hat function,
I have a hat function,

00:19:28.290 --> 00:19:34.940
I've a hat function, a hat
function, and is that the end?

00:19:34.940 --> 00:19:38.560
Is that the complete list
of my trial functions?

00:19:38.560 --> 00:19:40.660
One more, right?

00:19:40.660 --> 00:19:48.530
Because this condition is, all
my trial and test functions

00:19:48.530 --> 00:19:51.330
don't have to satisfy this.

00:19:51.330 --> 00:19:55.580
So I'm allowed, and should
have, another guy there.

00:19:55.580 --> 00:19:58.530
A half hat for that one.

00:19:58.530 --> 00:20:06.350
You may say, don't let that
clown into the finite element

00:20:06.350 --> 00:20:09.080
space but I think it should be.

00:20:09.080 --> 00:20:13.900
The solution won't
use much of it.

00:20:13.900 --> 00:20:18.450
Because the solution is
going to aim for zero slope.

00:20:18.450 --> 00:20:20.850
But it's going to need a
little-- You see why it needs

00:20:20.850 --> 00:20:22.360
a little bit, something here?

00:20:22.360 --> 00:20:26.150
Because this thing
has slope down.

00:20:26.150 --> 00:20:29.140
So if there's some
of that in there,

00:20:29.140 --> 00:20:35.390
there better be somebody
else to cancel it.

00:20:35.390 --> 00:20:39.580
If our approximation is going
to have about zero slope.

00:20:39.580 --> 00:20:41.340
OK, so then.

00:20:41.340 --> 00:20:44.000
Can you construct a matrix K?

00:20:44.000 --> 00:20:48.110
Let's see, what's
the (2,3) entry?

00:20:48.110 --> 00:20:52.487
So if I call this number
one, this number two,

00:20:52.487 --> 00:20:53.570
oh, I've already numbered.

00:20:53.570 --> 00:20:57.730
So number two and number
three, so that trial

00:20:57.730 --> 00:20:59.790
function against that one.

00:20:59.790 --> 00:21:00.450
What do I?

00:21:00.450 --> 00:21:05.340
What's my formula for the (2,3)
entry of the stiffness matrix?

00:21:05.340 --> 00:21:08.690
It's some integral, right?

00:21:08.690 --> 00:21:10.920
And what do I integrate?

00:21:10.920 --> 00:21:12.830
I integrate, yeah.

00:21:12.830 --> 00:21:15.560
And I've got to
have to remember.

00:21:15.560 --> 00:21:22.470
So I do, yeah, my weak form--
I've integrated by parts,

00:21:22.470 --> 00:21:28.670
so my weak form is the integral
of u'*v'*dx equals the integral

00:21:28.670 --> 00:21:33.010
of f times. v dx.

00:21:33.010 --> 00:21:34.120
That's my weak form.

00:21:34.120 --> 00:21:41.070
I did two integrations by
parts and the integrated term

00:21:41.070 --> 00:21:41.870
will go away.

00:21:41.870 --> 00:21:43.430
Because of those zeroes.

00:21:43.430 --> 00:21:52.020
OK, so K_(2,3) will come from
this side when I'm using phi_2

00:21:52.020 --> 00:21:55.380
and phi_3, because
I'm taking the phis,

00:21:55.380 --> 00:22:00.460
the phis and the V's both
the same hat function.

00:22:00.460 --> 00:22:08.270
OK, so what do I get
for that? phi_2' is?

00:22:08.270 --> 00:22:12.030
So this is it, and
it overlaps this one.

00:22:12.030 --> 00:22:15.290
So when it overlaps this
phi_2 is coming down

00:22:15.290 --> 00:22:17.480
and phi_3 is going up.

00:22:17.480 --> 00:22:23.210
And the slope is 1/h, let's say.

00:22:23.210 --> 00:22:30.240
So I think I'm integrating us a
negative slope, is that right?

00:22:30.240 --> 00:22:32.840
Times a positive slope.

00:22:32.840 --> 00:22:38.960
And I'm really only integrating
over one h interval.

00:22:38.960 --> 00:22:44.550
The two overlap only here,
where this one's coming down

00:22:44.550 --> 00:22:46.020
and that one's going up.

00:22:46.020 --> 00:22:49.920
So I think-- dx.

00:22:49.920 --> 00:22:52.170
And the great thing is, of
course, we have a constant.

00:22:52.170 --> 00:22:54.880
So I have minus one
over h squared times h,

00:22:54.880 --> 00:22:58.230
I think minus one over h.

00:22:58.230 --> 00:23:00.990
That would be K_(2,3).

00:23:00.990 --> 00:23:08.070
That's a simple example.

00:23:08.070 --> 00:23:19.050
And then at the end we will
see it, we'll see this one,

00:23:19.050 --> 00:23:21.850
I think we'll get some matrix.

00:23:21.850 --> 00:23:24.340
We'll have this 1/h
outside, I think

00:23:24.340 --> 00:23:27.980
we'll have something like two,
minus one; two, minus one;

00:23:27.980 --> 00:23:31.330
minus one and then only
a one from the half-hat.

00:23:31.330 --> 00:23:39.190
I think it would be that matrix
that would be K. I think.

00:23:39.190 --> 00:23:42.440
Maybe with more,
greater size if we

00:23:42.440 --> 00:23:43.920
have a whole bunch of elements.

00:23:43.920 --> 00:23:47.590
But that pattern.

00:23:47.590 --> 00:23:49.260
You're pretty much into this?

00:23:49.260 --> 00:23:54.420
Yeah, I mean we're doing
a lot in this course.

00:23:54.420 --> 00:23:59.090
I'm really grateful
you guys stay with it,

00:23:59.090 --> 00:24:06.770
and kept to these new ideas,
through doing exercises

00:24:06.770 --> 00:24:07.420
and so on.

00:24:07.420 --> 00:24:11.510
Because there's a lot here.

00:24:11.510 --> 00:24:13.440
Well, I thought I'd
put an example up,

00:24:13.440 --> 00:24:16.650
to open up, just
to remind you what

00:24:16.650 --> 00:24:18.360
that language is about there.

00:24:18.360 --> 00:24:23.340
And to be ready for any
question in that topic.

00:24:23.340 --> 00:24:26.600
Or any question whatever.

00:24:26.600 --> 00:24:28.990
So I jumped in with
finite elements,

00:24:28.990 --> 00:24:37.320
but I'm ready also to talk
about that area of the course.

00:24:37.320 --> 00:24:38.470
AUDIENCE: [INAUDIBLE]

00:24:38.470 --> 00:24:44.630
PROFESSOR STRANG:
Yeah. x+iy stuff, OK.

00:24:44.630 --> 00:24:47.940
Basically, any
function of x+iy, yeah.

00:24:47.940 --> 00:24:49.570
Any function.

00:24:49.570 --> 00:24:53.630
So strictly, yeah, I mean a
mathematician would say what,

00:24:53.630 --> 00:24:54.640
any function?

00:24:54.640 --> 00:25:02.200
That's, you've opened the door
to crazy things saying that.

00:25:02.200 --> 00:25:07.700
So what I really mean is, we
have these powers of x+iy,

00:25:07.700 --> 00:25:09.840
and then we have
combinations of them.

00:25:09.840 --> 00:25:12.430
So the only requirement
would be that if I

00:25:12.430 --> 00:25:14.760
want to take an
infinite combination

00:25:14.760 --> 00:25:19.170
it should, the series
should, add up to something.

00:25:19.170 --> 00:25:23.180
If it has a nice
Taylor series then

00:25:23.180 --> 00:25:25.450
those are the best
functions there are.

00:25:25.450 --> 00:25:27.730
Functions with
nice Taylor series.

00:25:27.730 --> 00:25:29.290
I'll just say it.

00:25:29.290 --> 00:25:30.880
Having used those words.

00:25:30.880 --> 00:25:37.130
Suppose I take that function.

00:25:37.130 --> 00:25:39.850
There's a function, that's
a function-- z is x+iy.

00:25:46.270 --> 00:25:50.160
But z is shorter to write.

00:25:50.160 --> 00:25:52.640
So it's not a
polynomial, obviously.

00:25:52.640 --> 00:25:54.770
But it is a function of x+iy.

00:26:00.620 --> 00:26:02.180
Well, tell me this.

00:26:02.180 --> 00:26:05.510
Where does that
function go wrong?

00:26:05.510 --> 00:26:10.820
So e^z is a function that
never goes wrong, right? e^z,

00:26:10.820 --> 00:26:14.270
that series always converges.

00:26:14.270 --> 00:26:18.600
Can you tell me the series, if
I expand that into a series,

00:26:18.600 --> 00:26:22.960
what series am I looking at?

00:26:22.960 --> 00:26:28.900
This is not on the
exam, so to speak.

00:26:28.900 --> 00:26:32.130
Do you know one over
one plus something,

00:26:32.130 --> 00:26:39.730
what's the series for that?

00:26:39.730 --> 00:26:43.370
Well the constant term, when
z is zero is certainly a one.

00:26:43.370 --> 00:26:47.330
I think the trick, it's
this is geometric series,

00:26:47.330 --> 00:26:51.500
and because it's a
z squared it's that.

00:26:51.500 --> 00:26:56.160
That would be the
geometric series.

00:26:56.160 --> 00:26:58.630
With constant ratio z squared.

00:26:58.630 --> 00:27:02.350
If I multiply that
by that, 1 plus z

00:27:02.350 --> 00:27:04.330
squared times that,
everything will cancel

00:27:04.330 --> 00:27:05.840
and I'll get the one.

00:27:05.840 --> 00:27:08.800
That's it.

00:27:08.800 --> 00:27:14.550
So there is a Taylor
series for this function.

00:27:14.550 --> 00:27:19.780
Now, the reason I chose that
example is, you could tell me,

00:27:19.780 --> 00:27:23.490
it doesn't converge if
z is too large, right?

00:27:23.490 --> 00:27:28.040
Is this an analytic function?

00:27:28.040 --> 00:27:30.840
Where is this a good
function and where does it

00:27:30.840 --> 00:27:33.030
have problems?

00:27:33.030 --> 00:27:38.540
If z is less than one, and I
really mean magnitude of z,

00:27:38.540 --> 00:27:40.810
so let me draw the z-plane.

00:27:40.810 --> 00:27:44.910
Here's the real part of z
that you usually call x,

00:27:44.910 --> 00:27:49.370
and the imaginary part of
z that you usually call y,

00:27:49.370 --> 00:27:55.190
because z is x+iy, and where
will this series converge?

00:27:55.190 --> 00:28:01.840
It'll converge out as
far as this circle.

00:28:01.840 --> 00:28:05.210
This is the Taylor
series around zero.

00:28:05.210 --> 00:28:05.710
Right?

00:28:05.710 --> 00:28:10.640
The constant term
I found at z=0.

00:28:10.640 --> 00:28:15.550
Then that series, this
function, is great.

00:28:15.550 --> 00:28:18.860
It's an analytic
function, everything,

00:28:18.860 --> 00:28:22.480
it gives us a solution to
Laplace's-- This'll be,

00:28:22.480 --> 00:28:26.530
the real and imaginary parts
of that will be the u and the s

00:28:26.530 --> 00:28:28.930
that solve Laplace's equation.

00:28:28.930 --> 00:28:33.450
Out to, at least in this circle.

00:28:33.450 --> 00:28:38.180
But something, there's a problem
at the edge of the circle.

00:28:38.180 --> 00:28:42.400
Now, here's my point.

00:28:42.400 --> 00:28:47.550
If I think of one over
one plus x squared,

00:28:47.550 --> 00:28:49.310
look at that for a minute.

00:28:49.310 --> 00:28:51.930
That function has no
problems at all, right?

00:28:51.930 --> 00:28:55.500
One over one plus x squared,
you can let x be anything?

00:28:55.500 --> 00:28:58.880
It's no trouble.

00:28:58.880 --> 00:29:00.840
But one over one
plus z squared, when

00:29:00.840 --> 00:29:04.190
we look in the complex
plane, ah, we find a problem.

00:29:04.190 --> 00:29:07.740
And where is the problem
with this function?

00:29:07.740 --> 00:29:12.530
At z equals, so everybody's
looking at this guy.

00:29:12.530 --> 00:29:16.730
There's a problem with
that function at z=i.

00:29:16.730 --> 00:29:20.200
And it happens to
be not an accident,

00:29:20.200 --> 00:29:22.430
it's right there on the circle.

00:29:22.430 --> 00:29:26.820
It's hiding in the complex--
it's not on the real axis.

00:29:26.820 --> 00:29:29.630
So the real person
didn't notice it.

00:29:29.630 --> 00:29:33.280
But the complex person said
ah, that's the problem.

00:29:33.280 --> 00:29:37.310
There's a singularity there, and
of course it's called a pole,

00:29:37.310 --> 00:29:42.840
and people in so many parts of
science are interested in that.

00:29:42.840 --> 00:29:46.500
Is there any other place
that there's a problem?

00:29:46.500 --> 00:29:47.990
At minus i.

00:29:47.990 --> 00:29:50.870
When z is minus i we'll
also get a problem.

00:29:50.870 --> 00:29:56.760
So this is a function with
two poles, those two poles

00:29:56.760 --> 00:30:05.510
and they're the reason that
the series couldn't make it.

00:30:05.510 --> 00:30:09.330
Going out this way the series
doesn't meet any problems.

00:30:09.330 --> 00:30:14.170
But the series always
goes out in a circle,

00:30:14.170 --> 00:30:18.150
and the first circle, the first
guy, the first problem it hits,

00:30:18.150 --> 00:30:21.480
the series stops converging.

00:30:21.480 --> 00:30:24.400
By the way, let me
ask you a question.

00:30:24.400 --> 00:30:29.760
Suppose I instead did the
Taylor series around this point?

00:30:29.760 --> 00:30:31.680
Now, what do I mean by that?

00:30:31.680 --> 00:30:33.630
That's the point one, let's say.

00:30:33.630 --> 00:30:36.620
What do I mean by that, the
Taylor series around one?

00:30:36.620 --> 00:30:41.820
I'll rewrite the function as one
plus, well now, what do I do?

00:30:41.820 --> 00:30:46.680
I want it in z
minus one squared.

00:30:46.680 --> 00:30:47.720
Oh, gosh.

00:30:47.720 --> 00:30:54.660
I'm getting beyond what
you will care about.

00:30:54.660 --> 00:31:00.820
Again, if I expanded, if I wrote
the power series in powers of z

00:31:00.820 --> 00:31:05.490
minus one, what
would it work in?

00:31:05.490 --> 00:31:07.900
And then I'll stop
with this example.

00:31:07.900 --> 00:31:12.950
The circle would reach
out until it hit a pole.

00:31:12.950 --> 00:31:14.700
And it can't make
it past that pole.

00:31:14.700 --> 00:31:18.630
So it would be a circle of
radius square root of two,

00:31:18.630 --> 00:31:20.700
there would be a circle there.

00:31:20.700 --> 00:31:24.690
If we were going to
discuss, and this is really

00:31:24.690 --> 00:31:26.050
Chapter 5 of the book.

00:31:26.050 --> 00:31:34.030
I mention it, because you've
got a book that explains this.

00:31:34.030 --> 00:31:36.690
If I thought the center
of the universe was there,

00:31:36.690 --> 00:31:39.470
and then the poles
are still here,

00:31:39.470 --> 00:31:42.370
the circle will make
it out to those poles.

00:31:42.370 --> 00:31:46.870
So I can do Taylor series, I
can sort of hook together Taylor

00:31:46.870 --> 00:31:49.200
series all over the place.

00:31:49.200 --> 00:31:52.470
And they'll all quit
when they reach a pole,

00:31:52.470 --> 00:31:54.380
but when I put all
those circles together

00:31:54.380 --> 00:31:57.990
I can get all the
rest of the plane.

00:31:57.990 --> 00:32:01.750
OK, so that's something
about, I don't

00:32:01.750 --> 00:32:06.220
know how I got onto that
department, but it's amazing.

00:32:06.220 --> 00:32:10.580
This, so the real and imaginary
parts of that would be a flow,

00:32:10.580 --> 00:32:12.080
would give me a flow.

00:32:12.080 --> 00:32:15.390
I don't know if it'd be
easy to compute it or not,

00:32:15.390 --> 00:32:18.310
maybe I won't tackle that here.

00:32:18.310 --> 00:32:21.240
But we could find
the real part of that

00:32:21.240 --> 00:32:24.280
and the imaginary part
of that, and we would

00:32:24.280 --> 00:32:30.420
have a genuine flow field.

00:32:30.420 --> 00:32:34.060
Satisfying Laplace's equation
with the two orthogonal,

00:32:34.060 --> 00:32:38.550
the streamlines orthogonal
to the equipotentials.

00:32:38.550 --> 00:32:41.840
We could totally
do that example.

00:32:41.840 --> 00:32:43.480
OK, let me, yeah, thanks.

00:32:43.480 --> 00:32:48.378
AUDIENCE: You say one test
question is based on x+iy?

00:32:48.378 --> 00:32:49.336
PROFESSOR STRANG: Yeah.

00:32:49.336 --> 00:32:50.377
Well, this sort of stuff.

00:32:50.377 --> 00:32:55.180
But, so u would be
the, yeah that's right.

00:32:55.180 --> 00:33:01.480
Yeah, so a test question
would be something like,

00:33:01.480 --> 00:33:06.630
one way or another, you would
end up with a u, and an s,

00:33:06.630 --> 00:33:10.200
and the u+is, if
they're a good pair,

00:33:10.200 --> 00:33:14.370
would be some function
of this magic z.

00:33:14.370 --> 00:33:16.609
Yeah, yeah.

00:33:16.609 --> 00:33:17.150
That's right.

00:33:17.150 --> 00:33:23.140
So whatever.

00:33:23.140 --> 00:33:25.670
We know examples, of course.

00:33:25.670 --> 00:33:30.050
For example, this could be
x squared minus y squared.

00:33:30.050 --> 00:33:33.840
And the s that goes
with that is 2xy,

00:33:33.840 --> 00:33:38.550
and the function that's involved
there when I throw in the i

00:33:38.550 --> 00:33:43.020
is simply z squared.

00:33:43.020 --> 00:33:45.750
OK, that would be
an example where

00:33:45.750 --> 00:33:50.640
the real and imaginary parts
of this give us the good u,

00:33:50.640 --> 00:33:55.810
its good friend
s, and the picture

00:33:55.810 --> 00:33:59.880
of streamlines
and equipotentials

00:33:59.880 --> 00:34:01.820
meeting at right angles.

00:34:01.820 --> 00:34:06.880
Just, a beautiful picture, all
coming out of this function.

00:34:06.880 --> 00:34:11.990
So probably the quiz will
have some other function.

00:34:11.990 --> 00:34:20.250
But you'll still have a u and
an s and a function of x+iy.

00:34:20.250 --> 00:34:23.750
So if it's not this one,
which I don't think it is.

00:34:23.750 --> 00:34:26.480
It won't be be this one.

00:34:26.480 --> 00:34:29.034
AUDIENCE: [INAUDIBLE]

00:34:29.034 --> 00:34:29.950
PROFESSOR STRANG: Yes.

00:34:29.950 --> 00:34:35.800
Because first of all, I wouldn't
have mentioned it if I was.

00:34:35.800 --> 00:34:39.010
And secondly, that's
a little too messy,

00:34:39.010 --> 00:34:43.220
I think, to get
a good handle of,

00:34:43.220 --> 00:34:45.360
to take the real and
imaginary parts of that.

00:34:45.360 --> 00:34:47.400
It's not impossible, of course.

00:34:47.400 --> 00:34:49.210
We could completely do it.

00:34:49.210 --> 00:34:57.120
There'd be some ratio
of two polynomials.

00:34:57.120 --> 00:35:02.380
Here we have just
simple polynomials.

00:35:02.380 --> 00:35:04.620
OK, does that help
with that question?

00:35:04.620 --> 00:35:05.500
Yeah.

00:35:05.500 --> 00:35:11.040
What else is on your mind?

00:35:11.040 --> 00:35:12.380
Any thoughts?

00:35:12.380 --> 00:35:14.100
Yeah, thanks.

00:35:14.100 --> 00:35:15.490
Curl, OK.

00:35:15.490 --> 00:35:21.520
Well, so I didn't really
do three dimensions.

00:35:21.520 --> 00:35:26.940
But curl is important.

00:35:26.940 --> 00:35:31.770
And we did see, in two
dimensions the key fact

00:35:31.770 --> 00:35:35.410
that all this
stuff-- Let me just

00:35:35.410 --> 00:35:39.120
write down what the great
connections are between these.

00:35:39.120 --> 00:35:41.850
Because I can't let
the whole semester go

00:35:41.850 --> 00:35:46.870
without writing down that,
what is it, the grad--

00:35:46.870 --> 00:35:51.920
Is it the curl of a gradient?

00:35:51.920 --> 00:36:00.010
The curl of any gradient
of u is always zero.

00:36:00.010 --> 00:36:01.820
Whatever it is.

00:36:01.820 --> 00:36:03.470
Whatever u is.

00:36:03.470 --> 00:36:09.550
And this comes from, let
me put the other one down

00:36:09.550 --> 00:36:14.070
and then I'll just say why.

00:36:14.070 --> 00:36:17.070
The other one is like the
transpose of this one.

00:36:17.070 --> 00:36:20.070
So if I transpose, so
this is the zero operator.

00:36:20.070 --> 00:36:22.970
Curl times gradient
gives the zero.

00:36:22.970 --> 00:36:25.670
So if I just transpose
I still have zero.

00:36:25.670 --> 00:36:27.190
So if it's a
transposed gradient,

00:36:27.190 --> 00:36:31.190
I have minus divergence, and
actually if I transpose curl

00:36:31.190 --> 00:36:33.310
I get curl again.

00:36:33.310 --> 00:36:39.240
Of any, now I should put in,
what should the curl act on?

00:36:39.240 --> 00:36:42.550
It acts on a w, I guess is.

00:36:42.550 --> 00:36:49.760
No, divergence w, it
acts on an S, sorry.

00:36:49.760 --> 00:36:54.227
OK, and the minus
sign, of course,

00:36:54.227 --> 00:36:55.810
isn't going to matter
because I've got

00:36:55.810 --> 00:36:57.870
a zero on the right-hand side.

00:36:57.870 --> 00:37:04.500
So S. Yeah, so if I take any
field-- I mean this is like,

00:37:04.500 --> 00:37:08.030
real proper vector calculus.

00:37:08.030 --> 00:37:13.100
To check these, I call them
identities and maybe sometimes

00:37:13.100 --> 00:37:16.670
people indicate an identity
meaning it's always

00:37:16.670 --> 00:37:19.470
true for every u,
or for every S.

00:37:19.470 --> 00:37:21.640
They'll use the
triple equals sign.

00:37:21.640 --> 00:37:24.070
Just to say they're
really equal.

00:37:24.070 --> 00:37:34.580
OK, so we could define the curl,
but you've met it elsewhere

00:37:34.580 --> 00:37:39.220
and maybe this isn't
the time to do that.

00:37:39.220 --> 00:37:42.990
What's the key fact, the
key little math business

00:37:42.990 --> 00:37:45.550
that makes all these true?

00:37:45.550 --> 00:37:52.200
So there's sort of a formal
math fact that makes them true.

00:37:52.200 --> 00:37:55.330
And then there's the
physical understanding

00:37:55.330 --> 00:38:03.030
of gradients being directions
out with no rotation.

00:38:03.030 --> 00:38:05.240
So the physical
understanding of that.

00:38:05.240 --> 00:38:08.100
But the math, the
formal math fact

00:38:08.100 --> 00:38:11.640
is the fact that the second
derivative of u with respect

00:38:11.640 --> 00:38:16.660
to x and y is equal to what?

00:38:16.660 --> 00:38:21.000
It's equal to second derivative
with respect to y and x, yep.

00:38:21.000 --> 00:38:25.160
So you would find if you
wrote out all the terms here,

00:38:25.160 --> 00:38:28.270
or all the terms here,
you would find that

00:38:28.270 --> 00:38:32.920
just by using that fact,
they all cancel each other.

00:38:32.920 --> 00:38:34.870
And the book, of
course, does that.

00:38:34.870 --> 00:38:42.740
So we simply didn't do 3-D in
the vector calculus section.

00:38:42.740 --> 00:38:45.870
So I'll stop there with
that, because it's really

00:38:45.870 --> 00:38:50.280
saying that the curl is
tremendously important.

00:38:50.280 --> 00:38:59.450
It measures vorticity and flow,
and it's-- Being able to-- You

00:38:59.450 --> 00:39:02.710
know that like, you take
the Navier-Stokes equations?

00:39:02.710 --> 00:39:09.460
Well, the pressure and
the velocity are the,

00:39:09.460 --> 00:39:13.260
I'd say primary variables
or the natural quantities

00:39:13.260 --> 00:39:17.160
to measure, pressure and
velocity for a fluid flow.

00:39:17.160 --> 00:39:22.410
But you could also use
these identities to set up

00:39:22.410 --> 00:39:25.590
in terms of other variables.

00:39:25.590 --> 00:39:32.890
Just rewrite the equation
and you get other things.

00:39:32.890 --> 00:39:39.950
Mentioning Navier-Stokes and
fluid flow reminds me to say,

00:39:39.950 --> 00:39:44.180
we keep using the example
of Laplace's equation.

00:39:44.180 --> 00:39:48.120
And a person could say wait
a minute, get beyond that.

00:39:48.120 --> 00:39:49.610
Right?

00:39:49.610 --> 00:39:54.050
So why are you always, when
you teach finite elements,

00:39:54.050 --> 00:39:56.600
why do you always start
with Laplace's equation?

00:39:56.600 --> 00:40:00.930
OK, well the main reason
is it's the simplest one.

00:40:00.930 --> 00:40:03.890
It's the one where you can
really see what's happening.

00:40:03.890 --> 00:40:06.160
More complicated
equations would be for,

00:40:06.160 --> 00:40:10.300
like, elasticity, plane
elasticity, or 3-D elasticity

00:40:10.300 --> 00:40:17.040
or other boundary value
problems could be quite messy.

00:40:17.040 --> 00:40:23.240
But Laplace's equation is
not totally a waste of time.

00:40:23.240 --> 00:40:26.510
First, it comes up when you
have these scalar unknowns.

00:40:26.510 --> 00:40:31.370
And then it also comes
up in numerical methods

00:40:31.370 --> 00:40:33.760
for Navier-Stokes.

00:40:33.760 --> 00:40:36.500
So the standard numerical
method for Navier-Stokes,

00:40:36.500 --> 00:40:42.020
which would come
in 18.086, ends up

00:40:42.020 --> 00:40:45.600
with Laplace's equation
for the pressure.

00:40:45.600 --> 00:40:51.910
So to have a fast Laplace
solver, as in today's lecture,

00:40:51.910 --> 00:40:53.870
pays off.

00:40:53.870 --> 00:40:56.400
So I'm just saying
Laplace's equation

00:40:56.400 --> 00:41:00.020
is important in itself,
it has the great advantage

00:41:00.020 --> 00:41:02.820
of being the simplest example
we could possibly think of.

00:41:02.820 --> 00:41:06.460
It's the example where
an x+iy trick works.

00:41:06.460 --> 00:41:11.960
And it actually comes up in
serious big computations,

00:41:11.960 --> 00:41:15.040
because the equation
for the pressure

00:41:15.040 --> 00:41:20.690
comes out to be a Laplace
or a Poisson equation.

00:41:20.690 --> 00:41:21.290
Now.

00:41:21.290 --> 00:41:23.000
I kept going there.

00:41:23.000 --> 00:41:23.690
Yeah, thank you.

00:41:23.690 --> 00:41:28.970
AUDIENCE: [INAUDIBLE]

00:41:28.970 --> 00:41:31.140
PROFESSOR STRANG: We
did, as a MATLAB problem.

00:41:31.140 --> 00:41:32.390
AUDIENCE: [INAUDIBLE]

00:41:32.390 --> 00:41:34.560
PROFESSOR STRANG: Sorry?

00:41:34.560 --> 00:41:35.730
And a first order, right.

00:41:35.730 --> 00:41:38.560
AUDIENCE: [INAUDIBLE]

00:41:38.560 --> 00:41:39.810
PROFESSOR STRANG: Huh.

00:41:39.810 --> 00:41:41.580
Yeah.

00:41:41.580 --> 00:41:44.351
I would do it the same way
but it wouldn't be symmetric,

00:41:44.351 --> 00:41:44.850
of course.

00:41:44.850 --> 00:41:48.070
That was the point about
that convection term,

00:41:48.070 --> 00:41:50.650
is-- The diffusion term would
be just what we've done, right?

00:41:50.650 --> 00:41:53.220
The diffusion term was
that second derivative.

00:41:53.220 --> 00:41:56.980
And what would it
look like in, as long

00:41:56.980 --> 00:42:01.340
as we're close to, what would
convection-diffusion in 2-D

00:42:01.340 --> 00:42:05.230
look like?

00:42:05.230 --> 00:42:08.000
Just, I mean part
of your interest

00:42:08.000 --> 00:42:12.730
is pass 18.085 and
get rid of it, right?

00:42:12.730 --> 00:42:16.590
But another part is
like, these are problems

00:42:16.590 --> 00:42:22.197
that if you're in Course
16, Course 2, others,

00:42:22.197 --> 00:42:23.280
you're going to meet this.

00:42:23.280 --> 00:42:27.430
So the diffusion
part is going to be,

00:42:27.430 --> 00:42:32.370
again in 2-D I'll have
some minus u, well,

00:42:32.370 --> 00:42:36.970
I made it simple because
I took c(x) to be one.

00:42:36.970 --> 00:42:39.460
It could have a c(x) in there.

00:42:39.460 --> 00:42:42.360
And what would the
convection term look like?

00:42:42.360 --> 00:42:49.210
I'd have a velocity, in
the x direction, say a V_x.

00:42:49.210 --> 00:42:52.820
And a velocity in
the y direction, V_y.

00:42:52.820 --> 00:42:56.340
V_y times-- That's
just a number.

00:42:56.340 --> 00:43:01.030
In the simplest case that
would be my river is traveling,

00:43:01.030 --> 00:43:09.675
or my flow is traveling
along, and equals zero.

00:43:09.675 --> 00:43:10.550
AUDIENCE: [INAUDIBLE]

00:43:10.550 --> 00:43:12.400
PROFESSOR STRANG: Sorry?

00:43:12.400 --> 00:43:16.870
Yeah I don't know which way the
river's traveling, actually.

00:43:16.870 --> 00:43:19.970
So those are just constants.

00:43:19.970 --> 00:43:23.070
They could have positive
or negative signs.

00:43:23.070 --> 00:43:27.800
The V_x and V_y is the
constant flow that's carrying,

00:43:27.800 --> 00:43:29.050
what am I doing here?

00:43:29.050 --> 00:43:32.990
The flow is flowing along,
and if those are constants

00:43:32.990 --> 00:43:36.300
it's just flowing
steady, steady flow.

00:43:36.300 --> 00:43:39.840
But it's diffusing
at the same time.

00:43:39.840 --> 00:43:43.640
And this would bring in
that same difficulties

00:43:43.640 --> 00:43:47.370
that we met in the MATLAB 1-D.
So the MATLAB 1-D problem just

00:43:47.370 --> 00:43:50.480
didn't have a y.

00:43:50.480 --> 00:43:53.260
I don't care, yeah the
sign I'm not worried about,

00:43:53.260 --> 00:43:56.490
it's just is that
there, now I'm in 2-D.

00:43:56.490 --> 00:43:58.380
And what would I expect to see?

00:43:58.380 --> 00:44:01.810
I'd expect to see some
trouble when V is large.

00:44:01.810 --> 00:44:06.960
When V is large, convection,
this is convection down here.

00:44:06.960 --> 00:44:09.390
This is the convection part.

00:44:09.390 --> 00:44:15.800
And if V is large, then so that
this should be a lower order

00:44:15.800 --> 00:44:19.590
term, is really fighting
against this higher order term.

00:44:19.590 --> 00:44:24.090
I expect numerical difficulties,
just the way we met.

00:44:24.090 --> 00:44:30.190
So anyway, if I did a MATLAB
example, stretched it to 2-D

00:44:30.190 --> 00:44:34.120
we see a whole lot
of interesting stuff.

00:44:34.120 --> 00:44:37.820
We'd see flow in,
flow out, yeah.

00:44:37.820 --> 00:44:40.180
But I just can't do everything.

00:44:40.180 --> 00:44:42.910
But that would have a weak
form, but your question

00:44:42.910 --> 00:44:45.150
about weak forms,
weak form, when

00:44:45.150 --> 00:44:48.380
you have this
anti-symmetric term

00:44:48.380 --> 00:44:52.870
for odd number of derivatives,
is not quite as beautiful.

00:44:52.870 --> 00:44:55.050
But you have to deal
with it, of course.

00:44:55.050 --> 00:44:56.860
Yep.

00:44:56.860 --> 00:45:01.810
OK, Ready for whatever.

00:45:01.810 --> 00:45:02.980
Any thoughts?

00:45:02.980 --> 00:45:05.360
Let's see, just
have a look again

00:45:05.360 --> 00:45:11.540
at the list of problem topics.

00:45:11.540 --> 00:45:14.130
To see if anything
occurs to you.

00:45:14.130 --> 00:45:16.480
I mean, not that it should.

00:45:16.480 --> 00:45:23.210
You know we're OK.

00:45:23.210 --> 00:45:27.290
I'm happy to call
it a day on that,

00:45:27.290 --> 00:45:32.720
and time for dinner for
everybody and see you at 7:30.