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PROFESSOR STRANG: So.

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Today's lecture is partly
cleaning up some pieces

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left from 1-D problems.

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3.1 was second
order equations, 3.2

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00:00:35,270 --> 00:00:38,280
was fourth order
equations for beams.

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And just a few more
comments to make.

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But then I do want to say
something about splines.

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And I included a
homework question that

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asked you to use the spline
command and get a result.

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00:00:59,910 --> 00:01:07,460
I'm not planning to make that
a serious topic in the course.

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In other words,
you're not like going

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to be responsible on an exam
for a discussion of splines.

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But it's such a major event, and
major requirement in scientific

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computing -- if you're
given some values,

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put a curve through them -- that
I have to say something about

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that.

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So that's interpolation.

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Because you'll have
this all the time.

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If your experiment
produces some values

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00:01:39,490 --> 00:01:44,550
at specific times or a
finite number of values,

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and you want to fit a
curve through those points.

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00:01:46,990 --> 00:01:48,320
How do you do it?

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00:01:48,320 --> 00:01:51,970
So splines give a
very, very good answer.

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00:01:51,970 --> 00:02:02,540
And then, the next and
big topic of the course

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00:02:02,540 --> 00:02:04,700
we'll make a beginning on.

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00:02:04,700 --> 00:02:09,170
Which is gradient, divergence,
leading to Laplace's equation.

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00:02:09,170 --> 00:02:14,430
Problems in 2-D. So partial
differential equations

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00:02:14,430 --> 00:02:15,600
are going to show up.

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00:02:15,600 --> 00:02:19,000
OK.

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00:02:19,000 --> 00:02:20,860
Some small points.

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00:02:20,860 --> 00:02:27,490
First point, about the
MATLAB homework for Monday.

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00:02:27,490 --> 00:02:34,880
A question came by email.

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I put the spike and also the
jump in c, in the coefficient,

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00:02:43,960 --> 00:02:45,910
not at mesh point.

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00:02:45,910 --> 00:02:47,570
If you've started
on that problem,

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you probably noticed that,
and maybe a little swearing

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00:02:55,100 --> 00:03:01,350
went on. [LAUGHTER] Because
it makes it not so easy.

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00:03:01,350 --> 00:03:05,160
And then the question
came, did I really

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00:03:05,160 --> 00:03:10,020
expect you to compute,
use the exact position

48
00:03:10,020 --> 00:03:15,720
and figure out-- When you
have some integrals to do,

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00:03:15,720 --> 00:03:20,120
they'll change at the place
where c changes values.

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00:03:20,120 --> 00:03:24,570
Where we have integrals of c,
if c has one of those jumps,

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00:03:24,570 --> 00:03:28,030
on the quiz the jump came
right at a neat point,

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00:03:28,030 --> 00:03:32,690
so it was clear. c was one on
one side and two on the other.

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Now the jump is going to
come at an in-between point.

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So the answer is yes.

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I had a nice email this
morning from somebody who

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has done that MATLAB homework.

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00:03:48,280 --> 00:03:55,130
And he said, quote,
it's a good problem.

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I learned more from it than
I did from the other problems

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in the pset.

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00:04:00,390 --> 00:04:06,240
So I'll blame it on him.

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00:04:06,240 --> 00:04:08,340
I'll stay with that
problem, and ask

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you to do your best to
deal with the calculation.

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It's second order and
I used linear elements.

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00:04:17,370 --> 00:04:20,610
I didn't want to push you
up to these cubic elements,

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00:04:20,610 --> 00:04:24,790
even though those would
give much better accuracy.

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00:04:24,790 --> 00:04:29,950
One reason I didn't put the
spike and the jump right

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at the node is that, if I did,
the finite element solution

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would be exactly right.

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00:04:38,390 --> 00:04:43,180
Because the correct solution is
going to be piecewise linear.

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And if I put the breaks between
pieces right at node points,

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well, then I have a function
that's in my finite element

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00:04:53,540 --> 00:04:55,360
space, in my linear space.

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00:04:55,360 --> 00:04:57,410
So it'll come out exactly.

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00:04:57,410 --> 00:05:00,940
So I thought well, let's
at least have some chance

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00:05:00,940 --> 00:05:05,550
to see the error between
the finite element

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00:05:05,550 --> 00:05:07,120
solution and the exact one.

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You see what I'm saying?

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00:05:08,730 --> 00:05:13,690
I'm just anticipating,
that the exact solution

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will-- Let's see.

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00:05:14,850 --> 00:05:18,820
Something happened at 1/3 and
something happened at 2/3.

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00:05:18,820 --> 00:05:21,460
I'm afraid I don't remember
the boundary conditions.

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00:05:21,460 --> 00:05:24,780
Does anybody remember what I
took for boundary conditions?

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00:05:24,780 --> 00:05:27,620
Probably free-fixed,
or something.

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00:05:27,620 --> 00:05:33,220
So maybe, if it was free-fixed,
probably the solution

85
00:05:33,220 --> 00:05:35,710
would be maybe
something like that.

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00:05:35,710 --> 00:05:36,520
I don't know.

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00:05:36,520 --> 00:05:38,010
Whatever.

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00:05:38,010 --> 00:05:42,530
Piecewise linear with breaks
where there's a jump in c,

89
00:05:42,530 --> 00:05:46,380
or where the point
load is hitting.

90
00:05:46,380 --> 00:05:48,110
Yeah.

91
00:05:48,110 --> 00:05:50,800
I don't know if that
picture is accurate.

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00:05:50,800 --> 00:05:56,990
But my point was, if I chose
those points to be also mesh

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00:05:56,990 --> 00:06:00,380
points, then the finite element
solution would be exact,

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00:06:00,380 --> 00:06:03,610
and we wouldn't learn
anything about accuracy.

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00:06:03,610 --> 00:06:04,210
OK.

96
00:06:04,210 --> 00:06:06,670
So anyway, for that
MATLAB problem,

97
00:06:06,670 --> 00:06:13,610
I'm hoping you can do
the integrals, which

98
00:06:13,610 --> 00:06:20,300
will mean noticing which
integral-- If that doesn't

99
00:06:20,300 --> 00:06:25,870
fall, so if the mesh
points are like this,

100
00:06:25,870 --> 00:06:30,230
in the finite element
integral over that mesh,

101
00:06:30,230 --> 00:06:32,530
over that interval,
you're going to have

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00:06:32,530 --> 00:06:34,900
to split it into two pieces.

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00:06:34,900 --> 00:06:36,600
That's it.

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00:06:36,600 --> 00:06:41,080
So see if you can do that split,
do the split into two pieces.

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00:06:41,080 --> 00:06:41,810
OK.

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00:06:41,810 --> 00:06:46,040
So that's a comment
on that homework.

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00:06:46,040 --> 00:06:47,480
OK.

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00:06:47,480 --> 00:06:49,740
What else do I need
to comment on, when

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00:06:49,740 --> 00:06:51,310
I'm sort of catching up here?

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00:06:51,310 --> 00:06:55,050
Oh, about boundary conditions.

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00:06:55,050 --> 00:06:57,130
I thought I would
just repeat clearly

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00:06:57,130 --> 00:07:04,870
on the board the rules
about and the difference

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00:07:04,870 --> 00:07:13,250
between fixed and free when
we're doing this weak form

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00:07:13,250 --> 00:07:17,580
approach to the problem.

115
00:07:17,580 --> 00:07:21,800
And then I gave the other
names: a fixed condition,

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00:07:21,800 --> 00:07:25,300
in this theory it's often
called an essential boundary

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00:07:25,300 --> 00:07:29,070
condition, but many
people just name

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00:07:29,070 --> 00:07:33,280
Dirichlet as the person's
name that's associated

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00:07:33,280 --> 00:07:34,700
with such conditions.

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00:07:34,700 --> 00:07:37,770
Like u(0)=0.

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Right, fixed.

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00:07:39,020 --> 00:07:43,360
And then free conditions
are natural conditions

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00:07:43,360 --> 00:07:45,850
in the finite element
method, and that

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00:07:45,850 --> 00:07:48,550
means that we don't
have to impose them.

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00:07:48,550 --> 00:07:49,600
Right.

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00:07:49,600 --> 00:07:51,770
I discussed this in
the earlier lecture.

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00:07:51,770 --> 00:07:54,180
I just thought I'd
bring it back here,

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00:07:54,180 --> 00:07:59,920
because it's easy to
describe in one line.

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00:07:59,920 --> 00:08:09,160
Now somebody asked, on
Wednesday, a good question.

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00:08:09,160 --> 00:08:17,650
Suppose u' is given,
but not given as zero.

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00:08:17,650 --> 00:08:28,490
So suppose we have a condition
like u'=A, let's say.

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00:08:28,490 --> 00:08:30,080
What to do?

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00:08:30,080 --> 00:08:32,320
So I'll just put question mark.

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00:08:32,320 --> 00:08:36,360
How do I deal with-- How do I?

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00:08:36,360 --> 00:08:40,890
How does the weak form approach
and the Galerkin approach

136
00:08:40,890 --> 00:08:47,840
deal with a non-zero,
natural condition?

137
00:08:47,840 --> 00:08:52,760
A non-zero Neumann condition,
that'd be the right word.

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00:08:52,760 --> 00:08:56,230
So this would be, in a
second order problem,

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00:08:56,230 --> 00:08:59,680
this would be a
Neumann condition.

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00:08:59,680 --> 00:09:02,780
And I'm thinking, what do
you do if A isn't zero.

141
00:09:02,780 --> 00:09:06,810
Somehow that has to show up in
your finite element equations,

142
00:09:06,810 --> 00:09:07,850
right?

143
00:09:07,850 --> 00:09:10,770
So I guess the right
way is to go back

144
00:09:10,770 --> 00:09:12,320
to the way they came from.

145
00:09:12,320 --> 00:09:15,140
So you remember how
the weak form came?

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00:09:15,140 --> 00:09:18,060
I took the
differential equation,

147
00:09:18,060 --> 00:09:20,320
I multiplied by
any test function,

148
00:09:20,320 --> 00:09:23,070
and I integrated by parts.

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00:09:23,070 --> 00:09:25,460
That's where the
weak form started.

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00:09:25,460 --> 00:09:28,680
So the integration
by parts gave me

151
00:09:28,680 --> 00:09:38,250
this nice symmetric integral.

152
00:09:38,250 --> 00:09:41,010
It's symmetric in
u and v. It only

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00:09:41,010 --> 00:09:44,760
requires one derivative
of u, so that I'm

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00:09:44,760 --> 00:09:48,720
allowed to use hat functions.

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00:09:48,720 --> 00:09:51,910
And I'm allowed to use
hat functions for v,

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00:09:51,910 --> 00:09:54,650
because their derivatives
have just a jump,

157
00:09:54,650 --> 00:09:58,890
and a jump function I can
integrate, no problem.

158
00:09:58,890 --> 00:10:04,110
Now what about this u'=A?

159
00:10:04,110 --> 00:10:08,640
And I guess we're
seeing it there.

160
00:10:08,640 --> 00:10:13,380
If u' was zero, if we
had a totally free end,

161
00:10:13,380 --> 00:10:16,540
u' was zero, nothing
happening there,

162
00:10:16,540 --> 00:10:21,690
then that term would drop out
of the integration by parts.

163
00:10:21,690 --> 00:10:24,460
And you see why I don't
have to impose anything

164
00:10:24,460 --> 00:10:27,960
on v, because that term
is already accounted

165
00:10:27,960 --> 00:10:31,120
for by the u' going away.

166
00:10:31,120 --> 00:10:38,330
Now, what about if u' is
given, but not given zero?

167
00:10:38,330 --> 00:10:41,500
Suppose it's given the
value A. All I want to say

168
00:10:41,500 --> 00:10:51,070
is, then this term has to stay.

169
00:10:51,070 --> 00:10:52,690
I have to pay attention to it.

170
00:10:52,690 --> 00:10:56,530
What happens when
Galerkin takes over?

171
00:10:56,530 --> 00:11:02,600
Galerkin will put in one
of his test functions.

172
00:11:02,600 --> 00:11:06,880
Maybe I just indicate that
by changing the v to a cap V.

173
00:11:06,880 --> 00:11:08,120
It'll be one of them.

174
00:11:08,120 --> 00:11:11,780
There'll be n of them.

175
00:11:11,780 --> 00:11:16,060
Each V_i, each test guy,
gives me an equation.

176
00:11:16,060 --> 00:11:22,140
And of course u is now going
to-- The Galerkin idea is that

177
00:11:22,140 --> 00:11:25,870
instead of any u, I
only have capital U's.

178
00:11:25,870 --> 00:11:32,130
Combinations of these phis.

179
00:11:32,130 --> 00:11:37,550
I'm not going to say anything
very big here, or very clear.

180
00:11:37,550 --> 00:11:46,800
All I want to say is that if u
prime is given at an end point,

181
00:11:46,800 --> 00:11:56,290
and given by A, then this whole
stuff is equaling something

182
00:11:56,290 --> 00:11:57,980
with an f, right?

183
00:11:57,980 --> 00:12:00,460
f*V_i*dx.

184
00:12:00,460 --> 00:12:03,520
I should remember the
other side of the equation.

185
00:12:03,520 --> 00:12:07,610
If u' is given, then
c at that end point,

186
00:12:07,610 --> 00:12:14,120
times the given value of
A, times the V will be --

187
00:12:14,120 --> 00:12:22,610
whatever value of V that is
-- will show up in equation i.

188
00:12:22,610 --> 00:12:25,180
That will be a term that
goes on the right-hand side.

189
00:12:25,180 --> 00:12:29,730
That's all I'm saying,
and all you would expect.

190
00:12:29,730 --> 00:12:35,740
That any time I'm
given some data,

191
00:12:35,740 --> 00:12:39,030
I'm given some non-zero
boundary conditions,

192
00:12:39,030 --> 00:12:42,530
or I'm given some
non-zero f in the inside,

193
00:12:42,530 --> 00:12:49,080
that stuff is going to show
up on the right-hand side.

194
00:12:49,080 --> 00:12:58,650
It'll show up as part of
the F. The vector of these.

195
00:12:58,650 --> 00:13:03,430
So up to now, F has just come
from little f(x), integrated

196
00:13:03,430 --> 00:13:07,820
against V. And all I'm
saying is that if we

197
00:13:07,820 --> 00:13:13,690
had one of these guys, then
that would contribute to big F,

198
00:13:13,690 --> 00:13:16,010
also.

199
00:13:16,010 --> 00:13:17,140
Yeah.

200
00:13:17,140 --> 00:13:22,580
You can see, it's not beautiful.

201
00:13:22,580 --> 00:13:25,590
But we have to realize
what we would do.

202
00:13:25,590 --> 00:13:28,520
OK, that's not my
favorite topic.

203
00:13:28,520 --> 00:13:32,420
But I'll just say
quit on that one.

204
00:13:32,420 --> 00:13:39,280
I'm not expecting you to do
problems or anything that

205
00:13:39,280 --> 00:13:40,900
involved that possibility.

206
00:13:40,900 --> 00:13:42,960
Just to see that
it could happen.

207
00:13:42,960 --> 00:13:44,250
OK.

208
00:13:44,250 --> 00:13:45,220
Quit.

209
00:13:45,220 --> 00:13:47,770
Now, interpolation.

210
00:13:47,770 --> 00:13:49,840
All right.

211
00:13:49,840 --> 00:13:54,030
So I guess if I had
to list problems

212
00:13:54,030 --> 00:13:59,725
that people face all the time
and need numerical guidance on,

213
00:13:59,725 --> 00:14:01,950
this is one of them.

214
00:14:01,950 --> 00:14:07,950
And so let's say we're in 1-D
-- 1-D is certainly a lot easier

215
00:14:07,950 --> 00:14:11,370
-- so, suppose I have this.

216
00:14:11,370 --> 00:14:15,390
I'll call this x, just to
have a name for the variable.

217
00:14:15,390 --> 00:14:21,360
Suppose I have some values.

218
00:14:21,360 --> 00:14:28,520
Say six values.

219
00:14:28,520 --> 00:14:30,720
And I've measured
them, I've worked hard

220
00:14:30,720 --> 00:14:32,660
to find those six values.

221
00:14:32,660 --> 00:14:37,310
But now you may say,
well I'm thinking

222
00:14:37,310 --> 00:14:43,320
I have some function F(x) here.

223
00:14:43,320 --> 00:14:45,770
But right now I don't
have a function.

224
00:14:45,770 --> 00:14:48,390
Right now I just have six
values of that function.

225
00:14:48,390 --> 00:14:55,030
You know, what is the stretching
constant, what is the c(x).

226
00:14:55,030 --> 00:14:57,650


227
00:14:57,650 --> 00:15:05,760
If you had a physical experiment
with an actual spring,

228
00:15:05,760 --> 00:15:09,630
you might measure, you might
put on different forces

229
00:15:09,630 --> 00:15:15,260
and measure the stretching,
and have six values of that.

230
00:15:15,260 --> 00:15:18,980
How do I fit those with a curve?

231
00:15:18,980 --> 00:15:23,720
In other words, how do I know--
Interpolation means, inter-

232
00:15:23,720 --> 00:15:29,440
means asking, what about
between the points?

233
00:15:29,440 --> 00:15:32,440
What value should it have there?

234
00:15:32,440 --> 00:15:37,100
OK, well there's one
simple rule would be,

235
00:15:37,100 --> 00:15:43,050
just interpolate
linearly between.

236
00:15:43,050 --> 00:15:46,970
OK.

237
00:15:46,970 --> 00:15:49,100
That's certainly pretty stable.

238
00:15:49,100 --> 00:15:51,880
It will not get out of control.

239
00:15:51,880 --> 00:16:00,590
But it's not very accurate.

240
00:16:00,590 --> 00:16:03,850
For the function that's
probably lying behind this,

241
00:16:03,850 --> 00:16:07,230
this isn't that great
of a representation.

242
00:16:07,230 --> 00:16:10,490
OK, you say, I want
something smoother.

243
00:16:10,490 --> 00:16:14,990
Well another idea that
naturally comes up

244
00:16:14,990 --> 00:16:24,910
is, the other extreme would be--
So that was completely local.

245
00:16:24,910 --> 00:16:28,700
Just, every two values
determine the broken line.

246
00:16:28,700 --> 00:16:30,930
The second idea that's
completely natural

247
00:16:30,930 --> 00:16:34,390
would be, fit a polynomial.

248
00:16:34,390 --> 00:16:36,270
Fit a polynomial
through those points,

249
00:16:36,270 --> 00:16:39,950
and then you have a nice,
totally simple function.

250
00:16:39,950 --> 00:16:41,370
Nice curve.

251
00:16:41,370 --> 00:16:46,410
And so what degree polynomial
would we be looking for here?

252
00:16:46,410 --> 00:16:53,460
If I have six points, I could
fit a polynomial of degree--

253
00:16:53,460 --> 00:16:54,830
What do you think?

254
00:16:54,830 --> 00:16:58,120
I want to have six
coefficients, so I

255
00:16:58,120 --> 00:17:02,100
can get six equations to
make the polynomial go

256
00:17:02,100 --> 00:17:03,660
through those six points.

257
00:17:03,660 --> 00:17:08,080
So my polynomial
P(x), which actually

258
00:17:08,080 --> 00:17:11,620
goes through those points, would
be some polynomial of the form,

259
00:17:11,620 --> 00:17:15,830
it's got some constant term,
and it's got some linear term.

260
00:17:15,830 --> 00:17:19,070
And how far am I going to go?

261
00:17:19,070 --> 00:17:19,920
Fifth, right.

262
00:17:19,920 --> 00:17:22,020
That'll give me
six coefficients.

263
00:17:22,020 --> 00:17:22,970
OK.

264
00:17:22,970 --> 00:17:26,740
So I could put a fifth
degree polynomial, that

265
00:17:26,740 --> 00:17:29,360
gives me a_0, a_1, up to a_5.

266
00:17:29,360 --> 00:17:33,620
Six coefficients
through six points.

267
00:17:33,620 --> 00:17:36,700
Well, six isn't too bad.

268
00:17:36,700 --> 00:17:44,080
But if six became
60, don't do it.

269
00:17:44,080 --> 00:17:45,040
That's the message.

270
00:17:45,040 --> 00:17:47,970
Don't do it.

271
00:17:47,970 --> 00:17:55,960
If I'm going on up to, say I
have an a_59 x to the 59th,

272
00:17:55,960 --> 00:17:58,070
And that fits my 60 points.

273
00:17:58,070 --> 00:18:01,030
And you say, well look,
I've got 60 values,

274
00:18:01,030 --> 00:18:04,570
that should be way better
than having only six.

275
00:18:04,570 --> 00:18:08,920
But the polynomial that
would come out of that,

276
00:18:08,920 --> 00:18:13,180
even if these values
were pretty smooth,

277
00:18:13,180 --> 00:18:17,460
the polynomial that fits -- well
I'd have to put in a whole lot

278
00:18:17,460 --> 00:18:20,070
more to get 60,
and I won't try --

279
00:18:20,070 --> 00:18:26,170
the polynomial would go crazy.

280
00:18:26,170 --> 00:18:28,950
Can I just draw something crazy?

281
00:18:28,950 --> 00:18:31,840
I mean, whatever.

282
00:18:31,840 --> 00:18:34,350
It's unstable.

283
00:18:34,350 --> 00:18:37,430
And there are classical
examples of that.

284
00:18:37,430 --> 00:18:41,290
In fact, the famous example
is the function one over one

285
00:18:41,290 --> 00:18:44,560
plus x squared.

286
00:18:44,560 --> 00:18:48,530
And later, there's a figure
in the book, I think it's

287
00:18:48,530 --> 00:18:56,780
in section 5.4, which
shows the result.

288
00:18:56,780 --> 00:19:00,770
That's a terrific function.

289
00:19:00,770 --> 00:19:04,090
It's infinitely differentiable.

290
00:19:04,090 --> 00:19:06,530
I would even call it
an analytic function.

291
00:19:06,530 --> 00:19:07,660
It's great.

292
00:19:07,660 --> 00:19:13,170
But if I tried to fit it
with a high-degree polynomial

293
00:19:13,170 --> 00:19:18,360
at points, that polynomial
gets out of hand.

294
00:19:18,360 --> 00:19:22,690
And a figure is in--
the figure there.

295
00:19:22,690 --> 00:19:34,220
So, MATLAB or any
computing system

296
00:19:34,220 --> 00:19:39,590
would have a subroutine
that does interpolation.

297
00:19:39,590 --> 00:19:43,450
And this is named
after Lagrange.

298
00:19:43,450 --> 00:19:46,100
So this is called
Lagrange interpolation,

299
00:19:46,100 --> 00:19:48,410
fitting a polynomial.

300
00:19:48,410 --> 00:19:53,260
And if the degree is small,
and maybe degree five

301
00:19:53,260 --> 00:19:58,620
would be okay, then
that's an important thing

302
00:19:58,620 --> 00:20:02,010
to be able to do.

303
00:20:02,010 --> 00:20:05,270
In other words, you're
finding these six coefficients

304
00:20:05,270 --> 00:20:06,520
from these six heights.

305
00:20:06,520 --> 00:20:10,210
You've got some six by six
matrix that connects the six

306
00:20:10,210 --> 00:20:13,700
heights to the six a's.

307
00:20:13,700 --> 00:20:18,760
But when you get up to 60, that
matrix stops behaving well.

308
00:20:18,760 --> 00:20:23,550
It becomes very ill conditioned,
and your coefficients

309
00:20:23,550 --> 00:20:25,100
go all over the place.

310
00:20:25,100 --> 00:20:29,440
OK, so my question
is, what do you do?

311
00:20:29,440 --> 00:20:34,850
And one answer, one good
answer is fit those points

312
00:20:34,850 --> 00:20:39,490
not with straight lines,
that's too crude; not

313
00:20:39,490 --> 00:20:43,640
with a single polynomial,
that's too unstable;

314
00:20:43,640 --> 00:20:45,790
but fit it with splines.

315
00:20:45,790 --> 00:20:50,030
So interpolation by splines.

316
00:20:50,030 --> 00:20:51,040
OK.

317
00:20:51,040 --> 00:20:57,440
So it's just sort of
appearing in this section

318
00:20:57,440 --> 00:21:07,470
3.2-- I mean by that,
and people often do mean,

319
00:21:07,470 --> 00:21:13,510
when they say spline,
they mean cubic spline.

320
00:21:13,510 --> 00:21:18,190
So you could have
splines of other degrees.

321
00:21:18,190 --> 00:21:22,290
But often, sort of the
natural choice is the cubic.

322
00:21:22,290 --> 00:21:27,800
So can I just briefly
describe what that does.

323
00:21:27,800 --> 00:21:32,020
What the cubic
spline would be like.

324
00:21:32,020 --> 00:21:35,490
So let me draw in
again these six points

325
00:21:35,490 --> 00:21:38,120
that I'm going to fit.

326
00:21:38,120 --> 00:21:46,000
And just say, for these few
minutes, what's a cubic spline.

327
00:21:46,000 --> 00:21:48,680
I touched on that
Wednesday, and now

328
00:21:48,680 --> 00:21:51,060
I just want to say
a little bit more.

329
00:21:51,060 --> 00:21:56,940
And then the homework
asked you to actually do it

330
00:21:56,940 --> 00:22:02,600
with values of a
particular function.

331
00:22:02,600 --> 00:22:05,540
And see how close does
the cubic spline come

332
00:22:05,540 --> 00:22:09,440
to the given function.

333
00:22:09,440 --> 00:22:12,260
OK, so what's a cubic spline?

334
00:22:12,260 --> 00:22:14,090
What does that
word, spline, what

335
00:22:14,090 --> 00:22:16,110
should that mean in your mind?

336
00:22:16,110 --> 00:22:22,020
So a cubic spline is
a cubic in each piece.

337
00:22:22,020 --> 00:22:24,800
A cubic in each piece.

338
00:22:24,800 --> 00:22:29,700
Now we've met cubics in each
piece as finite elements.

339
00:22:29,700 --> 00:22:32,950
And part of this
short discussion

340
00:22:32,950 --> 00:22:39,860
is to keep those two slightly
different ideas separate.

341
00:22:39,860 --> 00:22:47,030
The finite element idea
was also piecewise cubic.

342
00:22:47,030 --> 00:22:50,930
But it used values and slopes.

343
00:22:50,930 --> 00:22:52,560
So it was completely local.

344
00:22:52,560 --> 00:22:56,500
If I had a value, as I
have here, six values,

345
00:22:56,500 --> 00:23:02,540
and if I also had six slopes,
if I knew the slopes, then

346
00:23:02,540 --> 00:23:06,430
I would use those
cubic elements.

347
00:23:06,430 --> 00:23:12,980
And I would fit a-- If I knew
the height and slope there,

348
00:23:12,980 --> 00:23:14,770
and I knew the height
and slope there,

349
00:23:14,770 --> 00:23:16,560
there would be
exactly one cubic.

350
00:23:16,560 --> 00:23:19,340
Because I would have
four conditions,

351
00:23:19,340 --> 00:23:21,660
two conditions there,
two conditions there,

352
00:23:21,660 --> 00:23:22,470
would be four.

353
00:23:22,470 --> 00:23:25,990
I could fit a cubic between,
another cubic there,

354
00:23:25,990 --> 00:23:27,550
another cubic there.

355
00:23:27,550 --> 00:23:32,380
And because I'm using the
same slope from the left

356
00:23:32,380 --> 00:23:35,930
and from the right, the
slope would be good.

357
00:23:35,930 --> 00:23:37,830
It would be continuous.

358
00:23:37,830 --> 00:23:42,030
The second derivative,
the curvature,

359
00:23:42,030 --> 00:23:46,180
would not be continuous
with those cubic elements.

360
00:23:46,180 --> 00:23:47,570
And that's the difference.

361
00:23:47,570 --> 00:23:53,230
So the difference is,
splines have continuous,

362
00:23:53,230 --> 00:23:56,160
no jump in-- Let me
just put it this way.

363
00:23:56,160 --> 00:24:03,160
No jump in the function.

364
00:24:03,160 --> 00:24:05,930
I'll use S, maybe, for spline.

365
00:24:05,930 --> 00:24:09,070
No jump in its slope.

366
00:24:09,070 --> 00:24:12,490
And no jump in the
second derivative.

367
00:24:12,490 --> 00:24:14,460
So that's the difference.

368
00:24:14,460 --> 00:24:20,560
That the spline
functions, the only

369
00:24:20,560 --> 00:24:26,880
jumps you see for those are
jumps in the third derivative.

370
00:24:26,880 --> 00:24:29,820
So that makes this
extremely smooth.

371
00:24:29,820 --> 00:24:33,240
So if I just try to draw
now what a spline would do,

372
00:24:33,240 --> 00:24:37,330
it'll go through those points,
coming out of the spline fit

373
00:24:37,330 --> 00:24:39,000
MATLAB command.

374
00:24:39,000 --> 00:24:45,570
And it'll be as
smooth as I drew it.

375
00:24:45,570 --> 00:24:53,140
There is a change in the third
derivative at these points.

376
00:24:53,140 --> 00:24:58,230
Actually, have you ever seen
this word, "spline," before?

377
00:24:58,230 --> 00:25:05,730
It came out of
naval engineering.

378
00:25:05,730 --> 00:25:15,460
When people were designing
the shape of the ship.

379
00:25:15,460 --> 00:25:22,930
A naval architect is fitting
the proposed shape of a ship--

380
00:25:22,930 --> 00:25:30,340
this was before MATLAB,
yeah, before life started.

381
00:25:30,340 --> 00:25:37,610
[LAUGHTER] They had
little physical, slightly

382
00:25:37,610 --> 00:25:43,310
bendable-- I'll
call them splines.

383
00:25:43,310 --> 00:25:44,677
I guess that's what they were.

384
00:25:44,677 --> 00:25:46,010
That's where the word came from.

385
00:25:46,010 --> 00:25:50,070
Some physical thing which was
like a curve that you use-- I

386
00:25:50,070 --> 00:25:54,950
don't know if you guys ever
did mechanical drawing.

387
00:25:54,950 --> 00:25:58,070
That was a freshman
subject when I came to MIT.

388
00:25:58,070 --> 00:25:59,120
Mechanical drawing.

389
00:25:59,120 --> 00:26:04,960
I was terrible, terrible,
at mechanical drawing.

390
00:26:04,960 --> 00:26:06,570
I don't know.

391
00:26:06,570 --> 00:26:13,040
I had a friend who helped.
l probably helped him

392
00:26:13,040 --> 00:26:14,480
in some other course.

393
00:26:14,480 --> 00:26:16,730
Anyway, whatever.

394
00:26:16,730 --> 00:26:20,010
So there were physical things,
curves you used to use.

395
00:26:20,010 --> 00:26:22,450
And these spline
curves were used,

396
00:26:22,450 --> 00:26:27,700
and maybe still are
used, by naval architects

397
00:26:27,700 --> 00:26:29,740
in creating a drawing.

398
00:26:29,740 --> 00:26:33,120
But I'm assuming that
those things are now

399
00:26:33,120 --> 00:26:37,510
all computerized, and the
spline command is used.

400
00:26:37,510 --> 00:26:39,320
Anyway, the result is that.

401
00:26:39,320 --> 00:26:42,860
Now, I have to draw one
picture of a spline.

402
00:26:42,860 --> 00:26:44,470
Of the most important spline.

403
00:26:44,470 --> 00:26:48,960
And then I'm done with splines.

404
00:26:48,960 --> 00:26:56,320
So, what I'm going to
draw is now a B-spline.

405
00:26:56,320 --> 00:27:02,850
And that's a basic spline.

406
00:27:02,850 --> 00:27:04,810
It could be one
of our functions,

407
00:27:04,810 --> 00:27:09,710
it could be one of
our functions phi(x).

408
00:27:09,710 --> 00:27:12,430
One of our trial functions.

409
00:27:12,430 --> 00:27:15,170
It could be, and let
me comment on that.

410
00:27:15,170 --> 00:27:22,600
So let me remind myself,
good or bad, with a question.

411
00:27:22,600 --> 00:27:24,170
And let me try to answer that.

412
00:27:24,170 --> 00:27:27,140
But let me first
draw the B-spline.

413
00:27:27,140 --> 00:27:32,070
OK, so it's like a hat function.

414
00:27:32,070 --> 00:27:38,330
Actually a hat function
is a linear spline.

415
00:27:38,330 --> 00:27:42,450
A hat function is
that low level spline

416
00:27:42,450 --> 00:27:44,870
that's linear between pieces.

417
00:27:44,870 --> 00:27:53,940
Now the B-spline is going
to have the value one there.

418
00:27:53,940 --> 00:27:57,340
It's going to have no
jump in the function.

419
00:27:57,340 --> 00:27:59,730
So the function will
go through there.

420
00:27:59,730 --> 00:28:01,420
No jump in the slope.

421
00:28:01,420 --> 00:28:04,370
No jump in the
second derivative.

422
00:28:04,370 --> 00:28:07,180
And I want to get down to zero.

423
00:28:07,180 --> 00:28:13,660
I want to get down to zero.

424
00:28:13,660 --> 00:28:16,840
I want it to be as
local as I can make it.

425
00:28:16,840 --> 00:28:22,750
So I want it to get to zero.

426
00:28:22,750 --> 00:28:27,280
Here's the point.

427
00:28:27,280 --> 00:28:31,070
With those cubic
finite elements,

428
00:28:31,070 --> 00:28:33,110
I got them down to zero.

429
00:28:33,110 --> 00:28:35,830
They came in here
with zero slope,

430
00:28:35,830 --> 00:28:42,340
and then they continued
as zero, no problem.

431
00:28:42,340 --> 00:28:50,390
If I'm wanting the
second derivative also

432
00:28:50,390 --> 00:28:53,560
to be zero, to be
continuous, I won't be

433
00:28:53,560 --> 00:28:55,360
able to do it in one interval.

434
00:28:55,360 --> 00:28:59,310
It's going to take me a
total of four intervals.

435
00:28:59,310 --> 00:29:01,870
This one can come
down to some point,

436
00:29:01,870 --> 00:29:07,770
here, where another
cubic starts.

437
00:29:07,770 --> 00:29:10,130
Maybe I should do the one here.

438
00:29:10,130 --> 00:29:14,450
Going up, you remember
the allowed cubic spline

439
00:29:14,450 --> 00:29:21,680
would be an x cubed over six,
some multiple of x cubed.

440
00:29:21,680 --> 00:29:24,980
I don't know what multiple it'll
take, maybe x cubed over six

441
00:29:24,980 --> 00:29:25,570
is right.

442
00:29:25,570 --> 00:29:28,310
It'll come up to
some point here.

443
00:29:28,310 --> 00:29:36,990
Now I've got to get it
beginning to curve downwards.

444
00:29:36,990 --> 00:29:40,780
So I'm going to have to
change the third derivative.

445
00:29:40,780 --> 00:29:45,730
Change to another cubic
there, change to another cubic

446
00:29:45,730 --> 00:29:49,420
there, and a final cubic here.

447
00:29:49,420 --> 00:29:52,250
So there's a picture
of a B-spline.

448
00:29:52,250 --> 00:29:59,500
And we could figure out, by
requiring all these continuity

449
00:29:59,500 --> 00:30:03,470
conditions, we could figure
out the formula for the cubic

450
00:30:03,470 --> 00:30:05,130
in these four pieces.

451
00:30:05,130 --> 00:30:07,350
And it would be
symmetric, of course,

452
00:30:07,350 --> 00:30:09,200
across that center point.

453
00:30:09,200 --> 00:30:13,980
But I think I
won't try to do it.

454
00:30:13,980 --> 00:30:18,580
I'll just leave that idea, then.

455
00:30:18,580 --> 00:30:20,080
There's a figure
in the book showing

456
00:30:20,080 --> 00:30:24,910
a picture of the B-spline.

457
00:30:24,910 --> 00:30:28,880
So those are functions,
extremely valuable

458
00:30:28,880 --> 00:30:32,010
in this interpolation problem.

459
00:30:32,010 --> 00:30:35,620
Because all splines are
combinations of B-splines.

460
00:30:35,620 --> 00:30:38,570
Yeah, now let me pull
this topic together.

461
00:30:38,570 --> 00:30:44,260
Every spline, every
spline function,

462
00:30:44,260 --> 00:30:52,700
is combination of
these B-splines.

463
00:30:52,700 --> 00:30:58,810
Let B_i(x) be the one
centered on node i.

464
00:30:58,810 --> 00:31:03,450
And then there's a neighbor
centered on node i+1.

465
00:31:03,450 --> 00:31:07,200
And a neighbor to the
left centered on node i-1.

466
00:31:07,200 --> 00:31:09,220
What's the point?

467
00:31:09,220 --> 00:31:19,280
The point is that, at
a typical node, i+1,

468
00:31:19,280 --> 00:31:23,030
three of these functions
will be non-zero.

469
00:31:23,030 --> 00:31:28,110
With these very local hat
functions at that node,

470
00:31:28,110 --> 00:31:31,240
the only one of the phi
functions that wasn't zero

471
00:31:31,240 --> 00:31:32,950
is the one I've drawn.

472
00:31:32,950 --> 00:31:35,770
All the others were
zero there, right?

473
00:31:35,770 --> 00:31:38,650
All the other hats were zero.

474
00:31:38,650 --> 00:31:40,200
There was just the one.

475
00:31:40,200 --> 00:31:44,100
But now there'll be a
B-spline starting from here,

476
00:31:44,100 --> 00:31:47,780
going up to here,
coming down from here.

477
00:31:47,780 --> 00:31:51,390
There'll be the B-spline
starting from here, going up

478
00:31:51,390 --> 00:31:54,540
to here, up, down, so on.

479
00:31:54,540 --> 00:31:57,880
So at a typical
node, I'm getting

480
00:31:57,880 --> 00:32:01,700
the one centered at
that node, and also

481
00:32:01,700 --> 00:32:05,440
the one to the left,
which is on its way down,

482
00:32:05,440 --> 00:32:08,500
and also the one to the
right, which is on its way up.

483
00:32:08,500 --> 00:32:12,460
In other words,
it's not as local

484
00:32:12,460 --> 00:32:17,880
as the other construction.

485
00:32:17,880 --> 00:32:26,090
So I would say good,
because it's smooth, but bad

486
00:32:26,090 --> 00:32:30,020
because it's not fully local.

487
00:32:30,020 --> 00:32:32,420
Not completely local.

488
00:32:32,420 --> 00:32:35,230
It's reasonably local,
in that there are only

489
00:32:35,230 --> 00:32:39,400
three functions that are
affecting these points.

490
00:32:39,400 --> 00:32:44,090
If I use those polynomials,
they weren't local at all.

491
00:32:44,090 --> 00:32:50,630
All the points, it was--
A typical x to the fifth

492
00:32:50,630 --> 00:32:52,970
is not zero at
any of the points.

493
00:32:52,970 --> 00:32:56,220
These B-splines are
zero at a lot of points,

494
00:32:56,220 --> 00:32:59,940
but three of them,
one, two, and three,

495
00:32:59,940 --> 00:33:03,170
will be non-zero at
a typical mesh point.

496
00:33:03,170 --> 00:33:13,550
OK, so they're not popular, as
a result, for finite elements.

497
00:33:13,550 --> 00:33:16,750
I'm just wanting to be sure
you make the distinction.

498
00:33:16,750 --> 00:33:20,900
They're very popular for
the interpolation job.

499
00:33:20,900 --> 00:33:24,750
They're very popular for
the interpolation job.

500
00:33:24,750 --> 00:33:30,680
I've got some combination
of these at particular mesh

501
00:33:30,680 --> 00:33:31,210
points.

502
00:33:31,210 --> 00:33:36,470
It's supposed to agree with
F at those mesh points.

503
00:33:36,470 --> 00:33:38,450
This is my system to solve.

504
00:33:38,450 --> 00:33:41,880
I create these functions,
these B-splines.

505
00:33:41,880 --> 00:33:45,290
I'm given F at typical points.

506
00:33:45,290 --> 00:33:51,640
And I choose a combination
which matches F at those points.

507
00:33:51,640 --> 00:33:54,550
Yeah, OK.

508
00:33:54,550 --> 00:34:00,990
Is there a question
or discussion on this?

509
00:34:00,990 --> 00:34:05,550
I don't like to not say anything
about such an important problem

510
00:34:05,550 --> 00:34:09,380
as putting curves
through points.

511
00:34:09,380 --> 00:34:16,121
But I don't want to make
it a course on splines.

512
00:34:16,121 --> 00:34:16,620
Yes?

513
00:34:16,620 --> 00:34:18,620
AUDIENCE: [UNINTELLIGIBLE]

514
00:34:18,620 --> 00:34:24,720
PROFESSOR STRANG: I don't
know that I'll-- Oh, OK, yes.

515
00:34:24,720 --> 00:34:25,730
All right.

516
00:34:25,730 --> 00:34:30,010
Two or three comments, and
that suggests a good question,

517
00:34:30,010 --> 00:34:32,700
a very good question.

518
00:34:32,700 --> 00:34:36,240
Can I make a separate
comment that if I

519
00:34:36,240 --> 00:34:40,910
go into two dimensions,
this gets much tougher.

520
00:34:40,910 --> 00:34:44,160
What happens if I had
a function of x and y?

521
00:34:44,160 --> 00:34:46,940
So that I've got
points on a surface,

522
00:34:46,940 --> 00:34:49,930
and I'm trying to
fit a surface to it.

523
00:34:49,930 --> 00:34:55,630
Just one message first:
that's not as easy.

524
00:34:55,630 --> 00:35:00,970
It's pretty easy if those points
are on a rectangular grid.

525
00:35:00,970 --> 00:35:04,650
So this is like typical.

526
00:35:04,650 --> 00:35:06,830
And then I'll come to
your Nyquist question,

527
00:35:06,830 --> 00:35:09,400
which is a very good one.

528
00:35:09,400 --> 00:35:14,350
Can I just-- because we're
coming in to 2-D now.

529
00:35:14,350 --> 00:35:17,310
Suppose I-- There's
two dimensions.

530
00:35:17,310 --> 00:35:18,830
I have a grid.

531
00:35:18,830 --> 00:35:20,890
Let's suppose it's a nice grid.

532
00:35:20,890 --> 00:35:27,080
And at every point, at every
grid point, I have a height.

533
00:35:27,080 --> 00:35:28,620
And I'm fitting a surface.

534
00:35:28,620 --> 00:35:30,310
Right?

535
00:35:30,310 --> 00:35:38,480
My function is now a function of
x and y. x is here, y is here,

536
00:35:38,480 --> 00:35:44,400
F is the surface coming out of
board, in the third dimension.

537
00:35:44,400 --> 00:35:48,300
And fitting those points,
if they're regularly spaced

538
00:35:48,300 --> 00:35:50,660
like that, my life would be OK.

539
00:35:50,660 --> 00:35:56,080
I could use sort of
products of splines in 1-D.

540
00:35:56,080 --> 00:36:04,880
I could use products
of basis-- of splines

541
00:36:04,880 --> 00:36:09,270
in the x-direction times
splines in the y-direction.

542
00:36:09,270 --> 00:36:13,260
And it would be
pretty successful.

543
00:36:13,260 --> 00:36:18,680
It would be, not quite
as nice, but almost OK.

544
00:36:18,680 --> 00:36:28,650
But if I had irregularly spaced
points from a general grid,

545
00:36:28,650 --> 00:36:30,220
it's not as easy.

546
00:36:30,220 --> 00:36:33,880
And I won't-- People have
obviously had to figure out how

547
00:36:33,880 --> 00:36:38,480
to do it, and to repeat again,
that's what the CAD/CAM world

548
00:36:38,480 --> 00:36:42,010
is having to do all the
time, is fit a curve,

549
00:36:42,010 --> 00:36:44,190
fit a surface through points.

550
00:36:44,190 --> 00:36:45,950
It's significant.

551
00:36:45,950 --> 00:36:49,250
Now, you asked about Nyquist.

552
00:36:49,250 --> 00:36:51,420
So that's a good question.

553
00:36:51,420 --> 00:36:59,150
Can I just say, I
have to say what-- So

554
00:36:59,150 --> 00:37:00,830
what's a function called?

555
00:37:00,830 --> 00:37:04,210
Band-limited.

556
00:37:04,210 --> 00:37:07,330
How many have heard
Nyquist's name?

557
00:37:07,330 --> 00:37:10,900
Okay, some of you may know
a lot more than I about it.

558
00:37:10,900 --> 00:37:18,140
But let me just get some context
for band-limited functions.

559
00:37:18,140 --> 00:37:21,010
Okay.

560
00:37:21,010 --> 00:37:23,180
This is actually
a topic that will

561
00:37:23,180 --> 00:37:27,180
belong in the third part of our
course, in the Fourier part.

562
00:37:27,180 --> 00:37:35,240
So this is a Fourier idea.

563
00:37:35,240 --> 00:37:38,930
So I'll come back
to it, actually.

564
00:37:38,930 --> 00:37:47,740
So right here I'm just
going to say, very briefly,

565
00:37:47,740 --> 00:37:51,890
how it might connect us
to what I've said today.

566
00:37:51,890 --> 00:37:57,340
But then let's make
a plan to, when

567
00:37:57,340 --> 00:38:04,560
we've got the idea of
Fourier coefficients.

568
00:38:04,560 --> 00:38:11,770
What is a band-limited function?

569
00:38:11,770 --> 00:38:16,920
You know the Fourier
idea is to take F(x),

570
00:38:16,920 --> 00:38:23,970
and write it as some combination
of pure frequencies. e^(i*K*x),

571
00:38:23,970 --> 00:38:27,420
let me say. e^(i*K*x).

572
00:38:27,420 --> 00:38:30,570
So this is Fourier
that's coming.

573
00:38:30,570 --> 00:38:33,300
I take the function
F(x), and I write it--

574
00:38:33,300 --> 00:38:36,160
I could think of
it as a combination

575
00:38:36,160 --> 00:38:41,390
of pure exponentials,
pure frequencies.

576
00:38:41,390 --> 00:38:44,210
That would be a Fourier series.

577
00:38:44,210 --> 00:38:47,660
So that's a Fourier
series because I'm

578
00:38:47,660 --> 00:38:50,590
using-- K has integer values.

579
00:38:50,590 --> 00:38:55,110
The frequencies are zero,
one, two, three, whatever.

580
00:38:55,110 --> 00:39:01,820
Now that we'll use,
so that Fourier series

581
00:39:01,820 --> 00:39:05,030
comes for functions
that are periodic.

582
00:39:05,030 --> 00:39:06,640
They repeat every 2pi.

583
00:39:06,640 --> 00:39:10,410
Because those functions,
if I increase x by 2pi,

584
00:39:10,410 --> 00:39:11,650
don't change.

585
00:39:11,650 --> 00:39:13,970
So I'm repeating every 2pi.

586
00:39:13,970 --> 00:39:21,920
Now I have to say a word about
the other possibility, which

587
00:39:21,920 --> 00:39:26,150
would be to have
all frequencies.

588
00:39:26,150 --> 00:39:27,950
I'll integrate now, dK.

589
00:39:27,950 --> 00:39:32,150
Instead of summing on
K, I'll integrate on K.

590
00:39:32,150 --> 00:39:35,010
What does band-limited mean?

591
00:39:35,010 --> 00:39:41,770
Band-limited means that only
frequencies in a certain range,

592
00:39:41,770 --> 00:39:44,970
say a range around
zero, are included.

593
00:39:44,970 --> 00:39:49,560
So a band-limited function would
be a function whose frequencies

594
00:39:49,560 --> 00:39:55,490
go from some value, say
minus omega to omega,

595
00:39:55,490 --> 00:39:58,380
instead of going from
minus infinity to infinity.

596
00:39:58,380 --> 00:40:05,760
This would be
band-limited frequencies

597
00:40:05,760 --> 00:40:12,280
between minus omega and omega.

598
00:40:12,280 --> 00:40:18,730
So that's another kind
of smooth function.

599
00:40:18,730 --> 00:40:22,910
That's another
way-- Functions that

600
00:40:22,910 --> 00:40:27,790
have only low frequencies are
associated with smoothness.

601
00:40:27,790 --> 00:40:31,660
High frequencies are associated
with fast oscillations.

602
00:40:31,660 --> 00:40:36,160
So the class of
band-limited functions

603
00:40:36,160 --> 00:40:41,750
gives me another, a Fourier
way, to talk about smoothness.

604
00:40:41,750 --> 00:40:45,180
So for us, smoothness was
something about how many

605
00:40:45,180 --> 00:40:47,760
derivatives were continuous.

606
00:40:47,760 --> 00:40:50,900
That's the sort of
smoothness in the x domain.

607
00:40:50,900 --> 00:40:53,040
How many derivatives.

608
00:40:53,040 --> 00:40:55,320
Smoothness in the
frequency domain

609
00:40:55,320 --> 00:40:59,260
is, how fast do the
frequencies drop off.

610
00:40:59,260 --> 00:41:03,200
And here, band-limited
means they drop like a shot.

611
00:41:03,200 --> 00:41:09,270
Band-limited means that the
frequencies in the function,

612
00:41:09,270 --> 00:41:13,660
that these e^(i*K*x)'s are not
there for high frequencies.

613
00:41:13,660 --> 00:41:18,400
High frequencies are out
for a band-limited function.

614
00:41:18,400 --> 00:41:26,070
And then Shannon has a
formula for the natural way

615
00:41:26,070 --> 00:41:32,300
to fit-- So our same
interpolation problem.

616
00:41:32,300 --> 00:41:36,210
So now, completing the
answer to your question.

617
00:41:36,210 --> 00:41:38,820
So I have these points.

618
00:41:38,820 --> 00:41:45,400
So Shannon could fit a function,
and it would look smooth,

619
00:41:45,400 --> 00:41:47,590
through those points.

620
00:41:47,590 --> 00:41:53,010
And his function
would be band-limited.

621
00:41:53,010 --> 00:41:55,340
So it would be smooth again.

622
00:41:55,340 --> 00:42:02,430
It would be-- yeah,
it would be smooth.

623
00:42:02,430 --> 00:42:06,250
In some way, this Shannon
band-limited stuff

624
00:42:06,250 --> 00:42:12,180
is the limit of splines as
the spline degree goes way up.

625
00:42:12,180 --> 00:42:16,480
So we did hat functions,
degree one splines.

626
00:42:16,480 --> 00:42:23,740
Cubic splines I recommended as
a pretty reliable construction.

627
00:42:23,740 --> 00:42:26,820
But you could do fifth degrees
splines, seventh degree

628
00:42:26,820 --> 00:42:29,320
splines, you could keep going.

629
00:42:29,320 --> 00:42:32,120
And in the limit,
you would get this.

630
00:42:32,120 --> 00:42:36,030
So maybe that's
some partial answer

631
00:42:36,030 --> 00:42:42,350
to the connection between
splines and this Fourier world.

632
00:42:42,350 --> 00:42:43,690
OK.

633
00:42:43,690 --> 00:42:45,090
Thanks.

634
00:42:45,090 --> 00:42:48,640
So these are topics now--
Wow, today's lecture

635
00:42:48,640 --> 00:42:52,850
is kind of-- Can I do one
really important thing now

636
00:42:52,850 --> 00:42:54,330
in today's lecture?

637
00:42:54,330 --> 00:42:56,290
For you to remember?

638
00:42:56,290 --> 00:42:58,050
Gradient and divergence?

639
00:42:58,050 --> 00:42:59,700
I don't want you to
spend the weekend

640
00:42:59,700 --> 00:43:10,660
without thinking about gradient
and divergence. [LAUGHTER] OK.

641
00:43:10,660 --> 00:43:14,410
Here's the idea.

642
00:43:14,410 --> 00:43:19,500
For lots and lots of
applications, for a region,

643
00:43:19,500 --> 00:43:29,370
let's say, in the plane,
I have the same-- What

644
00:43:29,370 --> 00:43:31,960
physical example
shall I pick now?

645
00:43:31,960 --> 00:43:35,530
Maybe I'll let u
be the temperature.

646
00:43:35,530 --> 00:43:37,420
So instead of
being displacement,

647
00:43:37,420 --> 00:43:41,400
let me make it temperature. u.

648
00:43:41,400 --> 00:43:42,830
OK.

649
00:43:42,830 --> 00:43:46,310
Then I have a
temperature gradient.

650
00:43:46,310 --> 00:43:47,290
A slope.

651
00:43:47,290 --> 00:43:51,480
But now, the whole point is,
that's a function of x and y.

652
00:43:51,480 --> 00:43:55,600
Then I have a
temperature gradient.

653
00:43:55,600 --> 00:43:59,010
And if I'm consistent with the
notation, that'll be e(x,y).

654
00:43:59,010 --> 00:44:01,860


655
00:44:01,860 --> 00:44:08,440
And then you'll expect
that there's some c(x,y).

656
00:44:08,440 --> 00:44:13,640
Some operator c that tells
me how much heat flows.

657
00:44:13,640 --> 00:44:17,100
This will tell me
something about the the,

658
00:44:17,100 --> 00:44:18,960
the thermal conductivity.

659
00:44:18,960 --> 00:44:20,180
Right?

660
00:44:20,180 --> 00:44:27,300
Really, as I speak
about this framework,

661
00:44:27,300 --> 00:44:31,470
I'm just uttering
the correct words.

662
00:44:31,470 --> 00:44:34,260
Having started with
temperature, this thing

663
00:44:34,260 --> 00:44:36,180
should be a
temperature gradient.

664
00:44:36,180 --> 00:44:39,360
Then I should have some
physical thermal conductivity,

665
00:44:39,360 --> 00:44:43,110
different for different
metals or different materials.

666
00:44:43,110 --> 00:44:48,560
And I'll have a
heat flow, w(x,y).

667
00:44:48,560 --> 00:44:53,150
And everybody knows that
w will be c times e.

668
00:44:53,150 --> 00:44:57,610
And then there will
be some A transpose.

669
00:44:57,610 --> 00:45:01,860
Of course, there will be
some A transpose here,

670
00:45:01,860 --> 00:45:03,920
and some A here.

671
00:45:03,920 --> 00:45:07,740
And up here I'll have
a balance equation.

672
00:45:07,740 --> 00:45:13,730
OK.

673
00:45:13,730 --> 00:45:20,650
I just want to think,
what's the operator A?

674
00:45:20,650 --> 00:45:24,190
If we can focus
on that question,

675
00:45:24,190 --> 00:45:28,340
then that's what's going
to occupy us for, certainly

676
00:45:28,340 --> 00:45:31,010
the whole of next week.

677
00:45:31,010 --> 00:45:35,960
So I've actually used
the word gradient.

678
00:45:35,960 --> 00:45:39,280
We have functions
of two variables.

679
00:45:39,280 --> 00:45:42,730
We're looking for the
change, the rate of change,

680
00:45:42,730 --> 00:45:44,890
the steepness of
those functions.

681
00:45:44,890 --> 00:45:52,640
So this A, Au, is going to
give me two derivatives.

682
00:45:52,640 --> 00:45:56,650
I've got two variables, there
are two first derivatives.

683
00:45:56,650 --> 00:46:01,160
Both of them are important.

684
00:46:01,160 --> 00:46:03,500
That's what the A is.

685
00:46:03,500 --> 00:46:09,000
For the next big
example in the course.

686
00:46:09,000 --> 00:46:11,300
The final major
example of the course

687
00:46:11,300 --> 00:46:15,800
is, when A acts on a
function of two variables,

688
00:46:15,800 --> 00:46:19,420
because I'm in a
region in the plane,

689
00:46:19,420 --> 00:46:23,450
to find its rate of change.

690
00:46:23,450 --> 00:46:26,730
And this is called the gradient.

691
00:46:26,730 --> 00:46:34,030
The shorthand is
the gradient of u.

692
00:46:34,030 --> 00:46:37,720
So we have to understand that.

693
00:46:37,720 --> 00:46:41,870
We have to understand
what the gradient is.

694
00:46:41,870 --> 00:46:45,460
And, of course, we want
to know its transpose.

695
00:46:45,460 --> 00:46:48,900
So can I just think,
what should be

696
00:46:48,900 --> 00:46:54,410
the transpose of the gradient?

697
00:46:54,410 --> 00:46:55,000
OK.

698
00:46:55,000 --> 00:46:57,510
I'll take that picture out.

699
00:46:57,510 --> 00:46:58,470
OK.

700
00:46:58,470 --> 00:47:03,740
Thinking now about the
transpose of the gradient.

701
00:47:03,740 --> 00:47:06,070
OK.

702
00:47:06,070 --> 00:47:14,560
So A itself is, you could
say, is d/dx and d/dy.

703
00:47:14,560 --> 00:47:20,000
You notice how I'm
separating out A from Au.

704
00:47:20,000 --> 00:47:22,030
When this acts on
a function, this

705
00:47:22,030 --> 00:47:26,040
is the gradient operator
that acts on a function, u,

706
00:47:26,040 --> 00:47:29,700
to produce du/dx and du/dy.

707
00:47:29,700 --> 00:47:30,510
OK.

708
00:47:30,510 --> 00:47:33,670
Now what's the
transpose of this?

709
00:47:33,670 --> 00:47:35,480
OK.

710
00:47:35,480 --> 00:47:38,000
You can guess what it
should be, and then we'll

711
00:47:38,000 --> 00:47:40,960
see that yes, that
guess is correct.

712
00:47:40,960 --> 00:47:44,860
So what should A
transpose look like?

713
00:47:44,860 --> 00:47:54,540
If there's any justice, A
transpose should be-- OK,

714
00:47:54,540 --> 00:47:58,560
this is two by one.

715
00:47:58,560 --> 00:48:06,310
The transpose you would expect
to be a row, a row vector.

716
00:48:06,310 --> 00:48:08,930
I should have the
transpose of that there.

717
00:48:08,930 --> 00:48:11,160
And what is the
transpose of that?

718
00:48:11,160 --> 00:48:13,660
Just tell me.

719
00:48:13,660 --> 00:48:15,370
Because we have an idea.

720
00:48:15,370 --> 00:48:19,790
What should be the
transpose of d/dx?

721
00:48:19,790 --> 00:48:20,650
Negative d/dx.

722
00:48:20,650 --> 00:48:24,040


723
00:48:24,040 --> 00:48:26,710
And what should be
the transpose of d/dy?

724
00:48:26,710 --> 00:48:28,690
It should be negative d/dy.

725
00:48:28,690 --> 00:48:30,550
Those are two different pieces.

726
00:48:30,550 --> 00:48:32,460
This is not run together.

727
00:48:32,460 --> 00:48:35,020
There's a big space in there.

728
00:48:35,020 --> 00:48:37,750
That's two pieces.

729
00:48:37,750 --> 00:48:44,810
So what is A transpose applied
to a-- So, heat flow w.

730
00:48:44,810 --> 00:48:50,840
I want to say, what is A
transpose applied to w?

731
00:48:50,840 --> 00:48:54,900
You're going to see this
again, but we'll just

732
00:48:54,900 --> 00:48:58,210
take these minutes to show
it for the first time.

733
00:48:58,210 --> 00:49:03,030
So A transpose-- Wait a minute.

734
00:49:03,030 --> 00:49:08,090
Is w a function,
or is it a vector?

735
00:49:08,090 --> 00:49:11,540
Yeah we've got to get that
straight before we start.

736
00:49:11,540 --> 00:49:14,220
Here's an ordinary
function, a scalar function.

737
00:49:14,220 --> 00:49:18,330
Just whatever, x
squared plus y squared.

738
00:49:18,330 --> 00:49:21,520
What is e?

739
00:49:21,520 --> 00:49:25,660
Suppose this is x
squared plus y squared.

740
00:49:25,660 --> 00:49:27,750
Let's have a specific example.

741
00:49:27,750 --> 00:49:31,590
What would e then be?

742
00:49:31,590 --> 00:49:33,630
It's got two components, right?

743
00:49:33,630 --> 00:49:38,910
It's got an x derivative and
a y derivative. e is a vector,

744
00:49:38,910 --> 00:49:41,870
[2x, 2y].

745
00:49:41,870 --> 00:49:44,270
Then I multiply by c.

746
00:49:44,270 --> 00:49:46,730
So this w has two components.

747
00:49:46,730 --> 00:49:50,870
This has got a
w_1(x,y), w_2(x,y).

748
00:49:50,870 --> 00:49:53,910


749
00:49:53,910 --> 00:49:56,670
So just keep things straight.

750
00:49:56,670 --> 00:50:00,340
So it's that that I want
to apply A transpose to.

751
00:50:00,340 --> 00:50:08,550
So A transpose w is
minus d/dx, minus d/dy.

752
00:50:08,550 --> 00:50:10,210
And everything's
coming out right

753
00:50:10,210 --> 00:50:14,450
because it's applied to
a function w_1 and w_2.

754
00:50:14,450 --> 00:50:17,740
w is a vector field.

755
00:50:17,740 --> 00:50:20,350
It's not a scalar field,
it's a vector field.

756
00:50:20,350 --> 00:50:24,050
And the result is just
what it should be:

757
00:50:24,050 --> 00:50:31,290
minus dw_1/dx minus dw_2/dy.

758
00:50:31,290 --> 00:50:37,580
OK, good.

759
00:50:37,580 --> 00:50:40,620
This has got to come out of
integration by parts, right?

760
00:50:40,620 --> 00:50:43,290
So we'll have to think: what
does integration by parts

761
00:50:43,290 --> 00:50:45,910
mean in two variables?

762
00:50:45,910 --> 00:50:50,570
And it's a famous formula
named after Gauss and Green.

763
00:50:50,570 --> 00:50:52,330
The Green's formula, often.

764
00:50:52,330 --> 00:50:56,520
But do you recognize
what I'm looking at here?

765
00:50:56,520 --> 00:51:01,180
This is so important
it has a name.

766
00:51:01,180 --> 00:51:02,650
And what's the name?

767
00:51:02,650 --> 00:51:04,260
And then we're ready to go.

768
00:51:04,260 --> 00:51:09,470
What's the name of, if I take
a vector field, like [2x, 2y].

769
00:51:09,470 --> 00:51:11,840
Let me take [2x, 2y].

770
00:51:11,840 --> 00:51:17,570
as a specific example.

771
00:51:17,570 --> 00:51:21,480
Suppose w is [2x,2y].

772
00:51:21,480 --> 00:51:26,150
That was multiplied by,
let me take c to be one.

773
00:51:26,150 --> 00:51:33,140
Then what is A transpose w?

774
00:51:33,140 --> 00:51:34,800
Specifically.

775
00:51:34,800 --> 00:51:39,010
What am I getting out of it?

776
00:51:39,010 --> 00:51:43,860
What do I get from here?

777
00:51:43,860 --> 00:51:48,060
That's minus the x
derivative of the first guy,

778
00:51:48,060 --> 00:51:50,060
and the x derivative
of that guy is two,

779
00:51:50,060 --> 00:51:52,830
so I'm getting a minus two.

780
00:51:52,830 --> 00:51:58,650
And this is minus the y
derivative of the second guy,

781
00:51:58,650 --> 00:52:00,970
so that's another minus two.

782
00:52:00,970 --> 00:52:02,250
So I'm getting a number.

783
00:52:02,250 --> 00:52:03,990
It happens to be a number here.

784
00:52:03,990 --> 00:52:09,340
I chose such a simple function
it came out to be a number.

785
00:52:09,340 --> 00:52:11,100
And what's the name?

786
00:52:11,100 --> 00:52:15,550
So this is minus the what of w?

787
00:52:15,550 --> 00:52:20,470
Just tell me, what's
the name everybody uses

788
00:52:20,470 --> 00:52:22,890
for that operation?

789
00:52:22,890 --> 00:52:24,770
The divergence.

790
00:52:24,770 --> 00:52:28,040
Minus the divergence of w.

791
00:52:28,040 --> 00:52:37,950
So I what I'm saying
here is that this-- I'm

792
00:52:37,950 --> 00:52:41,240
saying it because
somehow I remember

793
00:52:41,240 --> 00:52:43,470
studying vector calculus.

794
00:52:43,470 --> 00:52:48,070
And in that process, I
learned about the gradient,

795
00:52:48,070 --> 00:52:51,360
and I learned about
the divergence.

796
00:52:51,360 --> 00:52:56,120
But I never learned that one
was the transpose of the other.

797
00:52:56,120 --> 00:53:03,420
I think, looking back,
that was criminal.

798
00:53:03,420 --> 00:53:07,500
To describe those-- With
a minus sign, of course.

799
00:53:07,500 --> 00:53:16,230
I learned Green's formula, but
now we'll see what it means.

800
00:53:16,230 --> 00:53:17,880
OK, that's next week's job.

801
00:53:17,880 --> 00:53:20,501
Have a great weekend
and see you Monday.

802
00:53:20,501 --> 00:53:21,000