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PROFESSOR STRANG: OK, so.

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00:00:22,870 --> 00:00:24,710
This is a big day.

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00:00:24,710 --> 00:00:27,480
Part One of the
course is completed,

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00:00:27,480 --> 00:00:30,040
and I have your quizzes
for you, and that

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00:00:30,040 --> 00:00:33,390
was a very successful
result, I'm very pleased.

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00:00:33,390 --> 00:00:35,860
I hope you are, too.

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00:00:35,860 --> 00:00:39,990
Quiz average of 85, that's on
the first part of the course.

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00:00:39,990 --> 00:00:45,380
And then the second part --
so this is Chapter 3 now --

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00:00:45,380 --> 00:00:49,370
Starts in one dimension with
an equation of a type that

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00:00:49,370 --> 00:00:52,120
we've already seen a little bit.

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00:00:52,120 --> 00:00:54,240
So there's some
more things to say

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00:00:54,240 --> 00:00:58,060
about the equation
and the framework,

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00:00:58,060 --> 00:01:05,350
but then we get to make a start
on the finite element approach

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00:01:05,350 --> 00:01:06,720
to solving it.

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00:01:06,720 --> 00:01:12,290
We could of course-- In 1-D
finite differences are probably

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00:01:12,290 --> 00:01:14,450
the way to go, actually.

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00:01:14,450 --> 00:01:21,850
In one dimension the special
success of finite elements

26
00:01:21,850 --> 00:01:27,130
doesn't really show up that much
because finite elements have

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00:01:27,130 --> 00:01:30,400
been, I mean one great
reason for their success

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00:01:30,400 --> 00:01:33,420
is that they handle
different geometries.

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00:01:33,420 --> 00:01:35,420
They're flexible; you
could have regions

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00:01:35,420 --> 00:01:39,980
in the plane, three-dimensional
bodies of different shapes.

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00:01:39,980 --> 00:01:41,740
Finite differences
doesn't really

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00:01:41,740 --> 00:01:46,070
know what to do on a
curved boundary in in 2-

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00:01:46,070 --> 00:01:50,540
or 3-D. Finite elements
copes much better.

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00:01:50,540 --> 00:01:55,120
So, we'll make a start
today, more Friday

35
00:01:55,120 --> 00:01:57,620
on one-dimensional
finite elements

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00:01:57,620 --> 00:02:02,300
and then, a couple
of weeks later

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00:02:02,300 --> 00:02:05,450
will be the real
thing, 2-D and 3-D.

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00:02:05,450 --> 00:02:09,160
OK, so, ready to
go on Chapter 3?

39
00:02:09,160 --> 00:02:11,560
So, that's our equation
and everybody sees

40
00:02:11,560 --> 00:02:15,000
right away what's the
framework, that that's

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00:02:15,000 --> 00:02:20,390
A transpose; in some way
this is A transpose C A,

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00:02:20,390 --> 00:02:23,510
but what's new of course is that
we're dealing with functions,

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00:02:23,510 --> 00:02:25,230
not vectors.

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00:02:25,230 --> 00:02:30,170
So we're dealing with, you could
say, operators, not matrices.

45
00:02:30,170 --> 00:02:35,950
And nevertheless, the big
picture is still as it was.

46
00:02:35,950 --> 00:02:40,720
So let me take u(x) to be
the displacements again.

47
00:02:40,720 --> 00:02:44,440
So I'm thinking more of
mechanics than electronics

48
00:02:44,440 --> 00:02:45,370
here.

49
00:02:45,370 --> 00:02:48,640
Displacements, and
then we have du--

50
00:02:48,640 --> 00:02:52,910
the e(x) will be
du/dx, that'll be

51
00:02:52,910 --> 00:03:00,700
the stretching, the elongation
and, of course at that step

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00:03:00,700 --> 00:03:04,940
you already see the
big new item, the fact

53
00:03:04,940 --> 00:03:10,330
that the A, the one that
gets us from u to du/dx,

54
00:03:10,330 --> 00:03:14,260
instead of being a difference
matrix which it has been,

55
00:03:14,260 --> 00:03:18,880
our matrix A is
now a derivative.

56
00:03:18,880 --> 00:03:21,730
A is d/dx.

57
00:03:21,730 --> 00:03:27,560
So maybe I'll just
take out that arrow.

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00:03:27,560 --> 00:03:29,950
So A is d/dx.

59
00:03:29,950 --> 00:03:37,330
OK, but if we dealt OK
with difference matrices,

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00:03:37,330 --> 00:03:39,570
we're going to deal
OK with derivatives.

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00:03:39,570 --> 00:03:46,720
Then, of course, this is the
C part, that produces w(x).

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00:03:46,720 --> 00:03:53,590
And it's a multiplication
by this possibly varying,

63
00:03:53,590 --> 00:03:58,370
possibly jumping
stiffness constant c(x).

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00:03:58,370 --> 00:04:05,790
So w(x) is c(x) e(x), that's our
old w=Ce, this is Hooke's Law.

65
00:04:05,790 --> 00:04:08,880
I'll put Hooke's
Law, but that's,

66
00:04:08,880 --> 00:04:13,590
or whose-ever law it is.

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00:04:13,590 --> 00:04:15,840
It's like a diagonal
matrix; I hope

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you see that it's like
a diagonal matrix.

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This function u is kind
of like of a vector

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00:04:23,590 --> 00:04:28,040
but a continuum vector
instead of just a fixed,

71
00:04:28,040 --> 00:04:30,040
finite number of values.

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00:04:30,040 --> 00:04:34,590
Then at each value we
used to multiply by c_i,

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00:04:34,590 --> 00:04:40,740
now our values are continuous
with x, so we multiply by c(x).

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00:04:40,740 --> 00:04:44,730
And then you're going to
expect that, going up here,

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00:04:44,730 --> 00:04:51,650
there's going to be an A
transpose w f, and of course

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00:04:51,650 --> 00:04:54,740
that A transpose
we have to identify

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00:04:54,740 --> 00:04:58,690
and that's the first point
of the lecture, really.

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00:04:58,690 --> 00:05:01,750
To identify what is A transpose.

79
00:05:01,750 --> 00:05:05,590
What do I mean by A transpose?

80
00:05:05,590 --> 00:05:09,080
And I've got to say
right away that I'm

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00:05:09,080 --> 00:05:15,760
a little, the notation, writing
a transpose of a derivative is

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00:05:15,760 --> 00:05:18,950
like, that's not legal.

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00:05:18,950 --> 00:05:21,570
Because we think of the
transpose of a matrix,

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00:05:21,570 --> 00:05:25,470
you sort of flip it
over the main diagonal,

85
00:05:25,470 --> 00:05:30,120
but obviously it's got to be
something more to it than that.

86
00:05:30,120 --> 00:05:33,630
And so that's a
central math part

87
00:05:33,630 --> 00:05:39,170
of this lecture is what's really
going on when you transpose?

88
00:05:39,170 --> 00:05:41,150
Because then we can
copy what's going on

89
00:05:41,150 --> 00:05:43,030
and it's quite
important to get it.

90
00:05:43,030 --> 00:05:46,970
Because the transpose,
well, other notations

91
00:05:46,970 --> 00:05:50,290
and other words for it
would be-- A notation

92
00:05:50,290 --> 00:05:51,970
might be a star.

93
00:05:51,970 --> 00:05:55,370
Star would be way more
common than transpose,

94
00:05:55,370 --> 00:05:57,200
I'll just stay with
transpose because I

95
00:05:57,200 --> 00:06:04,410
want to keep pressing the
parallel with A transpose C A.

96
00:06:04,410 --> 00:06:08,180
And the name for it
would be the adjoint.

97
00:06:08,180 --> 00:06:13,870
And adjoint methods, and adjoint
operator, those appear a lot.

98
00:06:13,870 --> 00:06:16,510
And you'll see them
appear in finite elements.

99
00:06:16,510 --> 00:06:24,340
So this is a good
thing to catch on to.

100
00:06:24,340 --> 00:06:25,060
Why?

101
00:06:25,060 --> 00:06:30,030
Why should the transpose or
the adjoint of the derivative

102
00:06:30,030 --> 00:06:32,000
be minus the derivative?

103
00:06:32,000 --> 00:06:36,610
And by the way, just
while we're fixing this,

104
00:06:36,610 --> 00:06:43,870
this is a key fact then,
which is certainly--

105
00:06:43,870 --> 00:06:47,640
We have a very strong hint from
centered difference, right?

106
00:06:47,640 --> 00:06:50,810
If I think of derivatives,
if I associate them

107
00:06:50,810 --> 00:06:54,100
with differences, the
centered difference matrix,

108
00:06:54,100 --> 00:07:01,800
so the A matrix may be centered,
would be-- Just to remind us,

109
00:07:01,800 --> 00:07:05,600
a centered difference has
ones and minus one, one,

110
00:07:05,600 --> 00:07:08,280
zeroes on the diagonal, right?

111
00:07:08,280 --> 00:07:10,430
Minus one, one.

112
00:07:10,430 --> 00:07:16,210
Takes that difference
at every row.

113
00:07:16,210 --> 00:07:18,280
Except possibly boundary rows.

114
00:07:18,280 --> 00:07:21,220
And of course as soon as you
look at that matrix you see,

115
00:07:21,220 --> 00:07:23,590
yeah, it's anti-symmetric.

116
00:07:23,590 --> 00:07:26,100
That's an anti-symmetric matrix.

117
00:07:26,100 --> 00:07:32,160
So A transpose is minus A
for centered differences

118
00:07:32,160 --> 00:07:35,580
and therefore we're
not so surprised

119
00:07:35,580 --> 00:07:38,330
to see a minus sign
up here when we

120
00:07:38,330 --> 00:07:40,910
go to the continuous
case, the derivative.

121
00:07:40,910 --> 00:07:45,320
But, we still have
to say what it means.

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00:07:45,320 --> 00:07:47,660
So that's what I'll do next, OK?

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00:07:47,660 --> 00:07:49,790
So this is a good thing to know.

124
00:07:49,790 --> 00:07:51,930
And I was just going
to comment, what

125
00:07:51,930 --> 00:07:55,530
would be the transpose
of the second derivative?

126
00:07:55,530 --> 00:07:57,490
I won't even write this down.

127
00:07:57,490 --> 00:08:04,380
If the derivative transpose
sort of flips its sign to minus,

128
00:08:04,380 --> 00:08:06,540
what would you guess
for this transpose

129
00:08:06,540 --> 00:08:10,550
of second derivative, our
more familiar d second

130
00:08:10,550 --> 00:08:12,190
by dx squared?

131
00:08:12,190 --> 00:08:14,820
Well we'll have two minus signs.

132
00:08:14,820 --> 00:08:16,550
So it'll come out plus.

133
00:08:16,550 --> 00:08:21,040
So second derivatives,
even order derivatives

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00:08:21,040 --> 00:08:24,030
are sort of like symmetric guys.

135
00:08:24,030 --> 00:08:28,410
Odd order derivatives, first
and third and fifth derivatives,

136
00:08:28,410 --> 00:08:30,850
well, God forbid we ever
meet a fifth derivative,

137
00:08:30,850 --> 00:08:36,910
but first derivative
anyway, is anti-symmetric.

138
00:08:36,910 --> 00:08:38,840
Except for boundary conditions.

139
00:08:38,840 --> 00:08:42,560
So I really have to
emphasize that the boundary

140
00:08:42,560 --> 00:08:44,030
conditions come in.

141
00:08:44,030 --> 00:08:45,580
And you'll see them come in.

142
00:08:45,580 --> 00:08:47,200
They have to come in.

143
00:08:47,200 --> 00:08:52,150
OK, so what meaning can I
assign to the transpose,

144
00:08:52,150 --> 00:08:58,790
or what was the real thing
happening when we flipped

145
00:08:58,790 --> 00:09:00,930
the matrix across its diagonal?

146
00:09:00,930 --> 00:09:14,050
I claim that we really define
the transpose by this rule,

147
00:09:14,050 --> 00:09:17,810
by-- We know what
inner products are.

148
00:09:17,810 --> 00:09:20,320
I'll do vectors first, we
know about inner products,

149
00:09:20,320 --> 00:09:23,650
dot products, we know what the
dot product of two vectors is.

150
00:09:23,650 --> 00:09:27,390
So, this is the
transpose of A. How

151
00:09:27,390 --> 00:09:29,650
am I going to define
the transpose of A?

152
00:09:29,650 --> 00:09:36,170
Well, I look at the dot
product of Au with w.

153
00:09:36,170 --> 00:09:40,250
I'll use a dot here for once,
I may erase it and replace it.

154
00:09:40,250 --> 00:09:46,860
If at the dot product of Au
with w, then that equals,

155
00:09:46,860 --> 00:09:49,700
for all u and w,
all vectors u and w,

156
00:09:49,700 --> 00:09:56,720
that equals the dot product
of u with something.

157
00:09:56,720 --> 00:10:01,790
Because u is coming-- If I write
out what the dot product is,

158
00:10:01,790 --> 00:10:05,810
I see u_1 multiplies something,
u_2 multiplies something.

159
00:10:05,810 --> 00:10:13,420
And what goes in
that little space?

160
00:10:13,420 --> 00:10:15,210
This is just an identity.

161
00:10:15,210 --> 00:10:18,760
I mean, it's like,
you'll say no big deal.

162
00:10:18,760 --> 00:10:22,330
But I'm saying there is at
least a small deal here.

163
00:10:22,330 --> 00:10:22,930
OK.

164
00:10:22,930 --> 00:10:28,710
So if I write it
this way, you'll

165
00:10:28,710 --> 00:10:32,850
tell me right away this should
be the same as u transpose

166
00:10:32,850 --> 00:10:33,900
times something.

167
00:10:33,900 --> 00:10:38,400
And again, so I'm asking for the
same something on both lines.

168
00:10:38,400 --> 00:10:41,930
What is that something?

169
00:10:41,930 --> 00:10:44,100
A transpose w.

170
00:10:44,100 --> 00:10:48,130
Whatever A transpose
is, it's the matrix

171
00:10:48,130 --> 00:10:50,220
that makes this right.

172
00:10:50,220 --> 00:10:51,960
That's really my message.

173
00:10:51,960 --> 00:10:54,550
That A transpose
is-- The reason we

174
00:10:54,550 --> 00:10:56,810
flipped the matrix
across the diagonal

175
00:10:56,810 --> 00:11:01,110
is that it makes that
equation correct.

176
00:11:01,110 --> 00:11:03,510
And I'm writing the
same thing here.

177
00:11:03,510 --> 00:11:04,940
OK.

178
00:11:04,940 --> 00:11:09,590
So again, if we knew
what dot products were,

179
00:11:09,590 --> 00:11:11,930
what inner product
of vectors were,

180
00:11:11,930 --> 00:11:15,940
then A transpose
is the matrix that

181
00:11:15,940 --> 00:11:18,780
makes this identity correct.

182
00:11:18,780 --> 00:11:22,310
And of course if
you write it all out

183
00:11:22,310 --> 00:11:24,580
in terms of i, j,
every component,

184
00:11:24,580 --> 00:11:26,640
you find it is correct.

185
00:11:26,640 --> 00:11:31,190
So that defines the
transpose of a matrix.

186
00:11:31,190 --> 00:11:34,590
And of course it coincides with
flipping across the diagonal.

187
00:11:34,590 --> 00:11:39,610
Now, how about the
transpose of a derivative.

188
00:11:39,610 --> 00:11:43,520
OK, so I'm going to
follow the same rule.

189
00:11:43,520 --> 00:11:46,080
Here A is now going
to be the derivative,

190
00:11:46,080 --> 00:11:48,230
and A transpose is
going to be whatever

191
00:11:48,230 --> 00:11:50,470
it takes to make this true.

192
00:11:50,470 --> 00:11:51,930
But what do I mean?

193
00:11:51,930 --> 00:11:54,620
Now I have functions, so
I have to think again,

194
00:11:54,620 --> 00:11:57,870
what do I mean by the inner
product, the dot product?

195
00:11:57,870 --> 00:12:02,950
So for this to make
sense I need to say,

196
00:12:02,950 --> 00:12:05,410
and it's a very
important thing anyway,

197
00:12:05,410 --> 00:12:07,830
and it's the right
natural choice,

198
00:12:07,830 --> 00:12:14,520
I need to say the dot
product, or the inner product

199
00:12:14,520 --> 00:12:19,170
is a better word, of functions.

200
00:12:19,170 --> 00:12:21,270
Of two functions.

201
00:12:21,270 --> 00:12:24,760
Say e(x) and w(x).

202
00:12:24,760 --> 00:12:30,260
If I have two functions, what do
I mean by their inner product?

203
00:12:30,260 --> 00:12:32,570
Well, really I just
think back what

204
00:12:32,570 --> 00:12:36,100
did we mean in the
finite dimensional case,

205
00:12:36,100 --> 00:12:40,520
I multiplied each e by a w,
each component of e by w,

206
00:12:40,520 --> 00:12:43,780
and I added, so what
am I going to do here?

207
00:12:43,780 --> 00:12:49,410
Maybe my notation
should be parentheses

208
00:12:49,410 --> 00:12:54,410
with a comma would be better
than a dot, for functions.

209
00:12:54,410 --> 00:12:55,800
So I have a function.

210
00:12:55,800 --> 00:12:59,400
I'm in function space now.

211
00:12:59,400 --> 00:13:04,100
We moved out of R^n today,
into function space.

212
00:13:04,100 --> 00:13:06,390
Our vectors have
become functions.

213
00:13:06,390 --> 00:13:09,420
And now what's the dot
product of two vectors?

214
00:13:09,420 --> 00:13:13,710
Well, what am I going to do?

215
00:13:13,710 --> 00:13:15,660
I'm going to do
what I have to do.

216
00:13:15,660 --> 00:13:20,970
I'm going to multiply each
e by its corresponding w,

217
00:13:20,970 --> 00:13:26,240
and now they depend on
this continuous variable x,

218
00:13:26,240 --> 00:13:28,840
so that's e(x) times w(x).

219
00:13:28,840 --> 00:13:32,010
And what do I do now?

220
00:13:32,010 --> 00:13:33,580
Integrate.

221
00:13:33,580 --> 00:13:38,420
Here I added e_i
times w_i, of course.

222
00:13:38,420 --> 00:13:41,200
Over here I have functions.

223
00:13:41,200 --> 00:13:43,610
I integrate dx.

224
00:13:43,610 --> 00:13:47,500
Over whatever the region
of the problem is.

225
00:13:47,500 --> 00:13:52,050
And then our example's in
1-D, it'll zero to one.

226
00:13:52,050 --> 00:13:54,370
If these are functions
of two variables

227
00:13:54,370 --> 00:13:58,990
I'd be integrating over some 2-D
region, but we're in 1-D today.

228
00:13:58,990 --> 00:14:15,110
OK, so you see that I'm prepared
to say-- This now makes sense.

229
00:14:15,110 --> 00:14:20,320
I now want to say, I'm going
to let A be the derivative,

230
00:14:20,320 --> 00:14:24,190
and I'm going to figure out
what A transpose has to be.

231
00:14:24,190 --> 00:14:29,930
So if A is the derivative,
so now is this key step.

232
00:14:29,930 --> 00:14:31,780
Why is the transpose that?

233
00:14:31,780 --> 00:14:40,950
OK, so I look at the
derivative, du/dx, with w,

234
00:14:40,950 --> 00:14:52,670
so that's this integral, from
zero to one of du/dx*w(x)dx,

235
00:14:52,670 --> 00:14:54,860
so that's my left side.

236
00:14:54,860 --> 00:14:58,070
Now I want to get u by itself.

237
00:14:58,070 --> 00:15:01,020
I want to get the
dot product, so I

238
00:15:01,020 --> 00:15:07,410
want to get another integral
here that has u(x) by itself.

239
00:15:07,410 --> 00:15:09,130
Times something,
and that something

240
00:15:09,130 --> 00:15:10,900
is what I'm looking for.

241
00:15:10,900 --> 00:15:15,230
That something will
be A transpose w.

242
00:15:15,230 --> 00:15:17,600
Right?

243
00:15:17,600 --> 00:15:19,340
Do you see what I'm doing?

244
00:15:19,340 --> 00:15:24,240
This is is the dot
product, this is Auw,

245
00:15:24,240 --> 00:15:28,680
so I've written out what
Au inner product with w is.

246
00:15:28,680 --> 00:15:32,680
And now I want to get u out by
itself and what it multiplies

247
00:15:32,680 --> 00:15:36,020
here will be the A
transpose w, and my rule

248
00:15:36,020 --> 00:15:42,510
will be extended to the function
case and I'll be ready to go.

249
00:15:42,510 --> 00:15:48,670
Now do you recognize, this
is a basic calculus step,

250
00:15:48,670 --> 00:15:51,420
what rule of calculus
am I going to use?

251
00:15:51,420 --> 00:15:53,660
We're back to 18.01.

252
00:15:53,660 --> 00:15:56,240
I have the integral of
a derivative times w,

253
00:15:56,240 --> 00:15:57,580
and what do I want to do?

254
00:15:57,580 --> 00:16:01,150
I want to get the
derivative off of u.

255
00:16:01,150 --> 00:16:02,890
What happens?

256
00:16:02,890 --> 00:16:04,710
What's it called?

257
00:16:04,710 --> 00:16:06,140
Integration by parts.

258
00:16:06,140 --> 00:16:08,060
Very important thing.

259
00:16:08,060 --> 00:16:10,060
Very important.

260
00:16:10,060 --> 00:16:13,420
You miss its
importance in calculus.

261
00:16:13,420 --> 00:16:16,480
It gets sometimes introduced
as a rule, or a trick

262
00:16:16,480 --> 00:16:21,490
to find some goofy integral,
but it's really the real thing.

263
00:16:21,490 --> 00:16:23,630
So what is integration by parts?

264
00:16:23,630 --> 00:16:24,770
What's the rule?

265
00:16:24,770 --> 00:16:28,280
You take the
derivative off of u,

266
00:16:28,280 --> 00:16:31,600
you put it onto the other
one just what we hope for,

267
00:16:31,600 --> 00:16:34,920
and then you also
have to remember

268
00:16:34,920 --> 00:16:40,290
that there is a minus sign.

269
00:16:40,290 --> 00:16:42,510
Integration by parts
has a minus sign.

270
00:16:42,510 --> 00:16:44,750
And usually you'd
see it out there

271
00:16:44,750 --> 00:16:49,090
but here I've left
more room for it there.

272
00:16:49,090 --> 00:16:54,290
So I have identified
now A transpose w.

273
00:16:54,290 --> 00:16:59,990
A transpose w has-- If this
is Au, inner product with w,

274
00:16:59,990 --> 00:17:02,950
then this is u inner
product with A transpose w,

275
00:17:02,950 --> 00:17:04,490
it had to be what was.

276
00:17:04,490 --> 00:17:08,910
And so that one integration by
parts brought out a minus sign.

277
00:17:08,910 --> 00:17:10,890
If I was looking at
second derivatives

278
00:17:10,890 --> 00:17:13,560
there would probably be
somewhere two integration

279
00:17:13,560 --> 00:17:17,430
by parts; I'd have minus
twice, I'd be back to plus.

280
00:17:17,430 --> 00:17:20,510
And you're going to ask
about boundary conditions.

281
00:17:20,510 --> 00:17:22,910
And you're right to ask
about boundary conditions.

282
00:17:22,910 --> 00:17:26,680
I even circled that, because
that is so important.

283
00:17:26,680 --> 00:17:33,200
So what we've done so far
is to get the interior

284
00:17:33,200 --> 00:17:35,670
of the interval right.

285
00:17:35,670 --> 00:17:39,140
Between zero and one,
if A is the derivative,

286
00:17:39,140 --> 00:17:42,020
then A transpose is
minus the derivative.

287
00:17:42,020 --> 00:17:43,750
That's all we've done.

288
00:17:43,750 --> 00:17:46,290
We have not got the
boundary conditions yet.

289
00:17:46,290 --> 00:17:50,490
And we can't go on without that.

290
00:17:50,490 --> 00:17:53,470
OK, so I'm ready
now to say something

291
00:17:53,470 --> 00:17:57,110
about boundary conditions.

292
00:17:57,110 --> 00:18:00,660
And it will bring up this
square versus rectangular also.

293
00:18:00,660 --> 00:18:04,660


294
00:18:04,660 --> 00:18:07,940
We're getting the rules
straight before we

295
00:18:07,940 --> 00:18:09,820
tackle finite elements.

296
00:18:09,820 --> 00:18:15,090
OK, let me take an example of
a matrix and its transpose.

297
00:18:15,090 --> 00:18:18,800
Just so you see how boundary
conditions-- Suppose

298
00:18:18,800 --> 00:18:23,130
I have a free-fixed problem.

299
00:18:23,130 --> 00:18:26,840
Suppose I have a
free-fixed line of springs.

300
00:18:26,840 --> 00:18:30,470
What's the matrix A for that?

301
00:18:30,470 --> 00:18:32,830
Well-- Question?

302
00:18:32,830 --> 00:18:33,330
Yes.

303
00:18:33,330 --> 00:18:36,480
AUDIENCE: [INAUDIBLE]

304
00:18:36,480 --> 00:18:37,740
PROFESSOR STRANG: Yeah.

305
00:18:37,740 --> 00:18:44,770
That's-- Yes, when I learned it,
it was also that stupid trick.

306
00:18:44,770 --> 00:18:50,036
So you would like me to put
plus, can I put plus, whatever.

307
00:18:50,036 --> 00:18:51,410
What do you want
me to call that?

308
00:18:51,410 --> 00:18:52,640
An integrated term?

309
00:18:52,640 --> 00:18:54,490
It would be, yeah.

310
00:18:54,490 --> 00:18:55,910
I even remember what it is.

311
00:18:55,910 --> 00:19:03,170
As you do better than me. u
times w at the, is that good?

312
00:19:03,170 --> 00:19:06,860
Yeah, I think.

313
00:19:06,860 --> 00:19:11,530
So it's really this part
that I'm now coming to.

314
00:19:11,530 --> 00:19:14,090
It's the boundary part
that I'm now coming to.

315
00:19:14,090 --> 00:19:18,630
And let me say, so I'm glad
you asked that question

316
00:19:18,630 --> 00:19:21,500
because I made it
seem unimportant,

317
00:19:21,500 --> 00:19:23,360
where that's not true at all.

318
00:19:23,360 --> 00:19:27,400
The boundary condition is
part of the definition of A,

319
00:19:27,400 --> 00:19:30,000
and part of the
definition of A transpose.

320
00:19:30,000 --> 00:19:33,830
Just the way, I'm about
to say free-fixed,

321
00:19:33,830 --> 00:19:38,350
I had to tell you that for
you to know what A was.

322
00:19:38,350 --> 00:19:40,570
Until I tell you the
boundary condition,

323
00:19:40,570 --> 00:19:42,410
you don't know what
the boundary rows are.

324
00:19:42,410 --> 00:19:45,900
You only know the
inside of the matrix.

325
00:19:45,900 --> 00:19:47,880
Or one possible inside.

326
00:19:47,880 --> 00:19:49,790
So I'm thinking
my inside is going

327
00:19:49,790 --> 00:19:53,120
to be minus one,
one; minus one, one;

328
00:19:53,120 --> 00:19:59,770
so on, as giving me
finite differences.

329
00:19:59,770 --> 00:20:01,750
Minus one, one.

330
00:20:01,750 --> 00:20:03,850
But.

331
00:20:03,850 --> 00:20:04,570
Oh no, let's see.

332
00:20:04,570 --> 00:20:07,467
So I'm doing free-fixed.

333
00:20:07,467 --> 00:20:08,050
Is that right?

334
00:20:08,050 --> 00:20:10,390
Am I doing free-fixed?

335
00:20:10,390 --> 00:20:16,360
OK, so am I taking
free at the left end?

336
00:20:16,360 --> 00:20:17,170
Yes.

337
00:20:17,170 --> 00:20:19,600
Alright, so if I'm
free at the left

338
00:20:19,600 --> 00:20:23,840
and fixed at the right
end, what's my A?

339
00:20:23,840 --> 00:20:27,320
We're getting better
at this, right?

340
00:20:27,320 --> 00:20:28,700
Minus one, one.

341
00:20:28,700 --> 00:20:29,660
Minus one, one.

342
00:20:29,660 --> 00:20:30,610
Minus one, one.

343
00:20:30,610 --> 00:20:37,080
Minus one, and the one
here gets chopped off.

344
00:20:37,080 --> 00:20:40,250
You could say if you want
the fifth row of A_0,

345
00:20:40,250 --> 00:20:44,750
remembering A_0 as the
hint on the quiz, where

346
00:20:44,750 --> 00:20:48,570
it had five rows for the
full thing, free-free.

347
00:20:48,570 --> 00:20:55,920
And then when an end got fixed,
the fifth column got removed,

348
00:20:55,920 --> 00:20:59,470
and that's my free-fixed matrix.

349
00:20:59,470 --> 00:21:02,180
At the left hand
end, at the zero end,

350
00:21:02,180 --> 00:21:04,282
it's got the
difference in there.

351
00:21:04,282 --> 00:21:05,740
Difference,
difference, difference.

352
00:21:05,740 --> 00:21:08,870
But here at the
right-hand end, it's

353
00:21:08,870 --> 00:21:12,030
the fixing, the setting
u, whatever it would be,

354
00:21:12,030 --> 00:21:17,110
u_5 to zero, or maybe it's u_4.
if this is u nought, one, two,

355
00:21:17,110 --> 00:21:17,980
three.

356
00:21:17,980 --> 00:21:20,820
Setting u_4 to zero
knocked that out.

357
00:21:20,820 --> 00:21:22,520
OK.

358
00:21:22,520 --> 00:21:26,130
All I want to do
is transpose that.

359
00:21:26,130 --> 00:21:30,350
And you'll see something that
we maybe didn't notice before.

360
00:21:30,350 --> 00:21:32,440
So I transpose it,
that's minus one,

361
00:21:32,440 --> 00:21:36,270
one becomes a column, minus
one, one becomes a column.

362
00:21:36,270 --> 00:21:38,480
Minus one, one becomes a column.

363
00:21:38,480 --> 00:21:41,390
Minus one, all there is.

364
00:21:41,390 --> 00:21:50,570
That row becomes, so this
was, so-- Have I got it right?

365
00:21:50,570 --> 00:21:52,140
Yes.

366
00:21:52,140 --> 00:21:55,730
What's happened?

367
00:21:55,730 --> 00:21:58,370
A transpose, what are
the boundary conditions

368
00:21:58,370 --> 00:22:00,930
going with A transpose?

369
00:22:00,930 --> 00:22:03,120
The boundary conditions
that went with A

370
00:22:03,120 --> 00:22:06,120
were, let me say
first, what were

371
00:22:06,120 --> 00:22:08,590
the boundary conditions with A?

372
00:22:08,590 --> 00:22:12,140
Those are going to be
boundary conditions on u.

373
00:22:12,140 --> 00:22:19,600
So A has boundary
conditions on u.

374
00:22:19,600 --> 00:22:26,340
And A transpose has
boundary conditions on w.

375
00:22:26,340 --> 00:22:31,130
Because A transpose acts
on w, and A acts on u.

376
00:22:31,130 --> 00:22:33,820
So there was no choice.

377
00:22:33,820 --> 00:22:36,230
So now what was the
boundary condition here?

378
00:22:36,230 --> 00:22:40,300
The boundary condition
was u_4=0, right?

379
00:22:40,300 --> 00:22:44,730
That was what I meant by
that guy getting fixed.

380
00:22:44,730 --> 00:22:49,450
Now-- And no boundary
condition at u_0, it was free.

381
00:22:49,450 --> 00:22:51,020
Now, what are the
boundary conditions

382
00:22:51,020 --> 00:22:52,800
that go with A transpose?

383
00:22:52,800 --> 00:22:55,410
And remember, A transpose
is multiplying w.

384
00:22:55,410 --> 00:22:59,550
I'm going to put w here, so
what are the boundary conditions

385
00:22:59,550 --> 00:23:02,990
that go with A transpose?

386
00:23:02,990 --> 00:23:05,830
This thing, nothing
got knocked off.

387
00:23:05,830 --> 00:23:09,220
The boundary condition came
up here for A transpose,

388
00:23:09,220 --> 00:23:13,940
the boundary
condition was w_0=0.

389
00:23:13,940 --> 00:23:21,290
That got knocked
out, w_0 to be zero.

390
00:23:21,290 --> 00:23:22,230
No surprise.

391
00:23:22,230 --> 00:23:25,800
Free-fixed, this is the
free end at the left,

392
00:23:25,800 --> 00:23:29,000
this is the fixed
end at the right.

393
00:23:29,000 --> 00:23:32,720
Did you ever notice that
the matrix does it for you?

394
00:23:32,720 --> 00:23:37,110
I mean, when you transpose that
matrix it just automatically

395
00:23:37,110 --> 00:23:43,780
built in the correct
boundary conditions on w by,

396
00:23:43,780 --> 00:23:48,130
you started with the conditions
on u, you transpose the matrix

397
00:23:48,130 --> 00:23:51,700
and you've discovered what the
boundary conditions on w are.

398
00:23:51,700 --> 00:23:58,910
And I'm going to do the same
for the continuous problem.

399
00:23:58,910 --> 00:24:01,360
I'm going to do the same for
the continuous problem, so

400
00:24:01,360 --> 00:24:04,980
the continuous free-fixed.

401
00:24:04,980 --> 00:24:08,250


402
00:24:08,250 --> 00:24:14,410
OK, what's the boundary
condition on u?

403
00:24:14,410 --> 00:24:17,930
If it's free-fixed, I just
want you to repeat this,

404
00:24:17,930 --> 00:24:21,160
on the interval zero
to one for functions

405
00:24:21,160 --> 00:24:24,380
u(x), w(x) instead
of for vectors.

406
00:24:24,380 --> 00:24:26,240
What's the boundary
condition on u,

407
00:24:26,240 --> 00:24:30,010
if I have a free-fixed
problem? u(1)=0.

408
00:24:30,010 --> 00:24:35,120


409
00:24:35,120 --> 00:24:38,860
u of one equals zero.

410
00:24:38,860 --> 00:24:40,280
So this is the
boundary conditions

411
00:24:40,280 --> 00:24:42,960
that goes with A in
the free-fixed case.

412
00:24:42,960 --> 00:24:54,710
And this is part
of A. That is part

413
00:24:54,710 --> 00:25:00,440
of A. I don't know what A
is until I know its boundary

414
00:25:00,440 --> 00:25:01,260
condition.

415
00:25:01,260 --> 00:25:04,269
Just the way I don't
know what this matrix is.

416
00:25:04,269 --> 00:25:05,810
It could have been
A_0, it could have

417
00:25:05,810 --> 00:25:08,720
lost one column, it could have
lost two columns, whatever.

418
00:25:08,720 --> 00:25:12,830
I don't know until I've told
you the boundary condition on u.

419
00:25:12,830 --> 00:25:16,800
And then transposing is going
to tell me, automatically,

420
00:25:16,800 --> 00:25:19,950
without any further input,
the boundary condition that

421
00:25:19,950 --> 00:25:21,890
goes on the adjoint.

422
00:25:21,890 --> 00:25:25,110
So what's the boundary
condition on w that

423
00:25:25,110 --> 00:25:28,160
goes as part of A transpose?

424
00:25:28,160 --> 00:25:30,940
Well, you're going to tell me.

425
00:25:30,940 --> 00:25:36,590
Tell me. w(0) should be zero.

426
00:25:36,590 --> 00:25:39,650
It came out
automatically, naturally.

427
00:25:39,650 --> 00:25:43,940
This is a big distinction
between boundary conditions.

428
00:25:43,940 --> 00:25:48,080
I would call that an
essential boundary condition.

429
00:25:48,080 --> 00:25:51,350
I had to start with it,
I had to decide on that.

430
00:25:51,350 --> 00:25:54,290
And then this, I call a
natural boundary condition.

431
00:25:54,290 --> 00:25:58,110
Or there are even
two guys' names,

432
00:25:58,110 --> 00:26:01,580
which are associated
with these two types.

433
00:26:01,580 --> 00:26:05,260
So maybe a first chance to
just mention these names.

434
00:26:05,260 --> 00:26:10,050
Because you'll often
see, reading some paper,

435
00:26:10,050 --> 00:26:15,840
maybe a little on the
mathematical side,

436
00:26:15,840 --> 00:26:18,060
you'll see that word used.

437
00:26:18,060 --> 00:26:20,870
The guy's name with this
sort of boundary condition

438
00:26:20,870 --> 00:26:22,730
is a French name.

439
00:26:22,730 --> 00:26:25,720
Not so easy to say, Dirichlet.

440
00:26:25,720 --> 00:26:28,340
I'll say it more
often in the future.

441
00:26:28,340 --> 00:26:33,020
Anyway, I would call that a
Dirichlet condition and you

442
00:26:33,020 --> 00:26:35,160
would say it's a fixed
boundary condition.

443
00:26:35,160 --> 00:26:37,310
And if you were
doing heat flow you

444
00:26:37,310 --> 00:26:39,270
would say it's a
fixed temperature.

445
00:26:39,270 --> 00:26:40,290
Whatever.

446
00:26:40,290 --> 00:26:45,990
Fixed, is really the
word to remember there.

447
00:26:45,990 --> 00:26:51,460
OK, and then I guess I better
give Germany a shot here, too.

448
00:26:51,460 --> 00:26:55,580
So the natural
boundary condition

449
00:26:55,580 --> 00:27:04,520
is associated with
the name of Neumann.

450
00:27:04,520 --> 00:27:09,960
So if I said a Dirichlet
problem, a total Dirichlet

451
00:27:09,960 --> 00:27:13,290
problem, I would be
speaking about fixed-fixed.

452
00:27:13,290 --> 00:27:15,410
And if I spoke about
a Neumann problem

453
00:27:15,410 --> 00:27:17,710
I would be talking
about free-free.

454
00:27:17,710 --> 00:27:22,650
And this problem is Dirichlet at
one end, Neumann at the other.

455
00:27:22,650 --> 00:27:26,340
Anyway, so essential
and natural.

456
00:27:26,340 --> 00:27:28,980
And now of course I'm
hoping that that's

457
00:27:28,980 --> 00:27:38,300
going to make this
boundary term go away.

458
00:27:38,300 --> 00:27:41,010
OK, now I'm paying
attention to this thing

459
00:27:41,010 --> 00:27:43,070
that you made me write. uw.

460
00:27:43,070 --> 00:27:46,120
OK, what happens there?

461
00:27:46,120 --> 00:27:48,280
uw?

462
00:27:48,280 --> 00:27:52,240
Oh, right this
isn't bad because it

463
00:27:52,240 --> 00:27:54,800
shows that there's a--
Boundary condition,

464
00:27:54,800 --> 00:27:59,400
I've got some little
deals going with them

465
00:27:59,400 --> 00:28:06,550
But do you see that
that becomes zero?

466
00:28:06,550 --> 00:28:10,170
Why is it zero at
the top end, at one?

467
00:28:10,170 --> 00:28:14,720
When I take u times w at
one, why do I get zero?

468
00:28:14,720 --> 00:28:17,920
Because u(1) is zero.

469
00:28:17,920 --> 00:28:18,820
Good.

470
00:28:18,820 --> 00:28:20,540
And at the bottom end.

471
00:28:20,540 --> 00:28:25,730
When I take uw at the other
boundary, why do I get zero?

472
00:28:25,730 --> 00:28:27,500
Because of w.

473
00:28:27,500 --> 00:28:30,780
You see that w was needed.

474
00:28:30,780 --> 00:28:35,350
That w(0) was needed because
there was no controlling u(0).

475
00:28:35,350 --> 00:28:38,260
I had no control of u
at the left-hand end,

476
00:28:38,260 --> 00:28:39,990
because it was free.

477
00:28:39,990 --> 00:28:43,290
So the control has
to come from w.

478
00:28:43,290 --> 00:28:46,360
And so w naturally
had to be zero,

479
00:28:46,360 --> 00:28:51,650
because I wasn't controlling
u at that left-hand, free end.

480
00:28:51,650 --> 00:28:58,140
So one way or the other,
the integration by parts

481
00:28:58,140 --> 00:28:59,970
is the key.

482
00:28:59,970 --> 00:29:07,320
So that's said what
I critically wanted

483
00:29:07,320 --> 00:29:13,240
to say about transposing,
taking the adjoint,

484
00:29:13,240 --> 00:29:16,800
except I was just going to add
a comment about this square A

485
00:29:16,800 --> 00:29:20,270
versus rectangular.

486
00:29:20,270 --> 00:29:24,040
And this was a case
of square, right?

487
00:29:24,040 --> 00:29:26,455
This was a case,
this free-fixed case,

488
00:29:26,455 --> 00:29:29,480
this example I happened
to pick was square.

489
00:29:29,480 --> 00:29:32,680
A_0, the free-free guy that
was a hint on the quiz,

490
00:29:32,680 --> 00:29:34,090
was rectangular.

491
00:29:34,090 --> 00:29:37,550
The fixed-fixed, which
was also on the quiz,

492
00:29:37,550 --> 00:29:40,120
was also rectangular.

493
00:29:40,120 --> 00:29:44,120
It was what, four by
three or something.

494
00:29:44,120 --> 00:29:47,990
This A is four by four.

495
00:29:47,990 --> 00:29:57,410
And what is especially
nice when it's square?

496
00:29:57,410 --> 00:30:00,540
If our problem happens
to give a square matrix,

497
00:30:00,540 --> 00:30:04,770
in the truss case if the
number of displacement unknowns

498
00:30:04,770 --> 00:30:09,410
happens to equal the number
of bars, so m equals n,

499
00:30:09,410 --> 00:30:14,870
I have a square matrix A. And
this guy's invertible, so it's

500
00:30:14,870 --> 00:30:16,070
all good.

501
00:30:16,070 --> 00:30:18,780
Oh, that's maybe
the whole point.

502
00:30:18,780 --> 00:30:20,890
That if it's
rectangular I wouldn't

503
00:30:20,890 --> 00:30:22,620
talk about its inverse.

504
00:30:22,620 --> 00:30:27,900
But this is a square matrix,
so A itself has an inverse.

505
00:30:27,900 --> 00:30:30,790
Instead of having,
as I usually have,

506
00:30:30,790 --> 00:30:34,170
to deal with A transpose
C A all at once,

507
00:30:34,170 --> 00:30:40,390
let me put this comment because
it's just a small one up here.

508
00:30:40,390 --> 00:30:41,760
Right under these words.

509
00:30:41,760 --> 00:30:43,910
Square versus rectangular.

510
00:30:43,910 --> 00:30:49,410
Square A, and let's
say invertible,

511
00:30:49,410 --> 00:30:51,370
otherwise we're in
the unstable case,

512
00:30:51,370 --> 00:30:55,690
so you know what I
mean, in the network

513
00:30:55,690 --> 00:31:02,670
problems the number of nodes
matches the number of edges?

514
00:31:02,670 --> 00:31:10,410
In the spring problem we
have free-fixed situations.

515
00:31:10,410 --> 00:31:12,660
Anyway, A comes out square.

516
00:31:12,660 --> 00:31:15,940
Whatever the application.

517
00:31:15,940 --> 00:31:19,400
If it comes out square,
what is especially good?

518
00:31:19,400 --> 00:31:21,170
It comes out square,
what's especially

519
00:31:21,170 --> 00:31:23,790
good is that it has an inverse.

520
00:31:23,790 --> 00:31:30,270
So that in this square case,
this K inverse is A transpose C

521
00:31:30,270 --> 00:31:35,260
A inverse, can be split.

522
00:31:35,260 --> 00:31:40,300
This allows us to
separate, to do what

523
00:31:40,300 --> 00:31:45,740
you better not do otherwise.

524
00:31:45,740 --> 00:31:48,550
In other words, we--
Our three steps,

525
00:31:48,550 --> 00:31:52,550
which usually mash together
and we can't separate them

526
00:31:52,550 --> 00:31:55,270
and we have to deal with
the whole matrix at once.

527
00:31:55,270 --> 00:31:58,940
In this square case,
they do separate.

528
00:31:58,940 --> 00:32:02,000
And so that's worth noticing.

529
00:32:02,000 --> 00:32:09,280
It means that we
can solve backwards,

530
00:32:09,280 --> 00:32:13,490
we can solve these
three one at a time.

531
00:32:13,490 --> 00:32:15,940
The inverses can
be done separately.

532
00:32:15,940 --> 00:32:19,170
When A and A
transpose are square,

533
00:32:19,170 --> 00:32:23,050
then from this
equation I can find w.

534
00:32:23,050 --> 00:32:25,630
From knowing w I
can find e, just

535
00:32:25,630 --> 00:32:29,010
by inverting C. By
knowing e I can find u,

536
00:32:29,010 --> 00:32:32,030
just by inverting A.
You see the three steps?

537
00:32:32,030 --> 00:32:35,330
You could invert that, and then
you can invert the middle step,

538
00:32:35,330 --> 00:32:37,920
and you can invert
A. And you've got u.

539
00:32:37,920 --> 00:32:41,540
So the square case
is worth noticing.

540
00:32:41,540 --> 00:32:46,450
It's special enough
that in this case

541
00:32:46,450 --> 00:32:48,600
we would have an easy problem.

542
00:32:48,600 --> 00:32:54,790
And this case is called,
for trusses and mechanics,

543
00:32:54,790 --> 00:32:55,840
there's a name for this.

544
00:32:55,840 --> 00:32:57,870
And unfortunately
it's a little long.

545
00:32:57,870 --> 00:33:01,600
Statically, that's
not the key word,

546
00:33:01,600 --> 00:33:07,280
determinate is the key word.

547
00:33:07,280 --> 00:33:13,380
Statically determinate,
that goes with square.

548
00:33:13,380 --> 00:33:16,490
And you can guess what
the rectangular matrix A,

549
00:33:16,490 --> 00:33:19,740
what word would I use, what's
the opposite of determinate?

550
00:33:19,740 --> 00:33:23,420
It's got to be indeterminate.

551
00:33:23,420 --> 00:33:26,590
Rectangular A will
be indeterminate.

552
00:33:26,590 --> 00:33:30,000
And all that is
referring to is the fact

553
00:33:30,000 --> 00:33:40,980
that in the determinate case the
forces determine the stresses.

554
00:33:40,980 --> 00:33:44,830
You don't have to take
that, get all three

555
00:33:44,830 --> 00:33:48,280
together, mix them,
invert, go backwards.

556
00:33:48,280 --> 00:33:48,780
All that.

557
00:33:48,780 --> 00:33:53,450
You just can do them one at the
time in this determinate case.

558
00:33:53,450 --> 00:33:58,090
And now I guess I'd better
say, so here's the matrix case.

559
00:33:58,090 --> 00:34:02,710
But now in this chapter I always
have to do the continuous part.

560
00:34:02,710 --> 00:34:08,990
So let me just stay
with free-fixed,

561
00:34:08,990 --> 00:34:11,690
what is this balance equation?

562
00:34:11,690 --> 00:34:17,590
So this is my force balance.

563
00:34:17,590 --> 00:34:23,680
I didn't give it its moment
but its moment has come now.

564
00:34:23,680 --> 00:34:27,800
So the force balance
equation is -dw/dx,

565
00:34:27,800 --> 00:34:30,050
because A transpose is minus
a derivative, equal f(x).

566
00:34:30,050 --> 00:34:37,540
And my free boundary
condition, my free end,

567
00:34:37,540 --> 00:34:43,170
gave me w of what was it?

568
00:34:43,170 --> 00:34:46,150
The Neumann guy gave me w(0)=0.

569
00:34:46,150 --> 00:34:49,270


570
00:34:49,270 --> 00:34:52,430
And what's the point?

571
00:34:52,430 --> 00:34:54,030
Do you see what I'm saying?

572
00:34:54,030 --> 00:34:57,940
I'm saying that this free-fixed
is a beautiful example

573
00:34:57,940 --> 00:34:59,840
of determinate.

574
00:34:59,840 --> 00:35:03,110
Square matrix A
in the matrix case

575
00:35:03,110 --> 00:35:07,285
and the parallel in
the continuous case

576
00:35:07,285 --> 00:35:09,480
is-- I can solve that for w(x).

577
00:35:09,480 --> 00:35:12,040


578
00:35:12,040 --> 00:35:14,690
I can solve directly for w(x).

579
00:35:14,690 --> 00:35:18,650


580
00:35:18,650 --> 00:35:24,920
Without involving, you see
I didn't have to know c(x).

581
00:35:24,920 --> 00:35:26,740
I hadn't even got that far.

582
00:35:26,740 --> 00:35:28,890
I'm just going backwards now.

583
00:35:28,890 --> 00:35:35,200
I can solve this, just the
way I can invert that matrix.

584
00:35:35,200 --> 00:35:39,040
Inverting the matrix here is the
same as solving the equation.

585
00:35:39,040 --> 00:35:41,420
You see I have a first
order, first derivative?

586
00:35:41,420 --> 00:35:44,750
I mean, it's a trivial
equation, right?

587
00:35:44,750 --> 00:35:48,350
It's the equation you solved in
the final problem of the quiz,

588
00:35:48,350 --> 00:35:50,370
where an f was a delta function.

589
00:35:50,370 --> 00:35:55,670
It was simple because it was
a square, determinate problem

590
00:35:55,670 --> 00:35:58,890
with one condition on w.

591
00:35:58,890 --> 00:36:08,330
When both conditions are on u,
then it's not square any more.

592
00:36:08,330 --> 00:36:10,760
OK for that point?

593
00:36:10,760 --> 00:36:13,170
Determinate versus
indeterminate.

594
00:36:13,170 --> 00:36:14,020
OK.

595
00:36:14,020 --> 00:36:19,170
So that's sort of, and
I could do examples.

596
00:36:19,170 --> 00:36:20,750
Maybe I've asked
you on the homework

597
00:36:20,750 --> 00:36:23,080
to take a particular f(x).

598
00:36:23,080 --> 00:36:25,120
I hope it was a
free-fixed problem,

599
00:36:25,120 --> 00:36:28,960
if I was feeling good that
day, because free-fixed you'll

600
00:36:28,960 --> 00:36:31,430
be able to get w(x) right away.

601
00:36:31,430 --> 00:36:34,070
If it was fixed-fixed
then I apologize,

602
00:36:34,070 --> 00:36:39,130
it's going to take you a
little bit longer to get to w.

603
00:36:39,130 --> 00:36:41,490
To get to u.

604
00:36:41,490 --> 00:36:41,990
OK.

605
00:36:41,990 --> 00:36:47,920
But this, of course,
I just integrate.

606
00:36:47,920 --> 00:36:55,110
Inverting a difference matrix
is just integrating a function.

607
00:36:55,110 --> 00:36:56,040
Good.

608
00:36:56,040 --> 00:37:03,150
OK, so this lecture so
far was the transition

609
00:37:03,150 --> 00:37:09,380
from vectors and
matrices to functions

610
00:37:09,380 --> 00:37:11,130
and continuous problems.

611
00:37:11,130 --> 00:37:13,760
And then, of course
we're going to get deep

612
00:37:13,760 --> 00:37:18,260
into that because we got partial
differential equations ahead.

613
00:37:18,260 --> 00:37:21,810
But today let's stay
in one dimension

614
00:37:21,810 --> 00:37:23,220
and introduce finite elements.

615
00:37:23,220 --> 00:37:26,710
OK.

616
00:37:26,710 --> 00:37:34,000
Ready for finite elements,
so that's now a major step.

617
00:37:34,000 --> 00:37:38,040
Finite differences, maybe
I'll mention this, probably

618
00:37:38,040 --> 00:37:41,550
in this afternoon's review
session, where I'll just

619
00:37:41,550 --> 00:37:43,070
be open to homework problems.

620
00:37:43,070 --> 00:37:46,430
I'll say something more
about truss examples,

621
00:37:46,430 --> 00:37:50,070
and I might say something about
finite differences for this.

622
00:37:50,070 --> 00:37:55,620
But really, it's finite elements
that get introduced right now.

623
00:37:55,620 --> 00:37:57,790
So let me do that.

624
00:37:57,790 --> 00:38:02,690
Finite elements and
introducing them.

625
00:38:02,690 --> 00:38:06,290
OK.

626
00:38:06,290 --> 00:38:09,690
So the prep, the getting
ready for finite elements

627
00:38:09,690 --> 00:38:12,120
is to get hold of
something called

628
00:38:12,120 --> 00:38:23,040
the weak form of the equation.

629
00:38:23,040 --> 00:38:28,150
So that's going to be a
statement of the-- Finite

630
00:38:28,150 --> 00:38:30,740
elements aren't appearing yet.

631
00:38:30,740 --> 00:38:32,700
Matrices are not appearing yet.

632
00:38:32,700 --> 00:38:38,090
I'm talking about the
differential equation.

633
00:38:38,090 --> 00:38:41,340
But what do I mean
by this weak form?

634
00:38:41,340 --> 00:38:44,680
OK, let me just go
for it directly.

635
00:38:44,680 --> 00:38:46,310
You see the equation up there?

636
00:38:46,310 --> 00:38:47,420
Let me copy it.

637
00:38:47,420 --> 00:38:52,360
So here's the strong form.

638
00:38:52,360 --> 00:38:55,770
The strong form is, you would
say, the ordinary equation.

639
00:38:55,770 --> 00:39:00,280
Strong form is what
our equation is, minus

640
00:39:00,280 --> 00:39:11,730
d/dx of c(x) du/dx equal f(x).

641
00:39:11,730 --> 00:39:14,180
OK, that's the strong form.

642
00:39:14,180 --> 00:39:15,650
That's the equation.

643
00:39:15,650 --> 00:39:19,110
Now, how do I get
to the weak form?

644
00:39:19,110 --> 00:39:23,310
Let me just go to it directly
and then over the next days

645
00:39:23,310 --> 00:39:27,330
we'll see why it's so
natural and important.

646
00:39:27,330 --> 00:39:31,390
If I go for it directly,
what I do is this.

647
00:39:31,390 --> 00:39:35,350
I multiply both sides of
the equation by something

648
00:39:35,350 --> 00:39:37,340
I'll call a test function.

649
00:39:37,340 --> 00:39:39,370
And I'll try to
systematically use

650
00:39:39,370 --> 00:39:43,090
the letter v for the
test function. u will

651
00:39:43,090 --> 00:39:48,880
be the solution. v
isn't the solution,

652
00:39:48,880 --> 00:39:54,440
v is like any function that I
test this equation in this way.

653
00:39:54,440 --> 00:39:58,320
I'm just multiplying both sides
by the same thing, some v(x).

654
00:39:58,320 --> 00:40:00,570
Any v(x).

655
00:40:00,570 --> 00:40:03,360
We'll see if there are
any limitations, OK?

656
00:40:03,360 --> 00:40:14,450
And I integrate.

657
00:40:14,450 --> 00:40:15,980
OK.

658
00:40:15,980 --> 00:40:18,910
So you're going to
say nope, no problem.

659
00:40:18,910 --> 00:40:22,240
I integrate from zero to one.

660
00:40:22,240 --> 00:40:30,730
Alright, this would
be true for f.

661
00:40:30,730 --> 00:40:34,050
So now I'll erase
the word strong form,

662
00:40:34,050 --> 00:40:37,500
because the strong form
isn't on the board anymore.

663
00:40:37,500 --> 00:40:40,110
It's the weak form now
that we're looking at.

664
00:40:40,110 --> 00:40:44,250
And this is for any, and
I'll put "any" in quotes

665
00:40:44,250 --> 00:40:51,690
just because eventually I'll
say a little more about this.

666
00:40:51,690 --> 00:40:56,470
I'll write the
equation this way.

667
00:40:56,470 --> 00:41:04,830
And you might think, OK, if
this has to hold for every v(x),

668
00:41:04,830 --> 00:41:09,370
I could let v(x) be
concentrated in a little area.

669
00:41:09,370 --> 00:41:11,090
And this would
have to hold, then

670
00:41:11,090 --> 00:41:13,650
I could try another
v(x), concentrated

671
00:41:13,650 --> 00:41:16,130
around other points.

672
00:41:16,130 --> 00:41:23,360
You can maybe feel that if
this holds for every v(x),

673
00:41:23,360 --> 00:41:25,950
then I can get back
to the strong form.

674
00:41:25,950 --> 00:41:28,640
If this holds for every
v(x), then somehow

675
00:41:28,640 --> 00:41:31,310
that had better be
the same as that.

676
00:41:31,310 --> 00:41:37,160
Because if this was
f(x)+1 and this is f(x),

677
00:41:37,160 --> 00:41:41,650
then I wouldn't have
the equality any more.

678
00:41:41,650 --> 00:41:43,200
Should I just say that again?

679
00:41:43,200 --> 00:41:46,410
It's just like, at this
point it's just a feeling,

680
00:41:46,410 --> 00:41:51,370
that if this is
true for every v(x),

681
00:41:51,370 --> 00:41:55,360
then that part had
better equal that part.

682
00:41:55,360 --> 00:41:57,500
That'll be my way back
to the strong form.

683
00:41:57,500 --> 00:41:59,610
It's a little bit
like climbing a hill.

684
00:41:59,610 --> 00:42:03,690
Going downhill was easy, I just
multiplied by v and integrated.

685
00:42:03,690 --> 00:42:05,500
Nobody objected to that.

686
00:42:05,500 --> 00:42:08,270
I'm saying I'll be
able to get back

687
00:42:08,270 --> 00:42:11,740
to the strong form
with a little patience.

688
00:42:11,740 --> 00:42:14,160
But I like the weak form.

689
00:42:14,160 --> 00:42:15,700
That's the whole point.

690
00:42:15,700 --> 00:42:19,120
You've got to begin
to like the weak form.

691
00:42:19,120 --> 00:42:22,120
If you begin to take
it in and think OK.

692
00:42:22,120 --> 00:42:25,270
Now, why do I like it?

693
00:42:25,270 --> 00:42:28,020
What am I going to
do to that left side?

694
00:42:28,020 --> 00:42:30,100
The right side's cool, right?

695
00:42:30,100 --> 00:42:31,530
It looks good.

696
00:42:31,530 --> 00:42:35,520
Left side does not
look good to me.

697
00:42:35,520 --> 00:42:38,370
When you see something like
that, what do you think?

698
00:42:38,370 --> 00:42:42,310
Today's lecture has
already said what to think.

699
00:42:42,310 --> 00:42:46,160
What should I do to
make that look better?

700
00:42:46,160 --> 00:42:48,410
I should, yep.

701
00:42:48,410 --> 00:42:50,540
Integrate by parts.

702
00:42:50,540 --> 00:42:52,360
If I integrate by
parts, you see what

703
00:42:52,360 --> 00:42:56,870
I don't like about it as it is,
is two derivatives are hitting

704
00:42:56,870 --> 00:43:01,180
u, and v is by itself.

705
00:43:01,180 --> 00:43:05,370
And I want it to be symmetric.

706
00:43:05,370 --> 00:43:07,340
I'm going to integrate
this by parts,

707
00:43:07,340 --> 00:43:10,090
this is minus the
derivative of something.

708
00:43:10,090 --> 00:43:15,900
Times v. And when I integrate
by parts, I'm going to have,

709
00:43:15,900 --> 00:43:17,450
it'll be an integral.

710
00:43:17,450 --> 00:43:19,980
And can you integrate
by parts now?

711
00:43:19,980 --> 00:43:24,840
I mean, you probably haven't
thought about integration

712
00:43:24,840 --> 00:43:26,120
by parts for a while.

713
00:43:26,120 --> 00:43:29,490
Just think of it as taking
the derivative off of this,

714
00:43:29,490 --> 00:43:32,620
so it leaves that by itself.

715
00:43:32,620 --> 00:43:41,000
And putting it onto v, so
it's dv/dx, and remembering

716
00:43:41,000 --> 00:43:43,250
the minus sign, but
we have a minus sign

717
00:43:43,250 --> 00:43:45,480
so now it's coming up plus.

718
00:43:45,480 --> 00:43:46,580
That's the weak form.

719
00:43:46,580 --> 00:43:50,870
Can I put a circle
around the weak form?

720
00:43:50,870 --> 00:43:53,670
Well, that wasn't
exactly a circle.

721
00:43:53,670 --> 00:43:54,360
OK.

722
00:43:54,360 --> 00:43:57,880
But that's the weak form.

723
00:43:57,880 --> 00:44:01,310
For every v, this
is-- I could give you

724
00:44:01,310 --> 00:44:10,040
a physical interpretation but
I won't do it just this minute.

725
00:44:10,040 --> 00:44:16,210
This is going to hold for
any v. That's the weak form.

726
00:44:16,210 --> 00:44:17,600
OK.

727
00:44:17,600 --> 00:44:18,650
Good.

728
00:44:18,650 --> 00:44:25,000
Now, why did I want to do that?

729
00:44:25,000 --> 00:44:29,580
The person who reminds me
about boundary conditions

730
00:44:29,580 --> 00:44:31,380
should remind me again.

731
00:44:31,380 --> 00:44:33,610
That when I did this
integration by parts,

732
00:44:33,610 --> 00:44:38,480
there should have been also--
What's the integrated part now,

733
00:44:38,480 --> 00:44:43,550
that has to be evaluated
at zero and one?

734
00:44:43,550 --> 00:44:47,160
This c, so it's that
times that, right?

735
00:44:47,160 --> 00:44:53,110
It's that c(x) du/dx times v(x).

736
00:44:53,110 --> 00:44:56,920


737
00:44:56,920 --> 00:44:57,849
Maybe minus.

738
00:44:57,849 --> 00:44:58,640
Yeah, you're right.

739
00:44:58,640 --> 00:44:59,960
Minus.

740
00:44:59,960 --> 00:45:01,070
Good.

741
00:45:01,070 --> 00:45:03,870
What do I want this to come out?

742
00:45:03,870 --> 00:45:05,550
Zero, of course.

743
00:45:05,550 --> 00:45:08,010
I don't want to think
about this anymore.

744
00:45:08,010 --> 00:45:11,750
Alright, so now I'm doing
this free-fixed problem still.

745
00:45:11,750 --> 00:45:16,710
So what's the deal about
the free-fixed f problem?

746
00:45:16,710 --> 00:45:22,830
Well, let's see.

747
00:45:22,830 --> 00:45:26,680
OK, I got the two ends and
I want them to be zero.

748
00:45:26,680 --> 00:45:36,840
OK, now at the free
end, I'm not controlling

749
00:45:36,840 --> 00:45:40,130
v. I wasn't controlling
u and I'm not

750
00:45:40,130 --> 00:45:43,130
going to be controlling
its friend v.

751
00:45:43,130 --> 00:45:45,880
So this had to be zero.

752
00:45:45,880 --> 00:45:52,210
So this part will be
zero at the free end.

753
00:45:52,210 --> 00:45:56,220
That boundary condition has
just appeared again naturally.

754
00:45:56,220 --> 00:45:59,250
I had to have it because
I had no control over v.

755
00:45:59,250 --> 00:46:01,830
And what about at the fixed end.

756
00:46:01,830 --> 00:46:06,930
At the fixed end,
which-- Is that?

757
00:46:06,930 --> 00:46:07,650
At the free end.

758
00:46:07,650 --> 00:46:11,320
Now, what's up at the fixed end?

759
00:46:11,320 --> 00:46:13,960
What was the fixed end?

760
00:46:13,960 --> 00:46:17,230
That's where u was zero.

761
00:46:17,230 --> 00:46:20,110
I'm going to make v also zero.

762
00:46:20,110 --> 00:46:22,910
So there's, when
I said any v(x),

763
00:46:22,910 --> 00:46:32,230
I better put in with v=0
at the Dirichlet point,

764
00:46:32,230 --> 00:46:37,070
at fixed point, at fixed end.

765
00:46:37,070 --> 00:46:40,050
I need that.

766
00:46:40,050 --> 00:46:44,450
I need to know that v
is zero at that end.

767
00:46:44,450 --> 00:46:46,730
I had u=0.

768
00:46:46,730 --> 00:46:50,350
Here's why I'm fine.

769
00:46:50,350 --> 00:46:55,000
So I'm saying that any time
I have a Dirichlet condition,

770
00:46:55,000 --> 00:46:59,030
a fixed condition that
tells me u, I think of v

771
00:46:59,030 --> 00:47:02,500
and you'll begin to think of
v as a little movement away

772
00:47:02,500 --> 00:47:13,170
from u. u is the solution.

773
00:47:13,170 --> 00:47:16,350
Now, remind me,
this was free-fixed.

774
00:47:16,350 --> 00:47:21,030
So the u might have been
something like this.

775
00:47:21,030 --> 00:47:22,600
I just draw that.

776
00:47:22,600 --> 00:47:24,140
That's my u.

777
00:47:24,140 --> 00:47:27,370
This guy was fixed, right?

778
00:47:27,370 --> 00:47:29,220
By u.

779
00:47:29,220 --> 00:47:34,050
Now, I'm thinking of
v's as, the letter

780
00:47:34,050 --> 00:47:36,070
v is very fortunate
because it stands

781
00:47:36,070 --> 00:47:38,310
for virtual displacement.

782
00:47:38,310 --> 00:47:41,480
A virtual displacement is
a little displacement away

783
00:47:41,480 --> 00:47:46,940
from u, but it has to satisfy
the zero-- the fixed condition

784
00:47:46,940 --> 00:47:48,210
that u satisfied.

785
00:47:48,210 --> 00:47:50,660
In other words, the
little virtual v

786
00:47:50,660 --> 00:47:53,340
can't move away from zero.

787
00:47:53,340 --> 00:48:02,680
So I get this term is
zero at the fixed end.

788
00:48:02,680 --> 00:48:07,060
OK.

789
00:48:07,060 --> 00:48:10,640
That's the little
five minute time-out

790
00:48:10,640 --> 00:48:13,040
to check the boundary
condition part.

791
00:48:13,040 --> 00:48:16,090
The net result is
that that term's gone

792
00:48:16,090 --> 00:48:21,560
and I've got the weak
form as I wanted.

793
00:48:21,560 --> 00:48:26,950
OK, three minutes to
start to tell you how to

794
00:48:26,950 --> 00:48:30,400
use the weak form.

795
00:48:30,400 --> 00:48:39,620
So this is called
Galerkin's method.

796
00:48:39,620 --> 00:48:50,250
And it starts with
the weak form.

797
00:48:50,250 --> 00:48:51,700
So he's Russian.

798
00:48:51,700 --> 00:48:54,170
Russia gets into
the picture now.

799
00:48:54,170 --> 00:48:56,770
We had France and Germany
with the boundary conditions,

800
00:48:56,770 --> 00:49:00,480
now we've got Russia with
this fundamental principle

801
00:49:00,480 --> 00:49:04,320
of how to turn a
continuous problem

802
00:49:04,320 --> 00:49:06,730
into a discrete problem.

803
00:49:06,730 --> 00:49:09,930
That's what
Galerkin's idea does.

804
00:49:09,930 --> 00:49:13,900
Instead of a function unknown
I want to have n unknowns.

805
00:49:13,900 --> 00:49:18,210
I want to get a discrete
equation which will eventually

806
00:49:18,210 --> 00:49:26,440
be KU=F. So I'm going to
get to an equation KU=F,

807
00:49:26,440 --> 00:49:29,320
but not by finite
difference, right?

808
00:49:29,320 --> 00:49:31,330
I could, but I'm not.

809
00:49:31,330 --> 00:49:36,130
I'm doing it this weak,
Galerkin, finite element way.

810
00:49:36,130 --> 00:49:42,780
OK, so if I tell you the
Galerkin idea then next time we

811
00:49:42,780 --> 00:49:46,370
bring in, we have libraries
of finite elements.

812
00:49:46,370 --> 00:49:48,850
But you have to get
the principle straight.

813
00:49:48,850 --> 00:49:51,240
So it's Galerkin's idea.

814
00:49:51,240 --> 00:50:01,220
Galerkin's idea was was
choose trial functions.

815
00:50:01,220 --> 00:50:10,370
Let me call them call them u_1?

816
00:50:10,370 --> 00:50:16,360
No, I won't call them--
Have to get the names right.

817
00:50:16,360 --> 00:50:21,550
Phi.

818
00:50:21,550 --> 00:50:23,280
OK, the Greeks get a shot.

819
00:50:23,280 --> 00:50:24,000
OK.

820
00:50:24,000 --> 00:50:31,050
Trial functions,
phi_1(x) up to phi_n(x).

821
00:50:31,050 --> 00:50:35,840
OK, so that's a choice you make.

822
00:50:35,840 --> 00:50:37,790
And we have a free choice.

823
00:50:37,790 --> 00:50:40,710
And it's a fundamental choice
for all of applied math here.

824
00:50:40,710 --> 00:50:43,430
You choose some functions,
and if you choose them well

825
00:50:43,430 --> 00:50:45,560
you get a great method,
if you choose them badly

826
00:50:45,560 --> 00:50:47,010
you got a lousy method.

827
00:50:47,010 --> 00:50:49,200
OK, so you choose
trial functions,

828
00:50:49,200 --> 00:50:51,120
and now what's the
idea going to be?

829
00:50:51,120 --> 00:50:58,270
Your approximate U,
approximate solution

830
00:50:58,270 --> 00:51:03,410
will be some
combination of this.

831
00:51:03,410 --> 00:51:09,160
So combinations of those, let
me call the coefficients U's,

832
00:51:09,160 --> 00:51:10,770
because those are the unknowns.

833
00:51:10,770 --> 00:51:12,590
Plus U_n*phi_n.

834
00:51:12,590 --> 00:51:15,360


835
00:51:15,360 --> 00:51:16,910
So those are the unknowns.

836
00:51:16,910 --> 00:51:23,820
The n unknowns.

837
00:51:23,820 --> 00:51:28,100
I'll even remove
that for the moment.

838
00:51:28,100 --> 00:51:31,740
You see, these are
functions of x.

839
00:51:31,740 --> 00:51:34,030
And these are numbers.

840
00:51:34,030 --> 00:51:38,330
So our unknown, our
n unknown numbers

841
00:51:38,330 --> 00:51:41,720
are the coefficients to be
decided of the functions

842
00:51:41,720 --> 00:51:43,230
we chose.

843
00:51:43,230 --> 00:51:47,260
OK, now I need n equations.

844
00:51:47,260 --> 00:51:50,500
I've got n unknowns now,
they're the unknown coefficients

845
00:51:50,500 --> 00:51:52,180
of these functions.

846
00:51:52,180 --> 00:51:58,480
I need an equation so I get
n equations by choose test

847
00:51:58,480 --> 00:52:08,780
functions, V_1, V_2, up to V_n.

848
00:52:08,780 --> 00:52:11,280
Each V will give me an equation.

849
00:52:11,280 --> 00:52:14,640
So I'll have n equations at
the end, I have n unknowns,

850
00:52:14,640 --> 00:52:16,390
I'll have a square matrix.

851
00:52:16,390 --> 00:52:20,390
And that'll be a linear system.

852
00:52:20,390 --> 00:52:22,630
I'll get to KU=F.

853
00:52:22,630 --> 00:52:24,740
But do you see how
I'm getting there?

854
00:52:24,740 --> 00:52:28,710
I'm getting there by
using the weak form,

855
00:52:28,710 --> 00:52:32,180
by using Galerkin's idea
of picking some trial

856
00:52:32,180 --> 00:52:35,020
functions, and some
test functions,

857
00:52:35,020 --> 00:52:37,500
and putting them
into the weak form.

858
00:52:37,500 --> 00:52:41,700
So Galerkin's idea is,
take these functions

859
00:52:41,700 --> 00:52:43,160
and these functions.

860
00:52:43,160 --> 00:52:47,030
And apply the weak form
just to those guys.

861
00:52:47,030 --> 00:52:50,420
Not to, the real weak form,
the continuous weak form,

862
00:52:50,420 --> 00:52:54,910
was for a whole lot of
V. We'll get n equations

863
00:52:54,910 --> 00:53:01,570
by picking n V's, and we'll get
n unknowns by picking n phis.

864
00:53:01,570 --> 00:53:04,420
So this method, this idea,
was a hundred years older

865
00:53:04,420 --> 00:53:06,340
than finite elements.

866
00:53:06,340 --> 00:53:11,210
The finite element idea
was a particular choice

867
00:53:11,210 --> 00:53:16,250
of these guys, a particular
choice of the phis

868
00:53:16,250 --> 00:53:21,900
and the V's as
simple polynomials.

869
00:53:21,900 --> 00:53:24,520
And you might think
well, why didn't Galerkin

870
00:53:24,520 --> 00:53:26,760
try those first, maybe he did.

871
00:53:26,760 --> 00:53:34,650
But the key is that now,
with the computing power

872
00:53:34,650 --> 00:53:37,550
we now have compared
to Galerkin,

873
00:53:37,550 --> 00:53:40,050
we can choose
thousands of functions.

874
00:53:40,050 --> 00:53:41,770
If we keep them simple.

875
00:53:41,770 --> 00:53:46,330
So that's really what the
finite element brought,

876
00:53:46,330 --> 00:53:49,440
finite element brought
is, keep the functions

877
00:53:49,440 --> 00:53:53,440
as simple polynomials
and take many of them.

878
00:53:53,440 --> 00:53:56,610
Where Galerkin, who
didn't have MATLAB,

879
00:53:56,610 --> 00:54:00,580
he probably didn't even
have a desk computer,

880
00:54:00,580 --> 00:54:04,130
he used pencil and paper,
he took one function.

881
00:54:04,130 --> 00:54:05,620
Or maybe two.

882
00:54:05,620 --> 00:54:08,820
I mean, that took him a day.

883
00:54:08,820 --> 00:54:12,740
But we take thousands of
functions, simple functions,

884
00:54:12,740 --> 00:54:17,840
and we'll see on Friday the
steps that get us to KU=F.

885
00:54:17,840 --> 00:54:21,280
So this is the prep
for finite elements.