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PROFESSOR STRANG:
Okay, so this is I

10
00:00:24,250 --> 00:00:27,310
could say delta function day.

11
00:00:27,310 --> 00:00:30,560
Break from linear
algebra mostly.

12
00:00:30,560 --> 00:00:34,250
So we're looking on another
type of right-hand side.

13
00:00:34,250 --> 00:00:36,720
Before in the
differential equation

14
00:00:36,720 --> 00:00:39,720
and in the difference equation.

15
00:00:39,720 --> 00:00:44,750
So the right-hand sides up
to now, the one we looked at

16
00:00:44,750 --> 00:00:51,580
was a uniform constant load
second derivative equal one.

17
00:00:51,580 --> 00:00:55,320
Now a point load.

18
00:00:55,320 --> 00:00:58,300
Well in a way, we're now solving
a whole bunch of problems

19
00:00:58,300 --> 00:01:02,470
because the point load can
be in different places.

20
00:01:02,470 --> 00:01:05,780
So instead of solving
one problem with one

21
00:01:05,780 --> 00:01:10,790
on the right-hand side, we're
solving with a delta function.

22
00:01:10,790 --> 00:01:15,010
Now a delta function is, you
probably have seen and heard

23
00:01:15,010 --> 00:01:19,220
the words and seen the symbol,
but maybe not done much

24
00:01:19,220 --> 00:01:22,090
with a delta function.

25
00:01:22,090 --> 00:01:25,620
It takes a little practice
but it's really worth it.

26
00:01:25,620 --> 00:01:31,490
It's a great model of maybe what
can't quite happen physically,

27
00:01:31,490 --> 00:01:36,470
to have a load acting exactly
at a point and nowhere else.

28
00:01:36,470 --> 00:01:40,040
So the delta function
is, I drew it's picture,

29
00:01:40,040 --> 00:01:46,880
the delta function is zero, this
is delta of x is zero except

30
00:01:46,880 --> 00:01:50,600
at that one point,
the origin, x=0,

31
00:01:50,600 --> 00:01:53,910
and then all along
back to zero again.

32
00:01:53,910 --> 00:02:00,300
So nothing's happening, no
load except at that one point.

33
00:02:00,300 --> 00:02:05,510
And let me just, so
there's no hesitation

34
00:02:05,510 --> 00:02:13,840
in when I change from x to x-a,
what does that do to a graph?

35
00:02:13,840 --> 00:02:21,440
If I have a function of x and
I instead shift the function

36
00:02:21,440 --> 00:02:27,600
to f(x-a), I shift x to
x-a, well in this case,

37
00:02:27,600 --> 00:02:30,700
and in all cases, it will
just shift the graph.

38
00:02:30,700 --> 00:02:37,150
So if I drew a picture
of delta, of x-a,

39
00:02:37,150 --> 00:02:41,360
the load now would
happen when this is zero,

40
00:02:41,360 --> 00:02:45,660
because it's delta at
zero is the impulse,

41
00:02:45,660 --> 00:02:48,490
and now this is zero at x=a.

42
00:02:48,490 --> 00:02:52,440
In other words, the load
moved to the point a.

43
00:02:52,440 --> 00:02:59,080
So there is the shifting
load, but the load

44
00:02:59,080 --> 00:03:05,030
could fall anywhere
between zero and one.

45
00:03:05,030 --> 00:03:07,940
So delta of x, the load
actually falls at zero.

46
00:03:07,940 --> 00:03:10,750
Well we don't quite want
that load at the boundary.

47
00:03:10,750 --> 00:03:14,900
So let's think of the point
a, the load point as somewhere

48
00:03:14,900 --> 00:03:17,410
between zero and one.

49
00:03:17,410 --> 00:03:22,610
Can I just take a little
time to recall the main fact

50
00:03:22,610 --> 00:03:24,580
about delta functions?

51
00:03:24,580 --> 00:03:29,960
When I say recall, it could
very well be new to you.

52
00:03:29,960 --> 00:03:32,990
So that's what the
delta function--

53
00:03:32,990 --> 00:03:36,110
that's my best graph
of the delta function.

54
00:03:36,110 --> 00:03:39,120
But of course I'm, in
using the word function,

55
00:03:39,120 --> 00:03:43,470
I'm kind of breaking the rules
because no function-- I mean

56
00:03:43,470 --> 00:03:46,200
the function is, functions
can be zero there,

57
00:03:46,200 --> 00:03:48,060
can be zero there,
but they're not

58
00:03:48,060 --> 00:03:52,810
supposed to be infinite at
a single point in between,

59
00:03:52,810 --> 00:03:54,300
but this one is.

60
00:03:54,300 --> 00:04:00,170
Let me go back to delta of
x to match these figures.

61
00:04:00,170 --> 00:04:04,450
Of course, they would also
just shift along by a.

62
00:04:04,450 --> 00:04:09,490
Maybe no harm in that.

63
00:04:09,490 --> 00:04:16,160
Sorry, I'll stay there and
now I want to integrate.

64
00:04:16,160 --> 00:04:22,260
And that's when a delta
function comes into its own.

65
00:04:22,260 --> 00:04:26,330
Its value of infinity is
a little bit uncertain.

66
00:04:26,330 --> 00:04:27,670
What does that mean?

67
00:04:27,670 --> 00:04:31,270
But when we integrate
it, what's the key fact

68
00:04:31,270 --> 00:04:32,790
about delta function?

69
00:04:32,790 --> 00:04:37,330
That the integral of
a delta function from,

70
00:04:37,330 --> 00:04:39,470
let's say, let's
integrate the whole thing,

71
00:04:39,470 --> 00:04:42,310
we can safely start
way at the far left

72
00:04:42,310 --> 00:04:44,340
and go away to the far
right because it's zero

73
00:04:44,340 --> 00:04:49,300
all the time there except
at one point, and you know.

74
00:04:49,300 --> 00:04:53,030
So what's the area
under that spike?

75
00:04:53,030 --> 00:04:54,939
It is one.

76
00:04:54,939 --> 00:04:55,480
That's right.

77
00:04:55,480 --> 00:04:58,380
So that's the fact, the
sort of central fact

78
00:04:58,380 --> 00:05:00,190
about a delta function.

79
00:05:00,190 --> 00:05:02,030
That the area is one.

80
00:05:02,030 --> 00:05:04,710
Oh well, let me, while
I'm really writing down

81
00:05:04,710 --> 00:05:14,010
the central fact, let me write
it more specifically, more

82
00:05:14,010 --> 00:05:14,830
generally.

83
00:05:14,830 --> 00:05:20,040
Suppose I integrate, and this
is delta functions now really

84
00:05:20,040 --> 00:05:24,640
showing up, if I integrate a
delta function against some,

85
00:05:24,640 --> 00:05:28,200
times some nice function.

86
00:05:28,200 --> 00:05:31,630
Now have you ever
thought about that?

87
00:05:31,630 --> 00:05:35,990
What would be the answer if I
integrate the delta function

88
00:05:35,990 --> 00:05:38,540
against some nice function?

89
00:05:38,540 --> 00:05:43,980
So I'm still getting zero from
this term all the way along

90
00:05:43,980 --> 00:05:47,510
until I hit the spike and
then after it goes back

91
00:05:47,510 --> 00:05:48,680
to zero again.

92
00:05:48,680 --> 00:05:54,030
So, whatever, it's gotta
be at the spike, at x=0,

93
00:05:54,030 --> 00:05:57,790
because I put the spike
here at zero, the impulse.

94
00:05:57,790 --> 00:06:02,980
So what do you think's
the answer for that one?

95
00:06:02,980 --> 00:06:04,790
Yeah, It's the function.

96
00:06:04,790 --> 00:06:10,000
So, yes, tell me again
and I'll write it down. g,

97
00:06:10,000 --> 00:06:12,240
it's a value of this function g.

98
00:06:12,240 --> 00:06:16,260
We don't care what it is to the
left and to the right at zero

99
00:06:16,260 --> 00:06:20,390
because it's really at zero
that this thing turns on

100
00:06:20,390 --> 00:06:24,690
and its value at
that point is just--

101
00:06:24,690 --> 00:06:31,260
gives us the amplitude of
the impulse, which is g(0).

102
00:06:31,260 --> 00:06:34,400
And of course if g is the
constant function one,

103
00:06:34,400 --> 00:06:36,310
I'm back to that formula.

104
00:06:36,310 --> 00:06:39,850
But this is maybe the
thing to watch for.

105
00:06:39,850 --> 00:06:43,100
Actually there's a lot built
into that little thing.

106
00:06:43,100 --> 00:06:46,580
We'll come back to that.

107
00:06:46,580 --> 00:06:51,580
So that's delta
functions integrated

108
00:06:51,580 --> 00:06:53,590
and now here are some pictures.

109
00:06:53,590 --> 00:06:57,640
These are the good pictures.

110
00:06:57,640 --> 00:07:03,540
So here's one integral
of the delta function.

111
00:07:03,540 --> 00:07:06,220
It's a step function.

112
00:07:06,220 --> 00:07:08,760
And the step of course
will occur at the point

113
00:07:08,760 --> 00:07:14,290
a if the integral of the
delta function at a point a

114
00:07:14,290 --> 00:07:17,220
will be the step function.

115
00:07:17,220 --> 00:07:20,060
Where the action happens.

116
00:07:20,060 --> 00:07:23,620
The jump happens, I could
call it a jump function.

117
00:07:23,620 --> 00:07:25,880
At that point a.

118
00:07:25,880 --> 00:07:28,840
Because, just for
the reason we said.

119
00:07:28,840 --> 00:07:32,480
That if we integrate,
the integral is zero.

120
00:07:32,480 --> 00:07:35,940
And then as soon as our
integral passes this point,

121
00:07:35,940 --> 00:07:40,410
so this is integral of the,
this is-- I integrated.

122
00:07:40,410 --> 00:07:44,600
I integrate to get
to this picture.

123
00:07:44,600 --> 00:07:48,930
I start with that delta function
and I integrate and it suddenly

124
00:07:48,930 --> 00:07:52,190
jumps to one as soon
as the integral goes

125
00:07:52,190 --> 00:07:56,030
past the spike, the impulse.

126
00:07:56,030 --> 00:07:57,390
So a step function.

127
00:07:57,390 --> 00:07:59,710
Very handy function,
step function.

128
00:07:59,710 --> 00:08:02,610
Sometimes called a
Heaviside function

129
00:08:02,610 --> 00:08:06,910
named after the guy who--
the electrical engineer

130
00:08:06,910 --> 00:08:13,190
I think who first sort of work
out the rules for using these.

131
00:08:13,190 --> 00:08:18,380
Let's integrate one more time
because we have second order

132
00:08:18,380 --> 00:08:22,810
equations, second derivatives,
so we better integrate twice

133
00:08:22,810 --> 00:08:26,160
to see what sort
of answer we get.

134
00:08:26,160 --> 00:08:28,200
Now integrate the step function.

135
00:08:28,200 --> 00:08:31,100
So again, the integral is
zero all the way to the left,

136
00:08:31,100 --> 00:08:34,870
so I'm still getting zero,
but now beyond this point

137
00:08:34,870 --> 00:08:36,810
I'm integrating one.

138
00:08:36,810 --> 00:08:40,810
And the integral of one is x.

139
00:08:40,810 --> 00:08:44,410
So now that I would
call a ramp function.

140
00:08:44,410 --> 00:08:48,760
That's a nice short word
for this valuable function.

141
00:08:48,760 --> 00:08:56,060
A ramp function is the function
that's zero and then x.

142
00:08:56,060 --> 00:09:00,600
So tell me about
that ramp function.

143
00:09:00,600 --> 00:09:02,660
Just think about it.

144
00:09:02,660 --> 00:09:08,070
What happens to its
derivative at the point a?

145
00:09:08,070 --> 00:09:11,700
As I run along and I
hit this key point,

146
00:09:11,700 --> 00:09:18,230
what happens to the
derivative of the ramp?

147
00:09:18,230 --> 00:09:20,870
What does the derivative do?

148
00:09:20,870 --> 00:09:22,820
Focus on that ramp now.

149
00:09:22,820 --> 00:09:27,300
What does the derivative
do at that point?

150
00:09:27,300 --> 00:09:28,620
It jumps.

151
00:09:28,620 --> 00:09:29,800
The derivative jumps.

152
00:09:29,800 --> 00:09:33,250
The slope is the derivative,
the slope jumps from zero

153
00:09:33,250 --> 00:09:34,590
and here the slope is one.

154
00:09:34,590 --> 00:09:36,620
And of course that's
what that's telling us.

155
00:09:36,620 --> 00:09:38,280
Here's the picture
of the derivative.

156
00:09:38,280 --> 00:09:41,660
What does the second
derivative do?

157
00:09:41,660 --> 00:09:46,600
Well, since I
integrated twice I guess

158
00:09:46,600 --> 00:09:49,100
going back two
steps I'll find out

159
00:09:49,100 --> 00:09:51,200
what the second derivative is.

160
00:09:51,200 --> 00:09:55,960
So the first derivative
takes a jump.

161
00:09:55,960 --> 00:09:59,990
The second derivative is
the derivative of that jump,

162
00:09:59,990 --> 00:10:01,660
so it's got the impulse.

163
00:10:01,660 --> 00:10:04,720
So the second derivative,
it's a straight line here,

164
00:10:04,720 --> 00:10:06,780
second derivative
a straight line.

165
00:10:06,780 --> 00:10:09,380
This is straight line
here, second derivative

166
00:10:09,380 --> 00:10:13,130
of a straight line
is a straight line.

167
00:10:13,130 --> 00:10:17,610
But at that point the
first derivative jumps,

168
00:10:17,610 --> 00:10:21,010
the second derivative
has that delta function.

169
00:10:21,010 --> 00:10:24,530
In other words,
that's that stuff.

170
00:10:24,530 --> 00:10:30,360
If I keep integrating -- and
I don't need higher integrals

171
00:10:30,360 --> 00:10:35,250
in today's lecture -- another
integral would be what?

172
00:10:35,250 --> 00:10:37,600
If I integrate this
function, then it's

173
00:10:37,600 --> 00:10:38,880
running along the zero.

174
00:10:38,880 --> 00:10:41,380
What's the integral of this?

175
00:10:41,380 --> 00:10:44,680
Doesn't quite turn that steeply.

176
00:10:44,680 --> 00:10:48,070
What's that curve there?

177
00:10:48,070 --> 00:10:49,925
If I've integrated the ramp.

178
00:10:49,925 --> 00:10:50,800
Here is the integral.

179
00:10:50,800 --> 00:10:55,340
First, the next step up, the
integral of the ramp would be?

180
00:10:55,340 --> 00:10:59,630
It'll be x squared, yeah,
it'll be a parabola.

181
00:10:59,630 --> 00:11:02,480
x squared over two,
the integral of that.

182
00:11:02,480 --> 00:11:05,470
And now what do I get
when I integrate this one?

183
00:11:05,470 --> 00:11:07,950
I get something very important.

184
00:11:07,950 --> 00:11:13,200
Not important today, but
important in a few weeks.

185
00:11:13,200 --> 00:11:15,920
And very useful in computing.

186
00:11:15,920 --> 00:11:19,100
These have turned out to
be just the right thing.

187
00:11:19,100 --> 00:11:21,140
So again, I'm integrating that.

188
00:11:21,140 --> 00:11:24,700
Everybody can tell
me, what is that?

189
00:11:24,700 --> 00:11:27,460
What's that curve now?

190
00:11:27,460 --> 00:11:29,410
It's the next
integral of course.

191
00:11:29,410 --> 00:11:35,250
The area under that will
be x cubed over six.

192
00:11:35,250 --> 00:11:37,060
So now that is a function.

193
00:11:37,060 --> 00:11:39,180
Yeah, it's worth maybe
just for practice.

194
00:11:39,180 --> 00:11:42,110
What's the deal
with that function?

195
00:11:42,110 --> 00:11:44,870
That's pretty smooth function.

196
00:11:44,870 --> 00:11:50,370
Because it certainly passes--
right, it meets at that point.

197
00:11:50,370 --> 00:11:54,560
The first derivative
meets at that point.

198
00:11:54,560 --> 00:11:57,940
The second derivative
meets at that point.

199
00:11:57,940 --> 00:12:00,480
The third derivative does what?

200
00:12:00,480 --> 00:12:02,150
Of this line.

201
00:12:02,150 --> 00:12:05,020
The third derivative, take
three steps back down the line

202
00:12:05,020 --> 00:12:08,481
and you see that the
third derivative jumps.

203
00:12:08,481 --> 00:12:08,980
Right?

204
00:12:08,980 --> 00:12:12,120
The third derivative of that
is the third derivative,

205
00:12:12,120 --> 00:12:17,570
would be, shall I-- for C,
for cubic spline or something,

206
00:12:17,570 --> 00:12:21,280
the third derivative
will be zero there.

207
00:12:21,280 --> 00:12:24,700
And the third derivative
of that is exactly

208
00:12:24,700 --> 00:12:29,860
like back to that, back to
that, back to one is one.

209
00:12:29,860 --> 00:12:34,750
So the third derivative. so
the cubic spline's so smooth

210
00:12:34,750 --> 00:12:38,640
your eye doesn't see that.

211
00:12:38,640 --> 00:12:42,390
They're very useful for
drawing many, many purposes.

212
00:12:42,390 --> 00:12:45,550
CAD programs would use
such things constantly

213
00:12:45,550 --> 00:12:50,460
because they're convenient,
they have nice pieces

214
00:12:50,460 --> 00:12:53,070
that you can fit together
and they fit together

215
00:12:53,070 --> 00:12:55,180
very smoothly.

216
00:12:55,180 --> 00:12:59,930
But they really are
two separate functions.

217
00:12:59,930 --> 00:13:02,270
So that's up to cubic spline.

218
00:13:02,270 --> 00:13:04,770
But our focus is--

219
00:13:04,770 --> 00:13:09,550
These would solve, what
equations would those solve?

220
00:13:09,550 --> 00:13:13,290
Well, that takes how many
derivatives to get to a delta?

221
00:13:13,290 --> 00:13:17,820
So what would be the equation?

222
00:13:17,820 --> 00:13:22,630
What would be the
right-hand side?

223
00:13:22,630 --> 00:13:25,520
Let me take the
fourth derivative.

224
00:13:25,520 --> 00:13:27,530
I'll just ask the
question that way.

225
00:13:27,530 --> 00:13:29,270
What would be the
fourth derivative

226
00:13:29,270 --> 00:13:31,820
of that cubic spline?

227
00:13:31,820 --> 00:13:32,800
A delta, right?

228
00:13:32,800 --> 00:13:34,740
Four steps back.

229
00:13:34,740 --> 00:13:40,620
So what is, physically,
what are we seeing here?

230
00:13:40,620 --> 00:13:44,100
Do you recognize what
kind-- If I ask now

231
00:13:44,100 --> 00:13:47,230
people in mechanics,
When will we

232
00:13:47,230 --> 00:13:49,700
meet a fourth order equation?

233
00:13:49,700 --> 00:13:53,560
Fourth derivative equals a load.

234
00:13:53,560 --> 00:13:58,110
Anybody know the
physical situation

235
00:13:58,110 --> 00:14:02,410
where fourth derivative?

236
00:14:02,410 --> 00:14:03,590
Beams, yeah.

237
00:14:03,590 --> 00:14:05,790
It's the equation for a beam.

238
00:14:05,790 --> 00:14:10,710
A beam has-- The
bending of a beam.

239
00:14:10,710 --> 00:14:11,530
So it's a beam.

240
00:14:11,530 --> 00:14:15,260
This eraser isn't too
very much like a beam,

241
00:14:15,260 --> 00:14:20,380
but anyway I put the chalk
on it, well nothing happened.

242
00:14:20,380 --> 00:14:22,680
Sit on it, whatever.

243
00:14:22,680 --> 00:14:28,430
It'll bend and that bending will
be given by a beam equation.

244
00:14:28,430 --> 00:14:31,050
So later we'll meet
the beam equation.

245
00:14:31,050 --> 00:14:40,150
So most equations of physics,
mechanics, biology, everything

246
00:14:40,150 --> 00:14:45,780
are second order, Newton's
Laws often the reason.

247
00:14:45,780 --> 00:14:49,730
But we get up to
fourth order sometimes.

248
00:14:49,730 --> 00:14:52,260
And very seldom get higher.

249
00:14:52,260 --> 00:14:53,380
Hopefully.

250
00:14:53,380 --> 00:14:58,500
Beams or plates, that
table would be a plate

251
00:14:58,500 --> 00:15:05,530
and it would have a
fourth order equation.

252
00:15:05,530 --> 00:15:07,850
Let's start solving
this problem.

253
00:15:07,850 --> 00:15:11,620
What's the solution,
what's the general solution

254
00:15:11,620 --> 00:15:13,200
to that equation?

255
00:15:13,200 --> 00:15:15,830
Minus the second derivative,
so notice the minus

256
00:15:15,830 --> 00:15:19,840
that I like, and the load
has now moved to the point a.

257
00:15:19,840 --> 00:15:25,920
So the solution u(x), let's
write down all solutions.

258
00:15:25,920 --> 00:15:27,390
Tell me one solution, first.

259
00:15:27,390 --> 00:15:29,280
One particular solution.

260
00:15:29,280 --> 00:15:32,410
What is one function
for which minus

261
00:15:32,410 --> 00:15:36,480
the second derivative
would be the delta?

262
00:15:36,480 --> 00:15:38,030
That's what we've
got over there.

263
00:15:38,030 --> 00:15:42,110
So just bring that
blackboard over here.

264
00:15:42,110 --> 00:15:45,120
Change its sign
because that minus,

265
00:15:45,120 --> 00:15:46,750
and what are you
going to tell me?

266
00:15:46,750 --> 00:15:51,250
Minus a ramp.

267
00:15:51,250 --> 00:15:53,170
Minus a ramp.

268
00:15:53,170 --> 00:15:57,830
And the ramp, of course,
will ramp up at the point

269
00:15:57,830 --> 00:16:04,820
a so that it's the second
derivative of that,

270
00:16:04,820 --> 00:16:07,650
the second derivative
of R will be delta.

271
00:16:07,650 --> 00:16:12,270
The minus is correct and
the point is correct.

272
00:16:12,270 --> 00:16:17,590
Now does that solve our problem?

273
00:16:17,590 --> 00:16:20,610
No.

274
00:16:20,610 --> 00:16:22,240
The ramp is going upwards.

275
00:16:22,240 --> 00:16:23,570
It's not zero.

276
00:16:23,570 --> 00:16:25,650
What am I forgetting?

277
00:16:25,650 --> 00:16:27,180
What do I not yet have?

278
00:16:27,180 --> 00:16:30,520
There's more to this solution.

279
00:16:30,520 --> 00:16:34,360
Just as there was
for a uniform load.

280
00:16:34,360 --> 00:16:36,850
What was the more?

281
00:16:36,850 --> 00:16:47,400
Constant and-- and I want
two homogeneous solutions,

282
00:16:47,400 --> 00:16:52,270
null solutions, two solutions
with second derivative equal

283
00:16:52,270 --> 00:16:53,030
zero.

284
00:16:53,030 --> 00:16:57,340
One of them is C and
the other one is Dx.

285
00:16:57,340 --> 00:16:59,440
That's the whole solution.

286
00:16:59,440 --> 00:17:04,450
So what I want to-- I
mean we need that C+Dx.

287
00:17:04,450 --> 00:17:06,030
We've got two
boundary conditions

288
00:17:06,030 --> 00:17:08,340
to satisfy, just as before.

289
00:17:08,340 --> 00:17:10,820
So I need two constants,
that'll do it perfectly

290
00:17:10,820 --> 00:17:13,430
and I'll get an exact answer.

291
00:17:13,430 --> 00:17:18,170
And so this is a ramp.

292
00:17:18,170 --> 00:17:19,350
Oh yeah.

293
00:17:19,350 --> 00:17:23,760
Before I go further, how
would I think about this?

294
00:17:23,760 --> 00:17:27,160
This is a ramp that
turns which way?

295
00:17:27,160 --> 00:17:28,270
Down.

296
00:17:28,270 --> 00:17:29,840
Right?

297
00:17:29,840 --> 00:17:35,330
With that minus sign, that ramp
turns down at the point x=a.

298
00:17:35,330 --> 00:17:37,630
Right?

299
00:17:37,630 --> 00:17:43,840
It's derivative goes
from zero to minus one.

300
00:17:43,840 --> 00:17:49,790
The slope of this guy drops by
one because of the minus sign.

301
00:17:49,790 --> 00:18:00,340
Sorry the slope of the ramp
function, of minus the ramp.

302
00:18:00,340 --> 00:18:03,630
It goes at zero, drops by one.

303
00:18:03,630 --> 00:18:07,070
And what this is going
to do is take that ramp

304
00:18:07,070 --> 00:18:11,700
and adjust it to go
through the fixed ends.

305
00:18:11,700 --> 00:18:13,000
Oh, let's just do it.

306
00:18:13,000 --> 00:18:13,780
Let's just do it.

307
00:18:13,780 --> 00:18:15,530
What are C and D?

308
00:18:15,530 --> 00:18:16,480
What are C and D?

309
00:18:16,480 --> 00:18:18,660
My point a-- let
me draw a graph,

310
00:18:18,660 --> 00:18:21,110
that's always the best thing.

311
00:18:21,110 --> 00:18:26,150
Always draw a graph
of these solutions.

312
00:18:26,150 --> 00:18:29,730
So let me put in the point a.

313
00:18:29,730 --> 00:18:34,830
So I'm drawing now a picture of
the solution from zero to one.

314
00:18:34,830 --> 00:18:40,470
I'll graph it.

315
00:18:40,470 --> 00:18:42,260
What do I have here?

316
00:18:42,260 --> 00:18:47,160
Shall we just plug in
the boundary conditions

317
00:18:47,160 --> 00:18:48,260
and find C and D?

318
00:18:48,260 --> 00:18:50,060
That's the direct way.

319
00:18:50,060 --> 00:18:52,880
What is C?

320
00:18:52,880 --> 00:18:55,030
C I'm going to plug in.

321
00:18:55,030 --> 00:18:57,760
Hopefully I might find it
from just the first boundary

322
00:18:57,760 --> 00:18:59,090
condition.

323
00:18:59,090 --> 00:19:02,690
If I'm starting from zero,
well this guy certainly

324
00:19:02,690 --> 00:19:04,770
starts at zero, right?

325
00:19:04,770 --> 00:19:09,010
The ramp hasn't done
anything until it gets to a.

326
00:19:09,010 --> 00:19:10,820
And this guy is certainly zero.

327
00:19:10,820 --> 00:19:12,880
So what is C?

328
00:19:12,880 --> 00:19:15,760
Gone, right.

329
00:19:15,760 --> 00:19:19,200
Now what is D?

330
00:19:19,200 --> 00:19:22,810
Well alright, what's D?

331
00:19:22,810 --> 00:19:23,310
Let's see.

332
00:19:23,310 --> 00:19:30,020
Let me draw the-- So there's
a Dx, and D won't be zero.

333
00:19:30,020 --> 00:19:34,480
I want that thing to
be zero at point one.

334
00:19:34,480 --> 00:19:39,390
So I want to determine
D. Let me determine D.

335
00:19:39,390 --> 00:19:45,180
So what is minus
the ramp at x=1?

336
00:19:45,180 --> 00:19:46,637
I'm plugging in x=1.

337
00:19:46,637 --> 00:19:47,220
Is that right?

338
00:19:47,220 --> 00:19:49,360
I'm going straight forward here.

339
00:19:49,360 --> 00:19:52,660
Plugging in x=1 into
this boundary condition,

340
00:19:52,660 --> 00:19:54,780
ready for this guy.

341
00:19:54,780 --> 00:19:56,070
What's the ramp?

342
00:19:56,070 --> 00:20:02,980
So it's minus and
the ramp is, well

343
00:20:02,980 --> 00:20:07,310
if the ramp is shifted over
then that's shifted over.

344
00:20:07,310 --> 00:20:12,220
So at x=1, what's the ramp?

345
00:20:12,220 --> 00:20:15,400
How high has that ramp gone?

346
00:20:15,400 --> 00:20:17,340
1-a.

347
00:20:17,340 --> 00:20:17,840
Right?

348
00:20:17,840 --> 00:20:19,030
The ramp is x-a.

349
00:20:19,030 --> 00:20:21,930


350
00:20:21,930 --> 00:20:25,860
At the point x=1 it will be 1-a.

351
00:20:25,860 --> 00:20:29,890
So I think I get one,
-(1-a) out of that.

352
00:20:29,890 --> 00:20:35,500
Minus the ramp
plus D times what?

353
00:20:35,500 --> 00:20:37,880
One, I'm plugging in x=1.

354
00:20:37,880 --> 00:20:40,610
And that's supposed to equal?

355
00:20:40,610 --> 00:20:43,310
Zero, good.

356
00:20:43,310 --> 00:20:49,370
So I'm doing this sort of the
systematic way of writing down

357
00:20:49,370 --> 00:20:51,230
the general solution.

358
00:20:51,230 --> 00:20:55,060
Discovering that D,
what do I discover D is?

359
00:20:55,060 --> 00:20:56,680
Put it on the other side.

360
00:20:56,680 --> 00:21:00,520
D is 1-a.

361
00:21:00,520 --> 00:21:08,420
And of course, don't forget
that it's multiplying the x.

362
00:21:08,420 --> 00:21:13,010
Let me just draw the picture.

363
00:21:13,010 --> 00:21:15,580
Here's how I think about it.

364
00:21:15,580 --> 00:21:20,690
The solution is, away from x=a,
what does the solution look

365
00:21:20,690 --> 00:21:22,730
like?

366
00:21:22,730 --> 00:21:28,000
To the left of x=a what's
my graph going to be?

367
00:21:28,000 --> 00:21:31,630
It's going to be?

368
00:21:31,630 --> 00:21:33,690
A straight line, right?

369
00:21:33,690 --> 00:21:38,400
To the left of here
there is no load.

370
00:21:38,400 --> 00:21:43,370
The equation is second
derivative equals zero.

371
00:21:43,370 --> 00:21:45,730
The solution to that
is a straight line.

372
00:21:45,730 --> 00:21:49,130
In other words, until I get to
a, this thing hasn't started.

373
00:21:49,130 --> 00:21:51,360
It's only this straight line.

374
00:21:51,360 --> 00:21:53,830
The solution does
something like that.

375
00:21:53,830 --> 00:21:55,650
It's a straight line.

376
00:21:55,650 --> 00:21:59,180
And I guess, actually,
that's what it is.

377
00:21:59,180 --> 00:22:02,710
Because the C isn't here and
that's all we've got left.

378
00:22:02,710 --> 00:22:06,830
So that's that straight line.

379
00:22:06,830 --> 00:22:10,820
What is it for the second half?

380
00:22:10,820 --> 00:22:14,660
Tell me what the solution
looks like in the second half.

381
00:22:14,660 --> 00:22:17,260
In between a and one.

382
00:22:17,260 --> 00:22:19,600
It's going downhill.

383
00:22:19,600 --> 00:22:22,030
Why?

384
00:22:22,030 --> 00:22:24,770
Because it gotta
get back to zero.

385
00:22:24,770 --> 00:22:28,690
And how's it going downhill?

386
00:22:28,690 --> 00:22:31,030
It has to be linear.

387
00:22:31,030 --> 00:22:34,280
In this region,
has to be linear.

388
00:22:34,280 --> 00:22:37,200
Why?

389
00:22:37,200 --> 00:22:38,800
How do I know it's linear here?

390
00:22:38,800 --> 00:22:43,040
Because one way is to say
the equation in that region

391
00:22:43,040 --> 00:22:46,380
is second derivative equal zero.

392
00:22:46,380 --> 00:22:49,250
Second derivative equal
zero, straight line.

393
00:22:49,250 --> 00:22:50,830
This is my solution.

394
00:22:50,830 --> 00:22:55,820
It's (1-a)x here and it's
whatever it is to get back

395
00:22:55,820 --> 00:22:58,320
to zero.

396
00:22:58,320 --> 00:23:02,100
What will it take
to get back to zero?

397
00:23:02,100 --> 00:23:03,540
Let's see.

398
00:23:03,540 --> 00:23:07,090
Well we could plug in, we've
got one expression here.

399
00:23:07,090 --> 00:23:09,350
Or I could just look at that.

400
00:23:09,350 --> 00:23:12,870
I could say, okay what's the
equation for the straight line

401
00:23:12,870 --> 00:23:17,510
that's at this point, what
is the, yeah, it's (1-a)x.

402
00:23:17,510 --> 00:23:23,230


403
00:23:23,230 --> 00:23:26,360
I want it to be linear.

404
00:23:26,360 --> 00:23:31,520
I want it to get to zero.

405
00:23:31,520 --> 00:23:32,390
Let's see.

406
00:23:32,390 --> 00:23:36,460
If I want that, it would
be great to have 1-x times

407
00:23:36,460 --> 00:23:37,460
something.

408
00:23:37,460 --> 00:23:39,910
I have to figure out what.

409
00:23:39,910 --> 00:23:44,160
Because with the 1-x at
x=1, that'll drop off.

410
00:23:44,160 --> 00:23:45,460
That's linear.

411
00:23:45,460 --> 00:23:49,960
What number, what's
the key here?

412
00:23:49,960 --> 00:23:58,230
That slope, I want to
match them up there.

413
00:23:58,230 --> 00:24:01,750
And that's the point x=a.

414
00:24:01,750 --> 00:24:04,640
This is supposed to
match that at x=a.

415
00:24:04,640 --> 00:24:09,380
Do you have an idea
for what I should take?

416
00:24:09,380 --> 00:24:15,740
What do I put right there? a.

417
00:24:15,740 --> 00:24:22,120
Look at the symmetry in those
two sides. (1-a)x going up.

418
00:24:22,120 --> 00:24:24,700
(1-x)a going down.

419
00:24:24,700 --> 00:24:28,990
At x=a it hits
that point, right.

420
00:24:28,990 --> 00:24:30,750
So we've solved it.

421
00:24:30,750 --> 00:24:34,130
We could think about
this different ways.

422
00:24:34,130 --> 00:24:39,210
I could have got
that 1-x, let's see,

423
00:24:39,210 --> 00:24:42,290
I could have got it
from the formula.

424
00:24:42,290 --> 00:24:47,040
In a way I like to get it from
the picture, I see it, sort of,

425
00:24:47,040 --> 00:24:49,000
I see the point.

426
00:24:49,000 --> 00:24:51,410
What happened at that point?

427
00:24:51,410 --> 00:24:54,160
What are the jump conditions?

428
00:24:54,160 --> 00:24:56,390
This is another
way to ask, to see

429
00:24:56,390 --> 00:24:58,410
how the delta function works.

430
00:24:58,410 --> 00:25:00,740
What are they jump conditions?

431
00:25:00,740 --> 00:25:03,530
I want to know, when I
ask about jump conditions,

432
00:25:03,530 --> 00:25:06,570
I want to know what are
the conditions on u(x),

433
00:25:06,570 --> 00:25:08,030
the displacement?

434
00:25:08,030 --> 00:25:12,850
What are the conditions
on the slope, u'(x)?

435
00:25:12,850 --> 00:25:19,790
That'll be the strain when
we're speaking about elasticity.

436
00:25:19,790 --> 00:25:23,020
Just for u(x),
what's the statement

437
00:25:23,020 --> 00:25:25,480
about u(x) from the
left and from the right

438
00:25:25,480 --> 00:25:32,760
at that critical point,
the point of the load.

439
00:25:32,760 --> 00:25:38,020
From the left and from
the right u(x) is?

440
00:25:38,020 --> 00:25:42,000
The same. u(x) matches
up. u(x) from the left

441
00:25:42,000 --> 00:25:44,500
is that height. u(x)
from the right is that.

442
00:25:44,500 --> 00:25:48,860
I want to write down
those jump conditions.

443
00:25:48,860 --> 00:25:53,050
Because that's another
way to see this. u(x),

444
00:25:53,050 --> 00:26:02,200
u(a) from the left
should equal u--

445
00:26:02,200 --> 00:26:04,500
do you want me to
say u is continuous?

446
00:26:04,500 --> 00:26:12,650
I'll just say it in
words. u(x) is continuous,

447
00:26:12,650 --> 00:26:15,200
that just means it
doesn't jump, at x=a.

448
00:26:15,200 --> 00:26:18,050


449
00:26:18,050 --> 00:26:22,770
So that's, you could say
that's a non-jump condition.

450
00:26:22,770 --> 00:26:24,390
The function itself
doesn't jump.

451
00:26:24,390 --> 00:26:25,020
Why not?

452
00:26:25,020 --> 00:26:28,190
Because we're talking about
some elastic bar on which

453
00:26:28,190 --> 00:26:30,160
we put a point load.

454
00:26:30,160 --> 00:26:32,600
The thing isn't going to break.

455
00:26:32,600 --> 00:26:36,510
The displacement is
going to be continuous.

456
00:26:36,510 --> 00:26:42,150
But what's the condition
on u'(x), the derivative,

457
00:26:42,150 --> 00:26:43,560
the slope?

458
00:26:43,560 --> 00:26:46,390
So that's the function
and now tell me

459
00:26:46,390 --> 00:26:48,400
what's the deal on the slope?

460
00:26:48,400 --> 00:26:51,350
What's the comparison
between the-- I

461
00:26:51,350 --> 00:26:54,420
have a slope of whatever
it is going along here

462
00:26:54,420 --> 00:26:56,980
and I have a slope
of-- a new slope.

463
00:26:56,980 --> 00:27:03,130
So u'(x), the
slope jumps, right?

464
00:27:03,130 --> 00:27:06,710
And how much does it jump?

465
00:27:06,710 --> 00:27:08,120
Minus one.

466
00:27:08,120 --> 00:27:09,860
It drops by one.

467
00:27:09,860 --> 00:27:13,420
The slope, because of my minus.

468
00:27:13,420 --> 00:27:17,849
So this tells me that--
Yeah, let me write that down.

469
00:27:17,849 --> 00:27:18,640
u'(x) drops by one.

470
00:27:18,640 --> 00:27:26,430


471
00:27:26,430 --> 00:27:32,550
This is another way to say
what the equation is asking.

472
00:27:32,550 --> 00:27:36,540
The equation is looking for
two pieces of straight lines

473
00:27:36,540 --> 00:27:41,260
that meet at a but their
slope drops by one.

474
00:27:41,260 --> 00:27:42,800
By the way, what
were the slopes?

475
00:27:42,800 --> 00:27:45,880
It's good to graph
the slopes, too.

476
00:27:45,880 --> 00:27:52,610
Let me graph the slopes.

477
00:27:52,610 --> 00:27:58,590
The slope u', the
derivative du/dx.

478
00:27:58,590 --> 00:28:00,790
What's the slope here?

479
00:28:00,790 --> 00:28:03,280
Slope is 1-a at
this point, right?

480
00:28:03,280 --> 00:28:06,350
The derivative is
1-a along here.

481
00:28:06,350 --> 00:28:09,790
So slope is 1-a.

482
00:28:09,790 --> 00:28:14,680
And now at x=a the slope
changes to this one.

483
00:28:14,680 --> 00:28:17,730
And what's the slope
of that second part?

484
00:28:17,730 --> 00:28:18,770
Minus a.

485
00:28:18,770 --> 00:28:19,660
Look.

486
00:28:19,660 --> 00:28:21,190
It did it right.

487
00:28:21,190 --> 00:28:24,430
Minus a is the slope along here.

488
00:28:24,430 --> 00:28:28,330
Do you see 1-a?

489
00:28:28,330 --> 00:28:29,520
It dropped by one.

490
00:28:29,520 --> 00:28:36,610
The one disappeared to
leave a slope of minus a.

491
00:28:36,610 --> 00:28:41,970
I guess if I just imagine a bar.

492
00:28:41,970 --> 00:28:45,110
I'm fixing it at both ends.

493
00:28:45,110 --> 00:28:50,870
There's a bar.

494
00:28:50,870 --> 00:28:53,450
I'm just thinking
for people who like

495
00:28:53,450 --> 00:28:57,510
to see a physical picture
of what's happening,

496
00:28:57,510 --> 00:29:03,490
that's what this is, we'll do
it properly very, very soon.

497
00:29:03,490 --> 00:29:05,500
I've got a bar.

498
00:29:05,500 --> 00:29:08,700
It's a very light bar.

499
00:29:08,700 --> 00:29:12,320
Its weight is not
a problem here.

500
00:29:12,320 --> 00:29:14,460
But it's got a
load at the point.

501
00:29:14,460 --> 00:29:16,340
So I'll measure x
going downwards.

502
00:29:16,340 --> 00:29:22,900
And at the point x=a I'm
hanging a heavy load.

503
00:29:22,900 --> 00:29:24,640
A load.

504
00:29:24,640 --> 00:29:30,940
How do I draw a load?

505
00:29:30,940 --> 00:29:40,110
Maybe I'll make a big
weight or something.

506
00:29:40,110 --> 00:29:49,180
What's going to happen to
this dumb bar when I do that?

507
00:29:49,180 --> 00:29:50,472
Just tell me physically.

508
00:29:50,472 --> 00:29:51,430
What's going to happen?

509
00:29:51,430 --> 00:29:54,810
What's going to
happen above the load?

510
00:29:54,810 --> 00:30:00,300
It's going to stretch,
right, tension.

511
00:30:00,300 --> 00:30:02,740
The load is going to
pull the bar down,

512
00:30:02,740 --> 00:30:05,350
it's going to stretch this part.

513
00:30:05,350 --> 00:30:07,340
And because nothing
special is happening,

514
00:30:07,340 --> 00:30:09,780
it's going to
stretch it linearly.

515
00:30:09,780 --> 00:30:13,180
And then what's going to
happen below the load?

516
00:30:13,180 --> 00:30:15,810
Compression.

517
00:30:15,810 --> 00:30:19,390
So the slope will go negative.

518
00:30:19,390 --> 00:30:23,730
And nothing special happened
so the slope will be negative

519
00:30:23,730 --> 00:30:25,240
but it'll be constant.

520
00:30:25,240 --> 00:30:27,600
The slope will drop
from this to this.

521
00:30:27,600 --> 00:30:34,650
The displacement, that point
will go down a little bit.

522
00:30:34,650 --> 00:30:38,687
That little bit it goes down
is actually the height of this,

523
00:30:38,687 --> 00:30:40,020
because that's the displacement.

524
00:30:40,020 --> 00:30:41,940
It'll go down a little bit.

525
00:30:41,940 --> 00:30:44,990
It'll stretch above,
it'll compress below,

526
00:30:44,990 --> 00:30:52,700
and we see that in that
picture of the displacement.

527
00:30:52,700 --> 00:30:55,070
The displacement's all down.

528
00:30:55,070 --> 00:30:56,000
Right?

529
00:30:56,000 --> 00:31:01,320
Displacement-- You know, nature
is still going to-- All the bar

530
00:31:01,320 --> 00:31:02,340
is going to move down.

531
00:31:02,340 --> 00:31:07,310
That's why this function
doesn't, this function,

532
00:31:07,310 --> 00:31:09,340
the displacement
function is positive.

533
00:31:09,340 --> 00:31:10,880
It goes all down.

534
00:31:10,880 --> 00:31:15,500
But the slope function
is positive here,

535
00:31:15,500 --> 00:31:19,780
so tension is positive
slope, stretch.

536
00:31:19,780 --> 00:31:24,740
And compression is negative.

537
00:31:24,740 --> 00:31:31,360
Well all that to
solve this equation.

538
00:31:31,360 --> 00:31:38,450
Maybe while we're on a roll,
let's solve the free-fixed guy.

539
00:31:38,450 --> 00:31:41,440
So this is our-- might
as well be systematic.

540
00:31:41,440 --> 00:31:43,160
This is the fixed-fixed problem.

541
00:31:43,160 --> 00:31:46,770
Let me below it solve
the free-fixed problem.

542
00:31:46,770 --> 00:31:50,490
So it'll be minus u'',
that's the second derivative,

543
00:31:50,490 --> 00:31:53,840
equals delta at x-a.

544
00:31:53,840 --> 00:31:55,680
Same setup.

545
00:31:55,680 --> 00:32:01,730
But now the top end is,
so it's free at the top.

546
00:32:01,730 --> 00:32:04,290
What does that mean?

547
00:32:04,290 --> 00:32:12,110
Slope is zero at the top but
it's still fixed at the bottom.

548
00:32:12,110 --> 00:32:21,490
So this will be now free-fixed.

549
00:32:21,490 --> 00:32:24,040
Let me go straight
to the picture.

550
00:32:24,040 --> 00:32:27,700
Let me go straight to
the picture of u(x).

551
00:32:27,700 --> 00:32:32,770
So there is x=0, there's
x=1, here's the load at a.

552
00:32:32,770 --> 00:32:37,050
What's up?

553
00:32:37,050 --> 00:32:38,940
And while you're
thinking about that,

554
00:32:38,940 --> 00:32:43,300
let me draw a picture
to match this picture.

555
00:32:43,300 --> 00:32:52,640
A bar fixed at the bottom
but not at the top.

556
00:32:52,640 --> 00:32:59,120
And it's got its load
here hanging down.

557
00:32:59,120 --> 00:33:08,240
But let's do it math first, and
then check with the picture.

558
00:33:08,240 --> 00:33:09,890
What have we got,
two or three ways now

559
00:33:09,890 --> 00:33:11,490
to try to get the answer?

560
00:33:11,490 --> 00:33:14,590
The systematic way
would be to write down

561
00:33:14,590 --> 00:33:22,710
this solution and plug in
the two boundary conditions.

562
00:33:22,710 --> 00:33:24,780
That'd be a straightforward way.

563
00:33:24,780 --> 00:33:27,350
Yeah, we could
even start by that.

564
00:33:27,350 --> 00:33:33,170
So u(x) is the
particular solution,

565
00:33:33,170 --> 00:33:46,350
the ramp plus any Cx+D. And just
plug in x=0 that'll be easy.

566
00:33:46,350 --> 00:33:51,550
If I plug in x=0 in
the free condition,

567
00:33:51,550 --> 00:33:53,240
what does that tell me?

568
00:33:53,240 --> 00:33:58,820
At x=0, this corner, this ramp
hasn't started so the slope is

569
00:33:58,820 --> 00:34:00,040
zero.

570
00:34:00,040 --> 00:34:01,730
The slope of the
constant is zero.

571
00:34:01,730 --> 00:34:05,280
What do I learn from this
boundary condition? u'(0)=0.

572
00:34:05,280 --> 00:34:08,650


573
00:34:08,650 --> 00:34:10,300
That C is zero.

574
00:34:10,300 --> 00:34:12,740
Before I learned that
D was zero, but now

575
00:34:12,740 --> 00:34:18,540
from that condition I'm
going to learn C is zero.

576
00:34:18,540 --> 00:34:21,160
Do the picture for me.

577
00:34:21,160 --> 00:34:24,090
Do the picture for me.

578
00:34:24,090 --> 00:34:27,770
What's the graph of--
this is a graph of u(x).

579
00:34:27,770 --> 00:34:30,990


580
00:34:30,990 --> 00:34:36,000
Remember now it
starts from zero slope

581
00:34:36,000 --> 00:34:38,330
because it's free at the top.

582
00:34:38,330 --> 00:34:44,880
What does the graph look
like in the first part?

583
00:34:44,880 --> 00:34:49,370
It's a straight line, has to be
a straight line because there's

584
00:34:49,370 --> 00:34:51,050
no force.

585
00:34:51,050 --> 00:34:54,260
And what kind of a line?

586
00:34:54,260 --> 00:34:57,920
It's going to be horizontal
because it starts off

587
00:34:57,920 --> 00:34:59,290
horizontal.

588
00:34:59,290 --> 00:35:05,080
The slope has to be zero at zero
and nothing changes until a.

589
00:35:05,080 --> 00:35:08,960
So it comes along there.

590
00:35:08,960 --> 00:35:11,220
Right?

591
00:35:11,220 --> 00:35:14,110
Now I've started out
with the right, left,

592
00:35:14,110 --> 00:35:18,950
the correct boundary condition
at zero, which was no slope.

593
00:35:18,950 --> 00:35:22,330
And now what's it going
to do the other half?

594
00:35:22,330 --> 00:35:25,200
From a to one.

595
00:35:25,200 --> 00:35:31,090
It's going to be again, it'll
be a straight line, right?

596
00:35:31,090 --> 00:35:33,380
Because there's no force there.

597
00:35:33,380 --> 00:35:37,680
And what happens at--
all the action of course

598
00:35:37,680 --> 00:35:41,900
is at this point a,
and what action is it?

599
00:35:41,900 --> 00:35:45,000
Tell me what sort of a line.

600
00:35:45,000 --> 00:35:49,940
How do I finish the picture?

601
00:35:49,940 --> 00:35:52,370
What do I do?

602
00:35:52,370 --> 00:35:53,770
I start here, right?

603
00:35:53,770 --> 00:36:01,070
Because the bar's not falling
apart. u is continuous.

604
00:36:01,070 --> 00:36:03,780
I don't get a gap suddenly.

605
00:36:03,780 --> 00:36:07,390
And now what do I do from there?

606
00:36:07,390 --> 00:36:11,130
Only thing I can possibly do,
because I have to end up here

607
00:36:11,130 --> 00:36:16,942
and it has to be a
straight line, that's it.

608
00:36:16,942 --> 00:36:18,900
That's what the picture
will have to look like.

609
00:36:18,900 --> 00:36:30,110
What does that correspond to
in the picture for the bar?

610
00:36:30,110 --> 00:36:33,300
Well what happens with this bar?

611
00:36:33,300 --> 00:36:38,740
Above the weight, what
happens to this top part

612
00:36:38,740 --> 00:36:42,630
of the bar in that picture?

613
00:36:42,630 --> 00:36:45,020
And what happens to the
lower part of the bar?

614
00:36:45,020 --> 00:36:51,300
So this was at the point x=a,
this is x=0, this is x=1.

615
00:36:51,300 --> 00:36:57,890
What happens above the
bar, above the weight?

616
00:36:57,890 --> 00:37:01,780
It just-- A rigid
motion, just goes down.

617
00:37:01,780 --> 00:37:04,620
Because what happens
below the weight?

618
00:37:04,620 --> 00:37:07,510
The same compression or
compression still happening.

619
00:37:07,510 --> 00:37:08,950
This is still squeezed.

620
00:37:08,950 --> 00:37:12,140
Shall I try to draw it?

621
00:37:12,140 --> 00:37:14,830
So this is after the weight.

622
00:37:14,830 --> 00:37:21,300
This got squeezed but this
part did not get squeezed.

623
00:37:21,300 --> 00:37:25,970
And that's what
we're seeing here.

624
00:37:25,970 --> 00:37:28,120
A fixed displacement.

625
00:37:28,120 --> 00:37:30,970
So this means,
that picture means

626
00:37:30,970 --> 00:37:36,600
that all the pieces
of the bar here

627
00:37:36,600 --> 00:37:39,370
got moved down by the same
amount, whatever this,

628
00:37:39,370 --> 00:37:41,860
we don't know that number yet.

629
00:37:41,860 --> 00:37:49,240
And then below it
they got compressed.

630
00:37:49,240 --> 00:37:56,430
Well we're almost there but we
don't yet have that solution.

631
00:37:56,430 --> 00:38:02,130
Come back to this picture.
u(x) is continuous, got it.

632
00:38:02,130 --> 00:38:04,690
And what's the real
condition that's

633
00:38:04,690 --> 00:38:09,980
going to determine where
we are, what those heights

634
00:38:09,980 --> 00:38:12,390
are, the numbers in there.

635
00:38:12,390 --> 00:38:16,060
It's gotta look like
that, but we get more than

636
00:38:16,060 --> 00:38:19,970
that, we gotta know what are
the actual, what is that height.

637
00:38:19,970 --> 00:38:20,950
What is this?

638
00:38:20,950 --> 00:38:22,050
What's the slope?

639
00:38:22,050 --> 00:38:27,250
Here the slope is zero.

640
00:38:27,250 --> 00:38:34,230
Here the slope is what?

641
00:38:34,230 --> 00:38:36,170
What's the slope
in the second part?

642
00:38:36,170 --> 00:38:37,080
That's the key.

643
00:38:37,080 --> 00:38:40,610
And you know what it
has to be because what

644
00:38:40,610 --> 00:38:42,740
happens to the slope?

645
00:38:42,740 --> 00:38:46,910
If I have the second
derivative as a delta function

646
00:38:46,910 --> 00:38:56,420
with that minus sign,
the slope drops by one.

647
00:38:56,420 --> 00:39:02,690
And the slope here is zero, so
the slope here is minus one.

648
00:39:02,690 --> 00:39:07,620
And now it has to get through
there, so what is the function?

649
00:39:07,620 --> 00:39:11,990
What's the function that
has a slope of minus one

650
00:39:11,990 --> 00:39:22,030
and comes down to zero?

651
00:39:22,030 --> 00:39:24,480
It's gotta have a minus
x in it and what's

652
00:39:24,480 --> 00:39:30,800
the constant to make
it come out right?

653
00:39:30,800 --> 00:39:34,510
What do I write
now here for u(x)?

654
00:39:34,510 --> 00:39:35,010
1-x.

655
00:39:35,010 --> 00:39:37,830


656
00:39:37,830 --> 00:39:41,230
That has a slope of minus one,
the derivative is minus one,

657
00:39:41,230 --> 00:39:44,140
at x=1 it comes to
zero, that's it.

658
00:39:44,140 --> 00:39:48,680
And what do I write,
what's u(x) up here?

659
00:39:48,680 --> 00:39:54,550
And therefore, right there?

660
00:39:54,550 --> 00:39:59,070
What's the displacement
there, of all this bit that

661
00:39:59,070 --> 00:40:03,200
moves down, how much
does it move down?

662
00:40:03,200 --> 00:40:04,360
1-a.

663
00:40:04,360 --> 00:40:05,680
Why 1-a?

664
00:40:05,680 --> 00:40:06,780
That's the right answer.

665
00:40:06,780 --> 00:40:07,279
1-a.

666
00:40:07,279 --> 00:40:09,750


667
00:40:09,750 --> 00:40:12,430
Why's that?

668
00:40:12,430 --> 00:40:15,600
Because it had to
match up at x=a.

669
00:40:15,600 --> 00:40:19,230
At x=a, this and that match up.

670
00:40:19,230 --> 00:40:23,590
At x=a, that slope, that
function and that function

671
00:40:23,590 --> 00:40:24,370
match up.

672
00:40:24,370 --> 00:40:31,570
So the slope picture is
zero and-- Oh, I'm sorry,

673
00:40:31,570 --> 00:40:36,500
can't draw it because I'm
at the bottom of the board.

674
00:40:36,500 --> 00:40:39,120
The slope picture, maybe
I can draw it here,

675
00:40:39,120 --> 00:40:42,740
the slope picture
is zero along here

676
00:40:42,740 --> 00:40:45,300
and then it drops by one to 1-a.

677
00:40:45,300 --> 00:40:47,600
So that's a picture of u'.

678
00:40:47,600 --> 00:40:51,470
Zero and minus one.

679
00:40:51,470 --> 00:40:59,930
This is the thing to look at.

680
00:40:59,930 --> 00:41:03,150
That's hard work, when
you're seeing delta functions

681
00:41:03,150 --> 00:41:03,790
the first time.

682
00:41:03,790 --> 00:41:06,590
But of course the functions
did not get complicated.

683
00:41:06,590 --> 00:41:11,220
We kept a clean example.

684
00:41:11,220 --> 00:41:18,240
And which we matched up with a
figure and we've got the answer

685
00:41:18,240 --> 00:41:20,440
and we've got a couple
of ways to do it.

686
00:41:20,440 --> 00:41:25,510
One is this standard,
systematic, plug-in boundary

687
00:41:25,510 --> 00:41:26,650
condition way.

688
00:41:26,650 --> 00:41:31,700
The other way is this.
u(x) does something here,

689
00:41:31,700 --> 00:41:34,460
then the slope has
to drop by one.

690
00:41:34,460 --> 00:41:40,120
And that's the key to everything
with a boundary condition.

691
00:41:40,120 --> 00:41:43,410
So in a way, we have a piece
to the left and a piece

692
00:41:43,410 --> 00:41:44,660
to the right.

693
00:41:44,660 --> 00:41:47,140
Two constants here,
two constants here,

694
00:41:47,140 --> 00:41:50,750
and somewhere there are
four conditions that

695
00:41:50,750 --> 00:41:52,940
settle those four constants.

696
00:41:52,940 --> 00:41:55,450
You know, we could have
a straight line here,

697
00:41:55,450 --> 00:41:57,990
a straight line here,
that's two and two.

698
00:41:57,990 --> 00:42:00,280
But what are the
four conditions that

699
00:42:00,280 --> 00:42:01,690
settle those four constants?

700
00:42:01,690 --> 00:42:05,460
Well we have a boundary
condition here, that's one.

701
00:42:05,460 --> 00:42:07,890
Boundary condition here is two.

702
00:42:07,890 --> 00:42:11,970
We need two more
conditions to settle

703
00:42:11,970 --> 00:42:19,410
the two pairs of constants,
and there they are.

704
00:42:19,410 --> 00:42:27,540
Two conditions at the
jump, at the discontinuity.

705
00:42:27,540 --> 00:42:36,270
Now I've got to do
the discrete case.

706
00:42:36,270 --> 00:42:39,170
Are you up for
the discrete case?

707
00:42:39,170 --> 00:42:47,320
The case where we're doing--
We have a difference equation,

708
00:42:47,320 --> 00:42:52,480
so we're doing Ku equal
a column of the identity.

709
00:42:52,480 --> 00:43:03,460
Column of I. Let me
take a specific column.

710
00:43:03,460 --> 00:43:06,420
Say [0, 1, 0, 0, 0].

711
00:43:06,420 --> 00:43:12,550
Let's suppose we have five--
I'm going to draw a picture now.

712
00:43:12,550 --> 00:43:15,680
We have five, because
I made it five by five.

713
00:43:15,680 --> 00:43:21,260
One, two, three, four, five,
here is zero and here is six.

714
00:43:21,260 --> 00:43:25,720
So h is 1/(5+1), 1/6,
that's the delta x.

715
00:43:25,720 --> 00:43:36,650
And my equation says-- So
what does my equation say?

716
00:43:36,650 --> 00:43:46,130
Remember what K is. u is then
u_1, u_2, u_3, u_4, and u_5,

717
00:43:46,130 --> 00:43:48,080
the unknowns.

718
00:43:48,080 --> 00:44:04,130
K is our old friend with twos
and minus ones and minus ones.

719
00:44:04,130 --> 00:44:07,140
I'm going to find the solution.

720
00:44:07,140 --> 00:44:15,280
And this'll be the solution
that has a load at this point.

721
00:44:15,280 --> 00:44:18,240
This is like my point a, right?

722
00:44:18,240 --> 00:44:21,400
Here in the continuous
case, a could run anywhere

723
00:44:21,400 --> 00:44:23,050
between zero and one.

724
00:44:23,050 --> 00:44:27,420
In the discrete case, I've
got five possible load points

725
00:44:27,420 --> 00:44:29,120
and I've picked the second one.

726
00:44:29,120 --> 00:44:31,830
Five columns of the
identity matrix, five

727
00:44:31,830 --> 00:44:36,050
places to put that
one, I put it there.

728
00:44:36,050 --> 00:44:42,660
Now can I draw the picture here?

729
00:44:42,660 --> 00:44:45,000
Which should we do first?

730
00:44:45,000 --> 00:44:46,950
Should we do free-fixed?

731
00:44:46,950 --> 00:44:52,280
Because that came out even
easier than fixed-fixed.

732
00:44:52,280 --> 00:44:55,920
Notice the solution
here had two parts.

733
00:44:55,920 --> 00:44:59,660
This is the way I would
write that answer.

734
00:44:59,660 --> 00:45:01,550
Because you could
draw a picture,

735
00:45:01,550 --> 00:45:06,560
but if you want to write the
formula, what would I do?

736
00:45:06,560 --> 00:45:10,410
I would break it
into two pieces.

737
00:45:10,410 --> 00:45:16,210
1-a up to the point
a because that's

738
00:45:16,210 --> 00:45:18,120
what it was running along here.

739
00:45:18,120 --> 00:45:23,480
And then down here it
was 1-x, x >=a.

740
00:45:23,480 --> 00:45:31,580


741
00:45:31,580 --> 00:45:33,830
That's important to mention.

742
00:45:33,830 --> 00:45:38,820
You have to have some guidance
on how to write the answer.

743
00:45:38,820 --> 00:45:42,210
And when the answer
has two parts,

744
00:45:42,210 --> 00:45:44,490
this is a good way to
write it, in two parts.

745
00:45:44,490 --> 00:45:46,770
It's a little too--
you're compressing

746
00:45:46,770 --> 00:45:50,580
it too much to write, to
use that ramp function.

747
00:45:50,580 --> 00:45:56,310
Better to split it apart
into before a and after a.

748
00:45:56,310 --> 00:45:59,660
What's going to
happen over here?

749
00:45:59,660 --> 00:46:04,860
Oh yeah, can we take a
shot at this problem?

750
00:46:04,860 --> 00:46:11,880
And let me mention again,
in the review that'll

751
00:46:11,880 --> 00:46:17,130
be in here this afternoon and
every Wednesday afternoon I'll

752
00:46:17,130 --> 00:46:18,880
just be ready for questions.

753
00:46:18,880 --> 00:46:22,040
Please bring questions.

754
00:46:22,040 --> 00:46:25,360
They can be questions
on the homework.

755
00:46:25,360 --> 00:46:28,200
Even better if they're
questions on other problems,

756
00:46:28,200 --> 00:46:33,060
questions on the lecture.

757
00:46:33,060 --> 00:46:42,000
Questions are essential to
make that help session helpful.

758
00:46:42,000 --> 00:46:46,470
What do you think's
cooking here?

759
00:46:46,470 --> 00:46:54,060
At a typical-- Somewhere
in the middle here,

760
00:46:54,060 --> 00:46:57,770
I'm going to draw the u's.

761
00:46:57,770 --> 00:47:00,560
Shall I just draw them?

762
00:47:00,560 --> 00:47:02,640
And now what's my condition?

763
00:47:02,640 --> 00:47:04,710
I gotta put the
boundary conditions on.

764
00:47:04,710 --> 00:47:07,800
Oh, I have put the
boundary conditions on it.

765
00:47:07,800 --> 00:47:11,350
By putting that two
there, I'm up to here.

766
00:47:11,350 --> 00:47:15,280
Okay, let's do that one.

767
00:47:15,280 --> 00:47:20,780
When I chose K and
put a two in there

768
00:47:20,780 --> 00:47:24,620
I was picking the fixed-fixed
boundary condition.

769
00:47:24,620 --> 00:47:29,010
So can I just say it's
going to be beautiful.

770
00:47:29,010 --> 00:47:32,530
The solution over there is
going to look like this.

771
00:47:32,530 --> 00:47:37,720
The solution over here is
going to be up, up, up.

772
00:47:37,720 --> 00:47:42,960
It's going to be a straight
line but only points in a line

773
00:47:42,960 --> 00:47:46,660
and it'll be straight line down.

774
00:47:46,660 --> 00:47:50,280
That value, that
value, that value.

775
00:47:50,280 --> 00:47:53,760
Those will be u_1,
u_2, u_3, u_4, and u_5.

776
00:47:53,760 --> 00:47:58,470


777
00:47:58,470 --> 00:48:05,690
And once more, this is
going to drop by one again.

778
00:48:05,690 --> 00:48:08,620
Actually I didn't have
to redraw the picture.

779
00:48:08,620 --> 00:48:11,140
It falls right on.

780
00:48:11,140 --> 00:48:22,540
In case x is 2/6 so that
it fits that picture,

781
00:48:22,540 --> 00:48:28,250
I'm claiming we have another
extremely lucky case.

782
00:48:28,250 --> 00:48:34,380
If we can use the
word lucky for math.

783
00:48:34,380 --> 00:48:37,560
I'm claiming that the
way the-- You remember

784
00:48:37,560 --> 00:48:41,180
for the uniform load
with a one, when

785
00:48:41,180 --> 00:48:44,840
we had second derivative
equal one, the solution

786
00:48:44,840 --> 00:48:49,160
was a perfect parabola
and the discrete solution,

787
00:48:49,160 --> 00:48:52,570
the difference equation
was right on the parabola

788
00:48:52,570 --> 00:48:54,700
for this fixed-fixed case.

789
00:48:54,700 --> 00:48:57,540
It's going to happen again.

790
00:48:57,540 --> 00:48:59,580
It won't always happen.

791
00:48:59,580 --> 00:49:04,330
Those are the only two important
right-hand sides I know.

792
00:49:04,330 --> 00:49:07,440
They're the two most important
right-hand sides and those

793
00:49:07,440 --> 00:49:09,900
are the two lucky ones.

794
00:49:09,900 --> 00:49:13,570
If we have a constant that
lies right on a parabola,

795
00:49:13,570 --> 00:49:21,660
if we have a delta function,
it lies right on a ramp.

796
00:49:21,660 --> 00:49:23,300
And there it is.

797
00:49:23,300 --> 00:49:26,630
So that's what the
solution looks like.

798
00:49:26,630 --> 00:49:31,720
Now, I have to figure out what
these numbers are, I guess.

799
00:49:31,720 --> 00:49:33,870
Yes, what are those numbers?

800
00:49:33,870 --> 00:49:35,060
Oh, well.

801
00:49:35,060 --> 00:49:40,900
Actually, if it falls right
on, I know the numbers.

802
00:49:40,900 --> 00:49:46,890
So a is 2/6.

803
00:49:46,890 --> 00:49:49,540
So let me keep 2/6.

804
00:49:49,540 --> 00:49:51,550
So a is 2/6.

805
00:49:51,550 --> 00:49:53,140
That's that value.

806
00:49:53,140 --> 00:49:58,870
So let me say what I think u is.

807
00:49:58,870 --> 00:50:00,590
So this was a picture of u.

808
00:50:00,590 --> 00:50:04,990
That's u_1, 2, 3,
4, and 5 and now

809
00:50:04,990 --> 00:50:06,780
I think it lies right on that.

810
00:50:06,780 --> 00:50:19,160
So it's going to be (1-2/6)x
going up and (1-x)2/6 going

811
00:50:19,160 --> 00:50:21,380
down.

812
00:50:21,380 --> 00:50:28,780
My point is that I'll be able
to figure out what that-- this

813
00:50:28,780 --> 00:50:33,420
is u, this is the u.

814
00:50:33,420 --> 00:50:38,140
You're going to say, why?

815
00:50:38,140 --> 00:50:40,530
Let me pause before
putting in numbers

816
00:50:40,530 --> 00:50:46,300
and say why is it, how do
I know that the solution is

817
00:50:46,300 --> 00:50:53,150
right on the function,
the continuous solution.

818
00:50:53,150 --> 00:50:59,150
Well, can I draw
a set of pictures

819
00:50:59,150 --> 00:51:03,420
just like those
guys for discrete?

820
00:51:03,420 --> 00:51:07,700
Yeah, let me just draw
those for discrete here.

821
00:51:07,700 --> 00:51:12,890
That shows you the magic.

822
00:51:12,890 --> 00:51:24,630
So there is a-- I'm going
to draw a vector now.

823
00:51:24,630 --> 00:51:27,220
I'm going to have to lift the
chalk, it won't be a function

824
00:51:27,220 --> 00:51:29,290
and it'll be the delta vector.

825
00:51:29,290 --> 00:51:32,600
So it'll be the
delta vector, delta

826
00:51:32,600 --> 00:51:37,730
with-- So there is point one,
zero, one, two, up to six.

827
00:51:37,730 --> 00:51:41,250
It'll be the delta vector.

828
00:51:41,250 --> 00:51:45,630
Well if I just draw
the delta vector,

829
00:51:45,630 --> 00:51:49,000
the delta vector
has a one there.

830
00:51:49,000 --> 00:51:51,360
So this is the delta vector.

831
00:51:51,360 --> 00:51:52,810
Do I need?

832
00:51:52,810 --> 00:51:57,610
Well you can see that
the delta vector is now

833
00:51:57,610 --> 00:52:01,140
going to be the
vector of all zeroes

834
00:52:01,140 --> 00:52:06,110
and it's got a one at the key--
at the impulse and then zero.

835
00:52:06,110 --> 00:52:07,940
So it's a discrete impulse.

836
00:52:07,940 --> 00:52:09,610
That would be a better word.

837
00:52:09,610 --> 00:52:10,870
Discrete impulse.

838
00:52:10,870 --> 00:52:13,520
Impulse at zero.

839
00:52:13,520 --> 00:52:16,700
So let's stay with
an impulse at zero.

840
00:52:16,700 --> 00:52:20,960
Alright.

841
00:52:20,960 --> 00:52:25,710
What's my next picture?

842
00:52:25,710 --> 00:52:28,780
Again let me put in zero.

843
00:52:28,780 --> 00:52:31,390
One, two, three, onwards.

844
00:52:31,390 --> 00:52:35,450
Minus one, so on.

845
00:52:35,450 --> 00:52:36,570
What do I want to do now?

846
00:52:36,570 --> 00:52:38,460
What do I draw second?

847
00:52:38,460 --> 00:52:40,890
I always look over here.

848
00:52:40,890 --> 00:52:44,080
What did I draw
second over here?

849
00:52:44,080 --> 00:52:46,450
The step.

850
00:52:46,450 --> 00:52:51,620
Now why did I draw
a step function?

851
00:52:51,620 --> 00:52:54,510
How did I get from here to here?

852
00:52:54,510 --> 00:52:56,260
I integrate.

853
00:52:56,260 --> 00:52:57,890
I took the integral.

854
00:52:57,890 --> 00:53:02,530
So how will I get from
here to this picture?

855
00:53:02,530 --> 00:53:07,300
I don't integrate, I add, sum.

856
00:53:07,300 --> 00:53:12,110
So coming along from the left,
all these all along here,

857
00:53:12,110 --> 00:53:14,560
this sum is all zero
because it was always zero.

858
00:53:14,560 --> 00:53:19,070
So it's zero, zero, zero, zero.

859
00:53:19,070 --> 00:53:21,660
And then, whoops, wait a minute.

860
00:53:21,660 --> 00:53:24,480
It says it a one there?

861
00:53:24,480 --> 00:53:26,630
Yeah, I think it must be.

862
00:53:26,630 --> 00:53:30,100
So here it wasn't a zero, wrong.

863
00:53:30,100 --> 00:53:31,880
Here it's a one.

864
00:53:31,880 --> 00:53:33,070
And what is it next?

865
00:53:33,070 --> 00:53:34,620
What's next to it?

866
00:53:34,620 --> 00:53:37,720
One, because I'm adding
more and more zeroes

867
00:53:37,720 --> 00:53:39,880
but I have that one now, okay.

868
00:53:39,880 --> 00:53:43,060
A discrete step.

869
00:53:43,060 --> 00:53:46,380
It's a discrete step,
zeroes and then ones.

870
00:53:46,380 --> 00:53:48,230
Now comes the second.

871
00:53:48,230 --> 00:53:52,860
So what am I going to call that?

872
00:53:52,860 --> 00:53:54,670
A step, right?

873
00:53:54,670 --> 00:54:01,620
It'll be a step
function, step vector.

874
00:54:01,620 --> 00:54:07,410
If the sums of the delta
vector gave me the step vector,

875
00:54:07,410 --> 00:54:11,530
how do I go the other way?

876
00:54:11,530 --> 00:54:13,990
What do I do to the
step vector to get back

877
00:54:13,990 --> 00:54:18,730
to the delta vector?

878
00:54:18,730 --> 00:54:20,180
Differences, right?

879
00:54:20,180 --> 00:54:23,500
Sums in one direction,
differences in the other.

880
00:54:23,500 --> 00:54:31,010
So the differences of the step
vector are the delta vector.

881
00:54:31,010 --> 00:54:33,780
The step is the
sum of the deltas

882
00:54:33,780 --> 00:54:37,600
and the delta is the
differences of the step.

883
00:54:37,600 --> 00:54:39,870
Now for the crucial next guy.

884
00:54:39,870 --> 00:54:42,450
What's it going to be?

885
00:54:42,450 --> 00:54:44,850
I add.

886
00:54:44,850 --> 00:54:46,090
Wait a minute.

887
00:54:46,090 --> 00:54:49,680
What's up?

888
00:54:49,680 --> 00:54:55,680
I'm looking for that picture.

889
00:54:55,680 --> 00:54:58,550
Do I get it?

890
00:54:58,550 --> 00:55:02,450
Yeah, I hope so.

891
00:55:02,450 --> 00:55:03,780
Oh, look, we ran out of time.

892
00:55:03,780 --> 00:55:05,940
I don't have to do
this, but I will.

893
00:55:05,940 --> 00:55:12,720
So as I add I get zeroes
and then it's one,

894
00:55:12,720 --> 00:55:15,200
and then I add on one more one.

895
00:55:15,200 --> 00:55:16,880
Look.

896
00:55:16,880 --> 00:55:18,420
You see what's happening.

897
00:55:18,420 --> 00:55:21,090
I run along at
zero but I'm going

898
00:55:21,090 --> 00:55:26,330
to look at the book to see
whether that jump should

899
00:55:26,330 --> 00:55:28,190
come here or here.

900
00:55:28,190 --> 00:55:32,070
So I've got a little bit
of this to finish next time

901
00:55:32,070 --> 00:55:35,970
and I'm open for any
questions this afternoon.

902
00:55:35,970 --> 00:55:38,810
Okay, thanks and sorry
to keep you late.