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00:00:01,792 --> 00:00:05,020
MAGDALENA TODA: Welcome
to our review of 13.1.

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00:00:05,020 --> 00:00:07,410
How many of you didn't
get your exams back?

4
00:00:07,410 --> 00:00:10,380
I have your exam, and yours.

5
00:00:10,380 --> 00:00:11,370
And you have to wait.

6
00:00:11,370 --> 00:00:12,607
I don't have it with me.

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00:00:12,607 --> 00:00:15,825
I have it in my office.

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00:00:15,825 --> 00:00:19,591
If you have questions
about the score,

9
00:00:19,591 --> 00:00:23,028
why don't you go ahead and
email me right after class.

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00:00:23,028 --> 00:00:31,375
Chapter 13 is a very
physical chapter.

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00:00:31,375 --> 00:00:34,262
It has a lot to do with
mechanical engineering,

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00:00:34,262 --> 00:00:36,970
with mechanics,
physics, electricity.

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00:00:36,970 --> 00:00:40,670


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You're going to see things,
weird things like work.

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You've already seen work.

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00:00:47,110 --> 00:00:49,780
Do you remember the definition?

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00:00:49,780 --> 00:00:52,962
So we define the work
as a path integral

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00:00:52,962 --> 00:00:55,120
along the regular curve.

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00:00:55,120 --> 00:00:58,600
And by regular curve-- I'm
sorry if I'm repeating myself,

20
00:00:58,600 --> 00:01:02,095
but this is part of the
deal-- R is the position

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00:01:02,095 --> 00:01:10,005
vector in R3 that is class C1.

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That means differentiable and
derivatives are continuous.

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00:01:15,243 --> 00:01:18,624
Plus you are not
allowed to stop.

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So no matter how drunk,
the bug has to keep flying,

25
00:01:24,580 --> 00:01:27,800
and not even for a
fraction of a second is he

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00:01:27,800 --> 00:01:31,090
or she allowed to
have velocity 0.

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00:01:31,090 --> 00:01:34,200
At no point I want
to have velocity 0.

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And that's the position vector.

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00:01:36,300 --> 00:01:43,620
And then you have some force
field acting on you-- no,

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00:01:43,620 --> 00:01:47,530
acting on the particle
at every moment.

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00:01:47,530 --> 00:01:54,010
So you have an F that is
acting at location xy.

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00:01:54,010 --> 00:01:58,140
Maybe if you are in space,
let's talk about the xyz,

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00:01:58,140 --> 00:02:02,130
where x is a function of
t, y is a functional of t,

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00:02:02,130 --> 00:02:06,520
z is a function of t,
which is the same as saying

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00:02:06,520 --> 00:02:11,910
that R of t, which is the given
position vector, is x of t

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00:02:11,910 --> 00:02:12,705
y of t.

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00:02:12,705 --> 00:02:15,580
Let me put angular bracket,
although I hate them,

38
00:02:15,580 --> 00:02:20,140
because you like angular
brackets for vectors.

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00:02:20,140 --> 00:02:22,910
F is also a nice function.

40
00:02:22,910 --> 00:02:24,490
How nice?

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00:02:24,490 --> 00:02:26,580
We discussed a
little bit last time.

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00:02:26,580 --> 00:02:28,860
It really doesn't
have to be continuous.

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00:02:28,860 --> 00:02:30,650
The book assumes it continues.

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00:02:30,650 --> 00:02:33,422
It has to be
integrable, so maybe it

45
00:02:33,422 --> 00:02:35,720
could be piecewise continuous.

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00:02:35,720 --> 00:02:42,596
So I had nice enough, was
it continues piecewise.

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00:02:42,596 --> 00:02:46,390


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00:02:46,390 --> 00:02:51,480
And we define the work as
being the path integral over c.

49
00:02:51,480 --> 00:02:54,730
I keep repeating, because
that's going to be on the final

50
00:02:54,730 --> 00:02:55,790
as well.

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00:02:55,790 --> 00:02:58,780
So all the notions
that are important

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00:02:58,780 --> 00:03:02,650
should be given enough
attention in this class.

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00:03:02,650 --> 00:03:03,430
Hi.

54
00:03:03,430 --> 00:03:09,270
So do you guys remember
how we denoted F?

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00:03:09,270 --> 00:03:15,015
F was, in general, three
components in our F1, F2, F3.

56
00:03:15,015 --> 00:03:18,827
They are functions of
the position vector,

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00:03:18,827 --> 00:03:22,010
or the position xyz.

58
00:03:22,010 --> 00:03:24,740
And the position is
a function of time.

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00:03:24,740 --> 00:03:28,380
So all in all, after
you do all the work,

60
00:03:28,380 --> 00:03:35,230
keep in mind that when you
multiply with a dot product,

61
00:03:35,230 --> 00:03:40,090
the integral will give you what?

62
00:03:40,090 --> 00:03:42,420
A time integral?

63
00:03:42,420 --> 00:03:47,210
From a time T0 to a time
T1, you are here at time T0

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00:03:47,210 --> 00:03:49,330
and you are here at time T1.

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00:03:49,330 --> 00:03:51,870


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00:03:51,870 --> 00:03:55,647
Maybe your curve is
piecewise, differentiable,

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00:03:55,647 --> 00:03:56,730
you don't know what it is.

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00:03:56,730 --> 00:04:01,890
But let's assume just a
very nice, smooth arc here.

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00:04:01,890 --> 00:04:03,030
Of what?

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00:04:03,030 --> 00:04:11,010
Of F1 times what is that?
x prime of t plus F2 times

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00:04:11,010 --> 00:04:18,160
y prime of t, plus F3
times z prime of t dt.

72
00:04:18,160 --> 00:04:21,440
So keep in mind that
Mr. dR is your friend.

73
00:04:21,440 --> 00:04:23,420
And he was-- what was he?

74
00:04:23,420 --> 00:04:29,110
Was defined as the
velocity vector multiplied

75
00:04:29,110 --> 00:04:32,380
by the infinitesimal element dt.

76
00:04:32,380 --> 00:04:35,840
Say again, the
velocity vector prime

77
00:04:35,840 --> 00:04:41,610
was a vector in F3 quantified
by the infinitesimal element dt.

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00:04:41,610 --> 00:04:46,140
So we reduce this Calc
3 notion path integral

79
00:04:46,140 --> 00:04:53,200
to a Calc 1 notion, which was a
simple integral from t0 to t1.

80
00:04:53,200 --> 00:04:55,160
And we've done a
lot of applications.

81
00:04:55,160 --> 00:04:56,630
What else have we done?

82
00:04:56,630 --> 00:05:00,430
We've done some
integral of this type

83
00:05:00,430 --> 00:05:03,850
over another curve, script c.

84
00:05:03,850 --> 00:05:06,030
I'm repeating mostly for Alex.

85
00:05:06,030 --> 00:05:10,040
You're caught in the process.

86
00:05:10,040 --> 00:05:14,310
And there are two or three
people who need an update.

87
00:05:14,310 --> 00:05:19,380
Maybe I have another
function of g and ds.

88
00:05:19,380 --> 00:05:25,296
And this is an integral that
in the end will depend on s.

89
00:05:25,296 --> 00:05:27,700
But s itself depends on t.

90
00:05:27,700 --> 00:05:31,410
So if I were to re-express
this in terms of d,

91
00:05:31,410 --> 00:05:34,910
how would I re-express
the whole thing?

92
00:05:34,910 --> 00:05:40,555
g of s, of t, whatever that
is, then Mr. ds was what?

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00:05:40,555 --> 00:05:42,040
STUDENT: s prime of t.

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00:05:42,040 --> 00:05:42,956
MAGDALENA TODA: Right.

95
00:05:42,956 --> 00:05:47,556
So this was the-- that s
prime of t was the speed.

96
00:05:47,556 --> 00:05:51,298
The speed of the arc of a curve.

97
00:05:51,298 --> 00:05:59,310
So you have an R of
t and R3, a vector.

98
00:05:59,310 --> 00:06:04,150
And the speed was,
by definition,

99
00:06:04,150 --> 00:06:07,245
arc length element was
by definition integral

100
00:06:07,245 --> 00:06:09,170
from 2t0 to t.

101
00:06:09,170 --> 00:06:14,745
Of the speed R prime
magnitude d tau.

102
00:06:14,745 --> 00:06:17,710
I'll have you put tau
because I'm Greek,

103
00:06:17,710 --> 00:06:19,090
and it's all Greek to me.

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00:06:19,090 --> 00:06:23,360
So the tau, some people call
the tau the dummy variable.

105
00:06:23,360 --> 00:06:24,930
I don't like to call it dumb.

106
00:06:24,930 --> 00:06:27,680
It's a very smart variable.

107
00:06:27,680 --> 00:06:31,160
It goes from t0 to t, so what
you have is a function of t.

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00:06:31,160 --> 00:06:32,900
This guy is speed.

109
00:06:32,900 --> 00:06:38,880
So when you do that
here, ds becomes speed,

110
00:06:38,880 --> 00:06:43,230
R prime of t times dt.

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00:06:43,230 --> 00:06:45,290
This was your old friend ds.

112
00:06:45,290 --> 00:06:50,750
And let me put it on top
of this guy with speed.

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00:06:50,750 --> 00:06:53,633
Because he was so
important to you,

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00:06:53,633 --> 00:06:56,339
you cannot forget about him.

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00:06:56,339 --> 00:07:03,846
So that was review of--
reviewing of 13.1 and 13.2

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00:07:03,846 --> 00:07:10,235
There were some things in
13.3 that I pointed out

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00:07:10,235 --> 00:07:12,400
to you are important.

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00:07:12,400 --> 00:07:16,990
13.3 was independence of path.

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Everybody write, magic-- no.

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00:07:18,920 --> 00:07:20,800
Magic section.

121
00:07:20,800 --> 00:07:22,470
No, have to be serious.

122
00:07:22,470 --> 00:07:31,950
So that's independence of path
of certain type of integrals,

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00:07:31,950 --> 00:07:35,080
of some integrals.

124
00:07:35,080 --> 00:07:37,730
And an integral like
that, a path integral

125
00:07:37,730 --> 00:07:41,060
is independent of path.

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00:07:41,060 --> 00:07:44,530
When would such an animal--
look at this pink animal,

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00:07:44,530 --> 00:07:47,030
inside-- when would
this not depend

128
00:07:47,030 --> 00:07:51,234
on the path you are taking
between two given points?

129
00:07:51,234 --> 00:07:55,130
So I can move on another
arc and another arc

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00:07:55,130 --> 00:07:59,320
and another regular arc, and
all sorts of regular arcs.

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00:07:59,320 --> 00:08:02,398
It doesn't matter
which path I'm taking--

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00:08:02,398 --> 00:08:04,120
STUDENT: If that
force is conservative.

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00:08:04,120 --> 00:08:05,995
MAGDALENA TODA: If the
force is conservative.

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00:08:05,995 --> 00:08:07,165
Excellent, Alex.

135
00:08:07,165 --> 00:08:12,770
And what did it mean for a
force to be conservative?

136
00:08:12,770 --> 00:08:15,880
How many of you
know-- it's no shame.

137
00:08:15,880 --> 00:08:17,350
Just raise hands.

138
00:08:17,350 --> 00:08:19,750
If you forgot what it is,
don't raise your hand.

139
00:08:19,750 --> 00:08:23,350
But if you remember what it
means for a force F force

140
00:08:23,350 --> 00:08:27,340
field-- may the force be
with you-- be conservative,

141
00:08:27,340 --> 00:08:30,157
then what do you do?

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00:08:30,157 --> 00:08:33,150
Say F is conservative
by definition.

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00:08:33,150 --> 00:08:38,751


144
00:08:38,751 --> 00:08:50,084
When, if and only, F
there is a so-called--

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00:08:50,084 --> 00:08:50,750
STUDENT: Scalar.

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00:08:50,750 --> 00:08:51,916
MAGDALENA TODA: --potential.

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00:08:51,916 --> 00:08:53,527
Scalar potential, thank you.

148
00:08:53,527 --> 00:08:54,110
I'll fix that.

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00:08:54,110 --> 00:09:01,770
A scalar potential function f.

150
00:09:01,770 --> 00:09:07,130


151
00:09:07,130 --> 00:09:09,850
Instead of there is, I
didn't want to put this.

152
00:09:09,850 --> 00:09:11,650
Because a few
people told me they

153
00:09:11,650 --> 00:09:14,570
got scared about
the symbolistics.

154
00:09:14,570 --> 00:09:17,146
This means "there exists."

155
00:09:17,146 --> 00:09:22,380
OK, smooth potential
such that-- at least

156
00:09:22,380 --> 00:09:26,800
is differential [INAUDIBLE]
1 such that the nabla of f--

157
00:09:26,800 --> 00:09:28,240
what the heck is that?

158
00:09:28,240 --> 00:09:32,180
The gradient of this little
f will be the given F.

159
00:09:32,180 --> 00:09:38,500
And we saw all sorts of wizards
here, like, Harry Potter,

160
00:09:38,500 --> 00:09:42,930
[INAUDIBLE] well,
there are many,

161
00:09:42,930 --> 00:09:47,670
Alex, Erin, many,
many-- Matthew.

162
00:09:47,670 --> 00:09:49,540
So what did they do?

163
00:09:49,540 --> 00:09:50,961
They guessed the
scalar potential.

164
00:09:50,961 --> 00:09:53,370
I had to stop because
there are 10 of them.

165
00:09:53,370 --> 00:09:55,940
It's a whole school
of Harry Potter.

166
00:09:55,940 --> 00:09:58,760
How do they find the little f?

167
00:09:58,760 --> 00:10:00,790
Through witchcraft.

168
00:10:00,790 --> 00:10:01,860
No.

169
00:10:01,860 --> 00:10:02,735
Normally you should--

170
00:10:02,735 --> 00:10:04,818
STUDENT: I've actually
done it through witchcraft.

171
00:10:04,818 --> 00:10:05,506
Tell you that?

172
00:10:05,506 --> 00:10:06,505
MAGDALENA TODA: You did.

173
00:10:06,505 --> 00:10:08,770
I think you can do it
through witchcraft.

174
00:10:08,770 --> 00:10:15,080
But practically everybody
has the ability to guess.

175
00:10:15,080 --> 00:10:17,512
Why do we have the ability
to guess and check?

176
00:10:17,512 --> 00:10:21,210
Because our brain does
the integration for you.

177
00:10:21,210 --> 00:10:24,160
Whether you tell your
brain to stop or not,

178
00:10:24,160 --> 00:10:27,250
when your brain, for example,
sees is kind of function--

179
00:10:27,250 --> 00:10:30,340
and now I'm gonna
test your magic skills

180
00:10:30,340 --> 00:10:32,040
on a little harder one.

181
00:10:32,040 --> 00:10:36,040
I didn't want to do an
R2 value vector function.

182
00:10:36,040 --> 00:10:38,070
Let me go to R3.

183
00:10:38,070 --> 00:10:42,070
But I know that you have
your witchcraft handy.

184
00:10:42,070 --> 00:10:47,650
So let's say somebody
gave you a force field

185
00:10:47,650 --> 00:10:55,620
that is yz i plus xzj plus xyk.

186
00:10:55,620 --> 00:10:58,900
And you're going to jump and
say this is a piece of cake.

187
00:10:58,900 --> 00:11:03,680
I can see the scalar potential
and just wave my magic wand,

188
00:11:03,680 --> 00:11:06,950
and I get it.

189
00:11:06,950 --> 00:11:08,099
STUDENT: [INAUDIBLE]

190
00:11:08,099 --> 00:11:09,390
MAGDALENA TODA: Oh my god, yes.

191
00:11:09,390 --> 00:11:11,000
Guys, you saw it fast.

192
00:11:11,000 --> 00:11:12,780
OK, I should be proud of you.

193
00:11:12,780 --> 00:11:14,300
And I am proud of you.

194
00:11:14,300 --> 00:11:18,610
I've had made classes
where the students couldn't

195
00:11:18,610 --> 00:11:21,930
see any of the scalar
potentials that I gave them,

196
00:11:21,930 --> 00:11:23,930
that I asked them to guess.

197
00:11:23,930 --> 00:11:25,380
How did you deal with it?

198
00:11:25,380 --> 00:11:27,500
You integrate this
with respect to F?

199
00:11:27,500 --> 00:11:29,927
In the back of
your mind you did.

200
00:11:29,927 --> 00:11:31,510
And then you guessed
one, and then you

201
00:11:31,510 --> 00:11:34,120
said, OK so should be xyz.

202
00:11:34,120 --> 00:11:36,790
Does it verify my
other two conditions?

203
00:11:36,790 --> 00:11:38,460
And you say, oh yeah, it does.

204
00:11:38,460 --> 00:11:42,550
Because of I prime with respect
to y, I have exactly xz.

205
00:11:42,550 --> 00:11:46,510
If I prime with respect to c I
have exactly xy, so I got it.

206
00:11:46,510 --> 00:11:49,560
And even if somebody
said xyz plus 7,

207
00:11:49,560 --> 00:11:51,160
they would still be right.

208
00:11:51,160 --> 00:11:56,060
In the end you can have
any xyz plus a constant.

209
00:11:56,060 --> 00:11:58,260
In general it's not
so easy to guess.

210
00:11:58,260 --> 00:12:02,020
But there are lots of examples
of conservative forces where

211
00:12:02,020 --> 00:12:06,590
you simply cannot see the scalar
potential or cannot deduce it

212
00:12:06,590 --> 00:12:09,850
like in a few seconds.

213
00:12:09,850 --> 00:12:12,970
Expect something easy,
though, like that,

214
00:12:12,970 --> 00:12:14,945
something that you can see.

215
00:12:14,945 --> 00:12:15,820
Let's see an example.

216
00:12:15,820 --> 00:12:19,260
Assume this is your force field
acting on a particle that's

217
00:12:19,260 --> 00:12:23,510
moving on a curving space.

218
00:12:23,510 --> 00:12:29,130
And it's stubborn and it
decides to move on a helix,

219
00:12:29,130 --> 00:12:32,560
because it's a-- I don't
know what kind of particle

220
00:12:32,560 --> 00:12:34,696
would move on a
helix, but suppose

221
00:12:34,696 --> 00:12:39,490
a lot of particles, just a
little train or a drunken bug

222
00:12:39,490 --> 00:12:40,690
or something.

223
00:12:40,690 --> 00:12:45,070
And you were moving
on another helix.

224
00:12:45,070 --> 00:12:52,170
Now suppose that helix will
be R of t equals cosine t

225
00:12:52,170 --> 00:13:01,110
sine t and t where you
have t as 0 to start with.

226
00:13:01,110 --> 00:13:02,352
What do I have at 0?

227
00:13:02,352 --> 00:13:05,640
The point 1, 0, 0.

228
00:13:05,640 --> 00:13:09,120
That's the point,
let's call it A.

229
00:13:09,120 --> 00:13:12,870
And let's call this B. I
don't know what I want to do.

230
00:13:12,870 --> 00:13:16,580
I'll just do a
complete rotation,

231
00:13:16,580 --> 00:13:18,510
just to make my life easier.

232
00:13:18,510 --> 00:13:23,120
And this is B. And that
will be A at t equals 0

233
00:13:23,120 --> 00:13:25,160
and B equals 2 pi.

234
00:13:25,160 --> 00:13:32,840


235
00:13:32,840 --> 00:13:35,352
So what will this be at B?

236
00:13:35,352 --> 00:13:37,120
STUDENT: 1, 0, 2 pi.

237
00:13:37,120 --> 00:13:40,430
MAGDALENA TODA: 1, 0, and 2 pi.

238
00:13:40,430 --> 00:13:44,320
So you perform a complete
rotation and come back.

239
00:13:44,320 --> 00:13:49,750
Now, if your force is
conservative, you are lucky.

240
00:13:49,750 --> 00:13:53,100
Because you know the theorem
that says in that case

241
00:13:53,100 --> 00:13:57,820
the work integral will
be independent of path.

242
00:13:57,820 --> 00:14:03,910
And due to the theorem in-- what
section was that again-- 13.3,

243
00:14:03,910 --> 00:14:06,890
independence of path,
you know that this

244
00:14:06,890 --> 00:14:11,720
is going to be-- let me rewrite
it one more time with gradient

245
00:14:11,720 --> 00:14:16,150
of f instead of big F.

246
00:14:16,150 --> 00:14:19,615
And this will become what,
f of the q-- not the q.

247
00:14:19,615 --> 00:14:24,065
In the book it's f of q minus
f of q. f of B minus f of A,

248
00:14:24,065 --> 00:14:24,565
right?

249
00:14:24,565 --> 00:14:28,050


250
00:14:28,050 --> 00:14:29,030
What does this mean?

251
00:14:29,030 --> 00:14:33,410
You have to measure
the-- to evaluate

252
00:14:33,410 --> 00:14:39,440
the coordinates of
this function xyz

253
00:14:39,440 --> 00:14:50,080
where t equals 2 pi minus
xyz where t equals what?

254
00:14:50,080 --> 00:14:52,404
0.

255
00:14:52,404 --> 00:14:56,830
And now I have to be
careful, because I

256
00:14:56,830 --> 00:14:58,200
have to evaluate them.

257
00:14:58,200 --> 00:15:06,632
So when t is 0 I have x
is 1, y is 0, and t is 0.

258
00:15:06,632 --> 00:15:08,100
In the end it doesn't matter.

259
00:15:08,100 --> 00:15:12,530
I can get 0-- I
can get 0 for this

260
00:15:12,530 --> 00:15:14,700
and get 0 for that as well.

261
00:15:14,700 --> 00:15:20,600
So when this is 2 pi I get
x equals 1, y equals 0,

262
00:15:20,600 --> 00:15:23,210
and t equals 2 pi.

263
00:15:23,210 --> 00:15:26,650
So in the end, both products
are 0 and I got a 0.

264
00:15:26,650 --> 00:15:31,890
So although the [INAUDIBLE]
works very hard-- I mean,

265
00:15:31,890 --> 00:15:36,450
works hard in our perception
to get from a point

266
00:15:36,450 --> 00:15:38,730
to another-- the work is 0.

267
00:15:38,730 --> 00:15:39,380
Why?

268
00:15:39,380 --> 00:15:42,470
Because it's a vector
value thing inside.

269
00:15:42,470 --> 00:15:46,560
And there are some
annihilations going on.

270
00:15:46,560 --> 00:15:50,473
So that reminds me
of another example.

271
00:15:50,473 --> 00:15:52,828
So we are done
with this example.

272
00:15:52,828 --> 00:15:55,654
Let's go back to our washer.

273
00:15:55,654 --> 00:15:57,560
I was just doing
laundry last night

274
00:15:57,560 --> 00:16:01,360
and I was thinking of
the washer example.

275
00:16:01,360 --> 00:16:04,960
And I thought of a small
variation of the washer

276
00:16:04,960 --> 00:16:09,430
example, just assuming that
I would give you a pop quiz.

277
00:16:09,430 --> 00:16:12,120
And I'm not giving you
a pop quiz right now.

278
00:16:12,120 --> 00:16:14,940
But if I gave you
a pop quiz now,

279
00:16:14,940 --> 00:16:20,850
I would ask you example
two, the washer.

280
00:16:20,850 --> 00:16:24,412


281
00:16:24,412 --> 00:16:27,770
It is performing
a circular motion,

282
00:16:27,770 --> 00:16:30,000
and I want to know
the work performed

283
00:16:30,000 --> 00:16:36,320
by the centrifugal force
between various points.

284
00:16:36,320 --> 00:16:48,180
So have the circular motion,
the centrifugal force.

285
00:16:48,180 --> 00:16:50,892
This is the
centrifugal, I'm sorry.

286
00:16:50,892 --> 00:16:54,065
I'll take the centrifugal force.

287
00:16:54,065 --> 00:16:56,480
And that was last
time we discussed

288
00:16:56,480 --> 00:17:04,960
that, that was extending
the radius of the initial--

289
00:17:04,960 --> 00:17:07,220
the vector value position.

290
00:17:07,220 --> 00:17:11,560
So you have that in
every point, xi plus yj.

291
00:17:11,560 --> 00:17:14,670
And you want F to
be able xi plus yj.

292
00:17:14,670 --> 00:17:20,368
But it points outside
from the point

293
00:17:20,368 --> 00:17:22,965
on the circular trajectory.

294
00:17:22,965 --> 00:17:26,040


295
00:17:26,040 --> 00:17:31,350
And I asked you, find
out what you performed

296
00:17:31,350 --> 00:17:38,686
by F in one full rotation.

297
00:17:38,686 --> 00:17:43,760


298
00:17:43,760 --> 00:17:48,975
We gave the equation of motion,
being cosine t y sine t,

299
00:17:48,975 --> 00:17:51,560
if you remember from last time.

300
00:17:51,560 --> 00:17:57,390
And then W2, let's
say, is performed by F

301
00:17:57,390 --> 00:18:01,910
from t equals 0 to t equals pi.

302
00:18:01,910 --> 00:18:03,496
I want that as well.

303
00:18:03,496 --> 00:18:12,570
And W2 performed by F from
t-- that makes t0 to t

304
00:18:12,570 --> 00:18:20,030
equals pi-- t equals 0
to t equals pi over 4.

305
00:18:20,030 --> 00:18:21,830
These are all very
easy questions,

306
00:18:21,830 --> 00:18:24,970
and you should be able to
answer them in no time.

307
00:18:24,970 --> 00:18:28,430
Now, let me tell you something.

308
00:18:28,430 --> 00:18:29,710
We are in plane, not in space.

309
00:18:29,710 --> 00:18:30,790
But it doesn't matter.

310
00:18:30,790 --> 00:18:35,440
It's like the third quadrant
would be 0, piece of cake.

311
00:18:35,440 --> 00:18:39,997
Your eye should be so
well-trained that when

312
00:18:39,997 --> 00:18:41,580
you look at the force
field like that,

313
00:18:41,580 --> 00:18:44,159
and people talk about what
you should ask yourself,

314
00:18:44,159 --> 00:18:44,950
is it conservative?

315
00:18:44,950 --> 00:18:48,390


316
00:18:48,390 --> 00:18:51,010
And it is conservative.

317
00:18:51,010 --> 00:18:53,830
And that means little f is what?

318
00:18:53,830 --> 00:18:56,900


319
00:18:56,900 --> 00:18:59,700
Nitish said that yesterday.

320
00:18:59,700 --> 00:19:00,860
Why did you go there?

321
00:19:00,860 --> 00:19:02,630
You want to sleep today?

322
00:19:02,630 --> 00:19:05,510
I'm just teasing you.

323
00:19:05,510 --> 00:19:08,470
I got so comfortable with
you sitting in the front row.

324
00:19:08,470 --> 00:19:10,070
STUDENT: I took his spot.

325
00:19:10,070 --> 00:19:12,071
STUDENT: She doesn't like
you sitting over here.

326
00:19:12,071 --> 00:19:13,070
MAGDALENA TODA: It's OK.

327
00:19:13,070 --> 00:19:13,920
It's fine.

328
00:19:13,920 --> 00:19:16,710
I still give him credit
for what he said last time.

329
00:19:16,710 --> 00:19:19,570
So do you guys remember,
he gave us this answer?

330
00:19:19,570 --> 00:19:23,070
x squared plus y squared over
2, and he found the scalar

331
00:19:23,070 --> 00:19:26,980
potential through witchcraft
in about a second and a half?

332
00:19:26,980 --> 00:19:28,190
OK.

333
00:19:28,190 --> 00:19:31,430
We are gonna conclude something.

334
00:19:31,430 --> 00:19:36,420
Do you remember that I found the
answer by find the explanation?

335
00:19:36,420 --> 00:19:39,160
I got W to be 0.

336
00:19:39,160 --> 00:19:45,340
But if I were to find another
explanation why the work would

337
00:19:45,340 --> 00:19:50,180
be 0 in this case, it
would have been 0 anyway

338
00:19:50,180 --> 00:19:53,220
for any force field.

339
00:19:53,220 --> 00:19:58,117
Even if I took the F
to be something else.

340
00:19:58,117 --> 00:20:03,320
Assume that F would be
G. Really wild, crazy,

341
00:20:03,320 --> 00:20:06,970
but still differentiable
vector value function.

342
00:20:06,970 --> 00:20:08,700
G differential.

343
00:20:08,700 --> 00:20:15,050
Would the work that we
want be the same for G?

344
00:20:15,050 --> 00:20:15,727
STUDENT: Yeah.

345
00:20:15,727 --> 00:20:16,560
MAGDALENA TODA: Why?

346
00:20:16,560 --> 00:20:18,310
STUDENT: Because of
displacement scenario.

347
00:20:18,310 --> 00:20:20,680
MAGDALENA TODA: Since
it's conservative,

348
00:20:20,680 --> 00:20:23,150
you have a closed loop.

349
00:20:23,150 --> 00:20:25,800
So the closed loop
will say, thick F

350
00:20:25,800 --> 00:20:30,340
at that terminal point minus
thick F at the initial point.

351
00:20:30,340 --> 00:20:33,620
But if a loop motion,
your terminal point

352
00:20:33,620 --> 00:20:35,290
is the initial point.

353
00:20:35,290 --> 00:20:36,040
Duh.

354
00:20:36,040 --> 00:20:40,902
So you have the
same point, the P

355
00:20:40,902 --> 00:20:44,490
equals qe if it's
a closed curve.

356
00:20:44,490 --> 00:20:48,294
So for a closed curve--
we also call that a loop.

357
00:20:48,294 --> 00:20:50,210
With a basketball, it
would have been too easy

358
00:20:50,210 --> 00:20:54,920
and you would have gotten a
dollar for free like that.

359
00:20:54,920 --> 00:20:57,930
So any closed curve
is called a loop.

360
00:20:57,930 --> 00:21:01,610
If your force field is
conservative-- attention,

361
00:21:01,610 --> 00:21:05,390
you might have examples
like that in the exams--

362
00:21:05,390 --> 00:21:08,990
then it doesn't matter
who little f is,

363
00:21:08,990 --> 00:21:12,510
if p equals q you get 0 anyway.

364
00:21:12,510 --> 00:21:15,750
But the reason why I
said you would get 0

365
00:21:15,750 --> 00:21:20,950
on the example of last time
was a slightly different one.

366
00:21:20,950 --> 00:21:24,367
What does the engineer
say to himself?

367
00:21:24,367 --> 00:21:25,700
STUDENT: Force is perpendicular.

368
00:21:25,700 --> 00:21:26,360
MAGDALENA TODA: Yeah.

369
00:21:26,360 --> 00:21:27,000
Very good.

370
00:21:27,000 --> 00:21:28,990
Whenever the force
is perpendicular

371
00:21:28,990 --> 00:21:33,180
to the trajectory, I'm going
to get 0 for the force.

372
00:21:33,180 --> 00:21:36,710
Because at every
moment the dot product

373
00:21:36,710 --> 00:21:41,160
between the force and the
displacement direction,

374
00:21:41,160 --> 00:21:45,190
which would be like dR, the
tangent to the displacement,

375
00:21:45,190 --> 00:21:46,750
would be [INAUDIBLE].

376
00:21:46,750 --> 00:21:50,190
And cosine of [INAUDIBLE] is 0.

377
00:21:50,190 --> 00:21:50,770
Duh.

378
00:21:50,770 --> 00:21:52,810
So that's another reason.

379
00:21:52,810 --> 00:21:59,822
Reason of last time
was F perpendicular

380
00:21:59,822 --> 00:22:05,330
to the R prime
direction, R prime

381
00:22:05,330 --> 00:22:11,030
being the velocity-- look,
when I'm moving in a circle,

382
00:22:11,030 --> 00:22:13,030
this is the force.

383
00:22:13,030 --> 00:22:14,520
And I'm moving.

384
00:22:14,520 --> 00:22:17,660
This is my velocity, is
the tangent to the circle.

385
00:22:17,660 --> 00:22:22,460
And the velocity and the normal
are always perpendicular,

386
00:22:22,460 --> 00:22:23,110
at every point.

387
00:22:23,110 --> 00:22:24,000
That's why I have 0.

388
00:22:24,000 --> 00:22:26,600


389
00:22:26,600 --> 00:22:32,100
So note that even if I
didn't take a close look,

390
00:22:32,100 --> 00:22:36,020
why would the answer
be from 0 to pi?

391
00:22:36,020 --> 00:22:38,670
Still?

392
00:22:38,670 --> 00:22:42,685
0 because of that.

393
00:22:42,685 --> 00:22:43,640
0.

394
00:22:43,640 --> 00:22:47,000
How about from 0 to pi over 4?

395
00:22:47,000 --> 00:22:49,890
Still 0.

396
00:22:49,890 --> 00:22:52,170
And of course if somebody
would not believe them,

397
00:22:52,170 --> 00:22:54,860
if somebody would not
understand the theory,

398
00:22:54,860 --> 00:22:57,605
they would do the work and
they would get to the answer

399
00:22:57,605 --> 00:23:01,046
and say, oh my
god, yeah, I got 0.

400
00:23:01,046 --> 00:23:02,660
All right?

401
00:23:02,660 --> 00:23:03,670
OK.

402
00:23:03,670 --> 00:23:09,796
Now, what if somebody--
and I want to spray this.

403
00:23:09,796 --> 00:23:11,730
Can I go ahead and
erase the board

404
00:23:11,730 --> 00:23:15,160
and move onto example
three or whatever?

405
00:23:15,160 --> 00:23:16,420
Yes?

406
00:23:16,420 --> 00:23:17,210
OK.

407
00:23:17,210 --> 00:23:19,081
All right.

408
00:23:19,081 --> 00:23:21,520
STUDENT: Could you say
non-conservative force?

409
00:23:21,520 --> 00:23:23,570
MAGDALENA TODA: Yeah,
that's what I-- exactly.

410
00:23:23,570 --> 00:23:26,371
You are a mind reader.

411
00:23:26,371 --> 00:23:31,838
You are gonna guess my mind.

412
00:23:31,838 --> 00:23:44,263


413
00:23:44,263 --> 00:23:46,800
And I'm going to
pick a nasty one.

414
00:23:46,800 --> 00:23:49,845
And since I'm doing
review anyway,

415
00:23:49,845 --> 00:23:50,970
you may have one like that.

416
00:23:50,970 --> 00:23:55,250
And you may have both one that
involves a conservative force

417
00:23:55,250 --> 00:24:00,110
field and one that does not
involve a conservative force

418
00:24:00,110 --> 00:24:00,690
field.

419
00:24:00,690 --> 00:24:07,450
And we can ask you, find us the
work belong to different path.

420
00:24:07,450 --> 00:24:11,690
And I've done this
type of example before.

421
00:24:11,690 --> 00:24:15,450
Let's take F of
x and y in plane.

422
00:24:15,450 --> 00:24:29,050
In our two I take xyi
plus x squared y of j.

423
00:24:29,050 --> 00:24:34,650
And the problem would
involve my favorite picture,

424
00:24:34,650 --> 00:24:39,000
y equals x squared and y
equals x, our two paths.

425
00:24:39,000 --> 00:24:40,520
One is the straight path.

426
00:24:40,520 --> 00:24:43,700
One is the [INAUDIBLE] path.

427
00:24:43,700 --> 00:24:46,280
They go from 0,
0 to 1, 1 anyway.

428
00:24:46,280 --> 00:24:49,220


429
00:24:49,220 --> 00:24:58,250
And I'm asking you to
find W1 along path one

430
00:24:58,250 --> 00:25:01,250
and W2 along path two.

431
00:25:01,250 --> 00:25:06,280
And of course,
example three, if this

432
00:25:06,280 --> 00:25:10,270
were conservative
you would say, oh,

433
00:25:10,270 --> 00:25:11,895
it doesn't matter
what path I'm taking,

434
00:25:11,895 --> 00:25:14,930
I'm still getting
the same answer.

435
00:25:14,930 --> 00:25:17,499
But is this conservative?

436
00:25:17,499 --> 00:25:18,425
STUDENT: No.

437
00:25:18,425 --> 00:25:20,177
Because you said it wasn't.

438
00:25:20,177 --> 00:25:21,260
MAGDALENA TODA: Very good.

439
00:25:21,260 --> 00:25:22,870
So how do you know?

440
00:25:22,870 --> 00:25:26,220
That's one test
when you are in two.

441
00:25:26,220 --> 00:25:30,708
There is the magic test that
says-- let's say this is M,

442
00:25:30,708 --> 00:25:36,700
and let's say this is N. You
would have to check if M sub--

443
00:25:36,700 --> 00:25:37,271
STUDENT: y.

444
00:25:37,271 --> 00:25:38,020
MAGDALENA TODA: y.

445
00:25:38,020 --> 00:25:38,519
Very good.

446
00:25:38,519 --> 00:25:39,640
I'm proud of you.

447
00:25:39,640 --> 00:25:42,260
You're ready for
3350, by the way.

448
00:25:42,260 --> 00:25:44,030
Is equal to N sub x.

449
00:25:44,030 --> 00:25:46,600
M sub y is x.

450
00:25:46,600 --> 00:25:49,180
N sub x is 2xy.

451
00:25:49,180 --> 00:25:51,530
They are not equal.

452
00:25:51,530 --> 00:25:55,240
So that's me crying that I have
to do the work twice and get--

453
00:25:55,240 --> 00:25:57,634
probably I'll get two
different examples.

454
00:25:57,634 --> 00:26:00,970


455
00:26:00,970 --> 00:26:03,510
If you read the book--
I'm afraid to ask

456
00:26:03,510 --> 00:26:09,320
how many of you opened the
book at section 13.2, 13.3.

457
00:26:09,320 --> 00:26:12,000
But did you read
it, any of them?

458
00:26:12,000 --> 00:26:13,000
STUDENT: Nitish read it.

459
00:26:13,000 --> 00:26:15,400
MAGDALENA TODA: Oh, good.

460
00:26:15,400 --> 00:26:20,140
There is another criteria for
a force to be conservative.

461
00:26:20,140 --> 00:26:22,120
If you are, it's piece of cake.

462
00:26:22,120 --> 00:26:23,296
You do that, right?

463
00:26:23,296 --> 00:26:24,337
MAGDALENA TODA: Yes, sir?

464
00:26:24,337 --> 00:26:25,620
STUDENT: Curl has frequency 0.

465
00:26:25,620 --> 00:26:27,494
MAGDALENA TODA: The curl
criteria, excellent.

466
00:26:27,494 --> 00:26:29,000
The curl has to be zero.

467
00:26:29,000 --> 00:26:38,977
So if F in R 3 is
conservative, then you'll

468
00:26:38,977 --> 00:26:40,060
get different order curve.

469
00:26:40,060 --> 00:26:42,052
Curl F is 0.

470
00:26:42,052 --> 00:26:44,640
Now let's check what
the heck was curl.

471
00:26:44,640 --> 00:26:47,610
You see, mathematics
is not a bunch

472
00:26:47,610 --> 00:26:51,910
of these joint discussions
like other sciences.

473
00:26:51,910 --> 00:26:55,340
In mathematics, if you don't
know a section or you skipped

474
00:26:55,340 --> 00:26:58,450
it, you are sick, you
have a date that day,

475
00:26:58,450 --> 00:27:02,700
you didn't study, then it's
all over because you cannot

476
00:27:02,700 --> 00:27:06,650
understand how to work out the
problems and materials if you

477
00:27:06,650 --> 00:27:08,000
skip the section.

478
00:27:08,000 --> 00:27:12,020
Curl was the one
where we learned

479
00:27:12,020 --> 00:27:16,150
that we used the determinant.

480
00:27:16,150 --> 00:27:17,300
That's the easiest story.

481
00:27:17,300 --> 00:27:20,740
It came with a t-shirt,
but that t-shirt really

482
00:27:20,740 --> 00:27:25,800
doesn't help because
it's easier to,

483
00:27:25,800 --> 00:27:28,270
instead of memorizing
the formula,

484
00:27:28,270 --> 00:27:31,440
you set out the determinant.

485
00:27:31,440 --> 00:27:33,865
So you have the operator
derivative with respect

486
00:27:33,865 --> 00:27:40,760
to x, y z followed by what?

487
00:27:40,760 --> 00:27:43,100
F1, F2, F3.

488
00:27:43,100 --> 00:27:46,945
Now in your case, I'm
asking you if you did it

489
00:27:46,945 --> 00:27:51,680
for this F, what is
the third component?

490
00:27:51,680 --> 00:27:52,510
STUDENT: The 0.

491
00:27:52,510 --> 00:27:54,800
MAGDALENA TODA: The
0, so this guy is 0.

492
00:27:54,800 --> 00:27:59,740
This guy is X squared
Y, and this guy is xy.

493
00:27:59,740 --> 00:28:01,690
And it should be
a piece of cake,

494
00:28:01,690 --> 00:28:03,990
but I want to do
it one more time.

495
00:28:03,990 --> 00:28:08,520
I times the minor derivative
of 0 with respect to y

496
00:28:08,520 --> 00:28:11,540
is 0 minus derivative
of x squared

497
00:28:11,540 --> 00:28:15,410
y respect to 0, all
right, plus j minus

498
00:28:15,410 --> 00:28:17,860
j because I'm alternating.

499
00:28:17,860 --> 00:28:19,965
You've known enough
in your algebra

500
00:28:19,965 --> 00:28:22,840
to know why I'm expanding
along the first row.

501
00:28:22,840 --> 00:28:25,700
I have a minus, all
right, then the x

502
00:28:25,700 --> 00:28:33,600
of 0, 0 derivative of xy
respect to the 0 plus k times

503
00:28:33,600 --> 00:28:37,550
the minor corresponding
to k derivative 2xy.

504
00:28:37,550 --> 00:28:45,050


505
00:28:45,050 --> 00:28:46,325
Oh, and the derivative--

506
00:28:46,325 --> 00:28:49,062


507
00:28:49,062 --> 00:28:52,504
STUDENT: Yeah, this
is the n equals 0.

508
00:28:52,504 --> 00:28:54,170
MAGDALENA TODA: Oh,
yeah, that's the one

509
00:28:54,170 --> 00:28:58,700
where it's not a because
that's not conservative.

510
00:28:58,700 --> 00:28:59,830
So what do you get.

511
00:28:59,830 --> 00:29:04,950
You get 2xy minus x, right?

512
00:29:04,950 --> 00:29:07,100
But I don't know how to
write it better than that.

513
00:29:07,100 --> 00:29:08,100
Well, it doesn't matter.

514
00:29:08,100 --> 00:29:09,510
Leave it like that.

515
00:29:09,510 --> 00:29:18,080
So this would be 0 if it only
if x would be 0, but otherwise y

516
00:29:18,080 --> 00:29:18,860
was 1/2.

517
00:29:18,860 --> 00:29:22,850
But in general, it
is not a 0, good.

518
00:29:22,850 --> 00:29:30,470
So F is not
conservative, and then we

519
00:29:30,470 --> 00:29:32,400
can say goodbye
to the whole thing

520
00:29:32,400 --> 00:29:39,680
here and move on to
computing the works.

521
00:29:39,680 --> 00:29:42,020
What is the only
way we can do that?

522
00:29:42,020 --> 00:29:46,491
By parameterizing
the first path,

523
00:29:46,491 --> 00:29:48,926
but I didn't say which
one is the first path.

524
00:29:48,926 --> 00:29:52,578
This is the first path, so
x of t equals t, and y of t

525
00:29:52,578 --> 00:29:55,257
equals t is your
parameterization.

526
00:29:55,257 --> 00:30:03,910
The simplest one, and then
W1 will be integral of-- I'm

527
00:30:03,910 --> 00:30:10,135
too lazy to write down x of t,
y of t, but this is what it is.

528
00:30:10,135 --> 00:30:14,540
Times x prime of
t plus x squared

529
00:30:14,540 --> 00:30:21,550
y times y prime of t dt where--

530
00:30:21,550 --> 00:30:31,372
STUDENT: Isn't that just
xy dx y-- never mind.

531
00:30:31,372 --> 00:30:33,565
MAGDALENA TODA: This is F2.

532
00:30:33,565 --> 00:30:36,560
And this is x prime,
and this is y prime

533
00:30:36,560 --> 00:30:40,394
because this thing is
just-- I have no idea.

534
00:30:40,394 --> 00:30:41,852
STUDENT: Right,
but what I'm asking

535
00:30:41,852 --> 00:30:46,719
is that not the same as just
F1 dx because we're going

536
00:30:46,719 --> 00:30:49,630
to do a chain rule anyway.

537
00:30:49,630 --> 00:30:53,750
MAGDALENA TODA: If I put
the x, I cannot put this.

538
00:30:53,750 --> 00:30:57,440
OK, this times that is dx.

539
00:30:57,440 --> 00:30:59,900
This guy times this guy is dx.

540
00:30:59,900 --> 00:31:01,400
STUDENT: But then
you can't use your

541
00:31:01,400 --> 00:31:03,980
MAGDALENA TODA: Then I
cannot use the t's then.

542
00:31:03,980 --> 00:31:06,486
STUDENT: All right, there we go.

543
00:31:06,486 --> 00:31:11,160
MAGDALENA TODA: All right, so
I have integral from 0 to 1 t,

544
00:31:11,160 --> 00:31:15,390
t times 1 t squared.

545
00:31:15,390 --> 00:31:18,260
If I make a mistake, that would
be a silly algebra mistake

546
00:31:18,260 --> 00:31:18,760
[INAUDIBLE].

547
00:31:18,760 --> 00:31:21,595


548
00:31:21,595 --> 00:31:23,980
All right, class.

549
00:31:23,980 --> 00:31:33,540
t cubed times 1dt,
how much is this?

550
00:31:33,540 --> 00:31:38,002
t cubed over 3 plus t
to the fourth over 4.

551
00:31:38,002 --> 00:31:39,252
STUDENT: It's just 2-- oh, no.

552
00:31:39,252 --> 00:31:45,100


553
00:31:45,100 --> 00:31:48,120
MAGDALENA TODA: Very good.

554
00:31:48,120 --> 00:31:52,500
Do not expect that we kill you
with computations on the exams,

555
00:31:52,500 --> 00:31:55,010
but that's not what we want.

556
00:31:55,010 --> 00:31:58,250
We want to test if you have
the basic understanding of what

557
00:31:58,250 --> 00:32:03,267
this is all about, not to
kill you with, OK, that.

558
00:32:03,267 --> 00:32:05,350
I'm not going to say that
in front of the cameras,

559
00:32:05,350 --> 00:32:06,805
but everybody knows that.

560
00:32:06,805 --> 00:32:08,270
There are professors
who would like

561
00:32:08,270 --> 00:32:09,690
to kill you with computations.

562
00:32:09,690 --> 00:32:12,180
Now, we're living in
a different world.

563
00:32:12,180 --> 00:32:15,590
If I gave you a long
polynomial sausage here

564
00:32:15,590 --> 00:32:17,660
and I ask you to
work with it, that

565
00:32:17,660 --> 00:32:21,090
doesn't mean that I'm smart
because MATLAB can do it.

566
00:32:21,090 --> 00:32:25,100
Mathematica, you get some
very nice simplifications

567
00:32:25,100 --> 00:32:28,500
over there, so I'm
just trying to see

568
00:32:28,500 --> 00:32:35,730
if rather than being able
to compute with no error,

569
00:32:35,730 --> 00:32:40,670
you are having the basic
understanding of the concept.

570
00:32:40,670 --> 00:32:44,520
And the rest can been done
by the mathematical software,

571
00:32:44,520 --> 00:32:49,650
which, nowadays, most
mathematicians are using.

572
00:32:49,650 --> 00:32:52,780
If you asked me 15
years ago, I think

573
00:32:52,780 --> 00:32:57,660
I knew colleagues at all the
ranks in academia who would not

574
00:32:57,660 --> 00:33:01,594
touch Mathematica or
MATLAB or Maple say

575
00:33:01,594 --> 00:33:04,810
that's like tool from
evil or something,

576
00:33:04,810 --> 00:33:08,072
but now everybody uses.

577
00:33:08,072 --> 00:33:10,760
Engineers use mostly
MATLAB as I told you.

578
00:33:10,760 --> 00:33:15,910
Mathematicians use both
MATLAB and Mathematica.

579
00:33:15,910 --> 00:33:19,450
Some of them use Maple,
especially the ones who

580
00:33:19,450 --> 00:33:23,002
have demos for K-12
level teachers,

581
00:33:23,002 --> 00:33:26,135
but MATLAB is a wonderful
tool, very pretty powerful

582
00:33:26,135 --> 00:33:27,583
in many ways.

583
00:33:27,583 --> 00:33:30,830
If you are doing any kind
of linear algebra project--

584
00:33:30,830 --> 00:33:33,760
I noticed three or four of you
are taking linear algebra-- you

585
00:33:33,760 --> 00:33:39,070
can always rely on MATLAB being
the best of all of the above.

586
00:33:39,070 --> 00:33:40,230
OK, W2.

587
00:33:40,230 --> 00:33:42,970


588
00:33:42,970 --> 00:33:49,830
For W2, I have a parabola, and
it's, again, a piece of cake.

589
00:33:49,830 --> 00:33:54,545
X prime will be 1,
y prime will be 2t.

590
00:33:54,545 --> 00:33:56,485
When I write down
the whole thing,

591
00:33:56,485 --> 00:33:58,667
I have to pay a little
bit of attention

592
00:33:58,667 --> 00:34:02,480
when I substitute
especially when I'm

593
00:34:02,480 --> 00:34:04,595
taking an exam under pressure.

594
00:34:04,595 --> 00:34:08,860


595
00:34:08,860 --> 00:34:13,909
x squared is t
squared, y is t squared

596
00:34:13,909 --> 00:34:17,270
times y prime, which is 2t.

597
00:34:17,270 --> 00:34:19,570
So now this is x prime.

598
00:34:19,570 --> 00:34:21,139
This is y prime.

599
00:34:21,139 --> 00:34:24,704
Let me change colors.

600
00:34:24,704 --> 00:34:26,924
All politicians change colors.

601
00:34:26,924 --> 00:34:29,300
But I'm not a
politician, but I'm

602
00:34:29,300 --> 00:34:34,030
thinking it's useful for you
to see who everybody was.

603
00:34:34,030 --> 00:34:38,690
This is the F1 in terms of t.

604
00:34:38,690 --> 00:34:46,214
That's the idea of what that
is, and this is F2 in terms of t

605
00:34:46,214 --> 00:34:48,121
as well.

606
00:34:48,121 --> 00:34:49,810
Oh, my God, another answer?

607
00:34:49,810 --> 00:34:53,540
Absolutely, I'm going to
get an another answer.

608
00:34:53,540 --> 00:34:57,276
Is it obviously to everybody
I'm going to get another answer?

609
00:34:57,276 --> 00:34:58,080
STUDENT: Yeah.

610
00:34:58,080 --> 00:35:01,495
MAGDALENA TODA: So I don't
have to put the t's here,

611
00:35:01,495 --> 00:35:03,790
but I thought it was
sort of neat to see

612
00:35:03,790 --> 00:35:05,950
that t goes from 0 to 1.

613
00:35:05,950 --> 00:35:08,760
And what do I get?

614
00:35:08,760 --> 00:35:16,355
This whole lot of them is t
cubed plus 2 t to the fifth.

615
00:35:16,355 --> 00:35:19,040


616
00:35:19,040 --> 00:35:26,002
So when I do the integration,
I get t to the 4 over 4 plus--

617
00:35:26,002 --> 00:35:27,335
shut up, Magdalena, get people--

618
00:35:27,335 --> 00:35:29,745


619
00:35:29,745 --> 00:35:30,620
STUDENT: [INAUDIBLE].

620
00:35:30,620 --> 00:35:32,040
MAGDALENA TODA: Very good.

621
00:35:32,040 --> 00:35:36,010
Yeah, he's done
the simplification.

622
00:35:36,010 --> 00:35:37,570
STUDENT: You get
the same values.

623
00:35:37,570 --> 00:35:40,330


624
00:35:40,330 --> 00:35:45,340
Plug in 1, you get 7/12 again.

625
00:35:45,340 --> 00:35:48,190
MAGDALENA TODA: So I'm
asking you-- OK, what was it?

626
00:35:48,190 --> 00:35:56,830
Solve 0, 1-- so I'm
asking why do you

627
00:35:56,830 --> 00:36:00,888
think we get the same value?

628
00:36:00,888 --> 00:36:03,310
Because the force
is not conservative,

629
00:36:03,310 --> 00:36:06,870
and I went on another path.

630
00:36:06,870 --> 00:36:10,075
I went on one path, and
I went on another path.

631
00:36:10,075 --> 00:36:15,656
And look, obviously my
expression was different.

632
00:36:15,656 --> 00:36:18,870
It's like one of those
math games or UIL games.

633
00:36:18,870 --> 00:36:20,970
And look at the algebra.

634
00:36:20,970 --> 00:36:23,521
The polynomials are different.

635
00:36:23,521 --> 00:36:25,960
What was my luck here?

636
00:36:25,960 --> 00:36:27,143
I took 1.

637
00:36:27,143 --> 00:36:27,726
STUDENT: Yeah.

638
00:36:27,726 --> 00:36:29,490
MAGDALENA TODA: I
could have taken 2.

639
00:36:29,490 --> 00:36:35,990
So if instead of 1, I would
have taken another number,

640
00:36:35,990 --> 00:36:38,330
then the higher the power,
the bigger the number

641
00:36:38,330 --> 00:36:39,490
would have been.

642
00:36:39,490 --> 00:36:40,407
I could have taken 2--

643
00:36:40,407 --> 00:36:42,114
STUDENT: You could
have taken negative 1,

644
00:36:42,114 --> 00:36:44,240
and you still wouldn't
have got the same answer.

645
00:36:44,240 --> 00:36:48,940
MAGDALENA TODA: Yeah, there
are many reasons why that is.

646
00:36:48,940 --> 00:36:53,900
But anyway, know that when you
take 1, 1 to every power is 1.

647
00:36:53,900 --> 00:36:55,470
And yeah, you were lucky.

648
00:36:55,470 --> 00:36:58,430
But in general, keep in
mind that if the force is

649
00:36:58,430 --> 00:37:01,500
conservative, in
general, in most examples

650
00:37:01,500 --> 00:37:04,510
you're not going to get the
same answer for the work

651
00:37:04,510 --> 00:37:10,680
because it does depend on
the path you want to take.

652
00:37:10,680 --> 00:37:18,210
I think I have reviewed quite
everything that I wanted.

653
00:37:18,210 --> 00:37:27,330


654
00:37:27,330 --> 00:37:29,870
So I should be ready
to move forward.

655
00:37:29,870 --> 00:37:32,630


656
00:37:32,630 --> 00:37:42,326
So I'm saying we are done
with sections 13.1, 13.2,

657
00:37:42,326 --> 00:37:48,920
and 13.3, which was my
favorite because it's not

658
00:37:48,920 --> 00:37:50,970
just the integral of
the path that I like,

659
00:37:50,970 --> 00:37:54,600
but it's the so-called
fundamental theorem of calculus

660
00:37:54,600 --> 00:38:04,870
3, which says, fundamental
theorem of the path integral

661
00:38:04,870 --> 00:38:12,030
saying that you have f of the
endpoint minus f of the origin,

662
00:38:12,030 --> 00:38:14,430
where little f is
that scalar potential

663
00:38:14,430 --> 00:38:17,310
as the linear function
was concerned.

664
00:38:17,310 --> 00:38:24,380
I'm going to call it the
fundamental theorem of path

665
00:38:24,380 --> 00:38:26,060
integral.

666
00:38:26,060 --> 00:38:29,230
Last time I told you the
fundamental theorem of calculus

667
00:38:29,230 --> 00:38:31,800
is Federal Trade Commission.

668
00:38:31,800 --> 00:38:34,620
We refer to that in Calc 1.

669
00:38:34,620 --> 00:38:39,080
But this one is the fundamental
theorem of path integral.

670
00:38:39,080 --> 00:38:42,815
Remember it because at
least one problem out of 15

671
00:38:42,815 --> 00:38:44,648
or something on the
final, and there are not

672
00:38:44,648 --> 00:38:45,564
going to be very many.

673
00:38:45,564 --> 00:38:48,815
It's going to ask you to
know that result. This is

674
00:38:48,815 --> 00:38:51,950
an important theorem.

675
00:38:51,950 --> 00:38:55,970
And another important theorem
that is starting right now

676
00:38:55,970 --> 00:38:57,810
is Green's theorem.

677
00:38:57,810 --> 00:39:02,690
Green's theorem is
a magic result. I

678
00:39:02,690 --> 00:39:04,905
have a t-shirt with it.

679
00:39:04,905 --> 00:39:06,410
I didn't bring it today.

680
00:39:06,410 --> 00:39:08,290
Maybe I'm going to bring
it next time First,

681
00:39:08,290 --> 00:39:12,376
I want you to see
the result, and then

682
00:39:12,376 --> 00:39:15,230
I'll bring the t-shirt
to the exam, so OK.

683
00:39:15,230 --> 00:39:18,080


684
00:39:18,080 --> 00:39:24,870
Assume that you have a
soup called Jordan curve.

685
00:39:24,870 --> 00:39:27,960


686
00:39:27,960 --> 00:39:32,380
You see, mathematicians don't
follow mathematical objects

687
00:39:32,380 --> 00:39:34,370
by their names.

688
00:39:34,370 --> 00:39:37,370
We are crazy people, but
we don't have a big ego.

689
00:39:37,370 --> 00:39:41,890
We would not say a theorem
of myself or whatever.

690
00:39:41,890 --> 00:39:45,340
We never give our names to that.

691
00:39:45,340 --> 00:39:50,930
But all through calculus you
saw all sorts of results.

692
00:39:50,930 --> 00:39:57,090
Like you see the Jordan
curve is a terminology,

693
00:39:57,090 --> 00:40:00,200
but then you see
everywhere the Linus rule.

694
00:40:00,200 --> 00:40:02,360
Did Linus get to
call it his own rule?

695
00:40:02,360 --> 00:40:06,320
No, but Euler's
number, these are

696
00:40:06,320 --> 00:40:08,515
things that were
discovered, and in honor

697
00:40:08,515 --> 00:40:11,640
of that particular
mathematician,

698
00:40:11,640 --> 00:40:13,290
we call them names.

699
00:40:13,290 --> 00:40:15,593
We call them the name
of the mathematician.

700
00:40:15,593 --> 00:40:18,430


701
00:40:18,430 --> 00:40:22,560
Out of curiosity for
0.5 extra credit points,

702
00:40:22,560 --> 00:40:25,140
find out who Jordan was.

703
00:40:25,140 --> 00:40:32,810
Jordan curve is a closed
curve that, in general,

704
00:40:32,810 --> 00:40:34,595
could be piecewise continuous.

705
00:40:34,595 --> 00:40:40,610


706
00:40:40,610 --> 00:40:43,230
So you have a closed
loop over here.

707
00:40:43,230 --> 00:40:50,170
So in general, I could
have something like that

708
00:40:50,170 --> 00:40:54,102
that does not enclose.

709
00:40:54,102 --> 00:40:56,542
That encloses a
domain without holes.

710
00:40:56,542 --> 00:41:04,350


711
00:41:04,350 --> 00:41:07,540
Holes are functions
of the same thing.

712
00:41:07,540 --> 00:41:10,240
STUDENT: So doesn't it
need to be continuous?

713
00:41:10,240 --> 00:41:12,440
MAGDALENA TODA:
No, I said it is.

714
00:41:12,440 --> 00:41:13,775
STUDENT: You said, piecewise.

715
00:41:13,775 --> 00:41:15,170
MAGDALENA TODA: Ah, piecewise.

716
00:41:15,170 --> 00:41:16,448
This is piecewise.

717
00:41:16,448 --> 00:41:17,762
STUDENT: Oh, so it's piecewise.

718
00:41:17,762 --> 00:41:18,262
OK.

719
00:41:18,262 --> 00:41:20,430
MAGDALENA TODA: So you
have a bunch of arcs.

720
00:41:20,430 --> 00:41:23,050
Finitely many, let's
say, in your case.

721
00:41:23,050 --> 00:41:26,230
Finitely many arcs,
they have corners,

722
00:41:26,230 --> 00:41:29,740
but you can see define the
integral along such a path.

723
00:41:29,740 --> 00:41:33,676


724
00:41:33,676 --> 00:41:37,530
Oh, and also for another
0.5 extra credit,

725
00:41:37,530 --> 00:41:40,140
find out who Mr.
Green was because he

726
00:41:40,140 --> 00:41:43,490
has several theorems that
are through mathematics

727
00:41:43,490 --> 00:41:46,360
and free mechanics and
variation calculus.

728
00:41:46,360 --> 00:41:50,860
There are several identities
that are called Greens.

729
00:41:50,860 --> 00:41:52,420
There is this famous
Green's theorem,

730
00:41:52,420 --> 00:41:54,890
but there are Green's
first identity,

731
00:41:54,890 --> 00:41:57,844
Green's second identity,
and all sorts of things.

732
00:41:57,844 --> 00:42:01,825
And find out who Mr.
Green was, and as a total,

733
00:42:01,825 --> 00:42:03,826
you have 1 point extra credit.

734
00:42:03,826 --> 00:42:08,870
And you can turn in a regular
essay like a two-page thing.

735
00:42:08,870 --> 00:42:12,602
You want biography of these
mathematicians if you want,

736
00:42:12,602 --> 00:42:15,500
just a few paragraphs.

737
00:42:15,500 --> 00:42:19,210
So what does Green's theorem do?

738
00:42:19,210 --> 00:42:26,200
Green's theorem is
a remarkable result

739
00:42:26,200 --> 00:42:31,083
which links the path integral
to the double integral.

740
00:42:31,083 --> 00:42:38,268
It's a remarkable
and famous result.

741
00:42:38,268 --> 00:42:48,330
And that links the path
integral on the closed

742
00:42:48,330 --> 00:43:07,262
curve to a double integral
over the domain enclosed.

743
00:43:07,262 --> 00:43:09,761
I can see the domain
inside, but you

744
00:43:09,761 --> 00:43:15,160
have to understand it's
enclosed by the curve.

745
00:43:15,160 --> 00:43:20,740


746
00:43:20,740 --> 00:43:24,444
All right, and assume
that you have--

747
00:43:24,444 --> 00:43:35,970
M and N are C1 functions of
x and y, what does it mean?

748
00:43:35,970 --> 00:43:37,620
M is a function of xy.

749
00:43:37,620 --> 00:43:40,300
N is a function of xy in plane.

750
00:43:40,300 --> 00:43:43,360
Both of them are differentiable
with continuous derivative.

751
00:43:43,360 --> 00:43:46,692


752
00:43:46,692 --> 00:43:47,830
They are differentiable.

753
00:43:47,830 --> 00:43:49,515
You can take the
partial derivatives,

754
00:43:49,515 --> 00:43:51,840
and all the partial
derivatives are continuous.

755
00:43:51,840 --> 00:43:55,496
That's what we mean
by being C1 functions.

756
00:43:55,496 --> 00:43:58,610
And there the magic
happens, so let me show you

757
00:43:58,610 --> 00:44:02,320
where the magic happens.

758
00:44:02,320 --> 00:44:06,360
This in the box,
the path integral

759
00:44:06,360 --> 00:44:20,637
over c of M dx plus Ndy is
equal to the double integral

760
00:44:20,637 --> 00:44:22,390
over the domain enclosed.

761
00:44:22,390 --> 00:44:24,250
OK, this is the c.

762
00:44:24,250 --> 00:44:27,160
On the boundary you
go counterclockwise

763
00:44:27,160 --> 00:44:29,415
like any respectable
mathematician

764
00:44:29,415 --> 00:44:33,880
would go in a trigonometric
sense, just counterclockwise.

765
00:44:33,880 --> 00:44:36,776
And this is the domain
being closed by c.

766
00:44:36,776 --> 00:44:40,060


767
00:44:40,060 --> 00:44:44,260
And you put here the
integral, which is magic.

768
00:44:44,260 --> 00:44:46,240
This is easy to
remember for you.

769
00:44:46,240 --> 00:44:48,170
This is not easy to
remember unless I

770
00:44:48,170 --> 00:44:49,980
take the t-shirt to
the exam, and you

771
00:44:49,980 --> 00:44:52,068
cheat by looking at my t-shirt.

772
00:44:52,068 --> 00:44:54,589
No, by the time of the
exam, I promised you

773
00:44:54,589 --> 00:44:59,310
you are going to have at least
one week, seven days or more,

774
00:44:59,310 --> 00:45:03,310
10-day period in which
we will study samples,

775
00:45:03,310 --> 00:45:05,692
various samples of old finals.

776
00:45:05,692 --> 00:45:08,320
I'm going to go ahead and
send you some by email.

777
00:45:08,320 --> 00:45:11,440
Do you mind?

778
00:45:11,440 --> 00:45:13,676
In the next week
after this week, we

779
00:45:13,676 --> 00:45:15,400
are going to start reviewing.

780
00:45:15,400 --> 00:45:20,240
And by dA I mean dxdy, the
usual area limit in Cartesian

781
00:45:20,240 --> 00:45:23,710
coordinates the way you
are used to it the most.

782
00:45:23,710 --> 00:45:27,320


783
00:45:27,320 --> 00:45:29,720
And then, Alex is looking
at it and said, well,

784
00:45:29,720 --> 00:45:32,380
then I tell her that
the most elegant way

785
00:45:32,380 --> 00:45:34,690
is to put it with dxdy.

786
00:45:34,690 --> 00:45:38,480
This is what we call a
one form in mathematics.

787
00:45:38,480 --> 00:45:39,750
What is a one form.

788
00:45:39,750 --> 00:45:43,945
It is a linear combination of
this infinitesimal elements

789
00:45:43,945 --> 00:45:47,490
dxdy in plane with some
scalar functions of x

790
00:45:47,490 --> 00:45:49,491
and y in front of her.

791
00:45:49,491 --> 00:45:50,750
OK, so what do we do?

792
00:45:50,750 --> 00:45:52,505
We integrate the one form.

793
00:45:52,505 --> 00:45:57,460
The book doesn't talk about one
forms because the is actually

794
00:45:57,460 --> 00:46:00,727
written for the average
student, the average freshman

795
00:46:00,727 --> 00:46:04,920
or the average
sophomore, but I think

796
00:46:04,920 --> 00:46:08,250
we have an exposure to
the notion of one form,

797
00:46:08,250 --> 00:46:10,770
so I can get a little bit
more elegant and more rigorous

798
00:46:10,770 --> 00:46:12,290
in my speech.

799
00:46:12,290 --> 00:46:15,752
If you are a graduate
student, you most likely

800
00:46:15,752 --> 00:46:18,140
would know this is a one form.

801
00:46:18,140 --> 00:46:22,140
That's actually the
definition of a one form.

802
00:46:22,140 --> 00:46:23,475
And you'll say, what is this?

803
00:46:23,475 --> 00:46:27,284
This is actually two
form, but you are

804
00:46:27,284 --> 00:46:28,450
going to say, wait a minute.

805
00:46:28,450 --> 00:46:30,610
I have a scalar
function, whatever

806
00:46:30,610 --> 00:46:34,430
that is, from the integration
in front of the dxdy

807
00:46:34,430 --> 00:46:39,740
you want but you never said
that dxdy is a two form.

808
00:46:39,740 --> 00:46:44,990
Actually, I did, and I
didn't call it a two form.

809
00:46:44,990 --> 00:46:47,280
Do you remember that
I introduced to you

810
00:46:47,280 --> 00:46:50,434
some magic wedge product?

811
00:46:50,434 --> 00:46:53,640


812
00:46:53,640 --> 00:46:57,750
And we said, this is
a tiny displacement.

813
00:46:57,750 --> 00:46:59,750
Dx infinitesimal is small.

814
00:46:59,750 --> 00:47:02,360
Imagine how much the
video we'll there

815
00:47:02,360 --> 00:47:04,960
is an infinitesimal
displacement dx

816
00:47:04,960 --> 00:47:07,570
and an infinitesimal
displacement dy,

817
00:47:07,570 --> 00:47:10,700
and you have some
sort of a sign area.

818
00:47:10,700 --> 00:47:15,130
So we said, we don't
just take dxdy,

819
00:47:15,130 --> 00:47:19,070
but we take a product
between dxdy with a wedge,

820
00:47:19,070 --> 00:47:21,700
meaning that if I
change the order,

821
00:47:21,700 --> 00:47:24,140
I'm going to have minus dy here.

822
00:47:24,140 --> 00:47:29,140
This is typical exterior
derivative theory-- exterior

823
00:47:29,140 --> 00:47:31,456
derivative theory.

824
00:47:31,456 --> 00:47:34,940
And it's a theory that
starts more or less

825
00:47:34,940 --> 00:47:36,450
at the graduate level.

826
00:47:36,450 --> 00:47:39,510
And many people get their
master's degree in math

827
00:47:39,510 --> 00:47:43,452
and never get to see it, and
I pity them, but this life.

828
00:47:43,452 --> 00:47:47,470
On the other hand, when
you have dx, which dx--

829
00:47:47,470 --> 00:47:50,546
the area between dx and dx is 0.

830
00:47:50,546 --> 00:47:53,400
So we're all very happy
I get rid of those.

831
00:47:53,400 --> 00:47:55,850
When I have the sign
between the displacement,

832
00:47:55,850 --> 00:47:57,570
dy and itself is 0.

833
00:47:57,570 --> 00:47:59,990
So these are the
basic properties

834
00:47:59,990 --> 00:48:05,420
that we started
about the sign area.

835
00:48:05,420 --> 00:48:07,720
I want to show you what happens.

836
00:48:07,720 --> 00:48:16,360
I'm going to-- yeah,
I'm going to erase here.

837
00:48:16,360 --> 00:48:22,690


838
00:48:22,690 --> 00:48:25,856
I'm going to show
you later I'm going

839
00:48:25,856 --> 00:48:32,700
to prove this theorem to you
later using these tricks that I

840
00:48:32,700 --> 00:48:35,096
just showed you here.

841
00:48:35,096 --> 00:48:53,656
I will provide proof
to this formula, OK?

842
00:48:53,656 --> 00:48:57,500
And let's take a look at
that, and we say, well,

843
00:48:57,500 --> 00:49:00,010
can I memorize that by
the time of the final?

844
00:49:00,010 --> 00:49:01,660
Yes, you can.

845
00:49:01,660 --> 00:49:13,070
What is beautiful about
this, it can actually

846
00:49:13,070 --> 00:49:18,510
help you solve problems that you
didn't think would be possible.

847
00:49:18,510 --> 00:49:20,920
For example, example
1, and I say,

848
00:49:20,920 --> 00:49:26,400
that would be one of
the most basic ones.

849
00:49:26,400 --> 00:49:38,710
Find the geometric meaning of
the integral over a c where

850
00:49:38,710 --> 00:49:39,890
c is a closed loop.

851
00:49:39,890 --> 00:49:41,920
OK, c is a loop.

852
00:49:41,920 --> 00:49:47,388
Piecewise define Jordan
curve-- Jordan curve.

853
00:49:47,388 --> 00:49:49,645
And I integrate out
of something weird.

854
00:49:49,645 --> 00:49:51,144
And you say, oh, my God.

855
00:49:51,144 --> 00:49:51,950
Look at her.

856
00:49:51,950 --> 00:49:59,360
She picked some weird function
where the path from the dx

857
00:49:59,360 --> 00:50:05,970
is M, and the path in front of
dy is N, the M and N functions.

858
00:50:05,970 --> 00:50:07,780
Why would pick like that?

859
00:50:07,780 --> 00:50:11,260
You wouldn't know yet, but
if you apply Green's theorem,

860
00:50:11,260 --> 00:50:14,040
assuming you believe
it's true, you

861
00:50:14,040 --> 00:50:18,271
have double integral over the
domain enclosed by this loop.

862
00:50:18,271 --> 00:50:24,820
The loop is enclosing
this domain of what?

863
00:50:24,820 --> 00:50:32,080
Now, I'm trying to shut up,
and I'm want you to talk.

864
00:50:32,080 --> 00:50:35,542
What am I going to
write over here?

865
00:50:35,542 --> 00:50:36,940
STUDENT: 1 plus 1.

866
00:50:36,940 --> 00:50:40,600
MAGDALENA TODA: 1 plus
1, how fun is that?

867
00:50:40,600 --> 00:50:46,130
Y minus 1, 1 plus 1 equals
2 last time I checked,

868
00:50:46,130 --> 00:50:49,370
and this is dA.

869
00:50:49,370 --> 00:50:52,900
And what do you think
this animal would be?

870
00:50:52,900 --> 00:50:56,090
The cast of 2 always can escape.

871
00:50:56,090 --> 00:51:00,566
So if we don't want
it, just kick it out.

872
00:51:00,566 --> 00:51:04,480
What is the remaining
double integral for d of DA?

873
00:51:04,480 --> 00:51:07,310
You have seen this guy all
through the Calculus 3 course.

874
00:51:07,310 --> 00:51:09,850
You're tired of it.

875
00:51:09,850 --> 00:51:13,780
You said, I cannot wait for
this semester to be over

876
00:51:13,780 --> 00:51:19,060
because this is the double
integral of 1dA over d.

877
00:51:19,060 --> 00:51:21,940
What in the world is that?

878
00:51:21,940 --> 00:51:24,364
That is the--

879
00:51:24,364 --> 00:51:25,330
STUDENT: --area.

880
00:51:25,330 --> 00:51:26,700
MAGDALENA TODA: Area, very good.

881
00:51:26,700 --> 00:51:31,150
This is the area of the
domain d inside the curve.

882
00:51:31,150 --> 00:51:34,980
The shaded area is this.

883
00:51:34,980 --> 00:51:39,060
So you have discovered
something beautiful

884
00:51:39,060 --> 00:51:46,530
that the area of the domain
enclosed by a Jordan curve

885
00:51:46,530 --> 00:51:51,040
equals 1/2 because you pull
the two out in front here,

886
00:51:51,040 --> 00:51:56,320
it's going to be 1/2 of the path
integrals over the boundary.

887
00:51:56,320 --> 00:51:58,590
This is called
boundary of a domain.

888
00:51:58,590 --> 00:52:00,380
c is the boundary of the domain.

889
00:52:00,380 --> 00:52:04,990


890
00:52:04,990 --> 00:52:06,940
Some mathematicians--
I don't know

891
00:52:06,940 --> 00:52:10,950
how far you want to go with your
education, but in a few years

892
00:52:10,950 --> 00:52:13,300
you might become
graduate students.

893
00:52:13,300 --> 00:52:18,700
And even some engineers use this
notation boundary of d, del d.

894
00:52:18,700 --> 00:52:22,230
That means the boundaries,
the frontier of a domain.

895
00:52:22,230 --> 00:52:24,430
The fence of a ranch.

896
00:52:24,430 --> 00:52:27,050
That is the del d, but
don't tell the rancher

897
00:52:27,050 --> 00:52:30,598
because he will take his gun
out and shoot you thinking

898
00:52:30,598 --> 00:52:33,860
that you are off the hook or
you are after something weird.

899
00:52:33,860 --> 00:52:38,340
So that's the boundary
of the domain.

900
00:52:38,340 --> 00:52:42,444
And then you have
minus ydx plus xdy.

901
00:52:42,444 --> 00:52:46,210


902
00:52:46,210 --> 00:52:48,210
MAGDALENA TODA: We discover
something beautiful.

903
00:52:48,210 --> 00:52:50,210
Something important.

904
00:52:50,210 --> 00:52:52,760
And now I'm asking,
with this exercise--

905
00:52:52,760 --> 00:52:59,480
one which I could even--
I could even call a lemma.

906
00:52:59,480 --> 00:53:03,964
Lemma is not quite a
theorem, because it's based--

907
00:53:03,964 --> 00:53:05,130
could be based on a theorem.

908
00:53:05,130 --> 00:53:09,460
It's a little result that can
be proved in just a few lines

909
00:53:09,460 --> 00:53:12,480
without being something
sophisticated based

910
00:53:12,480 --> 00:53:15,800
on something you
knew from before.

911
00:53:15,800 --> 00:53:19,950
So this is called a lemma.

912
00:53:19,950 --> 00:53:26,250
When you have a sophisticated
area to compute--

913
00:53:26,250 --> 00:53:30,020
or even can you prove-- if you
believe in Green's theorem,

914
00:53:30,020 --> 00:53:33,410
can you prove that the
area inside the circle

915
00:53:33,410 --> 00:53:34,780
is pi r squared?

916
00:53:34,780 --> 00:53:39,966
Can you prove that the
area inside of an ellipse

917
00:53:39,966 --> 00:53:41,490
is-- I don't know what.

918
00:53:41,490 --> 00:53:44,370
Do you know the area
inside of an ellipse?

919
00:53:44,370 --> 00:53:47,561
Nobody taught me in school.

920
00:53:47,561 --> 00:53:50,800
I don't know why
it's so beautiful.

921
00:53:50,800 --> 00:53:55,150
I learned what an ellipse
was in eleventh grade

922
00:53:55,150 --> 00:53:59,190
in high school and again
a review as a freshman

923
00:53:59,190 --> 00:54:01,310
analytic geometry.

924
00:54:01,310 --> 00:54:03,070
So we've seen conics again--

925
00:54:03,070 --> 00:54:04,945
STUDENT: I think we did
conics in 10th grade.

926
00:54:04,945 --> 00:54:06,684
We might have seen it.

927
00:54:06,684 --> 00:54:08,100
MAGDALENA TODA:
But nobody told me

928
00:54:08,100 --> 00:54:10,170
like-- I give you an ellipse.

929
00:54:10,170 --> 00:54:12,040
Compute the area inside.

930
00:54:12,040 --> 00:54:13,170
I had no idea.

931
00:54:13,170 --> 00:54:15,420
And I didn't know
the formula until I

932
00:54:15,420 --> 00:54:17,820
became an assistant professor.

933
00:54:17,820 --> 00:54:19,490
I was already in my thirties.

934
00:54:19,490 --> 00:54:23,970
That's a shame to see that
thing for the first time OK.

935
00:54:23,970 --> 00:54:27,900
So let's see if we believe
this lemma, and the Green's

936
00:54:27,900 --> 00:54:28,736
theorem of course.

937
00:54:28,736 --> 00:54:31,624
But let's apply the
lemma, primarily

938
00:54:31,624 --> 00:54:33,492
from the Green's theorem.

939
00:54:33,492 --> 00:54:36,600
Can we actually prove
that the area of the disk

940
00:54:36,600 --> 00:54:40,890
is pi r squared and the
area of the ellipse--

941
00:54:40,890 --> 00:54:43,330
inside the ellipse
will be god knows what.

942
00:54:43,330 --> 00:54:47,180
And we will discover
that by ourselves.

943
00:54:47,180 --> 00:54:49,385
I think that's the
beauty of mathematics.

944
00:54:49,385 --> 00:54:53,630
Because every now and then
even if you discover things

945
00:54:53,630 --> 00:54:56,280
that people have known
for hundreds of years,

946
00:54:56,280 --> 00:54:58,080
it still gives you
the satisfaction

947
00:54:58,080 --> 00:55:01,840
that you did something by
yourself-- all on yourself.

948
00:55:01,840 --> 00:55:06,350
Like, when you feel
build a helicopter or you

949
00:55:06,350 --> 00:55:07,730
build a table.

950
00:55:07,730 --> 00:55:09,898
There are many more
beautiful tables

951
00:55:09,898 --> 00:55:12,050
that were built before
you, but still it's

952
00:55:12,050 --> 00:55:14,980
a lot of satisfaction that
you do all by yourself.

953
00:55:14,980 --> 00:55:16,850
It's the same with mathematics.

954
00:55:16,850 --> 00:55:23,386
So let's see what we can
do now for the first time.

955
00:55:23,386 --> 00:55:24,590
Not for the first time.

956
00:55:24,590 --> 00:55:28,030
We do it in other ways.

957
00:55:28,030 --> 00:55:37,608
Can you prove using the lemma
or Green's theorem-- which

958
00:55:37,608 --> 00:55:43,125
is the same thing-- either
one-- that the area of the disk

959
00:55:43,125 --> 00:55:47,850
of radius r-- this is the r.

960
00:55:47,850 --> 00:55:52,330
so this the radius
r is pi r squared.

961
00:55:52,330 --> 00:55:56,290


962
00:55:56,290 --> 00:55:57,530
I hope so.

963
00:55:57,530 --> 00:55:59,700
And the answer is, I hope so.

964
00:55:59,700 --> 00:56:00,660
And that's all.

965
00:56:00,660 --> 00:56:03,540


966
00:56:03,540 --> 00:56:09,460
Area of the disk of radius r.

967
00:56:09,460 --> 00:56:10,240
Oh my god.

968
00:56:10,240 --> 00:56:12,620
So you go, well.

969
00:56:12,620 --> 00:56:19,420
If I knew the parameterization
of that boundary C,

970
00:56:19,420 --> 00:56:20,600
it would be a piece of cake.

971
00:56:20,600 --> 00:56:25,640
Because I would just-- I know
how to do a path integral now.

972
00:56:25,640 --> 00:56:27,620
I've learned in the
previous sections,

973
00:56:27,620 --> 00:56:30,460
so this should be easy.

974
00:56:30,460 --> 00:56:32,690
Can we do that?

975
00:56:32,690 --> 00:56:33,330
So let's see.

976
00:56:33,330 --> 00:56:36,360


977
00:56:36,360 --> 00:56:38,290
Without computing
the double integral,

978
00:56:38,290 --> 00:56:41,270
because I can always do
that with polar coordinates.

979
00:56:41,270 --> 00:56:42,880
And we are going to do that.

980
00:56:42,880 --> 00:56:47,640


981
00:56:47,640 --> 00:56:49,544
Let's do that as
well, as practice.

982
00:56:49,544 --> 00:56:53,801
Because so you
review for the exam.

983
00:56:53,801 --> 00:56:57,100


984
00:56:57,100 --> 00:57:00,940
But another way to
do it would be what?

985
00:57:00,940 --> 00:57:06,640
1/2 integral over the circle.

986
00:57:06,640 --> 00:57:13,900
And how do I parametrize a
circle of fixed radius r?

987
00:57:13,900 --> 00:57:14,830
Who tells me?

988
00:57:14,830 --> 00:57:18,020
x of t will be--
that was Chapter 10.

989
00:57:18,020 --> 00:57:20,872
Everything is a
circle in mathematics.

990
00:57:20,872 --> 00:57:21,800
STUDENT: r cosine t.

991
00:57:21,800 --> 00:57:22,925
MAGDALENA TODA: r cosine t.

992
00:57:22,925 --> 00:57:24,210
Excellent.

993
00:57:24,210 --> 00:57:25,786
y of t is?

994
00:57:25,786 --> 00:57:26,640
STUDENT: r sine t.

995
00:57:26,640 --> 00:57:29,420
MAGDALENA TODA: r sine t.

996
00:57:29,420 --> 00:57:32,360
So, finally I'm going to
go ahead and use this one.

997
00:57:32,360 --> 00:57:37,065
And I'm going to say, well,
minus y to be plugged in.

998
00:57:37,065 --> 00:57:39,800


999
00:57:39,800 --> 00:57:43,000
This is minus y.

1000
00:57:43,000 --> 00:57:44,580
Multiply by dx.

1001
00:57:44,580 --> 00:57:47,550
Well, you say, wait a
minute. dx with respect.

1002
00:57:47,550 --> 00:57:48,750
What is dx?

1003
00:57:48,750 --> 00:57:52,410
dx is just x prime dt.

1004
00:57:52,410 --> 00:57:54,690
Dy is just y prime dt.

1005
00:57:54,690 --> 00:57:56,370
And t goes out.

1006
00:57:56,370 --> 00:57:57,520
It's banished.

1007
00:57:57,520 --> 00:57:59,620
No, he's the most important guy.

1008
00:57:59,620 --> 00:58:02,970
So t goes from something
to something else.

1009
00:58:02,970 --> 00:58:05,260
We will see that later.

1010
00:58:05,260 --> 00:58:06,970
What is x prime dt?

1011
00:58:06,970 --> 00:58:12,850
X prime is minus r sine
theta-- sine t, Magdalena.

1012
00:58:12,850 --> 00:58:15,740
Minus r sine t.

1013
00:58:15,740 --> 00:58:18,260
That was x prime.

1014
00:58:18,260 --> 00:58:19,430
Change the color.

1015
00:58:19,430 --> 00:58:23,240
Give people some
variation in their life.

1016
00:58:23,240 --> 00:58:32,056
Plus r cosine t,
because this x--

1017
00:58:32,056 --> 00:58:32,930
STUDENT: [INAUDIBLE].

1018
00:58:32,930 --> 00:58:39,310


1019
00:58:39,310 --> 00:58:46,440
MAGDALENA TODA: --times
the y, which is r cosine t.

1020
00:58:46,440 --> 00:58:49,150
So it suddenly became beautiful.

1021
00:58:49,150 --> 00:58:52,480
It looks-- first it looks ugly,
but now it became beautiful.

1022
00:58:52,480 --> 00:58:52,980
Why?

1023
00:58:52,980 --> 00:58:54,365
How come it became beautiful?

1024
00:58:54,365 --> 00:58:56,740
STUDENT: Because you got sine
squared plus cosine square.

1025
00:58:56,740 --> 00:58:58,281
MAGDALENA TODA:
Because I got a plus.

1026
00:58:58,281 --> 00:59:01,850
If you pay attention, plus sine
squared plus cosine squared.

1027
00:59:01,850 --> 00:59:04,970
So I have, what is sine
squared plus cosine squared?

1028
00:59:04,970 --> 00:59:07,804
I heard that our
students in trig--

1029
00:59:07,804 --> 00:59:11,859
Poly told me-- who still don't
know that this is the most

1030
00:59:11,859 --> 00:59:13,650
important thing you
learn in trigonometry--

1031
00:59:13,650 --> 00:59:15,150
is Pythagorean theorem.

1032
00:59:15,150 --> 00:59:15,860
Right?

1033
00:59:15,860 --> 00:59:24,236
So you have 1/2 integral

1034
00:59:24,236 --> 00:59:27,075
STUDENT: r squared--

1035
00:59:27,075 --> 00:59:28,450
MAGDALENA TODA:
r-- no, I'm lazy.

1036
00:59:28,450 --> 00:59:30,920
I'm going slow-- r.

1037
00:59:30,920 --> 00:59:32,730
dt.

1038
00:59:32,730 --> 00:59:34,916
T from what to what?

1039
00:59:34,916 --> 00:59:36,780
From 0 times 0.

1040
00:59:36,780 --> 00:59:39,880
I'm starting whatever
I want, actually.

1041
00:59:39,880 --> 00:59:44,214
I go counterclockwise
I'm into pi.

1042
00:59:44,214 --> 00:59:45,986
STUDENT: Why is
that not r squared?

1043
00:59:45,986 --> 00:59:47,808
It should be r squared.

1044
00:59:47,808 --> 00:59:49,176
MAGDALENA TODA: I'm sorry, guys.

1045
00:59:49,176 --> 00:59:50,090
I'm sorry.

1046
00:59:50,090 --> 00:59:53,550
I don't know what
I am-- r squared.

1047
00:59:53,550 --> 00:59:57,060
1/2 r squared times 2 pi.

1048
00:59:57,060 --> 01:00:00,740


1049
01:00:00,740 --> 01:00:03,410
So we have pi r squared.

1050
01:00:03,410 --> 01:00:06,870
And if you did not
tell me it's r squared,

1051
01:00:06,870 --> 01:00:09,710
we wouldn't have
gotten the answer.

1052
01:00:09,710 --> 01:00:10,210
That's good.

1053
01:00:10,210 --> 01:00:14,130


1054
01:00:14,130 --> 01:00:16,257
What's the other way to do it?

1055
01:00:16,257 --> 01:00:18,530
If a problem on
the final would ask

1056
01:00:18,530 --> 01:00:22,172
you prove in two different
ways that the rubber

1057
01:00:22,172 --> 01:00:25,130
disk is pi r squared using
Calc 3, or whatever--

1058
01:00:25,130 --> 01:00:26,130
STUDENT: Would require--

1059
01:00:26,130 --> 01:00:28,350
MAGDALENA TODA: The
double integral, right?

1060
01:00:28,350 --> 01:00:28,850
Right?

1061
01:00:28,850 --> 01:00:31,141
STUDENT: Could have done
Cartesian coordinates as well.

1062
01:00:31,141 --> 01:00:33,105
If that counts as a second way.

1063
01:00:33,105 --> 01:00:33,980
MAGDALENA TODA: Yeah.

1064
01:00:33,980 --> 01:00:34,830
You can-- OK.

1065
01:00:34,830 --> 01:00:36,360
What could this be?

1066
01:00:36,360 --> 01:00:37,140
Oh my god.

1067
01:00:37,140 --> 01:00:42,280
This would be minus 1
to 1 minus square root

1068
01:00:42,280 --> 01:00:45,966
1 minus x squared to square
root 1 minus x squared.

1069
01:00:45,966 --> 01:00:46,849
Am i right guys?

1070
01:00:46,849 --> 01:00:47,390
STUDENT: Yep.

1071
01:00:47,390 --> 01:00:48,700
MAGDALENA TODA: 1 dy dx.

1072
01:00:48,700 --> 01:00:51,165
Of course it's a pain.

1073
01:00:51,165 --> 01:00:53,706
STUDENT: You could double that
and set the bottoms both equal

1074
01:00:53,706 --> 01:00:55,074
to 0.

1075
01:00:55,074 --> 01:00:55,990
MAGDALENA TODA: Right.

1076
01:00:55,990 --> 01:01:01,420
So we can do by symmetry--

1077
01:01:01,420 --> 01:01:02,720
STUDENT: Yeah.

1078
01:01:02,720 --> 01:01:05,060
MAGDALENA TODA: I'm--
shall I erase or leave it.

1079
01:01:05,060 --> 01:01:07,790
Are you understand
what Alex is saying?

1080
01:01:07,790 --> 01:01:12,192
This is 2i is the integral
that you will get.

1081
01:01:12,192 --> 01:01:13,650
STUDENT: Just write
it next to it--

1082
01:01:13,650 --> 01:01:14,990
MAGDALENA TODA: I tell
you four times, you

1083
01:01:14,990 --> 01:01:16,490
see, Alex, because you have--

1084
01:01:16,490 --> 01:01:16,820
STUDENT: Oh, yeah.

1085
01:01:16,820 --> 01:01:19,069
MAGDALENA TODA: --symmetry
with respect to the x-axis,

1086
01:01:19,069 --> 01:01:21,420
and symmetry with
respect to y-axis.

1087
01:01:21,420 --> 01:01:26,725
And you can take 0
to 1 and 0 to that.

1088
01:01:26,725 --> 01:01:29,210
And you have x from 0 to 1.

1089
01:01:29,210 --> 01:01:33,740
You have y from 0 to stop.

1090
01:01:33,740 --> 01:01:35,440
Square root of 1 minus x square.

1091
01:01:35,440 --> 01:01:37,465
Like the strips.

1092
01:01:37,465 --> 01:01:41,600
And you have 4
times that A1, which

1093
01:01:41,600 --> 01:01:44,925
would be the area of
the first quadratic.

1094
01:01:44,925 --> 01:01:46,470
You can do that, too.

1095
01:01:46,470 --> 01:01:46,970
It's easier.

1096
01:01:46,970 --> 01:01:50,100
But the best way to do that is
not in Cartesian coordinates.

1097
01:01:50,100 --> 01:01:52,770
The best way is to do
it in polar coordinates.

1098
01:01:52,770 --> 01:01:56,570
Always remember
your Jacobian is r.

1099
01:01:56,570 --> 01:02:00,892
So if you have
Jacobian r-- erase.

1100
01:02:00,892 --> 01:02:03,420
Let's put r here again.

1101
01:02:03,420 --> 01:02:08,270
And then dr d theta.

1102
01:02:08,270 --> 01:02:10,280
But now you say, wait
a minute, Magdalena.

1103
01:02:10,280 --> 01:02:11,780
You said r is fixed.

1104
01:02:11,780 --> 01:02:12,439
Yes.

1105
01:02:12,439 --> 01:02:13,980
And that's why I
need to learn Greek,

1106
01:02:13,980 --> 01:02:15,762
because it's all Greek to me.

1107
01:02:15,762 --> 01:02:18,860
Instead of r I put
rho as a variable.

1108
01:02:18,860 --> 01:02:23,560
And I say, rho is
between 0 and r.

1109
01:02:23,560 --> 01:02:25,300
r is fixed.

1110
01:02:25,300 --> 01:02:27,200
That's my [INAUDIBLE].

1111
01:02:27,200 --> 01:02:32,130
Big r is not usually written
as a variable from 0 to some.

1112
01:02:32,130 --> 01:02:33,485
I cannot use that.

1113
01:02:33,485 --> 01:02:37,260
So I have to us a Greek letter,
whether I like it or not.

1114
01:02:37,260 --> 01:02:39,580
And theta is from 0 to 2 pi.

1115
01:02:39,580 --> 01:02:41,750
And I still get the same thing.

1116
01:02:41,750 --> 01:02:47,050
I get r-- rho squared
over 2 between 0 and r.

1117
01:02:47,050 --> 01:02:48,610
And I have 2 pi.

1118
01:02:48,610 --> 01:02:53,350
And in the end that means
pi r squared, and I'm back.

1119
01:02:53,350 --> 01:02:56,260
And you say, wait,
this is Example 4.

1120
01:02:56,260 --> 01:02:57,462
Whatever example.

1121
01:02:57,462 --> 01:02:59,180
Is it Example 4, 5?

1122
01:02:59,180 --> 01:03:01,090
You say, this is
a piece of cake.

1123
01:03:01,090 --> 01:03:05,610
I have two methods showing
me that area of the disk

1124
01:03:05,610 --> 01:03:06,965
is so pi r squared.

1125
01:03:06,965 --> 01:03:08,204
It's so trivial.

1126
01:03:08,204 --> 01:03:12,140
Yeah, then let's move
on and do the ellipse.

1127
01:03:12,140 --> 01:03:14,740
Or we could have been
smart and done the ellipse

1128
01:03:14,740 --> 01:03:16,710
from the beginning.

1129
01:03:16,710 --> 01:03:18,600
And then the circular
disk would have

1130
01:03:18,600 --> 01:03:23,010
been just a trivial, particular
example of the ellipse.

1131
01:03:23,010 --> 01:03:25,010
But let's do the ellipse
with this magic formula

1132
01:03:25,010 --> 01:03:26,740
that I just taught you.

1133
01:03:26,740 --> 01:03:29,830


1134
01:03:29,830 --> 01:03:34,250
In the finals-- I'm going to
send you a bunch of finals.

1135
01:03:34,250 --> 01:03:36,630
You're going to be
amused, because you're

1136
01:03:36,630 --> 01:03:38,460
going to look at
them and you say,

1137
01:03:38,460 --> 01:03:41,650
regardless of the year and
semester when the final was

1138
01:03:41,650 --> 01:03:44,220
given for Calc 3,
there was always

1139
01:03:44,220 --> 01:03:49,040
one of the problems at the
end using direct application

1140
01:03:49,040 --> 01:03:50,960
of Green's theorem.

1141
01:03:50,960 --> 01:03:53,290
So Green's theorem
is an obsession,

1142
01:03:53,290 --> 01:03:55,214
and not only at Tech.

1143
01:03:55,214 --> 01:03:58,780
I was looking UT Austin,
A&M, other schools--

1144
01:03:58,780 --> 01:04:05,990
California Berkley-- all the
Calc 3 courses on the final

1145
01:04:05,990 --> 01:04:10,620
have at least one application--
direct application

1146
01:04:10,620 --> 01:04:12,360
applying principal.

1147
01:04:12,360 --> 01:04:12,860
OK.

1148
01:04:12,860 --> 01:04:17,250


1149
01:04:17,250 --> 01:04:18,960
So what did I say?

1150
01:04:18,960 --> 01:04:21,715
I said that we have
to draw an ellipse.

1151
01:04:21,715 --> 01:04:25,426
How do we draw an ellipse
without making it up?

1152
01:04:25,426 --> 01:04:26,845
That's the question.

1153
01:04:26,845 --> 01:04:28,737
STUDENT: Draw a circle.

1154
01:04:28,737 --> 01:04:30,156
MAGDALENA TODA: Draw a circle.

1155
01:04:30,156 --> 01:04:32,060
Good answer.

1156
01:04:32,060 --> 01:04:33,580
OK.

1157
01:04:33,580 --> 01:04:35,060
All right.

1158
01:04:35,060 --> 01:04:40,010
And guys this
started really bad.

1159
01:04:40,010 --> 01:04:43,433
So I'm doing what I can.

1160
01:04:43,433 --> 01:04:46,391


1161
01:04:46,391 --> 01:04:49,349
I should have tried
more coffee today,

1162
01:04:49,349 --> 01:04:52,460
because I'm getting
insecure and very shaky.

1163
01:04:52,460 --> 01:04:52,960
OK.

1164
01:04:52,960 --> 01:04:58,100
So I have the ellipse
in standard form

1165
01:04:58,100 --> 01:05:01,620
of center O, x squared over
x squared plus y squared

1166
01:05:01,620 --> 01:05:05,300
over B squared equals 1.

1167
01:05:05,300 --> 01:05:07,890
And now you are going to
me who is A and who is B?

1168
01:05:07,890 --> 01:05:08,920
What are they called?

1169
01:05:08,920 --> 01:05:10,314
Semi--

1170
01:05:10,314 --> 01:05:11,105
STUDENT: Semiotics.

1171
01:05:11,105 --> 01:05:12,188
MAGDALENA TODA: Semiotics.

1172
01:05:12,188 --> 01:05:15,480
A and B. Good.

1173
01:05:15,480 --> 01:05:18,900
Find the area.

1174
01:05:18,900 --> 01:05:21,810
I don't like-- OK.

1175
01:05:21,810 --> 01:05:27,080
Let's put B inside, and let's
put C outside the boundary.

1176
01:05:27,080 --> 01:05:42,690
So area of the ellipse domain
D will be-- by the lemma-- 1/2

1177
01:05:42,690 --> 01:05:46,010
integral over C.

1178
01:05:46,010 --> 01:05:47,250
This is C. Is not f.

1179
01:05:47,250 --> 01:05:48,200
Don't confuse it.

1180
01:05:48,200 --> 01:05:50,616
It is my beautiful
script C. I've

1181
01:05:50,616 --> 01:05:52,420
tried to use it many times.

1182
01:05:52,420 --> 01:05:55,240
Going to be minus y dx plus xdy.

1183
01:05:55,240 --> 01:05:58,380


1184
01:05:58,380 --> 01:05:59,490
Again, why was that?

1185
01:05:59,490 --> 01:06:04,530
Because we said this
is M and this is N,

1186
01:06:04,530 --> 01:06:09,110
and Green's theorem will give
you double integral of N sub x

1187
01:06:09,110 --> 01:06:10,570
minus M sub y.

1188
01:06:10,570 --> 01:06:13,910
So you have 1 minus
minus 1, which is 2.

1189
01:06:13,910 --> 01:06:15,910
And 2 knocked that out.

1190
01:06:15,910 --> 01:06:16,410
OK.

1191
01:06:16,410 --> 01:06:19,090
That's how we prove it.

1192
01:06:19,090 --> 01:06:19,720
OK.

1193
01:06:19,720 --> 01:06:24,210
Problem is that I do not the
parametrization of the ellipse.

1194
01:06:24,210 --> 01:06:28,220
And if somebody doesn't help me,
I'm going to be in big trouble.

1195
01:06:28,220 --> 01:06:32,620


1196
01:06:32,620 --> 01:06:34,420
And I'll start
cursing and I'm not

1197
01:06:34,420 --> 01:06:37,070
allowed to curse in
front of the classroom.

1198
01:06:37,070 --> 01:06:40,760
But you can help me on
that, because this reminds

1199
01:06:40,760 --> 01:06:46,040
you of a famous Greek identity.

1200
01:06:46,040 --> 01:06:48,520
The fundamental trig identity.

1201
01:06:48,520 --> 01:06:51,610
If this would be cosine
squared of theta,

1202
01:06:51,610 --> 01:06:55,260
and this would be sine squared
of theta, as two animals,

1203
01:06:55,260 --> 01:06:56,710
their sum would be 1.

1204
01:06:56,710 --> 01:07:00,630
And whenever you have sums
of sum squared thingies,

1205
01:07:00,630 --> 01:07:03,834
then you have to think trig.

1206
01:07:03,834 --> 01:07:06,710
So, what would be
good as a parameter?

1207
01:07:06,710 --> 01:07:07,300
OK.

1208
01:07:07,300 --> 01:07:10,240
What would be good
as a parametrization

1209
01:07:10,240 --> 01:07:12,214
to make this come true?

1210
01:07:12,214 --> 01:07:15,012
STUDENT: You have the cosine
of theta would equal x over x.

1211
01:07:15,012 --> 01:07:15,970
MAGDALENA TODA: Uh-huh.

1212
01:07:15,970 --> 01:07:18,126
So then x would be A times--

1213
01:07:18,126 --> 01:07:19,410
STUDENT: The cosine of theta.

1214
01:07:19,410 --> 01:07:21,590
MAGDALENA TODA:
Do you like theta?

1215
01:07:21,590 --> 01:07:23,690
You don't, because
you're not Greek.

1216
01:07:23,690 --> 01:07:25,050
That's the problem.

1217
01:07:25,050 --> 01:07:26,840
If you were Greek,
you would like it.

1218
01:07:26,840 --> 01:07:29,260
We had a colleague who
is not here anymore.

1219
01:07:29,260 --> 01:07:30,790
Greek from Cypress.

1220
01:07:30,790 --> 01:07:38,250
And he could claim that the
most important-- most important

1221
01:07:38,250 --> 01:07:40,230
alphabet is the
Greek one, and that's

1222
01:07:40,230 --> 01:07:44,210
why the mathematicians
adopted it.

1223
01:07:44,210 --> 01:07:45,150
OK?

1224
01:07:45,150 --> 01:07:47,150
B sine t.

1225
01:07:47,150 --> 01:07:48,270
How do you check?

1226
01:07:48,270 --> 01:07:49,290
You always think, OK.

1227
01:07:49,290 --> 01:07:51,460
This over that is cosine.

1228
01:07:51,460 --> 01:07:53,490
This over this is sine.

1229
01:07:53,490 --> 01:07:54,340
I square them.

1230
01:07:54,340 --> 01:07:56,210
I get exactly that
and I get a 1.

1231
01:07:56,210 --> 01:07:56,710
Good.

1232
01:07:56,710 --> 01:07:57,670
I'm in good shape.

1233
01:07:57,670 --> 01:08:01,380
I know that this
implicit equation--

1234
01:08:01,380 --> 01:08:04,710
this is an implicit
equation-- happens if and only

1235
01:08:04,710 --> 01:08:11,080
if I have this system of
the parametrization with t

1236
01:08:11,080 --> 01:08:17,000
between-- anything I want,
including the basic 0 to 2

1237
01:08:17,000 --> 01:08:18,420
pi interval.

1238
01:08:18,420 --> 01:08:22,380
And then if I were to move
all around for time real t

1239
01:08:22,380 --> 01:08:26,274
I would wind around that the
circle infinitely many times.

1240
01:08:26,274 --> 01:08:29,270
Between time equals
minus infinity--

1241
01:08:29,270 --> 01:08:33,060
that nobody remembers-- and
time equals plus infinity--

1242
01:08:33,060 --> 01:08:36,180
that nobody will
ever get to know.

1243
01:08:36,180 --> 01:08:38,550
So those are the values of it.

1244
01:08:38,550 --> 01:08:41,366
All the real values, actually.

1245
01:08:41,366 --> 01:08:44,960
I only needed from 0 to 2
pi to wind one time around.

1246
01:08:44,960 --> 01:08:46,660
And this is the idea.

1247
01:08:46,660 --> 01:08:48,514
I wind one time around.

1248
01:08:48,514 --> 01:08:51,149
Now people-- you're going
to see mathematicians

1249
01:08:51,149 --> 01:08:52,740
are not the greatest people.

1250
01:08:52,740 --> 01:09:01,254
I've seen engineers and
physicists use a lot this sign.

1251
01:09:01,254 --> 01:09:02,420
Do you know what this means?

1252
01:09:02,420 --> 01:09:04,420
STUDENT: It means
one full revolution.

1253
01:09:04,420 --> 01:09:06,550
MAGDALENA TODA: It
means a full revolution.

1254
01:09:06,550 --> 01:09:10,410
You're going to have
a loop-- loops, that's

1255
01:09:10,410 --> 01:09:11,240
whatever you want.

1256
01:09:11,240 --> 01:09:13,380
Here and goes counterclockwise.

1257
01:09:13,380 --> 01:09:15,720
And they put this
little sign showing

1258
01:09:15,720 --> 01:09:21,790
I'm going counterclockwise on
a closed curved, or a loop.

1259
01:09:21,790 --> 01:09:22,509
All right.

1260
01:09:22,509 --> 01:09:24,439
Don't think they are crazy.

1261
01:09:24,439 --> 01:09:27,479
This was used in lots
of scientific papers

1262
01:09:27,479 --> 01:09:30,810
in math, physics, and
engineering, and so on.

1263
01:09:30,810 --> 01:09:31,310
OK.

1264
01:09:31,310 --> 01:09:34,850


1265
01:09:34,850 --> 01:09:36,660
Let's do it then.

1266
01:09:36,660 --> 01:09:38,279
Can we do it by ourselves?

1267
01:09:38,279 --> 01:09:39,210
I think so.

1268
01:09:39,210 --> 01:09:39,810
That's see.

1269
01:09:39,810 --> 01:09:42,370
1/2 is 1.

1270
01:09:42,370 --> 01:09:45,310
And I don't like
the pink marker.

1271
01:09:45,310 --> 01:09:47,270
Integral log.

1272
01:09:47,270 --> 01:09:51,930
Time from 0 to 2 pi
should be measured.

1273
01:09:51,930 --> 01:09:55,740
y minus B sine t.

1274
01:09:55,740 --> 01:10:01,730


1275
01:10:01,730 --> 01:10:04,345
dx-- what tells me that?

1276
01:10:04,345 --> 01:10:06,810
STUDENT: B minus--

1277
01:10:06,810 --> 01:10:07,223


1278
01:10:07,223 --> 01:10:08,306
MAGDALENA TODA: Very good.

1279
01:10:08,306 --> 01:10:09,800
Minus A sine t.

1280
01:10:09,800 --> 01:10:10,796
How hard is that?

1281
01:10:10,796 --> 01:10:16,274
It's a piece of cake Plus x--

1282
01:10:16,274 --> 01:10:18,179
STUDENT: A cosine.

1283
01:10:18,179 --> 01:10:19,262
MAGDALENA TODA: Very good.

1284
01:10:19,262 --> 01:10:21,760
A cosine t.

1285
01:10:21,760 --> 01:10:23,090
TImes--

1286
01:10:23,090 --> 01:10:25,630
STUDENT: B cosine t.

1287
01:10:25,630 --> 01:10:28,790
MAGDALENA TODA: --B cosine t.

1288
01:10:28,790 --> 01:10:30,130
And dt.

1289
01:10:30,130 --> 01:10:32,750
And this thing-- look at it.

1290
01:10:32,750 --> 01:10:33,460
It's huge.

1291
01:10:33,460 --> 01:10:35,700
It looks huge, but it's
so beautiful, because--

1292
01:10:35,700 --> 01:10:36,569
STUDENT: AB.

1293
01:10:36,569 --> 01:10:37,360
MAGDALENA TODA: AB.

1294
01:10:37,360 --> 01:10:38,805
Why is it AB?

1295
01:10:38,805 --> 01:10:44,030
It's AB because sine squared
plus cosine squared inside

1296
01:10:44,030 --> 01:10:46,180
becomes 1.

1297
01:10:46,180 --> 01:10:49,990
And I have plus AB,
plus AB, AB out.

1298
01:10:49,990 --> 01:10:52,020
Kick out the AB.

1299
01:10:52,020 --> 01:10:57,090
Kick out the A and
the B and you get

1300
01:10:57,090 --> 01:11:01,750
something beautiful-- sine
squared t plus cosine squared

1301
01:11:01,750 --> 01:11:03,420
t is your old friend.

1302
01:11:03,420 --> 01:11:04,840
And he says, I'm 1.

1303
01:11:04,840 --> 01:11:08,049
Look how beautiful
life is for you.

1304
01:11:08,049 --> 01:11:09,498
Finally, we proved it.

1305
01:11:09,498 --> 01:11:10,947
What did we prove?

1306
01:11:10,947 --> 01:11:11,913
We are almost there.

1307
01:11:11,913 --> 01:11:12,879
We got a 1/2.

1308
01:11:12,879 --> 01:11:15,780


1309
01:11:15,780 --> 01:11:18,600
A constant value kick out, AB.

1310
01:11:18,600 --> 01:11:21,504


1311
01:11:21,504 --> 01:11:22,472
STUDENT: Times 2 pi.

1312
01:11:22,472 --> 01:11:23,597
MAGDALENA TODA: Times 2 pi.

1313
01:11:23,597 --> 01:11:26,840


1314
01:11:26,840 --> 01:11:28,010
Good.

1315
01:11:28,010 --> 01:11:30,110
2 goes away.

1316
01:11:30,110 --> 01:11:33,300
And we got a magic thing that
nobody taught us in school,

1317
01:11:33,300 --> 01:11:34,730
because they were mean.

1318
01:11:34,730 --> 01:11:37,360
They really didn't want
us to learn too much.

1319
01:11:37,360 --> 01:11:38,760
That's the thingy.

1320
01:11:38,760 --> 01:11:40,540
AB pi.

1321
01:11:40,540 --> 01:11:45,220
AB pi is what we were
hoping for, because, look.

1322
01:11:45,220 --> 01:11:47,820
I mean it's almost
too good to be true.

1323
01:11:47,820 --> 01:11:53,610
Well, it's a disk of radius
r, A and B are equal.

1324
01:11:53,610 --> 01:11:55,870
And they are the
radius of the disk.

1325
01:11:55,870 --> 01:11:58,730
And that's why we
have pi r squared

1326
01:11:58,730 --> 01:12:00,972
as a particular
example of the disk

1327
01:12:00,972 --> 01:12:04,924
of the area of this ellipse.

1328
01:12:04,924 --> 01:12:07,641
When I saw it the first
time, I was like, well,

1329
01:12:07,641 --> 01:12:12,500
I'm glad that I lived to be
30 or something to learn this.

1330
01:12:12,500 --> 01:12:17,510
Because nobody had shown it
to me in K-12 or in college.

1331
01:12:17,510 --> 01:12:22,534
And I was a completing-- I was
a PhD and I didn't know it.

1332
01:12:22,534 --> 01:12:25,378
And then I said, oh,
that's why-- pi AB.

1333
01:12:25,378 --> 01:12:26,800
Yes, OK.

1334
01:12:26,800 --> 01:12:27,770
All right.

1335
01:12:27,770 --> 01:12:31,870
So it's so easy to
understand once you-- well.

1336
01:12:31,870 --> 01:12:33,493
Once you learn the section.

1337
01:12:33,493 --> 01:12:34,909
If you don't learn
the section you

1338
01:12:34,909 --> 01:12:38,970
will not be able to understand.

1339
01:12:38,970 --> 01:12:39,470
OK.

1340
01:12:39,470 --> 01:12:39,970
All right.

1341
01:12:39,970 --> 01:12:42,300
I'm going to go
ahead and erase this.

1342
01:12:42,300 --> 01:12:44,950
And I'll show you
an example that

1343
01:12:44,950 --> 01:12:49,500
was popping up like an obsession
with the numbers changed

1344
01:12:49,500 --> 01:12:53,190
in most of the final exams
that happen in the last three

1345
01:12:53,190 --> 01:12:58,900
years, regardless of
who wrote the exam.

1346
01:12:58,900 --> 01:13:04,590
Because this problem really
matches the learning outcomes,

1347
01:13:04,590 --> 01:13:08,530
oh, just about any university--
any good university

1348
01:13:08,530 --> 01:13:10,690
around the world.

1349
01:13:10,690 --> 01:13:12,240
So you'll say, wow.

1350
01:13:12,240 --> 01:13:13,110
It's so easy.

1351
01:13:13,110 --> 01:13:16,754
I could not believe it
that-- how easy it is.

1352
01:13:16,754 --> 01:13:25,664
But once you see it, you
will-- you'll say, wow.

1353
01:13:25,664 --> 01:13:26,660
It's easy.

1354
01:13:26,660 --> 01:13:34,626


1355
01:13:34,626 --> 01:13:35,126
OK.

1356
01:13:35,126 --> 01:13:41,102


1357
01:13:41,102 --> 01:13:44,120
[CHATTER]

1358
01:13:44,120 --> 01:13:46,034
Let's try this one.

1359
01:13:46,034 --> 01:13:48,519
You have a circle.

1360
01:13:48,519 --> 01:13:57,770
and the circle will be
a circle radius r given

1361
01:13:57,770 --> 01:14:03,590
and origin 0 of 4, 9, 0, and 0.

1362
01:14:03,590 --> 01:14:08,520


1363
01:14:08,520 --> 01:14:17,290
And I'm going to
write-- I'm going

1364
01:14:17,290 --> 01:14:19,970
to give you-- first I'm going
to give you a very simple one.

1365
01:14:19,970 --> 01:14:31,171


1366
01:14:31,171 --> 01:14:37,989
Compute in the
simplest possible way.

1367
01:14:37,989 --> 01:14:41,570
If you don't want to
parametrize the circle--

1368
01:14:41,570 --> 01:14:43,400
you can always
parametrize the circle.

1369
01:14:43,400 --> 01:14:44,302
Right?

1370
01:14:44,302 --> 01:14:45,420
But you don't want to.

1371
01:14:45,420 --> 01:14:49,270
You want to do it the
fastest possible way

1372
01:14:49,270 --> 01:14:51,440
without parameterizing
the circle.

1373
01:14:51,440 --> 01:14:53,880
Without writing down
what I'm writing down.

1374
01:14:53,880 --> 01:14:55,110
You are in a hurry.

1375
01:14:55,110 --> 01:14:58,980
You have 20-- 15 minutes
left of your final.

1376
01:14:58,980 --> 01:15:00,700
And you're looking
at me and say, I

1377
01:15:00,700 --> 01:15:02,220
hope I get an A in this final.

1378
01:15:02,220 --> 01:15:05,887
So what do you have to
remember when you look at that?

1379
01:15:05,887 --> 01:15:09,660


1380
01:15:09,660 --> 01:15:14,770
M and M. M and M.
No, M and N. OK.

1381
01:15:14,770 --> 01:15:18,690
And you have to remember
that you are over a circle

1382
01:15:18,690 --> 01:15:20,150
so you have a closed loop.

1383
01:15:20,150 --> 01:15:21,780
And that's a Jordan curve.

1384
01:15:21,780 --> 01:15:24,130
That's enclosing a disk.

1385
01:15:24,130 --> 01:15:28,070
So you have a relationship
between the path

1386
01:15:28,070 --> 01:15:34,650
integral along the C and the
area along the D-- over D.

1387
01:15:34,650 --> 01:15:36,350
Which is of what?

1388
01:15:36,350 --> 01:15:38,720
Is N sub x minus M sub y.

1389
01:15:38,720 --> 01:15:41,300
So let me write it
in this form, which

1390
01:15:41,300 --> 01:15:46,120
is the same thing my students
mostly prefer to write it as.

1391
01:15:46,120 --> 01:15:48,710
N sub x minus M sub y.

1392
01:15:48,710 --> 01:15:51,790
The t-shirt I have
has it written

1393
01:15:51,790 --> 01:15:56,960
like that, because it was
bought from nerdytshirt.com

1394
01:15:56,960 --> 01:16:01,220
And it was especially
created to impress nerds.

1395
01:16:01,220 --> 01:16:04,300
And of course if you
look at the del notation

1396
01:16:04,300 --> 01:16:07,380
that gives you that kind
of snobbish attitude

1397
01:16:07,380 --> 01:16:11,670
that you aren't a scientist.

1398
01:16:11,670 --> 01:16:12,220
OK.

1399
01:16:12,220 --> 01:16:16,332
So what is this
going to be then?

1400
01:16:16,332 --> 01:16:19,390
Double integral over d.

1401
01:16:19,390 --> 01:16:22,390
And sub x is up
here so it gave 5.

1402
01:16:22,390 --> 01:16:24,750
And sub y is a piece of cake.

1403
01:16:24,750 --> 01:16:36,530
3 dx dy equals 2 out times
the area of the disk, which

1404
01:16:36,530 --> 01:16:38,466
is something you know.

1405
01:16:38,466 --> 01:16:40,886
And I'm not going to ask you
to prove that all over again.

1406
01:16:40,886 --> 01:16:42,760
So you have to say 2.

1407
01:16:42,760 --> 01:16:46,586
I know the area of the
disk-- pi r squared.

1408
01:16:46,586 --> 01:16:48,002
And that's the answer.

1409
01:16:48,002 --> 01:16:49,418
And you leave the room.

1410
01:16:49,418 --> 01:16:50,362
And that's it.

1411
01:16:50,362 --> 01:16:52,260
It's almost too
easy to believe it,

1412
01:16:52,260 --> 01:16:58,280
but it was always there in
the simplest possible way.

1413
01:16:58,280 --> 01:17:02,610
And now I'm wondering, if I
were to give you something hard,

1414
01:17:02,610 --> 01:17:08,040
because-- you know my theory
that when you practice

1415
01:17:08,040 --> 01:17:11,810
at something in
the classroom you

1416
01:17:11,810 --> 01:17:16,650
have to be working on harder
things in the classroom

1417
01:17:16,650 --> 01:17:19,310
to do better in the exam.

1418
01:17:19,310 --> 01:17:22,990
So let me cook up
something ugly for you.

1419
01:17:22,990 --> 01:17:25,830
The same kind of disk.

1420
01:17:25,830 --> 01:17:28,250
And I'm changing the functions.

1421
01:17:28,250 --> 01:17:33,055
And I'll make it
more complicated.

1422
01:17:33,055 --> 01:17:36,140


1423
01:17:36,140 --> 01:17:40,478
Let's see how you
perform on this one.

1424
01:17:40,478 --> 01:17:47,240


1425
01:17:47,240 --> 01:17:49,710
We avoided that one,
probably, on finals

1426
01:17:49,710 --> 01:17:52,478
because I think the
majority of students

1427
01:17:52,478 --> 01:17:58,060
wouldn't have understood what
theorem they needed to apply.

1428
01:17:58,060 --> 01:17:59,690
It looks a little bit scary.

1429
01:17:59,690 --> 01:18:01,910
But let's say that I've
given you the hint,

1430
01:18:01,910 --> 01:18:05,003
apply Greens theorem
on the same path

1431
01:18:05,003 --> 01:18:10,380
integral, which is a circle
of origin 0 and radius r.

1432
01:18:10,380 --> 01:18:14,410
I now draw counterclockwise.

1433
01:18:14,410 --> 01:18:18,490
You apply Green's theorem and
you say, I know how to do this,

1434
01:18:18,490 --> 01:18:21,130
because now I know the theorem.

1435
01:18:21,130 --> 01:18:27,270
This is M. This is N. And I--
my t-shirt did not say M and N.

1436
01:18:27,270 --> 01:18:30,870
It said P and Q. Do you
want to put P and Q?

1437
01:18:30,870 --> 01:18:31,750
I put P and Q.

1438
01:18:31,750 --> 01:18:34,740
So I can-- I can have this
like it is on my t-shirt.

1439
01:18:34,740 --> 01:18:39,230
So this is going
to be P sub x-- no.

1440
01:18:39,230 --> 01:18:39,730
Q sub x.

1441
01:18:39,730 --> 01:18:40,730
Sorry.

1442
01:18:40,730 --> 01:18:44,560
M and N. So the second
one with respect to x.

1443
01:18:44,560 --> 01:18:48,560
The one that sticks to the y
is prime root respect to x.

1444
01:18:48,560 --> 01:18:54,344
The one that sticks to dx is
prime root with respect to y.

1445
01:18:54,344 --> 01:18:56,885
And I think one
time-- the one time

1446
01:18:56,885 --> 01:19:01,140
when that my friend and
colleague wrote that,

1447
01:19:01,140 --> 01:19:02,680
he did it differently.

1448
01:19:02,680 --> 01:19:05,595
He wrote something
like, just-- I'll

1449
01:19:05,595 --> 01:19:09,750
put-- I don't remember what.

1450
01:19:09,750 --> 01:19:10,800
He put this one.

1451
01:19:10,800 --> 01:19:13,506


1452
01:19:13,506 --> 01:19:17,470
Then the student
was used to dx/dy

1453
01:19:17,470 --> 01:19:19,250
and got completely confused.

1454
01:19:19,250 --> 01:19:25,625
So pay attention to
what you are saying.

1455
01:19:25,625 --> 01:19:29,400
Most of us write it
in x and y first.

1456
01:19:29,400 --> 01:19:32,960
And we can see that the
derivative with respect

1457
01:19:32,960 --> 01:19:38,870
to x of q, because that is
the one next to be the y.

1458
01:19:38,870 --> 01:19:42,446
When he gave it to me
like that, he messed up

1459
01:19:42,446 --> 01:19:44,760
everybody's notations.

1460
01:19:44,760 --> 01:19:46,000
No.

1461
01:19:46,000 --> 01:19:47,190
Good students steal data.

1462
01:19:47,190 --> 01:19:49,770
So you guys have to
put it in standard form

1463
01:19:49,770 --> 01:19:52,825
and pay attention to
what you are doing.

1464
01:19:52,825 --> 01:19:53,770
All right.

1465
01:19:53,770 --> 01:19:57,140
So that one form can
be swapped by people

1466
01:19:57,140 --> 01:19:58,620
who try to play games.

1467
01:19:58,620 --> 01:20:02,570


1468
01:20:02,570 --> 01:20:08,312
Now in this one-- So you
have q sub x minus b sub y.

1469
01:20:08,312 --> 01:20:15,990
You have 3x squared minus
minus, or just plus, 3y squared.

1470
01:20:15,990 --> 01:20:16,490
Good.

1471
01:20:16,490 --> 01:20:17,460
Wonderful.

1472
01:20:17,460 --> 01:20:20,370
Am I happy, do you
think I'm happy?

1473
01:20:20,370 --> 01:20:22,740
Why would I be so happy?

1474
01:20:22,740 --> 01:20:25,790
Why is this a happy thing?

1475
01:20:25,790 --> 01:20:27,740
I could have had
something more wild.

1476
01:20:27,740 --> 01:20:28,270
I don't.

1477
01:20:28,270 --> 01:20:30,210
I'm happy I don't.

1478
01:20:30,210 --> 01:20:31,760
Why am I so happy?

1479
01:20:31,760 --> 01:20:34,360
Let's see.

1480
01:20:34,360 --> 01:20:39,850
3 out over the disk.

1481
01:20:39,850 --> 01:20:41,710
Is this ringing a bell?

1482
01:20:41,710 --> 01:20:48,850


1483
01:20:48,850 --> 01:20:49,880
Yeah.

1484
01:20:49,880 --> 01:20:53,150
It's r squared if I
do this in former.

1485
01:20:53,150 --> 01:20:58,640
So if I do this in former,
its going to be rdr, d theta.

1486
01:20:58,640 --> 01:21:01,340
So life is not as
hard as you believe.

1487
01:21:01,340 --> 01:21:04,035
It can look like
a harder problem,

1488
01:21:04,035 --> 01:21:06,250
but in reality, it's not really.

1489
01:21:06,250 --> 01:21:11,970
So I have 3 times-- now, I
have r squared, I have r cubed.

1490
01:21:11,970 --> 01:21:16,780
r cubed dr d theta, r between.

1491
01:21:16,780 --> 01:21:20,560


1492
01:21:20,560 --> 01:21:30,710
r was between 0 and big
R. Theta will always

1493
01:21:30,710 --> 01:21:34,150
be between 0 and 2 pi.

1494
01:21:34,150 --> 01:21:42,316
So, I want you, without
me to compute the answer

1495
01:21:42,316 --> 01:21:44,454
and tell me what you got.

1496
01:21:44,454 --> 01:21:46,839
STUDENT: Just say it?

1497
01:21:46,839 --> 01:21:48,270
MAGDALENA TODA: Yep.

1498
01:21:48,270 --> 01:21:52,580
STUDENT: 3/2, pi
r to the fourth.

1499
01:21:52,580 --> 01:21:54,350
MAGDALENA TODA: So
how did you do that?

1500
01:21:54,350 --> 01:21:56,610
You said, r to the
4 over 4, coming

1501
01:21:56,610 --> 01:22:00,470
from integration times the 2 pi,
coming from integration times

1502
01:22:00,470 --> 01:22:01,966
3.

1503
01:22:01,966 --> 01:22:05,312
Are you guys with me?

1504
01:22:05,312 --> 01:22:08,690
Is everybody with me on this?

1505
01:22:08,690 --> 01:22:12,130
OK so, we will simplify
the answer, we'll do that.

1506
01:22:12,130 --> 01:22:16,696
What regard is the
radius of the disk?

1507
01:22:16,696 --> 01:22:18,320
STUDENT: How did he
solve that integral

1508
01:22:18,320 --> 01:22:20,008
without switching the poles?

1509
01:22:20,008 --> 01:22:26,456


1510
01:22:26,456 --> 01:22:28,936
MAGDALENA TODA: It would
have been a killer.

1511
01:22:28,936 --> 01:22:30,920
Let me write it out.

1512
01:22:30,920 --> 01:22:32,408
[LAUGHTER]

1513
01:22:32,408 --> 01:22:34,910
Because you want to
write it out, of course.

1514
01:22:34,910 --> 01:22:40,550
OK, 3 integral, integral
x squared plus y

1515
01:22:40,550 --> 01:22:45,450
squared, dy/dx, just to make
my life a little bit funnier,

1516
01:22:45,450 --> 01:22:50,280
and then y between minus
square root-- you're

1517
01:22:50,280 --> 01:22:52,060
looking for trouble, huh?

1518
01:22:52,060 --> 01:22:59,580
Y squared minus x squared to
r squared minus s squared.

1519
01:22:59,580 --> 01:23:01,250
And again, you could
do what you just

1520
01:23:01,250 --> 01:23:04,536
said, split into four integrals
over four different domains,

1521
01:23:04,536 --> 01:23:07,440
or two up and down.

1522
01:23:07,440 --> 01:23:11,950
And minus r and are
you guys with me?

1523
01:23:11,950 --> 01:23:14,890
And then, when you go
and integrate that,

1524
01:23:14,890 --> 01:23:22,960
you integrate with respect
to y-- [INAUDIBLE].

1525
01:23:22,960 --> 01:23:25,300
Well he's right,
so you can get x

1526
01:23:25,300 --> 01:23:27,835
squared y plus y cubed over 3.

1527
01:23:27,835 --> 01:23:30,770


1528
01:23:30,770 --> 01:23:34,240
Between those points, minus 12.

1529
01:23:34,240 --> 01:23:36,120
And from that moment,
that would just

1530
01:23:36,120 --> 01:23:39,563
leave it and go for a walk.

1531
01:23:39,563 --> 01:23:43,260
I will not have the
patience to do this.

1532
01:23:43,260 --> 01:23:44,800
Just a second, Matthew.

1533
01:23:44,800 --> 01:23:46,680
For this kind of
stuff, of course

1534
01:23:46,680 --> 01:23:50,410
I could put this in Maple.

1535
01:23:50,410 --> 01:23:54,150
You know Maple has these
little interactive fields,

1536
01:23:54,150 --> 01:23:55,940
like little squares?

1537
01:23:55,940 --> 01:23:58,940
And you go inside there
and add your endpoints.

1538
01:23:58,940 --> 01:24:02,740
And even if it looks very ugly,
Maple will spit you the answer.

1539
01:24:02,740 --> 01:24:05,668
If you know your
syntax and do it right,

1540
01:24:05,668 --> 01:24:07,620
even if you don't
switch to polar

1541
01:24:07,620 --> 01:24:10,060
coordinates or put
it in Cartesian.

1542
01:24:10,060 --> 01:24:12,980
Give it the right data, and
it's going to spit the answer.

1543
01:24:12,980 --> 01:24:13,942
Yes, Matthew?

1544
01:24:13,942 --> 01:24:16,106
STUDENT: I was
out of the room, I

1545
01:24:16,106 --> 01:24:19,242
was wondering why
it's now y cubed.

1546
01:24:19,242 --> 01:24:21,450
MAGDALENA TODA: Because if
you integrate with respect

1547
01:24:21,450 --> 01:24:24,140
to y first--

1548
01:24:24,140 --> 01:24:27,460
STUDENT: Because when I
walked out, it was negative y.

1549
01:24:27,460 --> 01:24:29,084
MAGDALENA TODA: If
I didn't put minus.

1550
01:24:29,084 --> 01:24:30,250
STUDENT: It's a new problem.

1551
01:24:30,250 --> 01:24:32,120
That's what he's confused about.

1552
01:24:32,120 --> 01:24:34,700
He walked out of the room
during the previous problem

1553
01:24:34,700 --> 01:24:36,420
and came back after this one.

1554
01:24:36,420 --> 01:24:37,709
And now he's confused.

1555
01:24:37,709 --> 01:24:40,000
MAGDALENA TODA: You don't
care about what I just asked?

1556
01:24:40,000 --> 01:24:40,380
STUDENT: Oh.

1557
01:24:40,380 --> 01:24:40,880
No.

1558
01:24:40,880 --> 01:24:44,125


1559
01:24:44,125 --> 01:24:45,942
I like the polar coordinates.

1560
01:24:45,942 --> 01:24:47,650
MAGDALENA TODA: Let
me ask you a question

1561
01:24:47,650 --> 01:24:50,100
before I talk any further.

1562
01:24:50,100 --> 01:24:53,250
I was about to put a plus here.

1563
01:24:53,250 --> 01:24:56,460
What would have been the problem
if I had put a plus here?

1564
01:24:56,460 --> 01:24:59,810


1565
01:24:59,810 --> 01:25:02,560
If I worked this out,
I would have gotten

1566
01:25:02,560 --> 01:25:05,930
x squared minus y squared.

1567
01:25:05,930 --> 01:25:08,300
Would that have been
the end of the world?

1568
01:25:08,300 --> 01:25:10,040
No.

1569
01:25:10,040 --> 01:25:16,156
But it would have complicated
my life a little bit more.

1570
01:25:16,156 --> 01:25:20,990
Let's do that one as well.

1571
01:25:20,990 --> 01:25:22,422
STUDENT: I was
just curious of how

1572
01:25:22,422 --> 01:25:25,247
you do any of these problems
when you can't switch to polar.

1573
01:25:25,247 --> 01:25:27,830
MAGDALENA TODA: Right, let's see
what-- because Actually, even

1574
01:25:27,830 --> 01:25:32,440
in this case, life is not so
hard, not as hard as you think.

1575
01:25:32,440 --> 01:25:34,750
The persistence in that matters.

1576
01:25:34,750 --> 01:25:37,620
You never give up on a
problem that freaks you out.

1577
01:25:37,620 --> 01:25:41,240
That's the definition
of a mathematician.

1578
01:25:41,240 --> 01:25:48,380
3x squared minus 3y
squared over dx/dy.

1579
01:25:48,380 --> 01:25:50,270
Do it slowly because
I'm not in a hurry.

1580
01:25:50,270 --> 01:25:55,670
We are almost done with 13.4.

1581
01:25:55,670 --> 01:25:57,407
This is OK, right?

1582
01:25:57,407 --> 01:25:59,355
Just the minus sign again?

1583
01:25:59,355 --> 01:26:01,303
STUDENT: Well not
the minus sign.

1584
01:26:01,303 --> 01:26:03,738
I was just wondering because
in the previous problem

1585
01:26:03,738 --> 01:26:07,147
you were doing the ellipse, you
started out with the equation

1586
01:26:07,147 --> 01:26:08,608
with the negative y--

1587
01:26:08,608 --> 01:26:10,556
MAGDALENA TODA:
For this one that's

1588
01:26:10,556 --> 01:26:14,844
just the limit that says that
this is the go double integral

1589
01:26:14,844 --> 01:26:18,920
of the area of the domain.

1590
01:26:18,920 --> 01:26:23,120
It's just a consequence--
or correlate if you want.

1591
01:26:23,120 --> 01:26:27,715
It's a consequence
of Green's theorem.

1592
01:26:27,715 --> 01:26:31,100
When you forget that consequence
of Green's theorem and we say

1593
01:26:31,100 --> 01:26:32,005
goodbye to that.

1594
01:26:32,005 --> 01:26:35,810
But while you were out,
this is Green's theorem.

1595
01:26:35,810 --> 01:26:38,635
The real Green's theorem,
the one that was a teacher.

1596
01:26:38,635 --> 01:26:41,310
There are several
Greens I can give you.

1597
01:26:41,310 --> 01:26:43,280
The famous Green
theorem is the one

1598
01:26:43,280 --> 01:26:46,850
I said when you have--
this is what we apply here.

1599
01:26:46,850 --> 01:26:50,972
The integral of M dx plus M dy.

1600
01:26:50,972 --> 01:27:00,820
You have a double integral of
M sub x minus M sub y over c.

1601
01:27:00,820 --> 01:27:03,450


1602
01:27:03,450 --> 01:27:07,690
So I'm assuming we would have
had this case of maybe me not

1603
01:27:07,690 --> 01:27:10,990
paying attention, or
being mean and not wanting

1604
01:27:10,990 --> 01:27:14,484
to give you a simple problem.

1605
01:27:14,484 --> 01:27:17,661
And what do you
do in such a case?

1606
01:27:17,661 --> 01:27:19,660
It's not obvious to
everybody, but you will see.

1607
01:27:19,660 --> 01:27:22,050
It's so pretty at some
point, if you know

1608
01:27:22,050 --> 01:27:24,034
how to get out of the mess.

1609
01:27:24,034 --> 01:27:27,010


1610
01:27:27,010 --> 01:27:30,820
I was already thinking, but
I'm using polar coordinates.

1611
01:27:30,820 --> 01:27:35,620
So that's arc of sine, so I
have to go back to the basics.

1612
01:27:35,620 --> 01:27:40,000
If I go back to the
basics, ideas come to me.

1613
01:27:40,000 --> 01:27:41,760
Right?

1614
01:27:41,760 --> 01:27:45,930
So, OK.

1615
01:27:45,930 --> 01:27:51,560
r-- let's put dr d theta,
just to get rid of it,

1616
01:27:51,560 --> 01:27:53,780
because it's on my nerves.

1617
01:27:53,780 --> 01:28:00,370
This is 0 to 2 pi,
this is 0 to r.

1618
01:28:00,370 --> 01:28:03,340
And now, you say,
OK, in our mind,

1619
01:28:03,340 --> 01:28:06,960
because we are lazy
people, plug in those

1620
01:28:06,960 --> 01:28:10,970
and pull out what you can.

1621
01:28:10,970 --> 01:28:14,970
One 3 out equals for what?

1622
01:28:14,970 --> 01:28:18,400
Inside, you have r squared.

1623
01:28:18,400 --> 01:28:20,720
Do you agree?

1624
01:28:20,720 --> 01:28:29,510
And times your favorite
expression, which is cosine

1625
01:28:29,510 --> 01:28:32,550
squared theta, minus
i squared theta.

1626
01:28:32,550 --> 01:28:34,520
And you're going to ask me why.

1627
01:28:34,520 --> 01:28:36,070
You shouldn't ask me why.

1628
01:28:36,070 --> 01:28:40,305
You just square these
and subtract them,

1629
01:28:40,305 --> 01:28:44,050
and see what in the world
you're going to get.

1630
01:28:44,050 --> 01:28:48,290
Because you get r squared
times cosine squared,

1631
01:28:48,290 --> 01:28:49,510
minus i squared.

1632
01:28:49,510 --> 01:28:51,434
I'm too lazy to write
down the argument.

1633
01:28:51,434 --> 01:28:52,850
But you know we
have trigonometry.

1634
01:28:52,850 --> 01:28:55,670


1635
01:28:55,670 --> 01:28:58,170
Yes, you see why it's
important for you

1636
01:28:58,170 --> 01:29:02,010
to learn trigonometry
when you are little.

1637
01:29:02,010 --> 01:29:05,730
You may be 50 or
60, in high school,

1638
01:29:05,730 --> 01:29:09,188
or you may be freshman year.

1639
01:29:09,188 --> 01:29:11,980
I don't care when, but you
have to learn that this is

1640
01:29:11,980 --> 01:29:14,306
the cosine of the double angle.

1641
01:29:14,306 --> 01:29:16,240
How many of you remember that?

1642
01:29:16,240 --> 01:29:18,540
Maybe you learned that?

1643
01:29:18,540 --> 01:29:19,460
Remember that?

1644
01:29:19,460 --> 01:29:21,400
OK.

1645
01:29:21,400 --> 01:29:25,800
I don't blame you at all
when you don't remember,

1646
01:29:25,800 --> 01:29:30,920
because since I've been
the main checker of finals

1647
01:29:30,920 --> 01:29:39,490
for the past five years--
it's a lot of finals.

1648
01:29:39,490 --> 01:29:40,690
Yeah, the i is there.

1649
01:29:40,690 --> 01:29:42,840
That's exactly what
I wanted to tell you,

1650
01:29:42,840 --> 01:29:46,450
that's why I left some room.

1651
01:29:46,450 --> 01:29:52,566
This data would be t.

1652
01:29:52,566 --> 01:29:56,370
The double angle formula did
not appear on many finals.

1653
01:29:56,370 --> 01:29:58,110
And I was thinking
it's a period.

1654
01:29:58,110 --> 01:29:59,960
When I ask the
instructors, generally they

1655
01:29:59,960 --> 01:30:06,550
say students have trouble
remembering or understanding

1656
01:30:06,550 --> 01:30:10,130
this later on, by
avoiding the issue,

1657
01:30:10,130 --> 01:30:14,012
you sort of bound to it for
the first time in Cal 2,

1658
01:30:14,012 --> 01:30:16,868
because there are any
geometric formulas.

1659
01:30:16,868 --> 01:30:21,634
And then, you bump again
inside it in Cal 3.

1660
01:30:21,634 --> 01:30:23,310
And it never leaves you.

1661
01:30:23,310 --> 01:30:27,560
So this, just knowing this
will help you so much.

1662
01:30:27,560 --> 01:30:30,852
Let me put the r nicely here.

1663
01:30:30,852 --> 01:30:34,123
And now finally, we know
how to solve it, because I'm

1664
01:30:34,123 --> 01:30:35,289
going to go ahead and erase.

1665
01:30:35,289 --> 01:30:44,660


1666
01:30:44,660 --> 01:30:49,250
So why it is good for us is
that-- as Matthew observed

1667
01:30:49,250 --> 01:30:52,810
a few moments ago,
whenever you have

1668
01:30:52,810 --> 01:30:56,730
a product of a function, you
not only in a function in theta

1669
01:30:56,730 --> 01:31:01,170
only, your life becomes easier
because you can separate them

1670
01:31:01,170 --> 01:31:03,050
between the rhos.

1671
01:31:03,050 --> 01:31:04,190
In two different products.

1672
01:31:04,190 --> 01:31:05,950
So that's would be this theorem.

1673
01:31:05,950 --> 01:31:11,270
And you have 3 times-- the
part that depends only on r,

1674
01:31:11,270 --> 01:31:13,990
and the part that depends
only on theta, let's

1675
01:31:13,990 --> 01:31:14,890
put them separate.

1676
01:31:14,890 --> 01:31:22,170
We need theta, and
dr. And what do you

1677
01:31:22,170 --> 01:31:24,520
integrate when you integrate?

1678
01:31:24,520 --> 01:31:25,470
r cubed.

1679
01:31:25,470 --> 01:31:29,100
Attention, do not do rr.

1680
01:31:29,100 --> 01:31:30,850
From 0 to r.

1681
01:31:30,850 --> 01:31:32,360
OK?

1682
01:31:32,360 --> 01:31:33,651
STUDENT: And cosine theta?

1683
01:31:33,651 --> 01:31:39,550
MAGDALENA TODA: And then you
have a 0 to 2 pi, cosine 2.

1684
01:31:39,550 --> 01:31:42,126
now, let me give
you-- Let me tell you

1685
01:31:42,126 --> 01:31:44,875
what it is, because when
I was young, I was naive

1686
01:31:44,875 --> 01:31:48,100
and I always started with that.

1687
01:31:48,100 --> 01:31:52,490
You should always start with the
part, the trig part in theta.

1688
01:31:52,490 --> 01:31:54,030
Because that becomes 0.

1689
01:31:54,030 --> 01:31:56,552
So no matter how
ugly this is, I've

1690
01:31:56,552 --> 01:31:59,790
had professors who are
playing games with us,

1691
01:31:59,790 --> 01:32:03,800
and they were giving us
some extremely ugly thing

1692
01:32:03,800 --> 01:32:06,470
that would take you forever
for you to integrate.

1693
01:32:06,470 --> 01:32:09,290
Or sometimes, it would have
been impossible to integrate.

1694
01:32:09,290 --> 01:32:11,570
But then, the whole
thing would have been 0

1695
01:32:11,570 --> 01:32:14,295
because when you
integrate cosine 2 theta,

1696
01:32:14,295 --> 01:32:16,610
it goes to sine theta.

1697
01:32:16,610 --> 01:32:20,600
Sine 2 theta at 2 pi and 0
are the same things, 0 minus 0

1698
01:32:20,600 --> 01:32:21,120
equals z.

1699
01:32:21,120 --> 01:32:23,800
So the answer is z.

1700
01:32:23,800 --> 01:32:27,040
I cannot tell you how many
professors I've had who will

1701
01:32:27,040 --> 01:32:28,569
play this game with us.

1702
01:32:28,569 --> 01:32:30,360
They give us something
that discouraged us.

1703
01:32:30,360 --> 01:32:34,010
No, it's not a piece of cake,
compared to what I have.

1704
01:32:34,010 --> 01:32:37,450
Some integral value
will go over two lines,

1705
01:32:37,450 --> 01:32:40,490
with a huge polynomial
or something.

1706
01:32:40,490 --> 01:32:44,000
But in the end, the integral
was 0 for such a result. Yes?

1707
01:32:44,000 --> 01:32:45,250
STUDENT: So I have a question.

1708
01:32:45,250 --> 01:32:49,700
Could we take that force and
prove that it was conservative?

1709
01:32:49,700 --> 01:32:54,940
MAGDALENA TODA: So now
that I'm questioning this,

1710
01:32:54,940 --> 01:32:59,660
I'm not questioning
you, but I-- is

1711
01:32:59,660 --> 01:33:05,630
the force, that is with you--
what is the original force

1712
01:33:05,630 --> 01:33:08,360
that Alex is talking about?

1713
01:33:08,360 --> 01:33:17,000
If I take y cubed i plus x cubed
j-- and you have to be careful.

1714
01:33:17,000 --> 01:33:18,915
Is this conservative?

1715
01:33:18,915 --> 01:33:22,670


1716
01:33:22,670 --> 01:33:25,130
STUDENT: Yeah.

1717
01:33:25,130 --> 01:33:27,740
MAGDALENA TODA: Really?

1718
01:33:27,740 --> 01:33:30,936
Why would we pick
a conservative?

1719
01:33:30,936 --> 01:33:33,814
STUDENT: Y squared plus
x squared over 2 is--

1720
01:33:33,814 --> 01:33:35,605
MAGDALENA TODA: Why is
it not conservative?

1721
01:33:35,605 --> 01:33:38,860


1722
01:33:38,860 --> 01:33:40,560
IT doesn't pass the hole test.

1723
01:33:40,560 --> 01:33:43,590


1724
01:33:43,590 --> 01:33:48,280
So p sub y is not
equal to q sub x.

1725
01:33:48,280 --> 01:33:52,090
If you primed this with respect
to y, you get that dy squared.

1726
01:33:52,090 --> 01:33:54,640
Prime this with this respect
to x, you get 3x squared.

1727
01:33:54,640 --> 01:33:57,430
So it's not concerned with him.

1728
01:33:57,430 --> 01:34:00,642
And still, I'm
getting-- it's a loop,

1729
01:34:00,642 --> 01:34:04,520
and I'm getting a 0, sort
of like I would expect it

1730
01:34:04,520 --> 01:34:07,530
I had any dependence of that.

1731
01:34:07,530 --> 01:34:09,250
What is the secret here?

1732
01:34:09,250 --> 01:34:14,360
STUDENT: That is conservative,
given a condition.

1733
01:34:14,360 --> 01:34:16,730
MAGDALENA TODA: Yes,
given a condition

1734
01:34:16,730 --> 01:34:21,170
that your x and y are moving
on the serpent's circle.

1735
01:34:21,170 --> 01:34:25,900
And that happens, because this
is a symmetric expression,

1736
01:34:25,900 --> 01:34:28,420
and x and y are
moving on a circle,

1737
01:34:28,420 --> 01:34:31,090
and one is the cosine theta
and one is sine theta.

1738
01:34:31,090 --> 01:34:35,160
So in the end, it
simplifies out.

1739
01:34:35,160 --> 01:34:40,280
But in general, if I would
have this kind of problem--

1740
01:34:40,280 --> 01:34:43,950
if somebody asked me is this
conservative, the answer is no.

1741
01:34:43,950 --> 01:34:46,125
Let me give you a
few more examples.

1742
01:34:46,125 --> 01:34:57,620


1743
01:34:57,620 --> 01:35:14,765
One example that maybe will look
hard to most people is here.

1744
01:35:14,765 --> 01:35:36,545


1745
01:35:36,545 --> 01:35:49,750
The vector value function
given by f of x, y incline,

1746
01:35:49,750 --> 01:35:51,950
are two values.

1747
01:35:51,950 --> 01:35:54,966
No, I mean define two
values of [INAUDIBLE].

1748
01:35:54,966 --> 01:36:17,040


1749
01:36:17,040 --> 01:36:19,170
A typical exam problem.

1750
01:36:19,170 --> 01:36:22,980
And I saw it at
Texas A&M, as well.

1751
01:36:22,980 --> 01:36:27,610
So maybe some people like this
kind of a, b, c, d problem.

1752
01:36:27,610 --> 01:36:28,836
Is f conservative?

1753
01:36:28,836 --> 01:36:37,220


1754
01:36:37,220 --> 01:36:38,397
STUDENT: Yep

1755
01:36:38,397 --> 01:36:39,855
MAGDALENA TODA:
You already did it?

1756
01:36:39,855 --> 01:36:41,160
Good for you guys.

1757
01:36:41,160 --> 01:36:45,335
So if I gave you one that
has three components what

1758
01:36:45,335 --> 01:36:47,480
did you have to do?

1759
01:36:47,480 --> 01:36:49,860
Compute the curl.

1760
01:36:49,860 --> 01:36:52,460
You can, of course, compute
the curl also on this one

1761
01:36:52,460 --> 01:36:55,115
and have 0 for the
third component.

1762
01:36:55,115 --> 01:37:01,820
But the simplest thing
is to do f1 and f2.

1763
01:37:01,820 --> 01:37:07,040
f1 prime with respect to y
equals f2 prime with respect

1764
01:37:07,040 --> 01:37:08,090
to x.

1765
01:37:08,090 --> 01:37:12,870
So I'm going to
make a smile here.

1766
01:37:12,870 --> 01:37:16,310
And you realize that the authors
of such a problem, whether they

1767
01:37:16,310 --> 01:37:21,270
are at Tech or at Texas
A&M. They do that on purpose

1768
01:37:21,270 --> 01:37:28,390
so that you can use this
result to the next level.

1769
01:37:28,390 --> 01:38:04,070
And they're saying compute
the happy u over the curve

1770
01:38:04,070 --> 01:38:23,070
x cubed and y cubed equals 8 on
the path that connects points

1771
01:38:23,070 --> 01:38:29,375
2, 1 and 1, 2 in [INAUDIBLE].

1772
01:38:29,375 --> 01:38:39,275


1773
01:38:39,275 --> 01:38:46,205
Does this integral depend on f?

1774
01:38:46,205 --> 01:38:51,140


1775
01:38:51,140 --> 01:38:52,112
State why.

1776
01:38:52,112 --> 01:38:58,260


1777
01:38:58,260 --> 01:39:04,540
And you see, they don't tell
you find the scalar potential.

1778
01:39:04,540 --> 01:39:06,920
Which is bad, and
many of you will

1779
01:39:06,920 --> 01:39:09,115
be able to see it
because you have

1780
01:39:09,115 --> 01:39:14,000
good mathematical intuition,
and a computer process

1781
01:39:14,000 --> 01:39:17,290
planning in the background
over all the other processes.

1782
01:39:17,290 --> 01:39:18,900
We are very visual people.

1783
01:39:18,900 --> 01:39:22,310
If you realize that every time
just there with each other

1784
01:39:22,310 --> 01:39:25,720
through the classroom, there
are hundreds of distractions.

1785
01:39:25,720 --> 01:39:28,165
There's the screen,
there is somebody

1786
01:39:28,165 --> 01:39:31,210
who's next to you
who's sneezing,

1787
01:39:31,210 --> 01:39:34,568
all sorts of distractions.

1788
01:39:34,568 --> 01:39:37,860
Still, your computer
unit can still

1789
01:39:37,860 --> 01:39:41,200
function, trying to
integrate and find the scalar

1790
01:39:41,200 --> 01:39:42,450
potential, which is a miracle.

1791
01:39:42,450 --> 01:39:46,250
I don't know how we managed
to do that after all.

1792
01:39:46,250 --> 01:39:49,840
If you don't manage to do that,
what do you have to set up?

1793
01:39:49,840 --> 01:39:54,870
You have to say, find is
there-- well, you know there is.

1794
01:39:54,870 --> 01:39:59,900
So you're not going to question
the existence of the scalar

1795
01:39:59,900 --> 01:40:03,680
potential You know it exists,
but you don't know what it is.

1796
01:40:03,680 --> 01:40:09,750
What is f such that f sub
x would be 6xy plus 1,

1797
01:40:09,750 --> 01:40:14,486
and m sub y will be 3x squared?

1798
01:40:14,486 --> 01:40:18,315
And normally, you would
have to integrate backwards.

1799
01:40:18,315 --> 01:40:21,350
Now, I'll give you 10 seconds.

1800
01:40:21,350 --> 01:40:25,190
If in 10 seconds, you don't
find me a scalar potential,

1801
01:40:25,190 --> 01:40:27,120
I'm going to make you
integrate backwards.

1802
01:40:27,120 --> 01:40:31,360
So this is finding the scalar
potential by integration.

1803
01:40:31,360 --> 01:40:34,455
The way you should, if
you weren't very smart.

1804
01:40:34,455 --> 01:40:37,970
But I think you're
smart enough to smell

1805
01:40:37,970 --> 01:40:42,410
the potential-- Very good.

1806
01:40:42,410 --> 01:40:44,030
But what if you don't?

1807
01:40:44,030 --> 01:40:46,050
OK I'm asking.

1808
01:40:46,050 --> 01:40:50,475
So we had one or two
student who figured it out.

1809
01:40:50,475 --> 01:40:51,225
What if you don't?

1810
01:40:51,225 --> 01:40:55,240
If you don't, you can still do
perfectly fine on this problem.

1811
01:40:55,240 --> 01:41:01,120
Let's see how we do it
without seeing or guessing.

1812
01:41:01,120 --> 01:41:03,310
His brain was running
in the background.

1813
01:41:03,310 --> 01:41:05,060
He came up with the answer.

1814
01:41:05,060 --> 01:41:05,655
He's happy.

1815
01:41:05,655 --> 01:41:09,640
He can move on to
the next level.

1816
01:41:09,640 --> 01:41:12,270
STUDENT: Integrate both
sides with respect to r.

1817
01:41:12,270 --> 01:41:15,970
MAGDALENA TODA: Right, and
then mix and match them.

1818
01:41:15,970 --> 01:41:17,665
Make them in work.

1819
01:41:17,665 --> 01:41:20,550
So try to integrate
with respect to x.

1820
01:41:20,550 --> 01:41:25,970
6y-- or plus 1, I'm sorry guys.

1821
01:41:25,970 --> 01:41:28,970
And once you get it,
you're going to get--

1822
01:41:28,970 --> 01:41:32,130
STUDENT: 3x squared y plus x.

1823
01:41:32,130 --> 01:41:34,690
MAGDALENA TODA: And
plus a c of what?

1824
01:41:34,690 --> 01:41:37,970
And then take this fellow and
prime it with respect to y.

1825
01:41:37,970 --> 01:41:41,480
And you're going to
get-- it's not hard.

1826
01:41:41,480 --> 01:41:44,260
You're going to get dx
squared plus nothing,

1827
01:41:44,260 --> 01:41:50,170
plus c from the y, and it's
good because I gave you

1828
01:41:50,170 --> 01:41:51,330
a simple one.

1829
01:41:51,330 --> 01:41:54,390
So sometimes you can
have something here,

1830
01:41:54,390 --> 01:41:57,260
but in this case, it was just 0.

1831
01:41:57,260 --> 01:42:00,400
So c is kappa as a constant.

1832
01:42:00,400 --> 01:42:04,771
So instead of why we teach
found with a plus kappa here,

1833
01:42:04,771 --> 01:42:07,937
and it still does it.

1834
01:42:07,937 --> 01:42:12,810
So on such a problem,
I don't know,

1835
01:42:12,810 --> 01:42:18,066
but I think I would give equal
weights to it, B and C. Compute

1836
01:42:18,066 --> 01:42:20,874
the path integral
over the curve.

1837
01:42:20,874 --> 01:42:24,200
This is horrible, as
an increasing curve.

1838
01:42:24,200 --> 01:42:26,620
But I know that
there is a path that

1839
01:42:26,620 --> 01:42:28,680
connects the points 2, 1 and 1.

1840
01:42:28,680 --> 01:42:30,430
What I have to pay
attention to in my mind

1841
01:42:30,430 --> 01:42:33,390
is that these points
actually are on the curve.

1842
01:42:33,390 --> 01:42:36,170
And they are, because I
have 8 times 1 equals 8,

1843
01:42:36,170 --> 01:42:37,810
1 times 8 equals 8.

1844
01:42:37,810 --> 01:42:41,355
So while I was writing it,
I had to think a little bit

1845
01:42:41,355 --> 01:42:42,580
on the problem.

1846
01:42:42,580 --> 01:42:45,440
If you were to
draw-- well that's

1847
01:42:45,440 --> 01:42:48,450
for you have to find
out when you go home.

1848
01:42:48,450 --> 01:42:52,162
What do you think
this is going to be?

1849
01:42:52,162 --> 01:42:55,080


1850
01:42:55,080 --> 01:42:58,002
Actually, we have to
do it now, because it's

1851
01:42:58,002 --> 01:43:01,770
a lot simpler than
you think it is.

1852
01:43:01,770 --> 01:43:06,510
x and y will be positive,
I can also restrict that.

1853
01:43:06,510 --> 01:43:09,060
It looks horrible, but
it's actually much easier

1854
01:43:09,060 --> 01:43:09,990
than you think.

1855
01:43:09,990 --> 01:43:15,770
So how do I compute that path
integral that makes the points?

1856
01:43:15,770 --> 01:43:19,263
I'm going to have
fundamental there.

1857
01:43:19,263 --> 01:43:22,756


1858
01:43:22,756 --> 01:43:27,350
Which has f of x at q
minus f, with p, which

1859
01:43:27,350 --> 01:43:31,280
says that little f is here.

1860
01:43:31,280 --> 01:43:41,120
3x squared y plus
x at 2, 1 minus 3x

1861
01:43:41,120 --> 01:43:47,300
squared y plus x at 1, 2.

1862
01:43:47,300 --> 01:43:53,450
So all I have to do is
go ahead and-- do you

1863
01:43:53,450 --> 01:43:56,980
see what I'm actually doing?

1864
01:43:56,980 --> 01:43:57,790
It's funny.

1865
01:43:57,790 --> 01:44:02,460
Which one is the origin, and
which one is the endpoint?

1866
01:44:02,460 --> 01:44:03,830
The problem doesn't tell you.

1867
01:44:03,830 --> 01:44:07,302
It tells you only you are
connecting the two points.

1868
01:44:07,302 --> 01:44:10,164
But which one is the alpha,
and which one is the omega?

1869
01:44:10,164 --> 01:44:10,955
Where do you start?

1870
01:44:10,955 --> 01:44:12,550
You start here or
you start here?

1871
01:44:12,550 --> 01:44:16,460


1872
01:44:16,460 --> 01:44:17,070
OK.

1873
01:44:17,070 --> 01:44:19,150
Sort of arbitrary.

1874
01:44:19,150 --> 01:44:22,230
How do you handle this problem?

1875
01:44:22,230 --> 01:44:26,150
Depending on the direction--
pick one direction you move on

1876
01:44:26,150 --> 01:44:28,530
along the r, it's up to you.

1877
01:44:28,530 --> 01:44:31,690
And then you get an answer, and
if you change the direction,

1878
01:44:31,690 --> 01:44:34,250
what's going to happen
to the integral?

1879
01:44:34,250 --> 01:44:37,970
It's just change the
sign and that's all.

1880
01:44:37,970 --> 01:44:43,270
3 times 4, times 1, plus 2--
guys, keep an eye on my algebra

1881
01:44:43,270 --> 01:44:48,104
please, because I
don't want to mess up.

1882
01:44:48,104 --> 01:44:49,879
Am I right, here?

1883
01:44:49,879 --> 01:44:50,420
STUDENT: Yes.

1884
01:44:50,420 --> 01:44:52,300
MAGDALENA TODA: So how much?

1885
01:44:52,300 --> 01:44:53,990
14, is it?

1886
01:44:53,990 --> 01:44:56,315
STUDENT: It's 7.

1887
01:44:56,315 --> 01:44:57,315
MAGDALENA TODA: Minus 7.

1888
01:44:57,315 --> 01:45:06,491


1889
01:45:06,491 --> 01:45:06,990
Good.

1890
01:45:06,990 --> 01:45:07,880
Wonderful.

1891
01:45:07,880 --> 01:45:11,670
So we know what to get,
and we know this does not

1892
01:45:11,670 --> 01:45:13,180
depend on the fact.

1893
01:45:13,180 --> 01:45:16,770
How much blah, blah,
blah does the instructor

1894
01:45:16,770 --> 01:45:20,887
expect for you to get full
credit on the problem?

1895
01:45:20,887 --> 01:45:22,220
STUDENT: Just enough to explain.

1896
01:45:22,220 --> 01:45:23,845
MAGDALENA TODA: Just
enough to explain.

1897
01:45:23,845 --> 01:45:29,810
About 2 lines or 1 line saying
you can say anything really.

1898
01:45:29,810 --> 01:45:34,470
You can say this is the theorem
that either shows independence

1899
01:45:34,470 --> 01:45:35,861
of that integral.

1900
01:45:35,861 --> 01:45:43,310
If the force F vector value
function is conservative,

1901
01:45:43,310 --> 01:45:46,746
then this is what
you have to write.

1902
01:45:46,746 --> 01:45:49,252
This doesn't depend
on the path c.

1903
01:45:49,252 --> 01:45:51,450
And you apply the
fundamental theorem

1904
01:45:51,450 --> 01:45:54,290
of path integrals for
the scalar potential.

1905
01:45:54,290 --> 01:45:57,760
And that scalar potential
depends on the endpoints

1906
01:45:57,760 --> 01:45:59,660
that you're taking.

1907
01:45:59,660 --> 01:46:02,150
And the value of
the work depends--

1908
01:46:02,150 --> 01:46:06,290
the work depends only on the
scalar potential and the two

1909
01:46:06,290 --> 01:46:07,510
points.

1910
01:46:07,510 --> 01:46:08,320
That's enough.

1911
01:46:08,320 --> 01:46:09,780
That's more than enough.

1912
01:46:09,780 --> 01:46:13,816
What if somebody's
not good with wording?

1913
01:46:13,816 --> 01:46:17,170
I'm not going to write
her all that explanation.

1914
01:46:17,170 --> 01:46:21,650
I'm just going to say whatever.

1915
01:46:21,650 --> 01:46:25,100
I'm going to give
her the theorem

1916
01:46:25,100 --> 01:46:27,210
in mathematical compressed way.

1917
01:46:27,210 --> 01:46:30,960
And I don't care if she
understands it or not.

1918
01:46:30,960 --> 01:46:34,620
Even if you write this
formula with not much wording,

1919
01:46:34,620 --> 01:46:36,870
I still give you credit.

1920
01:46:36,870 --> 01:46:38,560
But I would prefer
that you give me

1921
01:46:38,560 --> 01:46:41,640
some sort of-- some
sort of explanation.

1922
01:46:41,640 --> 01:46:42,802
Yes, sir.

1923
01:46:42,802 --> 01:46:44,093
STUDENT: You said answer was 0.

1924
01:46:44,093 --> 01:46:45,801
Then it would have
been path independent?

1925
01:46:45,801 --> 01:46:50,860


1926
01:46:50,860 --> 01:46:53,630
MAGDALENA TODA: No, the
answer would not be for sure 0

1927
01:46:53,630 --> 01:46:56,770
if it was a longer loop.

1928
01:46:56,770 --> 01:46:58,870
If it were a longer
closed curve,

1929
01:46:58,870 --> 01:47:03,570
that way where it
starts, it ends.

1930
01:47:03,570 --> 01:47:07,470
Even if I take a weekly
road between the two points,

1931
01:47:07,470 --> 01:47:09,170
I still get 7, right?

1932
01:47:09,170 --> 01:47:11,290
That's the whole idea.

1933
01:47:11,290 --> 01:47:12,530
Am I clear about that?

1934
01:47:12,530 --> 01:47:14,710
Are we clear about that?

1935
01:47:14,710 --> 01:47:21,360
Let me ask you though,
how do you find out?

1936
01:47:21,360 --> 01:47:25,760
Because I don't know how
many of you figured out

1937
01:47:25,760 --> 01:47:28,526
what kind of curve that is.

1938
01:47:28,526 --> 01:47:32,600
And it looks like an enemy
to you, but there is a catch.

1939
01:47:32,600 --> 01:47:38,520
It's an old friend of
yours and you don't see it.

1940
01:47:38,520 --> 01:47:40,379
So what is the curve?

1941
01:47:40,379 --> 01:47:41,265
What is the curve?

1942
01:47:41,265 --> 01:47:46,650
And what is this arc of a
curve between 2, 1 and 1, 2?

1943
01:47:46,650 --> 01:47:48,230
Can we find out what that is?

1944
01:47:48,230 --> 01:47:49,350
Of course, or cubic.

1945
01:47:49,350 --> 01:47:50,450
It's a fake cubic.

1946
01:47:50,450 --> 01:47:53,780
It's a fake cubic--

1947
01:47:53,780 --> 01:47:56,224
STUDENT: To function together?

1948
01:47:56,224 --> 01:47:58,150
MAGDALENA TODA: Let's
see what this is.

1949
01:47:58,150 --> 01:48:03,060
xy cubed minus 2 cubed equals 0.

1950
01:48:03,060 --> 01:48:06,315
We were in fourth grade--
well, our teachers--

1951
01:48:06,315 --> 01:48:13,640
I think our teachers teach us
when we were little that this,

1952
01:48:13,640 --> 01:48:16,660
if you divided by a
minus- I wasn't little.

1953
01:48:16,660 --> 01:48:19,236
I was in high school.

1954
01:48:19,236 --> 01:48:21,132
Well, 14-year-old.

1955
01:48:21,132 --> 01:48:22,080
STUDENT: A cubed.

1956
01:48:22,080 --> 01:48:23,280
STUDENT: A squared.

1957
01:48:23,280 --> 01:48:24,370
MAGDALENA TODA: A squared.

1958
01:48:24,370 --> 01:48:26,010
STUDENT: Minus 2AB.

1959
01:48:26,010 --> 01:48:27,080
Plus 2AB.

1960
01:48:27,080 --> 01:48:28,960
MAGDALENA TODA: Very good.

1961
01:48:28,960 --> 01:48:30,870
Plus AB, not 2AB.

1962
01:48:30,870 --> 01:48:31,630
STUDENT: Oh, darn.

1963
01:48:31,630 --> 01:48:33,750
MAGDALENA TODA: Plus B squared.

1964
01:48:33,750 --> 01:48:34,890
Suppose you don't believe.

1965
01:48:34,890 --> 01:48:36,730
That proves this.

1966
01:48:36,730 --> 01:48:38,060
Let's multiply.

1967
01:48:38,060 --> 01:48:41,660
A cubed plus A squared
B plus AB squared.

1968
01:48:41,660 --> 01:48:44,020
I'm done with the
first multiplication.

1969
01:48:44,020 --> 01:48:50,366
Minus BA squared minus
AB squared minus B cubed.

1970
01:48:50,366 --> 01:48:52,290
Do they cancel out?

1971
01:48:52,290 --> 01:48:53,733
Yes.

1972
01:48:53,733 --> 01:48:55,180
Good.

1973
01:48:55,180 --> 01:48:57,605
Cancel out.

1974
01:48:57,605 --> 01:49:00,040
And cancel out.

1975
01:49:00,040 --> 01:49:01,560
Out, poof.

1976
01:49:01,560 --> 01:49:02,850
We've proved it, why?

1977
01:49:02,850 --> 01:49:08,700
Because maybe some of you--
nobody gave it to proof before.

1978
01:49:08,700 --> 01:49:11,590


1979
01:49:11,590 --> 01:49:17,510
So as an application,
what is this?

1980
01:49:17,510 --> 01:49:18,010
There.

1981
01:49:18,010 --> 01:49:19,100
Who is A and who is B?

1982
01:49:19,100 --> 01:49:23,790
A is xy, B is 2.

1983
01:49:23,790 --> 01:49:34,440
So you have xy minus 2 times
all this fluffy guy, xy

1984
01:49:34,440 --> 01:49:42,381
squared plus 2xy plus--

1985
01:49:42,381 --> 01:49:44,510
STUDENT: 4.

1986
01:49:44,510 --> 01:49:45,260
MAGDALENA TODA: 4.

1987
01:49:45,260 --> 01:49:49,292
And I also said, because
I was sneaky, that's why.

1988
01:49:49,292 --> 01:49:54,510
To make your life easier
or harder. xy is positive.

1989
01:49:54,510 --> 01:49:57,783
When I said xy was positive,
what was I intending?

1990
01:49:57,783 --> 01:50:03,320
I was intending for you to see
that this cannot be 0 ever.

1991
01:50:03,320 --> 01:50:07,678
So the only possible
for you to have 0 here

1992
01:50:07,678 --> 01:50:10,070
is when xy equals 2.

1993
01:50:10,070 --> 01:50:14,200
And xy equals 2 is a
much simpler curve.

1994
01:50:14,200 --> 01:50:17,720
And I want to know
if you realize

1995
01:50:17,720 --> 01:50:22,160
that this will have the points
2,1 and 1, 2 staring at you.

1996
01:50:22,160 --> 01:50:23,370
Have a nice day today.

1997
01:50:23,370 --> 01:50:25,201
Take care.

1998
01:50:25,201 --> 01:50:26,632
And good luck.

1999
01:50:26,632 --> 01:50:31,880


2000
01:50:31,880 --> 01:50:34,075
What is it?

2001
01:50:34,075 --> 01:50:34,950
STUDENT: [INAUDIBLE].

2002
01:50:34,950 --> 01:50:36,760
MAGDALENA TODA:
Some sort of animal.

2003
01:50:36,760 --> 01:50:38,380
It's a curve, a linear curve.

2004
01:50:38,380 --> 01:50:42,276
It's not a line.

2005
01:50:42,276 --> 01:50:43,250
What is it?

2006
01:50:43,250 --> 01:50:47,633
Talking about conics because
I was talking a little bit

2007
01:50:47,633 --> 01:50:49,581
with Casey about conics.

2008
01:50:49,581 --> 01:50:52,503
Is this a conic?

2009
01:50:52,503 --> 01:50:53,477
Yeah.

2010
01:50:53,477 --> 01:50:55,430
What is a conic?

2011
01:50:55,430 --> 01:50:59,970
A conic is any kind of
curve that looks like this.

2012
01:50:59,970 --> 01:51:04,530
In general form--
oh my god, ABCD.

2013
01:51:04,530 --> 01:51:08,190
Now I got my ABC
plus f equals 0.

2014
01:51:08,190 --> 01:51:10,030
This is a conic in plane.

2015
01:51:10,030 --> 01:51:13,650
My conic is missing
everything else.

2016
01:51:13,650 --> 01:51:16,020
And B is 0.

2017
01:51:16,020 --> 01:51:18,830
And there is a way where
you-- I showed you how you

2018
01:51:18,830 --> 01:51:21,730
know what kind of conic it is.

2019
01:51:21,730 --> 01:51:28,130
A, A, B, B, C. A is
positive is-- no, A is 0,

2020
01:51:28,130 --> 01:51:31,740
B is-- it should be 2 here.

2021
01:51:31,740 --> 01:51:34,158
So you split this in half.

2022
01:51:34,158 --> 01:51:37,110
1/2, 1/2, and 0.

2023
01:51:37,110 --> 01:51:41,086
The determinant of this is
negative, the discriminant.

2024
01:51:41,086 --> 01:51:43,653
That's why we call it
discriminant about the conic.

2025
01:51:43,653 --> 01:51:44,900
So it cannot be an ellipse.

2026
01:51:44,900 --> 01:51:46,491
So what the heck is it?

2027
01:51:46,491 --> 01:51:47,365
STUDENT: [INAUDIBLE].

2028
01:51:47,365 --> 01:51:48,698
MAGDALENA TODA: Well, I'm silly.

2029
01:51:48,698 --> 01:51:50,180
I should have pulled out for y.

2030
01:51:50,180 --> 01:51:53,040


2031
01:51:53,040 --> 01:51:56,725
And I knew that it
goes down like 1/x.

2032
01:51:56,725 --> 01:52:00,590
But I'm asking you, why in
the world is that a conic?

2033
01:52:00,590 --> 01:52:01,670
Because you say, wait.

2034
01:52:01,670 --> 01:52:03,230
Wait a minute.

2035
01:52:03,230 --> 01:52:09,900
I know this curve since I was
five year old in kindergarten.

2036
01:52:09,900 --> 01:52:13,150
And this is the point 2, 1.

2037
01:52:13,150 --> 01:52:16,510


2038
01:52:16,510 --> 01:52:17,240
It's on it.

2039
01:52:17,240 --> 01:52:22,930
And there is a symmetric
point for your pleasure here.

2040
01:52:22,930 --> 01:52:25,120
1, 2.

2041
01:52:25,120 --> 01:52:26,640
And between the
two points, there

2042
01:52:26,640 --> 01:52:31,530
is just one arc of a curve.

2043
01:52:31,530 --> 01:52:34,120
And this is the path that
you are dragging some object

2044
01:52:34,120 --> 01:52:35,390
with force f.

2045
01:52:35,390 --> 01:52:37,980
You are computing
the work of a-- maybe

2046
01:52:37,980 --> 01:52:40,990
you're computing the work of
a neutron between those two

2047
01:52:40,990 --> 01:52:42,930
locations.

2048
01:52:42,930 --> 01:52:43,786
It's a--

2049
01:52:43,786 --> 01:52:44,642
STUDENT: Hyperbola?

2050
01:52:44,642 --> 01:52:46,000
MAGDALENA TODA: Hyperbola.

2051
01:52:46,000 --> 01:52:47,030
Why Nitish?

2052
01:52:47,030 --> 01:52:47,768
Yes, sir.

2053
01:52:47,768 --> 01:52:49,518
STUDENT: I was just
wondering, couldn't we

2054
01:52:49,518 --> 01:52:51,696
have gone to xy equals 2 plane?

2055
01:52:51,696 --> 01:52:52,820
STUDENT: Yeah, way quicker.

2056
01:52:52,820 --> 01:52:55,259
STUDENT: x cubed, y
cubed equals 2 cubed.

2057
01:52:55,259 --> 01:52:56,550
Then you'd just do both sides--

2058
01:52:56,550 --> 01:52:57,250
MAGDALENA TODA:
That's what I did.

2059
01:52:57,250 --> 01:52:57,865
STUDENT: The cubed root.

2060
01:52:57,865 --> 01:52:59,400
MAGDALENA TODA:
Didn't I do that?

2061
01:52:59,400 --> 01:53:02,840
No, because in
general, it's not--

2062
01:53:02,840 --> 01:53:07,482
you cannot say if and only
if xy equals 2 in general.

2063
01:53:07,482 --> 01:53:10,600
You have to write to
decompose the polynomial.

2064
01:53:10,600 --> 01:53:12,450
You were lucky
this was positive.

2065
01:53:12,450 --> 01:53:15,049
STUDENT: Well, because
we divided by x cubed.

2066
01:53:15,049 --> 01:53:16,590
We could have just
divided everything

2067
01:53:16,590 --> 01:53:18,610
by x cubed, and then taken
the cube root of both sides.

2068
01:53:18,610 --> 01:53:20,401
MAGDALENA TODA: He's
saying the same thing.

2069
01:53:20,401 --> 01:53:23,931
But in mathematics, we don't--
let me show you something.

2070
01:53:23,931 --> 01:53:25,472
STUDENT: It would
work for this case,

2071
01:53:25,472 --> 01:53:26,860
but not necessarily
for all cases?

2072
01:53:26,860 --> 01:53:27,735
MAGDALENA TODA: Yeah.

2073
01:53:27,735 --> 01:53:39,140
Let me show you some other
example where you just-- how

2074
01:53:39,140 --> 01:53:41,130
do you solve this equation?

2075
01:53:41,130 --> 01:53:46,020
By the way, a math
field test is coming.

2076
01:53:46,020 --> 01:53:48,270
No, only if you're a math major.

2077
01:53:48,270 --> 01:53:50,510
Sorry, junior or senior.

2078
01:53:50,510 --> 01:53:53,360
In one math field test,
you don't have to take it.

2079
01:53:53,360 --> 01:53:56,880
But some people who
go to graduate school,

2080
01:53:56,880 --> 01:54:01,495
if they take the math field
test, that replaces the GRE,

2081
01:54:01,495 --> 01:54:03,475
if the school agrees.

2082
01:54:03,475 --> 01:54:06,450
So there was this questions,
how many roots does it have

2083
01:54:06,450 --> 01:54:07,770
and what kind?

2084
01:54:07,770 --> 01:54:11,380
Two are imaginary
and one is real.

2085
01:54:11,380 --> 01:54:14,741
But everybody said
it only had one root.

2086
01:54:14,741 --> 01:54:17,030
How can it have one root
if it's a cubic equation?

2087
01:54:17,030 --> 01:54:18,580
So one root.

2088
01:54:18,580 --> 01:54:20,596
x1 is 1.

2089
01:54:20,596 --> 01:54:23,081
The other two are imaginary.

2090
01:54:23,081 --> 01:54:24,330
This is the case in this also.

2091
01:54:24,330 --> 01:54:26,450
You have some imaginary roots.

2092
01:54:26,450 --> 01:54:31,440
So those roots
are funny, but you

2093
01:54:31,440 --> 01:54:35,740
would have to
solve this equation

2094
01:54:35,740 --> 01:54:42,100
because this is x minus 1
times x squared plus x plus 1.

2095
01:54:42,100 --> 01:54:45,580
So the roots are minus
1, plus minus square root

2096
01:54:45,580 --> 01:54:52,230
of b squared minus 4ac
over 2, which are minus 1

2097
01:54:52,230 --> 01:54:57,290
plus minus square
root of 3i over 2.

2098
01:54:57,290 --> 01:55:01,230
Do you guys know
how they are called?

2099
01:55:01,230 --> 01:55:05,600
You know them because in
some countries we learn them.

2100
01:55:05,600 --> 01:55:07,982
But do you know the notations?

2101
01:55:07,982 --> 01:55:09,190
STUDENT: What they call them?

2102
01:55:09,190 --> 01:55:10,065
MAGDALENA TODA: Yeah.

2103
01:55:10,065 --> 01:55:12,190


2104
01:55:12,190 --> 01:55:14,150
There is a Greek letter.

2105
01:55:14,150 --> 01:55:15,872
STUDENT: Iota.

2106
01:55:15,872 --> 01:55:17,330
MAGDALENA TODA: In
India, probably.

2107
01:55:17,330 --> 01:55:19,060
In my country, it was omega.

2108
01:55:19,060 --> 01:55:19,874
But I don't think--

2109
01:55:19,874 --> 01:55:20,874
STUDENT: In India, iota.

2110
01:55:20,874 --> 01:55:22,875


2111
01:55:22,875 --> 01:55:25,810
MAGDALENA TODA: But we call
them omega and omega squared.

2112
01:55:25,810 --> 01:55:28,640
Because one is the
square of the other.

2113
01:55:28,640 --> 01:55:30,140
They are, of course,
both imaginary.

2114
01:55:30,140 --> 01:55:35,662
And we call this the
cubic roots of unity.

2115
01:55:35,662 --> 01:55:39,110


2116
01:55:39,110 --> 01:55:41,970
You say Magdalena, why would
you talk about imaginary numbers

2117
01:55:41,970 --> 01:55:43,880
when everything is real?

2118
01:55:43,880 --> 01:55:44,500
OK.

2119
01:55:44,500 --> 01:55:48,140
It's real for the time being
while you are still with me.

2120
01:55:48,140 --> 01:55:50,330
The moment you're going
to say goodbye to me

2121
01:55:50,330 --> 01:55:55,120
and you know in 3350 your
life is going to change.

2122
01:55:55,120 --> 01:55:57,340
In that course,
they will ask you

2123
01:55:57,340 --> 01:56:02,571
to solve this equation just like
we asked all our 3350 students.

2124
01:56:02,571 --> 01:56:05,050
To our surprise,
the students don't

2125
01:56:05,050 --> 01:56:06,880
know what imaginary roots are.

2126
01:56:06,880 --> 01:56:07,940
Many, you know.

2127
01:56:07,940 --> 01:56:10,376
You will refresh your memory.

2128
01:56:10,376 --> 01:56:12,000
But the majority of
the students didn't

2129
01:56:12,000 --> 01:56:15,140
know how to get to
those imaginary numbers.

2130
01:56:15,140 --> 01:56:20,360
You're going to need to not
only use them, but also express

2131
01:56:20,360 --> 01:56:22,557
these in terms of trigonometry.

2132
01:56:22,557 --> 01:56:25,540


2133
01:56:25,540 --> 01:56:31,730
So just out of curiosity, since
I am already talking to you,

2134
01:56:31,730 --> 01:56:34,885
and since I've preparing you a
little bit for the differential

2135
01:56:34,885 --> 01:56:38,662
equations class where you
have lots of electric circuits

2136
01:56:38,662 --> 01:56:41,430
and applications
of trigonometry,

2137
01:56:41,430 --> 01:56:45,780
these imaginary numbers
can also be put-- they

2138
01:56:45,780 --> 01:56:50,726
are in general of the form
a plus ib. a plus minus ib.

2139
01:56:50,726 --> 01:56:54,868
And we agree that in
3350 you have to do that.

2140
01:56:54,868 --> 01:56:56,945
Out of curiosity,
is there anybody

2141
01:56:56,945 --> 01:57:02,926
who knows the trigonometric
form of these complex numbers?

2142
01:57:02,926 --> 01:57:06,300
STUDENT: Isn't it r e to the j--

2143
01:57:06,300 --> 01:57:10,170


2144
01:57:10,170 --> 01:57:14,240
MAGDALENA TODA: So you would
have exactly what he says here.

2145
01:57:14,240 --> 01:57:18,485
This number will
be-- if it's plus.

2146
01:57:18,485 --> 01:57:20,949
r e to the i theta.

2147
01:57:20,949 --> 01:57:26,220
He knows a little bit
more than most students.

2148
01:57:26,220 --> 01:57:34,280
And that is cosine
theta plus i sine theta.

2149
01:57:34,280 --> 01:57:36,810
Can you find me the
angle theta if I

2150
01:57:36,810 --> 01:57:42,870
want to write cosine theta
plus i sine theta or cosine

2151
01:57:42,870 --> 01:57:46,390
theta minus i sine theta?

2152
01:57:46,390 --> 01:57:50,210
Can you find me
the angle of theta?

2153
01:57:50,210 --> 01:57:50,965
Is it hard?

2154
01:57:50,965 --> 01:57:53,290
Is it easy?

2155
01:57:53,290 --> 01:57:54,990
What in the world is it?

2156
01:57:54,990 --> 01:57:59,810


2157
01:57:59,810 --> 01:58:01,400
Think like this.

2158
01:58:01,400 --> 01:58:04,440
We are done with this
example, but I'm just

2159
01:58:04,440 --> 01:58:08,170
saying some things that
will help you in 3350.

2160
01:58:08,170 --> 01:58:11,860
If you want cosine
theta to be minus 1/2

2161
01:58:11,860 --> 01:58:20,553
and you want sine theta to be
root 3 over 2, which quadrant?

2162
01:58:20,553 --> 01:58:22,900
Which quadrant are you in?

2163
01:58:22,900 --> 01:58:24,120
STUDENT: Second.

2164
01:58:24,120 --> 01:58:25,620
MAGDALENA TODA: The
second quadrant.

2165
01:58:25,620 --> 01:58:27,030
Very good.

2166
01:58:27,030 --> 01:58:28,030
All right.

2167
01:58:28,030 --> 01:58:31,465
So think cosine.

2168
01:58:31,465 --> 01:58:36,420
If cosine would be a half and
sine would be root 3 over 2,

2169
01:58:36,420 --> 01:58:38,390
it would be in first quadrant.

2170
01:58:38,390 --> 01:58:40,420
And what angle would that be?

2171
01:58:40,420 --> 01:58:40,922
STUDENT: 60.

2172
01:58:40,922 --> 01:58:41,755
STUDENT: That's 60--

2173
01:58:41,755 --> 01:58:45,720
MAGDALENA TODA: 60 degrees,
which is pi over 3, right?

2174
01:58:45,720 --> 01:58:52,900
But pi over 3 is your
friend, so he's happy.

2175
01:58:52,900 --> 01:58:54,900
Well, he is there somewhere.

2176
01:58:54,900 --> 01:58:58,900


2177
01:58:58,900 --> 01:59:00,770
STUDENT: 120.

2178
01:59:00,770 --> 01:59:05,300
MAGDALENA TODA: Where you
are here, you are at what?

2179
01:59:05,300 --> 01:59:07,200
How much is 120-- very good.

2180
01:59:07,200 --> 01:59:09,210
How much is 120 pi?

2181
01:59:09,210 --> 01:59:10,769
STUDENT: 4 pi?

2182
01:59:10,769 --> 01:59:11,560
MAGDALENA TODA: No.

2183
01:59:11,560 --> 01:59:11,850
STUDENT: 2 pi over 3.

2184
01:59:11,850 --> 01:59:13,155
MAGDALENA TODA: 2 pi over 3.

2185
01:59:13,155 --> 01:59:14,460
Excellent.

2186
01:59:14,460 --> 01:59:16,840
So 2 pi over 3.

2187
01:59:16,840 --> 01:59:21,130
This would be if you
were to think about it--

2188
01:59:21,130 --> 01:59:22,390
this is in radians.

2189
01:59:22,390 --> 01:59:24,320
Let me write radians.

2190
01:59:24,320 --> 01:59:28,470
In degrees, that's 120 degrees.

2191
01:59:28,470 --> 01:59:38,820
So to conclude my detour
to introduction to 3350.

2192
01:59:38,820 --> 01:59:47,104
When they will ask you to solve
this equation, x cubed minus 1,

2193
01:59:47,104 --> 01:59:49,580
you have to tell them like that.

2194
01:59:49,580 --> 01:59:53,142
They will ask you to put
it in trigonometric form.

2195
01:59:53,142 --> 02:00:04,140
x1 is 1, x2 is cosine of 2 pi
over 3 plus i sine 2 pi over 3.

2196
02:00:04,140 --> 02:00:07,330
And the other one
is x3 equals cosine

2197
02:00:07,330 --> 02:00:15,682
of 2 pi over 3 minus
i sine of 2 pi over 3.

2198
02:00:15,682 --> 02:00:16,840
The last thing.

2199
02:00:16,840 --> 02:00:18,660
Because I should let you go.

2200
02:00:18,660 --> 02:00:20,015
There was no break.

2201
02:00:20,015 --> 02:00:23,440
I squeezed your brains
really bad today.

2202
02:00:23,440 --> 02:00:26,370
We still have like 150 minutes.

2203
02:00:26,370 --> 02:00:29,190
I stole from you--
no, I stole really big

2204
02:00:29,190 --> 02:00:33,330
because we would have-- yeah,
we still have 15 minutes.

2205
02:00:33,330 --> 02:00:37,294
But the break was 10 minutes,
so I didn't give you a break.

2206
02:00:37,294 --> 02:00:40,030
What would this be if
you wanted to express it

2207
02:00:40,030 --> 02:00:43,030
in terms of another angle?

2208
02:00:43,030 --> 02:00:47,142
That's the last thing
I'm asking of you.

2209
02:00:47,142 --> 02:00:48,603
STUDENT: [INAUDIBLE].

2210
02:00:48,603 --> 02:00:50,551
MAGDALENA TODA: Not minus.

2211
02:00:50,551 --> 02:00:53,170
Like cosine of an angle
plus i sine of an angle.

2212
02:00:53,170 --> 02:00:55,942
You would need to go to
another quadrant, right?

2213
02:00:55,942 --> 02:00:57,570
And which quadrant?

2214
02:00:57,570 --> 02:00:58,407
STUDENT: 4.

2215
02:00:58,407 --> 02:00:59,990
MAGDALENA TODA:
You've said it before.

2216
02:00:59,990 --> 02:01:02,570
That would be 4 pi over 3.

2217
02:01:02,570 --> 02:01:05,230
And 4 pi over 3.

2218
02:01:05,230 --> 02:01:10,160
Keep in mind these things
with imaginary numbers because

2219
02:01:10,160 --> 02:01:13,540
in 3350, they will rely on
you knowing these things.

2220
02:01:13,540 --> 02:01:15,709


2221
02:01:15,709 --> 02:01:17,750
STUDENT: Then you apply
Euler's formula up there.

2222
02:01:17,750 --> 02:01:21,429


2223
02:01:21,429 --> 02:01:22,470
MAGDALENA TODA: Oh, yeah.

2224
02:01:22,470 --> 02:01:24,261
By the way, this is
called Euler's formula.

2225
02:01:24,261 --> 02:01:27,535


2226
02:01:27,535 --> 02:01:30,930
STUDENT: In middle
school, they teach you,

2227
02:01:30,930 --> 02:01:33,840
and they tell you when
discriminant is small,

2228
02:01:33,840 --> 02:01:35,885
there's no solutions.

2229
02:01:35,885 --> 02:01:36,760
MAGDALENA TODA: Yeah.

2230
02:01:36,760 --> 02:01:37,890
STUDENT: And you
go to [INAUDIBLE].

2231
02:01:37,890 --> 02:01:40,393
MAGDALENA TODA: When the
discriminant is less than 0,

2232
02:01:40,393 --> 02:01:42,357
there are no real solutions.

2233
02:01:42,357 --> 02:01:44,321
But you have in pairs
imaginary solutions.

2234
02:01:44,321 --> 02:01:46,285
They always come in pairs.

2235
02:01:46,285 --> 02:01:50,230


2236
02:01:50,230 --> 02:01:52,496
Do you want me to
show you probably

2237
02:01:52,496 --> 02:01:55,240
the most important problem
in 3350 in 2 minutes,

2238
02:01:55,240 --> 02:01:58,568
and then I'll let you go?

2239
02:01:58,568 --> 02:02:01,003
STUDENT: Sure.

2240
02:02:01,003 --> 02:02:07,690
MAGDALENA TODA: So somebody
gives you the equation

2241
02:02:07,690 --> 02:02:10,390
of the harmonic oscillator.

2242
02:02:10,390 --> 02:02:12,010
And you say, what
the heck is that?

2243
02:02:12,010 --> 02:02:17,420
You have a little spring
and you pull that spring.

2244
02:02:17,420 --> 02:02:19,360
And it's going to come back.

2245
02:02:19,360 --> 02:02:21,860
You displace it, it comes back.

2246
02:02:21,860 --> 02:02:24,360
It oscillates back and forth,
oscillates back and forth.

2247
02:02:24,360 --> 02:02:27,950
If you were to write the
solutions of the harmonic

2248
02:02:27,950 --> 02:02:29,800
oscillator in electric
circuits, there

2249
02:02:29,800 --> 02:02:31,300
would be oscillating functions.

2250
02:02:31,300 --> 02:02:36,530
So it has to do with sine and
cosine, so they must be trig.

2251
02:02:36,530 --> 02:02:38,910
If somebody gives
you this equation,

2252
02:02:38,910 --> 02:02:57,056
let's say ax squared-- y
double prime of x minus b.

2253
02:02:57,056 --> 02:02:59,040
Plus.

2254
02:02:59,040 --> 02:03:04,000
Equals to 0.

2255
02:03:04,000 --> 02:03:09,470
And here is a y equals 0.

2256
02:03:09,470 --> 02:03:12,500
Why would that
show up like that?

2257
02:03:12,500 --> 02:03:19,370
Well, Hooke's law tells
you that there is a force.

2258
02:03:19,370 --> 02:03:21,680
And there is a
force and the force

2259
02:03:21,680 --> 02:03:23,430
is mass times acceleration.

2260
02:03:23,430 --> 02:03:27,485
And acceleration is like this
type of second derivative

2261
02:03:27,485 --> 02:03:30,830
of the displacement.

2262
02:03:30,830 --> 02:03:37,580
And F and the displacement
are proportional,

2263
02:03:37,580 --> 02:03:41,230
when you write F
equals displacement,

2264
02:03:41,230 --> 02:03:43,980
let's call it y of x.

2265
02:03:43,980 --> 02:03:47,535
When you have y of x, x is time.

2266
02:03:47,535 --> 02:03:49,297
That's the displacement.

2267
02:03:49,297 --> 02:03:50,005
That's the force.

2268
02:03:50,005 --> 02:03:50,630
That's the k.

2269
02:03:50,630 --> 02:03:53,060
So you have a certain
Hooke's constant.

2270
02:03:53,060 --> 02:03:54,880
Hooke's law constant.

2271
02:03:54,880 --> 02:03:56,870
So when you write
this, Hooke's law

2272
02:03:56,870 --> 02:03:58,590
is going to become like that.

2273
02:03:58,590 --> 02:04:04,930
Mass times y double prime of
x equals-- this is the force.

2274
02:04:04,930 --> 02:04:06,725
k times y of x.

2275
02:04:06,725 --> 02:04:09,990


2276
02:04:09,990 --> 02:04:16,540
But it depends because
you can have plus minus.

2277
02:04:16,540 --> 02:04:18,580
So you can have plus or minus.

2278
02:04:18,580 --> 02:04:20,146
And these are
positive functions.

2279
02:04:20,146 --> 02:04:27,500


2280
02:04:27,500 --> 02:04:31,920
You have two equations
in that case.

2281
02:04:31,920 --> 02:04:39,642
One equation is the form y
double prime plus-- give me

2282
02:04:39,642 --> 02:04:40,470
a number.

2283
02:04:40,470 --> 02:04:43,405
Cy equals 0.

2284
02:04:43,405 --> 02:04:49,570
And the other one would be y
double prime minus cy equals 0.

2285
02:04:49,570 --> 02:04:51,371
All right.

2286
02:04:51,371 --> 02:04:54,976
Now, how hard is to
guess your solutions?

2287
02:04:54,976 --> 02:04:59,916


2288
02:04:59,916 --> 02:05:02,170
Can you guess the
solutions with naked eyes?

2289
02:05:02,170 --> 02:05:04,760


2290
02:05:04,760 --> 02:05:05,635
STUDENT: e to the x--

2291
02:05:05,635 --> 02:05:09,080


2292
02:05:09,080 --> 02:05:13,920
MAGDALENA TODA: So if you have--
you have e to the something.

2293
02:05:13,920 --> 02:05:17,780
If you didn't have a c, it
would make your life easier.

2294
02:05:17,780 --> 02:05:18,700
Forget about the c.

2295
02:05:18,700 --> 02:05:20,910
The c will act the
same in the end.

2296
02:05:20,910 --> 02:05:25,796
So here, what are the
possible solutions?

2297
02:05:25,796 --> 02:05:27,012
STUDENT: e to the--

2298
02:05:27,012 --> 02:05:29,220
MAGDALENA TODA: e to the t
is one of them. e to the x

2299
02:05:29,220 --> 02:05:32,750
is one of them, right?

2300
02:05:32,750 --> 02:05:35,540
So in the end, to
solve such a problem

2301
02:05:35,540 --> 02:05:37,300
they teach you the method.

2302
02:05:37,300 --> 02:05:39,660
You take the equation.

2303
02:05:39,660 --> 02:05:42,080
And for that, you associate
the so-called characteristic

2304
02:05:42,080 --> 02:05:44,480
equation.

2305
02:05:44,480 --> 02:05:47,250
For power 2, you put r squared.

2306
02:05:47,250 --> 02:05:51,474
Then you minus n for-- this
is how many times is it prime?

2307
02:05:51,474 --> 02:05:52,250
No times.

2308
02:05:52,250 --> 02:05:53,080
0 times.

2309
02:05:53,080 --> 02:05:55,250
So you put a 1.

2310
02:05:55,250 --> 02:05:58,010
If it's prime one times,
y prime is missing.

2311
02:05:58,010 --> 02:06:01,770
It's prime 1 time,
you would put minus r.

2312
02:06:01,770 --> 02:06:02,880
Equals 0.

2313
02:06:02,880 --> 02:06:06,950
And then you look at
the two roots of that.

2314
02:06:06,950 --> 02:06:07,820
And what are they?

2315
02:06:07,820 --> 02:06:08,710
Plus minus 1.

2316
02:06:08,710 --> 02:06:11,490
So r1 is 1, r2 is 2.

2317
02:06:11,490 --> 02:06:13,370
And there is a
theorem that says--

2318
02:06:13,370 --> 02:06:15,260
STUDENT: r2 is minus 1.

2319
02:06:15,260 --> 02:06:17,125
MAGDALENA TODA: r2 is minus 1.

2320
02:06:17,125 --> 02:06:19,620
Excuse me.

2321
02:06:19,620 --> 02:06:23,970
OK, there's a theorem that
says all the solutions

2322
02:06:23,970 --> 02:06:28,160
of this equation come as
linear combinations of e

2323
02:06:28,160 --> 02:06:31,230
to the r1t and e to the r2t.

2324
02:06:31,230 --> 02:06:33,106
So linear combination
means you can

2325
02:06:33,106 --> 02:06:39,150
take any number a and any
number b, or c1 and c2, anything

2326
02:06:39,150 --> 02:06:40,140
like that.

2327
02:06:40,140 --> 02:06:44,650
So all the solutions of
this will look like e

2328
02:06:44,650 --> 02:06:48,230
to the t with an a in
front plus e to the minus

2329
02:06:48,230 --> 02:06:50,055
t with a b in front.

2330
02:06:50,055 --> 02:06:53,300
Could you have seen
that with naked eye?

2331
02:06:53,300 --> 02:06:54,350
Well, yeah.

2332
02:06:54,350 --> 02:06:57,270
I mean, you are smart
and you guessed one.

2333
02:06:57,270 --> 02:06:59,250
An you said e to
the t satisfied.

2334
02:06:59,250 --> 02:07:02,320
Because if you put e to the
p and prime it as many times

2335
02:07:02,320 --> 02:07:04,870
as you want, you
still get e to the t.

2336
02:07:04,870 --> 02:07:06,050
So you get 0.

2337
02:07:06,050 --> 02:07:09,180
But nobody thought of-- or maybe
some people thought about e

2338
02:07:09,180 --> 02:07:10,483
to the minus t.

2339
02:07:10,483 --> 02:07:11,066
STUDENT: Yeah.

2340
02:07:11,066 --> 02:07:11,880
I was about to go
through that one.

2341
02:07:11,880 --> 02:07:13,030
MAGDALENA TODA: You were about.

2342
02:07:13,030 --> 02:07:13,800
STUDENT: That's for a selection.

2343
02:07:13,800 --> 02:07:16,008
MAGDALENA TODA: So even if
you take e to the minus t,

2344
02:07:16,008 --> 02:07:17,240
you get the same answer.

2345
02:07:17,240 --> 02:07:19,790
And you get this thing.

2346
02:07:19,790 --> 02:07:24,040
All right, all the combinations
will satisfy the same equation

2347
02:07:24,040 --> 02:07:24,660
as well.

2348
02:07:24,660 --> 02:07:26,630
This is a superposition
principle.

2349
02:07:26,630 --> 02:07:28,622
With this, it was easy.

2350
02:07:28,622 --> 02:07:31,610
But this is the so-called
harmonic oscillator equation.

2351
02:07:31,610 --> 02:07:36,100


2352
02:07:36,100 --> 02:07:40,290
So either you have it simplified
y double prime plus y equals 0,

2353
02:07:40,290 --> 02:07:46,250
or you have some constant c.

2354
02:07:46,250 --> 02:07:48,695
Well, what do you
do in that case?

2355
02:07:48,695 --> 02:07:50,766
Let's assume you have 1.

2356
02:07:50,766 --> 02:07:53,946
Who can guess the solutions?

2357
02:07:53,946 --> 02:07:55,920
STUDENT: 0 and cosine--

2358
02:07:55,920 --> 02:07:57,982
MAGDALENA TODA: No, 0
is the trivial solution

2359
02:07:57,982 --> 02:07:59,160
and it's not going to count.

2360
02:07:59,160 --> 02:08:03,860
You can get it from the
combination of the--

2361
02:08:03,860 --> 02:08:05,350
STUDENT: y equals sine t.

2362
02:08:05,350 --> 02:08:06,850
MAGDALENA TODA:
Sine t is a solution

2363
02:08:06,850 --> 02:08:11,665
because sine t prime is cosine.

2364
02:08:11,665 --> 02:08:13,770
When you prime it
again, it's minus sine.

2365
02:08:13,770 --> 02:08:16,680
When you add sine and
minus sine, you get 0.

2366
02:08:16,680 --> 02:08:19,410
So you just guessed
1 and you're right.

2367
02:08:19,410 --> 02:08:20,670
Make a face.

2368
02:08:20,670 --> 02:08:21,790
Do you see another one?

2369
02:08:21,790 --> 02:08:22,770
STUDENT: Cosine t.

2370
02:08:22,770 --> 02:08:23,728
MAGDALENA TODA: Cosine.

2371
02:08:23,728 --> 02:08:26,155


2372
02:08:26,155 --> 02:08:28,600
They are independent,
linear independent.

2373
02:08:28,600 --> 02:08:31,220
And so the multitude
of solutions

2374
02:08:31,220 --> 02:08:34,270
for that-- I taught you
a whole chapter in 3350.

2375
02:08:34,270 --> 02:08:37,140
Now you don't have
to take it anymore--

2376
02:08:37,140 --> 02:08:39,717
is going to be a equals sine t--

2377
02:08:39,717 --> 02:08:41,050
STUDENT: How about e to the i t?

2378
02:08:41,050 --> 02:08:42,300
MAGDALENA TODA: Plus b sine t.

2379
02:08:42,300 --> 02:08:43,440
I tell you in a second.

2380
02:08:43,440 --> 02:08:46,530
All right, we have to
do an e to the i t.

2381
02:08:46,530 --> 02:08:47,520
OK.

2382
02:08:47,520 --> 02:08:51,350
So you guessed that all the
solutions will be combinations

2383
02:08:51,350 --> 02:08:56,020
like-- on the monitor when you
have cosine and sine, if you

2384
02:08:56,020 --> 02:08:58,870
add them up-- multiply
and add them up,

2385
02:08:58,870 --> 02:09:02,390
you get something like the
monitor thing at the hospital.

2386
02:09:02,390 --> 02:09:04,480
So any kind of
oscillation like that

2387
02:09:04,480 --> 02:09:07,686
is a combination of this kind.

2388
02:09:07,686 --> 02:09:13,050
Maybe with some different
phases and amplitudes.

2389
02:09:13,050 --> 02:09:16,830
You have cosine of 70 or
cosine of 5t or something.

2390
02:09:16,830 --> 02:09:18,650
But let me show
you what they are

2391
02:09:18,650 --> 02:09:24,640
going to show you [INAUDIBLE]
for the harmonic oscillator

2392
02:09:24,640 --> 02:09:27,270
equation how the method goes.

2393
02:09:27,270 --> 02:09:29,250
You solve for the
characteristic equation.

2394
02:09:29,250 --> 02:09:34,600
So you have r squared
plus 1 equals 0.

2395
02:09:34,600 --> 02:09:38,860
Now, here's where most of
the students in 3350 fail.

2396
02:09:38,860 --> 02:09:40,468
They understand that.

2397
02:09:40,468 --> 02:09:43,880
And some of them say, OK,
this has no solutions.

2398
02:09:43,880 --> 02:09:46,990
Some of them even say this
has solutions plus minus 1.

2399
02:09:46,990 --> 02:09:48,870
I mean, crazy stuff.

2400
02:09:48,870 --> 02:09:51,490
Now, what are the
solutions of that?

2401
02:09:51,490 --> 02:09:53,455
Because the theory
in this case says

2402
02:09:53,455 --> 02:09:57,330
if your solutions are
imaginary, then y1

2403
02:09:57,330 --> 02:10:00,980
would be e to the ax cosine bx.

2404
02:10:00,980 --> 02:10:05,450
And y2 will be e
to the ax sine bx

2405
02:10:05,450 --> 02:10:09,446
where your imaginary
solutions are a plus minus ib.

2406
02:10:09,446 --> 02:10:13,990
It has a lot to do with
Euler's formula in a way.

2407
02:10:13,990 --> 02:10:21,000
So if you knew the theory in
3350 and not be just very smart

2408
02:10:21,000 --> 02:10:24,130
and get these by yourselves
by guessing them,

2409
02:10:24,130 --> 02:10:26,520
how are you supposed
to know that?

2410
02:10:26,520 --> 02:10:30,580
Well, r squared
equals minus 1, right?

2411
02:10:30,580 --> 02:10:34,036
The square root of minus 1 is i.

2412
02:10:34,036 --> 02:10:34,910
STUDENT: Or negative.

2413
02:10:34,910 --> 02:10:36,159
MAGDALENA TODA: Or negative i.

2414
02:10:36,159 --> 02:10:40,600
So r1 is 0 plus minus i.

2415
02:10:40,600 --> 02:10:42,460
So who is a?

2416
02:10:42,460 --> 02:10:44,181
a is 0.

2417
02:10:44,181 --> 02:10:45,710
Who is b?

2418
02:10:45,710 --> 02:10:46,720
b is 1.

2419
02:10:46,720 --> 02:10:53,010
So the solutions are e to
the 0x equal cosine 1x and e

2420
02:10:53,010 --> 02:10:59,040
to the 0x sine 1x, which
is cosine x, sine x.

2421
02:10:59,040 --> 02:11:03,210
Now you know why you can
do everything formalized

2422
02:11:03,210 --> 02:11:06,170
and you get all these
solutions from a method.

2423
02:11:06,170 --> 02:11:10,286
This method is an
entire chapter.

2424
02:11:10,286 --> 02:11:12,400
It's so much easier than in 350.

2425
02:11:12,400 --> 02:11:14,670
So much easier than Calculus 3.

2426
02:11:14,670 --> 02:11:16,420
You will say this is easy.

2427
02:11:16,420 --> 02:11:17,720
It's a pleasure.

2428
02:11:17,720 --> 02:11:22,790
You spend about one
fourth of the semester

2429
02:11:22,790 --> 02:11:24,562
just on this method.

2430
02:11:24,562 --> 02:11:26,270
So now you don't have
to take it anymore.

2431
02:11:26,270 --> 02:11:29,080
You can learn it all by
yourself and you're going

2432
02:11:29,080 --> 02:11:33,060
to be ready for the next thing.

2433
02:11:33,060 --> 02:11:34,760
So I'm just giving you courage.

2434
02:11:34,760 --> 02:11:40,350
If you do really, really well in
Calc 3, 3350 will be a breeze.

2435
02:11:40,350 --> 02:11:42,290
You can breeze through that.

2436
02:11:42,290 --> 02:11:46,170
You only have the probability
in stats for most engineers

2437
02:11:46,170 --> 02:11:48,595
to take.

2438
02:11:48,595 --> 02:11:54,210
Math is not so complicated.